The strain-energy density for plane stress must be the

A material is subjected to a general state of plane stress. Express the strain energy density in terms of the elastic constants E, G, and n and the stress components \(\sigma_{x}\), \(\sigma_{y}, \text { and } \tau_{x y}\).

The strain-energy density for plane stress must be the same whether the state of stress is represented by \(\sigma_{x}\), \(\sigma_{y}\) , and \(\tau_{x y}\). or by the principal stresses \(\sigma_{1}\) and \(\sigma_{2}\). This being the case, equate the strain–energy expressions for each of these two cases and show that G = E/[2(1 + v)].

The A-36 steel bar consists of two segments, one of circular cross section of radius r, and one of square cross section. If the bar is subjected to the axial loading of P, determine the dimensions a of the square segment so that the strain energy within the square segment is the same as in the circular segment.

Determine the torsional strain energy in the A992 steel shaft. The shaft has a radius of 50 mm.

Using bolts of the same material and cross-sectional area, two possible attachments for a cylinder head are shown. Compare the strain energy developed in each case, and then explain which design is better for resisting an axial shock or impact load.

If P = 60 kN, determine the total strain energy stored in the truss. Each member has a cross-sectional area of \(2.5\left(10^{3}\right) \mathrm{mm}^{2}\) and is made of A-36 steel.

Determine the maximum force P and the corresponding maximum total strain energy stored in the truss without causing any of the members to have permanent deformation. Each member has the cross-sectional area of \(2.5\left(10^{3}\right) \mathrm{mm}^{2}\) and is made of A-36 steel.

Determine the torsional strain energy in the A992 steel shaft. The shaft has a radius of 40 mm.

Determine the torsional strain energy in the A-36 steel shaft. The shaft has a radius of 40 mm.

The shaft assembly is fixed at C. The hollow segment BC has an inner radius of 20 mm and outer radius of 40 mm, while the solid segment AB has a radius of 20 mm. Determine the torsional strain energy stored in the shaft. The shaft is made of 2014-T6 aluminum alloy. The coupling at B is rigid.

Determine the total axial and bending strain energy in the A992 steel beam. \(A=2850 \mathrm{~mm}^{2}, I=28.9\left(10^{6}\right) \mathrm{mm}^{4}\).

If P= 10kip, the total strain energy in the truss. Each member has a diameter of 2 in. and is made of A992 steel.

Determine the maximum force P and the corresponding maximum total strain energy that can be stored in the truss without causing any of the members to have permanent deformation. Each member of the truss has a diameter of 2 in. and is made of A-36 steel.

Consider the thin-walled tube of Fig. 5–26. Use the formula for shear stress, \(\tau_{\mathrm{avg}}=T / 2 t A_{m}\), Eq. 5–18, and the general equation of shear strain energy, Eq. 14–11, to show that the twist of the tube is given by Eq. 5–20. Hint: Equate the work done by the torque T to the strain energy in the tube, determined from integrating the strain energy for a differential element, Fig. 14–4, over the volume of material.

Determine the bending strain energy in the A992 steel beam. \(I=156 \mathrm{in}^{4}\).

Determine the bending strain energy in the beam. EI is constant.

Determine the bending strain energy in the simply supported beam. EI is constant.

The steel beam is supported on two springs, each having a stiffness of k = 8 MN/m. Determine the strain energy in each of the springs and the bending strain energy in the beam. \(E_{\mathrm{st}}=200 \mathrm{GPa}, I=5\left(10^{6}\right) \mathrm{mm}^{4}\).

Determine the bending strain energy in the 2-in.-diameter A-36 steel rod.

Determine the total strain energy in the steel assembly. Consider the axial strain energy in the two 0.5-in.-diameter rods and the bending strain energy in the beam, which has a moment of inertia of \(I=43.4 \mathrm{in}^{4}\). \(E_{\mathrm{st}}=29\left(10^{3}\right) \mathrm{ksi}\).

Determine the bending strain energy in the beam. EI is constant.

The bolt has a diameter of 10 mm, and the arm AB has a rectangular cross section that is 12 mm wide by 7 mm thick. Determine the strain energy in the arm due to bending and in the bolt due to axial force. The bolt is tightened so that it has a tension of 500 N. Both members are made of A-36 steel. Neglect the hole in the arm.

Determine the bending strain energy in the cantilevered beam. Solve the problem two way (a) Apply Eq. 14–17. (b) The load w dx acting on a segment dx of the beam is displaced a distance y, where \(y=w\left(-x^{4}+4 L^{3} x-3 L^{4}\right) /(24 E I)\), the equation of the elastic curve. Hence the internal strain energy in the differential segment dx of the beam is equal to the external work, i.e., \(d U_{i}=\frac{1}{2}(w d x)(-y)\). Integrate this equation to obtain the total strain energy in the beam. EI is constant.

Determine the bending strain energy in the simply supported beam. Solve the problem two ways. (a) Apply Eq. 14–17. (b) The load w dx acting on the segment dx of the beam is displaced a distance y, where \(y=w\left(-x^{4}+2 L x^{3}-L^{3} x\right) /(24 E I)\), the equation of the elastic curve. Hence the internal strain energy in the differential segment dx of the beam is equal to the external work, i.e., \(d U_{i}=\frac{1}{2}(w d x)(-y)\). Integrate this equation to obtain the total strain energy in the beam. EI is constant.

Determine the vertical displacement of joint D. AE is constant.

Determine the horizontal displacement of joint C. AE is constant.

Determine the horizontal displacement of joint A. Each bar is made of A992 steel and has a cross-sectional area of \(1.5 \mathrm{in}^{2}\).

Determine the vertical displacement of joint C. AE is constant.

Determine the vertical displacement of point C of the A992 steel beam. \(I=80\left(10^{6}\right) \mathrm{mm}^{4}\).

Determine the vertical displacement of end B of the cantilevered 6061-T6 aluminum alloy rectangular beam. Consider both shearing and bending strain energy.

Determine the vertical displacement of point B on the A992 steel beam. \(I=80\left(10^{6}\right) \mathrm{mm}^{4}\).

Determine the slope at point A of the beam. EI is constant.

The A992 steel bars are pin connected at C and D. If they each have the same rectangular cross section, with a height of 200 mm and a width of 100 mm, determine the vertical displacement at B. Neglect the axial load in the bars.

The A992 steel bars are pin connected at C. If they each have a diameter of 2 in., determine the vertical displacement at E.

Determine the slope of the beam at the pin support A. Consider only bending strain energy. EI is constant.

The cantilevered beam has a rectangular cross- sectional area A, a moment of inertia I, and a modulus of elasticity E. If a load P acts at point B as shown, determine the displacement at B in the direction of P, accounting for bending, axial force, and shear.

The rod has a circular cross section with a moment of inertia I. If a vertical force P is applied at A, determine the vertical displacement at this point. Only consider the strain energy due to bending. The modulus of elasticity is E.

The rod has a circular cross section with a moment of inertia I. If a vertical force P is applied at A, determine the vertical displacement at this point. Only consider the strain energy due to bending. The modulus of elasticity is E.

Determine the vertical displacement of point B on the 2014-T6 aluminum beam.

The rod has a circular cross section with a polar moment of inertia J and moment of inertia I. If a vertical force P is applied at A, determine the vertical displacement at this point. Consider the strain energy due to bending and torsion. The material constants are E and G.

Determine the vertical displacement of end B of the frame. Consider only bending strain energy. The frame is made using two A-36 steel W460 x 68 wide-flange sections.

A bar is 4 m long and has a diameter of 30 mm. Determine the total amount of elastic energy that it can absorb from an impact loading if (a) it is made of steel for which \(E_{\mathrm{st}}=200 \mathrm{GPa}, \sigma_{Y}=800 \mathrm{MPa}\), and (b) it is made from an aluminum alloy for which\(E_{\mathrm{al}}=70 \mathrm{GPa}\), \(\sigma_{Y}=405 \mathrm{MPa}\).

Determine the diameter of a red brass C83400 bar that is 8 ft long if it is to be used to absorb \(800 \mathrm{ft} \cdot \mathrm{lb}\) of energy in tension from an impact loading. No yielding occurs.

Determine the speed v of the 50-Mg mass when it is just over the top of the steel post, if after impact, the maximum stress developed in the post is 550 MPa. The post has a length of L = 1 m and a cross-sectional area of \(0.01 \mathrm{~m}^{2}\). \(E_{\mathrm{st}}=200 \mathrm{GPa}, \sigma_{Y}=600 \mathrm{MPa}\).

The collar has a weight of 50 lb and falls down the titanium bar. If the bar has a diameter of 0.5 in., determine the maximum stress developed in the bar if the weight is (a) dropped from a height of h = 1 ft, (b) released from a height h = 0, and (c) placed slowly on the flange at A. \(E_{\mathrm{ti}}=16\left(10^{3}\right) \mathrm{ksi}, \sigma_{Y}=60 \mathrm{ksi}\).

The collar has a weight of 50 lb and falls down the titanium bar. If the bar has a diameter of 0.5 in., determine the largest height h at which the weight can be released and not permanently damage the bar after striking the flange at A. \(E_{\mathrm{ti}}=16\left(10^{3}\right) \mathrm{ksi}, \sigma_{Y}=60 \mathrm{ksi}\).

A steel cable having a diameter of 0.4 in. wraps over a drum and is used to lower an elevator having a weight of 800 lb. The elevator is 150 ft below the drum and is descending at the constant rate of 2 ft/s when the drum suddenly stops. Determine the maximum stress developed in the cable when this occurs. \(E_{\mathrm{st}}=29\left(10^{3}\right) \mathrm{ksi}, \sigma_{Y}=50 \mathrm{ksi}\).

A steel cable having a diameter of 0.4 in. wraps over a drum and is used to lower an elevator having a weight of 800 lb. The elevator is 150 ft below the drum and is descending at the constant rate of 3 ft / s when the drum suddenly stops. Determine the maximum stress developed in the cable when this occurs. \(E_{\mathrm{st}}=29\left(10^{3}\right) \mathrm{ksi}, \sigma_{Y}=50 \mathrm{ksi}\).

The A-36 steel bolt is required to absorb the energy of a 2-kg mass that falls h = 30 mm. If the bolt has a diameter of 4 mm, determine its required length L so the stress in the bolt does not exceed 150 MPa.

The A-36 steel bolt is required to absorb the energy of a 2-kg mass that falls h = 30 mm. If the bolt has a diameter of 4 mm and a length of L = 200 mm, determine if the stress in the bolt will exceed 175 MPa.

The A-36 steel bolt is required to absorb the energy of a 2-kg mass that falls along the 4-mm-diameter bolt shank that is 150 mm long. Determine the maximum height h of release so the stress in the bolt does not exceed 150 MPa.

A cylinder having the dimensions shown is made from magnesium Am 1004-T61. If it is struck by a rigid block having a weight of 800 lb and traveling at 2 ft/s, determine the maximum stress in the cylinder. Neglect the mass of the cylinder.

The composite aluminum 2014-T6 bar is made from two segments having diameters of 7.5 mm and 15 mm. 14 Determine the maximum axial stress developed in the bar if the 10-kg collar is dropped from a height of h = 100 mm.

The composite aluminum 2014-T6 bar is made from two segments having diameters of 7.5 mm and 15 mm. Determine the maximum height h from which the 10-kg collar should be dropped so that it produces a maximum axial stress in the bar of \(\sigma_{\max }=300 \mathrm{MPa}\).

When the 100-lb block is at h = 3 ft above the cylindrical post and spring assembly, it has a speed of = 20 ft/s. If the post is made of 2014-T6 aluminum and the spring has the stiffness of k = 250 kip/in., determine the required minimum diameter d of the post to the nearest \(\frac{1}{8}\) in. so that it will not yield when it is struck by the block.

The collar has a mass of 5 kg and falls down the titanium Ti-6A1-4V bar. If the bar has a diameter of 20 mm, determine the maximum stress developed in the bar if the weight is (a) dropped from a height of h = 1 m, (b) released from a height \(h \approx 0\) and (c) placed slowly on the flange at A.

The collar has a mass of 5 kg and falls down the titanium Ti-6A1-4V bar. If the bar has a diameter of 20 mm, determine if the weight can be released from rest at any point along the bar and not permanently damage the bar after striking the flange at A.

The tugboat has a weight of 120 000 lb and is traveling forward at 2 ft/s when it strikes the 12-in.-diameter fender post AB used to protect a bridge pier. If the post is made from treated white spruce and is assumed fixed at the river bed, determine the maximum horizontal distance the top of the post will move due to the impact. Assume the tugboat is rigid and neglect the effect of the water.

The W10 x 12 beam is made from A-36 steel and is cantilevered from the wall at B. The spring mounted on the beam has a stiffness of k = 1000 lb/in. If a weight of 8 lb is dropped onto the spring from a height of 3 ft, determine the maximum bending stress developed in the beam.

The weight of 175 lb is dropped from a height of 4 ft from the top of the A992 steel beam. Determine the maximum deflection and maximum stress in the beam if the supporting springs at A and B each have a stiffness of k = 500 lb/in. The beam is 3 in. thick and 4 in. wide.

The weight of 175 lb, is dropped from a height of 4 ft from the top of the A992 steel beam. Determine the load factor n if the supporting springs at A and B each have a stiffness of k = 500 lb/in. The beam is 3 in. thick and 4 in. wide.

Determine the maximum height h from which an 80-lb weight can be dropped onto the end of the A-36 steel W6 x 12 beam without exceeding the maximum elastic stress.

The 80-lb weight is dropped from rest at a height h=4ft on to the end of the A-36 steel W6 x 12 beam. Determine the maximum bending stress developed in the beam.

The 75-lb block has a downward velocity of 2 ft/s when it is 3 ft from the top of the beam. Determine the maximum bending stress in the beam due to the impact, and calculate the maximum deflection of its end D. \(E_{\mathrm{w}}=1.9\left(10^{3}\right) \mathrm{ksi}\). Assume the material will not yield.

The 75-lb block has a downward velocity of 2 ft > s when it is 3 ft from the top of the beam. Determine the maximum bending stress in the beam due to the impact, and calculate the maximum deflection of point B. \(E_{\mathrm{w}}=1.9\left(10^{3}\right) \mathrm{ksi}\).

The overhang beam is made of 2014-T6 aluminum. If the 75-kg block has a speed of = 3 m/s at h = 0.75 m, determine the maximum bending stress in the beam.

The overhang beam is made of 2014-T6 aluminum. Determine the maximum height h from which the 100-kg block can be dropped from rest ( =0), without causing the beam to yield.

A 40-lb weight is dropped from a height of h = 2 ft onto the center of the cantilevered A992 steel beam. If the beam is a W10 x 15, determine the maximum bending stress in the beam.

If the maximum allowable bending stress for the W10 x 15 structural A992 steel beam is \(\sigma_{\text {allow }}=20 \mathrm{ksi}\), determine the maximum height h from which a 50-lb weight can be released from rest and strike the center of the beam.

A 40-lb weight is dropped from a height of h = 2 ft onto the center of the cantilevered A992 steel beam. If the beam is a W10 x 15, determine the vertical displacement of its end B due to the impact.

The car bumper is made of polycarbonate-polybutylene terephthalate. If E = 2.0 GPa, determine the maximum deflection and maximum stress in the bumper if it strikes the rigidpostwhenthecariscoastingatv = 0.75 m/s.Thecarhas a mass of 1.80 Mg, and the bumper can be considered simply supported on two spring supports connected to the rigid frame of the car. For the bumper take \(I=300\left(10^{6}\right) \mathrm{mm}^{4}\), \(c=75 \mathrm{~mm}, \sigma_{Y}=30 \mathrm{MPa} \text { and } k=1.5 \mathrm{MN} / \mathrm{m}\).

Determine the vertical displacement of joint A. Each A992 steel member has a cross-sectional area of \(400 \mathrm{~mm}^{2}\).

Determine the horizontal displacement of joint B. Each A992 steel member has a cross-sectional area of \(2 \mathrm{~in}^{2}\).

Determine the vertical displacement of joint B. Each A992 steel member has a cross-sectional area of \(2 \mathrm{~in}^{2}\).

Determine the vertical displacement of joint B. Each A992 steel member has a cross-sectional area of \(1.5 \mathrm{~in}^{2}\).

Determine the vertical displacement of joint E. Each A992 steel member has a cross-sectional area of \(1.5 \mathrm{~in}^{2}\).

Determine the horizontal displacement of joint B. Each A-36 steel member has a cross-sectional area of \(2 \mathrm{~in}^{2}\).

Determine the vertical displacement of joint B. Each A-36 steel member has a cross-sectional area of \(2 \mathrm{~in}^{2}\).

Determine the horizontal displacement of joint B of the truss. Each A992 steel member has a cross-sectional area of \(400 \mathrm{~mm}^{2}\).

Determine the vertical displacement of joint C of the truss. Each A992 steel member has a cross-sectional area of \(400 \mathrm{~mm}^{2}\).

Determine the horizontal displacement of joint C. Each A-36 steel member has a cross-sectional area of \(400 \mathrm{~mm}^{2}\).

Determine the vertical displacement of joint D. Each A-36 steel member has a cross-sectional area of \(400 \mathrm{~mm}^{2}\).

Determine the vertical displacement of joint A. The truss is made from A992 steel rods having a diameter of 30 mm.

Determine the vertical displacement of joint D. The truss is made from A992 steel rods having a diameter of 30 mm.

Determine the horizontal displacement of joint D. Each A-36 steel member has a cross-sectional area of \(300 \mathrm{~mm}^{2}\).

Determine the horizontal displacement of joint E. Each A-36 steel member has a cross-sectional area of \(300 \mathrm{~mm}^{2}\).

Determine the displacement at point C. EI is constant.

The beam is made of southern pine for which \(E_{\mathrm{p}}=13 \mathrm{GPa}\). Determine the displacement at A.

Determine the displacement at point C. EI is constant.

Determine the slope at point C. EI is constant.

Determine the slope at point A. EI is constant.

Determine the displacement of point C of the beam made from A992 steel and having a moment of inertia of \(I=53.8 \mathrm{in}^{4}\).

Determine the slope at B of the beam made from A992 steel and having a moment of inertia of \(I=53.8 \mathrm{in}^{4}\).

The beam is made of Douglas fir. Determine the slope at C.

Determine the displacement at pulley B. The A992 steel shaft has a diameter of 30 mm.

The A992 steel beam has a moment of inertia of \(I=125\left(10^{6}\right) \mathrm{mm}^{4}\). Determine the displacement at point D.

The A992 steel beam has a moment of inertia of \(I=125\left(10^{6}\right) \mathrm{mm}^{4}\). Determine the slope at A.

The A992 structural steel beam has a moment of inertia of \(I=125\left(10^{6}\right) \mathrm{mm}^{4}\). Determine the slope at B.

Determine the displacement at point C of the shaft. EI is constant.

Determine the slope at A of the shaft. EI is constant.

Determine the slope of end C of the overhang beam. EI is constant.

Determine the displacement of point D of the overhang beam. EI is constant.

Determine the slope at A of the 2014-T6 aluminum shaft having a diameter of 100 mm.

Determine the displacement at point C of the 2014-T6 aluminum shaft having a diameter of 100 mm.

Determine the displacement at point C of the 2014-T6 aluminum shaft having a diameter of 100 mm.

Determine the displacement at point C of the W14 x 26 beam made from A992 steel.

Determine the slope at A of the W14 x 26 beam made from A992 steel.

Determine the slope at A. EI is constant.

Determine the slope at C of the overhang white spruce beam.

Determine the displacement at point D of the overhang white spruce beam.

Determine the maximum deflection of the beam caused only by bending, and caused by bending and shear. Take E = 3G.

The beam is made of oak, for which \(E_{\mathrm{o}}=11 \mathrm{GPa}\). Determine the slope and displacement at point A.

Determine the slope of the shaft at the bearing support A. EI is constant.

Determine the horizontal and vertical displacements of point C. There is a fixed support at A. EI is constant.

Beam AB has a square cross section of 100 mm by 100 mm. Bar CD has a diameter of 10 mm. If both members are made of A992 steel, determine the vertical displacement of point B due to the loading of 10 kN.

Beam AB has a square cross section of 100 mm by 100 mm. Bar CD has a diameter of 10 mm. If both members are made of A992 steel, determine the slope at A due to the loading of 10 kN.

Bar ABC has a rectangular cross section of 300 mm by 100 mm. Attached rod DB has a diameter of 20 mm. If both members are made of A-36 steel, determine the vertical displacement of point C due to the loading. Consider only the effect of bending in ABC and axial force in DB.

Bar ABC has a rectangular cross section of 300 mm by 100 mm. Attached rod DB has a diameter of 20 mm. If both members are made of A-36 steel, determine the slope at A due to the loading. Consider only the effect of bending in ABC and axial force in DB.

The L-shaped frame is made from two segments, each of length L and flexural stiffness EI. If it is subjected to the uniform distributed load, determine the horizontal displacement of point C.

The L-shaped frame is made from two segments, each of length L and flexural stiffness EI. If it is subjected to the uniform distributed load, determine the vertical displacement of point B.

Determine the vertical displacement of the ring at point B. EI is constant.

Determine the horizontal displacement at the roller at A due to the loading. EI is constant.

Solve Prob. 14–73 using Castigliano’s theorem.

Solve Prob. 14–74 using Castigliano’s theorem.

Solve Prob. 14–75 using Castigliano’s theorem.

Solve Prob. 14–76 using Castigliano’s theorem.

Solve Prob. 14–77 using Castigliano’s theorem.

Solve Prob. 14–78 using Castigliano’s theorem.

Solve Prob. 14–81 using Castigliano’s theorem.

Solve Prob. 14–82 using Castigliano’s theorem.

Solve Prob. 14–85 using Castigliano’s theorem.

Solve Prob. 14–86 using Castigliano’s theorem.

Solve Prob. 14–90 using Castigliano’s theorem.

Solve Prob. 14–91 using Castigliano’s theorem.

Solve Prob. 14–92 using Castigliano’s theorem.

Solve Prob. 14–93 using Castigliano’s theorem.

Solve Prob. 14–95 using Castigliano’s theorem.

Solve Prob. 14–96 using Castigliano’s theorem.

Solve Prob. 14–97 using Castigliano’s theorem.

Solve Prob. 14–98 using Castigliano’s theorem.

Solve Prob. 14–108 using Castigliano’s theorem.

Solve Prob. 14–119 using Castigliano’s theorem.

Solve Prob. 14–120 using Castigliano’s theorem.

Solve Prob. 14–105 using Castigliano’s theorem.