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.**

*Image/Mathematical drawings are created in Geogebra.*