Euler Buckling

The following derivation is adapted from Leonhard Euler’s (1707 – 1783) derivation.
See [Timoshenko] for more information about the history behind Euler Buckling and many other important concepts in MC300.

External moment: $P*\delta$
Internal moment: $-EI*\frac{d^2\delta}{dz^2}$

If $P$ is small, $-EI*\frac{d^2\delta}{dz^2}>>P*\delta$ , where we can see that both sides of the equality are dependent on $\delta$ . The column will remain straight until $P$ reaches a critical value. Thus, classical Euler Buckling is associated with instability rather than strength.

$P*\delta-M=0$
OR

(1) $\begin{equation*} P*\delta+EI*\frac{d^2\delta}{dz^2}=0 \end{equation*}$

Eq. 1 has a “+” in anticipation of the solution being a sine function. The second derivative of a sine function has the opposite sign of the function itself, so the two terms of eq. 1 will indeed have opposite signs and will sum to zero.

Eq. 1 is a $2^{nd} Order ODE$ with constant coefficients.
The solution to eq. 1 is:

$\begin{equation*} \delta(z)=A\sin\bigg(\sqrt{\frac{P}{EI}}*z\bigg)+B\cos\bigg(\sqrt{\frac{P}{EI}}*z\bigg) \end{equation*}$

We can employ pin-pin boundary conditions ( $\delta(z=0)=0$ and $\delta(z=L)=0$ ) and see that $0=B$ and $0=A\sin\bigg(\sqrt{\frac{P}{EI}}*L\bigg)$ .
For a nontrivial solution, we need to therefore satisfy $\sin\bigg(\sqrt{\frac{P}{EI}}*L\bigg)=0$ . This is true when $\sqrt{\frac{P}{EI}}*L=0,\pi,2\pi...n\pi$ . The dominant buckling mode for our columns will occur when $\sqrt{\frac{P}{EI}}*L=\pi$ .

So, $P_{cr}=\frac{\pi^2 EI}{L^2}$ , where the subscript “cr” stand for “critical,” and our expression is valid for a pin-pin column that buckles elastically.

A more general expression, that is valid for other boundary conditions, can be written:

(2) $\begin{equation*} P_{cr}=\frac{\pi^2 E I}{(L_e)^2} \end{equation*}$

In eq. 2, $L_e$ is the “effective length.”

It is important to know what factors influence the effective length, $L_e$ . Noting that this variable is squared in eq. 2, we can see that choosing the wrong $L_e$ can result in extreme error. Some important factors that influence $L_e$ include the kind of connection at the bottom and top of the column (e.x. moment connection or “simple” connection), the size of the moment-connected beam that is intersecting at these locations, as well as whether the structure, as a whole, is “braced.”

For very slender columns, the above equation for column strength is approximately correct. For very short columns, failure will occur via yielding of the entire cross-section, rather than buckling.

S. Timoshenko, History of strength of materials, Courier Dover Publications, 1983.
[Bibtex]

@book{Timoshenko1,
  title={History of strength of materials},
  author={Timoshenko, Stephen},
  year={1983},
  publisher={Courier Dover Publications}
}

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