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Theory of Elasticity

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Theory of Elasticity

Chapter 11

Bending of Thin Plates

薄板弯曲

Theory of Elasticity

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Chapter

- 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

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- 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）

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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

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Review: bending of beams

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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）

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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

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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。

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Physical Equation Reduced to 3

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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.

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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)

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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:

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u, v in terms of w

uz=0=0， vz=0=0

u-ε Relations

εx ,εy ,γxyin terms of w

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Physical Equations

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x ,y ,τxyin terms of w

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τxz,τyzin terms of w

The equilibrium equation

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zin terms of w

If body force fz≠0:

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The governing equation of the classical theory of bending of thin elastic plates:

Flexural rigidity of the plate

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Geometrical Equations

Physical Equations

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Equilibrium Equations

Boundary Cond. (load:q)

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+edges B.C.

薄板的弹性曲面微分方程

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Another method to get the equation

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Beam

Thin plate

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History of the Equation

Bernoulli, 1798:

Lagrange, 1811:

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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。)

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x

z

y

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Stress distribution

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应力分量

和内力、载荷关系

名称

数值

最大

最大

较小

最小

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+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

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Built-in or clamped edge固定边界

At a clamped edge parallel to the y axis:

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Simply Supported edge简支边界

Free to rotate

The bending moment and the deflection along the edge must be zero.

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Free edge自由边界

Only 2 are allowed for an equation of 4th order

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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:

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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:

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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

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- 9-1

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