theory of elasticity

Theory of Elasticity

Chapter 11Bending of Thin Plates

薄板弯曲

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

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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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11.1 Some Concepts and Assumptions

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Thin plate （薄板）

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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。

Physical Equation Reduced to 3

or

or

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Chapter Page

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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11.2 Differential Equation of Deflection弹性曲面的微分方程

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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: xyyx ,,

0,, zxzyz

Primary stess Components: xyyx ,,

。xyxy

xyy

yxx

E

12

,E

1

,E

1

Secondary stess Components: zzyzx ,

)(,,, wfvu

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11.2 Differential Equation of Deflection

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u, v in terms of w

0,0 zyzx 。y

w

z

v,

x

w

z

u

zy

wv,z

x

wu

uz=0=0 ， vz=0=0

xyyx ,,

εx , εy , γxy in terms of w

。zyx

w2

y

u

x

v

,y

w

y

u,

x

w

x

u

2

xy

2

2

y2

2

x

u-ε Relations

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Chapter Page

x , y , τ xy in terms of w

Physical Equations

。zyx

w2

y

u

x

v

,y

w

y

u,

x

w

x

u

2

xy

2

2

y2

2

x

。yx

w

1

Ez

,x

w

y

w

1

Ez

,y

w

x

w

1

Ez

2

xy

2

2

2

2

2y

2

2

2

2

2x

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τ xz , τ yz in terms of w

。wy4

z12

E

,wx4

z12

E

22

22zx

22

22zx

The equilibrium equation

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z in terms of w

。y,xFw3

zz

412

E 432

2z

。wz

1z

2

1

16

E

w8

z3

1

2z

412

E

42

2

43

32

2z

If body force fz≠0:

)(,,,,,,,,,, zyzxxy wfvu xyyxzyx

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

q2

zz

qwD 4 ，2

3

112

ED

。wz

1z

2

1

16

E

w8

z3

1

2z

412

E

42

2

43

32

2z

qw112

E 42

3

Flexural rigidity of the plate

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Chapter Page

)(, wfvu )(,, wfxyyx

)(,, wfxyyx

)(, zyzx wf

Geometrical Equations

Physical Equations

Equilibrium Equations

Boundary Cond. (load:q))(z wf

qwD 4 +edges B.C.

11 16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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History of the Equation

Bernoulli, 1798:

Beam Thin plate

Lagrange, 1811:

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11.3 Internal Forces of Thin Plate

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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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Chapter Page

x

xM

y

x

zxy

xyM

xz

Fsx

。yx

w

1

Ez

,x

w

y

w

1

Ez

,y

w

x

w

1

Ez

2

xy

2

2

2

2

2y

2

2

2

2

2x

。wy4

z12

E

,wx4

z12

E

22

22zy

22

22zx

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Stress distribution

。yx

w

1

Ez

,x

w

y

w

1

Ez

,y

w

x

w

1

Ez

2

xy

2

2

2

2

2y

2

2

2

2

2x

。wy4

z12

E

,wx4

z12

E

22

22zy

22

22zx

。 2

2

xx dzzM

。 2

2

xyxy dzzM

。 2

2

xzSx dzF

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Chapter Page

。 2

2

xx dzzM

。 2

2

xyxy dzzM

。 2

2

xzSx dzF

。

2

2

2

2

2

3

2

2

22

2

2

2

2x

y

w

x

w

112

E

dzzy

w

x

w

-1

EM

。yx

w

112

E

dzzyx

w

1

EM

23

2

2

22

xy

。

。

wx-112

E

dz4

zwx-12

EF

22

3

2

2

222

2Sx

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，2

3

112

ED

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。wy

-DF,wx

-DF

,yx

w1DMM

,x

w

y

wDM,

y

w

x

wDM

2Sy

2Sx

2

yxxy

2

2

2

2

y2

2

2

2

x

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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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11.4 Boundary Conditions

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qwD 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

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Chapter Page

Built-in or clamped edge 固定边界

。0x

w,0w

0x0x

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.

。0M,0w0yy0y

。0x

w

y

w,0w

0y2

2

2

2

0y

。0y

w,0w

0y2

2

0y

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Free edge 自由边界

Chapter Page

。0F,0M,0MbySybyyxbyy

x

MFF yxsysy

t

0,0

x

MFFM yxsysy

ty

0yx

w2

y

w

,0x

w

y

w

by

2

3

3

3

by

2

2

2

2

Only 2 are allowed for an equation of 4th order

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11.5 Examples: Simple supported rectangular plate

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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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11.5 Examples:Simple supported rectangular plate

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

qwD 4 Choose to represent w by the double Fourier series:

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

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11.5 Examples: Simple supported rectangular plate

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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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Homework

• 9-1

Chapter Page 11 32

theory of elasticity

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