EXTERNAL WORK

The external work  performed by a system of concentrated node, or joint, loads  is:

Within each element " i ", the external work  performed by the body forces because of gravitational loads is:

Application of the displacement assumptions given by Equation (1),

                                (1)

N The shape functions used to calculate the strain energy within the element

Integration over the volume of the element, and summation over all elements produces the following equation for the energy because of body forces:

The external work  performed because of element surface stresses (tractions) , for a typical surface "j" is of the form:

Application of the displacement assumptions given by Equation (1), integration over the surface of the element, and summation over all surface elements produces the following equation for the energy because of surface tractions:

Therefore, the total external work performed on any system of structural elements is:

Reference

Three-Dimensional Static and Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward L. Wilson