A fundamental equation for the bending of beams is:
The second moments of area of the beam sections most commonly met with are (about the central axis XX):
The neutral axis of any section, where bending produces no strain and therefore no stress, always passes through the centroid of the section. For the symmetrical sections listed above this means that for vertical loading the neutral axis is the horizontal axis of symmetry.
For example, let the maximum bending moment on a beam be 120 Nm. If the beam section is rectangular 18 mm wide and 36 mm deep, the maximum
bending stress is calculated as follows:
Second moment of area of section about the neutral axis,
Maximum distance from neutral axis
In another example, a cantilever is of tubular construction with internal and external diameters of 100 mm and 120 mm. If the length of the cantilever is 800 mm, the maximum load which it can carry at its free end if the maximum stress is not to exceed 50 MPa (assuming the weight of the beam is ignored) is determined as follows:
The second moment of area of the section is
If W is the load (in kN) at the free end of the cantilever, the bending moment at a point distance x from the free end is Wx with a maximum value where the cantilever is built into the wall, given by:
W kN x 0.8 m = 0.8 W kNm
Equating this to the calculated maximum permissible bending moment gives:
