Show by integration that the strain energy in the tapered

Determine the modulus of resilience for each of the following grades of structural steel: (a) ASTM A709 Grade 50: \(\sigma_{Y}=50 \mathrm{ksi}\) (b) ASTM A913 Grade 65: \(\sigma_{Y}=65 \mathrm{ksi}\) (c) ASTM A709 Grade 100: \(\sigma_{Y}=100 \mathrm{ksi}\)

Determine the modulus of resilience for each of the following aluminum alloys: (a) 1100-H14: E = 70 GPa: \(\sigma_{Y}=55 \mathrm{ksi}\) (b) 2014-T6: E = 72 GPa: \(\sigma_{Y}=220 \mathrm{ksi}\) (c) 6061-T6: E = 69 GPa: \(\sigma_{Y}=150 \mathrm{ksi}\)

Determine the modulus of resilience for each of the following metals: (a) Stainless steel AISI 302 (annealed): E = 190 GPa \(\sigma_{Y}=260 \mathrm{ksi}\) (b) Stainless steel AISI 302 (cold-rolled): E = 190 GPa \(\sigma_{Y}=520 \mathrm{ksi}\) (c) Malleable cast iron: E = 165 GPa \(\sigma_{Y}=230 \mathrm{ksi}\)

Determine the modulus of resilience for each of the following alloys: (a) Titanium: \(E=16.5 \times 10^{6} \mathrm{psi}\) \(\sigma_{Y}=120 \mathrm{ksi}\) (b) Magnesium: \(E=6.5 \times 10^{6} \mathrm{psi}\) \(\sigma_{Y}=29 \mathrm{ksi}\) (c) Cupronickel (annealed): \(E=20 \times 10^{6} \mathrm{psi}\) \(\sigma_{Y}=16 \mathrm{ksi}\)

The stress-strain diagram shown has been drawn from data obtained during a tensile test of a specimen of structural steel. Using \(E=29 \times 10^{6} \mathrm{psi}\), determine (a) the modulus of resilience of the steel, (b) the modulus of toughness of the steel.

The stress-strain diagram shown has been drawn from data obtained during a tensile test of an aluminum alloy. Using E = 72 GPa, determine (a) the modulus of resilience of the alloy, (b) the modulus of toughness of the alloy.

The load-deformation diagram shown has been drawn from data obtained during a tensile test of a specimen of an aluminum alloy. Knowing that the cross-sectional area of the specimen was \(600 \mathrm{mm}^{2}\) and that the deformation was measured using a 400-mm gage length, determine by approximate means (a) the modulus of resilience of the alloy, (b) the modulus of toughness of the alloy.

The load-deformation diagram shown has been drawn from data obtained during a tensile test of a \(\frac{5}{8}-\mathrm{in} .\)-diameter rod of structural steel. Knowing that the deformation was measured using an 18-in. gage length, determine by approximate means (a) the modulus of resilience of the steel, (b) the modulus of toughness of the steel.

Using \(E=29 \times 10^{6} \mathrm{psi}\), determine (a) the strain energy of the steel rod ABC when P = 8 kips, (b) the corresponding strain energy density in portions AB and BC of the rod.

Using E = 200 GPa, determine (a) the strain energy of the steel rod ABC when P = 25 kN, (b) the corresponding strain-energy density in portions AB and BC of the rod.

A 30-in. length of aluminum pipe of cross-sectional area \(1.85 \mathrm{in}^{2}\) is welded to a fixed support A and to a rigid cap B. The steel rod EF, of 0.75-in. diameter, is welded to cap B. Knowing that the modulus of elasticity is \(29 \times 10^{6} \mathrm{psi}\) for the steel and \(10.6 \times 10^{6} \mathrm{psi}\) for the aluminum, determine (a) the total strain energy of the system when P = 8 kips, (b) the corresponding strain-energy density of the pipe CD and in the rod EF.

A single 6-mm-diameter steel pin B is used to connect the steel strip DE to two aluminum strips, each of 20-mm width and 5-mm thickness. The modulus of elasticity is 200 GPa for the steel and 70 GPa for the aluminum. Knowing that for the pin at B the allowable shearing stress is \(\tau_{\text {all }}=85 \mathrm{MPa}\), determine, for the loading shown, the maximum strain energy that can be acquired by the assembled strips.

Rods AB and BC are made of a steel for which the yield strength is \(\sigma_{Y}=300 \mathrm{MPa}\) and the modulus of elasticity is E = 200 GPa. Determine the maximum strain energy that can be acquired by the assembly without causing any permanent deformation when the length a of rod AB is (a) 2 m, (b) 4 m.

Rod BC is made of a steel for which the yield strength is \(\sigma_{Y}=300 \mathrm{MPa}\) and the modulus of elasticity is E = 200 GPa. Knowing that a strain energy of 10 J must be acquired by the rod when the axial load P is applied, determine the diameter of the rod for which the factor of safety with respect to permanent deformation is six.

The assembly ABC is made of a steel for which E = 200 GPa and \(\sigma_{Y}=320 \mathrm{MPa}\). Knowing that a strain energy of 5 J must be acquired by the assembly as the axial load P is applied, determine the factor of safety with respect to permanent deformation when (a) x = 300 mm, (b) x = 600 mm.

Show by integration that the strain energy of the tapered rod AB is \(U=\frac{1}{4} \frac{P^{2} L}{E A_{\min }}\) where \(A_{\min }\) is the cross-sectional area at end B.

Using \(E=10.6 \times 10^{6} \mathrm{psi}\), determine by approximate means the maximum strain energy that can be acquired by the aluminum rod shown if the allowable normal stress is \(\sigma_{all}=22 \mathrm{ksi}\).

In the truss shown, all members are made of the same material and have the uniform cross-sectional area indicated. Determine the strain energy of the truss when the load P is applied.

In the truss shown, all members are made of the same material and have the uniform cross-sectional area indicated. Determine the strain energy of the truss when the load P is applied.

In the truss shown, all members are made of the same material and have the uniform cross-sectional area indicated. Determine the strain energy of the truss when the load P is applied.

In the truss shown, all members are made of the same material and have the uniform cross-sectional area indicated. Determine the strain energy of the truss when the load P is applied.

Each member of the truss shown is made of aluminum and has the cross-sectional area shown. Using E = 72 GPa, determine the strain energy of the truss for the loading shown.

Each member of the truss shown is made of aluminum and has the cross-sectional area shown. Using \(E=10.5 \times 10^{6} \mathrm{psi}\), determine the strain energy of the truss for the loading shown.

Taking into account only the effect of normal stresses, determine the strain energy of the prismatic beam AB for the loading shown.

Taking into account only the effect of normal stresses, determine the strain energy of the prismatic beam AB for the loading shown.

Taking into account only the effect of normal stresses, determine the strain energy of the prismatic beam AB for the loading shown.

Taking into account only the effect of normal stresses, determine the strain energy of the prismatic beam AB for the loading shown.

Using \(E=29 \times 10^{6} \mathrm{psi}\), determine the strain energy due to bending for the steel beam and loading shown. (Neglect the effect of shearing stresses.)

Using \(E=29 \times 10^{6} \mathrm{psi}\), determine the strain energy due to bending for the steel beam and loading shown. (Neglect the effect of shearing stresses.)

Using E = 200 GPa, determine the strain energy due to bending for the steel beam and loading shown. (Neglect the effect of shearing stresses.)

Using E = 200 GPa, determine the strain energy due to bending for the steel beam and loading shown. (Neglect the effect of shearing stresses.)

Assuming that the prismatic beam AB has a rectangular cross section, show that for the given loading the maximum value of the strain-energy density in the beam is \(u_{\max }=\frac{45}{8} \frac{U}{V}\) where U is the strain energy of the beam and V is its volume.

In the assembly shown, torques \(\mathbf{T}_{A} \text { and } \mathbf{T}_{B}\) are exerted on disks A and B, respectively. Knowing that both shafts are solid and made of aluminum (G = 73 GPa), determine the total strain energy acquired by the assembly.

The design specifications for the steel shaft AB require that the shaft acquire a strain energy of 400 in ? lb as the 25-kip ? in. torque is applied. Using \(G=11.2 \times 10^{6} \mathrm{psi}\), determine (a) the largest inner diameter of the shaft that can be used, (b) the corresponding maximum shearing stress in the shaft.

Show by integration that the strain energy in the tapered rod AB is \(U=\frac{7}{48} \frac{T^{2} L}{G J_{\min }}\) where \(J_{\min }\) is the polar moment of inertia of the rod at end B.

The state of stress shown occurs in a machine component made of a brass for which \(\sigma_{Y}=160 \mathrm{MPa}\). Using the maximum-distortion-energy criterion, determine the range of values of \(\sigma_{z}\) for which yield does not occur.

The state of stress shown occurs in a machine component made of a brass for which \(\sigma_{Y}=160 \mathrm{MPa}\). Using the maximum-distortion-energy criterion, determine whether yield occurs when (a) \(\sigma_{z}= +45 \mathrm{MPa}\), (b) \(\sigma_{z}= -45 \mathrm{MPa}\).

The state of stress shown occurs in a machine component made of a grade of steel for which \(\sigma_{Y}=65 \mathrm{ksi}\). Using the maximum-distortion-energy criterion, determine the range of values of \(\sigma_{y}\) for which the factor of safety associated with the yield strength is equal to or larger than 2.2.

The state of stress shown occurs in a machine component made of a grade of steel for which \(\sigma_{Y}=65 \mathrm{ksi}\). Using the maximum-distortion-energy criterion, determine the factor of safety associated with the yield strength when (a) \(\sigma_{y}= +16 \mathrm{ksi}\), (b) \(\sigma_{y}= -16 \mathrm{ksi}\).

Determine the strain energy of the prismatic beam AB, taking into account the effect of both normal and shearing stresses.

A vibration isolation support is made by bonding a rod A, of radius \(R_{1}\), and a tube B, of inner radius \(R_{2}\), to a hollow rubber cylinder. Denoting by G the modulus of rigidity of the rubber, determine the strain energy of the hollow rubber cylinder for the loading shown.

A 5-kg collar D moves along the uniform rod AB and has a speed \(v_{0}=6 \mathrm{~m} / \mathrm{s}\) when it strikes a small plate attached to end A of the rod. Using E = 200 GPa and knowing that the allowable stress in the rod is 250 MPa, determine the smallest diameter that can be used for the rod.

The 18-lb cylindrical block E has a horizontal velocity \(v_{0}\) when it strikes squarely the yoke BD that is attached to the \(\frac{7}{8}-\mathrm{in}\).-diameter rods AB and CD. Knowing that the rods are made of a steel for which \(\sigma_{Y}=50 \mathrm{ksi} \text { and } E=29 \times 10^{6} \mathrm{psi}\), determine the maximum allowable speed \(v_{0}\) if the rods are not to be permanently deformed.

The cylindrical block E has a speed \(v_{0}=16 \mathrm{ft} / \mathrm{s}\) when it strikes squarely the yoke BD that is attached to the \(\frac{7}{8}-\mathrm{in}\)-diameter rods AB and CD. Knowing that the rods are made of a steel for which \(\sigma_{Y}=50 \mathrm{ksi} \text { and } E=29 \times 10^{6} \mathrm{psi}\), determine the weight of block E for which the factor of safety is five with respect to permanent deformation of the rods.

The 35-kg collar D is released from rest in the position shown and is stopped by a plate attached at end C of the vertical rod ABC. Knowing that E = 200 GPa for both portions of the rod, determine the distance h for which the maximum stress in the rod is 250 MPa.

The 15-kg collar D is released from rest in the position shown and is stopped by a plate attached at end C of the vertical rod ABC. Knowing that E = 200 GPa for both portions of the rod, determine (a) the maximum deflection of end C, (b) the equivalent static load, (c) the maximum stress that occurs in the rod.

The 48-kg collar G is released from rest in the position shown and is stopped by plate BDF that is attached to the 20-mm-diameter steel rod CD and to the 15-mm-diameter steel rods AB and EF. Knowing that for the grade of steel used \(\sigma_{all}=180 \mathrm{MPa}\) and E = 200 GPa, determine the largest allowable distance h.

A 25-lb block C moving horizontally with at velocity \(\mathbf{v}_{0}\) hits the post AB squarely as shown. Using \(E=29 \times 10^{6} \mathrm{psi}\), determine the largest speed \(v_{0}\) for which the maximum normal stress in the post does not exceed 18 ksi.

Solve Prob. 11.48, assuming that the post AB has been rotated 90° about its longitudinal axis. A 25-lb block C moving horizontally with at velocity \(\mathbf{v}_{0}\) hits the post AB squarely as shown. Using \(E=29 \times 10^{6} \mathrm{psi}\), determine the largest speed \(v_{0}\) for which the maximum normal stress in the post does not exceed 18 ksi.

An aluminum tube having the cross section shown is struck squarely in its midsection by a 6-kg block moving horizontally with a speed of 2 m/s. Using E = 70 GPa, determine (a) the equivalent static load, (b) the maximum stress in the beam, (c) the maximum deflection at the midpoint C of the beam.

Solve Prob. 11.50, assuming that the tube has been replaced by a solid aluminum bar with the same outside dimensions as the tube. An aluminum tube having the cross section shown is struck squarely in its midsection by a 6-kg block moving horizontally with a speed of 2 m/s. Using E = 70 GPa, determine (a) the equivalent static load, (b) the maximum stress in the beam, (c) the maximum deflection at the midpoint C of the beam.

The 2-kg block D is dropped from the position shown onto the end of a 16-mm-diameter rod. Knowing that E = 200 GPa, determine (a) the maximum deflection of end A, (b) the maximum bending moment in the rod, (c) the maximum normal stress in the rod.

The 10-kg block D is dropped from a height h = 450 mm onto the aluminum beam AB. Knowing that E = 70 GPa, determine (a) the maximum deflection of point E, (b) the maximum stress in the beam.

The 4-lb block D is dropped from the position shown onto the end of a \(\frac{5}{8}-\mathrm{in}\).-diameter rod. Knowing that \(E=29 \times 10^{6} \mathrm{psi}\), determine (a) the maximum deflection at point A, (b) the maximum bending moment in the rod, (c) the maximum normal stress in the rod.

A 160-lb diver jumps from a height of 20 in. onto end C of a diving board having the uniform cross section shown. Assuming that the diver’s legs remain rigid and using \(E=1.8 \times 10^{6} \mathrm{psi}\), determine (a) the maximum deflection at point C, (b) the maximum normal stress in the board, (c) the equivalent static load.

A block of weight W is dropped from a height h onto the horizontal beam AB and hits it at point D. (a) Show that the maximum deflection ym at point D can be expressed as \(y_{m}=y_{s t}\left(1+\sqrt{1+\frac{2 h}{y_{s t}}}\right)\) where \(y_{s t}\) represents the deflection at D caused by a static load W applied at that point and where the quantity in parenthesis is referred to as the impact factor. (b) Compute the impact factor for the beam of Prob. 11.52.

A block of weight W is dropped from a height h onto the horizontal beam AB and hits point D. (a) Denoting by \(y_{m}\) the exact value of the maximum deflection at D and by \(y_{m}^{\prime}\) the value obtained by neglecting the effect of this deflection on the change in potential energy of the block, show that the absolute value of the relative error is \(\left(y_{m}^{\prime}-y_{m}\right) / y_{m}\), never exceeding \(y_{m}^{\prime} / 2 h\). (b) Check the result obtained in part a by solving part a of Prob. 11.52 without taking \(y_{m}\) into account when determining the change in potential energy of the load, and comparing the answer obtained in this way with the exact answer to that problem.

Using the method of work and energy, determine the deflection at point D caused by the load P.

Using the method of work and energy, determine the deflection at point D caused by the load P.

Using the method of work and energy, determine the slope at point D caused by the couple \(\mathbf{M}_{0}\).

Using the method of work and energy, determine the slope at point D caused by the couple \(\mathbf{M}_{0}\).

Using the method of work and energy, determine the deflection at point C caused by the load P.

Using the method of work and energy, determine the deflection at point C caused by the load P.

Using the method of work and energy, determine the slope at point B caused by the couple \(\mathbf{M}_{0}\).

Using the method of work and energy, determine the slope at point D caused by the couple \(\mathbf{M}_{0}\).

The 20-mm diameter steel rod BC is attached to the lever AB and to the fixed support C. The uniform steel lever is 10 mm thick and 30 mm deep. Using the method of work and energy, determine the deflection of point A when L = 600 mm. Use E = 200 GPa and G = 77.2 GPa.

Torques of the same magnitude T are applied to the steel shafts AB and CD. Using the method of work and energy, determine the length L of the hollow portion of shaft CD for which the angle of twist at C is equal to 1.25 times the angle of twist at A.

Two steel shafts, each of 0.75-in.-diameter, are connected by the gears shown. Knowing that \(G=11.2 \times 10^{6} \mathrm{psi}\) and that shaft DF is fixed at F, determine the angle through which end A rotates when a 750-lb · in. torque is applied at A. (Neglect the strain energy due to the bending of the shafts.)

The 20-mm-diameter steel rod CD is welded to the 20-mm-diameter steel shaft AB as shown. End C of rod CD is touching the rigid surface shown when a couple \(\mathbf{T}_{B}\) is applied to a disk attached to shaft AB. Knowing that the bearings are self aligning and exert no couples on the shaft, determine the angle of rotation of the disk when \(T_{B}=400 \mathrm{~N} \cdot \mathrm{m}\). Use E = 200 GPa and G = 77.2 GPa. (Consider the strain energy due to both bending and twisting in shaft AB and to bending in arm CD.)

The thin-walled hollow cylindrical member AB has a noncircular cross section of nonuniform thickness. Using the expression given in Eq. (3.50) of Sec. 3.10 and the expression for the strain-energy density given in Eq. (11.17), show that the angle of twist of member AB is \(\phi=\frac{T L}{4 \mathfrak{Q}^{2} G} \oint \frac{d s}{t}\) where ds is the length of a small element of the wall cross section and A is the area enclosed by center line of the wall cross section.

Each member of the truss shown has a uniform cross-sectional area A. Using the method of work and energy, determine the horizontal deflection of the point of application of the load P.

Each member of the truss shown has a uniform cross-sectional area A. Using the method of work and energy, determine the horizontal deflection of the point of application of the load P.

Each member of the truss shown is made of steel and has a uniform cross-sectional area of \(5 \mathrm{in}^{2}\). Using \(E=29 \times 10^{6} \mathrm{psi}\), determine the vertical deflection of joint B caused by the application of the 20-kip load.

Each member of the truss shown is made of steel. The cross-sectional area of member BC is \(800 \mathrm{~mm}^{2}\), and for all other members the cross-sectional area is \(400 \mathrm{~mm}^{2}\). Using E = 200 GPa, determine the deflection of point D caused by the 60-kN load.

Each member of the truss shown is made of steel and has a cross-sectional area of \(5 \mathrm{in}^{2}\). Using \(E=29 \times 10^{6} \mathrm{psi}\), determine the vertical deflection of point C caused by the 15-kip load.

The steel rod BC has a 24-mm diameter and the steel cable ABDCA has a 12-mm diameter. Using E = 200 GPa, determine the deflection of joint D caused by the 12-kN load.

Using the information in Appendix D, compute the work of the loads as they are applied to the beam (a) if the load P is applied first, (b) if the couple M is applied first.

Using the information in Appendix D, compute the work of the loads as they are applied to the beam (a) if the load P is applied first, (b) if the couple M is applied first.

For the beam and loading shown, (a) compute the work of the loads as they are applied successively to the beam, using the information provided in Appendix D, (b) compute the strain energy of the beam by the method of Sec. 11.2A and show that it is equal to the work obtained in part a.

For the beam and loading shown, (a) compute the work of the loads as they are applied successively to the beam, using the information provided in Appendix D, (b) compute the strain energy of the beam by the method of Sec. 11.2A and show that it is equal to the work obtained in part a.

For the beam and loading shown, (a) compute the work of the loads as they are applied successively to the beam, using the information provided in Appendix D, (b) compute the strain energy of the beam by the method of Sec. 11.2A and show that it is equal to the work obtained in part a.

For the beam and loading shown, (a) compute the work of the loads as they are applied successively to the beam, using the information provided in Appendix D, (b) compute the strain energy of the beam by the method of Sec. 11.2A and show that it is equal to the work obtained in part a.

For the prismatic beam shown, determine the deflection of point D.

For the prismatic beam shown, determine the deflection of point D.

For the prismatic beam shown, determine the deflection of point D.

For the prismatic beam shown, determine the slope at point D.

For the prismatic beam shown, determine the slope at point D.

For the prismatic beam shown, determine the slope at point D.

For the prismatic beam shown, determine the slope at point A.

For the prismatic beam shown, determine the slope at point B.

For the beam and loading shown, determine the deflection of point B. Use \(E=29 \times 10^{6} \mathrm{psi}\).

For the beam and loading shown, determine the deflection of point A. Use \(E=29 \times 10^{6} \mathrm{psi}\).

For the beam and loading shown, determine the deflection at point B. Use E = 200 GPa.

For the beam and loading shown, determine the deflection at point B. Use E = 200 GPa.

For the beam and loading shown, determine the slope at end A. Use E = 200 GPa.

For the beam and loading shown, determine the deflection at point D. Use E = 200 GPa.

For the beam and loading shown, determine the slope at end A. Use \(E=29 \times 10^{6} \mathrm{psi}\).

For the beam and loading shown, determine the deflection at point C. Use \(E=29 \times 10^{6} \mathrm{psi}\).

For the truss and loading shown, determine the horizontal and vertical deflection of joint C.

For the truss and loading shown, determine the horizontal and vertical deflection of joint C.

Each member of the truss shown is made of steel and has the cross-sectional area shown. Using \(E=29 \times 10^{6} \mathrm{psi}\), determine the deflection indicated. Vertical deflection of joint C.

Each member of the truss shown is made of steel and has the cross-sectional area shown. Using \(E=29 \times 10^{6} \mathrm{psi}\), determine the deflection indicated. Horizontal deflection of joint C.

Each member of the truss shown is made of steel and has a cross-sectional area of \(500 \mathrm{~mm}^{2}\). Using E = 200 GPa, determine the deflection indicated. Vertical deflection of joint B.

Each member of the truss shown is made of steel and has a cross-sectional area of \(500 \mathrm{~mm}^{2}\). Using E = 200 GPa, determine the deflection indicated. Horizontal deflection of joint B.

A uniform rod of flexural rigidity EI is bent and loaded as shown. Determine (a) the vertical deflection of point A, (b) the horizontal deflection of point A.

For the uniform rod and loading shown and using Castigliano’s theorem, determine the deflection of point B.

For the beam and loading shown and using Castigliano’s theorem, determine (a) the horizontal deflection of point B, (b) the vertical deflection of point B.

Two rods AB and BC of the same flexural rigidity EI are welded together at B. For the loading shown, determine (a) the deflection of point C, (b) the slope of member BC at point C.

Three rods, each of the same flexural rigidity EI, are welded to form the frame ABCD. For the loading shown, determine the deflection of point D.

Three rods, each of the same flexural rigidity EI, are welded to form the frame ABCD. For the loading shown, determine the angle formed by the frame at point D.

Determine the reaction at the roller support and draw the bending-moment diagram for the beam and loading shown.

Determine the reaction at the roller support and draw the bending-moment diagram for the beam and loading shown.

Determine the reaction at the roller support and draw the bending-moment diagram for the beam and loading shown.

Determine the reaction at the roller support and draw the bending-moment diagram for the beam and loading shown.

Determine the reaction at the roller support and draw the bending-moment diagram for the beam and loading shown.

For the uniform beam and loading shown, determine the reaction at each support.

Three members of the same material and same cross-sectional area are used to support the load P. Determine the force in member BC.

Three members of the same material and same cross-sectional area are used to support the load P. Determine the force in member BC.

Three members of the same material and same cross-sectional area are used to support the load P. Determine the force in member BC.

Three members of the same material and same cross-sectional area are used to support the load P. Determine the force in member BC.

Knowing that the eight members of the indeterminate truss shown have the same uniform cross-sectional area, determine the force in member AB.

Knowing that the eight members of the indeterminate truss shown have the same uniform cross-sectional area, determine the force in member AB.