Lecture Note 5

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Lecture Note 5 Bending of Thin Plates First Semester, Academic Year 2012 Department of Mechanical Engineering Chulalongkorn University Objectives Analyze bending of thin plates Analyze for stress and deformation in plates subjected to bendingandtwistingbending and twisting Analyze for stress and deformation in plates subjected to distributed transverse loads Analyze for stress and deformation in plates subjected to combined loads Describe energy method for analyzing thin plates 2

Topics

Thin plates subjected to bending, twisting, distributed transverse and in-plane loading BendingofthinplateswithasmallinitialcurvatureBending of thin plates with a small initial curvature Energy method Characteristics ShtfthitilSheets of thin materials Resist membrane forces and bending

Pure Bending

Descriptions TheneutralplanedoesnotdeformandisusedastheTheneutral planedoes not deform and is used as the reference plane. Planes remain planes Strains can be determined in terms of z and the radii of curvature r. Direct stresses vary linearly across the thickness.

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Pure Bending

Strain 5

 1 (xxyz xz E    

  

21 )() 1 1 (

x xy yyxz y

Ez z E   21 )() 1y yxEz

Pure Bending

Applied boundary Moments /2 2/2 /2 2/2

11 ()() 1 11 ()() 1

t xxxt xyxy t yyyt

Ez MyzydzMdzD Ez MxzxdzMdzD

     

 

   

  

6

1yxyx  ³ 2Flexural rigidity 12(1)Et D

1 () 1 ()

x xy y yx

MD MD

   

 

Pure Bending

Deflection 222 222 222 222

11 ()() 11 ()()

x xxy y

www MDD xxy www MDD yyx

    

      7

222 22 22If 0: 111 If : (1)

y yyx y xy xy

yyx ww M yx M MMM D

  

     

Bending & Tw isting

Description 8

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Bending & Tw isting

Equilibrium 



0 cossinsincosnnxyxyxyFMACMABMBCMABMBC 9

  

  

 

  

22 cossinsin2 0 sincoscossin sin2cos2 2

0 0

nxyxy ttxyxyxy x

n y ttxy

MMMM FMACMABMBCMAB

F F

MBC MM MM

Bending & Tw isting

Principal Moments   

sin2cos2 2 0 Principal moments and curvatures

xy txy t

MM MM M 10

 2 tan2xy xy

M MM

Bending & Tw isting

Shear #1  



 

xyxyMyyzdz Mxxzdz  

 

  

  

^/2 ^/2 ^/2 /2

xyxy t xy x

xyt t yxyt

Mxxzdz Mzdz GzdzM

Bending & Tw isting

Shear #2    , ww uzv xy vu  

    ² 2

x xy

y xy w z xy   

     



232/2/22 /2/2 322

2 6 (1) 12(1)

tt xyxyttwGtw MGzdzGzdz xyxy Etww D xyxy

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Transverse Load

Description 13

Shear strains are ignored.

Transverse Load

Equilibrium #1  

 

  

  

/2 /2 /2 /2 0

t xxzt t yyzt

Qdz Qdz F 14



      



0 ()()0 0

z yx xxyy yx

F QQ QxyQyQyxQxqxy xy QQ q xy

Transverse Load

Equilibrium #2  

 

 

 

/2 /2 /2 /2

t xxt t yyt

Mzdz Mxdz   

     

 

/2 /2 222

0 ()()

t xyyxxyt x xyy xyxyyy

MMxdz M MM MxyMyMyxMx xy QQyyy 15

         ()()0 222 0 0

yx yxx xyy y xyx x

QQyyy QyxyQxQqx yx MM Q xy MM Q yx

Transverse Load

Equilibrium #3   

22 200xyyxyyy yMMMMQ Q xyyxyy       

22 2 2222

00 From

xyxyxxx x yx xyyxyx

xyyxyy MMMMQ Q yxxyxx QQ q xy MMMM 16

    ²²

222 222

xyyxyx xyyx

q xyyxxy MMM q xyxy

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Transverse Load

Displacement #1   

22222 2222(), (), (1)xyxywwwww MDMDMD xyxyyx  ^

        

222 22 22222222 222222 424

From 2 ()2(1)() ()2(1)

xyyx

yyy MMM q xyxy wwwww DDDq xyxyxxyyyx www  

44 ()wwq 17

 4222()2(1) xxyx    

2422 444 4224

() 2

Dyyxy wwwq Dxxyy   ²²

2 22()q w Dxy

Transverse Load

Displacement #2 22MMww     ²² ²² 22

() ()

xyx x yxy y

MMww QD xyxxy MMww QD yxyxy 18

Transverse Load

BC #1: Simple Support Free to rotate, no deflection Alonganedgex 

     

2 2 22 22

Along an edge 0, 0 M()0

BC BC

x ww w yy ww D xy   ² 20, ()0BCBCw w x

Transverse Load

BC #2: Built-in Notorotation,nodeflectionNo to rotation, no deflection Along an edgex   0, 0BCw w x

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Transverse Load

BC #3: Free No bending moments, twisting moments or vertical shearing forces Along an edge ()0()0()0x MMQ 

       

33 32 22

()0, ()0, ()0 ()0 ((2))0

xBCxyBCxBC xy xBC BC

MMQ M Q y ww xxy ww 21

  22()()0xBCBCww M xy

Transverse Load

BC #3: Free No bending moments, twisting moments or vertical shearing forces Along an edge ()0()0()0x MMQ 

       

33 32 22

()0, ()0, ()0 ()0 ((2))0

xBCxyBCxBC xy xBC BC

MMQ M Q y ww xxy ww 22

  22()()0xBCBCww M xy

Example

Transverse Load #1 A simply supported plate of dimension a×bis subjected to a uniform transverse load q.Determine the deflection and bending moment distributions. 23

Example

Transverse Load #2 444 4224

The deflection must satisfy 2wwwq Dxxyy   2 2 2 2

Boundary conditions 0 and at 0 and 0 and at y0 and y

Dxxyy w wxxa x w wb y

       24

11 11

Thus sinsin sinsin

mn mn mn mn

mxny wA ab mxny qa ab

 

   

 

 

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Example

Transverse Load #3 00sinsinab abmxny qdxdy ab mxnymxny

 

 

  

0011 0

sinsinsinsin 4 For sinsin

ab mn mn mn a

mxnymxny adxdy abab ab a mxmx fdx aa a

 

 

   

  

25

00 when and when 2 For sinsin 0 when and whe 2

b

a fmmfmm nyny gdy bb b gnnf



   



n nn

Example

Transverse Load #4 444 422420wwwq Dxxyy   4224 2 422

()2()()()0 ()()0 1

mn mn mn mn

ammnn A aabbD amn A abD

  

 

   

 

26

422211 11

1 sinsin (/)(/) sinsin^mn

mn mn mn

amxny w abDmanb mxny qa ab

  

   

  

  

Example

Transverse Load #5 200416 sinsinab mnqmxnyq adxdy ababmn 

 

62221,3,51,3,5 max62221,3,51,3,5

16sin(/)sin(/) (/)(/) 16sin(/2)sin(/2) at /2,/2 (/)(/)

mn mn

ababmn qmxanyb w Dmnmanb qmn wxayb Dmnmanb

    

   

     

 

,,,, max

(/)(/) 0.0443

mnmanb wq

  

4 3 if ,0.3a ab Et

Example

Transverse Load #6 22 42221,3,51,3,5 22

(/)(/)16 sinsin (/)(/) (/)(/)

x mn

manbqmxny M abmnmanb b

 

 

   



22 42221,3,51,3,5 ,max,max ,max,max

(/)(/)16 sinsin (/)(/) at /2,/2 0

y mn xy xy

manbqmxny M abmnmanb MMxayb MM

 

 

    



2 .0479 if,0.3 1212yx

qaab MzMz 33 ,max,max22 2 ,max,max2

, 66 , at 2 0.2871 if,0.3

yx xy yx xy xy

tt MMt z tt a qab t

  

  

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Combined Loadings

Descriptions Combined loads TldTransverse loads In-plane forces 29

Combined Loadings

Equilibrium #1 0yxxNN xy   0xyyNN xy   30

2 20()cos()cos() ()0

x xxx yx yxyx

Nwww FNxyxNy xxxx N NyxNx y 

     



Combined Loadings

Equilibrium #2 2 xyNwww 31

2 2

()()()() (( ))xyz

xy xyzxyxy xy xy yx yxzyx

Nwww NNxyxNy xxxyy Nww Nxyxy xyxy Nww NNxyxy xyyx

N

  

        

Combined Loadings

Equilibrium #3 2 Nwww 32

2 2 2 2 2

()()()() ()()

x xzxx x x y yzy

xz

Nwww NNxyxNy xxxx Nww Nxyxy xxx Nww NNxyxy yy

N y

  

        

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Combined Loadings

Equilibrium #4 ()()()()

zz zxyzyxzxzyz

RF RNNNN 



2 22 22 222 22

)2 (2)

xyxy zxy yx xy zxyxy

z

NNwww RxyNxyxy xyxyyx NNwwww NxyxyNxyxy xxyyxy www RNNNxy xyxy

R

  

        

  33

fromyxx

xyxy NN xy

   

 0 and 0 and in transverse loadxyyNN xy   444222 4224221 2(2)xyxywwwwww qNNN Dxyxxyyxy  

Initial Curvature

Descriptions 010 444 111 4224

Assume total deflection as the sume of initial 2

wwww www   4224 222 010101 22 2 0

()()()1 (22) Initial curvature is equivalent to the application of

xyxy x

xxyy wwwwww qNNN Dxyxy qN

     

22 000 222yxywww NN xyxy   34

0 11 126222211

sinsin, if is compressive, 0 sinsin, (/)(/)

mnxyxy mn mnx mnmn mnx

xyxy mxny wANNN ab ANmxny wBB abDamnambN

  

   

    

 

Energy Method

Descriptions 222 22 2222 2222

1 (2) 2 From (),(),(1)

xyxy xyxy

www UMMMxy xyxy wwww MDMDMD xyyx

 

     

Energy Method

Bending and Twisting 22222 222 2222 22222 22

()()22(1)() 2 For a rectangular plate ab

Dwwwww Uxy xyxyxy ab Dwwwww

    



22 222200 2 2

(()2(1)() 2 For bending only, 0 ( 2

ab xy

Dwwwww Udxdy xyxyxy M Dw U x

      



222 2 22200()2abwww dxdy yxy  



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Energy Method

Transverse Load 00()abVwqxy Vwqdxdy

 



37

Energy Method

In-plane Loads #1 22 21 (1()) 2

axw w x xa

 

    38

2 0 2 0

2 1 (1()) 2 1 () 2 ()

a a xx

x w adx x w aadx x VNyNaay

 

       

 

Energy Method

In-plane Loads #2 21 ()abw VNdxdy



39

2 00 2 00

() 2 1 () 2

xx ab yy

VNdxdy x w VNdxdy y

    

  Energy Method

In-plane Loads #3 1 2 1 2

xyxy xyxy

ww VNx xy ww NyV 

     40

00 22 000

2 1 2 2 1 ()()2 2

xy ab xyxy abb xyxyxy

x xy

yy xy ww VNdxdy xy wwww VVVVNNNdxdy xyxy

      

  

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Example

Energy #1 A simply supported plate of dimension a×bis subjected to a uniform transverse load q.Determine the deflection and bending moment distributions. 1 22222 22 222200sinsin (()2(1)() 2^mn

m ab

mxny wA ab Dwwwww UVdxdy xyxyxy

 

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