A length of railway track is welded so that it cannot expand lengthways

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

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

Problem

A length of railway track is welded so that it cannot expand lengthways. The rail's cross-section has area

A = 69.3\,\mathrm{cm^2}

and second moment of area about the axis for sideways bending

I = 340\,\mathrm{cm^4}

. The steel has

E = 200\,\mathrm{GPa}

, thermal expansion coefficient

\alpha = 12\times10^{-6}\,\mathrm{K^{-1}}

, and yield stress

\sigma_Y = 250\,\mathrm{MPa}

. The rail is normally held sideways at every sleeper, spaced

0.65\,\mathrm{m}

apart, and each held point may be treated as a pinned support (

K_e = 1

) for sideways bending between sleepers. ((a)) The rail is fully restrained lengthways and warms by

\Delta T = 25\,\mathrm{K}

. Find the magnitude of the compressive stress this produces (no lengthways strain is allowed). ((b)) Treating each length between held points as a pinned strut, find the unsupported length

L

at which the Euler buckling stress, using this stress-form and the given

I

and

A

, equals the thermal compressive stress of part (a). This is the length of unsupported rail that would just buckle sideways at this temperature rise. ((c)) Find the Euler critical stress for a pinned length equal to the actual sleeper spacing,

0.65\,\mathrm{m}

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