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Consider a steel girder, such as the kind that forms the skeleton of a skyscraper, at a temperature of 33.1°C. To what temperature must the girder be cooled for it to contract by 0.109% of its initial length. The coefficient of linear expansion of steel is 1.20 × 10-5 (C°)–1.
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Method 1
α = coefficient of linear expansion
ΔT = temperature change
Fractional change in length = αΔT
In this case, a decrease of 0.109% is a fractional change of -0.109/100 = -0.00109
-0.00109 = 1.20x10^-5 ΔT
ΔT = -90.8 °C
Since the initial temperature was 33.1 °C, the girder must be cooled to 33.1-90.8 = -57.7°C
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Method 2.
Li = initial length
Lf = final length
α = coefficient of linear expansion
ΔT = temperature change
One possible formula is:
Lf = Li(1 + αΔT) (equation 1)
Since the length reduces by 0.109%:
Lf = Li – (0.109/100)Li
= Li(1 – 0.00109)
= 0.99891Li
Substituting into equation 1:
0.99891Li = Li(1 + αΔT)
0.99891 = 1 + αΔT
ΔT= (0.99891 - 1) / (1.20 x 10^-5)
= - 0.00109 / (1.20x10^-5)
= -90.8°C
Since the initial temperature was 33.1 °C, the girder must be cooled to:
33.1 - 90.8 = -57.7°C