Implementation
The implementation is based on the principle of virtual work, which states that the sum of virtual work from internal strains and external loads equals zero:
The beam elements are formulated in terms of the stress resultants (normal force, bending moments and twisting moment).
The normal force is defined as
Because the local coordinates are defined with their origin at the centroid of the cross section, any surface integral of an odd power of a local coordinate evaluates to zero.
The beam bending moments are defined as
Myl is present only in 3D, and so is the torsional moment Mxl described below. The torsional stiffness of the beam is defined using the torsional constant J given by
In a similar way as for the bending part a torsional moment is then defined as
Using the beam moment and normal force the expression for the virtual work becomes very compact:
For 2D, the first and fourth terms are omitted. For the case of Timoshenko beam, there is also a shear stress contribution added,
where the second term is present only in 3D.
A special feature of some unsymmetrical cross sections is that they twist under a transversal load that is applied to beam centerline. As an example, this would be the case for a U-profile under self-weight, loaded in the stiff direction. It is only a load applied at the shear center which causes a pure deflection without twist. This effect can be incorporated by supplying the coordinates of the shear center in the local coordinate system (ey, ez). A given transversal load (fxl, fyl, fzl), which is defined as acting along the centerline, is then augmented by a twisting moment given by