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Chapter 7: Problem 143

For the given state of plane strain, use Mohr's circle to determine \((a)\) the orientation and magnitude of the principal strains, \((b)\) the maximum in-plane strain, \((c)\) the maximum shearing strain. \begin{equation}\begin{array}{ccc} \hline \epsilon_{x} & \epsilon_{y} & \gamma_{x y} \\ \hline+60 \mu & +240 \mu & -50 \mu \\ +400 \mu & +200 \mu & +375 \mu \\ +300 \mu & +60 \mu & +100 \mu \\ -180 \mu & -260 \mu & +315 \mu \\ \hline \end{array}\end{equation}

Short Answer

Expert verified
Principal strains are 512.13 碌 and 87.87 碌; maximum in-plane strain is 512.13 碌; max shearing strain is 424.26 碌.

Step by step solution

01

Identify the Given Strain Components

We are focusing on the plane strain components from the provided table. Assume we are dealing with the row \(\epsilon_{x} = 400 \mu, \epsilon_{y} = 200 \mu, \gamma_{xy} = 375 \mu\), where \( \gamma_{xy} \) is the engineering shear strain.
02

Calculate Average Normal Strain

The average normal strain \(\epsilon_{avg}\) is computed using the formula: \[ \epsilon_{avg} = \frac{\epsilon_x + \epsilon_y}{2} = \frac{400 + 200}{2} = 300 \, \mu \]
03

Calculate the Radius of Mohr's Circle

The radius \(R\) of Mohr's circle is found using the formula: \[ R = \sqrt{\left(\frac{\epsilon_x - \epsilon_y}{2}\right)^2 + \left(\frac{\gamma_{xy}}{2}\right)^2} = \sqrt{\left(\frac{400 - 200}{2}\right)^2 + \left(\frac{375}{2}\right)^2} \]This simplifies to: \[ R = \sqrt{100^2 + 187.5^2} \approx 212.13 \, \mu \]
04

Determine the Principal Strains

The principal strains, \(\epsilon_1\) and \(\epsilon_2\), are given by:\[ \epsilon_1 = \epsilon_{avg} + R = 300 + 212.13 = 512.13 \, \mu \]\[ \epsilon_2 = \epsilon_{avg} - R = 300 - 212.13 = 87.87 \, \mu \]
05

Find the Orientation of Principal Strains

The orientation angle \(\theta_p\) for the principal strains relative to the \(x\)-axis can be found using the formula:\[ \tan(2\theta_p) = \frac{\gamma_{xy}}{\epsilon_x - \epsilon_y} = \frac{375}{400 - 200} = \frac{375}{200} \]Calculate \(2\theta_p\) and then find \(\theta_p\) by taking the arctangent.
06

Calculate the Maximum In-Plane Strain

The maximum in-plane strain is \(\epsilon_1\) calculated previously, which is \(512.13 \, \mu\).
07

Calculate the Maximum Shearing Strain

The maximum shearing strain \(\gamma_{max}\) is equal to the diameter of Mohr's circle:\[ \gamma_{max} = 2R = 2 \times 212.13 = 424.26 \, \mu \]

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Key Concepts

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

Mohr's Circle
Mohr's Circle is an essential graphical representation used in mechanics of materials to illustrate the state of stress or strain at a point. In the context of strain analysis, Mohr's Circle helps visualize how normal and shear strains transform as one rotates the coordinate system.
By plotting the normal strains on the horizontal axis and shear strains on the vertical axis, students can easily determine principal strains and shearing strains using the circle's geometry.
Understanding Mohr鈥檚 Circle steps for application helps solve complex problems of stress transformation without getting tangled in heavy mathematical derivations.
Principal Strains
Principal strains are the normal strains that occur on a particular set of planes where shearing strain is zero. In simpler terms, they are the maximum and minimum normal strains that happen at a specific orientation.
These strains are denoted as \(\epsilon_1\) and \(\epsilon_2\). The principal strains are crucial for engineering as they provide insights into material deformation under load.
By finding the principal strains, you are essentially determining the safest and most dangerous positions for the material when it is subjected to stress.
Shearing Strain
Shearing strain measures the deformation of an object due to shear stress, leading to a change in the angle between two lines originally perpendicular. It is not a direct measure of the length change, but rather how angles between material lines deviate from 90掳.
Calculating the maximum shearing strain using Mohr's Circle involves finding the maximum horizontal or vertical distance on the circle's representation.
This concept is critical in understanding material failure, as excessive shear can lead to issues like metal yielding or soil slipping.
Plane Strain
Plane strain conditions arise when a strain is assumed to occur in a two-dimensional plane, while out-of-plane strains are neglected. This assumption simplifies many structural analysis problems.
Common in geotechnical engineering and situations involving thick walls, such as dams or tunnels, plane strain assumes the thickness is large enough to prevent deformation out of the plane.
Understanding plane strain is vital in mechanical engineering because it dictates how we simplify and solve real-world structural challenges.
Mechanical Engineering
Mechanical Engineering is a broad and dynamic field focusing on the design, analysis, and manufacturing of machines and mechanical systems. It incorporates principles of physics and materials science to solve problems and create innovative solutions.
In mechanical engineering, strain analysis including the use of Mohr's Circle and understanding of principal and shearing strains, is fundamental in ensuring that structures like bridges, engines, and buildings can withstand operational forces.
The application of these concepts ensures safety, efficiency, and sustainability in engineering projects.

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Most popular questions from this chapter

Determine the largest in-plane normal strain, knowing that the following strains have been obtained by the use of the rosette shown: \\[ \begin{array}{c} \epsilon_{1}=-50 \times 10^{-6} \text {in./in. } \quad \epsilon_{2}=+360 \times 10^{-6} \text {in./in. } \\ \epsilon_{3}=+315 \times 10^{-6} \text {in./in. } \end{array} \\]

A spherical gas container made of steel has a 20 -ft outer diameter and a wall thickness of \(\frac{7}{16}\) in. Knowing that the internal pressure is 75 psi, determine the maximum normal stress and the maximum shearing stress in the container.

Two members of uniform cross section \(50 \times 80 \mathrm{mm}\) are glued together along plane \(a-a\) that forms an angle of \(25^{\circ}\) with the horizontal. Knowing that the allowable stresses for the glued joint are \(\sigma=800 \mathrm{kPa}\) and \(\tau=600 \mathrm{kPa},\) determine the largest centric load \(\mathbf{P}\) that can be applied.

A cylindrical storage tank contains liquefied propane under a pressure of \(1.5 \mathrm{MPa}\) at a temperature of \(38^{\circ} \mathrm{C}\). Knowing that the tank has an outer diameter of \(320 \mathrm{mm}\) and a wall thickness of \(3 \mathrm{mm},\) determine the maximum normal stress and the maximum shearing stress in the tank.

Two wooden members of \(80 \times 120\) -mm uniform rectangular cross section are joined by the simple glued scarf splice shown. Knowing that \(\beta=22^{\circ}\) and that the maximum allowable stresses in the joint are, respectively, \(400 \mathrm{kPa}\) in tension (perpendicular to the splice) and \(600 \mathrm{kPa}\) in shear (parallel to the splice), determine the largest centric load \(\mathbf{P}\) that can be applied.
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