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Laminated plates

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definitions Unidirectional: orientation direction is identical for all laminates Angle-ply: Fiber orientation alternates between +theta to –theta from the stress normal …/q/-q/q/-q/… Cross ply: 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**

7 ply stack: This stack is symmetrical about the centerline of the 4th ply 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 +3 2 +45 +2 3 -45 +1 4 90 5 -1 6 -2 7 -3

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**Symmetric laminate examples**

8 ply stack: 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 + 3 ½ h 2 +45 +2 ½ h 3 -45 + 1 ½ h 4 90 + ½ h 5 - ½ h 6 - 1 ½ h 7 - 2 ½ h 8 -3 ½ h

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**Antisymmetric q(z)=-q(-z) q/-q/q/-q is antisymmetric**

Ply orientation is antisymmetric about the centerline of the laminate 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 Unsymmetric: q1/q2/q3; random placement of angles**

Quasi-isotropic laminate: equal angles between adjacent lamina, increment is p/n +60/0/-60 +45/0/-45/-90 0/+60/-60 0/+45/-45/90

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**Quasi-isotropic laminates**

Quasi-isotropic laminate: equal angles between adjacent lamina, increment is p/n +60/0/-60 +45/0/-45/-90 0/+60/-60 0/+45/-45/90 A common quasi-isotropic symmetrical stacking sequence is: [0/±45/90]s

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Code examples [0/45/90]s: 1st angle is for the outermost ply, last angle is for the innermost ply; s subscript means symmetrical The bar over 90 indicates that the plan of symmetry passes midway through the thickness of the 90 degree laminate ±45 implies adjacent +45 and -45 degrees laminates 04: four adjacent zero degree plies (±45)2: two grouped, adjacent ±45 degree plies

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**In-class examples Sketch the following: [0/45/90]s [0/-45/90]s**

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**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 Lamination theory

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**Step-by-step procedure**

Calculate stiffness matrices for the laminate Calculate the midplane strains and curatures for the laminate due to specific applied forces Calculate in-plane stresses, exx, eyy, gxy, for each lamina Calculate in-plane strains, sxx, syy, txy, for each lamina

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**Assumptions Laminate is thin, wide: w>>t**

Perfect interlaminar bonds Strain distribution in the thickness direction is linear All laminas are macroscopically homogeneous and behave linearly elastically

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**Laminate strains Since strain distribution in z direction is linear,**

Geometric midplane = xy axes, z axis defines thickness direction Total laminate thickness = h Thickness of various lamina are ti Total lamina = N

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**Laminate strains Since strain distribution in z direction is linear,**

e0ii = midplane normal strains g0xy = midplane shear strain Kii bending curvatures of laminate Kxy = twisting curvature of laminate

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