753_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1 - 09Ch09.qxd 1:33 PM Page 747 747 SECTION 9.6 Moment-Area Method

# 753_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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SECTION 9.6Moment-Area Method747(uBis positive when counterclockwise)DEFLECTIONQfirst moment of areas A1and A2with respect topoint QA1x1+A2x2M0L22EIPL33EIuBA0M0LEIPL22EIuB/AuBuAA0uA0A0A1+A2M0LEIPL22EI(dBis positive when upward)FINAL RESULTSTo match the sign conventions for uBand dBused inAppendix G, change the signs as follows.(These results agree with Cases 4 and 6, Table G-1.)dBPL33EIM0L22EI(positive downward);uBPL22EIM0LEI(positive clockwise);tB/AQdBdBM0L22EIPL33EIProblem 9.6-4Determine the angle of rotation uBand the deflection dBat the free end of a cantilever beam ABwith a uniform load of intensity qacting over the middle third of the length (see figure). B Solution 9.6-4 Cantilever beam with partial uniform load DIAGRAM M EI D EFLECTION d B Q 23 qL 4 648 EI (Downward) ; Q A 1 x 1 + A 2 x 2 + A 3 x 3 23 qL 4 648 EI Q first moment of area A 0 with respect to point B u A u B 7 qL 3 162 EI (clockwise) ; u B / A u B u A A A A 1 + A 2 + A 3 7 qL 3 162 EI x 3 2 L 3 + 2 3 a L 3 b 8 L 9 A 3 1 2 a L 3 b a qL 2 9 EI b qL 3 54 EI B A q L 3 L 3 L 3 A NGLE OF ROTATION Use absolute values of areas. Appendix D, Cases 1, 6, and 18: A 2 a L 3 b a qL 2 18 EI b qL 3 54 EI x 2 2 L 3 + L 6 5 L 6 x 1 L 3 + 3 4 a L 3 b 7 L 12 A 1 1 3 a L 3 b a qL 2 18 EI b qL 3 162 EI 09Ch09.qxd 9/27/08 1:33 PM Page 747