# Triangular and trapezoidal loads on beams

> A two-way slab loads its beams with triangles and trapezoids set out by the 45-degree rule. Here is where the shapes come from, the equivalent UDL, and a worked example.

**Category:** Fundamentals  
**Author:** Sam Rivera (Structural engineer · Educator)  
**Published:** 2026-05-25

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A two-way slab does not hand its beams a tidy uniform strip; it hands them triangles and trapezoids. Drawing lines at 45 degrees inward from each corner splits the panel so the short edge beams take triangular loads and the long edge beams take trapezoidal loads. For design you use the real shape or an equivalent uniform load. StructLoads collects the slab load onto each beam, and this page explains the shapes it is distributing.

## Where the shapes come from

When a slab spans two ways, supported on all four edges, its [load](https://en.wikipedia.org/wiki/Structural_load) shares to every edge beam rather than just two. The standard idealisation, rooted in basic [structural principles](https://www.designingbuildings.co.uk/wiki/Structural_principles), is to draw a line at 45 degrees inward from each corner. On a rectangular panel those lines carve out triangles against the short edges and trapezoids against the long edges, meeting at a central ridge. Each [beam](https://www.designingbuildings.co.uk/wiki/Beam) carries the region next to it, so the load shape follows the region shape.

## Triangle on the short edge, trapezoid on the long edge

The two shapes and which beam gets them.

| Beam | Region shape | Peaks where |
| --- | --- | --- |
| Short edge beam | Triangle | At midspan, zero at the ends |
| Long edge beam | Trapezoid | Flat across the middle, sloping at the ends |

On a square panel all four beams get triangles, because the 45 degree lines meet at the centre. As the panel gets longer, the long edge shapes become trapezoids while the short edge shapes stay triangles. This is the load side of the distinction in [one-way vs two-way slabs](/blogs/one-way-vs-two-way-slabs).

## A worked example

Take a 6 by 8 metre panel carrying 8 kN per square metre. The 45 degree lines from the corners give each short, 6 metre, edge beam a triangle of base 6 metres and height 3 metres, an area of 9 square metres, so a load of 72 kN spread as a triangle peaking at midspan. The long, 8 metre, edge beams take trapezoids carrying the rest. The four beam loads plus any direct column areas add back to the panel total of 6 times 8 times 8, which is 384 kN, the conservation check.

## Using an equivalent UDL

Carrying a triangle or trapezoid through a hand calculation is awkward, so for quick beam sizing you replace it with an equivalent uniform load that gives the same maximum bending moment, using standard factors. This is convenient and common for preliminary design, as an extension of the simple [UDL on a beam from tributary area](/blogs/udl-on-a-beam-from-tributary-area). The factor comes straight from the standard [AISC beam diagrams and formulas](https://engineering.purdue.edu/~ce474/Docs/DA6-BeamFormulas.pdf): a simple beam under a symmetric triangle peaking at midspan carries a maximum moment of WL/6, against WL/8 for the same total load spread uniformly, which is the four-thirds equivalence factor. The equivalent UDL matches the peak moment but not the shear or reactions exactly, so the real shape is better when those govern.

## Reactions from the real shape

If you keep the true shape, the beam reactions split by the load centroid, not 50/50, unless the shape is symmetric. A symmetric triangle or trapezoid still splits evenly because its centroid is at midspan, so most two-way edge beams do give equal end reactions. The general lever-arm method is in [how to calculate beam reactions from a slab](/blogs/how-to-calculate-beam-reactions-from-a-slab).

## Key takeaways: triangular and trapezoidal beam loads

A two-way slab loads its short edge beams with triangles and its long edge beams with trapezoids, set out by the 45 degree rule. Use the real shape for accuracy or an equivalent UDL for quick sizing. The shapes appear only for two-way panels; one-way slabs give uniform strips. StructLoads distributes the slab load to each beam as a preliminary model a qualified engineer confirms.

## Quick answers

### Why do two-way slabs put triangular and trapezoidal loads on beams?
Because the slab shares its load to all four edges. Drawing lines at 45 degrees inward from each corner divides the panel into four regions: the short edge beams receive triangular regions and the long edge beams receive trapezoidal regions. Each beam carries the load of the region next to it, so the load shape follows the region shape.

### What is the 45-degree rule for slab load distribution?
It is the standard way to split a two-way panel between its edge beams: draw a line at 45 degrees inward from each corner. On a rectangular panel these lines form triangles against the short edges and trapezoids against the long edges, meeting at a central ridge. Each edge beam carries the load of the shape adjacent to it.

### How do you convert a triangular or trapezoidal load to a UDL?
You replace it with a uniform load that produces the same maximum bending moment, using standard equivalence factors, or you analyse the actual shape. The equivalent UDL is convenient for quick beam sizing, but the real triangle or trapezoid gives more accurate shear and reactions, so use the equivalent knowingly for preliminary work.

### When do you not get triangular or trapezoidal loads?
You do not get them on a one-way slab, where the slab spans in one direction and delivers a uniform strip to two beams instead. The triangular and trapezoidal shapes appear only when the slab spans two ways on a roughly square panel supported on all four edges. The slab proportions decide which case applies.