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EXAMPLE 10.8 Spring-Propped Cantilever Beam The built-in beam shown in Fig. 10.10 is supported at one end by a spring of constant k. Determine the redundant reaction. Prove that 5P OR А B D - - - Nr Rp 1+ BEI KL² 2P- P-5P Гу 10 1+ 3E1
Solution The expressions for the moments are M = Rp.X O s.rs L M = -Rox + P* kot + P(x - 1) 1 x SL LP FIGURE 10.10. Example 10.8. A propped can- tilever beam with load P. B NIN 늘 Nr IRO Applying Castigliano's theorem to obtain the deflection at point D. 8p = BulaR, we have 1 1 Ro Rox(x dx) + Rot PX dx = ΕΙ ΕΙ K 12 do Ex / ) + from which R, = P/(1 + 3EI/L). Equilibrium of vertical forces yields 5P RA = P - 16(1 + 3EI/KL) (10.17) Note that were the right end rigidly supported, 8 would be equated to zero.
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