ch01-03 stress & strain & properties

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03 Solutions 46060 5/7/10 8:45 AM Page 12 © 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. *3–16. Determine the elongation of the square hollow bar when it is subjected to the axial force P = 100 kN. If this axial force is increased to P = 360 kN and released, find the permanent elongation of the bar. The bar is made of a metal alloy having a stress–strain diagram which can be approximated as shown. s (MPa) 500 250 P 600 mm 50 mm 0.00125 0.05 5 mm P (mm/mm) P 50 mm 5 mm Normal Stress and Strain: The cross-sectional area of the hollow bar is A = 0.05 2 - 0.04 2 = 0.9(10 - 3 )m 2 . When P = 100 kN, s 1 = P A = 100(103 ) 0.9(10 - 3 = 111.11 MPa ) From the stress–strain diagram shown in Fig. a, the slope of the straight line OA which represents the modulus of elasticity of the metal alloy is E = 250(106 ) - 0 0.00125 - 0 = 200 GPa Since s 1 6 250 MPa, Hooke’s Law can be applied. Thus s 1 = Ee 1 ; 111.11(10 6 ) = 200(10 9 )e 1 e 1 = 0.5556(10 - 3 ) mm>mm Thus, the elongation of the bar is d 1 = e 1 L = 0.5556(10 - 3 )(600) = 0.333 mm Ans. When P = 360 kN, s 2 = P A = 360(103 ) 0.9(10 - 3 = 400 MPa ) From the geometry of the stress–strain diagram, Fig. a, e 2 - 0.00125 400 - 250 = 0.05 - 0.00125 500 - 250 e 2 = 0.0305 mm>mm When P = 360 kN is removed, the strain recovers linearly along line BC, Fig.a, parallel to OA. Thus, the elastic recovery of strain is given by s 2 = Ee r ; 400(10 6 ) = 200(10 9 )e r e r = 0.002 mm>mm The permanent set is e P = e 2 - e r = 0.0305 - 0.002 = 0.0285 mm>mm Thus, the permanent elongation of the bar is d P = e P L = 0.0285(600) = 17.1 mm Ans. 12
03 Solutions 46060 5/7/10 8:45 AM Page 13 © 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 3–16. Continued 13
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