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09-04 Chap Gere

# 09-04 Chap Gere - 588 CHAPTER 9 Deflections of Beams...

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588 CHAPTER 9 Deflections of Beams Use the method of superposition. (a) D EFLECTION d B AT THE FREE END (1) Part CB of the beam: (2) Part AC of the beam: ( d B ) 2 5 d C 1 u C ¢ L 2 5 7 PL 3 24 EI 2 u C 5 P ( L / 2) 2 2 EI 2 1 ( PL / 2)( L / EI 2 5 3 PL 2 8 EI 2 d C 5 P ( L / 3 3 EI 2 1 ( PL / L / 2 2 EI 2 5 5 PL 3 48 EI 2 ( d B ) 1 5 P 3 EI 1 ¢ L 2 3 5 PL 3 24 EI 1 (3) Total deflection at point B (b) P RISMATIC BEAM Ratio: (c) G RAPH OF RATIO r 5 d B d 1 5 1 8 ¢ 1 1 7 I 1 I 2 d 1 5 PL 3 3 EI 1 d B 5 ( d B ) 1 1 ( d B ) 2 5 PL 3 24 EI 1 ¢ 1 1 7 I 1 I 2 B C I 1 P L 2 1 0.5 0 12345 I 2 I 1 r Problem 9.7-2 The cantilever beam ACB shown in the figure supports a uniform load of intensity q throughout its length. The beam has moments of inertia I 2 and I 1 in parts AC and CB , respectively. (a) Using the method of superposition, determine the deflection d B at the free end due to the uniform load. (b) Determine the ratio r of the deflection d B to the deflection d 1 at the free end of a prismatic cantilever with moment of inertia I 1 carrying the same load. (c) Plot a graph of the deflection ratio r versus the ratio I 2 / I 1 of the moments of inertia. (Let I 2 / I 1 vary from 1 to 5.) B C A I 1 I 2 q L 2 L 2 C A I 2 P L 2 PL 2 Nonprismatic Beams Problem 9.7-1 The cantilever beam ACB shown in the figure has moments of inertia I 2 and I 1 in parts AC and CB , respectively. (a) Using the method of superposition, determine the deflection d B at the free end due to the load P . (b) Determine the ratio r of the deflection d B to the deflection d 1 at the free end of a prismatic cantilever with moment of inertia I 1 carrying the same load. (c) Plot a graph of the deflection ratio r versus the ratio I 2 / I 1 of the moments of inertia. (Let I 2 / I 1 vary from 1 to 5.) Solution 9.7-1 Cantilever beam (nonprismatic) B C A I 1 I 2 P L 2 L 2 r 1 1.00 2 0.56 3 0.42 4 0.34 5 0.30 I 2 I 1

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Solution 9.7-2 Cantilever beam (nonprismatic) SECTION 9.7 Nonprismatic Beams 589 Use the method of superposition (a) D EFLECTION d B AT THE FREE END (1) Part CB of the beam: (2) Part AC of the beam: ( d B ) 2 5 d C 1 u C ¢ L 2 5 15 qL 4 128 EI 2 5 7 qL 3 48 EI 2 u C 5 q ( L / 2) 3 6 EI 2 1 ( qL / 2)( L / 2 2 EI 2 1 ( qL 2 / 8)( L / EI 2 d C 5 q ( L / 4 8 EI 2 1 ¢ qL 2 ( L / 3 3 EI 2 1 ¢ qL 2 8 ¢ L 2 2 2 EI 2 5 17 qL 4 384 EI 2 ( d B ) 1 5 q 8 EI 1 ¢ L 2 4 5 qL 4 128 EI 1 (3) Total deflection at point B (b) P RISMATIC BEAM Ratio: (c) G RAPH OF RATIO r 5 d B d 1 5 1 16 ¢ 1 1 15 I 1 I 2 d 1 5 qL 4 8 EI 1 d B 5 ( d B ) 1 1 ( d B ) 2 5 qL 4 128 EI 1 ¢ 1 1 15 I 1 I 2 I 1 B C q L 2 I 2 C A q L 2 qL 2 qL 2 8 I 2 I 1 1 1 2 3 45 0.5 O r Problem 9.7-3 A simple beam ABCD has moment of inertia I near the supports and moment of inertia 2 I in the middle region, as shown in the figure. A uniform load of intensity q acts over the entire length of the beam.
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09-04 Chap Gere - 588 CHAPTER 9 Deflections of Beams...

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