________________________________________________________________________ PROBLEMS (*12.65 and *12.66) A load P is supported at joint B of a structure consisting of three bars of equal axial rigidity AE (Fig. P12.65 and P12.66). Use Castigliano's theorem to determine the force in each bar. SOLUTION (*12.65) Consider RD as redundant. C D 4 A ∑F x ∂FBD =1 ∂RD FBD =RD FBC 3 3 ft 4.5 ft B P 4 ∑F 3 FAB y B = 0: FBC = 1.25 P − 1.25RD P = 0 : FAB = −0.75P + 0.75 RD where ∂FBC ∂RD = −1.25 Thus, ∂FAB ∂RD = 0.75 ∂Fi =0 ∂RD = (1.25 P − 1.25 RD )(7.5)(−1.25) + (−0.75 P + 0.75 RD )(4.5)(0.75) + RD (3)1 = 0 = −14.25P + 17.25 RD = 0 AEδ D = ∑ Fi Li or RD = 0.826 P Then FBD = 0.826 P FAB = −0.131P FBC = 0.22 P Continued on next slide 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. SOLUTION (*12.66) RD =FAB=R D C 0.5L 0.6L 0.8L vD=0 R FBC B 3 P 4 5 4 3 0.5L B P A Figure (a) FAB Figure (b) Structure is statically indeterminate to the 1st degree. We have δ D = ∂U ∂R = 0. That is δD = ∂F ∂F ∂F 1 [ FAB LAB AB + FBC LBC BC + FBD LBD BD ] = 0 ∂R ∂R ∂R AE Joint B. Equilibrium results in: FAB = 0.8R − 0.8P FBC = 0.6 P − 0.6 R FBD = R (1) (2) Equation (1) is thus δ D = (0.8R − 0.8P )(0.8L)(0.8) + (0.6 P − 0.6 R)(0.6 L)(−0.6) + R(0.5L)(1) = 0 Solving R = 0.593P = FBD Equations (2) give then FAB = −0.32 P FBC = 0.244 P 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.