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Equilibrium Equations and Initial Recap
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Alright class, let's briefly recap what we discussed last time regarding the equilibrium of the hollow cylinder. We derived simplified forms of the equilibrium equations needed for our analysis. Who can remind me what those equations look like?
I remember one of them was an expression for stress components in terms of displacement.
Exactly! We expressed σ and τ in terms of u, along with the simplifications. Can anyone tell me what variables these stresses depend on?
σ depends on r only, while τ might depend on both r and θ.
Right! Remember this key point: the uniformity in the stress equation simplifies our calculations significantly. Let's move forward!
Mathematical Form of u
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Now let's analyze how we express the longitudinal strain mathematically in our equations. Can anyone define what we mean by longitudinal strain?
It's the change in length of an object over its original length, usually represented as ε.
Nice! And what does that imply for our equations?
It implies that u' becomes a constant in our equations, allowing us to write it as ε.
Good observation! This is crucial for our subsequent steps. Always remember that constants make our integration process much simpler.
Boundary Conditions Application
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To progress in solving our equations, we need to apply certain boundary conditions. What boundary conditions can we consider for our hollow cylinder?
We have pressure applied on the inner surface and zero pressure on the outer surface.
Correct! This relationship is fundamental because it leads us to express the stress at the boundary in terms of the internal pressure P. Now, how do we express this mathematically?
By using the equation σn = tapp at r = r1 for inner and σ = 0 at r = r2 for outer surfaces.
Exactly! These conditions help us eliminate uncertainties in our constants during integration.
Final Solution and Interpretation
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Now that we’ve applied our boundary conditions, what can we conclude about our stress components from equation (28)?
We see that σrr and σθθ vary with internal pressure and that they individually differ but sum up to be a constant.
Precisely! This shows us that while σrr and σθθ can vary independently through the cylinder's thickness, their sum remains constant. This insight will be crucial in understanding the behavior of materials under stress!
And if there's no internal pressure, both stress components vanish!
Wonderful! Understanding these relationships deepens our grasp of how cylindrical structures behave under various forces.
Introduction & Overview
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Quick Overview
Standard
The content elaborates on the mathematical relationships governing stress and displacement in a hollow cylinder subjected to torsion and inflation. Key equations are derived to facilitate the analysis of longitudinal strain and stress components.
Detailed
In section 4.1, we delve into the mathematical formulation pertinent to the behavior of a hollow cylinder under conditions of extension, torsion, and inflation. The equations related to stress and displacement are systematically derived, emphasizing the relationships between various strain components. The simplified forms of the equilibrium equations and stress distributions are discussed, followed by applications of boundary conditions to solve for unknown constants through mathematical integration. Overall, this section prepares students with the analytical tools necessary for examining complex mechanical systems, setting a fundamental basis for the study of solid mechanics.
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Summary of Stress Components
Chapter 1 of 4
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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 through the tube’s thickness.
Detailed Explanation
In this section, we derive the relationship between two stress components: radial stress (σ_rr) and hoop stress (σ_θθ). By adding the equations that describe these stress components, we find that while their individual values change across the thickness of a hollow cylinder, their sum remains constant. This constant behavior highlights the uniformity in the internal response of the cylinder under pressure.
Examples & Analogies
Imagine blowing up a balloon. As the balloon inflates, the pressure inside creates stress on the walls of the balloon. While the pressure might feel the same at different points on the surface, the material can stretch differently in various areas. This analogy illustrates how the sum of the stress on the balloon's surface is consistent, even as each area responds differently.
Equilibrium Equation in Terms of Stress
Chapter 2 of 4
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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_rr is a function of r alone.
Detailed Explanation
Here, we transition to using our derived constant from the previous chunk to express the equilibrium of forces in terms of stress components. Specifically, since we have established that σ_rr depends solely on the radial distance (r), we can simplify our calculations by treating σ_rr as a function of r only. This makes the equilibrium calculations more straightforward by reducing the number of variables we need to manage.
Examples & Analogies
Think of this like measuring how the weight of water in a tub affects the pressure at different points on the tub's bottom. If the tub is uniformly filled, we can predict the pressure using just the depth (distance from the surface) without worrying about other aspects like the shape of the tub, because the pressure increases uniformly as you go deeper.
Constant Nature of Strain Components
Chapter 3 of 4
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Chapter Content
Thus, we get that is also a constant, say C. Further, from the definition of strain components, we know that and . Thus, we get:
(16)
Integrating (15) twice, we finally get:
(17)
Detailed Explanation
In this part, we find that the quantity we derived behaves like a constant under the defined conditions. By exploring the strain components related to these stress components through integration, we can establish a mathematical expression that describes the behavior of the hollow cylinder when subjected to internal pressures. This leads to a more comprehensive understanding of how the material deforms under loads.
Examples & Analogies
It's similar to stretching a rubber band: if you know how much you stretched (strain), you can figure out how much force you applied (stress). Here, we analyze the relationship mathematically, representing how consistent stretching over a certain area leads to predictable material behavior.
Boundary Conditions and Their Importance
Chapter 4 of 4
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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 conditions. Similarly, we need to identify the boundary conditions for our deformation problem.
Detailed Explanation
Boundary conditions are essential constraints applied to the differential equations we generate from our models. They define the behavior of our system under specific conditions, such as forces or constraints applied at the edges of the structure. In this hollow cylinder example, the internal pressure and lack of external pressure at the outer surface guide how we calculate the unknown constants that originate from integrating our equations.
Examples & Analogies
Consider a race track where cars can only drive in from certain points (boundary conditions). The way cars behave as they enter and move around the track can be predicted based on where the track is laid out, just as knowing the pressures on our cylinder helps us predict how it behaves under stress.
Key Concepts
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Stress Components: Measurements of internal forces that are critical for analyzing material behavior.
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Boundary Conditions: Essential for determining the constants in our mathematical expressions.
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Longitudinal Strain: A measure of deformation that simplifies our equations significantly.
Examples & Applications
When calculating the stress in a hollow cylinder subject to a tensile load, the equations derived provide a direct relationship to how the material will deform.
For a cylinder under internal fluid pressure, the boundary conditions applied will yield a specific distribution of stress that influences design choices.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
To measure strain, keep things plain; length they change along their lane.
Stories
Imagine a cylinder at rest, under pressure it feels the test. Stresses rise, but if pressure's gone, they vanish under tension, leaving no fawn.
Memory Tools
STR Stress Terms: S for σrr, T for τ, R for σθθ — remember them in review!
Acronyms
PREP for Pressure Resulting Equations and Parameters — helps with deriving!
Flash Cards
Glossary
- Longitudinal Strain (ε)
The change in length per unit original length, representing deformation along the axis of the cylinder.
- Boundary Conditions
Conditions necessary for solving differential equations, determined by the physical constraints of the problem.
- Stress Components
Quantitative measures of internal forces within a material, represented as σrr, σθθ, and τ.
- Internal Pressure (P)
Pressure applied from within the hollow cylinder that affects stress distribution.
- Equilibrium Equations
Mathematical expressions that represent the balance of forces within a structural entity.
Reference links
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