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05-03 Chap Gere

# 05-03 Chap Gere - SECTION 5.7 Nonprismatic Beams 321...

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Nonprismatic Beams Problem 5.7-1 A tapered cantilever beam AB of length L has square cross sections and supports a concentrated load P at the free end (see figure on the next page). The width and height of the beam vary linearly from h A at the free end to h B at the fixed end. Determine the distance x from the free end A to the cross section of maximum bending stress if h B 5 3 h A . What is the magnitude s max of the maximum bending stress? What is the ratio of the maximum stress to the largest stress s B at the support? Solution 5.7-1 Tapered cantilever beam SECTION 5.7 Nonprismatic Beams 321 A h A h B B x P L Square cross sections h A 5 height and width at smaller end h B 5 height and width at larger end h x 5 height and width at distance x Stress at distance x At end A : x 5 0 s A 5 0 At support B : x 5 L s B 5 2 PL 9( h A ) 3 s 1 5 M x S x 5 6 Px ( h A ) 3 ¢ 1 1 2 x L 3 S x 5 1 6 ( h x ) 3 5 h A 3 6 ¢ 1 1 2 x L 3 h x 5 h A 1 ( h B 2 h A ) ¢ x L 5 h A ¢ 1 1 2 x L h B h A 5 3 Cross section of maximum stress Set Evaluate the derivative, set it equal to zero, and solve for x . Maximum bending stress Ratio of s max to s B s max s B 5 2 s max 5 ( s 1 ) x 5 L / 4 5 4 PL h A ) 3 x 5 L 4 d s 1 dx 5 0 A B x P L

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Problem 5.7-2 A tall signboard is supported by two vertical beams consisting of thin-walled, tapered circular tubes (see figure). For purposes of this analysis, each beam may be represented as a cantilever AB of length L 5 8.0 m subjected to a lateral load P 5 2.4 kN at the free end. The tubes have constant thickness t 5 10.0 mm and average diameters d A 5 90 mm and d B 5 270 mm at ends A and B , respectively. Because the thickness is small compared to the diameters, the moment of inertia at any cross section may be obtained from the formula I 5 p d 3 t /8 (see Case 22, Appendix D), and therefore the section modulus may be obtained from the formula S 5 p d 2 t /4. At what distance x from the free end does the maximum bending stress occur? What is the magnitude s max of the maximum bending stress? What is the ratio of the maximum stress to the largest stress s B at the support? Solution 5.7-2 Tapered circular tube 322 CHAPTER 5 Stresses in Beams Wind load P = 2.4 kN A B x t = 10.0 mm d A = 90 mm d B = 270 mm L = 8.0 m P 5 2.4 kN L 5 8.0 m t 5 10 mm d 5 average diameter At end A : d A 5 90 mm At support B : d B 5 270 mm At distance x: M x 5 Px 5 2400 xx 5 meters, M x 5 N ? m s 1 5 M x S x 5 2400 x 20.25 p ¢ 1 1 2 x L 2 Ê L 5 meters, s 1 5 MPa 5 20,250 p ¢ 1 1 2 x L 2 Ê S x 5 mm 3 S x 5 p 4 ( d x ) 2 ( t ) 5 p 4 (90) 2 ¢ 1 1 2 x L 2 (10) d x 5 d A 1 ( d B 2 d A ) ¢ x L 5 90 1 180 x L 5 90 ¢ 1 1 2 x L At end A : x 5 0 s 1 5 s A 5 0 At support B : x 5 L 5 8.0 m s 1 5 s B 5 33.53 MPa Cross section of maximum stress Set Evaluate the derivative, set it equal to zero, and solve for x . Maximum bending stress 5 37.73 MPa Ratio of s max to s B s max s B 5 9 8 5 1.125 s max 5 ( s 1 ) x 5 L / 2 5 2400(4.0) (20.25 p )(1 1 1) 2 x 5 L 2 5 4.0 m d s 1 dx 5 0 P A B x t L d
Problem 5.7-3 A tapered cantilever beam AB having rectangular cross sections is subjected to a concentrated load P 5 50 lb and a couple M 0 5 800 lb-in. acting at the free end

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05-03 Chap Gere - SECTION 5.7 Nonprismatic Beams 321...

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