/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 The inner surface of a hollow cy... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 69

The inner surface of a hollow cylinder is subjected to tangential and axial stresses of 40,000 and \(24,000 \mathrm{psi}\), respectively. Determine the maximum shear stress at the inner surface, if the cylinder is pressurized to \(10,000 \mathrm{psi}\).

Short Answer

Expert verified
The maximum shear stress at the inner surface of the cylinder is \( \tau max \) psi.

Step by step solution

01

Defining the Variables

Define the variables present: Axial stress is denoted \( \sigma a \) = 24,000 psi, tangential stress is \( \sigma t \) = 40,000 psi, and the pressure inside the cylinder is \( p \) = 10,000 psi.
02

Applying Pressure Effect

Account for the effect of the internal pressure of the cylinder by adding it to the axial and tangential stresses: \( \sigma a' = \sigma a - p = 24000 psi - 10000 psi = 14000 psi \) and \( \sigma t' = \sigma t - p = 40000 psi - 10000 psi = 30000 psi \) where \( \sigma a' \) and \( \sigma t' \) are the new axial and tangential stresses respectively.
03

Calculate Principal Stresses

Calculate the principal stresses ( \( \sigma 1 \) and \( \sigma 2 \) ) using the following relation: \( \sigma 1 = (\sigma a' + \sigma t')/2 + 鈭歿((\sigma a' - \sigma t')/2)^2} \) and \( \sigma 2 = (\sigma a' + \sigma t')/2 - 鈭歿((\sigma a' - \sigma t')/2)^2} \)
04

Derive the Maximum Shear Stress

Derive the maximum shear stress by using the formula: \( \tau max = (\sigma 1 - \sigma 2)/2 \)

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Most popular questions from this chapter

A \(250-\mathrm{mm}\) length of steel tubing (with properties of \(E=207 \times 10^{9} \mathrm{~Pa}\) and \(\alpha=12 \times 10^{-6}\) per degree Celsius) having a cross-sectional area of \(625 \mathrm{~mm}^{2}\) is installed with "fixed" ends so that it is stress-free at \(26^{\circ} \mathrm{C}\). In operation, the tube is heated throughout to a uniform \(249^{\circ} \mathrm{C}\). Careful measurements indicate that the fixed ends separate by \(0.20 \mathrm{~mm}\). What loads are exerted on the ends of the tube, and what are the resultant stresses?

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.

Figure P4.70 shows a triaxial stress element having a critical three- dimensional stress state where \(\sigma_{x}=60,000, \sigma_{y}=-30,000, \sigma_{z}=-15,000, \tau_{\mathrm{xy}}=9000\). \(\tau_{\mathrm{yz}}=\) \(-2000\), and \(\tau_{\mathrm{zx}}=3500 \mathrm{psi}\). Calculate the first, second, and third stress invariants. Then solve the characteristic equation for the principal normal stresses. Also, calculate the maximum shear stress and draw the resulting Mohr circle representation of the state of stress.

Is using force flow to study stress in a component an art or a science? Can the concept of force flow be used to study problems where numerical values for loads are indeterminate?

Two rectangular beams are made of steel having a tensile yield strength of \(550 \mathrm{MPa}\) and an assumed idealized stress-strain curve. Beam A has a uniform 25 \(\mathrm{mm} \times 12.5-\mathrm{mm}\) section. Beam B has a \(25 \mathrm{~mm} \times 12.5-\mathrm{mm}\) section that blends symmetrically into a \(37.5 \mathrm{~mm} \times 12.5-\mathrm{mm}\) section with fillets giving a stress concentration factor of \(2.5\). The beams are loaded in bending in such a way that \(Z=I / c=b h^{2} / 6=12.5(25)^{2} / 6=1302 \mathrm{~mm}^{3}\). (a) For each beam, what moment, \(M\), causes (1) initial yielding and (2) complete yielding? (b) Beam \(\mathrm{A}\) is loaded to cause yielding to a depth of \(6.35 \mathrm{~mm}\). Determine and plot the distribution of residual stresses that remain after the load is removed.
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