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Chapter 4: Problem 67

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?

Short Answer

Expert verified
The maximum shear stress at the inner surface of the cylinder is 200 MPa.

Step by step solution

01

Identify the principal stresses

The principal stresses are given by the tangential and axial stresses. They are \(\sigma_1 = 600 \, MPa\) and \(\sigma_2 = 200 \, MPa\).
02

Calculate the center of the Mohr's Circle

The center of the Mohr's Circle (C) is the average of the principal stresses. \[C = (\sigma_1 + \sigma_2) / 2 = (600 \, MPa + 200 \, MPa)/2 = 400 \, MPa\]
03

Calculate the radius of the Mohr's Circle

The radius (R) can be obtained by \[R = \sqrt{((\sigma_1 - \sigma_2)/2)^2} = \sqrt{((600 \, MPa - 200 \, MPa)/2)^2} = 200 \, MPa\]
04

Calculate the Maximum Shear Stress

The maximum shear stress is equal to the radius of the Mohr鈥檚 Circle, therefore, \(\tau_{max} = R = 200 \, MPa.\)

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Most popular questions from this chapter

Two rectangular beams are made of steel having a tensile yield strength of \(80 \mathrm{ksi}\) and an assumed idealized stress-strain curve. Beam A has a uniform \(1 \times 0.5\)-in. section. Beam B has a \(1 \times 0.5\)-in. section that blends symmetrically into a \(1.5 \times 0.5\)-in. section with fillets giving a stress concentration factor of 3 . The beams are loaded in bending in such a way that \(Z=I / c=b h^{2} / 6=0.5(1)^{2} / 6=\frac{1}{12}\) in. \(^{3}\) (a) For each beam, what moment, \(M\), causes (1) initial yielding and (2) complete yielding? (b) Beam A is loaded to cause yielding to a depth of \(\frac{1}{4}\) in. Determine and plot the distribution of residual stresses that remain after the load is removed.

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Figure P4.65 shows a cylinder internally pressurized to a pressure of \(7000 \mathrm{psi}\). The pressure causes tangential and axial stresses in the outer surface of 30,000 and \(20,000 \mathrm{psi}\), respectively. Determine the maximum shear stress at the outer surface.

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