Lecture 30: Radius of Gyration & Buckling

Lecture 30:

Radius of Gyration & Buckling

The radius of gyration (r) describes the way in which the area of a cross-section is distributed around its centroidal axis. If the area is concentrated far from the centroidal axis it will have a greater value of r and a greater resistance to buckling. A cross-section can have more than one radius of gyration and most sections have at least two. If this is the case, the section tends to buckle around the axis with the smallest value. The radius of gyration is defined as:

r = sqrt (I/A)

where

r = radius of gyration
I = moment of inertia
A = area of the cross section

All things being equal, a circular pipe is the most efficient column section to resist buckling. This is because it has an equal radius of gyration in all directions and it has the its area distributed as far away as possible from the centroid.

The steel columns shown below all have areas of 3-1/8 in². The safe loads for an 8 ft length are shown. The only difference between them is the way in which the cross-sectional area is distributed about the centroid.

radius of gyration values for typical column shapes

Buckling

Buckling is very similar to bending. Thus, the shape of the cross-section is very important. The shape of the column also effects the way in which it will buckle. Imagine for a moment a single sheet of paper (A4 or 8.5 x 11). If one would try to simply stand it on edge it would be impossible unless the paper was folded. This simple act of folding the paper actually increases the cross-sectional moment of Inertia and thus the stiffness of the newly formed column. The stiffness of another paper column could be futher increased by taking the paper and taping the long edges together to create a tube. Now, the paper will be very stiff since the material of the paper is distributed evenly as far away from the neutral axis as possible.

The load at which a column will begin to buckle is known as the Critical Buckling Load (or critical load). A number of qualities of the column must be known in order to determine this value. The Swiss mathematician Leonard Euler (1707 - 1783) derived a formula in 1744 (known as the Euler Buckling Formula) to determine the load at which a perfect column will buckle. It was a very important step in the history of technology and remains important for column design today. The equation is only accurate for columns which approach the perfect conditions for which he derrived the equation.

N_cr = ¼²E I / (l_k²)

In which the terms are defined as follows:

N_cr = Critical Buckling Load
E = Modulus of Elasticity
I = Moment of Inertia
l_k = Effective Buckling Length

This equation can be modified by dividing both sides by the area of the column so that the stress at which the column will buckle can be determined:

f_cr = sigma_cr = N_cr / A
= ¼² E I /A(l_k²)

now, knowing that r² = I/A, this equation becomes:

= ¼² E /(l_k/r)²

Lambda, or the slenderness ratio is a value with which one can gage the relative resistance of a column cross-section to buckling. Or, stated otherwise, the relative ease in which a column WILL buckle. It is defined as

lambda = l_k / r

Where l_k is the buckling length and r is the radius of gyration.

Thus, the critical buckling stress can be expressed as,

f_cr= ¼² E / (lambda)²

The buckling length of a column depends on its physical length and its end conditions. Euler discovered that if a column is hinged at both ends it will buckle in the form of a sine curve with the inflection points at the hinges. This would be the case in which the buckling length of a column is identical to its length. This is not the case if the ends of the column are both fixed. The determination of the buckling lengths for various column end conditions and frames is given below:

buckling length values for typical end restraints of columns and frames

The magnitude of the internal forces is important to know in order to size a column. A small amount of tensile stress has little effect on a wood or steel column, but could cause problems in a concrete or masonry column.

Questions for Thought

What are the relationships between the various columnar elements within the human skeleton? What are the various end conditions? What is the buckling length of the columns in a fassade near you?

Additional Reading

Schodeck, Daniel. "Structures." Chapter 7.