Mechanics of Materials in SI Units

Russell C. Hibbeler

Chapter 10

Strain Transformation - all with Video Answers

Educators

Chapter Questions

Problem 1

Prove that the sum of the normal strains in perpendicular directions is constant, i.e., $\boldsymbol{\epsilon}_x+\boldsymbol{\epsilon}_y=\boldsymbol{\epsilon}_{x^{\prime}}+\boldsymbol{\epsilon}_{y^{\prime}}$.

Mahnoor Amin

Mahnoor Amin

Numerade Educator

Problem 2

The state of strain at the point on the arm has components of $\epsilon_x=200\left(10^{-6}\right), \epsilon_y=-300\left(10^{-6}\right)$, 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.

Mahnoor Amin

Numerade Educator

Problem 3

The state of strain at the point on the pin leaf has components of $\epsilon_x=200\left(10^{-6}\right), \epsilon_y=180\left(10^{-6}\right)$, 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.

Mahnoor Amin

Numerade Educator

Problem 4

Solve Prob. 10-3 for an element oriented $\theta=30^{\circ}$ clockwise.

Mahnoor Amin

Numerade Educator

Problem 5

The state of strain at the point on the leaf of the caster assembly has components of $\epsilon_x=-400\left(10^{-6}\right)$, $\epsilon_y=860\left(10^{-6}\right)$, 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.

Mahnoor Amin

Numerade Educator

Problem 6

The state of strain at a point on the bracket has components of $\epsilon_x=150\left(10^{-6}\right), \epsilon_y=200\left(10^{-6}\right), \gamma_{x y}=$ $-700\left(10^{-6}\right)$. 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.

Mahnoor Amin

Numerade Educator

Problem 7

Solve Prob. 10-6 for an element oriented $\theta=30^{\circ}$ clockwise.

Mahnoor Amin

Numerade Educator

Problem 8

The state of strain at the point on the spanner wrench has components of $\epsilon_x=260\left(10^{-6}\right), \epsilon_y=320\left(10^{-6}\right)$, 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.

Hast Aggarwal

Numerade Educator

Problem 9

The state of strain at the point on the member has components of $\boldsymbol{\epsilon}_x=180\left(10^{-6}\right), \quad \epsilon_y=-120\left(10^{-6}\right)$, 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.

Hast Aggarwal

Numerade Educator

Problem 10

The state of strain at the point on the support has components of $\epsilon_x=350\left(10^{-6}\right), \quad \epsilon_y=400\left(10^{-6}\right)$, $\gamma_{x y}=-675\left(10^{-6}\right)$. 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.

Chai Santi

Numerade Educator

Problem 11

Due to the load $\mathbf{P}$, the state of strain at the point on the bracket has components of $\epsilon_x=500\left(10^{-6}\right)$, $\epsilon_y=350\left(10^{-6}\right)$, 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.

Mahnoor Amin

Numerade Educator

Problem 12

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

Chai Santi

Numerade Educator

Problem 13

The state of plane strain on the element is $\epsilon_x=-300\left(10^{-6}\right), \epsilon_y=0$, 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.

Chai Santi

Numerade Educator

Problem 14

The state of strain at the point on a boom of a shop crane has components of $\epsilon_s=250\left(0^{-6}\right), \varepsilon_y=300\left(10^{-6}\right)$, 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.

Hast Aggarwal

Numerade Educator

Problem 15

Consider the general case of plane strain where $\boldsymbol{\epsilon}_{\mathrm{s},} \boldsymbol{\epsilon}_{\mathrm{y}}$, and $\gamma_{x y}$ are known. Write a computer program that can be used to determine the normal and shear strain, $\boldsymbol{c}_3$ and $\gamma_{r^{\prime} y^{\prime}}$, 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.

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Problem 16

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.

Chai Santi

Numerade Educator

Problem 17

Solve Prob. 10-3 using Mohr's circle.

Chai Santi

Numerade Educator

Problem 18

Solve Prob. 10-4 using Mohr's circle.

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Problem 20

Solve Prob. 10-5 using Mohr's circle.

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Problem 20

Solve Prob. 10-8 using Mohr's circle.

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Problem 21

Solve Prob. 10-7 using Mohr's circle.

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Problem 22

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 $x-y$ plane, (b) the maximum shear strain in the $x-y$ plane, and (c) the absolute maximum shear strain.

Mahnoor Amin

Numerade Educator

Problem 23

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

Chai Santi

Numerade Educator

Problem 24

The strain at point $A$ on the pressure-vessel wall has components $\varepsilon_x=480\left(10^{-6}\right), \varepsilon_y=720\left(10^{-6}\right), \gamma_{x y}=$ $650\left(10^{-6}\right)$. 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.

Mahnoor Amin

Numerade Educator

Problem 25

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

Mahnoor Amin

Numerade Educator

Problem 26

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

Mahnoor Amin

Numerade Educator

Problem 27

$\mathbf{1 0 - . ~ T h e ~} 60^{\circ}$ strain rosette is mounted on the surface of the bracket. The following readings are obtained for each gage: $\epsilon_a=-780\left(10^{-6}\right), \epsilon_b=400\left(10^{-6}\right)$, 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.

Chai Santi

Numerade Educator

Problem 28

${ }^* $. The $45^{\circ}$ strain rosette is mounted on a steel shaft. The following readings are obtained from each gage: $\epsilon_a=800\left(10^{-6}\right), \epsilon_b=520\left(10^{-6}\right), \epsilon_c=-450\left(10^{-6}\right)$. Determine the in-plane principal strains.

Mahnoor Amin

Numerade Educator

Problem 29

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 $\theta_a=40^{\circ}$, $e_a=160\left(10^{-6}\right), \theta_b=125^{\circ}, \epsilon_b=100\left(10^{-6}\right), \theta_c=220^{\circ}$, $\epsilon_c=80\left(10^{-6}\right)$.

Chai Santi

Numerade Educator

Problem 30

For the case of plane stress, show that Hooke's law can be written as

$$
\sigma_x=\frac{E}{\left(1-v^2\right)}\left(\epsilon_x+v \varepsilon_y\right), \quad \sigma_y=\frac{E}{\left(1-v^2\right)}\left(\epsilon_y+\nu \varepsilon_s\right)
$$

Chai Santi

Numerade Educator

Problem 31

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.

Hast Aggarwal

Numerade Educator

Problem 32

A bar of copper alloy is loaded in a tension machine and it is determined that $\epsilon_x=940\left(10^{-6}\right)$ and $\sigma_x=100 \mathrm{MPa}, \sigma_y=0, \sigma_z=0$. Determine the modulus of elasticity, $E_{\mathrm{cs}}$ and the dilatation, $\varepsilon_{\mathrm{cu}}$, of the copper. $v_{c u}=0.35$.

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Problem 33

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 $\varepsilon_3=2.75\left(10^{-6}\right)$, determine the modulus of elasticity $E$ and the change in the rod's diameter. $p=0.23$.

Chai Santi

Numerade Educator

Problem 34

The principal strains at a point on the aluminum fuselage of a jet aircraft are $e_1=780\left(10^{-6}\right)$ and $\epsilon_2-400\left(10^{-6}\right)$. Determine the associated principal stresses at the point in the same plane. $E_{a l}=70 \mathrm{GPa}$. Hint: See Prob. 10-30.

Mahnoor Amin

Numerade Educator

Problem 35

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 elasticity $E$ and Poisson's ratio is $\nu$.

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Problem 36

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 $E_{\mathrm{at}}=200 \mathrm{GPa}$ and $v_{\mathrm{st}}=0.3$.

Chai Santi

Numerade Educator

Problem 37

Determine the bulk modulus for each of the following materials: (a) rubber, $E_{\mathrm{r}}=2.8 \mathrm{MPa}, \nu_{\mathrm{r}}=0.48$, and (b) glass, $E_{\mathrm{g}}=56 \mathrm{GPa}, v_{\mathrm{g}}=0.24$.

Chai Santi

Numerade Educator

Problem 38

The strain in the $x$ direction at point $A$ on the steel beam is measured and found to be $\boldsymbol{c}_x=-100\left(10^{-6}\right)$. Determine the applied load $P$. What is the shear strain $\gamma_{s y}$ at point $A$ ? $E_{\mathrm{za}}=200 \mathrm{GPa}, v_{\mathrm{nt}}=0.3$.

Chai Santi

Numerade Educator

Problem 39

The principal strains in a plane, measured experimentally at a point on the aluminum fuselage of a jet aircraft, are $\epsilon_1-630\left(10^{-6}\right)$ and $e_2-350\left(10^{-6}\right)$. If this is a case of plane stress, determine the associated principal stresses at the point in the same plane. $E_{21}=70 \mathrm{GPa}$ and $\nu_{21}=0.33$.

Chai Santi

Numerade Educator

Problem 40

The smooth rigid-body cavity is filled with liquid $6061-T 6$ aluminum. When cooled it is 0.3 mm from the top of the cavity. If the top of the cavity is covered and the temperature is increased by $110^{\circ} \mathrm{C}$, determine the stress components $\sigma_{\mathrm{x}}, \sigma_{\mathrm{y}}$ and $\sigma_z$ in the aluminum. Hint Use Eqs 10-18 with an additional strain term of $a \Delta T$ (Eq. 4-4).

Chai Santi

Numerade Educator

Problem 41

The smooth rigid-body cavity is filled with liquid $6061-\mathrm{T} 6$ aluminum. When cooled it is 0.3 mm from the top of the cavity. If the top of the cavity is not covered and the temperature is increased by $110^{\circ} \mathrm{C}$, determine the strain components $\epsilon_x, \epsilon_y$ and $\epsilon_z$ in the aluminum.

Chai Santi

Numerade Educator

Problem 42

The block is fitted between the fixed supports. If the glued joint can resist a maximum shear stress of $\tau_{\text {allow }}=14 \mathrm{MPa}$. determine the temperature rise that will cause the joint to fail. Take $E=70 \mathrm{GPa}, v=0.2$, and $a=11\left(10^{-6}\right) /{ }^{\circ} \mathrm{C}$.

Chai Santi

Numerade Educator

Problem 43

Two strain gauges $a$ and $b$ are attached to a plate made from a material having a modulus of elasticity of $E=70$ GPa and Poisson's ratio $v=0.35$. If the gauges give a reading of $\varepsilon_a=450\left(10^{-6}\right)$ and $\varepsilon_b=100\left(10^{-6}\right)$, determine the intensities of the uniform distributed load $w_x$ and $w_y$ acting on the plate. The thickness of the plate is 25 mm .

Chai Santi

Numerade Educator

Problem 44

Two strain gauges $a$ 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 $w_y=-175 \mathrm{kN} / \mathrm{m}$. If the gauges give a reading of $\boldsymbol{\epsilon}_a=450\left(10^{-6}\right)$ and $\boldsymbol{e}_b=100\left(10^{-6}\right)$, determine the modulus of elasticity $E$, shear modulus $G$, and Poisson's ratio $v$ for the material.

Chai Santi

Numerade Educator

Problem 45

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 $v$.

Hast Aggarwal

Numerade Educator

Problem 46

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 caps are attached can be eliminated by proper choice of the thickness $t_h$ and $t_c$ of the caps and cylinder, respectively. This requires the radial expansion to be the same for both the hemispheres and cylinder. Show that this ratio is $t_c / t_h-(2-\nu) /(1-v)$. Assume that the vessel is made of the same material and both the cylinder and bemispheres have the same inner radius. If the cylinder is to have a thickness of 12 mm , what is the required thickness of the hemispheres? Take $v=0.3$.

Mahnoor Amin

Numerade Educator

Problem 47

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 $d r-r e_1-p r^2\left(1-\frac{1}{2} v\right) / E t$ and the increase in its length is $\Delta L=p L r\left(\frac{1}{2}-v\right) / E r$. Using these results, show that the change in internal volume becomes $d V=\pi r^2\left(1+\epsilon_1\right)^2\left(1+\epsilon_2\right) L-\pi r^2 L$. Since $\epsilon_1$ and $\epsilon_2$ are small quantities, show further that the change in volume per unit volume, called volumetric sirain, can be written as $d V / V=p r(2.5-2 v) / E t$.

Chai Santi

Numerade Educator

Problem 48

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 $\nu$, determine the effective modulus of elasticity of the rubber under the confined condition.

Mahnoor Amin

Numerade Educator

Problem 49

Initially, gaps between the A-36 steel plate and the rigid constraint are as shown. Determine the normal stresses $\sigma_s$ and $\sigma_y$ developed in the plate if the temperature is increased by $\Delta T=55^{\circ} \mathrm{C}$. To solve, add the thermal strain $\alpha \Delta T$ to the equations for Hooke's Law.

Mahnoor Amin

Numerade Educator

Problem 50

The steel shaft has a radias or conm. Deternunt 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. $E_{2 t}=200 \mathrm{GP}^2, v_{\mathrm{xd}}=0.3$.

Mahnoor Amin

Numerade Educator

Problem 51

The shaft has a radius of 15 mm and is made of L 2 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.

Mahnoor Amin

Numerade Educator

Problem 52

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.

Hast Aggarwal

Numerade Educator

Problem 53

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 $A B$, 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 $A B$ if it has a diameter of 100 mm ? The inner radius of the vessel is 400 mm , and its thickness is 10 mm . $E_{\text {st }}=200$ GPa and $\nu_{x=}=0.3$.

Chai Santi

Numerade Educator

Problem 54

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.

Chai Santi

Numerade Educator

Problem 55

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 $\Delta V=\left(2 p \pi r^4 / E t\right)(1-\nu)$. Use a small-strain analysis.

Chai Santi

Numerade Educator

Problem 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 $v$, 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 $10 \mathrm{~mm} . E_{\mathrm{at}}=200 \mathrm{GiPa}^2, \nu_{\mathrm{m}}=0.3$.

Chai Santi

Numerade Educator

Problem 57

Estimate the increase in volume of the pressure vessel in Proh. 10-56.

Mahnoor Amin

Numerade Educator

Problem 58

A soft material is placed within the confines of a rigid cylinder which rests on a rigid support. Assuming that $\epsilon_x=0$ 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 $p=0.3$ for the material.

Mahnoor Amin

Numerade Educator

Problem 58

A soft material is placed within the confines of a rigid cylinder which rests on a rigid support. Assuming that $\epsilon_x=0$ 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 $p=0.3$ for the material.

Mahnoor Amin

Numerade Educator

Problem 59

A material is subjected to plane stress, Express the distortion energy theory of failure in terms of $\sigma_x, \sigma_y$, and $\tau_{x y}$ -

Chai Santi

Numerade Educator

Problem 60

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

Chai Santi

Numerade Educator

Problem 61

A bar with a square cross-sectional area is made of a material having a yield stress of $\sigma_Y=840 \mathrm{MPa}$. If the bar is subjected to a bending moment of $10 \mathrm{kN}-\mathrm{m}$, determine the required size of the bar according to the maximum-distortion-energy theory. Use a factor of safety of 1.5 with respect to yielding.

Mahnoor Amin

Numerade Educator

Problem 62

Solve Prob. 10-61 using the maximum-shear-stress theory.

Chai Santi

Numerade Educator

Problem 63

Derive an expression for an equivalent bending moment $M_{\varepsilon}$ 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 $T$.

Chai Santi

Numerade Educator

Problem 64

Derive an expression for an equivalent bending moment $M_c$ 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.

Chai Santi

Numerade Educator

Problem 65

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 $T$.

Chai Santi

Numerade Educator

Problem 66

. An aluminum alloy 6061 -T6 is to be used for a solid drive shaft such that it transmits 33 kW at $2400 \mathrm{rev} / \mathrm{min}$. Using a factor of safety of 2 with respect to yielding, determine the smallest-diameter shaft that can be selected based on the maximum-shear-stress theory.

Mahnoor Amin

Numerade Educator

Problem 67

Solve Prob. 10-66 using the maximum-distortionenergy theory.

Chai Santi

Numerade Educator

Problem 68

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.

Mahnoor Amin

Numerade Educator

Problem 69

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 $\sigma_{\text {all }}=28 \mathrm{MPa}$.

Narayan Hari

Numerade Educator

Problem 70

A bar with a circular cross-sectional area is made of SAE 1045 carbon steel having a yield stress of $\sigma_Y=1000 \mathrm{MPa}$. If the bar is subjected to a torque of $3.75 \mathrm{kN} \cdot \mathrm{m}$ and a bending moment of $7 \mathrm{kN} \cdot \mathrm{m}$, 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.

Chai Santi

Numerade Educator

Problem 71

The plate is made of hard copper, which yields at $\sigma_y=735 \mathrm{MPa}$. Using the maximum-shear-stress theory, determine the tensile stress $\sigma_x$ that can be applied to the plate if a tensile stress $\sigma_y=0.5 \sigma_s$ is also applied.

Chai Santi

Numerade Educator

Problem 72

Solve Prob. 10-71 using the maximum-distortionenergy theory.

Chai Santi

Numerade Educator

Problem 73

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-shearstress theory.

Chai Santi

Numerade Educator

Problem 74

Solve Prob. 10-73 using the maximum-distortionenergy theory

Chai Santi

Numerade Educator

Problem 75

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 $\sigma_Y=700 \mathrm{MP}$.

Mahnoor Amin

Numerade Educator

Problem 76

Solve Prob. 10-75 using the maximum shear stress theory.

Mahnoor Amin

Numerade Educator

Problem 77

If the $\mathrm{A}-36$ 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.

Chai Santi

Numerade Educator

Problem 78

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.

Chai Santi

Numerade Educator

Problem 79

If the 50 -mm diameter shaft is made from brittle material having an ultimate strength of $\sigma_{\mathrm{ul}}=350 \mathrm{MPa}$ 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.

Chai Santi

Numerade Educator

Problem 80

If the $50-\mathrm{mm}$ diameter shaft is made from cast iron having tensile and compressive ultimate strengths of $\left(\sigma_{\mathrm{uk}}\right)_c=350 \mathrm{MPa}$ and $\left(\sigma_{\mathrm{mi}}\right)_c=525 \mathrm{MPa}$, respectively, determine if the shaft fails in accordance with Mohr's failure criterion.

Mahnoor Amin

Numerade Educator

Problem 81

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-shear-stress theory.

Chai Santi

Numerade Educator

Problem 82

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.

Chai Santi

Numerade Educator

Problem 83

The yield stress for heat-treated beryllium copper is $\sigma_Y=900 \mathrm{MPa}$. If this material is subjected to plane stress and elastic failure occurs when one principal stress is 1000 MPa, what is the smallest magnitude of the other principal stress? Use the maximum-distortion-energy theory-

Chai Santi

Numerade Educator

Problem 84

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.

Mahnoor Amin

Numerade Educator

Problem 85

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.

Mahnoor Amin

Numerade Educator

Problem 86

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 $T$ that can be applied according to the maximum shear stress theory. Use a factor of safety of 15 against yielding.

Chai Santi

Numerade Educator

Problem 87

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 $T$ that can be applied according to the maximum distortion energy theory. Use a factor of safety of 1.5 against yielding.

Chai Santi

Numerade Educator

Problem 88

The principal stresses acting at a point on a thinwalled 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.

Chai Santi

Numerade Educator

Problem 89

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.

Chai Santi

Numerade Educator

Problem 90

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.

Chai Santi

Numerade Educator

Problem 91

The internal loadings at a critical section along the steel drive shaft of a ship are calculated to be a torque of $3.45 \mathrm{kN} \cdot \mathrm{m}$ a bending moment of $2.25 \mathrm{kN} \cdot \mathrm{m}$ and an axial thrust of 12.5 kN . If the yield points for tension and shear are $\sigma_Y=700 \mathrm{MPa}$ and $\tau_Y-350 \mathrm{MPa}$, respectively, determine the required diameter of the shaft using the maximum-shear-stress theory.

Chai Santi

Numerade Educator

Problem 92

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.

Mahnoor Amin

Numerade Educator

Problem 93

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

Mahnoor Amin

Numerade Educator