## 46 Application to deflection problems

Generally, deflection problems are most readily solved by the complementary energy approach, although for linearly elastic systems there is no difference between the methods of complementary and potential energy since, as we have seen, complementary and strain energy then become completely interchangeable. We shall illustrate the method by reference to the deflections of frames and beams which may or may not possess linear elasticity.

Let us suppose that we require to find the deflection A2 of the load P2 in the simple pin-jointed framework consisting, say, of k members and supporting loads P\, P2, ■ . ■, P„, as shown in Fig. 4.7. From Eqs (4.14) the total complementary energy of the framework is given by k

It must be pointed out that it is not absolutely necessary to remove the actual loads during the application of P{. The force in each member would then be calculated in terms of the actual loading and P{. Ft follows by substituting Pf = 0 and dFj/dPf is found by differentiation with respect to P{. Obviously the two approaches yield the same expressions for F, and dFjdPf, although the latter is arithmetically clumsier.

### Example 4.2

Calculate the vertical deflection of the point B and the horizontal movement of D in the pin-jointed framework shown in Fig. 4.8(a). All members of the framework are

E F
Fig. 4.8 (a) Actual loading of framework; (b) determination of vertical deflection of B; (c) determination of horizontal deflection of D.

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