882_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

882_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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Problem 11.5-9 The column shown in the figure is fixed at the base and free at the upper end. A compressive load P acts at the top of the column with an eccentricity e from the axis of the column. Beginning with the differential equation of the deflection curve, derive formulas for the maximum deflection of the column and the maximum bending moment M max in the column. d 876 CHAPTER 11 Columns Solution 11.5-9 Fixed-free column e ± eccentricity of load P ± deflection at the end of the column ± deflection of the column at distance x from base D IFFERENTIAL EQUATION (E Q . 11.3) G ENERAL SOLUTION ²± C 1 sin kx + C 2 cos kx + e + d ² œœ + k 2 k 2 ( e + d ) ² œœ ± k 2 ( e + d ³² ) EI ² œœ ± M ± P ( e + d ) k 2 ± P EI ² d B . C . B . C . B . C . M AXIMUM DEFLECTION M AXIMUM BENDING MOMENT ( AT BASE OF COLUMN ) NOTE: ± e (sec kL ) (1 ³ cos kx ) ( e + d ) (1 ³ cos kx ) M max ± P ( e + d ) ± Pe sec kL ; d ± e (sec kL ³ 1) ; or d ± e (sec kL ³ 1) 3 ² ( L ) ± d d ± ( e + d )(1 ³ cos kL ) ( e + d )(1 ³ cos kx ) 2 ² ¿ (0)
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