Integrating the stresses in beams to obtain the section forces

Figure 161: Expansion of a beam element

In beam elements the section forces can be requested at the end nodes. To this end the stresses in the expanded faces at the end nodes are integrated. How this is done can be explained by looking at Figure 161.

The stresses in the expanded element are at first determined in the integration points (e.g. the Gauss-Kronrod points, cf. Figure 76). Then, they are expanded to the nodes of the element. Consequently, the stresses are available at all 20 nodes of the element in Figure 161. In order to obtain the section force the following local coordinate systems are introduced:

The system I-II-III in the faces denotes the positive direction of the section forces. The location of the integration points in the corresponding $ \xi_l,\eta_l$ system is obtained from the local element coordinate system $ \xi-\eta-\zeta$ through the above face-dependent relationships.

In order to get the section forces the stresses are calculated in the integration points of the positive and negative face by interpolation from the stresses at the nodes belonging to the respective face. The integration point scheme depends on the beam section.