II.1.8.1 In-plane and flexural response

Besides thermo-elastic or hygro-elastic loading, the composite classes of FeResPost allow the definition of different types of mechanical loads:

• By specifying normal forces ${\left\{N\right\}}_{\text{load}}$ and bending moments ${\left\{M\right\}}_{\text{load}}$.

• By specifying average strains ${\left\{{ϵ}^0\right\}}_{\text{load}}$ and curvatures ${\left\{\kappa \right\}}_{\text{load}}$.

• By specifying average stresses and flexural stresses.

The type of loading is specified component-by-component. This means that a single loading may have some components imposed as normal forces and bending moments, with other components imposed as average strains, and other components as average stresses or flexural stresses. The mechanical part of loading is also characterized by a direction $\lambda$ with respect to the laminate axes. The subscript “load” indicates that the components are given in loading axes. The laminate response is calculated as described below.

1.

The solver first checks if average or flexural stresses are imposed. If such components of the loading are found, they are converted to in-plane forces and bending moments with the following equations:

$${\left\{N\right\}}_{\text{load}}=h{\left\{{\sigma }^0\right\}}_{\text{load}},$$

$${\left\{M\right\}}_{\text{load}}=\frac{h^2}{6}{\left\{{\sigma }^f\right\}}_{\text{load}},$$

in which $h$ is the laminate thickness.

2.

The mechanical part of loading is characterized by a direction with respect to the laminate axes. This direction is given by an angle $\lambda$. In order to have laminate properties and loading given in the same coordinate system, the laminate stiffness matrices and CTE vectors are calculated in this new coordinate system. (It is more convenient for the elimination of components imposed as average strains or curvatures.) More precisely, the stiffness matrices and CTE vector are rotated with the following expressions:

$${\left[A\right]}_{\text{load}}={\left[T_{-}\right]}_{(\lambda )}\cdot {\left[A\right]}_{\text{lam}}\cdot {\left[T_{-}^{\prime }\right]}_{(\lambda )},$$

$${\left[B\right]}_{\text{load}}={\left[T_{-}\right]}_{(\lambda )}\cdot {\left[B\right]}_{\text{lam}}\cdot {\left[T_{-}^{\prime }\right]}_{(\lambda )},$$

$${\left[D\right]}_{\text{load}}={\left[T_{-}\right]}_{(\lambda )}\cdot {\left[D\right]}_{\text{lam}}\cdot {\left[T_{-}^{\prime }\right]}_{(\lambda )},$$

$${\left\{\alpha Eh\right\}}_{\text{load}}={\left[T_{-}\right]}_{(\lambda )}\cdot {\left\{\alpha Eh\right\}}_{\text{lam}},$$

$$\dots$$

$${\left\{\beta E{h}^3\right\}}_{\text{load}}={\left[T_{-}\right]}_{(\lambda )}\cdot {\left\{\beta E{h}^3\right\}}_{\text{lam}}.$$

The calculation of CTE and CME related quantities is done only if the corresponding temperature or moisture contributions have been defined in the loading. The system of equations looks like:

$$\begin{array}{cc}& \left[\begin{array}{cc}\hfill {\left[A\right]}_{\text{load}}\hfill & \hfill {\left[B\right]}_{\text{load}}\hfill \\ \hfill {\left[B\right]}_{\text{load}}\hfill & \hfill {\left[D\right]}_{\text{load}}\hfill \end{array}\right]\cdot \left\{\begin{array}{c}\hfill {\left\{{ϵ}^0\right\}}_{\text{load}}\hfill \\ \hfill {\left\{\kappa \right\}}_{\text{load}}\hfill \end{array}\right\}=\left\{\begin{array}{c}\hfill {\left\{N\right\}}_{\text{load}}\hfill \\ \hfill {\left\{M\right\}}_{\text{load}}\hfill \end{array}\right\}\\ & +\left\{\begin{array}{c}\hfill {\left\{\alpha Eh\right\}}_{\text{load}}\hfill \\ \hfill {\left\{\alpha E{h}^2\right\}}_{\text{load}}\hfill \end{array}\right\}(T_0-T_{\text{ref}})+\left\{\begin{array}{c}\hfill {\left\{\alpha E{h}^2\right\}}_{\text{load}}\hfill \\ \hfill {\left\{\alpha E{h}^3\right\}}_{\text{load}}\hfill \end{array}\right\}{T}_{,z}\\ & +\left\{\begin{array}{c}\hfill {\left\{\beta Eh\right\}}_{\text{load}}\hfill \\ \hfill {\left\{\beta E{h}^2\right\}}_{\text{load}}\hfill \end{array}\right\}(H_0-H_{\text{ref}})+\left\{\begin{array}{c}\hfill {\left\{\beta E{h}^2\right\}}_{\text{load}}\hfill \\ \hfill {\left\{\beta E{h}^3\right\}}_{\text{load}}\hfill \end{array}\right\}{H}_{,z}.& \text{(II.1.88)}\end{array}$$

(Here again the CTE and CME related terms are optional.) In fact, one can write a single set of 6 equations with 6 unknowns. The general form of this system is

$$\sum _{j=1}^n{K}_{ij}{u}_j=b_i,i=1...n.$$

3.

Consider a case in which one component of vector $u_j$ is constrained to a certain value. For example $u_k=a$. This equation replaces the $k^{\text{th}}$ equation of the system:

$$\begin{array}{llll}\hfill \sum _{j=1}^n{K}_{ij}{u}_j& =b_i,i=1...n,i\ne k& \hfill & \\ \hfill u_k& =a.& \hfill & \end{array}$$

The unknown $u_k$ can easily be eliminated from the linear system:

$$\begin{array}{llll}\hfill \sum _{j=1,j\ne k}^n{K}_{ij}{u}_j& =b_i-K_{ik}a,i=1...n,i\ne k& \hfill & \\ \hfill u_k& =a.& \hfill & \end{array}$$

The first line above corresponds to a new linear system of $n-1$ equations with $n-1$ unknowns. The set of two lines defines the algebraic operations that are performed in FeResPost when an average strain or curvature component is imposed.

In fact, the operation can be simplified. It is sufficient to replace line $k$ in the linear system of equations by the constraint equation $u_k=a$ and perform the “usual” Gaussian elimination to solve the linear system of equations.

4.

When all the components of loading imposed as average strains or curvature have been eliminated from the linear system, a classical Gaussian elimination algorithm calculates the other unknowns of the system.

Then the components of tensors $\left\{{ϵ}^0\right\}$ and $\left\{\kappa \right\}$ are known in loading axes.

5.

The normal forces and bending moments are then calculated in loading axes with the following equations:

$$\begin{array}{cc}& {\left\{N\right\}}_{\text{load}}={\left[A\right]}_{\text{load}}\cdot {\left\{{ϵ}^0\right\}}_{\text{load}}+{\left[B\right]}_{\text{load}}\cdot {\left\{\kappa \right\}}_{\text{load}}-{\left\{\alpha Eh\right\}}_{\text{load}}(T_0-T_{\text{ref}})\\ & -{\left\{\alpha E{h}^2\right\}}_{\text{load}}{T}_{,z}-{\left\{\beta Eh\right\}}_{\text{load}}(H_0-H_{\text{ref}})-{\left\{\beta E{h}^2\right\}}_{\text{load}}{H}_{,z},& \text{ (II.1.89)}\end{array}$$

$$\begin{array}{cc}& {\left\{M\right\}}_{\text{load}}={\left[B\right]}_{\text{load}}\cdot {\left\{{ϵ}^0\right\}}_{\text{load}}+{\left[D\right]}_{\text{load}}\cdot {\left\{\kappa \right\}}_{\text{load}}-{\left\{\alpha E{h}^2\right\}}_{\text{load}}(T_0-T_{\text{ref}})\\ & -{\left\{\alpha E{h}^3\right\}}_{\text{load}}{T}_{,z}-{\left\{\beta E{h}^2\right\}}_{\text{load}}(H_0-H_{\text{ref}})-{\left\{\beta E{h}^3\right\}}_{\text{load}}{H}_{,z}.& \text{(II.1.90)}\end{array}$$

(The CTE and CME related terms are optional.)

6.

If $\lambda$ is the angle characterizing the loading orientation with respect to the laminate axes, a rotation of $-\lambda$ of the two vectors gives the average strain and curvature tensors in laminate axes:

$${\left\{{ϵ}^0\right\}}_{\text{lam}}={\left[T_{+}^{\prime }\right]}_{(-\lambda )}\cdot {\left\{{ϵ}^0\right\}}_{\text{load}},$$

$${\left\{\kappa \right\}}_{\text{lam}}={\left[T_{+}^{\prime }\right]}_{(-\lambda )}\cdot {\left\{\kappa \right\}}_{\text{load}}.$$

Similarly, the normal forces and bending moments components are re-expressed in laminate axes:

$${\left\{N\right\}}_{\text{lam}}={\left[T_{-}\right]}_{(-\lambda )}\cdot {\left\{N\right\}}_{\text{load}},$$

$${\left\{M\right\}}_{\text{lam}}={\left[T_{-}\right]}_{(-\lambda )}\cdot {\left\{M\right\}}_{\text{load}}.$$

7.

For each ply, stresses and strains are calculated (if required) as follows:

(a)

The laminate average strain and curvature tensors are rotated to ply axes. If the ply is characterized by an angle $\xi$ with respect to the laminate axes, the two tensors are rotated by the same angle $\xi$:

$${\left\{{ϵ}^0\right\}}_{\text{ply}}={\left[T_{+}^{\prime }\right]}_{(\xi )}\cdot {\left\{{ϵ}^0\right\}}_{\text{lam}},$$

$${\left\{\kappa \right\}}_{\text{ply}}={\left[T_{+}^{\prime }\right]}_{(\xi )}\cdot {\left\{\kappa \right\}}_{\text{lam}}.$$

Note that, even though the components of these two tensors are now given in one of the plies coordinate system, they correspond to strain or curvature of the laminate at mid-thickness.

(b)

At the different stations through the thickness at which strains and stresses are required, the strain components are calculated with:

$${\left\{ϵ\right\}}_{\text{ply}}(z)={\left\{{ϵ}^0\right\}}_{\text{ply}}+z{\left\{\kappa \right\}}_{\text{ply}}.$$

(II.1.91)

In FeResPost $z$ may have the values $z_{\text{inf}}$, $z_{\text{mid}}$, $z_{\text{sup}}$. Then the stress components are given by:

$$\begin{array}{cc}& {\left\{\sigma \right\}}_{\text{ply}}(z)={\left[C\right]}_{\text{ply}}\cdot {\left\{ϵ\right\}}_{\text{ply}}(z)\\ & -{\left[C\right]}_{\text{ply}}\cdot ({\left\{\alpha \right\}}_{\text{ply}}(T_0-T_{\text{ref}}+z{T}_{,z})\\ & +{\left\{\beta \right\}}_{\text{ply}}(H_0-H_{\text{ref}}+z{H}_{,z})).& \text{(II.1.92)}\end{array}$$

(Here again the CTE and CME related terms are optional.) A particular version of the ply strain tensor that corresponds to ply stresses, but without thermo-elastic or moisture contribution, is calculated as follows:

$${\left\{{ϵ}^{\text{Mech}}\right\}}_{\text{ply}}(z)={\left[c\right]}_{\text{ply}}\cdot {\left\{\sigma \right\}}_{\text{ply}}(z).$$

This version of the strain tensor is called the “Mechanical Strain Tensor” or “Equivalent Strain Tensor”. This is the version of the strain tensor that is used for the strain-based failure criteria. Note however that a “Total Strain” version of the criteria is also available.

At the end of the calculations, the laminate object which has been used to perform those calculations stores a few results:

• The average strain and curvature of the laminate in laminate axes ${\left\{{ϵ}^0\right\}}_{\text{lam}}$ and ${\left\{\kappa \right\}}_{\text{lam}}$.

• The laminate in-plane membrane forces and bending moments in laminate axes ${\left\{N\right\}}_{\text{lam}}$ and ${\left\{M\right\}}_{\text{lam}}$.

• The ply results in ply axes ${\left\{ϵ\right\}}_{\text{ply}}(z_i)$, ${\left\{\sigma \right\}}_{\text{ply}}(z_i)$ and ${\left\{{ϵ}^{\text{Mech}}\right\}}_{\text{ply}}(z_i)$, $z_i$ being the different stations through the thickness for which the ply results have been calculated.

The ply results may be used later to calculate failure indices or reserve factors.