2009+11Torsion

# 2009 11Torsion - Introduction to Solid Mechanics-Vm211 Introduction to Solid Mechanics-Vm211 CHAPTER OUTLINE Chapter 11 TORSION 1 2 3 4 5 Chapter

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1 SJTU Introduction to Solid Mechanics-Vm211 Chapter 11 TORSION Chapter 11 Torsion SJTU Introduction to Solid Mechanics-Vm211 CHAPTER OUTLINE 1. Torsional Deformation of a Circular Shaft 2. The Torsion Formula 3. Power Transmission 4. Angle of Twist 5. Statically Indeterminate Torque-Loaded Members SJTU Introduction to Solid Mechanics-Vm211 TORSION Today’s Objectives: a) Determine torsional deformation of a perfectly elastic circular shaft b) Determine support reactions when they cannot be determined solely from the moment equilibrium equation c) Determine the maximum power that can be transmitted by a shaft SJTU Introduction to Solid Mechanics-Vm211 ± Torsion is a moment that twists/deforms a member about its longitudinal axis ± By observation, if angle of rotation is small , length of shaft and its radius remain unchanged TORSIONAL DEFORMATION OF A CIRCULAR SHAFT SJTU Introduction to Solid Mechanics-Vm211 TORSION FORMULA Assumptions – Linear and elastic deformation – Plane section remains plane and undistorted Fig. 5.1 Fig. 5.2 SJTU Introduction to Solid Mechanics-Vm211 TORSIONAL DEFORMATION OF A CIRCULAR SHAFT ± By definition, shear strain is Let Δ x dx and Δφ = d φ BD = ρ d = dx γ = ( π /2) lim θ C A along CA B A along BA = d dx Since d / dx = / = max /c = max c ( ) Equation:

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2 SJTU Introduction to Solid Mechanics-Vm211 THE TORSION FORMULA ± For solid shaft, shear stress varies from zero at shaft’s longitudinal axis to maximum value at its outer surface. ± Due to proportionality of triangles, or using Hooke’s law and Eq., τ = max ρ c ( ) T = dA c dA c dA A AA = = ∫∫ 2 max max 2 ) ( J the polar moment of inertia SJTU Introduction to Solid Mechanics-Vm211 THE TORSION FORMULA ± The integral in the equation can be represented as the polar moment of inertia J , of shaft’s x-sectional area computed about its longitudinal axis max = Tc J SJTU Introduction to Solid Mechanics-Vm211 THE TORSION FORMULA max = Tc J max = max. shear stress in shaft, at the outer surface T = resultant internal torque acting at x-section, from method of sections & equation of moment equilibrium applied about longitudinal axis J = polar moment of inertia of x-sectional area c = outer radius of the shaft SJTU Introduction to Solid Mechanics-Vm211 THE TORSION FORMULA ± Shear stress at intermediate distance = T J The above two equations are referred to as the torsion formula Used only if shaft is circular, its material homogenous, and it behaves in an linear-elastic manner SJTU Introduction to Solid Mechanics-Vm211 THE TORSION FORMULA Solid shaft ± J can be determined using area element in the form of a differential ring or annulus having thickness d and circumference 2 πρ .
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## This note was uploaded on 08/09/2011 for the course EE 211 taught by Professor Liuxila during the Summer '09 term at Shanghai Jiao Tong University.

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2009 11Torsion - Introduction to Solid Mechanics-Vm211 Introduction to Solid Mechanics-Vm211 CHAPTER OUTLINE Chapter 11 TORSION 1 2 3 4 5 Chapter

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