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Mechanical Behavior of Materials

Norman E. Dowling, Stephen L. Kampe, Milo V. Kral

Chapter 8

Fracture of Cracked Members - all with Video Answers

Educators

Chapter Questions

01:26

Problem 1

Look at Fig. 8.32, and consider the data for this AISI 4340 steel heat treated to yield strengths of 800 and $1600 \mathrm{MPa}$. For each of these yield strengths, calculate the transition crack length $a_{t}$, and comment on the significance of the values obtained.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:34

Problem 2

For the two tempers of $300-\mathrm{M}$ steel in Table 8.1, perform the following tasks:
(a) Calculate the transition crack lengths, and plot stress versus crack length, showing the yielding alone lines, and the fracture alone curves, as in Fig. 8.6.
(b) Then compare these plots, and comment on the engineering use of this steel at these two tempers.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
02:06

Problem 3

For each metal in Table 8.1, do the following:
(a) Calculate the transition crack length $a_{t}$.
(b) Plot these as data points on a logarithmic scale, versus yield strength $\sigma_{o}$ on a linear scale, using different symbols for steels, aluminum alloys, and titanium alloys.
(c) Comment on the values obtained and on any trends with yield strength.

Anand Jangid
Anand Jangid
Numerade Educator
01:32

Problem 4

Using Tables 8.1 and 8.2, perform these tasks:
(a) Calculate transition crack lengths $a_{t}$ for the following materials: steels AISI 1144, ASTM A517, and 300-M (both tempers); aluminum alloys $2219-T 851$ and $7075-\mathrm{T} 651$; polymers ABS and epoxy; soda-lime glass; and the ceramic $\mathrm{Si}_{3} \mathrm{~N}_{4}$. Refer to Table $3.4$ or $3.3$ for tensile properties for the ceramics and polymers. For brittle materials where the yield strength $\sigma_{o}$ is not available, replace it with the ultimate tensile strength $\sigma_{u}$.
(b) Comment on the values obtained and any trends observed for the different classes of material. Which particular materials do you think are likely to be internally flawed?

Manik Pulyani
Manik Pulyani
Numerade Educator
02:31

Problem 5

Define the following concepts in your own words: (a) Modes I, II, and III, (b) crack-tip singularity, (c) stress intensity factor $K$, (d) strain energy release rate $G$, and (e) fracture toughness $K_{I c}$.

Hubert Agamasu
Hubert Agamasu
Numerade Educator
03:22

Problem 6

For center-cracked plates in tension, as in Fig. 8.12(a), accurate values of $F$ from numerical results are given in the Tada (2000) handbook, as listed in Table P8.6.
(a) Compare these values with the expression for $F$ from Fig. 8.12(a) that is recommended for any $\alpha$. What is its accuracy for $\alpha \leq 0.9$ ?
(b,c) Two approximations for $F$ that are sometimes employed for center-cracked plates are
$$
F=\sqrt{\sec \frac{\pi \alpha}{2}}, \quad F=\sqrt{\frac{2}{\pi \alpha} \tan \frac{\pi \alpha}{2}} \quad(\mathrm{~b}, \mathrm{c})
$$ where the arguments of the trigonometric functions are in radians. Compare each of these with the numerical values, and characterize the accuracy of each for $\alpha \leq 0.9$.

Amy Jiang
Amy Jiang
Numerade Educator
02:48

Problem 7

An engineering member is to be made of AISI $4130 \mathrm{steel}\left(\sigma_{u}=1150 \mathrm{MPa}\right)$. The member is a plate loaded in tension, and it may have a crack in one edge as shown in Fig. $8.12(\mathrm{c})$. The dimensions are width $b=50$ and $t=10 \mathrm{~mm}$, and the crack may be as long as $a=8.0 \mathrm{~mm}$. The member must resist a tension force of $P=90 \mathrm{kN}$. Determine: (a) the safety factor against brittle fracture, (b) the safety factor against fully plastic yielding, and (c) the overall (controlling) safety factor.

Chai Santi
Chai Santi
Numerade Educator
01:16

Problem 8

A center-cracked plate of ASTM A470-8 (Cr-Mo-V) steel has dimensions, as defined in Fig. $8.12$ (a), of $b=45$ and $t=5.0 \mathrm{~mm}$, and it is subject to a tension force of $P=75 \mathrm{kN}$.
(a) What are the safety factors against brittle fracture and against fully plastic yielding if the crack length is $a=6.0 \mathrm{~mm}$ ?
(b) Proceed as in (a) but use $a=23 \mathrm{~mm}$.
(c) Assume that this plate is an engineering component and comment on its safety for the two different crack lengths.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:29

Problem 9

A bending member with a rectangular cross section has dimensions, as defined in Fig. $8.13$ (a), of $b=60$ and $t=20 \mathrm{~mm}$, and a through-thickness edge crack of length $a$ is present. The beam is made of 7475 - T7351 aluminum and is subjected to a bending moment of $M=1500 \mathrm{~N} \cdot \mathrm{m}$.
(a) If the crack length is $a=7.0 \mathrm{~mm}$, what is the stress intensity factor $K$ ? What is the safety factor against brittle fracture?
(b) Repeat (a) for a crack length of $30 \mathrm{~mm}$.

João Gabriel Alencar Caribé
João Gabriel Alencar Caribé
Numerade Educator
02:31

Problem 10

For the same bending member with an edge crack as in Prob. 8.9, proceed as follows:
(a) What is the critical crack length for fracture?
(b) What crack length can be allowed if a safety factor of $3.0$ against brittle fracture is required?
(c) For the crack length found in (b), what is the safety factor against fully plastic yiclding?

Km Neeraj
Km Neeraj
Numerade Educator
05:13

Problem 11

A tension member made of 2014-T651 aluminum has dimensions, as defined in Fig. $8.12(\mathrm{c})$, of $b=30$ and $t=4.0 \mathrm{~mm}$. A safety factor of $3.0$ against failure by either brittle fracture or fully plastic yielding is required.
(a) If there is a through-thickness crack in one edge of length $a=6.0 \mathrm{~mm}$, what is the highest tension force $P$ that can be permitted in service.
(b) If the force in service is $P=6.0 \mathrm{kN}$, what is the largest crack length that can safely exist in the member?

Satpal Satpal
Satpal Satpal
Numerade Educator
01:07

Problem 12

A bending member with a rectangular cross section is made of ABS plastic and has dimensions, as defined in Fig. $8.13$ (a), of $b=24$ and $t=8.0 \mathrm{~mm}$. For a safety factor against brittle fracture of $3.5$, what is the largest through-thickness edge crack length $a$ that can be allowed if (a) $M=12.5 \mathrm{~N} \cdot \mathrm{m}$, and (b) $M=3.0 \mathrm{~N} \cdot \mathrm{m}$ ? Finally, (c) what are the safety factors against fully plastic yielding for case (a) and for case (b)?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:09

Problem 13

An engineering member is made of $300-\mathrm{M}\left(650^{\circ} \mathrm{C}\right.$ temper) steel. It is in the shape of a plate loaded in tension and may have a crack in one edge, as shown in Fig. 8.12(c). The dimensions are width $b=110 \mathrm{~mm}$ and thickness $t=20 \mathrm{~mm}$, and the member must resist a tension force of $P=230 \mathrm{kN}$. Determine the length $a$ of the largest edge crack that can be permitted such that the safety factor against brittle fracture is not less than $3.5$, and also the safety factor against fully plastic yielding is not less than $2.5$.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:09

Problem 14

An engineering member is to be made of an aluminum alloy. It is in the shape of a plate loaded in tension that may have a crack in one edge, as shown in Fig. 8.12(c). The dimensions are width $b=30 \mathrm{~mm}$ and thickness $t=4.0 \mathrm{~mm}$, and the crack may be as long as $a=6.0 \mathrm{~mm}$. The member must resist a tension force of $P=7.5 \mathrm{kN}$.
(a) For a safety factor of $2.8$ against brittle fracture, what minimum fracture toughness $K_{I c}$ is required?
(b) For a safety factor of $2.0$ against fully plastic yielding, what minimum yield strength $\sigma_{o}$ is required?
(c) Given your results from (a) and (b), select an aluminum alloy from Table $8.1$ that meets both requirements.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:06

Problem 15

A tension member has width $b=45 \mathrm{~mm}$ and thickness $t=12 \mathrm{~mm}$. An axial force $P$ is applied, and the member may contain an edge crack as deep as $a=7.0 \mathrm{~mm}$, as in Fig. 8.12(c).
(a) Estimate the force $P$ at failure if the material is the ASTM A517-F steel of Table 8.1.
(b) Also estimate the force $P$ at failure if the material is the AISI 4130 steel from Table 8.1.
(c) Which material would be the best choice if the member is to be used in an engineering application? Why?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:07

Problem 16

Bending members, as in Fig. 8.13(a), of depth $b=50 \mathrm{~mm}$ and thickness $t=10 \mathrm{~mm}$ are made of 18-Ni maraging steel (vacuum melted). In service, the bending moment may be as high as $M=3.5 \mathrm{kN} \cdot \mathrm{m}$, and members with edge cracks larger than $a=1.0 \mathrm{~mm}$ are normally found in inspection and scrapped.
(a) Estimate the moment $M$ necessary to cause failure in this situation. What is the safety factor?
(b) Assume that some of these members were accidentally not inspected and found their way into actual service with cracks as large as $a=5.0 \mathrm{~mm}$. Replacement is expensive. Assume that you are the engineer who must make the decision on replacement. What would you decide? Support your decision with additional calculations as needed.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:50

Problem 17

A beam with a rectangular cross section has dimensions, as defined in Fig. 8.13(a), of $b=50$ and $t=20 \mathrm{~mm}$. The beam is made of $2219-\mathrm{T} 851$ aluminum and is subjected to a bending moment of $M=900 \mathrm{~N} \cdot \mathrm{m}$. A through-thickness edge crack of length as large as $a=4.0 \mathrm{~mm}$ may be present. Safety factors of $2.0$ against yielding and $3.5$ against brittle fracture are needed.
(a) Are the safety factor requirements met?
(b) If not, what new beam depth $b$ is needed, assuming that $t$ and the other values given remain unchanged?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:19

Problem 18

For a round shaft with a circumferential crack, as in Fig. 8.14, derive equations for (a) the fully plastic force $P_{o}$, and (b) the fully plastic moment $M_{o}$. Express these as functions of crack length $a$, shaft radius $b$, and yield strength $\sigma_{o}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
06:04

Problem 19

A circular shaft of $50 \mathrm{~mm}$ diameter is subjected to bending and contains a circumferential surface crack of depth $a=5.0 \mathrm{~mm}$, as in Fig. 8.14. The shaft is made of the ASTM A517-F steel of Table 8.1. Estimate the bending moment $M$ that will cause the shaft to fail.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:09

Problem 20

A thin-walled tube, as in Fig. P8.20, is loaded with an internal pressure $p$ and has a longitudinal through-wall crack of length $2 a$. Stress intensity factors for this case from Tada (2000) are
$$
\begin{aligned}
&K=F\left(p r_{\mathrm{avg}} / t\right) \sqrt{\pi a}, \quad \text { where } F=F(\lambda), \quad \lambda=a / \sqrt{r_{\mathrm{avg}} t} \\
&F=\sqrt{1+1.25 \lambda^{2}} \text { for } \lambda \leq 1, \quad \text { and } F=0.6+0.9 \lambda \quad \text { for } 1 \leq \lambda \leq 5
\end{aligned}
$$ The material is titanium $6 \mathrm{Al}-4 \mathrm{~V}$ alloy (annealed), the pressure is $p=20 \mathrm{MPa}$, and the tube dimensions are $r_{\mathrm{avg}}=25$ and $t=2.0 \mathrm{~mm}$. A crack of tip-to-tip length $2 a=10 \mathrm{~mm}$ may be present. What is the safety factor against fracture? What is the safety factor against yielding with no crack present? Is the tube safe to use if failure could present a safety hazard?

Manik Pulyani
Manik Pulyani
Numerade Educator
07:57

Problem 21

A structural member has dimensions and area moment of inertia as shown in Fig. P8.21, and it contains a crack of length $a=15 \mathrm{~mm}$, as also shown. This member is made of A 572 structural steel (Fig. 8.37) and may be subjected in service to dynamic loading at temperatures as low as $-30^{\circ} \mathrm{C}$.
(a) What bending moment about the $x$-axis will cause brittle fracture of the beam, where the sense of the moment is such that the crack is subjected to a tensile stress? (Suggestions: Evaluate $K$ approximately by noting that the cracked flange of the beam is essentially an edge-cracked tension member. Also, verify that $K_{I c} \approx$ $40 \mathrm{MPa} \sqrt{\mathrm{m}}$ from Fig. 8.37.)
(b) The structural design code used for this beam permits a moment of $176 \mathrm{kN} \cdot \mathrm{m}$ to be applied in service, which is based on a safety factor of $1.67$ against yielding. Compare this value with your result from (a) and comment on the difference.

Ajay Singhal
Ajay Singhal
Numerade Educator
01:41

Problem 22

A stiffener in aircraft structure is a T-section, as shown in Fig. P8.22, and is made of 7075 -T651 aluminum. A crack of length $a$ may be present in the bottom of the web as shown. A bending moment of $180 \mathrm{~N} \cdot \mathrm{m}$ is applied about the $x$-axis, such that the crack is subjected to tensile stresses.
(a) To enable stress calculations, determine the $y$-centroid of the T-section and also its area moment of inertia about the centriodal $x$-axis. (Answers: $c=25.36 \mathrm{~mm}$, $\bar{I}_{x}=31,538 \mathrm{~mm}^{4}$.)
(b) If the crack has length $a=1.5 \mathrm{~mm}$, what is the safety factor against brittle fracture?
(c) What is the largest crack length $a$ that can be permitted if a safety factor against brittle fracture of $3.0$ is considered adequate?
(d) Consider the possibility of changing the material to the more expensive $7475-T 7351$ aluminum alloy. What are some possible advantages and disadvantages of making this change? Support your comments with calculations where possible.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
03:17

Problem 23

A tube having inner radius $r_{1}=50 \mathrm{~mm}$ and outer radius $r_{2}=55 \mathrm{~mm}$ is subjected to a bending moment of $8.0 \mathrm{kN} \cdot \mathrm{m}$. It is made of annealed titanium $6 \mathrm{Al}-4 \mathrm{~V}$. As shown in Fig. P8.23, the tube has a through-wall crack of width $2 a=12 \mathrm{~mm}$, located at an angle $\theta=50^{\circ}$ relative to the bending axis. Estimate the safety factor, considering both brittle fracture and fully plastic yielding. Use reasonable approximations as needed to reach a solution.

Dheeraj Sharma
Dheeraj Sharma
Numerade Educator
03:54

Problem 24

A large part in a turbine-generator unit operates near room temperature and is made of ASTM A470-8 steel. A surface crack has been found that is roughly a semi-ellipse, with surface length $2 c=50 \mathrm{~mm}$ and depth $a=15 \mathrm{~mm}$. The stress normal to the plane of the crack is $250 \mathrm{MPa}$, and the member width and thickness are large compared with the crack size. What is the safety factor against brittle fracture? Should the power plant continue to operate if failure of this part is likely to cause costly damage to the remainder of the unit?

Km Neeraj
Km Neeraj
Numerade Educator
01:16

Problem 25

A solid circular shaft $40 \mathrm{~mm}$ in diameter is made of the steel $300-\mathrm{M}\left(300^{\circ} \mathrm{C}\right.$ temper). It is subjected to a bending moment of $3.5 \mathrm{kN} \cdot \mathrm{m}$ and may contain a half-circular surface crack, as in Fig. 8.17(d).
(a) What crack size $a_{c}$ will cause brittle fracture?
(b) What crack size $a$ must be found by inspection to achieve a safety factor in stress of $3.2$ against brittle fracture?
(c) Calculate the ratio of the crack size from (a) to that from (b), and comment on the significance of this value.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:04

Problem 26

A beam with a rectangular cross section, as in Fig. 8.17(c), is made of 2219-T851 aluminum and must withstand a bending moment of $M=160 \mathrm{~N} \cdot \mathrm{m}$. The thickness is $b=10 \mathrm{~mm}$, and a quarter-circular corner crack as large as $a=2.0 \mathrm{~mm}$ may be present.
(a) What beam depth $t$ is required for a safety factor $3.0$ against brittle fracture?
(b) For the beam depth $t$ as calculated in (a), is the design adequate with respect to possible fully plastic failure? (Suggestion: Make a conservative estimate of $M_{o}$ by assuming that the crack extends across the full thickness $b$.)

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:06

Problem 27

A round rod of silicon nitride ceramic is loaded as a simply supported beam under a uniformly distributed force, as in Fig. A.4(b). The rod diameter is $10 \mathrm{~mm}$, the length between supports is $120 \mathrm{~mm}$, and the distributed force is $w=3.0 \mathrm{~N} / \mathrm{mm}$. (a) If a half-circular surface crack as deep as $0.5 \mathrm{~mm}$ may be present, what is the safety factor against brittle fracture?
(b) If a safety factor of $4.0$ is required, what is the largest permissible depth for a half-circular surface crack?

Manik Pulyani
Manik Pulyani
Numerade Educator
02:20

Problem 28

Solid circular shafts made of 18-Ni maraging steel (vacuum melted) are subjected in service to bending, with a moment $M=15.0 \mathrm{kN} \cdot \mathrm{m}$. Half-circular surface cracks, as in Fig. $8.17$ (d), may exist in the part. From nondestructive inspection, it is expected that no cracks larger than $a=5.0 \mathrm{~mm}$ are present.
(a) What shaft diameter is required to resist yielding with a safety factor of $2.0$ if no crack is present?
(b) For an inspection-size crack, what shaft diameter is required to resist brittle fracture with a safety factor of $3.5$ ?
(c) What shaft diameter should actually be used?

Chai Santi
Chai Santi
Numerade Educator
01:06

Problem 29

A shaft of diameter $50 \mathrm{~mm}$ has a circumferential surface crack, as in Fig. 8.14, of depth $a=5.0 \mathrm{~mm}$. The shaft is made of the $18-\mathrm{Ni}$ maraging steel (air melted) of Table 8.1.
(a) If the shaft is loaded with a bending moment of $1.5 \mathrm{kN} \cdot \mathrm{m}$, what is the safety factor against brittle fracture?
(b) If an axial tensile force of $120 \mathrm{kN}$ is combined with the bending moment, what is the safety factor?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:18

Problem 30

A solid circular shaft has a diameter of $60 \mathrm{~mm}$ and is made of 7075 -T651 aluminum. The shaft contains a half-circular surface crack, as in Fig. $8.17(\mathrm{~d})$, of depth $a=10 \mathrm{~mm}$, and it is subjected to a bending moment of $M=500 \mathrm{~N} \cdot \mathrm{m}$. What is the largest axial force $P$ that can be applied along with $M$ such that the safety factor against brittle fracture is not less than $4.0$ ?

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:43

Problem 31

For an eccentrically loaded tension member with an edge crack as in Fig. $8.22$ (left):
(a) Develop an equation for the fully plastic limit load, $P_{o}=f\left(a, b, t, e, \sigma_{o}\right.$ ).
(b) Show that your solution correctly reduces to the equations of Fig. A.16(c) and (d) as special cases.

Chai Santi
Chai Santi
Numerade Educator
04:15

Problem 32

A tension member made of the AISI 4130 steel of Table $8.1$ has dimensions, as defined in Fig. $8.12$ (c), of $b=50$ and $t=9.0 \mathrm{~mm}$. A safety factor of $3.5$ against brittle fracture is required.
(a) If there is a through-thickness crack in one edge of length $a=4.0 \mathrm{~mm}$, what is the highest tension force $P$ that can be permitted in service?
(b) The force $P$ may be off the center of the width dimension $b$ of the member (eccentric) by as much as $5.0 \mathrm{~mm}$. In this case, and for the same $a=4.0 \mathrm{~mm}$, what is the highest force $P$ that can be permitted in service?

Chai Santi
Chai Santi
Numerade Educator
04:33

Problem 33

A tube made of soda-lime glass has an inner radius $r_{1}=38 \mathrm{~mm}$ and a wall thickness $t=4.0 \mathrm{~mm}$, and it is subjected to an internal pressure of $0.56 \mathrm{MPa}$. A half-circular surface crack may be present as shown in Fig. P8.33, and a safety factor of $3.0$ against brittle fracture is required. What is the largest crack depth $a$ that can be allowed to be present?

Supratim Pal
Supratim Pal
Numerade Educator
08:53

Problem 34

For the cracked soda-lime glass tube of Fig. P8.33, the inner radius is $r_{1}=38 \mathrm{~mm}$, the wall thickness is $t=4.0 \mathrm{~mm}$, and the crack depth is $a=2.0 \mathrm{~mm}$. What internal pressure will cause fracture of the tube?

Surjit Tewari
Surjit Tewari
Numerade Educator
02:13

Problem 35

A plate with a round hole made of Ti-6Al-4V (annealed) has dimensions, as defined in Fig. $8.20$, of half-width $b=40 \mathrm{~mm}$, hole radius $c=10 \mathrm{~mm}$, and thickness $t=6.0 \mathrm{~mm}$. A tensile force $P=60 \mathrm{kN}$ is applied, and there are edge cracks on either side of the hole of length $l=0.50 \mathrm{~mm}$. The elastic stress concentration factor from Fig. A.11(a) is $k_{t n}=2.42$. What is the safety factor against brittle fracture? (Suggestion: Start by determining the value of the crack length $l^{\prime}$ where $K_{A}=K_{B}$.)

Manik Pulyani
Manik Pulyani
Numerade Educator
05:48

Problem 36

A cylindrical pressure vessel has an inner diameter of $150 \mathrm{~mm}$ and a wall thickness of $5 \mathrm{~mm}$, and it contains a pressure of $20 \mathrm{MPa}$. The safety factor against yielding must be at least $X_{o}=2$. Also, a leak-before-break criterion must be met, with a safety factor of at least $X_{a}=9$ on crack length, requiring $c_{c} \geq X_{a} t$.
(a) Is the vessel safe if it is made from $300-\mathrm{M}$ steel ( $\left(300^{\circ} \mathrm{C}\right.$ temper)?
(b) From ASTM A517-F steel?
(c) What minimum fracture toughness is required for the material in this application?
(d) What is the safety factor on $K$ relative to $K_{I c}$ due to the $X_{a}=9$ requirement?

Chai Santi
Chai Santi
Numerade Educator
01:53

Problem 37

A thick-walled tube having inner radius $r_{1}=30 \mathrm{~mm}$ and outer radius $r_{2}=50 \mathrm{~mm}$ contains a pressure of $200 \mathrm{MPa}$. It is made of the AISI 4130 steel of Table 8.1. As shown pressure in Fig. P8.37, a longitudinal crack is present, with width $2 c=10 \mathrm{~mm}$ and depth $a=3.0 \mathrm{~mm}$. Estimate the safety factors against brittle fracture and against yielding.

Km Neeraj
Km Neeraj
Numerade Educator
01:19

Problem 38

A disc having inner radius $r_{1}=110 \mathrm{~mm}$ and outer radius $r_{2}=440 \mathrm{~mm}$ rotates at 60 revolutions/sec. It is made of ASTM A470-8 steel. As shown in Fig. P8.38, the disc has a quarter-circular corner crack of depth $a=8.0 \mathrm{~mm}$ at the inner radius. Estimate the safety factors against brittle fracture and against yielding.

Surjit Tewari
Surjit Tewari
Numerade Educator
01:05

Problem 39

Assume that each of the rotor steels of Fig. $8.35$ except A217 is being considered for use at room temperature $\left(22^{\circ} \mathrm{C}\right)$. The design will be such that the highest stress does not exceed half of the respective yield strength in each case. Assuming a flaw geometry that is a half-circular surface crack in a semi-infinite body, determine the largest permissible crack size $a$ for each material if a safety factor of $2.0$ against brittle fracture is required. Also comment on how this information might affect the choice among these steels.

Narayan Hari
Narayan Hari
Numerade Educator
01:07

Problem 40

A bending member has dimensions, as defined in Fig. 8.13(a), of width $b=50 \mathrm{~mm}$ and thickness $t=20 \mathrm{~mm}$. A through-thickness crack in the edge subjected to tension stress may be as long as $a=10 \mathrm{~mm}$. What moment is expected to cause failure if the material is AISI 4340 steel (Fig. 8.32) with a yield strength of (a) $800 \mathrm{MPa}$ and (b) $1600 \mathrm{MPa}$ ? In each case, consider both brittle fracture and fully plastic yielding as possible failure modes. Then (c) comment on whether or not it is beneficial to use the higher strength steel in this case.

Manik Pulyani
Manik Pulyani
Numerade Educator
07:38

Problem 41

Two plates of A533B-1 steel (Fig. 8.33) are butted together and then welded from one side, with the weld only penetrating halfway, as shown in Fig. P8.41. A uniform tension stress is applied during service in a pressure vessel. Considering both brittle fracture and fully plastic yielding as possible failure modes, estimate the strength of this joint, as affected by the cracklike flaw that exists, for temperatures of (a) $-75^{\circ} \mathrm{C}$ and (b) $200^{\circ} \mathrm{C}$. Express your answers as values of gross stress, $S_{g}=P /(b t)$, calculated as if the joint were solid. Properties for $-75^{\circ} \mathrm{C}$ can be read from Fig. $8.33$ as $K_{I c} \approx 52 \mathrm{MPa} \sqrt{\mathrm{m}}$ and $\sigma_{o} \approx 550 \mathrm{MPa}$. For $200^{\circ} \mathrm{C}$, the yield strength is $\sigma_{o}=400 \mathrm{MPa}$ and the (upper shelf) fracture toughness is $K_{I c}=200 \mathrm{MPa} \sqrt{\mathrm{m}}$. The weld metal has similar properties to the plates. Then (c) comment on the suitability of this steel for use at these two temperatures.

Keshav Singh
Keshav Singh
Numerade Educator
01:36

Problem 42

Consider 300-M steel with properties for $650^{\circ} \mathrm{C}$ and $300^{\circ} \mathrm{C}$ tempers as listed in Table 8.1. Solid circular shafts for an engineering application are currently being made from the $650^{\circ} \mathrm{C}$ temper material and have a diameter of $45 \mathrm{~mm}$. The shafts are loaded with a bending moment of $4.0 \mathrm{kN}$.m, and nondestructive inspection assures that there are no cracks deeper than $a=2.0 \mathrm{~mm}$. Also, the design requires a safety factor of at least $2.3$ against yielding.
(a) Is the current design adequate? Assume that any cracks present are half-circular surface cracks as in Fig. 8.17(d).
(b) It has been suggested that weight and cost savings can be realized by changing to the $300^{\circ} \mathrm{C}$ temper material with a higher yield strength, and then using a smaller shaft diameter. What minimum shaft diameter would you recommend for the $300^{\circ} \mathrm{C}$ temper material? Would you recommend a change to the $300^{\circ} \mathrm{C}$ temper material?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:23

Problem 43

Consider rapid cooling (thermal shock) of the glass and ceramics listed in Table P8.43, with additional data in Table 8.2. Sudden cooling of a thin surface layer of material causes a stress, as given by Eq. $5.41$.
(a) Assume that a piece of each material contains a small half-circular surface crack of depth $a=1.0 \mathrm{~mm}$, and calculate the surface temperature change $\Delta T$ necessary to cause fracture for each. Which material is the most resistant to thermal shock? Which is the least?
(b) Apply the method of Section $3.7$ to determine the combination of materials properties giving the function $f_{2}$ that controls the resistance to thermal shock. Rank the materials according to $f_{2}$. Comment on the effects of each of the properties $K_{I c}$, $\alpha, E$, and $v$, and rationalize how each affects the resistance to thermal shock.

Manik Pulyani
Manik Pulyani
Numerade Educator
07:38

Problem 44

A solid round shaft is to be made from the AISI 4340 steel of Fig. 8.32. It must resist a bending moment of $M=3.8 \mathrm{kN} \cdot \mathrm{m}$, with a safety factor of $2.0$ against yielding. Also, a half-circular surface crack of depth $a=1.0 \mathrm{~mm}$ may be present, and a safety factor of $3.0$ against brittle fracture is needed. Some combinations of yield strength and fracture toughness are given in Fig. 8.32.
(a) For material heat treated to a yield strength of $800 \mathrm{MPa}$, what shaft diameter is required to resist yielding if the possible crack is ignored? Also, what shaft diameter is required to resist fracture due to the $1.0 \mathrm{~mm}$ crack.
(b) Which value from (a) should be chosen to avoid failure by either cause?
(c) Repeat (a) for the additional combinations of yield strength and fracture toughness given in Fig. 8.32. What combination of yield strength and shaft diameter gives the most efficient design, such that the diameter, and thus the weight, is minimized?
(d) Repeat the previous analysis for $a=0.50 \mathrm{~mm}$ and for $a=2.0 \mathrm{~mm}$. Is the choice of a yield strength sensitive to the crack size that might be present?

Keshav Singh
Keshav Singh
Numerade Educator
02:08

Problem 45

In Fig. 8.40, the fracture toughness of rolled plates of aluminum alloys is seen to vary with orientation. Explain the physical reasons for this behavior and why the toughness for the $L-T$ orientation is the highest and that for $S-L$ is the lowest.

Lottie Adams
Lottie Adams
Numerade Educator
01:36

Problem 46

Write a paragraph explaining the significance of the data for unirradiated and irradiated A533B-1 steel of Fig. 8.41.

AG
Ankit Gupta
Numerade Educator
02:18

Problem 47

Consider the choice of a steel for an oil pipeline in a cold climate, such as Alaska or Siberia. What are the desirable characteristics of a material for this application? What types of test data should be available on candidate materials to serve as a basis for the decision?

Ethan Brown
Ethan Brown
Numerade Educator
03:38

Problem 48

A shaft of diameter $20 \mathrm{~mm}$ has a circumferential surface crack, as in Fig. 8.14, of depth $a=1.5 \mathrm{~mm}$. The shaft is made of the AISI 4130 steel of Table $8.1$, and it is loaded with a bending moment of $150 \mathrm{~N} \cdot \mathrm{m}$, combined with a torque of $300 \mathrm{~N} \cdot \mathrm{m}$. What is the safety factor against brittle fracture? Noting that $K_{I I I c}$ is unknown, a reasonable and probably conservative assumption is to employ a relationship of the same form as Eq. $8.33$ and assume that $K_{I I I c}=K_{I c} / 2$.

Chai Santi
Chai Santi
Numerade Educator
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Problem 49

For the situation of Ex. $8.4$ under the applied stress given, do the following:
(a) Determine whether or not plane strain applies and whether or not LEFM is applicable.
(b) Estimate the plastic zone size, $2 r_{o \sigma}$ or $2 r_{o \varepsilon}$, whichever applies.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:09

Problem 50

A double-edge-cracked plate of 7075 -T651 aluminum has dimensions, as defined in Fig. $8.12$ (b), of $b=15.9 \mathrm{~mm}, t=6.35 \mathrm{~mm}$, large $h$, and sharp precracks with $a=5.7 \mathrm{~mm}$. Under tension load, failure by sudden fracture occurred at a force of $P_{\max }=55.6 \mathrm{kN}$. Prior to this, there was a small amount of slow-stable crack growth, with the $P-v$ curve being similar to Fig. 8.28, Type I, and crossing the $5 \%$ slope deviation at $P_{Q}=50.3 \mathrm{kN}$.
(a) Calculate $K_{Q}$ corresponding to $P_{Q}$.
(b) At the $K_{Q}$ point, determine whether or not plane strain applies and whether or not LEFM is applicable.
(c) What is the significance of the $K_{Q}$ calculated?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:10

Problem 51

A fracture toughness test was conducted on AISI 4340 steel having a yield strength of $1380 \mathrm{MPa}$. The standard compact specimen used had dimensions, as defined in Fig. 8.16, of $b=50.8 \mathrm{~mm}, t=12.95 \mathrm{~mm}$, and a sharp precrack to $a=25.4 \mathrm{~mm}$. Failure occurred suddenly at $P_{Q}=P_{\max }=15.03 \mathrm{kN}$, with the $P-v$ curve resembling Type III of Fig. 8.28.
(a) Calculate $K_{Q}$ at fracture.
(b) Does this value qualify as a valid (plane strain) $K_{I c}$ value?
(c) Estimate the plastic zone size at fracture.

Narayan Hari
Narayan Hari
Numerade Educator
01:03

Problem 52

Data are given in Table P8.52 for compact specimens of 7075 -T651 aluminum in the same sizes as those photographed in Fig. 8.47. All had dimensions, as defined in Fig. 8.16, of $b=50.8$ and $h=30.5 \mathrm{~mm}$, and initial sharp precracks and thickness as tabulated. For each test, perform the following tasks:
(a) Calculate $K_{Q}$, and determine whether or not $K_{Q}$ qualifies as a valid (plane strain) $K_{I c \cdot}$
(b) Estimate the plastic zone size at $K_{Q}$, using $2 r_{o \sigma}$ or $2 r_{o \varepsilon}$, whichever applies.
(c) Determine whether analysis by LEFM is applicable.
(d) Plot $K_{Q}$ versus thickness $t$, and comment on the trend observed and its relationship to the fracture surfaces in Fig. 8.47.
(e) For each test, employ the crack length $a_{i}$ and Fig. A.16(c) in Appendix A to estimate the fully plastic force. Then compare these values to the highest forces $P_{\max }$ reached prior to fracture. What is the significance of the trend observed?

Hossam Mohamed
Hossam Mohamed
Numerade Educator
04:22

Problem 53

The combinations of crack length and stress corresponding to failure in Fig. $8.5$ are given in Table P8.53.
(a) Plot these data, the line for $\sigma_{o}=518 \mathrm{MPa}$, and the curve for $K_{c}=S \sqrt{\pi a}=$ $66 \mathrm{MPa} \sqrt{\mathrm{m}}$, just as they appear in Fig. 8.5.
(b) Also plot a revised curve for $K_{c}=66 \mathrm{MPa} \sqrt{\mathrm{m}}$, where the plastic zone adjustment is used.
(c) Comment on the success of curve (b) in predicting the behavior.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator