Euler-Bernoulli and Timoshenko beams
In Ashes, beams can be modelled using either Euler-Bernoulli or Timoshenko beam theory. This can be set in the Analysis parameters window, under the Analysis tab with the Default element type parameter.
Euler-Bernoulli is widely used to compute the behaviour of slender beams under loading. It assumes that the plane sections of the beam remain plane and perpendicular to the neutral axis after deformation. This implies that no shear deformation occurs across the beam's cross section. This is a good approximation for long slender beams but ca lead to inaccuracies for short deep beams.
For such beams, Timoshenko beam theory might be more accurate as it accounts for bending and shear deformation. This makes it more suitable for shorter beams. This requires the introduction of the shear area, which is a modified cross-sectional area to account for the distribution of shear stresses across the beam's cross-section. In classical beam theory, it is assumed that shear stresses are uniformly distributed, but in reality, shear stresses vary across the section, with higher values near the neutral axis and lower values near the edges. The shear area takes this variation into account by reducing the total cross-sectional area.
Finding accurate values for the shear area can be difficult. It is common to consider the geometrical area
$$A$$
of the cross section and apply a Shear correction factor to obtain the shear area $$A_S$$
. The shear correction factor is defined as $$K=A_S/A$$
In Ashes, based on the work by Bell (1987b), the following shear correction factors are used:
- for Circular hollow cross sections, $$K = 0.5$$
- for Circular solid cross sections, $$K=0.9$$
- for Rectangular solid cross sections, $$K = 5/6$$
For Circular shape and Rectangular shape cross sections (i.s. cross sections that are not defined by their geometry but by their structural properties), the shear area can be entered by the user.