One of the simplest modes of elastic/plastic deformation and failure is buckling. Buckling is similar to bending in that the key consideration is the cross section of the member. If the cross sectional moment of inertia is increased (by the optimization of the material toward compressive and tensile loadings comparatively) then the new column will act stiffer and in turn will prove more resistant to buckling. Folding a sheet of paper, or rolling it into a tube can be seen as examples of making a simple 8.5 x 11 sheet of paper more resistant to bending under compressive loading. This is a simple exercise in modeling improved stiffness for a structural member as it relates to much heavier construction elements such as steel and engineered wood products.
The load at which a column begins to break is called the Critical Buckling Load or Euler Load. Leonard Euler (1707-1783) is most responsible for determining what this value was, and in 1744 he derived a formula that could accurately identify at what load a column would buckle. The equation is as follows, and applies to columns of any size:
Where F is the maximum load on a column, E is the modulus of elasticity, I the area moment of inertia, L the unsupported length and K (perhaps the most important consideration) the column effective length factor. K depends primarily on the end condition supports, which have much to do with the effective stability of a column. If the ends are fixed, a column or beam may be more resistant to moving than they would if pinned or hinged.
Some values of K are as follows (and as shown above in the photo):
For both ends pinned (hinged, free to rotate): K=1.0
For both ends fixed: K = 0.5
For one end fixed and the other end pinned: K=0.7
For one end fixed and the other end free to move laterally: K=2.0
One of the key considerations here is that columns buckling result in elastic instability, which by definition, is a mode of failure.
The Slenderness Ratio
Represented by the Greek letter lambda, λ , the slenderness ratio is one of the oldest forms used for classifying types of columns. Mathematically, it is the fraction of column length divided by a column's radius of gyration. The slenderness ratio is typically used for the following cases:
Steel
1. Short steel columns (lambda less than or equal to 50)
2. Average length steel columns (greater than 50, less than or equal to 200)
3. Long steel columns (greater than 200)
Concrete
1. Short concrete column (lambda less than or equal to 10)2. Long concrete column (greater than 10)
Wood
1. Short wood columns (lambda less than 10)2. Average length wood columns and long columns (greater than 10)
Self-buckling
Yes, buckling is a result of ALL weights, not just loads applied on a column. Therefore, internal stresses, if large enough, can cause a column to buckle. For a free-standing column with a particular density, Young's modulus of E, and radius r, the critical height for a column to buckle under it's own weight is the following:
Where g is gravity's acceleration, I is inertia and B is the Bessel function (which is usually 1.87).
One would typically see the Bessel function in electromagnetic wave equations or any calculation involving propagation or oscillation. Therefore, since columns tend to behave in a similar fashion, this second-order differential function would naturally make sense as part of the formula.
