Bathe-Wilson Beam

SAP2000 describes its beam elements:

Biaxial bending, torsion, axial deformation, and biaxial shear are all accounted for in the beam-column formulation (Bathe and Wilson, 1976)

The citation appears to refer to Numerical Methods in Finite Element Analysis, Bathe KJ and Wilson EL (1976). However, there doesn’t appear to be any element formulation matching that decription in this book. The only mentions of beam elements is Tables 3.1 and 3.2 that give the standard Euler formulation.

More recent texts by Bathe, e.g. Finite Element Procedures in Engineering Analysis, Bathe KJ (1982) and Finite Element Procedures, Bathe KJ (2016) do contain a beam formulation identical to the Timoshenko Beam, except for including the following alternative rule for determining the shear correction factor, \(k\):

The shear correction factor \(k\) can be evaluated using the condition that the constant shear stress … acting on \(A_{s}\) [the shear area] must yield the same shear strain energy as the actual shearing stress acting on the actual cross-sectional area \(A\) of the beam.

\[ k = \frac{V^{2}}{A \int_{A} \tau^{2}\,dA} \]

Most FEA packages will use this formula for \(k\) and still refer to their elements as Timoshenko beams, even though Timoshenko used \(k\) equal to the ratio of average shear stress to peak shear stress.