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ch9-1 - 9 Deflections of Beams Differential Equations of...

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Differential Equations of the Deflection Curve The beams described in the problems for Section 9.2 have constant flexural rigidity EI. Problem 9.2-1 The deflection curve for a simple beam AB (see figure) is given by the following equation: v (7 L 4 10 L 2 x 2 3 x 4 ) Describe the load acting on the beam. Solution 9.2-1 Simple beam q 0 x 360 LEI 9 Deflections of Beams y x A B L Take four consecutive derivatives and obtain: From Eq. (9-12c): The load is a downward triangular load of maximum intensity q 0 . q EIv –– q 0 x L v –– q 0 x LEI v q 0 x 360 LEI (7 L 4 10 L 2 x 2 3 x 4 ) L q 0 Problem 9.2-2 The deflection curve for a simple beam AB (see figure) is given by the following equation: v sin L x (a) Describe the load acting on the beam. (b) Determine the reactions R A and R B at the supports. (c) Determine the maximum bending moment M max . q 0 L 4 4 EI Probs. 9.2-1 and 9.2-2
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Solution 9.2-2 Simple beam 548 CHAPTER 9 Deflections of Beams (a) L OAD (E Q . 9-12c) The load has the shape of a sine curve, acts downward, and has maximum intensity q 0 . q EIv –– q 0 sin x L v ‡¿ q 0 EI sin x L v q 0 L EI cos x L v q 0 L 2 2 EI sin x L v ¿ q 0 L 3 3 EI cos x L v q 0 L 4 4 EI sin x L (b) R EACTIONS (E Q . 9-12b) At x 0: At x L : ; (c) M AXIMUM BENDING MOMENT (E Q . 9-12a) For maximum moment, M max q 0 L 2 2 x L 2 ; M EIv q 0 L 2 2 sin x L R B q 0 L V R B q 0 L V R A q 0 L V EIv q 0 L cos x L L q 0 Problem 9.2-3 The deflection curve for a cantilever beam AB (see figure) is given by the following equation: v (10 L 3 10 L 2 x 5 Lx 2 x 3 ) Describe the load acting on the beam. Solution 9.2-3 Cantilever beam q 0 x 2 120 LEI x y B A L Take four consecutive derivatives and obtain: From Eq. (9-12c): The load is a downward triangular load of maximum intensity q 0 . q EIv –– q 0 ¢ 1 x L v –– q 0 LEI ( L x ) v q 0 x 2 120 LEI (10 L 3 10 L 2 x 5 L x 2 x 3 ) q 0 L Probs. 9.2-3 and 9.2-4
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Problem 9.2-4 The deflection curve for a cantilever beam AB (see figure) is given by the following equation: v (45 L 4 40 L 3 x 15 L 2 x 2 x 4 ) (a) Describe the load acting on the beam. (b) Determine the reactions R A and M A at the support. Solution 9.2-4 Cantilever beam q 0 x 2 360 L 2 EI SECTION 9.2 Differential Equations of the Deflection Curve 549 (a) L OAD (E Q . 9-12c) The load is a downward parabolic load of maximum intensity q 0 . q EIv –– q 0 ¢ 1 x 2 L 2 v ‡¿ q 0 L 2 EI ( L 2 x 2 ) v q 0 3 L 2 EI ( 2 L 3 3 L 2 x x 3 ) v q 0 12 L 2 EI (3 L 4 8 L 3 x 6 L 2 x 2 x 4 ) v ¿ q 0 60 L 2 EI (15 L 4 x 20 L 3 x 2 10 L 2 x 3 x 5 ) v q 0 x 2 360 L 2 EI (45 L 4 40 L 3 x 15 L 2 x 2 x 4 ) (b) R EACTIONS R A AND M A (E Q . 9-12b AND E Q . 9-12a) At x 0: At x 0: N OTE : Reaction R A is positive upward. Reaction M A is positive clockwise (minus means M A is counterclockwise). M M A q 0 L 2 4 M EIv q 0 12 L 2 (3 L 4 8 L 3 x 6 L 2 x 2 x 4 ) V R A 2 q 0 L 3 V EIv q 0 3 L 2 ( 2 L 3 3 L 2 x x 3 ) L q 0
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Deflection Formulas Problems 9.3-1 through 9.3-7 require the calculation of deflections using the formulas derived in Examples 9-1, 9-2, and 9-3. All beams have constant flexural rigidity EI.
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