The centroid concept for a distributed load is used to locate the point where the equivalent single force acts.

The centroid concept for a distributed load is used to locate the point where the equivalent single force acts.

• A. At The Midpoint Of The Span
• B. The Centroid Of The Area Of The Loading Curve
• C. At The Point Where The Load Begins
• D. At The End Of The Loading Region

Answer: (B)

When you replace a distributed load with a single equivalent force, the force acts at the centroid of the load distribution. Think of the load w(x) along the beam as a shape whose area represents the total load. The resultant force equals that total area, Q = ∫ w(x) dx, and its line of action must pass through the centroid of that area to produce the same moment balance as the actual distribution. The location is given by x_R = (∫ x w(x) dx) / (∫ w(x) dx). If the loading is uniform over a span, the centroid is at the midpoint of that span; if the load is nonuniform, it shifts toward regions with greater intensity. For discrete loads, the equivalent position is a weighted average of the load positions. In short, the equivalent single force acts at the centroid of the area under the loading curve.

When you replace a distributed load with a single equivalent force, the force acts at the centroid of the load distribution. Think of the load w(x) along the beam as a shape whose area represents the total load. The resultant force equals that total area, Q = ∫ w(x) dx, and its line of action must pass through the centroid of that area to produce the same moment balance as the actual distribution. The location is given by x_R = (∫ x w(x) dx) / (∫ w(x) dx). If the loading is uniform over a span, the centroid is at the midpoint of that span; if the load is nonuniform, it shifts toward regions with greater intensity. For discrete loads, the equivalent position is a weighted average of the load positions. In short, the equivalent single force acts at the centroid of the area under the loading curve.