A 1.5-m-long tubular steel shaft of 38-mm outer diameter

Determine the torque T that causes a maximum shearing stress of 70 MPa in the steel cylindrical shaft shown.

For the cylindrical shaft shown, determine the maximum shearing stress caused by a torque of magnitude \(T=800 \mathrm{~N} \cdot \mathrm{m}\).

(a) Determine the torque T that causes a maximum shearing stress of 45 MPa in the hollow cylindrical steel shaft shown. (b) Determine the maximum shearing stress caused by the same torque T in a solid cylindrical shaft of the same cross-sectional area.

(a) Determine the maximum shearing stress caused by a 40-kip?in. torque T in the 3-in.-diameter solid aluminum shaft shown. (b) Solve part a, assuming that the solid shaft has been replaced by a hollow shaft of the same outer diameter and of 1-in. inner diameter.

(a) For the 3-in.-diameter solid cylinder and loading shown, determine the maximum shearing stress. (b) Determine the inner diameter of the 4-in.-diameter hollow cylinder shown, for which the maximum stress is the same as in part a.

A torque \(T=3 \mathrm{kN} \cdot \mathrm{m}\) is applied to the solid bronze cylinder shown. Determine (a) the maximum shearing stress, (b) the shearing stress at point D, which lies on a 15-mm-radius circle drawn on the end of the cylinder, (c) the percent of the torque carried by the portion of the cylinder within the 15-mm radius.

The solid spindle AB is made of a steel with an allowable shearing stress of 12 ksi, and sleeve CD is made of a brass with an allowable shearing stress of 7 ksi. Determine (a) the largest torque T that can be applied at A if the allowable shearing stress is not to be exceeded in sleeve CD, (b) the corresponding required value of the diameter \(d_{s}\) of spindle AB.

The solid spindle AB has a diameter \(d_{s}=1.5 \mathrm{in}\). and is made of a steel with an allowable shearing stress of 12 ksi, while sleeve CD is made of a brass with an allowable shearing stress of 7 ksi. Determine the largest torque T that can be applied at A.

The torques shown are exerted on pulleys A, B, and C. Knowing that both shafts are solid, determine the maximum shearing stress in (a) shaft AB, (b) shaft BC.

The shafts of the pulley assembly shown are to be redesigned. Knowing that the allowable shearing stress in each shaft is 8.5 ksi, determine the smallest allowable diameter of (a) shaft AB, (b) shaft BC.

Knowing that each of the shafts AB, BC, and CD consist of a solid circular rod, determine (a) the shaft in which the maximum shearing stress occurs, (b) the magnitude of that stress.

Knowing that an 8-mm-diameter hole has been drilled through each of the shafts AB, BC, and CD, determine (a) the shaft in which the maximum shearing stress occurs, (b) the magnitude of that stress.

Under normal operating conditions, the electric motor exerts a torque of \(2.4 \mathrm{kN} \cdot \mathrm{m}\) on shaft AB. Knowing that each shaft is solid, determine the maximum shearing stress in (a) shaft AB, (b) shaft BC, (c) shaft CD.

In order to reduce the total mass of the assembly of Prob. 3.13, a new design is being considered in which the diameter of shaft BC will be smaller. Determine the smallest diameter of shaft BC for which the maximum value of the shearing stress in the assembly will not be increased.

The allowable shearing stress is 15 ksi in the 1.5-in.-diameter steel rod AB and 8 ksi in the 1.8-in.-diameter brass rod BC. Neglecting the effect of stress concentrations, determine the largest torque T that can be applied at A.

The allowable shearing stress is 15 ksi in the steel rod AB and ksi in the brass rod BC. Knowing that a torque of magnitude \(T=10 \text { kip } \cdot \text { in. }\) is applied at A, determine the required diameter of (a) rod AB, (b) rod BC.

The solid shaft shown is formed of a brass for which the allowable shearing stress is 55 MPa. Neglecting the effect of stress concentrations, determine the smallest diameters \(d_{A B} \text { and } d_{B C}\) for which the allowable shearing stress is not exceeded.

Solve Prob. 3.17 assuming that the direction of \(\mathbf{T}_{C}\) is reversed.

The solid rod AB has a diameter \(d_{A B}=60 \mathrm{~mm}\) and is made of a steel for which the allowable shearing stress is 85 MPa. The pipe CD, which has an outer diameter of 90 mm and a wall thickness of 6 mm, is made of an aluminum for which the allowable shearing stress is 54 MPa. Determine the largest torque T that can be applied at A.

The solid rod AB has a diameter \(d_{A B}=60 \mathrm{~mm}\). The pipe CD has an outer diameter of 90 mm and a wall thickness of 6 mm. Knowing that both the rod and the pipe are made of steel for which the allowable shearing stress is 75 MPa, determine the largest torque T that can be applied at A.

A torque of magnitude \(T=1000 \mathrm{~N} \cdot \mathrm{m}\) is applied at D as shown. Knowing that the allowable shearing stress is 60 MPa in each shaft, determine the required diameter of (a) shaft AB, (b) shaft CD.

A torque of magnitude \(T=1000 \mathrm{~N} \cdot \mathrm{m}\) is applied at D as shown. Knowing that the allowable shearing stress is 60 MPa in each shaft, determine the required of (a) shaft AB, (b) shaft CD.

Under normal operating conditions a motor exerts a torque of magnitude \(T_{F}\) at F. The shafts are made of a steel for which the allowable shearing stress is 12 ksi and have diameters \(d_{C D E}=0.900 \mathrm{in}\) and \(d_{F G H}=0.800 \mathrm{in}\) Knowing that \(r_{D}=6.5 \mathrm{in}\). and \(r_{G}=4.5 \mathrm{in}\)., determine the largest allowable value of \(T_{F}\).

Under normal operating conditions a motor exerts a torque of magnitude \(T_{F}=1200 \text { lb-in. }\) at F. Knowing that \(r_{D}=8 \text { in., } r_{G}=3 \text { in. }\), and the allowable shearing stress is 10.5 ksi in each shaft, determine the required diameter of (a) shaft CDE, (b) shaft FGH.

The two solid shafts are connected by gears as shown and are made of a steel for which the allowable shearing stress is 7000 psi. Knowing the diameters of the two shafts are, respectively, \(d_{B C}=1.6 \mathrm{in} . \text { and } d_{E F}=1.25 \mathrm{in} .\) determine the largest torque \(\mathbf{T}_{C}\) that can be applied at C.

The two solid shafts are connected by gears as shown and are made of a steel for which the allowable shearing stress is 8500 psi. Knowing that a torque of magnitude \(T_{C}=5 \mathrm{kip} \cdot \mathrm{in}\). is applied at C and that the assembly is in equilibrium, determine the required diameter of (a) shaft BC, (b) shaft EF.

For the gear train shown, the diameters of the three solid shafts are: \(d_{A B}=20 \mathrm{~mm} \quad d_{C D}=25 \mathrm{~mm} \quad d_{E F}=40 \mathrm{~mm}\) Knowing that for each shaft the allowable shearing stress is 60 MPa, determine the largest torque T that can be applied.

A torque \(T=900 \mathrm{~N} \cdot \mathrm{m}\) is applied to shaft AB of the gear train shown. Knowing that the allowable shearing stress is 80 MPa, determine the required diameter of (a) shaft AB, (b) shaft CD, (c) shaft EF.

While the exact distribution of the shearing stresses in a hollow-cylindrical shaft is as shown in Fig. P3.29a, an approximate value can be obtained for \(\tau_{\max }\) by assuming that the stresses are uniformly distributed over the area A of the cross section, as shown in Fig. P3.29b, and then further assuming that all of the elementary shearing forces act at a distance from O equal to the mean radius \(\frac{1}{2}\left(c_{1}+c_{2}\right)\) of the cross section. This approximate value is \(\tau_{0}=T / A r_{m}\), where T is the applied torque. Determine the ratio \(\tau_{\max } / \tau_{0}\) of the true value of the maximum shearing stress and its approximate value \(\tau_{0}\) for values of \(c_{1} / c_{2}\) respectively equal to 1.00, 0.95, 0.75, 0.50, and 0.

(a) For a given allowable shearing stress, determine the ratio T/w of the maximum allowable torque T and the weight per unit length w for the hollow shaft shown. (b) Denoting by \((T / w)_{0}\) the value of this ratio for a solid shaft of the same radius \(c_{2}\), express the ratio T/w for the hollow shaft in terms of \((T / w)_{0}\) and \(c_{1} / c_{2}\).

Determine the largest allowable diameter of a 3-m-long steel rod (G = 77.2 GPa) if the rod is to be twisted through \(30^{\circ}\) without exceeding a shearing stress of 80 MPa.

The ship at A has just started to drill for oil on the ocean floor at a depth of 5000 ft. Knowing that the top of the 8-in.-diameter steel drill pipe \(\left(G=11.2 \times 10^{6} \mathrm{psi}\right)\) rotates through two complete revolutions before the drill bit at B starts to operate, determine the maximum shearing stress caused in the pipe by torsion.

(a) For the solid steel shaft shown, determine the angle of twist at A. Use \(G=11.2 \times 10^{6} \mathrm{psi}\). (b) Solve part a, assuming that the steel shaft is hollow with a 1.5-in. outer radius and a 0.75-in. inner radius.

(a) For the aluminum pipe shown (G = 27 GPa), determine the torque \(\mathbf{T}_{0}\) causing an angle of twist of \(2^{\circ}\). (b) Determine the angle of twist if the same torque \(\mathbf{T}_{0}\) is applied to a solid cylindrical shaft of the same length and cross-sectional area.

The electric motor exerts a \(500 \mathrm{~N} \cdot \mathrm{m} \text {-torque }\) on the aluminum shaft ABCD when it is rotating at a constant speed. Knowing that G = 27 GPa and that the torques exerted on pulleys B and C are as shown, determine the angle of twist between (a) B and C, (b) B and D.

The torques shown are exerted on pulleys A and B. Knowing that the shafts are solid and made of steel (G = 77.2 GPa), determine the angle of twist between (a) A and B, (b) A and C.

The aluminum rod BC (G = 26 GPa) is bonded to the brass rod AB (G = 39 GPa). Knowing that each rod is solid and has a diameter of 12 mm, determine the angle of twist (a) at B, (b) at C.

The aluminum rod AB (G = 27 GPa) is bonded to the brass rod BD (G = 39 GPa). Knowing that portion CD of the brass rod is hollow and has an inner diameter of 40 mm, determine the angle of twist at A.

The solid spindle AB has a diameter \(d_{s}=1.75 \mathrm{in}\) and is made of a steel with \(G=11.2 \times 10^{6} \mathrm{psi}\) and \(\tau_{\text {all }}=12 \mathrm{ksi}\), while sleeve CD is made of a brass with G 5 5.6 3 106 psi and tall 5 7 ksi. Determine (a) the largest torque T that can be applied at A if the given allowable stresses are not to be exceeded and if the angle of twist of sleeve CD is not to exceed \(0.375^{\circ}\), (b) the corresponding angle through which end A rotates.

The solid spindle AB has a diameter \(d_{s}=1.5 \mathrm{in}\). and is made of a steel with \(G=11.2 \times 10^{6} \mathrm{psi} \text { and } \tau_{\text {all }}=12 \mathrm{ksi}\), while sleeve CD is made of a brass with \(G=5.6 \times 10^{6} \mathrm{psi} \text { and } \tau_{\text {all }}=7 \mathrm{ksi}\). Determine the largest angle through which end A can be rotated.

Two shafts, each of \(\frac{7}{8}-\text { in. }\) diameter, are connected by the gears shown. Knowing that \(G=11.2 \times 10^{6} \mathrm{psi}\) and that the shaft at F is fixed, determine the angle through which end A rotates when a \(1.2 \text { kip·in. }\) torque is applied at A.

Two solid steel shafts, each of 30-mm diameter, are connected by the gears shown. Knowing that G = 77.2 GPa, determine the angle through which end A rotates when a torque of magnitude \(T=200 \mathrm{~N} \cdot \mathrm{m}\) is applied at A.

A coder F, used to record in digital form the rotation of shaft A, is connected to the shaft by means of the gear train shown, which consists of four gears and three solid steel shafts each of diameter d. Two of the gears have a radius r and the other two a radius nr. If the rotation of the coder F is prevented, determine in terms of T, l, G, J, and n the angle through which end A rotates.

For the gear train described in Prob. 3.43, determine the angle through which end A rotates when \(T=5 \text { lb-in., } l=2.4 \text { in., }\) \(d=\frac{1}{16} \text { in., } G=11.2 \times 10^{6} \mathrm{psi}, \text { and } n=2\).

The design specifications of a 1.2-m-long solid circular transmission shaft require that the angle of twist of the shaft not exceed \(4^{\circ}\) when a torque of \(750 \mathrm{~N} \cdot \mathrm{m}\) is applied. Determine the required diameter of the shaft, knowing that the shaft is made of a steel with an allowable shearing stress of 90 MPa and a modulus of rigidity of 77.2 GPa.

The solid cylindrical rod BC of length L = 24 in. is attached to the rigid lever AB of length a = 15 in. and to the support at C. Design specifications require that the displacement of A not exceed 1 in. when a 100-lb force P is applied at A. For the material indicated, determine the required diameter of the rod. Steel: \(\tau_{\text {all }}=15 \mathrm{ksi}, G=11.2 \times 10^{6} \mathrm{psi}\)

The solid cylindrical rod BC of length L = 24 in. is attached to the rigid lever AB of length a = 15 in. and to the support at C. Design specifications require that the displacement of A not exceed 1 in. when a 100-lb force P is applied at A. For the material indicated, determine the required diameter of the rod. Aluminum: \(\tau_{\text {all }}=10 \mathrm{ksi}, G=3.9 \times 10^{6} \mathrm{psi}\)

The design of the gear-and-shaft system shown requires that steel shafts of the same diameter be used for both AB and CD. It is further required that \(\tau_{\max } \leq 60 \mathrm{MPa}\) and that the angle \(\phi_{D}\) through which end D of shaft CD rotates not exceed \(1.5^{\circ}\). Knowing that G = 77.2 GPa, determine the required diameter of the shafts.

The electric motor exerts a torque of \(800 \mathrm{~N} \cdot \mathrm{m}\) on the steel shaft ABCD when it is rotating at a constant speed. Design specifications require that the diameter of the shaft be uniform from A to D and that the angle of twist between A and D not exceed \(1.5^{\circ}\). Knowing that \(\tau_{\max } \leq 60 \mathrm{MPa}\) and G = 77.2 GPa, determine the minimum diameter shaft that can be used.

A hole is punched at A in a plastic sheet by applying a 600-N force P to end D of lever CD, which is rigidly attached to the solid cylindrical shaft BC. Design specifications require that the displacement of D should not exceed 15 mm from the time the punch first touches the plastic sheet to the time it actually penetrates it. Determine the required diameter of shaft BC if the shaft is made of a steel with G = 77.2 GPa and \(\tau_{\text {all }}=80 \mathrm{MPa}\).

The solid cylinders AB and BC are bonded together at B and are attached to fixed supports at A and C. Knowing that the modulus of rigidity is \(3.7 \times 10^{6} \mathrm{psi}\) for aluminum and \(5.6 \times 10^{6} \mathrm{psi}\) for brass, determine the maximum shearing stress (a) in cylinder AB, (b) in cylinder BC.

Solve Prob. 3.51, assuming that cylinder AB is made of steel, for which \(G=11.2 \times 10^{6} \mathrm{psi}\).

The composite shaft shown consists of a 0.2-in.-thick brass jacket \(\left(G=5.6 \times 10^{6} \mathrm{psi}\right)\) bonded to a 1.2-in.-diameter steel core \(\left(G_{\text {steel }}=11.2 \times 10^{6} \mathrm{psi}\right)\). Knowing that the shaft is subjected to \(5 \text { kip·in. }\) torques, determine (a) the maximum shearing stress in the brass jacket, (b) the maximum shearing stress in the steel core, (c) the angle of twist of end B relative to end A.

The composite shaft shown consists of a 0.2-in.-thick brass jacket \(\left(G=5.6 \times 10^{6} \mathrm{psi}\right)\) bonded to a 1.2-in.-diameter steel core \(\left(G_{\text {steel }}=11.2 \times 10^{6} \mathrm{psi}\right)\). Knowing that the shaft is being subjected to the torques shown, determine the largest angle through which it can be twisted if the following allowable stresses are not to be exceeded: \(\tau_{\text {steel }}=15 \mathrm{ksi} \text { and } \tau_{\text {brass }}=8 \mathrm{ksi}\).

Two solid steel shafts (G = 77.2 GPa) are connected to a coupling disk B and to fixed supports at A and C. For the loading shown, determine (a) the reaction at each support, (b) the maximum shearing stress in shaft AB, (c) the maximum shearing stress in shaft BC.

Solve Prob. 3.55, assuming that the shaft AB is replaced by a hollow shaft of the same outer diameter and 25-mm inner diameter.

Two solid steel shafts are fitted with flanges that are then connected by bolts as shown. The bolts are slightly undersized and permit a \(1.5^{\circ}\) rotation of one flange with respect to the other before the flanges begin to rotate as a single unit. Knowing that G = 77.2 GPa, determine the maximum shearing stress in each shaft when a torque of T of magnitude \(500 \mathrm{~N} \cdot \mathrm{m}\) is applied to the flange indicated. The torque T is applied to flange B.

Two solid steel shafts are fitted with flanges that are then connected by bolts as shown. The bolts are slightly undersized and permit a \(1.5^{\circ}\) rotation of one flange with respect to the other before the flanges begin to rotate as a single unit. Knowing that G = 77.2 GPa, determine the maximum shearing stress in each shaft when a torque of T of magnitude \(500 \mathrm{~N} \cdot \mathrm{m}\) is applied to the flange indicated. The torque T is applied to flange C.

The steel jacket CD has been attached to the 40-mm-diameter steel shaft AE by means of rigid flanges welded to the jacket and to the rod. The outer diameter of the jacket is 80 mm and its wall thickness is 4 mm. If 500-N?m torques are applied as shown, determine the maximum shearing stress in the jacket.

A torque T is applied as shown to a solid tapered shaft AB. Show by integration that the angle of twist at A is \(\phi=\frac{7 T L}{12 \pi G c^{4}}\)

The mass moment of inertia of a gear is to be determined experimentally by using a torsional pendulum consisting of a 6-ft steel wire. Knowing that \(G=11.2 \times 10^{6} \mathrm{psi}\), determine the diameter of the wire for which the torsional spring constant will be \(4.27 \mathrm{lb} \cdot \mathrm{ft} / \mathrm{rad}\).

A solid shaft and a hollow shaft are made of the same material and are of the same weight and length. Denoting by n the ratio \(c_{1} / c_{2}\), show that the ratio \(T_{s} / T_{h}\) of the torque \(T_{s}\) in the solid shaft to the torque \(T_{h}\) in the hollow shaft is (a) \(\sqrt{\left(1-n^{2}\right)} /\left(1+n^{2}\right)\) if the maximum shearing stress is the same in each shaft, (b) \(\left(1-n^{2}\right) /\left(1+n^{2}\right)\) if the angle of twist is the same for each shaft.

An annular plate of thickness t and modulus G is used to connect shaft AB of radius \(r_{1}\) to tube CD of radius \(r_{2}\). Knowing that a torque T is applied to end A of shaft AB and that end D of tube CD is fixed, (a) determine the magnitude and location of the maximum shearing stress in the annular plate, (b) show that the angle through which end B of the shaft rotates with respect to end C of the tube is \(\phi_{B C}=\frac{T}{4 \pi G t}\left(\frac{1}{r_{1}^{2}}-\frac{1}{r_{2}^{2}}\right)\)

Determine the maximum shearing stress in a solid shaft of 1.5-in. diameter as it transmits 75 hp at a speed of (a) 750 rpm, (b) 1500 rpm.

Determine the maximum shearing stress in a solid shaft of 12-mm diameter as it transmits 2.5 kW at a frequency of (a) 25 Hz, (b) 50 Hz.

Using an allowable shearing stress of 4.5 ksi, design a solid steel shaft to transmit 12 hp at a speed of (a) 1200 rpm, (b) 2400 rpm.

Using an allowable shearing stress of 50 MPa, design a solid steel shaft to transmit 15 kW at a frequency of (a) 30 Hz, (b) 60 Hz.

While a steel shaft of the cross section shown rotates at 120 rpm, a stroboscopic measurement indicates that the angle of twist is \(2^{\circ}\) in a 4-m length. Using G = 77.2 GPa, determine the power being transmitted.

Determine the required thickness of the 50-mm tubular shaft of Concept Application 3.7, if it is to transmit the same power while rotating at a frequency of 30 Hz.

A steel drive shaft is 6 ft long and its outer and inner diameters are respectively equal to 2.25 in. and 1.75 in. Knowing that the shaft transmits 240 hp while rotating at 1800 rpm, determine (a) the maximum shearing stress, (b) the angle of twist of the shaft \(\left(G=11.2 \times 10^{6} \mathrm{psi}\right)\).

The hollow steel shaft shown (G = 77.2 GPa, \(\tau_{\text {all }}=50 \mathrm{MPa}\)) rotates at 240 rpm. Determine (a) the maximum power that can be transmitted, (b) the corresponding angle of twist of the shaft.

A steel pipe of 3.5-in. outer diameter is to be used to transmit a torque of 3000 lb?ft without exceeding an allowable shearing stress of 8 ksi. A series of 3.5-in.-outer-diameter pipes is available for use. Knowing that the wall thickness of the available pipes varies from 0.25 in. to 0.50 in. in 0.0625-in. increments, choose the lightest pipe that can be used.

The design of a machine element calls for a 40-mm-outer-diameter shaft to transmit 45 kW. (a) If the speed of rotation is 720 rpm, determine the maximum shearing stress in shaft a. (b) If the speed of rotation can be increased 50% to 1080 rpm, determine the largest inner diameter of shaft b for which the maximum shearing stress will be the same in each shaft.

Three shafts and four gears are used to form a gear train that will transmit power from the motor at A to a machine tool at F. (Bearings for the shafts are omitted in the sketch.) The diameter of each shaft is as follows: \(d_{A B}=16 \mathrm{~mm}, d_{C D}=20 \mathrm{~mm}, d_{E F}=28 \mathrm{~mm}\). Knowing that the frequency of the motor is 24 Hz and that the allowable shearing stress for each shaft is 75 MPa, determine the maximum power that can be transmitted.

Three shafts and four gears are used to form a gear train that will transmit 7.5 kW from the motor at A to a machine tool at F. (Bearings for the shafts are omitted in the sketch.) Knowing that the frequency of the motor is 30 Hz and that the allowable stress for each shaft is 60 MPa, determine the required diameter of each shaft.

The two solid shafts and gears shown are used to transmit 16 hp from the motor at A operating at a speed of 1260 rpm, to a machine tool at D. Knowing that each shaft has a diameter of 1 in., determine the maximum shearing stress (a) in shaft AB, (b) in shaft CD.

The two solid shafts and gears shown are used to transmit 16 hp from the motor at A operating at a speed of 1260 rpm to a machine tool at D. Knowing that the maximum allowable shearing stress is 8 ksi, determine the required diameter (a) of shaft AB, (b) of shaft CD.

The shaft-disk-belt arrangement shown is used to transmit 3 hp from point A to point D. (a) Using an allowable shearing stress of 9500 psi, determine the required speed of shaft AB. (b) Solve part a, assuming that the diameters of shafts AB and CD are, respectively, 0.75 in. and 0.625 in.

A 5-ft-long solid steel shaft of 0.875-in. diameter is to transmit 18 hp between a motor and a machine tool. Determine the lowest speed at which the shaft can rotate, knowing that \(G=11.2 \times 10^{6}\) psi, that the maximum shearing stress must not exceed 4.5 ksi, and the angle of twist must not exceed \(3.5^{\circ}\).

A 2.5-m-long steel shaft of 30-mm diameter rotates at a frequency of 30 Hz. Determine the maximum power that the shaft can transmit, knowing that G 5 77.2 GPa, that the allowable shearing stress is 50 MPa, and that the angle of twist must not exceed \(7.5^{\circ}\).

The design specifications of a 1.2-m-long solid transmission shaft require that the angle of twist of the shaft not exceed \(4^{\circ}\) when a torque of \(750 \mathrm{~N} \cdot \mathrm{m}\) is applied. Determine the required diameter of the shaft, knowing that the shaft is made of a steel with an allowable shearing stress of 90 MPa and a modulus of rigidity of 77.2 GPa.

A 1.5-m-long tubular steel shaft (G = 77.2 GPa) of 38-mm outer-diameter \(d_{1}\) and 30-mm inner diameter \(d_{2}\) is to transmit 100 kW between a turbine and a generator. Knowing that the allowable shearing stress is 60 MPa and that the angle of twist must not exceed \(3^{\circ}\), determine the minimum frequency at which the shaft can rotate.

A 1.5-m-long tubular steel shaft of 38-mm outer diameter \(d_{1}\) is to be made of a steel for which \(\tau_{\text {all }}=65 \mathrm{MPa}) and G = 77.2 GPa. Knowing that the angle of twist must not exceed \(4^{\circ}\) when the shaft is subjected to a torque of \(600 \mathrm{~N} \cdot \mathrm{m}\), determine the largest inner diameter \(d_{2}\) that can be specified in the design.

The stepped shaft shown must transmit 40 kW at a speed of 720 rpm. Determine the minimum radius r of the fillet if an allowable stress of 36 MPa is not to be exceeded.

The stepped shaft shown rotates at 450 rpm. Knowing that r = 0.5 in., determine the maximum power that can be transmitted without exceeding an allowable shearing stress of 7500 psi.

Knowing that the stepped shaft shown transmits a torque of magnitude \(T=2.50 \text { kip } \cdot i n\)., determine the maximum shearing stress in the shaft when the radius of the fillet is (a) \(r=\frac{1}{8} \mathrm{in}\)., (b) \(r=\frac{3}{16} \mathrm{in}\).

The stepped shaft shown must rotate at a frequency of 50 Hz. Knowing that the radius of the fillet is r = 8 mm and the allowable shearing stress is 45 MPa, determine the maximum power that can be transmitted.

The stepped shaft shown must transmit 45 kW. Knowing that the allowable shearing stress in the shaft is 40 MPa and that the radius of the fillet is r = 6 mm, determine the smallest permissible speed of the shaft.

A torque of magnitude \(T=200 \mathrm{lb} \cdot \mathrm{in}\). is applied to the stepped shaft shown, which has a full quarter-circular fillet. Knowing that D = 1 in., determine the maximum shearing stress in the shaft when (a) d = 0.8 in., (b) d = 0.9 in.

In the stepped shaft shown, which has a full quarter-circular fillet, the allowable shearing stress is 80 MPa. Knowing that D = 30 mm, determine the largest allowable torque that can be applied to the shaft if (a) d = 26 mm, (b) d = 24 mm.

In the stepped shaft shown, which has a full quarter-circular fillet, D = 1.25 in. and d = 1 in. Knowing that the speed of the shaft is 2400 rpm and that the allowable shearing stress is 7500 psi, determine the maximum power that can be transmitted by the shaft.

The solid circular shaft shown is made of a steel that is assumed to be elastoplastic with \(\tau_{Y}=145 \mathrm{MPa}\). Determine the magnitude T of the applied torques when the plastic zone is (a) 16 mm deep, (b) 24 mm deep.

A 1.25-in. diameter solid rod is made of an elastoplastic material with\tau_{Y}=5 \mathrm{ksi}. Knowing that the elastic core of the rod is 1 in. in diameter, determine the magnitude of the applied torque T.

The solid shaft shown is made of a mild steel that is assumed to be elastoplastic with \(G=11.2 \times 10^{6}\) and \(\tau_{Y}=21 \mathrm{ksi}\). Determine the maximum shearing stress and the radius of the elastic core caused by the application of a torque of magnitude (a) \(T=100 \text { kip·in. }\), (b) \(T=140 \mathrm{kip} \cdot \mathrm{in}\).

The solid shaft shown is made of a mild steel that is assumed to be elastoplastic with G = 77.2 GPa and \(\tau_{Y}=145 \mathrm{MPa}\). Determine the maximum shearing stress and the radius of the elastic core caused by the application of a torque of magnitude (a) \(T=600 \mathrm{~N} \cdot \mathrm{m}\), (b) \(T=1000 \mathrm{~N} \cdot \mathrm{m}\).

The solid shaft shown is made of a mild steel that is assumed to be elastoplastic with \(\tau_{Y}=145 \mathrm{MPa}\). Determine the radius of the elastic core caused by the application of a torque equal to \(1.1 T_{Y}\), where \(T_{Y}\) is the magnitude of the torque at the onset of yield.

It is observed that a straightened paper clip can be twisted through several revolutions by the application of a torque of approximately \(60 \mathrm{~N} \cdot \mathrm{m}\). Knowing that the diameter of the wire in the paper clip is 0.9 mm, determine the approximate value of the yield stress of the steel.

The solid shaft shown is made of a mild steel that is assumed to be elastoplastic with G = 77.2 GPa and \(\tau_{Y}=145 \mathrm{MPa}\). Determine the angle of twist caused by the application of a torque of magnitude (a) \(T=600 \mathrm{~N} \cdot \mathrm{m}\), (b) \(T=1000 \mathrm{~N} \cdot \mathrm{m}\).

For the solid circular shaft of Prob. 3.94, determine the angle of twist caused by the application of a torque of magnitude (a) \(T=80 \text { kip·in., }\), (b) \(T=130 \text { kip } \cdot \text { in }\).

For the solid shaft of Prob. 3.98, determine (a) the magnitude of the torque T required to twist the shaft through an angle of \(15^{\circ}\), (b) the radius of the corresponding elastic core.

A 3-ft-long solid shaft has a diameter of 2.5 in. and is made of a mild steel that is assumed to be elastoplastic with \(\tau_{Y}=21\) ksi and \(G=11.2 \times 10^{6} \mathrm{psi}\). Determine the torque required to twist the shaft through an angle of \(\text { (a) } 2.5^{\circ},(b) 5^{\circ}\).

An 18-mm-diameter solid circular shaft is made of a material that is assumed to be elastoplastic with \(\tau_{Y}=145 \mathrm{MPa}\) and G = 77.2 GPa. For a 1.2-m length of the shaft, determine the maximum shearing stress and the angle of twist caused by a\(200-\mathrm{N} \cdot \mathrm{m} \text { torque }\).

A 0.75-in.-diameter solid circular shaft is made of a material that is assumed to be elastoplastic with \(\tau_{Y}=20 \mathrm{ksi}\) and \(G=11.2 \times 10^{6} \mathrm{psi}\). For a 4-ft length of the shaft, determine the maximum shearing stress and the angle of twist caused by a \(\text { 1800-lb-in. }\) torque.

The shaft AB is made of a material that is elastoplastic with \(\tau_{Y}=90 \mathrm{MPa}\) and G = 30 GPa. For the loading shown, determine (a) the radius of the elastic core of the shaft, (b) the angle of twist at end B.

A solid circular rod is made of a material that is assumed to be elastoplastic. Denoting by \(T_{Y} \text { and } \phi_{Y}\), respectively, the torque and the angle of twist at the onset of yield, determine the angle of twist if the torque is increased to \(\text { (a) } T=1.1 T_{Y},(b) T=1.25 T_{Y}\), (c) \(T=1.3 T_{Y}\).

A hollow shaft is 0.9 m long and has the cross section shown. The steel is assumed to be elastoplastic with \(\tau_{Y}=180 \mathrm{MPa}\) and G = 77.2 GPa. Determine (a) the angle of twist at which the section first becomes fully plastic, (b) the corresponding magnitude of the applied torque.

A hollow shaft is 0.9 m long and has the cross section shown. The steel is assumed to be elastoplastic with \(\tau_{Y}=180 \mathrm{MPa}\) and G = 77.2 GPa. Determine the applied torque and the corresponding angle of twist (a) at the onset of yield, (b) when the plastic zone is 10 mm deep.

A steel rod is machined to the shape shown to form a tapered solid shaft to which a torque is of magnitude \(T=75 \text { kip } \cdot \text { in. }\) is applied. Assuming the steel to be elastoplastic with \(\tau_{Y}=21 \mathrm{ksi}\) and \(G=11.2 \times 10^{6} \mathrm{psi}\), determine (a) the radius of the elastic core in portion AB of the shaft, (b) the length of portion CD that remains fully elastic.

If the torque applied to the tapered shaft of Prob. 3.108 is slowly increased, determine (a) the magnitude T of the largest torque that can be applied to the shaft, (b) the length of the portion CD that remains fully elastic.

A solid brass rod of 1.2-in. diameter is subjected to a torque that causes a maximum shearing stress of 13.5 ksi in the rod. Using the \(\tau-\gamma\) diagram shown for the brass rod used, determine (a) the magnitude of the torque, (b) the angle of twist in a 24-in. length of the rod.

A solid brass rod of 0.8-in. diameter and 30-in. length is twisted through an angle of 108. Using the \(\tau-\gamma\) diagram shown for the brass rod used, determine (a) the magnitude of the torque applied to the rod, (b) the maximum shearing stress in the rod.

A 50-mm diameter cylinder is made of a brass for which the stress-strain diagram is as shown. Knowing that the angle of twist is \(5^{\circ}\) in a 725-mm length, determine by approximate means the magnitude T of torque applied to the shaft.

Three points on the nonlinear stress-strain diagram used in Prob. 3.112 are (0, 0), (0.0015, 55 MPa), and (0.003, 80 MPa). By fitting the polynomial \(T=A+B \gamma+C \gamma^{2}\) through these points, the following approximate relation has been obtained. \(T=46.7 \times 10^{9} \gamma-6.67 \times 10^{12} \gamma^{2}\) Solve Prob. 3.112 using this relation, Eq. (3.2), and Eq. (3.23).

The solid circular drill rod AB is made of a steel that is assumed to be elastoplastic with \(\tau_{Y}=22 \mathrm{ksi}\) and \(G=11.2 \times 10^{6} \mathrm{psi}\). Knowing that a torque \(T=75 \mathrm{kip} \cdot \mathrm{in}\). is applied to the rod and then removed, determine the maximum residual shearing stress in the rod.

In Prob. 3.114, determine the permanent angle of twist of the rod.

The solid shaft shown is made of a steel that is assumed to be elastoplastic with \(\tau_{Y}=145 \mathrm{MPa}\) and G = 77.2 GPa. The torque is increased in magnitude until the shaft has been twisted through \(6^{\circ}\); the torque is then removed. Determine (a) the magnitude and location of the maximum residual shearing stress, (b) the permanent angle of twist.

After the solid shaft of Prob. 3.116 has been loaded and unloaded as described in that problem, a torque \(\mathbf{T}_{1}\) of sense opposite to the original torque T is applied to the shaft. Assuming no change in the value of \(\phi_{Y}\), determine the angle of twist \(\phi_{1}\) for which yield is initiated in this second loading and compare it with the angle \(\phi_{Y}\) for which the shaft started to yield in the original loading.

The hollow shaft shown is made of a steel that is assumed to be elastoplastic with \(\tau_{Y}=145 \mathrm{MPa}\) and G = 77.2 GPa. The magnitude T of the torques is slowly increased until the plastic zone first reaches the inner surface of the shaft; the torques are then removed. Determine the magnitude and location of the maximum residual shearing stress in the rod.

In Prob. 3.118, determine the permanent angle of twist of the rod.

A torque T applied to a solid rod made of an elastoplastic material is increased until the rod is fully plastic and then removed. (a) Show that the distribution of residual shearing stresses is as represented in the figure. (b) Determine the magnitude of the torque due to the stresses acting on the portion of the rod located within a circle of radius \(c_{0}\).

Determine the smallest allowable square cross section of a steel shaft of length 20 ft if the maximum shearing stress is not to exceed 10 ksi when the shaft is twisted through one complete revolution. Use \(G=11.2 \times 10^{6} \mathrm{psi}\).

Determine the smallest allowable length of a stainless steel shaft of \(\frac{3}{8} \times \frac{3}{4} \text {-in. }\) cross section if the shearing stress is not to exceed 15 ksi when the shaft is twisted through \(15^{\circ}\). Use \(G=11.2 \times 10^{6} \mathrm{psi}\).

Using \(\tau_{\text {all }}=70 \mathrm{MPa}\) and G = 27 GPa, determine for each of the aluminum bars shown the largest torque T that can be applied and the corresponding angle of twist at end B.

Knowing that the magnitude of the torque T is \(200 \mathrm{~N} \cdot \mathrm{m}\) and that G = 27 GPa, determine for each of the aluminum bars shown the maximum shearing stress and the angle of twist at end B.

Determine the largest torque T that can be applied to each of the two brass bars shown and the corresponding angle of twist at B, knowing that \(\tau_{\text {all }}=12 \mathrm{ksi} \text { and } G=5.6 \times 10^{6} \mathrm{psi}\).

Each of the two brass bars shown is subjected to a torque of magnitude \(T=12.5 \text { kip } \cdot i n\). Knowing that \(G=5.6 \times 10^{6} \mathrm{psi}\), determine for each bar the maximum shearing stress and the angle of twist at B.

The torque T causes a rotation of \(0.6^{\circ}\) at end B of the aluminum bar shown. Knowing that b = 15 mm and G = 26 GPa, determine the maximum shearing stress in the bar.

The torque T causes a rotation of \(2^{\circ}\) at end B of the stainless steel bar shown. Knowing that b = 20 mm and G = 75 GPa, determine the maximum shearing stress in the bar.

Two shafts are made of the same material. The cross section of shaft A is a square of side b and that of shaft B is a circle of diameter b. Knowing that the shafts are subjected to the same torque, determine the ratio \(\tau_{A} / \tau_{B}\) of maximum shearing stresses occurring in the shafts.

Shafts A and B are made of the same material and have the same cross-sectional area, but A has a circular cross section and B has a square cross section. Determine the ratio of the maximum torques \(T_{A} \text { and } T_{B}\) when the two shafts are subjected to the same maximum shearing stress \(\left(\tau_{A}=\tau_{B}\right)\). Assume both deformations to be elastic.

Shafts A and B are made of the same material and have the same length and cross-sectional area, but A has a circular cross section and B has a square cross section. Determine the ratio of the maximum values of the angles \(\phi_{A} \text { and } \phi_{B}\) when the two shafts are subjected to the same maximum shearing stress \(\left(\tau_{A}=\tau_{B}\right)\). Assume both deformations to be elastic.

Shafts A and B are made of the same material and have the same cross-sectional area, but A has a circular cross section and B has a square cross section. Determine the ratio of the angles \(\phi_{A} \text { and } \phi_{B}\) through which shafts A and B are respectively twisted when the two shafts are subjected to the same torque \(\left(\tau_{A}=\tau_{B}\right)\). Assume both deformations to be elastic.

A torque of magnitude \(T=2 \text { kip } \cdot i n\). is applied to each of the steel bars shown. Knowing that \(\tau_{\text {all }}=6 \mathrm{ksi}\), determine the required dimension b for each bar.

A torque of magnitude \(T=300 \mathrm{~N} \cdot \mathrm{m}\) is applied to each of the aluminum bars shown. Knowing that \(\tau_{\text {all }}=60 \mathrm{MPa}\), determine the required dimension b for each bar.

A 1.25-m-long steel angle has an \(\text { L127 } \times 76 \times 6.4\) cross section. From Appendix C we find that the thickness of the section is 6.4 mm and that its area is \(1250 \mathrm{~mm}^{2}\). Knowing that \(\tau_{\text {all }}=60 \mathrm{MPa}\) and that G = 77.2 GPa, and ignoring the effect of stress concentrations, determine (a) the largest torque T that can be applied, (b) the corresponding angle of twist.

A 36-kip?in. torque is applied to a 10-ft-long steel angle with an \(\mathrm{L} 8 \times 8 \times 1\) cross section. From Appendix C we find that the thickness of the section is 1 in. and that its area is \(15 \mathrm{in}^{2}\). Knowing that \(G=11.2 \times 10^{6} \mathrm{psi}\), determine (a) the maximum shearing stress along line a-a, (b) the angle of twist.

A 4-m-long steel member has a W310 X 60 cross section. Knowing that G = 77.2 GPa and that the allowable shearing stress is 40 MPa, determine (a) the largest torque T that can be applied, (b) the corresponding angle of twist. Refer to Appendix C for the dimensions of the cross section and neglect the effect of stress concentrations. (Hint: consider the web and flanges separately and obtain a relation between the torques exerted on the web and a flange, respectively, by expressing that the resulting angles of twist are equal.)

An 8-ft-long steel member with a W8 X 31 cross section is subjected to a \(\text { 5-kip·in. }\) torque. The properties of the rolled-steel section are given in Appendix C. Knowing that \(G=11.2 \times 10^{6} \mathrm{psi}\), determine (a) the maximum shearing stress along line a-a, (b) the maximum shearing stress along line b-b, (c) the angle of twist. (See hint of Prob. 3.137.)

A \(\text { 5-kip·ft. }\) torque is applied to a hollow aluminum shaft having the cross section shown. Neglecting the effect of stress concentrations, determine the shearing stress at points a and b.

A torque \(T=750 \mathrm{kN} \cdot \mathrm{m}\) is applied to the hollow shaft shown that has a uniform 8-mm wall thickness. Neglecting the effect of stress concentrations, determine the shearing stress at points a and b.

A \(750 \mathrm{N} \cdot \mathrm{m}\) is applied to a hollow shaft having the cross section shown and a uniform 6-mm wall thickness. Neglecting the effect of stress concentrations, determine the shearing stress at points a and b.

A hollow member having the cross section shown is formed from sheet metal of 2-mm thickness. Knowing that the shearing stress must not exceed 3 MPa, determine the largest torque that can be applied to the member.

A hollow member having the cross section shown is formed from sheet metal of 2-mm thickness. Knowing that the shearing stress must not exceed 3 MPa, determine the largest torque that can be applied to the member.

A \(90-\mathrm{N} \cdot \mathrm{m}\) torque is applied to a hollow shaft having the cross section shown. Neglecting the effect of stress concentrations, determine the shearing stress at points a and b.

A hollow member having the cross section shown is to be formed from sheet metal of 0.06-in. thickness. Knowing that a \(\text { 1250-lb }\). torque will be applied to the member, determine the smallest dimension d that can be used if the shearing stress is not to exceed 750 psi.

A hollow member having the cross section shown is to be formed from sheet metal of 0.06-in. thickness. Knowing that a \(\text { 1250-lb }\). torque will be applied to the member, determine the smallest dimension d that can be used if the shearing stress is not to exceed 750 psi.

A cooling tube having the cross section shown is formed from a sheet of stainless steel of 3-mm thickness. The radii \(c_{1}=150 \mathrm{~mm}\) and \(c_{2}=100 \mathrm{~mm}\) are measured to the center line of the sheet metal. Knowing that a torque of magnitude \(T=3 \mathrm{kN} \cdot \mathrm{m}\) is applied to the tube, determine (a) the maximum shearing stress in the tube, (b) the magnitude of the torque carried by the outer circular shell. Neglect the dimension of the small opening where the outer and inner shells are connected.

A hollow cylindrical shaft was designed to have a uniform wall thickness of 0.1 in. Defective fabrication, however, resulted in the shaft having the cross section shown. Knowing that a \(\text { 15-kip-in. }\) torque is applied to the shaft, determine the shearing stresses at points a and b.

Equal torques are applied to thin-walled tubes of the same length L, same thickness t, and same radius c. One of the tubes has been slit lengthwise as shown. Determine (a) the ratio \(\tau_{b} / \tau_{a}\) of the maximum shearing stresses in the tubes, (b) the ratio \(\phi_{b} / \phi_{a}\) of the angles of twist of the tubes.

A hollow cylindrical shaft of length L, mean radius \(c_{m}\), and uniform thickness t is subjected to a torque of magnitude T. Consider, on the one hand, the values of the average shearing stress \(\tau_{\text {ave }}\) and the angle of twist \(\phi\) obtained from the elastic torsion formulas developed in Secs. 3.1C and 3.2 and, on the other hand, the corresponding values obtained from the formulas developed in Sec. 3.10 for thin-walled shafts. (a) Show that the relative error introduced by using the thin-walled-shaft formulas rather than the elastic torsion formulas is the same for \(\tau_{\text {ave }}\) and \(\phi\) and that the relative error is positive and proportional to the ratio \(t / c_{m}\). (b) Compare the percent error corresponding to values of the ratio \(t / c_{m}\) of 0.1, 0.2, and 0.4.

A steel pipe of 12-in. outer diameter is fabricated from \(\frac{1}{4} \text {-in. }\)-thick plate by welding along a helix that forms an angle of \(45^{\circ}\) with a plane parallel to the axis of the pipe. Knowing that the maximum allowable tensile stress in the weld is 12 ksi, determine the largest torque that can be applied to the pipe.

A torque of magnitude \(T=120 \mathrm{~N} \cdot \mathrm{m}\) is applied to shaft AB of the gear train shown. Knowing that the allowable shearing stress is 75 MPa in each of the three solid shafts, determine the required diameter of (a) shaft AB, (b) shaft CD, (c) shaft EF.

Two solid shafts are connected by gears as shown. Knowing that G = 77.2 GPa for each shaft, determine the angle through which end A rotates when \(T_{A}=1200 \mathrm{~N} \cdot \mathrm{m}\).

In the bevel-gear system shown, \(\alpha=18.43^{\circ}\). Knowing that the allowable shearing stress is 8 ksi in each shaft and that the system is in equilibrium, determine the largest torque \(\mathbf{T}_{A}\) that can be applied at A.

The design specifications for the gear-and-shaft system shown require that the same diameter be used for both shafts and that the angle through which pulley A will rotate when subjected to a \(\text { 2-kip·in. }\) torque \(\mathbf{T}_{A}\) while pulley D is held fixed will not exceed \(7.5^{\circ}\). Determine the required diameter of the shafts if both shafts are made of a steel with \(G=11.2 \times 10^{6}\) psi and \(\tau_{\text {all }}=12 \mathrm{ksi}\).

A torque of magnitude \(T=4 \mathrm{kN} \cdot \mathrm{m}\) is applied at end A of the composite shaft shown. Knowing that the modulus of rigidity is 77.2 GPa for the steel and 27 GPa for the aluminum, determine (a) the maximum shearing stress in the steel core, (b) the maximum shearing stress in the aluminum jacket, (c) the angle of twist at A.

Ends A and D of the two solid steel shafts AB and CD are fixed, while ends B and C are connected to gears as shown. Knowing that the allowable shearing stress is 50 MPa in each shaft, determine the largest torque T that can be applied to gear B.

As the hollow steel shaft shown rotates at 180 rpm, a stroboscopic measurement indicates that the angle of twist of the shaft is \(3^{\circ}\). Knowing that G = 77.2 GPa, determine (a) the power being transmitted, (b) the maximum shearing stress in the shaft.

Knowing that the allowable shearing stress is 8 ksi for the stepped shaft shown, determine the magnitude T of the largest torque that can be transmitted by the shaft when the radius of the fillet is (a) \(r=\frac{3}{16} \mathrm{in}\), (b) \(r=\frac{1}{4} \text { in. }\)

A hollow brass shaft has the cross section shown. Knowing that the shearing stress must not exceed 12 ksi and neglecting the effect of stress concentrations, determine the largest torque that can be applied to the shaft.

Two solid brass rods AB and CD are brazed to a brass sleeve EF. Determine the ratio \(d_{2} / d_{1}\) for which the same maximum shearing stress occurs in the rods and in the sleeve.

The shaft AB is made of a material that is elastoplastic with \(\tau_{Y}=12.5 \mathrm{ksi}\) and \(G=4 \times 10^{6} \mathrm{psi}\). For the loading shown, determine (a) the radius of the elastic core of the shaft, (b) the angle of twist of the shaft.

Shaft AB consists of n homogeneous cylindrical elements, which can be solid or hollow. Its end A is fixed, while its end B is free, and it is subjected to the loading shown. The length of element i is denoted by \(L_{i}\), its outer diameter by \(OD_{i}\), its inner diameter by \(ID_{i}\), its modulus of rigidity by \(G_{i}\), and the torque applied to its right end by \(\mathbf{T}_{i}\), the magnitude \(T_{i}\) of this torque being assumed to be positive if \(\mathbf{T}_{i}\) is counterclockwise from end B and negative otherwise. (Note that \(I D_{i}=0\) if the element is solid.) (a) Write a computer program that can be used to determine the maximum shearing stress in each element, the angle of twist of each element, and the angle of twist of the entire shaft. (b) Use this program to solve Probs. 3.35, 3.36, and 3.38.

The assembly shown consists of n cylindrical shafts, which can be solid or hollow, connected by gears and supported by brackets (not shown). End \(A_{1}\) of the first shaft is free and is subjected to a torque \(\mathbf{T}_{0}\), while end \(B_{n}\) of the last shaft is fixed. The length of shaft \(A_{i} B_{i}\) is \(L_{i}\), its outer diameter \(OD_{i}\), its inner diameter \(ID_{i}\), and its modulus of rigidity \(G_{i}\). (Note that IDi 5 0 if the element is solid.) The radius of gear \(A_{i}\) is \(a_{i}\), and the radius of gear \(B_{i}\) is \(b_{i}\). (a) Write a computer program that can be used to determine the maximum shearing stress in each shaft, the angle of twist of each shaft, and the angle through which end \(A_{i}\) rotates. (b) Use this program to solve Probs. 3.41 and 3.44.

Shaft AB consists of n homogeneous cylindrical elements, which can be solid or hollow. Both of its ends are fixed, and it is subjected to the loading shown. The length of element i is denoted by \(L_{i}\), its outer diameter by \(OD_{i}\), its inner diameter by \(ID_{i}\), its modulus of rigidity by \(G_{i}\), and the torque applied to its right end by \(T_{i}\), the magnitude \(T_{i}\) of this torque being assumed to be positive if \(\mathbf{T}_{i}\) is observed as counterclockwise from end B and negative otherwise. Note that \(I D_{i}=0\) if the element is solid and also that \(T_{1}=0\). Write a computer program that can be used to determine the reactions at A and B, the maximum shearing stress in each element, and the angle of twist of each element. Use this program (a) to solve Prob. 3.55 and (b) to determine the maximum shearing stress in the shaft of Sample Problem 3.7.

The homogeneous, solid cylindrical shaft AB has a length L, a diameter d, a modulus of rigidity G, and a yield strength \(\tau_{Y}\). It is subjected to a torque T that is gradually increased from zero until the angle of twist of the shaft has reached a maximum value \(\phi_{m}\) and then decreased back to zero. (a) Write a computer program that, for each of 16 values of \(\phi_{m}\) equally spaced over a range extending from 0 to a value 3 times as large as the angle of twist at the onset of yield, can be used to determine the maximum value \(T_{m}\) of the torque, the radius of the elastic core, the maximum shearing stress, the permanent twist, and the residual shearing stress both at the surface of the shaft and at the interface of the elastic core and the plastic region. (b) Use this program to obtain approximate answers to Probs. 3.114, 3.115, 3.116.

The exact expression is given in Prob. 3.64 for the angle of twist of the solid tapered shaft AB when a torque T is applied as shown. Derive an approximate expression for the angle of twist by replacing the tapered shaft by n cylindrical shafts of equal length and of radius \(r_{i}=\left(n+i-\frac{1}{2}\right)) \((c / n)\), where \(i=1,2, \ldots, n\). Using for T, L, G, and c values of your choice, determine the percentage error in the approximate expression when (a) n = 4, (b) n = 8, (c) n = 20, and (d) n = 100.

A torque T is applied as shown to the long, hollow, tapered shaft AB of uniform thickness t. Derive an approximate expression for the angle of twist by replacing the tapered shaft by n cylindrical rings of equal length and of radius \(r_{i}=\left(n+i-\frac{1}{2}\right)) \((c / n)\), where \(i=1,2, \ldots, n\). Using for T, L, G, and c values of your choice, determine the percentage error in the approximate expression when (a) n = 4, (b) n = 8, (c) n = 20, and (d) n = 100.