II.1.6.3 Out-of-plane shear stress partial derivative equations

Note that the global equilibrium equation (II.1.36) and (II.1.37) do not contain the components $M_{xx,y}$ and $M_{yy,x}$ of the bending moments tensor. Similarly, the local equilibrium equations do not contain the components ${\sigma }_{xx,y}$ and ${\sigma }_{yy,x}$ of the Cauchy stress tensor. Then, these components can be considered as zero without modifying the result of the developments. The corresponding lines and columns could be removed from the equations (II.1.39).

In fact, one can do better than that. The local equilibrium equations (II.1.34) and (II.1.35) are rewritten as follows:

The substitution of (II.1.39) in (II.1.40) leads to the following expression:

This yields a new expression of the relation between bending moment gradients and out-of-plane shear stress. A new matrix is first calculated as follows:

${\left[J\right]}_{\text{lam}}(z)$ is a $2\times 6$ matrix that relates the out-of-plane shear stress component partial derivatives with respect to $z$ to the in-plane bending moment components:

The matrix ${\left[J\right]}_{\text{lam}}(z)$ depends on $z$ for two reasons: because of the triangular distribution of strains through the thickness, and because material moduli depend on plies material and orientation. In a given ply of index $k$, one has:

in which the components of the two matrices ${\left[J_0^k\right]}_{\text{lam}}$ and ${\left[J_1^k\right]}_{\text{lam}}$ are constant. Similarly, one can write a polynomial expression for ${\left[F\right]}_{\text{lam}}(z)$ if one splits the definition by plies:

The following two relations hold: