Holooly Plus Logo
Holooly Plus Logo

Question 4.3: What must be the length of a 6-mm-diameter aluminum (G = 27 ...

What must be the length of a 6-mm-diameter aluminum (G = 27 GPa) wire so that it could be twisted through one complete revolution without exceeding a shear stress of 42 MPa?

Given: Cross section of wire, desired deformation, limit on shear stress.
Find: Required length of wire.
Assume: Hooke’s law applies.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

The angle of twist of the wire after “one complete revolution” is 2π. The angle of twist is defined as

\phi =\frac{TL}{JG},

and, using the definition of maximum shear stress, this can also be written as

\phi =\frac{\tau _{max}\sout{J}}{c}\frac{L}{\sout{J}G}=\frac{\tau _{max}L}{cG},

which we rearrange to solve for the wire length L:

L=\frac{\phi  cG}{\tau _{max}}=\frac{2\pi \cdot (0.003  \textrm{m})(27\times 10^9  \textrm{Pa})}{42\times 10^6  \textrm{Pa}}

L = 12.12 m.

Related Answered Questions

Question: 4.3.1

If the state of stress for an element is shown in Figure 4.23a, we may also express the state of stress on a wedge of angle α = 22.5 ̊. Because this wedge (ABC) is part of the original element, the stresses on faces AC and BC are known. They appear again on the wedge in Figure 4.19b. The unknown ...

Verified Answer:

Face AB has area dA, in m² . The area correspondin...
Question: 4.2.1

Aneurysm, a ballooning or dilation of a blood vessel, often afflicts the abdominal aorta, a large vessel supplying blood to the abdomen, pelvis, and legs. While aneurysms can develop and grow gradually, the rupture (rapid expansion and tearing) of an aneurysm is usually catastrophic. Although the ...

Verified Answer:

We’d like to model the artery as a pressure vessel...
Question: 4.9

For the given state of plane stress, construct Mohr’s circle, determine the principal stresses, and determine the maximum shearing stress and the corresponding normal stress (Figure 4.41). ...

Verified Answer:

We outline the steps used to construct Mohr’s circ...
Question: 4.5

A solid aluminum alloy shaft 60 mm in diameter and 1000 mm long is to be replaced by a tubular steel shaft of the same outer diameter such that the new shaft will exceed neither (1) twice the maximum shear stress nor (2) the angle of twist of the aluminum shaft. What should be the inner radius of ...

Verified Answer:

We design first for strength and then for stiffnes...
Question: 4.4

Find the required fillet radius for the juncture of a 6-in.-diameter shaft with a 4-in.-diameter segment if the shaft transmits 110 hp at 100 rpm and the maximum shear stress is limited to 8000 psi. ...

Verified Answer:

We make use of the relationship between applied to...
Question: 4.11

A compressed-air tank is supported by two cradles as shown in Figure 4.54. Relative to the effects of the air pressure inside the tank, the effects of the cradle supports are negligible. The cylindrical body of the tank has a 30 in outer diameter and is fabricated from a 3/8-in. steel plate by ...

Verified Answer:

First, we validate our assumption that thin-walled...
Question: 4.10

A state of plane stress consists of a tensile stress σo = 8 ksi exerted on vertical surfaces and unknown shear stresses τo (Figure 4.49). Determine (a) the magnitude of the shear stress τo for which the maximum normal stress is 10 ksi, and (b) the corresponding maximum shear stress. ...

Verified Answer:

We assume a sense (sign) for the unknown shear str...
Question: 4.8

Calculate the tensile stresses (circumferential and longitudinal) developed in the walls of a cylindrical pressure vessel with inside diameter 18 in. and wall thickness 1/4 in. The vessel is subjected to an internal gage pressure of 300 psi and a simultaneous external axial tensile load of 50,000 ...

Verified Answer:

Does thin-walled theory apply? Is the thickness [l...
Question: 4.7

Calculate the torsional stiffness kt of the rubber bushing shown in Figure 4.40. Assume that the rubber is bonded both to the steel shaft and to the outer steel tube, which is in turn attached to a machine housing. Assume that the metal parts do not deform, and that the shear modulus of rubber is G ...

Verified Answer:

Axisymmetric torque is resisted by constant shear ...
Question: 4.6

Two shafts (G = 28 GPa) A and B are joined and subjected to the torques shown in Figure 4.36. Section A has a solid circular cross section with diameter 40 mm and is 160 mm long; B has a solid circular cross section with diameter 20 mm and is 120 mm long. Find (a) the maximum shear stress in ...

Verified Answer:

Our strategy is to use the method of sections to f...