Laminated Composite Theoretical Principle

•Composite Material

ØParticulate Composites (Particles+ Matrix)

ØLaminated Composites (Layers)

ØFibrous-Matrix LaminatedComposites (Layers – “fiber + Matrix”)

ØCore Stiffened LaminatedComposites

•Fibrous-Matrix LaminatedComposites are the most commons usedfor high performance structural components. 

•1-axis direction is the fiberdirection

•2-axis direction is the transverse tothe fiber direction in the plane of theply

•3-axis direction is thethrough-thickness direction of theply.

•The fiber orientation, theta, is defined relative to thebase x-axisusing right hand rule to definepositive theta

•Plies are numbered 1 ton with the 1st ply defined as the most negative z ply and the nth ply as the most positive z ply

•The z-coordinate value for thekth ply is always defined as themost positive z-coordinate interface for that ply.

•A symmetric laminate is defined as a laminate which is composed of plies such that the thickness,angle, and material of the plies are symmetric about the middle surface of thelaminate.

•Symmetric laminates, the[B] matrix is zero and exhibit no extensional – bending or shear –twisting coupling behaviors.

[0/45/90/45/0]

[45/-45/90/0/0/90/-45/45]

•An anti-symmetriclaminate is defined as a laminate for which every +q ply and -q ply on the negative z-half of the laminate there exist a -q ply and +q ply respectively on the positive z-half of the laminate with the same thickness and material at the same stacking sequence location.

•In addition, 0 plies and 90 plies must be symmetric about the middle surface of the laminate.

[0/90/-45/45/90/0]

[0/45/90/-45/45/90/-45/0]

•A balanced laminateis defined as a laminate for which every +q ply there exists a -q ply of the same thickness and material.  The definition of a balanced laminate does not define where in the laminate stacking sequence the plies exist, just that there are same number of +q plies and -q plies in total for the laminate.  Balanced laminates havezero A14and A24 components and exhibit no extensional – shear coupling behavior. In addition, if a balanced laminate is also anti-symmetric, then the laminate will additionally have zero D14 and D24 components and will also not exhibit bending – twisting coupling behavior.

[45/-45/-30/30]

[22.5/-22.5/90/-22.5/22.5]

•A cross-plylaminateis defined as a laminate composed of only 0 plies and 90 plies of the same thickness and material.  cross-ply laminates havezero A14, A24, D14, and D24components and exhibit no extensional – shear or bending -twisting couplingbehaviors.  

[0/90/0/90/0]

[0/0/90/90/0/0/90]s

•An angle-ply laminateis defined as a laminate composedof only +q plies and -q plies of the same thickness and material.  In general angle-ply laminates have fully populated [A], [B], and [D] matrices. 

[45/-45/-30/30]

[-30/30/60/30/-30]

•A general laminateis defined as a laminate which does not fall into any of the previous laminate definitions.  General laminatesgenerally exhibit fully populated [A], [B], and [D] matrices and therefore all types of coupling typically exist including;

ØExtension – shear coupling (A14 and A24 terms)

ØExtension – bending coupling ([B] matrix terms)

ØShear – twisting coupling ([B] matrix terms)

ØBending – twisting coupling (D14 and D24 terms)

[0/45/90/22.5/0/45]

[90/-45/0/90/-45/0]

Orthotropic Compliance Matrix [S]

Orthotropic Stiffness Matrix [C]

正交各向异性材料平面应力问题的应力应变关系

应力作用在2方向引起的横向变形和应力作用在1方向引起的相同

4个独立的常数，E1,E2,n12和G12求解问题

•PlaneStress Stiffness Matrix (Material System)

•PlaneStress Stiffness Matrix (Global System)

Q—>Q

Laminate [ABD] Matrix

•[A] matrix (zk – zk-1) term is always positive and equal to the thickness of the ply. Therefore the [A] matrix for a laminated plate is stacking sequence independent

•[B] matrix (zk2 – zk-12) term is negative for negativez-coordinate plies and positive for positive z-coordinate plies and the terms are symmetric about the middle surface. Therefore, the [B] matrix is zero for any symmetric laminate.

•[D] matrix (zk3 – zk-13) term is always positive and increases significantly for plies further away from the middle surface.  Therefore, the [D] matrix is stacking sequence dependent.

求解步骤

1、

2、

3、

4、求ABD矩阵