3104:
bend significantly and the material of the column will diverge from a linear stress-strain behavior. The stress-strain behavior of materials is not strictly linear even below the yield point, hence the modulus of elasticity decreases as stress increases, and significantly so as the stresses approach the material's yield strength. This reduced material rigidity reduces the buckling strength of the structure and results in a buckling load less than that predicted by the assumption of linear elastic behavior.
3111:, which is less than the elastic modulus, in place of the elastic modulus of elasticity. The tangent is equal to the elastic modulus and then decreases beyond the proportional limit. The tangent modulus is a line drawn tangent to the stress-strain curve at a particular value of strain (in the elastic section of the stress-strain curve, the tangent modulus is equal to the elastic modulus). Plots of the tangent modulus of elasticity for a variety of materials are available in standard references.
2781:. If the beam is not supported in the lateral direction (i.e., perpendicular to the plane of bending), and the flexural load increases to a critical limit, the beam will experience a lateral deflection of the compression flange as it buckles locally. The lateral deflection of the compression flange is restrained by the beam web and tension flange, but for an open section the twisting mode is more flexible, hence the beam both twists and deflects laterally in a failure mode known as
357:. At that load, the introduction of the slightest lateral force will cause the column to fail by suddenly "jumping" to a new configuration, and the column is said to have buckled. This is what happens when a person stands on an empty aluminum can and then taps the sides briefly, causing it to then become instantly crushed (the vertical sides of the can may be understood as an infinite series of extremely thin columns). The formula derived by Euler for long slender columns is
4175:
2762:
253:
3168:
wave to travel along the column to the other (free) end and back down as a relief wave. Maximum buckling occurs near the impact end at a wavelength much shorter than the length of the rod, and at a stress many times the buckling stress of a statically loaded column. The critical condition for buckling amplitude to remain less than about 25 times the effective rod straightness imperfection at the buckle wavelength is
4054:
231:
that member, the member will continue to support the load that caused it to buckle. If the buckled member is part of a larger assemblage of components such as a building, any load applied to the buckled part of the structure beyond that which caused the member to buckle will be redistributed within the structure. Some aircraft are designed for thin skin panels to continue carrying load even in the buckled state.
3985:
1652:
4005:
3975:
245:
2560:
boundary conditions, when a plate is loaded with a critical stress and buckles, the edges perpendicular to the load cannot deform out-of-plane and will therefore continue to carry the stresses. This creates a non-uniform compressive loading along the ends, where the stresses are imposed on half of the effective width on either side of the specimen, given by the following:
4063:
4015:
747:
34:
4123:
3614:
1406:
283:. A load at any other point in the cross section is known as an eccentric load. A short column under the action of an axial load will fail by direct compression before it buckles, but a long column loaded in the same manner will fail by springing suddenly outward laterally (buckling) in a bending mode. The buckling mode of
4422:
Aerothermal heating can lead to buckling of surface panels on super- and hypersonic aerospace vehicles such as high-speed aircraft, rockets and reentry vehicles. If buckling is caused by aerothermal loads, the situation can be further complicated by enhanced heat transfer in areas where the structure
2676:
314:
columns may be classified as short columns if the ratio of the length to least dimension of the cross section is equal to or less than 10. The dividing line between intermediate and long timber columns cannot be readily evaluated. One way of defining the lower limit of long timber columns would be to
4242:
consists of a thin rim kept under high compressive stress by the (roughly normal) inward pull of a large number of spokes. It can be considered as a loaded column that has been bent into a circle. If spoke tension is increased beyond a safe level or if part of the rim is subject to a certain lateral
1397:
is a 3-dimensional structure defined as having a width of comparable size to its length, with a thickness that is very small in comparison to its other two dimensions. Similar to columns, thin plates experience out-of-plane buckling deformations when subjected to critical loads; however, contrasted
759:
Another insight that may be gleaned from this equation is the effect of length on critical load. Doubling the unsupported length of the column quarters the allowable load. The restraint offered by the end connections of a column also affects its critical load. If the connections are perfectly rigid
741:
The boundary conditions have a considerable effect on the critical load of slender columns. The boundary conditions determine the mode of bending of the column and the distance between inflection points on the displacement curve of the deflected column. The inflection points in the deflection shape
3103:
The buckling strength of a member is less than the elastic buckling strength of a structure if the material of the member is stressed beyond the elastic material range and into the non-linear (plastic) material behavior range. When the compression load is near the buckling load, the structure will
299:
column is one whose slenderness ratio does not exceed 50; an intermediate length steel column has a slenderness ratio ranging from about 50 to 200, and its behavior is dominated by the strength limit of the material, while a long steel column may be assumed to have a slenderness ratio greater than
3167:
If a column is loaded suddenly and then the load released, the column can sustain a much higher load than its static (slowly applied) buckling load. This can happen in a long, unsupported column used as a drop hammer. The duration of compression at the impact end is the time required for a stress
2559:
allows for an increase of the number of sine waves produced by buckling along the length, but also increases the resistance from the buckling along the width. This creates the preference of the plate to buckle in such a way to equal the number of curvatures both along the width and length. Due to
230:
in the material of which the structure is composed. Further loading may cause significant and somewhat unpredictable deformations, possibly leading to complete loss of the member's load-carrying capacity. However, if the deformations that occur after buckling do not cause the complete collapse of
2744:
displacement, the fundamental path is shown. It demonstrates the plate's similarity to a column under buckling; however, past the buckling load, the fundamental path bifurcates into a secondary path that curves upward, providing the ability to be subjected to higher loads past the critical load.
3426:
964:
Since structural columns are commonly of intermediate length, the Euler formula has little practical application for ordinary design. Issues that cause deviation from the pure Euler column behaviour include imperfections in geometry of the column in combination with plasticity/non-linear stress
755:
A conclusion from the above is that the buckling load of a column may be increased by changing its material to one with a higher modulus of elasticity (E), or changing the design of the column's cross section so as to increase its moment of inertia. The latter can be done without increasing the
287:
is considered a failure mode, and it generally occurs before the axial compression stresses (direct compression) can cause failure of the material by yielding or fracture of that compression member. However, intermediate-length columns will fail by a combination of direct compressive stress and
3367:
Often it is very difficult to determine the exact buckling load in complex structures using the Euler formula, due to the difficulty in determining the constant K. Therefore, maximum buckling load is often approximated using energy conservation and referred to as an energy method in structural
4400:
The
Federal Railroad Administration issued a Safety Advisory on July 11, 2012 alerting railroad operators to inspect tracks for "buckling-prone conditions." The Advisory included a brief summary of four derailments that had occurred between June 23 to July 4 that appeared to be "heat related
2752:
Flexural-torsional buckling can be described as a combination of bending and twisting response of a member in compression. Such a deflection mode must be considered for design purposes. This mostly occurs in columns with "open" cross-sections and hence have a low torsional stiffness, such as
2743:
As the loaded stress increases, the effective width continues to shrink; if the stresses on the ends ever reach the yield stress, the plate will fail. This is what allows the buckled structure to continue supporting loadings. When the axial load over the critical load is plotted against the
3371:
The first step in this method is to assume a displacement mode and a function that represents that displacement. This function must satisfy the most important boundary conditions, such as displacement and rotation. The more accurate the displacement function, the more accurate the result.
742:
of the column are the points at which the curvature of the column changes sign and are also the points at which the column's internal bending moments of the column are zero. The closer the inflection points are, the greater the resulting axial load capacity (bucking load) of the column.
1926:
4292:
4322:
Sun kink can lead to railroads drastically reducing the speed of trains, leading to delays and cancellations. This is done to avoid derailment. Intensifying heat waves due to climate change doubled the number of hours of heat related delays in 2023, compared to 2018.
2785:. In wide-flange sections (with high lateral bending stiffness), the deflection mode will be mostly twisting in torsion. In narrow-flange sections, the bending stiffness is lower and the column's deflection will be closer to that of lateral bucking deflection mode.
41:
aircraft. Thin skin panels buckle at very low loads. In the case shown here, the weight of the forward fuselage structure ahead of the nose undercarriage is sufficient to cause the panels to buckle. Buckled panels are still effective in carrying shear by diagonal
965:
strain behaviour of the column's material. Consequently, a number of empirical column formulae have been developed that agree with test data, all of which embody the slenderness ratio. Due to the uncertainty in the behavior of columns, for design, appropriate
1647:{\displaystyle {\frac {\partial ^{4}w}{\partial x^{4}}}+2{\frac {\partial ^{4}w}{\partial x^{2}\partial y^{2}}}+{\frac {\partial ^{4}w}{\partial y^{4}}}={\frac {12\left(1-\nu ^{2}\right)}{Et^{3}}}\left(-N_{x}{\frac {\partial ^{2}w}{\partial x^{2}}}\right)}
2515:
272:(sometimes expressed with the Greek letter lambda, λ). This ratio affords a means of classifying columns and their failure mode. The slenderness ratio is important for design considerations. All the following are approximate values used for convenience.
3420:
In this method, there are two equations used (for small deformations) to approximate the "strain" energy (the potential energy stored as elastic deformation of the structure) and "applied" energy (the work done on the system by external forces).
2565:
1313:
3375:
The method assumes that the system (the column) is a conservative system in which energy is not dissipated as heat, hence the energy added to the column by the applied external forces is stored in the column in the form of strain energy.
1398:
to column buckling, plates under buckling loads can continue to carry loads, called local buckling. This phenomenon is incredibly useful in numerous systems, as it allows systems to be engineered to provide greater loading capacities.
763:
Since the radius of gyration is defined as the square root of the ratio of the column's moment of inertia about an axis to its cross sectional area, the above Euler formula may be reformatted by substituting the radius of gyration
2192:
1789:
3158:
Diagonal tension may also result in a pulling force on any fasteners such as rivets that are used to fasten the web to the supporting members. Fasteners and sheets must be designed to resist being pulled off their supports.
756:
weight of the column by distributing the material as far from the principal axis of the column's cross section as possible. For most purposes, the most effective use of the material of a column is that of a tubular section.
2970:
1056:
3127:
Because of the thin skins typically used in aerospace applications, skins may buckle at low load levels. However, once buckled, instead of being able to transmit shear forces, they are still able to carry load through
307:
column is one having a ratio of unsupported length to least dimension of the cross section equal to or less than 10. If the ratio is greater than 10, it is considered a long column (sometimes referred to as a slender
2307:
750:
A demonstration model illustrating the different "Euler" buckling modes. The model shows how the boundary conditions affect the critical load of a slender column. The columns are identical, apart from the boundary
3147:
Sheets under diagonal tension are supported by stiffeners that as a result of sheet buckling carry a distributed load along their length, and may in turn result in these structural members failing under buckling.
4247:") like a three-dimensional Euler column. If this is a purely elastic deformation the rim will resume its proper plane shape if spoke tension is reduced or a lateral force from the opposite direction is applied.
895:
4409:
Pipes and pressure vessels subject to external overpressure, caused for example by steam cooling within the pipe and condensing into water with subsequent massive pressure drop, risk buckling due to compressive
3609:{\displaystyle {\begin{aligned}U_{\text{strain}}&={\frac {E}{2}}\int I(x)(w_{xx}(x))^{2}\,\mathrm {d} x\\U_{\text{applied}}&={\frac {P_{\text{crit}}}{2}}\int (w_{x}(x))^{2}\,\mathrm {d} x\end{aligned}}}
2399:
2353:, and the number of lengthwise curvatures. For an increasing number of such curvatures, the aspect ratio produces a varying buckling coefficient; but each relation provides a minimum value for each
3132:(DT) stresses in the web. This results in a non-linear behaviour in the load carrying behaviour of these details. The ratio of the actual load to the load at which buckling occurs is known as the
3416:
430:
3431:
3140:
of the wrinkles. Although they may buckle, thin sheets are designed to not permanently deform and return to an unbuckled state when the applied loading is removed. Repeated buckling may lead to
2811:
when determining lateral-torsional buckling. The reason for this factor is to allow for non-uniform moment diagrams when the ends of a beam segment are braced. The conservative value for
1220:
3119:
Sections that are made up of flanged plates such as a channel, can still carry load in the corners after the flanges have locally buckled. Crippling is failure of the complete section.
78:. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have
2765:
Lateral-torsional buckling of an I-beam with vertical force in center: a) longitudinal view, b) cross section near support, c) cross section in center with lateral-torsional buckling
4275:, forcing adjacent pieces to push against each other. If the stress is sufficient, the pavement can lift and crack without warning. Traversing a buckled section can be jarring to
3210:
2708:
2075:
653:
4113:
3755:
209:
2737:
931:
4219:
4167:
3002:
1139:
973:
which estimates the critical buckling load based on an assumed small initial curvature, hence an eccentricity of the axial load. The
Rankine Gordon formula, named for
3819:
3312:
3232:
2863:
315:
set it as the smallest value of the ratio of length to least cross sectional area that would just exceed a certain constant K of the material. Since K depends on the
2224:
792:
624:
3089:
3060:
3031:
2068:
1706:
1173:
1112:
1085:
682:
595:
462:
3645:
1728:
3958:
2351:
987:
959:
3688:
709:
4045:
3879:
3859:
3839:
3795:
3775:
3720:
3665:
3352:
3332:
3292:
3272:
3252:
2557:
2537:
2391:
2371:
2329:
2041:
2017:
1995:
1973:
1951:
1778:
1753:
1677:
1379:
1355:
1335:
1213:
1193:
812:
562:
540:
514:
488:
2671:{\displaystyle {\frac {b_{\text{eff}}}{b}}\approx {\sqrt {{\frac {\sigma _{cr}}{\sigma _{y}}}\left(1-1.022{\sqrt {\frac {\sigma _{cr}}{\sigma _{y}}}}\right)}}}
1401:
For a rectangular plate, supported along every edge, loaded with a uniform compressive force per unit length, the derived governing equation can be stated by:
760:(not allowing rotation of its ends), the critical load will be four times that for a similar column where the ends are pinned (allowing rotation of its ends).
2229:
4893:
Spottswood, S. Michael; Beberniss, Timothy J.; Eason, Thomas G.; Perez, Ricardo A.; Donbar, Jeffrey M.; Ehrhardt, David A.; Riley, Zachary B. (March 2019).
819:
4513:
Elishakoff, I. Li Y-W. and
Starnes, J.H. Jr., Non Classical Problems in the Theory of Elastic Stability, Cambridge University Press, 2001, XVI +pp.336;
3934:
Single-degree-of-freedom (SDoF) rigid link models depicting four distinct types of buckling phenomena. The spring in each model is unstressed when
4329:
2753:
channels, structural tees, double-angle shapes, and equal-leg single angles. Circular cross sections do not experience such a mode of buckling.
3930:. Simple models for each of these types of buckling behaviour are shown in the figures below, along with the associated bifurcation diagrams.
3722:, it is possible to identify four fundamental forms of buckling found in structural models with one degree of freedom. We start by expressing
202:
4846:
5097:
5069:
4562:
4708:
4414:. Design rules for calculation of the required wall thickness or reinforcement rings are given in various piping and pressure vessel codes.
1921:{\displaystyle w=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }w_{mn}\sin \left({\frac {m\pi x}{a}}\right)\sin \left({\frac {n\pi y}{b}}\right)}
3151:
Thicker plates may only partially form a diagonal tension field and may continue to carry some of the load through shear. This is known as
3381:
362:
2519:
From the derived equations, it can be seen the close similarities between the critical stress for a column and for a plate. As the width
977:
and Perry
Hugesworth Gordon (1899 – 1966), is also based on experimental results and suggests that a column will buckle at a load
5139:
4665:
4518:
195:
339:, that gives the maximum axial load that a long, slender, ideal column can carry without buckling. An ideal column is one that is:
5029:
Rajesh, Dhakal; Maekawa, Koichi (2002). "Reinforcement
Stability and Fracture of Cover Concrete in Reinforced Concrete Members".
353:
When the applied load reaches the Euler load, sometimes called the critical load, the column comes to be in a state of unstable
4438:
2539:
shrinks, the plate acts more like a column as it increases the resistance to buckling along the plate's width. The increase of
4718:
4362:
565:
87:
5129:
4682:
3173:
974:
731:
4811:
4735:
3861:
is stationary with respect to the coordinate measuring the degree(s) of freedom and that these points are only stable if
5144:
4444:
4642:
4027:
Bifurcation diagrams (blue) for the above models with the energy function (red) animated at different values of load,
2820:
can be taken as 1, regardless of beam configuration or loading, but in some cases it may be excessively conservative.
158:
715:
Examination of this formula reveals the following facts with regard to the load-bearing ability of slender columns.
5134:
148:
38:
3725:
5113:
The complete theory and example experimental results for long columns are available as a 39-page PDF document at
3916:
2789:
163:
55:
20:
3107:
A more accurate approximation of the buckling load can be had by the use of the tangent modulus of elasticity, E
2510:{\displaystyle \sigma _{cr}=k_{cr}{\frac {\pi ^{2}E}{12\left(1-\nu ^{2}\right)\left({\frac {b}{t}}\right)^{2}}}}
4389:
284:
4432:
3841:
in its direction. Using the axioms of elastic instability theory, namely that equilibrium is any point where
336:
83:
5042:
4786:
3886:
2396:
Given stress is found by the load per unit area, the following expression is found for the critical stress:
970:
168:
3136:
of a sheet. High buckling ratios may lead to excessive wrinkling of the sheets which may then fail through
4760:
4378:
4351:
2774:
517:
354:
138:
47:
4243:
force, the wheel spontaneously fails into a characteristic saddle shape (sometimes called a "taco" or a "
3926:
3906:
3896:
1756:
1358:
735:
724:
491:
316:
71:
4868:
4761:
Derailment of Amtrak Auto Train P052-18 on the CSXT Railroad Near
Crescent City, Florida April 18, 2002
1308:{\displaystyle h_{\text{crit}}=\left({\frac {9B^{2}}{4}}\,{\frac {EI}{\rho gA}}\right)^{\frac {1}{3}}}
4955:
4906:
4371:
3294:
is the smaller lateral dimension of a rectangular rod. Because the buckle wavelength depends only on
720:
227:
59:
5016:
2686:
632:
4838:
4256:
3137:
2778:
183:
153:
118:
4068:
5149:
4986:
4944:"Experiments on High-Temperature Hypersonic Fluid–Structure Interaction with Plastic Deformation"
4924:
4895:"Exploring the response of a thin, flexible panel to shock-turbulent boundary-layer interactions"
4472:
3141:
2793:
2715:
902:
265:
223:
143:
123:
75:
4180:
4128:
3155:(IDT). This behavior was studied by Wagner and these beams are sometimes known as Wagner beams.
2373:. This minimum value can then be used as a constant, independent from both the aspect ratio and
1731:
4530:
Kato, K. (1915). "Mathematical
Investigation on the Mechanical Problems of Transmission Line".
2980:
1117:
5093:
5065:
4714:
4661:
4558:
4514:
4272:
2808:
276:
269:
113:
3804:
3297:
3217:
5046:
5038:
4963:
4914:
4591:
4385:
4260:
2199:
1114:
is the maximum compressive load. This formula typically produces a conservative estimate of
966:
767:
603:
320:
248:
A column under a concentric axial load exhibiting the characteristic deformation of buckling
4047:(black). Note, the load is on the vertical axis. All graphs are in non-dimensional form.
3067:
3038:
3009:
2046:
1684:
1158:
1090:
1063:
661:
574:
440:
256:
The eccentricity of the axial force results in a bending moment acting on the beam element.
3621:
1713:
1382:
280:
63:
3937:
2334:
2070:
can be separated providing the equation for the critical compressive loading of a plate:
936:
4959:
4910:
4291:
4174:
3670:
691:
4991:
4579:
4449:
4308:
4030:
3864:
3844:
3824:
3780:
3760:
3705:
3650:
3337:
3317:
3277:
3257:
3237:
2542:
2522:
2376:
2356:
2314:
2187:{\displaystyle N_{x,cr}=k_{cr}{\frac {\pi ^{2}Et^{3}}{12\left(1-\nu ^{2}\right)b^{2}}}}
2026:
2002:
1980:
1958:
1936:
1763:
1738:
1662:
1364:
1340:
1320:
1198:
1178:
797:
547:
525:
499:
473:
332:
4457: – method of increasing rigidity and structural integrity of materials or objects
4053:
3881:
is a local minimum and unstable if otherwise (e.g. maximum or a point of inflection).
2761:
2023:
The previous equation can be substituted into the earlier differential equation where
252:
5123:
4928:
4239:
1394:
1150:
173:
323:
parallel to the grain, it can be seen that this arbitrary limit would vary with the
4316:
4264:
4014:
3984:
1215:, will buckle under its own weight if its height exceeds a certain critical value:
3091:, absolute value of maximum moment at three-quarter point of the unbraced segment,
1784:
The solution to the deflection can be expanded into two harmonic functions shown:
568:, whose value depends on the conditions of end support of the column, as follows.
4463:
4441: – The absorption and transmission of neutrons by nuclear reactor materials
4411:
4304:
4296:
4004:
3974:
2965:{\displaystyle C_{b}={\frac {12.5M_{\max }}{2.5M_{\max }+3M_{A}+4M_{B}+3M_{C}}}}
2792:
will mitigate the effects of lateral-torsional buckling by virtue of their high
331:
The theory of the behavior of columns was investigated in 1757 by mathematician
300:
200 and its behavior is dominated by the modulus of elasticity of the material.
4919:
4894:
4595:
4454:
4342:
4315:. It is more common for rails to move laterally, often pulling the underlying
4280:
4276:
2830:
3033:, absolute value of maximum moment at quarter point of the unbraced segment,
1051:{\displaystyle {\frac {1}{F_{\max }}}={\frac {1}{F_{e}}}+{\frac {1}{F_{c}}}}
244:
108:
5114:
4062:
746:
33:
4393:
4244:
3062:, absolute value of maximum moment at centerline of the unbraced segment,
304:
133:
4766:(Report). National Transportation Safety Board. 5 August 2003. p. v
4122:
2770:
2302:{\displaystyle k_{cr}=\left({\frac {mb}{a}}+{\frac {a}{mb}}\right)^{2}}
324:
128:
5051:
4968:
4943:
4839:"Misaligned Track from Heat 'Probable Cause' In Green Line Derailment"
4707:
Li, Dingquing; Hyslip, James; Sussmann, Ted; Chrismer, Steven (2002).
2858:
is not given in tables, it can be obtained via the following formula:
2311:
The buckling coefficient is influenced by the aspect of the specimen,
4358:
4338:
890:{\displaystyle \sigma ={\frac {F}{A}}={\frac {\pi ^{2}E}{(l/r)^{2}}}}
261:
67:
27:
327:
of the timber. The value of K is given in most structural handbooks.
3334:, this same formula holds for thin cylindrical shells of thickness
226:
that develop in the structure are well below those needed to cause
4290:
4173:
4121:
4061:
4052:
4013:
4003:
3983:
3973:
2760:
745:
465:
296:
251:
243:
32:
4557:] (in German). Wein, Austria: J. Springer. pp. 107–109.
1385:
of the first kind of order −1/3, which is equal to 1.86635086...
311:
178:
3924:
bifurcation. All but the first of these examples is a form of
3690:
refer to the first and second derivatives of the displacement.
5090:
Resolution of
Twentieth Century Conundrum in Elastic Stability
4435: – Formula to quantify column buckling under a given load
4367:
4347:
4268:
3978:
A tied truss model with inclined links and horizontal spring.
969:
are introduced into these formulae. One such formula is the
658:
For one end fixed and the other end free to move laterally,
4736:"Amtrak Passengers Face Record Delays From Extreme Weather"
3004:, absolute value of maximum moment in the unbraced segment,
520:(second moment of area) of the cross section of the column,
16:
Sudden change in shape of a structural component under load
5002:
Nenezich, M. (2004). "Thermoplastic
Continuum Mechanics".
4475: – Pattern of buckling used in mechanical engineering
2807:
is a modification factor used in the equation for nominal
4942:
Daub, Dennis; Esser, Burkard; Gülhan, Ali (April 2020).
4683:"Derailments May Increase as 'Sun Kinks' Buckle Tracks"
279:(centroid) of its cross section, it is called an axial
90:
are used to determine the buckling stress of a column.
5092:. Singapore: World Scientific/Imperial College Press.
2829:
is always equal to or greater than 1, never less. For
1953:, number of half sine curvatures that occur lengthwise
933:
is the stress that causes buckling in the column, and
723:
of the material of the column and not the compressive
4183:
4131:
4071:
4033:
3940:
3867:
3847:
3827:
3807:
3783:
3763:
3728:
3708:
3673:
3653:
3624:
3429:
3384:
3340:
3320:
3300:
3280:
3260:
3240:
3220:
3176:
3070:
3041:
3012:
2983:
2866:
2718:
2689:
2568:
2545:
2525:
2402:
2379:
2359:
2337:
2317:
2232:
2202:
2078:
2049:
2029:
2005:
1983:
1975:, number of half sine curvatures that occur widthwise
1961:
1939:
1792:
1766:
1741:
1716:
1687:
1665:
1409:
1367:
1343:
1323:
1223:
1201:
1181:
1161:
1120:
1093:
1066:
990:
939:
905:
822:
800:
770:
694:
664:
635:
606:
577:
550:
528:
502:
476:
443:
365:
4532:
Journal of the Japan
Society of Mechanical Engineers
4468:
Pages displaying wikidata descriptions as a fallback
4459:
Pages displaying wikidata descriptions as a fallback
4326:
These accidents were deemed to be sun kink-related (
3411:{\displaystyle U_{\text{applied}}=U_{\text{strain}}}
727:
of the column determines the column's buckling load.
425:{\displaystyle F_{c}={\frac {\pi ^{2}EI}{(KL)^{2}}}}
4812:"Heat could warp train tracks, causing derailments"
4787:"Passenger Train Cars Plunge Off Overheated Tracks"
4990:
4213:
4161:
4107:
4039:
3952:
3873:
3853:
3833:
3813:
3789:
3769:
3749:
3714:
3682:
3659:
3639:
3608:
3410:
3346:
3326:
3306:
3286:
3266:
3246:
3226:
3204:
3083:
3054:
3025:
2996:
2964:
2731:
2702:
2670:
2551:
2531:
2509:
2385:
2365:
2345:
2323:
2301:
2218:
2186:
2062:
2035:
2011:
1989:
1967:
1945:
1920:
1772:
1747:
1722:
1700:
1671:
1646:
1373:
1349:
1329:
1307:
1207:
1187:
1167:
1133:
1106:
1079:
1050:
953:
925:
889:
806:
786:
703:
676:
647:
618:
589:
556:
534:
508:
482:
456:
424:
5081:Analysis and Design of Flight Vehicle Structures
4501:Analysis and Design of Flight Vehicle Structures
4311:, and can fail by buckling, a phenomenon called
3647:is the displacement function and the subscripts
2989:
2906:
2891:
1126:
1001:
4874:(Report). Federal Register. p. 41881-41882
4271:is absorbed in the road surface, causing it to
1155:A free-standing, vertical column, with density
571:For both ends pinned (hinged, free to rotate),
275:If the load on a column is applied through the
3777:is the strain energy stored in the structure,
2833:or overhangs where the free end is unbraced, C
4555:The buckling resistance of members and frames
4551:Die Knickfestigkeit von Stäben und Stabwerken
3884:These four forms of elastic buckling are the
3095:The result is the same for all unit systems.
203:
8:
629:For one end fixed and the other end pinned,
5064:(Fourth ed.). United States: Thomson.
4582:(1998). "The Shape of the Tallest Column".
4259:materials, primarily with concrete, since
2788:The use of closed sections such as square
2769:When a simply supported beam is loaded in
210:
196:
92:
5050:
5043:10.1061/(ASCE)0733-9445(2002)128:10(1253)
4967:
4918:
4637:Lindberg, H. E.; Florence, A. L. (1987).
4494:
4492:
4490:
4488:
4203:
4188:
4182:
4151:
4136:
4130:
4088:
4076:
4070:
4032:
3939:
3866:
3846:
3826:
3806:
3782:
3762:
3727:
3707:
3672:
3652:
3623:
3594:
3593:
3587:
3568:
3547:
3541:
3528:
3512:
3511:
3505:
3483:
3451:
3438:
3430:
3428:
3402:
3389:
3383:
3339:
3319:
3299:
3279:
3259:
3239:
3219:
3193:
3175:
3075:
3069:
3046:
3040:
3017:
3011:
2988:
2982:
2953:
2937:
2921:
2905:
2890:
2880:
2871:
2865:
2723:
2717:
2694:
2688:
2652:
2639:
2632:
2610:
2597:
2591:
2589:
2575:
2569:
2567:
2544:
2524:
2498:
2484:
2468:
2439:
2432:
2423:
2407:
2401:
2378:
2358:
2338:
2336:
2316:
2293:
2273:
2255:
2237:
2231:
2207:
2201:
2175:
2160:
2134:
2121:
2114:
2105:
2083:
2077:
2054:
2048:
2028:
2004:
1982:
1960:
1938:
1896:
1864:
1845:
1835:
1824:
1814:
1803:
1791:
1765:
1740:
1715:
1692:
1686:
1664:
1630:
1612:
1605:
1599:
1578:
1558:
1537:
1525:
1507:
1500:
1488:
1475:
1457:
1450:
1435:
1417:
1410:
1408:
1366:
1342:
1322:
1294:
1266:
1265:
1253:
1243:
1228:
1222:
1200:
1180:
1160:
1125:
1119:
1098:
1092:
1071:
1065:
1040:
1031:
1020:
1011:
1000:
991:
989:
943:
938:
915:
904:
878:
866:
849:
842:
829:
821:
799:
778:
769:
693:
663:
634:
605:
576:
549:
527:
501:
475:
448:
442:
413:
386:
379:
370:
364:
5115:http://lindberglce.com/tech/buklbook.htm
4418:Super- and hypersonic aerospace vehicles
4025:
3932:
1708:, uniformly distributed compressive load
4484:
260:The ratio of the effective length of a
95:
4617:
4615:
4613:
4611:
4609:
4607:
4605:
711:is the effective length of the column.
4658:A general theory of elastic stability
4656:Thompson, J.M.T.; Hunt, G.W. (1973).
4584:SIAM Journal on Mathematical Analysis
4377:July 8, 2010 CSX train derailment in
4279:drivers, described as running over a
7:
5018:The Stability of Elastic Equilibrium
4466: – method in strucural analysis
3205:{\displaystyle \sigma L=\rho c^{2}h}
1337:is the acceleration due to gravity,
2837:is equal to 1. Tables of values of
268:of its cross section is called the
222:Buckling may occur even though the
4330:List of rail accidents (2000–2009)
3808:
3744:
3595:
3513:
2846:for simply supported beams exist.
1836:
1815:
1623:
1609:
1518:
1504:
1481:
1468:
1454:
1428:
1414:
74:or the wrinkling of a plate under
14:
5031:Journal of Structural Engineering
4008:Link model with transverse spring
3988:Link model with rotational spring
335:. He derived the formula, termed
4867:Robert C. Lauby (16 July 2012).
4423:deforms towards the flow-field.
5004:Journal of Aerospace Structures
4849:from the original on 2018-02-04
4785:Braun, Stephen (30 July 2002).
4439:Geometric and material buckling
4018:Link model with inclined spring
3694:Single-degree-of-freedom models
3274:is the elastic wave speed, and
2196:where the buckling coefficient
1361:of the beam cross section, and
542:, unsupported length of column,
54:is the sudden change in shape (
4899:Journal of Sound and Vibration
4713:. CRC Press. p. 430-431.
4681:Magill, Bobby (31 July 2014).
4549:Ratzersdorfer, Julius (1936).
4328:more information available at
4255:Buckling is a failure mode in
4102:
4093:
3995:Unstable-symmetric bifurcation
3634:
3628:
3584:
3580:
3574:
3561:
3502:
3498:
3492:
3476:
3473:
3467:
2703:{\displaystyle b_{\text{eff}}}
1087:is the Euler maximum load and
875:
860:
730:The buckling load is directly
648:{\displaystyle K\approx 0.699}
566:column effective length factor
410:
400:
346:made of a homogeneous material
319:and the allowable compressive
1:
4845:. American University Radio.
3968:Stable-symmetric bifurcation
975:William John Macquorn Rankine
4997:(2nd ed.). McGraw-Hill.
4818:. Columbia, SC. 15 July 2010
4626:. Chatto and Windus, London.
4108:{\displaystyle P^{C}=c/(2L)}
3750:{\displaystyle V=U-P\Delta }
2777:, and the bottom side is in
4993:Theory of Elastic Stability
4734:Kim, Minho (17 July 2024).
4643:Martinus Nijhoff Publishers
3153:incomplete diagonal tension
2849:If an appropriate value of
2748:Flexural-torsional buckling
2732:{\displaystyle \sigma _{y}}
1195:, and cross-sectional area
926:{\displaystyle \sigma =F/A}
159:Metal-induced embrittlement
88:Johnson's parabolic formula
5166:
5060:Segui, Willian T. (2007).
4837:Lucero, Kat (2012-07-07).
4645:. pp. 11–56, 297–298.
4405:Pipes and pressure vessels
4214:{\displaystyle P^{C}=kL/2}
4162:{\displaystyle P^{C}=kL/2}
3254:is the length of the rod,
2783:lateral-torsional buckling
2757:Lateral-torsional buckling
1148:
961:is the slenderness ratio.
468:(vertical load on column),
149:Liquid metal embrittlement
66:, such as the bowing of a
25:
18:
4920:10.1016/j.jsv.2018.11.035
4869:Notice of Safety Advisory
4596:10.1137/s0036141097314537
3821:is the distance moved by
2997:{\displaystyle M_{\max }}
1679:, out-of-plane deflection
1381:is the first zero of the
1134:{\displaystyle F_{\max }}
349:free from initial stress.
164:Stress corrosion cracking
37:Buckled skin panels on a
21:Buckling (disambiguation)
5140:Mechanical failure modes
4390:West Hyattsville station
725:strength of the material
96:Mechanical failure modes
26:Not to be confused with
5088:Elishakoff, I. (2004).
5083:. Indianapolis: Jacobs.
4503:. Indianapolis: Jacobs.
4445:Perry–Robertson formula
3998:Asymmetric bifurcation
3887:saddle-node bifurcation
3814:{\displaystyle \Delta }
3307:{\displaystyle \sigma }
3227:{\displaystyle \sigma }
971:Perry Robertson formula
169:Sulfide stress cracking
5015:Koiter, W. T. (1945).
4989:; Gere, J. M. (1961).
4660:. London: John Wiley.
4639:Dynamic Pulse Buckling
4622:Bulson, P. S. (1970).
4388:train derailment near
4379:Waxhaw, North Carolina
4352:Crescent City, Florida
4300:
4295:Railway tracks in the
4221:
4215:
4169:
4163:
4115:
4109:
4057:
4041:
4019:
4009:
3989:
3979:
3954:
3875:
3855:
3835:
3815:
3791:
3771:
3751:
3716:
3700:total potential energy
3684:
3661:
3641:
3610:
3412:
3348:
3328:
3308:
3288:
3268:
3248:
3234:is the impact stress,
3228:
3206:
3085:
3056:
3027:
2998:
2966:
2766:
2733:
2704:
2672:
2553:
2533:
2511:
2387:
2367:
2347:
2325:
2303:
2220:
2219:{\displaystyle k_{cr}}
2188:
2064:
2037:
2013:
1991:
1969:
1947:
1922:
1840:
1819:
1774:
1749:
1724:
1702:
1673:
1648:
1375:
1351:
1331:
1309:
1209:
1189:
1169:
1135:
1108:
1081:
1052:
955:
927:
891:
808:
788:
787:{\displaystyle Ar^{2}}
752:
705:
678:
649:
620:
619:{\displaystyle K=0.50}
591:
558:
536:
518:area moment of inertia
510:
484:
464:, maximum or critical
458:
426:
257:
249:
139:Hydrogen embrittlement
48:structural engineering
43:
5079:Bruhn, E. F. (1973).
4624:Theory of Flat Plates
4499:Bruhn, E. F. (1973).
4433:Euler's critical load
4294:
4216:
4177:
4164:
4125:
4110:
4065:
4056:
4042:
4017:
4007:
3987:
3977:
3955:
3927:pitchfork bifurcation
3914:bifurcation; and the
3876:
3856:
3836:
3816:
3792:
3772:
3752:
3717:
3698:Using the concept of
3685:
3662:
3642:
3611:
3413:
3349:
3329:
3309:
3289:
3269:
3249:
3229:
3207:
3086:
3084:{\displaystyle M_{C}}
3057:
3055:{\displaystyle M_{B}}
3028:
3026:{\displaystyle M_{A}}
2999:
2967:
2773:, the top side is in
2764:
2734:
2705:
2673:
2554:
2534:
2512:
2388:
2368:
2348:
2326:
2304:
2221:
2189:
2065:
2063:{\displaystyle N_{x}}
2038:
2014:
1992:
1970:
1948:
1923:
1820:
1799:
1775:
1757:modulus of elasticity
1750:
1725:
1703:
1701:{\displaystyle N_{x}}
1674:
1649:
1376:
1359:second moment of area
1352:
1332:
1310:
1210:
1190:
1170:
1168:{\displaystyle \rho }
1136:
1109:
1107:{\displaystyle F_{c}}
1082:
1080:{\displaystyle F_{e}}
1053:
956:
928:
892:
809:
789:
749:
738:of the cross section.
736:second moment of area
706:
679:
677:{\displaystyle K=2.0}
650:
621:
600:For both ends fixed,
592:
590:{\displaystyle K=1.0}
559:
537:
511:
492:modulus of elasticity
485:
459:
457:{\displaystyle F_{c}}
427:
337:Euler's critical load
317:modulus of elasticity
255:
247:
84:Euler's critical load
36:
5130:Elasticity (physics)
4372:Kensington, Maryland
4299:affected by sun kink
4229:Engineering examples
4181:
4129:
4069:
4031:
3938:
3865:
3845:
3825:
3805:
3781:
3761:
3726:
3706:
3671:
3651:
3640:{\displaystyle w(x)}
3622:
3427:
3382:
3338:
3318:
3298:
3278:
3258:
3238:
3218:
3174:
3068:
3039:
3010:
2981:
2864:
2716:
2687:
2566:
2543:
2523:
2400:
2377:
2357:
2335:
2315:
2230:
2200:
2076:
2047:
2027:
2003:
1997:, length of specimen
1981:
1959:
1937:
1790:
1764:
1739:
1723:{\displaystyle \nu }
1714:
1685:
1663:
1407:
1365:
1341:
1321:
1221:
1199:
1179:
1159:
1118:
1091:
1064:
988:
937:
903:
820:
798:
768:
692:
662:
633:
604:
575:
548:
526:
500:
474:
441:
363:
60:structural component
19:For other uses, see
5145:Structural analysis
4960:2020AIAAJ..58.1423D
4911:2019JSV...443...74S
4710:Railway Geotechnics
4384:July 6, 2012 WMATA
4283:at highway speeds.
4048:
3961:
3953:{\displaystyle q=0}
2794:torsional stiffness
2346:{\displaystyle {b}}
2019:, width of specimen
954:{\displaystyle l/r}
154:Mechanical overload
4473:Yoshimura buckling
4319:(sleepers) along.
4309:expand when heated
4301:
4263:is more flexible.
4222:
4211:
4170:
4159:
4116:
4105:
4058:
4037:
4026:
4020:
4010:
3990:
3980:
3950:
3933:
3912:unstable-symmetric
3871:
3851:
3831:
3811:
3787:
3767:
3747:
3712:
3683:{\displaystyle xx}
3680:
3657:
3637:
3606:
3604:
3408:
3344:
3324:
3304:
3284:
3264:
3244:
3224:
3202:
3081:
3052:
3023:
2994:
2962:
2767:
2729:
2700:
2668:
2549:
2529:
2507:
2383:
2363:
2343:
2321:
2299:
2216:
2184:
2060:
2033:
2009:
1987:
1965:
1943:
1918:
1770:
1745:
1720:
1698:
1669:
1644:
1371:
1347:
1327:
1305:
1205:
1185:
1175:, Young's modulus
1165:
1131:
1104:
1077:
1048:
951:
923:
887:
804:
784:
753:
704:{\displaystyle KL}
701:
674:
645:
616:
587:
554:
532:
506:
480:
454:
422:
343:perfectly straight
266:radius of gyration
258:
250:
44:
5135:Materials science
5099:978-981-4583-53-4
5071:978-0-495-24471-4
5037:(10): 1253–1262.
4987:Timoshenko, S. P.
4969:10.2514/1.J059150
4791:Los Angeles Times
4580:C. Maeve McCarthy
4564:978-3-662-24075-5
4226:
4225:
4040:{\displaystyle P}
4024:
4023:
3904:bifurcation; the
3874:{\displaystyle V}
3854:{\displaystyle V}
3834:{\displaystyle P}
3790:{\displaystyle P}
3770:{\displaystyle U}
3715:{\displaystyle V}
3660:{\displaystyle x}
3556:
3550:
3531:
3459:
3441:
3405:
3392:
3347:{\displaystyle h}
3327:{\displaystyle h}
3287:{\displaystyle h}
3267:{\displaystyle c}
3247:{\displaystyle L}
2960:
2809:flexural strength
2739:, yielding stress
2710:, effective width
2697:
2666:
2659:
2658:
2616:
2584:
2578:
2552:{\displaystyle a}
2532:{\displaystyle b}
2505:
2492:
2386:{\displaystyle m}
2366:{\displaystyle m}
2324:{\displaystyle a}
2286:
2268:
2182:
2036:{\displaystyle n}
2012:{\displaystyle b}
1990:{\displaystyle a}
1968:{\displaystyle n}
1946:{\displaystyle m}
1912:
1880:
1773:{\displaystyle t}
1748:{\displaystyle E}
1672:{\displaystyle w}
1637:
1585:
1532:
1495:
1442:
1374:{\displaystyle B}
1350:{\displaystyle I}
1330:{\displaystyle g}
1302:
1287:
1263:
1231:
1208:{\displaystyle A}
1188:{\displaystyle E}
1046:
1026:
1006:
885:
837:
807:{\displaystyle I}
557:{\displaystyle K}
535:{\displaystyle L}
509:{\displaystyle I}
483:{\displaystyle E}
420:
277:center of gravity
270:slenderness ratio
235:Forms of buckling
220:
219:
114:Corrosion fatigue
5157:
5103:
5084:
5075:
5056:
5054:
5025:
5023:
5011:
4998:
4996:
4974:
4973:
4971:
4954:(4): 1423–1431.
4939:
4933:
4932:
4922:
4890:
4884:
4883:
4881:
4879:
4873:
4864:
4858:
4857:
4855:
4854:
4834:
4828:
4827:
4825:
4823:
4808:
4802:
4801:
4799:
4797:
4782:
4776:
4775:
4773:
4771:
4765:
4757:
4751:
4750:
4748:
4746:
4731:
4725:
4724:
4704:
4698:
4697:
4695:
4693:
4678:
4672:
4671:
4653:
4647:
4646:
4634:
4628:
4627:
4619:
4600:
4599:
4578:Cox, Steven J.;
4575:
4569:
4568:
4546:
4540:
4539:
4527:
4521:
4511:
4505:
4504:
4496:
4469:
4460:
4366:derailment, off
4346:derailment, off
4220:
4218:
4217:
4212:
4207:
4193:
4192:
4168:
4166:
4165:
4160:
4155:
4141:
4140:
4114:
4112:
4111:
4106:
4092:
4081:
4080:
4049:
4046:
4044:
4043:
4038:
3962:
3959:
3957:
3956:
3951:
3902:stable-symmetric
3880:
3878:
3877:
3872:
3860:
3858:
3857:
3852:
3840:
3838:
3837:
3832:
3820:
3818:
3817:
3812:
3796:
3794:
3793:
3788:
3776:
3774:
3773:
3768:
3756:
3754:
3753:
3748:
3721:
3719:
3718:
3713:
3689:
3687:
3686:
3681:
3666:
3664:
3663:
3658:
3646:
3644:
3643:
3638:
3615:
3613:
3612:
3607:
3605:
3598:
3592:
3591:
3573:
3572:
3557:
3552:
3551:
3548:
3542:
3533:
3532:
3529:
3516:
3510:
3509:
3491:
3490:
3460:
3452:
3443:
3442:
3439:
3417:
3415:
3414:
3409:
3407:
3406:
3403:
3394:
3393:
3390:
3353:
3351:
3350:
3345:
3333:
3331:
3330:
3325:
3313:
3311:
3310:
3305:
3293:
3291:
3290:
3285:
3273:
3271:
3270:
3265:
3253:
3251:
3250:
3245:
3233:
3231:
3230:
3225:
3211:
3209:
3208:
3203:
3198:
3197:
3163:Dynamic buckling
3130:diagonal tension
3123:Diagonal tension
3099:Plastic buckling
3090:
3088:
3087:
3082:
3080:
3079:
3061:
3059:
3058:
3053:
3051:
3050:
3032:
3030:
3029:
3024:
3022:
3021:
3003:
3001:
3000:
2995:
2993:
2992:
2971:
2969:
2968:
2963:
2961:
2959:
2958:
2957:
2942:
2941:
2926:
2925:
2910:
2909:
2896:
2895:
2894:
2881:
2876:
2875:
2738:
2736:
2735:
2730:
2728:
2727:
2709:
2707:
2706:
2701:
2699:
2698:
2695:
2677:
2675:
2674:
2669:
2667:
2665:
2661:
2660:
2657:
2656:
2647:
2646:
2634:
2633:
2617:
2615:
2614:
2605:
2604:
2592:
2590:
2585:
2580:
2579:
2576:
2570:
2558:
2556:
2555:
2550:
2538:
2536:
2535:
2530:
2516:
2514:
2513:
2508:
2506:
2504:
2503:
2502:
2497:
2493:
2485:
2478:
2474:
2473:
2472:
2448:
2444:
2443:
2433:
2431:
2430:
2415:
2414:
2392:
2390:
2389:
2384:
2372:
2370:
2369:
2364:
2352:
2350:
2349:
2344:
2342:
2330:
2328:
2327:
2322:
2308:
2306:
2305:
2300:
2298:
2297:
2292:
2288:
2287:
2285:
2274:
2269:
2264:
2256:
2245:
2244:
2225:
2223:
2222:
2217:
2215:
2214:
2193:
2191:
2190:
2185:
2183:
2181:
2180:
2179:
2170:
2166:
2165:
2164:
2140:
2139:
2138:
2126:
2125:
2115:
2113:
2112:
2097:
2096:
2069:
2067:
2066:
2061:
2059:
2058:
2042:
2040:
2039:
2034:
2018:
2016:
2015:
2010:
1996:
1994:
1993:
1988:
1974:
1972:
1971:
1966:
1952:
1950:
1949:
1944:
1927:
1925:
1924:
1919:
1917:
1913:
1908:
1897:
1885:
1881:
1876:
1865:
1853:
1852:
1839:
1834:
1818:
1813:
1779:
1777:
1776:
1771:
1754:
1752:
1751:
1746:
1729:
1727:
1726:
1721:
1707:
1705:
1704:
1699:
1697:
1696:
1678:
1676:
1675:
1670:
1653:
1651:
1650:
1645:
1643:
1639:
1638:
1636:
1635:
1634:
1621:
1617:
1616:
1606:
1604:
1603:
1586:
1584:
1583:
1582:
1569:
1568:
1564:
1563:
1562:
1538:
1533:
1531:
1530:
1529:
1516:
1512:
1511:
1501:
1496:
1494:
1493:
1492:
1480:
1479:
1466:
1462:
1461:
1451:
1443:
1441:
1440:
1439:
1426:
1422:
1421:
1411:
1380:
1378:
1377:
1372:
1356:
1354:
1353:
1348:
1336:
1334:
1333:
1328:
1314:
1312:
1311:
1306:
1304:
1303:
1295:
1293:
1289:
1288:
1286:
1275:
1267:
1264:
1259:
1258:
1257:
1244:
1233:
1232:
1229:
1214:
1212:
1211:
1206:
1194:
1192:
1191:
1186:
1174:
1172:
1171:
1166:
1140:
1138:
1137:
1132:
1130:
1129:
1113:
1111:
1110:
1105:
1103:
1102:
1086:
1084:
1083:
1078:
1076:
1075:
1057:
1055:
1054:
1049:
1047:
1045:
1044:
1032:
1027:
1025:
1024:
1012:
1007:
1005:
1004:
992:
960:
958:
957:
952:
947:
932:
930:
929:
924:
919:
896:
894:
893:
888:
886:
884:
883:
882:
870:
858:
854:
853:
843:
838:
830:
813:
811:
810:
805:
793:
791:
790:
785:
783:
782:
710:
708:
707:
702:
683:
681:
680:
675:
654:
652:
651:
646:
625:
623:
622:
617:
596:
594:
593:
588:
563:
561:
560:
555:
541:
539:
538:
533:
515:
513:
512:
507:
489:
487:
486:
481:
463:
461:
460:
455:
453:
452:
431:
429:
428:
423:
421:
419:
418:
417:
398:
391:
390:
380:
375:
374:
212:
205:
198:
93:
5165:
5164:
5160:
5159:
5158:
5156:
5155:
5154:
5120:
5119:
5110:
5100:
5087:
5078:
5072:
5059:
5028:
5021:
5014:
5001:
4985:
4982:
4980:Further reading
4977:
4941:
4940:
4936:
4892:
4891:
4887:
4877:
4875:
4871:
4866:
4865:
4861:
4852:
4850:
4836:
4835:
4831:
4821:
4819:
4810:
4809:
4805:
4795:
4793:
4784:
4783:
4779:
4769:
4767:
4763:
4759:
4758:
4754:
4744:
4742:
4733:
4732:
4728:
4721:
4706:
4705:
4701:
4691:
4689:
4687:Climate Central
4680:
4679:
4675:
4668:
4655:
4654:
4650:
4636:
4635:
4631:
4621:
4620:
4603:
4577:
4576:
4572:
4565:
4548:
4547:
4543:
4529:
4528:
4524:
4512:
4508:
4498:
4497:
4486:
4482:
4467:
4458:
4429:
4420:
4407:
4363:Capitol Limited
4337:April 18, 2002
4289:
4253:
4238:A conventional
4236:
4231:
4184:
4179:
4178:
4132:
4127:
4126:
4072:
4067:
4066:
4029:
4028:
3936:
3935:
3863:
3862:
3843:
3842:
3823:
3822:
3803:
3802:
3797:is the applied
3779:
3778:
3759:
3758:
3724:
3723:
3704:
3703:
3696:
3669:
3668:
3649:
3648:
3620:
3619:
3603:
3602:
3583:
3564:
3543:
3534:
3524:
3521:
3520:
3501:
3479:
3444:
3434:
3425:
3424:
3398:
3385:
3380:
3379:
3365:
3360:
3336:
3335:
3316:
3315:
3296:
3295:
3276:
3275:
3256:
3255:
3236:
3235:
3216:
3215:
3189:
3172:
3171:
3165:
3125:
3117:
3110:
3101:
3071:
3066:
3065:
3042:
3037:
3036:
3013:
3008:
3007:
2984:
2979:
2978:
2949:
2933:
2917:
2901:
2897:
2886:
2882:
2867:
2862:
2861:
2857:
2845:
2836:
2828:
2819:
2806:
2759:
2750:
2719:
2714:
2713:
2690:
2685:
2684:
2648:
2635:
2622:
2618:
2606:
2593:
2571:
2564:
2563:
2541:
2540:
2521:
2520:
2480:
2479:
2464:
2457:
2453:
2449:
2435:
2434:
2419:
2403:
2398:
2397:
2375:
2374:
2355:
2354:
2333:
2332:
2313:
2312:
2278:
2257:
2254:
2250:
2249:
2233:
2228:
2227:
2226:, is given by:
2203:
2198:
2197:
2171:
2156:
2149:
2145:
2141:
2130:
2117:
2116:
2101:
2079:
2074:
2073:
2050:
2045:
2044:
2025:
2024:
2001:
2000:
1979:
1978:
1957:
1956:
1935:
1934:
1898:
1892:
1866:
1860:
1841:
1788:
1787:
1762:
1761:
1737:
1736:
1732:Poisson's ratio
1712:
1711:
1688:
1683:
1682:
1661:
1660:
1626:
1622:
1608:
1607:
1595:
1591:
1587:
1574:
1570:
1554:
1547:
1543:
1539:
1521:
1517:
1503:
1502:
1484:
1471:
1467:
1453:
1452:
1431:
1427:
1413:
1412:
1405:
1404:
1391:
1383:Bessel function
1363:
1362:
1339:
1338:
1319:
1318:
1276:
1268:
1249:
1245:
1242:
1238:
1237:
1224:
1219:
1218:
1197:
1196:
1177:
1176:
1157:
1156:
1153:
1147:
1121:
1116:
1115:
1094:
1089:
1088:
1067:
1062:
1061:
1036:
1016:
996:
986:
985:
983:
935:
934:
901:
900:
874:
859:
845:
844:
818:
817:
796:
795:
774:
766:
765:
690:
689:
660:
659:
631:
630:
602:
601:
573:
572:
546:
545:
524:
523:
498:
497:
472:
471:
444:
439:
438:
409:
399:
382:
381:
366:
361:
360:
291:In particular:
242:
237:
216:
31:
24:
17:
12:
11:
5:
5163:
5161:
5153:
5152:
5147:
5142:
5137:
5132:
5122:
5121:
5118:
5117:
5109:
5108:External links
5106:
5105:
5104:
5098:
5085:
5076:
5070:
5057:
5026:
5012:
4999:
4981:
4978:
4976:
4975:
4934:
4885:
4859:
4829:
4803:
4777:
4752:
4740:New York Times
4726:
4719:
4699:
4673:
4666:
4648:
4629:
4601:
4590:(3): 547–554.
4570:
4563:
4541:
4522:
4506:
4483:
4481:
4478:
4477:
4476:
4470:
4461:
4452:
4450:Rail stressing
4447:
4442:
4436:
4428:
4425:
4419:
4416:
4406:
4403:
4398:
4397:
4382:
4375:
4357:July 29, 2002
4355:
4288:
4285:
4252:
4249:
4235:
4234:Bicycle wheels
4232:
4230:
4227:
4224:
4223:
4210:
4206:
4202:
4199:
4196:
4191:
4187:
4171:
4158:
4154:
4150:
4147:
4144:
4139:
4135:
4118:
4117:
4104:
4101:
4098:
4095:
4091:
4087:
4084:
4079:
4075:
4059:
4036:
4022:
4021:
4011:
4000:
3999:
3996:
3992:
3991:
3981:
3970:
3969:
3966:
3949:
3946:
3943:
3870:
3850:
3830:
3810:
3786:
3766:
3746:
3743:
3740:
3737:
3734:
3731:
3711:
3695:
3692:
3679:
3676:
3656:
3636:
3633:
3630:
3627:
3601:
3597:
3590:
3586:
3582:
3579:
3576:
3571:
3567:
3563:
3560:
3555:
3546:
3540:
3537:
3535:
3527:
3523:
3522:
3519:
3515:
3508:
3504:
3500:
3497:
3494:
3489:
3486:
3482:
3478:
3475:
3472:
3469:
3466:
3463:
3458:
3455:
3450:
3447:
3445:
3437:
3433:
3432:
3401:
3397:
3388:
3364:
3361:
3359:
3356:
3343:
3323:
3303:
3283:
3263:
3243:
3223:
3201:
3196:
3192:
3188:
3185:
3182:
3179:
3164:
3161:
3134:buckling ratio
3124:
3121:
3116:
3113:
3108:
3100:
3097:
3093:
3092:
3078:
3074:
3063:
3049:
3045:
3034:
3020:
3016:
3005:
2991:
2987:
2956:
2952:
2948:
2945:
2940:
2936:
2932:
2929:
2924:
2920:
2916:
2913:
2908:
2904:
2900:
2893:
2889:
2885:
2879:
2874:
2870:
2853:
2841:
2834:
2824:
2815:
2802:
2790:hollow section
2758:
2755:
2749:
2746:
2741:
2740:
2726:
2722:
2711:
2693:
2664:
2655:
2651:
2645:
2642:
2638:
2631:
2628:
2625:
2621:
2613:
2609:
2603:
2600:
2596:
2588:
2583:
2574:
2548:
2528:
2501:
2496:
2491:
2488:
2483:
2477:
2471:
2467:
2463:
2460:
2456:
2452:
2447:
2442:
2438:
2429:
2426:
2422:
2418:
2413:
2410:
2406:
2382:
2362:
2341:
2320:
2296:
2291:
2284:
2281:
2277:
2272:
2267:
2263:
2260:
2253:
2248:
2243:
2240:
2236:
2213:
2210:
2206:
2178:
2174:
2169:
2163:
2159:
2155:
2152:
2148:
2144:
2137:
2133:
2129:
2124:
2120:
2111:
2108:
2104:
2100:
2095:
2092:
2089:
2086:
2082:
2057:
2053:
2032:
2021:
2020:
2008:
1998:
1986:
1976:
1964:
1954:
1942:
1916:
1911:
1907:
1904:
1901:
1895:
1891:
1888:
1884:
1879:
1875:
1872:
1869:
1863:
1859:
1856:
1851:
1848:
1844:
1838:
1833:
1830:
1827:
1823:
1817:
1812:
1809:
1806:
1802:
1798:
1795:
1782:
1781:
1769:
1759:
1744:
1734:
1719:
1709:
1695:
1691:
1680:
1668:
1642:
1633:
1629:
1625:
1620:
1615:
1611:
1602:
1598:
1594:
1590:
1581:
1577:
1573:
1567:
1561:
1557:
1553:
1550:
1546:
1542:
1536:
1528:
1524:
1520:
1515:
1510:
1506:
1499:
1491:
1487:
1483:
1478:
1474:
1470:
1465:
1460:
1456:
1449:
1446:
1438:
1434:
1430:
1425:
1420:
1416:
1390:
1389:Plate buckling
1387:
1370:
1346:
1326:
1301:
1298:
1292:
1285:
1282:
1279:
1274:
1271:
1262:
1256:
1252:
1248:
1241:
1236:
1227:
1204:
1184:
1164:
1149:Main article:
1146:
1143:
1128:
1124:
1101:
1097:
1074:
1070:
1043:
1039:
1035:
1030:
1023:
1019:
1015:
1010:
1003:
999:
995:
981:
967:safety factors
950:
946:
942:
922:
918:
914:
911:
908:
881:
877:
873:
869:
865:
862:
857:
852:
848:
841:
836:
833:
828:
825:
803:
781:
777:
773:
744:
743:
739:
728:
713:
712:
700:
697:
687:
686:
685:
673:
670:
667:
656:
644:
641:
638:
627:
615:
612:
609:
598:
586:
583:
580:
553:
543:
531:
521:
505:
495:
479:
469:
451:
447:
416:
412:
408:
405:
402:
397:
394:
389:
385:
378:
373:
369:
351:
350:
347:
344:
333:Leonhard Euler
329:
328:
309:
301:
241:
238:
236:
233:
218:
217:
215:
214:
207:
200:
192:
189:
188:
187:
186:
181:
176:
171:
166:
161:
156:
151:
146:
141:
136:
131:
126:
121:
116:
111:
106:
98:
97:
15:
13:
10:
9:
6:
4:
3:
2:
5162:
5151:
5148:
5146:
5143:
5141:
5138:
5136:
5133:
5131:
5128:
5127:
5125:
5116:
5112:
5111:
5107:
5101:
5095:
5091:
5086:
5082:
5077:
5073:
5067:
5063:
5058:
5053:
5048:
5044:
5040:
5036:
5032:
5027:
5024:(PhD Thesis).
5020:
5019:
5013:
5009:
5005:
5000:
4995:
4994:
4988:
4984:
4983:
4979:
4970:
4965:
4961:
4957:
4953:
4949:
4945:
4938:
4935:
4930:
4926:
4921:
4916:
4912:
4908:
4904:
4900:
4896:
4889:
4886:
4870:
4863:
4860:
4848:
4844:
4840:
4833:
4830:
4817:
4813:
4807:
4804:
4792:
4788:
4781:
4778:
4762:
4756:
4753:
4741:
4737:
4730:
4727:
4722:
4716:
4712:
4711:
4703:
4700:
4688:
4684:
4677:
4674:
4669:
4667:9780471859918
4663:
4659:
4652:
4649:
4644:
4640:
4633:
4630:
4625:
4618:
4616:
4614:
4612:
4610:
4608:
4606:
4602:
4597:
4593:
4589:
4585:
4581:
4574:
4571:
4566:
4560:
4556:
4552:
4545:
4542:
4537:
4533:
4526:
4523:
4520:
4519:0-521-78210-4
4516:
4510:
4507:
4502:
4495:
4493:
4491:
4489:
4485:
4479:
4474:
4471:
4465:
4462:
4456:
4453:
4451:
4448:
4446:
4443:
4440:
4437:
4434:
4431:
4430:
4426:
4424:
4417:
4415:
4413:
4412:hoop stresses
4404:
4402:
4395:
4391:
4387:
4383:
4380:
4376:
4373:
4370:tracks, near
4369:
4365:
4364:
4360:
4356:
4353:
4350:tracks, near
4349:
4345:
4344:
4340:
4336:
4335:
4334:
4332:
4331:
4324:
4320:
4318:
4314:
4310:
4306:
4298:
4293:
4286:
4284:
4282:
4278:
4274:
4270:
4266:
4262:
4258:
4250:
4248:
4246:
4241:
4240:bicycle wheel
4233:
4228:
4208:
4204:
4200:
4197:
4194:
4189:
4185:
4176:
4172:
4156:
4152:
4148:
4145:
4142:
4137:
4133:
4124:
4120:
4119:
4099:
4096:
4089:
4085:
4082:
4077:
4073:
4064:
4060:
4055:
4051:
4050:
4034:
4016:
4012:
4006:
4002:
4001:
3997:
3994:
3993:
3986:
3982:
3976:
3972:
3971:
3967:
3964:
3963:
3947:
3944:
3941:
3931:
3929:
3928:
3923:
3919:
3918:
3917:transcritical
3913:
3909:
3908:
3903:
3899:
3898:
3897:supercritical
3893:
3889:
3888:
3882:
3868:
3848:
3828:
3800:
3784:
3764:
3741:
3738:
3735:
3732:
3729:
3709:
3701:
3693:
3691:
3677:
3674:
3654:
3631:
3625:
3616:
3599:
3588:
3577:
3569:
3565:
3558:
3553:
3544:
3538:
3536:
3525:
3517:
3506:
3495:
3487:
3484:
3480:
3470:
3464:
3461:
3456:
3453:
3448:
3446:
3435:
3422:
3418:
3399:
3395:
3386:
3377:
3373:
3369:
3363:Energy method
3362:
3357:
3355:
3341:
3321:
3301:
3281:
3261:
3241:
3221:
3212:
3199:
3194:
3190:
3186:
3183:
3180:
3177:
3169:
3162:
3160:
3156:
3154:
3149:
3145:
3143:
3139:
3135:
3131:
3122:
3120:
3114:
3112:
3105:
3098:
3096:
3076:
3072:
3064:
3047:
3043:
3035:
3018:
3014:
3006:
2985:
2977:
2976:
2975:
2972:
2954:
2950:
2946:
2943:
2938:
2934:
2930:
2927:
2922:
2918:
2914:
2911:
2902:
2898:
2887:
2883:
2877:
2872:
2868:
2859:
2856:
2852:
2847:
2844:
2840:
2832:
2827:
2823:
2818:
2814:
2810:
2805:
2801:
2797:
2795:
2791:
2786:
2784:
2780:
2776:
2772:
2763:
2756:
2754:
2747:
2745:
2724:
2720:
2712:
2691:
2683:
2682:
2681:
2678:
2662:
2653:
2649:
2643:
2640:
2636:
2629:
2626:
2623:
2619:
2611:
2607:
2601:
2598:
2594:
2586:
2581:
2572:
2561:
2546:
2526:
2517:
2499:
2494:
2489:
2486:
2481:
2475:
2469:
2465:
2461:
2458:
2454:
2450:
2445:
2440:
2436:
2427:
2424:
2420:
2416:
2411:
2408:
2404:
2394:
2380:
2360:
2339:
2318:
2309:
2294:
2289:
2282:
2279:
2275:
2270:
2265:
2261:
2258:
2251:
2246:
2241:
2238:
2234:
2211:
2208:
2204:
2194:
2176:
2172:
2167:
2161:
2157:
2153:
2150:
2146:
2142:
2135:
2131:
2127:
2122:
2118:
2109:
2106:
2102:
2098:
2093:
2090:
2087:
2084:
2080:
2071:
2055:
2051:
2030:
2006:
1999:
1984:
1977:
1962:
1955:
1940:
1933:
1932:
1931:
1928:
1914:
1909:
1905:
1902:
1899:
1893:
1889:
1886:
1882:
1877:
1873:
1870:
1867:
1861:
1857:
1854:
1849:
1846:
1842:
1831:
1828:
1825:
1821:
1810:
1807:
1804:
1800:
1796:
1793:
1785:
1767:
1760:
1758:
1742:
1735:
1733:
1717:
1710:
1693:
1689:
1681:
1666:
1659:
1658:
1657:
1654:
1640:
1631:
1627:
1618:
1613:
1600:
1596:
1592:
1588:
1579:
1575:
1571:
1565:
1559:
1555:
1551:
1548:
1544:
1540:
1534:
1526:
1522:
1513:
1508:
1497:
1489:
1485:
1476:
1472:
1463:
1458:
1447:
1444:
1436:
1432:
1423:
1418:
1402:
1399:
1396:
1388:
1386:
1384:
1368:
1360:
1344:
1324:
1315:
1299:
1296:
1290:
1283:
1280:
1277:
1272:
1269:
1260:
1254:
1250:
1246:
1239:
1234:
1225:
1216:
1202:
1182:
1162:
1152:
1151:Self-buckling
1145:Self-buckling
1144:
1142:
1122:
1099:
1095:
1072:
1068:
1058:
1041:
1037:
1033:
1028:
1021:
1017:
1013:
1008:
997:
993:
980:
976:
972:
968:
962:
948:
944:
940:
920:
916:
912:
909:
906:
897:
879:
871:
867:
863:
855:
850:
846:
839:
834:
831:
826:
823:
815:
801:
779:
775:
771:
761:
757:
748:
740:
737:
733:
729:
726:
722:
718:
717:
716:
698:
695:
688:
671:
668:
665:
657:
642:
639:
636:
628:
613:
610:
607:
599:
584:
581:
578:
570:
569:
567:
551:
544:
529:
522:
519:
503:
496:
493:
477:
470:
467:
449:
445:
437:
436:
435:
432:
414:
406:
403:
395:
392:
387:
383:
376:
371:
367:
358:
356:
348:
345:
342:
341:
340:
338:
334:
326:
322:
318:
313:
310:
306:
302:
298:
294:
293:
292:
289:
286:
282:
278:
273:
271:
267:
264:to the least
263:
254:
246:
239:
234:
232:
229:
225:
213:
208:
206:
201:
199:
194:
193:
191:
190:
185:
182:
180:
177:
175:
174:Thermal shock
172:
170:
167:
165:
162:
160:
157:
155:
152:
150:
147:
145:
142:
140:
137:
135:
132:
130:
127:
125:
122:
120:
117:
115:
112:
110:
107:
105:
102:
101:
100:
99:
94:
91:
89:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
40:
35:
29:
22:
5089:
5080:
5062:Steel Design
5061:
5034:
5030:
5017:
5007:
5003:
4992:
4951:
4948:AIAA Journal
4947:
4937:
4902:
4898:
4888:
4876:. Retrieved
4862:
4851:. Retrieved
4842:
4832:
4820:. Retrieved
4815:
4806:
4794:. Retrieved
4790:
4780:
4768:. Retrieved
4755:
4743:. Retrieved
4739:
4729:
4709:
4702:
4690:. Retrieved
4686:
4676:
4657:
4651:
4638:
4632:
4623:
4587:
4583:
4573:
4554:
4550:
4544:
4535:
4531:
4525:
4509:
4500:
4421:
4408:
4401:incidents."
4399:
4361:
4341:
4327:
4325:
4321:
4312:
4302:
4265:Radiant heat
4254:
4237:
3925:
3921:
3915:
3911:
3905:
3901:
3895:
3891:
3885:
3883:
3799:conservative
3798:
3699:
3697:
3617:
3423:
3419:
3378:
3374:
3370:
3366:
3213:
3170:
3166:
3157:
3152:
3150:
3146:
3133:
3129:
3126:
3118:
3106:
3102:
3094:
2973:
2860:
2854:
2850:
2848:
2842:
2838:
2825:
2821:
2816:
2812:
2803:
2799:
2798:
2787:
2782:
2768:
2751:
2742:
2679:
2562:
2518:
2395:
2310:
2195:
2072:
2022:
1929:
1786:
1783:
1655:
1403:
1400:
1392:
1316:
1217:
1154:
1059:
978:
963:
898:
816:
762:
758:
754:
732:proportional
714:
516:, smallest
433:
359:
352:
330:
290:
274:
259:
221:
103:
79:
51:
45:
4464:Wood method
4305:rail tracks
4303:Similarly,
4297:Netherlands
4287:Rail tracks
3965:Limit Point
3907:subcritical
3892:limit point
2831:cantilevers
2775:compression
1780:, thickness
751:conditions.
355:equilibrium
72:compression
56:deformation
5124:Categories
5052:10092/4229
4853:2019-01-21
4720:148228880X
4480:References
4455:Stiffening
4343:Auto-Train
4281:speed hump
4277:automobile
3922:asymmetric
3368:analysis.
3144:failures.
2043:equals 1.
984:given by:
721:elasticity
285:deflection
5150:Mechanics
4929:125479249
4905:: 74–89.
4386:Metrorail
4267:from the
3809:Δ
3801:load and
3745:Δ
3739:−
3559:∫
3462:∫
3302:σ
3222:σ
3187:ρ
3178:σ
3115:Crippling
2721:σ
2650:σ
2637:σ
2627:−
2608:σ
2595:σ
2587:≈
2466:ν
2462:−
2437:π
2405:σ
2158:ν
2154:−
2119:π
1903:π
1890:
1871:π
1858:
1837:∞
1822:∑
1816:∞
1801:∑
1718:ν
1624:∂
1610:∂
1593:−
1556:ν
1552:−
1519:∂
1505:∂
1482:∂
1469:∂
1455:∂
1429:∂
1415:∂
1278:ρ
1163:ρ
907:σ
847:π
824:σ
640:≈
384:π
288:bending.
109:Corrosion
4847:Archived
4427:See also
4394:Maryland
4313:sun kink
4257:pavement
3138:yielding
308:column).
305:concrete
303:A short
295:A short
224:stresses
184:Yielding
134:Fracture
104:Buckling
52:buckling
42:tension.
4956:Bibcode
4907:Bibcode
4878:18 July
4822:18 July
4796:18 July
4770:18 July
4745:18 July
4692:18 July
4261:asphalt
4245:pringle
3530:applied
3391:applied
3142:fatigue
2779:tension
2771:bending
1930:where
1357:is the
734:to the
325:species
240:Columns
228:failure
129:Fouling
124:Fatigue
80:buckled
58:) of a
5096:
5068:
4927:
4816:WIS 10
4717:
4664:
4561:
4517:
4359:Amtrak
4339:Amtrak
4273:expand
3894:; the
3757:where
3618:where
3440:strain
3404:strain
3358:Theory
3214:where
2974:where
2680:where
1656:where
1317:where
1060:where
899:where
434:where
321:stress
312:Timber
262:column
144:Impact
70:under
68:column
62:under
28:buckle
5022:(PDF)
4925:S2CID
4872:(PDF)
4843:DCist
4764:(PDF)
4553:[
4538:: 41.
4307:also
4251:Roads
2630:1.022
1395:plate
643:0.699
466:force
297:steel
119:Creep
76:shear
5094:ISBN
5066:ISBN
4880:2024
4824:2024
4798:2024
4772:2024
4747:2024
4715:ISBN
4694:2024
4662:ISBN
4559:ISBN
4515:ISBN
4317:ties
3667:and
3549:crit
3314:and
2884:12.5
1230:crit
794:for
719:The
614:0.50
281:load
179:Wear
86:and
64:load
39:B-52
5047:hdl
5039:doi
5035:128
4964:doi
4915:doi
4903:443
4592:doi
4368:CSX
4348:CSX
4333:):
4269:sun
3920:or
3910:or
3900:or
3890:or
2990:max
2907:max
2899:2.5
2892:max
2696:eff
2577:eff
1887:sin
1855:sin
1127:max
1002:max
982:max
672:2.0
585:1.0
46:In
5126::
5045:.
5033:.
5006:.
4962:.
4952:58
4950:.
4946:.
4923:.
4913:.
4901:.
4897:.
4841:.
4814:.
4789:.
4738:.
4685:.
4641:.
4604:^
4588:29
4586:.
4536:19
4534:.
4487:^
4392:,
3960:.
3702:,
3354:.
2796:.
2451:12
2393:.
2331:/
2143:12
1755:,
1730:,
1541:12
1393:A
1141:.
814::
564:,
490:,
82:.
50:,
5102:.
5074:.
5055:.
5049::
5041::
5010:.
5008:4
4972:.
4966::
4958::
4931:.
4917::
4909::
4882:.
4856:.
4826:.
4800:.
4774:.
4749:.
4723:.
4696:.
4670:.
4598:.
4594::
4567:.
4396:.
4381:.
4374:.
4354:.
4209:2
4205:/
4201:L
4198:k
4195:=
4190:C
4186:P
4157:2
4153:/
4149:L
4146:k
4143:=
4138:C
4134:P
4103:)
4100:L
4097:2
4094:(
4090:/
4086:c
4083:=
4078:C
4074:P
4035:P
3948:0
3945:=
3942:q
3869:V
3849:V
3829:P
3785:P
3765:U
3742:P
3736:U
3733:=
3730:V
3710:V
3678:x
3675:x
3655:x
3635:)
3632:x
3629:(
3626:w
3600:x
3596:d
3589:2
3585:)
3581:)
3578:x
3575:(
3570:x
3566:w
3562:(
3554:2
3545:P
3539:=
3526:U
3518:x
3514:d
3507:2
3503:)
3499:)
3496:x
3493:(
3488:x
3485:x
3481:w
3477:(
3474:)
3471:x
3468:(
3465:I
3457:2
3454:E
3449:=
3436:U
3400:U
3396:=
3387:U
3342:h
3322:h
3282:h
3262:c
3242:L
3200:h
3195:2
3191:c
3184:=
3181:L
3109:t
3077:C
3073:M
3048:B
3044:M
3019:A
3015:M
2986:M
2955:C
2951:M
2947:3
2944:+
2939:B
2935:M
2931:4
2928:+
2923:A
2919:M
2915:3
2912:+
2903:M
2888:M
2878:=
2873:b
2869:C
2855:b
2851:C
2843:b
2839:C
2835:b
2826:b
2822:C
2817:b
2813:C
2804:b
2800:C
2725:y
2692:b
2663:)
2654:y
2644:r
2641:c
2624:1
2620:(
2612:y
2602:r
2599:c
2582:b
2573:b
2547:a
2527:b
2500:2
2495:)
2490:t
2487:b
2482:(
2476:)
2470:2
2459:1
2455:(
2446:E
2441:2
2428:r
2425:c
2421:k
2417:=
2412:r
2409:c
2381:m
2361:m
2340:b
2319:a
2295:2
2290:)
2283:b
2280:m
2276:a
2271:+
2266:a
2262:b
2259:m
2252:(
2247:=
2242:r
2239:c
2235:k
2212:r
2209:c
2205:k
2177:2
2173:b
2168:)
2162:2
2151:1
2147:(
2136:3
2132:t
2128:E
2123:2
2110:r
2107:c
2103:k
2099:=
2094:r
2091:c
2088:,
2085:x
2081:N
2056:x
2052:N
2031:n
2007:b
1985:a
1963:n
1941:m
1915:)
1910:b
1906:y
1900:n
1894:(
1883:)
1878:a
1874:x
1868:m
1862:(
1850:n
1847:m
1843:w
1832:1
1829:=
1826:n
1811:1
1808:=
1805:m
1797:=
1794:w
1768:t
1743:E
1694:x
1690:N
1667:w
1641:)
1632:2
1628:x
1619:w
1614:2
1601:x
1597:N
1589:(
1580:3
1576:t
1572:E
1566:)
1560:2
1549:1
1545:(
1535:=
1527:4
1523:y
1514:w
1509:4
1498:+
1490:2
1486:y
1477:2
1473:x
1464:w
1459:4
1448:2
1445:+
1437:4
1433:x
1424:w
1419:4
1369:B
1345:I
1325:g
1300:3
1297:1
1291:)
1284:A
1281:g
1273:I
1270:E
1261:4
1255:2
1251:B
1247:9
1240:(
1235:=
1226:h
1203:A
1183:E
1123:F
1100:c
1096:F
1073:e
1069:F
1042:c
1038:F
1034:1
1029:+
1022:e
1018:F
1014:1
1009:=
998:F
994:1
979:F
949:r
945:/
941:l
921:A
917:/
913:F
910:=
880:2
876:)
872:r
868:/
864:l
861:(
856:E
851:2
840:=
835:A
832:F
827:=
802:I
780:2
776:r
772:A
699:L
696:K
684:.
669:=
666:K
655:.
637:K
626:.
611:=
608:K
597:.
582:=
579:K
552:K
530:L
504:I
494:,
478:E
450:c
446:F
415:2
411:)
407:L
404:K
401:(
396:I
393:E
388:2
377:=
372:c
368:F
211:e
204:t
197:v
30:.
23:.
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