statics of building structures i., erasmusfast10.vsb.cz/koubova/sobsi_theme4_three_mom_eq.pdfstatics...
TRANSCRIPT
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Department of Structural Mechanics
Faculty of Civil Engineering, VŠB-Technical University of Ostrava
Statics of Building Structures I., ERASMUS
Continuous beam
• Basic properties of a continuous beam
• Solution of a continuous beam by the Force method
• Symmetry of a continuous beam
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2 / 40Basic properties of a continuous beam
Supports of the transversally loaded
continuous beam
Continuous beam
2) 1, (0, supports fixed ofnumber
spans ofnumber
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:acyindetermin statical of Degree
support) (fixed
node endat rotation against b)
movement, rticalagainst ve a)
: restrained isIt
loading. sversal with tranbeamdirect ateindetermin statically a is beam Continuos
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vp 1
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3 / 40Solution of a continuous beam by the Force method
First 3 steps of the Force method in
solution of a continuous beam
Continuous beam, derivation of the “Three moments equation”
.conditions naldeformatio writing4)
supports), fixed(at reactionsor nsinteractio
moment with links removed of replacing 3)
hinges), (inserting links internal of removal 2)
acy indetermin statical of degree ofion determinat 1)
:equation moments Three derive tosteps basic method, Force The
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4 / 40Solution of a continuous beam by the Force method
Derivation of the Three moments equation
1,1, :condition nalDeformatio rrrr
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5 / 40
Calulation of rotations at the ends of beams
Simply supported beam as a member of a statically indeterminate structure
Deformations of a simply supported beams at the ends
abababbaababbaabbabababa MMMM ,,,,0,,,,,,,0,,,
:rotation clockwise For the
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6 / 40
Derivation of 3moments (Clapeyron’s) equation
equation each in moments bendingunknown 3maximally are there
,acy indetermin statical of degree its toequal is beam continuous afor equations ofnumber
rotation), (clockwise #5 and #4 slideson picturesat marking toscorrespond ofsign
:Notes
:conversion andon substitutiAfter
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7 / 40Solution of a continuous beam by the Force method
Derivation of the Three moments equation
:support For
:is beam theof endright For
:is beam theof endleft For
.( is equations ofnumber
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11
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![Page 8: Statics of Building Structures I., ERASMUSfast10.vsb.cz/koubova/SoBSI_theme4_three_mom_eq.pdfStatics of Building Structures I., ERASMUS ... Derivation of 3moments (Clapeyron’s) equation](https://reader030.vdocuments.mx/reader030/viewer/2022040404/5e8fb5573ee29626a93ca778/html5/thumbnails/8.jpg)
8 / 40Solution of a continuous beam by the Force method
Derivation of the 3moments equation, the beam with a cantilever
s.cantilever theof loading fromdirectly calculated becan moments Those
below) depicted case in the negative are(they
loaded are ends cantilever when zero-non are supportsouter above moments Bending
![Page 9: Statics of Building Structures I., ERASMUSfast10.vsb.cz/koubova/SoBSI_theme4_three_mom_eq.pdfStatics of Building Structures I., ERASMUS ... Derivation of 3moments (Clapeyron’s) equation](https://reader030.vdocuments.mx/reader030/viewer/2022040404/5e8fb5573ee29626a93ca778/html5/thumbnails/9.jpg)
9 / 40Solution of a continuous beam by the Force method
Derivation of the 3moments equation, fixed end
0,0
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equation. moments Three theof solvingby calculated becan moments The
zero.-non moments bendingbut zero are , ends fixedat Rotations
![Page 10: Statics of Building Structures I., ERASMUSfast10.vsb.cz/koubova/SoBSI_theme4_three_mom_eq.pdfStatics of Building Structures I., ERASMUS ... Derivation of 3moments (Clapeyron’s) equation](https://reader030.vdocuments.mx/reader030/viewer/2022040404/5e8fb5573ee29626a93ca778/html5/thumbnails/10.jpg)
10 / 40
Variable cross-section within span
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11 / 40
3moments equation – constant cross-section on all beam
Assumption:
materially and geometrically invariable cross-section
on all beam, ie. E·I = konst.
Then the Three moments equation has form:
0)(2 1,1,1,1,1,11,1,1,1 rrrrrrrrrrrrrrrrrrr lZlZlMllMlM
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12 / 40Solution of a continuous beam by the Force method
Loading members
Formulas for loading members of
the Three moments equation
beam supportedsimply aon determined is
a Rotation
:loading thermaland forcefor members Loading
a,0b,b,0a,
0,,
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,
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6
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13 / 40
Internal forces of the continuous beam
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14 / 40
Reactions of a continuous beam
1,1,
1
rrV
rrVrR
rR : writtenbecan r support aat reaction verticalaFor
sideright at 1rr,span theb)
sideleft at r ,-rspan thea)
:separatesr support The
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15 / 40
Loading members for loading by displacement of supports
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: is end fixed at the 1)("support right theof rotationa clockwiseFor
: is end fixed at the "support left theofrotation clockwiseFor
:is )( nt displaceme alFor vertic
:span) 1ithin constant w(but spansin section -crossdifferent with beam continuousfor equation moments Three The
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16 / 40Solution of a continuous beam by the Force method
Problem definition and solution
of the example 4.1 (part one)
Example 4.1
loading Thermal
(a), picture toaccording loading Force
:definition Problem
mhmhh
mImII
6,03,0
1040,106
3,24,32,1
44
3,2
44
4,32,1
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17 / 40Solution of a continuous beam by the Force method
Solution of example 4.1 (part two)
Example 4.1, displacement of supports
kPaEm,m , l,,hmI
m,lm,,m, l,h,hmII
)mm(,w), mm(w
,,
-
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-
,,
7
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104,246601040
138230106
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,
:definition Problem
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18 / 40
Example 4.1, displacement of supports, solution
.10129032,1101,3
05,3,10234375,010
4,6
25,3
,10234375,0104,6
25,3,10714286,010
6,2
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19 / 40
Example 4.1, displacement of supports, solution continuation
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kNl
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l
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is equations 3 of system theofSolution
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20 / 40Solution of a continuous beam by the Force method
Problem definition and solution
of the example 4.2
Example 4.2
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21 / 40Employment of the symmetry of a continuous beam
Symmetry of geometry and supports of continuous beam
Symmetry of a continuous beam
)cantileveror fixed ended,roller or ended(pin
same are beam continuous a of endsboth - supports ofsymmetry b)
sections-cross identical andlenght identical have spans lsymmetrica -geometry ofsymmetry a)
:assumes beam continuous a ofSymmetry
spans ofnumber even b)
spans ofnumber odd a)
: withbeams lsymmetricafor symmetry of axis ofposition different is There
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22 / 40Employment of the symmetry of a continuous beam
Symmetrical, antisymmetrical and
general loading
Loading of a symmetrical continuous beam
spans ofnumber even loading, ricalantisymmet AS
spans ofnumber even loading, lsymmetrica SS
spans ofnumber even loading, ricalantisymmet AL
spans ofnumber odd loading, lsymmetrica SL
:
ryantisymmetor symmetry
ofcharacter without loading c) ad
forces ofdirection oposite of pictures mirrored b) ad
forces ofdirection same of pictures mirrored a) ad
general c)
A- ricalantisymmet b)
S - lsymmetrica a)
:
onsAbbraviati
:by formed
is sidesboth on Loading
becan beam continuous lsymmetrica a of loading The
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23 / 40Employment of the symmetry of a continuous beam
Utilization of a symmetry for symmetrical and antisymmetrical
loading of a continuous beam.
Symmetrical and antisymmetrical loading
2
1 , ´,:AS ,
2 ́ , ́ :AL
2
1 , ´, :SS ,
2 ́ , ́ :SL
3223322
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sAL
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sSS
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sSL
s
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24 / 40Employment of the symmetry of a continuous beam
Problem definition and solution
of the example 4.3 (part one)
Example 4.3
beam. lsymmetrica
same theof states loading ricalantisymmet
and lsymmetrica theof solutions of
ionsuperpositby given issolution Final
moments..) bending (shear, forces internal of
evaluation including ,separately solved are
beam continuous a of states loadingBoth
loading ricalantisymmet b)
loading symetrical a)
:into decomposed becan beam
continuous lsymmetrica a of Loading
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25 / 40Employment of the symmetry of a continuous beam
Solution of example 4.3 (part two)
Example 4.3
Resulting diagrams of internal forces - by superposition of SL+AL
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26 / 40
Donau-wald bridge, Winzer, Germany
Examples of real continuous beam structures
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Donau-wald bridge, Winzer, Germany
Examples of real continuous beam structures
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Bogenberg bridge, Bogen, Germany
Examples of real continuous beam structures
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Kingstone Bridge, Glasgow, Scotland
Examples of real continuous beam structures
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Kingstone Bridge, Glasgow, Scotland
Examples of real continuous beam structures
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Kingstone Bridge, Glasgow, Scotland
Examples of real continuous beam structures
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Bridge in Nusle quarter, Prague
Examples of real continuous beam structures
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Bridge in Nusle quarter, Prague
Examples of real continuous beam structures
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Construction of highway D47, Ostrava
Examples of real continuous beam structures
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Construction of highway D47, Ostrava
Examples of real continuous beam structures
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Construction of highway D47, Ostrava
Examples of real continuous beam structures
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Energetic Research Centre, VŠB-TU Ostrava
Examples of real continuous beam structures
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Energetic Research Centre, VŠB-TU Ostrava
Examples of real continuous beam structures
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Energetic Research Centre, VŠB-TU Ostrava
Examples of real continuous beam structures
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Energetic Research Centre, VŠB-TU Ostrava
Examples of real continuous beam structures