09-01ChapGere.0008

# 09-01ChapGere.0008 - AB loaded by a couple M at the...

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Deflections by Integration of the Bending-Moment Equation Problems 9.3-8 through 9.3-16 are to be solved by integrating the second-order differential equation of the deflection curve (the bending-moment equation). The origin of coordinates is at the left-hand end of each beam, and all beams have constant flexural rigidity EI. Problem 9.3-8 Derive the equation of the deflection curve for a cantilever beam AB supporting a load P at the free end (see figure). Also, determine the deflection d B and angle of rotation u B at the free end. ( Note: Use the second-order differential equation of the deflection curve.) Solution 9.3-8 Cantilever beam (concentrated load) 554 CHAPTER 9 Deflections of Beams x y B A P L B ENDING - MOMENT EQUATION (E Q . 9-12a) B . C . EIv 52 P Lx 2 2 1 Px 3 6 1 C 2 v ¿ (0) 5 0 Ê C 2 5 0 EIv ¿ 52 PLx 1 Px 2 2 1 C 1 EIv 5 M 52 P ( L 2 x ) B . C . (These results agree with Case 4, Table G-1.) u B 52 v ¿ ( L ) 5 PL 2 2 EI d B 52 v ( L ) 5 PL 3 3 EI v ¿ 52 Px 2 EI (2 L 2 x ) v 52 Px 2 6 EI (3 L 2 x ) v (0) 5 0 Ê C 1 5 0 Problem 9.3-9 Derive the equation of the deflection curve for a simple beam
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Unformatted text preview: AB loaded by a couple M at the left-hand support (see figure). Also, determine the maximum deflection d max . ( Note: Use the second-order differential equation of the deflection curve.) Solution 9.3-9 Simple beam (couple M ) y x A M B L B ENDING-MOMENT EQUATION (E Q . 9-12a) B . C . B . C . v 5 2 M x 6 LEI (2 L 2 2 3 Lx 1 x 2 ) v ( L ) 5 Ê ∴ C 1 5 2 M L 3 v (0) 5 Ê ∴ C 2 5 EIv 5 M ¢ x 2 2 2 x 3 6 L ≤ 1 C 1 x 1 C 2 EIv ¿ 5 M ¢ x 2 x 2 2 L ≤ 1 C 1 EIv – 5 M 5 M ¢ 1 2 x L ≤ M AXIMUM DEFLECTION Set and solve for x : Substitute x 1 into the equation for v : (These results agree with Case 7, Table G-2.) 5 M L 2 9 Ï 3 EI d max 5 2 ( v ) x 5 x 1 x 1 5 L ¢ 1 2 Ï 3 3 ≤ v ¿ 5 v ¿ 5 2 M 6 LEI (2 L 2 2 6 Lx 1 3 x 2 ) A-PDF Split DEMO : Purchase from www.A-PDF.com to remove the watermark...
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## This note was uploaded on 09/20/2009 for the course COE 3001 taught by Professor Armanios during the Spring '08 term at Georgia Tech.

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