Compression Spring Buckling Calculator

The deflection at which a compression spring goes unstable, from the published stability criterion, with all four end conditions computed side by side — and the critical free length 2.63·D/α worked out from the material’s own moduli.

Last updated 24 Sep 2026

Compression spring buckling and stability

Free length, diameter, end condition → critical deflection

Free length L₀ (mm)Buckling is a geometry problem: it depends on the free length and the mean diameter and on nothing else about the spring — not the wire, not the coil count, not the load.

Outside diameter OD (mm)Mean diameter D = OD − d. The stability criterion is the ratio L₀/D, so the mean diameter is the denominator of the whole page.

Wire diameter d (mm)Used only to get the mean diameter from the outside diameter. The buckling criterion does not otherwise depend on the wire.

End conditionThis is the whole page. The same spring is stable guided and unstable unguided, and the four published constants span a factor of four in allowable free length.

Wire materialThe criterion depends on E and G through C₂′ = 2π²(E−G)/(2G+E), so it is slightly different for stainless and very different for phosphor bronze. The familiar 2.63 is the carbon-steel value of √C₂′.

Working deflection (mm)How far the spring is actually compressed in service. Buckling is a deflection criterion, not a load one: a spring that is stable at 20 per cent of its free length may buckle at 40.

Not a circuit: the published stability chart, computed from the criterion rather than traced from a figure. The horizontal axis is the slenderness ratio L₀/D and the vertical axis is the deflection at which the spring goes unstable, as a percentage of its free length. The right-hand curve is both ends squared, ground and resting on flat parallel surfaces — guided, α = 0.5. The left-hand one is one end on a flat surface and the other pivoting, α = 0.707. Each curve ends at a vertical wall, and to the left of its own wall the spring cannot buckle at any deflection whatever: those walls are at 5.25 and 3.71 times the mean diameter, which differ by a factor of √2 and not by the factor of two often claimed. The moving vertical line is your spring's slenderness and the moving horizontal one is your working deflection; where they cross tells you which side of which curve you are on. Both lines move in one per cent steps, which is finer than the stroke width.

26.1%Example

A 100 mm free length spring on a 16 mm outside diameter with 2 mm wire — a 14 mm mean diameter — squared and ground at both ends on flat parallel surfaces, compressed 20 mm

A spring buckles like a column, with its own two constants

λ = αL₀/D · y_cr = L₀·C₁′[1 − √(1 − C₂′/λ²)] · C₁′ = E/(2(E−G)) · C₂′ = 2π²(E−G)/(2G+E) · (L₀)_cr = √C₂′·D/α ≈ 2.63D/α

λ: effective slenderness ratio: the free length over the mean diameter, times the end-condition constant. The spring’s counterpart of a column’s L/r
α: end-condition constant. 0.5 both ends on flat parallel surfaces, 0.707 one end flat and one pivoting, 1 both pivoting, 2 one clamped and one free
y_cr: the deflection at which the spring goes unstable. Note that it is a DEFLECTION criterion: the same spring is stable lightly compressed and unstable further down
C₁′: E/(2(E−G)), about 0.811 for carbon steel. It is the maximum possible critical deflection as a fraction of free length, reached exactly at the critical length
C₂′: 2π²(E−G)/(2G+E), about 6.888 for carbon steel. When C₂′/λ² exceeds 1 the square root has no real value and no deflection can buckle the spring
2.63: not a constant of nature: it is √C₂′ for carbon steel spring wire. Stainless and bronze give slightly different numbers, which this page computes

Worked example

A 100 mm free length spring on a 16 mm outside diameter with 2 mm wire — a 14 mm mean diameter — squared and ground at both ends on flat parallel surfaces, compressed 20 mm
Slenderness first: L₀/D = 100/14 = 7.143. The wire diameter does not appear again; buckling is a geometry problem in the free length and the mean diameter
The end condition sets α. Both ends squared, ground and on flat parallel surfaces is the best case there is, α = 0.5, so the effective slenderness is λ = 0.5 × 7.143 = 3.571
The two material constants, computed from E = 206,800 and G = 79,300 N/mm²: C₁′ = E/(2(E−G)) = 0.8110 and C₂′ = 2π²(E−G)/(2G+E) = 6.8877. Note that √C₂′ = 2.6244 — that is where the familiar 2.63 comes from, and it is a property of these two moduli rather than a universal number
Is it absolutely stable? Only if C₂′/λ² exceeds 1. Here it is 6.8877/12.7551 = 0.5400, which is below 1, so there IS a deflection at which this spring buckles and the calculation continues
The critical deflection: y_cr = L₀·C₁′[1 − √(1 − C₂′/λ²)] = 100 × 0.8110 × (1 − 0.6782) = 26.09 mm, which is 26.1 per cent of the free length
Your working deflection is 20 mm, which is 20 per cent — so this spring is stable, with a margin of 6.1 percentage points and a safety factor of 1.30 on deflection
NOW TAKE THE GUIDE AWAY, which is the point of the page. Let one end pivot — a ball seat, a swinging lever, a washer that can tip — and α becomes 0.707. The effective slenderness rises to 5.050 and the critical deflection falls to 11.81 mm, 11.8 per cent. Your 20 mm is now above it: the identical spring buckles
The two lengths to remember. Below (L₀)_cr = 2.63D/α no deflection can buckle the spring at all: that is 73.5 mm guided and 52.0 mm with one end pivoting. And those two differ by 0.5/0.707, a factor of 1.414 — not a factor of two. The factor of two is between guided and BOTH ends pivoted, which is a third case at 36.7 mm

The four published end conditions, for this page’s own spring

The end condition is the whole of this page and it is the thing a drawing almost never states. Note carefully what the four constants do and do not say. Guided (α = 0.5) against one end pivoting (α = 0.707) is a factor of 0.5/0.707 = 1/√2, about 1.41 — NOT a factor of two, which is a claim worth correcting because it is often made. The factor of two is between guided and BOTH ends pivoted (α = 1), and there is another factor of two below that to one end clamped and the other free (α = 2), which is four times worse than guided. The practical reading: a spring sitting between two flat washers is the 0.5 case only if both washers stay flat and parallel and the spring cannot slide sideways on them. A spring over a shoulder bolt, or on a ball seat, or pushing on a lever that swings, is not. These dimensions come from a published standard’s table, not from a formula. The standard itself is cited below and the printed values are attributed to the catalogue they were taken from; a different publisher may round differently in the last digit. This page sizes a part; it does not certify one. Where the answer carries a consequence — a load path, a lifting duty, a pressure boundary, a fastener holding something that can fall — confirm it against the design code that governs the application, and against the manufacturer’s own rating, before relying on it.
End condition	α	Critical free length	… in mm here	Buckles at this % of a 100 mm free length	… in mm
Both ends squared and ground, resting on flat parallel surfaces	0.500	5.249·D	73.5 mm	26.1 %	26.1 mm
One end on a flat surface, the other pivoting	0.707	3.712·D	52.0 mm	11.8 %	11.8 mm
Both ends pivoted (a ball seat at each end)	1.000	2.624·D	36.7 mm	5.7 %	5.7 mm
One end clamped, the other completely free	2.000	1.312·D	18.4 mm	1.4 %	1.4 mm

The 2.63 is a material property, not a universal constant

The familiar rule “a guided spring is stable below 5.26 times its mean diameter” comes from 2.63·D/0.5, and the 2.63 is √C₂′ with C₂′ = 2π²(E−G)/(2G+E). That is a function of the two moduli, so it is 2.624 for carbon steel spring wire, 2.719 for 302 stainless and 2.564 for phosphor bronze. Nobody is going to redraw the published chart for each of those, and the differences are small enough that the carbon steel figure is a reasonable default — but they go the direction that surprises people: stainless, which is SOFTER in shear, is stable a little FURTHER than carbon steel, because what the criterion depends on is the ratio of bending stiffness to shear stiffness and not on either alone. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
Material	E (GPa)	G (GPa)	√C₂′ — the “2.63”	C₁′	Guided critical length	… for a 14 mm mean diameter
Music wire	206.8	79.3	2.6244	0.8110	5.249·D	73.5 mm
Hard-drawn wire	206.8	79.3	2.6244	0.8110	5.249·D	73.5 mm
Oil-tempered wire	206.8	79.3	2.6244	0.8110	5.249·D	73.5 mm
Chrome silicon	203.4	77.2	2.6386	0.8059	5.277·D	73.9 mm
Chrome vanadium	203.4	77.2	2.6386	0.8059	5.277·D	73.9 mm
302 stainless	193.0	69.0	2.7193	0.7782	5.439·D	76.1 mm
Phosphor bronze	103.4	41.4	2.5637	0.8339	5.127·D	71.8 mm

A deflection criterion, four end conditions, and what guiding a spring costs

A compression spring buckles like a column, and the criterion is a deflection and not a load. That is the first thing to get straight, because column buckling is taught as a load problem. For a spring the useful statement is: at a given slenderness L₀/D there is a deflection, as a fraction of the free length, beyond which the spring goes unstable. A spring that sits happily at 20 per cent of its free length may throw itself sideways at 40. So the check has to be made at the maximum working deflection, not at the nominal one, and it has to be repeated if the machine ever gets heavier.

The end condition is the whole page. Four constants are published: α = 0.5 for a spring between two flat parallel surfaces, 0.707 for one end flat and one pivoting, 1 for both pivoting, and 2 for one end clamped and the other free. They enter as a multiplier on the slenderness, so the allowable free length is 2.63D/α and the four cases span a factor of four. It is worth correcting a claim that is often made: guided against one-end-pivoting is a factor of 0.5/0.707, about 1.41, and NOT a factor of two. The factor of two is between guided and both ends pivoted, which is a different case again. Either way the practical point stands: the same spring is comfortable guided and unstable unguided, and drawings almost never say which it is.

2.63 is not a constant of nature. It is √C₂′ with C₂′ = 2π²(E−G)/(2G+E), which for carbon steel spring wire at E = 206.8 and G = 79.3 GPa comes out at 2.6244. Change the material and it changes: this page computes it for each of seven wires, and the direction surprises people, because the criterion depends on the ratio of bending stiffness to shear stiffness rather than on either alone. The same applies to C₁′ = E/(2(E−G)) = 0.8110, which sets the ceiling of the stability curve: a spring exactly at its critical length buckles at 81 per cent of its free length, and nothing buckles later than that.

The cure is a rod or a bore — and the cure has a cost. Guiding a spring inside a bore or over a rod removes the buckling problem rather than reducing it, and it is almost always the right answer. But a guided spring rubs. As the spring compresses, the coils grow in diameter and move axially along the guide, so there is relative sliding under contact pressure at every cycle. The consequences are real and they are not in any stability calculation: fretting at the contact, which is a crack initiation site; wear of the rod or the bore, producing debris; and the removal of any protective coating, which starts corrosion. Fretted spring wire fails in fatigue at a small fraction of its unfretted life, because the initiation stage is skipped altogether. Allow diametral clearance — enough for the coil diameter to grow as it compresses, which it does — keep the guide smooth, and treat any spring that rubs as having a finite life even if the stress calculation says otherwise.

What the criterion assumes. A perfectly straight spring, of uniform pitch and diameter, loaded exactly on its own axis, with ends that are square within tolerance and seats that are flat and parallel. Every departure from that moves the real buckling deflection DOWN, exactly as an initial bow lowers a column’s buckling load. Springs have a published squareness tolerance; seats are machined to a tolerance; the load usually arrives slightly off centre. So treat the calculated critical deflection as an upper bound, leave margin under it, and where the answer is close, guide the spring rather than arguing with the arithmetic.

Frequently asked questions

Does a spring buckle at a load or at a deflection?

At a deflection. The published criterion gives the critical deflection as a fraction of the free length, as a function of the slenderness ratio L₀/D and the end condition. The load only matters because it is what produces the deflection. This is why the check must be made at the maximum working deflection the machine can reach rather than at the nominal one, and why a spring that has always been fine can start buckling when something in the machine changes.

What is the difference between a guided and an unguided spring?

A factor of 1.41 in allowable free length between guided (α = 0.5, both ends squared and ground on flat parallel surfaces) and one end pivoting (α = 0.707), and a factor of 2 between guided and both ends pivoting (α = 1). One end clamped and the other free is α = 2, four times worse than guided. The real question is usually which of these your installation actually is — a spring against a swinging lever, on a ball seat, or under a cap that can lift is not the 0.5 case however square its ends are.

Where does the 2.63 come from?

It is √C₂′, where C₂′ = 2π²(E−G)/(2G+E). For carbon steel spring wire at E = 206.8 GPa and G = 79.3 GPa that is 2.6244. It is a material property, not a universal constant: this page computes it for stainless, bronze and the alloy steels too, and they differ. When the free length is below 2.63D/α the term C₂′/λ² exceeds 1, the square root in the criterion has no real value, and no deflection can buckle the spring at all.

What does guiding a spring do to its fatigue life?

It shortens it, sometimes drastically, and this is the cost nobody quotes. A spring in a bore or over a rod slides against the guide every cycle, because the coils move axially and grow in diameter as the spring compresses. That contact fretts the wire, and fretting produces surface damage that acts as a ready-made crack initiation site — which removes the initiation stage of fatigue life entirely. It also strips coatings and generates debris. Allow generous diametral clearance, keep the guide surface smooth, and treat a rubbing spring as having a finite life whatever the stress calculation says.

Can I stop a spring buckling without guiding it?

Three ways, all geometric. Shorten it below the critical free length for your end condition, which is the strongest fix because it removes buckling at any deflection. Increase the mean diameter — this page computes the diameter that would make your length absolutely stable. Or split it into two shorter springs in series with a rigid plate between them, which is the same fix as shortening and often the only one that fits. Improving the end condition, where the installation allows it, is free and is worth doing first.

Is this the same as column buckling of a structural member?

Mathematically it is the same kind of instability, but the constants are different and the two are not interchangeable. A helical spring is a shear-flexible column: it is far more compliant in shear relative to bending than a solid member is, which is why the criterion involves both E and G rather than just E, and why the answer comes out as a critical deflection rather than a critical load. Structural column buckling of building members is a separate subject and is not covered here.

Related calculators

References

R. G. Budynas and J. K. Nisbett, Shigley’s Mechanical Engineering Design, chapter 10 “Mechanical Springs”. The source for the four-way end-condition table (Table 10-1, itself attributed there to Associated Spring’s Design Handbook), for the Bergsträsser factor, for the buckling constants C₁′ and C₂′, for the torsion-spring constant 10.8, and for the S_ut = A/d^m constants. Two of its published worked solutions (problems 10-35 and 10-37) were reproduced here to four significant figures as a check on the initial-tension band, and the end-condition table was checked cell for cell against RoyMech’s independent printing.
RoyMech, Spring Stability. The independent printing of C₁′ = E/(2(E−G)), C₂′ = 2π²(E−G)/(2G+E), the effective slenderness ratio λ_eff = αL₀/D and the four end-condition constants α = 0.5, 0.707, 1 and 2. The 2.63 in the familiar rule is not taken from either source: it is computed here as √C₂′, which gives 2.6244 for carbon steel and a different number for stainless.
Spring Manufacturers Institute, Handbook of Spring Design, and Associated Spring / Barnes Group, Design Handbook. The origin of the end-condition table, of the preferred initial-tension band plotted against spring index, and of the stability curves. Both are cited by name and NOT reproduced; the values used here were taken from the named catalogue and textbook printings below, which is this site’s standing policy on copyrighted design data.
MW Components, How to Select a Compression Spring and How to Select an Extension Spring. The source for the working-deflection window: “reserve at least the first and last 15 to 20 per cent of the range”, and “if critical force-versus-deflection linearity is required, only the centre 20 to 80 per cent of the available deflection range should be employed”. The Engineer‘s published “Top Tips for Compression Spring Design” gives the same 20 to 80 per cent window independently.
EN 13906-1, -2 and -3, Cylindrical helical springs made from round wire and bar — Calculation and design (Part 1 compression, Part 2 extension, Part 3 torsion). Cited by number, not reproduced. It is the European counterpart to the treatment on these pages and it is the document a European drawing will name; the closed forms used here (k = Gd⁴/8D³n, the Wahl correction, the bending rate of a torsion spring) are the same ones, and every figure this site prints is either computed from a definition or attributed to a named catalogue.
RoyMech, Helical Spring Design — Compression and Extension Springs and Spring Materials. Used as the independent check on the four-way end-condition table (which agrees with Shigley’s Table 10-1 cell for cell), on the SI A and m constants, and on the shear moduli. RoyMech’s own published position is worth quoting because it is the honest one: “do not use a fixed percentage of tensile strength as a universal spring allowable”.