Stability module
With the help of the stability module of eLamX² it is possible to model plane plates from given laminates and to determine their eigenvalues and eigenmodes of buckling by means of the “linearized prebuckling” method, as described in the theory section. Due to the approaches used, the calculations only provide correct results for symmetrical laminates. They are based on the principle of virtual displacements using global trial functions by means of a Ritz approach. It is also possible to add stiffening elements to the panel. Based on the laminate defined in the main menu of eLamX², the
Due to the assumed symmetry of the laminate, the calculations in the stability module of eLamX² are based exclusively on the bending stiffness matrix
The model sizes are entered separately for the panel made of the defined laminate and the additionally applied stiffening elements.
First, a flat rectangular panel is modeled from the specified laminate by specifying the length and width. The desired boundary conditions must then be specified. The default setting is a simple support on all sides. In addition, further boundary conditions can be selected by clicking on the drop-down menu, which are indicated by corresponding symbols.
- SS - simple support on both sides
- CC - fixed clamping on both sides
- CF - fixed clamping on one side (negative coordinate end) and free support on one side (positive coordinate end)
- FF - free on both sides
- SC - simple support on one side (negative coordinate end) and fixed clamping on one side (positive coordinate end)
- SF - simple support on one side (negative coordinate end) and free on one side (positive coordinate end)
In principle, any combination of boundary conditions (as well FC, CS and FS) can be calculated with the stability module of eLamX² (eventually the layup has to be rotated and the shear load must be inverted). Furthermore, the membrane section loads on the plate must be specified in the form of
To increase the buckling loads of the plate, stiffening elements can be added. These are modeled as one-dimensional isotropic Bernoulli beams and therefore take bending and torsional stiffness into account. The supporting influence of any stringer flange is not taken into account. According to the model concept used, the centers of gravity of the stringer cross-sections are located in the center or symmetry plane of the plate. It is assumed that local buckling of the stringer segments does not occur before the stability failure of the plate.
The dropbox D matrix options allows to set various options regarding the bending stiffness matrix
With the option Original, the original bending stiffness matrix
To ensure closed-form analytical solutions, orthotropic or quasiorthotropic material behavior is assumed in many calculation methods and literature. In these methods, the influence of the bending-torsion coupling through the terms
With this option, the bending stiffness matrix
This option allows to treat asymmetrical laminates with an approximate solution. For symmetric laminates, the option has no effect compared to the Original option, as
Pressing the Calculate button returns the eigenvalues and eigenmodes of the model.
In the case of an asymmetrical laminate structure, the following warning message is displayed, unless the option for using the
Despite the warning message, the stability module of eLamX² can also be called up for asymmetrical laminates. In this case, the user is responsible for checking the meaningfulness of the results.
The results of the calculations with the stability module of eLamX² are eigenvalues and eigenvectors. First, the critical load vector of the buckling of the plate under consideration is given. The critical buckling loads are the product of the applied loads and the smallest positive eigenvalue of the linearized buckling problem.
If none of the calculated buckling values is positive, the smallest value is the critical eigenvalue. This behavior occurs, for example, when the plate is loaded with pure axial tension.
In addition to the critical buckling loads, the output of all calculated eigenvalues of the stability problem is sorted by amount. The graphical output of the corresponding eigenmode of buckling is controlled via the combo box shown.
It is also possible to specify the scaling of the resulting deflection of the mode shapes in their graphical output. By default, this value is one. This corresponds to the normalization of the deflection to the maximum value occurring in the plate.
In the info section, the effective aspect ratio
with the length
In addition to the numerical output of the eigenvalues, the resulting eigenform for the selected eigenvalue is displayed graphically.
The representation is three-dimensional according to the Cartesian coordinate system shown. Here x and y denote the axes of the plate plane. The z-axis points in the thickness direction of the laminate. The stiffening elements are shown in gray, as it is assumed that the stringer segments will not buckle before the panel fails. The stiffeners are positioned in the graphical output as shown in the model. Their center of gravity is in the middle plane of the plate. The model can be moved, rotated and zoomed to any position by holding down the mouse buttons.
- Rotate
- Zoom
- Move
These buttons can be used to edit the view of the rectangular plate's inherent shapes. The top three buttons can be used to display the three planes of the three-dimensional Cartesian coordinate system. In addition, an isometric display and an adaptation of the display of the bulge shapes to the existing window size are possible.
- View of the x-y plane
- View of the x-z plane
- View of the y-z plane
- Isometric representation
- Adaptation to window size
- Display of the acting forces
- Show the legend or information box
The stiffening wizard is used to define stiffening elements. You must first select the cross-section of the stiffener. The following cross-sections are available for this:
- Arbitrary stiffener cross-section by defining the modulus of elasticity E, the second-order moment of inertia I, shear modulus G and torsional area moment J
- I-cross-section defined by height, width, modulus of elasticity E and shear modulus G
- T-cross-section defined by base height, base width, web height, web width, modulus of elasticity E and shear modulus G
The further modeling of the stiffening elements depends on the respective type.
In general, the direction of the stiffener must be specified. The orientation indicates the axis of the plate to which the stiffening element should run parallel. In addition to the mechanical and geometric characteristics of the stiffening elements, their position must also be specified. This value is absolute and therefore not linked to the size of the plate. They refer to the coordinate system shown in the 3D view. If the dimensions of the plate are changed, the position of the stiffening elements remains constant. Only arrangements of stiffening elements within the plate geometry make sense. If stringers are nevertheless defined outside the plate boundaries, these are initially displayed in the table of stiffening elements and in the 3D view, but are not taken into account during the calculation.
It is possible to subsequently change the position, direction or the entered data within the table display or the properties window in the main window. Right-click on the stiffener to edit, copy or delete it.
If a stiffener without torsional stiffness is to be modeled, this is done by setting G, J or both to zero. Each stiffening element can have different material and geometry parameters.