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 nil without modifying the result of the developments. The corresponding lines and columns could be removed from the equations (II.1.39).

Actually, 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 allows to find a new expression of the relation between bending moment gradients and out-of-plane shear stress. One first calculates a new matrix as follows:

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

The matrix ${\left[J\right]}_{\text{lam}}\left(z\right)$ 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 ${\left[F\right]}_{\text{lam}}\left(z\right)$ if one splits the definition by plies:

Of course, one has the two relations: