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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 …/ /- /… 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 4 th ply The stack order is a mirror reflection about the centerline We will need the average height of the ply above the centerline. Ply #orientationDistance above ctrln 10+3 2+45+2 3-45+1 4900 5-45 6+45-2 70-3

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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 #orientationDistance above ctrln 10+ 3 ½ h 2+45+2 ½ h 3-45+ 1 ½ h 490+ ½ h 590- ½ h 6-45- 1 ½ h 7+45- 2 ½ h 80-3 ½ h

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+ Antisymmetric Ply orientation is antisymmetric about the centerline of the laminate For each ply with fiber orientation angle, above the midplane, this is a ply of fiber orientation angle, – with identical material and thickness and equal distance below the midplane (z)=- (-z) is antisymmetric is symmetric

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+ Unsymmetric laminate Unsymmetric: ; random placement of angles Quasi-isotropic laminate: equal angles between adjacent lamina, increment is /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 /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 : 1 st 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 0 4 : 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 [0/±45/90]s

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+ Lamination theory 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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+ 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, xx, yy, xy, for each lamina 4. Calculate in-plane strains, xx, yy, xy, 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 t i Total lamina = N

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+ Laminate strains Since strain distribution in z direction is linear, 0 ii = midplane normal strains 0 xy = midplane shear strain K ii bending curvatures of laminate K xy = twisting curvature of laminate

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