Solution: The beam is subjected to a moment of 15 kip # ft.

Draw the shear and moment diagrams for the shaft and determine the shear and moment throughout the shaft as a function of x for \(0 \leq x<3 \mathrm{ft}, \ 3 \mathrm{ft}<x<5 \mathrm{ft}\), and \(5 \mathrm{ft}<x<6 \mathrm{ft}\). The bearings at A and B exert only vertical reactions on the shaft.

Draw the shear and moment diagrams for the beam, and determine the shear and moment in the beam as functions of x for \(0 \leq x<4 \mathrm{ft}, \ 4 \mathrm{ft}<x<10 \mathrm{ft}\), and \(10 \mathrm{ft}<x<14 \mathrm{ft}\).

Draw the shear and moment diagrams for the beam, and determine the shear and moment throughout the beam as functions of x for \(0 \leq x \leq 6 \mathrm{ft} \text { and } 6 \mathrm{ft} \leq x \leq 10 \mathrm{ft}\).

Express the shear and moment in terms of x for \(0<x<3 \mathrm{~m} \text { and } 3 \mathrm{~m}<x<4.5 \mathrm{~m}\), and then draw the shear and moment diagrams for the simply supported beam.

Express the internal shear and moment in the cantilevered beam as a function of x and then draw the shear and moment diagrams.

Draw the shear and moment diagrams for the shaft. The bearings at A and B exert only vertical reactions on the shaft. Also, express the shear and moment in the shaft as a function of x within the region \(125 \mathrm{~mm}<x<725 \mathrm{~mm}\).

Express the internal shear and moment in terms of x for \(0 \leq x<L / 2, \text { and } L / 2<x<L\), and then draw the shear and moment diagrams.

Draw the shear and moment diagrams for the beam, and determine the shear and moment throughout the beam as functions of x for \(0 \leq x \leq 6 \mathrm{ft} \text { and } 6 \mathrm{ft} \leq x \leq 9 \mathrm{ft}\).

If the force applied to the handle of the load binder is 50 lb, determine the tensions \(T_{1}\) and \(T_{2}\) in each end of the chain and then draw the shear and moment diagrams for the arm ABC.

Draw the shear and moment diagrams for the shaft. The bearings at A and D exert only vertical reactions on the shaft.

The crane is used to support the engine, which has a weight of 1200 lb. Draw the shear and moment diagrams of the boom ABC when it is in the horizontal position.

Draw the shear and moment diagrams for the beam.

Draw the shear and moment diagrams for the beam.

Draw the shear and moment diagrams for the beam.

Members ABC and BD of the counter chair are rigidly connected at B and the smooth collar at D is allowed to move freely along the vertical post. Draw the shear and moment diagrams for member ABC.

A reinforced concrete pier is used to support the stringers for a bridge deck. Draw the shear and moment diagrams for the pier. Assume the columns at A and B exert only vertical reactions on the pier.

Draw the shear and moment diagrams for the beam and determine the shear and moment in the beam as functions of x, where \(4 \mathrm{ft}<x<10 \mathrm{ft}\).

The industrial robot is held in the stationary position shown. Draw the shear and moment diagrams of the arm ABC if it is pin connected at A and connected to a hydraulic cylinder (two-force member) BD. Assume the arm and grip have a uniform weight of 1.5 lb/in. and support the load of 40 lb at C.

Determine the placement distance a of the roller support so that the largest absolute value of the moment is a minimum. Draw the shear and moment diagrams for this condition.

Draw the shear and moment diagrams for the beam.

Draw the shear and moment diagrams for the beam.

Draw the shear and moment diagrams for the overhanging beam.

The 150-lb man sits in the center of the boat, which has a uniform width and a weight per linear foot of 3 lb/ft. Determine the maximum internal bending moment. Assume that the water exerts a uniform distributed load upward on the bottom of the boat.

Draw the shear and moment diagrams for the beam.

The footing supports the load transmitted by the two columns. Draw the shear and moment diagrams for the footing if the soil pressure on the footing is assumed to be uniform.

Draw the shear and moment diagrams for the beam.

Draw the shear and moment diagrams for the beam.

Draw the shear and moment diagrams for the beam.

Draw the shear and moment diagrams for the beam.

Draw the shear and moment diagrams for the beam.

The support at A allows the beam to slide freely along the vertical guide so that it cannot support a vertical force. Draw the shear and moment diagrams for the beam.

The smooth pin is supported by two leaves A and B and subjected to a compressive load of 0.4 kN/m caused by bar C. Determine the intensity of the distributed load \(w_{0}\) of the leaves on the pin and draw the shear and moment diagram for the pin.

The shaft is supported by a smooth thrust bearing at A and smooth journal bearing at B. Draw the shear and moment diagrams for the shaft.

Draw the shear and moment diagrams for the cantilever beam.

Draw the shear and moment diagrams for the beam.

Draw the shear and moment diagrams for the rod. Only vertical reactions occur at its ends A and B.

Draw the shear and moment diagrams for the beam.

The beam is used to support a uniform load along CD due to the 6-kN weight of the crate. Also, the reaction at the bearing support B can be assumed uniformly distributed along its width. Draw the shear and moment diagrams for the beam.

Draw the shear and moment diagrams for the double overhanging beam.

Draw the shear and moment diagrams for the simply supported beam.

The compound beam is fixed at A, pin connected at B, and supported by a roller at C. Draw the shear and moment diagrams for the beam.

Draw the shear and moment diagrams for the compound beam.

The compound beam is fixed at A, pin connected at B, and supported by a roller at C. Draw the shear and moment diagrams for the beam.

Draw the shear and moment diagrams for the beam.

A short link at B is used to connect beams AB and BC to form the compound beam. Draw the shear and moment diagrams for the beam if the supports at A and C are considered fixed and pinned, respectively.

The truck is to be used to transport the concrete column. If the column has a uniform weight of w (force/length), determine the equal placement a of the supports from the ends so that the absolute maximum bending moment in the column is as small as possible. Also, draw the shear and moment diagrams for the column.

An A-36 steel strip has an allowable bending stress of 165 MPa. If it is rolled up, determine the smallest radius r of the spool if the strip has a width of 10 mm and a thickness of 1.5 mm. Also, find the corresponding maximum internal moment developed in the strip.

Determine the moment M that will produce a maximum stress of 10 ksi on the cross section.

Determine the maximum tensile and compressive bending stress in the beam if it is subjected to a moment of \(M=4 \mathrm{kip} \cdot \mathrm{ft}\).

The beam is constructed from four pieces of wood, glued together as shown. If \(M=10 \mathrm{kip} \cdot \mathrm{ft}\), determine the maximum bending stress in the beam. Sketch a three-dimensional view of the stress distribution acting over the cross section.

The beam is constructed from four pieces of wood, glued together as shown. If \(M=10 \mathrm{kip} \cdot \mathrm{ft}\), determine the resultant force this moment exerts on the top and bottom boards of the beam.

The beam is made from three boards nailed together as shown. If the moment acting on the cross section is \(M=600 \mathrm{N} \cdot \mathrm{m}\), determine the maximum bending stress in the beam. Sketch a three-dimensional view of the stress distribution and cover the cross section.

The beam is made from three boards nailed together as shown. If the moment acting on the cross section is \(M=600 \mathrm{N} \cdot \mathrm{m}\), determine the resultant force the bending stress produces on the top board.

If the built-up beam is subjected to an internal moment of \(M=75 \mathrm{kN} \cdot \mathrm{m}\), determine the maximum tensile and compressive stress acting in the beam.

If the built-up beam is subjected to an internal moment of \(M=75 \mathrm{kN} \cdot \mathrm{m}\), determine the amount of this internal moment resisted by plate A.

The beam is subjected to a moment M. Determine the percentage of this moment that is resisted by the stresses acting on both the top and bottom boards of the beam.

Determine the moment M that should be applied to the beam in order to create a compressive stress at point D of \(\sigma_{D}=10 \mathrm{MPa}\). Also sketch the stress distribution acting over the cross section and calculate the maximum stress developed in the beam.

The beam is made from three boards nailed together as shown. If the moment acting on the cross section is \(M=1 \mathrm{kip} \cdot \mathrm{ft}\), determine the maximum bending stress in the beam. Sketch a three-dimensional view of the stress distribution acting over the cross section.

If \(M=1 \mathrm{kip} \cdot \mathrm{ft}\), determine the resultant force the bending stresses produce on the top board A of the beam.

The beam is subjected to a moment of \(15 \mathrm{kip} \cdot \mathrm{ft}\). Determine the resultant force the bending stress produces on the top flange A and bottom flange B. Also calculate the maximum bending stress developed in the beam.

The beam is subjected to a moment of \(15 \mathrm{kip} \cdot \mathrm{ft}\). Determine the percentage of this moment that is resisted by the web D of the beam.

The beam is subjected to a moment of \(M=40 \mathrm{kN} \cdot \mathrm{m}\). Determine the bending stress at points A and B. Sketch the results on a volume element acting at each of these points.

The steel shaft has a diameter of 2 in. It is supported on smooth journal bearings A and B, which exert only vertical reactions on the shaft. Determine the absolute maximum bending stress in the shaft if it is subjected to the pulley loadings shown.

The beam is made of steel that has an allowable stress of \(\sigma_{\text {allow }}=24 \mathrm{ksi}\). Determine the largest internal moment the beam can resist if the moment is applied (a) about the z axis, (b) about the y axis.

A shaft is made of a polymer having an elliptical cross section. If it resists an internal moment of \(M=50 \mathrm{~N} \cdot \mathrm{m}\), determine the maximum bending stress in the material (a) using the flexure formula, where \(I_{z}=\frac{1}{4} \pi(0.08 \mathrm{~m})(0.04 \mathrm{~m})^{3}\), (b) using integration. Sketch a three-dimensional view of the stress distribution acting over the cross-sectional area. Here \(I_{x}=\frac{1}{4} \pi(0.08 \mathrm{~m})(0.04 \mathrm{~m})^{3}\).

Solve Prob. 6–65 if the moment \(M=50 \mathrm{~N} \cdot \mathrm{m}\) is applied about the y axis instead of the x axis. Here \(I_{y}=\frac{1}{4} \pi(0.04 \mathrm{~m})(0.08 \mathrm{~m})^{3}\).

The shaft is supported by smooth journal bearings at A and B that only exert vertical reactions on the shaft. If d = 90 mm, determine the absolute maximum bending stress in the beam, and sketch the stress distribution acting over the cross section.

The shaft is supported by smooth journal bearings at A and B that only exert vertical reactions on the shaft. Determine its smallest diameter d if the allowable bending stress is \(\sigma_{\text {allow }}=180 \mathrm{MPa}\). \

The axle of the freight car is subjected to a wheel loading of 20 kip. If it is supported by two journal bearings at C and D, determine the maximum bending stress developed at the center of the axle, where the diameter is 5.5 in.

The strut on the utility pole supports the cable having a weight of 600 lb. Determine the absolute maximum bending stress in the strut if A, B, and C are assumed to be pinned.

The boat has a weight of 2300 lb and a center of gravity at G. If it rests on the trailer at the smooth contact A and can be considered pinned at B, determine the absolute maximum bending stress developed in the main strut of the trailer which is pinned at C. Consider the strut to be a box-beam having the dimensions shown.

Determine the absolute maximum bending stress in the 1.5-in.-diameter shaft. The shaft is supported by a thrust bearing at A and a journal bearing at B.

Determine the smallest allowable diameter of the shaft. The shaft is supported by a thrust bearing at A and a journal bearing at B. The allowable bending stress is \(\sigma_{\text {allow }}=22 \mathrm{ksi}\).

The pin is used to connect the three links together. Due to wear, the load is distributed over the top and bottom of the pin as shown on the free-body diagram. If the diameter of the pin is 0.40 in., determine the maximum bending stress on the cross-sectional area at the center section a–a. For the solution it is first necessary to determine the load intensities \(w_{1}\) and \(w_{2}\).

The shaft is supported by a thrust bearing at A and journal bearing at D. If the shaft has the cross section shown, determine the absolute maximum bending stress in the shaft.

A timber beam has a cross section which is originally square. If it is oriented as shown, determine the dimension \(h^{\prime}\) so that it can resist the maximum moment possible. By what factor is this moment greater than that of the beam without its top or bottom flattened?

If the beam is subjected to an internal moment of \(M=2 \mathrm{kip} \cdot \mathrm{ft}\), determine the maximum tensile and compressive stress in the beam. Also, sketch the bending stress distribution on the cross section.

If the allowable tensile and compressive stress for the beam are \(\left(\sigma_{\text {allow }}\right)_{t}=2 \mathrm{ksi} \text { and }\left(\sigma_{\text {allow }}\right)_{c}=3 \mathrm{ksi}\), respectively, determine the maximum moment M that can be applied on the cross section.

If the beam is subjected to an internal moment of \(M=2 \mathrm{kip} \cdot \mathrm{ft}\), determine the resultant force of the bending stress distribution acting on the top board A.

If the beam is subjected to a moment of \(M=100 \mathrm{kN} \cdot \mathrm{m}\), determine the bending stress at points A, B, and C. Sketch the bending stress distribution on the cross section.

If the beam is made of material having an allowable tensile and compressive stress of \(\left(\sigma_{\text {allow }}\right)_{t}=125 \mathrm{MPa} \text { and }\left(\sigma_{\text {allow }}\right)_{c}=150 \mathrm{MPa}\), respectively, determine the maximum moment M that can be applied to the beam.

The shaft is supported by a smooth thrust bearing at A and smooth journal bearing at C. If d = 3 in., determine the absolute maximum bending stress in the shaft.

The shaft is supported by a thrust bearing at A and journal bearing at C. If the material has an allowable bending stress of \(\sigma_{\text {allow }}=24 \mathrm{ksi}\), determine the required minimum diameter d of the shaft to the nearest \(\frac{1}{16} \mathrm{in}\).

If the intensity of the load w = 15 kN/m, determine the absolute maximum tensile and compressive stress in the beam.

If the allowable bending stress is \(\sigma_{\text {allow }}=150 \mathrm{MPa}\), determine the maximum intensity w of the uniform distributed load.

The beam is subjected to the triangular distributed load with a maximum intensity of \(w_{0}=300 \mathrm{lb} / \mathrm{ft}\). If the allowable bending stress is \(\sigma_{\text {allow }}=1.40 \mathrm{ksi}\), determine the required dimension b of its cross section to the nearest \(\frac{1}{8} \text { in. }\) Assume the support at A is a pin and B is a roller.

The beam has a rectangular cross section with b = 4 in. Determine the largest maximum intensity \(w_{0}\) of the triangular distributed load that can be supported if the allowable bending stress is \(\sigma_{\text {allow }}=1.40 \mathrm{ksi}\).

Determine the absolute maximum bending stress in the beam. Each segment has a rectangular cross section with a base of 4 in. and height of 12 in.

If the compound beam in Prob. 6–42 has a square cross section of side length a, determine the minimum value of a if the allowable bending stress is \(\sigma_{\text {allow }}=150 \mathrm{MPa}\).

If the beam in Prob. 6–28 has a rectangular cross section with a width b and a height h, determine the absolute maximum bending stress in the beam.

Determine the absolute maximum bending stress in the 80-mm-diameter shaft which is subjected to the concentrated forces. There is a journal bearing at A and a thrust bearing at B.

Determine, to the nearest millimeter, the smallest allowable diameter of the shaft which is subjected to the concentrated forces. There is a journal bearing at A and a thrust bearing at B. The allowable bending stress is \(\sigma_{\text {allow }}=150 \mathrm{MPa}\).

Determine the absolute maximum bending stress in the beam, assuming that the support at B exerts a uniformly distributed reaction on the beam. The cross section is rectangular with a base of 3 in. and height of 6 in.

Determine the absolute maximum bending stress in the 2-in.-diameter shaft. There is a journal bearing at A and a thrust bearing at B.

Determine the smallest diameter of the shaft to the nearest \(\frac{1}{8} \text { in }\). There is a journal bearing at A and a thrust bearing at B. The allowable bending stress is \(\sigma_{\text {allow }}=22 \mathrm{ksi}\).

A log that is 2 ft in diameter is to be cut into a rectangular section for use as a simply supported beam. If the allowable bending stress is \(\sigma_{\text {allow }}=8 \mathrm{ksi}\), determine the required width b and height h of the beam that will support the largest load possible. What is this load?

A log that is 2 ft in diameter is to be cut into a rectangular section for use as a simply supported beam. If the allowable bending stress is \(\sigma_{\text {allow }}=8 \mathrm{ksi}\), determine the largest load P that can be supported if the width of the beam is b = 8 in.

If the beam in Prob. 6–3 has a rectangular cross section with a width of 8 in. and a height of 16 in., determine the absolute maximum bending stress in the beam.

The simply supported truss is subjected to the central distributed load. Neglect the effect of the diagonal lacing and determine the absolute maximum bending stress in the truss. The top member is a pipe having an outer diameter of 1 in. and thickness of \(\frac{3}{16} \text { in }\)., and the bottom member is a solid rod having a diameter of \(\frac{1}{2} \text { in }\).

If d = 450 mm, determine the absolute maximum bending stress in the overhanging beam.

If the allowable bending stress is \(\sigma_{\text {allow }}=6 \mathrm{MPa}\), determine the minimum dimension d of the beam’s cross-sectional area to the nearest mm.

The beam has a rectangular cross section as shown. Determine the largest intensity w of the uniform distributed load so that the bending stress in the beam does not exceed \(\sigma_{\text {max}}=10 \mathrm{MPa}\).

The beam has the rectangular cross section shown. If w = 1 kN/m, determine the maximum bending stress in the beam. Sketch the stress distribution acting over the cross section.

The member has a square cross section and is subjected to the moment \(M=850 \mathrm{~N} \cdot \mathrm{m}\). Determine the stress at each corner and sketch the stress distribution. Set \(\theta=45^{\circ}\).

The member has a square cross section and is subjected to the moment \(M=850 \mathrm{~N} \cdot \mathrm{m}\) as shown. Determine the stress at each corner and sketch the stress distribution. Set \(\theta=30^{\circ}\).

Consider the general case of a prismatic beam subjected to bending-moment components \(\mathbf{M}_{y} \text { and } \mathbf{M}_{z}\) when the x, y, z axes pass through the centroid of the cross section. If the material is linear elastic, the normal stress in the beam is a linear function of position such that \(\sigma=a+b y+c z\). Using the equilibrium conditions \(0=\int_{A} \sigma d A, M_{y}=\int_{A} z \sigma d A, M_{z}=\int_{A}-y \sigma d A\), determine the constants a, b, and c, and show that the normal stress can be determined from the equation \(\sigma=\left[-\left(M_{z} I_{y}+M_{y} I_{y z}\right) y+\left(M_{y} I_{z}+M_{z} I_{y z}\right) z\right] /\left(I_{y} I_{z}-I_{y z}^{2}\right)\), where the moments and products of inertia are defined in Appendix A.

Determine the bending stress at point A of the beam, and the orientation of the neutral axis. Using the method in Appendix A, the principal moments of inertia of the cross section are \(I_{z^{\prime}}=8.828 \text { in }{ }^{4} \text { and } I_{y^{\prime}}=2.295 \text { in }^{4}\), where \(z^{\prime} \text { and } y^{\prime}\) are the principal axes. Solve the problem using Eq. 6–17.

Determine the bending stress at point A of the beam using the result obtained in Prob. 6–106. The moments of inertia of the cross-sectional area about the z and y axes are \(I_{z}=I_{y}=5.561 \mathrm{in}^{4}\) and the product of inertia of the cross sectional area with respect to the z and y axes is \(I_{y z}=-3.267 \mathrm{in}^{4}\). (See Appendix A.)

The steel shaft is subjected to the two loads. If the journal bearings at A and B do not exert an axial force on the shaft, determine the required diameter of the shaft if the allowable bending stress is \(\sigma_{\text {allow }}=180 \mathrm{MPa}\).

The 65-mm-diameter steel shaft is subjected to the two loads. If the journal bearings at A and B do not exert an axial force on the shaft, determine the absolute maximum bending stress developed in the shaft.

For the section, \(I_{z^{\prime}}=31.7\left(10^{-6}\right) \mathrm{m}^{4}, I_{y^{\prime}}=114\left(10^{-6}\right) \mathrm{m}^{4}\), \(I_{y^{\prime} z^{\prime}}=-15.8\left(10^{-6}\right) \mathrm{m}^{4}\). Using the techniques outlined in Appendix A, the member’s cross-sectional area has principal moments of inertia of \(I_{z}=28.8\left(10^{-6}\right) \mathrm{m}^{4} \text { and } I_{y}=117\left(10^{-6}\right) \mathrm{m}^{4}\), calculated about the principal axes of inertia y and z, respectively. If the section is subjected to the moment \(M=15 \mathrm{kN} \cdot \mathrm{m}\), determine the stress at point A using Eq. 6–17.

Solve Prob. 6–111 using the equation developed in Prob. 6–106.

The box beam is subjected to a moment of \(M=15 \mathrm{kip} \cdot \mathrm{ft}\). Determine the maximum bending stress in the beam and the orientation of the neutral axis.

Determine the maximum magnitude of the bending moment M so that the bending stress in the member does not exceed 15 ksi.

The shaft is subjected to the vertical and horizontal loadings of two pulleys D and E as shown. It is supported on two journal bearings at A and B which offer no resistance to axial loading. Furthermore, the coupling to the motor at C can be assumed not to offer any support to the shaft. Determine the required diameter d of the shaft if the allowable bending stress is \(\sigma_{\text {allow }}=180 \mathrm{MPa}\).

For the section, \(I_{y^{\prime}}=31.7\left(10^{-6}\right) \mathrm{m}^{4}, I_{z^{\prime}}=114\left(10^{-6}\right) \mathrm{m}^{4}\), \(I_{y^{\prime} z^{\prime}}=15.8\left(10^{-6}\right) \mathrm{m}^{4}\). Using the techniques outlined in Appendix A, the member’s cross-sectional area has principal moments of inertia of \(I_{y}=28.8\left(10^{-6}\right) \mathrm{m}^{4}\) and \(I_{z}=117\left(10^{-6}\right) \mathrm{m}^{4}\), calculated about the principal axes of inertia y and z, respectively. If the section is subjected to a moment of \(M=2500 \mathrm{~N} \cdot \mathrm{m}\), determine the stress produced at point A, using Eq. 6–17.

Solve Prob. 6–116 using the equation developed in Prob. 6–106. For the section, \(I_{y^{\prime}}=31.7\left(10^{-6}\right) \mathrm{m}^{4}, I_{z^{\prime}}=114\left(10^{-6}\right) \mathrm{m}^{4}\), \(I_{y^{\prime} z^{\prime}}=15.8\left(10^{-6}\right) \mathrm{m}^{4}\). Using the techniques outlined in Appendix A, the member’s cross-sectional area has principal moments of inertia of \(I_{y}=28.8\left(10^{-6}\right) \mathrm{m}^{4}\) and \(I_{z}=117\left(10^{-6}\right) \mathrm{m}^{4}\), calculated about the principal axes of inertia y and z, respectively. If the section is subjected to a moment of \(M=2500 \mathrm{~N} \cdot \mathrm{m}\), determine the stress produced at point A, using Eq. 6–17.

If the applied distributed loading of w = 4 kN/m can be assumed to pass through the centroid of the beam’s cross-sectional area, determine the absolute maximum bending stress in the joist and the orientation of the neutral axis. The beam can be considered simply supported at A and B.

Determine the maximum allowable intensity w of the uniform distributed load that can be applied to the beam. Assume w passes through the centroid of the beam’s cross-sectional area, and the beam is simply supported at A and B. The allowable bending stress is \(\sigma_{\text {allow }}=165 \mathrm{MPa}\).

The composite beam is made of steel (A) bonded to brass (B) and has the cross section shown. If it is subjected to a moment of \(M=6.5 \mathrm{kN} \cdot \mathrm{m}\), determine the maximum bending stress in the brass and steel. Also, what is the stress in each material at the seam where they are bonded together? \(E_{\mathrm{br}}=100 \mathrm{GPa}, E_{\mathrm{st}}=200 \mathrm{GPa}\).

The composite beam is made of steel (A) bonded to brass (B) and has the cross section shown. If the allowable bending stress for the steel is \(\left(\sigma_{\text {allow }}\right)_{\mathrm{st}}=180 \mathrm{MPa}\), and for the brass \(\left(\sigma_{\text {allow }}\right)_{\mathrm{br}}=60 \mathrm{MPa}\), determine the maximum moment M that can be applied to the beam. \(E_{\mathrm{br}}=100 \mathrm{GPa}, E_{\mathrm{st}}=200 \mathrm{GPa} .\).

Segment A of the composite beam is made from 2014-T6 aluminum alloy and segment B is A-36 steel. If w = 0.9 kip/ft, determine the absolute maximum bending stress in the aluminum and steel. Sketch the stress distribution on the cross section.

Segment A of the composite beam is made from 2014-T6 aluminum alloy and segment B is A-36 steel. The allowable bending stress for the aluminum and steel are \(\left(\sigma_{\text {allow }}\right)_{\mathrm{al}}=15 \mathrm{ksi} \text { and }\left(\sigma_{\text {allow }}\right)_{\mathrm{st}}=22 \mathrm{ksi}\). Determine the maximum allowable intensity w of the uniform distributed load.

The white spruce beam is reinforced with A-992 steel straps at its center and sides. Determine the maximum stress developed in the wood and steel if the beam is subjected to a bending moment of \(M_{z}=10 \mathrm{kip} \cdot \mathrm{ft}\). Sketch the stress distribution acting over the cross section.

The wooden section of the beam is reinforced with two steel plates as shown. Determine the maximum moment M that the beam can support if the allowable stresses for the wood and steel are \(\left(\sigma_{\text {allow }}\right)_{\mathrm{w}}=6 \mathrm{MPa}, \text { and }\left(\sigma_{\text {allow }}\right)_{\text {st }}=150 \mathrm{MPa}\), respectively. Take \(E_{\mathrm{w}}=10 \mathrm{GPa} \text { and } E_{\mathrm{st}}=200 \mathrm{GPa}\).

The wooden section of the beam is reinforced with two steel plates as shown. If the beam is subjected to a moment of \(M=30 \mathrm{kN} \cdot \mathrm{m}\), determine the maximum bending stresses in the steel and wood. Sketch the stress distribution over the cross section. Take \(E_{\mathrm{w}}=10 \mathrm{GPa} \text { and } E_{\mathrm{st}}=200 \mathrm{GPa}\).

The Douglas Fir beam is reinforced with A-992 steel straps at its sides. Determine the maximum stress in the wood and steel if the beam is subjected to a moment of \(M_{z}=80 \mathrm{kN} \cdot \mathrm{m}\). Sketch the stress distribution acting over the cross section.

The steel channel is used to reinforce the wood beam. Determine the maximum stress in the steel and in the wood if the beam is subjected to a moment of \(M=850 \mathrm{lb} \cdot \mathrm{ft} . E_{\mathrm{st}}=29\left(10^{3}\right) \mathrm{ksi}, E_{\mathrm{w}}=1600 \mathrm{ksi}\).

A wood beam is reinforced with steel straps at its top and bottom as shown. Determine the maximum bending stress developed in the wood and steel if the beam is subjected to a moment of \(M=150 \mathrm{kN} \cdot \mathrm{m}\). Sketch the stress distribution acting over the cross section. Take \(E_{\mathrm{w}}=10 \mathrm{GPa}, E_{\mathrm{st}}=200 \mathrm{GPa}\).

A bimetallic strip is made from pieces of 2014-T6 aluminum and C83400 red brass, having the cross section shown. A temperature increase causes its neutral surface to be bent into a circular arc having a radius of 16 in. Determine the moment that must be acting on its cross section due to the thermal stress.

Determine the maximum uniform distributed load \(w_{0}\) that can be supported by the reinforced concrete beam if the allowable tensile stress for the steel is \(\left(\sigma_{\text {st }}\right)_{\text {allow }}=28 \mathrm{ksi}\) and the allowable compressive stress for the concrete is \(\left(\sigma_{\text {conc }}\right)_{\text {allow }}=3 \mathrm{ksi}\). Assume the concrete cannot support a tensile stress. Take \(E_{\mathrm{st}}=29\left(10^{3}\right) \mathrm{ksi}, E_{\text {conc }}=3.6\left(10^{3}\right) \mathrm{ksi}\).

The composite beam is made of A-36 steel (A) bonded to C83400 red brass (B) and has the cross section shown. If it is subjected to a moment of \(M=6.5 \mathrm{kN} \cdot \mathrm{m}\), determine the maximum stress in the brass and steel. Also, what is the stress in each material at the seam where they are bonded together?

The composite beam is made of A-36 steel (A) bonded to C83400 red brass (B) and has the cross section shown. If the allowable bending stress for the steel is \(\left(\sigma_{\text {allow }}\right)_{\mathrm{st}}=180 \mathrm{MPa}\) and for the brass \(\left(\sigma_{\text {allow }}\right)_{\mathrm{br}}=60 \mathrm{MPa}\), determine the maximum moment M that can be applied to the beam.

If the beam is subjected to a moment of \(M=45 \mathrm{kN} \cdot \mathrm{m}\), determine the maximum bending stress in the A-36 steel section A and the 2014-T6 aluminum alloy section B.

The Douglas Fir beam is reinforced with A-36 steel straps at its sides. Determine the maximum stress in the wood and steel if the beam is subjected to a bending moment of \(M_{z}=4 \mathrm{kN} \cdot \mathrm{m}\). Sketch the stress distribution acting over the cross section.

For the curved beam in Fig. 6–40a, show that when the radius of curvature approaches infinity, the curved-beam formula, Eq. 6–24, reduces to the flexure formula, Eq. 6–13.

The curved member is subjected to the moment of \(M=50 \mathrm{kN} \cdot \mathrm{m}\). Determine the percentage error introduced in the calculation of maximum bending stress using the flexure formula for straight members.

The curved member is made from material having an allowable bending stress of \(\sigma_{\text {allow }}=100 \mathrm{MPa}\). Determine the maximum allowable moment M that can be applied to the member.

The curved beam is subjected to a moment of \(M=40 \mathrm{lb} \cdot \mathrm{ft}\). Determine the maximum bending stress in the beam. Also, sketch a two-dimensional view of the stress distribution acting on section a–a.

The curved beam is made from material having an allowable bending stress of \(\sigma_{\text {allow }}=24 \mathrm{ksi}\). Determine the maximum moment M that can be applied to the beam.

If P = 3 kN, determine the bending stress at points A, B, and C of the cross section at section a–a. Using these results, sketch the stress distribution on section a–a.

If the maximum bending stress at section a–a is not allowed to exceed \(\sigma_{\text {allow }}=150 \mathrm{MPa}\), determine the maximum allowable force P that can be applied to the end E.

The elbow of the pipe has an outer radius of 0.75 in. and an inner radius of 0.63 in. If the assembly is subjected to the moments of \(M=25 \mathrm{lb} \cdot \mathrm{in}\)., determine the maximum stress at section a–a.

The curved bar used on a machine has a rectangular cross section. If the bar is subjected to a couple as shown, determine the maximum tensile and compressive stresses acting at section a–a. Sketch the stress distribution on the section in three dimensions.

The curved bar used on a machine has a rectangular cross section. If the bar is subjected to a couple as shown, determine the maximum tensile and compressive stresses acting at section a–a. Sketch the stress distribution on the section in three dimensions.

The steel rod has a circular cross section. If it is gripped at its ends and a couple moment of \(M=12 \mathrm{lb} \cdot \mathrm{in}\) is developed at each grip, determine the stress acting at points A and B and at the centroid C.

The member has a circular cross section. If it is subjected to a moment of \(M=5 \mathrm{kN} \cdot \mathrm{m}\), determine the stress at points A and B. Is the stress at point \(A^{\prime}\), which is located on the member near the wall, the same as that at A? Explain.

The member has a circular cross section. If the allowable bending stress is \(\sigma_{\text {allow }}=100 \mathrm{MPa}\), determine the maximum moment M that can be applied to the member.

The curved bar used on a machine has a rectangular cross section. If the bar is subjected to a couple as shown, determine the maximum tensile and compressive stress acting at section a–a. Sketch the stress distribution on the section in three dimensions.

The bar is subjected to a moment of \(M=100 \mathrm{~N} \cdot \mathrm{m}\). Determine the maximum bending stress in the bar and sketch, approximately, how the stress varies over the critical section.

The allowable bending stress for the bar is \(\sigma_{\text {allow }}=200 \mathrm{MPa}\). Determine the maximum moment M that can be applied to the bar.

The bar has a thickness of 1 in. and the allowable bending stress is \(\sigma_{\text {allow }}=30 \mathrm{ksi}\). Determine the maximum moment M that can be applied.

The bar has a thickness of 1 in. and is subjected to a moment of \(3 \mathrm{kip} \cdot \mathrm{ft}\). Determine the maximum bending stress in the bar.

The bar has a thickness of 0.5 in. and the allowable bending stress is \(\sigma_{\text {allow }}=20 \mathrm{ksi}\). Determine the maximum moment M that can be applied.

If the radius of each notch on the plate is r = 10 mm, determine the largest moment M that can be applied. The allowable bending stress is \(\sigma_{\text {allow }}=180 \mathrm{MPa}\).

The stepped bar has a thickness of 10 mm. Determine the maximum moment that can be applied to its ends if the allowable bending stress is \(\sigma_{\text {allow }}=150 \mathrm{MPa}\).

The bar has a thickness of 0.5 in. and is subjected to a moment of \(600 \mathrm{lb} \cdot \mathrm{ft}\). Determine the maximum bending stress in the bar.

Determine the shape factor for the wide-flange beam.

The wide-flange member is made from an elastic perfectly plastic material. Determine the shape factor for the beam.

The rod has a circular cross section. If it is made of an elastic perfectly plastic material, determine the shape factor.

The rod has a circular cross section. If it is made of an elastic perfectly plastic material where \(\sigma_{Y}=345 \mathrm{MPa}, determine the maximum elastic moment and plastic moment that can be applied to the cross section.

The beam is made of an elastic perfectly plastic material. Determine the plastic moment \(\mathbf{M}_{p}\) that can be supported by a beam having the cross section shown. \(\sigma_{Y}=30 \mathrm{ksi}\).

Determine the plastic moment \(\mathbf{M}_{p}\) that can be supported by a beam having the cross section shown. \(\sigma_{Y}=30 \mathrm{ksi}\).

Determine the shape factor for the beam.

The beam is made of elastic perfectly plastic material. Determine the maximum elastic moment and the plastic moment that can be applied to the cross section. Take \(\sigma_{Y}=36 \mathrm{ksi}\).

Determine the shape factor for the beam.

The beam is made of an elastic perfectly plastic material for which \(\sigma_{Y}=200 \mathrm{MPa}\). If the largest moment in the beam occurs within the center section a–a, determine the magnitude of each force P that causes this moment to be (a) the largest elastic moment and (b) the largest plastic moment.

The beam is made of elastic perfectly plastic material for which \(\sigma_{Y}=345 \mathrm{MPa}\). Determine the maximum elastic moment and the plastic moment that can be applied to the cross section.

Determine the shape factor of the cross section.

The rod has a circular cross section. If it is made of an elastic perfectly plastic material, determine the shape factor for the rod.

The rod has a circular cross section. If it is made of an elastic perfectly plastic material, determine the maximum elastic moment and plastic moment that can be applied to the cross section. Take r = 3 in., \(\sigma_{Y}=36 \mathrm{ksi}\).

Determine the shape factor of the cross section.

The beam is made of elastic perfectly plastic material. Determine the maximum elastic moment and the plastic moment that can be applied to the cross section. Take a = 50 mm and \(\sigma_{Y}=230 \mathrm{MPa}\).

Determine the shape factor for the member having the tubular cross section.

Determine the shape factor of the cross section.

The box beam is made of an elastic perfectly plastic material for which \(\sigma_{Y}=250 \mathrm{MPa}\). Determine the residual stress in the top and bottom of the beam after the plastic moment \(\mathbf{M}_{p}\) is applied and then released.

The beam is made of an elastic perfectly plastic material for which \(\sigma_{Y}=250 \mathrm{MPa}\). Determine the residual stress in the beam at its top and bottom after the plastic moment \(\mathbf{M}_{p}\) is applied and then released.

The plexiglass bar has a stress–strain curve that can be approximated by the straight-line segments shown. Determine the largest moment M that can be applied to the bar before it fails.

The stress–strain diagram for a titanium alloy can be approximated by the two straight lines. If a strut made of this material is subjected to bending, determine the moment resisted by the strut if the maximum stress reaches a value of (a) \(\sigma_{A}\) and (b) \(\sigma_{B}\).

A beam is made from polypropylene plastic and has a stress–strain diagram that can be approximated by the curve shown. If the beam is subjected to a maximum tensile and compressive strain of \(\epsilon=0.02 \mathrm{~mm} / \mathrm{mm}\), determine the moment M.

The bar is made of an aluminum alloy having a stress–strain diagram that can be approximated by the straight line segments shown. Assuming that this diagram is the same for both tension and compression, determine the moment the bar will support if the maximum strain at the top and bottom fibers of the beam is \(\epsilon_{\max }=0.05\).

The beam is made of phenolic, a structural plastic, that has the stress–strain curve shown. If a portion of the curve can be represented by the equation \(\sigma=\left(5\left(10^{6}\right) \epsilon\right)^{1 / 2} \mathrm{MPa}\), determine the magnitude w of the distributed load that can be applied to the beam without causing the maximum strain in its fibers at the critical section to exceed \(\epsilon_{\max }=0.005 \mathrm{~mm} / \mathrm{mm}\).