________________________________________________________________________ PROBLEMS (*12.40 and 12.41) Apply the work-energy method to determine the slope at point C of the beam supported and loaded as shown in Fig. P12.40 and Fig. P12.41. SOLUTION (*12.40) M0 A 3EI M0/L C B EI x Bending moment : M = − x’ M0/L M0 x L Part AC. U AC = ∫ M L2 0 x = - M0 M 02 x 2 dx 3(2 EI ) M 02 6 EIL2 ∫ L2 0 x 2 dx = 1 M 02 L 144 EI Part BC. U BC M 02 M 02 dx ' = =∫ L 2 2 EI 2 EIL2 L Total strain energy: U = U AC M 02 7 M 02 L L 3 3 ∫L 2 x ' dx ' = 6EIL2 [ L − ( 2 ) ] = 48 EI 11 M o2 L + U BC = 72 EI L 2 Hence 1 M 0θ B = U , 2 θB = 11 M 0 L 36 EI SOLUTION (12.41) P PL A EI 1.5 EI C P Bending moment : M = − Px ' B Part AC. M x -PL M2 P2 L 2 dx ' = x ' dx ' L 2 3EI 3EI ∫L 2 7 P 2 L3 P2 3 L 3 = [L − ( ) ] = 72 EI 9 EI 2 U AC = ∫ x’ L M2 P2 L 2 2 1 P 2 L3 dx ' = x ' dx ' = 0 2 EI 2 EI ∫0 48 EI 2 3 17 P L Total strain energy: U = U AC + U BC = 144 EI So, 1 17 PL3 PvB = U , vB = 2 72 EI Part BC. U BC = ∫ L2 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.