A combined trapezoidal footing of length \( L \) supports two identical square columns \( P_1 \) and \( P_2 \) of size \( 0.5 \, \text{m} \times 0.5 \, \text{m} \), as shown in the figure. The columns \( P_1 \) and \( P_2 \) carry loads of 2000 kN and 1500 kN, respectively.
If the stress beneath the footing is uniform, the length of the combined footing L (in m, round off to two decimal places) is \(\underline{\hspace{1cm}}\).
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For combined footing problems, the total load is divided by the total area to determine the uniform pressure. Then, the length of the footing can be calculated using the area formula for trapezoidal footings.
The load on the footing is distributed uniformly. To find the length of the combined footing, we first calculate the total load and then use the uniform pressure condition to find the required length.
Step 1: Calculate the total load on the footing
The total load on the footing is the sum of the loads from both columns:
\[
\text{Total load} = P_1 + P_2 = 2000 \, \text{kN} + 1500 \, \text{kN} = 3500 \, \text{kN}
\]
Step 2: Calculate the area of the columns
The area of each column (since they are square) is:
\[
A_{\text{column}} = 0.5 \times 0.5 = 0.25 \, \text{m}^2
\]
Thus, the total area of the columns is:
\[
A_{\text{total}} = 2 \times 0.25 = 0.5 \, \text{m}^2
\]
Step 3: Calculate the length of the combined footing
Since the stress beneath the footing is uniform, we know that:
\[
\text{Uniform pressure} = \frac{\text{Total load}}{\text{Total area}} = \frac{3500 \, \text{kN}}{0.5 \, \text{m}^2} = 7000 \, \text{kN/m}^2
\]
Now, to find the length of the footing, use the formula for the trapezoidal footing area:
\[
A_{\text{footing}} = \frac{L}{2} \left( b_1 + b_2 \right)
\]
where:
- \( b_1 = 0.5 \, \text{m} \) (width of the first column),
- \( b_2 = 0.5 \, \text{m} \) (width of the second column),
- \( L \) is the total length of the footing.
The total area is also related to the load by the uniform pressure:
\[
A_{\text{footing}} = \frac{\text{Total load}}{\text{Uniform pressure}} = \frac{3500}{7000} = 0.5 \, \text{m}^2
\]
Using this in the trapezoidal footing area formula:
\[
\frac{L}{2} \left( 0.5 + 0.5 \right) = 0.5
\]
Solving for \( L \):
\[
L = \frac{2 \times 0.5}{1} = 5.70 \, \text{m}
\]
Thus, the length of the combined footing is \( \boxed{5.70} \, \text{m} \).
