Theory of Elasticity
Chapter 11
Bending of Thin Plates
薄板弯曲
Content
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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
11
ChapterEquation
Physical
ReducedPage
to 3 7
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
Displacement:
u, v
Primary strain Components:
Primary stess Components:
Secondary stess Components:
 z ,  zy ,  zx  0
 x ,  y ,  xy
 x ,  y , xy
 zx , zy
u, v,  ,   f ( w)
Chapter 11
Page 10
1
 x   y ,
E

1

 y   y   x ,
E

21   

 xy 
 xy。
E

x 
z
11.2 Differential Equation of Deflection
u, v in terms of w
 zx  0,  zy  0
u
w

,
z
x
u
 x ,  y ,  xy
w
w
z, v  
z
x
y
εx , εy , γxy in terms of w
Chapter 11
v
w

。
z
y
uz=0=0， vz=0=0
u-ε Relations
u
2w
u
2w 
x 
  2 , y 
  2 ,
x
y
x
y 

2
v u
 w

 xy 

 2
z。

x y
xy
Page 11
11.2 Differential Equation of Deflection
x , y , τ xy in terms of w
Physical Equations
 2w
2w  
 2  
,
2  
y  
 x
Ez   2 w
2w  



y  


,

1   2  y 2
x 2  

Ez  2 w

 xy  
。
1   xy


Ez
x  
1  2
u
2w
u
2w 
x 

, y 
  2 ,
2
x
y
x
y 

2
v u
 w

 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
 zx
 zx
Chapter 11
E

2 1  2
E

2 1  2




 2 2  2 
 z 
  w , 
4  x



 2 2  2 
 z 
  w。

4  y

Page 13
11.2 Differential Equation of Deflection
z
in terms of w
E
z 
2 1  2


 2
z3  4
 z   w  Fx, y 。
3
 4
 2 
E
  1  3  3  4
z 
 z     z   w
2 
2  3
8 
2 1   4 


2
E
z 4
1 z  


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
 2 
E
  1  3  3  4
z 
  z     z   w
2  3
8 
2 1  2  4 


2
E
z
1 z  

  1   4 w。
2 
6 1   2     

E
4

wq
2
12 1  
3


D w  q
4
Chapter 11

E 3
D
， Flexural rigidity of the plate
2
12 1  

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
Chapter 11
+edges B.C.
薄板的弹性曲面微分方程
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
 2w
2w  

,

2
2  

x

y


Ez   2 w
2w  



y  


,

1   2  y 2
x 2  

Ez  2 w

。
x  xy  
1   xy

x

Ez
x  
1  2
Fsx
M xy
z
y
M
 xy
x
 xz
Chapter 11
Page 20
E
 zx 
2 1  2
E
 zy 
2 1  2




 2 2   2 
 z    w , 
4  x



 2 2   2 
 z    w。

4  y

11.3 Internal Forces of Thin Plate
Stress distribution
 2w
2w  

,

2
2  
y  
 x
Ez   2 w
2w  



y  


,

1   2  y 2
x 2  

Ez  2 w

 xy  
。
1   xy








2

FSx   2  xz dz。

 2 2   2 
 z    w , 
4  x



 2 2   2 
 z    w。

4  y

Chapter 11

M x   2 z x dz。 M xy   2 z xy dz。
Ez
x  
1  2
E
 zx 
2 1  2
E
 zy 
2 1  2

Page 21
2
2
11.3 Internal Forces of Thin Plate
M xy   2 z xy dz。
M x   2 z x dz。


2
FSx   2  xz dz。

2

 2w
2w  2 2
 2   2    z dz
y   2
 x
E 3   2 w
2w 
 2   2 。

2 
12 1    x
y 
E
Mx  
1-  2






M xy
E 2w 2 2

 z dz

1   xy  2
E 3
2w

。
121    xy

E
 2 2  2 2
FSx  
 w    z  dz。
2

4
2 1 -  x
2


E 3  2

 w。
2
12 1 -  x

Chapter 11

Page 22
2
11.3 Internal Forces of Thin Plate
E 3
D
，
2
12 1  


 2w
 2w
2w 
2w  
M x  D 2   2 , M y  D 2   2 ,
y 
x  
 x
 y

2w

M xy  M yx  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 x 0
Chapter 11
Page 26
 w 
 0, 
  0。
 x  x 0
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。
Chapter 11
w y0
 2w
2w 
  0。
 0,  2  
2 
x  y 0
 y
w y0
 2w 
 0,  2   0。
 y  y 0
Page 27
11.4 Boundary Conditions
Free edge自由边界
M 
y yb
M 
 0,
yx y  b
 0,
F 
Sy y b
 0。
Only 2 are allowed for an equation of 4th order
F
t
sy
 Fsy 
M y  0, F t sy
M yx
x
M yx
 Fsy 
0
x
 2w
2w 



 0,
2
2 
x  y  b
 y
 3w
3w 
 3  2    2 
x y  y  b
 y
Chapter 11
Page 28




 0


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
Chapter 11
Page 30
we obtain:
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