Classic Laminate Theory: how to derive the reduced stiffness matrix

Those of you who work with composite materials will be familiar with the Classic Laminate Theory. According to this theory, from the properties of the material, the stiffness matrix can be easily obtained. Then, using very tedious expressions, the coefficients of the reduced stiffness matrix can be calculated. The thing is that, based on my own experience, those expressions are introduced without any explanation, so most of the people just use them, ignoring where they come from. So, if you’re curious or you just want to understand a bit more about this theory, keep reading this post!


The first thing that we have to consider is the rotation of our coordinate system . We need to know how to express our new coordinate system in terms of the original one. Following Figure 1 and using basic trigonometry, the relationship can be found.

coordinate_system

Figure 1 Rotation of the original coordinate system

1

Then, using basic concepts of Strength of Materials and Continuum Mechanics, the stress tensor can be written as follows:

2

Hence, when i=1 and j=1, the stress in direction 11′ should be:

3

Following the same approach, the stress in direction 22′ can be obtained when i=2 and j=2:

4

Finally, the stress in direction 12′ can be found when i=1 and j=2:

5

Then, rearranging the expressions using a matrix format, the stresses in the new coordinate system are given by:

6

From that last expression, the “famous” transformation matrix can be identified:

7

Now, the reduced stiffness matrix can be obtained doing some simple calculations. Yes ladies and gentlemen, this is where those tedious and long expressions for each component of the reduced stiffness matrix come from.

8

To conclude, I would like to say that the way of calculating each coefficient of the reduced stiffness matrix is totally up to you, but from my experience, it’s usually easier to work with the matrices rather than using the long expressions. Trust me, it’s very common to make mistakes while introducing all the terms in the equations!

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