The tapered shaft is confined by the fixed supports at A

The solid shaft of radius r is subjected to a torque T. Determine the radius \(r^\prime\) 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 \(r^\prime\) 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.

The solid shaft is fixed to the support at C and subjected to the 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 copper pipe has an outer diameter of 40 mm and an inner diameter of 37 mm. If it is tightly secured to the wall at A and three torques are applied to it as shown, 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 at C and three torques are applied to it as shown, 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 shaft has a diameter of 0.75 in. If it is subjected to the torques shown, determine the maximum shear stress developed in regions BC and DE of the shaft. The bearings at A and F allow free rotation of the shaft.

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

The solid shaft is fixed to the support at C and subjected to the torsional loadings shown. 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 coupling is used to connect the two shafts together. Assuming that the shear stress in the bolts is uniform, determine the number of bolts necessary to make the maximum shear stress in the shaft equal to the shear stress in the bolts. Each bolt has a diameter d.

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 developed in each section of the pipe when the couple shown is applied to the handles of the wrench.

The 150-mm-diameter shaft is supported by a smooth journal bearing at E and a smooth thrust bearing at F. Determine the maximum shear stress developed in each segment of the shaft.

If the tubular shaft is made from material having an allowable shear stress of \(\tau_\text{allow} = 85 \text{ MPa}\), determine the required minimum wall thickness of the shaft to the nearest millimeter. The shaft has an outer diameter of 150 mm.

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 \(\frac{1}{8}\) in. if the allowable shear stress is \(\tau_\text{allow} = 10 \text{ ksi}\).

The solid shaft is made of material that has an allowable shear stress of \(\tau_\text{allow} = 10 \text{ MPa}\). Determine the required diameter of the shaft to the nearest millimeter.

The solid shaft has a diameter of 40 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.

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 shaft consists of solid 80-mm-diameter rod segments AB and CD, and the tubular segment BC which has an outer diameter of 100 mm and inner diameter of 80 mm. If the material has an allowable shear stress of \(\tau_\text{allow} = 75 \text{ MPa}\), determine the maximum allowable torque T that can be applied to the shaft.

The shaft consists of rod segments AB and CD , and the tubular segment BC. If the torque \(T = 10~ \mathrm{kN \cdot m}\) is applied to the shaft, determine the required minimum diameter of the rod and the maximum inner diameter of the tube. The outer diameter of the tube is 120 mm, and the material has an allowable shear stress of \(\tau_\text{allow} = 75 \text{ MPa}\).

If the 40-mm-diameter rod is subjected to a uniform distributed torque of \(t_0 = 1.5~ \mathrm{kN \cdot m/m}\), determine the shear stress developed at point C.

If the rod is subjected to a uniform distributed torque of \(t_0 = 1.5~ \mathrm{kN \cdot m/m}\), determine the rod’s minimum required diameter d if the material has an allowable shear stress of \(\tau_\text{allow} = 75 \text{ MPa}\).

If the 40-mm diameter rod is made from a material having an allowable shear stress of \(\tau_\text{allow} = 75 \text{ MPa}\), determine the maximum allowable intensity \(t_0\) of the uniform distributed torque.

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 at C and a uniformly distributed torque is applied to it as shown, 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 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 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.

A cylindrical spring consists of a rubber annulus bonded to a rigid ring and shaft. If the ring is held fixed and a torque T is applied to the shaft, determine the maximum shear stress in the rubber.

The assembly consists of the solid rod AB, tube BC, and the lever arm. If the rod and the tube are made of material having an allowable shear stress of \(\tau_\text{allow} = 12 \text{ ksi}\), determine the maximum allowable torque T that can be applied to the end of the rod and from there the couple forces P that can be applied to the lever arm. The diameter of the rod is 2 in., and the outer and inner diameters of the tube are 4 in. and 2 in., respectively.

The assembly consists of the solid rod AB, tube BC, and the lever arm. If a torque of \(T = 20~ \mathrm{kip \cdot in.}\) is applied to the rod and couple forces of P = 5 kip are applied to the lever arm, determine the required diameter for the rod, and the outer and inner diameters of the tube, if the ratio of the inner diameter \(d_i\), to outer diameter \(d_o\), is required to be \(d_i / d_o = 0.75\). The rod and the tube are made of material having an allowable shear stress of \(\tau_\text{allow} = 12 \text{ ksi}\).

The steel shafts are connected together using a fillet weld as shown. Determine the average shear stress in the weld along section a–a if the torque applied to the shafts is \(T = 60~ \mathrm{N \cdot m}\). Note: The critical section where the weld fails is along section a–a.

The shaft has a diameter of 80 mm. Due to friction at its surface within the hole, it is subjected to a variable torque described by the function \(t = (25xe^{x^2})~ \mathrm{N \cdot m/m}\), where x is in meters. Determine the minimum torque \(T_0\) needed to overcome friction and cause it to twist. Also, determine the absolute maximum stress in the shaft.

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 developed 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 developed in the 20-mm-diameter transmission shaft at A.

The motor M is connected to the speed reducer C by the tubular shaft and coupling. If the motor supplies 20 hp and rotates the shaft at a rate of 600 rpm, determine the minimum inner and outer diameters \(d_i\) and \(d_o\) of the shaft if \(d_i /d_o = 0.75\). The shaft is made from a material having an allowable shear stress of \(\tau_\text{allow} = 12 \text{ ksi}\).

The motor M is connected to the speed reducer C by the tubular shaft and coupling. The shaft has an outer and inner diameter of 1 in. and 0.75 in., respectively, and is made from a material having an allowable shear stress of \(\tau_\text{allow} = 12 \text{ ksi}\), when the motor supplies 20 hp of power. Determine the smallest allowable angular velocity of the shaft.

The 25-mm-diameter shaft on the motor is made of a material having an allowable shear stress of \(\tau_\text{allow} = 75 \text{ MPa}\). If the motor is operating at its maximum power of 5 kW, determine the minimum allowable rotation of the shaft.

The drive shaft of the motor is made of a material having an allowable shear stress of \(\tau_\text{allow} = 75 \text{ MPa}\). If the outer diameter of the tubular shaft is 20 mm and the wall thickness is 2.5 mm, determine the maximum allowable power that can be supplied to the motor when the shaft is operating at an angular velocity of 1500 rev/min.

A ship has a propeller drive shaft that is turning at 1500 rev/min while developing 1800 hp. If it is 8 ft long and has a diameter of 4 in., determine the maximum shear stress in the shaft caused by torsion.

The motor A develops a power of 300 W and turns its connected pulley at 90 rev/min. Determine the required diameters of the steel shafts on the pulleys at A and B if the allowable shear stress is \(\tau_\text{allow} = 85 \text{ MPa}\).

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 developed in the shaft within regions CF and BC . The shaft is free to turn in its support bearings D and E.

Determine the absolute maximum shear stress developed in the shaft in Prob. 5–39 .

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 \(\tau_\text{allow} = 80 \text{ MPa}\).

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 can have if the allowable shear stress is \(\tau_\text{allow} = 80 \text{ MPa}\).

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 rod has a diameter of 0.5 in. and weight of 5 lb/ft. Determine the maximum torsional stress in the rod at a section located at A due to the rod’s weight.

Solve Prob. 5–44 for the maximum torsional stress 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 allowable shear stress for the material is \(\tau_\text{allow} = 25 \text{ ksi}\).

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 developed in the shaft is \(\gamma_\max = Tc/JG\). What is the shear strain on an element located at point A, c/2 from the center of the shaft? Sketch the strain distortion of this element.

The A-36 steel axle is made from tubes AB and CD and a solid section BC. It is supported on smooth bearings that allow it to rotate freely. If the gears, fixed to its ends, are subjected to \(85 \text{-N}\cdot \text m\) torques, determine the angle of twist of gear A relative to gear D. The tubes have an outer diameter of 30 mm and an inner diameter of 20 mm. The solid section has a diameter of 40 mm.

The hydrofoil boat has an A992 steel propeller shaft that is 100 ft long. It is connected to an in-line diesel engine that delivers a maximum power of 2500 hp and causes the shaft to rotate at 1700 rpm. If the outer diameter of the shaft is 8 in. and the wall thickness is \(\frac{3}{8}\) in., determine the maximum shear stress developed in the shaft. Also, what is the “wind up,” or angle of twist in the shaft at full power?

The 60-mm-diameter shaft is made of 6061-T6 aluminum having an allowable shear stress of \(\tau_\text{allow} = 80 \text{ MPa}\). 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 \(\tau_\text{allow} = 80 \text{ MPa}\), 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 20-mm-diameter A-36 steel shaft is subjected to the torques shown. Determine the angle of twist of the end B.

The shaft is made of A992 steel with the allowable shear stress of \(\tau_\text{allow} = 75 \text{ MPa}\). 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 \(\tau_\text{allow} = 75 \text{ MPa}\), 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 A-36 steel axle is made from tubes AB and CD and a solid section BC. It is supported on smooth bearings that allow it to rotate freely. If the gears, fixed to its ends, are subjected to \(85 \text{-N}\cdot \text{m}\) torques, determine the angle of twist of the end B of the solid section relative to end C. The tubes have an outer diameter of 30 mm and an inner diameter of 20 mm. The solid section has a diameter of 40 mm.

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 \(\omega = 800 \text{ rev/min.}\), 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 \(\omega = 500 \text{ rev/min.}\), 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 two shafts are made of A992 steel. Each has a diameter of 1 in., and they are supported by bearings at A, B, and C, which allow free rotation. If the support at D is fixed, determine the angle of twist of end B when the torques are applied to the assembly as shown.

The two shafts are made of A992 steel. Each has a diameter of 1 in., and they are supported by bearings at A, B, and C, which allow free rotation. If the support at D is fixed, determine the angle of twist of end A when the torques are applied to the assembly as shown.

If the shaft is made of red brass C83400 copper with an allowable shear stress of \(\tau_\text{allow} = 20 \text{ MPa}\), determine the maximum allowable torques \(T_1\) and \(T_2\) that can be applied at A and B. Also, find the corresponding angle of twist of end A. Set L = 0.75 m.

If the shaft is made of red brass C83400 copper and is subjected to torques \(T_1=20~ \mathrm{kN \cdot m}\) and \(T_2=50~ \mathrm{kN \cdot m}\), determine the distance L so that the angle of twist at end A is zero.

The 8-mm·diameter A-36 steel bolt is screwed tightly into a block at A. Determine the couple forces F that should be applied to the wrench so that the maximum shear stress in the bolt becomes 18 MPa. Also, compute the corresponding displacement of each force F needed to cause this stress. Assume that the wrench is rigid.

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 \(\tau_\text{allow} = 140 \text{ MPa}\) 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 the shaft can have if it is restricted not to twist more than \(1^\circ\).

If the shaft is subjected to a uniform distributed torque \(t_0\), determine the angle of twist at A. The material has a shear modulus G. The shaft is hollow for exactly half its length.

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 \(t = (kx^2)~ \mathrm{N \cdot m/m}\), where x is in meters. If a torque of \(T = 50~ \mathrm{N \cdot m}\) 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.

Solve Prob. 5–69 if the distributed torque is \(t = (kx^{2/3})~ \mathrm{N \cdot m/m}\).

Consider the general problem of a circular shaft made from m segments, each having a radius of \(c_m\) and shear modulus \(G_m\). If there are n torques on the shaft as shown, write a computer program that can be used to determine the angle of twist of its end A. Show an application of the program using the values \(L_1 = 0.5 \text{ m}\), \(c_1=0.02 \text{ m\), \(G_1=30 \text{ GPa}\), \(L_2=1.5 \text{ m}\), \(c_2=0.05 \text{ m}\), \(G_2=15 \text{ GPa}\), \(T_1= -450~ \mathrm{N \cdot m}\), \(d_1=0.25 \text{ m}\), \(T_2= 600~ \mathrm{N \cdot m}\), \(d_2=0.8 \text{ m}\).

The 80-mm diameter shaft is made of 6061-T6 aluminum alloy and subjected to the torsional loading shown. Determine the angle of twist at end A.

The contour of the surface of the shaft is defined by the equation \(y = e^{ax}\), 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 rod ABC of radius c is embedded into a medium where the distributed torque reaction varies linearly from zero at C to \(t_0\) at B. If couple forces P are applied to the lever arm, determine the value of \(t_0\) for equilibrium. Also, find the angle of twist of end A. The rod is made from material having a shear modulus of G.

The A992 steel posts are “drilled” at constant angular speed into the soil using the rotary installer. If the post has an inner diameter of 200 mm and an outer diameter of 225 mm, determine the relative angle of twist of end A of the post with respect to end B when the post reaches the depth indicated. Due to soil friction, assume the torque along the post varies linearly as shown, and a concentrated torque of \(80~ \mathrm{kN \cdot m}\) acts at the bit.

A cylindrical spring consists of a rubber annulus bonded to a rigid ring and shaft. If the ring is held fixed and a torque T is applied to the rigid shaft, determine the angle of twist of the shaft. The shear modulus of the rubber is G. Hint: As shown in the figure, the deformation of the element at radius r can be determined from \(rd \theta = dr \gamma\). Use this expression, along with \(\tau=T/(2 \pi r^2 h)\) from Prob. 5–26 , to obtain the result.

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.\(G_\text{st} = 75 \text{ GPa}\).

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 \(500~ \mathrm{lb \cdot ft}\), determine the maximum shear stress in the shaft. \(G_\text{st} = 10.8(10^3) \text{ ksi}\).

The shaft is made of A-36 steel and is fixed at its ends A and D. Determine the torsional reactions at these supports.

The shaft is made of A-36 steel and is fixed at end D, while end A is allowed to rotate 0.005 rad when the torque is applied. Determine the torsional reactions at these supports.

The shaft is made from a solid steel section AB and a tubular portion made of steel and having a brass core. If it is fixed to a rigid support at A, and a torque of \(T=50 \ \mathrm{lb} \cdot \mathrm{ft}\) is applied to it at C, determine the angle of twist that occurs at C and compute the maximum shear stress and maximum shear strain in the brass and steel. Take \(G_{\mathrm{st}}=11.5\left(10^{3}\right) \mathrm{ksi}\), \(G_{\mathrm{br}}=5.6\left(10^{3}\right) \mathrm{ksi}\).

The motor A develops a torque at gear B of \(450 \ \mathrm{lb} \cdot \mathrm{ft}\), which is applied along the axis of the 2-in.-diameter steel shaft CD. This torque is to be transmitted to the pinion gears at E and F. If these gears are temporarily fixed, determine the maximum shear stress in segments CB and BD of the shaft. Also, what is the angle of twist of each of these segments? The bearings at C and D only exert force reactions on the shaft and do not resist torque. \(G_{\mathrm{st}}=12\left(10^{3}\right) \mathrm{ksi}\).

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 \(\left(\tau_{\text {allow }}\right)_{\mathrm{st}}=75 \ \mathrm{MPa}\), 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 as shown, determine the reactions at A and B.

Determine the rotation of the gear at E in Prob. 5–86.

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. \(G_{\mathrm{st}}=75 \ \mathrm{GPa}\), \(G_{\mathrm{br}}=39 \ \mathrm{GPa}\).

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

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 of \(800 \ \mathrm{lb} \cdot \mathrm{ft}\). The material is A-36 steel.

The A992 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 uniform distributed torque of \(60 \ \mathrm{lb} \cdot \mathrm{in} . / \mathrm{in}\). along segment CB, determine the absolute maximum shear stress in the shaft.

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 aluminum rod has a square cross section of 10 mm by 10 mm. If it is 8 m long, determine the torque T that is required to rotate one end relative to the other end by \(90^{\circ}\). \(G_{\mathrm{al}}=28 \ \mathrm{GPa}, \left(\tau_{Y}\right)_{\mathrm{al}}=240 \ \mathrm{MPa}\).

The shafts have elliptical and circular cross sections and are to be made from the same amount of a similar material. Determine the percent of increase in the maximum shear stress and the angle of twist for the elliptical shaft compared to the circular shaft when both shafts are subjected to the same torque and have the same length.

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 shown, determine the maximum shear stress within regions AC and BC , and the angle of twist \(\phi\) 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 \(\phi\) of end B relative to C.

If end B of the shaft, which has an equilateral triangle cross section, is subjected to a torque of \(T=900 \ \mathrm{lb} \cdot \mathrm{ft}\), determine the maximum shear stress developed 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 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 torque of \(80 \ \mathrm{lb} \cdot \mathrm{ft}\) at C, determine the reactions at the fixed supports. Also, what is the angle of twist at C? \(G_{\mathrm{al}}=3.8\left(10^{3}\right) \mathrm{ksi}\).

A torque of \(2 \ \mathrm{kip} \cdot \text { in }\). 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 \(3 \ \mathrm{kN} \cdot \mathrm{m}\) 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 copper 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 copper 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.

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

A torque T is applied to two tubes having the cross sections shown. Compare the shear flow developed in each tube.

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 average stress is not to exceed 12 ksi when a torque of \(T=20 \ \mathrm{kip} \cdot \text { in }\). is applied to the tube. Neglect stress concentrations at the corners. 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. Neglect stress concentrations at the corners. The mean dimensions of the tube are shown and the tube has a thickness of 0.125 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 tube is made of plastic, is 5 mm thick, and has the mean dimensions shown. Determine the average shear stress at points A and B if the tube is subjected to the torque of \(T=500 \ \mathrm{N} \cdot \mathrm{m}\). Show the shear stress on volume elements located at these points. Neglect stress concentrations at the corners.

The mean dimensions of the cross section of the leading edge and torsion box of an airplane wing can be approximated as shown. If the wing is made of 2014-T6 aluminum alloy having an allowable shear stress of \(\tau_{\text {allow }}=125 \ \mathrm{MPa}\) and the wall thickness is 10 mm, determine the maximum allowable torque and the corresponding angle of twist per meter length of the wing.

The mean dimensions of the cross section of the leading edge and torsion box of an airplane wing can be approximated as shown. If the wing is subjected to a torque of \(4.5 \ \mathrm{MN} \cdot \mathrm{m}\) and the wall thickness is 10 mm, determine the average shear stress developed in the wing and the angle of twist per meter length of the wing. The wing is made of 2014-T6 aluminum alloy.

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 step shaft is to be designed to rotate at 720 rpm while transmitting 30 kW of power. Is this possible? The allowable shear stress is \(\tau_{\text {allow }}=12 \ \mathrm{MPa}\) and the radius at the transition on the shaft is 7.5 mm.

The built-up shaft is designed to rotate at 540 rpm. If the radius at the transition on the shaft is r = 7.2 mm, and the allowable shear stress for the material is \(\tau_{\text {allow }}=55 \ \mathrm{MPa}\), determine the maximum power the shaft can transmit.

The transition at the cross sections of the step shaft has a radius of 2.8 mm. Determine the maximum shear stress developed in the shaft.

The steel used for the step shaft has an allowable shear stress of \(\tau_{\text {allow }}=8 \ \mathrm{MPa}\). If the radius at the transition between the cross sections is r = 2.25 mm, determine the maximum torque T that can be applied.

The step shaft is subjected to a torque of \(710 \ \mathrm{lb} \cdot \mathrm{in}\). If the allowable shear stress for the material is \(\tau_{\text {allow }}=12 \ \mathrm{ksi}\), determine the smallest radius at the junction between the cross sections that can be used to transmit the torque.

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.

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-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 \(100 \text { in. } \cdot \text { kip }\). If the material is elastic-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 as shown. 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 if the material has strain hardening as shown by the shear stress–strain diagram.

An 80-mm-diameter solid circular shaft is made of an elastic-perfectly plastic material having a yield shear stress of \(\tau_{Y}=125 \ \mathrm{MPa}\). Determine (a) the maximum elastic torque \(T_{Y}\); and (b) the plastic torque \(T_{p}\).

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 \(T_{Y}\).

If the step shaft is elastic-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.

The solid shaft is made from an elastic-plastic material as shown. Determine the torque T needed to form an elastic core in the shaft having a radius of \(\rho_{Y}=23 \ \mathrm{mm}\). If the shaft is 2 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.

A 1.5-in.-diameter shaft is made from an elastic-plastic material as shown. Determine the radius of its elastic core if it is subjected to a torque of \(T=200 \ \mathrm{lb} \cdot \mathrm{ft}\). If the shaft is 10 in. long, determine the angle of twist.

The tubular shaft is made of a strain-hardening material having a \(\tau-\gamma\) diagram as shown. Determine the torque T that must be applied to the shaft so that the maximum shear strain is 0.01 rad.

The shaft is made from a strain-hardening material having a \(\tau-\gamma\) diagram as shown. Determine the torque T that must be applied to the shaft in order to create an elastic core in the shaft having a radius of \(\rho_{c}=0.5 \ \mathrm{in}\).

The tube is made of elastic-perfectly plastic material, which has the \(\tau-\gamma\) diagram shown. Determine the torque T that just causes the inner surface of the shaft to yield. Also, find the residual shear-stress distribution in the shaft when the torque is removed.

The shear stress–strain diagram for a solid 50-mm-diameter 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?

The 2-m-long tube is made of an elastic-perfectly plastic material as shown. Determine the applied torque T that 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.

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 2-m-long tube is made from an elastic-plastic material as shown. Determine the applied torque T, which subjects the material of the tube’s outer edge to a shearing strain, of \(\gamma_{\max }=0.008 \ \mathrm{rad}\). What would be the permanent angle of twist of the tube when the torque is removed? Sketch the residual stress distribution of the tube.

The shaft is made of A992 steel and has an allowable shear stress of \(\tau_{\text {allow }}=75 \ \mathrm{MPa}\). When the shaft is rotating at 300 rpm, the motor supplies 8 kW of power, while gears A and B withdraw 5 kW and 3 kW, respectively. Determine the required minimum diameter of the shaft to the nearest millimeter. Also, find the rotation of gear A relative to C.

The shaft is made of A992 steel and has an allowable shear stress of \(\tau_{\text {allow }}=75 \ \mathrm{MPa}\). When the shaft is rotating at 300 rpm, the motor supplies 8 kW of power, while gears A and B withdraw 5 kW and 3 kW, respectively. If the angle of twist of gear A relative to C is not allowed to exceed 0.03 rad, determine the required minimum diameter of the shaft to the nearest millimeter.

The A-36 steel circular tube is subjected to a torque of \(10 \ \mathrm{kN} \cdot \mathrm{m}\). Determine the shear stress at the mean radius \(\rho=60 \mathrm{~mm}\) and compute the angle of twist of the tube if it is 4 m long and fixed at its far end. Solve the problem using Eqs. 5–7 and 5–15 and by using Eqs. 5–18 and 5–20.

A portion of an airplane fuselage can be approximated by the cross section shown. If the thickness of its 2014-T6-aluminum skin is 10 mm, determine the maximum wing torque T that can be applied if \(\tau_{\text {allow }}=4 \ \mathrm{MPa}\). Also, in a 4-m long section, determine the angle of twist.

The material of which each of three shafts is made has a yield stress of \(\tau_{Y}\) and a shear modulus of G. Determine which shaft geometry will resist the largest torque without yielding. What percentage of this torque can be carried by the other two shafts? Assume that each shaft is made of the same amount of material and that it has the same cross-sectional area A.

Segments AB and BC of the assembly are made from 6061-T6 aluminum and A992 steel, respectively. If couple forces P = 3 kip are applied to the lever arm, determine the maximum shear stress developed in each segment. The assembly is fixed at A and C.

Segments AB and BC of the assembly are made from 6061-T6 aluminum and A992 steel, respectively. If the allowable shear stress for the aluminum is \(\left(\tau_{\text {allow }}\right)_{a l}=12 \ \mathrm{ksi}\) and for the steel \(\left(\tau_{\text {allow }}\right)_{s t}=10 \ \mathrm{ksi}\), determine the maximum allowable couple forces P that can be applied to the lever arm. The assembly is fixed at A and C.

The tapered shaft is made from 2014-T6 aluminum alloy, and has a radius which can be described by the function \(r=0.02\left(1+x^{3 / 2}\right) \mathrm{m}\), where x is in meters. Determine the angle of twist of its end A if it is subjected to a torque of \(450 \mathrm{~N} \cdot \mathrm{m}\).

The 60-mm-diameter shaft rotates at 300 rev/min. This motion is caused by the unequal belt tensions on the pulley of 800 N and 450 N. Determine the power transmitted and the maximum shear stress developed in the shaft.