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Chapter 4: Problem 64

Determine the maximum shear stress at the outer surface of an internally pressurized cylinder where the internal pressure causes tangential and axial stresses in the outer surface of 300 and \(150 \mathrm{MPa}\), respectively.

Short Answer

Expert verified
The maximum shear stress at the outer surface of the internally pressurized cylinder is 75 MPa.

Step by step solution

01

Identify the Stress Elements

The state of stress at a point is described by two orthogonal stress planes: one is subjected to a tangential stress of \(\sigma_t = 300 MPa\), and the other one is subjected to an axial stress of \(\sigma_a = 150 MPa\). As the cylinder is not subjected to any shear stress initially (purely pressurized), the shear stress \(\tau = 0\).
02

Calculate the Average Normal Stress

The first step in calculating the maximum shear stress is to determine the average normal stress which is given by: \[\sigma_{avg} = \frac{\sigma_a + \sigma_t}{2}\] Substituting the given values, we can find the average normal stress, \[\sigma_{avg}=\frac{300MPa + 150MPa}{2}= 225 MPa\]
03

Calculate the Radius of Mohr's Circle

Next, calculate the radius of Mohr's Circle, which represents the maximum shear stress. The radius \(R\) is given by: \[R = \sqrt{\left(\frac{\sigma_a - \sigma_t}{2}\right)^2 + \tau^2}\] Given that \(\tau = 0\), the formula simplifies as: \[R=\frac{\sigma_a - \sigma_t}{2}\] Substituting the known values, \(R=\frac{300 MPa - 150 MPa}{2} = 75 MPa\) which is the maximum shear stress at the outer surface.

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Most popular questions from this chapter

Represent the surface stresses on a Mohr circle of an internally pressurized section of round steel tubing that is subjected to tangential and axial stresses at the surface of 400 and \(250 \mathrm{MPa}\), respectively. Superimposed on this is a torsional stress of \(200 \mathrm{MPa}\).

The inner surface of a hollow cylinder internally pressurized to \(100 \mathrm{MPa}\) experiences tangential and axial stresses of 600 and \(200 \mathrm{MPa}\), respectively. Make a Mohr circle representation of the stresses in the inner surface. What maximum shear stress exists at the inner surface?

A 30 -mm-diameter shaft transmits \(700 \mathrm{~kW}\) at \(1500 \mathrm{rpm}\). Bending and axial loads are negligible. (a) What is the nominal shear stress at the surface? (b) If a hollow shaft of inside diameter \(0.8\) times outside diameter is used, what outside diameter would be required to give the same outer surface stress? (c) How do weights of the solid and hollow shafts compare?

A bending moment of \(2000 \mathrm{~N} \cdot \mathrm{m}\) is applied to a \(40-\mathrm{mm}\)-diameter shaft. Estimate the bending stress at the shaft surface. If a hollow shaft of outside diameter \(1.15\) times inside diameter is used, determine the outside diameter required to give the same outer surface stress.

For a critical three-dimensional state of stress where, \(\sigma_{x}=45,000, \sigma_{y}=25,000\), \(\sigma_{z}=-50,000, \tau_{\mathrm{xy}}=4000, \tau_{\mathrm{yz}}=2000\), and \(\tau_{\mathrm{zx}}=-3500 \mathrm{psi}\), determine the principal stresses and draw the Mohr circle representation of the state of stress.
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