From Structural and Stress Analysis, Second Edition

16.4 STATICALLY INDETERMINATE BEAMS

Beams are statically indeterminate generally because of their support systems. In this category are propped cantilevers, fixed beams and continuous beams. A propped can tilever and some fixed beams were analysed in Section 13.6 using either the principle of superposition or moment-area methods. We shall now apply the methods described in Chapter 15 to some examples of statically indeterminate beams.

EXAMPLE 16.5

Calculate the support reaction at B in the propped cantilever shown in Fig. 16.14.


Figure 16.14: Propped cantilever of Ex. 16.5

In this example it is unnecessary to employ the procedures described in Section 16.2 to calculate the degree of statical indeterminacy since this is obvious by inspection. Thus the removal of the vertical support at B would result in a statically determinate cantilever beam so that we deduce that the degree of statical indeterminacy is 1. Furthermore, it is immaterial whether we use the principle of virtual work or complementary energy in the solution since, for linearly elastic systems, they result in the same equations (see Chapter 15). First, we shall adopt the complementary energy approach.

The total complementary energy, C, of the beam is given, from Eq. (i) of Ex. 15.8, by


in which ? B is the vertical displacement of the cantilever at B (in this case ? B = 0 since the beam is supported at B).

From the principle of the stationary value of the total complementary energy we have


which, by comparison with Eq. (iii)...

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