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:
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.
One would like to eliminate the six and partial derivative of bending moment tensor components in the previous expression. For this, one uses the global equilibrium equations (II.1.36) and (II.1.37). This leaves some arbitrary choice in the determination of dependence wrt out-of-plane shear. For example:
The choice gives more symmetry to the relation between and . Indeed, this choice leads to:
It seems however that the choice is more common.
If the approach is adopted, one can introduce a new matrix:
This matrix allows to write a simple relation between out-of-plane shear stresses in laminate and the total out-of-plane shear force:
The matrix introduced in equation (II.1.43) also allows to introduce a new matrix:
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 approach in which:
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:
(The non-diagonal components of the four matrices have been set to zero.) Note that the membrane-flexural coupling is maintained by this modification.
This new matrix has the same structure as . (, and compnents are also mutually decoupled.)
This new matrix also uncouples , and components of the equations.
and the relation between gradient of moments tensor, and out-of-plane shear forces can be written
The “Uncoupled X-Y” approach is an impoverished version of the “” approach.
Using “” approaches, one decides that laminate axes have a special physical meaning for the composite. This choice is arbitrary however. For example, one can also 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 to shear loading defined as follows:
In which is shear force magnitude. In this new coordinate system, the shear force vector has only one non zero component:
Then, one can assume a simple relation between bending moments and out-of-plane shear force:
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:
(II.1.49) |
One checks easily that this relation between shear forces and bending moments is non-linear. For example:
(II.1.50) |
(II.1.51) |
and
(II.1.52) |
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.
The main disadvantage of approach is that the laminate out-of-plane shear equations lose their objectivity wrt rotations of the laminate axes around axis as illustrated by the example described in section IV.3.5. (This example also allows to estimate the effects of the approximation on the precision of results given by the theory.)
On the other hand, approach leads to linear calculations, which is an advantage compared to the “resolution in shear force axes” approach. Actually, resolution in shear force axes approach is a little paradoxical wrt this aspect, 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, one expects the three approaches give good results however.
To simplify the notations, we rewrite equation (II.1.49) as follows:
Actually, this notation also applies to the approach except that the function is then linear:
Finally, we summarize below our recommendations regarding the calculation of out-of-plane shear in laminates:
But again, the three approaches are approximate, and none of the thre appraoch is better than the other ones as far as results accuracy is concerned.