Bending Stresses in Beams: The bending moment, M, along the length of the beam can be determined from the moment diagram.
Bending Stresses in Beams
The bending moment at any location along the beam can then be used to calculate the bending stress over the beam’s cross section at that location. The bending moment varies over the height of the cross section according to the flexure formula below:
where M is the bending moment at the location of interest along the beam’s length, Ic is the centroidal moment of inertia of the beam’s cross section, and y is the distance from the beam’s neutral axis to the point of interest along the height of the cross section. The negative sign indicates that a positive moment will result in a compressive stress above the neutral axis.
The bending stress is zero at the beam’s neutral axis, which is coincident with the centroid of the beam’s cross section. The bending stress increases linearly away from the neutral axis until the maximum values at the extreme fibers at the top and bottom of the beam.
where c is the centroidal distance of the cross section (the distance from the centroid to the extreme fiber).
If the beam is asymmetric about the neutral axis such that the distances from the neutral axis to the top and to the bottom of the beam are not equal, the maximum stress will occur at the farthest location from the neutral axis. In the figure below, the tensile stress at the top of the beam is larger than the compressive stress at the bottom.