Strain Energy Density

The strain energy density for a beam is a measure of how hard the beam is working. Strain energy density may be output for Beam, Bar, Tie and Strut element types.

The definition of strain energy density is:

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

For a linear material this is equivalent to:

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

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

$$\begin{aligned} &S E_{\theta}=\frac{F^{2}}{2 E A} \\ &S E_{b}=\frac{M^{2}}{2 E I} \end{aligned}$$

Values of strain energy density may be output 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.

The average strain energy density is the average density along the element or member.