Uniformly Varying (Triangular) Loads on Beams

Triangular (uniformly varying) loads show up constantly: soil pressure behind a retaining wall, water in a tank, wind on a sloped roof, and any load that ramps from zero to full over a length.

The resultant trick

A triangular load from 0 to w over length a is equivalent, for reactions only, to a single force wa/2 acting at 2a/3 from the zero end (the centroid of the triangle). A trapezoid splits into a rectangle plus a triangle.

That trick gets you reactions. It does not get you the shear and moment diagrams — those vary smoothly under a ramped load, so you need the shape.

Diagram shapes under a triangular load

From the load–shear–moment relationships:

• Load is linear → shear is parabolic
• Shear is parabolic → moment is cubic

So a triangular load produces curved shear and moment diagrams. Sketching them as straight lines between the ends is a classic exam error.

Simply supported, full-span triangular (0 at A, w at B)

Reactions:

R_A = wL / 6        R_B = wL / 3

(The load’s centroid is at 2L/3 from A, so B carries twice A.)

Maximum moment (where shear crosses zero, at x = L/√3 ≈ 0.577L):

M_max = wL² / 9√3 ≈ 0.0642 · wL²

Compare with the full uniform case wL²/8 = 0.125·wL² — the triangular load’s peak moment is barely half, because the load is concentrated toward one end.

Partial triangular loads

For a triangle over part of the span, integrate or use a calculator: the resultant gives reactions, then the cubic moment curve runs between the load boundaries. Hand-integrating a partial ramp is where most people lose an hour; this is exactly the case the beam calculator handles in one entry — pick “Linear varying”, set start and end magnitudes, done.

Retaining wall note

For a vertical wall retaining soil to height H, the pressure triangle runs from 0 at the top to K·γ·H at the base (K = earth pressure coefficient, γ = soil unit weight). The total thrust is K·γ·H²/2 per metre at H/3 from the base — the same triangle, applied horizontally.