Mechanics of Materials

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 the two perpendicular directions is constant, i.e., $\epsilon_3+\epsilon_y=\epsilon_{x^{+}}+\epsilon_{y^{+}}$

Mahnoor Amin

Mahnoor Amin

Numerade Educator

Problem 2

The state of strain at the point on the pin leaf has components of $\epsilon_3=200\left(10^{-6}\right), \quad \epsilon_y=180\left(10^{-6}\right)$, and $\gamma_{n y}=-300\left(10^{-6}\right)$. Use the strain transformation equations and determine the equivalent in-plane strains on an element oriented at an angle of $\theta=30^{\circ}$ counterclockwise from the original position. Sketch the deformed element due to these strains within the $x-y$ plane.

Mahnoor Amin

Numerade Educator

Problem 2

Consider the general arrangement 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 $\theta_e=40^{\circ}$, $\epsilon_4=160\left(10^{-6}\right), \quad \theta_b=125^{\circ}, \quad \epsilon_b=100\left(10^{-6}\right), \quad \theta_c=220^{\circ}$, $\epsilon_5=80\left(10^{-6}\right)$

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Problem 3

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

Prab. 10-2/3

Mahnoor Amin

Numerade Educator

Problem 4

The state of strain at the point on the spanner wrench has components of $\varepsilon_x=-260\left(10^{-6}\right), \epsilon_y=320\left(10^{-6}\right)$, and $\gamma_{s y}=180\left(10^{-6}\right)$. 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 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 the point has components of $\epsilon_{\mathrm{x}}=-100\left(10^{-6}\right), \quad \epsilon_{\mathrm{y}}=400\left(10^{-6}\right)$, and $\gamma_{x y}=-300\left(10^{-6}\right)$. Use the strain transformation equations to determine the equivalent in-plane strains on an element oriented at an angle of $60^{\circ}$ counterclockwise from the original position. Sketch the deformed element due to these strains within the $x-y$ plane.

Chai Santi

Numerade Educator

Problem 7

If a machine part is made of Ti-6A1-4V titanium and a critical point in the material is subjected to plane stress, such that the principal stresses are $\sigma_1$ 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.

Mahnoor Amin

Numerade Educator

Problem 7

The state of strain at the point has components of $\epsilon_1=100\left(10^{-6}\right), \quad \epsilon_y=300\left(10^{-6}\right)$, and $\gamma_{x y}=-150\left(10^{-6}\right)$. Use the strain transformation equations to determine the equivalent in-plane strains on an element oriented $\theta=30^{\circ}$ clockwise. Sketch the deformed element due to these strains within the $x-y$ plane.

Chai Santi

Numerade Educator

Problem 8

The state of strain at the point on the bracket has components $\epsilon_{\mathrm{x}}=400\left(10^{-6}\right), \epsilon_{\mathrm{y}}=-250\left(10^{-6}\right)$. $\gamma_{x y}=310\left(10^{-6}\right)$. 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.

Chai Santi

Numerade Educator

Problem 9

The state of strain at the point has components of $\epsilon_t=-100\left(10^{-6}\right), \quad \epsilon_y=-200\left(10^{-6}\right)$, 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 inplane 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 10

The state of strain at the point on the gear tooth has components of $\epsilon_{\mathrm{f}}=520\left(10^{-6}\right), \epsilon_{\mathrm{y}}=-760\left(10^{-6}\right)$, $\gamma_{0 y}=-750\left(10^{-6}\right)$. 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.

Chai Santi

Numerade Educator

Problem 11

The state of strain on the element has components of $\epsilon_x=500\left(10^{-6}\right), \epsilon_y=300\left(10^{-h}\right)$, and $\gamma_{x y}=-200\left(10^{-6}\right)$. Determine the equivalent state of strain on an element at the same point oriented $45^{\circ}$ clockwise with respect to the original clement.

Chai Santi

Numerade Educator

Problem 12

The state of strain on the element has components of $\epsilon_x=-400\left(10^{-6}\right), \epsilon_y=0, \gamma_{x y}=150\left(10^{-6}\right)$. 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 strain on the element has components of $\epsilon_x=-300\left(10^{-6}\right), \epsilon_y=0$, and $\gamma_{x y}=150\left(10^{-\phi}\right)$. 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 a point has components of $\epsilon_1=250\left(0^{-6}\right), \epsilon_y=300\left(10^{-6}\right)$, and $\gamma_{x y}=-180\left(10^{-6}\right)$. Use the strain transformation equations to determine (a) the inplane principal strains and (b) the maximum in-plane shear strain and associated 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 15

Consider the general case of plane strain where $\boldsymbol{\epsilon}_2, \epsilon_3$, and $\gamma_0$, 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.

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Problem 16

The state of strain on the element has components of $\quad \epsilon_{\mathrm{z}}=-300\left(10^{-6}\right), \quad \epsilon_{\mathrm{y}}=100\left(10^{-6}\right) . \quad \gamma_{\mathrm{xy}}=150\left(10^{-6}\right)$ 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. 10-2 using Mohr's circle.

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Problem 18

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

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Problem 19

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 21

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

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Problem 22

The strain at point $A$ on the bracket has components $\epsilon_{\mathrm{x}}=300\left(10^{-6}\right), \epsilon_{\mathrm{y}}=550\left(10^{-6}\right), \gamma_t=-650\left(10^{-6}\right), \epsilon_2=0$. 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 $\epsilon_x=450\left(10^{-6}\right), \quad \epsilon_x=825\left(10^{-6}\right), \quad \gamma_{x y}=275\left(10^{-6}\right), \quad \epsilon_z=0$. Determine (a) the principal strains at $A_{\text {, ( }}$ (b) the maximum shear strain in the $x-y$ plane, and (c) the absolute maximum shear strain.

Chai Santi

Numerade Educator

Problem 24

The $60^{\circ}$ strain rosette is mounted on the bracket. The following readings are obtained for each gage: $\epsilon_a=-100\left(10^{-6}\right), \epsilon_b=250\left(10^{-6}\right), \epsilon_c=150\left(10^{-6}\right)$. 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 25

The $45^{\circ}$ strain rosette is mounted on a machine element. The following readings are obtained from each gage: $\quad \epsilon_a=-450\left(10^{-6}\right), \quad \epsilon_b=-600\left(10^{-6}\right), \quad \epsilon_c=-300\left(10^{-6}\right)$. 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.

Chai Santi

Numerade Educator

Problem 26

The $60^{\circ}$ strain rosette is mounted on a beam. The following readings are obtained from each gage: $\epsilon_e=-500\left(10^{-6}\right)_c \epsilon_c=250\left(10^{-6}\right), \epsilon_c=400\left(10^{-6}\right)$. 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.

Chai Santi

Numerade Educator

Problem 27

The $45^{\circ}$ strain rosette is mounted on the link of the backhoe. The following readings are obtained from each gage: $\epsilon_5=650\left(10^{-6}\right), \epsilon_0=-300\left(10^{-6}\right), \epsilon_c=480\left(10^{-6}\right)$. Determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and associated average normal strain.

Chai Santi

Numerade Educator

Problem 28

The $45^{\circ}$ strain rosette is mounted on a stecl shaft. The following readings are obtained from each gage: $\epsilon_a=300\left(10^{-6}\right) . \quad \epsilon_6=-250\left(10^{-6}\right) \quad \epsilon_{\mathrm{c}}=-450\left(10^{-6}\right)$. 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.

Chai Santi

Numerade Educator

Problem 30

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

$$
\sigma_n=\frac{E}{\left(1-v^2\right)}\left(\epsilon_1+v \epsilon_y\right) . \quad \sigma_y=\frac{E}{\left(1-\nu^2\right)}\left(\epsilon_y+\nu \epsilon_1\right)
$$

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 $\sigma_x=14 \mathrm{ksi}$, $\sigma_y=0, \sigma_z=0$. Determine the modulus of elasticity. $E_{\text {cur }}$ and the dilatation, $e_{\mathrm{cu}}$, of the copper. $v_{\mathrm{cu}}=0.35$.

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Problem 33

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. $E_{21}=10\left(10^3\right) \mathrm{ksi}, \nu_{\mathrm{al}}=0.33$.

Chai Santi

Numerade Educator

Problem 34

The rod is made of aluminum 2014-T6. If it is subjected to the tensile load of 700 N and has a diameter of 20 mm , determine the absolute maximum shear strain in the rod at a point on its surface.

Km Neeraj

Numerade Educator

Problem 35

The rod is made of aluminum 2014-T6. If it is subjected to the tensile load of 700 N and has a diameter of 20 mm , determine the principal strains at a point on the surface of the rod.

TG

Troy Gabriele

Numerade Educator

Problem 36

The cross section of the rectangular beam is subjected to the bending moment $\mathbf{M}$. 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 $v$.

Hast Aggarwal

Numerade Educator

Problem 37

Determine the bulk modulus for each of the following materials: (a) rubber, $E_r=0.4 \mathrm{ksi}, v_r=0.48$, and (b) glass, $E_k=8\left(10^3\right) \mathrm{ksi}, v_g=0.24$.

Chai Santi

Numerade Educator

Problem 38

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 $0.2\left(10^{-3}\right) \mathrm{in}$. 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. Take $E_{\mathrm{st}}=29\left(10^3\right) \mathrm{ksi}, v_{\mathrm{st}}=0.3$.

Mahnoor Amin

Numerade Educator

Problem 39

The principal strains at a point on an aluminum plate are $\epsilon_1=780\left(10^{-6}\right)$ and $\epsilon_2=400\left(10^{-6}\right)$. Determine the associated principal stresses at the point in the same plane. Take $E_{\text {al }}=10\left(10^3\right) \mathrm{ksi}, \quad v_{\mathrm{al}}=0.33$.

Mahnoor Amin

Numerade Educator

Problem 40

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

Chai Santi

Numerade Educator

Problem 41

If a load of $P=3 \mathrm{kip}$ is applied to the A-36 structural-steel beam, determine the strain $\epsilon_5$, and $\gamma_{n y}$ at point $A$.

Chai Santi

Numerade Educator

Problem 42

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 $\Delta T=50^{\circ} \mathrm{C}$.

Chai Santi

Numerade Educator

Problem 43

The principal strains in a plane are $\epsilon_1=630\left(10^{-6}\right)$ and $\epsilon_2=350\left(10^{-6}\right)$. Determine the associated principal stresses at the point in the same plane. $E_{a l}=10\left(10^3\right) \mathrm{ksi}$ and $v_{a l}=0.33$.

Chai Santi

Numerade Educator

Problem 44

The principal stresses at a point are shown in the figure. If the material is aluminum for which $E_{a l}=10\left(10^3\right) \mathrm{ksi}$ and $v_{a l}=0.33$, determine the principal strains.

Chai Santi

Numerade Educator

Problem 45

The block is fitted between the fixed supports. If the glued joint can resist a maximum shear stress of $\mathrm{T}_{\text {allow }}=2 \mathrm{ksi}$, determine the temperature rise that will cause the joint to fail. Take $E=10\left(10^3\right) \mathrm{ksi}, v=0.2$.

Chai Santi

Numerade Educator

Problem 46

Two strain gages $a$ and $b$ are attached to the surface of the plate made from a material having a modulus of elasticity of $E=70 \mathrm{GPa}$ and Poisson's ratio $v=0.35$. If the gages give a reading of $\epsilon_3=450\left(10^{-6}\right)$ and $\epsilon_{>}=100\left(10^{-6}\right)$, 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 .

Prohs. 10-46

Chai Santi

Numerade Educator

Problem 47

Two strain gages $a$ and $b$ are attached to the surface of the plate which is subjected to the uniform distributed load $w_x=700 \mathrm{kN} / \mathrm{m}$ and $w_y=-175 \mathrm{kN} / \mathrm{m}$. If the gages give a reading of $\epsilon_3=450\left(10^{-6}\right)$ and $\epsilon_{\mathrm{b}}=100\left(10^{-6}\right)$, determine the modulus of clasticity $E$, shear modulus $G$, and Poisson's ratio $v$ for the material.

Chai Santi

Numerade Educator

Problem 48

The principal strains in a plane, measured experimentally at a point on the 2014-T6 aluminum fusclage of a jet aircraft, are $\epsilon_1=4,50\left(10^{-6}\right)$ and $\epsilon_2=-600\left(10^{-6}\right)$, Determine the associated principal stresses at the point in the same plane.

Chai Santi

Numerade Educator

Problem 49

The principal stresses at a point are shown in the figure. If the material is structural A992 steel, determine the principal strains.

Proh. 10-49

Chai Santi

Numerade Educator

Problem 50

The principal plane stresses and associated strains in a plane at a point are $\sigma_1=30 \mathrm{ksi}, \sigma_2=-10 \mathrm{ksi}$, $\epsilon_1=1.14\left(10^{-3}\right), \epsilon_2=-0.655\left(10^{-3}\right)$, Determine the modulus of elasticity and Poisson's ratio.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 51

A rod has a radius of 20 mm . If it is subjected to an axial load of 20 kN such that the axial strain in the rod is $\epsilon_4=218\left(10^{-6}\right)$, determine the modulus of elasticity $E$ and the change in its diameter. Take $v=0.35$.

Mahnoor Amin

Numerade Educator

Problem 52

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 $t_b$ and $t_c$ of the caps and eylinder, respectively. This requires the radial expansion to be the same for both the hemispheres and cylinder. Show that this ratio is $t_c / t_h=(2-v) /(1-v)$. Assume that the vessel is made of the same material and both the cylinder and hemispheres have the same inner radius. If the cylinder is to have a thickness of 0.5 in ., what is the required thickness of the hemispheres? Take $v=0.3$.

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Problem 53

A thin-walled cylindrical pressure vessel has an inner radius $r$, thickness $L$, 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 $\Delta L=p L r\left(\frac{1}{2}-v\right) / E t$. 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 $d V / V=\operatorname{pr}(2.5-2 v) / E x$.

Chai Santi

Numerade Educator

Problem 54

The rubber block is confined in the U-shape smooth rigid block. If the rubber has a modulus of elasticity $E$ and Poisson's ratio $v$, determine the effective modulus of elasticity of the rubber under the confined condition.

Mahnoor Amin

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 volumse 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, compute 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 . $E_{\mathrm{st}}=200 \mathrm{GPa} . v_{\mathrm{st}}=0.3$,

Chai Santi

Numerade Educator

Problem 57

Estimate the increase in volume of the tank in Prob, 10-56.

Probs. 10-56/57

Chai Santi

Numerade Educator

Problem 58

A soft material is placed within the confines of the rigid cylinder which rests on a rigid support. Assuming that $\epsilon_x=0$ and $\epsilon_y=0$, determine the factor by which the modulus of elasticity will be increased when a load is applied if $v=0.3$ for the material.

Chai Santi

Numerade Educator

Problem 59

A material is subjected to plane stress. Express the distortion energy theory of failure in terms of $\sigma_n, \sigma_y$, and $\mathrm{r}_{s y}$.

Chai Santi

Numerade Educator

Problem 60

A material is subjected to plane stress. Express the maximum shear stress theory of failure in terms of $\sigma_n, \sigma_n$ and $t_{s y}$. Assume that the principal stresses have different algebraic signs.

Chai Santi

Numerade Educator

Problem 61

An aluminum alloy 6061-T6 is to be used for a solid drive shaft such that it transmits 40 hp 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 62

Solve Prob, 10-61 using the maximum distortion energy theory.

Chai Santi

Numerade Educator

Problem 63

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

Chai Santi

Numerade Educator

Problem 64

Solve Prob. 10-64 using the maximum shear stress theory.

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 maximum shear stress as the combination of an applied moment $M$ and torque $T$. Assume that the principal stresses have opposite algebraic signs.

Chai Santi

Numerade Educator

Problem 66

A bar with a square cross section is made of a material having a yield stress of $\sigma_y=120 \mathrm{ksi}$. If the bar is subjected to a bending moment of 75 kip - in., 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.

Chai Santi

Numerade Educator

Problem 67

Solve Prob, 10-66 using the maximum shear stress theory.

Chai Santi

Numerade Educator

Problem 68

Cast iron when tested in tension and compression has an ultimate strength of $\left(\sigma_{\text {wi }}\right)_t=280 \mathrm{MPa}$ and $\left(\sigma_{\text {at }}\right)_c=420 \mathrm{MPa}$, respectively. Alsa, when subjected to pure torsion it can sustain an ultimate shear stress of $\mathrm{T}_{\mathrm{ah}}=168 \mathrm{MPa}$. Plot Mohr's circles for each case and establish the failure envelope. If a part made of this material is subjected to the state of plane stress shown, determine if it fails according to Mohr's failure criterion.

Chai Santi

Numerade Educator

Problem 69

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.

Chai Santi

Numerade Educator

Problem 70

Derive an expression for an equivalent bending moment $M_e$ that, if applied alone to a solid eylinder, 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 71

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 $\sigma_y=0.5 \sigma_n$ is also applied.

Chai Santi

Numerade Educator

Problem 72

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

Prohs. 10-71/72

Chai Santi

Numerade Educator

Problem 73

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 foree of 2 kN . Determine if it fails according to the maximum normal stress theory. The ultimate stress of the concrete is $\sigma_{i \mathrm{ith}}=28 \mathrm{MPa}$.

Narayan Hari

Numerade Educator

Problem 75

The element is subjected to the state of stress shown. If $\sigma_Y=36 \mathrm{ksi}$, determine the factor of safety for the loading based on the maximum shear stress theory.

Mahnoor Amin

Numerade Educator

Problem 76

Solve Prob. 10-75 using the maximum distortion energy theory.

Chai Santi

Numerade Educator

Problem 77

If the 3-in-diameter short rod is made from brittle material having an ultimate strength of $\sigma_{\text {eit }}=60 \mathrm{ksi}$, for both tension and compression. determine if the shaft fails according to the maximum normal stress theory. Use a factor of safety of 2 against rupture.

Mahnoor Amin

Numerade Educator

Problem 78

If the 3 -in-diameter shaft is made from cast fron having tensile and compressive ultimate strengths of $\left(\sigma_{u i}\right)_z=40 \mathrm{ksi}$ and $\left(\sigma_{\mathrm{ut}}\right)_c=80 \mathrm{ksi}$, respectively, determine if the shaft fails according to Mohr's failure criterion.

Chai Santi

Numerade Educator

Problem 79

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 1.5 against yielding.

Chai Santi

Numerade Educator

Problem 80

The shaft consists of a solid segment $A B$ and a bollow 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.

Probs 10-79/80

Chai Santi

Numerade Educator

Problem 81

If $\sigma_Y=50 \mathrm{ksi}$, determine the factor of safety for this state of stress against yielding, based on (a) the maximum shear stress theory and (b) the maximum distortion energy theory.

Mahnoor Amin

Numerade Educator

Problem 82

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.

Mahnoor Amin

Numerade Educator

Problem 83

Solve Proh, 10-82 using the maximum shear stress theory.

Prohs. 10-82/83

Mahnoor Amin

Numerade Educator

Problem 84

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

Mahnoor Amin

Numerade Educator

Problem 85

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

Mahnoor Amin

Numerade Educator

Problem 86

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.

Chai Santi

Numerade Educator

Problem 87

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 88

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.

Chai Santi

Numerade Educator

Problem 89

If the 2 -in.-diameter shaft is made from cast iron having tensile and compressive ultimate strengths of $\left(\sigma_{\text {ut }}\right)_r=50 \mathrm{ksi}$ and $\left(\sigma_{\text {ath }}\right)_c=75 \mathrm{ksi}$, respectively, determine if the shaft fails in according to Mohr's failure criterion.

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 minimum required wall thickness to the nearest millimeter using (a) the maximum shear stress theory and (b) the maximum distortion energy theory. Apply a factor of safety of 1.5 against yiclding.

Chai Santi

Numerade Educator

Problem 91

The element is subjected to the state of stress shown. If the material is machine steel having a yield stress of $\sigma_y=750 \mathrm{MPa}$, determine the factor of safety with respect to yielding using the maximum distortion energy theory.

Mahnoor Amin

Numerade Educator