stability of structures - solved examples

7/30/2019 Stability of structures - Solved examples

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0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0 = 0

0 > 0

= P/Pkr = P L/(4k).


6/134

L/2

F = 4k/L

22222222

dd dd '

Pk

0

cF

cos 0L/2 cos 0L/2

(; 0, ) =

1

2 k[2( 0)]2

P L(cos 0 cos ) 1

2 F L(sin 0 sin )

=d

d =

4k( 0) P L sin + 1

2F L cos

= 0 = 0

P = 4kL 0 +

12

cos

sin

= 20 L

P = 0

= 20

d

d= ( 0) P L sin + 4k0 cos

= 4k P L sin + 4k0(cos 1) = 0

= 0 PI,

P = 4

k

L

+ 0(cos 1)

sin PII

2 =d2

d2 ()2 = (4k P L cos 4k0 sin )()2 = 0


7/134


8/134


9/134


10/134

EI

k

k

pP =1

48

F L3

EI

pq =5

384

qL4

EI.

jP =1

8

F L2

kand jq =

1

16

qL3

k.

jP = pP

kP = 6EIL

jq = pq

kq = 12EIL

Pcr =4k

L PPkr =

24EI

L2and Pqcr =

48EI

L2.

k


11/134


12/134

sin = 0 (k1 4k2)L2 cos + 4k2L2 2k2Lu = 0

k2 = k k1 = k

P = kL 4 + ( 32

4)cos (, P)

cos = 1 +P

2kL 3

2

u

L,

P = kL4

2 + (8 3)

u

L ,(u, P)

= 0

u =P

3k

P = [4 + (32

4)cos ]kL

P =

kL

4 2 + (8 3) uL = 1 = 5

= 1

P = (4 52

cos )kL

P =1

3kL

2 + 5

u

L

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

PkL


13/134

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

PkL

u/L

0.333*(2+5*x)3*x

(Pkr =32

kL

= 5

P = (4 + 72

cos )kL

P =

kL 10

7u

L

0

1

2

3

4

5

6

7

8

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

PkL


14/134

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3

PkL

u/L

-10+7*x3*x

k1 = 5k2

0

U =1

2k1L

2(sin sin 0)2 + k2u2 + 12

[u 2L(cos 0 cos )]2

= k1L2(sin

sin 0)cos + k2[u

2L(cos 0

cos )](

2L sin ) = 0

u= 2k2u + k2[u 2L(cos 0 cos )] P = 0

u

u =k1

2k2

sin sin 0tan

L 2L(cos cos 0)

P = 3k2u 2k2L((cos cos 0) = 3k12

sin sin 0tan

L 8k2L((cos cos 0)


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16/134


17/134

= (1, 2) =1

2k21(1, 2) +

1

2k22(1, 2) P(1, 2)

= 0

1= k1

11

+ k221

P1

= 0

2= k1

12

+ k222

P2

= 0

11

= 2 + 21 +1

222 12

2

1 = 1 1

2

2

1 1

2

2

2 + 12

1= L

21 2 + 31

3

2212 +

3

21

22

1

232

12

= 1 12

21 1

222 + 12

22

= 2 + 22 12 +1

221

2= L

22 1 + 32

1

231 +

3

2212

3

21

22

1 = 2 = = 0 1 = 2 = 0

2 =2

21[(1)]

2 + 22

12(1)(2) +

2

22[(2)]

2

= ((1) (2))

22

1

212

212

22

1

(1)

(2)

,

K = [2/ij ]

2

> 0 K

1 = 2 = 0

2

21= k1

2121

+ k

11

2+ k2

2221

+ k

21

2 P

2

21

2

21= 5k 2P L

2

12= 4k + P L


18/134

2

22= 5k 2P L

P = k/L K K

Kx = x (K I)x = 0 det(K I) = 0

det

5 2 4

4 5 2

= 0

1 = 1 ja 2 = 9 3 = 1 = 3

< 1 K

= 1

3 33 3

1

2

=

0

0

1 = 2

Pkr = k/L

2222222222

= 3 1 11 1

12

=

00

1 = 2Pkr = 3k/L

2222222222 22222222

22 = 1

1 = 2 = 1 = 21

= 0 k + 16

3

2 + 12

2 1 P L 1

23

= 0

k 1 + 16

2

1 + 12

2 P L 1 + 1

22

= 0

= 0 tai PII =

1 + 16

2

kL

= 3 1 = 2 = 2 =1

1

= 0 k 3+ 32

3

2 + 52

2 (1 22) P L 3+ 92

3

= 0

3k 1 + 122 3 + 922 P L 3 +

92

2 = 0 = 0 tai PII I = 1 + 122 3kL


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0

0.5

1

1.5

2

2.5

3

3.5

4

-0.4 -0.2 0 0.2 0.4

PIIPII I

PII PII I

PII 1 = 2

2

21

PII

= k

+

1

63

+ k

2 +

1

222

+ k

+

1

63

0 + k(1)2 k

1 +1

62

2 +3

22

= k

3 +7

62 +

1

64

> 0

PII I 1 = 2, 1 = 2

2

21

PIII

= k

3+

3

23

3+ k

2 +

5

222

k

3+3

23

(2) + k(1 22)2

3k

1 +1

22

2 +15

22

= k 1 +7

2

2 +3

2

4 < 0, PII PII I


20/134

cv

E' a ' Eac

P

T

c

L

(v, ) =1

2k2A +

1

2k2B PD

= k(v2 + a2 sin2 ) P[v + L(1 cos )]= ka2(u2 + sin2 ) P

u +

1

(1 cos )

,

v = au a = L P = ka

ka2

= = u2 + sin2

u + 1

(1 cos )

u= 2u = 0 u =

2

= 2 sin cos 1

sin = 0

= sin

2cos 1

= 0

sin = 0 = 0 = 2 cos

u = /2 = 0

2

u2= 2,

2

u= 0,

2

2= 2cos 2

cos

u = /2 = 0

K = 2 0

0 2


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22/134

x

x x

x

da = dx E x, u

cy, v

dadv

u u + du

sin = dvda

= dvdx

= v

= arcsin v

= 1/R =

1

R= =

v1 (v)2 ,

d arcsin x

dx=

11 x2

da E x, u

cy, v

dadv

dx

da = dv2 + dx2 = dx(v)2 + 1


23/134

tan =dv

dx= v

= arctan v

=1

R=

a

not

x!!

=1

1 + v2

a

dv

dx

where

d arctan x

dx=

1

1 + x2

=1

1 + v2d2v

dx21

1 + v2(where da = dx

1 + v2)

=v

(1 + v2)

1 + v2=

v

(1 + v2)3/2

= v


24/134

Pcr

e ee eE'

v(4)1 + k2v1 = 0 k

2 = P/2EI

v(4)2 = 0

BC : v1L

2

= v1

L2

= v2

L2

= v2

L2

= 0

v1(0) = v2(0)

v1(0) = v2(0)

M1(0) = M2(0)

Q1(0) = Q2(0) + P v2(0)

Pz

M110

E

Q1#

P'

Q2

M2()'

v1 = C1 sin kx + C2 cos kx + C3x + C4

v1 = C1k cos kx C2k sin kx + C3v1 = C1k2 sin kx C2k2 cos kxv1 = C1k3 cos kx + C2k3 sin kxv2 = C5x

3 + C6x2 + C7x + C8

v2 = 3C5x2 + 2C6x + C7

v2 = 6C5x + 2C6

v2 = 6C5

Q1(0) = Q2(0) + P v2(0)

2EI v1 (0) = EI v2 (0) + P v2(0)2C1k

3 = 6C5 + 2k2C7C5 =

1

3

k3C1 +1

3

k2C7


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26/134

Pcr = 2, = 1

dd

dddd

dd

E'

= 1 (EA)1 = (EA)2 P1 = P2 = P/2v(4)1 + k

21v

1 = 0 k

21 =

P/22EI

= P4EI

v(4)2

k22v

2 = 0 k

22 =

P/2EI

= P2EI

k22 = 2k21 k2 = 2k1

BC : v1L

2

= v1

L2

= v2

L2

= v2

L2

= 0

v1(0) = v2(0)

v1(0) = v2(0)

M1(0) = M2(0)

Q1(0) = Q2(0) + P v2(0)

P2z

M110

E

Q1#

P'

Q2

M2()'

P2

z


v1 = C1k cos kx C2k sin kx + C3v1 = C1k2 sin kx C2k2 cos kxv1 = C1k3 cos kx + C2k3 sin kxv2 = C5 sinh kx + C6 cosh kx + C7x + C8

v2 = C5k cosh kx C6k sinh kx + C7v2 = C5k2 sinh kx C6k2 cosh kxv2 = C5k3 cosh kx + C6k3 sinh kx

Q1(0) = Q2(0) + P v2(0)

2EI v1 (0) = EI v2 (0) + P v1(0)2C1k

31 = C5k32 + 4k21(C1k1 + C3)

C5 = 1

k32

2k31C1 + 4k21C3

=1

2C1 +2

2k1C3


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28/134

'

(v) =1

2

L0

EI(v)2 P(v)2 dx,

P

dx + du

dv

= ,vv =

L0

(EI vv P vv )dx = 0,

v

v(0) = v(0) = 0v

=

L

0

EI vv L0

(EI v)v dx L

0

P vv +

L0

vdx

=

L

0

EI vv L

0

(EI v)v L

0

P vv +

L0

[(EI v) + (P v)] vdx

v(0) = v (0) = 0


29/134

M = EI v Q = (EI v)

= M(L)v (L) + [Q(L) P v(L)]v(L) +L0

[(EI v) + (P v)] vdx = 0,

v v(0) = v(0) = 0

(EI v) + (P v) = 0 x (0, L)

M(L) = 0

Q(L) P v(L) = 0

v(0) = 0

v(0) = 0

EI P

EI v(4) + P v = 0

EI v + P v = Cx + D, (C, D constants) v = A sin kx + B cos kx + Cx + D, k =

P

EI

v = Ak cos kx Bk sin kx + Cv = Ak2 sin kx Bk2 cos kxv = Ak3 cos kx + Bk3 sin kx

v(0) = 0 B + D = 0v(0) = 0 Ak + C = 0

v(L) = 0

A sin kL + B cos kL = 0

EI v(L)P v(L) = 0 EI(Ak3 cos kL+Bk3 sin kL)P(Ak cos kLBk sin kL+C) = 0

P = EI

L2 k =

L

A = 0 C = 0 B cos kL = 0 B = 0 cos kL = 0B = 0 v 0

cos kL = 0 kL = 2

+ n, n = 0, 1, 2,...


30/134

n =

2+ n

2,

0 = 22 Pcr =

2

4

EI

L2

n

vn = B(cos knx 1), kn = 1L

2

+ n

||vn||2E =L

0

EI(vn)2dx.

L0

EI vnvmdx = 0, kun n = m.

vn ||vn||E = E1 E1[E1] =

Nm

vn = Bk2n cos knx

E21 = EI B2k4nL0

cos2 knxdx

y = knx, dx =1

kndy rajat

x = 0 y = 0x = L y =

2+ n

E21 = EI B2k3n

2

+n0

cos2 ydy = EI B2k3n1

2

2

+ n

B2 = 2E21

EI k3n2

+ n = 2E21

EI4

16(1 + 2n)4

B = 4

2L3/2E1

2(1 + 2n)2

EI

vn(x) = bn(cos knx 1), Bn =4

2

2(1 + 2n)2E1L3/2

EI


31/134

L0

EI vnvmdx = 0, kun n = m.

L0

vnvmdx =

L0

cos

2(1 + 2n)

x

L

cos

2

(1 + 2m)x

L

merk. y =

2

x

L, dx =

2L

dy

=2L

L0

cos[(1 + 2n)y] cos[(1 + 2m)y]dy = 0, kun n = m.


32/134


33/134

x = L

2

Mt =

qL2

2tan2

1

(x = 0) Mk =qL2

2sin2

1

P/Pkr Mt/qL2 Mk/qL

2 v(0)/ qL4

EI

22

32

3.62

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

MqL2

P/Pcr

MtMk


34/134

P/PE

PE

e e

e eE

'c

v(4)1 + k

2v1 = 0 k2 = P

2EI

v(4)2 = 0

BC : v1L

2

= v1

L2

= 0

v2L2

= v2

L2

= 0

v1(0) = v2(0)

v1(0) = v2(0)M1(0) = M2(0)

Q1(0) = Q2(0) + P v2(0) + F

Pz

M110

E

Q1#

F

cP'Q2

M2(

)

'


v1 = C1k cos kx C2k sin kx + C3v1 = C1k2 sin kx C2k2 cos kxv1

=

C1k3 cos kx + C2k

3 sin kx

v2 = C5x3 + C6x

2 + C7x + C8

v2 = 3C5x2 + 2C6x + C7

v2 = 6C5x + 2C6

v2 = 6C5

Q1(0) = Q2(0) + P v2(0)


35/134

2EI v1 (0) = EI v2 (0) + P v2(0) + F2C1k

3 = 6C5 + 2k2C7 + FEI

C5 = 13

k3C1 +1

3k2C7 +

F

6EI

M1(0) = M2(0)

2EI v1(0) = EI v2(0)2C2k

2 = 2C6 C6 = k2C2

v1(0) = v2(0)

C1k + C3 = C7 C5 = 13

k3C1 +1

3k2(C1k + C3) +

F

6EI=

1

3k2C3 +

F

6EI

v1(0) = v2(0)C2 + C4 = C8

v1

L

2

= 0 C4 = C1 sin kL

2 C2 cos kL

2+ C3

L

2

v1

L

2

= 0 C3 = k(C1 cos kL

2+ C2 sin

kL

2)

v2

L

2

= 0

1

3k2C3 +

F

6EI

L

2

3 k2C2

L

2

2+ (C1k + C3)

L

2+ C2 + C4 = 0

kL2 kL cos kL2 1 + 124 (kL)2 + sin kL2 C1+

1 1

4(kL)2 cos kL

2 kL sin kL

2

1 +

1

24(kL)2

C2 = F L

3

48EI

v2

L

2

= 0

k2C3 +

F

2EI

L

2

2 2k2C2L

2+ C1k + C3 = 0

1

1 +1

4(kL)2

cos

kL

2

kC1 +

kL

1 +

1

4(kL)2

sin

kL

2

kC2 = F L

2

8EI

M1(x) = 2EI k

2(C1 sin kx + C2 cos kx) when L2 x 0M2(x) = EI(6C5x + 2C6) when 0 < x L2

C1 C2 [] C1 + [] C2 = FL348EI

[] kC1 + [] kC2 = FL28EI

C5 C6 C1 C2


36/134


37/134

v0(x) = v0sin(x/L)

P y E

v0 = L/1000

f

@@@@4

44

dd dd'

&%'$

M(x) + P[v(x) + v0(x)] = 0

v(x) + k2v(x) = k2v0(x), where k2 = PEI

vy(x) = A sinx

L

2L2

+ k2

A sin xL

= k2v0 sin xL

A = k2v0

k2 2L2

v(x) = C1 sin kx + C2 cos kx + C3x + C4 k2v0

k2 2L2

sinx

L

v(0) = C2 + C4 = 0

v(0) = k2C2 = 0 C2 = 0 C4 = 0v(L) = k2C1 sin kL = 0 C1 = 0()

v(L) = C3L = 0

() kL = n k

v(x) =

k2v0

k2

2

L2

sinx

L

,


38/134

M(x) = EI v(x) = EI k2v0

2

k2L2 2 sinx

L

(k2

= P/EI)

M

L

2

= P v0

2

PL2

EI 2

PE = 2EI/L2

M

L

2

= P v0

PPE

1M(L/2) P

PE

= PA

MW

= P

1

A 1

PPE

1v0W

A = (502 452) = 1492mm2I =

4(504 454) = 1.688 106mm4

W =1

50mm2 I = 33760mm

3

k2 = 1.41 103 , when P = 50 kN

= 33.5 15.4 MPaP

y

y = P1A

+ 1PPE

1v0W

PPE

1

y PA v0

WP = 0

P2 mA + PE + PEAv0W P + yPEA = 0

P

2

509.5P + 45945 = 0 P1 = 117.1 kN

P2 = 392.4 kN


39/134

n =117.1

50= 2.34

v0

= 0.49

i =

I

A= 33.6 mm

k =Lci

fyE

= 1.53

=

1 + (k

0.2) + 2k

22k = 0.852

fck = (

2 1/2k)fy = 66.99 MPaNRc = fck

A

m= 66.99 1492 = 99.97 kN

NRc

f =99.97

50= 2.0

= 0.21

f =118

50= 2.36


40/134

N

dd dd

Mt'' MtEE

NE

Ny Nz

Mz

xy

My xz

EI y = Ny + MzEI z = Nz My

y = C1erx, z = C2e

rx

EI r2C1e

rx = NC1erx

MC2re

rx

EI r2C2erx = NC2erx + MC1rerx


41/134

(45)1 C2(EI r2 N) = MC1r(45)2 EI r2C1 NC1 + Mr MC1r

EI r2 N = 0

(EI r2 N)2 + M2r2 = 02 = r2

EI 2 N = M 2 + MEI

+ NEI

= 0

EI 2 + N = M 2 MEI

+ NEI

= 0

1, 2 > 0

1,2 = MEI MEI2 4 NEI

2

y = A1 sin 1x + B1 cos 1x + C1 sin 2x + D1 cos 2x

z = A2 sin 1x + B2 cos 1x + C2 sin 2x + D2 cos 2x

441

EI z = EI A2(21)sin 1x EI B221 cos 1x EI C222 sin 2x EI D222 cos 2xNz = NA2 sin 1x NB2 cos 1x NC2 sin 2x ND2 cos 2x

My = MB11 sin 1x MA11 cos 1x + MD12 sin 2x MC12 cos 2x

EI z Nz My = 0 x

(EI 21 N)A2 + MB11 = 0(EI 21 N)B2 MA11 = 0(EI 22 N)C2 + MD12 = 0(EI 22 N)D2 MC12 = 0

A2 = MB11EI21+N = MB11M1 = B1


42/134

B2 =MA11EI2

1+N

= MA11M1 = A1

C2 =MD12EI2

2+N

= MD12M2 = D1D2 =

MC12EI2

2+N

= MC12M2 = C1

z = B1 sin 1x + A1 cos 1x D1 sin 2x + C1 cos 2x

y(0) = 0 B1 + D1 = 0z(0) = 0 A1 + C1 = 0y(L) = 0 A1 sin 1L + B1 cos 1L + C1 sin 2L + D1 cos 2L = 0z(L) = 0 B1 sin 1L + A1 cos 1L D1 sin 2L + C1 cos 2L = 0

2 2(cos 1L cos 2L + sin 1L sin 2L) = 2 2 cos(1L 2L) = 0

(1 2)L = n2

1 2 = 2

M

2EI2

NEI

1 = 2

N = M = 0

y = z = 0

2 1 = 0 (2 1)L = 2

2

M

2EI2

NEI

L = 2

M2EI

2 N

EI=

L

2

Mcr = 2EI

N

EI+

L

2

N > 0 Mcr

N < 0 Mcr N = 2EI/L2


43/134

L

Pcr Ibolt = Isleeve

Pcr

sleeve : 12 = M12L

3EI11 M21 L

6EI11

= M0 L3EI1

(1 +1

21)

bolt : 12 = M12L

3EI22

M21

L

6EI22

= M0L

3EI2(2 +

1

22)

12,sleeve = 12,bolt

M0 L3EI1

(1 +1

21) + M0

L

3EI2(2 +

1

22) = 0

M0 = 0 I1I2

(2 +1

22) + (1 +

1

21) = 0


44/134

I1I2

3

(kL)2

1

tanh(kL)2 1

(kL)2+

1

(kL)2 1

sinh(kL)2

+

3

(kL)1 1

(kL)1 1

tan(kL)1 +

1

sin(kL)1 1

(kL)1

= 0

I1I2

(kL)1(kL)2

1

tanh(kL)2 1

sinh(kL)2

+

1

sin(kL)1 1

tan(kL)1

= 0

Pcr = EI1k21

I1 = I2 (kL)1 = (kL)2 = kL

cosh kL 1sinh kL

=cos kL 1

sin kL kL 4.73 Pcr = 22.4EI

L2

2 = 2 = 1

I1I2

3

2+ (1 +

1

21) = 0

3(kL)1

1

sin(kL)1 1

tan(kL)1

= I1

I2

3

2

I1/I2 = 1

kL

4.057

Pcr = 16.5

EI

L2v1 = v2

M1 = EI1v1 = P v1 M0M2 = EI2v2 = P v2 + M0

EI1v1 + EI2v2 = 0 v1 = 0 v1 = Ax + B

v1

0 (since v1(0) = v1(L) = 0)


45/134

L/h = 20

dd P'

L

hc

T

EI

EI v(4) + P v = 0

v

EI

L

0

v(4)vdx + P

L

0

vvdx = 0

EIL0

v(3)vdx PL0

vvdx = 0

(M = EI v)

L

0

Mv + EI

L0

vvdx PL0

vvdx = 0

M(L)v(L) M(0)v(0) + EIL

0

vvdx PL

0

vvdx = 0

M(0) + P Rv(0) = 0

x = L

M(L) P Rv(L) = 0v(x) = v0 sin

xL

, v(x) = sin xL

P R[v(L)v(L) + v(0)v(0)] + EIL0

vvdx PL0

vvdx = 0

EI v0

L

4 L2

+ 2P Rv0

L

2 P v0

L

2 L2

= 0

v0

L

2 L2

2

EI

L2 P

1 4 R

L

= 0

P = 2

1 4RL

EI

L2

if R =L

40 P = 10

92

EI

L2


46/134

dd P'

L/2 L/2

2EI EI

v2, 2, 3

dd

P'

L/2 L/2

j j

'c

c

P = EI/L2 Kx =

Sx K S

K

(e)

=

EI

L

12L2

6L

12L2

6L

4

6L

2

12L2 6L

symm. 4

S(e) = N(e)

65L

110

65L

110

2L15

110

L30

65L

110

symm. 2L15

v1, v1, v2, v

2 N

(1) = N(2) = P =

EI/L2

1 = v(1)2 = v

(2)1 , 2 =

(1)2 =

(2)1 , 3 = (2)2

K11 = K(1)33 + K

(2)11 =

8 12L3

2EI +8 12

L3EI = 288

EI

L3

K12 = K(1)34 + K

(2)12 = 24

EI

L2

K13 = K(2)14 = 24

EI

L2

K22 = K(1)44 + K(2)22 = 24 EIL


47/134

K23 = K(2)24 = 4

EI

L

K33 = K(2)44 = 8

EI

L

S11 = S(1)33 + S

(2)11 = P

24

5L = 24EI

5L3

S12 = S(1)34 + S

(2)12 = 0

S13 = S(2)14 = P

1

10=

EI

10L2

S22 = S(1)44 + S

(2)22 = P

2L

15=

2EI

15L

S23 = S(2)24 = P

L

60= EI

60L

S33 = S(2)44 = P

L

15=

EI

15L

EI

L

288 24 2424 24 424 4 8

1/L

2

3

= EIL

245

0 110

0 215

160

110

160

115

1/L

2

3

,

1 1/L Li

det

288 245 24 24 110

24 24 215

4 + 160

24 110

4 + 160

8 115

= 0 kr = 26.32

Chk C

h k

1 = (h1) 2 = (h2)

1 = ex + Chk1

2 = ex + Chk2


48/134

C = (1 ex)hk1

2 = ex + (1 ex)h2h1

k

ex =2 1

h2h1

k1

h2h1

k

= 25.18

k = 4 h1 = L/2, h2 = L/10

h2/h1 = 0.2

1ex = 25.18


49/134

EA =

e e

c c

e ej1c

jv2'u2T2

c jv3'u3T

3c

j4c

EA =

u2 = u3 = 0

v2 = v3

1 = 4

2 = 3

1, v2, 2


50/134


51/134

1 4

dd

dd

dd

dd

'

(EA)iu = 0

u1 = C1x + C2u2 = C3x + C4

Ni = (EA)iu =

EAC1 L2 x < 0EAC3 0 < x L2

u1

L

2

= u2

L

2

= 0 C1L

2= C2, C3

L

2= C4

u1(0) = u2(0) C2 = C4 C1 = C3

N1 + P = N2

EAC1 + P = EAC3

C3 = C1 + PEA

C1 = 11 +

P

EA= C3

N1 = EAC1 = 1 +

P (compression)

N2 = EAC3 = 11 +

P (tension)

= 1

N1 =

N2 =

P/2

N1

P, N2

0


52/134


53/134

-3000

-2000

-1000

0

1000

2000

3000

1 1.5 2 2.5 3 3.5 4


54/134

e e

e eE

'

v1(0) v1(0)

K11 = K(1)33 + K

(2)11 =

122EI(L/2)3

+ 12EI(L/2)3

=288EI

L3

K12 = K(1)34 + K

(2)12 = 62EI(L/2)2 + 6EI(L/2)2 =

24EI

L2

K22 = K(1)44 + K

(2)22 =

42EIL/2

+ 4EIL/2

=24EI

L

N1 = P, N2 = 0 P = EIL2

S11 = S(1)33 =

3630(L/2)

P = 12EI

5L3

S12 = S(1)34 = 110P =

EI

10L2

S22 = S(1)44 =

215

L2

P = EI

15L

EI

L2 288 1L 2424 24L

v1

v1

=

EI

L2 125L

110

110

L15

v1v1

EI

L2

288 12

5 110

24 110

24 24 115

v1L

v1

=

0

0

2 480 + 42240 = 0 = 116, 1 = 363.9

= 114.04


55/134


56/134

Ln

= A2

+ B + C 0 = 392MPa , 0 =0, 002 Et()

Pcr = 20A(

K+ 1)/K K = (0L2nA/

2I)2

dd P'L = 9000D = 410

t = 10jdd dd L

a

L/a = 20

= () = A2 + B + C

= 0 = 0 C = 0 = 0 = 0 = 0, dd = 0

d

d

=0

= 2A0 + B = 0 B = 2A0

0 = A20 + B0 = A20 A =

020

= 0

20

2

+ 2

0

0 d

d = Et = 2

0

0

1

0

d

d

=0

= 200

= E Et = E

1 0

= ()

0=

0

2+ 2

0

0= 1

1

0

Et = E1

0

, when < 0


57/134

Pcr =2EtI

L2n, kr =

PcrA

, merk. =2I

L2n

Pkr = 2E2

20A 4E4

420A2 + 2E2 =

2E2

20A

1 +420A

2

2E2 1

=20A

K

1 + K 1

, K =

0L

2nA

2I

2

A (D t)t = 1.257 104 mm2, I /8(D t)3t = 2.513 108 mm4, L = 9.0 m Ln = 0.699L = 6.291 m K = 0.1608 Pcr = 4.742 kN

A = a2, I = 1/12 a4, L = 20a K = 0.947 cr = Pcr/A = 327


58/134

cr = Ln/i

Ln i =

I/A

Et

d

d = Et = Ey

y c ,

y c

(kr/y) (kr/y) [0, 1], [0, 200] c = 0, 9 E/y = 500

E/y = 200

Pcr =2EtI

L2n,

Et = Ey

y c cr =2Ey

ycI

L2nA

i =

I/A, = Ln/i

cr =2Ei2

L2n

y cry

ccr

2cr = 2E y cry ccr

2cry

(1 ccry

) = 2E

y(1 cr

y)

2c

cry

2

2 + 2E

y

cry

+ 2

E

y= 0

cry

=1

22c

2 + 2 E

y

2 + 2E

y

2 42c2 E

y

c = 0.9 E/y

Pcr =2EI

L2n

cry

=

E

y

2

2


59/134

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200

cry

tang. mod. E/y = 200tang. mod. E/y = 500

elastic E/y=200elastic E/y=500


60/134


61/134

v = PE1 + E2

+E2 E1E1 + E2

a

L (P ) = E2 E1E1 + E2

aP +4E1E2

E1 + E2a2

(P ) = E2 E1E1 + E2

aL

P + 4E1E2E1 + E2

a2L

=E2 E1E1 + E2

a

L, P1 = 4

a2

L

E1E2E1 + E2

(P ) = P + P1integrating

tt0

P P00 = (P P0) + P1( 0)

0 = 0

P = (P P0) + P1 P = P0 P1

E1 = E2 = 0E1 = E2 = E P = P1 = 2Ea2L = PEE1 = E2 = ET P = P1 = 2ETa

2

L = PT i < 0 = 0 i > 0 Ei = E

E1 = E, E2 = ET P1 = PR = 2ER a2L ER = 2(EET)/(E+ ET)P0 (PT, PR)

P0 (PT, PE)


62/134

EI kGA(v ) = 0

kGA(v ) P v = 0

Pcr =PE

1 + PE,

PE PE = 2EI /(4L2) = 1/(kGA) k = 1/

P'

M Q = 0

Q P v = 0

v

M = EI Q = kGA(v ) (49)

EI, kGA P

kGA

v + EI = 0(1 P)v = 0

v(0) = 0

Q(0) = 0

M(L) = EI (L) = 0Q(L) = P v(L)

rrrQ

! P'

v

v = Aerx, = Berx

[Ar + B(EIr2

1)] erx = 0

[(1 P)Ar2 Br] erx = 0


63/134

r EIr2 1(1 P)r2 r

A

B

=

0

0

A = 0 = B

det = 0

r2 1 + (1 P)(EIr2 1) = 0 r1,2 = 0 tai 1 + (1 P)(EIr2 1) = 0

r2 = P(1 P)EI

P > 0 (1 P) = 1 P/kGA P = EI/L2

P = EI

kGAL2 = 2(1 + )k1 I

AL2 (inserting G =

E

2(1 + ) , k1

= )

= 2(1 + )I

AL2, 0 1

2, 1, I/A


64/134

A1, B1, B3, B4

v(x) = A1 + B1x +

1 + EI2

B4 sin x 1 + EI2

B3 cos x(x) = B1 + B3 sin x + B4 cos x

1) v(0) = 0 A1 1+EI2 B3 = 02) (0) = 0 B1 + B4 = 03) (L) = 0 B3 cos L B4 sin L = 04) Q(L) = P v(L)

kGA[v(L) (L)] P v(L) = 0 (1 P)v(L) (L) = 0 (1 P) [(1 + (1 + EI2)cos L) B4 + (1 + EI2)B3 sin L]

(1 + cos L)B4 B3 sin L = 0 (1 P)B4 + cos LB4 (1 + cos L)B4 = 0 P B4 = 0 B4 = 0 B1 = 0

B3

cos L = 0 B3

= 0

cos L = 0 L = 2

P1 P =

2EI

4L2= PE

P = (1 P)PE (1 + PE)P = PE P = PE

1 + PE

v(x) = 1+EI2 B3(1 cos x)(x) = B3 sin x


65/134

Pcr

dd

dd

dd

ddd

P E

2Pc

2Pc

L

L L

j

j j

j j

M21 + M23 = 0

M32 + M34 + M35 = 0

21 = 23 = 2

32 = 34 = 35 = 3

M21 = A212

M23 = A232 + B323

M32 = A323 + B232

M34 = A343

M35 = A353

Aij, Bij

Aij

=2((kL)ij)

42 ((kL)ij) 2 ((kL)ij)6(EI)ij

Lij, B

ij=

((kL)ij)

42 ((kL)ij) 2 ((kL)ij)6(EI)ij

Lij

kL

(kL)221 =2P

2EIL2 =

P L2

EI= k2L2

(kL)223 =P

2EIL2 =

P L2

2EI= 1

2k2L2

(kL)234 =2P P

3EIL2 =

P L2

3EI= 1

3k2L2

(kL)235 =

2P

EI (2L)2

=

2

2P L2

EI = 22k2

L2


66/134

A21 + A23 B32

B23 A32 + A34 + A35

2

3

=

0

0

K

(A21 + A23)(A32 + A34 + A35) B232 = 0 kL = 3.564 Pkr = 12.705 EI

L2

g(x) = 0

x0g x0

g(x) g(x0) + g(x0)(x x0) + ...

xk+1 = xk g(xk)g(xk)

x0 x

g


67/134


68/134

(kL)

kL = 0

kL

det(K)


69/134

P

HEP

c Pcq

c c c c c c c c

L

2L

j

j j

j

EI

EI

EI

q = 180 kN/m2

H = 20 kN

L = 6 m

A = 36 103 mm2I = 650 106 mm4

W = 3, 25 106 mm3E = 210 GPa

m = 220 MPa

M23, M34, 12

M21 + M23 = 0 M21 = M23M32 + M34 = 0 M32 = M34

21 = 23

21 =

2L3EI

(k12L)M21 + 12

23 =L3EI

M23 L6EIM32 + 023

LEI

2

3(k12L) +

1

3M23 +L

6EIM34 12 + 023 = 0

32 = 34

32 = L6EIM23 + L3EIM32 + 03234 =

2L3EI

(k22L)M34 + 12

LEI

2

3(k22L) +

1

3

M34 +

L

6EIM23 + 12 032 = 0


70/134

Q21 + Q34 = H

Q21 = Q021

M212L

N1212

Q34 = Q

0

34 M34

2L N3412

HE

Q21'

Q34'

M23 M34 2L(N12 + N34)12 = 2LM N23 M34 2L(qL + 2P)12 = 2LM

L

EI

23

(k12L) +13

16

11

6

2

3(k

22L) + 1

31

1 1 2L(qL + 2P) LEI

M23

M34

EIL

12

=

0230322HL2

EI

023 =qL3

24EI, 032 =

qL3

24EI, =

H

qL, q =

EI

L3, P =

EI

L2,

23

(k12L) +13

16

116 23(k22L) + 13 1

1 1 2L(qL + 2P) LEI

M23

M34EIL

12

= 124

1242

qL2

k1 =

N12EI

N12 = Q23 + P = Q023

M23 + M32L

+ P = P +qL

2+

1

L(M34 M23)

k2 = N34EI

N34 = P Q32 = P + qL2

+ 1L

(M23 M34)

M23 =M23qL2

, M34 =M34qL2

, 12 =12EI

qL3

M23 = M23qL2 = M23EIL


71/134

M34 = M34qL2 = M34

EI

L

12 = 12qL3

EI= 12

N12 =

+

1

2+ M34 M23

EI

L2

N34 =

+

1

2+ M23 M34

EI

L2

2k1L = 2

+

1

2+ M34 M23

2k2L = 2 + 12 + M23 M34

M23E

Q23

T

qc c c c c c c c M(x)

'

xE

M(x) = Q23x + M23 12

qx2

Q(x) = Q23

qx

Q(x) = 0 x = Q23q

=1

2L +

M34 M23qL

H E

Q21'

N23'

N23 = H Q21= H Q021 +

M212L

+ N1212

= H M232L

+

12

+ M34 M23 + EIL2

12

P = 0 M23, M34 = 0

< m P = EI/L

= 0, 01


72/134


73/134


74/134

m = 220 =

0, 22 = 0, 23 max = 0, 22 Pmax = 0, 834

Pallowable = Pmaxh

=0, 834

1, 5MN = 0, 56MN


75/134


76/134

d b

2h L Ep

En In

dd

dd

dd

P'

dAA

b

P = PE1 + PE

, PE = 2EI

L2

= Q

=QS

bI

S =

h0

bydy =1

2bh2

I =1

12b(2h)3 =

2

3bh3

= Q

=S

bI=

S

bI

=

3

4bh

db

db = kn = knh

kn

= dbknh = 3d4knh2


77/134

kn

EEEEE

h2

T

c

v

1

2Epv

h

2= Q Q = knv

kn = 14

Eph

1 = 3d4Eph3

In

ffdd

ff v

v(0)

c

Qc

Ep

En, In

v(0) =2

EpQ, =

Ep

4EnIn

1/4

kn = Ep2

2 = 32

dEph2

= 32

h dEph3

2 > 132

h > 3 h > 2En = 210

Ep 8 13

EpEn

120

h h(80In)1/4

, In =D4n

64

h hDn

,


78/134

P E

y

Pcr = PE1 + PE

, where PE = 2

EIoL2

Io = 2

Izo + Aa

b

2

2= 281 106 mm4 PE = 5.84 MN

Ia, Aa

Ib, Ab b = 304

a = 985

Ia = Iz0 = 4.95 106 mm4Ib = 4.5 106 mm4

Ab = 2400 mm2

E = 210 GPa, G =E

2(1 + )= 80.8 GPa

=ab

12(EI)b+

a2

24(EI)a(1 ) +a

b

GA

b

= Pcr2

2

(EI)aa2

Pkr

= 0

Pcr

a1, a2, a3, 1, 2

(EI)b = a1(EI)o, (EI)a = a2(EI)o, (GA)b = a3 (EI)oL2

, ab

= 1, aL

= 2


79/134

=

22

121a1+

2224a2(1 ) +

1

a3

L2

(EI)o=

L2

(EI)o

Pcr =2(EI)o

L2

1 + L2

(EI)o

2(EI)oL2

= 2(EI)o1 + 2

=PE

1 + 2

= 2(EI)o

2L2(1 + 2)22L

2

2a2(EI)o=

222a2(1 + 2)

Pcr = 3.77 PE

Io = 2Iyo = 1160.6 106 mm4

Pcr = 2

(EI)oL2

= 3.33 MN

Py = y2Aa = 2.82

Py


80/134

Q y

= Q/

Q

d1, d2, d3, d4 D1, D2, D3, D4

d1 =b

3

2j b

2i + ak

d2 =b

3

2j +

b

2i + ak

d3 = b

3

2j +

b

2i + ak

d4 = b

3

2j b

2i + ak

d =a2 + b2

cos(d1, x) =d1 i|d1|

= b2d

= cos 2

cos(d2, x) =cos

2cos(d3, x) =

cos

2

cos(d4, x) = cos 2


81/134

cos(d1, y) =d1 j|d1|

=b

3

2d=

3

2cos

cos(d2, y) =

3

2cos

cos(d3, y) =

3

2 cos

cos(d4, y) =

3

2cos

y Q

2(D1 + D2) j = Q D1 cos(d1, y) + D2 cos(d2, y) = Q2

2(

D1 +

D2) i = 0 D1 cos(

d1, x) + D2 cos(

d2, x) = 0

D1 = D2 = Q2

3cos

y Q

(D1 + D2 + D2 + D4) j = Q (D1 + D2 D3 D4)

3

2cos = Q

(D1 + D2 + D2 + D4)

i = 0

(

D1 + D2 + D3

D4)cos

2= 0

D3 = D4 = Q2

3cos

S

=

D1

D2

D3

D4

=Q

2

3cos

1

1

11

=

D1

D2

D3

D4

=

d1/2 EAd

d1/2 EAd

d1/2 EAd

d1/2 EAd

D1

D2

D3

D4

=

S


82/134

aQ =

T

S

aQ =Q

23cos 1 1 1 1 Q

23cos

1

1

11

=Q2

4 3cos2 d

1/2 EAd

1 1 1 1

1

1

11

=

4dQ2

6EAd cos2

= 2dQ3EAda cos2

=2d3Q

3b2EAda

=

Q =2d3

3ab2EAd


83/134

L

PE dd dd ddj j jL L

P v0 = L

dd

dd @@@@@

@@@@

ddv0PE

L L

M21 + M23 = 0

21 = 23 M21 + 21 = M23 + 23 2M21 = 2

21 = 23 = ja = L3EI

(kL)

P = EI

L2 kL = M21 = 3EI

L(kL) = 3EI

L2() v0 = 3EI

L()

T = Q23 Q21 = 2

M21L

+ P

=

2EI

L2

3

(

)


84/134

TL2

EI

2

0 T 6EI/L2

P

6EI/L2 T

P = 8EI/L2 = 1/100

0.204EI/L2

P

0ij(kL)

dd 222dd 222ddv0 nv0PE

L L

v0 = L

M21 + M23 = 0

21 = 23 M21 + 021 = M23 + 023 2M21 = 023 021, 023 = 021 =

(kL)2

2 (kL)2

M21 = 1 + n2

(kL)2

2 (kL)23EI

L(kL)

kL =

, P = EI

L2

M21 = 3(1 + n)EI2L

(2

)(

)

T =2M21

L=

3(1 + n)EI

L2

(2 )(

)


85/134


86/134


87/134


88/134

'T

Ec2qL(1

)2qL

qkr + qc c c c c c c c c c c c c c c c c c c c

qLT 2qL(1 )T qLT

2

2 1 =5

384

q(2L)4

EIv (1

)

48

q(2L)4

EIv

= (q)


89/134

Pcr

tcT

b

EIzv(4) + P(v + zv) = 0

EIyw(4) + P(w yv) = 0

EI(4) GIt + P(zvv yvw + r2) = 0

r2 = y2v + z2v + (Iz + Iy)/A zv = yv = 0

EIzv(4) + P v = 0

EIyw(4) + P w = 0

EI(4) (GIt P r2) = 0

v, w

1 2

Pcr,z = Pcr,y = 2EIz

L2

v = A sinx

L, w = B sin

x

L

Iz = Iy, Iyz =

yzdA = 0

Iz = Iz cos2 + Iy sin2 2Izy sin cos = Iz = I


90/134

3 I = 0

(4) +P r2 GIt

EI = 0

(4)

+ k

2

= 0 , where k2

=

P r2

GIt

EI

P > GItr2

= A + Bx + Ccos kx + D sin kx

(0) = 0 A + C = 0(L) = 0

A + BL + Ccos kL + D sin kL = 0

(0) = 0 C = 0 A = 0(L) = 0 Ccos kL + D sin kL = 0 D sin kL = 0 B = 0

A = B = C = 0

sin kL = 0 k = nL

P r2 GItEI

=n

L

2

Pcr, =GIt + EI

2

L2

r2

Iz = Iy =bt

12(b2 + t2)

It = 0.78bt3

I = 0

Pcr, =

It

Iz + Iy

GA = 4.67t2

b2

+ t2

GA = 4.67GA

1 + (b/t)2

I = 0

3 I = 0

(P r2 GIt) = 0 Pcr, = GIt/r2


91/134

Pcr

b = 10t, = 0

tcT

b

b

EIzv(4) + P(v + zv) = 0

EIyw(4) + P(w yv) = 0

EI(4) GIt + P(zvv yvw + r2) = 0

b

c

T

b' E

yc

zEd =

b4Tc

zv = 0, EI = 0

yv = b/4Iy b

3t

12, Iz =

5

24b3t

It 23

t3b, r2 =IpA

+ y2v + z2v =

5

24b2

EIzv(4) + P v = 0

EIyw(4) + P[w yv] = 0

GIt + P[yvw + r2

] = 0

y

z y

Py = 42EIz

L2

w = B 1 cos2nx

L = C

1 cos2 nx

L


92/134

P = GItr2

GIt

r2 yv

yv r2(1 ) B

C = 0

0

= 42(r/L)2EIy/GIt A, B

1

yvr

22 (1 + ) + = 0

Iy = I Iz =52

I It =225

I = 0 G = E/2 GIt =125

EI

(yv/r)2 = 3

10

2 107

(1 + ) + 107

= 0

= 1256

2(b/L)2

1 =57

(1 + )

1

1 14

5(1 + )2

1 1

Py = 42EIz

L2

= 2502 r

L2 GIt

r2

=625

12

2b

L2

GIt

r2

Pz,,1 = 1GIt

r2

z

Pz = 42EIy

L2=

GItr2

=125

62

b

L

2GItr2

=2

5Py > Pz,,1

cr = 1 (L/b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100

cr

L/b


93/134

E

m

h

t

tcT

h

h

EIzv(4) + P[v + zv)

] = 0

EIyw(4) + P[w yv)] = 0

EIz(4)

GIt

+ P(zvv

yvw

+ r2)) = 0

(yv, zv) r2 = y2v + z

2v + (Iz + Iy)/A

Iy = 13h3t Iz = 712h3t It = ht3 A = 3ht


94/134

s = 0

s = h

s = h

s = 2hh2

B = B(s) =

s0

h(s)ds

h s h , B = 0h s 2h , B = h(h s)

yv = yB + Iz/Iy = 0 (symmetry)zv = zB + Iy/Iz

Iy

h/2h/2

h/2

y

Iy = y(s)B(s)t(s)ds = t2h

h

h

2

h(h

s)ds =

th4

4

zv = h3

1

4h4t

712

h3t

= 16

21h

r2 = y2v + z2v +Iz + Iy

A=

521

588h2

v

hhhhhhhhhhhh

37

h2 114

h2

12

h2

37

h

vv

s

v(s)t(s)ds = 0

C v

s

vtds +s

Ctds =

vtds + Cs

tds = 0


95/134

C = s

vtds

A= 3

14h2

s y = 0 z =

h/3

hhhhhhhhhhhh

314

h2

314h2

27

h2

2

7h2

vTs

v(s) =

h28

(14s + 13h) , 32

h s h2

33

hs , h2

s h2

h28

(14s 13h) , h2

s 32

h

I =

A

2vdA = 2t

h/20

9

49h2s2ds +

3h/2h/2

1

282h2(14s 13h)

= 5h5t

84

Pc

xT

Lv(0) = w(0) = (0) = 0

v(0) = w(0) = (0) = 0

v(L) = w(L) = (L) = 0

v = C1

cos

x

2L 1

, w = C2

cos

x

2L 1

, = C3

cos

x

2L 1

2

4L2EIz P

C1 P zvC3 = 0

2

4L2EIy P

C2 = 0

P zvC1 +

2

4L2

EI + GIt

P r2C3 = 0


96/134

Py =2

4L2EIz, Pz =

2

4L2EIy, P =

2

4L2EI + GIt

Py P 0 zvP

0 Pz P 0zvP 0 P P r2

C1

C2

C3

= 0

00

det[]

(Pz P)

(Py P)(P P r2) z2vP2

= 0

P1 = Pztai (r2 z2v)P2 (P + Pyr2)P + PyP = 0

P2,3 =(P + Pyr

2)

(P + Pyr2)2 4PyP(r2 z2v)2(r2

z2v

)

P1 = 1727 kN, P2 = 964 kN, P3 = 11454 kN

Pkr = P2 =

PkrA

= 321 MPa > m = 220 MPa

Pkr 660 kN


97/134

M

Mcr

M'&E

TTTTTTTTTTTTTTTTTTTTTT

dd

dd

M$%'

L = 50b

E ' b

EIyw

(4) M0z = 0GIt M0z w = 0

M0z /EIy

(M0z )2

EIy + M0z w

(4) = 0

GIt(4) M0z w(4) = 0

(M0z )2

EIy GIt(4) = 0 trial = erx

(M0z )2

EIyr2erx GItr4erx = 0

r2 = (M0z )2

EIyGIt

k2 = r2

= A1 sin kx + A2 cos kx + A3x + A4

w = B1 sin kx + B2 cos kx + B3x + B4

B1 = A1GI

tM0z

, B2 = A2GI

tM0z


98/134

z

w +5b

2 0 w

= 5b

2

w

B1A1

=B2A2

= 5b2

M0z = 2GIt5b

k =M0z

EIyGIt=

n

Ln halfwaves

M0z =n

L

EIyGIt =

2GIt5b

Iy =5b b3

12=

5

12b4

It =1

35b b3 = 5

3b4

L = 50b and if G = 0.4E

M0z =n

50bEb4

0.5 5/12 5/3 = 0.0331nEb3 = 0.267Eb3

n = 8.

2

GIt M0z

5b2

= 0

= 0 = 0 = Ax + BA = B = 0 0

GIt M0z

5b2

= 0 M0z =

2GIt5b


99/134

Mcr

b hh

b

M'&

E

dd

dd

M$%'

EIyw

(4) M0z = 0GIt M0z w = 0

w(0) = 0, w(0) = 0, (0) = 0

ee

ee

ee

ee

ee

ee

yee

zB z, wE

y, vc

dd

M''

c

w(L) = h2

(L)

Mz%% Mye

eueeu

M'' zE

yc

zT

zE

$

$$$$

Mx$$X

MzgggMc

w(L)

M

Mz MMy = EIyw

(L)MMx =

w(L)M


100/134

w(0) = 0 w(L) = h2

(L)

w(0) = 0 EIyw(L) = (L)M(0) = 0 GIt(L) = w(L)M

2 1

w(4) + k2w = 0, k2 =M2

EIyGIt w = A sin kx + B cos kx + Cx + D

w(0) = w(0) = 0 D = B = 0 w = A sin kx + Cx

2

= Esin kx + F x GItk2Esin kx Mk2A sin kx = 0 E = M

GItA

GIt(L) = w(L)M GIt(k M

GItA cos kx F) = (Ak cos kx + C)M

F = MGIt

C

w(L) = h2

(L) A sin kL + CL = h2

M

GIt(A sin kL + CL)

1 Mh2GIt

A sin kL +

1 Mh

2GIt

CL = 0

EIyw

(L) =

(L)M

EIyk

2A sin kL =M

GIt

(A sin kL + CL)

EIyk2 M

2

GIt

A sin kL + M

GItC = 0

k2 = M2/EIyGIt (M/GIt)C = 0 C = 0

1 Mh2GIt

sin kL

= 0

Mcr = min2GIt

h , EIyGIt

L


101/134

w(x) = A sin kx

(x) = MGIt

A sin kx

Mcr =2GIth

kL = w(L), (L) = 0 Mcr = 2GIth kL =


102/134

Mcr =

EIyGIt/L

= (k,h/L) k

k (1, 1) L/h = 20, = 0

M'&E

dd

dd

kM$%'

L b

h

=1

2

L0

GIt(

)2 + EIy(w)2 + 2(M0z )

w

dx

=1

2

L0

GIt(

)2 + EIy(w)2 + 2(M0z

+ M0z )w

dx

M0z = M(1 x/L) + kMx/L = M[1 + (k 1)x/L] M0z = (k 1)M/L = 0 sin x/L w = w0 sin x/L

L0

M0zwdx = 0

L0

x

Lcos2

x

Ldx =

L

4

= 12

GIt

L2

20L2 + EIy

L4

w20L2 + 2M

L2

0w0

L2 + (k 1) L4

0=

2GIt2L

0 + M

L

2w0

L

2+ (k 1) L

4

= 0

w0=

4EIy2L3

w0 + M

L

20

L

2+ (k 1) L

4

= 0

2EIy2L3

M2L

1 + k1

2

M2L1 + k1

2 GIt

2L

w0

0 =

0

0


103/134


104/134


105/134

Mcr

b = 10t, L = 20b, = 1/3 M

M'&E

dd

dd

M$%'

L

d

dd

b

bdd

dds

It = 23

t3b, Iy = 13

tb3, Iz = 112

tb3, yv = 24

b, zv = 0

z =1

Iz

y(y2 + z2)dA 2yv,

y3dA = 0,

yz2dA = 2

bt

6

2

4b

1

2b2 =

2

24tb4 z =

2b

EIyw(4) M = 0

GIt Mw zM = 0 = MGIt + zM

w

w(4) + M2

EIy(GIt + zM)w = 0

w = A sin kx + B cos kx + Cx + D k2 =M2

EIy(GIt + zM)

w(0) = 0 B + D = 0w(0) = 0 B = 0

w(L) = 0 A sin kL + CL = 0

w(L) = 0 Ak2 sin kL = 0 kL = n,

n = 1

M2 z 2

L2EIyM EIyGIt

2

L2= 0

M =

EIyGIt/L EIy =

2GIt

2 2 zL

2 = 0


106/134


107/134

Nx10

2Da2

h

h = 4h

w(x, y)

h = a/40 a

Nx

EEEEEEEEEEE

an+1

nNx

'''''''''''

c ay

E

a

x

w(x, y) = w0 sinx

asin

y

a

= U + V = Uplate + Ubeams + Vplate + Vbeams

Uplate =D

2

A

(w)2dA

w = w,xx + w,yy, and w,xx = w02

a2 sin xa sin ya = w,yy

Uplate = D2

4

a2w20

Vplate = Nx2

A

w2,xdA = Nx2

2

4w20

Ubeams =n

i=1

EI

2

a0

w2,xxdx =EI

4

4

a3w20

sin2

i

n + 1

Vbeams

=

n

i=1

xhh

2

a

0

w2,x

dx, where x

h = Nx


108/134

= Nx4

h

aw20

2

sin2i

n + 1

=

D

2

4

a2 Nx

2

2

4+

EI

4

4

a3

sin2

i

n + 1 Nx

4

h

a2

sin2i

n + 1

w20

Vpalkit Nx

Nx = 2Eh3

12a2, I =

3h4

12, D =

Eh3

12, when = 0

= 4 h = a/40

= Eh3

24

4

a2w20 1 + 3

h

2a

n

i=1

sin2i

n + 1

1

4+

h

2a

n

i=1

sin2i

n + 1 = 0 w0 = 0

2 = 0 2

w20= 0

= 1 + 3 h2a

sin2 i

n+114

+ h2a

sin2 i

n+1

10

= 4 h = a/40 n

n = 1

sin2i

n + 1= 1 = 1 +

45

14

+ 120

= 6

n = 2

sin2i

n + 1= 2 3

4=

3

2 = 1 +

4532

14

+ 12032

6.8

n = 5

sin2i

n + 1= 3 = 8.5

n = 9

sin2i

n + 1= 5 = 10


109/134

cr

w(x, y) = A sin(y/b) sin[(x y)/s]s

D cr =5.352D/b2t t

E E E E E E E E E E

' ' ' ' ' ' ' ' ' '

b

w(x, y) = A siny

bsin

s(x y)

s x w = 0 x = y

x = y + s

y

s

b

x

=D

2

A

(w)2dA + Nxy

A

w,xw,ydA

y = 0, y = b, x = y x = y + s

w,x = As

sin yb

cos s

(x y)


110/134


111/134


112/134

r

R

N =-P

Nr +

Nrr

dr

(r + dr)d Nrrd 2N d

2dr = 0

Nrr

+Nr N

r= 0

N0r N0

Dd4w

dr4 +2

r

d3w

dr3 1

r2d2w

dr2 1

r3dw

dr

= p


113/134

p N0r N0

prdrd =

N0r

dw

dr

rd +

N0r

dw

dr+

r

N0r

dw

dr

dr

(r + dr)d

= N0rdw

dr

drd +

rN0r

dw

dr rdrd +

rN0r

dw

dr (dr)2d

p = N0r

r

dw

dr+

d

dr

N0r

dw

dr

d4w

dr4+

2

r

d3w

dr3 1

r2d2w

dr2 1

r3dw

dr=

1

D

N0rr

dw

dr+

d

dr

N0r

dw

dr


114/134


115/134

+

R0

(rNr + N)rdr 2R0

N0r u +

d

dr(N0r u)r

dr

= D

R

0

(2r + 2 + 2r)rdr + 2

R

0

(N0r r + N0)rdr

2

R

0

N0r u + ddr

(N0r u)r dr +

r =du

dr+

1

2

dw

dr

2, =

u

r, r = d

2w

dr2, = 1

r

dw

dr

= D

R

0

(w)2 +

(w)2

r2+

2

rww

rdr

+2

R0

N0r

u +

1

2(w)2

(N0r u)

rdr

+2

R0

(N0 N0r )udr + R0

(rNr + N)rdr

N0r N0

Nr =E

12

(r + ), where r =du

dr

= u

N =E

12 ( + r) =ur

dNrdr

+Nr N

r= 0

u + r

u u 1r2

+1

r

u +

u

r u

r u

= 0

u + 1r

u 1r2

= 0

r2d2u

dr2 + r

du

dr u = 0

r = et t = lnrdu

dr=

du

dt

dt

dr=

1

r

du

dtd2u

dr2=

d

dr

1

r

du

dt

=

1

r2

d2u

dt2 du

dt

r2d2

udr2

+ r dudr

u = 0


116/134


117/134

=

R0

D

(w)2 +

w

r

2+

2

rww

P(w)2

rdr

w

w = w0

1 r

2

R2

w = 2w0 r

R2 w = 2 w0

R2

= D 4r

R4+ 4

r

R4+ 8

r

R4 P4r3

R4 drw20

d2

dw20= 0 P = 4(1 + ) D

R2

w = w0

1 r

2

R2

2

w = 4w0

R

1 r2

R2 r

R

w = 4w0R2

1 3 r

2

R2

=

32D

3R2 2

3P

w20

P = 16 DR2

D/R2


118/134

P

L

c

T

L

c

T

k1q1

k2

q2

P

V

Fi = Vqi

W = Fiqi = V

qi qi = V

W = PixBi

P = P sin q2i P cos q2jxB = (sin q1 + sin q2)Li + (cos q1 + cos q2)Lj

xB = (q1 cos q1 + q2 cos q2)Li (q1 sin q1 + q2 sin q2)Lj W = P L( cos q1 sin q2 sin q1 cos q2)q1 + 0 q2

Vq1

= P L sin(q1 + q2)Vq2

= 0 V = V(q1)

P


119/134


120/134


121/134


122/134


123/134


124/134


125/134

K = Lq

(qe, 0) Qq

(qe, 0)

= k1 0

0 k2

2 1

1 1

P L, attheequilibriumpoint qe =0

0

M =J

q(qe, 0, 0) =

1

3

5 1

1 1

mL3

q = estx

K+ s2M

x = 0

k1 = k2 = k P = k/L r2 = s2mL3

4

9r4 + k

2 5

3

r2 + k2(1 3 + 2) = 0

r2 = k2 + 5

3

2 4

3 + 20

9

8/9

r = 0

2

3 + 1 = 0 =3

2

5

2


126/134


127/134

x1

1 y1 = Sx1

k = 1, 2,... |(k+1 k)/k+1| < TOL

Kxk+1 = yk

yk = Sxk

(xk+1) =xTk+1yk

xTk+1yk+1

yk+1 =yk+1

(xTk+1yk+1)1/2

yT1 1 = 0

yk+1 S1, (xk+1) 1, when k


128/134


129/134

K = S = aS + (1 a)S (K S) = S K = S

= +

K p1S


130/134


131/134


132/134

x1

1 y1 = Sx1

k = 1, 2,... |(k+1 k)/k+1| < TOL

(K (xk)S)xk+1 = ykyk = Sxk

(xk+1) =xTk+1yk

xTk+1yk+1+ (xk)

yk+1 =

yk+1

(xTk+1yk+1)1/2

yT1 1 = 0

yk+1 S1, (xk+1) 1, when k


133/134


134/134

1e-20

1e-10

1

1 2 3 4 5 6 7

ln(virhe)

iteraatiokierros

kaanteisiteraatio

shift ++

+

++

+

++

Rayleigh

stability of structures - solved examples

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