Add any number of trapezoidal line loads. Each load begins at a and extends a length b, varying linearly
from wa at x=a to wb at x=a+b.
Global units
These selections apply to all inputs and results on this page.
Distributed loads
Enter one row per load segment. Positive w acts downward.
- Load varies linearly from wa at x=a to wb at x=a+b.
- All rows must satisfy 0 ≤ a, 0 ≤ b, and a + b ≤ L.
Numbers and units update together.
Results
| Parameter | Value |
|---|
| Reaction Force R₁ |
---
lbf |
| Reaction Force R₂ |
---
lbf |
| Shear @ x (Vₓ) |
---
lbf |
| Max Shear (Vmax) |
---
lbf |
| Moment @ x (Mₓ) |
---
lbf·in |
| Max Moment (Mmax) |
---
lbf·in |
| Slope @ x (θₓ) |
---
radian |
| Max Slope (θmax) |
---
radian |
| End Slope Left (θ₁) |
---
radian |
| End Slope Right (θ₂) |
---
radian |
| Deflection @ x (yₓ) |
---
inch |
| Max Deflection (ymax) |
---
inch |
| Bending Stress @ x (σₓ) |
---
psi |
| Max Bending Stress (σmax) |
---
psi |
How the multiple distributed-load solver works
This calculator handles any number of trapezoidal (linear) distributed loads on a simply supported beam
by superposition. Every row you enter defines a load that starts at position a, spans
a length b, and varies from wa at x=a to wb at x=a+b.
Internally, each trapezoid is decomposed into two patches so closed-form expressions for reactions, shear, moment,
slope, and deflection can be summed along the span.
Resultant of a trapezoidal line load
For one segment (length b, intensities wa→wb):
- Equivalent resultant: W = ((w_a + w_b)/2) · b
- Line of action (from left support): x_c = a + b · (w_a + 2 w_b) / (3 (w_a + w_b))
With several segments, total vertical reaction at the right support follows global equilibrium,
R₂ = (Σ W_i · x_{c,i}) / L, and R₁ = Σ W_i − R₂, where L is the beam length.
Shear/moment diagrams are obtained by integrating the piecewise line-load functions; slopes and deflections follow
the Euler–Bernoulli relation EI·y''(x) = M(x).
Assumptions & sign convention
- Small-deflection Euler–Bernoulli theory; constant E and I.
- Positive w acts downward; reactions are positive upward.
- Bending moment is positive when it compresses top fibers. Shear/deflection positive upward.
- Supports are ideal simple supports; self-weight can be modeled as an additional distributed load.
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