Mathematical form of u - 2
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Equilibrium Equations in Hollow Cylinders
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Today, we will discuss the equilibrium equations for a hollow cylinder that is not subject to any body force. Can anyone tell me why equilibrium is important in understanding the performance of materials?
Equilibrium helps ensure that the structure doesn't collapse or fail under load conditions.
Exactly! In this case, we express the stress components in terms of the displacement components. Let's review the equations we've derived in previous lectures, specifically equations (1) to (8).
I remember that the radial and hoop stresses are functions of the radius only, right?
Correct! This shows that certain stress components depend solely on the geometry of the cylinder and applied loads. Now, let’s follow up with a few questions.
How do we derive the relationship between the stress and displacement?
Great question! It involves substituting the stress equations into the differential equations that govern equilibrium. We will discuss that next.
To summarize, equilibrium equations are foundational as they guide our understanding of how materials behave under load and how to express stress in terms of displacement.
Mathematical Form of u
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Now that we understand the equilibrium equations, let's dive into the mathematical form of u, the displacement component. How is u related to the stress components we've discussed?
Isn't u a function of the radial distance r and the axis z?
Precisely! We denote it as u(r, z). This allows us to express the stress components σ_rr and σ_θθ. Let's rewrite equation (5) in a simplified manner. How can we relate the axial strain and longitudinal strain?
I think we can use the constants from earlier equations, correct?
Absolutely! By integrating these relationships, we can establish how the cylinder's deformation responds to internal pressures. Let’s summarize what we derived.
Boundary Conditions Impact
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We’ve discussed the mathematical forms; now let's talk about boundary conditions. Why are boundary conditions important in our analysis?
They define how the stresses and displacements behave at the edges of the material, right?
Exactly! They help us establish relationships that must hold, such as the internal pressure at the inner surface versus zero traction at the outer surface. Who can tell me the implications of these conditions?
With boundary conditions, we can solve the unknown integrating constants in our equations.
Correct! After applying these conditions, we found that the sum of stress components remains constant throughout the thickness. Let’s summarize why this is critical for engineers.
In summary, boundary conditions dictate how we solve for stresses and ensure stability in our design. These are crucial for safe engineering practices.
Introduction & Overview
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Quick Overview
Standard
In this section, we derive the mathematical representation of the displacement component, u, in a hollow cylinder subjected to pressure. Key concepts include the relationship between stress components, boundary conditions, and the implications of strain through various stages of analysis in solid mechanics.
Detailed
Detailed Summary
This section discusses the mathematical form of the displacement component, u, in a hollow cylinder subjected to axial strain and internal pressure. The analysis begins with the derivation of the equilibrium equations, which reveal how stress components can be expressed in terms of displacement. The longitudinal strain (u') is denoted by a constant parameter, ϵ, leading to simplified relations among the various stress components. As the analysis progresses, boundaries are established to relate the internal pressure on the cylinder to its stress state. Significantly, it is demonstrated that both radial (_r_r) and hoop (_θ_θ) stresses vary through the thickness of the cylinder, while also concluding that under certain conditions (such as zero internal pressure), the stresses can vanish altogether regardless of axial force and twisting moments. This highlights the critical dependency of stress states on boundary conditions within the framework of solid mechanics.
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Dependence on Variables
Chapter 1 of 4
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Chapter Content
As u and u are functions of z and r respectively, we can rewrite equation (5) as ...
Detailed Explanation
In this chunk, we are establishing that the functions u (axial displacement) and u (radial displacement) depend solely on the variables z (axial position) and r (radial position), respectively. This means that we can simplify the expressions related to these functions in our calculations. By acknowledging their dependence on different variables, we can address them separately in our equations, which will help us derive further relationships more easily.
Examples & Analogies
Imagine a tall glass filled with water. The height of the water (how full it is) depends on how much water you pour into it (the axial position or z), while the radius around the glass defines how far from the center you measure (the radial position or r). Just as the glass's height doesn't rely on its radius, our functions behave similarly.
Stress Components and Their Relationships
Chapter 2 of 4
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Chapter Content
From equation (2), we can infer that σ does not depend on z. Also, σ does not have any term that has dependence on θ. Thus, it is a function of r only.
Detailed Explanation
Here, we learn about how the stress components σ behave in relation to the variables we identified earlier. Specifically, we discover that the stress σ does not change in the axial direction (z), nor does it vary with the angle θ; it is solely a function of the radial distance (r). This simplification allows us to focus our analysis on how stress changes only with respect to radial displacement, making our equations less complex.
Examples & Analogies
Consider squeezing an inflatable balloon. The pressure inside (analogous to stress σ) will vary depending on how far you are from the center (radius r), but not with the height of the balloon (which could represent z). This helps us understand how stress is distributed in cylindrical objects.
Longitudinal Strain as a Constant
Chapter 3 of 4
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Chapter Content
Accordingly, the term dependent on z in the LHS must be a constant, or u’ must be a constant. As u’ denotes longitudinal strain in axial direction, we will denote it by ϵ, an unknown but constant parameter.
Detailed Explanation
In this section, we conclude that any term in our equations that depends on the variable z must remain constant throughout our analysis. This is crucial because it means that the longitudinal strain (denoted by u’) is not changing within the locations we are considering—it’s a constant value represented by ϵ. This idea simplifies our analysis and computation of strains and stresses further as we are now treating this term as a known constant in our equations.
Examples & Analogies
Think of a rubber band stretched between two fingers. If you hold it taut (axial displacement), the amount it stretches (longitudinal strain) stays the same no matter how far apart your fingers are—this helps illustrate that the strain is constant under certain conditions.
Integration for Further Analysis
Chapter 4 of 4
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Chapter Content
Integrating (15) twice, we finally get ...
Detailed Explanation
This segment shows that we are integrating the equation derived from earlier steps to find a solution that represents our stress in a holistic manner across the cylindrical structure. The process of integration allows us to find the total effect of stress along with the equations moving forward. It is essential for determining how stress varies throughout different sections and provides the groundwork for solving real-world engineering problems.
Examples & Analogies
Imagine you have a stack of books on a shelf. If you calculate the total weight acting on the shelf (like integrating stress), you consider each book's individual weight and how they influence each other when stacked, leading to a comprehensive understanding of the total load.
Key Concepts
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Stress Components: Expressed in terms of displacement; important for understanding material behavior.
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Longitudinal Strain (ϵ): A constant that represents axial deformation; crucial for material analysis.
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Boundary Conditions: Essential for deriving relationships in material behavior under load.
Examples & Applications
When a hollow cylinder is subjected to internal pressure, the radial stress increases linearly from the inner to the outer surface.
In a case where no internal pressure is applied, it can be observed that the radial and hoop stresses drop to zero, highlighting the material's response only to axial loads.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
For a cylinder that twists and turns, stress and strain are what we learn!
Stories
Imagine a hollow tube being pushed and pulled; it displaces but keeps its form, following nature’s rules.
Memory Tools
Remember 'BC' for Boundary Conditions—how boundaries shape our material's responses!
Acronyms
SCALES
Stress
Components
Axial
Longitudinal
Equilibrium
Strain—key concepts to remember!
Flash Cards
Glossary
- u
Displacement component in a material, particularly in the context of a hollow cylinder.
- σ
Stress component, representing internal forces within a material.
- ϵ
Longitudinal strain in the axial direction, treated as a constant parameter.
- Boundary Conditions
Constraints applied at the edges of a structure that help define its behavior under load.
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