Formula to quantify column buckling under a given load
Euler's critical load or Euler's buckling load is the compressive load at which a slender column will suddenly bend or buckle. It is given by the formula:[1]
where
, Euler's critical load (longitudinal compression load on column),
This formula was derived in 1744 by the SwissmathematicianLeonhard Euler.[2] The column will remain straight for loads less than the critical load. The critical load is the greatest load that will not cause lateral deflection (buckling). For loads greater than the critical load, the column will deflect laterally. The critical load puts the column in a state of unstable equilibrium. A load beyond the critical load causes the column to fail by buckling. As the load is increased beyond the critical load the lateral deflections increase, until it may fail in other modes such as yielding of the material. Loading of columns beyond the critical load are not addressed in this article.
Around 1900, J. B. Johnson showed that at low slenderness ratios an alternative formula should be used.
Assumptions of the model
The following assumptions are made while deriving Euler's formula:[3]
is the second moment of area (area moment of inertia),
is the area cross section.
For slender columns, the critical buckling stress is usually lower than the yield stress. In contrast, a stocky column can have a critical buckling stress higher than the yield, i.e. it yields prior to buckling.
Mathematical derivation
Pin ended column
The following model applies to columns simply supported at each end ().
Firstly, we will put attention to the fact there are no reactions in the hinged ends, so we also have no shear force in any cross-section of the column. The reason for no reactions can be obtained from symmetry (so the reactions should be in the same direction) and from moment equilibrium (so the reactions should be in opposite directions).
Using the free body diagram in the right side of figure 3, and making a summation of moments about point x:
The general solutions of this equation is: , where and are constants to be determined by boundary conditions, which are:
Left end pinned:
Right end pinned:
If , no bending moment exists and we get the trivial solution of .
However, from the other solution we get , for
Together with as defined before, the various critical loads are:
and depending upon the value of , different buckling modes are produced[4] as shown in figure 4. The load and mode for n=0 is the nonbuckled mode.
Theoretically, any buckling mode is possible, but in the case of a slowly applied load only the first modal shape is likely to be produced.
The critical load of Euler for a pin ended column is therefore:
and the obtained shape of the buckled column in the first mode is:
General approach
The differential equation of the axis of a beam[5] is:
For a column with axial load only, the lateral load vanishes and substituting , we get:
This is a homogeneous fourth-order differential equation and its general solution is
The four constants are determined by the boundary conditions (end constraints) on , at each end. There are three cases:
Pinned end:
and
Fixed end:
and
Free end:
and
For each combination of these boundary conditions, an eigenvalue problem is obtained. Solving those, we get the values of Euler's critical load for each one of the cases presented in Figure 2.