To determine the reaction at the left support, we need to analyze the beam using static equilibrium. The beam is 12 meters long and is simply supported with an overhang of 2 meters. It carries a uniformly distributed load (UDL) of 30 kN/m over the entire length.

Step 1: Determine the total load on the beam

The total length of the beam is 12 meters, and the UDL is 30 kN/m. Thus, the total load ($W_{\text{total}}$) is:

$W_{\text{total}} = 30 \, \text{kN/m} \times 12 \, \text{m} = 360 \, \text{kN}$

Step 2: Reaction forces at supports

Let $R_A$ be the reaction at the left support and $R_B$ be the reaction at the right support.

Since the beam is in static equilibrium, we apply the following equilibrium conditions:

• Sum of vertical forces = 0: $R_A + R_B = 360 \, \text{kN}$

• Sum of moments about one of the supports (we'll take moments about the left support $A$):

Taking moments about point $A$ eliminates $R_A$ and helps solve for $R_B$. The UDL acts at the center of the beam (6 meters from each end), but with an overhang, the total load is distributed over 12 meters, and the load acts on both the supported part and the overhang.

For this problem, the 360 kN load can be thought of as two parts:

1. A load on the 10-meter portion (the supported span) contributing directly to the reactions.
2. A 2-meter overhang contributing additional load.

We need more detailed balance of moments to accurately assess