## Info

Figure 2.5 Statically determinate and indeterminate beams

Statically determinate structures may also be called 'isostatic', while determinate single span beams may be called 'simply supported' or 'simple beams'.

Structures are classed as indeterminate when their support reactions cannot be calculated by considering only the two equations of equilibrium. For instance, the two-span beam shown in Figure 2.5 (b) has three support reactions, and this requires three equations to solve for the value of the reactions. The third equation may be generated by a variety of means that are the scope of specialist books on structural analysis [1]. Indeterminate structures are also called 'hyperstatic' or 'redundant', while monolithic beams with more than one span are called 'continuous beams'.

In statically determinate structures, the reactions are known absolutely; if one of the supports of the beam shown in Figure 2.5 (a) was to settle, the support reactions would not be affected, and in consequence the bending moments and shear forces in the beam would also not be changed. In indeterminate structures, the support reactions and the bending moments and shear forces in the beam depend on the rigidity of the supports. For instance, if the central support of the two-span beam shown in Figure 2.5 (b) was to settle, some of its load would be shed onto the end supports, and additional bending moments and shear forces would be set up in the beam.

In real life, no support is completely unyielding, and consequently the exact reactions and the bending moments in any indeterminate structure, and in particular in a continuous beam, are subject to a degree of uncertainty. The best one can do is to make assumptions as to the likely settlement of the supports and then calculate the support reactions and the bending moments in the beam that result from those assumptions.

A three-legged stool is determinate, as the load on each leg is independent of the unevenness of the ground. A four-legged table is indeterminate, because the load will always be unequally distributed between its legs, unless they are exactly the same length and the ground on which it rests is perfectly flat, impossible conditions in reality.

Most real structures are indeterminate, unless specific measures are taken to create determinacy, such as introducing hinges into the structure, or using simply supported beams. As a result of this indeterminacy, the external reactions of structures and their internal stresses cannot be known precisely. This is only a problem for those who believe that for a structure to be safe it must comply strictly with the limiting stresses or conditions set by the codes of practice. In reality, structures are safe or unsafe depending on the quality of their designers, and code compliance is a box that has to be ticked to provide some minimum standards of public safety.

In many structures, indeterminacy is a desirable attribute despite the uncertainty it creates, as it reduces the vulnerability of a structure to accidental damage. A discussion on the relative merits of determinacy and indeterminacy specifically for bridge structures may be found in 7.14.