## Free and Fixed Bending Moments

When a beam is free to rotate at both ends as shown in Figures 4.70(a) and (b) such that no bending moment can develop at the supports, then the bending moment diagram resulting from the applied loads on the beam is known as the Free Bending Moment

Diagram. Figure 4.70—Free Bending Moment Diagrams

When a beam is fixed at the ends (encastre) such that it cannot rotate, i.e. zero slope at the supports, as shown in Figure 4.71, then bending moments are induced at the supports and are called Fixed-End Moments. The bending moment diagram associated only with the fixed-end moments is called the Fixed Bending Moment Diagram.

Figure 4.71—Fixed Bending Moment Diagram

Figure 4.71—Fixed Bending Moment Diagram

Using the principle of superposition, this beam can be considered in two parts in order to evaluate the support reactions and the Final bending moment diagram:

(i) The fixed-reactions (moments and forces) at the supports

Figure 4.72

(ii) The free reactions at the supports and the bending moments throughout the length due to the applied load, assuming the supports to be pinned Figure 4.73

Combining (i)+(ii) gives the final bending moment diagram as shown in Figure 4.74:

Va~(rkid + VAûw)i VB ~ C^Bfixed + P'B ÛW) Ma-(Ma+0); Md-(Mb + 0) Figure 4.74

The values of MA and MB for the most commonly applied load cases are given in Appendix 2. These are standard Fixed-End Moments relating to single-span encastre beams and are used extensively in structural analysis.