Press a wooden ruler end to end between your palms. Long before the wood cracks, it bows sideways and stays bowed. That sideways escape has a formula, P = π²EI/(KL)², and Leonhard Euler, born in Basel in 1707, printed the mathematics behind it in 1744. A slender column does not fail when the material crushes. It fails when straight stops being the stable shape.

Look at where the length sits: in the denominator, squared. Double the unsupported length and the column keeps a quarter of its load. No grade of steel and no species of timber argues with that square. Every slim column, glazing mullion and roof-truss diagonal you have ever rendered is an Euler problem first and a material problem second.

What it is

What it is: Buckling is a loss of stability, not a loss of strength. Below a certain axial load, a straight strut that is nudged sideways springs back. Above it, the bent shape becomes an equilibrium too, and the strut takes it. That threshold is the Euler critical load. In P = π²EI/(KL)², E is Young’s modulus, the material’s stiffness. I is the second moment of area of the section, taken about its weakest axis. L is the unsupported length, and K is the effective-length factor that encodes the supports. K = 1 for a column pinned at both ends, 0.5 when both ends are fully fixed, and 2 for a free-standing cantilever. A cantilever post therefore behaves like a pinned column twice as long, and keeps a quarter of the load.

Why it works

Why it works: Write the bending equation for a pinned strut that has bowed by a small amount y. The internal bending moment is EI·y″. The axial load acting on that bow adds a moment of P·y. Equilibrium gives EI·y″ + P·y = 0, the same equation as a vibrating string. For pinned ends it has only one non-trivial solution worth knowing: a half sine wave, possible only when P reaches π²EI/L². Below that value, the only answer is y = 0, which means straight. At that value, straight and bent are equally valid, and the slightest imperfection chooses bent. This is an eigenvalue problem, and the buckling load is its smallest eigenvalue. Hold that phrase. It is exactly what your analysis software computes.

The consequence for form is the reason this desk loves the formula. I sits in the numerator, and I grows with the square of the distance between the material and the axis. Move the same steel outward, from a solid rod to a thin-walled tube, and the column gets stiffer against buckling without one extra kilogram. Shape beats material. The weakest axis governs, which is why an I-section column fails about its minor axis and why a tube, with no weak axis, is the honest shape for a strut.

The formula has assumptions, and the reference literature lists them plainly. The bar must start straight. The load must be purely axial. The section must be uniform and the material elastic, and the column must be long relative to its cross-section. A slender column usually buckles below yield, so Euler governs. A stocky column reaches yield first, and Euler’s number sits uselessly above the true capacity. Making the theory honest for that stocky range took the rest of the nineteenth century.

Origins

Origins: Euler left Basel in 1727 for St Petersburg and arrived in Berlin in 1741. His treatise Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes appeared in 1744, printed in Lausanne and Geneva by Marc-Michel Bousquet. According to the Euler Archive, it was written by April 1743. It is a book about curves that maximise or minimise a property, which makes it the founding text of the calculus of variations. Buckling arrives almost as an aside. On pages 267–268, after deriving the shapes of the elastic curve, Euler adds that these results “inservire possunt viribus columnarum dijudicandis”: they can serve in judging the strength of columns. A mathematician’s footnote became the first law of structural stability.

The builders turned it into form. In Scotland, Sir John Fowler and Sir Benjamin Baker designed the Forth Bridge, which opened on 4 March 1890 with two main spans of 518 m and 55,000 t of steel. New Steel Construction reports that Baker had researched the strength of tubular sections at Imperial College and used that principle in all the compression members of the bridge. The main tower legs are tubes 12 ft (about 3.7 m) across, and the bottom booms taper from 12 ft at the skewbacks to 6 ft at their tips. That is Euler’s I written in riveted plate at the scale of a landscape. The bridge also has an invisible hero in its most famous photograph. In Baker’s 1887 human-cantilever demonstration, the Japanese engineer Kaichi Watanabe sits in the middle, the suspended load between Fowler and Baker. His seat in that photograph made tension and compression legible to the public.

The Swiss thread runs through Zurich. Ludwig von Tetmajer, born in 1850 in Krompach, became professor of structural mechanics at the Zurich Polytechnikum, today’s ETH, in 1880. From 1881 he directed its materials-testing laboratory, the institution that became Empa. In 1886 he published the Tetmajersche Gleichung, which the German Wikipedia entry on Tetmajer describes as a supplement to Euler’s equation for the range where real bars stop obeying it. His later inquiry into the 1891 Münchenstein bridge collapse sharpened that work into a new design method. J. B. Johnson’s parabolic formula of 1893 covered the same low-slenderness range in the United States. Tetmajer’s real legacy is the laboratory: a place where a formula is checked against a bar that is actually squeezed until it gives.

←TODAY: In 2026 a Grasshopper definition returns a whole truss’s buckling factor before anyone prices the steel. →3012: The derivation, an eigenvalue of EI·y″ + P·y = 0, will outlive every file format that ever stored a column. Fulcrum: A slim column is defensible only when its maker can say which square decided it.

In practice

In practice: In a Zurich studio, buckling enters the model the moment someone drags a column slider thinner because the render looks better. Euler says that slider is the wrong control. Section depth and unsupported length carry far more weight than material grade. The Karamba3D team inside Grasshopper ships the right instrument. Grasshopper itself, which David Rutten released in 2007, made this kind of learn-while-you-build loop normal in practice. Run AnalyzeThII to get second-order normal forces N^II, then feed the model into Buckling Modes. Per the Karamba3D manual, the component checks the global instability of a sub-system such as a truss. It returns BLFacs, buckling load factors in ascending order: the factor by which the current forces could grow before the structure turns unstable. The manual also states that the method assumes small deflections up to the point of instability. Treat the first BLFac as a screening number for the architect’s desk, and the engineer’s code check as the verdict. PAZ has followed this thread before: in our timber shell piece, a single under-stiff node could govern the stability of an entire shell. The trade-off is plain. Every millimetre you take off a column for elegance costs buckling capacity roughly with the fourth power of the section size, and the render will never show it.

Atelier: Offices adopting AI-assisted massing will see generated structural grids with columns sized by eye. The model optimised the image, not the eigenvalue. Monday move: add a rule to your office’s Grasshopper template that no column or strut leaves the concept stage without a recorded slenderness L/r and a first BLFac next to its section label.

Hack: Watch the square on L take three quarters of a steel tube’s capacity away. The Python below runs in a GhPython component or any terminal. It computes the second moment of a 168.3 × 8 mm circular hollow section, then the pin-ended Euler load at 3 m and at 6 m.

import math
E, D, t = 210e9, 0.1683, 0.008             # steel Pa, tube diameter and wall in m
I = math.pi/64*(D**4 - (D - 2*t)**4)         # second moment of area, m^4
for L in (3.0, 6.0):
    print(L, "m:", round(math.pi**2*E*I/(1.0*L)**2/1e3), "kN")

You get roughly 2,990 kN at 3 m and roughly 750 kN at 6 m, a quarter, exactly as the square promised. Now compare both numbers with the squash load at 235 MPa, about 950 kN. At 3 m the steel yields long before Euler’s load, so you are in Tetmajer’s stocky range and Euler overstates the capacity. At 6 m Euler governs. Pair the script with a length slider and you can see the crossover move.

From the late 2070s, the columns I respect are not the thinnest ones. They are the ones whose makers wrote down the slenderness, the support condition and the eigenvalue, so the next engineer could rebuild the decision from the principle. Steel saved by honest geometry is the cheapest material a building ever uses. Open your current Rhino model, pick its slimmest column, and compute its buckling factor before you touch the slider again.