ch9-2 - SECTION 9.4 Differential Equations of the...

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SECTION 9.4 Differential Equations of the Deflection Curve 559 Differential Equations of the Deflection Curve 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 B and slope B at the free end. Use the third-order differential equation of the deflection curve (the shear-force equation). Solution 9.4-1 Cantilever beam (couple M 0 ) x y B A M 0 L S HEAR - FORCE EQUATION (E Q . 9-12 b). B . C . 1 M M 0 M M 0 C 1 C 1 x C 2 M 0 x C 2 B . C . 2 (0) 0 C 2 0 EIv M 0 x 2 2 C 3 v ¿ EIv ¿ EIv EIv C 1 EIv V 0 B . C . 3 v (0) 0 C 3 0 (upward) (counterclockwise) (These results agree with Case 6, Table G-1.) u B v ¿ ( L ) M 0 L EI B v ( L ) M 0 L 2 2 EI v ¿ M 0 x EI v M 0 x 2 2 EI Problem 9.4-2 A simple beam AB is subjected to a distributed load of intensity q q 0 sin x / L , where q 0 is the maximum intensity of the load (see figure). Derive the equation of the deflection curve, and then determine the deflection max at the midpoint of the beam. Use the fourth-order differential equation of the deflection curve (the load equation). Solution 9.4-2 Simple beam (sine load) A y x x L L B q = q 0 sin L OAD EQUATION (E Q . 9-12 c). B . C . 1 C 2 0 B . C . 2 C 1 0 EIv q 0 ¢ L 4 sin x L C 3 x C 4 EIv ¿ q 0 ¢ L 3 cos x L C 3 EIv ( L ) 0 EIv (0) 0 EIv M EIv q 0 ¢ L 2 sin x L C 1 x C 2 EIv q 0 ¢ L cos x L C 1 EIv –– q q 0 sin x L B . C . 3 v (0) 0 C 4 0 B . C . 4 v ( L ) 0 C 3 0 (These results agree with Case 13, Table G-2.) max v ¢ L 2 q 0 L 4 4 EI v q 0 L 4 4 EI sin x L
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560 CHAPTER 9 Deflections of Beams Problem 9.4-3 The simple beam AB shown in the figure has moments 2 M 0 and M 0 acting at the ends. Derive the equation of the deflection curve, and then determine the maximum deflection max . Use the third-order differential equation of the deflection curve (the shear-force equation). Solution 9.4-3 Simple beam with two couples y x A B M 0 2 M 0 L Reaction at support A : (downward) Shear force in beam: S HEAR - FORCE EQUATION (E Q . 9-12 b) B . C . 1 C 1 2 M 0 B . C . 2 v (0) 0 C 3 0 EIv M 0 x 3 2 L M 0 x 2 C 2 x C 3 EIv ¿ 3 M 0 x 2 2 L 2 M 0 x C 2 EIv (0) 2 M 0 EIv M EIv 3 M 0 x L C 1 EIv V 3 M 0 L V R A 3 M 0 L R A 3 M 0 L B . C . 3 v ( L ) 0 M AXIMUM DEFLECTION Set v 0 and solve for x : x 1 L and Maximum deflection occurs at . (downward) max v ¢ L 3 2 M 0 L 2 27 EI x 2 L 3 x 2 L 3 v ¿ M 0 2 LEI ( L x )( L 3 x ) v M 0 x 2 LEI ( L 2 2 Lx x 2 ) M 0 x 2 LEI ( L x ) 2 C 2 M 0 L 2 Problem 9.4-4 A simple beam with a uniform load is pin supported at one end and spring supported at the other. The spring has stiffness k 48 EI / L 3 . Derive the equation of the deflection curve by starting with the third-order differential equation (the shear-force equation).
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