1 - Mohr’s Circle Recap
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Introduction to Mohr's Circle
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Welcome to our recap on Mohr's Circle! Can anyone tell me what Mohr's Circle represents when we draw it in a stress context?
It represents the relationship between normal stress and shear stress on different planes.
Great! So, how do we get the values for the normal and shear stresses on any given plane?
We can use the relationships given by the stress equations for stress matrices.
Exactly! To remember these equations we can use the mnemonic 'Shear is on the side, normal at the base'. Let's move on to the graphical representation. What does it show us?
The circle shows the possible values of shear and normal stress for all angles!
Exactly, and what's the significance of the center of the circle?
That corresponds to the average normal stress.
Correct! Let's summarize today’s session: Mohr's Circle allows us to visualize stress states effectively, and we can derive important stress information from its properties.
Calculating Principal Stresses Using Mohr's Circle
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Now, let's apply Mohr's Circle to determine principal stresses. Does anyone remember how to derive them?
We plot the stress components and find the center and radius of the circle.
Correct! Given a stress matrix, we first find our σx and σy. What's the significance of the radius in Mohr's Circle?
The radius represents the maximum shear stress!
Right! Let’s say with our example the center is at 6√2. Using Pythagorean theorem, how would you calculate the radius?
By calculating R = √[(σy - σx)/2)^2 + τxy^2].
Perfect! So in our example, what are our calculated values?
The principal stresses would be σ1 = 6√2 + R and σ2 = 6√2 - R!
Excellent work today! In summary, we derived principal stresses from Mohr's Circle step-by-step which shows the efficiency of this method!
Understanding Limitations of Mohr's Circle
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We’ve seen Mohr's Circle at work, but are there scenarios when it may not be applicable?
It doesn’t apply if the principal stress axes aren't aligned with the coordinate axes!
That's right! Can anyone explain why?
Because the Mohr’s Circle assumes a specific format of the stress matrix that can only happen when aligned.
Exactly! In those cases, what method should we rely on instead?
We should use the algebraic method to find eigenvalues!
Correct! Always remember, Mohr's Circle provides great visual insights but has constraints. Let’s recap: when the axes don’t align, turn to algebra.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, the key principles of Mohr’s Circle are reviewed, including the mathematical formulation of stress components on various planes. It also highlights how to derive principal stresses and identify maximum shear stress, as well as touches upon the graphical and algebraic methods for stress analysis.
Detailed
Mohr’s Circle Recap
This section provides a comprehensive overview of Mohr's Circle, a crucial tool in understanding stress states in materials. It begins by discussing the basic configuration illustrated in Figure 1, where the stress components are plotted. The objective is to determine normal (3) and shear (4) stress on planes at an angle 6 from the x-axis. The relationship between these stresses is captured in a set of equations [Equation 1 and Equation 2].
Mohr's Circle allows the visualization of stress components, indicating that the locus of 3 and 4 forms a circle as per Figure 2. The principal stress and maximum shear stress can be easily identified from this graphical representation, which simplifies the otherwise complex algebraic methods of stress analysis. For instance, the radius of the circle, representing maximum shear stress, is given by [Equation 3], and the coordinates of the highest points correspond to principal stress values calculated via the center of the circle.
In the second section, the derivation of principal planes using angles (6 and 8) is discussed, emphasizing the vector relationship in two dimensions. This leads to the correlation between Mohr's Circle and the eigenvalue-eigenvector methodology for stress matrices, highlighting its limitations in general cases where principal axes may not coincide with coordinate axes. An example problem illustrates the step-by-step application of Mohr's Circle to extract principal stresses and shear stresses effectively. The section concludes by contrasting the intuitive graphical representation with the methodical approach for comprehensive understanding.
Youtube Videos
![Mohr's Circle: Normal and Tangential Stress, Principal Stress, Maximum Shear Stress [Solved Problem]](https://img.youtube.com/vi/mykHhyVeCyg/mqdefault.jpg)
Audio Book
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Introduction to Mohr’s Circle
Chapter 1 of 6
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Chapter Content
Let us have a quick recap of what we discussed about the Mohr’s circle. Figure 1 shows the square on which we had drawn the components of the stress matrix. Here e direction is coming out of the plane.
We wanted to find out σ and τ on a plane which is at an angle α in the counterclockwise direction from e plane (see Figure 1).
Detailed Explanation
Mohr's Circle is a graphical representation of the state of stress at a point. In this context, σ (normal stress) and τ (shear stress) are being analyzed for a plane that is oriented at an angle (α) to the reference direction (e plane). This introduction sets the stage for visualizing the relationships between different stress components through a circle.
Examples & Analogies
Imagine the Mohr's Circle as a clock where each hour represents a different orientation of a physical object. If you're trying to understand how that object responds to a force from various angles, just like viewing time from different hours, Mohr's Circle helps visualize how stress components change with orientation.
Understanding Mohr’s Circle
Chapter 2 of 6
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Chapter Content
We had figured out that... This set of equations is for the stress matrix which looks like the following... We also saw that the locus of σ and τ for all α-planes is a circle which is called the Mohr’s circle.
Detailed Explanation
The equations derived describe how you can determine the various stress components acting on different planes by plotting them on the Mohr's Circle. The circle itself is defined such that all possible normal and shear stresses can be visualized at once, aiding in easily identifying principal stresses and maximum shear.
Examples & Analogies
Think of the Mohr's Circle like a spinning tire. As the tire rotates, any point on its edge can represent different stresses at different angles on a plane. The center of the tire represents a neutral state (or average stress), and the farthest points on the edge are where the maximum stress occurs.
Finding Principal Stress Components
Chapter 3 of 6
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Chapter Content
From the above equations, we had also got the expression of Radius R. Also, as R=τ_max, we get: Also, σ on the plane having maximum shear (topmost and bottom-most points of the circle) was the σ corresponding to the center of the circle.
Detailed Explanation
Here, R symbolizes the radius of the Mohr’s Circle, which equates to the maximum shear stress τ_max. By identifying the topmost and bottom-most points of the circle, one can derive the principal stress values, which are key in understanding the nature of material failure.
Examples & Analogies
Consider pulling a rubber band. The farthest it stretches corresponds to the maximum tension (analogous to maximum shear stress in this context). The point at rest (the center) shows no tension, similar to calculating principal stress at the circle's center.
Planes of Principal Stresses
Chapter 4 of 6
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Chapter Content
We now want to find the angle of the principal planes with the x plane. We can see in Figure 2 that we need to go by an angle of (2φ) clockwise from the x plane in the Mohr’s circle to reach the first principal plane, i.e., where λ is acting.
Detailed Explanation
To determine the orientation of the principal stress planes relative to the x-axis, the angle must be calculated noting that it involves a rotation of 2φ in the Mohr's Circle representation. This understanding is vital for applying the results back to real-world situations where material orientations are essential.
Examples & Analogies
Think of this as adjusting the angle of a camera to capture the perfect shot. Just like you adjust the orientation for the best angle, here the orientation of stress planes needs to be determined to find the impact of stresses on material integrity.
Complexity of Finding Principal Stresses
Chapter 5 of 6
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Chapter Content
Earlier, when we were finding principal planes and principal stress components by algebraic techniques, we had to solve for the eigenvalues and eigenvector of the stress matrix.
Detailed Explanation
Finding principal stress through algebra entails computing eigenvalues, which can be quite complex and time-consuming, especially since it often deals with cubic equations. Mohr's Circle, by contrast, offers a more intuitive, graphical approach to the same problem, simplifying the concept of principal stresses.
Examples & Analogies
Solving for eigenvalues can be likened to trying to find the perfect flavor blend in a recipe by testing numerous combinations. Mohr's Circle, however, is like using a well-defined recipe that outlines the precise ratios without guesswork.
Example of Mohr’s Circle Analysis
Chapter 6 of 6
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Chapter Content
Let us consider an example to understand it better. The stress matrix for a given coordinate system is given: ...We can see from the stress matrix that e axis is aligned with the principal axis and hence we can use Mohr’s Circle analysis here.
Detailed Explanation
Through this example, the process demonstrates how to extract principal planes and stresses from a specific stress matrix. By plotting it on the Mohr's Circle, students will see a practical application of the previously discussed concepts, confirming their understanding of the intersection of theoretical and practical mechanics.
Examples & Analogies
Visualizing stress on an object, such as a beam under load, can be seen as measuring how much weight to lift at certain points. This analogy helps understand the varying stress states similar to finding out different levels of strain and stress across its length.
Key Concepts
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Mohr's Circle: Visual representation of stress states.
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Principal Stress: The maximum and minimum stresses at specific orientations.
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Radius: Indicates maximum shear stress in Mohr's Circle.
Examples & Applications
When given stress components σx = 12 MPa and σy = 6 MPa, the radius R can be calculated as the distance from the center to the edge of the circle.
If the first principal stress is calculated at an angle of φ = 30°, we would rotate back in the actual coordinate system to find true principal plane orientations.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Mohr’s Circle spins, drawing stress for wins!
Stories
Imagine a roundabout where cars represent stress, circulating to show us their maximum strengths in each direction!
Memory Tools
SOAP: S for Shear, O for Orientation, A for Average stress, P for Principal stress found.
Acronyms
CPR
Center
Principal
Radius – elements of our Mohr's analysis!
Flash Cards
Glossary
- Mohr's Circle
A graphical representation of the relationship between normal and shear stress as a function of orientation.
- Shear Stress (τ)
The stress component acting parallel to the surface of an object.
- Normal Stress (σ)
The stress component acting perpendicular to the surface.
- Principal Stress
The maximum or minimum values of normal stress that occur at specific orientations.
- Radius (R)
Represents maximum shear stress in Mohr's Circle.
Reference links
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