Laminate stiffness matrices

In the first-order shear deformation theory, stresses are assumed to be piecewise linear, that is linear within each ply. The above integrals can be replaced by the following sums:

Where N corresponds to the number of plies in the laminate. It is straightforward to replace the integrals for the normal, moment, and transverse shear resultants by sums over plies. Substituting the stress resultant definitions, the in-plane stress resultant definitions then become:

Similarly, the moment resultants can be written:

The last 2 equations can be re-written:

where:

which represent the terms of the membrane, membrane-bending coupling and bending stiffness matrices, respectively.

For symmetric laminates, [B] is zero.

The terms of the [A], [B], and [D] matrices are commonly grouped into a single 6 by 6 matrix denoted the [ABD] matrix. Once the [ABD] matrix has been calculated, it can be inverted to calculate the mid-plane strains and curvatures corresponding to user-specified values of the in-plane force and moment resultants:

Once the midplane strains and curvatures have been calculated, the strains at a ply level can be computed with:

These strains can be multiplied by the [] matrix to obtain the stresses. The strains can also be rotated from the XY coordinate system to obtain the strains in the 12 ply coordinate system. Finally, the stresses in the 12 coordinate system can be calculated by multiplying the strains by the matrix. The strains and stresses in the 12 ply coordinate system are required for the evaluation of the failure criteria and margins of safety.

For more information, see Laminate failure analysis nomenclature.

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First order shear deformation theory

Strain displacement relationship

Constitutive equations

Off-axis relationships

Shell stress resultants

Transverse shear stiffness matrix

Shear deformation theory references

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