Supports, reactions & determinacy

This is a static copy of the chapter for search engines. The interactive version has animated figures, check questions, and buttons that load the example into the calculator.

A bent bar in space has six degrees of freedom per point: three translations (u_x, u_y, u_z) and three rotations (θ_x, θ_y, θ_z). A support removes some of them and hands you a reaction in exchange. FrameLab lets you pick a ready-made support or tick the individual degrees of freedom yourself.

Support	Blocks	Reactions you get back	Typical use
Fixed (clamped)	all six	R_x, R_y, R_z, M_x, M_y, M_z	a bracket welded to a machine frame
Pin	u_x, u_y, u_z	three forces, no moment	a bolted base plate idealised as hinged
Roller — vertical support	u_y, u_z	R_y, R_z	a bearing that allows thermal expansion
Roller — horizontal support	u_x, u_z	R_x, R_z	a side guide against a wall
Custom	whatever you tick	the matching reactions	lateral bracing, a torsion collar, a guided end

Determinacy in the plane

For the in-plane problem the count is the familiar one. With r reaction components and a bar that has no internal hinges:

r = 3 — statically determinate. The three equilibrium equations ΣF_x = ΣF_y = ΣM = 0 give you everything, and the stiffnesses do not matter at all.
r > 3 — statically indeterminate. Two pinned bases give r = 4 and the frame is indeterminate to the first degree; two fixed bases give r = 6 and it is indeterminate to the third. Now the answer does depend on EI and on the ratio of member stiffnesses. FrameLab solves this automatically.
r < 3, or a bad arrangement (all reactions parallel, or all lines of action through one point) — a mechanism. You get an error instead of a result.

The trap: three parallel reactions

Counting alone is not enough. Three vertical rollers give r = 3 but cannot resist any horizontal load — the count is right and the frame still moves. FrameLab detects this the honest way: the stiffness matrix turns out to be singular and it reports a mechanism.

Out-of-plane stability is a separate question

A frame can be perfectly stable in its own plane and completely unrestrained out of it. If you apply an out-of-plane load and nothing blocks u_z, you will get a warning. And a bar that happens to be straight between two pins has one more subtlety: the pins block u_z but nothing stops the bar rotating about the line joining them. That is a torsional mechanism, and the cure is to tick θ about the bar axis at one support (use the custom support type) — physically, a fork bearing or a welded collar.

Example — the supports decide everything

The same portal frame with the same load, twice: once on two pins, once on two clamped bases. Compare the corner moment, the sag of the beam, and the horizontal reaction.