Beam finite elements of arbitrary cross sections are important members of finite element models for a variety of practical applications. In the effective use of such elements, at the modelling stage, the user is faced with the time consuming problem of accurate determination of beam cross sectional properties.

In the analysis of beams, the following local coordinate systems and notations are used (see Fig. below):

• N node of finite element mesh,
• G centroid,
• S shear centre,
• x, y, z local coordinates, where x is passing through element nodes,
• x', y', z' central axes, parallel to x, y, z (i.e. x' is the beam axis),
• y_G, z_G coordinates of G centroid, relative to N node,Fig. below
• y_S, z_S coordinates of S shear (torsion) centre, relative to G centroid,
• e_xmax, e_xmin, e_ymax, e_ymin, e_1max, e_1min, e_2max, e_2min distances of extreme fibres.

The cross sectional properties are related to internal forces, namely tension, bending, torsion (free or constrained) and shear. The general solution of elastic beam problems can be found in many textbooks, for example in references [3] and [4], only the definitions and final results will be presented.

Tension, bending

The well known formula calculating the normal stress distribution due to tension and bending:

The cross sectional properties appearing here are:

area: