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## Classical Lamination Theory W.rangsri

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**Definition of Composite Materials**Fibers and Matrix Phases Composite Composition and Properties Manufacturing Processes Applications Classical Lamination TheoryW.rangsri Intro. to Mechanics of Laminated Composite Materials Introduction to Composite Materials**Contents**• The Kirchhoff Hypothesis • Laminate Nomenclature • LaminateStrains and Displacements: The Kirchhoff Hypothesis • LaminateStrains • Laminate Stresses • Stress Distributions through the Thickness • Force and Moment Resultants • LaminateStiffness Matrix • LaminateStiffness: The ABD Matrix • Classification of Laminates • ElasticCouplings**Introduction**• Understand • how laminatesresponse to load • how the fiber angles of the individuallayers influence laminateresponse • how the stacking arrangement of the layers influences the response • how changingmaterialproperties in a group of layers changes response • how the stresses are influenced by theseparameters**The Kirchhoff Hypothesis**Laminate Nomenclature**The Kirchhoff Hypothesis**Laminate Nomenclature**The Kirchhoff Hypothesis**Laminate Strains and Displacements: The Kirchhoff Hypothesis • 1800s by Kirchoff • Greatlysimplifiedanalysis (accuratelypredict the response of beams, plate, and shells) • Hypothesisapply to other structures**The Kirchhoff Hypothesis**Laminate Strains and Displacements: The Kirchhoff Hypothesis M: Applied moments q: Distributedappliedloads N: Inplaneloads P: Point loads Laminateis flat. All layer are perfectlybondedtogether.**The Kirchhoff Hypothesis**Laminate Strains and Displacements: The Kirchhoff Hypothesis Kirchoffhypothesis assumes thatline AA’ remainsstraightand normalto the deformedgeometricmidplane and does not change length.**The Kirchhoff Hypothesis**Laminate Strains and Displacements: The Kirchhoff Hypothesis Kirchoffhypothesis assumes thatline AA’ remainsstraight and normal to the deformedgeometricmidplane and does not change length.**: Rotation of ref. surface about y**The Kirchhoff Hypothesis Laminate Strains and Displacements: The Kirchhoff Hypothesis z-x plane u°: Translation of point P° in x direction w°: Translation of point P° in z direction**: Rotation of ref. surface about x**The Kirchhoff Hypothesis Laminate Strains and Displacements: The Kirchhoff Hypothesis y-z plane v°: Translation of point P° in y direction w°: Translation of point P° in z direction**The Kirchhoff Hypothesis**Laminate Strains and Displacements: The Kirchhoff Hypothesis**The Kirchhoff Hypothesis**Laminate Strains (small deformations, small dispalcements) Tensor shear strain = curvature of the ref. surface = inverse of the radius of curvature**The Kirchhoff Hypothesis**Laminate Strains (small deformations, small dispalcements)**The Kirchhoff Hypothesis**Laminate Strains (small deformations, small dispalcements)**The Kirchhoff Hypothesis**Laminate Strains (small deformations, small dispalcements)**The Kirchhoff Hypothesis**Laminate Strains (small deformations, small dispalcements) A small segment of a four-layer laminate (0.6 mm thick) deformed such that at a point P0 on the reference surface the extensional strain in the x direction is 1000 x 10^-6. The radius of curvature R0, of the reference surface, is 0.2 m.**The Kirchhoff Hypothesis**Laminate Stresses • CLT => each point within the vol. of a laminateis in a state of plane stress. • the stresses varywith z • the strainsvarywith z • reducedstiffnessesvarywith z**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness**The Kirchhoff Hypothesis**Stress Distributions through the Thickness**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 0° 90° 90°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 0°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 90°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 0° 90°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness aluminum**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 0°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 90°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 0° 90°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 0°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 90°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 0° 90°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 0° 90°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 0° 90°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 0° 90°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness 0° 90°**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Stress Distributions through the Thickness aluminum**The Kirchhoff Hypothesis**Force and Moment Resultants**The Kirchhoff Hypothesis**Force and Moment Resultants**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Force and Moment Resultants**CLT Example 1 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Force and Moment Resultants**CLT Example 2 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Force and Moment Resultants**CLT Example 2 [0/90]S graphite-epoxy laminate (0.6 mm)**subjectedto known The Kirchhoff Hypothesis Force and Moment Resultants