1. Chapter 11 Bending of Thin Plates 薄板弯曲
Theory of Elasticity
Chapter 11
Bending of Thin Plates
薄板弯曲

2. Content Theory of Elasticity Introduction Mathematical Preliminaries
Chapter
Page
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 11 1

3. Bending of Thin Plates Theory of Elasticity
Chapter
Page
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） 11 2

4. 11.1 Some Concepts and Assumptions
Theory of Elasticity
Chapter
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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 11 3

5. 11.1 Some Concepts and Assumptions
Theory of Elasticity
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11.1 Some Concepts and Assumptions
Review: bending of beams 11 4

6. 11.1 Some Concepts and Assumptions
Theory of Elasticity
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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） 11 5

7. 11.1 Some Concepts and Assumptions
Theory of Elasticity
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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 11 6

8. 11.1 Some Concepts and Assumptions
Theory of Elasticity
Chapter
Page
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 7
Physical Equation Reduced to 3 11

9. 11.1 Some Concepts and Assumptions
Theory of Elasticity
Chapter
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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. 11 8

10. 11.1 Some Concepts and Assumptions
Theory of Elasticity
Chapter
Page
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) 11 9

11. 11.2 Differential Equation of Deflection 弹性曲面的微分方程
Theory of Elasticity
Chapter
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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: 11 10

12. 11.2 Differential Equation of Deflection
Theory of Elasticity
Chapter
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11.2 Differential Equation of Deflection
u, v in terms of w
uz=0=0， vz=0=0
u-ε Relations
εx , εy , γxy in terms of w 11 11

13. 11.2 Differential Equation of Deflection
Theory of Elasticity
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11.2 Differential Equation of Deflection
x , y , τ xy in terms of w
Physical Equations 11 12

14. 11.2 Differential Equation of Deflection
Theory of Elasticity
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11.2 Differential Equation of Deflection
τ xz , τ yz in terms of w
The equilibrium equation 11 13

15. 11.2 Differential Equation of Deflection
Theory of Elasticity
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11.2 Differential Equation of Deflection
 z in terms of w
If body force fz≠0: 11 14

16. 11.2 Differential Equation of Deflection
Theory of Elasticity
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11.2 Differential Equation of Deflection
The governing equation of the classical theory of bending of thin elastic plates:
Flexural rigidity of the plate 11 15

17. 11.2 Differential Equation of Deflection
Theory of Elasticity
Chapter
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11.2 Differential Equation of Deflection
Geometrical Equations
Physical Equations
Equilibrium Equations
Boundary Cond. (load:q)
+edges B.C.
薄板的弹性曲面微分方程 11 16

18. 11.2 Differential Equation of Deflection
Theory of Elasticity
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11.2 Differential Equation of Deflection
Another method to get the equation 11 17

19. 11.2 Differential Equation of Deflection
Theory of Elasticity
Chapter
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11.2 Differential Equation of Deflection
History of the Equation
Bernoulli, 1798:
Beam
Thin plate
Lagrange, 1811: 11 18

20. 11.3 Internal Forces of Thin Plate
Theory of Elasticity
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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。) 11 19

21. 11.3 Internal Forces of Thin Plate
Theory of Elasticity
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11.3 Internal Forces of Thin Plate y x z 11 20

22. 11.3 Internal Forces of Thin Plate
Theory of Elasticity
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11.3 Internal Forces of Thin Plate
Stress distribution 11 21

23. 11.3 Internal Forces of Thin Plate
Theory of Elasticity
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11.3 Internal Forces of Thin Plate 11 22

24. 11.3 Internal Forces of Thin Plate
Theory of Elasticity
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11.3 Internal Forces of Thin Plate 11 23

25. 11.3 Internal Forces of Thin Plate
Theory of Elasticity
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11.3 Internal Forces of Thin Plate
应力分量
和内力、载荷关系 名称 数值 最大 较小 最小 11 24

26. 11.4 Boundary Conditions Theory of Elasticity +edges B.C.
Chapter
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11.4 Boundary Conditions
+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 11 25

27. 11.4 Boundary Conditions Theory of Elasticity
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11.4 Boundary Conditions
Built-in or clamped edge固定边界
At a clamped edge parallel to the y axis: 11 26

28. 11.4 Boundary Conditions Theory of Elasticity
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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. 11 27

29. 11.4 Boundary Conditions Theory of Elasticity Free edge自由边界 Page
Chapter
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11.4 Boundary Conditions
Free edge自由边界
Only 2 are allowed for an equation of 4th order 11 28

30. 11.5 Examples: Simple supported rectangular plate
Theory of Elasticity
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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 load of q(x,y)
Solution:
Boundary conditions: 11 29

31. 11.5 Examples:Simple supported rectangular plate
Theory of Elasticity
Chapter
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11.5 Examples:Simple supported rectangular plate
The plate deflection must satisfy the following equation and the boundary conditions.
Choose to represent w by the double Fourier series:
All the boundary conditions are satisfied. Substituted into we obtain: 11 30

32. 11.5 Examples: Simple supported rectangular plate
Theory of Elasticity
Chapter
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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 11 31

33. Theory of Elasticity
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Homework
9-1 11 32