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Px = radial pressure at radius ‘x’ and px+dpx = radial pressure at radius ‘x+dx’, where dpx = change in radial pressure over ‘dx’ thickness. Consider failure of a thin ring having radius ‘x’ and thickness ‘dx’. Bursting force = pressure x projected area of the ______ surface onto horizontal plane.
Px = radial pressure at radius ‘x’ and px+dpx = radial pressure at radius ‘x+dx’, where dpx = change in radial pressure over ‘dx’ thickness. Consider failure of a thin ring having radius ‘x’ and thickness ‘dx’. Bursting force = pressure x projected area of the ______ surface onto horizontal plane.
curved
For equilibrium, Bursting force = Resisting ______.
For equilibrium, Bursting force = Resisting ______.
force
Lame’s equations for radial pressure and Hoop ______ apply to thick cylinders.
Lame’s equations for radial pressure and Hoop ______ apply to thick cylinders.
stress
The outer radius is denoted as ______ and the inner radius as r2.
The outer radius is denoted as ______ and the inner radius as r2.
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Boundary conditions specify that at radius x = r2, radial pressure px = ______.
Boundary conditions specify that at radius x = r2, radial pressure px = ______.
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Maximum hoop stress occurs at the ______ surface of the thick cylinder.
Maximum hoop stress occurs at the ______ surface of the thick cylinder.
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Longitudinal stress is caused by the force exerted by fluid pressure on the ______ of the thick cylinder.
Longitudinal stress is caused by the force exerted by fluid pressure on the ______ of the thick cylinder.
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Lame’s constants help determine the values of stresses and ______ in thick cylinders under internal or external pressure.
Lame’s constants help determine the values of stresses and ______ in thick cylinders under internal or external pressure.
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The radial stress will be ______.
The radial stress will be ______.
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To join the maximum hoop stress point and minimum hoop stress point defines the ______.
To join the maximum hoop stress point and minimum hoop stress point defines the ______.
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A thick cylinder subjected to external fluid pressure will have hoop, radial, and longitudinal stresses that are ______.
A thick cylinder subjected to external fluid pressure will have hoop, radial, and longitudinal stresses that are ______.
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The numerical difference between maximum and minimum hoop stresses in a thick cylinder is equal to the numerical difference of internal and ______.
The numerical difference between maximum and minimum hoop stresses in a thick cylinder is equal to the numerical difference of internal and ______.
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When a thick cylinder is subjected to internal fluid pressure only, hoop stress and longitudinal stress are ______.
When a thick cylinder is subjected to internal fluid pressure only, hoop stress and longitudinal stress are ______.
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Lame’s constants A and B are positive when the thick cylinder is subjected to ______ fluid pressure only.
Lame’s constants A and B are positive when the thick cylinder is subjected to ______ fluid pressure only.
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Volumetric strain in a thick cylinder refers to the ______ stored in the thick cylinder.
Volumetric strain in a thick cylinder refers to the ______ stored in the thick cylinder.
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The change in storage capacity of the thick cylinder can be calculated using the ______.
The change in storage capacity of the thick cylinder can be calculated using the ______.
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The Young's modulus of elasticity for the cylinder material is ______ GPa.
The Young's modulus of elasticity for the cylinder material is ______ GPa.
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Poisson's ratio of the cylinder material is ______.
Poisson's ratio of the cylinder material is ______.
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The internal fluid pressure acting at the inner radius r2 is ______ MPa.
The internal fluid pressure acting at the inner radius r2 is ______ MPa.
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The outer radius of the cylinder is ______ mm.
The outer radius of the cylinder is ______ mm.
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When the ends of the thick cylinder are closed, the longitudinal stress is ______ MPa.
When the ends of the thick cylinder are closed, the longitudinal stress is ______ MPa.
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The change in wall thickness when the ends are closed is ______ mm.
The change in wall thickness when the ends are closed is ______ mm.
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When the ends of the cylinder are opened, the change in wall thickness is ______ mm.
When the ends of the cylinder are opened, the change in wall thickness is ______ mm.
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Lame's constant A has a value of ______.
Lame's constant A has a value of ______.
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Longitudinal stress is the same at any point in the thick ______.
Longitudinal stress is the same at any point in the thick ______.
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The internal diameter of a thick cylinder is ______ mm.
The internal diameter of a thick cylinder is ______ mm.
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The radial stress is always ______ and its magnitude is equal to radial pressure.
The radial stress is always ______ and its magnitude is equal to radial pressure.
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The external diameter of the thick cylinder is ______ mm.
The external diameter of the thick cylinder is ______ mm.
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The internal fluid pressure acting on the cylinder is ______ MPa.
The internal fluid pressure acting on the cylinder is ______ MPa.
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At any point in the thick cylinder, the algebraic sum of the hoop stress and radial stress is equal to ______.
At any point in the thick cylinder, the algebraic sum of the hoop stress and radial stress is equal to ______.
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Longitudinal strain εl is constant at all points and is not a function of the radius ‘______’.
Longitudinal strain εl is constant at all points and is not a function of the radius ‘______’.
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The maximum hoop stress allowed is ______ MPa.
The maximum hoop stress allowed is ______ MPa.
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The pressure varies from p (at r2) to ______ (at r1).
The pressure varies from p (at r2) to ______ (at r1).
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The thickness of the shell determined was ______ mm.
The thickness of the shell determined was ______ mm.
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The designed internal fluid pressure of the first cylinder is ______ MPa.
The designed internal fluid pressure of the first cylinder is ______ MPa.
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After re-boring, the new inner radius must not exceed the previous hoop stress by more than ______%.
After re-boring, the new inner radius must not exceed the previous hoop stress by more than ______%.
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To calculate hoop stress, the external radius is denoted as ______.
To calculate hoop stress, the external radius is denoted as ______.
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Study Notes
Longitudinal Stress in Thick Cylinders
- Longitudinal stress remains constant throughout a thick cylinder despite radius variations.
- Plane sections normal to the cylinder's axis retain their flatness under internal fluid pressure.
- Longitudinal strain (εl) is uniform across all points, independent of the radius 'x'.
Lame’s Assumptions and Constants
- Material constants E (Young’s modulus) and μ (Poisson's ratio) are critical for stress calculations.
- According to Lame's assumptions, longitudinal stress (σl) is also constant within the cylinder.
- The algebraic sum of hoop stress and radial stress equals 2A, where A is Lame’s constant.
Pressure Variation and Stress Analysis
- Internal fluid pressure (p) applies at the cylinder's inner radius (r2), while the outer surface is under atmospheric pressure (datum).
- Radial stress (px) changes from internal pressure p at r2 to zero at r1.
- An elemental ring can be evaluated for bursting and resisting forces to analyze stability.
Lame's Equations
- Lame's equations address radial pressure and hoop stress for thick cylinders subjected to internal or external fluid pressures.
- Radial pressures at specific boundaries lead to varying hoop stresses.
- Conditions dictate boundary stresses at both inner and outer surfaces based on fluid pressure effects.
Stress Types in Thick Cylinders
- Maximum hoop stress occurs at the inner surface, while minimum hoop stress occurs at the outer surface.
- Radial stresses also vary, with maximum radial stress at the inner surface and minimum at the outer surface.
- Longitudinal stress depends on internal/external fluid pressures and their effects on the structure.
Strain Analysis
- Strains in the cylinder include hoop strain, longitudinal strain, and radial strain.
- The volumetric strain considers the storage capacity of the thick cylinder and changes due to internal pressures.
Lame’s Constants and Stress Conditions
- Lame’s constants A and B indicate whether the cylinder is under internal or external pressures, signifying tensile or compressive stresses.
- The stress state varies significantly depending on the enclosures (open or closed ends) of the cylinder and the pressures applied.
Practical Examples
- Given a thick cylinder of specific diameters and internal pressures, stresses can be calculated using Lame’s equations.
- Change in wall thickness, hoop strain, and impacts of pressures define the mechanical behavior of the cylinder.
Problem Solving and Analyses
- Calculating maximum hoop stress and radial stress using boundary conditions provides insights into cylinder performance.
- Changes in dimensions such as inner and outer diameters directly affect the resistance to pressures and overall stability.
- Evaluation of re-boring impacts on maximum hoop stress limits emphasizes the importance of design considerations in engineering applications.
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Description
Test your understanding of the properties of thick cylinders under longitudinal stress. This quiz covers important concepts such as the independence of longitudinal strain from the radial position and the behavior of plane sections under internal pressure. Perfect for engineering students looking to reinforce their knowledge.
