Distributed loads
Real loads are often spread out: snow on a roof, water pressure on a dam, people on a floor. Each spread load can be replaced by a single equivalent force.
Load intensity
A distributed load is described by its intensity w, in force per unit length (N/m or kN/m) for a beam, or force per unit area (Pa or kPa) for a surface. A floor designed for 3 kPa can carry 3 kN on every square metre.
Resultant equals area
The resultant of a distributed load equals the area under the load curve, \(F_R = \int w\,dx\), and acts through the centroid of that area. This turns a spread-out load into a single force for calculating reactions.
Uniform loads
A uniform load w over length L has resultant wL acting at the middle. A beam 5 m long carrying 4 kN/m has a 20 kN resultant at 2.5 m from either end.
Triangular loads
A load rising linearly from 0 to w over length L has resultant wL/2 acting at the centroid of the triangle, two thirds of the way from the zero end, or one third from the high end. A load from 0 to 6 kN/m over 3 m gives 9 kN acting 2 m from the zero end.
Combined shapes
Trapezoidal or irregular loads can be split into rectangles and triangles, each replaced by its own resultant. A load rising from 2 kN/m to 5 kN/m over 6 m splits into a 12 kN rectangle at 3 m and a 9 kN triangle at 4 m from the low end, totalling 21 kN.
Water pressure
Pressure in still water grows linearly with depth, \(p = \rho gh\), so the load on a vertical wall is triangular. On a 4 m deep wall, the pressure at the base is \(1000 \times 9.81 \times 4 \approx 39.2\) kPa, and the resultant on each metre of width is \(\tfrac{1}{2} \times 39.2 \times 4 \approx 78.5\) kN, acting 4/3 m above the base.
Design loads
Building codes specify distributed loads for design, such as live loads for occupants and furniture, snow loads depending on climate, and wind pressures depending on height and exposure. Engineers apply safety factors on top, so structures carry more than their rated loads.
Worked example
A 6 m flat roof beam carries snow at 2 kN/m plus its own weight at 0.5 kN/m, a uniform total of 2.5 kN/m. The resultant is 15 kN at the beam's midpoint, so each support carries 7.5 kN. If wind drift piles extra snow as a triangle from 0 at the left to 3 kN/m at the right, that adds 9 kN acting 4 m from the left end.
Key takeaways
Distributed loads have an intensity w in force per length or area. Their resultant equals the area under the load curve and acts at that area's centroid. A uniform load gives wL at the middle; a triangular load gives wL/2 at two thirds from the zero end. Split complex shapes into rectangles and triangles. Water pressure rises linearly with depth, giving triangular loads on walls and dams. Building codes set design loads for occupants, snow, and wind, with safety factors.
Flashcard
Force per length or area
Check your understanding
A uniform load of 4 kN/m acts over a 5 m beam. What is its resultant and where does it act?