Mechanics of aircraft materials 7 laminate plates theory cdio

Classical Lamination Theory

Dr Ly Hung Anh

Department of Aerospace Engineering – Faculty of Transportation Engineering

Basic Assumptions

1.Each ply (or layer or lamina) is in state of plane stress

2.Interlaminar stresses are ignored

3.Lamina are perfectly bonded to each other

a. No interfacial slip

b. No interfacial shear deformation

Coordinate system and stress resultantsfor a laminated plate

Basic Assumptions

1. Plate consists of orthotropic laminae

bonded together Principal material axes of laminae oriented along arbitrary directions

w.r.t x-y axes

2. Thickness of the plate, t, is much smaller than lengths along edges, a and b

3. Displacement u,v, and w are small compared to thickness, t

Basic Assumptions

4.In-plane strains, x , y , and xy are small compared to unity

5.Transverse shear strains, xz and yz

are negligible (=0)

6.Tangential displacement, u and v are linear functions of the z coordinate

7.Transverse normal strain, z are negligible (=0)

Basic Assumptions

8.Each ply obeys Hooke’s law

9.The plate thickness t is constant

10.Transverse shear stresses xz and yz

vanish on plate surfaces defined by z = t/2and z = -t/2

Basic Assumptions

(in global axes)

The “Reduced Transformed Stiffness Matrix” may change from ply to ply; it

depends on the material and orientation of the ply

Q  

(6)

The curvatures (the ’s) are hard to measure We seek to express stresses in terms of loads and moments.

Midplane strains and Plate curvatures are independent of the “z” coordinate  Withdraw them from the and  

Stress Resultants

Similar Results for

NyNxyMyand Mxy

Complete Set of Equations

Extensional Stiffnesses Matrix [A]

Bending Stiffnesses Matrix [D]

“Coupling” Stiffnesses Matrix [B]

Classifying Laminates

3.Quasi-Isotropic

Both geometric and material properties are symmetric about the mid-plane

Types of symmetric laminates include angle-ply and cross-ply laminates No bending-extensional coupling, Bij=0

Symmetric Laminate

Symmetric Cross-Ply Laminate

Problem 1

Determine the stiffness matrix of a Laminate

Determine the stiffness matrix for a [+45/-45/-45/+45] symmetric angle-ply laminate

consisting of 0.25 mm thick unidirectional AS/3501 graphite epoxy laminate.

Determine the stiffness matrix of a Laminate solution

-1 Find the value of the reduced stiffness matrix [Q] for each ply using its four elastic moduli E11, E22, G122 Find the value of the “transformed reduced stiffness

matrix” for each ply by using the [Q] and the angle of each ply

3 Find the coordinate of the top and bottom surface, zi, i=1…n of each ply

4 Find the three stiffness matrices [A], [B] and [D] using the results obtained from step 2 and 3

Q  

Antisymmetric Laminates

Plies of identical material and thickness at equal positive and negative distances

from middle surface Orientation is

antisymmetric, that is if at distance +z

the orientation is  then at distance -z the

orientation is -.

Antisymmetric Angle-Ply Laminate

Determine the stiffness matrix for a [-45/+45/-45/+45] anti-symmetric

angle-ply laminate consisting of 0.25 mm thick unidirectional AS/3501 graphite epoxy

laminae.

y x

Exploded view of a [-45/+45/-45/+45] antisymmetric laminate

Quasi-isotropic Laminates

It is possible to construct a laminate that exhibits some of the properties of isotropic behavior.

Three or more adjacent layers oriented at the same angle relative to each other [A] is isotropic [B] and [D] may or may not be.

One example:

Angle between adjacent laminae = /N where N is the number of lamina.

Laminate Compliances

Inverse of stiffness relationship!

Analyze a laminated composite

subjected to the applied forces and moments

Problem 2

1.Find the three stiffness matrices [A], [B] and [D] (problem 1)

2.Knowing the applied forces {N} and moments {M}, finding the mid-plane strains and curvatures (by using Eqs (11))

3.Knowing the location of each ply, find the global strains in each (by using Eqs (3))

4.Find the global stresses in each ply by using the stress-strain relationship in global axes

5.Find the local strains, the local stresses in each ply ; Use the failure criteria to study the strength of the laminate (“first ply failure” theory)

The [-45/+45/-45/+45] antisymmetric angle-ply laminateconsisting of 0.25 mm thick unidirectional AS/3501graphite epoxy laminae.

The laminate is subjected to a single axial force per unit

associated with the x and y axes in each lamina.

Step 1 – Step 4

Global strains in each ply

In-plane and flexural modulus of a symmetric laminate

In-plane Engineering Constants

Flexural Engineering Constants

Symmetric laminate, bending loads only