Resultant Force and Moment and Stress Relations

An imaginary cut (or section) through a load bearing member can expose the internal forces. The complex distribution of forces acting over this cut surface is equipollent to a force and a moment.

We may resolve this resultant force and moment into components along the coordinate axes to study their effects. If we assume the cut is made perpendicular to the x-axis (the member’s longitudinal axis), we get six distinct internal resultants, each corresponding to a specific mode of loading:

Components of the force are: 1. Axial force (P): This force acts perpendicular to the section. It tries to either stretch or compress it. 2. Shear forces and : These forces act parallel to the cut surface, and cause one portion of the member to slide relative to the adjacent portion.

Components of the moment are: 1. Torque or twisting moment (T) which tries to twist the member or rotate it along the x-axis 2. Bending moments and . These moment try to bend.

These six resultants aren’t arbitrary; they are the direct mathematical sum (the integral) of the stresses acting over the entire area of the cut.

If we consider a small element on the section, then

The sum of these forces give: Similarly, the moments generated by the stress distribution are obtained by integrating the moment of each elemental force about the section’s centroidal axes.