Formula used
Particle equilibrium
A particle is in static equilibrium when the net force vanishes:
- ΣFx = Σ(Fi cos θi) = 0
- ΣFy = Σ(Fi sin θi) = 0
The resultant is R = √( (ΣFx)² + (ΣFy)² ), with direction atan2(ΣFy, ΣFx).
Beam reactions
For a simply supported beam with a pin at A and a roller at B:
- ΣFx: Ax + ΣFx(applied) = 0
- ΣFy: Ay + By + ΣFy(applied) = 0
- ΣMA: By·L + Σ(x·Fy) + M0 = 0
Uniform load w over [a,b] becomes W = w(b−a) acting at x = a + (b−a)/2.
How to use this calculator
- Select a solver mode that matches your system.
- Enter forces or loads with consistent units.
- Use angles from the positive x-axis, counterclockwise.
- For beams, set L and load positions from A.
- Click Solve Equilibrium to compute reactions or residuals.
- Check residuals to confirm the setup is balanced.
- Export CSV for reports or PDF for sharing.
Example data table
| Scenario | Inputs | Key outputs |
|---|---|---|
| Particle check |
F1=50 N @ 0° F2=50 N @ 120° F3=50 N @ 240° |
ΣFx≈0 N, ΣFy≈0 N R≈0 N (equilibrium) |
| Beam reactions |
L=4 m P1=200 N down at x=1 m P2=100 N down at x=3 m M0=0 N·m |
By≈125 N, Ay≈175 N Residuals ≈ 0 |
| Distributed load |
L=6 m w=60 N/m down over [1,5] No point loads |
W=240 N at x=3 m By≈120 N, Ay≈120 N |
Downward loads are entered by choosing “Down” in the direction menu.
Practical guide to equilibrium solving
1) Why equilibrium matters
Static equilibrium is the starting checkpoint for many physical models. When net force and net moment are zero, the system does not accelerate. This calculator highlights imbalances early, saving time in labs, coursework, and preliminary design.
2) Particle versus rigid-body thinking
Particle equilibrium uses two equations, ΣFx and ΣFy, to test balance at a point. Rigid bodies add moments, so beams require ΣM in addition to force balance. Choosing the correct model avoids underconstrained or misleading results.
3) Component breakdown and sign conventions
Any force can be decomposed into x and y components using cosine and sine. The solver uses +x to the right and +y upward. Moments are positive counterclockwise. Keeping one consistent sign convention makes residuals meaningful.
4) Reaction forces for simply supported beams
In beam mode, the pin at A provides Ax and Ay, while the roller at B provides By. With beam length L, By follows from the moment equation about A. Ay then follows from ΣFy. Ax follows from ΣFx when horizontal loads exist.
5) Distributed loads as equivalent resultants
A uniform load w over a span (b−a) becomes a single force W = w(b−a) acting at the span midpoint. Converting distributed loads to resultants is common in statics, letting you compute reactions without building a full shear diagram.
6) Residuals as a quality metric
Residuals report how closely the computed solution satisfies equilibrium. In beam mode, residual ΣFx, ΣFy, and ΣM should be near zero. In particle mode, compare ΣFx and ΣFy against the tolerance. Larger residuals signal wrong inputs or units.
7) Typical data ranges
For classroom problems, point loads often range from 10 to 1000 N, beam lengths from 0.5 to 10 m, and moments from 1 to 10,000 N·m. Distributed loads commonly range from 5 to 500 N/m, depending on the scenario.
For precision work, keep units consistent end-to-end. If you mix centimeters with meters or kN with N, reactions can shift by factors of 10 to 1000. A good practice is to normalize all inputs first, then verify that output magnitudes match physical expectations. This check catches unit slips before they propagate into reports.
8) Workflow for reliable results
Start with a quick sketch, label distances, and list loads. Enter values with consistent units, then solve. If residuals are not small, recheck directions and positions. Export CSV for documentation, and PDF for sharing calculations with teammates.
FAQs
1) What does “tolerance” mean in particle mode?
Tolerance is the acceptable magnitude for ΣFx and ΣFy. If both sums fall within the tolerance, the solver reports equilibrium. Use a larger value when inputs are rounded or measured with uncertainty.
2) Which angle convention does the calculator use?
Angles are measured from the positive x-axis, increasing counterclockwise. Enter 0° for rightward, 90° for upward, 180° for leftward, and 270° for downward forces.
3) Can I enter downward loads as negative numbers?
Yes, but it is safer to enter magnitudes as positive and choose “Down” for direction. If you enter a negative magnitude, the solver flips the direction internally to keep the component math consistent.
4) What beam support model is implemented?
The beam solver assumes a pin at A and a roller at B. The pin provides Ax and Ay reactions, while the roller provides By. This setup is common for simply supported beams in statics problems.
5) How is a uniform distributed load handled?
The solver converts w over [a,b] into an equivalent resultant W = w(b−a) acting at x = a + (b−a)/2. That equivalent force contributes to ΣFy and ΣM about A.
6) Why do I see small nonzero residuals?
Small residuals usually come from floating-point rounding. If residuals are large, check units, load directions, and positions. Also confirm that your support model matches the physical constraints of the system.
7) When should I use CSV versus PDF export?
Use CSV when you want numeric fields for reports or spreadsheets. Use PDF when you want a clean snapshot of the computed results for printing, sharing, or attaching to homework and lab submissions.