Beam Elements: Thin and Thick Beam Formulation
Description
The Straus7 beam element supports both the thin beam and the thick beam theories.
The thin beam formulation of the element follows the classical beam bending theory. This yields accurate results for deflections of beams with moderate to large length to depth ratios. The assumption made in the formulation is that plane sections remain plane and normal to the tangent of the axial axis (i.e., the principal 3 direction).
For beams that are relatively deep in comparison with their length (e.g., length/depth ratio < 10.0), the assumption that the sections remain normal to the tangent of the axial axis of the beam is no longer valid. Significant deflection results from the transverse shear strains within the beam. This is particularly true of beams with thin webs such as I-beams of very short span. Beams of this type should be modelled using the thick beam formulation. The thick beam formulation is based on the Timoshenko beam theory.
Thin beam theory accounts for transverse deformation by cubic displacements only (i.e., transverse shear deformation is neglected). So for a thin beam the rotation of the beam is
.
In the thick beam theory, the section rotation is included in the formulation as an independent variable and the rotation of the beam is independent from the transverse deflection. (In a thin beam, the rotation and transverse deflection are coupled). The shear strain, 
, is the difference between the slope of the beam 
and the section rotation 
, thus
.
Expressions for the thick beam bending moment, 
, and shear force, 
, are:
In the thick beam formulation, the shear strain is assumed to be constant through the depth of the beam. For most beams, this is not the case; for example, rectangular solid sections have a parabolic variation of transverse shear strain between zero on the extreme fibres and maximum on the neutral axis. So to account for the total shear deformation in a beam, an effective shear area is calculated, over which a constant shear strain produces the correct total shear deflection as the real section. This shear area is usually given as a factor on the cross section area. This factor is the 
in the above equation. As an example, the value of 
for a rectangular section is 5/6. The resulting area, 
, is called the shear area and is the input required by Straus7.
Shear area may be defined in each of the principal axis directions of the beam section. The shear areas are denoted by Shear A1 and Shear A2 respectively.
For beams with thin webs such as I-beams, the shear strain will be essentially constant in the web and near zero in the flanges. In such cases, the shear area is approximately equal to the cross section area of the web carrying the shear.
Shear areas can be automatically calculated by Straus7 for any section, including standard sections, sections from the beam libraries, and used-defined section shapes constructed via Utility: Make BXS (see Special Topics: BXS Generator).
How to select thin beam or thick beam
To use the thin beam formulation, enter zero values for both Shear A1 and Shear A2 in Common Properties: Sections.
To use the thick beam formulation, enter non-zero values for either Shear A1 or Shear A2 or both in Common Properties: Sections. The shear areas are automatically calculated by setting the Assign Shear Areas option in Properties: Beam Cross Section Selection.
See Also