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					Theory of Elasticity

      Chapter 11
 Bending of Thin Plates
       薄板弯曲
           Content
•   Introduction
•   Mathematical Preliminaries
•   Stress and Equilibrium
•   Displacements and Strains
•   Material Behavior- Linear Elastic Solids
•   Formulation and Solution Strategies
•   Two-Dimensional Problems
•   Three-Dimensional Problems
•   Bending of Thin Plates(薄板弯曲)
•   Plastic deformation - Introduction
•   Introduction to Finite Element Method

    Chapter 11          Page 1
    Bending of Thin Plates
• 11.1 Some Concepts and Assumptions
(有关概念及假定)
• 11.2 Differential Equation of Deflection
(弹性曲面的微分方程)
• 11.3 Internal Forces of Thin Plate
(薄板截面上的内力)
• 11.4 Boundary Conditions(边界条件)
• 11.5 Examples(例题)



 Chapter 11       Page 2
 11.1 Some Concepts and Assumptions
Thin plate(薄板)
One dimension of which (the thickness)is small in comparison with the
other two.(1/8-1/5)>/b≥(1/80-1/100)
                                Middle surface(中面)
                                The plane of Z=0

                             Bending of thin plate(薄板弯曲)
                             Only transverse loads act on the plate.
                             (垂直于板面的载荷,横向)
                            Longitudinal loads: Plane stress State
                            Similar with Bending of elastic beams



   Chapter 11               Page 3
   11.1 Some Concepts and Assumptions

Review: bending of beams




   Chapter 11        Page 4
 11.1 Some Concepts and Assumptions
Assumptions(beam):
1, The plane sections normal to the longitudinal axis of the
beam remained plane (平面假设)
2, In the course “elementary strength of materials”: simple
stress state :only normal stress exists, no shearing stress.
Pure bending
(单向受力假设)




  Chapter 11              Page 5
  11.1 Some Concepts and Assumptions

Assumptions for bending of thin plate ( Kirchhoff)
            Besides of the basic assumptions of “Theory of
            elasticity”
                1,Straight lines normal to the middle surface will
                remain straight and the same length.变形前垂直于中
                面的直线变形后仍然保持直线,而且长度不变。

                2,Normal stresses transverse to the middle surface
                of the plate are small and the corresponding strain
                can be neglected.垂直于中面方向的应力分量z, τzx, τzy
                远小于其他应力分量,其引起的变形可以不计.

                3,The middle surface of the plate is initially plane and
                is not strained in bending.中面各点只有垂直中面的位
                移w,没有平行中面的位移


   Chapter 11             Page 6
 11.1 Some Concepts and Assumptions
1,Straight lines normal to the middle surface will
remain straight and the same length.变形前垂直于中
面的直线变形后仍然保持直线,而且长度不变。




                                                     or




                                                     or




                          Page
 Physical Equation Reduced to 3 7
  Chapter 11
    11.1 Some Concepts and Assumptions

2,Normal stresses transverse to the middle surface of the plate are
small and the corresponding strain can be neglected.垂直于中面方向的应
力分量z, τzx, τzy远小于其他应力分量,其引起的变形可以不计.




    Chapter 11              Page 8
 11.1 Some Concepts and Assumptions
3,The middle surface of the plate is initially plane and is not strained in
bending.中面各点只有垂直中面的位移w,没有平行中面的位移




           uz=0=0, vz=0=0, w=w(x, y)




   Chapter 11                  Page 9
 11.2 Differential Equation of Deflection
           弹性曲面的微分方程
Displacement Formulation
The equilibrium equation is expressed in terms of displacement. w

Besides w, the unknowns include
                                                         x 
                                                                1
                                                                   x   y ,
                                                                                
                                                                E
 Displacement:        u, v                                                      
                                  z ,  zy ,  zx  0    y   y   x ,
                                                                1               
 Primary strain Components:
                                   x ,  y ,  xy              E
                                                                21   
                                                                                
                                                                                
 Primary stess Components:        x ,  y , xy          xy             xy。
                                                                    E           
 Secondary stess Components:       zx , zy             z
                       u, v,  ,   f (w)
   Chapter 11                Page 10
   11.2 Differential Equation of Deflection

u, v in terms of w
                                                      uz=0=0, vz=0=0
                        u    w        v    w
  zx  0,  zy  0            ,             。
                        z    x        z    y
                            w          w
                      u      z, v      z
                            x          y
  x ,  y ,  xy                              u-ε Relations

εx , εy , γxy in terms of w               u    2w         u    2w 
                                   x         2 , y         2 ,
                                          x    x          y    y 
                                                                      
                                           v u        w
                                                        2
                                                                      
                                    xy          2      z。
                                           x y       xy           
                                                                      

      Chapter 11              Page 11
   11.2 Differential Equation of Deflection

 x , y , τ xy in terms of w
Physical Equations


                                               Ez        2w  2w  
                                       x              2 
                                                         x        ,
                                                                  2  
                                              1  2          y  
                                                  Ez   2 w   2w   
                                       y              2       ,
                                                1   2  y
                                                              x 2  
                                                                    
       u    2w         u    2w 
x         2 , y         2 ,              Ez  2 w             
       x    x          y    y      xy                。         
                                               1   xy             
                                                                       
        v u        w
                     2
                                   
 xy          2      z。
        x y       xy           
                                   


    Chapter 11              Page 12
 11.2 Differential Equation of Deflection

τ xz , τ yz in   terms of w
The equilibrium equation




                               E            2 2  2 
                    zx                   z      w, 
                             
                            2 1  2      
                                               4  x
                                                        
                                                         
                               E            2 2  2 
                    zx                   z      w。
                             
                            2 1  2      
                                               4  y
                                                        
                                                         


    Chapter 11                     Page 13
 11.2 Differential Equation of Deflection
z   in terms of w
                                              2 z3  4
                                              z   w  Fx, y 。
                                       E
                              z 
                                   21     4
                                          2 
                                                  3 
                                      E     2      1  3  3  4
                              z        2     z    z       w
                                         
                                   2 1   4      2  3    8 
                                                                  
                                                  2
                                      E     1 z    z 4
                                              1   w。
                                         
                                   6 1  2  2     
  If body force fz≠0:



u, v,  x ,  y ,  z , xy , zx , zy ,  x ,  y ,  xy  f ( w)
     Chapter 11             Page 14
  11.2 Differential Equation of Deflection
The governing equation of the classical theory of
bending of thin elastic plates:

       z z                 q
                            2
                                                     E      2    1  3  3  4
                                             z            z    z        w
                                                       
                                                  2 1  2  4   2  3    8 
                                                                                
                                                                2
                                                     E    1 z    z
                                                       2 
                                                              1   4 w。
                                                       
                                                  6 1   2     
           E   3
                   4w  q
          
        12 1   2      
                                       E 3
          D w  q  4            D
                                        
                                    12 1   2
                                               , Flexural rigidity of the plate
                                               

   Chapter 11                    Page 15
11.2 Differential Equation of Deflection

u, v  f ( w)
 x ,  y ,  xy  f ( w)            Geometrical Equations


 x ,  y , xy  f ( w)             Physical Equations


 zx , zy  f ( w)                  Equilibrium Equations



 z  f ( w)                         Boundary Cond. (load:q)


                            D 4 w  q      +edges B.C.

                     薄板的弹性曲面微分方程
Chapter 11              Page 16
  11.2 Differential Equation of Deflection
Another method to get the equation




   Chapter 11         Page 17
 11.2 Differential Equation of Deflection
History of the Equation

 Bernoulli, 1798:




        Beam                    Thin plate
  Lagrange, 1811:




  Chapter 11          Page 18
  11.3 Internal Forces of Thin Plate

Internal Forces:

Stress resultants: It is customary to integrate the stresses ovet the
constant plate thickness defining stress reslultants.薄板截面的每单
位宽度上,由应力向中面简化而合成的主矢量和主矩。



 Design requirement(薄板是按内力来设计的;)
 Dealing with the Boundary Conditions(在应用圣维南原理处理边界
 条件,利用内力的边界代替应力边界条件。)




   Chapter 11                 Page 19
    11.3 Internal Forces of Thin Plate
                                              Ez       2w      2w  
                                      x             x 2   y 2 ,
                                                                     
                                             1  2                  
                                                Ez   2 w       2w   
                    Fsx              y                          , 
                                              1   2  y 2
                                                                x 2  
                                                                      
                 M xy                           Ez  2 w                 
                               M   x  xy                。             
                                   x          1   xy                 
                                                                         
                           xy
          z                      x                           2 2   2 
                                                 E
                                        zx                  z    w,
y
                         xz                   
                                              2 1  2      
                                                                 4  x
                                                                         
                                                                          
                                                 E            2 2   2 
                                        zy                  z    w。
                                              
                                              2 1  2      
                                                                 4  y
                                                                         
                                                                          



    Chapter 11           Page 20
      11.3 Internal Forces of Thin Plate
                                                                        
 Stress distribution
                                         M x   2 z x dz。 M xy   2 z xy dz。
                                                                        
         Ez       2w      2w                     2                      2
 x             x 2   y 2 ,
                                
        1  2                                     
                            2w  
y  
           Ez   2 w
                 
         1   2  y 2
                        
                            x 2  
                                    
                                 ,     FSx      
                                                  
                                                      2
                                                              xz   dz。
                                                        2
           Ez  2 w                 
 xy                。             
         1   xy                 
                                    

          E            2 2   2 
 zx                  z    w,
        
       2 1  2      
                          4  x
                                  
                                   
          E            2 2   2 
 zy                  z    w。
       
       2 1  2      
                          4  y
                                  
                                   


     Chapter 11                         Page 21
    11.3 Internal Forces of Thin Plate
                                                                 
 M x   2 z x dz。         M xy   2 z xy dz。        FSx     
                                                                   2
                                                                              xz   dz。
                                                                
             2                            2                           2

                                                                         
         E     2w    2w  2 2                          E 2w 2 2
Mx           2   2    z dz
               x                            M xy               2 z dz
       1-  2         y   2
                                                       1   xy
          E 3   2 w    2w                            E 3  2 w
                 2   2 。                       
                
                2 
       12 1    x      y                          121    xy
                                                                        。

                                          
                              E     2 2  2 2
                  FSx               w    z  dz。
                               
                          2 1 -  x
                                 2         
                                           2
                                                  4
                                                   
                         E 3  2
                                   w。
                               
                       12 1 -  x
                               2




     Chapter 11                     Page 22
11.3 Internal Forces of Thin Plate
                      E 3
                D            ,
                       
                   12 1   2
                              
          2w       2w           2w 2w  
M x  D 2   2 , M y  D 2  
          x                      y      ,
                                           2  
                    y                x  
                          2w                   
                                                
M xy  M y x  D1         ,                
                          xy                  
          2                    2              
FSx  -D  w , FSy  -D  w。                    
        x                    y                
                                                


 Chapter 11         Page 23
11.3 Internal Forces of Thin Plate

应力分量           和内力、载荷关系       名称   数值
                                   最大

                                   最大

                                   较小

                                   最小




  Chapter 11        Page 24
11.4 Boundary Conditions

       D w  q
             4           +edges B.C.


 Simply Supported edge简支边界

 Free edge自由边界


 Built-in or clamped edge固定边界




Chapter 11         Page 25
   11.4 Boundary Conditions
Built-in or clamped edge固定边界

                      At a clamped edge parallel to the y axis:




                                            w 
                         w  x  0    0,       0。
                                            x  x 0




  Chapter 11          Page 26
   11.4 Boundary Conditions
Simply Supported edge简支边界

                                Free to rotate
                                The bending moment and
                                the deflection along the
                                edge must be zero.
                      w y0  0, M y y0  0。
                                      2w   2w 
                     w y0     0,  2  
                                      y          0。
                                               2 
                                            x  y 0

                                      2w 
                      w y0    0,  2   0。
                                      y 
                                           y 0


   Chapter 11        Page 27
 11.4 Boundary Conditions
Free edge自由边界
                 M y yb    0,       M      yx y b    0,    F 
                                                                    Sy y b    0。
                Only 2 are allowed for an equation of 4th order
                                                           M yx
                                    F   t
                                            sy    Fsy 
                                                          x
                                                              M yx
                              M y  0, F t sy          Fsy        0
                                                               x

                                             2w      2w                      
                                             y 2   x 2 
                                                                 0,           
                                                            yb                
                                                                                 
                                             3w          3w 
                                             3  2    2                  0
                                                                                 
                                             y          x y  y  b
                                                                                 



 Chapter 11        Page 28
  11.5 Examples: Simple supported rectangular plate

An application of plate theory to a specific problem
                       Problem: Calculating the deflection w of a
                                  simply supported rectangular
                                  plate as shown in the fig., which
                                  is loaded in the z direction by a
                        Solution: load of q(x,y)
                        Boundary conditions:




   Chapter 11              Page 29
   11.5 Examples:Simple supported rectangular plate

The plate deflection must satisfy the following equation and the
boundary conditions.

                         D w  q
                               4

Choose to represent w by the double Fourier series:




All the boundary conditions are satisfied. Substituted into   we obtain:




     Chapter 11                Page 30
 11.5 Examples: Simple supported rectangular plate




If q(x,y) were represented by Fourier series, It might be possible
to match coefficients. Expand q(x,y) in a Fourier series.




                                                        W



  Chapter 11                 Page 31
             Homework
• 9-1




Chapter 11      Page 32