II.1.8.1 In-plane and flexural response

Beside thermo-elastic or hygro-elastic loading, the composite classes of FeResPost allows the definition 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 $\lambda$ wrt laminate axes. The subscript “load” indicates that the components are given in loading axes. One explains below how the laminate response is calculated.

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 wrt 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:

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

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

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

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

$$\dots <!--FUNCTION\; APPLICATION-->$$

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

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:

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

(Here again the CTE and CME related terms are optional.) Actually, 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.

Now, one considers a case in which one component of vector $u_j$ is constrained to be 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 be easily eliminated from the linear

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

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

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

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

(The CTE and CME related terms are optional.)

6.

If $\lambda$ is the angle characterizing the loading orientation wrt laminate axes a rotation of $-\lambda$ of the two vectors gives the average strain and curvature tensors in laminate axes: ${\left\{{ϵ}^0\right\}}_{\text{lam}}$ and ${\left\{\kappa \right\}}_{\text{lam}}$.

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

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

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

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

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

7.

For each ply, one calculates (if required) the stresses and strains as follows:

(a)

One rotates the laminate average strain and curvature tensors to obtain them in ply axes. If the ply is characterized by an angle $\xi$ wrt laminate axes, the two tensors are rotated by the same angle $\xi$:

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

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

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:

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

(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:

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

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

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

This version of the strain tensor is called the “Mechanical Strain Tensor” or “Equivalent Strain Tensor”. This is the version of the strain tensors that is used for the strain failure criteria. Note however that a “Total Strain” version of the criteria is proposed as well.

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 ${\left\{ϵ\right\}}_{\text{ply}}\left(z_i\right)$, ${\left\{\sigma \right\}}_{\text{ply}}\left(z_i\right)$ and ${\left\{{ϵ}^{\text{Mech}}\right\}}_{\text{ply}}\left(z_i\right)$, $z_i$ 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.