Commissioning engineers run two independent checks on a new pedestrian footbridge…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Temperature, heat and the macroscopic behaviour of matter

Conditions of Validity, Ch Green functions: inverting the operator

Problem

Commissioning engineers run two independent checks on a new pedestrian footbridge before it opens. Check 1: load-deflection reciprocity. The deck cable, of length

L=12.0\,\mathrm{m}

, is modelled as a light, perfectly flexible cable under uniform tension

T=4.0\times10^{5}\,\mathrm{N}

, pinned at both supports (

u(0)=u(L)=0

). Self-weight sag is ignored, and only the incremental transverse response to a small point load is modelled, so a load

F

x=\xi

produces a deflection

u(x)

satisfying

-T\,u''(x)=F\,\delta(x-\xi)

. Check 2: thermal pulse test. A suspect weld in the steel anchor block is tested by depositing

Q=500\,\mathrm{J}

of resistive heat instantaneously at the weld, treated as a point source in an effectively infinite homogeneous solid with

\rho c=3.5\times10^{6}\,\mathrm{J\,m^{-3}\,K^{-1}}

and thermal diffusivity

D=1.1\times10^{-5}\,\mathrm{m^2\,s^{-1}}

. ((a)) A test actuator applies

F=750\,\mathrm{N}

x=3.0\,\mathrm{m}

. Using the closed-form Green function for the tensioned cable, find the deflection at the sensor at

x=9.0\,\mathrm{m}

. ((b)) Without repeating the boundary-value calculation, state the deflection that would be measured at

x=3.0\,\mathrm{m}

if the same

750\,\mathrm{N}

load instead acted at

x=9.0\,\mathrm{m}

, then verify it directly from the closed form. Name, in one sentence, the property of the operator that guarantees the two readings agree. ((c)) An embedded sensor sits

r=25\,\mathrm{mm}

from the weld. Find the temperature rise it registers

t=2.0\,\mathrm{s}

after the pulse. ((d)) Find the time at which the sensor's reading peaks, and the value of that peak.

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