Solution for σ and σ
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Introduction to Stress Components
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Today, we are going to talk about radial stress, which we denote as σ<sub>rr</sub>, and hoop stress, denoted as σ<sub>θ</sub>. Can anyone tell me what these stresses represent?
Isn't radial stress the force exerted in the direction of the radius?
Exactly! Radial stress acts inward or outward along the radius from the center of the cylinder. Now, what about hoop stress?
Hoop stress acts tangentially around the cylinder, right?
Correct! It is crucial for understanding how hollow cylinders behave under pressure. Let's remember: *Radial goes inwards or outwards, while hoop goes around!*
Equilibrium Equations and Stress Relationships
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We derived some key equilibrium equations last time. Can anyone recall what they help us find?
They help find displacement and determine the stress components?
Exactly! When we simplify these equations, we can express stress in terms of radial displacement. For instance, we noticed that σ<sub>rr</sub> is solely a function of r. Why do you think that is?
Because the stress state changes radially but doesn't depend on other dimensions like height?
Right again! It's a key part of how mechanical systems work. Remember *RHL*: Radial depends on Height and Length, but only in a radial way!
Application of Boundary Conditions
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Now, let’s discuss boundary conditions. When we apply pressure at the inner surface of the cylinder, how do we express that mathematically?
I think we write it as σ<sub>n</sub> = t<sub>app</sub>? Where t<sub>app</sub> is the applied traction.
Precisely! And what happens at the outer surface where there's no external force?
Then, the stress σ<sub>rr</sub> would be zero?
Exactly! Remember the condition: *Pressure gives internal traction; no pressure yields no tension.*
Final Solutions Derivation
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Using our results from before, let's derive the final stresses. Who remembers the integral equations we set up?
We integrated twice to get the final forms of the stresses!
Yes! And do we know what the results indicate about a positive internal pressure?
It indicates that σ<sub>rr</sub> is negative at the inner radius and zero at the outer radius. That makes sense!
Exactly! Remembering how these stresses behave helps us design better structures. Think of *Internal is Inward: Pressure is Negative*!
Graphs and Interpretation of Stresses
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Now let’s interpret our final plots showing stress variations. What do you see in terms of values and shapes?
The radial stress decreases as we move outward, while hoop stress seems to increase overall?
Yes! This relationship is critical. Can anyone describe why they mirror each other?
Because of the equilibrium conditions and how stresses need to maintain balance within the structure!
Well said! Keep in mind our *Mirror Principle*—stresses reflect each other across the radial plot.
Introduction & Overview
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Quick Overview
Standard
In this section, we derive the solutions for radial and hoop stress components in a hollow cylinder subjected to internal pressure, using equilibrium equations and boundary conditions. The discussion emphasizes how these stress components behave throughout the cylinder thickness and their dependency on applied parameters.
Detailed
Detailed Summary
In this section, we thoroughly analyze the stress components, specifically radial (C3rr) and hoop (C3θ), in a hollow cylinder under the influence of internal pressure. After a brief recap of previously derived equilibrium equations, we establish a mathematical framework to express these stress components in relation to displacement and boundary conditions.
Key Points Covered:
- Equilibrium Equations: Utilizing the derived equilibrium equations, we determine that the radial and hoop stresses vary through the thickness of the cylinder but their sum remains constant.
- Stress Components: By establishing expressions of stresses in terms of displacement, we identify that both stress components depend solely on the radial coordinate, r, under static conditions.
- Boundary Conditions: The application of crucial boundary conditions, specifically the internal pressure and zero traction conditions at the outer surface, allows us to derive necessary constants.
- Final Solutions for Stresses: Integrating the equations leads to explicit solutions for the stress components, highlighting their variations and behavior under different loading scenarios, particularly noting that axial force and twisting moments do not generate stress if there’s no internal pressure.
- Graphical Interpretation: The section concludes with a graphical representation of how these stresses vary across the cylinder’s thickness, illustrating fundamental principles of behavior under torsion and extension conditions.
This section is essential for understanding how internal pressures result in stress distributions within hollow cylinders, a crucial aspect in solid mechanics applications.
Audio Book
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Sum of Radial and Hoop Stresses
Chapter 1 of 5
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Chapter Content
Let us add equations (11) and (12):
(18)
Thus, the sum of radial and hoop stresses turns out to be a constant through the thickness of the tube, however, they individually vary throughout the tube’s thickness.
Detailed Explanation
In this chunk, we are looking at equations that describe the equilibrium of stresses in a hollow cylinder. By adding the expressions for radial stress (σ_rr) and hoop stress (σ_θθ), we find that their combined value remains constant across the thickness of the cylinder. This indicates a balance of forces, meaning that while each type of stress changes from the inside to the outside of the cylinder, their total remains steady. This property is essential in ensuring that the structure remains stable under pressure.
Examples & Analogies
Imagine a balloon being inflated. As you blow air into the balloon, the pressure inside increases. The rubber material at any point on the surface experiences tension (like hoop stress) in a circular manner and compression (like radial stress) pushing out. While the individual stresses may change as the balloon expands, the overall tension experienced over the surface (combined effect) remains balanced, allowing the balloon to hold its shape.
Equilibrium Equation in Terms of Stress Components
Chapter 2 of 5
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Chapter Content
Let us now solve the equilibrium equation (1) directly in terms of stress components as follows:
(using (18)). (19)
From equation (11), we know that σ is a function of r alone. Thus, the partial derivative with respect to r becomes total derivative, i.e.,
(using (18)) (20)
Detailed Explanation
This chunk demonstrates the process of rearranging the equilibrium equations to express them in terms of the stress components. By substituting the sum of stresses found in the previous chunk into the equilibrium equations, we simplify our calculations. Since σ_rr is defined only in terms of the radial position r, taking the derivative implies we can model its rate of change within the cylinder. This simplification is vital as it reduces complexity when addressing how stresses affect deformation.
Examples & Analogies
Think of the gauge pressure inside a bicycle tire as a function of its radius. As you move from the center (where the air is located) outwards to the tire surface, the inward pressure does not change based on height, but on the radial position. Therefore, as you analyze how tire pressure influences tire deformation, you can directly relate it to the radius, thus simplifying your study.
Application of Boundary Conditions
Chapter 3 of 5
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Chapter Content
Whenever we solve a differential equation, we get unknown integrating constants. To obtain those constants, one has to apply boundary condition. Similarly, we need to identify the boundary conditions for our deformation problem...
Detailed Explanation
Boundary conditions are crucial in solving differential equations, particularly in mechanical problems involving stress. They define how the physical system interacts with its environment. For instance, in this scenario, the internal pressure acting on the cylinder creates a known boundary condition, which helps to determine the constants that arise in our equations. It is also important to consider the conditions on the outer surface where no external pressure is applied, leading to a simplified system to solve.
Examples & Analogies
Consider a guitar string under tension. The points where the string is fixed (at the bridge and the nut) act as boundary conditions. The sound produced depends significantly on how these boundary points interact with the string, similarly affecting the stress distribution along its length whenever plucked. Just as boundary conditions provide necessary constraints to determine how the string vibrates, in engineering problems, they define how objects behave under loads.
Final Integration and Constants
Chapter 4 of 5
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Chapter Content
Upon plugging the boundary condition (26) in equation (20), we get the following set of equations:
(27)
solving which we get...
(28)
Detailed Explanation
In this chunk, we substitute the boundary conditions obtained earlier into our equations, leading to specific results for the unknown constants in our system. Through solving these conditions mathematically, we derive final expressions for the stresses, indicating how they change with respect to the radial position. This process emphasizes the significance of boundary conditions as they guide the solution process, ensuring it corresponds with the physical behaviors expected from the material under loading.
Examples & Analogies
Think about setting the height of water in a glass. The final height represents the outcome of the water pressure and how it reflects the system's constraints (like the glass's height). Just as the way water levels off directly relates to the constraints set by the glass, in our equations, the constants we solve for must align with the established physical boundaries, conclusively defining how stresses behave through the hollow cylinder.
Behavior of Stresses Under Internal Pressure
Chapter 5 of 5
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Chapter Content
For a positive internal pressure P, we always have A > 0 and B < 0. Using (20), we can now plot the variation in both σ and σ through the tube’s thickness...
(28)
Detailed Explanation
This final chunk touches on the results derived from earlier equations, showcasing that the stresses within the cylinder respond predictably to changes in internal pressure. The expressions provide insight into how hoop and radial stresses vary across the material's thickness. The resulting plots reveal the relationships between internal pressure, radius, and stress distributions, highlighting critical points where those stresses reach maximum tension or compression.
Examples & Analogies
Consider the way a soft drink can behaves when it’s shaken. The internal carbonation (our positive pressure P) increases the stress on the can's walls. Just like the varying stress on different parts of the can determines whether it holds together or bursts, our derived equations help predict how the hollow cylinder will hold up under pressure. The mathematical modeling allows engineers to understand and ensure stable structures, avoiding failures much like we try to avoid spilling our drink.
Key Concepts
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Radial Stress: The stress experienced in a radial direction within a hollow cylinder.
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Hoop Stress: The circumferential stress exerted due to internal pressure.
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Equilibrium: A condition where all forces acting on a body are balanced.
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Boundary Conditions: Critical specifications needed to solve differential equations relating to mechanical systems.
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Displacement: The movement occurring in a structure due to applied loads.
Examples & Applications
For a hollow cylinder subjected to uniform internal pressure, the hoop stress at any radial distance shows a maximum at the inner surface.
Utilizing boundary conditions, if the internal pressure is increased, the radial and hoop stresses can be recalculated, demonstrating their inverse relationship across the cylinder's radius.
Memory Aids
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Rhymes
For radial stress that goes side to side, It acts on edges, wide and wide.
Stories
Imagine a hollow cylinder at a party, with friends pushing from inside. The pressure is their excitement, creating hoop stresses as they circle around!
Memory Tools
Remember 'RHL' — Radial depends on Height and Length, but radial behavior is all about the radius!
Acronyms
BPO - Boundary conditions impact Pressure Outside.
Flash Cards
Glossary
- Radial Stress (σ<sub>rr</sub>)
The stress component acting along the radius of the cylinder, either inward or outward.
- Hoop Stress (σ<sub>θ</sub>)
The stress component acting tangentially around the cylinder, due to internal pressure.
- Equilibrium Equation
Mathematical expressions that represent forces in balance, crucial for solving stress in structures.
- Boundary Conditions
Constraints applied to a physical system regarding the values of stresses or displacements at its boundaries.
- Displacement
The change in position of a point or the deformation of a material under load.
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