Solution: Determine (a) the principal stresses and (b) the

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

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 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 stress components acting on the inclined plane AB. Solve the problem using the method of equilibrium described in Sec. 9.1.

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

Solve Prob. 9–7 using the stress transformation equations developed in Sec. 9.2. 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 stress components acting on the plane AB. Solve the problem using the method of equilibrium described in Sec. 9.1.

Solve Prob. 9–9 using the stress transformation equation developed in Sec. 9.2. Determine the stress components acting on the plane AB. Solve the problem using the method of equilibrium described in Sec. 9.1.

Determine the equivalent state of stress on an element at the same point oriented 60° 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° counterclockwise with respect to the element shown. Sketch the results 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 (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 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.

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 and sketch the results on the element.

A point on a thin plate is subjected to the two stress components. 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.

The stress along two planes at a point is indicated. Determine the normal stresses on plane b–b and the principal stresses.

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

The grains of wood in the board make an angle of 20° 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 grains of wood in the beam at point A make an angle of 25 with the horizontal as shown, determine the normal and shear stress that act perpendicular to the grains due to the loading.

The internal loadings at a section of the beam are shown. Determine the in-plane principal stresses at point A. Also compute the maximum in-plane shear stress at this point.

Solve Prob. 9–25 for point B. The internal loadings at a section of the beam are shown. Determine the in-plane principal stresses at point A. Also compute the maximum in-plane shear stress at this point.

Solve Prob. 9–25 for point C. The internal loadings at a section of the beam are shown. Determine the in-plane principal stresses at point A. Also compute the maximum in-plane shear stress at this point.

A rod has a circular cross section with a diameter of 2 in. It is subjected to a torque of \(12 \mathrm{kip} \cdot \mathrm{in}\). and a bending moment M. The greater principal stress at the point of maximum flexural stress is 15 ksi. Determine the magnitude of the bending moment.

The bell crank is pinned at A and supported by a short link BC. If it is subjected to the force of 80 N, determine the principal stresses at (a) point D and (b) point E. The crank is constructed from an aluminum plate having a thickness of 20 mm.

The beam has a rectangular cross section and is subjected to the loadings shown. Determine the principal stresses at point A and point B, which are located just to the left of the 20-kN load. Show the results on elements located at these points.

A paper tube is formed by rolling a cardboard strip in a spiral and then gluing the edges together as shown. Determine the shear stress acting along the seam, which is at 50° from the horizontal, when the tube is subjected to an axial compressive force of 200 N. The paper is 2 mm thick and the tube has an outer diameter of 100 mm.

Solve Prob. 9–31 for the normal stress acting perpendicular to the seam. A paper tube is formed by rolling a cardboard strip in a spiral and then gluing the edges together as shown. Determine the shear stress acting along the seam, which is at 50° from the horizontal, when the tube is subjected to an axial compressive force of 200 N. The paper is 2 mm thick and the tube has an outer diameter of 100 mm.

The 2-in.-diameter drive shaft AB on the helicopter is subjected to an axial tension of 10 000 lb and a torque of \(300 \mathrm{lb} \cdot \mathrm{ft}\). Determine the principal stresses and the maximum in-plane shear stress that act at a point on the surface of the shaft.

Determine the principal stresses in the cantilevered beam at points A and B.

The internal loadings at a cross section through the 6-in.-diameter drive shaft of a turbine consist of an axial force of 2500 lb, a bending moment of \(800 \mathrm{lb} \cdot \mathrm{ft}\), and a torsional moment of \(1500 \mathrm{lb} \cdot \mathrm{ft}\). Determine the principal stresses at point A. Also calculate the maximum in-plane shear stress at this point.

The internal loadings at a cross section through the 6-in.-diameter drive shaft of a turbine consist of an axial force of 2500 lb, a bending moment of \(800 \mathrm{lb} \cdot \mathrm{ft}\), and a torsional moment of \(1500 \mathrm{lb} \cdot \mathrm{ft}\). Determine the principal stresses at point B. Also calculate the maximum in-plane shear stress at this point.

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

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 60-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 outer surface of the pipe.

Solve Prob. 9–38 for point B, which is located on the outer surface of the pipe. 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 60-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 outer surface of the pipe.

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 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.

The box beam is subjected to the 26-kN force that is applied at the center of its width, 75 mm from each side. Determine the principal stresses at point A and show the results in an element located at this point. Use the shear formula to calculate the shear stress.

Solve Prob. 9–42 for point B. The box beam is subjected to the 26-kN force that is applied at the center of its width, 75 mm from each side. Determine the principal stresses at point A and show the results in an element located at this point. Use the shear formula to calculate the shear stress.

Solve Prob. 9–2 using Mohr’s circle. Determine the stress components acting on the inclined plane AB. Solve the problem using the method of equilibrium described in Sec. 9.1.

Solve Prob. 9–3 using Mohr’s circle. Determine the stress components acting on the inclined plane AB. Solve the problem using the method of equilibrium described in Sec. 9.1.

Solve Prob. 9–6 using Mohr’s circle. Determine the stress components acting on the inclined plane AB. Solve the problem using the method of equilibrium described in Sec. 9.1.

Solve Prob. 9–11 using Mohr’s circle. Determine the equivalent state of stress on an element at the same point oriented 60° clockwise with respect to the element shown. Sketch the results on the element.

Solve Prob. 9–15 using Mohr’s circle. 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.

Solve Prob. 9–16 using Mohr’s circle. 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.

Mohr’s circle for the state of stress is shown in Fig. 9–17a. 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 (a) the principal stresses and (b) the maximum in-plane shear stress and average normal stress. Specify the orientation of the element in each cas

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 60° clockwise from the element shown.

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

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

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 (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 (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 (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 (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.

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° with the horizontal as shown. 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 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 stresses, 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.

The frame supports the triangular distributed load shown. 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 35° with the horizontal as shown.

The frame supports the triangular distributed load shown. Determine the normal and shear stresses at point E that act perpendicular and parallel, respectively, to the grains. The grains at this point make an angle of 45° with the horizontal as shown.

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 stresses 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 stresses 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° 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° 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 stresses 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 stresses 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 post is fixed supported at its base and the loadings are applied at its end as shown. Determine (a) the maximum in-plane shear stress developed at A and (b) the principal stresses at A.

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.

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

Determine the principal stresses and the absolute maximum shear stress.

Determine the principal stresses and the absolute maximum shear stress.

Determine the principal stresses and the absolute maximum shear stress.

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 determine the maximum in-plane shear stress and the absolute maximum shear stress.

The solid shaft is subjected to a torque, bending moment, and shear force. Determine the principal stresses at points A and B and the absolute maximum shear stress.

The frame is subjected to a horizontal force and couple moment. Determine the principal stresses and the absolute maximum shear stress at point A. The cross-sectional area at this point is shown.

The bolt is fixed to its support at C. If a force of 18 lb is applied to the wrench to tighten it, determine the principal stresses and the absolute maximum shear stress in the bolt shank at point A. Represent the results on an element located at this point. The shank has a diameter of 0.25 in.

The bolt is fixed to its support at C. If a force of 18 lb is applied to the wrench to tighten it, determine the principal stresses and the absolute maximum shear stress developed in the bolt shank at point B. Represent the results on an element located at this point. The shank has a diameter of 0.25 in.