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728_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

728_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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722 CHAPTER 9 Deflections of Beams Deflections by Integration of the Shear Force and Load Equations The beams described in the problems for Section 9.4 have constant flexural rigidity EI. Also, the origin of coordinates is at the left-hand end of each beam . Problem 9.4-1 Derive the equation of the deflection curve for a cantilever beam AB when a couple M 0 acts counterclockwise at the free end (see figure). Also, determine the deflection d B and slope u B at the free end. Use the third-order differential equation of the deflection curve (the shear-force equation). x y B A M 0 L Solution 9.4-1 Cantilever beam (couple M 0 ) S HEAR - FORCE EQUATION (E Q . 9-12b). B . C . 1 M M 0 B . C . 2 n (0) 0 C 2 0 EI M 0 x 2 2 + C 3 EIv ¿ C 1 x + C 2 M 0 x + C 2 EIv M M 0 C 1 EIv ¿¿ C 1 EIv –¿ V 0 B . C . 3 n (0) 0 C 3 0 (These results agree with Case 6, Table G-1.) u B ¿ ( L ) M 0 L EI (counterclockwise) ; d B ( L ) M 0 L 2 2 EI (upward) ; ¿ M 0 x EI M 0 x 2 2 EI ; Problem 9.4-2 A simple beam AB is subjected to a distributed load of
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Unformatted text preview: intensity q ± q sin p x/L , where q is the maximum intensity of the load (see figure). Derive the equation of the deflection curve, and then determine the deflection d max at the midpoint of the beam. Use the fourth-order differential equation of the deflection curve (the load equation). A y p x L — L B q = q sin Solution 9.4-2 Simple beam (sine load) L OAD EQUATION (E Q . 9-12c). EI ³ – ± q a L p b 2 sin p x L + C 1 x + C 2 EI ³ –¿ ± q a L p b cos p x L + C 1 EI ³ –– ± ´ q ± ´ q 0 sin p x L B . C . 1 B . C . 2 EI ³ ± ´ q a L p b 4 sin p x L + C 3 x + C 4 EI ³ ¿ ± ´ q a L p b 3 cos p x L + C 3 EIv – ( L ) ± ‹ C 1 ± EIv – ± M EIv – (0) ± ‹ C 2 ± 09Ch09.qxd 9/27/08 1:31 PM Page 722...
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