The stress acting on two planes at a point is indicated.

Prove that the sum of the normal stresses \(\sigma_{x}+\sigma_{y}=\sigma_{x^{\prime}}+\sigma_{y^{\prime}}\) is constant. See Figs. 9–2a and 9–2b.

The state of stress at a point in a member is shown on the element. Determine the stress components acting on the inclined plane AB. Solve the problem using the method of equilibrium described in Sec. 9.1.

The state of stress at a point in a member is shown on the element. Determine the stress components acting on the inclined plane AB. Solve the problem using the method of equilibrium described in Sec. 9.1.

Determine the normal stress and shear stress acting on the inclined plane AB. Solve the problem using the method of equilibrium described in Sec. 9.1.

Determine the normal stress and shear stress acting on the inclined plane AB. Solve the problem using the stress transformation equations. Show the results on the sectional element.

Determine the normal stress and shear stress acting on the inclined plane AB. Solve the problem using the method of equilibrium described in Sec. 9.1.

Determine the normal stress and shear stress acting on the inclined plane AB. Solve the problem using the stress transformation equations. Show the result on the sectioned element.

Determine the equivalent state of stress on an element at the same point oriented \(30^{\circ}\) clockwise with respect to the element shown. Sketch the results on the element.

Determine the equivalent state of stress on an element at the same point oriented \(30^{\circ}\) counterclockwise with respect to the element shown. Sketch the results on the element.

Determine the equivalent state of stress on an element at the same point oriented \(60^{\circ}\) clockwise with respect to the element shown. Sketch the results on the element.

Determine the equivalent state of stress on an element at the same point oriented \(60^{\circ}\) counterclockwise with respect to the element shown. Sketch the results on the element.

Determine the equivalent state of stress on an element if it is oriented \(50^{\circ}\) counterclockwise from the element shown. Use the stress-transformation equations.

Determine the equivalent state of stress on an the element if it is oriented \(30^{\circ}\) clockwise from the element shown. Use the stress-transformation equations.

The state of stress at a point is shown on the element. Determine (a) the principal stress and (b) the maximum in-plane shear stress and average normal stress at the point. Specify the orientation of the element in each case. Show the results on each element.

The state of stress at a point is shown on the element. Determine (a) the principal stress and (b) the maximum in-plane shear stress and average normal stress at the point. Specify the orientation of the element in each case.

Determine the equivalent state of stress on an element at the point which represents (a) the principal stresses and (b) the maximum in-plane shear stress and the associated average normal stress. Also, for each case, determine the corresponding orientation of the element with respect to the element shown and sketch the results on the element.

Determine the equivalent state of stress on an element at the same point which represents (a) the principal stress, and (b) the maximum in-plane shear stress and the associated average normal stress. Also, for each case, determine the corresponding orientation of the element with respect to the element shown. Sketch the results on each element.

A point on a thin plate is subjected to the two successive states of stress shown. Determine the resultant state of stress represented on the element oriented as shown on the right.

Determine the equivalent state of stress on an element at the same point which represents (a) the principal stress, and (b) the maximum in-plane shear stress and the associated average normal stress. Also, for each case, determine the corresponding orientation of the element with respect to the element shown and sketch the results on the element.

Planes AB and BC at a point are subjected to the stresses shown. Determine the principal stresses acting at this point and find \(\sigma_BC\).

The stress acting on two planes at a point is indicated. Determine the shear stress on plane a–a and the principal stresses at the point.

The grains of wood in the board make an angle of \(20^{\circ}\) with the horizontal as shown. Determine the normal and shear stress that act perpendicular and parallel to the grains if the board is subjected to an axial load of 250 N.

The wood beam is subjected to a load of 12 kN. If a grain of wood in the beam at point A makes an angle of \(25^{\circ}\) with the horizontal as shown, determine the normal and shear stress that act perpendicular and parallel to the grain due to the loading.

The wood beam is subjected to a load of 12 kN. Determine the principal stress at point A and specify the orientation of the element.

The wooden block will fail if the shear stress acting along the grain is 550 psi. If the normal stress \(\sigma_x = 400 \ \mathrm{psi}\), determine the necessary compressive stress \(\sigma_y\) that will cause failure.

The bracket is subjected to the force of 3 kip. Determine the principal stress and maximum in-plane shear stress at point A on the cross section at section a–a . Specify the orientation of this state of stress and show the results on elements.

The bracket is subjected to the force of 3 kip. Determine the principal stress and maximum in-plane shear stress at point B on the cross section at section a–a . Specify the orientation of this state of stress and show the results on elements.

The 25-mm thick rectangular bar is subjected to the axial load of 10 kN. If the bar is joined by the weld, which makes an angle of \(60^{\circ}\) with the horizontal, determine the shear stress parallel to the weld and the normal stress perpendicular to the weld.

The 3-in. diameter shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B . Determine the principal stresses and maximum in-plane shear stress at a point on the outer surface of the shaft at section a – a .

The state of stress at a point in a member is shown on the element. Determine the stress components acting on the plane AB .

Determine the principal stress at point A on the cross section of the arm at section a–a . Specify the orientation of this state of stress and indicate the results on an element at the point.

Determine the maximum in-plane shear stress developed at point A on the cross section of the arm at section a–a . Specify the orientation of this state of stress and indicate the results on an element at the point.

The clamp bears down on the smooth surface at E by tightening the bolt. If the tensile force in the bolt is 40 kN, determine the principal stress at points A and B and show the results on elements located at each of these points. The cross-sectional area at A and B is shown in the adjacent figure.

Determine the principal stress and the maximum in-plane shear stress that are developed at point A in the 2-in.-diameter shaft. Show the results on an element located at this point. The bearings only support vertical reactions.

The square steel plate has a thickness of 10 mm and is subjected to the edge loading shown. Determine the maximum in-plane shear stress and the average normal stress developed in the steel.

The square steel plate has a thickness of 0.5 in. and is subjected to the edge loading shown. Determine the principal stresses developed in the steel.

The shaft has a diameter d and is subjected to the loadings shown. Determine the principal stress and the maximum in-plane shear stress that is developed at point A . The bearings only support vertical reactions.

A paper tube is formed by rolling a paper strip in a spiral and then gluing the edges together as shown. Determine the shear stress acting along the seam, which is at \(30^{\circ}\) from the vertical, when the tube is subjected to an axial force of 10 N. The paper is 1 mm thick and the tube has an outer diameter of 30 mm.

Solve Prob. 9–38 for the normal stress acting perpendicular to the seam.

The wide-flange beam is subjected to the 50-kN force. Determine the principal stresses in the beam at point A located on the web at the bottom of the upper flange. Although it is not very accurate, use the shear formula to calculate the shear stress.

Solve Prob. 9–40 for point B located on the web at the top of the bottom flange.

The drill pipe has an outer diameter of 3 in., a wall thickness of 0.25 in., and a weight of 50 lb/ft. If it is subjected to a torque and axial load as shown, determine (a) the principal stresses and (b) the maximum in-plane shear stress at a point on its surface at section a.

The nose wheel of the plane is subjected to a design load of 12 kN. Determine the principal stresses acting on the aluminum wheel support at point A .

Solve Prob. 9–3 using Mohr’s circle.

Solve Prob. 9–6 using Mohr’s circle.

Solve Prob. 9–14 using Mohr’s circle.

Solve Prob. 9–10 using Mohr’s circle.

Solve Prob. 9–12 using Mohr’s circle.

Solve Prob. 9–16 using Mohr’s circle.

Mohr’s circle for the state of stress in Fig. 9–15 a is shown in Fig. 9–15 b . Show that finding the coordinates of point \(P\left(\sigma_{x^{\prime}} , \tau_{x^{\prime} y^{\prime}}\right )\) on the circle gives the same value as the stress-transformation Eqs. 9–1 and 9–2 .

Determine the equivalent state of stress if an element is oriented \(45^{\circ}\) clockwise from the element shown.

Determine the equivalent state of stress if an element is oriented \(20^{\circ}\) clockwise from the element shown.

Determine the equivalent state of stress if an element is oriented \(30^{\circ}\) clockwise from the element shown.

Determine the equivalent state of stress which represents (a) the principal stress, and (b) the maximum in-plane shear stress and the associated average normal stress. For each case, determine the corresponding orientation of the element with respect to the element shown.

Determine the equivalent state of stress which represents (a) the principal stress, and (b) the maximum in-plane shear stress and the associated average normal stress. For each case, determine the corresponding orientation of the element with respect to the element shown.

Determine the principal stress, the maximum in-plane shear stress, and average normal stress. Specify the orientation of the element in each case.

Determine (a) the principal stresses and (b) the maximum in-plane shear stress and average normal stress. Specify the orientation of the element in each case.

Determine the equivalent state of stress if an element is oriented \(25^{\circ}\) counterclockwise from the element shown.

Determine (a) the principal stresses and (b) the maximum in-plane shear stress and average normal stress. Specify the orientation of the element in each case.

Determine the principal stresses, the maximum in-plane shear stress, and average normal stress. Specify the orientation of the element in each case.

Draw Mohr’s circle that describes each of the following states of stress.

The grains of wood in the board make an angle of \(20^{\circ}\) with the horizontal as shown. Using Mohr’s circle, determine the normal and shear stresses that act perpendicular and parallel to the grains if the board is subjected to an axial load of 250 N.

The post has a square cross-sectional area. If it is fixed supported at its base and a horizontal force is applied at its end as shown, determine (a) the maximum in-plane shear stress developed at A and (b) the principal stresses at A.

Determine the principal stress, the maximum in-plane shear stress, and average normal stress. Specify the orientation of the element in each case.

The thin-walled pipe has an inner diameter of 0.5 in. and a thickness of 0.025 in. If it is subjected to an internal pressure of 500 psi and the axial tension and torsional loadings shown, determine the principal stress at a point on the surface of the pipe.

Determine the principal stress and maximum in-plane shear stress at point A on the cross section of the pipe at section a – a.

Determine the principal stress and maximum in-plane shear stress at point B on the cross section of the pipe at section a – a .

The rotor shaft of the helicopter is subjected to the tensile force and torque shown when the rotor blades provide the lifting force to suspend the helicopter at midair. If the shaft has a diameter of 6 in., determine the principal stress and maximum in-plane shear stress at a point located on the surface of the shaft.

The pedal crank for a bicycle has the cross section shown. If it is fixed to the gear at B and does not rotate while subjected to a force of 75 lb, determine the principal stress in the material on the cross section at point C .

A spherical pressure vessel has an inner radius of 5 ft and a wall thickness of 0.5 in. Draw Mohr’s circle for the state of stress at a point on the vessel and explain the significance of the result. The vessel is subjected to an internal pressure of 80 psi.

The cylindrical pressure vessel has an inner radius of 1.25 m and a wall thickness of 15 mm. It is made from steel plates that are welded along the \(45^{\circ}\) seam. Determine the normal and shear stress components along this seam if the vessel is subjected to an internal pressure of 8 MPa.

Determine the normal and shear stresses at point D that act perpendicular and parallel, respectively, to the grains. The grains at this point make an angle of \(30^{\circ}\) with the horizontal as shown. Point D is located just to the left of the 10-kN force.

Determine the principal stress at point D , which is located just to the left of the 10-kN force.

If the box wrench is subjected to the 50 lb force, determine the principal stress and maximum in-plane shear stress at point A on the cross section of the wrench at section a – a . Specify the orientation of these states of stress and indicate the results on elements at the point.

If the box wrench is subjected to the 50 lb force, determine the principal stress and maximum in-plane shear stress at point B on the cross section of the wrench at section a – a . Specify the orientation of these states of stress and indicate the results on elements at the point.

The ladder is supported on the rough surface at A and by a smooth wall at B . If a man weighing 150 lb stands upright at C , determine the principal stresses in one of the legs at point D . Each leg is made from a 1-in.-thick board having a rectangular cross section. Assume that the total weight of the man is exerted vertically on the rung at C and is shared equally by each of the ladder’s two legs. Neglect the weight of the ladder and the forces developed by the man’s arms.

Draw the three Mohr’s circles that describe each of the following states of stress.

Draw the three Mohr’s circles that describe the following state of stress.

The stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

The stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

The stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

The stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

The state of stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

Consider the general case of plane stress as shown. Write a computer program that will show a plot of the three Mohr’s circles for the element, and will also calculate the maximum in-plane shear stress and the absolute maximum shear stress.

The solid cylinder having a radius r is placed in a sealed container and subjected to a pressure p. Determine the stress components acting at point A located on the center line of the cylinder. Draw Mohr’s circles for the element at this point.

The plate is subjected to a tensile force P = 5 kip. If it has the dimensions shown, determine the principal stresses and the absolute maximum shear stress. If the material is ductile it will fail in shear. Make a sketch of the plate showing how this failure would appear. If the material is brittle the plate will fail due to the principal stresses. Show how this failure occurs.

Determine the principal stresses and absolute maximum shear stress developed at point A on the cross section of the bracket at section a – a.

Determine the principal stresses and absolute maximum shear stress developed at point B on the cross section of the bracket at section a – a

The solid propeller shaft on a ship extends outward from the hull. During operation it turns at \(\omega=15 \mathrm{rad} / \mathrm{s}\) when the engine develops 900 kW of power. This causes a thrust of F = 1.23 MN on the shaft. If the shaft has an outer diameter of 250 mm, determine the principal stresses at any point located on the surface of the shaft.

The solid propeller shaft on a ship extends outward from the hull. During operation it turns at \(\omega=15 \mathrm{rad} / \mathrm{s}\) when the engine develops 900 kW of power. This causes a thrust of F = 1.23 MN on the shaft. If the shaft has an outer diameter of 250 mm, determine the maximum in-plane shear stress at any point located on the surface of the shaft.

The steel pipe has an inner diameter of 2.75 in. and an outer diameter of 3 in. If it is fixed at C and subjected to the horizontal 20-lb force acting on the handle of the pipe wrench at its end, determine the principal stresses in the pipe at point A, which is located on the surface of the pipe.

Solve Prob. 9–91 for point B, which is located on the surface of the pipe.

Determine the equivalent state of stress if an element is oriented \(40^{\circ}\) clockwise from the element shown. Use Mohr’s circle.

The crane is used to support the 350-lb load. Determine the principal stresses acting in the boom at points A and B. The cross section is rectangular and has a width of 6 in. and a thickness of 3 in. Use Mohr’s circle.

Determine the equivalent state of stress on an element at the same point which represents (a) the principal stresses, and (b) the maximum in-plane shear stress and the associated average normal stress. Also, for each case, determine the corresponding orientation of the element with respect to the element shown and sketch the results on the element.

The propeller shaft of the tugboat is subjected to the compressive force and torque shown. If the shaft has an inner diameter of 100 mm and an outer diameter of 150 mm, determine the principal stress at a point A located on the outer surface.

The box beam is subjected to the loading shown. Determine the principal stress in the beam at points A and B.

The state of stress at a point is shown on the element. Determine (a) the principal stresses and (b) the maximum in-plane shear stress and average normal stress at the point. Specify the orientation of the element in each case.

The state of stress at a point in a member is shown on the element. Determine the stress components acting on the inclined plane AB. Solve the problem using the method of equilibrium described in Sec. 9.1.