A steel chimney stack of solid circular cross-section

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Ordinary differential equations

Bending, Ringing, Breaking and Flowing, Ch Buckling, plates and elastic foundations

Problem

A steel chimney stack of solid circular cross-section, diameter

d = 50\,\mathrm{mm}

and length

L = 3\,\mathrm{m}

, is clamped rigidly at its base and free at its top (

E = 200\,\mathrm{GPa}

). ((a)) Starting from

EIw''''+Pw''=0

, show that the smallest buckling load is

P_{\rm cr} = \pi^2EI/4L^2

. At the clamped base

w=0

and

w'=0

. At the free top the bending moment vanishes,

w''(L)=0

, and so does the net sideways force on the top section; for a load that stays vertical as the top tilts, that force is

EIw'''+Pw'

, so

EIw'''(L)+Pw'(L)=0

. ((b)) Evaluate

P_{\rm cr}

for the stated stack.