761_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

761_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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Unformatted text preview: 09Ch09.qxd 9/27/08 1:35 PM Page 755 755 SECTION 9.7 Nonprismatic Beams Problem 9.7-2 The cantilever beam ACB shown in the figure supports a uniform q load of intensity q throughout its length. The beam has moments of inertia I2 and I1 in parts AC and CB, respectively. (a) Using the method of superposition, determine the deflection dB at the free end due to the uniform load. (b) Determine the ratio r of the deflection dB to the deflection d1 at the free end of a prismatic cantilever with moment of inertia I1 carrying the same load. (c) Plot a graph of the deflection ratio r versus the ratio I2/I1 of the moments of inertia. (Let I2 /I1 vary from 1 to 5.) A C I2 L — 2 B I1 L — 2 Solution 9.7-2 Cantilever beam (nonprismatic) Use the method of superposition (3) Total deflection at point B (a) DEFLECTION dB AT THE FREE END dB (dB)1 + (dB)2 (1) Part CB of the beam: (b) PRISMATIC BEAM d1 (dB)1 q L4 ab 8EI1 2 qL4 128EI1 Ratio: r dB d1 (c) GRAPH OF RATIO (2) Part AC of the beam: 4 dC q(L/2) + 8EI2 a a qL b (L/2)3 2 3EI2 2 qL L2 ba b 8 2 + 2EI2 17qL4 384EI2 3 uC q(L/2) (qL/2)(L/2)2 (qL2/8)(L/2) + + 6EI2 2EI2 EI2 7qL3 48EI2 (dB)2 L dC + uC a b 2 15qL4 128EI2 I2 I1 r 1 2 3 4 5 1.00 0.53 0.38 0.30 0.25 qL4 15I1 a1 + b 128EI1 I2 qL4 8EI1 15I1 1 a1 + b 16 I2 ; ; ...
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