Cross strain energy density

The strain energy density for a beam is a measure of how hard the beam is working. The definition of strain energy density is:

$$S E D=\int \sigma d \varepsilon$$

For a linear material this is equivalent to:

$$S E D=\frac{\sigma^{2}}{2 E}$$

A related quantity is the cross strain energy density. The cross terms are calculated from the forces in one analysis case (i) and the displacements in a different analysis case (j). This can be used to provide information about how to optimise a structure by indicating where stiffening will be most effective.

The definition of cross strain energy density is:

$$c S E D=\int \sigma_{i} d \varepsilon_{j}$$

This quantity is useful only for linear materials where this is equivalent to:

$$c S E D=\frac{\sigma_{i} \sigma_{j}}{2 E}$$

For beam elements it is more convenient to integrate this over the area (whole or partial) of the section and express the result as cross strain energy per unit length. So considering the axial stresses only the extensional (e) and bending (b) cross strain energies per length for the section are:

$$\begin{aligned} S E_{e} &=\frac{F_{i} F_{j}}{2 E A} \\ S E_{b} &=\frac{M_{i} M_{j}}{2 E I} \end{aligned}$$

Values of cross strain energy density may be calculated for the web, the flange or the total value, though web and flange values are only output for I section and rectangular hollow sections with unmodified section properties. In the case of rectangular hollow sections 'web' refers to the sides of the section and 'Flange' refers to the top and bottom of the section.

Note that unlike strain energy density which must always be a positive value the cross strain energy density may be negative.

These results differ from most GSA results in that they are not specific to a single analysis case so they are stored as an Element user module. Use the Tools > Cross strain energy density results command to generate the user modules.