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Mechanical

Resultant of a trapezoidal distributed load

Equivalent point load and its position for a load that varies linearly along a beam.

Ref: MEC-20
Rev: v1
Field: Mechanical
Result: F_R [kN]

F_{R} = \int w (x) dx, x_{R} = \frac{\int}{x} \frac{1}{F_{R}}

For parallel loads of intensity w in N/m with a non-zero resultant. For a linear load from w₁ to w₂ over length L, F_R = L(w₁ + w₂)/2 acting at x_R = L(w₁ + 2w₂)/(3(w₁ + w₂)) from the w₁ end. Retain the distribution for internal shear and moment.

A spread-out load can be replaced by a single force of the same total size acting at the load’s balance point.

Calculator

Solve for

F_RResultant forceComputed

w₁Intensity at start

kN/m

050

w₂Intensity at end

kN/m

050

LLoaded length

0.530

Position of resultant from the w₁ end2.333 m
Average intensity4 kN/m

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Resultant force (kN) current value 16 kN

LLoaded lengthKey driver

2120

w₁Intensity at start

12112

w₂Intensity at end

4104

For a uniform load the resultant acts at mid-length; for a triangle it acts at one third of the length from the heavy end.
The resultant preserves overall equilibrium but not internal shear and moment in the beam.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (kN)	Local step	Change in result
L Loaded length	0.5 to 30 m	2	120	118	+10%	+10 %
w₁ Intensity at start	0 to 50 kN/m	12	112	100	+10%	+2.5 %
w₂ Intensity at end	0 to 50 kN/m	4	104	100	+10%	+7.5 %

Sweep of Intensity at end

Intensity at end (kN/m)	Resultant force (kN)
0	4
6.25	16.5
12.5	29
18.75	41.5
25	54
31.25	66.5
37.5	79
43.75	91.5
50	104

Sweep one variable

Vary

Resultant force (kN)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

F_R

Resultant force

Result · kN

What the equation gives you. Everything else feeds it.

w₁

Intensity at start

kN/m · typical 0 to 50

Load intensity at x = 0. Adds to the total.

w₂

Intensity at end

kN/m · typical 0 to 50

Load intensity at x = L. Adds to the total.

Loaded length

m · typical 0.5 to 30

Length over which the load acts. Directly proportional.

Worked example with the values above

Average intensity: (2 + 6)/2 = 4 kN/m.
Multiply by the length 4 m.
F_R = 16 kN.

Assumptions and limits

Linear variation of load intensity.
Parallel loads.

Common mistakes

Placing the resultant at mid-span for a non-uniform load.
Mixing kN and N.

Statistics and uncertainty

Load intensities are often design estimates with ±10% or more, which dominates the structural uncertainty.

Where it is used

Beam support-reaction calculations.
Wind and pressure loading on panels.

Maths you need

Area under a curve: integration
Centroid: balance point

Related formulas

Practise this formula