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FME201 Solid & Structural Mechanics I Dr. Hussein Jama [email protected] Office 414 Lecture: Mon 11am -1pm (CELT) Tutorial Tue 12-1pm (E207) 7/20/2015 1 Lecture Outline This lecture is based on the book by R C Hibbeler Chapter 5 5.1Torsional deformation of a circular shaft 5.2 The Torsion Formula 5.3 Power of Transmission 5.4 Angle of twist 5.5 Statically indeterminate Torque Loaded Members* 5.8 Stress concentration 7/20/2015 2 Engineering Professor of the Year 7/20/2015 3 Engineering Professor of the Year 7/20/2015 4 Student complaints I'm currently studying for my mechanical engineering exam on this Friday and I'd like to hear stories of bad professors to lighten my frustration. The professor for this class is the least liked professor in the ME department. The only preparation for what's on the exam is basically the table of contents that he typed out. He also noted that it's an incomplete list so we have to figure out what else to study. Roughly 30% of the class fails and he doesn't care and has said so happily. 7/20/2015 5 Torsion Torque is a moment that twists a member about its longitudinal axis. External loads (T) produce internal loads which produce deformation, strain and stress. Application • Design of shafts in machinery 7/20/2015 6 Introduction Assumption: •The angle of rotation is small •The length of the shaft will remain unchanged •The radius of the shaft will remain unchanged. 7/20/2015 7 Torsion Before Torque After Torque 7/20/2015 8 5.1 Torsional deformation of a circular shaft f(x) = angle of twist (varies linearly along the length, 0 at x = 0, max at x = L) 7/20/2015 9 Torsional deformation cont.. Recall g = shear strain (rad) f df g x dx Notice, shear strain, g varies linearly with radial distance, , and is max on the outer surface!! 7/20/2015 10 Torsional deformation cont.. Notice, shear strain, g varies linearly with radial distance, , and is max on the outer surface!! Distance from center to point of interest g g max c Distance from center to outer fiver (i.e. outer radius) 7/20/2015 11 Stress due to torsion Deformation = shear strain Shear stress is proportional to shear strain (Hooke’s Law) Gg If you can visualize deformation, you can visualize stress 7/20/2015 12 Recap f = angle of twist varies from zero at fixed support to max at end. g = shear strain varies from zero at center to max at outer fiber 7/20/2015 13 5.2 The Torsion formula If linear elastic, Hooke’s law applies, t = Gy Therefore, stress follows same profile as strain!! max c 7/20/2015 14 Torsion formula Derivation – simple Torque balance. The torque produced by the stress distribution over the entire cross section must be equal to the resultant internal torque, or: T dA maxdA A c A T max c 2 dA A This is simply polar moment of inertia, J (an area property) 7/20/2015 15 Torsion formula Torque (N-m, Nmm or lb-in, lbft, etc) Tc max J Outer radius of shaft (m or in) Polar moment of inertia (m4 or in4) Max shear stress in shaft (MPa, psi/ksi, etc.) or 7/20/2015 T J 16 Polar moment of inertia This is a geometric property and is always positive. It has units of m^4 of mm^4 Solid shaft: J 7/20/2015 2 c 4 •Hollow shaft: J c 2 4 o ci4 17 Stress profile 7/20/2015 18 Stress profile – wood failure 7/20/2015 19 Example 5.3 7/20/2015 20 Solution Example 5.3 7/20/2015 21 Example 5.4 7/20/2015 22 Solution example 5.4 7/20/2015 23 5.3 Power transmission Nothing new, just calculate Torque, T, from power equation: Power (watts, ft-lb/s or hp) P = Tw Angular velocity (rad/s or Hz) Torque (N-m, lbft) w 2f f = Hz or rev/s Careful with units! 7/20/2015 24 Power transmission cont.. Shaft powered by 500 W electric motor spins at 10 Hz, find Torque in shaft. P = Tw 10 Hz (2 rad/rev) = 62.83 rad/s T = 500 N-m/s 62.83 rad/s = 7.96 N-m 7/20/2015 25 Class example Determine: 1. Torque throughout shaft 2. Stress throughout shaft 7/20/2015 26 Homework 5.5 5.7 5.15 5.19 5.25 7/20/2015 27 Recommended Texts 7/20/2015 Mechanics of Materials – 2nd Edition, Madhukar Vable – available online FREE Engineering Mechanics – Statics, R.C. Hibbler, Engineering Mechanics – Statics, D.J. McGill & W.W. King Mechanics of Materials , J.M. Gere & S.P. Timoshenko Mechanics of solids, Abdul Mubeen, Pearson Education Asia 28