konstrukcje metalowe wykład iii geometria przekrojufootbridge.pl/stud/z/sn1/kbi/w103.pdf ·...

Konstrukcje metalowe

Wykład III

Geometria przekroju

Spis treści

Podstawowe charakterystyki geometrycznerarr t 3

Zaawansowane charakterystyki geometrycznerarr t 27

Przykład obliczeniowy rarr t 58

Zagadnienia egzaminacyjne rarr t 88

Podstawowe charakterystyki geometryczne

Rys Autor

Pole powierzchni [m2]

A = int int dz dy

y0

z0

Rys Autor

Moment statyczny [m3]

Sy = int int z dz dy

Sz = int int y dz dy

y0

z0

y1

z1

∆z

∆y

CoG

Sy = Sz = 0 gdy osie przechodzą przez środek

ciężkości

∆y = Sz A

∆z = Sy A

Dyz = 0 dla osi głoacutewnych centralnych

Jy = (Jy1 + Jz1) + radic[(Jy1 - Jz1)2 + 4 Dy1z1

2 ] 2

Jz = (Jy1 + Jz1) - radic[(Jy1 - Jz1)2 + 4 Dy1z1

2 ] 2

tg α = Dy1z1 (Jy1 - Jy)

Rys Autor

Moment bezwładności [m4]

Jy = int int z2 dz dy

Jz = int int y2 dz dy

Moment dewiacji [m4]

Dyz = int int y z dz dy

y1

z1 z

y

α

Wskaźnik wytrzymałości przekroju [m3]

Wy top = Jy | zmax top|

Wy bottom = Jy | zmax bottom|

Wz left = Jy | ymax left|

Wz right = Jy | ymax right|

Dla przekroju bisymetrycznego

Wy top = Wy bottom = Wy

Wz left = Wz right = Wz

Promień bezwładności [m]

iy = radic (Jy A)

iz = radic (Jz A)

Rys Autor

y

z

ymax left

ymax right

zmax top

zmax bottom

Charakterystyki geometryczne ndash przekroacutej prostokątny

A = b h

Jy = b h3 12

Wy = Jy zmax = Jy (05h) = b h2 6

iy = radic (Jy A) = h (2 radic 3)

Rys Autor

Wskaźnik wytrzymałości przekroju



Dlaczego jest taki ważny

Naprężenia w zakresie sprężystym

Rys Autor

y

z

bull naprężenia zmieniają się wzdłuż osi z

σ (y z) = σ (z)

bull z = 0 rarr σ = 0

bull σ (zmax) = σ (ztop) = σtop

bull jest to funkcja liniowaσ (z)

σ(z) = σtop (z ztop)

σ (ztop) = σtop

σ (0) = 0

σ (0) = 0

σ (zbottom) = σbottom lt σtop

z

Wypadkowa siła osiowa


N = int int [σ(z)] dz dy = int int [σtop (z ztop)] dz dy = (σtop ztop) int int z dz dy =

= t 4 Sy = int int z dz dy śc Sy = 0 = (σtop ztop) 0 = 0

Wypadkowy moment zginający


My =-int int z [σ(z)] dz dy = int int z [σtop (z ztop)] dz dy =

= (σtop ztop) int int (z2) dz dy =

= t 5 Jy = int int z2 dz dy = (σtop ztop) Jy =

= t 6 Wy top = Jy | z top| = σtop Wy top

My = σmax Wy max

My max = fy Wy max

Jaka jest najlepsza geometria przekroju gdy mamy do

czynienia z następującymi ograniczeniami

A rarr jak najmniejsze (mała masa własna)

Wy rarr jak największe (duża nośność przekroju na

zginanie)

h rarr ograniczone

Na przykład

A = 2 a2

h le 3a

Rys Autor

b = 2a h = a

Jy = b h3 12 = 0167 a4

Wy = b h2 6 = 0333 a3

iy = radic (Jy A) = 0289 a

My max = 0333 a3 fy

Rys Autor

b = a h = 2a

Jy = b h3 12 = 0667 a4

Wy = b h2 6 = 0667 a3


My max = 0667 a3 fy

Rys Autor

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]

Sztywność własna Twierdzenie Steinera

b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a

M y 0333 a3 fy 0667 a3 fy 1000 a3 fy 1444 a3 fy 2111 a3 fy

Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor

W obliczeniach celowo został popełniony błąd

Jaki to błąd

Twierdzenie Steinera może być zastosowane jedynie wtedy

gdy istnieje sztywne połączenie między wszystkimi częściami

przekroju zapobiegające zmianie ich wzajemnej odległości

Brak połączenia ndash jedynie sztywność własna

Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor

Bak połączenia ndash jedynie sztywność własna

Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor

Twierdzenie Steinera Prawda

A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a

M y 0333 a3 fy 0667 a3 fy 1000 a3 fy 1444 a3 fy 2111 a3 fy 0111 a3 fy 0028 a3 fy

Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor

Najlepszy kształt przekroju w przypadku elementu zginanego

I

Dwuteownik Kratownica

Rys Autor

Rys discountsteelcom

Rys conference-truss-hirecouk

Dla przekroju prostokątnego roacutewnoległego do osi i oddalonego od niej sztywność własna

może być pominięta

Rys Autor

Sztywność własna= 0167 a4 SzW = 0167 a4 SzW = 0042 a4

Twierdzenie Steinera= 00 a4 TS = 2000 a4 TS = 3125 a4

Całość = 1000 + 00 Całość = 77 + 923 Całość = 13 + 987

IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor

Trzy prostokąty dla poziomych SzW + TS

Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)

Trzy prostokąty dla poziomych tylko TS

Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)

Oproacutecz zakresu sprężystego obliczenia można prowadzić w zakresie sprężysto-

plastycznym i plastycznym

Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w

części ściskanej i

rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy

Zaawansowane charakterystyki geometryczne

Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy

Przyjmijmy taki sam układ sił

przekrojowych jak poprzednio

N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1

int fy dz dy + intA2

int -fy dz dy =

= fy intA1

int dz dy - fy intA2

int dz dy =

= t 3 A = int int dz dy =

= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

Przy czystym zginaniu w przypadku pełnego

uplastycznienia przekroju konieczne staje się

znalezienie linii dzielącej pole przekroju na dwie

połowy (A1 = A2 = A 2)

Dla przekroju symetrycznego (oś sprężysta y = oś

symetrii) nowa oś pokrywa się ze starą

Dla przekroju niesymetrycznego nowa oś y nie jest

starą osią yA1

A2

yy

z

Rys Autor

Zakres sprężysty S1el(yel) = -S2el(yel)

A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl

Zakres plastyczny A1pl = A2pl

S1pl(ypl) ne -S2pl(ypl)l

Rys Autor

A1 = A2 = A 2

My = int int z [σ(z)] dz dy =

= intA1

int (z fy) dz dy + intA2

int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =

= t 4 Sy = int int z dz dy =

= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]

Można to zapisać jako

My = fy Wpl y

Wpl y (plastyczny wskaźnik wytrzymałości) jest specyficzną

charakterystyką geometryczną przekroju ważną dla pracy w zakresie

plastycznym

Dla przekroju symerycznego (y = y)

Wpl y = 2 (moment statyczny połowy przekroju względem osi y = y)

Ogoacutelnie (i j = y z)

Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z

Rys Autor Rys ijirdcom

Oproacutecz sprężystego i plastycznego wskaźnika wytrzymałości konieczne jest niekiedy

rozpatrywanie wskaźnika efektywnego Pojawia się on w sytuacji przekrojoacutew bardzo

smukłych (cienki i wysoki środnik) dla ktoacuterych może dojść do utraty stateczności lokalnej

ścianek ściskanych

Rys Autor

Część ktoacutera podlega niestateczności lokalnej jest pomijana

Przekroacutej efektywny to fikcyjny przekroacutej z ktoacuterego usunięto

fragmenty poacutełek środnika podlegające niestateczności

W przypadku rozpatrywania przekroju efektywnego

konieczne jest znalezienie efektywnego pola przekroju

efektywnego środka ciężkości efektywnego momentu

bezwładności i efektywnego wskaźnika wytrzymałości

Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie

Zgodnie z Mechaniką Teoretyczną

τ = V S y(z) Jy t(z)


τ = V AV

τ = max gdy AV = min AV = min dla t = min oraz S y = max

AV = Jy t(z = 0) S y(z = 0)

AV kolejna specyficzna charakterystyka geometryczna (bdquoczynne pole przekroju) ważna dla

ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)

Sy (połowa przekroju) asymp

asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =

= tw [h2 (h tw 12 + tf b 2)] [h (h tw 8 + tf b 2)] =

= tw h [(h tw 12 + tf b 2) (h tw 8 + tf b 2)] =

= h tw asymp tf b = A = tw h [(A 12 + A 2) (A 8 + A 2)] =

= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie

AV może być przybliżone jako pole części przekroju roacutewnoległej do kierunku

działania siły

Rys Autor

Vz

Vy

AV z = tw hAV y = 2 tf b

Skręcanie jest najbardziej skomplikowanym rodzajem obciążenia Należy je rozpatrywać

na kilka roacuteżne sposoby

bull przekroje okrągłe (pręty okrągłe)

bull przekroje zamknięte okrągłe (rury okrągłe)

bull przekroje zamknięte nieokrągłe (rury kwadratowe i prostokątne)

bull część przekrojoacutew otwartych ( L T )

bull pozostałe przekroje otwarte

Zroacuteżnicowanie polega na odmiennych deformacjach prętoacutew skręcanych odmiennych

rozkładach naprężeń lub odmiennych wzorach na charakterystyki geometryczne

TEd

∆x

∆x

∆x

Efektem momentu skręcającego

jest nie tylko rotacja przekroju ale i

jego deplanacja ndash początkowo

płaskie i roacutewnoległe przekroje stają

nie nie-płaskie i nieroacutewnoległe

Nie wszystkie kształty przekroju

podlegają deplanacji a dla części

tych ktoacutere podlegają deplanacja ma

pomijalnie małe wartościRys Autor

Przypadek ogoacutelny

Przekroacutej Deplanacja Moment skręcający

TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T

Bardzo mała Swobodna TEd = Tt Ed -

Skrępowana

podporami

TEd asymp Tt Ed -

Pozostałe Istotna Swobodna TEd = Tt Ed -

Skrępowana

podporami

TEd = Tt Ed + Tw Ed Dodatkowo pod uwagę

należy wziąć BEd

Tt Ed ndash moment skręcania St Venanta (swobodna deplanacja przekroju)

Tw Ed ndash moment skręcania skrępowanego (skrępowana deplanacja przekroju)

B Ed - bimoment

Deplanacja dwuteownika obie poacutełki

podlegają deplanacji w przeciwnych

kierunkach

W przypadku skręcania skrępowanego

pojawia się specyficzny rozkład naprężeń

w poacutelkach

σ

σ

Naprężenia te mogą być

przedstawione jako efekt

bimomentu

B = h M

M = |M1| = |M2|

Bimoment jest także brany pod

uwagę w przypadku analizy

przekrojoacutew cienkościennych

h

M1

M2

Rys Autor

W analogii do zginania wprowadzić można

moment bezwładności przy skręcaniu i

wskaźnik wytrzymałości przekroju przy

skręcaniu

WT = JT tmax

tmax = max ( t1 t2 t3 )

h1

h2

h3

t1

t2

t3

Rys Autor

Sposoacuteb deformacji pręta powoduje że konieczne jest rozpatrywanie dwu

odrębnych grup charakterystyk geometrycznych

Pierwsza grupa dotyczy skręcenia przekroju i nośnością z uwagi na moment

skręcający

Druga związana jest z deplanacją przekroju i nośnością z uwagi na bimoment

Dla dwuteownikoacutew wartości zestawione są w tablicach do projektowania

Rys europrofillu

Można też zastosować inne wzory przybliżone

JT asymp α (h1 t13 + h2 t2

3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3

Przekroacutej αDwuteownik

gorącowalcowany

120

Dwuteownik spawany ż

żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor

Dla przekrojoacutew zamkniętych moment bezwładności przy skręcaniu i wskaźnik

wytrzymałości przy skręcaniu jest liczony odmiennie

t

A0

Rys Autor

JT = 4 A02 int (ds t)

WT = 2 t A0

ds

Także dla przekrojoacutew okrągłych

JT = π r 4 2

WT = π r 3 2

Charakterystyki geometryczne związane z deplanacją przekroju są najbardziej

skomplikowane od strony obliczeniowej

W analogii do zginania można wprowadzić pojęcia wycinkowego momentu bezwładności

Jw wspoacutełrzędnej wycinkowej ω i wycinkowego wskaźnika wytrzymałości przekroju Ww

Ww = Jw ω

Przy zginaniu zgodnie ze wzorami z t 6

Wy = Jy z

Wz = Jz y

Jy i Jz maja jedną wartość dla całego przekroju Dla każdego punktu przekroju o

wspoacutełrzędnych (y z) można wyliczyć wartości Wy i Wz ndash aczkolwiek najczęściej

potrzebujemy Wy i Wz dla punktoacutew najbardziej odległych od środka ciężkości (Wy i Wz są

najmniejsze)

Analogicznie jest w przypadku Jw ndash ma jedną wartość dla całego przekroju zaś ω i Ww są

roacuteżne w roacuteżnych punktach Tu roacutewnież z najczęściej potrzebujemy Ww dla punktoacutew

położonych ekstremalnie (najmniejszego Ww)

Obliczenia wskaźnikoacutew wytrzymałości przy zginaniu dwukrotnie związane są z procedurą

polegającą na wstępnym przyjęciu pewnego punktu lub kierunku policzeniu na tej

podstawie pewnych charakterystyk geometrycznych a następnie zweryfikowaniu

wstępnego założenia przy pomocy tych charakterystyk

1 Środek ciężkości przyjmujemy dowolny punkt i dowolną orientację osi ndash wyliczamy

momenty statyczne względem tych osi ndash rzeczywiste położenie środka ciężkości jest

odległe od wstępnie przyjętego o proporcje między odpowiednim momentem statycznym a

polem powierzchni

2 Osie głoacutewne centralne w środku ciężkości przyjmujemy dowolnie zorientowane osie ndash

wyliczamy względem tych osi momenty bezwładności i momenty dewiacji ndash znajdujemy

położenie osi głoacutewnych centralnych i wartości odpowiadających im momentoacutew

bezwładności z warunku zerowania się momentoacutew dewiacji

Pętla bdquowstępne przyjęcie i weryfikacja na podstawie wynikoacutewrdquo występuje roacutewnież

dwukrotnie przy obliczaniu charakterystyk geometrycznych związanych z deplanacją

1a Sprowadzamy przekroacutej do linii środkowych gałęzi przekroju

1b Przyjmujemy tzw Biegun w środku ciężkości i Punkt Zerowy w innym

charakterystycznym miejscu najlepiej na przecięciu ktoacuterejś osi i linii przekroju

Rys Autor

B

0

1c Wykonujemy wykres wstępny wspoacutełrzędnej wycinkowej ω0 według zasady

bull w punkcie 0 ma wartość 0

bull krążymy po linii środkowej wokoacuteł bieguna zgodnie z ruchem wskazoacutewek zegara

przechodząc do najbliższego punktu węzłowego

bull w węźle wartość ω0 jest roacutewna (odległość gałęzi od B) (odległość węzła od 0)bull przechodzimy do następnego węzła zgodnie z ruchem wskazoacutewek zegara

bull w kolejnym węźle wartość ω0 jest roacutewna wartości w węźle poprzednim + iloczyn dla

nowego węzła

bull jeśli poruszamy się przeciwnie do zegara nowemu iloczynowi przypisujemy wartość

ujemną

bull pomiędzy węzłami wartości ω0 zmieniają się liniowo

Rys Autor

0

B

1d Liczymy wycinkowe momenty odśrodkowe

Jωy = int [ω0(s) y(s) t(s) ] ds

Jωz = int [ω0(s) z(s) t(s) ] ds

s ndash wspoacutełrzędna położenia na linii środkowej y z ndash odległości od środka ciężkości t ndash

grubości kolejnych gałęzi przekroju

1e Odległość punktu R zwanego w literaturze środkiem zginania ścinania skręcania

od środka ciężkości opisują wzory

y = -Jωy Jz

z = -Jωz Jy

1f Przyjmujemy nowy biegun w punkcie R i punkt zerowy w poprzednim miejscu 0 i

powtarzamy po raz drugi procedurę wyznaczenia wykresu wstępnego wspoacutełrzędnej

wycinkowej ω1

2a Wyliczamy wycinkowy moment statyczny

Sω = int [ω1(s) δ(s) ] ds

s(Q) = Sω A

s(Q) to odległość wzdłuż linii środkowej między punktem 0 a głoacutewnym punktem zerowym

Q Jeśli s(Q) gt 0 wędrujemy zgodnie z zegarem wokoacuteł R jeśli s(Q) lt 0 ndash przeciwnie do

zegara

2b dla bieguna w punkcie R i punktu zerowego w Q po raz trzeci powtarzamy procedurę

wyznaczenia wykresu wspoacutełrzędnej wycinkowej ω Tym razem jest to ostateczny wykres

3 Obliczamy wycinkowy moment bezwładności

Jω = int [ω(s)2 δ(s) ] ds

Charakterystyki geometryczne związane ze skręcaniem i deplanacją są ważne nie tylko w

przypadku skręcania ale i w przypadku przekrojoacutew cienkościennych (czasami taką metodą

liczone są przekroje aluminiowe lub stalowe zimnogięte) oraz dla spraw związanych z

utratą stateczności elementu przy ściskaniu czy zginaniu ( rarr wykład 6) Dla przekrojoacutew

bdquopodstawowychrdquo (dwuteowniki ceowniki) wycinkowy moment bezwładności i

wspoacutełrzędna wycinkowa są podane w tablicach do projektowania

Rys Tablice do projektowania konstrukcji

metalowych W Bogucki M Żyburtowicz

Arkady Wa-wa 1996

W razie potrzeby można użyć wzoroacutew przybliżonych

J Żmuda bdquoPodstawy projektowania konstrukcji metalowychrdquo TiT Opole 1992

Przykład obliczeniowy

2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor

Bez skręcania deplanacji i charakterystyk efektywnych

HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050

Oba ceowniki są ułożone horyzontalnie Oznacza to ze ich lokalna oś y

odpowiada globalnej osi z zaś lokalna oś z odpowiada globalnej soi y Należy o

tym pamiętać przy obliczaniu momentu bezwładności przekroju

A = Σ Ai = 2 A (L 150x150x15) + A (C 300) + A (HEB 220) + A (C 220) +

+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor

Środek ciężkości HEB

z3 = 0

Środek ciężkości 2 (L 150x150x15)


Środek ciężkości C 300


Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor


Środek ciężkości przekroju

Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +

+ [Jz (C 300) + A (C 300) z22-1] + [Jy (HEB 220) + A (HEB 220) z2

3-1] +

+ [Jz (C 220) + A (C 220) z24-1] + [2 Jy (L 100x100x10) + 2 A (L 100x100x10) z2

5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor

iy = radic ( Jy A) = 12776 cm

iz = radic ( Jz A) = 8927 cm

Wy el top = Jy ztop = 2 185364 cm3

Wy el bottom = Jy zbottom = 2 134960 cm3

Wz el = Jz ymax = 1 205291 cm3

Rys Autor

ztop = 237

zbottom = 243

ymax = 210

bdquoCzynne pole przekrojurdquo przy ścinaniu

Kształt przekroju przybliżono zestawem prostokątoacutew

Rys Autor

Vz

AV z

Vy

AV y

AV z = 2x (ramię L 150x150x15) + 2x (poacutełka C 300) + (środnik HEB 220) +

+ 2x (poacutełka C 220) + 2x (ramię L 100x100x10) =

= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y

AV y = 2x (ramię L 150x150x15) + (środnik C 300) + 2x (poacutełka HEB 220) +

+ (środnik C 220) + 2x (ramię L 100x100x10) =

= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698

Analiza plastyczna podział przekroju na dwa identyczne pola powierzchni

Przekroacutej nadal przybliżony jest zestawem prostokątoacutew

Oś pionowa z (= oś symetrii) dzieli przekroacutej na dwie identyczne połoacutewki

Oś pozioma ndash możliwe położenia

Rys Autor

1

2 3

4

5

1) Linia przechodzi przez ramiona L 150x150x15

2) Linia przechodzi przez środnik C 300

3) Linia przechodzi przez poacutełkę HEB 220

4) Linia przechodzi przez środnik HEB 220 i poacutełki

C 300

5) Linia przechodzi przez środnik HEB 220 poza

poacutełkami C300

Możliwość 1

Linia przechodzi przez ramiona L 150x150x15

W najniższym możliwym położeniu (po dolnych licach ramion otrzymamy wtedy

maksymalne pole części goacuternej)

Goacutera 2x L 150x150x15

Doacuteł C 300 + HEB 220 + 2x L 100x100x10 + C 220

Powierzchnia goacuterna = 2x 460 = 920

Powierzchnia dolna = 588 + 919 + 2x 192 + 374 = 2256

Nawet przy ekstremalnym położeniu maksymalizującym część goacuterną część goacuterna ma

zbyt małe pole

Możliwość 1 wykluczona

Możliwość 2

Linia przechodzi przez środnik C 300

W najniższym możliwym położeniu (po dolnym licu środnika otrzymamy wtedy


Goacutera 2x L 150x150x15 + środnik C 300

Doacuteł poacutełki C 300 + HEB 220 + 2x L 100x100x10 + C 220

Powierzchnia goacuterna = 2x 460 + 10 ∙ 300 = 1220



zbyt małe pole


Możliwość 5

Linia przechodzi przez środnik HEB 220 poza poacutełkami C300

Goacutera 2x L 150x150x15 + C 300 + goacuterna poacutełka HEB 220 + goacuterna część środnika HEB 220

Doacuteł 2x L 100x100x10 + C 220 + dolna poacutełka HEB 220 + dolna część środnika HEB 220

Założenie powierzchnia goacuterna = powierzchnia dolna

Powierzchnia (poacutelka goacuterna HEB 220) = powierzchnia (poacutełka dolna HEB 220) poacutełki możemy

pominąć

Powierzchnia goacuterna = 2x 460 + 588 + goacutera środnika = 1508 + goacutera środnika

Powierzchnia dolna= 2x 192 + 375 + doacuteł środnika = 759 + doacuteł środnika

Cały środnik (goacutera + doacuteł) asymp 17 za mało by doprowadzić do wyroacutewnania obu części


Do szczegoacutełowej analizy pozostały możliwości 3 i 4

Rys Autor

a a

b b

4) Linia przechodzi przez środnik HEB 220 i

poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)

3) Linia przechodzi przez poacutełkę HEB

220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300

część poacutełek C 300

cała goacuterna poacutełka HEB 220

część środnika HEB 220

2x (L 150x150x15)

web C 300


część goacuternej poacutełki HEB 220

brak środnika 220

A2 =

= A 2 = A1

brak poacutełki HEB 220



poacutełka dolna HEB 220

C 220

2x (L 100x100x10)


cały środnik HEB 220



C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski

Możliwość 4 położenie osi opisane jest przez zależność 225 mm le a le 1000 mm

rezultat a = -14 mm sprzeczność możliwość wykluczona

Możliwość 3 położenie osi opisane jest przez zależność 100 mm le a lt 225 mm

rezultat a = 19 mm OK możliwość przyjęta

19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +

+ Sy (środnik C 300) + Sy [ 2x (goacuterne części poacutełek C 300)] +

+ Sy (goacuterna część poacutełki HEB 220)

Sy (A2) = Sy (dolna część poacutełki HEB 220) +

+ Sy [ 2x (dolne części poacutełek C 300)] +

+ Sy (środnik HEB 220) + Sy (dolna poacutełka HEB 220) +

+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +

+ z2-1 A(środnik C 300) +

+ z3-1 [ 2x A(goacuterne części poacutełek C 300)] +

+ z4-1 A(goacuterna część poacutelki HEB 220)

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]

Sy (A2) = z5-1 A(dolna część poacutełki HEB 220) + z6-1 [ 2x A(dolne części poacutelek C 300)] +

+ z7-1 A(środnik HEB 220) + z8-1 A(poacutelka dolna HEB 220) + z9-1 A(C 220) +

+ z10-1 [2x A(L 100x100x10)]

Wz pl = |Sz (A1)| + |Sz (A2)| = 2x |Sz (A1)| = 2x |Sz (A2)|

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +

+ Sz (lewa część środnika C 300) +

+ Sz (lewa część poacutełki goacuternej HEB 220) +

+ Sz (lewa poacutełka C 300) +

+ Sz (lewa część środnika HEB 220) +

+ Sz (lewa część poacutełki dolnej HEB 220) +


+ Sz (lewa poacutelka C 220) +

+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +

+ y2-1 A(lewa część środnika C 300) +

+ y3-1 A(lewa część poacutełki goacuternej HEB 220) +

+ y4-1 A(lewa poacutełka C 300) +

+ y5-1 A(lewa część środnika HEB 220) +

+ y6-1 A(lewa część poacutełki dolnej HEB 220) +



+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor

Wy pl = |Sy (A1)| + |Sy (A2)| = 64214 [cm3] + 2 64066 [cm3] = 3 28280 [cm3]

Wz pl = |Sz (A1)| + |Sz (A2)| = 2x |Sz (A1)| = 2x |Sz (A2)| = 2 17518 [cm3]

Charakterystyka WartośćA [cm2] 3176

AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536

Wy el bottom [cm3] 2 13496

Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518

Sprawdzenie wynikoacutew

AV y + AV z asymp A OK

min (Wy el top Wy el bottom ) lt Wy pl OK

Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor

Rodzaje wskaźnikoacutew wytrzymałości przy zginaniu

Definicje deplanacji przekroju i bimomentu

Zagadnienia egzaminacyjne

Dziękuję za uwagę

copy 2019 dr inż Tomasz Michałowski

tmichaluskpkedupl

Spis treści

Podstawowe charakterystyki geometrycznerarr t 3

Zaawansowane charakterystyki geometrycznerarr t 27

Przykład obliczeniowy rarr t 58

Zagadnienia egzaminacyjne rarr t 88


Rys Autor


A = int int dz dy

y0

z0

Rys Autor




y0

z0

y1

z1

∆z

∆y

CoG


ciężkości

∆y = Sz A

∆z = Sy A



2 ] 2


2 ] 2


Rys Autor






y1

z1 z

y

α










iy = radic (Jy A)

iz = radic (Jz A)

Rys Autor

y

z

ymax left

ymax right

zmax top

zmax bottom


A = b h

Jy = b h3 12



Rys Autor






Rys Autor

y

z


σ (y z) = σ (z)





σ (ztop) = σtop

σ (0) = 0

σ (0) = 0


z











My = σmax Wy max

My max = fy Wy max





zginanie)

h rarr ograniczone

Na przykład

A = 2 a2

h le 3a

Rys Autor

b = 2a h = a

Jy = b h3 12 = 0167 a4

Wy = b h2 6 = 0333 a3


My max = 0333 a3 fy

Rys Autor

b = a h = 2a

Jy = b h3 12 = 0667 a4

Wy = b h2 6 = 0667 a3


My max = 0667 a3 fy

Rys Autor

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl


Rys Autor


A = int int dz dy

y0

z0

Rys Autor




y0

z0

y1

z1

∆z

∆y

CoG


ciężkości

∆y = Sz A

∆z = Sy A



2 ] 2


2 ] 2


Rys Autor






y1

z1 z

y

α










iy = radic (Jy A)

iz = radic (Jz A)

Rys Autor

y

z

ymax left

ymax right

zmax top

zmax bottom


A = b h

Jy = b h3 12



Rys Autor






Rys Autor

y

z


σ (y z) = σ (z)





σ (ztop) = σtop

σ (0) = 0

σ (0) = 0


z











My = σmax Wy max

My max = fy Wy max





zginanie)

h rarr ograniczone

Na przykład

A = 2 a2

h le 3a

Rys Autor

b = 2a h = a

Jy = b h3 12 = 0167 a4

Wy = b h2 6 = 0333 a3


My max = 0333 a3 fy

Rys Autor

b = a h = 2a

Jy = b h3 12 = 0667 a4

Wy = b h2 6 = 0667 a3


My max = 0667 a3 fy

Rys Autor

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Rys Autor




y0

z0

y1

z1

∆z

∆y

CoG


ciężkości

∆y = Sz A

∆z = Sy A



2 ] 2


2 ] 2


Rys Autor






y1

z1 z

y

α










iy = radic (Jy A)

iz = radic (Jz A)

Rys Autor

y

z

ymax left

ymax right

zmax top

zmax bottom


A = b h

Jy = b h3 12



Rys Autor






Rys Autor

y

z


σ (y z) = σ (z)





σ (ztop) = σtop

σ (0) = 0

σ (0) = 0


z











My = σmax Wy max

My max = fy Wy max





zginanie)

h rarr ograniczone

Na przykład

A = 2 a2

h le 3a

Rys Autor

b = 2a h = a

Jy = b h3 12 = 0167 a4

Wy = b h2 6 = 0333 a3


My max = 0333 a3 fy

Rys Autor

b = a h = 2a

Jy = b h3 12 = 0667 a4

Wy = b h2 6 = 0667 a3


My max = 0667 a3 fy

Rys Autor

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl



2 ] 2


2 ] 2


Rys Autor






y1

z1 z

y

α










iy = radic (Jy A)

iz = radic (Jz A)

Rys Autor

y

z

ymax left

ymax right

zmax top

zmax bottom


A = b h

Jy = b h3 12



Rys Autor






Rys Autor

y

z


σ (y z) = σ (z)





σ (ztop) = σtop

σ (0) = 0

σ (0) = 0


z











My = σmax Wy max

My max = fy Wy max





zginanie)

h rarr ograniczone

Na przykład

A = 2 a2

h le 3a

Rys Autor

b = 2a h = a

Jy = b h3 12 = 0167 a4

Wy = b h2 6 = 0333 a3


My max = 0333 a3 fy

Rys Autor

b = a h = 2a

Jy = b h3 12 = 0667 a4

Wy = b h2 6 = 0667 a3


My max = 0667 a3 fy

Rys Autor

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl










iy = radic (Jy A)

iz = radic (Jz A)

Rys Autor

y

z

ymax left

ymax right

zmax top

zmax bottom


A = b h

Jy = b h3 12



Rys Autor






Rys Autor

y

z


σ (y z) = σ (z)





σ (ztop) = σtop

σ (0) = 0

σ (0) = 0


z











My = σmax Wy max

My max = fy Wy max





zginanie)

h rarr ograniczone

Na przykład

A = 2 a2

h le 3a

Rys Autor

b = 2a h = a

Jy = b h3 12 = 0167 a4

Wy = b h2 6 = 0333 a3


My max = 0333 a3 fy

Rys Autor

b = a h = 2a

Jy = b h3 12 = 0667 a4

Wy = b h2 6 = 0667 a3


My max = 0667 a3 fy

Rys Autor

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl


A = b h

Jy = b h3 12



Rys Autor






Rys Autor

y

z


σ (y z) = σ (z)





σ (ztop) = σtop

σ (0) = 0

σ (0) = 0


z











My = σmax Wy max

My max = fy Wy max





zginanie)

h rarr ograniczone

Na przykład

A = 2 a2

h le 3a

Rys Autor

b = 2a h = a

Jy = b h3 12 = 0167 a4

Wy = b h2 6 = 0333 a3


My max = 0333 a3 fy

Rys Autor

b = a h = 2a

Jy = b h3 12 = 0667 a4

Wy = b h2 6 = 0667 a3


My max = 0667 a3 fy

Rys Autor

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl






Rys Autor

y

z


σ (y z) = σ (z)





σ (ztop) = σtop

σ (0) = 0

σ (0) = 0


z











My = σmax Wy max

My max = fy Wy max





zginanie)

h rarr ograniczone

Na przykład

A = 2 a2

h le 3a

Rys Autor

b = 2a h = a

Jy = b h3 12 = 0167 a4

Wy = b h2 6 = 0333 a3


My max = 0333 a3 fy

Rys Autor

b = a h = 2a

Jy = b h3 12 = 0667 a4

Wy = b h2 6 = 0667 a3


My max = 0667 a3 fy

Rys Autor

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl











My = σmax Wy max

My max = fy Wy max





zginanie)

h rarr ograniczone

Na przykład

A = 2 a2

h le 3a

Rys Autor

b = 2a h = a

Jy = b h3 12 = 0167 a4

Wy = b h2 6 = 0333 a3


My max = 0333 a3 fy

Rys Autor

b = a h = 2a

Jy = b h3 12 = 0667 a4

Wy = b h2 6 = 0667 a3


My max = 0667 a3 fy

Rys Autor

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl





zginanie)

h rarr ograniczone

Na przykład

A = 2 a2

h le 3a

Rys Autor

b = 2a h = a

Jy = b h3 12 = 0167 a4

Wy = b h2 6 = 0333 a3


My max = 0333 a3 fy

Rys Autor

b = a h = 2a

Jy = b h3 12 = 0667 a4

Wy = b h2 6 = 0667 a3


My max = 0667 a3 fy

Rys Autor

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

b = 2a h = a

Jy = b h3 12 = 0167 a4

Wy = b h2 6 = 0333 a3


My max = 0333 a3 fy

Rys Autor

b = a h = 2a

Jy = b h3 12 = 0667 a4

Wy = b h2 6 = 0667 a3


My max = 0667 a3 fy

Rys Autor

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

b = a h = 2a

Jy = b h3 12 = 0667 a4

Wy = b h2 6 = 0667 a3


My max = 0667 a3 fy

Rys Autor

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

b = 2 a 3 h = 3 a

Jy = b h3 12 = 1500 a4

Wy = b h2 6 = 1000 a3


My max = 1000 a3 fy

Rys Autor

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Jy = 2 [ b h3 12 ] + 2 [ A1 d2 ]


b = a h = a d = a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0167 a4 + 2000 a4 = 2167 a4

Wy = Jy z = 1444 a3


My max = 1444 a3 fy

Rys Autor

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

b = 2a h = 05a d = 125 a x = 15 a A1 = a2

Jy = 2 [ b h3 12 ] + 2 [A1 d2 ] =

= 0042 a4 + 3125 a4 = 3167 a4

Wy = Jy z = 2111 a3


My max = 2111 a3 fy

Rys Autor

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

A 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a


Jy 1000 4000 9000 13000 19000

Wy 1000 2000 3000 4333 6333

iy 1000 2000 3000 3602 4221

My max 1000 2000 3000 4333 6333

Rys Autor


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl


Jaki to błąd





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl





Jy = 2 [ b h3 12 ]

b = a h = a d = a x = 15 a

Jy = 0167 a4

Wy = Jy z = 0111 a3


My max = 0111 a3 fy

Rys Autor


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl


Jy = 2 [ b h3 12 ]

b = 2a h = 05a d = 125 a x = 15 a

Jy = 0042 a4

Wy = Jy z = 0028 a3


My max = 0028 a3 fy

Rys Autor


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl


A 2a2 2a2 2a2 2a2 2a2 2a2 2a2

Jy 0167 a4 0667 a4 1500 a4 2167 a4 3167 a4 0167 a4 0042 a4

Wy 0333 a3 0667 a3 1000 a3 1444 a3 2111 a3 0111 a3 0028 a3

iy 0289 a 0577 a 0866 a 1041 a 1258 a 0289 a 0145 a


Jy 1000 4000 9000 13000 19000 1000 0250

Wy 1000 2000 3000 4333 6333 0333 0083

iy 1000 2000 3000 3602 4221 1000 0500

My max 1000 2000 3000 4333 6333 0333 0083

Rys Autor


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl


I


Rys Autor





Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl



Rys Autor




IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

IPE A 600 [mm]

h b tf tw r

597 220 175 98 24

Jy = 82 920 cm4

Rys Autor


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [ (1753 22 12) + 175 22 (597 2ndash 175 2)2 ] =

= 79 161 cm4 (9547)


Jy = (597 ndash 2 175)3 098 12 +

+ 2 [175 22 (597 2ndash 175 2)2 ] = 79 142 cm4

(9544)



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl



Rys Autor

Zakres sprężysty

liniowy rozkład

naprężeń

Stały rozkład

naprężeń w


rozciąganej

Liniowo zmienne

Stałe

σtop = fy

Stałe

σbottom = fy

Zakres spręzysto-

plastycznyZakres

plastyczny

σbottom le fy

σtop le fy

σbottom = fy

σtop = fy


Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Rys Autor

Ogoacutelnie

My = intA

int σ(y z) z dz dy

and

N = intA

int σ(y z) dz dy



N = 0

oraz

My ne 0

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Rys Autor

N = intA

int σ(y z) dz dy =

= intA1


int -fy dz dy =

= fy intA1


int dz dy =


= A1 fy - A2 fy

N = 0 rarr A1 = A2

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Rys Autor








starą osią yA1

A2

yy

z

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Rys Autor


A1el ne A2el

S1el

S2el

yel

ypl

yel

ypl

A2pl

A1pl



Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Rys Autor

A1 = A2 = A 2


= intA1


int (-z fy) dz dy =

= fy ( intA1

int z dz dy - intA2

int z dz dy ) =


= fy [ |S(A1)y| + |S(A2)y|]

A1

A2

σ(y z) = fy

σ(y z) = -fy

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Rys Autor

My = fy [ |S(A1)y| + |S(A2)y|]


My = fy Wpl y



plastycznym




Wel i = Ji | jmax|

Wpl i gt Wel i

A1

A2 yy

z






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl






Rys Autor








Weff le Wel lt Wpl

Aeff le A

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Rys Autor

Ścinanie




τ = V AV


AV = Jy t(z = 0) S y(z = 0)


ścinania

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Ogoacutelnie

tf ltlt h

Jy asymp h3 tw 12 + 2 [tf b (h 2)2] =

= h2 (h tw 12 + tf b 2)


asymp tw (h 2) (h 4) + tf b (h 2) =

= h (h tw 8 + tf b 2)

Rys Autor

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

AV = Jy tw S y =




= tw h (A 2) [(1 6) +1] (A 2) [(1 4) + 1] =

= tw h [(1 6) +1] [(1 4) + 1] = tw h (1167 1250) asymp tw h

AV asymp tw h

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Uogoacutelnienie


działania siły

Rys Autor

Vz

Vy











TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl










TEd

∆x

∆x

∆x












TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl


TEd

Uwagi

Okrągły (pręt

rura)

Brak TEd = Tt Ed -

L

T


Skrępowana

podporami

TEd asymp Tt Ed -


Skrępowana

podporami


należy wziąć BEd



B Ed - bimoment



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl



kierunkach



w poacutelkach

σ

σ



bimomentu

B = h M

M = |M1| = |M2|




h

M1

M2

Rys Autor




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl




skręcaniu

WT = JT tmax


h1

h2

h3

t1

t2

t3

Rys Autor




skręcający



Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl


Rys europrofillu



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl



3 + h3 t33 + ) 3

h1

h2

h3

t1

t2

t3


gorącowalcowany

120


żebrami pionowymi

150

Kątownik 100

Ceownik 112

Teownik 140

Rys Autor



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl



t

A0

Rys Autor


WT = 2 t A0

ds


JT = π r 4 2

WT = π r 3 2





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl





Ww = Jw ω


Wy = Jy z

Wz = Jz y




najmniejsze)











polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl








polem powierzchni










Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl




Rys Autor

B

0







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl







nowego węzła


ujemną


Rys Autor

0

B








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl








y = -Jωy Jz

z = -Jωz Jy



wycinkowej ω1



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl



s(Q) = Sω A



zegara













Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl









Arkady Wa-wa 1996




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl




2x L 150x150x15

HEB 220

C 2202x L 100x100x10

C 300

Rys Autor


HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

HEB 220

y

z

220

220

125

189

C 220

y

z

220

65

125

9

125

1 125

214 + 586

80

40 + 40

Rys Autor

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

y

z

300

C 300

10

81 125

16

16

50 + 50

100

27 + 73

L 150x150x15

L 100x100x10

y

y

z

z

10

10

6

12

100

100

45 + 10545 + 105

150

150

8

1515

15

282 + 718282 + 718

Rys Autor

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

A

[cm2]

Jy

[cm4]

Jz

[cm4]

e + f

[cm]

HEB 220 910 8 090 2 840 -

C 220 374 2 690 197 214 + 586

C 300 588 8 030 495 270 + 730

L 100x100x10 192 177 177 282 + 718

L 150x150x15 460 1 100 1 100 450 + 1050





+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl


+ 2 A (L 100x100x10) =

= 2 ∙ 460 + 588 + 910 + 374 + 2 ∙ 192 =

= 3176 cm2

z1 = 220 2 + 10 + 45 = 165

z2 = 220 2 + 10 - 27 = 93

z4 = 220 2 + 214 = 1314

z5 = 220 2 + 282 = 1382

Rys Autor


z3 = 0





Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Sy = Σ Si y = 2 A (L 150x150x15) z1 +

+ A (C 300) z2 + A (HEB 220) z3 +

+ A (C 220) z4 + 2 A (L 100x100x10) z5 =

= 2 ∙ 460 ∙ 165 + 588 ∙ 93 +

+ 910 ∙ 0 - 374 ∙ 1314 - 2 ∙ 192 ∙ 1382 =

= 1042716 cm3

z3-1 = Sy A = 1042716 3176 = 328 cm

z1-1 = 165 - 328 = 1322

z2-1 = 93 - 328 = 602

z3-1 = 328

z4-1 = 1314 + 328 =

1642

z5-1 = 1382 + 328 = 171

Rys Autor



Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Jy = [2 Jy (L 150x150x15) + 2 A (L 150x150x15) z21-1] +


3-1] +


5-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 13222] + [495 + 588 ∙ 6022] + [8 090 + 910 ∙ 3282] +

+ [197 + 374 ∙ 16422] + [2 ∙ 177 + 2 ∙ 192 ∙ 1712] =

= 51 836840 cm4

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Jz = [2 Jz (L 150x150x15) + 2 A (L 150x150x15) y21-1] +

+ [Jy (C 300)] + [Jz (HEB 220)] +

+ [Jy (C 220)] + [2 Jy (L 100x100x10) +

+ 2 A (L 100x100x10) y25-1] =

= [2 ∙ 1 100 + 2 ∙ 460 ∙ 452] + [8 030] + [2 840] +

+ [2 690] + [2 ∙ 177 + 2 ∙ 192 ∙ 13822] =

= 25 311108 cm4

y1-1 = 45

y5-1 = 220 2 + 282 = 1382

Rys Autor






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl






Rys Autor

ztop = 237

zbottom = 243

ymax = 210



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl



Rys Autor

Vz

AV z

Vy

AV y



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl



= 2 ∙ (15 ∙ 15) + 2 ∙ (10 ∙ 16) + 22 ∙ 09 + 2x (8 ∙ 125) + 2 ∙ (10 ∙ 1) = 1368

Rys Autor

Vz

AV z

Vy

AV y



= 2x (15 ∙ 15) + 30x1 + 2 ∙ (22 ∙ 125) + 22 ∙ 09 + 2 ∙ (10 ∙ 1) = 1698





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl





Rys Autor

1

2 3

4

5





C 300


poacutełkami C300

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Możliwość 1









zbyt małe pole


Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Możliwość 2









zbyt małe pole


Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Możliwość 5






pominąć






Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl


Rys Autor

a a

b b


poacutełki C 300

(225 mm le a le 1000 mm a + b = 1000 mm)


220

(100 mm le a lt 225 mm a + b = 100 mm)

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

Rys Autor

a a

b b

4 3

A1 =

= A 2 = A2

2x (L 150x150x15)

środnik C 300




2x (L 150x150x15)

web C 300



brak środnika 220

A2 =

= A 2 = A1





C 220

2x (L 100x100x10)





C 220

2x (L 100x100x10)

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

4 3

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

22 ∙ 125

(a - 1 - 125) ∙ 09

2 ∙ (460)

30 ∙ 1

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

[22 - 125 - (a - 1)] ∙ 09

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

[125 - (a - 1)] ∙ 22

09 ∙ (22 - 2 ∙ 125)

(10 - a) ∙ 16

22 ∙ 125

374

2 ∙ (192)

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

4 3

92

30

(a - 1) ∙ 16

275

(a - 225) ∙ 09

92

30

(a - 1) ∙ 16

(a - 1) ∙ 22

0

0

(2175 - a) ∙ 09

(10 - a) ∙ 16

275

374

384

(225 - a) ∙ 22

1755

(10 - a) ∙ 16

275

374

384

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

4 3

1495

16a - 16

09a - 2025

122

16a - 16

22a - 22

19575 - 09a

16 - 16a

1033

495 - 22a

16 - 16a

12085

4 3

25a + 145875 236a + 98400

138875 - 25a 186450 - 236a

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

4 3

25a + 145875 =

= 138875 - 25a

236a + 98400 =

= 186450 - 236a

4 3

50a = -7000 472a = 88050

4 3

a = -14 a = 19

Wnioski





19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

19

81

Wy pl = |Sy (A1)| + |Sy (A2)|

Rys Autor

yel

68

ypl

Sy (A1) = Sy [2x (L 150x150x15)] +






+ Sy (C 220) + Sy [2x (L 100x100x10)]

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

z1-1 = 64

Rys Autor

ypl

Sy (A1) = z1-1 [2x A(L 150x150x15)] +




z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

Sy (A1) = 64 ∙ 2 ∙ 460 + 14 ∙ 30 ∙ 1 + 05 ∙ 2 ∙ 16 ∙ 09 + 05 ∙ 22 ∙ 09 = 64214

[cm3]

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

z1-1 = 64

Rys Autor

ypl

z2-1 = 14

z3-1 = z4-1 = 5

z5-1 = 2

z6-1 = 41

z7-1 = 101

z8-1 = 205z9-1 = 232 z10-1 = 239

|Sy (A2)| = 02 ∙ 04 ∙ 22 + 2 ∙ 41 ∙ 81 ∙ 16 + 101 ∙ 195 ∙ 09 + 205 ∙ 125 ∙ 22 +

+ 232 ∙ 374+ 239 ∙ 2 ∙ 192 = 2 64066 [cm3]



+ z10-1 [2x A(L 100x100x10)]


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl


y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = Sz (L 150x150x15) +








+ Sz (L 100x100x10)

zel = zpl

Rys Autor

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

y1-1 = 45

y2-1 = 75

y3-1 =

= y6-1 =

= y7-1 =

55

y4-1 = 142

y5-1 = 2

y8-1 = 104

y9-1 = 138

|Sz (A2)| = y1-1 A(L 150x150x15) +








+ y9-1 A(L 100x100x10) =

= 45 ∙ 46 + 75 ∙ 15 ∙ 1 + 55 ∙ 11 ∙ 125 +

+ 142 ∙ 9 ∙ 16 + 02 ∙ 195 ∙ 05 + 55 ∙ 11 ∙ 125 +

+ 55 ∙ 11 ∙ 09 + 104 ∙ 125 ∙ 7 + 138 ∙ 192 =

= 108759 [cm3]

zel = zpl

Rys Autor




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl




AV y [cm2] 1698

AV z [cm2] 1368

Jy [cm4] 51 836840

Jz [cm4] 25 311108

iy [cm] 1278

iz [cm] 893

Wy el top [cm3] 2 18536


Wz el [cm3] 1 20529

Wy pl [cm3] 3 28280

Wz pl [cm3] 2 17518




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl




Wz el lt Wz pl OK

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl

yel

ypl

zel = zpl

243

68

169

Rys Autor






tmichaluskpkedupl






tmichaluskpkedupl

konstrukcje metalowe wykład iii geometria przekrojufootbridge.pl/stud/z/sn1/kbi/w103.pdf ·...

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