II.1.9.1 Calculation of laminate loads and strains

We explain in this section how the laminate load response calculation can be accelerated. In particular, when the calculation of laminate load response is done repititively with similar loading, the benefit of simplifying the sequence of operations to estimate laminate loading becomes obvious.

One explains the calculation of laminate in-plane and flexural load response in section II.1.8.1. The sequence of operations results in the building of $6\times 6$ matrix $K$ that depends on laminate definition and loading angle. For all the calculations done with a common laminate and loading angle, the operations can be simplified as follows:

1.

The matrix $K$ is assembled.

2.

The components of loading that are specified as in-plane strain, or curvature lead to the imposition of constraints on matrix $K$. For example, if one imposes $u_k=a$, it is sufficient to replace the elements of line $k$ in matrix $K$ by 0, except $K_{kk}$=1. When all the constraints have been imposed, one obtains a new $6\times 6$ matrix that we call $K_{\text{constr}}$.

3.

Finally, this matrix is inversed, and on obtained the $6\times 6$ matrix $K_{\text{constr}}^{-1}$.

This $K_{\text{constr}}^{-1}$ matrix is the same for all the loadings that we apply with on the same laminate, with the same loading angle $\theta$, and with the same constraints. When calculating load response, this matrix is used as follows:

1.

For each laminate load, one assembles the 6 components vector $b_{\text{constr}}$ as explained in section II.1.8.1. The components usually correspond to shell forces or moments, but the “constrained components” (components specified as strains or curvature) are replaced by components of shell in-plane strains or curvatures.

2.

Shell in-plane strains and curvatures in loading axes are obtained by calculating the following matricial product:

$$\left\{\begin{array}{c}\hfill {\left\{{ϵ}^0\right\}}_{\text{load}}\hfill \\ \hfill {\left\{\kappa \right\}}_{\text{load}}\hfill \end{array}\right\}=K_{\text{constr}}^{-1}\cdot b_{\text{constr}}.$$

3.

Then, the shell strains and curvature components can be expressed in laminate axes by performing the following matricial operation: $$\begin{array}{cc}& \left\{\begin{array}{c}\hfill {\left\{{ϵ}^0\right\}}_{\text{lam}}\hfill \\ \hfill {\left\{\kappa \right\}}_{\text{lam}}\hfill \end{array}\right\}=\left[\begin{array}{cc}\hfill {\left[T_{+}^{\prime }\right]}_{\left(-\lambda \right)}\hfill & \hfill \left[0\right]\hfill \\ \hfill \left[0\right]\hfill & \hfill {\left[T_{+}^{\prime }\right]}_{\left(-\lambda \right)}\hfill \end{array}\right]\cdot \left\{\begin{array}{c}\hfill {\left\{{ϵ}^0\right\}}_{\text{load}}\hfill \\ \hfill {\left\{\kappa \right\}}_{\text{load}}\hfill \end{array}\right\},\\ & =\left[\begin{array}{cc}\hfill {\left[T_{+}^{\prime }\right]}_{\left(-\lambda \right)}\hfill & \hfill \left[0\right]\hfill \\ \hfill \left[0\right]\hfill & \hfill {\left[T_{+}^{\prime }\right]}_{\left(-\lambda \right)}\hfill \end{array}\right]\cdot K_{\text{constr}}^{-1}\cdot b_{\text{constr}}.& \text{(II.1.96)}\end{array}$$

Finally, the last expression leads to the definition of a new $6\times 6$ matrix

that allows to write

This matrix can be constructed once and for all for a given laminate, loading angle and loading characteristics.

A similar approach can be used for the simplification of the laminate out-of-plane shear response calculation:

1.

The construction of a $2\times 2$ $K_{\text{constr}}^{\prime -1}$ matrix allows to write:

$${\left[\Gamma \right]}_{\text{load}}=K_{\text{constr}}^{\prime -1}\cdot b_{\text{constr}}^{\prime },$$

in which $b_{\text{constr}}$ corresponds to the out-of-plane shear loads (forces or strains).

2.

Then

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

3.

And this leads to the definition of a new $2\times 2$ matrix that allows to write:

$$K_{\text{constr, lam}}^{\prime -1}={\left[S_{-}\right]}_{\left(\lambda \right)}\cdot K_{\text{constr}}^{\prime -1},$$

$${\left[\Gamma \right]}_{\text{lam}}=K_{\text{constr, lam}}^{\prime -1}\cdot b_{\text{constr}}^{\prime }.$$

(II.1.98)

Finally, now that the laminate strains and curvatures have been estimated in laminate axes, the corresponding laminate forces and moments are estimated as follows:

One can substitute (II.1.97) and (II.1.98) in expressions (99) and (100). This leads to the following expressions:

From equations (II.1.97), (II.1.98), (101) and (102), one identifies four matrices and four vectors that allow the calculation of laminate stress/strain state in laminate axes:

• The $6\times 6$ matrix $K_{\text{constr, lam}}^{-1}$,
• The $2\times 2$ matrix $K_{\text{constr, lam}}^{\prime -1}$,
• The $6\times 10$ matrix $K_{\text{constr, lam}}^{\text{NM}}$,
• The $2\times 6$ matrix $K_{\text{constr, lam}}^{\text{Q}}$,
• The 6-components vector $b_{\text{constr}}$,
• The 2-components vector $b_{\text{constr}}^{\prime }$,
• The 10-components vector $\left\{b_{\text{constr}}^{FM}\right\}$,
• The 6-components vector $\left\{b_{\text{constr}}^Q\right\}$.

The four matrices depend on laminate definition, angle $\theta$ of the loading wrt laminate axes, and on the set of components that are constrained to strain or curvature values. The four vectors also depend on the values of the particular loading that is examined. This means that the four matrices can be calculated once and for all the particular loadings. On the other-hand the four vectors must be re-estimated for each element defining the load.