Engineering Calc Tools

Statics

Rigid Body and Frame Equilibrium Solver

Support reactions and pin forces for any 2D rigid body, frame or machine: pins, rollers, fixed supports, cables, couples and distributed loads.

Tool no.
ECT-S24
Rev
1.0
Updated
Oct 2026
Method
ΣF = 0, ΣM = 0 for every member
Examples

Given

Every support, load, pin and corner needs a point. Use one length unit throughout.

One member per line, like A B C. Where two members share a point they are pinned together there. A beam or a single body is one member.

A pin · A fixed · B roller (pushes up; or roller x, roller -x, roller 120 for an angle) · C cable to E (or a direction cable 3 4 or angle cable 37).

C 0 -10 (Fx Fy, + right and up) · C 10 @ 270 (size and angle) · couple B 20 (counterclockwise; add cw for clockwise) · dist A B 4 (per unit length, straight down) or dist A B 0 6 (ramp).

Answers

Largest support reaction = —
Runs in your browser. Nothing you enter is uploaded.ΣF = 0 and ΣM = 0 for every member and pin
How to use this solver

Put a point at every support, pin, corner and load. List each rigid member by the points on it; members that share a point are pinned together there. Then add the supports and loads. The solver writes ΣFx = 0, ΣFy = 0 and ΣM = 0 for every member, plus force balance at every shared pin, and solves them together. It handles beams, brackets, cranes, frames and machines.

What each support provides

Pin (hinge)two force components, Ax and Ay
Roller, smooth surface, collarone force, perpendicular to the surface
Cable or linkone force along the cable (a cable can only pull)
FixedAx, Ay and a moment MA

Worked example

Overhanging beam: pin at A, roller at B (6 m), 4 kN/m over AB, 10 kN at the tip C (8 m) (the default)

  1. The distributed load is 4 × 6 = 24 kN acting at 3 m.
  2. ΣMA = 0: 6By − 24(3) − 10(8) = 0, so By = 25.33 kN.
  3. ΣFy = 0: Ay = 24 + 10 − 25.33 = 8.667 kN, and Ax = 0.

Frames and machines

A frame has at least one member with three or more forces on it, so you can’t treat every member as a two-force member as you would in a truss. Take the members apart at the pins, apply equal and opposite pin forces to the two sides, and write three equations for each member. The solver does exactly that, and recognises two-force members (two pins, no other loads), reporting them as tension or compression.

Frequently asked questions

What does “statically indeterminate” mean here?

There are more unknown reactions than equilibrium equations, so statics can’t find them all. For a beam, the beam solver handles this using the beam’s stiffness.

How do I include a pulley?

Replace the cable with its tension: two forces of size T along the cable directions, applied at the pulley’s centre point.