Buckling Buckling Church of the Province of West Africa Category="Materials science"

Mechanical failure modes
Buckling
Corrosion
Creep
Fatigue
Fracture
Melting
Thermal shock
Wear

In Engineering, buckling is a failure mode of a structural member characterised by a failure to react to the bending moment generated by a compressive load.

Buckling in columns

[image]The ratio of the length of a Column to the least Radius of gyration of its cross section is called the slenderness ratio (usually expressed with the Greek letter lambda - λ). This ratio affords a means of classifying columns. All the following are approximate values used for convenience.

If the load on a column is applied through the Center of gravity of its cross section it is called an Axial Load. 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 but a long column loaded in the same manner will fail by buckling (Bending), the buckling effect being so large that the effect of the direct load may be neglected. The intermediate length column will fail by a combination of direct stress and bending.

The 18th-century Mathematician Leonhard Euler derived a formula which gives the maximum axial load that a long, slender ideal column can carry without buckling. An ideal column is one which is perfectly straight, homogenous, and free from initial stress. The maximum load, sometimes called the critical load, causes the column to be in a state of unstable Equilibrium, that is, any increase in the loads or the introduction of the slightest lateral force will cause the column to fail by buckling. The Euler formula for columns is:

F =  (K π 2 E I )
–––––––––––––––
l 2

Where

F = maximum or critical Force (vertical load on column)
E = modulus of elasticity
I = area moment of inertia
l = unsupported length of column
K = a constant whose value depends upon the conditions of end support of the column, For both ends free to turn K = 1; for both ends fixed K = 4; for one end free to turn and the other end fixed K = 2 approximately, and for one end fixed and the other end free to move laterally K = 1/4.

Examination of this formula reveals the following interesting facts with regard to the load bearing ability of columns:

  1. that elasticity and not compressive strength of the materials of the column determines the critical load.
  2. the critical load is directly proportional to the moment of inertia of the cross-section.

The strength of a column may therefore be increased by distributing the material so as to increase the moment of inertia. This can be done without increasing the weight of the column by distributing the material as far from the principal axes of the transverse section as possible consistent with keeping the material thick enough to prevent local buckling. This bears out the well-known fact that a tubular section is much superior to a solid section for column service.

Another bit of information that may be gleaned from this equation is the effect of length upon critical load. For a given size column, doubling the unsupported length quarters the allowable load. The restraint offered by the end connections of a column also affects the critical load. If the connections are perfectly rigid, the critical load will be four times that for a similar column where there is no resistance to rotation (hinged at the ends).

Since the moment of inertia of a surface is its area multiplied by the square of a length called the radius of gyration, the above formula may be rearranged as follows. Using the Euler formula for hinged ends and substituting Ar2 for I the following formula results:

σ = F/A =  π 2 E
––––––––
(l/r) 2

where F/A is the allowable unit stress of the column and l/r is the slenderness ratio.

Since the structural column is generally an intermediate length column and it is impossible to obtain an ideal column, the Euler formula has little practical application for ordinary design. Consequently, a number of empirical column formulae have been developed to agree with test data, all of which embody the slenderness ratio. For design, appropriate safety factors are introduced into these formulae.

Buckling of surface materials

Buckling is also a failure mode in Pavement materials, primarily with concrete since asphalt is more flexible. Radiant heat from the Sun is absorbed in the road surface, causing it to expand and forcing adjacent pieces to push against each other. If the stress is great enough, the pavement can lift up and crack without warning. Going over a buckled section can be very jarring to Automobile drivers, described as running over a Speed bump at highway speeds.

Similarly, railroad tracks also expand when heated, and can fail by buckling. It is more common for rails to move laterally, often pulling the underlain ties (sleepers) along with them.

Local Buckling

Open structural sections may be considered to be several plates joined together. For example, a wide-flange shape (I-beam) can be thought of as two parallel plates (flanges) connected by a third perpendicular plate (web). It is possible that if any one of those three "plates" were sufficiently slender and loaded in compression, then the indivisual plate could buckle before the beam as a whole buckled (called global buckling). When this happens, it is referred to as local buckling since a local element buckled prior to the global buckling of the beam.

Lateral-Torsional Buckling

When a beam is loaded in flexure the load bearing side carries the load in compression whereas the non-load bearing side will be in tension. If the beam is not laterally supported and the flexural load increases to a critical limit, the beam will fail due to local buckling at the load bearing side. In wide-flange sections, if the top flange buckles laterally, the rest of the section will twist resulting in a failure mode known as lateral-torsional buckling.

See also