II.1.6.7 Calculation algorithm for shear stiffness

One describes below the calculation sequences that is used to calculate the laminate out-of-plane shear stiffness properties, and the out-of-plane shear stresses related to a given loading of the laminate.

The calculation sequence is described below. It involves two loops on the laminate layers.

1.

Calculate laminate in-plane and flexural properties. This is necessary because one needs the matrices ${\left[b\right]}_{\text{lam}}$ and ${\left[d\right]}_{\text{lam}}$ to calculate out-of-plane shear properties.

2.

One initializes the $2\times 2$ matrix ${\left[G\right]}_{\text{lam}}^{-1}$ to zero.

3.

Then for each layer $k$ with $1\le k\le N$, one performs the following sequence of operations:

(a)

One estimates the $3\times 3$ matrix of in-plane stiffness coefficients in laminate axes ${\left[C^k\right]}_{\text{lam}}$. For other calculations, one also need properties like the laminate thickness $t$ and the positions $z_k$ of different layer interfaces.

(b)

This matrix is used to calculate the two matrices ${\left[F_0^k\right]}_{\text{lam}}$ and ${\left[F_1^k\right]}_{\text{lam}}$. (See section II.1.6.3 for more details.) One has:

$${\left[F^k\right]}_{\text{lam}}\left(z\right)={\left[F_0^k\right]}_{\text{lam}}+z{\left[F_1^k\right]}_{\text{lam}}.$$

(c)

Then, one calculates two other $2\times 2$ matrices ${\left[X_0^k\right]}_{\text{lam}}$ and ${\left[X_1^k\right]}_{\text{lam}}$. (See section II.1.6.3.) One has:

$${\left[X^k\right]}_{\text{lam}}\left(z\right)={\left[X_0^k\right]}_{\text{lam}}+z{\left[X_1^k\right]}_{\text{lam}}.$$

(d)

Then one calculates the ${\left[Y_i^k\right]}_{\text{lam}}$ $2\times 2$ matrices:

$${\left[Y_0^k\right]}_{\text{lam}}={\left[Y_0^{k-1}\right]}_{\text{lam}}+\left({\left[Y_1^{k-1}\right]}_{\text{lam}}-{\left[X_0^k\right]}_{\text{lam}}\right)z_{k-1}+\left({\left[Y_2^{k-1}\right]}_{\text{lam}}-\frac{{\left[X_1^k\right]}_{\text{lam}}}{2}\right)z_{k-1}^2,$$

$${\left[Y_1^k\right]}_{\text{lam}}={\left[X_0^k\right]}_{\text{lam}},$$

$${\left[Y_2^k\right]}_{\text{lam}}=\frac{{\left[X_1^k\right]}_{\text{lam}}}{2}.$$

As the expression of ${\left[Y_0^k\right]}_{\text{lam}}$ is recursive, one needs another expression for the first value. The expression is:

$$\begin{array}{llll}\hfill {\left[Y_0^1\right]}_{\text{lam}}& =-{\left[Y_1^1\right]}_{\text{lam}}{z}_0-{\left[Y_2^1\right]}_{\text{lam}}{z}_0^2.& \hfill & \\ \hfill & =-{\left[X_0^1\right]}_{\text{lam}}{z}_0-\frac{1}{2}{\left[X_1^1\right]}_{\text{lam}}{z}_0^2.& \hfill & \end{array}$$

(e)

One calculates the ${\left[U_i^k\right]}_{\text{lam}}$ $2\times 2$ matrix. (See the end of section II.1.6.6 for the expressions to be used.) Then to ${\left[G\right]}_{\text{lam}}^{-1}$, one adds one term:

$${\left[G\right]}_{\text{lam}}^{-1}={\left[G\right]}_{\text{lam}}^{-1}+\sum _{i=0}^4\left({\left[U_i^k\right]}_{\text{lam}}\frac{z_k^{i+1}-z_{k-1}^{i+1}}{i+1}\right).$$

4.

At the end of the loop on layers, the shear stiffness matrix ${\left[G\right]}_{\text{lam}}$ is calculated by inversion of ${\left[G\right]}_{\text{lam}}^{-1}$.

One also defines an out-of-plane shear compliance matrix calculated as follows:

This matrix allows to calculate the laminate out-of-plane shear moduli:

Note that the values calculated above do not correspond to an out-of-plane shear stiffness of a material equivalent to the defined laminate. To convince yourself of this you can define a laminate with a single ply of orthotropic material. Then, you will observe that $G_{xz}=G_{yz}=5G∕6$ in which $G$ is material shear modulus. (The usual $5∕6$ factor in shell theory is recovered.)