1 Classical Laminated Plate Theory. 2 CONSTITUENTS STRUCTURE COMPOSITE STRUCTURAL ELEMENT ELEMENTARY...
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Transcript of 1 Classical Laminated Plate Theory. 2 CONSTITUENTS STRUCTURE COMPOSITE STRUCTURAL ELEMENT ELEMENTARY...
2
CONSTITUENTS
STRUCTURE COMPOSITE
STRUCTURALELEMENT
ELEMENTARYSTRUCTURE
Micro Mechanics
Ex
Ey
G
Experimental Data
Finite Element Analysis
Combining Constituents
4
Plane-Stress Assumption
• Fiber-reinforced materials are utilized in beams, plates, cylinders and other structures
• Typically one characteristic geometric dimension is an order of magnitude less than the other two
• Three of the six components of stress are generally much smaller than the other three
5
Plane Stress Inaccuracies
• Errors in analysis near edges– The stresses 3, 23, 13 lead to delaminations
– Bonded joints can not be modeled– Adhesive or cocured interface can not be
evaluated
• The stress components equated to zero are forgotten and no attempt is made to estimate their magnitude
• erroneously assumed that 3 is zero
6
Stress Transformation
xy
y
x
22
22
22
12
2
1
sincoscossincossin
cossin2cossin
cossin2sincos
22
22
22
nmmnmn
mn2mn
mn2nm
T
7
Compliance Transformation Equations
xy
y
x
662612
262212
161211
xy
y
x
SSS
SSS
SSS
4466
226612221166
3661222
366121126
422
226612
41122
3661222
366121116
4412
2266121112
422
226612
41111
mnSmnSS4S2S22S
mnSS2S2mnSS2S2S
mSmnSS2nSS
mnSS2S2mnSS2S2S
mnSmnSSSS
nSmnSS2mSS
8
Reduced Stiffness Transformation
xy
y
x
662612
262212
161211
xy
y
x
QQQ
QQQ
QQQ
44
6622
6612221166
3662212
366121126
422
226612
41122
3662212
366121116
4412
2266221112
422
226612
41111
mnQmnQ2Q2QQQ
mnQ2QQmnQ2QQQ
mQmnQ2Q2nQQ
mnQ2QQmnQ2QQQ
mnQmnQ4QQQ
nQmnQ2Q2mQQ
9
Classical Lamination TheoryThe influence of fiber direction, stacking arrangements, material properties, and more on structural response
10
Laminate Coordinate System
• Laminate thickness H• Layer thickness h
– not all layers same h
– kth layer - hk
• z-axis downward from geometric midplane– can be between layers– can be within a layer
• Fiber angles identified relative to x axis
11
Laminate Nomenclature
T
T2
s
45/45/0/0/45/45
45/45/0/45/45
0/45
• Layer 1 is the most -z• Layer N is the most +z• To catergorize a
laminate as symmetric a mirror about the geometric midplane– material properties– fiber orientation– thickness of layer
12
The Kirchhoff Hypothesis
• Mid 1800’s, simplified analysis
• Beams, plates, shells• metal, wood,
concrete, and other materials
13
Initially Flat Laminated Plate Acted Upon by Various Loads
• Loads– applied moments, M
– distributed loads, q
– inplane loads, N
– point loads, P
• Multiple layers of fiber reinforced material
• Fibers parallel to the plane of the plate
• Layers are perfectly bonded
14
Deformation of Lines Normal to Geometric Midplane
• Before deformation are straight
• Despite the deformations caused by the applied loads, line AA’ remains straight and normal to the deformed geometric midplane and does not change length
16
Implications of the Kichhoff Hypothesis in X-Z Plane
• No through-thickness strain• Small deformations• Two components of
translation– uo in x direction, horizontal
translation– wo in z direction, vertical
translation
• One component of rotation about y-axis
xwo
20
Laminate Strains
yx
)y,x(wz2
y
y,xu
x
y,xv
y
z,y,xu
x
z,y,xvz,y,x
0x
y,xw
x
y,xw
z
z,y,xu
x
z,y,xwz,y,x
0y
y,xw
y
y,xw
z
z,y,xv
y
z,y,xwz,y,x
0z
y,xw
z
z,y,xwz,y,x
y
)y,x(wz
y
y,xv
y
z,y,xvz,y,x
x
)y,x(wz
x
y,xu
x
z,y,xuz,y,x
o2oo
xy
oo
xz
oo
yz
o
z
2
o2o
y
2
o2o
x
21
Laminate Strains Composed of Two Parts
• Extensional Stain of the Reference Surface
• Curvature of the Reference Surface– inverse of the radius of curvature– involves more than just second derivative
• For small strains second derivative and curvature identical
22
Strain Notation
yx
y,xw2y,x
y
y,xu
x
y,xvy,x
y
y,xwy,x
y
y,xvy,x
x
y,xwy,x
x
y,xuy,x
o2oxy
oooxy
2
o2oy
ooy
2
o2ox
oox
23
Laminate Strains using Revised Notation
y,xzy,xz,y,x
0z,y,x
0z,y,x
0z,y,x
y,xzy,xz,y,x
y,xzy,xz,y,x
oxy
oxyxy
xz
yz
z
oy
oyy
ox
oxx
The yz, and xz are zero because the Kirchhoff hypothesisassumes that lines perpendicular to the reference surfacebefore deformation remain perpendicular after thedeformation; right angles in the thickness direction do notchange when the laminate deforms
31
Definitions of Stress Resultants
• Stress in each ply varies through the thickness
• It is convenient to define stresses in terms of equivalent forces acting at the middle surface
• Stresses at the edge can be broken into increments and summed
• The resulting integral is defined as the stress resultant, Ni [force per length]
32
Stress Resultant in X direction
2/h
2/h xx
2/h
2/h xx
x
dzN
dzyydz,0dzAs
ydzdirectionxinforceTotal
33
Stress and Moment Resultants
2/h
2/h xyxy
2/h
2/h yy
2/h
2/h xx
2/h
2/h xyxy
2/h
2/h yy
2/h
2/h xx
dzzM
dzzM
dzzM
dzN
dzN
dzN
bend
bend
twist
34
Putting the Resultants in Matrix Form and Summing
dzz
M
M
M
dz
N
N
N
k
n
0k
h
h
xy
y
x
xy
y
x
k
n
0k
h
h
xy
y
x
xy
y
x
k
1k
k
1k
35
Relating Stress to Strain
n
0k
h
h
h
h
2
oxy
oy
ox
k662616
262212
161211
oxy
oy
ox
k662616
262212
161211
xy
y
x
n
0k
h
h
h
hoxy
oy
ox
k662616
262212
161211
oxy
oy
ox
k662616
262212
161211
xy
y
x
k
1k
k
1k
k
1k
k
1k
dzz
QQQ
QQQ
QQQ
dzz
QQQ
QQQ
QQQ
M
M
M
dzz
QQQ
QQQ
QQQ
dz
QQQ
QQQ
QQQ
N
N
N
36
Performing the Integration
n
0k
31k
3k
oxy
oy
ox
k662616
262212
1612112
1k2k
oxy
oy
ox
k662616
262212
161211
xy
y
x
n
0k
21k
2k
oxy
oy
ox
k662616
262212
161211
1kkoxy
oy
ox
k662616
262212
161211
xy
y
x
hh3
1
QQQ
QQQ
QQQ
hh2
1
QQQ
QQQ
QQQ
M
M
M
hh2
1
QQQ
QQQ
QQQ
hh
QQQ
QQQ
QQQ
N
N
N
37
Defining Laminate Stiffness Terms
31k
3k
k
n
0kijij
21k
2k
k
n
0kijij
1kkk
n
0kijij
hhQ3
1D
hhQ2
1B
hhQA
38
Constitutive Equations in Matrix Form
xy
y
x
oxy
oy
ox
662616662616
262212262212
161211161211
662616662616
262212262212
161211161211
xy
y
x
xy
y
x
DDDBBB
DDDBBB
DDDBBB
BBBAAA
BBBAAA
BBBAAA
M
M
M
N
N
N
39
Symmetric Laminates
• For every layer to one side of the laminate reference surface with a specific thickness, material properties, and fiber orientation, there is another layer an identical distance on the opposite side
• All components of [B] are zero• 6x6 set of equations decouples into two 3x3
sets of equations
40
Balanced Laminates
• For every layer with a specified thickness, material properties, and fiber orientation, there is another layer with the identical thickness, material properties, but opposite fiber orientation somewhere in the laminate
• If a laminate is balanced, A16 and A26 are always zero– Q16 & Q26 from opposite orientation have
opposite signs