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physics
mechanics
Mechanics of Materials 7th edition James M. Gere, Barry J. Goodno - Solutions
A uniformly tapered aluminum-alloy tube AB of 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 extends from B halfway toward A.(a) Find the angle
For the thin nonprismatic steel pipe of constant thickness t and variable diameter d shown with applied torques at joints 2 and 3, determine the following.(a) Find reaction moment R1.(b) Find an expression for twist rotation (3 at joint 3. Assume that G is constant.(c) Draw the torsional moment
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). Segments AB and CD are prismatic with
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).(b) Determine the angle of twist ( between the ends of the bar.
A prismatic bar AB of 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 to zero at end B. Also, the length of the bar is L
A nonprismatic bar ABC of 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 T0/L at B. Segment BC has linearly distributed torque of
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 the wire inside the tube. At the other end A,
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 of diameter d1 = 40 mm is inside of, and concentric with, the tube. The bar passes through a
Two hollow tubes are connected by a pin at B which is inserted into a hole drilled through both tubes at B (see cross-section view at B). Tube BC fits snugly into tube AB but neglect any friction on the interface. Tube inner and outer diameters di (i = 1, 2, 3) and pin diameter dp are labeled in
A stepped shaft ABCD consisting of solid circular segments is subjected to three torques, as shown in the figure. The torques have magnitudes 12.5 k-in., 9.8 k-in., and 9.2 k-in. The length of each segment is 25 in. and the diameters of the segments are 3.5 in., 2.75 in., and 2.5 in. The material
A solid circular bar ABC consists of two segments, as shown in the figure. One segment has diameter d1 = 56 mm and length L1 = 1.45 m; the other segment has diameter d2 = 48 mm and length L2 = 1.2 m.What is the allowable torque Tallow if the shear stress is not to exceed 30 MPa and the angle of
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 = 1000 lb-in., T2 = T4 = 500 lb-in., and T3 = T5 = 800 lb-in. The tube has an outside diameter d2 = 1.0 in. The allowable shear stress is
A shaft of solid circular cross section consisting of two segments is shown in the first part of the figure. The left-hand segment has diameter 80 mm and length 1.2 m; the right-hand segment has diameter 60 mm and length 0.9 m.Shown in the second part of the figure is a hollow shaft made of the
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 outside diameter d if the shaft is hollow with an
A tapered bar AB of 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.For what ratio dB/dA will the angle of twist of the tapered bar be one-half the angle of twist of a prismatic bar
A tapered bar AB of 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 aluminum alloy having shear modulus of elasticity
A hollow aluminum shaft (see figure) has 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 0.54°/ft. The shear modulus of elasticity of the aluminum is G = 4.0 ( 106 psi.(a) Determine the maximum
A solid aluminum bar (G = 27 GPa) of diameter d = 40 mm is subjected to torques T = 300 N ( m 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 of properly oriented stress elements.(b)
A hollow steel bar (G = 80 GPa) is twisted by torques T (see figure). The twisting of the bar produces a maximum shear strain (max = 640 ( 10-6 rad. The bar has outside and inside diameters of 150 mm and 120 mm, respectively. (a) Determine the maximum tensile strain in the bar. (b) Determine the
A tubular bar with outside diameter d2 = 4.0 in. is twisted by torques T = 70.0 k-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) If the bar has length L = 48.0 in. and is made
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 N ( m. At this value of torque, a strain gage oriented at 45° to the axis of the bar gives a reading ϵ 339 ( 10-6.What is the shear modulus G of
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 steel (G = 78 GPa) transmits a torque T = 360 N ( m. 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. Determine the minimum required diameter d of the bar.
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.If the outside diameter d2 of the tube is 0.8 in., what is the inside diameter d1?
An aluminium tube has inside diameter d1 = 50 mm, shear modulus of elasticity G = 27 GPa, and torque T = 4.0 kN ( m. The allowable shear stress in the aluminum is 50 MPa and the allowable normal strain is 900 ( 10-6. Determine the required outside diameter d2.
A solid steel bar (G = 11.8 ( 106 psi) of diameter d = 2.0 in. is subjected to torques T = 8.0 k-in. 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 of properly oriented stress
A generator shaft in a small hydroelectric plant turns at 120 rpm and delivers 50 hp (see figure).(b) If the shear stress is limited to 4000 psi, what is the minimum permissible diameter dmin of the shaft?
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 and 180 kW, respectively. The lengths of the two parts of the shaft are L1 = 1.5 m and L2 = 0.9 m. Determine the required diameter d of the shaft if the
A motor drives a shaft at 12 Hz and delivers 20 kW of power (see figure).(b) If the maximum allowable shear stress is 40 MPa, what is the minimum permissible diameter dmin of the shaft?
The propeller shaft of a large ship has 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 can be transmitted without exceeding the allowable
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, what is the maximum power that can be transmitted?
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 minimum required outside diameter d.
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 outside diameter d?
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 the splice can transmit the same power as the
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 twist is 3.0°/m?
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 the motor and gear C is limited to 1.5°.
A solid circular bar ABCD with fixed supports is acted upon by torques T0 and 2T0 at the locations shown in the figure.Obtain a formula for the maximum angle of twist (max of the bar. (Use Eqs. 3-46 a and b of Example 3-9 to obtain the reactive torques.)
A solid steel bar of diameter d1 = 25.0 mm is enclosed by a steel tube of outer diameter d3 = 37.5 mm and inner diameter d2 = 30.0 mm (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 composite bar, which has a length L = 550
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. The composite bar, which has length L = 30.0
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 for the steel sleeve. The shear moduli of
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 in. for the steel sleeve. The shear moduli of
A steel shaft (Gs = 80 GPa) of total length L = 3.0 m is encased for one-third of its length by a brass sleeve (Gb = 40 GPa) 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. respectively.(a) Determine the allowable torque
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 = dA/10 is cast into the tube and extends
A hollow circular tube A (outer diameter dA, wall thickness tA) fits over the end of a circular tube B (dB, tB), as shown in the figure. The far ends of both tubes are fixed. Initially, a hole through tube B makes an angle ( with a line through two holes in tube A. Then tube B is twisted until the
A solid circular bar ABCD with fixed supports at ends A and D is acted upon by two equal and oppositely directed torques T0, as shown in the figure. The torques are applied at points B and C, each of which is located at distance x from one end of the bar. (The distance x may vary from zero to
A solid circular shaft AB of diameter d is fixed against rotation at both ends (see figure). A circular disk is attached to the shaft at the location shown.
A hollow steel shaft ACB of outside diameter 50 mm and inside diameter 40 mm is held against rotation at ends A and B (see figure). Horizontal forces P are applied at the ends of a vertical arm that is welded to the shaft at point C.Determine the allowable value of the forces P if the maximum
A stepped shaft ACB having solid circular cross sections with two different diameters is held against rotation at the ends (see figure).If the allowable shear stress in the shaft is 6000 psi, what is the maximum torque (T0)max that may be applied at section C? (Use Eqs. 3-45a and b of Example 3-9
A stepped shaft ACB having solid circular cross sections with two different diameters is held against rotation at the ends (see figure).If the allowable shear stress in the shaft is 43 MPa, what is the maximum torque (T0)max that may be applied at section C? (Use Eqs. 3-45 a and b of Example 3-9 to
A stepped shaft ACB is held against rotation at ends A and B and subjected to a torque T0 acting at section C (see figure). The two segments of the shaft (AC and CB) have diameters dA and dB, respectively, and polar moments of inertia IPA and IPB, respectively. The shaft has length L and segment AC
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 t0 at end B (see figure).Obtain formulas for the fixed-end torques TA and TB.
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.At what distance x from the left-hand end of the bar
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 the maximum shear stress is 4500 psi.(b) From the strain
A hollow circular tube A fits over the end of a solid circular bar B, as shown in the figure. The far ends of both bars are fixed. Initially, a hole through bar B makes an angle ( with a line through two holes in tube A. Then bar B is twisted until the holes are aligned, and a pin is placed through
A heavy flywheel rotating at n revolutions per minute is rigidly attached to the end of a shaft of diameter d (see figure). If the bearing at A suddenly freezes, what will be the maximum angle of twist ( of the shaft? What is the corresponding maximum shear stress in the shaft?(Let L = length of
A solid circular bar of copper (G = 45 GPa) with length L = 0.75 m and diameter d = 40 mm 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 the maximum shear stress is 32 MPa. (b) From the strain energy,
A stepped shaft of solid circular cross sections (see figure) has length L = 45 in., diameter d2 = 1.2 in., and diameter d1 = 1.0 in. The material is brass with G = 5.6 ( 106 psi.Determine the strain energy U of the shaft if the angle of twist is 3.0°.
A stepped shaft of solid circular cross sections (see figure) has length L = 0.80 m, diameter d2 = 40 mm, and diameter d1 = 30 mm. The material is steel with G = 80 GPa. Determine the strain energy U of the shaft if the angle of twist is 1.0°.
A cantilever bar of circular cross section and length L is fixed at one end and free at the other (see figure). The bar is loaded by a torque T at the free end and by a distributed torque of constant intensity t per unit distance along the length of the bar.(a) What is the strain energy U1 of the
Obtain a formula for the strain energy U of the statically indeterminate circular bar shown in the figure. The bar has fixed supports at ends A and B and is loaded by torques 2T0 and T0 at points C and D, respectively.Use Eqs. 3-46 a and b of Example 3-9, Section 3.8, to obtain the reactive torques.
A statically indeterminate stepped shaft ACB is fixed at ends A and B and loaded by a torque T0 at point C (see figure). The two segments of the bar are made of the same material, have lengths LA and LB, and have polar moments of inertia IPA and IPB.Determine the angle of rotation ( of the cross
Derive a formula for the strain energy U of the cantilever bar shown in the figure.The bar has circular cross sections and length L. It is subjected to a distributed torque of intensity t per unit distance. The intensity varies linearly from t = 0 at the free end to a maximum value t = t0 at the
A thin-walled hollow tube AB of conical shape has constant thickness t and average diameters dA and dB at the ends (see figure).(a) Determine the strain energy U of the tube when it is subjected to pure torsion by torques T.(b) Determine the angle of twist ( of the tube.Use the approximate formula
Calculate the shear force V and bending moment M at a cross section just to the left of the 1600-1b load acting on the simple beam AB shown in the figure.
Under cruising conditions the distributed load acting on the wing of a small airplane has the idealized variation shown in the figure.Calculate the shear force V and bending moment M at the inboard end of the wing.
A beam ABCD with a vertical arm CE is supported as a simple beam at A and D (see figure). A cable passes over a small pulley that is attached to the arm at E. One end of the cable is attached to the beam at point B.What is the force P in the cable if the bending moment in the beam just to the left
A simply supported beam AB supports a trapezoidally distributed load (see figure). The intensity of the load varies linearly from 50 kN/m at support A to 25 kN/m at support B.Calculate the shear force V and bending moment M at the midpoint of the beam.
Beam ABCD represents a reinforced-concrete foundation beam that supports a uniform load of intensity q1 = 3500 lb/ft (see figure). Assume that the soil pressure on the underside of the beam is uniformly distributed with intensity q2.(a) Find the shear force VB and bending moment MB at point B.(b)
The simply-supported beam ABCD is loaded by a weight W = 27 kN through the arrangement shown in the figure. The cable passes over a small frictionless pulley at B and is attached at E to the end of the vertical arm.Calculate the axial force N, shear force V, and bending moment M at section C, which
The centrifuge shown in the figure rotates in a horizontal plane (the xy plane) on a smooth surface about the z axis (which is vertical) with an angular acceleration α. Each of the two arms has weight w per unit length and supports a weight W = 2.0 wL at its end.Derive formulas for
Determine the shear force V and bending moment M at the midpoint C of the simple beam AB shown in the figure.
Determine the shear force V and bending moment M at the midpoint of the beam with overhangs (see figure). Note that one load acts downward and the other upward. Also clockwise moments Pb are applied at each support.
Calculate the shear force V and bending moment M at a cross section located 0.5 m from the fixed support of the cantilever beam AB shown in the figure.
Determine the shear force V and bending moment M at a cross section located 18 ft from the left-hand end A of the beam with an overhang shown in the figure.
The beam ABC shown in the figure is simply supported at A and B and has an overhang from B to C. The loads consist of a horizontal force P1 = 4.0 kN acting at the end of a vertical arm and a vertical force P2 = 8.0 kN acting at the end of the overhang.Determine the shear force V and bending moment
The beam ABCD shown in the figure has overhangs at each end and carries a uniform load of intensity q.For what ratio b/L will the bending moment at the midpoint of the beam be zero?
At full draw, an archer applies a pull of 130 N to the bowstring of the bow shown in the figure. Determine the bending moment at the midpoint of the bow.
A curved bar ABC is subjected to loads in the form of two equal and opposite forces P, as shown in the figure. The axis of the bar forms a semicircle of radius r.Determine the axial force N, shear force V, and bending moment M acting at a cross section defined by the angle (.
Draw the shear-force and bending-moment diagrams for a simple beam AB supporting two equal concentrated loads P (see figure).
Draw the shear-force and bending-moment diagrams for a cantilever beam AB supporting a linearly varying load of maximum intensity (see figure).
Draw the shear-force and bending-moment diagrams for this beam.
The beam AB shown in the figure supports a uniform load of intensity 3000 N/m acting over half the length of the beam. The beam rests on a foundation that produces a uniformly distributed load over the entire length.Draw the shear-force and bending-moment diagrams for this beam.
A cantilever beam AB supports a couple and a concentrated load, as shown in the figure.Draw the shear-force and bending-moment diagrams for this beam.
The cantilever beam AB shown in the figure is subjected to a triangular load acting throughout one-half of its length and a concentrated load acting at the free end.Draw the shear-force and bending-moment diagrams for this beam.
The uniformly loaded beam ABC has simple supports at A and B and an overhang BC (see figure).Draw the shear-force and bending-moment diagrams for this beam.
A beam ABC with an overhang at one end supports a uniform load of intensity 12 kN/m and a concentrated moment of magnitude 3 kN ( m at C (see figure).Draw the shear-force and bending-moment diagrams for this beam.
Consider the two beams below; they are loaded the same but have different support conditions. Which beam has the larger maximum moment?First, find support reactions, then plot axial force (N), shear (V) and moment (M) diagrams for both beams. Label all critical N, V & M values and also the distance
The three beams below are loaded the same and have the same support conditions. However, one has a moment release just to the left of C, the second has a shear release just to the right of C, and the third has an axial release just to the left of C. Which beam has the largest maximum moment?First,
A beam ABCD shown in the figure is simply supported at A and B and has an overhang from B to C. The loads consist of a horizonatal force P1 = 400 lb acting at the end of the vertical arm and a vertical force P2 = 900 lb acting at the end of the overhang.Draw the shear-force and bending-moment
A simple beam AB is subjected to a counterclockwise couple of moment M0 acting at distance a from the left-hand support (see figure).Draw the shear-force and bending-moment diagrams for this beam.
A simple beam AB is loaded by two segments of uniform load and two horizontal forces acting at the ends of a vertical arm (see figure).Draw the shear-force and bending-moment diagrams for this beam.
The two beams below are loaded the same and have the same support conditions. However, the location of internal axial, shear and moment releases is different for each beam (see figures). Which beam has the larger maximum moment?First, find support reactions, then plot axial force (N), shear (V) and
The beam ABCD shown in the figure has overhangs that extend in both directions for a distance of 4.2 m from the supports at B and C, whiich are 1.2 m apart.Draw the shear-force and bending-moment diagrams for this overhanging beam
A beam ABCD with a vertical arm CE is supported as a simple beam at A and B (see figure). A cable passes over a small pulley that is attached to the arm at E. One end of the cable is attached to the beam at point B. The tensile force in the cable is 1800 lb.Draw the shear-force and bending-moment
Beams ABC and CD are supported at A, C and D, and are joined by a hinge (or moment release) just to the left of C and a shear release just to the right of C.The support at A is a sliding support (hence reaction Ay = 0 for the loading shown below). Find all support reactions then plot shear (V) and
The simple beam AB shown in the figure supports a concentrated load and a segment of uniform load.Draw the shear-force and bending-moment diagrams for this beam.
The cantilever beam shown in the figure supports a concentrated load and a segment of uniform load.Draw the shear-force and bending-moment diagrams for this cantilever beam.
The simple beam ACB shown in the figure is subjected to a triangular load of maximum intensity 180 lb/ft and a concentrated moment of 300 lb-ft at A.Draw the shear-force and bending-moment diagrams for this beam.
A beam with simple supports is subjected to a trapezoidally distributed load (see figure). The intensity of the load varies from 1.0 kN/m at support A to 3.0 kN/m at support B.Draw the shear-force and bending-moment diagrams for this beam.
A beam of length L is being designed to support a uniform load of intensity q (see figure). If the supports of the beam are placed at the ends, creating a simple beam, the maximum bending moment in the beam is ql2/8. However, if the supports of the beam are moved symmetrically toward the middle of
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