Solution: The copper pipe has an outer diameter of 2.50 in.

The solid shaft of radius r is subjected to a torque T. Determine the radius r’ of the inner core of the shaft that resists one-half of the applied torque (T/2). Solve the problem two ways: (a) by using the torsion formula, (b) by finding the resultant of the shear-stress distribution.

The solid shaft of radius r is subjected to a torque T. Determine the radius r’ of the inner core of the shaft that resists one-quarter of the applied torque (T/4). Solve the problem two ways: (a) by using the torsion formula, (b) by finding the resultant of the shear-stress distribution.

A shaft is made of an aluminum alloy having an allowable shear stress of \(\tau_{\text {allow }}=100 \mathrm{MPa}\). If the diameter of the shaft is 100 mm, determine the maximum torque T that can be transmitted.What would be the maximum torque T’ if a 75-mm-diameter hole were bored through the shaft? Sketch the shear-stress distribution along a radial line in each case.

The copper pipe has an outer diameter of 40 mm and an inner diameter of 37 mm. If it is tightly secured to the wall and three torques are applied to it, determine the absolute maximum shear stress developed in the pipe.

The copper pipe has an outer diameter of 2.50 in. and an inner diameter of 2.30 in. If it is tightly secured to the wall and three torques are applied to it, determine the shear stress developed at points A and B. These points lie on the pipe’s outer surface. Sketch the shear stress on volume elements located at A and B.

The solid aluminum shaft has a diameter of 50 mm and an allowable shear stress of \(\tau_{\text {allow }}=60 \mathrm{MPa}\). Determine the largest torque \(T_{1}\) that can be applied to the shaft if it is also subjected to the other torsional loadings. It is required that \(\mathbf{T}_{1}\) act in the direction shown. Also, determine the maximum shear stress within regions CD and DE.

The solid aluminum shaft has a diameter of 50 mm. Determine the absolute maximum shear stress in the shaft and sketch the shear-stress distribution along a radial line of# the shaft where the shear stress is maximum. Set \(T_{1}=2000 \mathrm{~N} \cdot \mathrm{m}\).

The solid 30-mm-diameter shaft is used to transmit the torques applied to the gears. Determine the absolute maximum shear stress in the shaft.

The solid shaft is fixed to the support at C and subjected to the torsional loadings. Determine the shear stress at points A and B on the surface, and sketch the shear stress on volume elements located at these points.

The link acts as part of the elevator control for a small airplane. If the attached aluminum tube has an inner diameter of 25 mm and a wall thickness of 5 mm, determine the maximum shear stress in the tube when the cable force of 600 N is applied to the cables. Also, sketch the shear-stress distribution over the cross section.

The assembly consists of two sections of galvanized steel pipe connected together using a reducing coupling at B. The smaller pipe has an outer diameter of 0.75 in. and an inner diameter of 0.68 in., whereas the larger pipe has an outer diameter of 1 in. and an inner diameter of 0.86 in. If the pipe is tightly secured to the wall at C, determine the maximum shear stress in each section of the pipe when the couple is applied to the handles of the wrench.

The shaft has an outer diameter of 100 mm and an inner diameter of 80 mm. If it is subjected to the three torques, determine the absolute maximum shear stress in the shaft. The smooth bearings A and B do not resist torque.

The shaft has an outer diameter of 100 mm and an inner diameter of 80 mm. If it is subjected to the three torques, plot the shear stress distribution along a radial line for the cross section within region CD of the shaft. The smooth bearings at A and B do not resist torque.

A steel tube having an outer diameter of 2.5 in. is used to transmit 9 hp when turning at 27 rev/min. Determine the inner diameter d of the tube to the nearest 18 in. if the allowable shear stress is \(\tau_{\text {allow }}=10 \mathrm{ksi}\).

If the gears are subjected to the torques shown, determine the maximum shear stress in the segments AB and BC of the A-36 steel shaft. The shaft has a diameter of 40 mm.

If the gears are subjected to the torques shown, determine the required diameter of the A-36 steel shaft to the nearest mm if \(\tau_{\text {allow }}=60 \mathrm{MPa}\).

The rod has a diameter of 1 in. and a weight of 10 lb/ft. Determine the maximum torsional stress in the rod at a section located at A due to the rod’s weight.

The rod has a diameter of 1 in. and a weight of 15 lb/ft. Determine the maximum torsional stress in the rod at a section located at B due to the rod’s weight.

The copper pipe has an outer diameter of 3 in. and an inner diameter of 2.5 in. If it is tightly secured to the wall at C and a uniformly distributed torque is applied to it as shown, determine the shear stress at points A and B. These points lie on the pipe’s outer surface. Sketch the shear stress on volume elements located at A and B.

The copper pipe has an outer diameter of 3 in. and an inner diameter of 2.50 in. If it is tightly secured to the wall at C and it is subjected to the uniformly distributed torque along its entire length, determine the absolute maximum shear stress in the pipe. Discuss the validity of this result.

The 60-mm-diameter solid shaft is subjected to the distributed and concentrated torsional loadings shown. Determine the shear stress at points A and B, and sketch the shear stress on volume elements located at these points.

The 60-mm-diameter solid shaft is subjected to the distributed and concentrated torsional loadings shown. Determine the absolute maximum and minimum shear stresses on the shaft’s surface, and specify their locations, measured from the fixed end C.

The solid shaft is subjected to the distributed and concentrated torsional loadings shown. Determine the required diameter d of the shaft if the allowable shear stress for the material is \(\tau_{\text {allow }}=1.6 \mathrm{MPa}\)

The 60-mm-diameter solid shaft is subjected to the distributed and concentrated torsional loadings shown. Determine the absolute maximum and minimum shear stresses on the shaft’s surface and specify their locations, measured from the free end.

The solid shaft is subjected to the distributed and concentrated torsional loadings shown. Determine the required diameter d of the shaft if the allowable shear stress for the material is \(\tau_{\text {allow }}=60 \mathrm{MPa}\)

When drilling a well at constant angular velocity, the bottom end of the drill pipe encounters a torsional resistance \(T_{A}\). Also, soil along the sides of the pipe creates a distributed frictional torque along its length, varying uniformly from zero at the surface B to \(t_{A}\) at A. Determine the minimum torque \(T_{B}\) that must be supplied by the drive unit to overcome the resisting torques, and calculate the maximum shear stress in the pipe. The pipe has an outer radius \(r_{o}\) and an inner radius \(r_{i}\).

The drive shaft AB of an automobile is made of a steel having an allowable shear stress of \(\tau_{\text {allow }}=8 \mathrm{ksi}\). If the outer diameter of the shaft is 2.5 in. and the engine delivers 200 hp to the shaft when it is turning at 1140 rev/min, determine the minimum required thickness of the shaft’s wall.

The drive shaft AB of an automobile is to be designed as a thin-walled tube. The engine delivers 150 hp when the shaft is turning at 1000 rev>min. Determine the minimum thickness of the shaft’s wall if the shaft’s outer diameter is 2.5 in. The material has an allowable shear stress of \(\tau_{\text {allow }}=7 \mathrm{ksi}\).

The shaft is subjected to a distributed torque along its length of \(t=\left(10 x^{2}\right) \mathrm{N} \cdot \mathrm{m} / \mathrm{m}\) where x is in meters. If the maximum stress in the shaft is to remain constant at 80 MPa, determine the required variation of the radius c of the shaft for \(0 \leq x \leq 3 \mathrm{~m}\).

The motor delivers 50 hp while turning at a constant rate of 1350 rpm at A. Using the belt and pulley system this loading is delivered to the steel blower shaft BC. Determine to the nearest \(\frac{1}{8}\) in. the smallest diameter of this shaft if the allowable shear stress for steel is \(\tau_{\text {allow }}=12 \mathrm{ksi}\).

The solid steel shaft AC has a diameter of 25 mm and is supported by smooth bearings at D and E. It is coupled to a motor at C, which delivers 3 kW of power to the shaft while it is turning at 50 rev/s. If gears A and B remove 1 kW and 2 kW, respectively, determine the maximum shear stress in the shaft within regions AB and BC. The shaft is free to turn in its support bearings D and E.

The pump operates using the motor that has a power of 85 W. If the impeller at B is turning at 150 rev/min, determine the maximum shear stress in the 20-mm-diameter transmission shaft at A.

The gear motor can develop \(\frac{1}{10}\) hp when it turns at 10 300 rev/min. If the shaft has a diameter of \(\frac{1}{2}\) in., determine the maximum shear stress in the shaft.

The gear motor can develop \(\frac{1}{10}\) hp when it turns at 80 rev/min. If the allowable shear stress for the shaft is \(\tau_{\text {allow }}=4 \mathrm{ksi}\)., determine the smallest diameter of the shaft to the nearest \(\frac{1}{8}\) in. that can be used.

The gear motor can develop \(\frac{1}{4}\) hp when it turns at 600 rev/min. If the shaft has a diameter of \(\frac{1}{2}\) in., determine the maximum shear stress in the shaft.

The gear motor can develop 2 hp when it turns at 150 rev/min. If the allowable shear stress for the shaft is \(\tau_{\text {allow }}=8 \mathrm{ksi}\), determine the smallest diameter of the shaft to the nearest \(\frac{1}{8}\) in. that can be used.

The 6-hp reducer motor can turn at 1200 rev/min. If the allowable shear stress for the shaft is \(\tau_{\text {allow }}=6 \mathrm{ksi}\), determine the smallest diameter of the shaft to the nearest \(\frac{1}{16}\) in. that can be used.

The 6-hp reducer motor can turn at 1200 rev/min. If the shaft has a diameter of \(\frac{5}{8}\) in., determine the maximum shear stress in the shaft.

The solid steel shaft DF has a diameter of 25 mm and is supported by smooth bearings at D and E. It is coupled to a motor at F, which delivers 12 kW of power to the shaft while it is turning at 50 rev/s. If gears A, B, and C remove 3 kW, 4 kW, and 5 kW respectively, determine the maximum shear stress in the shaft within regions CF and BC. The shaft is free to turn in its support bearings D and E.

The solid steel shaft DF has a diameter of 25 mm and is supported by smooth bearings at D and E. It is coupled to a motor at F, which delivers 12 kW of power to the shaft while it is turning at 50 rev/s. If gears A, B, and C remove 3 kW, 4 kW, and 5 kW respectively, determine the absolute maximum shear stress in the shaft.

The A-36 steel tubular shaft is 2 m long and has an outer diameter of 50 mm. When it is rotating at 40 rad/s, it transmits 25 kW of power from the motor M to the pump P. Determine the smallest thickness of the tube if the allowable shear stress is \(\tau_{\text {allow }}=80 \mathrm{MPa}\)

The A-36 solid steel shaft is 2 m long and has a diameter of 60 mm. It is required to transmit 60 kW of power from the motor M to the pump P. Determine the smallest angular velocity the shaft if the allowable shear stress is \(\tau_{\text {allow }}=80 \mathrm{MPa}\)

The solid shaft has a linear taper from \(r_{A}\) at one end to \(r_{B}\) at the other. Derive an equation that gives the maximum shear stress in the shaft at a location x along the shaft’s axis.

The 1-in.-diameter bent rod is subjected to the load shown. Determine the maximum torsional stress in the rod at a section located through A.

The 1-in.-diameter bent rod is subjected to the load shown. Determine the maximum torsional stress in the rod at B.

A motor delivers 500 hp to the shaft, which is tubular and has an outer diameter of 2 in. If it is rotating at 200 rad/s, determine its largest inner diameter to the nearest \(\frac{1}{8}\) in. if the 5 allowable shear stress for the material is \(\tau_{\text {allow }}=25 \mathrm{ksi}\)

The propellers of a ship are connected to an A-36 steel shaft that is 60 m long and has an outer diameter of 340 mm and inner diameter of 260 mm. If the power output is 4.5 MW when the shaft rotates at 20 rad/s, determine the maximum torsional stress in the shaft and its angle of twist.

The solid shaft of radius c is subjected to a torque T at its ends. Show that the maximum shear strain in the shaft is \(\gamma_{\max }=T c / J G\). What is the shear strain on an element located at point A, c/2 from the center of the shaft? Sketch the shear strain distortion of this element.

The splined ends and gears attached to the A992 steel shaft are subjected to the torques shown. Determine the angle of twist of end B with respect to end A. The shaft has a diameter of 40 mm.

The A-36 steel shaft has a diameter of 50 mm and is subjected to the distributed and concentrated loadings shown. Determine the absolute maximum shear stress in the shaft and plot a graph of the angle of twist of the shaft in radians versus x.

The 60-mm-diameter shaft is made of 6061-T6 aluminum having an allowable shear stress of \(\tau_{\text {allow }}=80 \mathrm{MPa}\). Determine the maximum allowable torque T. Also, find the corresponding angle of twist of disk A relative to disk C.

The 60-mm-diameter shaft is made of 6061-T6 aluminum. If the allowable shear stress is \(\tau_{\text {allow }}=80 \mathrm{MPa}\), and the angle of twist of disk A relative to disk C is limited so that it does not exceed 0.06 rad, determine the maximum allowable torque T.

The 50-mm-diameter A992 steel shaft is subjected to the torques shown. Determine the angle of twist of the end A.

The shaft is made of A992 steel with the allowable shear stress of \(\tau_{\text {allow }}=75 \mathrm{MPa}\). If gear B supplies 15 kW of power, while gears A, C and D withdraw 6 kW, 4 kW and 5 kW, respectively, determine the required minimum diameter d of the shaft to the nearest millimeter. Also, find the corresponding angle of twist of gear A relative to gear D. The shaft is rotating at 600 rpm.

Gear B supplies 15 kW of power, while gears A, C and D withdraw 6 kW, 4 kW and 5 kW, respectively. If the shaft is made of steel with the allowable shear stress of \(\tau_{\text {allow }}=75 \mathrm{MPa}\), and the relative angle of twist between any two gears cannot exceed 0.05 rad, determine the required minimum diameter d of the shaft to the nearest millimeter. The shaft is rotating at 600 rpm.

The rotating flywheel-and-shaft, when brought to a sudden stop at D, begins to oscillate clockwise-counter- clockwise such that a point A on the outer edge of the fly- wheel is displaced through a 6-mm arc. Determine the maximum shear stress developed in the tubular A-36 steel shaft due to this oscillation. The shaft has an inner diameter of 24 mm and an outer diameter of 32 mm. The bearings at B and C allow the shaft to rotate freely, whereas the support at D holds the shaft fixed.

The turbine develops 150 kW of power, which is transmitted to the gears such that C receives 70% and D receives 30%. If the rotation of the 100-mm-diameter A-36 steel shaft is \(\omega=800 \mathrm{rev} / \mathrm{min}\), determine the absolute maximum shear stress in the shaft and the angle of twist of end E of the shaft relative to B. The journal bearing at E allows the shaft to turn freely about its axis.

The turbine develops 150 kW of power, which is transmitted to the gears such that both C and D receive an equal amount. If the rotation of the 100-mm-diameter A-36 steel shaft is \(\omega=500 \mathrm{rev} / \mathrm{min}\), determine the absolute maximum shear stress in the shaft and the rotation of end B of the shaft relative to E. The journal bearing at E allows the shaft to turn freely about its axis.

The shaft is made of A992 steel. It has a diameter of 1 in. and is supported by bearings at A and D, which allow free rotation. Determine the angle of twist of B with respect to D.

The shaft is made of A-36 steel. It has a diameter of 1 in. and is supported by bearings at A and D, which allow free rotation. Determine the angle of twist of gear C with respect to B.

The A992 steel shaft has a diameter of 50 mm and is subjected to the distributed loadings shown. Determine the absolute maximum shear stress in the shaft and plot a graph of the angle of twist of the shaft in radians versus x.

The turbine develops 300 kW of power, which is transmitted to the gears such that both B and C receive an equal amount. If the rotation of the 100-mm-diameter A992 steel shaft is \(\omega=600 \mathrm{rev} / \mathrm{min}\), determine the absolute maximum shear stress in the shaft and the rotation of end D of the shaft relative to A. The journal bearing at D allows the shaft to turn freely about its axis.

The device shown is used to mix soils in order to provide in-situ stabilization. If the mixer is connected to an A-36 steel tubular shaft that has an inner diameter of 3 in. and an outer diameter of 4.5 in., determine the angle of twist of the shaft at A relative to C if each mixing blade is subjected to the torques shown.

The device shown is used to mix soils in order to provide in-situ stabilization. If the mixer is connected to an A-36 steel tubular shaft that has an inner diameter of 3 in. and an outer diameter of 4.5 in, determine the angle of twist of the shaft at A relative to B and the absolute maximum shear stress in the shaft if each mixing blade is subjected to the torques shown

The 6-in.-diameter L-2 steel shaft on the turbine is supported on journal bearings at A and B. If C is held fixed and the turbine blades create a torque on the shaft that increases linearly from zero at C to \(2000 \mathrm{lb} \cdot \mathrm{ft}\) at D, determine the angle of twist of the shaft at D relative to C. Also, calculate the absolute maximum shear stress in the shaft. Neglect the size of the blades.

The A-36 hollow steel shaft is 2 m long and has an outer diameter of 40 mm. When it is rotating at 80 rad/s, it transmits 32 kW of power from the engine E to the generator G. Determine the smallest thickness of the shaft if the allowable shear stress is \(\tau_{\text {allow }}=140\) MPa and the shaft is restricted not to twist more than 0.05 rad.

The A-36 solid steel shaft is 3 m long and has a diameter of 50 mm. It is required to transmit 35 kW of power from the engine E to the generator G. Determine the smallest angular velocity of the shaft if it is restricted not to twist more than \(1^{\circ} .\)

The shaft of radius c is subjected to a distributed torque t, measured as torque/length of shaft. Determine the angle of twist at end A. The shear modulus is G.

The A-36 steel bolt is tightened within a hole so that the reactive torque on the shank AB can be expressed by the equation \(t=\left(k x^{2}\right) \mathrm{N} \cdot \mathrm{m} / \mathrm{m}\), where x is in meters. If a torque of \( T=50 \mathrm{~N} \cdot \mathrm{m}\) is applied to the bolt head, determine the constant k and the amount of twist in the 50-mm length of the shank. Assume the shank has a constant radius of 4 mm.

The A-36 steel bolt is tightened within a hole so that the reactive torque on the #shank AB can be expressed by the equation \(t=\left(k x^{2 / 3}\right) \mathrm{N} \cdot \mathrm{m} / \mathrm{m}\), where x is in meters. If a torque of \( T=50 \mathrm{~N} \cdot \mathrm{m}\) is applied to the bolt head, determine the constant k and the amount of twist in the 50-mm length of the shank. Assume the shank has a constant radius of 4 mm.

The tubular drive shaft for the propeller of a hovercraft is 6 m long. If the motor delivers 4 MW of power to the shaft when the propellers rotate at 25 rad/s, determine the required inner diameter of the shaft if the outer diameter is 250 mm. What is the angle of twist of the shaft when it is operating? Take \(\tau_{\text {allow }}=90 \mathrm{MPa}\) and G = 75 GPa.

The 60-mm-diameter solid shaft is made of 2014-T6 aluminum and is subjected to the distributed and concentrated torsional loadings shown. Determine the angle of twist at the free end A of the shaft.

The motor produces a torque of \(T=20 \mathrm{~N} \cdot \mathrm{m}\) on gear A. If gear C is suddenly locked so it does not turn, yet B can freely turn, determine the angle of twist of F with respect to E and F with respect to D of the L2-steel shaft, which has an inner diameter of 30 mm and an outer diameter of 50 mm. Also, calculate the absolute maximum shear stress in the shaft. The shaft is supported on journal bearings at G at H.

The shaft has a radius c and is subjected to a torque per unit length of \(t_{0}\), which is distributed uniformly over the shaft’s entire length L. If it is fixed at its far end A, determine the angle of twist \(\phi\) of end B. The shear modulus is G.

The 60-mm-diameter solid shaft is made of A-36 steel and is subjected to the distributed and concentrated torsional loadings shown. Determine the angle of twist at the free end A of the shaft due to these loadings.

The contour of the surface of the shaft is defined by the equation \(y=e^{a x}\), where a is a constant. If the shaft is subjected to a torque T at its ends, determine the angle of twist of end A with respect to end B. The shear modulus is G.

The steel shaft has a diameter of 40 mm and is fixed at its ends A and B. If it is subjected to the couple, determine the maximum shear stress in regions AC and CB of the shaft. \(G_{\text {st }}=75 \mathrm{GPa} .\)

The A992 steel shaft has a diameter of 60 mm and is fixed at its ends A and B. If it is subjected to the torques shown, determine the absolute maximum shear stress in the shaft.

The steel shaft is made from two segments: AC has a diameter of 0.5 in, and CB has a diameter of 1 in. If the shaft is fixed at its ends A and B and subjected to a torque of \(500 \mathrm{lb} \cdot \mathrm{ft}\), determine the maximum shear stress in the shaft \(G_{\mathrm{st}}=10.8\left(10^{3}\right) \mathrm{ksi}\).

The bronze C86100 pipe has an outer diameter of 1.5 in. and a thickness of 0.125 in. The coupling on it at C is being tightened using a wrench. If the torque developed at A is \(125 \mathrm{lb} \cdot \mathrm{in}\)., determine the magnitude F of the couple forces. The pipe is fixed supported at end B.

The bronze C86100 pipe has an outer diameter of 1.5 in. and a thickness of 0.125 in. The coupling on it at C is being tightened using a wrench. If the applied force is F = 20 lb, determine the maximum shear stress in the pipe.

The shaft is made of L2 tool steel, has a diameter of 40 mm, and is fixed at its ends A and B. If it is subjected to the torque, determine the maximum shear stress in regions AC and CB.

The shaft is made of L2 tool steel, has a diameter of 40 mm, and is fixed at its ends A and B. If it is subjected to the couple, determine the maximum shear stress in regions AC and CB.

The Am1004-T61 magnesium tube is bonded to the A-36 steel rod. If the allowable shear stresses for the magnesium and steel are \(\left(\tau_{\text {allow }}\right)_{\mathrm{mg}}=45 \mathrm{MPa}\]) and \(\left(\tau_{\text {allow }}\right)_{\mathrm{mg}}=75 \mathrm{MPa}\), respectively, determine the maximum allowable torque that can be applied at A. Also, find the corresponding angle of twist of end A.

The Am1004-T61 magnesium tube is bonded to the A-36 steel rod. If a torque of \(T=5 \mathrm{kN} \cdot \mathrm{m}\) is applied to end A, determine the maximum shear stress in each material. Sketch the shear stress distribution.

The two shafts are made of A-36 steel. Each has a diameter of 25 mm and they are connected using the gears fixed to their ends. Their other ends are attached to fixed supports at A and B. They are also supported by journal bearings at C and D, which allow free rotation of the shafts along their axes. If a torque of \(500 \mathrm{~N} \cdot \mathrm{m}\) is applied to the gear at E, determine the reactions at A and B.

The two shafts are made of A-36 steel. Each has a diameter of 25 mm and they are connected using the gears fixed to their ends. Their other ends are attached to fixed supports at A and B. They are also supported by journal bearings at C and D, which allow free rotation of the shafts along their axes. If a torque of \(500 \mathrm{~N} \cdot \mathrm{m}\) is applied to the gear at E, determine the rotation of this gear.

A rod is made from two segments: AB is steel and BC is brass. It is fixed at its ends and subjected to a torque of \(T=680 \mathrm{~N} \cdot \mathrm{m}\). If the steel portion has a diameter of 30 mm, determine the required diameter of the brass portion so the reactions at the walls will be the same. \(G_{\mathrm{st}}=75 \mathrm{GPa}\), \(G_{\mathrm{br}}=75 \mathrm{GPa}\).

Determine the absolute maximum shear stress in 5 the shaft of Prob. 5–88.

The two 3-ft-long shafts are made of 2014-16 aluminum. Each has a diameter of 1.5 in. and they are connected using the gears fixed to their ends. Their other ends are attached to fixed supports at A and B. They are also supported by bearings at C and D, which allow free rotation of the shafts along their axes. If a torque of \(600 \mathrm{~lb} \cdot \mathrm{ft}\) is applied to the top gear as shown, determine the maximum shear stress in each shaft.

The composite shaft consists of a mid-section that includes the 1-in.-diameter solid shaft and a tube that is welded to the rigid flanges at A and B. Neglect the thickness of the flanges and determine the angle of twist of end C of the shaft relative to end D. The shaft is subjected to a torque 5 of \(800 \mathrm{~lb} \cdot \mathrm{ft}\). The material is A-36 steel.

If the shaft is subjected to a uniform distributed torque of \(t=20 \mathrm{kN} \cdot \mathrm{m} / \mathrm{m}\), determine the maximum shear stress developed in the shaft. The shaft is made of 2014-T6 aluminum alloy and is fixed at A and C.

The tapered shaft is confined by the fixed supports at A and B. If a torque T is applied at its mid-point, determine the reactions at the supports.

The shaft of radius c is subjected to a distributed torque t, measured as torque/length of shaft. Determine the reactions at the fixed supports A and B.

The brass wire has a triangular cross section, 2 mm on a side. If the yield stress for brass is \(\tau_{Y}=205 \mathrm{MPa}\), determine the maximum torque T to which it can be subjected so that the wire will not yield. If this torque is applied to the 4-m-long segment, determine the greatest angle of twist of one end of the wire relative to the other end that will not cause permanent damage to the wire. \(G_{\mathrm{br}}=37 \mathrm{GPa}\).

If a = 25 mm and b = 15 mm, determine the maximum shear stress in the circular and elliptical shafts when the applied torque is \(T=80 \mathrm{~N} \cdot \mathrm{m}\). By what percentage is the shaft of circular cross section more efficient at withstanding the torque than the shaft of elliptical cross section?

It is intended to manufacture a circular bar to resist torque; however, the bar is made elliptical in the process of manufacturing, with one dimension smaller than the other by a factor k as shown. Determine the factor by which the maximum shear stress is increased.

The shaft is made of red brass C83400 and has an elliptical cross section. If it is subjected to the torsional loading, determine the maximum shear stress within regions AC and BC, and the angle of twist f of end B relative to end A.

Solve Prob. 5–98 for the maximum shear stress within regions AC and BC, and the angle of twist f of end B relative to C.

If end B of the shaft, which has an equilateral triangular cross section, is subjected to a torque of T = \(900 \mathrm{~lb} \cdot \mathrm{ft}\) , determine the maximum shear stress in the shaft. Also, find the angle of twist of end B. The shaft is made from 6061-T1 aluminum.

If the shaft has an equilateral triangle cross section and is made from an alloy that has an allowable shear stress of \(\tau_{\text {allow }}=12 \mathrm{ksi}\), determine the maximum allowable torque T that can be applied to end B. Also, find the corresponding angle of twist of end B.

The 2014-T6 aluminum strut is fixed between the two walls at A and B. If it has a 2 in. by 2 in. square cross section, and it is subjected to the torsional loading shown, determine the reactions at the fixed supports. Also, what is the angle of twist at C?

A torque of \(2 \text { kip } \cdot \text { in }\). is applied to the tube. If the wall thickness is 0.1 in., determine the average shear stress in the tube.

The 6061-T6 aluminum bar has a square cross section of 25 mm by 25 mm. If it is 2 m long, determine the maximum shear stress in the bar and the rotation of one end relative to the other end.

If the shaft is subjected to the torque of \(2 \text { kN } \cdot \text { m }\), determine the maximum shear stress developed in the shaft. Also, find the angle of twist of end B. The shaft is made from A-36 steel. Set a = 50 mm.

If the shaft is made from A-36 steel having an allowable shear stress of \(\tau_{\text {allow }}=75 \mathrm{MPa}\), determine the minimum dimension a for the cross section to the nearest millimeter. Also, find the corresponding angle of twist at end B.

If the solid shaft is made from red brass C83400 having an allowable shear stress of \(\tau_{\text {allow }}=4 \mathrm{ksi}\), determine the maximum allowable torque T that can be applied at B.

If the solid shaft is made from red brass C83400 and it is subjected to a torque \(T=6 \mathrm{kip} \cdot \mathrm{ft}\) at B, determine the maximum shear stress developed in segments AB and BC.

For a given maximum average shear stress, determine the factor by which the torque-carrying capacity is increased if the half-circular section is reversed from the dashed-line position to the section shown. The tube is 0.1 in. thick.

The plastic tube is subjected to a torque of \(150 \mathrm{~N} \cdot \mathrm{m}\). Determine the mean dimension a of its sides if the allowable shear stress is \(\tau_{\text {allow }}=60 \mathrm{MPa}\). Each side has a thickness of t = 3 mm.

The plastic tube is subjected to a torque of \(150 \mathrm{~N} \cdot \mathrm{m}\). Determine the average shear stress in the tube if the mean dimension a = 200 mm. Each side has a thickness of t = 3 mm.

Due to a fabrication error the inner circle of the tube is eccentric with respect to the outer circle. By what percentage is the torsional strength reduced when the eccentricity e is one-fourth of the difference in the radii?

Determine the constant thickness of the rectangular tube if the average shear stress is not to exceed 12 ksi when a torque of \(T=100 \mathrm{kip} \cdot \mathrm{in}\). is applied to the tube. The mean dimensions of the tube are shown.

Determine the torque T that can be applied to the rectangular tube if the average shear stress is not to exceed 12 ksi. The mean dimensions of the tube are shown and the tube has a thickness of 0.25 in.

The steel tube has an elliptical cross section of mean dimensions shown and a constant thickness of t = 0.2 in. If the allowable shear stress is \(\tau_{\text {allow }}=8 \mathrm{ksi}\), and the tube is to resist a torque of \(T=250 \mathrm{lb} \cdot \mathrm{ft}\), determine the necessary dimension b. The mean area for the ellipse is \(A_{m}=\pi b(0.5 b)\).

The plastic hexagonal tube is subjected to a torque of \(150 \mathrm{~N} \cdot \mathrm{m}\). Determine the mean dimension a of its sides if the allowable shear stress is \(\tau_{\text {allow }}=60 \mathrm{MPa}\). Each side has a thickness of t = 3 mm.

The 304 stainless steel tube has a thickness of 10 mm. If the allowable shear stress is \(\tau_{\text {allow }}=80 \mathrm{MPa}\), determine the maximum torque T that it can transmit. Also, what is the angle of twist of one end of the tube with respect to the other if the tube is 4 m long? The mean dimensions are shown.

The 304 stainless steel tube has a thickness of 10 mm. If the applied torque is \(T=50 \mathrm{~N} \cdot \mathrm{m}\), determine the average shear stress in the tube. The mean dimensions are shown.

The symmetric tube is made from a high-strength steel, having the mean dimensions shown and a thickness of 5 mm. If it is subjected to a torque of \(T=40 \mathrm{~N} \cdot \mathrm{m}\), determine the average shear stress developed at points A and B. Indicate the shear stress on volume elements located at these points.

The steel step shaft has an allowable shear stress of \(\tau_{\text {allow }}=8 \mathrm{MPa}\). If the transition between the cross sections has a radius r = 4 mm, determine the maximum torque T that can be applied.

The built-up shaft is to be designed to rotate at 450 rpm while transmitting 230 kW of power. Is this possible? The allowable shear stress is \(\tau_{\text {allow }}=150 \mathrm{MPa}\).

The built-up shaft is designed to rotate at 450 rpm. If the radius of the fillet weld connecting the shafts is r = 13.2 mm, and the allowable shear stress for the material \(\tau_{\text {allow }}=150 \mathrm{MPa}\) determine the maximum power the shaft can transmit.

The shaft is used to transmit 30 hp while turning at 600 rpm. Determine the maximum shear stress in the shaft. The segments are connected together using a fillet weld having a radius of 0.18 in.

The shaft is fixed to the wall at A and is subjected to the torques shown. Determine the maximum shear stress in the shaft. A fillet weld having a radius of 2.75 mm is used to connect the shafts at B.

A solid shaft has a diameter of 40 mm and length of 1 m. It is made from an elastic-plastic material having a yield stress of \(\tau_{Y}=100 \mathrm{MPa}\). Determine the maximum elastic torque \(T_{Y}\) and the corresponding angle of twist. What is the angle of twist if the torque is increased to \(T=1.2 T_{Y}\)? G = 80 GPa.

The stepped shaft is subjected to a torque T that produces yielding on the surface of the larger diameter segment. Determine the radius of the elastic core produced in the smaller diameter segment. Neglect the stress concentration at the fillet.

Determine the torque needed to twist a short 2-mm-diameter steel wire through several revolutions if it is made from steel assumed to be elastic perfectly plastic and having a yield stress of \(\tau_{Y}=50 \mathrm{MPa}\). Assume that the material becomes fully plastic.

A bar having a circular cross section of 3 in. diameter is subjected to a torque of \(100 \ \mathrm{in}. \cdot \ \mathrm{kip}\). If the material is elastic perfectly plastic, with \(\tau_{Y}=16 \ \mathrm{ksi}\), determine the radius of the elastic core.

The solid shaft is made of an elastic perfectly plastic material. Determine the torque T needed to form an elastic core in the shaft having a radius of \(\rho_{Y}=20 \mathrm{~mm}\). If the shaft is 3 m long, through what angle does one end of the shaft twist with respect to the other end? When the torque is removed, determine the residual stress distribution in the shaft and the permanent angle of twist.

The shaft is subjected to a maximum shear strain of 0.0048 rad. Determine the torque applied to the shaft.

A circular shaft having a diameter of 4 in. is subjected to a torque of \(250 \ \mathrm{kip} \cdot \mathrm{in}\). If the material is elastic perfectly plastic, with \(\tau_Y=16 \ \mathrm{ksi}\), determine the radius of the elastic core.

The hollow shaft has the cross section shown and is made of an elastic perfectly plastic material having a yield shear stress of \(\tau_{Y}\). Determine the ratio of the plastic torque \(T_{p}\) to the maximum elastic torque \(T_{Y}\).

The 2-m-long tube is made of an elastic perfectly plastic material as shown. Determine the applied torque T, which subjects the material at the tube’s outer edge to a shear strain of \(\gamma_{\max }=0.006 \ \mathrm{rad}\). What would be the permanent angle of twist of the tube when this torque is removed? Sketch the residual stress distribution in the tube.

The tube has a length of 2 m and is made of an elastic perfectly plastic material as shown. Determine the torque needed to just cause the material to become fully plastic. What is the permanent angle of twist of the tube when this torque is removed?

The 3-in.-diameter shaft is made of an elastic perfectly plastic material as shown. Determine the radius of its elastic core if it is subjected to a torque of \(T=18 \mathrm{kip} \cdot \mathrm{ft}\). If the shaft is 3 ft long, determine the angle of twist.

The shaft is made of an elastic perfectly plastic material as shown. Plot the shear-stress distribution acting along a radial line if it is subjected to a torque of \(T=20 \ \mathrm{kN} \cdot \mathrm{m}\). What is the residual stress distribution in the shaft when the torque is removed?

A torque is applied to the shaft having a radius of 80 mm. If the material obeys a shear stress–strain relation of \(\tau=500 \ \gamma^{1 / 4} \ \mathrm{MPa}\), determine the torque that must be applied to the shaft so that the maximum shear strain becomes 0.008 rad.

A tubular shaft has an inner diameter of 60 mm, an outer diameter of 80 mm, and a length of 1 m. It is made of an elastic perfectly plastic material having a yield stress of \(\tau_{\text {Y }}=150 \ \mathrm{MPa}\). Determine the maximum torque it can transmit. What is the angle of twist of one end with respect to the other end if the inner surface of the tube is about to yield? G = 75 GPa.

The stepped shaft is subjected to a torque T that produces yielding on the surface of the larger diameter segment. Determine the radius of the elastic core produced in the smaller diameter segment. Neglect the stress concentration at the fillet.

The tube has a length of 2 m and is made of an elastic perfectly plastic material as shown. Determine the torque needed to just cause the material to become fully plastic. What is the permanent angle of twist of the tube when this torque is removed?

A shaft of radius c = 0.75 in. is made from an elastic perfectly plastic material as shown. Determine the torque T that must be applied to its ends so that it has an elastic core of radius \(\rho=0.6 \text { in }\). If the shaft is 30 in. long, determine the angle of twist.

The shaft consists of two sections that are rigidly connected. If the material is elastic perfectly plastic as shown, determine the largest torque T that can be applied to the shaft. Also, draw the shear-stress distribution over a radial line for each section. Neglect the effect of stress concentration.

A steel alloy core is bonded firmly to the copper alloy tube to form the shaft shown. If the materials have the \(\tau-\gamma\) diagrams shown, determine the torque resisted by the core and the tube.

The shear stress–strain diagram for a solid 50-mmdiameter shaft can be approximated as shown in the figure. Determine the torque required to cause a maximum shear stress in the shaft of 125 MPa. If the shaft is 3 m long, what is the corresponding angle of twist?