II.1.6.8 Ply out-of-plane shear stresses

II.1.6.8.1 With $({\mu }_x,{\mu }_y)$ approach

When the $({\mu }_x,{\mu }_y)$ approach is adopted, ${\left[Y^k\right]}_{\text{lam}}(z)$ matrices allow the calculation of out-of-plane shear stress from the global out-of-plane shear force:

In practice, stresses in ply axes are of interest, rather than in laminate axes. If $\xi$ is the orientation of the ply in the laminate axes, then:

Thus:

A $2\times 2$ matrix ${\left[Y\right]}_{\text{ply}}^{\text{lam}}$ is stored for each station through the laminate thickness where out-of-plane shear stress might be requested. This is done at the top, mid and bottom surfaces in each ply, meaning that $3N$ $2\times 2$ matrices ${\left[Y^j\right]}_{\text{ply}}^{\text{lam}}$ are stored in the ClaLam object.

II.1.6.8.2 With “resolution in shear force axes” approach

The components of the bending moments gradient in laminate axes are first estimated with equation (II.1.49):

Then, the laminate out-of-plane shear stresses can be estimated by an expression like (II.1.43):

In this expression, ${\left[V\right]}_{\text{lam}}(z)$ is given by (II.1.44):

The three $2\times 6$ matrices ${\left[V_0^k\right]}_{\text{lam}}$, ${\left[V_1^k\right]}_{\text{lam}}$ and ${\left[V_2^k\right]}_{\text{lam}}$ are calculated recursively using the expressions (II.1.46), (II.1.45) and (II.1.45):

To resolve the recursion for ${\left[V_0^k\right]}_{\text{lam}}$, an estimate for the first ply is needed, given by (II.1.47):

Again, out-of-plane shear stresses in ply axes are of more interest than those in laminate axes.

A $2\times 6$ matrix ${\left[V\right]}_{\text{ply}}^{\text{lam}}$ is stored for each station through the laminate thickness where out-of-plane shear stress might be requested. This is done at the top, mid and bottom surfaces in each ply, meaning that $3N$ $2\times 6$ matrices ${\left[V^j\right]}_{\text{ply}}^{\text{lam}}$ are stored in the ClaLam object.