Solve Prob. 868 for point B.

A spherical gas tank has an inner radius of r = 1.5 m. If it is subjected to an internal pressure of p = 300 kPa, determine its required thickness if the maximum normal stress is not to exceed 12 MPa.

A pressurized spherical tank is to be made of 0.5-in.-thick steel. If it is subjected to an internal pressure of p = 200 psi, determine its outer radius if the maximum normal stress is not to exceed 15 ksi.

The thin-walled cylinder can be supported in one of two ways as shown. Determine the state of stress in the wall of the cylinder for both cases if the piston P causes the internal pressure to be 65 psi. The wall has a thickness of 0.25 in. and the inner diameter of the cylinder is 8 in.

The tank of the air compressor is subjected to an internal pressure of 90 psi. If the internal diameter of the tank is 22 in., and the wall thickness is 0.25 in., determine the stress components acting at point A. Draw a volume element of the material at this point, and show the results on the element.

The open-ended polyvinyl chloride pipe has an inner diameter of 4 in. and thickness of 0.2 in. If it carries flowing water at 60 psi pressure, determine the state of stress in the walls of the pipe.

If the flow of water within the pipe in Prob. 8–5 is stopped due to the closing of a valve, determine the state of stress in the walls of the pipe. Neglect the weight of the water. Assume the supports only exert vertical forces on the pipe.

A boiler is constructed of 8-mm thick steel plates that are fastened together at their ends using a butt joint consisting of two 8-mm cover plates and rivets having a diameter of 10 mm and spaced 50 mm apart as shown. If the steam pressure in the boiler is 1.35 MPa, determine (a) the circumferential stress in the boiler’s plate apart from the seam, (b) the circumferential stress in the outer cover plate along the rivet line a–a, and (c) the shear stress in the rivets.

The steel water pipe has an inner diameter of 12 in. and wall thickness 0.25 in. If the valve A is opened and the flowing water is under a gauge pressure of 250 psi, determine the longitudinal and hoop stress developed in the wall of the pipe.

The steel water pipe has an inner diameter of 12 in. and wall thickness 0.25 in. If the valve A is closed and the water pressure is 300 psi, determine the longitudinal and hoop stress developed in the wall of the pipe. Draw the state of stress on a volume element located on the wall.

The A-36-steel band is 2 in. wide and is secured around the smooth rigid cylinder. If the bolts are tightened so that the tension in them is 400 lb, determine the normal stress in the band, the pressure exerted on the cylinder, and the distance half the band stretches.

Two hemispheres having an inner radius of 2 ft and wall thickness of 0.25 in. are fitted together, and the inside gauge pressure is reduced to -10 psi. If the coefficient of static friction is \(\mu_{\mathrm{s}}=0.5\) between the hemispheres, determine (a) the torque T needed to initiate the rotation of the top hemisphere relative to the bottom one, (b) the vertical force needed to pull the top hemisphere off the bottom one, and (c) the horizontal force needed to slide the top hemisphere off the bottom one.

A pressure-vessel head is fabricated by gluing the circular plate to the end of the vessel as shown. If the vessel sustains an internal pressure of 450 kPa, determine the average shear stress in the glue and the state of stress in the wall of the vessel.

An A-36-steel hoop has an inner diameter of 23.99 in., thickness of 0.25 in., and width of 1 in. If it and the 24-in.-diameter rigid cylinder have a temperature of \(65^{\circ} \mathrm{F}\), determine the temperature to which the hoop should be heated in order for it to just slip over the cylinder. What is the pressure the hoop exerts on the cylinder, and the tensile stress in the ring when it cools back down to \(65^{\circ} \mathrm{F}\)?

The ring, having the dimensions shown, is placed over a flexible membrane which is pumped up with a pressure p. Determine the change in the internal radius of the ring after this pressure is applied. The modulus of elasticity for the ring is E.

The inner ring A has an inner radius \(r_{1}\) and outer radius \(r_{2}\). Before heating, the outer ring B has an inner radius \(r_{3}\) and an outer radius \(r_{4}\), and \(r_{2}>r_{3}\). If the outer ring is heated and then fitted over the inner ring, determine the pressure between the two rings when ring B reaches the temperature of the inner ring. The material has a modulus of elasticity of E and a coefficient of thermal expansion of \(\alpha\).

A closed-ended pressure vessel is fabricated by cross winding glass filaments over a mandrel, so that the wall thickness t of the vessel is composed entirely of filament and an expoxy binder as shown in the figure. Consider a segment of the vessel of width w and wrapped at an angle \(\theta\). If the vessel is subjected to an internal pressure p, show that the force in the segment is \(F_{\theta}=\sigma_{0} w t\), where \(\sigma_{0}\) is the stress in the filaments. Also, show that the stresses in the hoop and longitudinal directions are \(\sigma_{h}=\sigma_{0} \ \sin ^{2} \theta\) and \(\sigma_{1}=\sigma_{0} \quad \cos ^{2} \theta\), respectively. At what angle \(\theta\) (optimum winding angle) would the filaments have to be would so that the hoop and longitudinal stresses are equivalent?

In order to increase the strength of the pressure vessel, filament winding of the same material is wrapped around the circumference of the vessel as shown. If the pretension in the filament is T and the vessel is subjected to an internal pressure p, determine the hoop stresses in the filament and in the wall of the vessel. Use the free-body diagram shown, and assume the filament winding has a thickness \(t^{\prime}\) and width w for a corresponding length L of the vessel.

The vertical force P acts on the bottom of the plate having a negligible weight. Determine the shortest distance d to the edge of the plate at which it can be applied so that it produces no compressive stresses on the plate at section a–a. The plate has a thickness of 10 mm and P acts along the center line of this thickness.

Determine the maximum and minimum normal stress in the bracket at section a – a when the load is applied at x = 0.

Determine the maximum and minimum normal stress in the bracket at section a – a when the load is applied at x = 300 mm.

If the load has a weight of 600 lb, determine the maximum normal stress developed on the cross section of the supporting member at section a – a. Also, plot the normal stress distribution over the cross-section.

The clamp is made from members AB and AC, which are pin connected at A. If it exerts a compressive force at C and B of 180 N, determine the maximum compressive stress in the clamp at section a – a. The screw EF is subjected only to a tensile force along its axis.

The clamp is made from members AB and AC, which are pin connected at A. If it exerts a compressive force at C and B of 180 N, sketch the stress distribution acting over section a – a. The screw EF is subjected only to a tensile force along its axis.

The bearing pin supports the load of 700 lb. Determine the stress components in the support member at point A. The support is 0.5 in. thick.

The bearing pin supports the load of 700 lb. Determine the stress components in the support member at point B. The support is 0.5 in. thick.

The column is built up by gluing the two identical boards together. Determine the maximum normal stress developed on the cross section when the eccentric force of P = 50 kN is applied.

The column is built up by gluing the two identical boards together. If the wood has an allowable normal stress of \(\sigma_{\text {allow }}=6 \ \mathrm{MPa}\), determine the maximum allowable eccentric force P that can be applied to the column.

The cylindrical post, having a diameter of 40 mm, is being pulled from the ground using a sling of negligible thickness. If the rope is subjected to a vertical force of P = 500 N, determine the normal stress at points A and B. Show the results on a volume element located at each of these points.

Determine the maximum load P that can be applied to the sling having a negligible thickness so that the normal stress in the post does not exceed \(\sigma_{\text {allow }}=30 \ \mathrm{MPa}\). The post has a diameter of 50 mm.

The rib-joint pliers are used to grip the smooth pipe C. If the force of 100 N is applied to the handles, determine the state of stress at points A and B on the cross section of the jaw at section a – a. Indicate the results on an element at each point.

Determine the smallest distance d to the edge of the plate at which the force P can be applied so that it produces no compressive stresses in the plate at section a–a. The plate has a thickness of 20 mm and P acts along the centerline of this thickness.

The horizontal force of P = 80 kN acts at the end of the plate. The plate has a thickness of 10 mm and P acts along the centerline of this thickness such that d = 50 mm. Plot the distribution of normal stress acting along section a–a.

The control lever is subjected to a horizontal force of 20 lb on the handle. Determine the state of stress at points A and B. Sketch the results on differential elements located at each of these points. The assembly is pin-connected at C and attached to a cable at D.

The control lever is subjected to a horizontal force of 20 lb on the handle. Determine the state of stress at points E and F. Sketch the results on differential elements located at each of these points. The assembly is pin connected at C and attached to a cable at D.

The tubular shaft of the soil auger is subjected to the axial force and torque shown. If the auger is rotating at a constant rate, determine the state of stress at points A and B on the cross section of the shaft at section a – a.

The drill is jammed in the wall and is subjected to the torque and force shown. Determine the state of stress at point A on the cross section of drill bit at section a – a.

The drill is jammed in the wall and is subjected to the torque and force shown. Determine the state of stress at point B on the cross section of drill bit, in back, at section a – a.

The frame supports the distributed load shown. Determine the state of stress acting at point D. Show the results on a differential element at this point.

The frame supports the distributed load shown. Determine the state of stress acting at point E. Show the results on a differential element at this point.

The 500-kg engine is suspended from the jib crane at the position shown. Determine the state of stress at point A on the cross section of the boom at section a – a.

The 500-kg engine is suspended from the jib crane at the position shown. Determine the state of stress at point B on the cross section of the boom at section a – a. Point B is just above the bottom flange.

Determine the state of stress at point A on the cross section of the post at section a – a. Indicate the results on a differential element at the point.

Determine the state of stress at point B on the cross section of the post at section a – a. Indicate the results on a differential element at the point.

Determine the normal stress developed at points A and B. Neglect the weight of the block.

Sketch the normal stress distribution acting over the cross section at section a–a. Neglect the weight of the block.

The support is subjected to the compressive load P. Determine the absolute maximum possible and minimum possible normal stress acting in the material, for \(x \geq 0\).

The bent shaft is fixed in the wall at A. If a force F is applied at B, determine the stress components at points D and E. Show the results on a differential element located at each of these points. Take F = 12 lb and \(\theta=0^{\circ}\).

The bent shaft is fixed in the wall at A. If a force F is applied at B, determine the stress components at points D and E. Show the results on a differential element located at each of these points. Take F = 12 lb and \(\theta=90^{\circ}\).

The bent shaft is fixed in the wall at A. If a force F is applied at B, determine the stress components at points D and E. Show the results on a volume element located at each of these points. Take F = 12 lb and \(\theta=45^{\circ}\).

The coiled spring is subjected to a force P. If we assume the shear stress caused by the shear force at any vertical section of the coil wire to be uniform, show that the maximum shear stress in the coil is \(\tau_{\max }=P / A+P R r / J\), where J is the polar moment of inertia of the coil wire and A is its cross-sectional area.

A post having the dimensions shown is subjected to the bearing load P. Specify the region to which this load can be applied without causing tensile stress to be developed at points A, B, C, and D.

The vertebra of the spinal column can support a maximum compressive stress of \(\sigma_{\max }\), before undergoing a compression fracture. Determine the smallest force P that can be applied to a vertebra, if we assume this load is applied at an eccentric distance e from the centerline of the bone, and the bone remains elastic. Model the vertebra as a hollow cylinder with an inner radius \(r_{i}\) and outer radius \(r_{o}\).

The 1-in.-diameter rod is subjected to the loads shown. Determine the state of stress at point A, and show the results on a differential element located at this point.

The 1-in.-diameter rod is subjected to the loads shown. Determine the state of stress at point B, and show the results on a differential element located at this point.

Determine the state of stress at point A on the cross section of the post at section a – a. Indicate the results on a differential element at the point.

Determine the state of stress at point B on the cross section of the post at section a – a. Indicate the results on a differential element at the point.

The sign is subjected to the uniform wind loading. Determine the stress components at points A and B on the 100-mm-diameter supporting post. Show the results on a volume element located at each of these points.

The sign is subjected to the uniform wind loading. Determine the stress components at points C and D on the 100-mm-diameter supporting post. Show the results on a volume element located at each of these points.

If P = 60 kN, determine the maximum normal stress developed on the cross section of the column.

Determine the maximum allowable force P, if the column is made from material having an allowable normal stress of \(\sigma_{\text {allow }}=100 \ \mathrm{MPa}\).

The C-frame is used in a riveting machine. If the force at the ram on the clamp at D is P = 8 kN, sketch the stress distribution acting over the section a – a.

Determine the maximum ram force P that can be applied to the clamp at D if the allowable normal stress for the material is \(\sigma_{\text {allow }}=180 \ \mathrm{MPa}\).

The uniform sign has a weight of 1500 lb and is supported by the pipe AB, which has an inner radius of 2.75 in. and an outer radius of 3.00 in. If the face of the sign is subjected to a uniform wind pressure of \(p=150 \ \mathrm{lb} / \mathrm{ft}^{2}\), determine the state of stress at points C and D. Show the results on a differential volume element located at each of these points. Neglect the thickness of the sign, and assume that it is supported along the outside edge of the pipe.

Solve Prob. 8–63 for points E and F.

Determine the state of stress at point A on the cross section of the pipe at section a–a.

Determine the state of stress at point B on the cross section of the pipe at section a–a.

The metal link is subjected to the axial force of P = 7 kN. Its original cross section is to be altered by cutting a circular groove into one side. Determine the distance a the groove can penetrate into the cross section so that the tensile stress does not exceed \(\sigma_{\text {allow }}=175 \ \mathrm{MPa}\). Offer a better way to remove this depth of material from the cross section and calculate the tensile stress for this case. Neglect the effects of stress concentration.

The bar has a diameter of 40 mm. If it is subjected to a force of 800 N as shown, determine the stress components that act at point A and show the results on a volume element located at this point.

Solve Prob. 8–68 for point B.

The \(\frac{3}{4} \text {-in.-diameter }\) shaft is subjected to the loading shown. Determine the stress components at point A. Sketch the results on a volume element located at this point. The journal bearing at C can exert only force components \(\mathbf{C}_{y}\) and \(\mathbf{C}_{z}\) on the shaft, and the thrust bearing at D can exert force components \(\mathbf{D}_{x}, \mathbf{D}_{y}\), and \(\mathbf{D}_{z}\) on the shaft.

Solve Prob. 8–70 for the stress components at point B.

The hook is subjected to the force of 80 lb. Determine the state of stress at point A at section a – a. The cross section is circular and has a diameter of 0.5 in. Use the curved-beam formula to compute the bending stress.

The hook is subjected to the force of 80 lb. Determine the state of stress at point B at section a – a. The cross section has a diameter of 0.5 in. Use the curved-beam formula to compute the bending stress.

The eye hook has the dimensions shown. If it supports a cable loading of 80 kN, determine the maximum normal stress at section a – a and sketch the stress distribution acting over the cross section.

The 20-kg drum is suspended from the hook mounted on the wooden frame. Determine the state of stress at point E on the cross section of the frame at section a–a. Indicate the results on an element.

The 20-kg drum is suspended from the hook mounted on the wooden frame. Determine the state of stress at point F on the cross section of the frame at section b–b. Indicate the results on an element.

A bar having a square cross section of 30 mm by 30 mm is 2 m long and is held upward. If it has a mass of 5 kg/m, determine the largest angle \(\theta\), measured from the vertical, at which it can be supported before it is subjected to a tensile stress along its axis near the grip.

Solve Prob. 8–77 if the bar has a circular cross section of 30-mm diameter.

The gondola and passengers have a weight of 1500 lb and center of gravity at G. The suspender arm AE has a square cross-sectional area of 1.5 in. by 1.5 in., and is pin connected at its ends A and E. Determine the largest tensile stress developed in regions AB and DC of the arm.

The hydraulic cylinder is required to support a force of P = 100 kN. If the cylinder has an inner diameter of 100 mm and is made from a material having an allowable normal stress of \(\sigma_{\text {allow }}=150 \ \mathrm{MPa}\), determine the required minimum thickness t of the wall of the cylinder.

The hydraulic cylinder has an inner diameter of 100 mm and wall thickness of t = 4 mm. If it is made from a material having an allowable normal stress of \(\sigma_{\text {allow }}=150 \ \mathrm{MPa}\), determine the maximum allowable force P.

If the cross section of the femur at section a–a can be approximated as a circular tube as shown, determine the maximum normal stress developed on the cross section at section a–a due to the load of 75 lb.

Air pressure in the cylinder is increased by exerting forces P = 2 kN on the two pistons, each having a radius of 45 mm. If the cylinder has a wall thickness of 2 mm, determine the state of stress in the wall of the cylinder.

Determine the maximum force P that can be exerted on each of the two pistons so that the circumferential stress component in the cylinder does not exceed 3 MPa. Each piston has a radius of 45 mm and the cylinder has a wall thickness of 2 mm.

The wall hanger has a thickness of 0.25 in. and is used to support the vertical reactions of the beam that is loaded as shown. If the load is transferred uniformly to each strap of the hanger, determine the state of stress at points C and D on the strap at A . Assume the vertical reaction F at this end acts in the center and on the edge of the bracket as shown.

The wall hanger has a thickness of 0.25 in. and is used to support the vertical reactions of the beam that is loaded as shown. If the load is transferred uniformly to each strap of the hanger, determine the state of stress at points C and D on the strap at B. Assume the vertical reaction F at this end acts in the center and on the edge of the bracket as shown.