Mathematical form of u - 3
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Interactive Audio Lesson
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Introduction to Displacement in Hollow Cylinders
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Today, we are going to discuss the mathematical form of 'u', which represents displacement in hollow cylinders. Can anyone tell me why understanding displacement is crucial in solid mechanics?
I think it helps determine how structures respond under loads.
Exactly! Displacement tells us about the deformation our structure experiences. Now, 'u' is dependent on axial and radial coordinates. Can anyone recall how we express this mathematically?
Is it in terms of functions of 'r' and 'z'?
Correct! Remember that 'u' varies with both r for radial displacement and z for axial displacement. Let's break it down into the stress components and see how they relate to displacement.
Understanding Stress Components
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Now, shifting focus to stress components. Do you recall how stress, represented as σ, relates only to r when the cylinder is under static equilibrium?
Yes, that’s because it doesn't change with z!
Exactly! The expression for σ is thus only a function of r. This leads us to express the longitudinal strain in terms of a constant. What do we denote this constant as?
We denote it with the symbol epsilon, ϵ.
Well done! So, understanding this constant is crucial for solving our equations. What happens when we plug it into our equilibrium equations?
Applying Boundary Conditions
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Let’s discuss boundary conditions. Why are they important in our equations?
They help us find the unknown constants from our general solutions.
Absolutely! An example of this would be when a hollow cylinder is subjected to internal pressure. Can anyone explain what our boundary conditions would be in this case?
On the inner surface, the pressure gives us one condition. Outside, there's no pressure, that's another condition!
Exactly! Using these conditions, we derive our final equations. Next, let’s explore how our final solution for 'u' combines all these concepts.
Final Solution for u
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We've arrived at the final expression for 'u'. Can anyone recall what that looks like?
It integrates all our previous constants and components!
Great! Just to recap, the displacement formula is crucial for understanding material behavior. It reveals how external forces influence our cylinder. So, what did we learn about the significance of internal pressure on stress delivery?
It shows that without pressure, stress disappears despite the application of force!
Exactly! This knowledge is essential in designing materials effectively, ensuring they can handle the loads they will face.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
This section elaborates on the mathematical formulation of displacement, denoted as 'u', in hollow cylinders subjected to extension and torsion. It establishes how strain and stress components are interrelated, leading to the generation of equilibrium equations pivotal for structural analysis.
Detailed
Mathematical form of u
This section elaborates on the mathematical representation of displacement, denoted as 'u', in the context of solid mechanics, specifically pertaining to hollow cylinders under extension and torsion. In the previous lectures, the equilibrium equations were derived, and now we focus on how these equations help in calculating the displacement.
Overview of Displacement in Cylinders
The displacement 'u' varies in relation to the radial and axial dimensions of the cylinder. The mathematical form of 'u' is expressed as functions of radial coordinates (r) and axial coordinates (z). For instance, equations are developed indicating that the stress components do not depend on axial coordinates but only on the radial coordinates.
Key Mathematical Expressions
Key expressions are introduced, including the relationships among the various stress components, strain, and displacement. The derivation leads to defining constants, denoting longitudinal strain in the axial direction as an unknown constant parameter.
Integration and Boundary Conditions
The section progresses into the integration of stress and strain relationships and highlights the application of boundary conditions, which are essential for determining unknown constants in equilibrium equations. These boundary conditions are critical when considering the behavior of materials under external pressure and zero traction on outer surfaces.
Conclusion
The final solution for 'u' is derived, which is essential in understanding how the hollow cylinder behaves under varying loads. It concludes by demonstrating the relationships between applied forces, resulting strains, and their implications on material behavior.
Audio Book
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Mathematical Representation of u
Chapter 1 of 5
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Chapter Content
As u and u are functions of z and r respectively, we can rewrite equation (5) as
z
r
(9)
Detailed Explanation
This chunk introduces the idea that the variables u (displacement) are dependent on the coordinates z (longitudinal direction) and r (radial direction). The equation provided (equation 9) describes how to express these displacements in the context of the cylinder's equilibrium. Essentially, it sets the stage for analyzing the distribution of strain within the hollow cylinder based on its geometry.
Examples & Analogies
Imagine stretching a balloon; the amount it stretches depends on how far you are from the center (the radius) and along the length of the balloon. Similarly, in our hollow cylinder, the displacement changes based on the radial and longitudinal positions, which we capture mathematically.
Dependence of Stress on Radius
Chapter 2 of 5
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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 zz zz dependence on θ. Thus, it is a function of r only, i.e.,
(10)
Detailed Explanation
This chunk emphasizes that the stress σ in the context of the hollow cylinder is independent of the longitudinal position (z) and the angular position (θ). It states that σ is solely a function of the radial position (r). This simplification is critical because it allows us to focus on a one-dimensional analysis of stress as it varies only with the distance from the center of the cylinder.
Examples & Analogies
Think of a thick rubber band stretched from both ends. Regardless of where you pinch it, the stretch in the rubber band at any point solely depends on how far you are from the center and not along the length (z) or rotational angle (θ) of the band.
Longitudinal Strain Interpretation
Chapter 3 of 5
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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
This chunk discusses that since stress does not depend on z, the derivative of u with respect to z must be constant. This derivative, which represents axial strain (denoted as ϵ), is introduced as a key concept. It underlines that the strain in the axial direction is uniform throughout the length of the cylinder, which simplifies our calculations and analysis in further sections.
Examples & Analogies
Consider a rubber band being stretched lengthwise. After a certain point, no matter how much more you pull, the amount of stretch remains the same throughout its length. Similarly, the cylinder experiences constant axial strain, making it easier to analyze.
Derive Stress Expressions
Chapter 4 of 5
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Chapter Content
The expressions of σ and σ given in (3) and (4) now become
rr
θθ
(11)
(12)
Subtracting them, we get
(13)
Detailed Explanation
Here, we transform the equations of stress given earlier (equations 3 and 4) into a more usable form to capture the radial and hoop stresses. By subtracting one expression from the other (equation 12 from equation 11), we derive a new relationship that helps in solving equilibrium equations. This step is critical for subsequently understanding how these stress distributions affect the overall behavior of the hollow cylinder under pressure.
Examples & Analogies
Imagine two different rubber bands of various thicknesses being stretched. By examining the differences in their behavior as they are pulled, we can understand how stress varies within each band. This subtraction of stress expressions allows us to isolate how different factors contribute to overall stress.
Integrating the Equation
Chapter 5 of 5
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Chapter Content
Integrating (15) twice, we finally get
(17)
where C and D are the unknown integrating constants.
Detailed Explanation
This chunk discusses the integration of the previously derived equation from the differential equations showing stress. By integrating the relationship, we can find the displacement responses in the radial direction, capturing how the hollow cylinder deforms. The constants C and D are essential as they will help specify particular solutions based on boundary conditions in subsequent discussions.
Examples & Analogies
Think of a sponge being squeezed; the way it deforms continuously as pressure is applied can be likened to the integration of stress across the cylinder. The constants (C and D) will correspond to the sponge's individual characteristics like size and material type affecting how it expands or contracts under pressure.
Key Concepts
-
Displacement (u): Represents how much a point in the material has moved from its original position due to applied forces.
-
Stress (σ): Derived from forces acting on the material and influence the strain experienced.
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Strain (ϵ): Essential for understanding how materials deform when forces are applied.
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Boundary Conditions: These are necessary constraints used to solve for unknown variables in equations.
Examples & Applications
A hollow cylinder under internal pressure experiences different stress distributions than one without pressure.
In analyzing a hollow cylinder with no external forces apart from internal pressure, stress varies radially and has boundary conditions applied at the inner and outer surfaces.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
In a hollow cylinder with force at play, stress changes radially, come what may!
Stories
Imagine a hollow tube trying to hold a weight inside. The weight stresses the tube differently at the top and bottom, helping us see how forces affect its shape.
Memory Tools
Remember the 'S' for Stress relates to 'D' for Deformation, guiding equations to solve thoroughly.
Acronyms
B.E.S.T for Boundary, Equilibrium, Stress, and Tension helps recall essential factors in solving cylinder mechanics.
Flash Cards
Glossary
- Displacement (u)
The change in position of a point in a material, represented as a function of axial and radial coordinates.
- Stress (σ)
The force per unit area within materials, which arises from externally applied forces or induced by internal forces.
- Strain (ϵ)
The deformation per unit length; it is a measure of how much a material deforms under stress.
- Boundary conditions
Constraints or known values applied at the boundaries of a system to solve mathematical equations.
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