End Restraint and Buckling of a Cold-Formed Section with pyCUFSM

A signature curve is usually drawn for simply supported ends, but real members are rarely pinned at both ends. This page runs the same cold-formed section under four different end restraints and shows where that choice matters and where it makes no difference at all. The analysis is a live finite strip run using pyCUFSM, so you enter arbitrary geometry as a table rather than picking a fixed standard case, and you get buckling diagrams and contour plots alongside the numbers.

How it works

End restraint acts on the longitudinal shape of the buckle, not on the cross-section. In a finite strip analysis the displacement along the member is assumed to follow a series whose form depends on the boundary condition: a single half sine wave for simple supports, a shape with zero slope at both ends for clamped supports, and so on. From that one fact the whole result follows.

• Global buckling is strongly affected. It is a member-length phenomenon, so restraining the ends against rotation shortens the effective length and raises the load, in the familiar Euler way.
• Local buckling is almost unaffected. It occurs at a half-wavelength of the order of the plate width, so a member many times longer than that contains many local half-waves and the ends are simply too far away to matter.
• Distortional buckling sits in between, since its half-wavelength is intermediate. On a short member, end restraint can lift it noticeably; on a long one it cannot.

The practical consequence is that a signature curve computed for simple supports is a perfectly good source of local and distortional buckling loads regardless of the real end conditions, whereas the global branch has to be evaluated for the real restraint. This page quantifies that rather than asserting it.

Limits and caveats

For end restraints other than simple supports the horizontal axis stops being a pure half-wavelength axis and becomes closer to a physical member length, because the assumed longitudinal shape is no longer a single half sine wave. Compare the curves at equal x with that in mind: the comparison is meaningful, but the axis does not mean quite the same thing for every curve.

The clamped to free case is a cantilever and is fundamentally different from the other three. Its buckling load is far lower at a given length, and it is included to show the range, not because it is a common column condition.

Real end conditions are almost never ideal. A bolted base plate is neither pinned nor fixed, and restraint often differs about each axis and against warping. Where that is the case, the conventional approach is to compute local and distortional buckling from the simply supported curve, then evaluate the global branch for the restraint you can actually justify.

Does end restraint change local buckling?

Almost never in a member of realistic length. Local buckling forms at a half-wavelength of the order of the plate width, so a long member contains many local half-waves and the end conditions are too far away to influence them. You can read local buckling loads straight off the simply supported signature curve.

Then why bother analysing other end restraints at all?

Because global buckling is a member-length phenomenon. Restraining the ends against rotation shortens the effective length and raises the global buckling load in the familiar Euler way. If your member is governed by global buckling, the simply supported curve can be substantially conservative or, if you assume more fixity than exists, unsafe.

What about distortional buckling?

It sits between the two. Its half-wavelength is intermediate, so on a short member end restraint can lift the distortional load noticeably, while on a long member it cannot. Check it rather than assume, especially for shorter members.

Why does the clamped to free curve look so different?

Clamped to free is a cantilever, not a column with two supported ends. Its buckling load is far lower at a given length. It is included to show the full range of behaviour, not because it represents a common column condition.

Does the x-axis mean the same thing on every curve?

No. For simple supports the horizontal axis is a pure half-wavelength axis. For the other restraints the assumed longitudinal shape is no longer a single half sine wave, so the axis behaves more like a physical member length. Comparing curves at equal x is still meaningful, but keep that difference in mind.

How should I handle a real, imperfect end condition?

Real ends are rarely ideal. A bolted base plate is neither pinned nor fixed, and restraint often differs about each axis and against warping. The usual approach is to take local and distortional buckling from the simply supported curve, then evaluate the global branch for the restraint you can genuinely justify.

Can I run my own section?

Yes. This page runs a real finite strip analysis with pyCUFSM, so you enter arbitrary geometry as a table rather than choosing a fixed standard case, and you get diagrams and contour plots alongside the numerical results.