Theory of Elasticity

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Theory of Elasticity. Chapter 11 Bending of Thin Plates 薄板弯曲. Theory of Elasticity. Page. Chapter. Content. Introduction Mathematical Preliminaries Stress and Equilibrium Displacements and Strains Material Behavior- Linear Elastic Solids Formulation and Solution Strategies

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### Theory of Elasticity

Chapter 11

Bending of Thin Plates

Theory of Elasticity

Page

Chapter

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

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Theory of Elasticity

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Chapter

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）

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Theory of Elasticity

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Chapter

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

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Chapter

11.1 Some Concepts and Assumptions

Review: bending of beams

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Chapter

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）

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Chapter

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

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or

Theory of Elasticity

or

Page

Chapter

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。

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Physical Equation Reduced to 3

Theory of Elasticity

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Chapter

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.

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Chapter

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)

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Chapter

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:

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Chapter

11.2 Differential Equation of Deflection

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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Chapter

11.2 Differential Equation of Deflection

x ,y ,τxyin terms of w

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Chapter

11.2 Differential Equation of Deflection

τxz,τyzin terms of w

The equilibrium equation

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Chapter

11.2 Differential Equation of Deflection

zin terms of w

If body force fz≠0:

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Chapter

11.2 Differential Equation of Deflection

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

Theory of Elasticity

Equilibrium Equations

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Chapter

11.2 Differential Equation of Deflection

+edges B.C.

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Theory of Elasticity

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Chapter

11.2 Differential Equation of Deflection

Another method to get the equation

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Beam

Thin plate

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Chapter

11.2 Differential Equation of Deflection

History of the Equation

Bernoulli, 1798:

Lagrange, 1811:

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Chapter

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。)

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x

z

y

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Chapter

11.3 Internal Forces of Thin Plate

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Theory of Elasticity

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Chapter

11.3 Internal Forces of Thin Plate

Stress distribution

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Chapter

11.3 Internal Forces of Thin Plate

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Chapter

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

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Chapter

11.4 Boundary Conditions

Built-in or clamped edge固定边界

At a clamped edge parallel to the y axis:

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Chapter

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.

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Chapter

11.4 Boundary Conditions

Free edge自由边界

Only 2 are allowed for an equation of 4th order

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Chapter

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:

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Chapter

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:

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Chapter

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

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Chapter

Homework
• 9-1

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