Buckling - Wikiparency

# Buckling **Buckling** is the sudden change of shape that a structural member undergoes when the compressive load on it reaches a critical value: a straight [[Column|column]] bows sideways, a flat plate ripples, a thin tube folds into a pattern of diamonds. It is a failure of stability rather than of strength. Below the critical load the member sits in [[Mechanical_equilibrium|equilibrium]] and any small sideways disturbance dies away; at the critical load a second, bent equilibrium becomes available at the same load, and the straight one stops being stable. Nothing in the material has broken when this happens, and the stress may be a small fraction of the [[Yield_(engineering)|yield]] strength. Because buckling is governed by stiffness and geometry rather than by strength, the numbers that decide it are the [[Young's_modulus|elastic modulus]] `E`, the [[Second_moment_of_area|second moment of area]] `I` of the cross-section, and the length over which the member is free to bow. [[Leonhard_Euler|Leonhard Euler]] published the first solution in 1744, in the appendix on elastic curves to his book on variational methods,[^euler1744] and the result still carries his name: `P_cr = π² E I / (K L)²`. One consequence is that a long column of a high-strength alloy buckles at the same load as one of ordinary steel, because `E` is nearly the same for all steels while the strength is not. A second is that every practical remedy is geometric — shorten the member, fix its ends, move material away from the bending axis, or brace it partway along. The [[Compression_member|compression member]] is only the simplest case. Plates buckle out of plane, rings and pipes collapse inward under external pressure, open sections twist as they bow, beams in [[Bending|bending]] roll sideways, and thin shells in a spacecraft's skin crumple at a fraction of the load a perfect shell would carry. What unites these forms is the mathematics: each is an eigenvalue problem whose lowest eigenvalue is the critical load and whose eigenvector is the shape the member snaps into. The framework microsim *Buckling: a column fails before it yields* is built around the first of these. The reader loads a slender column and watches it bow out long before the stress reaches yield, then changes the length, the section, the end fixity and the material and follows the column's own point as it slides along the Euler hyperbola or drops onto the Johnson parabola that caps it at the yield stress. ## Forms of buckling Buckling appears wherever a thin or slender piece of structure is pushed rather than pulled, and each geometry has its own critical load, its own shape and its own name. The forms below share one arithmetic: a critical stress proportional to `E` times the square of a thinness ratio — thickness over width for a plate, radius of gyration over length for a column, wall over radius for a shell. They differ in the constant in front, which carries the boundary conditions, and in how brutally sensitive each form is to the imperfections a real part is built with. ### Columns A column is a straight member loaded along its axis. For a column that stays elastic, the critical load is Euler's: `P_cr = π² E I / (K L)²`, where `L` is the length between supports and `K` the effective-length factor that encodes what the ends do. The four ideal cases have theoretical values `K = 1` for pinned ends, `K = 2` for one end built in and the other entirely free, `K = 0.7` for one end built in and the other pinned, and `K = 0.5` for both ends built in.[^sim] Design codes recommend larger values than the theoretical ones, because a joint that a drawing calls fixed is never quite fixed. Dividing through by the area `A` and writing `r = √(I/A)` for the [[Radius_of_gyration|radius of gyration]] converts the load into a stress — force per unit area on the cut face, in the sense the portal's physics text defines it[^up12-3] — and the geometry into one number, the [[Slenderness_ratio|slenderness ratio]] `KL/r`: `σ_cr = π² E / (KL/r)²`. The microsim's default column makes the numbers concrete. A 1020 steel bar 40 mm square and 2 m long, pinned at both ends, has `I = 2.13 × 10⁻⁷ m⁴` and `r = 40/√12 = 11.55 mm`, so `KL/r = 173`. With `E = 207 GPa`,[^mat] the critical stress is 68 MPa and the critical load is 109 kN.[^sim] The same bar would need 472 kN to squash it at its 295 MPa yield stress: the column gives way at less than a quarter of the load the material could carry, and it does so while the steel is still perfectly elastic. Buckling about the weak axis is what matters, so the sim computes `I` for the smaller of the two principal axes; a section that is strong in one direction and thin in the other fails in the thin one.[^sim] The drawn column is a picture, not a measurement. The bowed shape on screen is exaggerated by a factor the reader sets, and the sim labels it ILLUSTRATIVE for that reason: a real column at its critical load is still visibly straight, its mid-height deflection being of the order of the imperfection it started with. The column's drawn height is likewise not to scale against its width, and the four end-condition mode shapes it draws — a half sine, a `1 − cos` curve, a versine and the transcendental curve of `tan kL = kL` — are drawing geometry rather than a computed deflection.[^sim] *Try: set the section to 40 × 40 and drag the length from 0.5 m to 6 m — the gold point labelled "this column" slides right along the column curve while P_cr falls as `1/L²`; then switch the end fixity from pinned-pinned to fixed-free and watch the same column lose three quarters of its capacity as K goes from 1 to 2.* #### Self-buckling A column standing under nothing but its own weight has a greatest possible height. The load now grows from zero at the top to the full weight at the base, so the governing differential equation is no longer one with constant coefficients; its solution is a [[Bessel_function|Bessel function]] of order one third, and the first zero of that function sets the critical length. A. G. Greenhill worked the problem out in 1881 and asked it in the form that still names it: how tall a pole, or a tree of given proportions, can be made before it falls over sideways.[^greenhill] The result is `L_max = (7.8373 · E I / (ρ g A))^(1/3)`, with `ρ` the density and `A` the area. A round steel rod 20 mm in diameter reaches 8.1 m by this formula, and because the length grows only as the cube root of `EI/ρA`, doubling the diameter buys a height increase of about 60 per cent rather than a factor of four. The same scaling constrains how tall trees can grow, which is the question Greenhill set out to answer. ### Thin-walled rings A ring or a long tube under uniform external pressure has no sideways load to speak of, and yet it collapses. The circle is in compression all round, and above a critical pressure an oval shape becomes available at the same pressure. For a long tube of mean diameter `D` and wall thickness `t` the elastic collapse pressure is `p_cr = 2 E / (1 − ν²) · (t/D)³`, a form that goes back to the nineteenth-century ring analyses and is reproduced in the standard treatment of elastic stability.[^timoshenko] The cube is severe. A steel pipe 300 mm in diameter with a 3 mm wall collapses at about 0.46 MPa — under five atmospheres of external pressure — while the same pipe would hold many times that pressure applied from the inside, where the wall is in tension and no stability question arises. [[Cylinder_stress|Hoop stress]] governs the inside case; stability governs the outside one. It is why a drinking straw is trivially crushed by suction and why the casing of an [[Oil_well|oil well]], a [[Submarine_hull|submarine hull]] and the tube of a condenser are all checked for external-pressure collapse rather than for strength alone. A thin cylinder pushed along its axis rather than squeezed from outside is the most imperfection-sensitive structure in the catalogue. The classical elastic critical stress, `σ_cr = 0.605 E t / R`, predicts 501 MPa for a 2 mm wall on a 0.5 m radius, and real cylinders commonly fail between a fifth and a half of that, because a dent of the order of the wall thickness is enough to set the collapse off. NASA's design criteria monograph on thin-walled circular cylinders codified the practice of multiplying the classical value by an empirical knockdown factor fitted to test data, and the practice survives, with revised factors, in current shell design.[^sp8007] The regular array of diamond facets such a cylinder collapses into is [[Yoshimura_buckling|Yoshimura buckling]], the same pattern a drink can takes when it is crushed lengthwise, and the reason a [[Shell_(structure)|shell]] is the one form where a test programme, not a formula, usually has the last word. ### Plate buckling A flat plate compressed in its own plane buckles into waves out of plane. Unlike a column, a plate has a second direction in which to carry load, so the buckled plate does not collapse: the middle goes slack and the material near the supported edges keeps taking load, a reserve known as post-buckling strength that thin-skinned aircraft structures are designed to use. The elastic critical stress is `σ_cr = k π² E / (12 (1 − ν²)) · (t/b)²`, where `b` is the width between supported edges and `k` a coefficient carrying the edge conditions and the aspect ratio: about 4.0 for a long plate simply supported on all four edges, and about 0.4 for one with a free edge.[^timoshenko] A 6 mm steel plate 360 mm wide, supported along both edges, buckles at 208 MPa, below the 295 MPa at which it would yield;[^mat] narrow the same plate to 300 mm and the elastic critical stress rises above yield, so the plate squashes instead of rippling. Design rules for [[Structural_steel|structural steel]] are written as limits on the width-to-thickness ratio for exactly this reason: keep `b/t` low enough and the section reaches its yield stress before it buckles. The [[I-beam|I-beam]] whose flange outstands are too wide, and the gusset plate whose unsupported edge is too long, are both failures of that rule, and the second of them is the subject of the [[Minnesota]] section below. ### Flexural-torsional buckling A column whose cross-section is open and thin-walled — a channel, an angle, a tee — has a third way to fail. Its shear centre does not coincide with its centroid, so as the column bows it also twists, and the two motions are coupled: the critical load of the combined flexural-torsional mode can be well below either the pure bending mode or the pure twisting mode taken alone. The coupling strength depends on the distance between centroid and shear centre and on the section's [[Torsion_constant|torsion constant]], which for an open thin-walled section is tiny — a consequence worked out in [[Torsion_(mechanics)|torsion]]. A closed tube of the same weight has a torsion constant hundreds of times larger and does not suffer the mode at all, which is one reason why compression members in towers and masts are so often tubes. ### Lateral-torsional buckling A deep, narrow beam bent about its strong axis can roll over sideways. The compression flange behaves like a column with nothing to lean against, and the beam escapes by displacing laterally and twisting at once. For a simply supported beam under uniform moment the classical critical moment is `M_cr = (π/L) √(E I_y G J)`, so the resistance is the geometric mean of the weak-axis bending stiffness `E I_y` and the torsional stiffness `G J`.[^timoshenko] Both are small for the deep, thin-webbed shapes that are most efficient in bending, which is why lateral restraint — a floor deck, a tie, a line of bracing — is what makes such a beam usable. Halving the unbraced length doubles `M_cr`; a beam fully braced along its compression flange reaches its full plastic moment instead. ### Plastic buckling The Euler curve is a hyperbola that runs to infinity as the column gets short, which is nonsense: no column can carry more than its squash load. Short columns yield before they buckle, and columns in between do both — part of the section goes plastic, the effective stiffness falls, and the member gives way at a load below both the elastic prediction and the squash load. The engineering fix is to cap the hyperbola with a parabola tangent to it. In the form the microsim uses, `σ_cr = S_y − (S_y (KL/r) / (2π))² / E`, the parabola meets the Euler curve at the transition slenderness `C_c = √(2 π² E / S_y)`, where the Euler stress is exactly half the yield stress.[^shigley] For 1020 steel with `S_y = 295 MPa` that transition sits at `KL/r = 118`, and at that point both curves read 147 MPa.[^sim] For 6061-T6 aluminium, which is nearly a third as stiff but of comparable strength, the transition falls at `KL/r = 70`:[^mat] an aluminium column of a given shape enters the plastic regime at a shorter length than a steel one. The sim divides the world at `C_c` and prints the regime beside the critical stress, and in the plastic regime it also prints the uncalculated Euler load so the gap between the two is visible.[^sim] The tangent-modulus arguments that justify the cap are the subject of a long literature; the parabola is the workaday version of them. *Try: load the `Slenderness_ratio` variant, which opens on a 60/50 mm tube 2.3 m long at exactly `KL/r = 118` — the point sits on the join where the red parabola meets the blue hyperbola at half the yield stress; shorten the tube and watch the point climb the parabola toward S_y, lengthen it and watch it fall away down the hyperbola.* ### Crippling Crippling is the local failure of a thin-walled section when its individual flat elements buckle and then, unlike an ideal plate, lose the ability to carry more. A stiffened aluminium panel's flanges ripple first, load sheds to the corners where the elements meet and are held straight, and the section fails when those corners yield. The result is a crippling stress for the section as a whole that lies between its plate-buckling stress and its yield stress, and that is estimated in aerospace practice from empirical curves fitted to element width-to-thickness ratios rather than from a closed-form solution.[^john009] Crippling is a short-column phenomenon: it sets the ceiling that the column curve is capped at for built-up thin sections, in the same place the Johnson parabola is capped at `S_y` for solid ones. ### Diagonal tension When the thin web of a deep beam buckles in shear, the beam does not fail. The web gives up carrying compression along one diagonal and continues to carry tension along the other, becoming in effect a field of parallel tension straps between the flanges and stiffeners, which then take the compression as struts. Herbert Wagner set out the theory of this tension field in 1929, and it reached English-language practice through a three-part NACA translation in 1931.[^wagner] The consequence for design is that a shear web may be allowed to buckle in normal service, provided the uprights are sized for the compression the tension field throws onto them and the flanges for the inward pull; a web designed this way is far lighter than one thick enough never to buckle. It is the reason the skin of an aircraft wing can be seen to ripple in flight without anything being wrong, and it is the oldest deliberate use of post-buckling strength in engineering. ### Dynamic buckling A load applied suddenly behaves differently from one applied slowly. A column struck on the end can briefly carry many times its static critical load, because the bending motion needs time to develop and the stress wave reaches the far end before the column has moved sideways; conversely, a pulse well below the static critical load, repeated or sustained long enough, can drive the column into a growing oscillation. The relevant measure is no longer a single critical load but a boundary in the plane of pulse amplitude against pulse duration. Dynamic buckling governs [[Impact_(mechanics)|impact]] energy absorption in vehicle crash structures, where the aim is the opposite of the usual one: the member is designed to buckle, in a controlled progressive fold that dissipates energy at a nearly constant force. ## Theory Every form above is one boundary value problem. Write the equilibrium of the slightly deflected member, and for most loads the only solution is the undeflected one; at particular loads a non-zero solution appears. Those loads are the eigenvalues and the shapes the eigenvectors, and the smallest eigenvalue is the critical load in practice because the structure reaches it first. For the pin-ended column, the equation is `E I y'' + P y = 0` with `y(0) = y(L) = 0`, whose non-trivial solutions are `y = C sin(nπx/L)` at `P = n² π² E I / L²`; the case `n = 1` is Euler's load and the higher ones are the modes a braced column can be pushed into. The same structure of argument — an [[Ordinary_differential_equation|ordinary differential equation]] or a variational statement, a [[Boundary_value_problem|boundary value problem]], a discrete spectrum — carries over to plates, shells and frames, where the algebra becomes a matrix eigenvalue problem solved by the [[Finite_element_method|finite element method]]. ### Energy method The equilibrium route says nothing about whether the equilibrium found is stable. The energy route does. Write the total potential energy of the loaded member as the strain energy stored in bending minus the work the load does as the member shortens: `Π = ∫ (E I /2) (y'')² dx − (P/2) ∫ (y')² dx`. Equilibrium is a stationary point of `Π`; the equilibrium is stable when `Π` is a minimum. For a trial shape `y(x)` the two integrals give a ratio — a [[Rayleigh_quotient|Rayleigh quotient]] — whose value is an estimate of the critical load, and because the true mode is the one that minimises it, any assumed shape gives an estimate that is too high. That one-sided error makes the method safe to use in reverse: pick a plausible shape, compute the quotient, and know that the real column buckles at or below the answer. Taking the parabola `y = 4δx(L − x)/L²` for a pin-ended column, for instance, returns `12 E I/L²` against Euler's `π² E I/L² = 9.87 E I/L²`, high by 22 per cent; taking a half sine returns the exact value because a half sine is the exact mode. The method generalises directly to [[Virtual_work|virtual work]] formulations and to the assembled stiffness matrices of a computer model, where the same quotient reappears as a generalised eigenvalue problem. *Try: open the `Euler's_critical_load` variant, which opens on a 2.5 m pinned 1020 steel column far out on the Euler branch at `KL/r = 217` and `P_cr = 70 kN` — step the end-fixity select through all four K factors printed on the equation line and watch the same column's critical load change by a factor of sixteen between fixed-free and fixed-fixed while nothing about the material or the section moves.* ### Single-degree-of-freedom models The whole of stability theory can be seen in a rigid bar on a spring. Stand a rigid link of length `L` upright on a pin with a rotational spring of stiffness `k` at its base, put a load `P` on top, and tip it through an angle `θ`. The spring's restoring moment is `kθ`; the load's overturning moment is `P L sin θ`. For small angles these balance at `P_cr = k/L`, independent of `θ`: the model has a [[Pitchfork_bifurcation|pitchfork bifurcation]] at that load, exactly as the real column does. Variants of the model — a linear spring at the top instead of a rotational one at the base, two links instead of one, a spring whose stiffness falls with deflection — reproduce the whole zoo of post-critical behaviour, stable-symmetric, unstable-symmetric and asymmetric, in a single equation; the discrete models A, B and C of the portal's aerospace-structures text are set up for exactly this purpose.[^john009] The models also explain why real structures never reach the theoretical load. Give the bar an initial lean `θ₀` instead of standing it perfectly straight and the bifurcation disappears: the deflection grows smoothly from the first increment of load, following `θ = θ₀/(1 − P/P_cr)`, and rises toward infinity only as `P` approaches the critical value. The imperfect column has no sharp event to fail at; it simply deflects more and more, and it fails when the bending stress it accumulates reaches yield, which happens at a load below `P_cr`. The relation can be inverted to measure the critical load without breaking anything: plotting the deflection against deflection-over-load gives a straight line whose slope is `P_cr`, the [[Southwell_plot|Southwell plot]], which is how a critical load is extracted from a test on a real, slightly crooked specimen.[^john009] Design formulas such as the [[Perry–Robertson_formula|Perry–Robertson formula]] build an assumed initial crookedness into the column curve for the same reason. ## Engineering examples Buckling is rarely the failure a non-specialist expects, because the member that fails is intact afterwards and the load that broke it was not large. The cases below span five orders of magnitude of size, and every one is a stiffness problem with a strength problem standing behind it. ### Bicycle wheels A [[Bicycle_wheel|bicycle wheel]] is a rim held in compression by spokes in tension. Every spoke pulls inward, so the rim carries a compressive hoop load all the way round, and the wheel is an assembly held together by prestress: a rider's weight is carried not by the spokes below the hub pushing, but by those above the hub pulling slightly less hard. Tightening the spokes raises the rim's compression, and past a critical tension the rim buckles laterally out of its plane — the taco the wheel becomes when it is overtensioned or struck sideways. The critical tension depends on the rim's lateral bending stiffness and its torsional stiffness together, the same pairing that governs lateral-torsional buckling of a beam, so a deep aerodynamic rim can hold much more spoke tension than a shallow box one. A wheel that is true and evenly tensioned distributes the hoop load uniformly; one with a few very tight spokes buckles locally at a fraction of the average tension, in the usual way of imperfection-sensitive structures. ### Roads Concrete [[Road_surface|pavement]] is cast in slabs with joints between them, and the joints exist because concrete expands when it is heated. Seal those joints with incompressible grit over several seasons and the slabs can no longer grow; a hot afternoon then puts the slab into compression along its length with nothing to relieve it, and a slab that cannot expand in plane buckles out of it. The result is a blowup: the pavement lifts at a joint, sometimes several centimetres in a few seconds, and sometimes with enough violence to throw a vehicle. The mechanism is that of a long plate compressed in its own plane and restrained at its edges, and the remedies are all geometric — joints wide enough and clean enough to close, or continuously reinforced slabs in which the steel spreads the movement into many fine cracks instead of a few wide joints. ### Rail tracks Continuously welded rail has no expansion joints at all, so a rise in temperature that the rail cannot accommodate by growing longer becomes a compressive stress directly: `σ = E α ΔT`. With `E = 207 GPa` and a nominal coefficient of thermal expansion of `12 × 10⁻⁶ K⁻¹` for steel, a rail 30 K above the temperature at which it was laid carries 75 MPa of compression.[^mat] A 60 kg/m rail has a cross-sectional area of about 7,640 mm², obtained by dividing its mass per metre by the density of steel,[^railcalc] so that stress is a compressive force of 570 kN in each rail, 1.1 MN in the pair — and it is there whether or not a train is. The track resists it as a very long column on an elastic foundation: the sleepers and the ballast shoulder supply the lateral restraint, the [[Railway_track|track]]'s own lateral stiffness supplies the rest, and the critical condition is a balance between the axial force and that restraint over a wavelength of several metres. When the balance fails the track buckles sideways into a lateral kink, often several metres long and tens of centimetres out of line, at a place where the ballast was disturbed or the alignment already slightly off. The countermeasures are the ones stability theory predicts: lay and destress the rail at a stress-free temperature near the middle of the local range, so that neither summer compression nor winter tension is extreme — the practice known as [[Rail_stressing|rail stressing]] — keep a full ballast shoulder to supply lateral restraint, watch for the disturbed track that follows tamping or renewal, and impose speed restrictions on hot days when a kink is likelier and a train's dynamic push can trigger one. [[Railroad_tie|Sleepers]] that are heavier and better keyed into the ballast raise the critical force directly. ### Pipes and pressure vessels A [[Pressure_vessel|pressure vessel]] under internal pressure is a strength problem; the same vessel under external pressure, or under vacuum inside, is a stability problem, and the two are not close in magnitude. The collapse pressure of a long cylinder falls as the cube of wall-over-diameter, so halving the wall of a pipe divides its collapse pressure by eight while dividing its burst pressure only by two. Vessels that must resist external pressure are therefore designed with stiffening rings at intervals — each ring shortens the length over which the shell is free to lobe, exactly as bracing shortens a column — rather than with a uniformly thicker wall. Vacuum jackets, condenser shells, [[Casing_(borehole)|well casing]] under formation pressure, and the [[Submarine_hull|pressure hull]] of a submarine are all ring-stiffened for this reason, and the rings themselves must be checked against the ring-collapse formula above. Silos and tanks have a further mode: a [[Grain_elevator|grain silo]] emptied too fast can develop enough internal vacuum, or enough downward friction drag from the flowing contents, to buckle its own wall. ### Super- and hypersonic aerospace vehicles Fast flight adds heat to the list of loads. A skin panel restrained at its edges by the structure behind it cannot expand when aerodynamic heating raises its temperature, so the restraint turns the [[Thermal_expansion|thermal expansion]] into compression in exactly the way it does in a welded rail, and the panel buckles at a temperature rise rather than at a load. Thermal buckling of [[Skin_(aeronautics)|skin]] panels is therefore a design condition for supersonic and hypersonic vehicles, alongside the mechanical compression a [[Monocoque|semi-monocoque]] skin already carries in bending. Because a buckled panel is stiff in a different way than a flat one, its natural frequencies change and its interaction with the airflow changes with them, which couples the stability problem to panel flutter. The structural answers are the familiar geometric ones — closer stiffener spacing to shorten the panel, [[Honeycomb_structure|honeycomb]] and [[Sandwich_panel|sandwich]] construction to buy bending stiffness per unit mass, and deliberate expansion provisions so the skin is not fully restrained — supported by a knockdown philosophy for the shells, since a thin curved skin is the most imperfection-sensitive shape in the catalogue.[^sp8007] ## Minnesota *This section is specific to Wikitube.* The [[I-35W_Mississippi_River_bridge|I-35W bridge]] over the Mississippi River in [[Minneapolis|Minneapolis]], [[Minnesota]], collapsed at about 6:05 p.m. on August 1, 2007, killing 13 people and injuring 145.[^ntsb] The National Transportation Safety Board determined the probable cause to be "the inadequate load capacity, due to a design error … of the gusset plates at the U10 nodes," which failed under the weight added by earlier modifications together with the traffic and the construction loads staged on the deck that afternoon.[^ntsb] A [[Gusset_plate|gusset plate]] is the plate that joins the members of a [[Truss_bridge|truss]] at a node, and the U10 plates joined a compression diagonal to an upper chord. The buckling interest is in what the plates looked like beforehand. Photographs taken in 1999 and 2003 show visible bowing in the plates at all four U10 nodes, measured on the inside plates at between 0.44 and 0.99 inch out of plane.[^ntsb] The unsupported edge in question was about 30 inches long on a plate 0.5 inch thick — a ratio of 60, where the design specification of the day required an edge to be stiffened above a ratio of 48.[^ntsb] That rule is the plate analogue of the column's `KL/r`: a cap on a slenderness, written so that a free edge reaches its material's strength before it goes unstable. The Safety Board also found that the 24 plates at the three critical node groups should have been about 1 inch thick, twice their specified 0.5 inch, for acceptable capacity.[^ntsb] The honest reading is the one the report itself gives, and it is the lesson the microsim's column curve teaches rather than the one a casual account would draw. The Board concluded that although the U10 plates would have required edge stiffeners under the specification, "the addition of stiffeners would not have made the U10 gusset plates adequate or prevented the gusset plates from yielding": the plates were beyond yield stress under dead load alone, and they failed at the strength end of the curve, not the stability end.[^ntsb] The report is careful even about the word, noting that "bowing" there describes the appearance of the plates and is not the technical term an inspector would use for the distortion.[^ntsb] What the bowing did do was give a visible warning that went unread for at least eight years; the Board's contributing cause names the practice of giving inadequate attention to gusset plates during inspections "for conditions of distortion, such as bowing."[^ntsb] A member that has buckled looks wrong long before it fails, which is the practical argument for the whole subject. ## See also - [[Euler's_critical_load]] - [[Slenderness_ratio]] - [[Johnson's_parabolic_formula]] - [[Perry–Robertson_formula]] - [[Self-buckling]] - [[Yoshimura_buckling]] - [[Radius_of_gyration]] - [[Second_moment_of_area]] - [[Bending]] (section 4) - [[Torsion_(mechanics)]] (section 6) - [[Engineering_tolerance]] (section 25) - [[Material_selection]] (section 30) - [[Structural_engineering]] ## References [^euler1744]: Euler, Leonhard (1744). *Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes*, Additamentum I, "De curvis elasticis." Lausanne and Geneva. (The appendix on elastic curves in which the critical load of a column first appears; place and publisher as recalled, the year standard in histories of mechanics.) [^john009]: Johnson. *Aerospace Structures*. Portal Book 009, pp. 303–318 (buckling and stability: the discrete models A, B and C, bifurcation, the imperfect column and the Southwell construction) and p. 349 (the Euler load `P_cr = π² E I / (K L)²`); cited by the sub-manual `02_mechanics` §10.1. (Author's given names are not carried in the portal's book list.) [^up12-3]: OpenStax (2016). *University Physics Volume 1*. Ling, S. J.; Sanny, J.; Moebs, W. Rice University. §12.3 "Stress, Strain, and Elastic Modulus," pp. 587–600 — stress as force per unit area, elastic strain and Young's modulus, the quantities the column curve is written in. https://openstax.org/details/books/university-physics-volume-1 (Portal Book 077). [^mat]: `solid.mech.MATERIALS` in the Wikitube framework library, rows `steel_1020` (`E = 207 GPa`, `S_y = 295 MPa`) and `al_6061_T6` (`E = 68.9 GPa`, `S_y = 276 MPa`), transcribed from Callister, *Materials Science and Engineering: An Introduction*, Appendix B; the values the microsim computes with. Steel's nominal coefficient of thermal expansion, `12 × 10⁻⁶ K⁻¹`, and density, 7,850 kg/m³, are textbook nominals used here for the rail and self-buckling arithmetic. [^sim]: Wikitube engineering-portal run, September 18, 2026: sim spec `specs/sims/Buckling.json` and the job E-B build report in `plans/reports_engrun_2026-09-18.md`. The report's hand check reproduces the numbers used here — `r = 40/√12 = 11.55 mm`, `KL/r = 173.2`, `σ_cr = π² × 207 GPa / 173.2² = 68.1 MPa`, `P_cr = 109 kN`, `C_c = 117.7` — and records the theoretical K factors (1, 2, 0.7, 0.5), the weak-axis `I_min` convention, the uncorrected Euler load printed in the Johnson regime, and the ILLUSTRATIVE items: the exaggerated deflection amplitude, the drawn column height, and the four drawn mode shapes as drawing geometry. [^shigley]: The Johnson parabola `σ_cr = S_y − (S_y (KL/r)/(2π))²/E` and the transition slenderness `C_c = √(2 π² E / S_y)` in the Shigley / AISC form, as carried by `design.beam.johnson` and `design.beam.transitionSlenderness` in the Wikitube framework library. Standard machine-design form; no page pinned for this article. [^greenhill]: Greenhill, A. G. (1881). "Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and of the greatest height to which a tree of given proportions can grow." *Proceedings of the Cambridge Philosophical Society* 4: 65–73. (Volume and pages as recalled; the result, `L_max = (7.8373 E I/(ρ g A))^(1/3)`, is standard and follows from the first zero of the Bessel function of order −1/3.) [^timoshenko]: Timoshenko, Stephen P.; Gere, James M. (1961). *Theory of Elastic Stability*, 2nd ed. New York: McGraw-Hill. The standard source for the plate-buckling coefficient `k`, the ring and long-cylinder collapse pressure, and the lateral-torsional critical moment `M_cr = (π/L) √(E I_y G J)`. (Chapter and page not re-checked for this article.) [^wagner]: Wagner, Herbert (February 1931). "Flat sheet metal girders with very thin metal web," Parts I–III. National Advisory Committee for Aeronautics, Technical Memorandums NACA-TM-604, NACA-TM-605 and NACA-TM-606; translated from the German of 1929. Part III, "the stress in uprights — diagonal tension fields," is the tension-field theory cited here. https://ntrs.nasa.gov/citations/19930094810 [^sp8007]: National Aeronautics and Space Administration (August 1968). *Buckling of Thin-Walled Circular Cylinders*. NASA Space Vehicle Design Criteria (Structures), NASA SP-8007. https://ntrs.nasa.gov/citations/19690013955 (Revision 2, 2020, is NASA/SP-8007-2020/REV 2.) The source of the empirical knockdown-factor practice referred to here; the classical value `σ_cr = 0.605 E t/R` is textbook. [^railcalc]: Rail area derived in this article rather than quoted: a 60 kg/m rail divided by the density of steel, 7,850 kg/m³, gives 7.64 × 10⁻³ m² ≈ 7,640 mm², which is within a per cent of the published section area of the common 60 kg/m sections. The force figures follow from `σ = E α ΔT` with the values in [^mat]. [^ntsb]: National Transportation Safety Board (November 14, 2008). *Collapse of I-35W Highway Bridge, Minneapolis, Minnesota, August 1, 2007*. Highway Accident Report NTSB/HAR-08/03, PB2008-916203. Washington, DC. Probable cause, p. 152; bowed gusset plates at the U10 nodes, p. 61 and the analysis at p. 143 (measured bowing 0.44–0.99 in, unsupported edge 30 in on a 0.5 in plate for a ratio of 60 against the specification limit of 48); the FHWA finding that 24 plates should have been about 1 in thick; and the conclusion that edge stiffeners "would not have made the U10 gusset plates adequate or prevented the gusset plates from yielding." https://www.ntsb.gov/investigations/AccidentReports/Reports/HAR0803.pdf  **Microsim — three.js (Wikitube framework):** *Buckling* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Buckling.html" data-title="Buckling"></div> *Built from `MICROSIM_GUIDE/specs/sims/Buckling.json`; part of the [[PORTAL_Engineering|Engineering portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Buckling) : [Wikitube](https://en.wikitube.io/wiki/Buckling) - skeleton pinned to revision 1362996017 (2026-09-18).