________________________________________________________________________ PROBLEM (12.45) For the beam loaded as shown in Fig. P12.45, using Castigliano’s theorem, determine: (a) The vertical deflection vB of the free end. (b) The slope θB of the free end. SOLUTION x C L M = − Px − C ∂M ∂P = − x (a) In this case C=0. L M ∂M 1 vB = ∫ dx = 0 EI ∂P EI ∫ L ∫ L 0 P ∂M ∂C = −1 (− Px)(− x)dx = PL3 ↓ 3EI (b) For θ B C=0: θB = ∫ L 0 M ∂M 1 dx = EI ∂C EI 0 PL2 (− Px)(−1)dx = 2 EI Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.