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
11.1 Some Concepts and Assumptions
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Review: bending of beams
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11.1 Some Concepts and Assumptions
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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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11.1 Some Concepts and Assumptions
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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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11.1 Some Concepts and Assumptions
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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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11.1 Some Concepts and Assumptions
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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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11.1 Some Concepts and Assumptions
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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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11.2 Differential Equation of Deflection
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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11.2 Differential Equation of Deflection
Chapter Page
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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11.2 Differential Equation of Deflection
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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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11.2 Differential Equation of Deflection
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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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11.2 Differential Equation of Deflection
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)(, 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
11.2 Differential Equation of Deflection
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Another method to get the equation
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11.2 Differential Equation of Deflection
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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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11.3 Internal Forces of Thin Plate
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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11.3 Internal Forces of Thin Plate
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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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11.3 Internal Forces of Thin Plate
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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11.3 Internal Forces of Thin Plate
,2
3
112
ED
Chapter Page
。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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11.3 Internal Forces of Thin Plate
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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
Chapter Page
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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11.4 Boundary Conditions
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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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11.4 Boundary Conditions
Chapter Page
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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11.4 Boundary Conditions
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
Chapter Page
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
Chapter Page
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
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