Problem 19 - An Introduction to the Mechanics of Solids in SI Units

Question

A cantilever bridge is built out from piers at either end and is to be joined in the middle. Discuss with a numerical example the problem of lining up the two halves of the span if the truss on one side of the roadway is in the sun and that on the other side is in the shade.

Step-by-Step Solution

Answer

Due to the difference in temperature, the side of the bridge in the sun will expand by 0.03m, meaning it needs to be joined at this length longer than the half in the shade in order to properly align the two halves.

Step 1: Understanding Thermal Expansion

When a material is heated or cooled, its length changes by an amount proportional to the original length and the change in temperature. This is expressed by the formula: \( \Delta L = \alpha L \Delta T \) where \( \Delta L \) is the change in length, \( \alpha \) is the coefficient of thermal expansion, \( L \) is the original length, and \( \Delta T \) is the change in temperature.

Step 2: Determine Temperature Difference

Next, we need to assign numerical values to the temperature difference that the two halves of the bridge are experiencing. Say, one half of the bridge (in the sun) has a temperature of \( 35^\circ C \) while the other half (in the shade) is at \( 25^\circ C \). Therefore, the difference in temperature, \( \Delta T \) is \( 35 - 25 = 10^\circ C \).

Step 3: Determine Material Coefficient and Length

Let's assume that the bridge is made of steel, which has a coefficient of linear expansion, \( \alpha \), of approximately \( 12 \times 10 ^{-6} / ^\circ C \). Lets also assume that the span of the bridge, \( L \), is 500 meters (half span therefore being 250 meters).

Step 4: Calculate Change in Length

Now we can calculate the change in length of each half of the bridge. For the side in the sun: \( \Delta L_{\text{sun}} = \alpha L \Delta T = 12 \times 10^{-6}/ ^\circ C \times 250 m \times 10 ^\circ C = 0.03 m \), and for the side in the shade, there is no change in length as the temperature has remained constant.

Step 5: Joining the Spans

Finally, to align the two halves of the bridge, an allowance will have to be made to account for the increased length of the half in the sun. In this case, the two halves should be joined such that the heated half is 0.03 m longer than the other half which is in the shade.

Key Concepts

Cantilever Bridge

A cantilever bridge is a unique type of structure where the main span is supported or anchored at only one end. This feature makes it quite different from typical bridges. Understanding cantilever bridges requires knowing some basic points:

• These bridges are generally built from both ends towards the center, which is known as building out from piers.
• They are constructed using two independent halves that eventually need to meet precisely in the middle.
• The design can handle different forms of stress, providing greater strength and stability compared to simple beam bridges.

This style of construction can pose challenges, particularly when it comes to thermal expansion. The halves of a cantilever bridge need to be precisely aligned so that they join seamlessly. However, external factors, such as sunlight or shade, can lead to misalignment if not adjusted for, as one half can expand or contract differently than the other.

Temperature Difference

Temperature difference is a key factor in thermal expansion, especially in structures like bridges. When different parts of a bridge are exposed to varying temperatures, it can lead to uneven expansion. In our example, one half of the cantilever bridge is exposed to the sun while the other is in the shade. Here's what happens:

• The sunlit side of the bridge gets heated up to a higher temperature compared to the shaded side.
• In our numerical example, the part in the sun reaches a temperature of \( 35^\circ C \) while the shaded half remains at \( 25^\circ C \).
• This creates a temperature difference of \( 10^\circ C \) between the two halves.

Such a temperature difference causes the materials in the two halves to expand differently. This disparity in expansion must be accounted for to ensure that the bridge halves can be properly joined.

Coefficient of Linear Expansion

The coefficient of linear expansion, represented by \( \alpha \), is a fundamental concept when dealing with the expansion of materials due to temperature changes. It quantifies how much a material expands per degree change in temperature. For instance:

• Every material has a unique \( \alpha \). In our example, the steel used in the bridge has a modest \( \alpha \) of \( 12 \times 10^{-6} / ^\circ C \).
• This number indicates how much a unit length of steel will expand for each degree increase in temperature.
• In practical terms, it allows engineers to predict how the length of a bridge will change as the temperature varies.

Using this coefficient in the formula \( \Delta L = \alpha L \Delta T \), where \( \Delta L \) is the change in length, \( L \) is the original length, and \( \Delta T \) is the temperature change, provides a way to calculate thermal expansion accurately. In our example, this calculation showed a change of 0.03 meters in the sunlit bridge half, leading engineers to need a compensatory adjustment when joining the two spans.