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Questions and Answers
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.
Boundary conditions specify that at radius x = r2, radial pressure px = ______.
Boundary conditions specify that at radius x = r2, radial pressure px = ______.
Maximum hoop stress occurs at the ______ surface of the thick cylinder.
Maximum hoop stress occurs at the ______ surface of the thick cylinder.
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.
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.
The radial stress will be ______.
The radial stress will be ______.
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 ______.
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 ______.
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 ______.
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 ______.
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.
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.
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 ______.
The Young's modulus of elasticity for the cylinder material is ______ GPa.
The Young's modulus of elasticity for the cylinder material is ______ GPa.
Poisson's ratio of the cylinder material is ______.
Poisson's ratio of the cylinder material is ______.
The internal fluid pressure acting at the inner radius r2 is ______ MPa.
The internal fluid pressure acting at the inner radius r2 is ______ MPa.
The outer radius of the cylinder is ______ mm.
The outer radius of the cylinder is ______ mm.
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.
The change in wall thickness when the ends are closed is ______ mm.
The change in wall thickness when the ends are closed is ______ mm.
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.
Lame's constant A has a value of ______.
Lame's constant A has a value of ______.
Longitudinal stress is the same at any point in the thick ______.
Longitudinal stress is the same at any point in the thick ______.
The internal diameter of a thick cylinder is ______ mm.
The internal diameter of a thick cylinder is ______ mm.
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.
The external diameter of the thick cylinder is ______ mm.
The external diameter of the thick cylinder is ______ mm.
The internal fluid pressure acting on the cylinder is ______ MPa.
The internal fluid pressure acting on the cylinder is ______ MPa.
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 ______.
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 ‘______’.
The maximum hoop stress allowed is ______ MPa.
The maximum hoop stress allowed is ______ MPa.
The pressure varies from p (at r2) to ______ (at r1).
The pressure varies from p (at r2) to ______ (at r1).
The thickness of the shell determined was ______ mm.
The thickness of the shell determined was ______ mm.
The designed internal fluid pressure of the first cylinder is ______ MPa.
The designed internal fluid pressure of the first cylinder is ______ MPa.
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 ______%.
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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