Analysis of Statically Determinate Structures 3

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 . 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 a). Using equilibrium equations, we can find the

internal forces:

Similarly, we can find out the internal forces in the portions x = 6 m → 10m (Figure b) and x = 10 m

→ 12 m (Figure c). For x = 6 m → 10 m :

and for x = 10 m → 12 m :

If we look at these expressions carefully, we see that:

•We measure x always from the same origin and in the same direction. As noted earlier, it is not absolutely

necessary to follow this convention, but it is easier this way.

•The internal force expressions change at points where concentrated forces/moments (including support reactions) act. We will see later that these forces also change if a distributed force changes its distribution.

Using singularity functions , we can combine different expressions for different segments of the beam

together into a single expression, which we will discuss later.

•We need to obtain mathematical expressions of internal forces first in order to plot the force variation

diagrams. Although these expressions provide adequate information on variation of internal forces, a pictorial

representation is always very useful.