Mechanics of Materials

R. 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., $\epsilon_{x}+\epsilon_{y}=\epsilon_{x^{\prime}}+\epsilon_{y^{\prime}}$.

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 $\epsilon_{x}=-400\left(10^{-6}\right)$ $\epsilon_{y}=860\left(10^{-6}\right), \quad$ and $\quad \gamma_{x y}=375\left(10^{-6}\right) .$ 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 $\epsilon_{x}=150\left(10^{-6}\right), \quad \epsilon_{y}=200\left(10^{-6}\right), \quad \gamma_{x y}=$ $-700\left(10^{-6}\right) .$ Use the strain transformation equations and determine the equivalent in-plane strains on an element oriented at an angle of $\theta=60^{\circ}$ counterclockwise from the original position. Sketch the deformed element within the $x-y$ plane due to these strains.

Mahnoor Amin

Numerade Educator

Problem 7

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

Chai Santi

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 $\epsilon_{x}=180\left(10^{-6}\right), \quad \epsilon_{y}=-120\left(10^{-6}\right), \quad$ 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 $\epsilon_{x}=500\left(10^{-6}\right)$
\[
\epsilon_{y}=350\left(10^{-6}\right), \quad \text { and } \quad \gamma_{x y}=-430\left(10^{-6}\right) . \quad \text { Use } \quad \text { the }
\]
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 $\boldsymbol{\epsilon}_{x}=-400\left(10^{-6}\right), \boldsymbol{\epsilon}_{y}=0, \gamma_{x y}=150\left(10^{-6}\right) .$ 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 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 $\epsilon_{x}=250\left(0^{-6}\right), \epsilon_{y}=300\left(10^{-6}\right),$ and $\gamma_{x y}=-180\left(10^{-6}\right) .$ Use the strain transformation equations to determine (a) the 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 $\epsilon_{x}, \epsilon_{y},$ and $\gamma_{x y}$ are known. Write a computer program that can be used to determine the normal and shear strain, $\epsilon_{x^{\prime}}$ 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.

Chai Santi

Numerade Educator

Problem 16

$* 10-16 .$ 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.

Hast Aggarwal

Numerade Educator

Problem 17

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

Hast Aggarwal

Numerade Educator

Problem 18

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

Chai Santi

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Problem 19

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

Chai Santi

Numerade Educator

Problem 20

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

Hast Aggarwal

Numerade Educator

Problem 21

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

Chai Santi

Numerade Educator

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

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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 $A,(\mathrm{b})$ 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 $\epsilon_{x}=480\left(10^{-6}\right), \epsilon_{y}=720\left(10^{-6}\right), \quad \gamma_{x y}=$ $650\left(10^{-6}\right) .$ 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

10-26. 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 $\epsilon_{c}=-360\left(10^{-6}\right) .$ Determine the in-plane principal strains.

Mahnoor Amin

Numerade Educator

Problem 27

10-27. The $60^{\circ}$ 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

$* 10-28 .$ 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 $\theta_{a}=40^{\circ}$ $\epsilon_{a}=160\left(10^{-6}\right), \quad \theta_{b}=125^{\circ}, \quad \epsilon_{b}=100\left(10^{-6}\right), \quad \theta_{c}=220^{\circ}$ $\boldsymbol{\epsilon}_{c}=80\left(10^{-6}\right)$

Chai Santi

Numerade Educator

Problem 30

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

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

The principal plane stresses and associated strains in a plane at a point are $\sigma_{1}=36$ ksi, $\sigma_{2}=16$ 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.

Chai Santi

Numerade Educator

Problem 33

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

Mahnoor Amin

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Problem 34

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

Chai Santi

Numerade Educator

Problem 35

10-35. The polyvinyl chloride bar is subjected to an axial force of 900 lb. If it has the original dimensions shown, determine the value of Poisson's ratio if the angle $\theta$ decreases by $\Delta \theta=0.01^{\circ}$ after the load is applied. $E_{\mathrm{pvc}}=800\left(10^{3}\right) \mathrm{psi}$

Mahnoor Amin

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Problem 36

*10-36. The spherical pressure vessel has an inner diameter of $2 \mathrm{m}$ and a thickness of $10 \mathrm{mm} .$ A strain gage having a length of $20 \mathrm{mm}$ is attached to it, and it is observed to increase in length by $0.012 \mathrm{mm}$ 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 $E_{\mathrm{st}}=200 \mathrm{GPa}$ and $\nu_{\mathrm{st}}=0.3$

Mahnoor Amin

Numerade Educator

Problem 37

10-37. Determine the bulk modulus for each of the following materials:
(a) rubber, $E_{\mathrm{r}}=0.4 \mathrm{ksi}, \nu_{\mathrm{r}}=0.48$ and (b) glass, $E_{\mathrm{g}}=8\left(10^{3}\right) \mathrm{ksi}, \nu_{\mathrm{g}}=0.24$

Mahnoor Amin

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

Mahnoor Amin

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Problem 39

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. $E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi}, \nu_{\mathrm{al}}=0.33$ Hint: See Prob. $10-30$

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 $\epsilon_{x}=200\left(10^{-6}\right) .$ Determine the applied load $P .$ What is the shear strain $\gamma_{x y}$ at point $A ?$

Chai Santi

Numerade Educator

Problem 41

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

Chai Santi

Numerade Educator

Problem 42

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

Mahnoor Amin

Numerade Educator

Problem 43

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

Mahnoor Amin

Numerade Educator

Problem 44

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

Mahnoor Amin

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 $\nu$

Hast Aggarwal

Numerade Educator

Problem 46

A single strain gage, placed in the vertical plane on the outer surface and at an angle $60^{\circ}$ to the axis of the pipe, gives a reading at point $A$ of $\epsilon_{A}=-250\left(10^{-6}\right) .$ Determine the principal strains in the pipe at this point. The pipe has an outer diameter of 1 in. and an inner diameter of 0.6 in. and is made of $\mathrm{C} 86100$ bronze.

Mahnoor Amin

Numerade Educator

Problem 47

A single strain gage, placed in the vertical plane on the outer surface and at an angle of $60^{\circ}$ to the axis of the pipe, gives a reading at point $A$ of $\epsilon_{A}=-250\left(10^{-6}\right)$ Determine the vertical force $P$ if the pipe has an outer diameter of 1 in and an inner diameter of 0.6 in. The pipe is made of $C 86100$ bronze.

Hast Aggarwal

Numerade Educator

Problem 48

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

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 $\sigma_{x}$ and $\sigma_{y}$ in the plate if the temperature is increased by $\Delta T=100^{\circ} \mathrm{F} .$ Hint: To solve, add the thermal strain $\alpha \Delta T$ to the equations for Hooke's law.

Mahnoor Amin

Numerade Educator

Problem 50

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

Mahnoor Amin

Numerade Educator

Problem 51

The shaft has a radius of $15 \mathrm{mm}$ and is made of $\mathrm{L} 2$ 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 $60 \mathrm{kN}$. 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 $100 \mathrm{mm} ?$ The inner radius of the vessel is $400 \mathrm{mm},$ and its thickness is $10 \mathrm{mm} . E_{\mathrm{st}}=200 \mathrm{GPa}$ and $\nu_{\mathrm{st}}=0.3$

Mahnoor Amin

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 $\nu,$ determine the strains in the circumferential and longitudinal directions. Using these results, calculate the increase in both the diameter and the length of a steel pressure vessel filled with air and having an internal gage pressure of 15 MPa. The vessel is $3 \mathrm{m}$ long, and has an inner radius of $0.5 \mathrm{m}$ and a thickness of $10 \mathrm{mm} . E_{\mathrm{st}}=200 \mathrm{GPa}, \nu_{\mathrm{st}}=0.3$

Mahnoor Amin

Numerade Educator

Problem 57

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

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 $\boldsymbol{\epsilon}_{x}=0$ 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 $\nu=0.3$ for the material.

Mahnoor Amin

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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

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 $\sigma_{1}$ and $\sigma_{2}=-0.5 \sigma_{1},$ determine the magnitude of $\sigma_{1}$ that will cause yielding according to the maximum shear stress theory.

Chai Santi

Numerade Educator

Problem 62

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

Chai Santi

Numerade Educator

Problem 63

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

Mahnoor Amin

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Problem 64

Solve Prob. $10-63$ using the maximum distortion energy theory.

Mahnoor Amin

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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

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

Mahnoor Amin

Numerade Educator

Problem 67

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

Mahnoor Amin

Numerade Educator

Problem 68

If the material is machine steel having a yield stress of $\sigma_{Y}=700$ 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 $50 \mathrm{mm}$ is subjected to a torque of $500 \mathrm{N} \cdot \mathrm{m}$ and an axial compressive force of $2 \mathrm{kN}$. Determine if it fails according to the maximum normal stress theory. The ultimate stress of the concrete is $\sigma_{\mathrm{ult}}=28 \mathrm{MPa}$.

Chai Santi

Numerade Educator

Problem 70

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 $T$

Chai Santi

Numerade Educator

Problem 71

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

Mahnoor Amin

Numerade Educator

Problem 72

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

Mahnoor Amin

Numerade Educator

Problem 73

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

Mahnoor Amin

Numerade Educator

Problem 74

If a machine part is made of titanium (Ti-6A1-4V) and a critical point in the material is subjected to plane stress, such that the principal stresses are $\sigma_{1}$ 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 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 $\sigma_{Y}=700 \mathrm{MPa}$.

Mahnoor Amin

Numerade Educator

Problem 76

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

Mahnoor Amin

Numerade Educator

Problem 77

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

Mahnoor Amin

Numerade Educator

Problem 78

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

Mahnoor Amin

Numerade Educator

Problem 79

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

Mahnoor Amin

Numerade Educator

Problem 80

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

Mahnoor Amin

Numerade Educator

Problem 81

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

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 Prob. $10-82$ using the maximum shear stress theory.

Mahnoor Amin

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 1.5 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 $\mathrm{A}-36$ 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

$\mathbf{1 0 - 8 9} .$ If $\sigma_{Y}=50 \mathrm{ksi},$ determine the factor of safety for this loading based on (a) the maximum shear stress theory and (b) the maximum distortion energy theory.

Mahnoor Amin

Numerade Educator

Problem 90

The gas tank is made from A-36 steel and has an inner diameter of $1.50 \mathrm{m}$. If the tank is designed to withstand a pressure of 5 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 $2300 \mathrm{lb} \cdot \mathrm{ft}$ a bending moment of $1500 \mathrm{lb} \cdot \mathrm{ft}$, and an axial thrust of $2500 \mathrm{lb}$. If the yield points for tension and shear are $\sigma_{Y}=100 \mathrm{ksi}$ and $\tau_{Y}=50 \mathrm{ksi},$ respectively, determine the required diameter of the shaft using the maximum shear stress theory.

Hast Aggarwal

Numerade Educator

Problem 92

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

Mahnoor Amin

Numerade Educator