Mechanics of Materials 9th Edition solutions

Mechanics of Materials 9
Prove that the sum of the normal stresses \(\sigma_{x}+\sigma_{y}=\sigma_{x^{\prime}}+\sigma_{y^{\prime}}\) is constant. See Figs. 92a and 92b.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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 t

Mechanics of Materials 9
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 respe

Mechanics of Materials 9
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.

Mechanics of Materials 9
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 respe

Mechanics of Materials 9
Planes AB and BC at a point are subjected to the stresses shown. Determine the principal stresses acting at this point and find \(\sigma_BC\).

Mechanics of Materials 9
The stress acting on two planes at a point is indicated. Determine the shear stress on plane aa and the principal stresses at the point.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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 aa . Specify the orientation of this state of stress and show the results on elements.

Mechanics of Materials 9
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 aa . Specify the orientation of this state of stress and show the results on elements.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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 .

Mechanics of Materials 9
The state of stress at a point in a member is shown on the element. Determine the stress components acting on the plane AB .

Mechanics of Materials 9
Determine the principal stress at point A on the cross section of the arm at section aa . Specify the orientation of this state of stress and indicate the results on an element at the point.

Mechanics of Materials 9
Determine the maximum in-plane shear stress developed at point A on the cross section of the arm at section aa . Specify the orientation of this state of stress and indicate the results on an element at the point.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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 ou

Mechanics of Materials 9
Solve Prob. 938 for the normal stress acting perpendicular to the seam.

Mechanics of Materials 9
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.

Mechanics of Materials 9
Solve Prob. 940 for point B located on the web at the top of the bottom flange.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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 .

Mechanics of Materials 9
Solve Prob. 93 using Mohrs circle.

Mechanics of Materials 9
Solve Prob. 96 using Mohrs circle.

Mechanics of Materials 9
Solve Prob. 914 using Mohrs circle.

Mechanics of Materials 9
Solve Prob. 910 using Mohrs circle.

Mechanics of Materials 9
Solve Prob. 912 using Mohrs circle.

Mechanics of Materials 9
Solve Prob. 916 using Mohrs circle.

Mechanics of Materials 9
Mohrs circle for the state of stress in Fig. 915 a is shown in Fig. 915 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. 91 and 92 .

Mechanics of Materials 9
Determine the equivalent state of stress if an element is oriented \(45^{\circ}\) clockwise from the element shown.

Mechanics of Materials 9
Determine the equivalent state of stress if an element is oriented \(20^{\circ}\) clockwise from the element shown.

Mechanics of Materials 9
Determine the equivalent state of stress if an element is oriented \(30^{\circ}\) clockwise from the element shown.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
Determine the principal stress, the maximum in-plane shear stress, and average normal stress. Specify the orientation of the element in each case.

Mechanics of Materials 9
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.

Mechanics of Materials 9
Determine the equivalent state of stress if an element is oriented \(25^{\circ}\) counterclockwise from the element shown.

Mechanics of Materials 9
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.

Mechanics of Materials 9
Determine the principal stresses, the maximum in-plane shear stress, and average normal stress. Specify the orientation of the element in each case.

Mechanics of Materials 9
Draw Mohrs circle that describes each of the following states of stress.

Mechanics of Materials 9
The grains of wood in the board make an angle of \(20^{\circ}\) with the horizontal as shown. Using Mohrs 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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
Determine the principal stress, the maximum in-plane shear stress, and average normal stress. Specify the orientation of the element in each case.

Mechanics of Materials 9
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.

Mechanics of Materials 9
Determine the principal stress and maximum in-plane shear stress at point A on the cross section of the pipe at section a a.

Mechanics of Materials 9
Determine the principal stress and maximum in-plane shear stress at point B on the cross section of the pipe at section a a .

Mechanics of Materials 9
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 su

Mechanics of Materials 9
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 .

Mechanics of Materials 9
A spherical pressure vessel has an inner radius of 5 ft and a wall thickness of 0.5 in. Draw Mohrs 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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
Determine the principal stress at point D , which is located just to the left of the 10-kN force.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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 t

Mechanics of Materials 9
Draw the three Mohrs circles that describe each of the following states of stress.

Mechanics of Materials 9
Draw the three Mohrs circles that describe the following state of stress.

Mechanics of Materials 9
The stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

Mechanics of Materials 9
The stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

Mechanics of Materials 9
The stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

Mechanics of Materials 9
The stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

Mechanics of Materials 9
The state of stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

Mechanics of Materials 9
Consider the general case of plane stress as shown. Write a computer program that will show a plot of the three Mohrs circles for the element, and will also calculate the maximum in-plane shear stress and the absolute maximum shear stress.

Mechanics of Materials 9
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 Mohrs circles for the element at this point.

Mechanics of Materials 9
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 t

Mechanics of Materials 9
Determine the principal stresses and absolute maximum shear stress developed at point A on the cross section of the bracket at section a a.

Mechanics of Materials 9
Determine the principal stresses and absolute maximum shear stress developed at point B on the cross section of the bracket at section a a

Mechanics of Materials 9
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 str

Mechanics of Materials 9
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-pl

Mechanics of Materials 9
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.

Mechanics of Materials 9
Solve Prob. 991 for point B, which is located on the surface of the pipe.

Mechanics of Materials 9
Determine the equivalent state of stress if an element is oriented \(40^{\circ}\) clockwise from the element shown. Use Mohrs circle.

Mechanics of Materials 9
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 Mohrs circle.

Mechanics of Materials 9
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 ele

Mechanics of Materials 9
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.

Mechanics of Materials 9
The box beam is subjected to the loading shown. Determine the principal stress in the beam at points A and B.

Mechanics of Materials 9
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.

Mechanics of Materials 9
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.