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Chapter 10: Problem 28

\(* 10-28 .\) The \(45^{\circ}\) strain rosette is mounted on a steel shaft. The following readings are obtained from each gage: \(\epsilon_{a}=800\left(10^{-6}\right), \epsilon_{b}=520\left(10^{-6}\right), \epsilon_{c}=-450\left(10^{-6}\right) .\) Determine the in-plane principal strains.

Short Answer

Expert verified
\(\epsilon_1 = 940 \cdot 10^{-6}\) and \(\epsilon_2 = -360 \cdot 10^{-6}\)

Step by step solution

01

Calculate the average normal strain

The average normal strain \(\overline{\epsilon}\) is calculated using the equation \(\overline{\epsilon} = (\epsilon_{a} + \epsilon_{b} + \epsilon_{c}) / 3\). Substituting the given values we get \(\overline{\epsilon} = (800 + 520 - 450) \cdot 10^{-6} / 3 = 290 \cdot 10^{-6}\)
02

Calculate the strains change

Next we calculate the strains change for each strain as \(\Delta \epsilon_a = \epsilon_a - \overline{\epsilon}\), \(\Delta \epsilon_b = \epsilon_b - \overline{\epsilon}\), \(\Delta \epsilon_c = \epsilon_c - \overline{\epsilon}\). Substituting the given values for each we get \(\Delta \epsilon_a = 800 \cdot 10^{-6} - 290 \cdot 10^{-6} = 510 \cdot 10^{-6}\), \(\Delta \epsilon_b = 520 \cdot 10^{-6} - 290 \cdot 10^{-6} = 230 \cdot 10^{-6}\) and \(\Delta \epsilon_c = -450 \cdot 10^{-6} - 290 \cdot 10^{-6} = -740 \cdot 10^{-6}\).
03

Calculate the principal strains

We can now calculate the principal strains using the formulas for two-dimensional strain: \(\epsilon_1 = \overline{\epsilon} + \sqrt{((\Delta \epsilon_a - \Delta \epsilon_c)/2)^2 + (\Delta \epsilon_b)^2)} and \(\epsilon_2 = \overline{\epsilon} - \sqrt{((\Delta \epsilon_a - \Delta \epsilon_c)/2)^2 + (\Delta \epsilon_b)^2}\Substituting the values we obtain: \(\epsilon_1 = 290 \cdot 10^{-6} + \sqrt{((510 \cdot 10^{-6} - -740 \cdot 10^{-6})/2)^2 + (230 \cdot 10^{-6})^2)} and \(\epsilon_2 = 290 \cdot 10^{-6} - \sqrt{((510 \cdot 10^{-6} - -740 \cdot 10^{-6})/2)^2 + (230 \cdot 10^{-6})^2}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Strain Rosette
When dealing with strain analysis, a strain rosette is an essential tool used to evaluate the multi-directional strain on structures. It consists of three or more strain gauges attached at different angles to measure strains in various directions. These readings help engineers understand how materials deform when subjected to external forces.
The 45-degree strain rosette, as applied to steel shafts, measures strain at angles set apart by 45 degrees, facilitating the calculation of principal strains and shear strains. It's particularly useful because real-world forces rarely act in a single direction, requiring a more comprehensive analysis of deformation in materials.
Principal Strains
Principal strains represent the maximum and minimum values of strain at a particular point within a material. They align with the principal axes where normal strains occur, and these values are critical in predicting material failure.
To determine principal strains from a strain rosette, you first need the average normal strain and then calculate deviations from this average for each strain measurement. Using equations involving these deviations, you can solve for the maximum and minimum strains. Principal strains help in understanding the extreme points of deformation in a stressed object, giving insights into potential points of failure.
Knowing where these strains occur in materials like steel shafts aids in stress management and overall safety of structures.
Steel Shaft
Steel shafts often serve as critical components in machinery, experiencing complex loading and stress conditions. Understanding how strains manifest along their surfaces is essential for ensuring their reliability and longevity.
Strain analysis in steel shafts, as performed using a strain rosette, assists in indicating potential weaknesses points of stress concentration. Engineers can use this data to predict failure modes and enhance design effectiveness. The shafts are subject to torsion, bending, axial forces, and sometimes all these simultaneously, making it crucial to analyze their strain distributions accurately.
Average Normal Strain
The average normal strain is a fundamental concept in deformation analysis. It represents the mean value of deformations measured by a strain rosette, giving an overview of overall change experienced by the material.
Mathematically, it is calculated by averaging the readings from the strain gauges. In our case, it reflects the general trend of deformation across the steel shaft surface. Calculating average normal strain is the first step before moving into more detailed analyses like finding principal strains.
The simplicity of an average measure helps in setting a baseline from which more complex, directional behaviors of strain can be explored.

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Most popular questions from this chapter

For the case of plane stress, show that Hooke's law can be written as \\[ \sigma_{x}=\frac{E}{\left(1-\nu^{2}\right)}\left(\epsilon_{x}+\nu \epsilon_{y}\right), \quad \sigma_{y}=\frac{E}{\left(1-\nu^{2}\right)}\left(\epsilon_{y}+\nu \epsilon_{x}\right) \\]

A thin-walled spherical pressure vessel having an inner radius \(r\) and thickness \(t\) is subjected to an internal pressure \(p .\) Show that the increase in the volume within the vessel is \(\Delta V=\left(2 p \pi r^{4} / E t\right)(1-\nu) .\) Use a small-strain analysis.

The steel shaft has a radius of 15 mm. Determine the torque \(T\) in the shaft if the two strain gages, attached to the surface of the shaft, report strains of \(\epsilon_{x^{\prime}}=-80\left(10^{-6}\right)\) and \(\epsilon_{y^{\prime}}=80\left(10^{-6}\right) .\) Also, determine the strains acting in the and \(y\) directions. \(E_{\mathrm{st}}=200 \mathrm{GPa}, \nu_{\mathrm{st}}=0.3\)

The strain at point \(A\) on the pressure-vessel wall has components \(\epsilon_{x}=480\left(10^{-6}\right), \epsilon_{y}=720\left(10^{-6}\right), \quad \gamma_{x y}=\) \(650\left(10^{-6}\right) .\) Determine (a) the principal strains at \(A,\) in the \(x-y\) plane, (b) the maximum shear strain in the \(x-y\) plane, and (c) the absolute maximum shear strain.

10-37. Determine the bulk modulus for each of the following materials: (a) rubber, \(E_{\mathrm{r}}=0.4 \mathrm{ksi}, \nu_{\mathrm{r}}=0.48\) and (b) glass, \(E_{\mathrm{g}}=8\left(10^{3}\right) \mathrm{ksi}, \nu_{\mathrm{g}}=0.24\)
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