Buckling Case Study Solution

Buckling:
Structures that are subjected to compressive axial loads may fail due to the lateral deflection this phenomenon is known as Buckling. It is demonstrated as by pressing the opposite edges of column towards one another. For small loads the process is elastic as buckling displacement disappears when the load is removed. Local buckling of plates or shells are indicated by bulges, waves or ripples and this are most common in thin structural members.
Buckling proceeds in a manner which may be either:
Stable- in which displacements increases as load increases but the structure can sustain loads
Unstable- in which deformation increases instantaneously and the load carrying capacity of the structure collapses
Neutral equilibrium- this is characterized

It is valid only for long and slender columns. When this formula is used for columns of lesser length then the critical stress value exceeds the yield strength of the material. Obviously, Euler’s solution is of no interest under these considerations.

Johnsons solution for buckling of intermediate columns:
For very short columns, the column will not buckle but simply compress. Failure will then be due to yielding of material. There is an intermediate range of slenderness ratio where neither Euler’s nor P/A model matches. Johnson’s solution is often used in intermediate range of columns and is given by σ_cr=σ_y-1/E (σ_y/2π)^2 (l/k)^2
Note from Johnson’s equation and Euler’s equation we find that equations are equal and tangent to each other at a specific slenderness ratio. Euler’s equation tends to infinity when slenderness ratio equal to zero, whereas Johnsons equations tends to σ_y for a slenderness ratio of zero.
The slenderness ratio at equal point is given by,
(l/k)_equal=√(((2π^2 E)/σ_y ) )
For the case E= 10.4Mpsi and σ_y= 40Ksi, we have the following plot at equal point at
(l/k)_equal=71

From observation Euler’s equation is good for slenderness ratio greater than the point value, whereas Johnsons equation is good for lesser slenderness