Question: A hollow steel shaft used in a

> A simply supported beam ABC is loaded by a vertical load P acting at the end of a bracket BDE (see figure). (a) Draw the shear-force and bending-moment diagrams for beam ABC. (b) Now assume that load P at E is directed to the right. The vertical dimensio

> A simple beam AB subjected to couples M1 and 3M1 acting at the third points is shown in the figure. Draw the shear-force and bending-moment diagrams for this beam. M1 3M1 A B L 3 3

> The simple beam AB shown in the figure is subjected to a concentrated load P and a clockwise couple M1 = PL/3 acting at the third points. Draw the shear-force and bending-moment diagrams for this beam. PL M1 = IP 3 B L 3

> Cantilever beam AB carries an upward uniform load of intensity q1 from x = 0 to L/2 (see Fig. a) and a downward uniform load of intensity q from x = L/2 to L. (a) Find q1 in terms of q if the resulting moment at A is zero. Draw V and M diagra

> The cantilever beam AB shown in the figure is subjected to a concentrated load P at the midpoint and a counterclockwise couple of moment M1 = PL/4 at the free end. Draw the shear-force and bending-moment diagrams for this beam. PL M1 = 4 A |B L 2 2

> A stepped shaft consisting of solid circular segments having diameters D1 = 2.0 in. and D2 = 2.4 in. (see figure) is subjected to torques T. The radius of the fillet is R = 0.1 in. If the allowable shear stress at the stress concentration is 6000 psi, wh

> A hollow circular tube having an inside diameter of 10.0 in. and a wall thickness of 1.0 in. (see figure) is subjected to a torque T = 1200 kip-in. Determine the maximum shear stress in the tube using (a) the approximate theory of thin-walled tubes, and

> A solid circular bar of steel (G = 11.4 × 106 psi) with length L = 30 in. and diameter d = 1.75 in. is subjected to pure torsion by torques T acting at the ends (see figure). (a) Calculate the amount of strain energy U stored in the bar when

> A solid circular bar ABCD with fixed supports is acted upon by torques To and 2To at the locations shown in the figure. (a) Obtain a formula for the maximum angle of twist φmax of the bar. (b) What is φmax if the applied torque To a

> A generator shaft in a small hydroelectric plant turns at 120 rpm and delivers 50 hp (see figure). (a) If the diameter of the shaft is d = 3.0 in., what is the maximum shear Ï„max in the shaft? (b) If the shear stress is limited to 4000 psi, wh

> A circular copper bar with diameter d = 3 in. is subjected to torques T = 30 kip-in. at its ends. Find the maximum shear, tensile, and compressive stresses in the tube and their corresponding strains. Assume that G = 6000 ksi. d= 3 in. T T -L = 80 i

> A solid copper bar of circular cross section has length L = 1.25 m and shear modulus of elasticity G = 45 GPa. The bar is designed to carry a 250 Nm? torque acting at the ends. If the allowable shear stress is 30 MPa and the allowable angle of twist betw

> Two tubes (AB, BC) of the same material are connected by three pins (pin diameter = dp) just left of B as shown in the figure. Properties and dimensions for each tube are given in the figure. Torque 2T is applied at x = 2L/5 and uniformly distributed tor

> A non prismatic bar ABC with a solid circular cross section is loaded by distributed torques (see figure). The intensity of the torques, that is, the torque per unit distance, is denoted t(x) and varies linearly from zero at A to a maximum value To/L at

> A magnesium-alloy wire of diameter d = 4 mm and length L rotates inside a flexible tube in order to open or close a switch from a remote location (see figure). A torque T is applied manually (either clockwise or counterclockwise) at end B, thus twisting

> A prismatic bar AB with a solid circular cross section (diameter d) is loaded by a distributed torque (see figure). The intensity of the torque, that is, the torque per unit distance, is denoted t(x) and varies linearly from a maximum value tA at end A t

> A prismatic bar AB of length L and solid circular cross section (diameter d) is loaded by a distributed torque of constant intensity t per unit distance (see figure). (a) Determine the maximum shear stress Ï„max in the bar. (b) Determine the an

> A mountain-bike rider going uphill applies torque T = Fd (F = 15 lb, d = 4 in.) to the end of the handlebars ABCD by pulling on the handlebar extenders DE. Consider the right half of the handlebar assembly only (assume the bars are fixed at the fork at&A

> A wood beam ABC with simple supports at A and B and an overhang BC has height h = 300 mm (see figure). The length of the main span of the beam is L = 3.6 m and the length of the overhang is L/3 = 1.2m. The beam supports a concentrated load P = 31 8 kN at

> For the thin non prismatic steel pipe of constant thickness t and variable diameter d shown with applied torques at joints 2 and 3, determine the following. (a) Find the reaction moment R1. (b) Find an expression for twist rotation φ3 at join

> A uniformly tapered aluminum-alloy tube AB with a circular cross section and length L is shown in the figure. The outside diameters at the ends are dA and dB = 2dA. A hollow section of length L/2 and constant thickness t = dA/10 is cast into the tube and

> The non prismatic, cantilever circular bar shown has an internal cylindrical hole from 0 to x, so the net polar moment of inertia of the cross section for segment 1 is (7/8) Ip. Torque T is applied at x and torque T/2 is applied at x= L. Assume that G is

> The bar shown in the figure is tapered linearly from end A to end B and has a solid circular cross section. The diameter at the smaller end of the bar is dA = 25 mm and the length is L = 300 mm. The bar is made of steel with shear modulus of elasticity G

> A tapered bar AB with a solid circular cross section is twisted by torques T = 36,000 lb-in (see figure). The diameter of the bar varies linearly from dA at the left-hand end to dB at the right-hand end. The bar has length L = 4.0 ft and is made of an al

> A tapered bar AB with a solid circular cross section is twisted by torques T (see figure). The diameter of the bar varies linearly from dA at the left-hand end to dB at the right-hand end. (a) Confirm that the angle of twist of the tapered bar is (b) For

> Four gears are attached to a circular shaft and transmit the torques shown in the figure. The allowable shear stress in the shaft is 10,000 psi. (a) What is the required diameter d of the shaft if it has a solid cross section? (b) What is the required ou

> Two sections of steel drill pipe, joined by bolted flange plates at B, are being tested to assess the adequacy of both the pipes. In the test, the pipe structure is fixed at A, a concentrated torque of 500 kNm? is applied at x = 0.5 m, and uniformly dist

> A solid steel shaft ABCDE turns freely in bearings at points A and E. The shaft is driven by the gear at C, which applies a torque T2 = 325 lb-ft. Gears at B and D are driven by the shaft and have resisting torques T1 = 200 lb-ft and T3 = 125 lb-ft, resp

> A square wood platform is 8 ft × 8ft in area and rests on masonry walls (see figure). The deck of the platform is constructed of 2-in. nominal thickness tongue-and-groove planks (actual thickness 1.5 in.; see Appendix G) supported on two 8-f

> A shaft with a solid, circular cross section consisting of two segments is shown in part a of the figure. The left-hand segment has a diameter of 80 mm and length of 1.2 m; the right-hand segment has a diameter of 60 mm and length of 0.9 m. S

> A hollow tube ABCDE constructed of monel metal is subjected to five torques acting in the directions shown in the figure. The magnitudes of the torques are T1 = 1000lb-in, T2 = T4 = 500 lb-in, and T3 = T5 = 800 lb-in. The tube has an outside diameter of

> A solid, circular bar ABC consists of two segments, as shown in the figure. One segment has a diameter of d1 = 56 mm and length of L1 = 1.45 m; the other segment has a diameter of d2 = 48 mm and length of L2 = 1.2 m. What is the allowable torque Tallow i

> A stepped shaft ABCD consisting of solid circular segments is subjected to three torques, as shown in the figure. The torques have magnitudes of 12.5 kip-in., 9.8 kip-in., and 9.2 kip-in. The length of each segment is 25 in. and the diameters

> A circular tube of outer diameter d3 = 70 mm and inner diameter d2 = 60 mm is welded at the right hand end to a fixed plate and at the left-hand end to a rigid end plate (see figure). A solid, circular bar with a diameter of d1 = 40 mm is inside of, and

> A stepped shaft ABC consisting of two solid circular segments is subjected to torques T1 and T2 acting in opposite directions, as shown in the figure. The larger segment of the shaft has a diameter of d1 = 2.25 in and length L1 = 30 in; the smaller segme

> A solid steel bar of circular cross section has diameter d = 2.5 in, L = 60 in, and shear modulus of elasticity G = 11.5 × 106 psi. The bar is subjected to torques T = 300 lb-ft at the ends. Calculate the angle of twist between the ends. Wha

> A circular aluminum tube subjected to pure torsion by torques T (see figure) has an outer radius r2 equal to 1.5 times the inner radius r1. (a) If the maximum shear strain in the tube is measured as 400 × 10-6 rad, what is the shear strain g

> Solve the preceding problem if the length L = 56 in, the inner radius r1 = 1.25 in, the angle of twist is 0.5 °, and the allowable shear strain is 0.0004 rad. Data from Problem 4: A circular steel tube of length L = 1.0m is loaded in torsio

> A circular steel tube of length L = 1.0m is loaded in torsion by torques T (see figure). (a) If the inner radius of the tube is r1 = 45 mm and the measured angle of twist between the ends is 0.5°, what is the shear strain γ1 (in ra

> A simply supported wood beam of rectangular cross section and span length 1.2 m carries a concentrated load P at mid span in addition to its own weight (see figure). The cross section has width 140 mm and height 240 mm. The weight density of the wood is

> A copper rod of length L = 18.0in. is to be twisted by torques T (see figure) until the angle of rotation between the ends of the rod is 3.0°. (a) If the allowable shear strain in the copper is 0.0006 rad, what is the maximum permissible diame

> A plastic bar of diameter d = 56 mm is to be twisted by torques T (see figure) until the angle of rotation between the ends of the bar is 4.0°. (a) If the allowable shear strain in the plastic is 0.012 rad, what is the minimum permissible length of the b

> A circular tube is subjected to torque T at its ends. The resulting maximum shear strain in the tube is 0.005. Calculate the minimum shear strain in the tube and the shear strain at the median line of the tube section. T T -Median line -2.5 in.- 3 i

> A circular tube of inner radius r1 and outer radius r2 is subjected to a torque produced by forces P = 900 lb (see figure part a). The forces have their lines of action at a distance b = 5.5 in from the outside of the tube. (a) If the allowable shear str

> A hollow aluminum tube used in a roof structure has an outside diameter d2 = 104 mm and an inside d1 = 82 mm (see figure). The tube is 2.75 m long, and the aluminum has shear modulus G = 28GPa. (a) If the tube is twisted in pure torsion by torques acting

> A solid brass bar of diameter d = 1.25 in is subjected to torques T1, as shown in part a of the figure. The allowable shear stress in the brass is 12 ksi. (a) What is the maximum permissible value of the torques T1? (b) If a hole of diameter 0.625 in. is

> A vertical pole of solid, circular cross section is twisted by horizontal forces P = 5 kN acting at the ends of a rigid horizontal arm AB (see figure part a). The distance from the outside of the pole to the line of action of each force is c = 125 mm (se

> A vertical pole of solid, circular cross section is twisted by horizontal forces P = 1100 lb acting at the ends of a rigid horizontal arm AB (see figure part a). The distance from the outside of the pole to the line of action of each force is c = 5.0 in

> Solve the preceding problem if the shaft has an outer diameter d2 = 150 mm and inner diameter d1 = 100 mm. Also, the steel has a shear modulus of elasticity G = 75 GPa, and the applied torque is 16 kNm? Data from Problem 15: A hollow steel shaft used

> A wood beam AB on simple supports with span length equal to 10 ft is subjected to a uniform load of intensity 125 lb/ft acting along the entire length of the beam, a concentrated load of magnitude 7500 lb acting at a point 3 ft from the right-hand suppor

> The steel axle of a large winch on an ocean liner is subjected to a torque of 1.65 kNm? (see figure). (a) What is the minimum required diameter dmin if the allowable shear stress is 48 MPa and the allowable rate of twist is 0.75°/m? (Assume th

> Three identical circular disks A, B, and C are welded to the ends of three identical solid circular bars (see figure). The bars lie in a common plane and the disks lie in planes perpendicular to the axes of the bars. The bars are welded at their intersec

> A propeller shaft for a small yacht is made of a solid steel bar 104 mm in diameter. The allowable stress in shear is 48 MPa, and the allowable rate of twist is 2.0° in 3.5 meters. (a) Assuming that the shear modulus of elasticity is G = 80 GP

> A circular tube of aluminum is subjected to torsion by torques T applied at the ends (see figure). The bar is 24 in. long, and the inside and outside diameters are 1.25 in. and 1.75 in., respectively. It is determined by measurement that the angle of twi

> A high-strength steel drill rod used for boring a hole in the earth has a diameter of 0.5 in. (see figure). The allowable shear stress in the steel is 40 ksi and the shear modulus of elasticity is 11,600 ksi. (a) What is the minimum required

> An aluminum bar of solid circular cross section is twisted by torques T acting at the ends (see figure). The dimensions and shear modulus of elasticity are L = 1.4 m, d = 32 mm, and G = 28 GPa. (a) Determine the torsional stiffness of the bar. (b) If the

> While removing a wheel to change a tire, a driver applies forces P = 25 lb at the ends of two of the arms of a lug wrench (see figure). The wrench is made of steel with shear modulus of elasticity G = 11.4 × 106 psi. Each arm of the wrench i

> When drilling a hole in a table leg, a furniture maker uses a hand-operated drill (see figure) with a bit of diameter d = 4.0 mm. (a) If the resisting torque supplied by the table leg is equal to 0.3 Nm? what is the maximum shear stress in the drill bit?

> A prospector uses a hand-powered winch (see figure) to raise a bucket of ore in his mine shaft. The axle of the winch is a steel rod of diameter d = 0.625 in. Also, the distance from the center of the axle to the center of the lifting rope is b = 4.0 in.

> A laminated plastic beam of square cross section is built up by gluing together three strips, each 10 mm × 30 mm in cross section (see figure). The beam has a total weight of 3.6 N and is simply supported with span length L = 360 mm. Conside

> A copper tube with circular cross section has length L = 1.25 m, thickness t = 2 mm, and shear modulus of elasticity G = 45 GPa. The bar is designed to carry a 300 Nm? torque acting at the ends. If the allowable shear stress is 25 MPa and the allowable a

> Repeat Problem 1, but now use a circular tube with outer diameter do = 2.5 in. and inner diameter di = 1.5 in. Data from Problem 1: A solid steel bar of circular cross section has diameter d = 2.5 in, L = 60 in, and shear modulus of elasticity G = 11.

> Two circular aluminum pipes of equal length L = 24 in. are loaded by torsional moments T (see figure). Pipe 1 has outside and inside diameters d2 = 3 in and d1 = 2.5 in, respectively. Pipe 2 has a constant outer diameter of d2 along its entire length&Aci

> A solid aluminum bar (G = 2 7 GPa) of diameter d = 40 mm is subjected to torques T = 300 Nm? acting in the directions shown in the figure. (a) Determine the maximum shear, tensile, and compressive stresses in the bar and show these stresses on sketches o

> A solid steel bar (G = 11.8 × 106 psi) of diameter d = 2.0 in. is subjected to torques T = 8.0 kip-in. acting in the directions shown in the figure. (a) Determine the maximum shear, tensile, and compressive stresses in the bar and show these

> An aluminum tube has inside diameter d1 = 50 mm, shear modulus of elasticity G = 27 GPa, v = 0.33, and torque T = 4.0 kNm? The allowable shear stress in the aluminum is 50 MPa, and the allowable normal strain is 900 × 10-6. (a) Determine the required out

> The normal strain in the 45° direction on the surface of a circular tube (see figure) is 880 × 10-6 when the torque T = 750 lb-in. The tube is made of copper alloy with G = 6.2 × 106 psi and n = 0.35. (a) If the outsi

> A solid circular bar of steel (G = 78 GPa) transmits a torque T = 360 Nm? The allowable stresses in tension, compression, and shear are 90 MPa, 70 MPa, and 40 MPa, respectively. Also, the allowable tensile strain is 220 × 10-6. (a) Determine the minimum

> A steel tube (G = 11.5 × 106 psi) has an outer diameter d2 = 2.0 in. and an inner diameter d1 = 1.5 in. When twisted by a torque T, the tube develops a maximum normal strain of 170 × 10-6. What is the magnitude of the applied torque T?

> A solid circular bar of diameter d = 50 mm (see figure) is twisted in a testing machine until the applied torque reaches the value T = 500 Nm?. At this value of torque, a strain gage oriented at 45° to the axis of the bar gives a reading &Icir

> A laminated wood beam on simple supports (figure part a) is built up by gluing together four 2 in. × 4 in boards (actual dimensions) to form a solid beam 4 in. × 8 in in cross section, as shown in the figure part b. The allowabl

> A tubular bar with outside diameter d2 = 4.0 in. is twisted by torques T = 70.0 kip-in. (see figure). Under the action of these torques, the maximum tensile stress in the bar is found to be 6400 psi. (a) Determine the inside diameter d1 of the bar. (b) I

> A hollow steel bar (G = 8 0GPa) is twisted by torques T (see figure). The twisting of the bar produces a maximum shear strain γmax = 640 × 10-6 rad. The bar has outside and inside diameters of 150 mm and 120 mm, respectively. (a)

> A hollow aluminum shaft (see figure) has an outside diameter d2 = 4.0 in. and inside diameter d1 = 2.0 in. When twisted by torques T, the shaft has an angle of twist per unit distance equal to 0.548 /ft. The shear modulus of elasticity of the aluminum is

> A circular steel tube with an outer diameter of 75 mm and inner diameter of 65 mm is subjected to torques T at its ends. Calculate the maximum permissible torque Tmax if the allowable normal strain is εa = 5 Ã&

> The shaft ABC shown in the figure is driven by a motor that delivers 300 kW at a rotational speed of 32 Hz. The gears at B and C take out 120 kW and 180 kW, respectively. The lengths of the two parts of the shaft are L1 = 1.5 m an

> A motor delivers 275 hp at 1000 rpm to the end of a shaft (see figure). The gears at B and C take out 125 and 150 hp, respectively. Determine the required diameter d of the shaft if the allowable shear stress is 7500 psi and the angle of twist between th

> What is the maximum power that can be delivered by a hollow propeller shaft (outside diameter 50 mm, inside diameter 40 mm, and shear modulus of elasticity 80 GPa) turning at 600 rpm if the allowable shear stress is 100 MPa and the allowable rate of twis

> A propeller shaft of solid circular cross section and diameter d is spliced by a collar of the same material (see figure). The collar is securely bonded to both parts of the shaft. What should be the minimum outer diameter d1 of the collar in order that

> A tubular shaft being designed for use on a construction site must transmit 120 kW at 1.75 Hz. The inside diameter of the shaft is to be one-half of the outside diameter. If the allowable shear stress in the shaft is 45 MPa, what is the minimum required

> A hollow circular shaft for use in a pumping station is being designed with an inside diameter equal to 0.75 times the outside diameter. The shaft must transmit 400 hp at 400 rpm without exceeding the allowable shear stress of 6000 psi. Determine the min

> A beam of rectangular cross section (width b and height h) supports a uniformly distributed load along its entire length L. The allowable stresses in bending and shear are σallow and τallow, respectively. (a) If the beam is simply supported, what is the

> The drive shaft for a truck (outer diameter 60 mm and inner diameter 40 mm) is running at 2500 rpm (see figure). (a) If the shaft transmits 150 kW, what is the maximum shear stress in the shaft? (b) If the allowable shear stress is 30 MPa, wh

> The propeller shaft of a large ship has an outside diameter 18 in. and inside diameter 12 in., as shown in the figure. The shaft is rated for a maximum shear stress of 4500 psi. (a) If the shaft is turning at 100 rpm, what is the maximum horsepower that

> A solid steel shaft ABC with diameter d = 40 mm is driven at A by a motor that transmits 75 kW to the shaft at 15 Hz. The gears at B and C drive machinery requiring power equal to 50 kW and 25 kW, respectively. Compute the maximum

> A motor driving a solid circular steel shaft with diameter d = 1.5 in. transmits 50 hp to a gear at B. The allowable shear stress in the steel is 6000 psi. Calculate the required speed of rotation (number of revolutions per minute) so that the shear stre

> A motor drives a shaft at 12 Hz and delivers 20 kW of power (see figure). (a) If the shaft has a diameter of 30 mm, what is the maximum shear stress τmax in the shaft? (b) If the maximum allowable shear stress is 40 MPa, what is th

> Two pipes (L1 = 2.5 m and L2 = 1.5 m) are joined at B by flange plates (thickness tf = 14 mm) with five bolts (dbf = 1 3 mm) arranged in a circular pattern (see figure). Also, each pipe segment is attached to a wall (at A and C, see figure) using a base

> A uniformly tapered aluminum-alloy tube AB of circular cross section and length L is fixed against rotation at A and B, as shown in the figure. The outside diameters at the ends are dA and dB = 2dA. A hollow section of length L/2 and constant thickness t

> A steel shaft (Gs = 8 0GPa) of total length L = 3.0 m is encased for one-third of its length by a brass sleeve (Gb = 4 0GPa) that is securely bonded to the steel (see figure). The outer diameters of the shaft and sleeve are d1 = 70 mm and d2 = 90 mm, res

> The composite shaft shown in the figure is manufactured by shrink-fitting a steel sleeve over a brass core so that the two parts act as a single solid bar in torsion. The outer diameters of the two parts are d1 = 1.6 in. for the brass core and d2 = 2.0 i

> The composite shaft shown in the figure is manufactured by shrink-fitting a steel sleeve over a brass core so that the two parts act as a single solid bar in torsion. The outer diameters of the two parts are d1 = 40 mm for the brass core and d2 = 50 mm f

> A steel beam of length L = 16 in and cross- sectional dimensions b = 0.6 in and h = 2 in (see figure) supports a uniform load of intensity q = 240 lb/in, which includes the weight of the beam. Calculate the shear stresses in the beam (at the cross sectio

> A rectangular beam with semicircular notches, as shown in part b of the figure, has dimensions h = 120 mm and h1 = 100 mm. The maximum allowable bending stress in the plastic beam is σmax = 6 MPa, and the bending moment is M = 150 Nm? Determin

> A solid steel bar of diameter d1 = 1.50 in. is enclosed by a steel tube of outer diameter d3 = 2.25 in. and inner diameter d2 = 1.75 in. (see figure). Both bar and tube are held rigidly by a support at end A and joined securely to a rigid plate at end B.

> A solid steel bar of diameter d1 = 25.0mm is enclosed by a steel tube of outer diameter d3 = 37.5 mm and inner diameter d2 = 30.0mm (see figure). Both bar and tube are held rigidly by a support at end A and joined securely to a rigid plate at end B. The

> A circular bar AB with ends fixed against rotation has a hole extending for half of its length (see figure). The outer diameter of the bar is d2 = 3.0 in., and the diameter of the hole is d1 = 2.4 in. The total length of the bar is L = 50 in. (a) At what

> A circular bar AB of length L is fixed against rotation at the ends and loaded by a distributed torque t(x) that varies linearly in intensity from zero at end A to to at end B (see figure). (a) Obtain formulas for the fixed-end torques TA and TB. (b) Fin

> Two sections of steel drill pipe, joined by bolted flange plates at B, are subjected to a concentrated torque 4000 kip-in. at x = 3 ft, and a uniformly distributed torque to = 50 kip-ft/ft is applied on pipe BC. Let G = 11,800 ksi and assume that pipes A

> A solid circular aluminum bar AB is fixed at both ends and loaded by a uniformly distributed torque 150 Nm/m? The bar has diameter d = 30 mm. Calculate the reactive torques at the supports and the angle of twist at mid span. Assume that G = 28 GPa.