A long straight strut of bending stiffness

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids

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

Problem

A long straight strut of bending stiffness EIEI, pinned at both ends a length LL apart, rests along its whole length against an elastic bed that pushes back on any sideways deflection ww with a force kfwk_fw per unit length, the way ballast holds a rail. Treating the bed's push as a load q=−kfwq=-k_fw in the beam-column equation EIw′′′′+Pw′′=qEIw''''+Pw''=q gives EIw′′′′+Pw′′+kfw=0EIw''''+Pw''+k_fw=0. ((a)) Show that w=Asin⁡(nπx/L)w=A\sin(n\pi x/L) meets both pinned-end conditions for any whole number nn, and find the axial load PnP_n at which it is a bent equilibrium. ((b)) A welded rail with EI=1.03×106 N m2EI=1.03\times10^6\,\mathrm{N\,m^2} for sideways bending lies on a bed with kf=1.0×105 N m−2k_f=1.0\times10^5\,\mathrm{N\,m^{-2}}, between two points held against sideways movement L=20 mL=20\,\mathrm{m} apart. Find which nn appears first as the load rises, and its buckling load. ((c)) Show that for a strut much longer than its buckled half-wave the buckling load approaches 2kfEI2\sqrt{k_fEI} whatever its length, and find the half-wavelength of the buckled shape in that limit. The rail carries a thermal compressive force of 552 kN552\,\mathrm{kN}. Does the bed keep it straight?

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