1. **Problem Statement:** We have a beam with a uniformly distributed load (UDL) $w = 10$ kN/m over the first 2 meters from the left support A, and a point load of 50 kN downward at the middle-right of the beam. The beam has supports at A (left) and B (right). We need to find the reactions at supports A and B. 2. **Given:** - Uniformly distributed load $w = 10$ kN/m over length $L_1 = 2$ m - Point load $P = 50$ kN at some point on the beam - Beam length is not fully specified, but we assume the total length $L$ is known or can be considered as $L = 2 + x$ meters, where $x$ is the remaining length to support B. 3. **Step 1: Calculate the total load from the UDL** The total load from the UDL is: $$ W = w \times L_1 = 10 \times 2 = 20 \text{ kN} $$ This load acts at the midpoint of the 2 m segment, i.e., 1 m from support A. 4. **Step 2: Define reaction forces at supports** Let $R_A$ be the reaction at support A and $R_B$ be the reaction at support B. 5. **Step 3: Apply equilibrium equations** - Sum of vertical forces: $$ R_A + R_B = W + P = 20 + 50 = 70 \text{ kN} $$ - Sum of moments about A (taking counterclockwise as positive): Assuming the point load is at distance $d$ from A (to be specified or assumed), the moment equation is: $$ R_B \times L = W \times 1 + P \times d $$ 6. **Step 4: Solve for $R_B$ and $R_A$** Rearranged moment equation: $$ R_B = \frac{W \times 1 + P \times d}{L} = \frac{20 \times 1 + 50 \times d}{L} $$ Then, $$ R_A = 70 - R_B $$ 7. **Note:** To fully solve, the total length $L$ and the position $d$ of the point load must be known. If the point load is at the midpoint of the beam, and the beam length is $L = 2 + 2 = 4$ m, then $d = 3$ m (2 m + 1 m into the second segment). 8. **Example with $L=4$ m and $d=3$ m:** $$ R_B = \frac{20 \times 1 + 50 \times 3}{4} = \frac{20 + 150}{4} = \frac{170}{4} = 42.5 \text{ kN} $$ $$ R_A = 70 - 42.5 = 27.5 \text{ kN} $$ **Final answer:** $$ R_A = 27.5 \text{ kN}, \quad R_B = 42.5 \text{ kN} $$ --- "slug": "beam reactions", "subject": "structural engineering", "desmos": {"latex": "y=0", "features": {"intercepts": true, "extrema": true}}, "q_count": 1