Stress resultants

These are the stress resultants (also called membrane forces, shear forces, and bending moment) that may be used to determine the detailed stress state in the structural element.

If the element has been extracted from a beam, the width and thickness are comparable in size.

More explicitly, for a beam of height t and width b, Similarly the shear force resultants are The bending moment vector due to stresses in the cross-section A perpendicular to the x1-axis is given by Expanding this expression we have, We can write the bending moment resultant components as For plates and shells, the x1 and x2 dimensions are much larger than the size in the x3 direction.

Integration over the area of cross-section would have to include one of the larger dimensions and would lead to a model that is too simple for practical calculations.

Following the same procedure as for beams, and keeping in mind that the resultants are now per unit length, we have We can write the above as where the membrane forces are defined as and the shear forces are defined as For the bending moment resultants, we have where r = x3 e3.