4.3 (SI units) A bridge to be fabricated of steel girders is designed to be 500 m long and 12 m wide at ambient temperature (assumed \( 20^{\circ} \mathrm{C} \) ). Expansion joints will be provided to compensate for the change in length in the girders as the temperature varies. Each expansion joint can compensate for a maximum of 20 mm of change in length. From historical records it is estimated that the minimum and maximum temperatures in the region will be \( -35^{\circ} \mathrm{C} \) and \( 40^{\circ} \mathrm{C} \), respectively. Determine (a) the minimum number of expansion joints required and (b) the length that each bridge section should be fabricated. The coefficient of thermal expansion \( \alpha=12\left(10^{-6}\right) / \mathrm{C} \) for steel.

To determine the number of expansion joints required and the appropriate length for each bridge section, we'll analyze the thermal expansion behavior of the steel girders used in the bridge. ### **Given Data:** - **Total Bridge Length (\( L \))**: 500 meters - **Ambient Temperature (\( T_{\text{ambient}} \))**: \( 20^{\circ} \mathrm{C} \) - **Minimum Temperature (\( T_{\text{min}} \))**: \( -35^{\circ} \mathrm{C} \) - **Maximum Temperature (\( T_{\text{max}} \))**: \( 40^{\circ} \mathrm{C} \) - **Coefficient of Thermal Expansion (\( \alpha \))**: \( 12 \times 10^{-6} \, \text{°C}^{-1} \) - **Maximum Compensation per Joint**: 20 mm ### **Temperature Variation (\( \Delta T \)):** The maximum temperature change relative to the ambient temperature is: \[ \Delta T = \max\left(|T_{\text{max}} - T_{\text{ambient}}|, |T_{\text{min}} - T_{\text{ambient}}|\right) = \max(20, 55) = 55^{\circ} \mathrm{C} \] ### **Thermal Expansion Calculation:** The total possible change in length (\( \Delta L \)) due to thermal expansion or contraction is given by: \[ \Delta L = \alpha \times L \times \Delta T = 12 \times 10^{-6} \times 500 \times 55 = 0.33 \, \text{meters} = 330 \, \text{mm} \] ### **(a) Minimum Number of Expansion Joints Required:** Each expansion joint can accommodate up to 20 mm of length change. To ensure the entire bridge can accommodate the total thermal expansion without exceeding the joint capacity: \[ \text{Number of Joints} \geq \frac{\Delta L}{\text{Maximum per Joint}} = \frac{330 \, \text{mm}}{20 \, \text{mm/joint}} = 16.5 \] Since we can't have half a joint, we round up: \[ \text{Minimum Number of Expansion Joints} = 17 \] **However**, it's important to note that the number of joints relates to the number of bridge sections. If there are \( n \) expansion joints, there are \( n + 1 \) sections. Each section can handle up to 20 mm of expansion. To ensure the total expansion is covered: \[ (n + 1) \times 20 \geq 330 \implies n \geq 16 \] Thus, **16 expansion joints** are sufficient because: \[ 16 \times 20 = 320 \, \text{mm} \quad (\text{for 17 sections: } 17 \times 20 = 340 \, \text{mm}) \] ### **(b) Length of Each Bridge Section:** With 16 expansion joints, the bridge is divided into 17 sections. The length of each section (\( L' \)) is: \[ L' = \frac{500 \, \text{meters}}{17} \approx 29.41 \, \text{meters} \] This ensures that each section's thermal expansion does not exceed the 20 mm limit per joint. ### **Final Answers:** (a) **16 expansion joints** are required. (b) Each bridge section should be fabricated to a length of approximately **29.4 meters**.