Questions and Answers of Mechanics Of Materials

Determine the variation of the radius \(r\) of the cantilevered beam that supports the uniform distributed load so that it has a constant maximum bending stress \(\sigma_{\max }\) throughout its
Determine the variation in the width \(b\) as a function of \(x\) for the cantilevered beam that supports a uniform distributed load along its centerline so that it has the same maximum bending
The end gear connected to the shaft is subjected to the loading shown. If the bearings at \(A\) and \(B\) exert only \(y\) and \(z\) components of force on the shaft, determine the equilibrium torque
The end gear connected to the shaft is subjected to the loading shown. If the bearings at \(A\) and \(B\) exert only \(y\) and \(z\) components of force on the shaft, determine the equilibrium torque
The 50 -mm-diameter shaft is supported by journal bearings at \(A\) and \(B\). If the pulleys \(C\) and \(D\) are subjected to the loadings shown, determine the absolute maximum bending stress in the
The pulleys fixed to the shaft are loaded as shown. If the journal bearings at \(A\) and \(B\) exert only horizontal and vertical forces on the shaft, determine the required diameter of the shaft to
The pulleys fixed to the shaft are loaded as shown. If the journal bearings at \(A\) and \(B\) exert only horizontal and vertical forces on the shaft, determine the required diameter of the shaft to
The two pulleys fixed to the shaft are loaded as shown. If the journal bearings at \(A\) and \(B\) exert only vertical forces on the shaft, determine the required diameter of the shaft to the nearest
The shaft is supported by journal bearings at \(A\) and \(B\) that exert force components only in the \(x\) and \(z\) directions on the shaft. If the allowable normal stress for the shaft is
The shaft is supported by bearings at \(A\) and \(B\) that exert force components only in the \(x\) and \(z\) directions on the shaft. If the allowable normal stress for the shaft is \(\sigma_{\text
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. 16 ksi 10 ksi
The wood beam is subjected to a load of \(12 \mathrm{kN}\). Determine the principal stresses at point \(A\) and specify the orientation of the element. -2 m- 201 12 kN m 25 75 mm 4 m- 300 mm 200 mm mm
The internal loadings at a section of the beam consist of an axial force of \(500 \mathrm{~N}\), a shear force of \(800 \mathrm{~N}\), and two moment components of \(30 \mathrm{~N} \cdot \mathrm{m}\)
Determine the principal stresses at point \(A\), which is located at the bottom of the web. Show the results on an element located at this point. -0.6 m- + 0.3 m 150 kN/m
The square steel plate has a thickness of \(10 \mathrm{~mm}\) and is subjected to the edge loading shown. Determine the maximum in-plane shear stress and the average normal stress developed in the
The square steel plate has a thickness of \(0.5 \mathrm{in}\). and is subjected to the edge loading shown. Determine the principal stresses developed in the steel. 4 in. 4 in. -16 lb/in. -16 lb/in.
Determine the principal stresses acting at point \(A\) of the supporting frame. Show the results on a properly oriented element located at this point. B 150 mm -800 mm- 300 mm 12 mm B 130 mm- -15 mm
Determine the principal stresses acting at point \(B\), which is located just on the web, below the horizontal segment on the cross section. Show the results on a properly oriented element located at
Determine the maximum in-plane shear stress in the box beam at point \(A\). Show the results on an element located at this point. 10 kip 4 kip A B* -2 ft- -1.5 ft- || -2 ft- 4 in. HA 4 in. B +6 in!
Determine the principal stresses in the box beam at point \(B\).Show the results on an element located at this point. 10 kip 4 kip A B* -2 ft- -1.5 ft -2 ft- ft- 0.5 ft 4 in. HA in. 4 in. B 6 in. 3
Determine the principal stresses and also the maximum in-plane shear stress that are developed at point \(A\). For each case, show the results on an element located at this point. The rod has a
The post has a square cross-sectional area. If it is fixed 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)
Determine the principal stresses at point \(A\) on the cross section of the hanger at section \(a-a\). Specify the orientation of this state of stress and indicate the result on an element at the
Determine the principal stresses at point \(A\) on the cross section of the hanger at section \(b-b\). Specify the orientation of the state of stress and indicate the results on an element at the
The stair tread of the escalator is supported on two of its sides by the moving pin at \(A\) and the roller at \(B\). If a man having a weight of \(300 \mathrm{lb}\) stands in the center of the
The cantilevered rectangular bar is subjected to the force of 5 kip. Determine the principal stresses at point \(A\). 1.5 in. 1.5 in.. 1.5 in. A 1 in. 1.5 in. B 3 in. 1 in. 15 in. 3 in. 5 kip
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
The propane gas tank has an inner diameter of \(1500 \mathrm{~mm}\) and wall thickness of \(15 \mathrm{~mm}\). If the pressure in the tank is \(2 \mathrm{MPa}\), determine the absolute maximum shear
Consider the general case of plane strain where \(\epsilon_{x}, \epsilon_{y}\), and \(\gamma_{x y}\) are known. Write a computer program that can be used to determine the normal and shear strain,
The \(45^{\circ}\) strain rosette is mounted on the link of the backhoe. The following readings are obtained from each gauge: \(\epsilon_{a}=650\left(10^{-6}\right),
The \(45^{\circ}\) strain rosette is mounted on a steel shaft. The following readings are obtained from each gauge: gage: \(\epsilon_{a}=300\left(10^{-6}\right),
Consider the general orientation of three strain gages at a point as shown. Write a computer program that can be used to determine the principal in-plane strains and the maximum in-plane shear strain
For the case of plane stress, show that Hooke's law can be written as\[\sigma_{x}=\frac{E}{\left(1-u^{2}\right)}\left(\epsilon_{x}+u \epsilon_{y}\right), \quad
A bar of copper alloy is loaded in a tension machine and it is determined that \(\epsilon_{x}=940\left(10^{-6}\right)\) and \(\sigma_{x}=14 \mathrm{ksi}, \sigma_{y}=0\), \(\sigma_{z}=0\). Determine
The rod is made of aluminum 2014-T6. If it is subjected to the tensile load of \(700 \mathrm{~N}\) and has a diameter of \(20 \mathrm{~mm}\), determine the absolute maximum shear strain in the rod at
The rod is made of aluminum 2014-T6. If it is subjected to the tensile load of \(700 \mathrm{~N}\) and has a diameter of \(20 \mathrm{~mm}\), determine the principal strains at a point on the surface
The cross section of the rectangular beam is subjected to the bending moment M. Determine an expression for the increase in length of lines \(A B\) and \(C D\). The material has a modulus of
The principal strains at a point on an aluminum plate are \(\epsilon_{1}=780\left(10^{-6}\right)\) and \(\epsilon_{2}=400\left(10^{-6}\right)\). Determine the associated principal stresses at the
The 6061-T6 aluminum alloy plate fits snugly into the rigid constraint. Determine the normal stresses \(\sigma_{x}\) and \(\sigma_{y}\) developed in the plate if the temperature is increased by
The principal strains in a plane are \(\epsilon_{1}=630\left(10^{-6}\right)\) and \(\epsilon_{2}=350\left(10^{-6}\right)\). Determine the associated principal stresses at the point in the same plane.
The principal stresses at a point are shown in the figure. If the material is aluminum for which \(E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi}\) and \(u_{\mathrm{al}}=0.33\), determine the
The block is fitted between the fixed supports. If the glued joint can resist a maximum shear stress of \(\tau_{\text {allow }}=2 \mathrm{ksi}\), determine the temperature rise that will cause the
Two strain gages \(a\) and \(b\) are attached to the surface of the plate made from a material having a modulus of elasticity of \(E=70 \mathrm{GPa}\) and Poisson's ratio \(u=0.35\). If the gages
Two strain gages \(a\) and \(b\) are attached to the surface of the plate which is subjected to the uniform distributed load \(w_{x}=700 \mathrm{kN} / \mathrm{m}\) and \(w_{y}=-175 \mathrm{kN} /
The principal strains in a plane, measured experimentally at a point on the 2014-T6 aluminum fuselage of a jet aircraft, are \(\epsilon_{1}=450\left(10^{-6}\right)\) and
The principal stresses at a point are shown in the figure. If the material is structural A992 steel, determine the principal strains. 10 ksi 30 ksi 40 ksi
The principal plane stresses and associated strains in a plane at a point are \(\sigma_{1}=30 \mathrm{ksi}, \sigma_{2}=-10 \mathrm{ksi}, \epsilon_{1}=1.14\left(10^{-3}\right)\),
A rod has a radius of \(20 \mathrm{~mm}\). If it is subjected to an axial load of \(20 \mathrm{kN}\) such that the axial strain in the rod is \(\epsilon_{x}=218\left(10^{-6}\right)\), determine the
The cylindrical pressure vessel is fabricated using hemispherical end caps in order to reduce the bending stress that would occur if flat ends were used.The bending stresses at the seam where the
A thin-walled cylindrical pressure vessel has an inner radius \(r\), thickness \(t\), and length \(L\). If it is subjected to an internal pressure \(p\), show that the increase in its inner radius is
The rubber block is confined in the U-shape smooth rigid block. If the rubber has a modulus of elasticity \(E\) and Poisson's ratio \(v\), determine the effective modulus of elasticity of the rubber
Derive an expression for an equivalent torque \(T_{e}\) that, if applied alone to a solid bar with a circular cross section, would cause the same maximum shear stress as the combination of an applied
A bar with a square cross section is made of a material having a yield stress of \(\sigma_{Y}=120 \mathrm{ksi}\). If the bar is subjected to a bending moment of \(75 \mathrm{kip} \cdot\) in.,
Cast iron when tested in tension and compression has an ultimate strength of \(\left(\sigma_{\mathrm{ult}}\right)_{t}=280 \mathrm{MPa}\) and \(\left(\sigma_{\mathrm{ult}}\right)_{c}=420
Derive an expression for an equivalent bending moment \(M_{e}\) that, if applied alone to a solid cylinder, would cause the same maximum shear stress as the combination of an applied moment \(M\) and
The plate is made of hard copper, which yields at \(\sigma_{Y}=105 \mathrm{ksi}\). Using the maximum shear stress theory, determine the tensile stress \(\sigma_{Y}\) that can be applied to the plate
The element is subjected to the state of stress shown. If \(\sigma_{Y}=36 \mathrm{ksi}\), determine the factor of safety for the loading based on the maximum shear stress theory. 12 ksi 8 ksi 4 ksi
If the 3-in.-diameter short rod is made from brittle material having an ultimate strength of \(\sigma_{\text {ult }}=60 \mathrm{ksi}\), for both tension and compression, determine if the shaft fails
If the 3-in.-diameter shaft is made from cast iron having tensile and compressive ultimate strengths of \(\left(\sigma_{\mathrm{ult}}\right)_{t}=40 \mathrm{ksi}\), and
The shaft consists of a solid segment \(A B\) and a hollow segment \(B C\), which are rigidly joined by the coupling at \(B\). If the shaft is made from A-36 steel, determine the maximum torque \(T\)
The shaft consists of a solid segment \(A B\) and a hollow segment \(B C\), which are rigidly joined by the coupling at \(B\). If the shaft is made from A-36 steel, determine the maximum torque \(T\)
If \(\sigma_{Y}=50 \mathrm{ksi}\), determine the factor of safety for this state of stress against yielding, based on(a) the maximum shear stress theory and(b) the maximum distortion energy theory.
The state of plane stress at a critical point in a steel machine bracket is shown. If the yield stress for steel is \(\sigma_{Y}=36 \mathrm{ksi}\), determine if yielding occurs using the maximum
The state of stress acting at a point on a wrench is shown. Determine the smallest yield stress for steel that might be selected for the part, based on the maximum distortion energy theory. 150 MPa
The state of stress acting at a point on a wrench is shown. Determine the smallest yield stress for steel that might be selected for the part, based on the maximum shear stress theory. 150 MPa 300
If the A-36 steel pipe has outer and inner diameters of \(30 \mathrm{~mm}\) and \(20 \mathrm{~mm}\), respectively, determine the factor of safety against yielding of the material at point \(A\)
If the A-36 steel pipe has an outer and inner diameter of \(30 \mathrm{~mm}\) and \(20 \mathrm{~mm}\), respectively, determine the factor of safety against yielding of the material at point \(A\)
The principal stresses acting at a point on a thin-walled cylindrical pressure vessel are \(\sigma_{1}=p r / t, \sigma_{2}=p r / 2 t\), and \(\sigma_{3}=0\). If the yield stress is \(\sigma_{Y}\),
The gas tank is made from A-36 steel and has an inner diameter of \(1.50 \mathrm{~m}\). If the tank is designed to withstand a pressure of \(5 \mathrm{MPa}\), determine the minimum required wall
The element is subjected to the state of stress shown. If the material is machine steel having a yield stress of \(\sigma_{Y}=750 \mathrm{MPa}\), determine the factor of safety with respect to
The gas storage tank is fabricated by bolting together two half cylindrical thin shells and two hemispherical shells as shown. If the tank is designed to withstand a pressure of \(3 \mathrm{MPa}\),
The gas storage tank is fabricated by bolting together two half cylindrical thin shells and two hemispherical shells as shown. If the tank is designed to withstand a pressure of \(3 \mathrm{MPa}\),
The cylindrical tank is fabricated by welding a strip of thin plate helically, making an angle \(\theta\) with the longitudinal axis of the tank. If the strip has a width \(w\) and thickness \(t\),
The staves or vertical members of the wooden tank are held together using semicircular hoops having a thickness of 0.5 in. and a width of 2 in. Determine the normal stress in hoop \(A B\) if the tank
A wood pipe having an inner diameter of \(3 \mathrm{ft}\) is bound together using steel hoops each having a cross-sectional area of \(0.2 \mathrm{in}^{2}\). If the allowable stress for the hoops is
The 304 stainless steel band initially fits snugly around the smooth rigid cylinder. If the band is then subjected to a nonlinear temperature drop of \(\Delta T=20 \sin ^{2} \theta{ }^{\circ}
The spherical gas tank is fabricated by bolting together two hemispherical thin shells of thickness \(30 \mathrm{~mm}\). If the gas contained in the tank is under a pressure of \(2 \mathrm{MPa}\),
The spherical gas tank is fabricated by bolting together two hemispherical thin shells. If the 8 -m inner diameter tank is to be designed to withstand a pressure of \(2 \mathrm{MPa}\), determine the
Determine the state of stress acting at point \(E\). Show the results on a differential element at this point. A 3 90 kN 30 kN B D |-1.5m+ E -1.5 m 1.5 m--1.5 m- 0.5 m 40 mnt D 100 mm E 60 mm 80 mm
The joint is subjected to a force of \(P=200 \mathrm{lb}\) and \(F=150 \mathrm{lb}\). Determine the state of stress at points \(A\) and \(B\), and sketch the results on differential elements located
The C-clamp applies a compressive stress on the cylindrical block of 80 psi. Determine the maximum tensile and compressive stress developed in the clamp. t 4 in. 4.5 in. | 11 in. 0.25 in. 0.75 in.
The control lever is subjected to a horizontal force of \(20 \mathrm{lb}\) on the handle. Determine the state of stress at points \(A\) and \(B\). Sketch the results on differential elements located
The control lever is subjected to a horizontal force of \(20 \mathrm{lb}\) on the handle. Determine the state of stress at points \(E\) and \(F\). Sketch the results on differential elements located
The coping saw has an adjustable blade that is tightened with a tension of \(40 \mathrm{~N}\). Determine the state of stress in the frame at points \(A\) and \(B\). 100 mm -75 mm- 8 mm 3 mm: A 3 mm B
The rod has a diameter of \(40 \mathrm{~mm}\). Determine the stress components that act at point \(B\), and show the results on a volume element located at this point. 1500 N 300 mm 600 N 100 Nm 800
Determine the state of stress at point \(A\) on the cross section of the post at section \(a-a\). Indicate the results on a differential element at the point. 5ft 400 lb a 1.5 ft 300 lb a 2.5 in. 2
Determine the state of stress at point \(B\) on the cross section of the post at section \(a-a\). Indicate the results on a differential element at the point. 5 ft 400 lb 1.5 ft 300 lb a a 2.5 in.- 2
The wide-flange beam is subjected to the loading shown. Determine the stress components at points \(A\) and \(B\) and show the results on a volume element at each of these points. 2500 lb 3000 lb 500
If the \(75-\mathrm{kg}\) man stands in the position shown, determine the state of stress at point \(A\) on the cross section of the plank at section \(a-a\). The center of gravity of the man is at
The eccentric force \(\mathbf{P}\) is applied at a distance \(e_{y}\) from the centroid on the concrete support shown. Determine the range along the \(y\) axis where \(\mathbf{P}\) can be applied on
Determine the state of stress at point \(A\) on the cross section of the post at section \(a-a\). Indicate the results on a differential element at the point. 100 mm 100 mm 4 kN 3 kN 400 mm 400 mm a
Determine the state of stress at point \(B\) on the cross section of the post at section \(a-a\). Indicate the results on a differential element at the point. 100 mm 100 mm 4 kN 50 mm 50 mm 50 mm B
The bar has a diameter of \(40 \mathrm{~mm}\). If it is subjected to the two force components at its end as shown, determine the state of stress at point \(A\), and show the results on a differential
The gas storage tank is fabricated by bolting together two half cylindrical thin shells and two hemispherical shells as shown. If the tank is designed to withstand a pressure of \(3 \mathrm{MPa}\),
The gas storage tank is fabricated by bolting together two half cylindrical thin shells and two hemispherical shells as shown. If the tank is designed to withstand a pressure of \(3 \mathrm{MPa}\),
The cylindrical tank is fabricated by welding a strip of thin plate helically, making an angle \(\theta\) with the longitudinal axis of the tank. If the strip has a width \(w\) and thickness \(t\),
Determine the maximum shear stress in the T-beam at the section where the internal shear is maximum. A 20 kN/m 30 kN/m -3 m- C 200 mm -B +1.5 m 1.5m-| 200 mm 20 mm --20 mm
Determine the shear stress at point C in the T-beam where the internal shear is maximum. Show the result on a volume element at this point. A 20 kN/m 30 kN/m 3 m- 200 mm B -1.5 m- 200 mm 20 mm -20 mm
The shaft is supported by a thrust bearing at A and a journal bearing at B. If the shaft is made from a material having an allowable shear stress of τallow = 75 MPa, determine the maximum value for
The strut is subjected to a vertical shear of V = 130 kN. Plot the intensity of the shear-stress distribution acting over the cross-sectional area, and determine the resultant shear force developed

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