________________________________________________________________________ PROBLEMS (10.159 and *10.160) A two-span continuous beam ABC is loaded as shown in Figs. P10.159 and P10.160. Find the reactions RA , RB , and RC for the beam. SOLUTION (10.159) Consider RB as redundant. P -tC/B= tA/B P B A RA 2EI a C a 2a tangent at B RB M EI tC/B RC a A1 A A1 -Pa/2 M C x 4a/3 B A A2 A2 C 1 Pa 1 (2a) = Pa 2 2 2 2 1 A2 = M B (2a) = M B a 2 2 2 EIt A / B = − A1 (a) − A2 ( ⋅ 2a) 3 2 EIt B / C = − A1 (a) − A2 ( ⋅ 2a ) 3 A1 = x -MB We have t A / B = − tC / B : − from which Pa 3 2 Pa 3 4M B a 2 − M B a 2 = −( − ) 4 3 2 3 1 M B = Pa 8 Segment BC: M B = RC (2a) − Pa 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. Thus, Pa = RC (2a ) − Pa , 8 Segment AB: Pa − = − RA (2a ) + Pa , 8 1 Statics: RB = P ↑ 8 − RC = 7 P↑ 16 RA = 9 P↓ 16 w tC/B A tA/B M L tangent at B RA B L/2 C RC RB L/2 wL2/8 A1 L/3 A2 2L/3 A3 x -RCL/2 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 (*10.160) Assume RC as redundant. 2 wL2 wL3 ( L) = 3 8 12 R L2 1 RC L ( L) = − C A2 = − 2 2 4 RC L2 1 RC L L ( )=− A3 = − 2 2 2 8 A1 = 2L L wL4 RC L3 EIt A / B = A1 ( ) − A2 ( ) = − 2 3 24 6 3 R L L EItC / B = A3 ( ) = − C 3 24 We have R L3 wL4 RC L3 2tC / B = −t A / B : C = − , 12 24 6 Statics: 3 5 RB = wL ↑ RA = wL ↑ 4 12 RC = 1 wL ↓ 6 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.