Calculation Module¶

Generall¶

The calculation module is used to calculate loads or strains, as well as stresses within the specified laminate according to the classical laminated plate 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 cutting loads or strains and the stresses of the layers, the calculations are updated automatically. Among other things, hygrothermal effects can also be taken into account! Cite {Barth2009}.

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

construction¶

p =. {{thumbnail (calculation.png, size = 500, title = calculation window)}}

1 - ABP 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 color-coded when the calculation module is called up on the basis of classical laminate theory shown separately within the ABP matrix.

2 - cutting loads, distortions and hygrothermal loads¶

Both membrane sectional loads and moments as well as elongations and curvatures can be specified as loads on the defined composite. Either the load or the distortion can be specified in the input fields under 2a and 2b for each coordinate direction. In addition, a temperature difference and the percentage change in the relative humidity can be specified in the fields in window section 2c. According to the classical laminate theory applies to mechanical and hygrothermal loads and distortions

$$ \ begin {pmatrix} \ underline {n} \\ \ underline {m} \ end {pmatrix} _ {mech} = \ begin {bmatrix} \ mathbf {A} & \ mathbf {B} \\ \ mathbf { B} & \ mathbf {D} \ end {bmatrix} \ begin {pmatrix} \ underline {\ varepsilon} \\ \ underline {\ kappa} \ end {pmatrix} - \ begin {pmatrix} \ underline {n} \\ \ underline {m} \ end {pmatrix} _ {hygrotherm} $$

If no effects of the 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 view is the standard setting in the calculation module of eLamX².

3 - hygrothermal cutting loads¶

At this point, the resulting hygrothermal cutting loads are output based on the specified temperature difference and the percentage difference in humidity on the entire laminate. They are not input data. The selection of the check boxes 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 relate to the fiber angle, the specified x-value reflecting the size in the 0 ° direction and the y-value reflecting the size in the 90 ° direction of the laminate. The fiber orientations of the individual layers are shown by the hatching shown. Horizontal lines correspond to zero degrees and vertical lines to 90 degrees fiber angles. If the fiber angles of adjacent layers are very close together, the hatching is output in different colors.

p =. {{thumbnail (tension distribution.png, size = 500, title = call up 3D failure body of the single layer)}}

5 - Delete button¶

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

6 - Representation of the expansions and curvatures on the basis of 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 is used for a better understanding of the coupling effects within the ABD matrix.

p =. {{thumbnail (darstellung_eps_kappa.png, size = 500, title = call up 3D failure body of the single layer)}}

7 - Layer sizes in the local fiber coordinate system¶

After a calculation, the stresses or strains of each layer of the composite are calculated in this part of the window, depending on the selection. The ID