Module 5: Laminate Theory
  Lecture 16: Introduction to Classical Plate Theory
 


Introduction:

In this lecture we are going to introduce the concept of laminate and its analysis based on Classical Laminate Theory. Further, we will introduce the notations to designate a laminate and will explain in detail the development of the classical laminate theory.

As we have studied earlier, laminate is defined as stacking of two or more laminae with same or different fibre orientation with respect to global direction. The laminae may be made of same or different material and have individual thicknesses.

Stacking Sequence Notation:

A laminate is designated by using a special nomenclature. In this nomenclature, the fibre orientation of all layers stacked in the laminate is given. In the following the main steps are given to designate a laminate.

  1. The stacking of layers starts from the top of the laminate.

  2. The stacking sequence gives the orientation of fibres with respect to global axis in degrees.

  3. The stacking sequence is enclosed in square brackets symbol,

  4. The distinct layers or groups of layers are separated with a slash symbol, /.

  5. For repeated groups or layers, subscript n is used to designate.

  6. The symmetric laminate is designated by subscript S on the square bracket, that is, by .

  7. The total stacking sequence is designated by subscript T, that is, by . However, in general, this is not used for denoting a complete stacking sequence.

To help the readers to understand the designation of stacking sequence of laminates, in the following Table 5.1, some laminate sequences, their description and total number of laminae in that laminate are given. A laminate with coordinate system and ply numbering is shown in Figure 5.1(a).

Note: In some of the books on composites and research articles the coordinate systems used have z direction positive in upward direction. In that case the stacking of layers in a laminate starts from the bottom. Accordingly, the ply top and bottom coordinate designation also changes. However, the end results remain unchanged.

Table 5.1:  Sample laminate stacking sequence notations and their description

Laminate

Description

Layers

One layer each of  and

3

One layer of  and

2

Two layer of

2

Two layers of (in a group of two layers) and one layer of

3

Symmetric with   and   layers

4

Symmetric with  and  and layers

8

Symmetric with two groups of  and two groups of   and    layers

16

Symmetric with one layer of   and one layer of

4

Symmetric with  and layers 

12

 
Laminate Coordinate System:

The coordinate systems for global and principal material directions for laminae are same as given earlier. Here, we introduce the coordinates in the thickness direction to get the z coordinate of the top and bottom of each ply. For example, the bottom coordinate of the kth ply is  and the top coordinate of the ply is . Thus, the bottom coordinate of the first ply is  and the top coordinate of top ply is . The total thickness of the laminate is taken as 2H. Thus, the bottom most coordinate of the laminate is –H and top most coordinate is H. The lamina thickness coordinate notations are shown in Figure 5.1(b).

Figure 5.1: (a) Stacking of laminae in a laminate and (b) coordinate designation for laminate