Page 1 Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Torsion Introduction Click here to check the Animation Assumptions Basic Assumptions Torsion Formula Stress Formula Stresses on Inclined Planes Angle of twist Angle of Twist in Torsion Maximum Stress Torsion of Circular Elastic Bars: Formulae Table of Formulae Page 2 Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Torsion Introduction Click here to check the Animation Assumptions Basic Assumptions Torsion Formula Stress Formula Stresses on Inclined Planes Angle of twist Angle of Twist in Torsion Maximum Stress Torsion of Circular Elastic Bars: Formulae Table of Formulae Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Introduction: Detailed methods of analysis for determining stresses and deformations in axially loaded bars were presented in the first two chapters. Analogous relations for members subjected to torque about their longitudinal axes are developed in this chapter. The constitutive relations for shear discussed in the preceding chapter will be employed for this purpose. The investigations are confined to the effect of a single type of action, i.e., of a torque causing a twist or torsion in a member. The major part of this chapter is devoted to the consideration of members having circular cross sections, either solid or tubular. Solution of such elastic and inelastic problems can be obtained using the procedures of engineering mechanics of solids. For the solution of torsion problems having noncircular cross sections, methods of the mathematical theory of elasticity (or finite elements) must be employed. This topic is briefly discussed in order to make the reader aware of the differences in such solutions from that for circular members. Further, to lend emphasis to the difference in the solutions discussed, this chapter is subdivided into four distinct parts. It should be noted, however, that in practice, members for transmitting torque, such as shafts for motors, torque tubes for power equipment, etc., are predominantly circular or tubular in cross section. Therefore, numerous applications fall within the scope of the formulas derived in this chapter. In this section, discussion is limited to torsion of circular bars only. Top Page 3 Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Torsion Introduction Click here to check the Animation Assumptions Basic Assumptions Torsion Formula Stress Formula Stresses on Inclined Planes Angle of twist Angle of Twist in Torsion Maximum Stress Torsion of Circular Elastic Bars: Formulae Table of Formulae Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Introduction: Detailed methods of analysis for determining stresses and deformations in axially loaded bars were presented in the first two chapters. Analogous relations for members subjected to torque about their longitudinal axes are developed in this chapter. The constitutive relations for shear discussed in the preceding chapter will be employed for this purpose. The investigations are confined to the effect of a single type of action, i.e., of a torque causing a twist or torsion in a member. The major part of this chapter is devoted to the consideration of members having circular cross sections, either solid or tubular. Solution of such elastic and inelastic problems can be obtained using the procedures of engineering mechanics of solids. For the solution of torsion problems having noncircular cross sections, methods of the mathematical theory of elasticity (or finite elements) must be employed. This topic is briefly discussed in order to make the reader aware of the differences in such solutions from that for circular members. Further, to lend emphasis to the difference in the solutions discussed, this chapter is subdivided into four distinct parts. It should be noted, however, that in practice, members for transmitting torque, such as shafts for motors, torque tubes for power equipment, etc., are predominantly circular or tubular in cross section. Therefore, numerous applications fall within the scope of the formulas derived in this chapter. In this section, discussion is limited to torsion of circular bars only. Top Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Basic Assumptions a. Plane sections perpendicular to the axis of a circular member before application of torque remains plane even after application of torque. b. Shear strains vary linearly from the central axis reaching a maximum value at the outer surface. c. For linearly elastic material, Hooke's law is valid. Hence shear stress is proportional to shear strain. Top Page 4 Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Torsion Introduction Click here to check the Animation Assumptions Basic Assumptions Torsion Formula Stress Formula Stresses on Inclined Planes Angle of twist Angle of Twist in Torsion Maximum Stress Torsion of Circular Elastic Bars: Formulae Table of Formulae Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Introduction: Detailed methods of analysis for determining stresses and deformations in axially loaded bars were presented in the first two chapters. Analogous relations for members subjected to torque about their longitudinal axes are developed in this chapter. The constitutive relations for shear discussed in the preceding chapter will be employed for this purpose. The investigations are confined to the effect of a single type of action, i.e., of a torque causing a twist or torsion in a member. The major part of this chapter is devoted to the consideration of members having circular cross sections, either solid or tubular. Solution of such elastic and inelastic problems can be obtained using the procedures of engineering mechanics of solids. For the solution of torsion problems having noncircular cross sections, methods of the mathematical theory of elasticity (or finite elements) must be employed. This topic is briefly discussed in order to make the reader aware of the differences in such solutions from that for circular members. Further, to lend emphasis to the difference in the solutions discussed, this chapter is subdivided into four distinct parts. It should be noted, however, that in practice, members for transmitting torque, such as shafts for motors, torque tubes for power equipment, etc., are predominantly circular or tubular in cross section. Therefore, numerous applications fall within the scope of the formulas derived in this chapter. In this section, discussion is limited to torsion of circular bars only. Top Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Basic Assumptions a. Plane sections perpendicular to the axis of a circular member before application of torque remains plane even after application of torque. b. Shear strains vary linearly from the central axis reaching a maximum value at the outer surface. c. For linearly elastic material, Hooke's law is valid. Hence shear stress is proportional to shear strain. Top Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Torsion Formula Since shear strains varies linearly across the section, max R C ? ?= where ? is the shear strains at a point of raidus R, C is the radius of the member. A Torque, T = R dA ? t ? A = GR dA ? ? where t = G ?, the shear stress at any point at a distance R (Refer Figure 6.1) Figure 6.1 Hence writing in terms of shear stresses. max A 2 max R T = RdA C = R dA C t t ? ? 2 p RdA I = ? the Polar moment of Inertia of the circular section. max p I T = C t ? Page 5 Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Torsion Introduction Click here to check the Animation Assumptions Basic Assumptions Torsion Formula Stress Formula Stresses on Inclined Planes Angle of twist Angle of Twist in Torsion Maximum Stress Torsion of Circular Elastic Bars: Formulae Table of Formulae Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Introduction: Detailed methods of analysis for determining stresses and deformations in axially loaded bars were presented in the first two chapters. Analogous relations for members subjected to torque about their longitudinal axes are developed in this chapter. The constitutive relations for shear discussed in the preceding chapter will be employed for this purpose. The investigations are confined to the effect of a single type of action, i.e., of a torque causing a twist or torsion in a member. The major part of this chapter is devoted to the consideration of members having circular cross sections, either solid or tubular. Solution of such elastic and inelastic problems can be obtained using the procedures of engineering mechanics of solids. For the solution of torsion problems having noncircular cross sections, methods of the mathematical theory of elasticity (or finite elements) must be employed. This topic is briefly discussed in order to make the reader aware of the differences in such solutions from that for circular members. Further, to lend emphasis to the difference in the solutions discussed, this chapter is subdivided into four distinct parts. It should be noted, however, that in practice, members for transmitting torque, such as shafts for motors, torque tubes for power equipment, etc., are predominantly circular or tubular in cross section. Therefore, numerous applications fall within the scope of the formulas derived in this chapter. In this section, discussion is limited to torsion of circular bars only. Top Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Basic Assumptions a. Plane sections perpendicular to the axis of a circular member before application of torque remains plane even after application of torque. b. Shear strains vary linearly from the central axis reaching a maximum value at the outer surface. c. For linearly elastic material, Hooke's law is valid. Hence shear stress is proportional to shear strain. Top Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Torsion Formula Since shear strains varies linearly across the section, max R C ? ?= where ? is the shear strains at a point of raidus R, C is the radius of the member. A Torque, T = R dA ? t ? A = GR dA ? ? where t = G ?, the shear stress at any point at a distance R (Refer Figure 6.1) Figure 6.1 Hence writing in terms of shear stresses. max A 2 max R T = RdA C = R dA C t t ? ? 2 p RdA I = ? the Polar moment of Inertia of the circular section. max p I T = C t ? Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras max p TC I t= and p TR I t= TopRead More

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