Buckling
[2] Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress of a column.
Further loading may cause significant and somewhat unpredictable deformations, possibly leading to complete loss of the member's load-carrying capacity.
Some aircraft are designed for thin skin panels to continue carrying load even in the buckled state.
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.
Since structural columns are commonly of intermediate length, the Euler formula has little practical application for ordinary design.
Consequently, a number of empirical column formulae have been developed that agree with test data, all of which embody the slenderness ratio.
is the first zero of the Bessel function of the first kind of order −1/3, which is equal to 1.86635086... A plate 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.
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:[6]
For an increasing number of such curvatures, the aspect ratio produces a varying buckling coefficient; but each relation provides a minimum value for each
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:[6]
This mostly occurs in columns with "open" cross-sections and hence have a low torsional stiffness, such as channels, structural tees, double-angle shapes, and equal-leg single angles.
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 lateral-torsional buckling.
In wide-flange sections (with high lateral bending stiffness), the deflection mode will be mostly twisting in torsion.
Cb is a modification factor used in the equation for nominal flexural strength 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.
However, once buckled, instead of being able to transmit shear forces, they are still able to carry load through diagonal tension (DT) stresses in the web.
Although they may buckle, thin sheets are designed to not permanently deform and return to an unbuckled state when the applied loading is removed.
Thicker plates may only partially form a diagonal tension field and may continue to carry some of the load through shear.
[7] 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 analysis.
, it is possible to identify four fundamental forms of buckling found in structural models with one degree of freedom.
A conventional bicycle wheel consists of a thin rim kept under high compressive stress by the (roughly normal) inward pull of a large number of spokes.
If spoke tension is increased beyond a safe level or if part of the rim is subject to a certain lateral force, the wheel spontaneously fails into a characteristic saddle shape (sometimes called a "taco" or a "pringle") 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.
Radiant heat from the sun is absorbed in the road surface, causing it to expand, forcing adjacent pieces to push against each other.
Similarly, rail tracks also expand when heated, and can fail by buckling, a phenomenon called sun kink.
[11] These accidents were deemed to be sun kink-related (more information available at List of rail accidents (2000–2009)): 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 incidents.
"[16] 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 hoop stresses.
Design rules for calculation of the required wall thickness or reinforcement rings are given in various piping and pressure vessel codes.
[17] If buckling is caused by aerothermal loads, the situation can be further complicated by enhanced heat transfer in areas where the structure deforms towards the flow-field.
