Self-buckling

A column can buckle due to its own weight with no other direct forces acting on it, in a failure mode called self-buckling. In conventional column buckling problems, the self-weight is often neglected since it is assumed to be small when compared to the applied axial loads. However, when this assumption is not valid, it is important to take the self-buckling into account.

Elastic buckling of a "heavy" column i.e., column buckling under its own weight, was first investigated by Greenhill at 1881.^[1] He found that a free-standing, vertical column, with density $\rho$, Young's modulus $E$, and cross-sectional area $A$, will buckle under its own weight if its height exceeds a certain critical value:

${\displaystyle l_{max}=\left(7.8373\,{\frac {EI}{\rho gA}}\right)^{1/3}}$

where $g$ is the acceleration due to gravity, $I$ is the second moment of area of the beam cross section.

One interesting example for the use of the equation was suggested by Greenhill in his paper. He estimated the maximal height of a pine tree, and found it cannot grow over 90 ft tall. This length sets the maximum height for trees on earth if we assume the trees to be prismatic and the branches are neglected.

Mathematical Derivation

A column exhibiting a compressive buckling load due to its own weight.

Suppose a uniform column fixed in a vertical direction at its lowest point, and carried to a height $l$, in which the vertical position becomes unstable and flexure begins. There is a body force $q$ per unit length ${\displaystyle q=\rho gA}$, where $A$ is the cross-sectional area of the column, $g$ is the acceleration due to gravity and $\rho$ is its mass density.

The column is slightly curved under its own weight, so the curve $w(x)$ describes the deflection of the beam in the $y$ direction at some position $x$.

Looking at any point on the column, we can write the moment equilibrium:

${\displaystyle M=-\int _{0}^{x}q(y-w)dx}$

where the right-hand side of the equation is the moment of the weight of BP about P.

According to Euler–Bernoulli beam theory: ${\displaystyle M=-EI{d^{2}w \over dx^{2}}}$

Where $E$ is the Young's modulus of elasticity of the substance, $I$ is the moment of inertia.

Therefore, the differential equation of the central line of BP is:

${\displaystyle EI{d^{2}w \over dx^{2}}=q\int _{0}^{x}(y-w)dx}$

Differentiating with respect to x, we get

${\displaystyle EI{d^{3}w \over dx^{3}}=q\int _{0}^{x}{\Bigl (}-{dw \over dx}{\Bigr )}dx}$ ,Or ${\displaystyle EI{d^{3}w \over dx^{3}}=-qx{dw \over dx}}$

We get that the governing equation is the third order linear differential equation with a variable coefficient. The way to solve the problem is to use new variables ${\displaystyle n,z,k}$ and $r$:

${\displaystyle k^{2}={4 \over 9}{q \over EI},}$ ${\displaystyle r^{2}=x^{3},}$ ${\displaystyle {\sqrt {x}}z={\operatorname {d} \!w \over \operatorname {d} \!x},}$ ${\displaystyle n^{2}={\operatorname {} \!1 \over \operatorname {} \!9}}$

Then, the equation transforms to the Bessel equation

${\displaystyle r^{2}{d^{2}z \over dr^{2}}+r{dz \over dr}+(k^{2}r^{2}-n^{2})z=0}$

The solution of the transformed equation is ${\displaystyle z=AJ_{\frac {1}{3}}(kr)+BJ_{-{\frac {1}{3}}}(kr)}$

Where ${\displaystyle J_{n}}$ is the Bessel function of the first kind. Then, the solution of the original equation is:

${\displaystyle {\operatorname {d} \!w \over \operatorname {d} \!x}={\sqrt {x}}{\Bigl (}AJ_{\frac {1}{3}}(kx^{\frac {3}{2}})+BJ_{-{\frac {1}{3}}}(kx^{\frac {3}{2}}){\Bigr )}}$

Now, we will use the boundary conditions:

• No moment at $x=0$ $\rightarrow$ ${\displaystyle {\operatorname {d^{2}} \!w \over \operatorname {d} \!x^{2}}(x=0)=0}$ $\rightarrow$ ${\displaystyle A=0}$
• Fixed at ${\displaystyle x=l}$ $\rightarrow$ ${\displaystyle w(x=l)=0}$ $\rightarrow$ ${\displaystyle J_{-{\frac {1}{3}}}(kl^{\frac {3}{2}})=0}$

From the second B.C., we get that the critical length in which a vertical column will buckle under its own weight is:

${\displaystyle l_{max}={\biggl (}{\frac {j_{\frac {1}{3}}}{k}}{\biggr )}^{2/3}={\Biggl (}{\frac {9(j_{\frac {1}{3}})^{2}}{4}}{\frac {EI}{q}}{\Biggl )}^{1/3}}$

Using ${\displaystyle {j_{\frac {1}{3}}}\cong 1.86635}$ , the first zero of the Bessel function of the first kind of order -1/3, ${\displaystyle l_{max}}$ can be approximated to:

${\displaystyle l_{max}=\left(7.8373\,{\frac {EI}{\rho gA}}\right)^{1/3}}$

See also

• Buckling
• Bending moment
• Euler's critical load
• Bending
• Euler–Bernoulli beam theory

External links

• Advanced Topic in Column Buckling, MIT Open-Course-Ware

References

1. "Greenhill, A. G. (1881). "Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and the greatest height to which a tree of given proportions can grow." Proc. Cambridge Philos. Soc., 4, 65–73." (PDF).