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Chapter 10: Problem 30

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) \\]

Short Answer

Expert verified
When Hooke's Law is rearranged for plane stress, we obtain: \(\sigma_{x}=\frac{E}{\left(1-\nu^{2}\right)}\left(\epsilon_{x}+\nu\epsilon_{y}\right)\) and \(\sigma_{y}=\frac{E}{\left(1-\nu^{2}\right)}\left(\epsilon_{y}+\nu\epsilon_{x}\right)\)

Step by step solution

01

Recall Hooke's law

Hooke's law for plane stress is typically written as \(\sigma_{x}=E(1-\nu)\epsilon_{x}-\nu \sigma_{y}\) and \(\sigma_{y}=E(1-\nu)\epsilon_{y}-\nu \sigma_{x}\)
02

Express \(\sigma_y\) in terms of \(\epsilon_y\)

We will begin by rearranging the equation for \(\sigma_{y}\), to give \(\sigma_{y}=(E\epsilon_{y}-\nu \sigma_{x})/(1-\nu)\)
03

Substitute for \(\sigma_{y}\) in equation for \(\sigma_{x}\)

Substitute the rearranged \(\sigma_{y}\) into the \(\sigma_{x}\) equation, yielding \(\sigma_{x}=E\epsilon_{x}-\nu\left((E\epsilon_{y}-\nu \sigma_{x})/(1-\nu)\right)\)
04

Simplify the equation for \(\sigma_{x}\)

Simplify the equation for \(\sigma_{x}\) to get \(\sigma_{x}=\frac{E}{\left(1-\nu^{2}\right)}\left(\epsilon_{x}+\nu\epsilon_{y}\right)\)
05

Repeat for equation for \(\sigma_{y}\)

Now repeat the steps for \(\sigma_{y}\), starting with expressing \(\sigma_{x}\) in terms of \(\epsilon_{x}\), and substituting in the equation for \(\sigma_{y}\). Similarly, it simplifies to \(\sigma_{y}=\frac{E}{\left(1-\nu^{2}\right)}\left(\epsilon_{y}+\nu\epsilon_{x}\right)\)

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Key Concepts

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

Elastic Modulus (E)
The Elastic Modulus, often symbolized as \( E \), is a fundamental property of materials that describes their stiffness. In simple terms, it measures how much a material will deform (change its shape) under an applied stress.
It is a constant for a given material and is represented by the slope of the linear portion of a stress-strain curve.
Imagine pulling on a rubber band. As you pull, it stretches. The amount it stretches relative to how much you pull is related to its Elastic Modulus.
  • If the material has a high \( E \), it means it is very stiff, and will stretch less under an applied stress.
  • If the material has a low \( E \), it is more flexible, and will stretch more.
Engineers use this property to predict how materials will behave when loaded. A material like steel has a high Elastic Modulus compared to rubber, meaning it won't deform as much under the same stress.
Poisson's Ratio (谓)
Poisson鈥檚 Ratio, denoted as \( u \), is another crucial factor to consider when dealing with material deformation. It describes how a material expands or contracts in directions perpendicular to the direction of loading.
For example, if you squeeze a rubber ball between your palms, it bulges out sideways. This lateral expansion is what Poisson鈥檚 Ratio measures.
  • If \( u \) is high, the material will experience a significant lateral expansion or contraction under axial stress.
  • If \( u \) is low, the lateral deformation is minimal.
The ratio is dimensionless and typically ranges between 0 and 0.5 for most materials. For example, cork has a very low Poisson鈥檚 Ratio, meaning it doesn鈥檛 change its lateral dimension much when compressed.
Stress-Strain Relationship
The Stress-Strain Relationship is a fundamental concept to understand how materials respond to applied forces. Stress (\( \sigma \)) is the internal force divided by the area it acts upon. Strain (\( \epsilon \)) is the deformation of the material divided by its original length.
This relationship helps in predicting the behavior of materials when subjected to different types of forces, such as tension, compression, or shear.
  • The linear portion of the stress-strain curve is where Hooke's Law applies. According to Hooke's Law, the stress is directly proportional to the strain, within the elastic limit of that material.
  • Understanding this relationship enables engineers to design structures that can withstand various load conditions without failure.
In the context of plane stress conditions, knowing the stress-strain relationship allows the calculation of stresses in materials under load, using both the Elastic Modulus and Poisson's Ratio.

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Most popular questions from this chapter

The strain at point \(A\) on the bracket has components \(\epsilon_{x}=300\left(10^{-6}\right), \quad \epsilon_{y}=550\left(10^{-6}\right)\) \(\gamma_{x y}=-650\left(10^{-6}\right), \epsilon_{z}=0 .\) 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.

10-26. The \(45^{\circ}\) strain rosette is mounted on the surface of a pressure vessel. The following readings are obtained for each gage: \(\epsilon_{a}=475\left(10^{-6}\right), \quad \epsilon_{b}=250\left(10^{-6}\right),\) and \(\epsilon_{c}=-360\left(10^{-6}\right) .\) Determine the in-plane principal strains.

The state of strain at the point on the support has components of \(\epsilon_{x}=350\left(10^{-6}\right), \quad \epsilon_{y}=400\left(10^{-6}\right)\) \(\gamma_{x y}=-675\left(10^{-6}\right) .\) Use the strain-transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case specify the orientation of the element and show how the strains deform the element within the \(x-y\) plane.

The thin-walled cylindrical pressure vessel of inner radius \(r\) and thickness \(t\) is subjected to an internal pressure \(p .\) If the material constants are \(E\) and \(\nu,\) determine the strains in the circumferential and longitudinal directions. Using these results, calculate the increase in both the diameter and the length of a steel pressure vessel filled with air and having an internal gage pressure of 15 MPa. The vessel is \(3 \mathrm{m}\) long, and has an inner radius of \(0.5 \mathrm{m}\) and a thickness of \(10 \mathrm{mm} . E_{\mathrm{st}}=200 \mathrm{GPa}, \nu_{\mathrm{st}}=0.3\)

The strain gage is placed on the surface of the steel boiler as shown. If it is 0.5 in. long, determine the pressure in the boiler when the gage elongates \(0.2\left(10^{-3}\right)\) in. The boiler has a thickness of 0.5 in. and inner diameter of 60 in. Also, determine the maximum \(x, y\) in-plane shear strain in the material. \(E_{\mathrm{st}}=29\left(10^{3}\right) \mathrm{ksi}, \nu_{\mathrm{st}}=0.3\)
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