If a solid shaft having a diameter d is subjected to a

Prove that the sum of the normal strains in perpendicular directions is constant.

The state of strain at the point 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 a point on a wrench has components \(\epsilon_{x}=120\left(10^{-6}\right), \epsilon_{y}=-180\left(10^{-6}\right), \gamma_{x y}=150\left(10^{-6}\right)\). 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 x–y plane.

The state of strain at the point on the gear tooth has components \(\epsilon_{x}=850\left(10^{-6}\right), \epsilon_{y}=480\left(10^{-6}\right), \gamma_{x y}=650\left(10^{-6}\right)\). 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 gear tooth has the components \(\epsilon_{x}=520\left(10^{-6}\right), \epsilon_{y}=-760\left(10^{-6}\right), \gamma_{x y}=-750\left(10^{-6}\right)\). 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.

A differential elements on the bracket is subjected to plane strain that has the following components: \(\epsilon_{x}=150\left(10^{-6}\right), \epsilon_{y}=200\left(10^{-6}\right), \gamma_{x y}=-700\left(10^{-6}\right)\) strain-transformation equations and determine the equivalent in plane strains on an element oriented at an angle of u = 60° 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 the point on the bracket has components \(\epsilon_{x}=-200\left(10^{-6}\right), \epsilon_{y}=-650\left(10^{-6}\right), \gamma_{x y}=-175\left(10^{-6}\right)\). Use The Strain-transformation equations to determine the equivalent in-plane strains on an element oriented at an angle of \(\theta=20^{\circ}\) 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 has components of \(\epsilon_{x}=180\left(10^{-6}\right), \epsilon_{y}=-120\left(10^{-6}\right), \gamma_{x y}=-100\left(10^{-6}\right)\). 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 \(\epsilon_{x}=350\left(10^{-6}\right), \epsilon_{y}=-400\left(10^{-6}\right), \gamma_{x y}=-675\left(10^{-6}\right)\). 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.

The state of strain on an element has components \(\epsilon_{x}=150\left(10^{-6}\right), \epsilon_{y}=-450\left(10^{-6}\right), \gamma_{x y}=200\left(10^{-6}\right)\). Determine the equivalent state of strain on an element at the same point oriented 30° counterclockwise with respect to the original element. Sketch the results on this element.

The state of strain on an element has components \(\epsilon_{x}=-400\left(10^{-6}\right), \epsilon_{y}=0\left(10^{-6}\right), \gamma_{x y}=150\left(10^{-6}\right)\). 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 an element is \(\epsilon_{x}=-300\left(10^{-6}\right), \epsilon_{y}=0\left(10^{-6}\right), \gamma_{x y}=150\left(10^{-6}\right)\). 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.

The state of strain at the point on a boom of an hydraulic engine crane has components of \(\epsilon_{x}=250\left(10^{-6}\right), \epsilon_{y}=300\left(10^{-6}\right), \gamma_{x y}=-180\left(10^{-6}\right)\). 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 \(\boldsymbol{\epsilon}_{x}, \boldsymbol{\epsilon}_{y}, \text { and } \gamma_{x y}\) are known. Write a computer program that can be used to determine the normal and shear strain, \(\gamma_{x^{\prime} y^{\prime}}\), on the plane of an element oriented u 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 an 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 part (a) of Prob. 10–3 using Mohr’s circle.

Solve part (b) of Prob. 10–3 using Mohr’s circle.

Solve Prob. 10–4 using Mohr’s circle.

Solve Prob. 10–5 using Mohr’s circle.

Solve Prob. 10–7 using Mohr’s circle.

The strain at point A on the bracket has components \(\boldsymbol{\epsilon}_{x}=300\left(10^{-6}\right), \epsilon_{y}=550\left(10^{-6}\right), \gamma_{x y}=550\left(10^{-6}\right), \epsilon_{z}=0\). 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 \(\boldsymbol{\epsilon}_{x}=450\left(10^{-6}\right), \epsilon_{y}=825\left(10^{-6}\right), \gamma_{x y}=275\left(10^{-6}\right), \epsilon_{z}=0\). 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 steel bar is subjected to the tensile load of 500 lb. If it is 0.5 in. thick determine the three principal strains. \(E=29\left(10^{3}\right) \mathrm{ksi}, \nu=0.3\).

The \(45^{\circ}\) strain rosette is mounted on a machine element. The following readings are obtained from each gauge: \(\epsilon_{a}=650\left(10^{-6}\right), \epsilon_{b}=-300\left(10^{-6}\right), \epsilon_{c}=480\left(10^{-6}\right)\). Determine (a) the in-plane 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 \(60^{\circ}\) strain rosette is attached to point A on the surface of the support. Due to the loading the strain gauges give a reading of \(\epsilon_{a}=300\left(10^{-6}\right), \epsilon_{b}=-150\left(10^{-6}\right), \epsilon_{c}=-450\left(10^{-6}\right)\). Use Mohr’s circle and determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and the associated average normal strain. Specify the orientation of each element that has these states of strain with respect to the x axis.

The strain rosette is attached at the point on the surface of the pump. Due to the loading, the strain gauges give a reading of \(\epsilon_{a}=-250\left(10^{-6}\right), \epsilon_{b}=300\left(10^{-6}\right), \epsilon_{c}=-200\left(10^{-6}\right)\). Determine (a) the in-plane principal strains, and (b) the maximum in-plane shear strain. Specify the orientation of each element that has these states of strain with respect to the x axis.

The 60° strain rosette is mounted on a beam. The following readings are obtained from each gauge: \(\epsilon_{a}=-250\left(10^{-6}\right), \epsilon_{b}=-400\left(10^{-6}\right), \epsilon_{c}=280\left(10^{-6}\right)\). Determine (a) the in-plane principal strains and their orientation, and (b) the maximum in-plane shear strain and average normal strain. In each case show the deformed element due to these strains.

Consider the general orientation of three strain gauges 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 \(\theta_{a}=40^{\circ}\) \(\epsilon_{a}=160\left(10^{-6}\right), \theta_{b}=125^{\circ}, \epsilon_{b}=100\left(10^{-6}\right), \theta_{c}=220^{\circ}\) \(\epsilon_{c}=80\left(10^{-6}\right)\).

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 are at a point \(\sigma_{1}=36 \mathrm{ksi}, \sigma_{2}=16 \mathrm{ksi}\),\(\epsilon_{1}=1.02\left(10^{-3}\right), \epsilon_{2}=0.180\left(10^{-3}\right)\).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 \(\boldsymbol{\epsilon}_{x}=2.75\left(10^{-6}\right)\), determine the modulus of elasticity E and the change in its 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 u 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 u 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 gauge 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 \(E_{\mathrm{st}}=200 \mathrm{GPa} \text { and } \nu_{\mathrm{st}}=0.3 \text {. }\)

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

The strain gauge is placed on the surface of a thin- walled steel boiler as shown. If it is 0.5 in. long, determine the pressure in the boiler when the gauge elongates \(0.2\left(10^{-3}\right) \text { in. }\) 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 strain in the x direction at point A on the A-36 structural-steel beam is measured and found to be \(\epsilon_{x}=100\left(10^{-6}\right)\). Determine the applied load P. What is the shear strain \(\gamma_{x y}\) at point A?

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 \(??\epsilon_{x}\) and \(\gamma_{x y}\) at point A.

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 \(\nu_{\mathrm{al}}=0.33\), determine the principal strains.

A strain gauge a is attached in the longitudinal direction (x axis) on the surface of the gas tank. When the tank is pressurized, the strain gauge gives a reading of \(\epsilon_{a}=100\left(10^{-6}\right)\). Determine the pressure p in the tank. The tank has an inner diameter of 1.5 m and wall thickness of 25 mm. It is made of steel having a modulus of elasticity E = 200 GPa and Poisson’s ratio \(\nu=\frac{1}{3}\).

Strain gauge b is attached to the surface of the gas tank at an angle of \(45^{\circ}\) with x axis as shown. When the tank is pressurized, the strain gauge gives a reading of \(\epsilon_{b}=250\left(10^{-6}\right)\). Determine the pressure in the tank. The tank has an inner diameter of 1.5 m and wall thickness of 25 mm. It is made of steel having a modulus of elasticity E = 200 GPa and Poisson’s ratio \(\nu=\frac{1}{3}\).

A material is subjected to principal stresses \(\sigma_{x}\) and \(\sigma_{Y}\). Determine the orientation \(\theta\) of a strain gauge placed at the point so that its reading of normal strain responds only to \(\sigma_{Y}\) and \(\sigma_{x}\). The material constants are \(E \text { and } \nu\).

The principal strains in a plane, measured experimentally at a point on the aluminum fuselage of a jet aircraft, 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. \(E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi}\) and \(\nu_{\mathrm{al}}=0.33\).

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 \(\nu_{\mathrm{al}}=0.33\), determine the principal strains.

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 \(\Delta T=50^{\circ} \mathrm{C}\). To solve, add the thermal strain \(\alpha \Delta T\) to the equations for Hooke’s law.

Initially, gaps between the A-36 steel plate and the rigid constraint are as shown. Determine the normal stresses \(\sigma_{x}\) and \(\sigma_{y}\) developed in the plate if the temperature is increased by \(\Delta T=100^{\circ} \mathrm{F}\). 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 gauges, 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, compute 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’ and y’ direction if a torque \(T=2 \mathrm{kN} \cdot \mathrm{m}\) is applied to the shaft.

The metal 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 joint to fail. Take \(E=10\left(10^{3}\right) \mathrm{ksi}\) \(\nu=0.2, \text { and } \alpha=6.0\left(10^{-6}\right) /{ }^{\circ} \mathrm{F}\). Hint: Use Eq. 10-18 with an additional strain term of \(\alpha \Delta T \text { (Eq. 4-4) }\).

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 gauge 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 volume within the vessel is \(\Delta V=\left(2 p \pi r^{4} / E t\right)(1-\nu)\). 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 n, determine the strains in the circumferential and longitudinal directions. Using these results, compute the increase in both the diameter and the length of a steel pressure vessel filled with air and having an internal gauge 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 tank in Prob. 10–56.

A soft material is placed within the confines of a rigid cylinder which rests on a rigid support. Assuming that \(\boldsymbol{\epsilon}_{x}=0 \text { and } \epsilon_{y}=0\), determine the factor by which the 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 of \(\sigma_{x}, \sigma_{y}, \text { and } \tau_{x y}\). 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.

An aluminum alloy is to be used for a drive shaft such that it transmits 25 hp at 1500 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-distortion-energy theory. \(\sigma_{Y}=3.5 \mathrm{ksi}\)

If a shaft is made of a material for which \(\sigma_{Y}=50 \mathrm{ksi}\), determine the torsional shear stress required to cause yielding using the maximum-distortion-energy theory.

Solve Prob. 10–64 using 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.

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 principal plane stresses acting on a differential element are shown. 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 maximum shear stress as the combination of an applied moment M and torque T. Assume that the principal stresses are of opposite algebraic signs.

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 \(\sigma_{x}\) that can be applied to the plate if a tensile stress \(\sigma_{y}=0.5 \sigma_{x}\) is also applied.

Solve Prob. 10–71 using the maximum-distortion energy theory.

If the 2-in.-diameter shaft is made from brittle material having an ultimate strength of \(\sigma_{u l t}=50 \mathrm{ksi}\), 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 strengths of \(\left(\sigma_{u l t}\right)_{t}=50 \mathrm{ksi} \quad \text { and } \quad\left(\sigma_{u l t}\right)_{c}=75 \mathrm{ksi}\), respectively, determine if the shaft fails in accordance with Mohr’s failure criterion.

The components of plane stess at a critical point on an A-36 steel shell are shown. Determine if failure (yielding) has occurred on the basis of the maximum-shear-stress theory.

The components of plane stress at a critical point on an A-36 steel shell are shown. Determine if failure (yielding) has occurred on the basis of the maximum-distortion-energy theory.

If the A-36 steel pipe has outer and inner diameters of 30 mm and 20 mm, respectively, determine the factor of safety against yielding of the material at point A according to the maximum-shear-stress theory.

If the A-36 steel pipe has an outer and inner diameter of 30 mm and 20 mm, respectively, determine the factor of safety against yielding of the material at point A according to the maximum-distortion-energy theory.

The yield stress for heat-treated beryllium copper is \(\sigma_{Y}=130 \mathrm{ksi}\). If this material is subjected to plane stress and elastic failure occurs when one principal stress is 145 ksi, what is the smallest magnitude of the other principal stress? Use the maximum-distortion-energy theory.

The yield stress for a uranium alloy is \(\sigma_{Y}=160 \mathrm{MPa}\). If a machine part is made of this material and a critical point in the material is subjected to plane stress, such that the principal stresses are \(\sigma_{1} \text { and } \sigma_{2}=0.25 \sigma_{1}\), determine the magnitude of \(\sigma_{1}\) that will cause yielding according to the maximum-distortion energy theory.

Solve Prob. 10–80 using the maximum-shear-stress theory.

The state of stress acting at a critical point on the seat frame of an automobile during a crash is shown in the figure. Determine the smallest yield stress for a steel that can be selected for the member, based on the maximum-shear-stress theory.

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

A bar with a circular cross-sectional area is made of SAE 1045 carbon steel having a yield stress of s = 150 ksi. If the bar is subjected to a torque of \(30 \text { kip · in. }\) and a bending moment of \(56 \text { kip · in. }\), determine the required diameter of the bar according to the maximum-distortion-energy theory. Use a factor of safety of 2 with respect to yielding.

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

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 \(\sigma_{Y}??\), determine the maximum value of p based on (a) the maximum-shear-stress theory and (b) the maximum-distortion-energy theory.

If a solid shaft having a diameter d is subjected to a torque T and moment M, show that by the maximum-shear-stress theory the maximum allowable shear stress is \(\tau_{\text {allow }}=\left(16 / \pi d^{3}\right) \sqrt{M^{2}+T^{2}}\). Assume the principal stresses to be of opposite algebraic signs.

If a solid shaft having a diameter d is subjected to a torque T and moment M, show that by the maximum-normal-stress theory the maximum allowable principal stress is \(\tau_{\text {allow }}=\left(16 / \pi d^{3}\right) \sqrt{M^{2}+T^{2}}\).

The gas tank has an inner diameter of 1.50 m and a wall thickness of 25 mm. If it is made from A-36 steel and the tank is pressured to 5 MPa, determine the factor of safety against yielding using (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 \(2300 \mathrm{lb} \cdot \mathrm{ft}\), 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 \(\sigma_{Y}=100 \mathrm{ksi} \text { and } \tau_{Y}=50 \mathrm{ksi}\), respectively, determine the required diameter of the shaft using 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.

In the case of plane stress, where the in-plane principal strains are given by \(\epsilon_{1} \text { and } \epsilon_{2}\) , show that the third principal strain can be obtained from \(\epsilon_{3}=-[\nu /(1-\nu)]\left(\epsilon_{1}+\epsilon_{2}\right)\) where \(\nu\) is Poisson’s ratio for the material.

The plate is made of material having a modulus of elasticity E = 200 GPa and Poisson’s ratio \(v=\frac{1}{3}\). Determine the change in width a, height b, and thickness t when it is subjected to the uniform distributed loading shown.

The principal plane stresses acting at a point are shown in the figure. If the material is machine steel having a yield stress of \(\sigma_{Y}=500 \mathrm{MPa}\), determine the factor of safety with respect to yielding if the maximum-shear-stress theory is considered.

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 \(\sigma_{Y}=650 \mathrm{MPa}\).

The \(60^{\circ}\) strain rosette is mounted on a beam. The following readings are obtained for each gauge: \(\boldsymbol{\epsilon}_{a}=600\left(10^{-6}\right), \quad \epsilon_{b}=-700\left(10^{-6}\right), \quad \text { and } \quad \epsilon_{c}=350\left(10^{-6}\right)\). Determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case show the deformed element due to these strains.

The state of strain at the point on the bracket has components \(\epsilon_{x}=350\left(10^{-6}\right), \epsilon_{y}=-860\left(10^{-6}\right), \gamma_{x y}=250\left(10^{-6}\right)\). Use the strain-transformation equations to determine the equivalent in-plane strains on an element oriented at an angle of \(\theta=45^{\circ}\) clockwise from the original position. Sketch the deformed element within the x–y plane due to these strains.

The A-36 steel post is subjected to the forces shown. If the strain gauges a and b at point A give readings of \(\epsilon_{a}=300\left(10^{-6}\right) \text { and } \epsilon_{b}=175\left(10^{-6}\right) \text {, }\) determine the magnitudes of \(\mathbf{P}_{1} \text { and } \mathbf{P}_{2}\).

A differential element is subjected to plane strain that has the following components; \(\epsilon_{x}=950\left(10^{-6}\right)\) \(\epsilon_{y}=420\left(10^{-6}\right), \quad \gamma_{x y}=-325\left(10^{-6}\right)\). Use the strain-transformation equations and determine (a) the principal strains and (b) the maximum in plane shear strain and the associated average strain. In each case specify the orientation of the element and show how the strains deform the element.

The state of strain at the point on the bracket has components \(\epsilon_{x}=-130\left(10^{-6}\right)\) \(\epsilon_{y}=280\left(10^{-6}\right), \quad \gamma_{x y}=75\left(10^{-6}\right)\) . 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 plain strain on an element is \(\epsilon_{x}=400\left(10^{-6}\right), \quad \epsilon_{y}=200\left(10^{-6}\right), \quad \text { and } \quad \gamma_{x y}=-300\left(10^{-6}\right)\). 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 element at the point with respect to the original element. Sketch the results on the element.