Solved: The principal plane stresses and associated

Prove that the sum of the normal strains in perpendicular directions is constant, i.e., \(\epsilon_{x}+\epsilon_{y}=\epsilon_{x^{\prime}}+\epsilon_{y^{\prime}}\).

The state of strain at the point on the arm has components of \(\epsilon_{x}=200\left(10^{-6}\right), \epsilon_{y}=-300\left(10^{-6}\right), \text { and } \gamma_{x y}=400\left(10^{-6}\right)\). Use the strain transformation equations to determine the equivalent in-plane strains on an element oriented at an angle of \(30^{\circ}\) counterclockwise from the original position. Sketch the deformed element due to these strains within the x–y plane.

The state of strain at the point on the pin leaf has components of \(\epsilon_{x}=200\left(10^{-6}\right), \quad \epsilon_{y}=180\left(10^{-6}\right)\), and \(\gamma_{x y}=-300\left(10^{-6}\right)\). Use the strain transformation equations and determine the equivalent in-plane strains on an element oriented at an angle of \(\theta=60^{\circ}\) counterclockwise from the original position. Sketch the deformed element due to these strains within the x–y plane.

Solve Prob. 10–3 for an element oriented \(\theta=60^{\circ}\) clockwise. The state of strain at the point on the pin leaf has components of \(\epsilon_{x}=200\left(10^{-6}\right), \quad \epsilon_{y}=180\left(10^{-6}\right)\), and \(\gamma_{x y}=-300\left(10^{-6}\right)\). Use the strain transformation equations and determine the equivalent in-plane strains on an element oriented at an angle of \(\theta=60^{\circ}\) counterclockwise from the original position. Sketch the deformed element due to these strains within the x–y plane.

The state of strain at the point on the leaf of the caster assembly has components of \(\epsilon_{x}=-400\left(10^{-6}\right)\), \(\epsilon_{v}=860\left(10^{-6}\right), \quad \text { and } \gamma_{x y}=375\left(10^{-6}\right)\). Use the strain transformation equations to determine the equivalent in-plane strains on an element oriented at an angle of \(\(\theta=30^{\circ}\)\) counterclockwise from the original position. Sketch the deformed element due to these strains within the x–y plane.

The state of strain at a point on the bracket has components of \(\epsilon_{x}=150\left(10^{-6}\right), \quad \epsilon_{y}=200\left(10^{-6}\right), \quad \gamma_{x y}=-700\left(10^{-6}\right)\). Use the strain transformation equations and determine the equivalent in-plane strains on an element oriented at an angle of \(\(\theta=60^{\circ}\)\) counterclockwise from the original position. Sketch the deformed element within the x–y plane due to these strains.

Solve Prob. 10–6 for an element oriented \(\(\theta=30^{\circ}\)\) clockwise. The state of strain at a point on the bracket has components of \(\epsilon_{x}=150\left(10^{-6}\right), \quad \epsilon_{y}=200\left(10^{-6}\right), \quad \gamma_{x y}=-700\left(10^{-6}\right)\). Use the strain transformation equations and determine the equivalent in-plane strains on an element oriented at an angle of \(\(\theta=60^{\circ}\)\) counterclockwise from the original position. Sketch the deformed element within the x–y plane due to these strains.

The state of strain at the point on the spanner wrench has components of \(\epsilon_{x}=260\left(10^{-6}\right), \quad \epsilon_{y}=320\left(10^{-6}\right)\), and \(\gamma_{x y}=180\left(10^{-6}\right)\). Use the strain transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case specify the orientation of the element and show how the strains deform the element within the x–y plane.

The state of strain at the point on the member has components of \(\epsilon_{x}=180\left(10^{-6}\right), \quad \epsilon_{y}=-120\left(10^{-6}\right)\), and \(\gamma_{x y}=-100\left(10^{-6}\right)\). Use the strain transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case specify the orientation of the element and show how the strains deform the element within the x–y plane.

The state of strain at the point on the support has components of \(\epsilon_{x}=350\left(10^{-6}\right), \quad \epsilon_{y}=-400\left(10^{-6}\right)\), and \(\gamma_{x y}=-675\left(10^{-6}\right)\). Use the strain-transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case specify the orientation of the element and show how the strains deform the element within the x–y plane.

Due to the load P, the state of strain at the point on the bracket has components of \(\epsilon_{x}=500\left(10^{-6}\right), \quad \epsilon_{y}=-350\left(10^{-6}\right)\), and \(\gamma_{x y}=-430\left(10^{-6}\right)\). Use the strain transformation equations to determine the equivalent in-plane strains on an element oriented at an angle of \(\(\theta=30^{\circ}\)\) clockwise from the original position. Sketch the deformed element due to these strains within the x–y plane.

The state of strain on an element has components \(\epsilon_{x}=-400\left(10^{-6}\right), \quad \epsilon_{y}=0\), and \(\gamma_{x y}=150\left(10^{-6}\right)\). Determine the equivalent state of strain on an element at the same point oriented 30° clockwise with respect to the original element. Sketch the results on this element.

The state of plane strain on the element is \(\epsilon_{x}=-300\left(10^{-6}\right), \quad \epsilon_{y}=0\), and \(\gamma_{x y}=150\left(10^{-6}\right)\). Determine the equivalent state of strain which represents (a) the principal strains, and (b) the maximum in-plane shear strain and the associated average normal strain. Specify the orientation of the corresponding elements for these states of strain with respect to the original element.

The state of strain at the point on a boom of a shop crane has components of \(\epsilon_{x}=-250\left(0^{-6}\right), \quad \epsilon_{y}=300\left(10^{-6}\right)\), and \(\gamma_{x y}=180\left(10^{-6}\right)\). Use the strain transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case, specify the orientation of the element and show how the strains deform the element within the x–y plane.

Consider the general case of plane strain where \(\epsilon_{x}, \epsilon_{y}, \text { and } \gamma_{x y}\) are known. Write a computer program that can be used to determine the normal and shear strain, \(\epsilon_{x^{\prime}}\) and \(\gamma_{x^{\prime} y^{\prime}}\), on the plane of an element oriented \(\theta\) from the horizontal. Also, include the principal strains and the element’s orientation, and the maximum in-plane shear strain, the average normal strain, and the element’s orientation.

The state of strain on the element has components \(\epsilon_{x}=-300\left(10^{-6}\right), \epsilon_{y}=100\left(10^{-6}\right), \gamma_{x y}=150\left(10^{-6}\right)\). Determine the equivalent state of strain, which represents (a) the principal strains, and (b) the maximum in-plane shear strain and the associated average normal strain. Specify the orientation of the corresponding elements for these states of strain with respect to the original element.

Solve Prob. 10–3 using Mohr’s circle. The state of strain at the point on the pin leaf has components of \(\epsilon_{x}=200\left(10^{-6}\right), \quad \epsilon_{y}=180\left(10^{-6}\right)\), and \(\gamma_{x y}=-300\left(10^{-6}\right)\). Use the strain transformation equations and determine the equivalent in-plane strains on an element oriented at an angle of \(\theta=60^{\circ}\) counterclockwise from the original position. Sketch the deformed element due to these strains within the x–y plane.

Solve Prob. 10–4 using Mohr’s circle. Solve Prob. 10–3 for an element oriented \(\theta=60^{\circ}\) clockwise. The state of strain at the point on the pin leaf has components of \(\epsilon_{x}=200\left(10^{-6}\right), \quad \epsilon_{y}=180\left(10^{-6}\right)\), and \(\gamma_{x y}=-300\left(10^{-6}\right)\). Use the strain transformation equations and determine the equivalent in-plane strains on an element oriented at an angle of \(\theta=60^{\circ}\) counterclockwise from the original position. Sketch the deformed element due to these strains within the x–y plane.

Solve Prob. 10–5 using Mohr’s circle. The state of strain at the point on the leaf of the caster assembly has components of \(\epsilon_{x}=-400\left(10^{-6}\right)\), \(\epsilon_{v}=860\left(10^{-6}\right), \quad \text { and } \gamma_{x y}=375\left(10^{-6}\right)\). Use the strain transformation equations to determine the equivalent in-plane strains on an element oriented at an angle of \(\(\theta=30^{\circ}\)\) counterclockwise from the original position. Sketch the deformed element due to these strains within the x–y plane.

Solve Prob. 10–8 using Mohr’s circle. The state of strain at the point on the spanner wrench has components of \(\epsilon_{x}=260\left(10^{-6}\right), \quad \epsilon_{y}=320\left(10^{-6}\right)\), and \(\gamma_{x y}=180\left(10^{-6}\right)\). Use the strain transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case specify the orientation of the element and show how the strains deform the element within the x–y plane.

Solve Prob. 10–7 using Mohr’s circle. Solve Prob. 10–6 for an element oriented \(\(\theta=30^{\circ}\)\) clockwise. The state of strain at a point on the bracket has components of \(\epsilon_{x}=150\left(10^{-6}\right), \quad \epsilon_{y}=200\left(10^{-6}\right), \quad \gamma_{x y}=-700\left(10^{-6}\right)\). Use the strain transformation equations and determine the equivalent in-plane strains on an element oriented at an angle of \(\(\theta=60^{\circ}\)\) counterclockwise from the original position. Sketch the deformed element within the x–y plane due to these strains.

The strain at point A on the bracket has components \(\epsilon_{x}=300\left(10^{-6}\right), \quad \epsilon_{y}=550\left(10^{-6}\right)\), \(\gamma_{x y}=-650\left(10^{-6}\right), \epsilon_{z}=0\). Determine (a) the principal strains at A in the x9y plane, (b) the maximum shear strain in the x–y plane, and (c) the absolute maximum shear strain.

The strain at point A on a beam has components \(\epsilon_{x}=450\left(10^{-6}\right), \quad \epsilon_{y}=825left(10^{-6}\right)\), \(\gamma_{x y}=-275\left(10^{-6}\right), \epsilon_{z}=0\). Determine (a) the principal strains at A, (b) the maximum shear strain in the x–y plane, and (c) the absolute maximum shear strain.

The strain at point A on the pressure-vessel wall has components \(\epsilon_{x}=480\left(10^{-6}\right), \quad \epsilon_{y}=720 left(10^{-6}\right)\), \(\gamma_{x y}=-650\left(10^{-6}\right)\). Determine (a) the principal strains at A, in the x-y plane, (b) the maximum shear strain in the x-y plane, and (c) the absolute maximum shear strain.

The 45° strain rosette is mounted on the surface of a shell. The following readings are obtained for each gage: \(\epsilon_{a}=-200\left(10^{-6}\right), \epsilon_{b}=300\left(10^{-6}\right), \text { and } \epsilon_{c}=250\left(10^{-6}\right)\). Determine the in-plane principal strains.

The 45° strain rosette is mounted on the surface of a pressure vessel. The following readings are obtained for each gage: \(\epsilon_{a}=475\left(10^{-6}\right), \epsilon_{b}=250\left(10^{-6}\right), \text { and } \epsilon_{c}=-360\left(10^{-6}\right)\). Determine the in-plane principal strains.

The 60° strain rosette is mounted on the surface of the bracket. The following readings are obtained for each gage: \(\epsilon_{a}=-780\left(10^{-6}\right), \epsilon_{b}=400\left(10^{-6}\right), \text { and } \epsilon_{c}=500\left(10^{-6}\right)\). Determine (a) the principal strains and (b) the maximum in-plane shear strain and associated average normal strain. In each case show the deformed element due to these strains.

The 45° strain rosette is mounted on a steel shaft. The following readings are obtained from each gage: \(\epsilon_{a}=800\left(10^{-6}\right), \epsilon_{b}=520\left(10^{-6}\right), \text { and } \epsilon_{c}=-450\left(10^{-6}\right)\). Determine the in-plane principal strains.

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 at the point. Show an application of the program using the values \(\epsilon_{a}=160\left(10^{-6}\right), \quad \theta_{b}=125^{\circ}, \quad \epsilon_{b}=100\left(10^{-6}\right), \quad \theta_{c}=220^{\circ}\), \(\epsilon_{c}=80\left(10^{-6}\right)\).

For the case of plane stress, show that Hooke’s law can be written as \(\sigma_{x}=\frac{E}{\left(1-\nu^{2}\right)}\left(\epsilon_{x}+\nu \epsilon_{y}\right), \quad \sigma_{y}=\frac{E}{\left(1-\nu^{2}\right)}\left(\epsilon_{y}+\nu \epsilon_{x}\right)\)

Use Hooke’s law, Eq. 10–18, to develop the strain tranformation equations, Eqs. 10–5 and 10–6, from the stress tranformation equations, Eqs. 9–1 and 9–2.

The principal plane stresses and associated strains in a plane at a point are \(\sigma_{1}=36 \mathrm{ksi}, \quad \sigma_{2}=16 \mathrm{ksi}\), \(\epsilon_{1}=1.02\left(10^{-3}\right), \epsilon_{2}=0.180\left(10^{-3}\right)\). Determine the modulus of elasticity and Poisson’s ratio.

A rod has a radius of 10 mm. If it is subjected to an axial load of 15 N such that the axial strain in the rod is \(\epsilon_{x}=2.75\left(10^{-6}\right)\), determine the modulus of elasticity E and the change in the rod’s diameter.\(\nu=0.23\).

The polyvinyl chloride bar is subjected to an axial force of 900 lb. If it has the original dimensions shown, determine the change in the angle \(\theta\) after the load is applied. \(E_{\mathrm{pvc}}=800\left(10^{3}\right) \mathrm{psi}, \nu_{\mathrm{pvc}}=0.20\).

The polyvinyl chloride bar is subjected to an axial force of 900 lb. If it has the original dimensions shown, determine the value of Poisson’s ratio if the angle \(\theta\) decreases by \(\Delta \theta=0.01^{\circ}\) after the load is applied. \(E_{\mathrm{pvc}}=800\left(10^{3}\right) \mathrm{psi}\).

The spherical pressure vessel has an inner diameter of 2 m and a thickness of 10 mm. A strain gage having a length of 20 mm is attached to it, and it is observed to increase in length by 0.012 mm when the vessel is pressurized. Determine the pressure causing this deformation, and find the maximum in-plane shear stress, and the absolute maximum shear stress at a point on the outer surface of the vessel. The material is steel, for which \(E_{\mathrm{st}}=200 \mathrm{GPa} \text { and } \nu_{\mathrm{st}}=0.3\).

Determine the bulk modulus for each of the following materials: (a) rubber, \9E_{\mathrm{r}}=0.4 \mathrm{ksi}, \nu_{\mathrm{r}}=0.48\), and (b) glass, \(E_{\mathrm{g}}=8\left(10^{3}\right) \mathrm{ksi}, \nu_{\mathrm{g}}=0.24\).

The strain gage is placed on the surface of the steel boiler as shown. If it is 0.5 in. long, determine the pressure in the boiler when the gage elongates 0.2(10-3) in. The boiler has a thickness of 0.5 in. and inner diameter of 60 in. Also, determine the maximum x, y in-plane shear strain in the material. \(E_{\mathrm{st}}=29\left(10^{3}\right) \mathrm{ksi}, \nu_{\mathrm{st}}=0.3\).

The principal strains at a point on the aluminum fuselage of a jet aircraft are \(\epsilon_{1}=780\left(10^{-6}\right)\) and \(\epsilon_{2}=400\left(10^{-6}\right)\). Determine the associated principal stresses at the point in the same plane. \(\epsilon_{2}=400\left(10^{-6}\right)\). Hint: See Prob. 10–30.

The strain in the x direction at point A on the A-36 structural-steel beam is measured and found to be \(\epsilon_{x}=200\left(10^{-6}\right)\). Determine the applied load P. What is the shear strain \(\gamma_{x y}\) at point A?

If a load of P = 3 kip is applied to the A-36 structural-steel beam, determine the strain \(\epsilon_{\underline{x}}\) and \(\gamma_{x y}\) at point A.

The cube of aluminum is subjected to the three stresses shown. Determine the principal strains. Take \(E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi} \text { and } \nu_{\mathrm{al}}=0.33\).

The principal strains at a point on the aluminum surface of a tank are \(\epsilon_{1}=630\left(10^{-6}\right) \text { and } \epsilon_{2}=350\left(10^{-6}\right)\). If this is a case of plane stress, determine the associated principal stresses at the point in the same plane. \(E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi}\), \(\nu_{\mathrm{al}}=0.33\). Hint: See Prob. 10–30.

A uniform edge load of 500 lb>in. and 350 lb>in. is applied to the polystyrene specimen. If the specimen is originally square and has dimensions of a = 2 in., b = 2 in., and a thickness of t = 0.25 in., determine its new dimensions \(a^{\prime}, b^{\prime}, \text { and } t^{\prime}\) after the load is applied. \(E_{p}=597\left(10^{3}\right) \mathrm{psi}\) and \(\nu_{p}=0.25\).

A material is subjected to principal stresses \(\sigma_{x}\) and \(\sigma_{y}\). Determine the orientation \(\theta\) of the strain gage so that its reading of normal strain responds only to \(\sigma_{y}\) and not \(\sigma_{x}\). The material constants are E and \(\nu\).

A single strain gage, placed in the vertical plane on the outer surface and at an angle 60° to the axis of the pipe, gives a reading at point A of \(\epsilon_{A}=-250\left(10^{-6}\right)\). Determine the principal strains in the pipe at this point. The pipe has an outer diameter of 1 in. and an inner diameter of 0.6 in. and is made of C86100 bronze.

A single strain gage, placed in the vertical plane on the outer surface and at an angle of 60° to the axis of the pipe, gives a reading at point A of \(\epsilon_{A}=-250\left(10^{-6}\right)\). Determine the vertical force P if the pipe has an outer diameter of 1 in and an inner diameter of 0.6 in. The pipe is made of C86100 bronze.

If the material is graphite for which \(E_{\mathrm{g}}=800 \mathrm{ksi}\) and \(v_{\mathrm{g}}=0.23\), determine the principal strains.

Initially, gaps between the A-36 steel plate and the rigid constraint are as shown. Determine the normal stresses \(\sigma_{x} \text { and } \sigma_{y}\) in the plate if the temperature is increased by \(\Delta T=100^{\circ} \mathrm{F}\). Hint: To solve, add the thermal strain \(\alpha \Delta T\) to the equations for Hooke’s law.

The steel shaft has a radius of 15 mm. Determine the torque T in the shaft if the two strain gages, attached to the surface of the shaft, report strains of \(\epsilon_{x^{\prime}}=-80\left(10^{-6}\right)\) and \(\epsilon_{y^{\prime}}=80\left(10^{-6}\right)\). Also, determine the strains acting in the x and y directions. \(E_{\mathrm{st}}=200 \mathrm{GPa}, \nu_{\mathrm{st}}=0.3\).

The shaft has a radius of 15 mm and is made of L2 tool steel. Determine the strains in the \(x^{\prime}\) and \(y^{\prime}\) direction if a torque \(T=2 \mathrm{kN} \cdot \mathrm{m}\) is applied to the shaft.

The A-36 steel pipe is subjected to the axial loading of 60 kN. Determine the change in volume of the material after the load is applied.

Air is pumped into the steel thin-walled pressure vessel at C. If the ends of the vessel are closed using two pistons connected by a rod AB, determine the increase in the diameter of the pressure vessel when the internal gage pressure is 5 MPa. Also, what is the tensile stress in rod AB if it has a diameter of 100 mm? The inner radius of the vessel is 400 mm, and its thickness is 10 mm. \(E_{\mathrm{st}}=200 \mathrm{GPa}\) and \(\nu_{\mathrm{st}}=0.3\).

Determine the increase in the diameter of the pressure vessel in Prob. 10–53 if the pistons are replaced by walls connected to the ends of the vessel.

A thin-walled spherical pressure vessel having an inner radius r and thickness t is subjected to an internal pressure p. Show that the increase in the volume within the vessel is \9\Delta V=\left(2 p \pi r^{4} / E t\right)(1 \vdash \nu)\). Use a small-strain analysis.

The thin-walled cylindrical pressure vessel of inner radius r and thickness t is subjected to an internal pressure p. If the material constants are E and \(\nu\), determine the strains in the circumferential and longitudinal directions. Using these results, calculate the increase in both the diameter and the length of a steel pressure vessel filled with air and having an internal gage pressure of 15 MPa. The vessel is 3 m long, and has an inner radius of 0.5 m and a thickness of 10 mm. \(E_{\mathrm{st}}=200 \mathrm{GPa}, \nu_{\mathrm{st}}=0.3\).

Estimate the increase in volume of the pressure vessel in Prob. 10–56. The thin-walled cylindrical pressure vessel of inner radius r and thickness t is subjected to an internal pressure p. If the material constants are E and \(\nu\), determine the strains in the circumferential and longitudinal directions. Using these results, calculate the increase in both the diameter and the length of a steel pressure vessel filled with air and having an internal gage pressure of 15 MPa. The vessel is 3 m long, and has an inner radius of 0.5 m and a thickness of 10 mm. \(E_{\mathrm{st}}=200 \mathrm{GPa}, \nu_{\mathrm{st}}=0.3\).

A soft material is placed within the confines of a rigid cylinder which rests on a rigid support. Assuming that \(\epsilon_{x}=0 \text { and } \epsilon_{y}=0\), determine the factor by which the stiffness of the material, or the apparent modulus of elasticity, will be increased when a load is applied, if \(\nu=0.3\) for the material.

A material is subjected to plane stress. Express the distortion energy theory of failure in terms of \(\sigma_{x}, \sigma_{y}, \text { and } \tau_{x y}\).

A material is subjected to plane stress. Express the maximum shear stress theory of failure in terms \(\sigma_{x}, \sigma_{y}, \text { and } \tau_{x y}\). Assume that the principal stresses are of different algebraic signs.

The yield stress for a zirconium-magnesium alloy is \(\sigma_{Y}=15.3 \mathrm{ksi}\). If a machine part is made of this material and a critical point in the material is subjected to in-plane principal stresses \(\sigma_{1} \text { and } \sigma_{2}=-0.5 \sigma_{1}\), determine the magnitude of \(\sigma_{1}\) that will cause yielding according to the maximum shear stress theory.

Solve Prob. 10–61 using the maximum distortion energy theory. The yield stress for a zirconium-magnesium alloy is \(\sigma_{Y}=15.3 \mathrm{ksi}\). If a machine part is made of this material and a critical point in the material is subjected to in-plane principal stresses \(\sigma_{1} \text { and } \sigma_{2}=-0.5 \sigma_{1}\), determine the magnitude of \(\sigma_{1}\) that will cause yielding according to the maximum shear stress theory.

If a machine part is made of tool L2 steel and a critical point in the material is subjected to in-plane principal stresses \(\sigma_{1} \text { and } \sigma_{2}=-0.5 \sigma_{1}\), determine the magnitude of \(\sigma_{1}\) in ksi that will cause yielding according to the maximum shear stress theory.

Solve Prob. 10–63 using the maximum distortion energy theory. If a machine part is made of tool L2 steel and a critical point in the material is subjected to in-plane principal stresses \(\sigma_{1} \text { and } \sigma_{2}=-0.5 \sigma_{1}\), determine the magnitude of \(\sigma_{1}\) in ksi that will cause yielding according to the maximum shear stress theory.

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 energy of distortion as the combination of an applied bending moment M and torque T.

If a shaft is made of a material for which \(\sigma_{Y}=75 \mathrm{ksi}\), determine the maximum torsional shear stress required to cause yielding using the maximum distortion energy theory.

Solve Prob. 10–66 using the maximum shear stress theory. If a shaft is made of a material for which \(\sigma_{Y}=75 \mathrm{ksi}\), determine the maximum torsional shear stress required to cause yielding using the maximum distortion energy theory.

If the material is machine steel having a yield stress of \(\sigma_{Y}=700 \mathrm{MPa}\), determine the factor of safety with respect to yielding if the maximum shear stress theory is considered.

The short concrete cylinder having a diameter of 50 mm is subjected to a torque of \(500 \mathrm{N} \cdot \mathrm{m}\) and an axial compressive force of 2 kN. Determine if it fails according to the maximum normal stress theory. The ultimate stress of the concrete is \(\sigma_{\text {ult }}=28 \mathrm{MPa}\).

Derive an expression for an equivalent bending moment \(M_{e}\) that, if applied alone to a solid bar with a circular cross section, would cause the same energy of distortion as the combination of an applied bending moment M and torque T.

The plate is made of Tobin bronze, which yields at \(\sigma_{Y}=25 \mathrm{ksi}\). Using the maximum shear stress theory, determine the maximum tensile stress \(\sigma_{x}\) that can be applied to the plate if a tensile stress \(\sigma_{y}=1.5 \sigma_{x}\) is also applied.

The plate is made of Tobin bronze, which yields at \(\sigma_{Y}=25 \mathrm{ksi}\). Using the maximum distortion energy theory, determine the maximum tensile stress \(\sigma_{x}\) that can be applied to the plate if a tensile stress \(\sigma_{y}=1.5 \sigma_{x}\) is also applied.

An aluminum alloy is to be used for a solid drive shaft such that it transmits 30 hp at 1200 rev>min. Using a factor of safety of 2.5 with respect to yielding, determine the smallest-diameter shaft that can be selected based on the maximum shear stress theory. sY = 10 ksi.

If a machine part is made of titanium (Ti-6A1-4V) and a critical point in the material is subjected to plane stress, such that the principal stresses are \(\sigma_{1} \text { and } \sigma_{2}=0.5 \sigma_{1}\), determine the magnitude of \(\sigma_{1}\) in MPa that will cause yielding according to (a) the maximum shear stress theory, and (b) the maximum distortion energy theory.

The components of plane stress at a critical point on a thin steel shell are shown. Determine if failure (yielding) has occurred on the basis of the maximum distortion energy theory. The yield stress for the steel is \(\sigma_{Y}=700 \mathrm{MPa}\).

Solve Prob. 10–75 using the maximum shear stress theory. The components of plane stress at a critical point on a thin steel shell are shown. Determine if failure (yielding) has occurred on the basis of the maximum distortion energy theory. The yield stress for the steel is \(\sigma_{Y}=700 \mathrm{MPa}\).

The 304-stainless-steel cylinder has an inner diameter of 4 in. and a wall thickness of 0.1 in. If it is subjected to an internal pressure of p = 80 psi, axial load of 500 lb, and a torque of 70 lb · ft, determine if yielding occurs according to the maximum distortion energy theory.

The 304-stainless-steel cylinder has an inner diameter of 4 in. and a wall thickness of 0.1 in. If it is subjected to an internal pressure of p = 80 psi, axial load of 500 lb, and a torque of 70 lb · ft, determine if yielding occurs according to the maximum shear stress theory.

If the 2-in.-diameter shaft is made from brittle material having an ultimate stress of \(\sigma_{\text {ult}}=50 \mathrm{ksi}\), for both tension and compression, determine if the shaft fails according to the maximum normal stress theory. Use a factor of safety of 1.5 against rupture.

If the 2-in.-diameter shaft is made from cast iron having tensile and compressive ultimate stress of \(\left(\sigma_{\text {ult }}\right)_{t}=50 \mathrm{ksi} \text { and }\left(\sigma_{\text {ult }}\right)_{c}=75 \mathrm{ksi}\), respectively, determine if the shaft fails according to Mohr’s failure criterion.

If \(\sigma_{Y}=50 \mathrm{ksi}\), determine the factor of safety for this loading 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 distortion energy theory.

Solve Prob. 10–82 using the maximum shear stress 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 distortion energy theory.

The state of stress acting at a critical 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.

The state of stress acting at a critical point on a wrench is shown in the figure. Determine the smallest yield stress for steel that might be selected for the part, based on the maximum shear stress theory.

The shaft consists of a solid segment AB and a hollow segment BC, which are rigidly joined by the coupling at B. If the shaft is made from A-36 steel, determine the maximum torque T that can be applied according to the maximum shear stress theory. Use a factor of safety of 1.5 against yielding.

The shaft consists of a solid segment AB and a hollow segment BC, which are rigidly joined by the coupling at B. If the shaft is made from A-36 steel, determine the maximum torque T that can be applied according to the maximum distortion energy theory. Use a factor of safety of 1.5 against yielding.

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 sY , determine the maximum value of p based on (a) the maximum shear stress theory and (b) the maximum distortion energy theory.

If \(\sigma_{Y}=50 \mathrm{ksi}\), determine the factor of safety for this loading based on (a) the maximum shear stress theory and (b) the maximum distortion energy theory.

The gas tank is made from A-36 steel and has an inner diameter of 1.50 m. If the tank is designed to withstand a pressure of 5 MPa, determine the required minimum wall thickness to the nearest millimeter using (a) the maximum shear stress theory, and (b) maximum distortion energy theory. Apply a factor of safety of 1.5 against yielding.

The internal loadings at a critical section along the steel drive shaft of a ship are calculated to be a torque of \(1500 \mathrm{lb} \cdot \mathrm{ft}\), a bending moment of \(1500 \mathrm{lb} \cdot \mathrm{ft}\), and an axial thrust of 2500 lb. If the yield points for tension and shear are \(\sigma_{Y}=100 \mathrm{ksi}\) and \(\tau_{Y}=50 \mathrm{ksi}, respectively, determine the required diameter of the shaft using the maximum shear stress theory.

If the material is machine steel having a yield stress of \(\sigma_{Y}=750 \mathrm{MPa}\), determine the factor of safety with respect to yielding using the maximum distortion energy theory.

If the material is machine steel having a yield stress of \(\sigma_{Y}=750 \mathrm{MPa}\), determine the factor of safety with respect to yielding if the maximum shear stress theory is considered.