Thermal loads can be either a constant temperature rise in the whole element or a temperature gradient, varying over the surface element. To evaluate the equivalent nodal forces the temperatures have to be converted to strains, using the temperature coefficient of expansion. The strains are then related to the stress through the material matrix and then the internal stresses are integrated over the element. The thermal effects are the same in all directions so there are no shear strains introduced.

For in-plane effects (constant temperature) the strain is

$\boldsymbol{\varepsilon}_{p} = \begin{bmatrix} \varepsilon_{xx} & \varepsilon_{yy} & \gamma_{xy} \\ \end{bmatrix} = \begin{bmatrix} \alpha T & \alpha T & 0 \\ \end{bmatrix}$

For bending effects (temperature gradients) the strain and stress are

$\boldsymbol{\kappa}_{b} = \begin{bmatrix} \kappa_{xx} & \kappa_{yy} & \kappa_{xy} \\ \end{bmatrix} = \frac{1}{t}\begin{bmatrix} 4\alpha T & 4\alpha T & 0 \\ \end{bmatrix}$

These are converted to stresses through the material matrix

$\boldsymbol{\sigma}_{p} = \mathbf{C}_{p}\boldsymbol{\varepsilon}_{p}\qquad\boldsymbol{\sigma}_{b} = \mathbf{C}_{b}\boldsymbol{\kappa}_{b}$

Note: In GSA, loads can be applied to a list of members or elements. Any loads applied onto members will be automatically expanded into the appropriate elements loads in the solver in order to analyse the model.