Euler’s theory of column buckling is used to estimate the critical buckling load of column since the stress in the column remains elastic. The critical buckling load is the maximum load that a column can withstand when it is on the verge of buckling. The buckling failure occurs when the length of the column is greater when compared with its cross-section.

The Euler’s theory is based on certain assumptions related to the point of axial load application, column material, cross-section, stress limits, and column failure. The validity of Euler’s theory is subjected to a condition that failure occurs due to buckling.

This theory does not consider the effect of direct stress in column, the crookedness in column which is always present, and possible shifts of axial load application point from the center of the column cross-section. As a result, the theory may overestimate the critical buckling load. The Euler theory of column buckling was invented by Leonhard Euler in 1757.

Euler’s Theory

The Euler’s theory states that the stress in the column due to direct loads is small compared to the stress due to buckling failure. Based on this statement, a formula derived to compute the critical buckling load of column. So, the equation is based on bending stress and neglects direct stress due to direct loads on the column.

Assumptions

  • Initially, the column is perfectly straight.
  • The cross-section of the column is uniform throughout its length.  
  • The load is axial and passes through the centroid of the section.
  • The stresses in the column are within the elastic limit.
  • The materials of the column are homogenous and isotropic.
  • The self-weight of the column itself is neglected.
  • The failure of the column occurs due to buckling only.
  • Length of column is large compared to its cross-sectional dimensions.
  • The ends of the column are frictionless.
  • The shortening of column due to axial compression is negligible.

Limitations

  • The possibility of crookedness in column is not accounted for in this theory, and the load may not be axial.
  • The axial stress is not considered in the formula derived in Euler theory of column buckling, and the critical buckling load may be greater than the actual buckling load.

Critical Buckling Load

The maximum axial load that a column can support when it is on the verge of buckling is called the crippling load or critical buckling load (Pcr). At this stage, the ultimate stress in the column would less than the yield stress of the material, and the column is pinned at both ends:

Pcr= EI(PI/KL) ^2                                                           Equation 1

Where:

Pcr: Critical buckling load

E: Modulus of elasticity

I: Moment of inertia which is equal to cross-sectional area multiply radius of gyration.

L: Length of the slender column

K: Effective length factor which is based on the support conditions of the column as illustrated in Fig. 1.

End Conditions
Fig. : End Conditions

Failure of Columns

Buckling failure occurs when the cross-section of the column is small in comparison with its height. The buckling takes place about the axis having minimum radius of gyration or least moment of inertia. The formula of critical buckling load can be expressed in terms of radius of gyration:

Pcr= Ear^2(PI/KL) ^2                                                             Equation 2

Or

Mean compressive stress on column/E= (PI)^2/(KL/r)^2       Equation 3

Equation 3 is the most convenient form of presenting theoretical and experimental results for buckling problems. The ratio KL /r is called the slenderness ratio.

Failure of the column would occur in purely axial compression provided that the stress in the column reaches the yield stress of the material. However, when the critical buckling stress is less than the yield stress, then the column would fail by buckling before the yield stress is reached.

Buckling Failure of Slender Column
Fig.2: Buckling Failure of Slender Column
Failure of Long Column
Fig. 3: Failure of Long Column

Sign convention for Bending Moments

A bending moment which bends the column as to present convexity towards the initial centre line of the member would be regarded as positive.

Bending moment which bends the column as to present concavity towards the initial centre line of the member would be regarded as negative.

Sign Convention for Bending Moment
Fig. 4: Sign Convention for Bending Moment