Euler's critical load
Euler's critical load or Euler's buckling load is the compressive load at which a slender column will suddenly bend or buckle.
{\displaystyle P_{cr}={\frac {\pi ^{2}EI}{(KL)^{2}}}}
where This formula was derived in 1744 by the Swiss mathematician Leonhard Euler.
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.
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.
B. Johnson showed that at low slenderness ratios an alternative formula should be used.
The following assumptions are made while deriving Euler's formula:[3] 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.
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:
According to Euler–Bernoulli beam theory, the deflection of a beam is related with its bending moment by:
We get a classical homogeneous second-order ordinary differential equation.
The general solutions of this equation is:
are constants to be determined by boundary conditions, which are: If
, no bending moment exists and we get the trivial solution of
sin ( λ ℓ ) = 0
as defined before, the various critical loads are:
, different buckling modes are produced[4] as shown in figure 4.
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:
The differential equation of the axis of a beam[5] is:
This is a homogeneous fourth-order differential equation and its general solution is
are determined by the boundary conditions (end constraints) on
There are three cases: 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.
