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Extra Problem1 - Extra Problem 1 Hollow cylinder under...

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Extra Problem 1. Hollow cylinder under inner pressure Consider the problem of an hollow cylinder (with inner radius a , outer radius b and length L ) made of a linearly elastic homogeneous isotropic material (with Young’s modulus E and Poisson’s ratio ν ) subjected to an internal pressure p . The outer surface is assumed to be traction free and the two ends of the cylinder are assumed to be constrained by the presence of two (frictionless) rigid walls. Starting from an assumed form of the displacement field (in cylindrical coordinates), obtain the stress distribution everywhere in the cylinder. Solution The problem geometry and loading suggest that, of the three displacement components u r r , ! , z ( ) , u ! r , ! , z ( ) , u z r , ! , z ( ) ( ) , only the radial one is non-zero and depends only on r . So our initial guess for the displacement field is u r r , ! , z ( ) = f r ( ) u ! r , ! , z ( ) = 0 u z r , ! , z ( ) = 0 " # $ % $ . (1) Our goal is to find the expression of f r ( ) which would satisfy the equilibrium equations and the boundary conditions.
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