Calculation module

General

The Calculation module is used to calculate section loads and strains as well as stresses within the specified fiber composite laminate according to the Classical Laminate Theory ([1], [2]). In order to be able to directly consider the effects of the laminate structure on the occupation of the stiffness matrix of the composite, the section loads or strains and the stresses on the layers, the calculations are updated automatically. Among other things, hygrothermal effects can also be taken into account [3].

During a calculation, the system of linear equations listed below is solved using the Gaussian elimination method. The calculated unknowns are entered in the fields of the internal loads and distortions. In addition, the stresses and strains in each laminate layer are calculated. These can be viewed in the layer variables subwindow in the local fiber coordinate system.

Structure

1 - ABD matrix

Based on the stiffness matrices of the individual layers, the membrane stiffness matrix $\mathbf{A}$, coupling stiffness matrix $\mathbf{B}$ and bending stiffness matrix $\mathbf{D}$ of the composite are automatically calculated and displayed separately in color within the $\mathbf{ABD}$ matrix on the basis of the classical laminate theory when the Calculation module is called up.

2 - Section loads, distortions and hygrothermal loads

Membrane internal forces and moments as well as strains and curvatures can be specified as loads on the defined composite. For each coordinate direction, either the load or the distortion can be specified in the input fields under 2a and 2b. In addition, a temperature difference and the percentage change in relative humidity can be specified in the fields in window section 2c. According to classical laminate theory, the following applies to mechanical and hygrothermal loads and distortions

$${\left(\begin{array}{c} \underline{n} \\\ \underline{m} \end{array}\right)}_{mech} = \left[\begin{array}{cc} \mathbf{A} & \mathbf{B} \\\ \mathbf{B} & \mathbf{D} \end{array}\right] \left(\begin{array} \ \underline{\varepsilon} \\\ \underline{\kappa} \end{array}\right) - {\left(\begin{array}{c} \underline{n} \\\ \underline{m} \end{array}\right)}_{hygrotherm}$$

If no effects of hygrothermal loads are to be considered, the fields for the temperature difference and the percentage humidity difference must be filled with the values zero. This is the default setting in the eLamX² calculation module.

3 - Hygrothermal section loads

At this point, the hygrothermal section loads resulting from the specified temperature difference and the percentage change in relative humidity on the total laminate are output. They are not input data. The selection of the checkboxes for mechanical loads and the resulting distortions has no influence on the hygrothermal cutting loads. They result from the direction-dependent thermal conductivity and swelling coefficients of the individual layers as well as their stiffness matrices and the specified temperature and humidity difference.

4 - Stress and strain distribution button

This button can be used to display the stress and strain distribution within the laminate in local and global coordinates for the coordinate axes of the laminate plane. The output is only qualitative. The specified coordinate directions refer to the fiber angles, whereby the specified x-value reflects the size in the 0° direction and the y-value the size in the 90° direction of the laminate. The fiber orientations of the individual layers are represented by the hatching shown. Horizontal lines correspond to zero degrees and vertical lines to 90° fiber angles. If the grain angles of adjacent layers are very close to each other, the hatching is displayed in different colors.

5 - Delete button

Pressing the button deletes the specified loads and the calculated unknown internal forces and distortions, as well as the calculated stress distribution.

6 - Display of strains and curvatures using a square plate

This button can be used to open a window in which the calculated strains and curvatures are visualized on a square plate. This provides a better understanding of the coupling effects within the $\mathbf{ABD}$ matrix.

7 - Layer values in the local fiber coordinate system

After a calculation, the stresses or strains of each layer of the composite are given in this part of the window, depending on the selection. The evaluation is carried out on the top and bottom of the individual layer. The stresses in each layer are compared with their strengths and a reserve factor is calculated at the top and bottom of the layer using the selected failure criterion. In addition, the type of failure to be expected with regard to the selected failure criterion of each layer is specified.

Reserve factor formatting

In practical application, it might be helpful to round the reserve factor down instead of mathematical rounding. Therefore, a round down option for reserve factors can be activated in the formatting options of eLamX² (toolbar: Tools - Options, tab: Format) as shown below.

It shall be noted that eLamX² has to be restarted in order to fully apply formatting changes.

1 - Format specifier

As for other values, an individual format specifier can be defined for the reserve factors.

2 - Round down option

If the round down option is activated via the respective checkbox, the reserve factor is rounded towards zero on the last decimal. If this option is deactived, default mathematical rounding is performed.

Laminate information/engineering constants

This window can be opened by right-clicking on a laminate and then selecting Engineering constants.

1 - ABD matrix

Based on the stiffness matrices of the individual layers, the membrane stiffness matrix $\mathbf{A}$, coupling stiffness matrix $\mathbf{B}$ and bending stiffness matrix $\mathbf{D}$ of the composite are calculated automatically when the Calculation module is called up and displayed separately in color within the $\mathbf{ABD}$ matrix. This matrix is displayed in the form of a table so that all values can be copied out.

2 - Compliance matrix of the composite

The inverse of the stiffness matrix of the laminate is displayed here. It may be that for symmetrical laminates, terms of the inverse coupling stiffness matrix ($\mathbf{b}$) are different from zero, but very small. This results from numerical inaccuracies in the calculation with double precision within the program. This matrix is shown in the form of a table so that all values can be copied out.

3 - Engineering constants of the multilayer composite

The engineering constants of the multilayer composite are also calculated automatically according to [2] for membrane and bending loads. This is done both with and without consideration of transverse contraction restraint (QKB). In the case of asymmetrical laminates, where there is a coupling between membrane and bending deformation, the engineering constants determined in this way are of little use and should be used with caution.

4 - Expansion coefficients of the multilayer composite for heat and moisture

In this part of the window, the thermal expansion coefficients $\alpha^T_i$ and the swelling coefficients $\beta_i$ of the overall composite are output according to the unit system selected for specifying the values of the individual layer. The definition according to [2] also applies here and thus

$${\left( \begin{array}{c} \epsilon_x \\\ \epsilon_y \\\ \gamma_{xy} \end{array} \right)}_{T+H} = \Delta T \left( \begin{array}{c} \alpha^T_x \\\ \alpha^T_y \\\ \alpha^T_{xy} \end{array} \right) + \Delta c \left( \begin{array}{c} \beta_x \\\ \beta_y \\\ \beta_{xy} \end{array} \right)$$

5 - Non-dimensional parameters

This part of the window displays the non-dimensional parameters based on the $\mathbf{D}$ matrix, which are Seydel's orthotropy parameter

$$\beta_{D} = \frac{D_{12} + 2 \cdot D_{66}}{\sqrt{D_{11} \cdot D_{22}}} \ ,$$

the transverse contraction parameter

$$\nu_{D} = \frac{D_{12}}{\sqrt{D_{11} \cdot D_{22}}} \ ,$$

and the anisotropy parameters

$$\gamma_{D} = \frac{D_{16}}{\sqrt[4]{D_{11}^{3} \cdot D_{22}}} \ ,$$ $$\delta_{D} = \frac{D_{26}}{\sqrt[4]{D_{11} \cdot D_{22}^{3}}} \ .$$

Notes on the engineering constants

The engineering constants can be determined for two types of stress. Firstly for a membrane stress and secondly under a bending stress. For both, this is possible with and without transverse contraction restraint.

The procedure is explained on the basis of the explanations in [2] for symmetrical laminates. For the membrane state, only the $\mathbf{A}$ matrix is used form the $\mathbf{ABD}$ matrix

$$\left( \begin{array}{c} n \\\ m \end{array} \right) = \left[ \begin{array}{cc} \ \mathbf{A} & \mathbf{B} \\\ \mathbf{B} & \mathbf{D} \end{array} \right] \left( \begin{array}{c} \epsilon \\\ \kappa \end{array} \right)$$

which is permissible for symmetrical laminates ($\mathbf{B} = 0$). This results in

$$\begin{equation} \left( \begin{array}{c} n_x \\\ n_y \\\ n_{xy} \end{array} \right) = \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{16} \\\ A_{21} & A_{22} & A_{26} \\\ A_{61} & A_{62} & A_{66} \end{array} \right] \left( \begin{array}{c} \epsilon_{x} \\\ \epsilon_{y} \\\ \gamma_{xy} \end{array} \right) \end{equation}$$

The aim is now to arrive at an equation for the stress $\sigma_{x}$ that corresponds to the uniaxial elasticity law $\sigma = E \epsilon$. To do this, the equation is divided by the total thickness of the laminate. This corresponds to a homogenization of the material and you get

$$\begin{equation} \left( \begin{array}{c} \sigma_x \\\ \sigma_y \\\ \tau_{xy} \end{array} \right) = \frac{1}{t_{ges}} \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{16} \\\ A_{21} & A_{22} & A_{26} \\\ A_{61} & A_{62} & A_{66} \end{array} \right] \left( \begin{array}{c} \epsilon_{x} \\\ \epsilon_{y} \\\ \gamma_{xy} \end{array} \right) \ . \end{equation}$$

This equation corresponds to a single anisotropic layer over the entire thickness. There are two ways to arrive at an equation corresponding to the uniaxial law of elasticity. Firstly, $\epsilon_{y}$ and $\gamma_{xy}$ are set to zero, which corresponds to a transverse contraction restraint, and the following equations are obtained:

$$\left( \begin{array}{c} \sigma_x \\\ \sigma_y \\\ \tau_{xy} \end{array} \right) = \frac{1}{t_{ges}} \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{16} \\\ A_{21} & A_{22} & A_{26} \\\ A_{61} & A_{62} & A_{66} \end{array} \right] \left( \begin{array}{c} \epsilon_{x} \\\ 0 \\\ 0 \end{array} \right) \ .$$ $$\sigma_x = \frac{A_{11}}{t_{ges}}\epsilon_{x}$$

To obtain a modulus of elasticity without transverse contraction hindrance, the equation must be transformed and $\sigma_{y}$ and $\tau_{xy}$ are set to zero.

$$\left( \begin{array}{c} \epsilon_{x} \\\ \epsilon_{y} \\\ \gamma_{xy} \end{array} \right) = t_{ges} \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{16} \\\ A_{21} & A_{22} & A_{26} \\\ A_{61} & A_{62} & A_{66} \end{array} \right]^{-1} \left( \begin{array}{c} \sigma_x \\\ 0 \\\ 0 \end{array} \right)$$ $$\sigma_x = \frac{1}{(A^{-1})_{11} \cdot t_{ges}}\epsilon_{x}$$

All engineering constants can now be determined in this way. The constants without transverse contraction obstruction result in:

$$\begin{array} \ E_x & = & \frac{1}{(A^{-1})_{11} \cdot t_{ges}} \\\ E_y & = & \frac{1}{(A^{-1})_{22} \cdot t_{ges}} \\\ G_{xy} & = & \frac{1}{(A^{-1})_{66} \cdot t_{ges}} \\\ \nu_{xy} & = & -\frac{(A^{-1})_{12}}{(A^{-1})_{11}} \\\ \nu_{yx} & = & -\frac{(A^{-1})_{12}}{(A^{-1})_{22}} \end{array}$$

The constants with transverse contraction obstruction result in:

$$\begin{array} \ E_x & = & \frac{A_{11}}{t_{ges}} \\\ E_y & = & \frac{A_{22}}{t_{ges}} \\\ G_{xy} & = & \frac{A_{66}}{t_{ges}} \\\ \end{array}$$

If a transverse contraction hindrance is taken into account, transverse contraction numbers are not useful and are therefore not specified.

These homogenized material parameters are only permissible for membrane loads. An equivalent procedure using the $\mathbf{D}$-matrix is necessary for bending, as the layer position has a major influence on bending stresses, in contrast to membrane stresses. The derivation of the calculation rule is taken from [2].

The analogy to the beam is used for bending. Since the laminate panel has a rectangular cross-section, it is compared with the bending elasticity law of a rectangular beam and the following applies:

$$M = -EI \cdot w'' = - E \frac{b t^3}{12} \cdot w'' \rightarrow m = -E \frac{t^3}{12} \cdot w'' \ .$$

In addition, $w'' = -\kappa$ applies.

For the laminate panel, bending stresses are described using the $\mathbf{D}$ matrix.

$$\left( \begin{array}{c} m_x \\\ m_y \\\ m_{xy} \end{array} \right) = \left[ \begin{array}{ccc} D_{11} & D_{12} & D_{16} \\\ D_{21} & D_{22} & D_{26} \\\ D_{61} & D_{62} & D_{66} \end{array} \right] \left( \begin{array}{c} \kappa_{x} \\\ \kappa_{y} \\\ \kappa_{xy} \end{array} \right)$$

Now the procedure is equivalent to membrane stress. Assuming a transverse contraction constraint (in this case a transverse curvature constraint), the following results:

$$\left( \begin{array}{c} m_x \\\ m_y \\\ m_{xy} \end{array} \right) = \left[ \begin{array}{ccc} D_{11} & D_{12} & D_{16} \\\ D_{21} & D_{22} & D_{26} \\\ D_{61} & D_{62} & D_{66} \end{array} \right] \left( \begin{array}{c} \kappa_{x} \\\ 0 \\\ 0 \end{array} \right)$$ $$m_x = D_{11} \kappa_x$$ $$E_{x,b} = \frac{12}{t_{ges}^3}D_{11}$$

Thus all engineering constants with transverse contraction constraint result in:

$$\begin{eqnarray} E_{x,b} & = & \frac{12}{t_{ges}^3}D_{11} \\\ E_{y,b} & = & \frac{12}{t_{ges}^3}D_{22} \\\ G_{x,b} & = & \frac{12}{t_{ges}^3}D_{66} \\\ \end{eqnarray}$$

and without transverse contraction obstruction:

$$\begin{eqnarray} E_{x,b} & = & \frac{12}{(D^{-1})_{11} \cdot t_{ges}^3} \\\ E_{y,b} & = & \frac{12}{(D^{-1})_{22} \cdot t_{ges}^3} \\\ G_{x,b} & = & \frac{12}{(D^{-1})_{66} \cdot t_{ges}^3} \\\ \end{eqnarray}$$

Here too, transverse contraction coefficients are of little use, as these correspond more to transverse curvature coefficients.

This procedure generally only applies to symmetrical laminates. In the case of asymmetrical laminates, the inverse of the entire $\mathbf{ABD}$ matrix must be formed and then the corresponding terms used.

[1] J.N. Reddy. Mechanics of Laminated Composite Plates and Shells- Theory and Analysis. CRC Press, Boca Raton, 2. auflage edition, 2003.

[2] H. Schürmann. Konstruieren mit Faser-Kunststoff-Verbunden. Springer, 2003.

[3] M. Barth. Umsetzung eines Moduls zur Betrachtung von Temperatur und Feuchtigkeit in Faser-Kunststoff-Verbunden für das Programm eLamX. ILR –LFT G 08-27, Technische Universität Dresden, April 2009.