02-01 Chap Gere

02-01 Chap Gere - 2 Axially Loaded Numbers Changes in...

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63 Changes in Lengths of Axially Loaded Members Problem 2.2-1 The T-shaped arm ABC shown in the figure lies in a vertical plane and pivots about a horizontal pin at A. The arm has constant cross-sectional area and total weight W. A vertical spring of stiffness k supports the arm at point B. Obtain a formula for the elongation d of the spring due to the weight of the arm. Solution 2.2-1 T-shaped arm 2 Axially Loaded Numbers b b AB C k b F REE - BODY DIAGRAM OF ARM F 5 tensile force in the spring d 5 elongation of the spring d 5 F k 5 4 W 3 k F 5 4 W 3 F ( b ) 2 W 3 ¢ b 2 2 W 3 ¢ 3 b 2 2 W 3 (2 b ) 5 0 © M A 5 0 ` ~ b b A B C F W 3 W 3 W 3 Problem 2.2-2 A steel cable with nominal diameter 25 mm (see Table 2-1) is used in a construction yard to lift a bridge section weighing 38 kN, as shown in the figure. The cable has an effective modulus of elasticity E 5 140 GPa. (a) If the cable is 14 m long, how much will it stretch when the load is picked up? (b) If the cable is rated for a maximum load of 70 kN, what is the factor of safety with respect to failure of the cable?
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Solution 2.2-2 Bridge section lifted by a cable 64 CHAPTER 2 Axially Loaded Numbers A 5 304 mm 2 (from Table 2-1) W 5 38 kN E 5 140 GPa L 5 14 m (a) S TRETCH OF CABLE 5 12.5 mm d 5 WL EA 5 (38 kN)(14 m) (140 GPa)(304 mm 2 ) (b) F ACTOR OF SAFETY P ULT 5 406 kN (from Table 2-1) P max 5 70 kN n 5 P ULT P max 5 406 kN 70 kN 5 5.8 Problem 2.2-3 A steel wire and a copper wire have equal lengths and support equal loads P (see figure). The moduli of elasticity for the steel and copper are E s 5 30,000 ksi and E c 5 18,000 ksi, respectively. (a) If the wires have the same diameters, what is the ratio of the elongation of the copper wire to the elongation of the steel wire? (b) If the wires stretch the same amount, what is the ratio of the diameter of the copper wire to the diameter of the steel wire? P Steel wire P Copper wire Solution 2.2-3 Steel wire and copper wire P Steel wire P Copper wire Equal lengths and equal loads Steel: E s 5 30,000 ksi Copper: E c 5 18,000 ksi (a) R ATIO OF ELONGATIONS ( EQUAL DIAMETERS ) d c d s 5 E s E c 5 30 18 5 1.67 d c 5 PL E c A Ê d s 5 PL E s A (b) R ATIO OF DIAMETERS ( EQUAL ELONGATIONS ) d c 2 d s 2 5 E s E c Ê d c d s 5 B E s E c 5 B 30 18 5 1.29 E c ¢ p 4 d c 2 5 E s ¢ p 4 d s 2 d c 5 d s Ê PL E c A c 5 PL E s A s or Ê E c A c 5 E s A s
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Problem 2.2-4 By what distance h does the cage shown in the figure move downward when the weight W is placed inside it? Consider only the effects of the stretching of the cable, which has axial rigidity EA 5 10,700 kN. The pulley at A has diameter d A 5 300 mm and the pulley at B has diameter d B 5 150 mm. Also, the distance L 1 5 4.6 m, the distance L 2 5 10.5 m, and the weight W 5 22 kN. ( Note: When calculating the length of the cable, include the parts of the cable that go around the pulleys at A and B .) SECTION 2.2 Changes in Lengths of Axially Loaded Members 65 L 1 L 2 A B W Cage Solution 2.2-4 Cage supported by a cable L 1 L 2 A B W d A 5 300 mm d B 5 150 mm L 1 5 4.6 m L 2 5 10.5 m EA 5 10,700 kN W 5 22 kN L ENGTH OF CABLE 5 4,600 mm 1 21,000 mm 1 236 mm 1 236 mm 5 26,072 mm E
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02-01 Chap Gere - 2 Axially Loaded Numbers Changes in...

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