627_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

627_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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SECTION 7.6 Triaxial Stress 621 Problem 7.6-10 A solid steel sphere ( , = 0.3) is subjected to hydrostatic pressure such that its volume is reduced by 0.4%. (a) Calculate the pressure . (b) Calculate the volume modulus of elasticity for the steel. (c) Calculate the strain energy stored in the sphere if its diameter is d ± 150 mm U K p p ² E ± 210 GPa Solution 7.6-10 Steel sphere Hydrostatic Pressure. = Initial volume (a) P RESSURE Pressure ; p ± s 0 ± 700 MPa or s 0 ± Ee 3(1 ³ 2 ² ) ± 700 MPa Eq.(7-60): e ± 3 s 0 (1 ³ 2 ² ) E Dilatation: e ± ¢ V V 0 ± 0.004 ¢ V ± 0.004 V 0 V 0 E ± 210 GPa ²± 0.3 (b) V OLUME MODULUS OF ELASTICITY (c) S TRAIN ENERGY ( = diameter) = 150 mm r = 75 mm From Eq. (7-57b) with U ± uV 0 ± 2470 N # m ± 2470 J ; V 0 ± 4 p r 3 3 ± 1767 * 10 ³ 6 m 3 u ± 3(1 ³ 2 ² ) s 0 2 2 E ± 1.40 MPa s x ± s y ± s z ± s 0 : d d Eq. (7-63): K ± s 0 E ± 700 MPa 0.004 ± 175 GPa ; Problem 7.6-11 A solid bronze sphere (volume modulus of elasticity K ± 14.5 ´ 10 6 psi) is suddenly heated around its outer surface. The tendency of the heated part of the sphere to expand produces uniform tension in all directions at the center
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Unformatted text preview: of the sphere. If the stress at the center is 12,000 psi, what is the strain? Also, calculate the unit volume change e and the strain-energy density u at the center. Solution 7.6-11 Bronze sphere (heated) (tension at the center) S TRAIN AT THE CENTER OF THE SPHERE Combine the two equations: â ± s 3 K ± 276 * 10 ³ 6 ; Eq. (7-61): K ± E 3(1 ³ 2 ² ) Eq. (7-59): â ± s E (1 ³ 2 ² ) s ± 12,000 psi K ± 14.5 * 10 6 psi U NIT VOLUME CHANGE AT THE CENTER S TRAIN ENERGY DENSITY AT THE CENTER u ± 4.97 psi ; u ± 3(1 ³ 2 ² ) s 2 2 E ± s 2 2 K Eq. (7-57b) with s x ± s y ± s z ± s : Eq. (7-62): e ± s K ± 828 * 10 ³ 6 ; 07Ch07.qxd 9/27/08 1:22 PM Page 621...
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