Membrane properties

Consider a loading that consists of in-plane loads only. In this case, the relationship between load and strain can be rewritten:

Where the matrix [a] is the 3 × 3 submatrix corresponding to lines and columns 1 through 3 of the inverse of the [ABD] matrix. In order to define an equivalent laminate modulus in the X direction, apply the following stress state:

• Nx ≠ 0

• Ny = Nxy = 0

For this loading condition, one obtains:

Next define an equivalent stress where h corresponds to the total thickness of the laminate. Then, one can define an equivalent for a laminate:

To obtain an equivalent Poisson’s ratio, one writes:

Similarly, to define an equivalent Young’s modulus in the Y direction, one must apply the following loading stress state:

• Ny ≠ 0

• Nx = Nxy = 0

The equivalent stress can be written:

And consequently is given by:

To obtain the laminate equivalent shear modulus, , one must apply a stress state such that:

• Nxy ≠ 0

• Nx = Ny = 0

The resulting expression is:

Although best applicable to balanced and symmetric laminates, the previous equations could be equally applied to general laminates for which:

1. A fully populated [a] matrix is obtained

2. The terms of the [a] matrix are influenced by the presence of the [B] matrix

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