Torsion Lectures PDF

TORSION Torsion Torsion Turbine A and an electric generator B connected by a transmission shaft AB. The turbine exerts a twisting couple or torque T on the shaft, Shaft exerts an equal torque on the generator The generator reacts by exerting the equal and opposite torque T′ on t...

TORSION Torsion Torsion Turbine A and an electric generator B connected by a transmission shaft AB. The turbine exerts a twisting couple or torque T on the shaft, Shaft exerts an equal torque on the generator The generator reacts by exerting the equal and opposite torque T′ on the shaft The shaft reacts by exerting the torque T′ on the turbine. When a circular shaft is subjected to torsion, every cross section remains plane and undistorted. Torsion When a circular shaft is subjected to torsion, every cross section remains plane and undistorted.  Various cross sections along the shaft rotate through different angles - each cross section rotates as a solid rigid slab ρ - perpendicular distance from the force dF to the axis of the shaft sum of the moments of the shearing forces dF about the axis of the shaft Shaft AB subjected at A and B to is equal in magnitude to equal and opposite torques T and T′ the torque T Torsion ρ - perpendicular distance from the force dF to the axis of the shaft sum of the moments of the shearing forces dF about the axis of the shaft Shaft AB subjected at A and B to is equal in magnitude to equal and opposite torques T and T′ the torque T Condition that must be satisfied by the shearing stresses in any given cross section of the shaft – without showing their distribution in the cross section Torsion Condition that must be satisfied by the shearing stresses in any given cross section of the shaft – without showing their distribution in the cross section The torque applied to the shaft - produces shearing stresses τ on the faces perpendicular to the axis of the shaft. The existence of equal stresses on the faces formed by the two planes containing the axis of the shaft is required ϕ - angle of twist Within a certain range of values of T - ϕ is proportional to T and the length L of the shaft. The angle of twist for a shaft of the same material and same cross section, but twice as long, will be twice as large under the same torque T Torsion Every cross section remains plane - cross sections remain flat and undistorted While the various cross sections along the shaft rotate through different amounts - each cross section rotates as a solid rigid slab. All the equally spaced circles - rotate by the same amount relative to their neighbors Each of the straight lines - transformed into a curve (helix) Torsion o Detach a cylinder of radius ρ from the cylinder o Consider a small square element formed by two adjacent circles and two adjacent straight lines traced on the surface o Under loading - the element deforms into a rhombus o The circles defining two of the sides remain unchanged o The shearing strain γ must be equal to the angle between lines AB and A′B Torsion o The shearing strain γ at a given point of a shaft in torsion is proportional to the angle of twist ϕ and the distance ρ o The shearing strain in a circular shaft is zero at the axis of the shaft – and varies linearly with the distance from the axis of the shaft Torsion o The shearing strain γ at a given point of a shaft in torsion is proportional to the angle of twist ϕ and the distance ρ o The shearing strain in a circular shaft is zero at the axis of the shaft – and varies linearly with the distance from the axis of the shaft The stresses in the shaft will remain below both the proportional limit and the elastic limit - Hooke’s law will apply Torsion The stresses in the shaft will remain below both the proportional limit and the elastic limit - Hooke’s law will apply Multiplying by G Within the yield limit - the shearing stress in the shaft varies linearly with the distance ρ from the axis of the shaft Torsion Within the yield limit - the shearing stress in the shaft varies linearly with the distance ρ from the axis of the shaft Torsion – Angle of twist Consider a shaft of length L with a uniform cross section of radius c subjected to a torque T at its free end - the angle of twist ϕ Within the elastic range – based on Hooke’s law L, J, and G are constant for a given shaft - within the elastic range, the angle of twist ϕ is proportional to the torque T applied to the shaft Torsion – Practice problems A hollow cylindrical steel shaft is 1.5 m long and has inner and outer diameters respectively equal to 40 and 60 mm, see the figure. (a) What is the largest torque that can be applied to the shaft if the shearing stress is not to exceed 120 MPa? (b) What is the corresponding minimum value of the shearing stress in the shaft? The largest torque T that can be applied to the shaft is the torque for which τmax = 120 MPa. The polar moment of inertia Torsion – Angle of twist Torsion – Practice problems A hollow cylindrical steel shaft is 1.5 m long and has inner and outer diameters respectively equal to 40 and 60 mm, see the figure. (a) What is the largest torque that can be applied to the shaft if the shearing stress is not to exceed 120 MPa? (b) What is the corresponding minimum value of the shearing stress in the shaft? What torque should be applied to the end of the shaft to produce a twist of 2°? Use the modulus of rigidity of steel as G = 77 GPa. Torsion – Practice problems A hollow cylindrical steel shaft is 1.5 m long and has inner and outer diameters respectively equal to 40 and 60 mm, see the figure. (a) What is the largest torque that can be applied to the shaft if the shearing stress is not to exceed 120 MPa? (b) What is the corresponding minimum value of the shearing stress in the shaft? What torque should be applied to the end of the shaft to produce a twist of 2°? Use the modulus of rigidity of steel as G = 77 GPa. What angle of twist will create a shearing stress of 70 MPa on the inner surface of the hollow steel shaft? 1. Find the torque T corresponding to the given value of τ and determine the angle of twist ϕ corresponding to the value of T just found. 2. Use Hooke’s law to compute the shearing strain on the inner surface of the shaft and find ϕ. Torsion – Practice problems The horizontal shaft AD is attached to a fixed base at D and is subjected to the torques shown. A 44-mm-diameter hole has been drilled into portion CD of the shaft. Knowing that the entire shaft is made of steel for which G = 77 GPa, determine the angle of twist at end A. Solution Use free-body diagrams to determine the torque in each shaft segment AB, BC, and CD … use the below equation to determine the angle of twist at end A. Torsion – Practice problems The horizontal shaft AD is attached to a fixed base at D and is subjected to the torques shown. A 44-mm-diameter hole has been drilled into portion CD of the shaft. Knowing that the entire shaft is made of steel for which G = 77 GPa, determine the angle of twist at end A. Solution Use free-body diagrams to determine the torque in each shaft segment AB, BC, and CD … use the below equation to determine the angle of twist at end A. Torsion – Practice problems The horizontal shaft AD is attached to a fixed base at D and is subjected to the torques shown. A 44-mm-diameter hole has been drilled into portion CD of the shaft. Knowing that the entire shaft is made of steel for which G = 77 GPa, determine the angle of twist at end A. Solution Use free-body diagrams to determine the torque in each shaft segment AB, BC, and CD … use the below equation to determine the angle of twist at end A. Torsion – Practice problems The horizontal shaft AD is attached to a fixed base at D and is subjected to the torques shown. A 44-mm-diameter hole has been drilled into portion CD of the shaft. Knowing that the entire shaft is made of steel for which G = 77 GPa, determine the angle of twist at end A. Solution Use free-body diagrams to determine the torque in each shaft segment AB, BC, and CD … use the below equation to determine the angle of twist at end A. Torsion – Practice problems A steel shaft and an aluminum tube are connected to a fixed support and to a rigid disk as shown in the cross section. Knowing that the initial stresses are zero, determine the maximum torque T0 that can be applied to the disk if the allowable stresses are 120 MPa in the steel shaft and 70 MPa in the aluminum tube. Use G = 77 GPa for steel and G = 27 GPa for aluminum. Practice at home … Torsion – Transmission shafts The principal specifications of a transmission shaft – (i) the power to be transmitted and (ii) the speed of rotation of the shaft where ω is the angular velocity of the body in radians per second (rad/s). But ω = 2πf, where f is the frequency of the rotation (i.e., the number of revolutions per second) Torsion – Practice problems A shaft consisting of a steel tube of 50-mm outer diameter is to transmit 100 kW of power while rotating at a frequency of 20 Hz. Determine the tube thickness that should be used if the shearing stress is not to exceed 60 MPa. A tube thickness of 5 mm is required.