II.1.8.2 Out-of-plane shear response

Some of the quantities calculated above, and stored in the ClaLam object are used to estimate laminate shear loading response.

The different steps of the calculation are described below:

1.

The first step of the calculation is to resolve the loading in out-of-plane shear forces in loading axes ${\left\{Q\right\}}_{\text{load}}$ (or ${\left\{\Gamma \right\}}_{\text{load}}$). For this, one proceeds as in section II.1.5, but with the following differences: the conversion of average out-of-plane shear strain to out-of-plane shear force components requires the knowledge of out-of-plane shear stiffness matrix in loading axes. This one is readily obtained by transforming the corresponding matrix in laminate axes:

$${\left[G\right]}_{\text{load}}={\left[S_{+}\right]}_{\left(\lambda \right)}\cdot {\left[G\right]}_{\text{lam}}\cdot {\left[S_{-}\right]}_{\left(\lambda \right)}.$$

2.

The out-of-plane shear loading can be expressed by specifying the out-of-plane shear forces, or the out-of-plane average shear strain, or a combination of the two. In all cases, the components are specified in loading axes.

If out-of-plane average shear forces are specified, the resolution of the following linear system of equations allows to calculate the corresponding out-of-plane shear strains:

$$\begin{array}{cc}& {\left[G\right]}_{\text{load}}\cdot {\left\{\Gamma \right\}}_{\text{load}}={\left\{Q\right\}}_{\text{load}}+{\left\{{\alpha }_{s}Gh\right\}}_{\text{load}}\left(T_0-T_{\text{ref}}\right)\\ & +{\left\{{\alpha }_{s}G{h}^2\right\}}_{\text{load}}{T}_{,z}+{\left\{{\beta }_{s}Gh\right\}}_{\text{load}}\left(H_0-H_{\text{ref}}\right)+{\left\{{\beta }_{s}G{h}^2\right\}}_{\text{load}}{H}_{,z}.& \text{(II.1.93)}\end{array}$$

(The CTE and CME related terms are optional.) The resolution of this equation is done following the same approach as for the in-plane and bending loading. One performs a Gaussian elimination in a $2\times 2$ matrix. Constraints can be imposed if out-of-plane shear strains are specified for some components of the loading instead of out-of-plane lineic shear force.

3.

At this stage, whatever the type of loading applied to the laminate, ${\left\{\Gamma \right\}}_{\text{load}}$ is known. One can obtain the lineic out-of-plane shear forces with $$\begin{array}{cc}& {\left[Q\right]}_{\text{load}}={\left[G\right]}_{\text{load}}\cdot {\left\{\Gamma \right\}}_{\text{load}}-{\left\{{\alpha }_0^s\right\}}_{\text{load}}\left(T_0-T_{\text{ref}}\right)\\ & -{\left\{{\alpha }_1^s\right\}}_{\text{load}}{T}_{,z}-{\left\{{\beta }_0^s\right\}}_{\text{load}}\left(H_0-H_{\text{ref}}\right)-{\left\{{\beta }_1^s\right\}}_{\text{load}}{H}_{,z}.& \text{(II.1.94)}\end{array}$$

(The CTE and CME related terms are optional.) Once ${\left\{Q\right\}}_{\text{load}}$ and ${\left\{\Gamma \right\}}_{\text{load}}$ are known, the corresponding loading in laminate axes is obtained with:

$${\left\{Q\right\}}_{\text{lam}}={\left[S_{-}\right]}_{\left(\lambda \right)}\cdot {\left\{Q\right\}}_{\text{load}},$$

$${\left\{\Gamma \right\}}_{\text{lam}}={\left[S_{-}\right]}_{\left(\lambda \right)}\cdot {\left\{\Gamma \right\}}_{\text{load}}.$$

4.

For the calculation of ply out-of-plane shear stresses, one makes a distinction depending on the calculation approach that has been adopted:

(a)

With $\left({\mu }_x,{\mu }_y\right)$ approach ply out-of-plane shear stress components are calculated at the different requested locations by:

$${\left\{{\tau }_s^j\left(z\right)\right\}}_{\text{ply}}={\left[Y^j\left(z\right)\right]}_{\text{ply}}^{\text{lam}}\cdot {\left\{Q_s\right\}}_{\text{lam}},$$

in which the matrix ${\left[Y^j\right]}_{\text{ply}}^{\text{lam}}$ is relative to the station at which the stress is requested.

(b)

If the “resolution in shear force axes” approach is adopted, one first calculates an estimate of the gradient of bending moments tensor with equation (II.1.49):

$${\left\{\begin{array}{c}\hfill M_{xx,x}\hfill \\ \hfill M_{yy,x}\hfill \\ \hfill M_{xy,x}\hfill \\ \\ \hfill M_{xx,y}\hfill \\ \hfill M_{yy,y}\hfill \\ \hfill M_{xy,y}\hfill \end{array}\right\}}_{\text{lam}}=\frac{1}{Q_{xz}^{\text{lam}}\cdot Q_{xz}^{\text{lam}}+Q_{yz}^{\text{lam}}\cdot Q_{yz}^{\text{lam}}}\left\{\begin{array}{c}\hfill Q_{xz}^{\text{lam}}\cdot Q_{xz}^{\text{lam}}\cdot Q_{xz}^{\text{lam}}\hfill \\ \hfill Q_{yz}^{\text{lam}}\cdot Q_{yz}^{\text{lam}}\cdot Q_{xz}^{\text{lam}}\hfill \\ \hfill Q_{xz}^{\text{lam}}\cdot Q_{yz}^{\text{lam}}\cdot Q_{xz}^{\text{lam}}\hfill \\ \\ \hfill Q_{xz}^{\text{lam}}\cdot Q_{xz}^{\text{lam}}\cdot Q_{yz}^{\text{lam}}\hfill \\ \hfill Q_{yz}^{\text{lam}}\cdot Q_{yz}^{\text{lam}}\cdot Q_{yz}^{\text{lam}}\hfill \\ \hfill Q_{xz}^{\text{lam}}\cdot Q_{yz}^{\text{lam}}\cdot Q_{yz}^{\text{lam}}\hfill \end{array}\right\}.$$

Then, ply stresses are given by

$${\left\{{\tau }_s^j\left(z\right)\right\}}_{\text{ply}}={\left[V^j\left(z\right)\right]}_{\text{ply}}^{\text{lam}}\cdot {\left\{\nabla <!--FUNCTION\; APPLICATION-->M\right\}}_{\text{lam}},$$

in which the matrix ${\left[V^j\right]}_{\text{ply}}^{\text{lam}}$ is relative to the station at which the stress is requested.

5.

Finally, for the stations where out-of-plane shear stresses have been calculated the out-of-plain shear strain is also calculated using the corresponding ply material coefficients: $$\begin{array}{cc}& {\left\{{\gamma }_s\left(z\right)\right\}}_{\text{ply}}={\left[g\right]}_{\text{ply}}^{-1}\cdot {\left\{{\tau }_s\left(z\right)\right\}}_{\text{ply}}+\left({\left\{\alpha \right\}}_{\text{ply}}\left(T_0-T_{\text{ref}}+z{T}_{,z}\right)\right.\\ & \left.+{\left\{\beta \right\}}_{\text{ply}}\left(H_0-H_{\text{ref}}+z{H}_{,z}\right)\right).& \text{(II.1.95)}\end{array}$$

(The CTE and CME related terms are optional. One takes benefits of the “decoupling of out-of-plane shear” assumption.)