The post has a circular cross section of radius c.

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 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 inner 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.

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 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.

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 away 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 a wall thickness of 0.25 in. If the valve A is opened and the flowing water has a pressure of 250 psi as it passes point B, determine the longitudinal and hoop stress developed in the wall of the pipe at point B.

The steel water pipe has an inner diameter of 12 in. and a wall thickness of 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 at point B. 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.

The gas pipe line is supported every 20 ft by concrete piers and also lays on the ground. If there are rigid retainers at the piers that hold the pipe fixed, determine the longitudinal and hoop stress in the pipe if the temperature rises 60° F from the temperature at which it was installed. The gas within the pipe is at a pressure of \(600 \mathrm{lb} / \mathrm{in}^{2}\). The pipe has an inner diameter of 20 in. and thickness of 0.25 in. The material is A-36 steel.

A pressure-vessel head is fabricated by welding 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 weld 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° 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° 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 inner 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 \(r_{2}\). 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 \(\alpha\).

Two hemispheres having an inner radius of 2 ft and wall thickness of 0.25 in. are fitted together, and the inside 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.

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.

Determine the shortest 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 10 mm and P acts along the centerline of this thickness.

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

The plate has a thickness of 20 mm and the force P = 3 kN acts along the centerline of this thickness such that d = 150 mm. Plot the distribution of normal stress acting along section a–a.

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

The steel bracket is used to connect the ends of two cables. If the allowable normal stress for the steel is \(\sigma_{\text {allow }}=30 \mathrm{ksi}\), determine the largest tensile force P that can be applied to the cables. Assume the bracket is a rod having a diameter of 1.5 in.

The steel bracket is used to connect the ends of two cables. If the applied force P = 1.50 kip, determine the maximum normal stress in the bracket. Assume the bracket is a rod having a diameter of 1.5 in.

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

The column is built up by gluing the two 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 screw of the clamp exerts a compressive force of 500 lb on the wood blocks. Determine the maximum normal stress along section a–a. The cross section is rectangular, 0.75 in. by 0.50 in.

The screw of the clamp exerts a compressive force of 500 lb on the wood blocks. Sketch the stress distribution along section a–a of the clamp. The cross section is rectangular, 0.75 in. by 0.50 in.

The joint is subjected to the force system shown. Sketch the normal-stress distribution acting over section a–a if the member has a rectangular cross section of width 0.5 in. and thickness 1 in.

The joint is subjected to the force system shown. Determine the state of stress at points A and B, and sketch the results on differential elements located at these points. The member has a rectangular cross-sectional area of width 0.5 in. and thickness 1 in.

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.

The \(\frac{1}{2} \text { in }\).-diameter bolt hook is subjected to the load of F = 150 lb. Determine the stress components at point A on the shank. Show the result on a volume element located at this point.

The \(\frac{1}{2} \text { in }\).-diameter bolt hook is subjected to the load of F = 150 lb. Determine the stress components at point B on the shank. Show the result on a volume element located at this point.

The block is subjected to the eccentric load shown. Determine the normal stress developed at points A and B. Neglect the weight of the block.

The block is subjected to the eccentric load shown. Sketch the normal-stress distribution acting over the cross section at section a–a. Neglect the weight of the block.

The spreader bar is used to lift the 2000-lb tank. Determine the state of stress at points A and B, and indicate the results on a differential volume element.

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 the 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 the drill bit 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 rod has a diameter of 40 mm. If it is subjected to the force system shown, determine the stress components that act at point A, and show the results on a volume element located at this point.

Solve Prob. 8–40 for point B. The rod has a diameter of 40 mm. If it is subjected to the force system shown, determine the stress components that act at point A, and show the results on a volume element located at this point.

The beveled gear is subjected to the loads shown. Determine the stress components acting on the shaft at point A, and show the results on a volume element located at this point. The shaft has a diameter of 1 in. and is fixed to the wall at C.

The beveled gear is subjected to the loads shown. Determine the stress components acting on the shaft at point B, and show the results on a volume element located at this point. The shaft has a diameter of 1 in. and is fixed to the wall at C.

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 man has a mass of 100 kg and center of mass at G. If he holds himself in the position shown, determine the maximum tensile and compressive stress developed in the curved bar at section a–a. He is supported uniformly by two bars, each having a diameter of 25 mm. Assume the floor is smooth. Use the curved-beam formula to calculate the bending stress.

The solid rod is subjected to the loading shown. Determine the state of stress at point A, and show the results on a differential volume element located at this point.

The solid rod is subjected to the loading shown. Determine the state of stress at point B, and show the results on a differential volume element at this point.

The solid rod is subjected to the loading shown. Determine the state of stress at point C, and show the results on a differential volume element at this point.

The post has a circular cross section of radius c. Determine the maximum radius e at which the load P can be applied so that no part of the post experiences a tensile stress. Neglect the weight of the post.

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

The vertebra of the spinal column can support a maximum compressive stress of \(\sigma_{\text {max}}\), 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 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.

The frame supports a centrally applied distributed load of 1.8 kip/ft. Determine the state of stress at points A and B on member CD and indicate the results on a volume element located at each of these points. The pins at C and D are at the same location as the neutral axis for the cross section.

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 volume element located at this point.

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 volume element located at this 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.

The bearing pin supports the load of 900 lb. Determine the stress components in the support member at point A. Represent the state of stress at point A with a differential element.

The bearing pin supports the load of 900 lb. Determine the stress components in the support member at point B. Represent the state of stress at point B with a differential element.

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. 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.

The pin support is made from a steel rod and has a diameter of 20 mm. Determine the stress components at points A and B and represent the results on a volume element located at each of these points.

Solve Prob. 8–65 for points C and D. The pin support is made from a steel rod and has a diameter of 20 mm. Determine the stress components at points A and B and represent the results on a volume element located at each of these points.

The handle of the press is subjected to a force of 20 lb. Due to internal gearing, this causes the block to be subjected to a compressive force of 80 lb. Determine the normal-stress acting in the frame at points along the outside flanges A and B. Use the curved-beam formula to calculate the bending stress.

The bar has a diameter of 40 mm. Determine the state of stress at point A and show the results on a differential volume element located at this point.

Solve Prob. 8–68 for point B. The bar has a diameter of 40 mm. Determine the state of stress at point A and show the results on a differential volume element located at this point.

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 \(\frac{3}{4} \text { in }\).-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.

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 calculate 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 calculate the bending stress.