Thick Walled Cylinder

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Previous Equations: thin walled cylinder

Previous Equations: thin walled sphere

Assumptions for Thick Walled Cylinders

(1) LEHI behaviors and small strains

(2) Axisymmetry

(3) Uniform pressure

(4) Axial force passes through centroid

Axisymmetry

(1) Only variation in radial direction

(2) d/dO = d/dz = 0

Cylindrical Equilibrium Equation in Radial Direction

Consituitive Equations for LEHI Behavior/Small Strains

Linear Strain Displacement Equations

Compatibility Equation

(1) Between two strains

(2) Requirement of a continuum

(3) Violated? void space

Combine Compatibiltity and Constituitive Equation

Rewrite with Product Rule and Simplify

Constrained Tube Ends

(1) Ends dont move in axial direction

(2) Ezz = 0 or constant for all of r

Constriained Tube End Equation (const eq deriv)

Plug Cylindrical EE and Contrained Tube End into New Compatibility Eq

Rewrite Equilibrium Equation (reverse product rule)

Sub into Constant Eq

Expand

Multiply by r and Integrate

Radial Stress Dependence

(1) radial stress has inverse square dependance on raidal position

(2) no dependence on axial component

What are the boundaries?

The inner and outer surfaces (a and b)

Pressure at boundaries

Equation relating radial stress to pressure

Equation relating angular stress to pressure

Equation relating axial stress to pressure