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A steel rod with a cross-sectional area ...

A steel rod with a cross-sectional area of `150 mm^(2)` is stretched between two fixed points. The tensile load at `20^(@)C` is `5000N`.
`(a)` What will be the stress at `-20^(@)C` ?
`(b)` At what temperature will the stress be zero ?
(Assume `alpha=11.7mu m//m^(@)C` and `Y=200GN//m^(2)`)

Text Solution

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`(a)` Let `L` be the free length of the rod at `0^(@)C` and `L_(0)` be the final stretched length in each case.
If `L_(1)` and `L_(2)` are the free lengths at `+20^(@)C` and `-20^(@)C`, respectively, then using geometric conditions `delta_(2)=delta_(1)+2alphaLDeltaT`, where `delta_(1)` and `delta_(2)` are the load deformations at `+20^(@)C` and `-20^(@)C`, respectively.
Now, `(F_(2)L)/(AY)=(F_(1)L)/(AY)+2alphaLDeltaT`
or, `sigma_(2)=(F_(2))/(A)=(F_(1))/(A)+2alphaYDeltaT`
`:. sigma_(2)=(5000)/(150xx10^(-6))+2(11.7xx10^(-6))(2xx10^(11))(20)`
`=127xx10^(6)N//m^(2)`
`(b)` Let `T'` be the temperature at which the stress is zero, then
`alphaLT'=alphaLDeltaT+delta_(1)`
or `T'=DeltaT+(F_(1))/(AYalpha)`
`T'=20+(5000)/((150xx10^(-6))(2xx10^(11))(11.7xx10^(-6)))`
`=34.2^(@)C`
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