This guide aims to find the change in length of the steel deck of the span when the temperature increases from – 5.0 ° C to 18 ° C. The Humber Bridge in England has the longest single span of 1410 m in the world.
Linear thermal expansion is defined as the increase in the linear dimensions of any object due to temperature variations. Thermal expansion can affect the energy, volume, and area of any solid or fluid.
Expert Answer
To determine the change of length of the steel deck of the span, we will take the initial length of the span as $ l_o $.
\[ l_o = 1410 m \]
The initial temperature is $ – 5.0 ° C $ and after the temperature is raised, it becomes $- 18 ° C $ represented as $ T_1 $ and $ T_2 $, respectively.
\[ T_1 = – 5.0 ° C \]
\[ T_2 = 18.0 ° C \]
\[ \alpha = 1.2 \times 10 ^ { -5 } ( C )^{-1} \]
Temperature and change in length are directly related. When the temperature increases, the length of the solid also increases. According to Linear thermal expansion:
\[\Delta l = l _ o \times \alpha \times \Delta T \]
Delta T is the difference in temperature represented as:
\[ \Delta T = T _ 2 – T _ 1 \]
By putting the value of $ \Delta T $ in the equation:
\[ \Delta l = l_o \times \alpha \times ( T_2 – T_1 )\]
Where $\alpha$ is the certain coefficient of linear thermal expansion and $\Delta l$ is the change in length of the span when temperature $ T _ 1 $ increases to $ T _ 2 $.
By putting values of initial length, initial temperature, and final temperature in the above equation:
\[\Delta l = 1410 m \times 1 . 2 \times 10 ^ { -5 } ( C )^{-1} \times (18 ° C – ( – 5 . 0 ° C) )\]
\[\Delta l = 0. 39 m\]
Numerical Results
The change in length of the steel deck of the span is 0.39 m.
Example
Find the change in length of the steel deck of the Humber bridge when its temperature rises from 6 ° C to 14 ° C.
\[ l _ o = 1410 m \]
\[T _ 1 = 6 ° C \]
\[T _ 2 = 14 ° C \]
\[\alpha = 1 . 2 \times 10 ^ { -5 } ( C )^{-1}\]
According to Linear thermal expansion:
\[\Delta l = l _ o \times \alpha \times ( T _ 2 – T _ 1 )\]
By putting values:
\[\Delta l = 1410 m \times 1 . 2 \times 10 ^ {-5}(C )^{-1} \times ( 14 ° C – ( 6 ° C) ) \]
\[\Delta l = 0.14 m\]
The change in length of the span is 0.14 m.
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