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.
If is small, , where we can see that both sides of the equality are dependent on . The column will remain straight until reaches a critical value. Thus, classical Euler Buckling is associated with instability rather than strength.
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.
We can employ pin-pin boundary conditions ( and ) and see that and .
For a nontrivial solution, we need to therefore satisfy . This is true when . The dominant buckling mode for our columns will occur when .
So, , where the subscript “cr” stand for “critical,” and our expression is valid for a pin-pin column that buckles elastically.
In eq. 2, is the “effective length.”
It is important to know what factors influence the effective length, . Noting that this variable is squared in eq. 2, we can see that choosing the wrong can result in extreme error. Some important factors that influence 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.