II.1.6.5 Approximations with out-of-plane shear forces

Expression (41) shows that out-of-plane shear stresses in the laminate are related to partial derivatives of bending moment components, but not directly to out-of-plane shear force components. Intuitively, one would have expected a dependence of out-of-plane shear stresses on laminate out-of-plane shear forces:

• Such a choice allows a reduction of size for the matrices involved in out-of-plane shear stress calculation, as one has only two components for the out-of-plane shear forces instead of 6 independent components for the gradient of bending moments.

• FE solvers produce different result outputs in shell elements, as in-plane forces, in-plane bending moments and out-of-plane shear forces. They do not output gradient of in-plane bending moments however. The calculation of bending moment gradient would lead to a significant amplification of numerical errors, and to unacceptable out-of-plane shear stress results.

• This observation might explain why most laminate analysis software calculate out-of-plane shear stresses from the out-of-plane shear forces.

In the end, we propose a calculation method based on out-of-plane shear force components because this is what most software do. The rest of this section is devoted to the presentation of different approaches to calculate dependence on shear forces.

II.1.6.5.1 The “$({\mu }_x,{\mu }_y)$” approach

The goal is to eliminate the six $x$ and $y$ partial derivatives of bending moment tensor components from the previous expression, by using the global equilibrium equations (II.1.36) and (II.1.37). This leaves some arbitrary choice in the determination of dependence with respect to the out-of-plane shear. For example:

It seems however that the choice ${\mu }_x={\mu }_y=1$ is more common.

If the $({\mu }_x,{\mu }_y)$ approach is adopted, a new $2\times 2$ matrix can be introduced:

This matrix allows one to write a simple relation between out-of-plane shear stresses in laminate and the total out-of-plane shear force:

The ${\left[V\right]}_{\text{lam}}(z)$ $2\times 6$ matrix introduced in equation (II.1.43) also allows the introduction of a new $2\times 2$ matrix:

II.1.6.5.2 “Uncoupled X-Y” approach

The presentation of laminate out-of-plane shear theory in Nastran Reference Manual [Sof04a] is based on a kind of beam theory in which laminate shear response is calculated separately in directions X and Y. This corresponds to a simplification of our $({\mu }_x,{\mu }_y)$ approach in which:

• A total decoupling between the X and Y components for membrane and bending behaviour of the laminate is assumed when out-of-plane shear stresses are calculated.

• No ${\mu }_x$ and ${\mu }_y$ parameters are considered in the theory.

We investigated different ways to reproduce Nastran out-of-plane shear stress calculations with FeResPost and found a few modifications that allow the calculation of out-of-plane shear stresses very similar to those produced by Nastran. We modify the calculation method as follows:

1.

The $6\times 6$ ${\left[ABBD\right]}_{\text{lam}}$ matrix is modified in such a way that one no longer has a coupling between $XX$, $YY$ and $XY$ components. Practically, this is done by setting matrix components responsible for this coupling to zero:

$${\left[ABB{D}_2\right]}_{\text{lam}}=\left[\begin{array}{cc}\hfill \begin{array}{ccc}\hfill A_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill A_{22}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill A_{33}\hfill \end{array}\hfill & \hfill \begin{array}{ccc}\hfill B_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill B_{22}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill B_{33}\hfill \end{array}\hfill \\ & \\ \hfill \begin{array}{ccc}\hfill B_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill B_{22}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill B_{33}\hfill \end{array}\hfill & \hfill \begin{array}{ccc}\hfill D_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill D_{22}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill D_{33}\hfill \end{array}\hfill \end{array}\right].$$

(The non-diagonal components of the four $3\times 3$ matrices have been set to zero.) Note that the membrane-flexural coupling is maintained by this modification.

2.

Corresponding to this ${\left[ABB{D}_2\right]}_{\text{lam}}$ matrix, we calculate a compliance matrix:

$${\left[abb{d}_2\right]}_{\text{lam}}={\left[ABB{D}_2\right]}_{\text{lam}}^{-1}.$$

This new matrix has the same structure as ${\left[ABB{D}_2\right]}_{\text{lam}}$. ($XX$, $YY$ and $XY$ components are also mutually decoupled.)

3.

Uncoupling of $XX$, $YY$ and $XY$ components is also done for the $3\times 3$ material stiffness matrix ${\left[C\right]}_{\text{lam}}(z)$:

$${\left[C_2\right]}_{\text{lam}}(z)=\left[\begin{array}{ccc}\hfill C_{11}^{\text{lam}}(z)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill C_{22}^{\text{lam}}(z)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill C_{33}^{\text{lam}}(z)\hfill \end{array}\right].$$

4.

Then, a new version of the ${\left[F\right]}_{\text{lam}}(z)$ matrix is calculated:

$${\left[F_2\right]}_{\text{lam}}(z)={\left[C_2\right]}_{\text{lam}}(z)\cdot \left({\left[b_2\right]}_{\text{lam}}+z{\left[d_2\right]}_{\text{lam}}\right).$$

This new matrix also uncouples $XX$, $YY$ and $XY$ components of the equations.

5.

Finally, the $XY$ components are no longer needed in the calculation. Practically, this means that equation (II.1.40) becomes

$${\left\{\begin{array}{c}\hfill {\tau }_{xz,z}(z)\hfill \\ \hfill {\tau }_{yz,z}(z)\hfill \end{array}\right\}}_{\text{lam}}=\left[\begin{array}{cccccc}\hfill -1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill -1& \hfill 0\\ \hfill \end{array}\right]\cdot {\left\{\begin{array}{c}\hfill {\sigma }_{xx,x}(z)\hfill \\ \hfill {\sigma }_{yy,x}(z)\hfill \\ \hfill {\tau }_{xy,x}(z)\hfill \\ \\ \hfill {\sigma }_{xx,y}(z)\hfill \\ \hfill {\sigma }_{yy,y}(z)\hfill \\ \hfill {\tau }_{xy,y}(z)\hfill \end{array}\right\}}_{\text{lam}},$$

and the relation between gradient of moments tensor, and out-of-plane shear forces can be written

$${\left\{\begin{array}{c}\hfill M_{xx,x}\hfill \\ \hfill M_{yy,x}\hfill \\ \hfill M_{xy,x}\hfill \\ \\ \hfill M_{xx,y}\hfill \\ \hfill M_{yy,y}\hfill \\ \hfill M_{xy,y}\hfill \end{array}\right\}}_{\text{lam}}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ & \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\cdot {\left\{\begin{array}{c}\hfill Q_{xz}\hfill \\ \hfill Q_{yz}\hfill \end{array}\right\}}_{\text{lam}}.$$

6.

The uncoupling also affects the calculation of out-of-plane shear stiffness matrix. (See section II.1.6.6.) In the corresponding equations in sections II.1.6.3 and II.1.6.4, matrix ${\left[F\right]}_{\text{lam}}(z)$ is replaced by ${\left[F_2\right]}_{\text{lam}}(z)$. A consequence of this approximation is that the out-of-plane shear stiffness matrix ${\left[G\right]}_{\text{lam}}$ is diagonal.

The “Uncoupled X-Y” approach is an impoverished version of the “$({\mu }_x,{\mu }_y)$” approach.

II.1.6.5.3 “Resolution in shear force axes” approach

Using the “$({\mu }_x,{\mu }_y)$” approaches, a special physical meaning is implicitly assigned to the laminate axes. This choice is arbitrary, however. For example, one can write the relation between bending moments and out-of-plane shear force in a coordinate system related to the shear loading direction.

Let us define a coordinate system associated with the shear loading, defined as follows:

in which $Q$ is the shear force magnitude. In this new coordinate system, the shear force vector has only one non-zero component:

Then, a simple relation between bending moments and out-of-plane shear force can be assumed:

all the other components of bending moments gradient being zero. As the gradient of bending moments tensor is an order 3 tensor, previous relation can be written in laminate axes as follows:

It is easy to verify that this relation between shear forces and bending moments is non-linear. For example:

and

Clearly, the gradient of bending moments tensor in (II.1.52) is not the sum of corresponding tensors in (II.1.50) and (II.1.51), but the out-of-plane shear force in (II.1.52) is the sum of corresponding vectors in (II.1.50) and (II.1.51). This demonstrates the non-linearity of an approach based on a calculation in shear loading axes.

II.1.6.5.4 Comparison of the three approaches

The main disadvantage of the $({\mu }_x,{\mu }_y)$ approach is that the laminate out-of-plane shear equations lose their objectivity with respect to rotations of the laminate axes around axis $z$, as illustrated by the example described in section IV.3.5. (This example also allows estimation of the effects of the approximation on the precision of results given by the theory.)

On the other hand, the $({\mu }_x,{\mu }_y)$ approach leads to linear calculations, which is an advantage compared to the “resolution in shear force axes” approach. Indeed, the “resolution in shear force axes” approach is somewhat paradoxical in this respect, as in many cases the laminate out-of-plane shear stress will be the only non-linear response of an otherwise linear problem.

This means that none of the three approaches is perfect, and the imperfections result from the fact that both approaches are approximations of the reality. Both approaches are inaccurate, and it is not possible to decide which one is better. In practical problems, the three approaches are expected to give good results.

To simplify the notations, we rewrite equation (II.1.49) as follows:

In fact, this notation also applies to the $({\mu }_x,{\mu }_y)$ approach, except that the function is then linear:

Finally, we summarize below our recommendations regarding the calculation of out-of-plane shear in laminates:

• Our recommended approach is to use the “$({\mu }_x,{\mu }_y)$” approach with ${\mu }_x={\mu }_y=1$. This is the default option proposed by FeResPost.

• The “uncoupled X-Y” approach is a poorer version of the previous one. It should be used only when a very close match with Nastran results is needed.

• The advantage of the “resolution in shear force axes” approach is that it is objective. Results do not depend on the choice of laminate axes. Note however that this approach leads to a non-linear dependence of ply shear stresses on the laminate loading. Also, be aware that this approach affects only the calculation of ply out-of-plane shear stresses. The laminate out-of-plane shear stiffness matrix will be the same as with “$({\mu }_x,{\mu }_y)$” approach

Again, the three approaches are approximate, and none of them is better than the others as far as result accuracy is concerned.