Laminate Analysis Equation Section 9

Laminate Analysis Equation Section 9

A09 - Laminate Plate Analysis1

Laminate Analysis Equation Section 9

In this section, we consider plate analysis for layered orthotropic materials. Recall from plate theory, the following relationships:

Kinematics (see section A5 - plate theory):

(9.1)

The terms are the mid-surface extensional and shear strains for plate; while are the curvatures about the x and y axes, respectively, plus the twisting curvature (xy).

Constitutive (orthotropic laminae) - see section A8:

(9.2)

Force and Moment Resultants (see section A5 - plate theory)

(9.3)

Consider now a plate made of "N" layers (lamina) arranged as shown below.

= thickness of layer i =

= z centroid of layer i =

Now, substitute (9.2) and (9.1) into (9.3), and integrate over the thickness. The integral can be replace by a summation over all N layers.

(9.4)

or, summing over all N layers:

(9.5)

Both the mid-surface strains and curvatures {) are independent of z for the laminated plate, i.e., the mid-surface strains are at the laminated plate mid-surface (z=0) and the curvature of each lamina is the same. Hence, the last equation can be written as:

(9.6)

The moment resultants can be similarly written.

(9.7)

(9.8)

(9.9)

Now do the following:

  • integrate with respect to z for layer k,
  • multiply times the integral result for layer k, and
  • sum over all N layers.

The result can be written as follows:

(9.10)

and

(9.11)

where the A, B and D coefficients are given by

(9.12)

Using and , then above becomes:

(9.13)

Note that equations (9.10) and (9.11) could be combined and written as follows:

(9.14)

Determination of stress and strain .

Suppose we have given force and moment resultants and want to determine resultant stress and strain .

We have just derived the A, B and D matrices which relate the force and moment resultants to strains and curvatures:

The last equation can be inverted to obtain

(9.15)

Once we have the mid-surface strains and curvatures (of the laminated plate), we can use equations (9.2) and (9.1) to determine the stress for any layer k:

(9.16)

where z is the appropriate value for layer k ().

Alternate Approach. Rather then inverting [E], it may be more convenient to obtain equation (9.15) in another way (avoiding inversion of [E]). First consider equation (9.14) as two separate equations, i.e., take equations (9.10) and (9.11) written as

(a)

(b)

Now solve for from equation (b) to obtain

(c)

Substitute equation (c) into equation (b) to obtain

(d)

Combine equations (c) and (d) to obtain

or (9.17)

where

(9.18)

Solve the second equation in (9.17) for to obtain

(e)

Now substitute equation (e) into first equation in (9.17) to obtain

(f)

Combine (e) and (f) to obtain:

(9.19)

where

(9.20)

Note that with this alternate approach (eqns. (9.19) and (9.20)), one still has to invert matrices, but ONLY one (3x3) [D*] matrix instead of the (6x6) [E] matrix in equation (9.15). Generally, this is computationally faster.

Approximate (x,y) laminate material properties:

It turns out that the terms in the [A'] matrix represent compliances and can be associated with the material properties in the global (x-y) axes for the laminated plate (all layers combined). If the terms in [B'] are small compared to those in [A'], we can show that

(9.21)

where t is the total plate thickness (all layers).

From Engineering Mechanics of Composite Materials, Isaac Daniel & Ori Ishai: