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.

Euler Buckling

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}
    }