Euler's critical load

The critical load is the maximum load (unit: Newton, it is a force) which a column can bear while staying straight. It is given by the formula:[1]

Fig. 1: Column effective length factors for Euler's critical load. In practical design, it is recommended to increase the factors as shown above.

${\displaystyle P_{cr}={\frac {\pi ^{2}EI}{(KL)^{2}}}}$

where

${\displaystyle P_{cr}}$, Euler's critical load (longitudinal compression load on column),

${\displaystyle E}$, modulus of elasticity of column material,

${\displaystyle I}$, minimum area moment of inertia of the cross section of the column,

${\displaystyle L}$, unsupported length of column,

${\displaystyle K}$, column effective length factor

This formula was derived in 1757 by the Swiss mathematician Leonhard Euler. 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

Fig. 2: Critical stress vs slenderness ratio for steel, for E = 200 GPa, yield strength = 240 MPa.

The following assumptions are made while deriving Euler's formula:[2]

1. The material of the column is homogeneous and isotropic.
2. The compressive load on the column is axial only.
3. The column is free from initial stress.
4. The weight of the column is neglected.
5. The column is initially straight (no eccentricity of the axial load).
6. Pin joints are friction-less (no moment constraint) and fixed ends are rigid (no rotation deflection).
7. The cross-section of the column is uniform throughout its length.
8. The direct stress is very small as compared to the bending stress (the material is compressed only within the elastic range of strains).
9. The length of the column is very large as compared to the cross-sectional dimensions of the column.
10. The column fails only by buckling. This is true if the compressive stress in the column does not exceed the yield strength ${\displaystyle \sigma _{y}}$ (see figure 2):

    ${\displaystyle \sigma ={\frac {P_{cr}}{A}}={\frac {\pi ^{2}E}{(L_{e}/r)^{2}}}<\sigma _{y}}$

For slender columns, critical stress is usually lower than yield stress, and in the elastic range. In contrast, a stocky column would have a critical buckling stress higher than the yield, i.e. it yields in shortening prior the virtual elastic buckling onset.

Where:

${\textstyle {\frac {L_{e}}{r}}}$, slenderness ratio,

${\displaystyle L_{e}=KL}$, the effective length,

${\textstyle r={\sqrt {\operatorname {I} \over \operatorname {A} }}}$, radius of gyration,

${\displaystyle I}$, area moment of inertia,

${\displaystyle A}$, area cross section.

Mathematical derivation: Pin ended column

The following model applies to columns simply supported at each end (${\displaystyle K=1}$).

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 A:

${\displaystyle \Sigma M=0\Rightarrow M(x)+Pw=0}$

where w is the lateral deflection.

According to Euler–Bernoulli beam theory, the deflection of a beam is related with its bending moment by:

${\displaystyle M=-EI{\frac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}}$,

Fig. 3: Pin ended column under the effect of Buckling load

so:

${\displaystyle EI{\frac {d^{2}w}{dx^{2}}}+Pw=0}$

Let ${\displaystyle \lambda ^{2}={\frac {P}{EI}}}$, so:

${\displaystyle {\frac {d^{2}w}{dx^{2}}}+\lambda ^{2}w=0}$

We get a classical homogeneous second-order ordinary differential equation.

The general solutions of this equation is: ${\displaystyle w(x)=A\cos(\lambda x)+B\sin(\lambda x)}$, where ${\displaystyle A}$ and ${\displaystyle B}$ are constants to be determined by boundary conditions, which are:

• Left end pinned ${\displaystyle \rightarrow w(0)=0\rightarrow A=0}$
• Right end pinned ${\displaystyle \rightarrow w(l)=0\rightarrow B\sin(\lambda l)=0}$

Fig. 4: First three modes of buckling loads

If ${\displaystyle B=0}$, no bending moment exists and we get the trivial solution of ${\displaystyle w(x)=0}$.

However, from the other solution ${\displaystyle \sin(\lambda l)=0}$ we get ${\displaystyle \lambda _{n}l=n\pi }$, for ${\displaystyle n=0,1,2,\ldots }$

Together with ${\displaystyle \lambda ^{2}={\frac {P}{EI}}}$ as defined before, the various critical loads are:

${\displaystyle P_{n}={\frac {n^{2}\pi ^{2}EI}{l^{2}}}}$, for ${\displaystyle n=0,1,2,\ldots }$

and depending upon the value of ${\displaystyle n}$, different buckling modes are produced[3] 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:

${\displaystyle P_{cr}={\frac {\pi ^{2}EI}{l^{2}}}}$

and the obtained shape of the buckled column in the first mode is:

${\displaystyle w(x)=B\sin \left({\pi \over l}x\right)}$.

Mathematical derivation: General approach

Fig. 5: forces and moments acting on a column.

The differential equation of the axis of a beam[4] is:

${\displaystyle {\frac {d^{4}w}{dx^{4}}}+{\frac {P}{EI}}{\frac {d^{2}w}{dx^{2}}}={\frac {q}{EI}}}$

For a column with axial load only, the lateral load ${\displaystyle q(x)}$ vanishes and substituting ${\displaystyle \lambda ^{2}={\frac {P}{EI}}}$, we get:

${\displaystyle {\frac {d^{4}w}{dx^{4}}}+\lambda ^{2}{\frac {d^{2}w}{dx^{2}}}=0}$

This is a homogeneous fourth-order differential equation and its general solution is

${\displaystyle w(x)=A\sin(\lambda x)+B\cos(\lambda x)+Cx+D}$

The four constants ${\displaystyle A,B,C,D}$ are determined by the boundary conditions (end constraints) on ${\displaystyle w(x)}$, at each end. There are three cases:

1. Pinned end:

   ${\displaystyle w=0}$ and ${\displaystyle M=0\rightarrow {d^{2}w \over dx^{2}}=0}$

2. Fixed end:

   ${\displaystyle w=0}$ and ${\displaystyle {dw \over dx}=0}$

3. Free end:

   ${\displaystyle M=0\rightarrow {d^{2}w \over dx^{2}}=0}$ and ${\displaystyle V=0\rightarrow {d^{3}w \over dx^{3}}+\lambda ^{2}{dw \over dx}=0}$

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 1.

The review of column buckling results was conducted by Elishakoff and Bert[5]. Neuringer and Elishakoff[6][7] provided several interesting cases that might prove useful in classroom setting.

Closed-Form Solutions of Functionally-Graded Material, Inhomogeneous Columns

Functional grading and inhomogeneity in various directions provide for richness in solutions of column buckling. Closed-form solutions were derived in Refs.[8][9][10][11][12] by resorting to the semi-inverse method, namely postulating the mode shape either as a polynomial, exponential. Or trigonometric function and matching the space-wise varying, suitably chosen flexural rigidity so that the governing differential equation is satisfied. Closed-form solutions can serve as benchmark solutions against which the approximate techniques’ efficacy can be examined. Specifically, Refs [10][11] resort to the fourth-order polynomials whereas Ref.[12] uses fifth order polynomials. It is remarkable that the static deflection as well as the vibration mode of the uniform beam might serve as exact buckling modes of axially graded columns (Ref. [9]).It turns out that even 260 years after Euler’s work[13]—described in detail by Van den Broek[14]—the closed-form solutions might be available [13]. The entire monograph [15] is devoted to obtaining such closed-form solutions for eigenvalue problems in bars, columns, beams, and plates.

See also

• Buckling
• Bending moment
• Bending
• Euler–Bernoulli beam theory

References

1. "Column Buckling".
2. "Questions on Columns and Struts".
3. "Buckling of Columns" (PDF). Archived from the original (PDF) on 2015-05-28.
4. Timoshenko, S. P. & Gere, J. M. (1961). Theory of Elastic Stability, 2 ed., McGraw-Hill.
5. Elishakoff, I. and Bert, C.W., Columns, Encyclopedia of Vibration (S. Braun, ed.), Academic Press, pp. 236-243, 2001.
6. Neuringer, J. and Elishakoff, I., Interesting Instructional Problems in Column Buckling for Strength of Materials and Mechanics of Solids Courses, International Journal of Engineering Education, Vol. 14, 204-216, 1998
7. Elishakoff, I., Material That Ought to Find Its Place in Future Strength of Materials Textbooks, International Journal of Mechanical Engineering Education, Vol. 20(5), 886-890, 2004.
8. I. Elishakoff, I. and Rollot, O., A Note on New Closed-Form Solutions for Buckling of a Variable Stiffness Column by MATHEMATICA, Journal of Sound and Vibration, Vol. 224, 172-182, 1999
9. Elishakoff, I., Both Static Deflection and Vibration Mode of Uniform Beam Can Serve As a Buckling Mode of a Non-Uniform Column, Journal of Sound and Vibration, Vol. 232(2), 477-489, 2000.
10. Elishakoff, I., Euler's Problem Revisited: 222 Years Later, Meccanica, Vol. 36, 265-272, 2001.
11. Elishakoff, I., Inverse Buckling Problem for Inhomogeneous Columns, International Journal of Solids and Structures, Vol.38, 457-464, 2001.
12. Storch, J. and Elishakoff, I., Buckling of Axially Graded Columns: A Fifth Order Polynomial Mode Shape, AIAA Journal, Vol. 56(6), 2509-2513, 2018.
13. Euler L, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. apud Marcum-Michaelem, Bousquet; 1744 (in Latin) (see also, Leonhardi Euleri Opera Omnia, Ser. I 14, 2010).
14. Van den Broek, J.A., Euler's Classic Paper “On the Strength of Columns”, American Journal of Physics, Vol. 15(4),309-314,1947.
15. Elishakoff, I., Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions of Semi-Inverse Problems, CRC Press, Boca Raton, 2005, XIV + pp. 729; ISBN 0-8493-2892-6.