FME201 Solid & Structural Mechanics I Dr. Hussein Jama hussein.jama@uobi.ac.ke Office 414 Lecture: Mon 11am -1pm (CELT) Tutorial Tue 12-1pm (E207) 12/4/2018

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 12/4/2018

Engineering Professor of the Year 12/4/2018

Engineering Professor of the Year 12/4/2018

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. 12/4/2018

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 12/4/2018

Introduction Assumption: The angle of rotation is small The length of the shaft will remain unchanged The radius of the shaft will remain unchanged. 12/4/2018

Torsion Before Torque After Torque 12/4/2018

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) 12/4/2018

Torsional deformation cont.. Recall g = shear strain (rad) Notice, shear strain, g varies linearly with radial distance, r, and is max on the outer surface!! 12/4/2018

Torsional deformation cont.. Notice, shear strain, g varies linearly with radial distance, r, and is max on the outer surface!! Distance from center to point of interest Distance from center to outer fiver (i.e. outer radius) 12/4/2018

Stress due to torsion Deformation = shear strain Shear stress is proportional to shear strain (Hooke’s Law) If you can visualize deformation, you can visualize stress 12/4/2018

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 12/4/2018

5.2 The Torsion formula If linear elastic, Hooke’s law applies, t = Gy Therefore, stress follows same profile as strain!! 12/4/2018

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: This is simply polar moment of inertia, J (an area property) 12/4/2018

Torsion formula or Torque (N-m, N-mm or lb-in, lb-ft, etc) Outer radius of shaft (m or in) Polar moment of inertia (m4 or in4) Max shear stress in shaft (MPa, psi/ksi, etc.) or 12/4/2018

Polar moment of inertia This is a geometric property and is always positive. It has units of m^4 of mm^4 Solid shaft: Hollow shaft: 12/4/2018

Stress profile 12/4/2018

Stress profile – wood failure 12/4/2018

Example 5.3 12/4/2018

Solution Example 5.3 12/4/2018

Example 5.4 12/4/2018

Solution example 5.4 12/4/2018

5.3 Power transmission P = Tw Nothing new, just calculate Torque, T, from power equation: P = Tw Careful with units! Power (watts, ft-lb/s or hp) Angular velocity (rad/s or Hz) Torque (N-m, lb-ft) f = Hz or rev/s 12/4/2018

Power transmission cont.. Shaft powered by 500 W electric motor spins at 10 Hz, find Torque in shaft. P = Tw 10 Hz (2p rad/rev) = 62.83 rad/s T = 500 N-m/s 62.83 rad/s = 7.96 N-m 12/4/2018

Class example Determine: Torque throughout shaft Stress throughout shaft 12/4/2018

Homework 5.5 5.7 5.15 5.19 5.25 12/4/2018

Recommended Texts 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 12/4/2018