Solution for u and u
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Introduction to Longitudinal and Circumferential Displacements
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Today, we're diving into the displacements of a hollow cylinder, specifically u_z and u_θ. Who can tell me what u_z represents?
Isn't u_z the longitudinal displacement?
Exactly! u_z is the displacement in the axial direction. And what about u_θ?
That would be the circumferential displacement, right?
Correct! Now, both displacements depend on the mechanical state of the cylinder under loading, particularly the strain.
How do we determine the strain in this case?
Great question! The axial strain ϵ is defined as a constant within our equations, simplifying our analysis.
Mathematical Formulations of Strain
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Now, relating forces to strain is essential. Why do we consider ϵ as a constant?
Is it because it simplifies the integration process?
That's correct! It makes our calculations more manageable. We'll start integrating to find u.
What do we end up with after the integrations?
From our integrations, we'll identify relationships involving axial force F and the related displacements.
And the torque T relates to the angular displacement, right?
Absolutely! Each aspect is interconnected.
Boundary Conditions and Their Role
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Boundary conditions play a pivotal role in our analysis. What can someone tell me about them in our hollow cylinder case?
We have known pressures at the inner surface and no traction at the outer surface?
Correct! These conditions allow us to apply the necessary equations and solve for unknowns.
What do we represent with σ_n and τ?
Good inquiry! σ_n represents axial stress, and τ represents shear stress. Applying these gives us valuable insight into deformation.
And how do we derive the relationships for F and Ω?
We use torque equations and substitute the stress relations effectively.
Final Solutions and Their Interpretation
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Finally, what do we conclude from our displacement solutions u_z and u_θ?
They indicate how our hollow cylinder reacts to axial and circumferential loads.
Exactly! They also relate back to mechanical properties like Young's modulus and surface strains.
Can we apply this knowledge practically?
Definitely! This understanding is crucial in designing safer and efficient structures.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
This section explores the mathematical formulation of displacement solutions in a hollow cylinder subjected to axial force and torque. Key relationships between strain, force, and rotation are introduced, providing insights into the mechanics of materials and structures.
Detailed
In this section, the solutions for the displacements u (longitudinal) and u (circumferential) in a hollow cylinder are derived. It begins by establishing the relationship between axial displacement u and the axial strain ϵ, noting that ϵ is constant and relates to axial force F applied to the cylinder. The section further derives the total torque T and its relation to end-to-end rotation Ω, using integration techniques to present the underlying mechanics. The analysis reveals essential boundary conditions that lead to important relationships, showcasing how internal pressures and mechanical forces dictate the deformations in the cylinder.
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Mathematical Form of Displacement
Chapter 1 of 3
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Chapter Content
As u is only a function of z and axial strain u' is a constant, integrating axial strain leads to
(37)
Here we assumed that u vanishes when z=0 because the axial displacement of the cylindrical mid-section (z=0) is zero by symmetry.
Detailed Explanation
In this chunk, we discuss how the displacement 'u' depends only on the axial position 'z'. The axial strain, represented as 'u'', is constant across the length of the cylinder. When we integrate this strain with respect to 'z', we derive a relationship for 'u'. Additionally, we make an assumption that at the midpoint of the cylinder (where z = 0), there is no displacement due to symmetry. This means that any movement is balanced out at the center of the cylinder.
Examples & Analogies
Imagine a perfectly balanced seesaw where the center does not move while the ends go up and down. The midpoint's position remains unchanged even if the ends are lifted. Similarly, in our cylinder, the midpoint has no axial displacement even as other points may experience strain.
Relating Torque and End-to-End Rotation
Chapter 2 of 3
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Chapter Content
A typical cross-section of the hollow cylinder is shown in Figure 3. The cross-section normal points in the z direction. So, σ, τ and τ act on it. We had found τ to be zero. Thus, we need to analyze σ and τ only. If the traction on any point on the cross-section is represented by t, then the moment due to this traction about the cross-section center 'O' will be given by the integration of r × t for each small area element in the cross-section (r represents the position vector of the area element from the center). If Ω denotes the area of the cross-section, the moment M will be given by (39) The torque T is simply the component of moment along the axis, i.e., (using (8)) (40) The term is the polar moment of area (a geometrical quantity) and is denoted by J.
Detailed Explanation
Here, we are discussing how the torque (T) in the cylinder is related to the applied traction at any point in its cross-section. The moment (M) is derived from the traction times the distance from the center. Importantly, since we found that shear stress (τ) isn't present, our analysis focuses mainly on the normal stress (σ) and the resulting torque. The polar moment (J) is a measure of the cylinder's geometric resistance to twisting, which plays a critical role in determining how much torque is required to achieve a desired end-to-end rotation (Ω).
Examples & Analogies
Think of twisting a soda can. When you twist it, the forces on different points of its circular cross-section create moments trying to cause rotation about the center. The further out you apply force (imagine pressing downwards on the edge), the more effective your twist is, much like how torque functions in the cylinder.
Relating Axial Force and Axial Strain
Chapter 3 of 3
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Chapter Content
To find ϵ, let us obtain the axial force in the cross-section through the integration of σ, i.e., (42) Here, A denotes the cross-sectional area. As we know the value of C from equation (29), we are finally able to relate axial force F with axial strain ϵ. In the special case when P=0, the expression of C becomes simpler which yields (43) where E is the Young’s modulus of elasticity.
Detailed Explanation
In this chunk, we focus on deriving the relationship between axial force (F) and axial strain (ϵ) using the integration of normal stress (σ) over the cylinder's cross-section area (A). By applying our earlier findings, especially related to the constant 'C', we establish how the force is linked to strain. In situations where there is no internal pressure (P=0), we simplify our earlier equations to arrive at a clearer expression that involves material properties represented by Young's modulus (E), a measure of stiffness.
Examples & Analogies
Consider stretching a rubber band. The more force you apply (the axial force), the more it elongates (the strain). The relationship between the amount you can stretch it and the force you apply is characterized by the rubber band’s stiffness, similar to how we relate axial force and strain using Young’s modulus in our equations.
Key Concepts
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Displacement Components: u_z and u_θ represent the axial and circumferential displacements, respectively.
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Boundary Conditions: Conditions like pressure and traction influence the stress and displacement outcomes.
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Strain Relationships: Axial strain (ϵ) and corresponding axial forces (F) correlate with displacement solutions.
Examples & Applications
A hollow cylinder subjected to internal pressure where axial strain leads to calculated changes in length.
The rotation of a cylinder when acted upon by a twisting moment, illustrating the relationship between torque and end-to-end rotation.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
To find u_z, look up high, it's how the cylinder stretches, oh my!
Stories
Imagine a cylinder under pressure, it stretches like a balloon, creating movement in multiple directions, including lengthening and twisting.
Memory Tools
Remember: 'Cylinders Slide Past' to relate Circumferential and Longitudinal displacements.
Acronyms
PATT - Pressure Affects Torque and Tension - to recall what influences a hollow cylinder.
Flash Cards
Glossary
- Axial Strain (ϵ)
The change in length per unit length in the direction of the applied axial force.
- Circumferential Displacement (u_θ)
The displacement of a point on the circumference of the hollow cylinder.
- Longitudinal Displacement (u_z)
The displacement of a point along the axis of the hollow cylinder.
- Torque (T)
The rotational force exerted on the hollow cylinder causing twisting.
- Boundary Conditions
The conditions at the edges of the structure which affect its mechanical response.
Reference links
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