Transverse shear stiffness matrix

In the previous sections, the equations of the first-order shear deformation theory have been derived. Those are based on Mindlin assumption which results in constant through the-thickness transverse shear deformations. In this section, very specific assumptions are made in order to derive an expression for the transverse shear stiffness of a laminate. These are required because the calculation of the transverse shear matrix from the summation of the ply transverse shear matrix would result in a much too stiff predictions.

The following description is based partly on [6], section 3.6.8. Consider a plate in the XY plane:

Assuming that bending and shear occur only in the XY plane, without any gradient in the Y direction. For a symmetric and balanced laminate, this means that Nx, Ny and Nxy are zero, and so are their derivatives with respect to Y. In this case, equilibrium in the X direction is governed by:

Moment equilibrium about the y-direction gives:

where Mx and Qx are stress resultants. Since pure bending is assumed, strains vary linearly across the section. Also, the strains are related to stresses, and the mid-plane strains and curvatures can be calculated. Assuming that the in-plane stress resultants as well as My and Mxy are zero, then these relationships can be simplified as follows:

where abdij correspond to the i, j term of the inverse of the ABD matrix. For a symmetric laminate, abd14, abd24, and abd34 are 0. The σx stress component for ply k can now be rewritten:

The derivative of this equation with respect to X can now be equated to:

where:

A laminated composite shell consists of N layers for which varies varies from layer to layer.

The interlaminar shear stress in the in the first ply (k = 1) is given by:

Where z0 corresponds to the z position of the bottom surface of the laminate. It has been assumed that:

The expression of in the second ply (k = 2) can then be calculated:

Substituting the value of into the last equation into the last, one gets:

Let’s now calculate the expression of , that is the expression of in ply 3. This expression is given by:

Substituting the value of into the last equation, one gets:

where hj corresponds to the thickness of ply j.

Finally, the expression for the interlaminar shear stress τxz in ply k, , can be written:

The expression for the interlaminar shear stress τyz in ply k, , can be derived in a similar fashion:

where:

Once the stress distributions have been defined, the transverse shear flexibility of the laminate can be estimated. This is done by matching the laminate’s strain energy with the strain energy associated with the shear stress distributions predicted above. In other words, the transverse shear flexibility matrix of the laminate [F] is computed as:

Where the matrix corresponds to the inverse of the matrix relating the transverse strains to the transverse shear stresses.

Substituting the expressions for and into the last equation, the shear flexibility coefficients are defined:

Where the expressions in brackets are complicated terms that result from the integration process. The transverse shear stiffness of the laminate is then available as the inverse of the flexibility matrix:

Look up more details

First order shear deformation theory

Strain displacement relationship

Constitutive equations

Off-axis relationships

Shell stress resultants

Laminate stiffness matrices

Shear deformation theory references

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