Another alternative of studying internal force variations in a structural member is to express the internal force as a mathematical function of the longitudinal dimension ( x ). Thus, the axial force, shear force and bending moment at a section are expressed as P ( x ), V ( x ) and M ( x ), respectively, where x is the distance measured along the primary dimension from one end of the member (Figure 2.13). For this course, we will consider the left end of the member as origin unless otherwise specified. Note that equations involving these internal forces change if the direction for positive x or its origin changes.

Considering the example of Figure 2.7 again, let us obtain these internal force functions for the whole length. After obtaining the support reactions, we can investigate internal forces at different sections. Let us first consider the portion x = 0 → 6 m . Since no force or moment is acting between these two points, the internal force functions will be continuous in this section. We draw the free body diagram of the beam upto a distance x from the left end of the beam (Figure 2.14a). Using equilibrium equations, we can find the internal forces: