2 - Deriving formulas for normal and shear components on such planes
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Interactive Audio Lesson
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Understanding the cuboid element and stress representation
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Today, we are going to derive formulas for normal and shear components on various planes using Mohr's Circle. To start us off, can anyone explain why we are using a cuboid for stress representation?
I think it’s because the cuboid represents the balance of forces on all sides, making it easier to visualize stresses?
Exactly! A cuboid allows us to analyze stress components efficiently. Remember, we want to find normal and shear stresses when the plane normal is perpendicular to a principal stress direction. Why do you think this condition is important?
It’s important because that’s where we can assume some shear components are zero, simplifying our stress matrix!
Great observation! By doing this, we can derive clearer formulas for our stress components. What happens to the stress matrix in this case?
The shear components along the principal axes become zero, and we only have normal stresses to deal with.
Correct! Now, let’s move on to how we can derive the equations for the normal and shear components on the specific planes.
Deriving the stress components formulas
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To derive the stress components, we rotate the e1 plane by an angle, α, about the principal direction. Can anyone represent the normal vector of the arbitrary plane?
We can represent it using the rotated coordinates e1, e2, and e3!
Yes! Then, the normal stress C3 can be derived using the rotational equation: C3 = f(α). Does anyone remember the key trigonometric identities we should use?
We could use cos(2α) and sin(2α) to express the components!
Right! And that leads us to derive two new equations for normal and shear stresses. What’s the formula for the shear stresses then?
It should be τ = Rsin(2φ - 2α) using Mohr's Circle!
Exactly! Now, remember this visual interpretation through Mohr's Circle as we discuss stress transformation.
Visualizing stress components with Mohr's Circle
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Let's move on to visualize our equations using Mohr's Circle. Why do you think visualizing stress components in this way can be useful?
It allows us to see the relationship between different stress states at various angles!
Exactly! And by plotting the points of the stress components on the σ-τ plane, we can derive important characteristics of stress. So how do we plot the σ and τ points for a specific angle α?
We plot the point corresponding to the plane's normal, then measure the angle and draw the radius!
Nice work! The radius of Mohr's Circle corresponds to the maximum shear stress, where do we find the maximum and the minimum values of σ?
The maximum σ values are at the center of the circle, while the minimum values are at the edges of the circle.
Perfect! This visual tool is powerful in obtaining effective stress evaluations without complex calculations. Great engagement today, everyone!
Introduction & Overview
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Quick Overview
Standard
In this section, we derive the formulas for normal and shear components acting on arbitrary planes when the normal to these planes is perpendicular to one of the principal stress directions. This discussion is rooted in using Mohr's Circle for graphical interpretation.
Detailed
Deriving Formulas for Normal and Shear Components
This section delves into deriving the formulas for the normal (C3) and shear (C4) components on arbitrary planes using Mohr's Circle in solid mechanics. In a cuboid element, we focus on stress states assuming one principal stress axis aligned with the third axis of the coordinate system. The section begins by setting up a framework for the stress matrix, indicating that shear components are absent along this principal axis. The displacement to explore other planes involves defining a rotation through an angle α, allowing us to express stress components through trigonometric identities. By defining a scalar R for the right triangle representing the stress state, we subsequently find equations for C3 and C4, utilizing Mohr’s Circle as a visual representation. Thus, with a structured approach, the section ultimately provides insights for calculating stress components efficiently, reinforcing the graphical and mathematical perspectives of stress analysis.
Audio Book
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Overview of the Cuboidal Element
Chapter 1 of 4
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Chapter Content
Let us draw a cuboid element at the point of interest with its face normals along the coordinate system. As discussed earlier, the third coordinate axis (e3) is along the third principal direction. We will call e1, e2, and e3 axes as x, y, and z axes, respectively. On the e1 plane, we have normal component of traction denoted as σxx and shear component of traction denoted as τxy pointing towards y axis. The third component (τzx) is absent as per the stress representation.
Detailed Explanation
This chunk introduces the concept of a cuboidal element used to analyze stress states. We are looking at three axes of a coordinate system: x (e1), y (e2), and z (e3). In this specific setup, the z-axis is aligned with the principal stress direction, meaning it experiences no shear stress (τzx). The normal stress (σxx) acts directly on the e1 plane, while the shear stress (τxy) acts parallel to the e1 axis towards the y-axis. Understanding these components is fundamental as they set the stage for further analyzing stress on different planes.
Examples & Analogies
Think of a cuboidal box filled with water sitting on a table. The box represents our cuboidal element. The weight of the water pushing downwards on the bottom of the box is analogous to the normal stress (σxx). If we were to push one side of the box towards the other, the force we apply parallel to that side represents shear stress (τxy). In our analysis, we want to understand how these forces affect the box in different orientations.
Reducing to a Square Representation
Chapter 2 of 4
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Chapter Content
There is another simpler way to draw such a state of stress for whom there are no shear components in the third direction: we can just draw a square instead of a cube where the sides of the square denote the faces of the cuboid. The right edge of the square represents the (+e1) face. On the edges of the square, we have both the shear and the normal component of traction.
Detailed Explanation
This chunk provides an alternative visualization for the cuboidal element. Instead of a cube, we consider a 2D square to simplify the analysis, especially when the third direction (e3) has no shear components. Each side of the square corresponds to a face of the cuboid. The right side represents the face where the normal stress σxx is acting, and we can depict shear stress τ as acting on the square’s edges.
Examples & Analogies
Imagine a flat pizza. Each slice represents a side of our square. The toppings and sauce act like normal stress on the surface of the pizza. If you were to push down on the pizza, the force you apply would represent the shear stress. By simplifying our view to just the pizza (the square), we make calculations easier while still understanding how the stress is distributed.
Trigonometric Formulas for σ and τ
Chapter 3 of 4
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Chapter Content
Now, our goal is to calculate the normal and shear components of traction on planes whose normals are perpendicular to e3. Let n be the normal to this plane making an angle α with e1. The column representation of n will be specified. Using this, σ will be given by the derived formula and similarly for τ.
Detailed Explanation
In this segment, we focus on deriving the formulas for shear and normal stresses when rotated planes make angles with the principal directions. We assume the normal vector n makes an angle α with the e1 axis and will need to represent its orientation geometrically. By applying trigonometric functions and manipulating equations, we find expressions for σ and τ, which are the normal and shear components acting on the plane.
Examples & Analogies
Picture a street sign standing vertically. If a wind blows against it at an angle (just as n makes an angle with e1), the force acting on the sign can be decomposed into two parts: one straight pushing against the post (normal stress) and another causing it to sway or twist (shear stress). By knowing the angle of the wind, we can mathematically break down these components similar to how we analyze the stresses here.
Introducing Graphical Parameters
Chapter 4 of 4
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Chapter Content
If we look at equations (10) and (11), we notice that σ and τ both have cos(2α) and sin(2α) terms. Let us define a scalar R and relate it to a right-angled triangle formed with σ and τ.
Detailed Explanation
This chunk introduces the concept of defining a scalar R based on the geometric relationship between normal and shear stress components visualized as the sides of a right-angled triangle. By identifying these relationships, we simplify the process of calculating stresses across various orientations. The trigonometric identities will help us connect these relationships to our overall formulas for stress.
Examples & Analogies
Think of a triangle formed by a ramp leading up to a platform. The height of the platform could represent total stress while the distance along the base could model shear stress. By establishing this triangle using R as the hypotenuse, we can navigate through our stress calculations just like measuring the incline of a ramp to ensure we can safely bring items up to the platform.
Key Concepts
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Normal Stress (C3): The component of stress perpendicular to a given plane, calculated using a transformation equation.
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Shear Stress (C4): The component of stress acting parallel to a specific plane, derived using trigonometric identities through Mohr’s Circle.
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Mohr’s Circle: A graphical method to represent the relationship between normal and shear stress, allowing for easy visual assessment of stress states.
Examples & Applications
Example 1: For a given material under uniaxial stress, calculating C3 and C4 on various inclined planes using Mohr's Circle by determining the angles.
Example 2: Using Principal Stress values to find maximum shear stress from Mohr’s Circle, demonstrating its graphical capability.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
When stress gets high, we must not cry, Mohr's Circle will help us see the stress reply!
Stories
Imagine a watchmaker, adjusting the gears. Each change aligns them perfectly due to the angles of Mohr’s insights.
Memory Tools
To remember Mohr's Circle: 'SNOOT' - Stresses, Normal, On, Other, Transforms.
Acronyms
Use 'CIRCLE' to recall
Center
Intersection
Radius
Circle
Length of segments
Evaluate.
Flash Cards
Glossary
- Normal Stress (C3)
The stress component perpendicular to a specified plane.
- Shear Stress (C4)
The stress component parallel to a specified plane.
- Mohr's Circle
A graphical method to determine normal and shear stresses on any plane at a given point in material under stress.
- Principal Stress
The maximum and minimum normal stresses at a given point in a deformed body.
- Traction
The surface force per unit area acting on a material.
Reference links
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