Introduction to Solid Mechanics

Irving H. Shames, James M. Pitarresi

Chapter 13 SINGULARITY FUNCTIONS - all with Video Answers

Educators

Chapter Questions

Problem 1

What is the maximum shear stress for the solid shaft shown ia Fig. P.14.1? What is the total twist at $A$ ? Determine the torsional spring constant and the torsional strength of the shaft. (Take $G=1 \times 10^{11} \mathrm{~Pa}$.) Length $L$ Is 3 m .
Figure P.I4.1.

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Problem 2

In Problem 14.1, determine the amount of torque that can be applied to the shaft if the yield stress is 2.8 $\times 10^3 \mathrm{~Pa}$ and if a safety factor of 2 is employed.

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Problem 2

In Fig. P. 14.5 is shown a solid shift. Find the length of an equivalent shaft of the same material having the same torsional stiffoess. The diameter of the equivalent shaft is to be 75 mm .

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Problem 3

Evaluate the maximum torsional shear stress for the hollow shaft shown in Fig P.143. Determine the angle of twist at a position 3 ft from the left end. (Take $G=$ $15 \times 10^6$ psi.)
Figure P.14.3.

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00:54

Problem 4

Shown in Fig. P. 14.4 is a solid shaft of two sections welded together at $A$. The materials for the section have different shear moduli. What is the twist at $B$ at the end of the shaft?

Hast Aggarwal

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Problem 6

Plot $\tau_{\text {rasi }}$ versus $x$ in the series of shafts welded together as shown in Fig. P.14.6. Disregard stress concentrations at the juncture of the matcrials. What is the total twiss the left ead? Take the following data:
$$
\begin{aligned}
& G_A=10 \times 10^6 \mathrm{psi} \\
& G_s=20 \times 10^6 \mathrm{psi} \\
& G_C=15 \times 10^6 \mathrm{psi}
\end{aligned}
$$

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Problem 7

For the system shown in Fig. P.14.7,
(a). Find the supporting torques at $A$.
(b). Find the magnitude of the twist $|\Lambda| \|_{\mid}$at $B$.
Fgure P.14.7.

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01:12

Problem 8

A turbine is developing 7500 kW at 6000 rpm (see Fig, P.14.8). For an allowable shear stress of $1 \times 10^8 \mathrm{~Pa}_3$, what should be the diameter $D_1$ of the shaft transmitting torque from the turbine to the reduction gear system? If the reduction in speed is 100 to 1 , what should be the diameter $D_2$ of the shaft between the reduction gears and the gencrator?
Figure P.148.

Hast Aggarwal

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02:29

Problem 10

Deternine the horsepower transmitted by a $10-\mathrm{in}$. solid shaft of length 10 ft if it is twisted an angle of $2^{\circ}$ over its leagth. Compute the maximum shear stress. Take $G$ $=15 \times 10^6 \mathrm{psi}$. The shaft rotates at 150 rpm .

Anand Jangid

Numerade Educator

02:18

Problem 11

A torque $T$ of $150 \mathrm{~N}-\mathrm{m}$ is transmitted by a thaft $A$ (see Fig. P. 14.11) at an angular speed $\omega_A$ of 500 rpm . The diameter of shaft $A$ is .15 m . If the same maximum stress is to be maintained in shaft $B$ as in shaft $A$, what should its diameter be? Take $D_1=.6 \mathrm{~m}$ and $D_2=3 \mathrm{~m}$.
Figure P.14.11.

Kratika Bhadauria

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01:44

Problem 12

What is the limiting speed of a shaft transmitting $7.5 \times 10^5 \mathrm{~W}$ if the allowable stress is $1.4 \times 10^8 \mathrm{~Pa}$ and the diameter is .15 m ? If $G=1.4 \times 10^{11} \mathrm{~Pa}$, what is the minimam kagth to allow for a rotation of $6^{\circ}$ between the ends?

Hast Aggarwal

Numerade Educator

01:03

Problem 12

If the allowable shear stress is $1 \times 10^8 \mathrm{~Pa}$ for a hollow shaft having an outside diameter of .6 m and an inside diameter of 3 m , what is the maximum twist over a length of 4.5 m for $G=1 \times 10^{11} \mathrm{~Pa}$ ? What is the rate of twist?

Kratika Bhadauria

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Problem 13

Torques $C_1$ and $C_2$ are applied to two shafts connected to two gears as shown in Fig. P.14.13. End $A$ is connected through a rubber grommet to ground.
(a). Determine the rotation of end $B$ as a result of the torques.
(b). Determine the rotation of the upper shaft.

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Problem 14

Compute the supporting torques and the rotation of the left end of the shaft in Fig. P.14.14.

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02:29

Problem 15

Determine the horsepower transmitted by a $8-\mathrm{in}$. solid shaft of length 5 ft if it is twisted a tital angle of $1^*$ over its length. Compute the maximum shear stress. Take $G$ $=10 \times 10^6$ psi. The shaft rotates at 200 rpar .

Anand Jangid

Numerade Educator

Problem 16

Find the supporting torque $A$ in Fig. P.14.16. At $B$ the shaft is bonded to a rubber grommet giving a torsional stiffaess of $10^6 \mathrm{ft}-\mathrm{lb} / \mathrm{rad}$. Sbaft $C$ is steel with $G=10 \times$ $10^{\circ}$ psi. Tube $D$ is welded to shaft $C$ at $E$ through a rigid plate and to the base at $\boldsymbol{A}$. It is also steel with the same $G$ as $C$
Figure P.14.16.

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Problem 17

Find rotation at $A$ from the $500-\mathrm{ft}-\mathrm{lb}$ torque (see Fig. P.14.17). Take $G=15 \times 10^6$ psi for the shafts. Gears connect the two shafts with each otber.
Figure P.14.17.

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01:13

Problem 18

A sleeve is welded to a shaft at $C$ in Fig. P14.18. What is the twist angle at $D$ from the $500-\mathrm{N}-\mathrm{m}$ torque on sleeve CD? Use $G=10 \times 10^{11} \mathrm{~Pa}$ as the shear modulas for both sleeve and shaft. Neglect twisting deformation of that part of $D$ which is perpendicular to $A B$ at $C$.

Hast Aggarwal

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Problem 19

Find the supporting torques in the shaft shown in Fig. P.14.19. Take $G=15 \times 10^{\circ}$ psi. Make a simple plot of $\phi(x)$ and give maximum value of $\phi$. Note the $1^{1 "}$ cylindrical hole.
Figure P.14.19.

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Problem 20

An aluminum shaft $A$ is fixed to the wall at $B$ and is cronected to a steel sleeve $E$ through a rubber elastic grommet at $C$ in Fig. P.14.20. Find the rotation at $H$ due to the torque $T=500 \mathrm{~N}-\mathrm{m}$. Use the following data:
Figure P. 14.20.

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02:24

Problem 21

In Fig. P.14.21 is shown a shaft composed of an inside solid rod baving $G=7 \times 10^{10} \mathrm{~Pa}$ and an outer sleeve having $G=1.4 \times 10^{11} \mathrm{~Pa}$. What is the maximum stress wben shafts are connected at the right end only?

Chai Santi

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02:36

Problem 22

A steel shaft (Fig. P.14.22) having diameter $D_1$ $=6$ in. and an outer sleeve of aluminum having an inner diameter of 12 in . and a thickness $t$ of 2 in . is held by rigid end plates on which a torque $T$ of $5000 \mathrm{ft}-\mathrm{lb}$ is applied. What are the largest torsion strewses in each material? Take $L=10 \mathrm{ft}$, $G_{\mathrm{sa}}=15 \times 10^6$ pri and $G_{\mathrm{AL}}=10 \times 10^6 \mathrm{psi}$.
Figure P. 1422

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Problem 23

A steel shaft $A$ and an aluminum sleeve $B$ are connected at the right end (see Fig, P.14.23) by a rigid support and at the left end by a rigid plate C. A torque $T_1=500$ $\mathrm{ft}-\mathrm{lb}$ is applied to the plate and a second torque $T_2=800 \mathrm{ft}-\mathrm{lb}$ is applied to the sleeve as shown. What is the maximum torsional shear stress in each member'? What is the angle of twist at the rigid plate? Take $G_A=15 \times 10^6$ psi and $G_B=10$ $\times 10^{\circ}$ psi. (Hint: Consider free-body diagrams of sleeve $B$, shaft $A$, and end plate $C$.)
Figure P. 14.23.

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01:22

Problem 24

Consider the close-coiled spring shown in Fig. P.14.24. Show that if the shear force in the coil is sesumed to give a uniform stress, the maximum shear stress in the coil is given as
$$
\tau_{\operatorname{man}}=\frac{P}{A}+\frac{P R r}{J}
$$
where $A$ is the cross-sectional area of the wire making up the spring and $J$ is its polar moment of aren. Next, show that for $n$ complete coils of the spring, the elongation $\delta$ of the spring resulting from twisting of the wire is given as
$$
\delta=\frac{4 P R^1 n}{G r^4}
$$

What other deformation contributions give rise to further deflection?

Chai Santi

Numerade Educator

01:23

Problem 25

A steel rod having a shear modulus of $1 \times 10^{11}$ Pa is used to form a spring having a mean radius of 75 mm . The diameter of the

Narayan Hari

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01:05

Problem 25

m . What is the spring constant $K$ if the rod is 1.5 m long? See Problem 14.24 before solving this problem.

Raj Bala

Numerade Educator

01:22

Problem 26

A very good formulation for the maximum shear stress in helical springs (see Fig. P.14.24) is the Wahd formula, given as follows:
$$
\tau_{\max }=\frac{2 P R}{\pi r^3}\left[\frac{4(R / r)-1}{4(R / r)-4}+\frac{615}{R / r}\right]
$$

Determine the shear stress in Problem 14.25 for a $450-\mathrm{N}$ load using the approximate formula of Problem 14.24 and the Wahl formula. What is the percentage crror for the approximate formula?

Chai Santi

Numerade Educator

02:46

Problem 27

Considering torsional stresses only, compute the maximum shear stress in a "slinky" toy shown in Fig. P. 14.27 when it is hanging vertically under its own weight. A typical slinky weighs about $1 / 2 \mathrm{lb}$, has about 90 coils, and has a diameter $D$ of about 2.7 in . The rectangular cross section of the wire itself is $b=.01 \mathrm{in}$. and $a=0.025 \mathrm{in}$. For a rectangular cross-section, we will learn in Section 14.7 that $\tau_{\text {max }}=$ $\mathrm{M}_k / \mathrm{kba}^2$. Here, $\kappa \propto .280$.

Hubert Agamasu

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01:26

Problem 28

Shafts 1 and 2 shown in Fig. P.14.28 are circular shafts and have the same diameter. Investigate the effect of doubling the diameter of shaft 1 on the reactions.
Figure P.14.28.

Surendra Kumar

Numerade Educator

Problem 29

A shaft $B$ is shrunk-fit over a shaft $A$ (Fig. P.14.29) so as to transmit torque without slippage. Shaft $A$ has a value of $G_A=15 \times 10^6 \mathrm{psi}$, while skaft $B$ has a modulus $G_8=20 \times 10^6 \mathrm{psi}$. What is rate of twist a for a torque $T=$ $1000 \mathrm{ft}-\mathrm{lb}$ applied at the end as showa? Find the rauximum shear stress in each material. See result from Problem 14.62.
Figure P.14.29.

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Problem 30

What is the twist at the end of the shaft shown in Fig. P. 14.30 (the sleeve and inside core are made of different materials and are entirely fastened together), when $G$ for the sieeve is $1 \times 10^{11} \mathrm{~Pa}$ and $G$ for the core is $7 \times 10^{16} \mathrm{~Pa}$ ? Sce result from Problem 14,62.

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Problem 31

A $\operatorname{rod} A B$ is welded onto a solid cylindrical cantilever beam CB in Fig. P.14.31. What is the maximum principal stress at point $a$ in the cantilever beam? Neglect the weights of $C B$ and $A B$. Note: Point $a$ is on top of cantilever.

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00:41

Problem 32

Standard $10-\mathrm{in}$. steel pipe (see Appepdix IV-F) is shown in Fig. P. 14.32 . It is full of water. Show Mohr's circle for stresses parallel to the $x z$ plane at point $A$ in the diagram. What is the extremal value of shear stress parallel to the $x z$ plane for this point? Include the weight of the piping in yoar calculations.
Figure P.14.32.

Hast Aggarwal

Numerade Educator

02:28

Problem 33

For Fig. P.14.33, determine the principal stresses at point $a, 5$ ft from the support.

Hast Aggarwal

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01:33

Problem 34

In Problem 14.33, evaluate the total downward deflection of point $B$. (The shaft has a Young's modulus of 30 $\times 10^6$ psi and a Poisson ratio of 3 . The rod connecting the load to the shaft has a diameter of 1.5 in . and a Young's modalus of $30 \times 10^6 \mathrm{psi}$ )

Dominador Tan

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05:18

Problem 35

A thin-walled pipe system is shown in Fig. P. 14.35 supporting a force $P$. If $P$ is 5000 lb , what should the thickness $t$ be for a tensile yield stress of 50,000 pai and a safety factor of 17 Use the octahedral shear criterion. Do not consider weight of the pipe.
Figure P. 14.35 .

Chai Santi

Numerade Educator

Problem 36

What standard-size pipe should you employ in Fig. P. 14.36 for a tensile yield stress of 80,000 psi and a safety factor of 17 Do not coasider the weight of the pipe. Use Tresca yield criterion.

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01:34

Problem 37

Shaft $A B$ is supported by four beams all welded together at $B$ as shown in Fig. P.14.37. Each beam is of length $L_1$ and has a bending rigidity EI. The shaft is of length $L_2$ and has a torsional rigidity $G J$. What is the total twist at $A$ ?
Figure P.14.37.

Chai Santi

Numerade Educator

05:23

Problem 38

Redo Problen 14.37 , however, with the beams now fixed at the ends as shown in Fig. P.14.38.
Figure P.14.38.

Vipender Yadav

Numerade Educator

Problem 39

In Fig. P. 14.39 is shown a section of a thinwalled member made from an aluminum alloy. If the thickness is .050 in., what is the stress and rate of $t$ wist for a torque of 1000 in-lb? Take $G$ for this material as $4 \times 10^6$ psi. Assume no buckling.

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Problem 40

Shown in Fig. P. 14.40 is the cross section of a thin-walled member which is part of an airplane wing. If the allowable shear stress is $5.5 \times 10^7 \mathbf{P a}$, find the allowable shear flow for this section and the stresses away from corners. Sides $A B$ and $C D$ have a thickness of 1 mm and sides $A C$ and $B D$ have a thickness of 2 mm . The mean height of the section is 265 mm and $G$ for the material is $2.8 \times 10^{10} \mathrm{~Pa}$. Assume no buckling. Find the allowable torque transmitted and the rate of twist.
Figure P.14.40.

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03:21

Problem 41

Shown in Fig. P. 14.41 is a thin-walled member having the shape of an ellipse. For $G=5 \times 10^6$ psi, determine the shear stress and rate of twist for an applied torque of 500 in .-lb. The wall is .050 in . thick.
Figure P.14.41.

Anand Jangid

Numerade Educator

01:06

Problem 42

A torque of 3300 N -m is transmitted through a $50-\mathrm{mm}$-diameter solid shaft having a yield stress in shear of $1.1 \times 10^4 \mathrm{~Pa}$. Is there plastic deformation for this loading? If so, how far has the plastic action penetrated from the surface assuming elastic, perfectly plastic behavior? What is the maximum possible static moment?

Hast Aggarwal

Numerade Educator

01:50

Problem 43

Consider a hollow steel shaft with a 130 -mm outside diameter and a 70 -mm inside diameter. What is the maximum torque that this shaft can transmit if the shear yield stress is $1.4 \times 10^8 \mathrm{~Pa}$ and the behavior is elastic, perfectly-plastic?

Hast Aggarwal

Numerade Educator

00:58

Problem 44

Compute the torsional stiffness of the hollow shaft shown in Fig. P.14.44.

Hast Aggarwal

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Problem 45

Compare the torsional stiffness of the shapes shown in Fig. P.14.45. Each has a cross-sectional area of 5 in. ${ }^2$
Circle
$2: 1$
Ellipwe
Square
Equilateral triangle
Figure. P.1445.

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01:40

Problem 47

Compare the torsional stiffness of the two shapes shown in Fig. P.14.46.
(a)
(b)
Figure P.14.46.

Compute the torsional stiffpess of the beam shown in Fig. 14.27(c) of Example 14.12. Compare the value obtained with that of the example.

Kratika Bhadauria

Numerade Educator

Problem 48

Determine the resisting torques at the supports for the shaft shown in Fig. P14.48 using the second Castigliano theorem.

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Problem 49

Determithe the relationship between the force $F$ and vertical deflection $\Delta$ for the split ring londed as shown in Fig. P.14.49. Assume that the flexural rigidity is $E I$ and that the torsional rigidity is $G K_r$. Make use of the second Castigliano theorem.
Figure P.14.49.

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02:09

Problem 50

Two solid shatts 1 and 2 (see Fig. P.1450) are rigidly connected to a hollow shaft 3 . Shaft 2 is fixed at the right end, while shaft 3 is fixed at the left end. A torque $T=500 \mathrm{ft}$-ib is applied to the end of the shaft 1 . What are the supporting torques? Take the same $G$ for all shafts.
Figure P.14.50.

Hast Aggarwal

Numerade Educator

Problem 51

Do Problem 14.50 for the case of a single applied torque $T_A=500 \mathrm{ft} \cdot \mathrm{lb}$ clockwise at $A$ as you observe along the shaft from left to right. The torque is applied on the connecting plate between shaft 3 and shafts 1 and 2 .

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Problem 52

A shaft having a diameter $D$ of 3 in. (see Fig. P.14.52) is acted on by a torque $T$ of $300 \mathrm{ft}-\mathrm{lb}$ at one end, while at the other end is consected by a rubber material to a fixed rigid disk. The latter support gives a linear, elastic torsional spring constant $K_T$ to the end having the value of 700 ft -lbideg. What is the total twist $\phi$ at end $\boldsymbol{A}$ and the relative twist $\phi_{A B}$ between eads $A$ and $B$ ? The length $L$ is 10 ft and $G=10 \times 10^6 \mathrm{psi}$.
Figure P.14.52.

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Problem 53

A torque $T$ of 675 N - m is applied to a shaft as shown in Fig. P.14.53. The shaft is attached to two rubber grommets, which in turn are attached to fixed disks. Each gives a linear, elastic torsional spring constant $K_T=550 \mathrm{~N}$-m/deg. If $G=1$ $\times 10^{11} \mathrm{~Pa}$, find the twist of end $A$ and the relative twist between support 1 and support 2 .
Figure P.14.53.

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01:40

Problem 54

Compare the torsional stiffness of the hollow box shaft shown in Fig. P. 14.54 by using Bredt's fornula and by using the approximate method presented in Section 14.7.

Kratika Bhadauria

Numerade Educator

Problem 55

Find the supporting torques at $A$ and $B$ (see Fig P.14.55). What is the rotation of the shaft at A? Take $G=1.2$ $\times 10^{11} \mathrm{~Pa}$. What is the maximum sleear stress in the shaft? Sketch $\phi$. vs. position along skaft.

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Problem 56

Find the supporting torque at $A$ in Fig, P.14.56. At end $B$, the shaft is held by a rubber bushing giving a resisting torque of $10,000 \mathrm{ft}-\mathrm{lb}$ per radian. Take $G=30 \times 10^6$ psi for the shatt.
Figure P.14.56.

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01:55

Problem 57

What are the supporting torques in terms of $d$ and $t$ at $A$ and $B$ (see Fig, P.14.57)? Determine the length of a drilied hole at a function of diameter to have the torques at $A$ and $B$ equal. Do this for $t<1.5 \mathrm{~m}$. Determine $l$ for $d=60$ min from your equation for this case.
Figure P. 14.57.

Anand Jangid

Numerade Educator

00:54

Problem 58

The shaft shown in Fig. P. 14.58 is a shaft on which is applied a linearly varying torque distribution. What is the twist at end $A$ of the shaft? The digmeter of the shaft is .1 in and the modulus of shear is $1 \times 10^{11} \mathrm{~Pa}$. Assume that the theory developed thus far is valid for nonuaiform torques.

Hast Aggarwal

Numerade Educator

04:05

Problem 59

If the shaft in Problem 14.58 is loaded as shown in Fig. P.1459, would you expect the same twist at end $A$ of the shaft? Check to see if your supposition is correct. Soe comment at ead of Problem 14.58.

Ajay Singhal

Numerade Educator

Problem 60

Do Problem 14.59 assuming this time that the loading varies parabolically.

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Problem 61

Suppose in Problem 14.59 that the right side of the shaft were fixed in a rigid support. Determine the torques at the supports to resist the linearly varying load.

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01:29

Problem 62

Several shafts are shrink-fitted over a solid shaft (see Fig. P.14.62) so as to form a combined member which can transmit torque with no slippage between the members. Explain why the assumption of cross sections rotating as platelets still applies. If each material is linear and elastic, show using equilibrium, Hooke's law, and the assumption above that
$$
\begin{aligned}
T & =\alpha\left[\left(\int_0^{R_1} r^2 d A\right) G_1+\left(\int_{R_2}^{R_2} r^2 d A\right) G_2+\left(\int_{2_3}^{E_3} r^2 d A\right) G_3\right] \\
& =\alpha\left[G_2 J_1+G_2 J_2+G_3 J_3\right]
\end{aligned}
$$
where $\alpha$ is the rate of twist.

Hast Aggarwal

Numerade Educator

01:21

Problem 63

What is the maximum torque that can be applied to a $\frac{1}{d}$-in. bolt without yielding? Take the yield stress of the material as 36,000 psi.

Prabhu Ramji

Numerade Educator

06:07

Problem 64

You are to design for an automobile a hollow shaft baving an outside diameter of 75 mum to transmit a maximum torque of $1000 \mathrm{~N}-\mathrm{m}$. What is the thickness $t$ for a steel shaft with a yield stress of $3.5 \times 10^8 \mathrm{~Pa}$ ? Employ a safety factor of 5 .

Satpal Satpal

Numerade Educator

02:09

Problem 65

Stown in Fig. P.14.65 are two shafts connected by rigid gears. Determine the maximum shear stress in each shaft and the twist at $A$. (The lower shaft is supported by two bearings.)
Figure P.14.65.

Hast Aggarwal

Numerade Educator

Problem 66

What is the twist at $A$ for the shaft showe in Fig. P.14.66? The diameter varies linearly as indicated. Assume that the simple theory for uniform shatis is valid locally here. (Take $G=20 \times 10^6 \mathrm{psi}$.)

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01:26

Problem 67

In Broblem 14.66, assume that a uniform hoic 1 in. in diameter has been bored out of the center of the shaft. Compute the maximum stress.

Surendra Kumar

Numerade Educator

Problem 68

A shaft is formed by laminating very thin cylinders as shown in Fig P.14.68. The inside diameter (i.e., of the smallest cylinder) is 1 in. and the outside diameter is 3 in . We shall assume that the shear moduli of the larninae are different and can be expressed as a function of $r$ as follows:
$$
G=G_1(1+2 r)
$$

Determine the maximum shear stress and the twist at the end of the shaft for $G_1=15 \times 10^6 \mathrm{psi}$.
Figure P.14.68.

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Problem 69

Determine the resisting torques at the support for the shaft shown in Fig. P.14.69. (Take $G=10 \times 10^{\circ}$ psi.)
Figure P. 14.69.

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Problem 70

Determine the resisting torques at the supports for the shaft shown in Fig. P.14.70.
Figare P. 14.70.

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01:29

Problem 71

In Fig. P. 14.71 is shown a shaft made by welding two sections together. The moduli of the materials comprising the sections are given in the diagram. Determine the resisting torques at the supports.

Hast Aggarwal

Numerade Educator

01:55

Problem 72

Two solid shafts 1 and 2 (see Fig, P.14.72) are rigidly connected to a hollow shaft 3 . Shaft 2 is fixed to the right end, while shaft 3 is fixed at the left ead. A torque $T=500 \mathrm{ft}-1 \mathrm{~b}$ is applied to the end of shaft 1 . What are the supporting torques? Take the same $G$ for all shafts. Use energy method and compare with results of Probiem 14.50.
Figure P. 14.72

Hast Aggarwal

Numerade Educator

Problem 73

What are the principal stresses at the top of section $A$ in Fig. P.14.73 as a result of the force $P$ given as follows:
$$
P=500 \mathrm{i}+800 \mathrm{j} \mathrm{ib}
$$

Figure P.14.73.

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Problem 74

The "L" shaped frame ABC shown in Fig. P. 14.74 is fixed at $C$ and carries a vertical load $P=100 \mathrm{lb}$ at point $C$. The length of $B C$ is 6 in., and the length of $A B$ is 8 in. Both are prismatic, square beams of identical cross section and linear elastic material. Determine the dimension 6 of the cross section so that vertical displacement at point $C$ is Xin. Take $E$ $=30 \times 10^6$ psi and $v=0.33$. Do not include shear deforma. tion. Use energy method.

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Problem 75

Compute the torsional constant $\boldsymbol{K}_t$ for the shape shown in Fig. P. 14.75 using Bredt's formula and by approximation summing the torsional stiffnesses of outside and inside hexagonal boundaries. Hint. First show that
$$
K_i=\frac{4 A^2 t}{S}
$$
using Bredt's formula and Eq. 14.29. Take the crosssectional area
$$
A=\frac{3 \sqrt{3}}{2} b
$$

Finally, using the midline of the section of length $\bar{b}$ show that for the outside length
$$
b=\bar{b}+\frac{a}{\sqrt{3}}
$$
and for the inside length
$$
b^{\prime}=\bar{b}-\frac{a}{\sqrt{3}}
$$

Using software and Table 14.1 for a hexagon, you can show that
$$
K_r=4.203 a \bar{b}^3+1.40 a^3 b
$$

For $\bar{b} \gg a \quad$ explain why
$$
K_i=4.203 a b^3
$$

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01:26

Problem 76

Redo Problem 14.34 using the second Castigliano theorem. Consider bending and twisting energy only.

Chai Santi

Numerade Educator

01:14

Problem 77

Redo Problem 14,69 using the second Castigliano theorem.

Chai Santi

Numerade Educator

01:13

Problem 78

Plot the reacting torques versus the shaft diameter ratio $d_1 / d_2$ for the system shown in Fig. P.14.78. Consider ratios of the range 1 to 5. At what ratio does shate 1 take $95 \%$ of the load?

Hast Aggarwal

Numerade Educator