________________________________________________________________________ PROBLEM (12.76) Using the unit-load method, find the support reactions RA and MA for the beam of Fig. P12.76. SOLUTION Let RA and M A be redundant. P a MA b C L MB B RA RB P 1 lb B δP C x Mm dx EI θA = ∫ Mm ' dx EI b x MA B 1 lb ⋅ in. B A RA M = RA x − M A 1 lb x m=x m' =1 ( RA x − M A ) x 1 RA L3 M A L2 dx = ( ) − 0 EI EI 3 2 L ( R x − M )1 1 RA L2 A A =∫ dx = ( − M A L) 0 EI EI 2 v AR = ∫ θ AR m ' = −1 Pb 2 a b v AP ∫0 (− Px)[−(a + x)]dx = EI ( 2 + 3 ) 1 b Pb 2 ( Px )( 1) dx = − − = EI ∫0 2 EI Let now RA and M A are applied. 1 = EI θ AP x m = −1(a + x) M = − Px vA = ∫ B C 1 lb ⋅ in. L 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. Since v AP = v AR θ AP = θ AR or 1 RA L3 M A L2 Pb 2 a b ( + )= ( − ) 2 EI 2 3 EI 3 1 RA L2 Pb 2 ( = − M A L) 2 EI EI 3 Solving, Pb 2 Pab 2 RA = 3 (3a + b) ↑ MA = 2 L L 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.