753_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

753_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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Unformatted text preview: 09Ch09.qxd 9/27/08 1:33 PM Page 747 747 SECTION 9.6 Moment-Area Method A1 + A2 A0 uB/A uB uB A0 PL2 2EI M0 L EI uA M0 L EI uA A0 tB/A 0 PL3 3EI (dB is positive when upward) PL 2EI FINAL RESULTS To match the sign conventions for uB and dB used in Appendix G, change the signs as follows. uB DEFLECTION Q first moment of areas A1 and A2 with respect to point B A1x1 + A2x2 M0 L2 2EI dB 2 (uB is positive when counterclockwise) Q dB Q M0 L2 2EI PL3 3EI PL2 2EI M0 L EI dB PL3 3EI M0 L2 2EI ; (positive clockwise) ; (positive downward) (These results agree with Cases 4 and 6, Table G-1.) Problem 9.6-4 Determine the angle of rotation uB and the deflection dB at the free end q of a cantilever beam AB with a uniform load of intensity q acting over the middle third of the length (see figure). A B L — 3 L — 3 L — 3 Solution 9.6-4 Cantilever beam with partial uniform load A3 2L 2L + ab 3 33 A0 DIAGRAM 1 L qL2 a ba b 23 9EI x3 M EI A1 + A2 + A3 uB/A uA ANGLE OF ROTATION Use absolute values of areas. Appendix D, Cases 1, 6, and 18: A1 x1 A2 qL2 1L a ba b 33 18EI L 3L + ab 3 43 qL2 L a ba b 3 18EI uA uB 0 8L 9 7qL3 162EI A0 7qL3 162EI (clockwise) ; DEFLECTION Q 7L 12 x2 2L L + 3 6 5L 6 first moment of area A0 with respect to point B Q qL3 162EI qL3 54EI uB qL3 54EI A1x1 + A2x2 + A3x3 dB Q 23qL4 648EI 23qL4 648EI (Downward) ; ...
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