Stress Evaluation
Since the basic result quantities for beams are the integrated stresses in terms of section forces and moment, special considerations are needed for the evaluation of actual stresses.
The normal stress from axial force is constant over the section, and computed as
The normal stress from bending is computed in four user-selected points (ylk, zlk) in the cross section as
In 2D, only two points, specified by their local y-coordinates are used.
The total normal stress in these points is then
The peak normal stress in the section is defined as
When using the built in common cross sections, a special method is used for the Circular and Pipe sections. Since there are no extreme positions around a circle, a maximum bending stress is computed as
where do is the outer diameter. This stress then replaces the stress from the stress evaluation points in maximum stress expressions. This ensures that the correct peak stress is evaluated irrespective of where it appears along the circumference.
The shear stress from twist in general has a complex distribution over the cross section. The maximum shear stress due to torsion is defined as
where Wt is the torsional section modulus. This result is available only in 3D.
The section shear forces are computed in two different ways depending on the beam formulation. For Euler-Bernoulli theory, the section forces proportional to the third derivative of displacement, or equivalently, the second derivative of the rotation.
where Tlz is available only in 3D. In the case of Timoshenko theory shear force is computed directly from the shear strain.
The average shear stresses are computed from the shear forces as
(8-1)
Since the shear stresses are not constant over the cross section, the maximum shear stresses are also available, using section dependent correction factors:
(8-2)
As the directions and positions of maximum shear stresses from shear and twist are not known in a general case, upper bounds to the shear stress components are defined as
The maximum von Mises equivalent stress for the cross section is then defined as
Since the maximum values for the different stress components in general occur at different positions over the cross section, the equivalent stress thus computed is a conservative approximation.