WORLD FOR CIVIL

Topic 7.2: Columns & Buckling - II

III. Intermediate Columns

There are a number of semi-empirical formulas for buckling in columns in the intermediate length range. One of these is the J.B. Johnson Formula. We will not derive this formula, but make several comments. The J.B. Johnson formula is the equation of a parabola with the following characteristics. For a graph of stress versus slenderness ratio, the parabola has its vertex at the value of the yield stress on the y-axis. Additionally, the parabola is tangent to the Euler curve at a value of the slenderness ratio, such that the corresponding stress is one-half of the yield stress.

In the diagram below, we have a steel member with a yield stress of 40,000 psi. Notice the parabolic curve beginning at the yield stress and arriving tangent to the Euler curve at 1/2 the yield stress.

We first note that at the point where the Johnson formula and Euler's formula are tangent, we can relate the stress to Euler's formula as follows (where C represents the slenderness ratio when the stress is 1/2 the yield stress):

From this we find an expression for C (critical slenderness ratio) of:

For our particular case, where we have a steel member with a yield stress of 40,000 psi, and a Young's modulus of 30 x 10⁶ psi., we find C = sqrt(2 * 3.14² * 30 x 10⁶ / 40,000 psi) = 122. If our actual beam has a slenderness ratio greater than the critical slenderness ratio we may use Euler’s formula. If on the other hand our actual slenderness ratio is smaller than the critical slenderness ratio, we may use the J.B. Johnson Formula.

Example: As an example let us now take a 20 foot long W12 x 58 steel column (made of same steel as above), and calculate the critical stress using the J.B. Johnson formula. (Beam information and Johnson formula shown below.)

-	-	-	Flange	Flange	Web	Cross	Section	Info.	Cross	Section	Info.
Designation	Area	Depth	Width	thick	thick	x-x axis	x-x axis	x-x axis	y-y axis	y-y axis	y-y axis
-	A-in2	d - in	wf - in	tf - in	tw - in	I - in4	S -in3	r - in	I - in4	S -in3	r - in
W 12x58	17.10	12.19	10.014	0.641	0.359	476.0	78.1	5.28	107.00	21.40	2.51

J.B. Johnson's formula:

For our beam the slenderness ratio = (20 ft * 12 in/ft)/2.51in = 95.6 (where 2.51 in. is the smallest radius of gyration, about y-y axis). Inserting values we find:

Critical Stress = [ 1 - (95.6²/2* 122²)]*40,000 psi. = 27,720 psi. This is the critical stress that would produce buckling. Note we did not have a safety factor in this problem. As a result we really would not want to load the column to near the critical stress, but at a lower 'allowable' stress.

The Critical Load will equal the product of the critical stress and the area, or P_cr = 27,720 psi. * 17.10 in2 = 474,012 lb.

VI. The Secant Formula:

Another useful formula is known as the Secant formula. We will not go through the derivation of this relationship, but focus on its application.

The Secant formula may be used for both axially loaded and eccentrically loaded columns. It may be used with pinned-pinned (L_e = L), and with fixed-free (Le = 2L) end columns, but not with other end conditions.

The Secant formula gives the maximum compressive stress in the column as a function of the average axial stress (P/A), the slenderness ratio (L/r), the eccentricity ratio (ec/r²), and Young’s Modulus for the material.

If, for a given column, the load, P, and eccentricity of the load, e, are known, then the maximum compressive stress can be calculated. Once we have the maximum compressive stress due to the load, we can compare this stress with the allowable stress for the material and decide if the column will be able to carry the load.

On the other hand, if we know the allowable compressive stress for the column, we may use the Secant formula to determine the maximum load we can safely apply to the column. In this case we will be solving for P, and we take note that the equation is a transcendental equation when solved for P. Thus, the easiest method of solution is to simply try different values of P, until we find a satisfactory fit. See following example.

Please Select 7.2a: Secant Formula - Example 1

The eccentricity ratio has a normal range from 0 to 3, with most values being less than 1. When the eccentricity value is zero (corresponding to an axial loading) we have the special case that the maximum load is the critical load:

and the corresponding stress is the critical stress or

This is one way to look at axial loads. On the other hand a common practice with axially loaded structural steel columns is to use an eccentricity ratio of .25 to account for the effects of imperfections, etc. Then the allowable stress does not have to be reduced to account for column imperfections, etc., as this is taken into account in eccentricity ratio.

V. Empirical Design Formulas for Columns:

A number of empirical design formulas have been developed for materials such as structural steel, aluminum and wood, and may be found in such publications as the Manual of Steel Construction, Mechanic, Specifications for Aluminum Structures, Aluminum Construction Manuel, Timber Construction Manual, and National Design Specifications for Wood Constuction.

1. Structural Steel:

Please Select 7.2b: Structural Steel - Example 2 for structural steel example.
Please Select 7.2c: Structural Steel Column Selection - Example 3 for structural steel example.

2. Aluminum

Please Select 7.2d: Aluminum Column - Example 4 for Aluminum example.

3. Wood Columns

Please Select 7.2e: Wood Column - Example 5 for wood column example.

VI Short Eccentrically Loaded Columns

An eccentrically loaded short column is shown in the diagram, with the force, P, acting a distance, e, from the centroid of the column cross sectional area. We may replace the eccentrically acting force, P , with an axial force, P, plus a moment whose value will be M = P x e. Next we calculate the compressive stress due to the axial force, P, which will simply be P/A. Then we calculate the bending stress due to the moment P x e, which gives (Pe)c/I where the bending stress will be a compressive maximum on the right side of the column and a tensile maximum on the left side of the column (and zero at the centroid). Finally, we add the two stress and obtain Total Maximum Compressive Stress = (P/A)(1 + A e c₁/I) (right side of column), and the Total Minimum Compressive Stress = (P/A)(1 - A e c₁/I) (left side of column). And in fact, if the bending stress is large enough, the left side on the column may be in tension.