All points in the length of the beam, where the shear and bending moment are not equal to zero, and at locations other than the extreme fiber or neutral axis are subject to both shearing stresses and bending stresses. The combination of these stresses is of such a nature that maximum normal and shearing stresses at a point in a beam exist on planes that are inclined with respect to the axis of the beam. It can be shown that maximum and minimum normal stresses exist on two perpendicular planes. These planes are commonly called the principal planes, and the stresses that act on them are called principal stresses. The principal stresses in a beam subjected to shear and bending may be calculated using the following formula:
where fpr is the principal stress and f and v are the bending and shear stresses, respectively, calculated from Equation (4-1).
The orientation of the principal planes may be calculated using the following formula:
taα=2v/f (4-3)
where α is the angle measured from the horizontal.