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Laminated plates
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definitions
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Unidirectional: orientation direction is identical for all
laminates
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Angle-ply:
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Cross ply:
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Fiber orientation alternates between +theta to –theta from the
stress normal
…/q/-q/q/-q/…
Fibers in alternating plies are at right angles to each other
…/0/90/0/90/…
Symmetric: fiber orientation in the stack is symmetric about
the center line of the laminate
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Symmetric laminate examples
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7 ply stack:
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This stack is symmetrical about
the centerline of the 4th ply
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The stack order is a ‘mirror’
reflection about the centerline
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We will need the average height
of the ply above the centerline.
Ply #
orientation
Distance
above
ctrln
1
0
+3
2
+45
+2
3
-45
+1
4
90
0
5
-45
-1
6
+45
-2
7
0
-3
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Symmetric laminate examples
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8 ply stack:
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This stack is symmetrical about
the interface between ply 4 and
ply 5

The stack order is a ‘mirror’
reflection about the centerline

We will need the average height
of the ply above the centerline.
Ply #
orientation Distance
above
ctrln
1
0
+3½h
2
+45
+2 ½ h
3
-45
+1½h
4
90
+½h
5
90
-½h
6
-45
-1½h
7
+45
-2½h
8
0
-3 ½ h
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Antisymmetric
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Ply orientation is antisymmetric about the centerline of the
laminate
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For each ply with fiber orientation angle, q, above the
midplane, this is a ply of fiber orientation angle, –q, with
identical material and thickness and equal distance below
the midplane
 q(z)=-q(-z)
 q/-q/q/-q
is antisymmetric
 q/-q/-q/q is symmetric
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Unsymmetric laminate
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Unsymmetric: q1/q2/q3; random placement of angles
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Quasi-isotropic laminate: equal angles between adjacent
lamina, increment is p/n
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+60/0/-60
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+45/0/-45/-90
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0/+60/-60
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0/+45/-45/90
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Quasi-isotropic laminates
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Quasi-isotropic laminate: equal angles between adjacent
lamina, increment is p/n
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+60/0/-60
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+45/0/-45/-90
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0/+60/-60
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0/+45/-45/90
A common quasi-isotropic symmetrical stacking sequence is:
[0/±45/90]s
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Code examples
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[0/45/90]s: 1st angle is for the outermost ply, last angle is for
the innermost ply; s subscript means symmetrical
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The bar over 90 indicates that the plan of symmetry passes
midway through the thickness of the 90 degree laminate
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±45 implies adjacent +45 and -45 degrees laminates
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04: four adjacent zero degree plies
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(±45)2: two grouped, adjacent ±45 degree plies
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In-class examples
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Sketch the following:
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[0/45/90]s
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[0/-45/90]s
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[0/±45/90]s
Young B35 classroom (30 student seats and 1 teacher station)
Thursday 4/3 3:30pm-4:45pm
Thursday 4/10 3:30pm-4:45pm
Peggy M. Akridge
Manager, Student Computing Services
UK Analytics and Technologies
University of Kentucky
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Lamination theory
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Step-by-step procedure
1.
Calculate stiffness matrices for the laminate
2.
Calculate the midplane strains and curatures for the
laminate due to specific applied forces
3.
Calculate in-plane stresses, exx, eyy, gxy, for each lamina
4.
Calculate in-plane strains, sxx, syy, txy, for each lamina
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Assumptions
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Laminate is thin, wide: w>>t
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Perfect interlaminar bonds
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Strain distribution in the thickness direction is linear
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All laminas are macroscopically homogeneous and behave
linearly elastically
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Laminate strains
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Since strain distribution in z direction is linear,
e xx = e + z × kxx
0
xx
• Geometric midplane = xy axes, z axis
defines thickness direction
• Total laminate thickness = h
• Thickness of various lamina are ti
• Total lamina = N
e yy = e + z × kyy
0
yy
g xy = g + z × kxy
0
xy
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Laminate strains
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Since strain distribution in z direction is linear,
e xx = e + z × kxx
0
xx
•
•
•
•
e0ii = midplane normal strains
g0xy = midplane shear strain
Kii bending curvatures of laminate
Kxy = twisting curvature of laminate
e yy = e + z × kyy
0
yy
g xy = g + z × kxy
0
xy
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e xx = e xx0 + z × kxx
e yy = e yy0 + z × kyy
g xy = g xy0 + z × kxy
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