A pinned aluminium-alloy strut of length ( ) has a second moment of area that…

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 pinned aluminium-alloy strut of length L=2 mL = 2\,\mathrm{m} (E=70 GPaE = 70\,\mathrm{GPa}) has a second moment of area that increases linearly along its length, from I0=2.0×10−7 m4I_0 = 2.0\times10^{-7}\,\mathrm{m^4} at one pinned end (x=0x=0) to 2I02I_0 at the other (x=Lx=L), so I(x)=I0(1+x/L)I(x) = I_0(1 + x/L). ((a)) Using the trial shape w(x)=sin⁡(πx/L)w(x) = \sin(\pi x/L), which satisfies the geometric conditions at both pins, evaluate the energy quotient Pcr≤∫0LEI(x)(w′′)2 dx∫0L(w′)2 dxP_{\rm cr} \le \frac{\int_0^L EI(x)(w'')^2\,\mathrm{d}x}{\int_0^L (w')^2\,\mathrm{d}x} to obtain an upper bound on the buckling load of the tapered strut, in terms of EE, I0I_0 and LL. ((b)) Evaluate this bound numerically for the stated strut, and compare it with the Euler loads of uniform struts of the same length with I=I0I=I_0 throughout and with I=2I0I=2I_0 throughout.

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