Column design and analysis

In a design situation, the expected load on a column is known along with the required column length. The designer must specify the following parameters:

1. The manner of attaching the column ends to the structure, which affects end fixity.
2. The general shape of the column cross section (round, square, rectangular, hollow tube, etc.).
3. The material selected for the column.
4. The design factor appropriate for the application.
5. The final dimensions of the column.

It is often desirable to propose and analyze several alternative designs to reach an optimum solution. Design software significantly facilitates this iterative process.

It is assumed that for each design trial, the designer specifies items 1 through 4. For simple cross sections such as solid round or square shapes, the final dimensions can be computed using classical buckling equations such as the Euler formula or the J. B. Johnson formula. If a closed-form algebraic solution is not possible, iterative calculations are required.

In a design situation, the cross-sectional dimensions are initially unknown. This makes it impossible to compute the radius of gyration and therefore the slenderness ratio, KL/r. Without knowing KL/r, it cannot be determined whether the column behaves as a long column (Euler) or a short column (Johnson).

This difficulty is overcome by assuming the column is either long or short and proceeding with the corresponding formula. Once the cross-sectional dimensions are determined, the actual value of KL/r is calculated and compared with Cc. If the assumption was correct, the solution is valid. If not, the alternate formula must be used and the calculations repeated.

Design Factor

Under typical industrial conditions, a design factor of N = 3 is recommended. For smooth, well-controlled loading, values as low as N = 2 may be justified. For applications involving shock or impact, N = 4 or higher should be used, and testing is recommended.

Column Analysis

Procedure for analyzing straight, centrally loaded columns:
1. Compute the actual slenderness ratio, KL/r.
2. Compute the critical slenderness ratio, Cc.
3. Compare KL/r with Cc to determine column type.
4. If KL/r > Cc, the column is long and Euler’s equation applies.

The equation gives the critical load, Pcr, at which the column would begin to buckle.
An alternative form of the Euler formula is often desirable. Note that:

But, from the definition of the radius of gyration, r,

Then

This form of the Euler equation aids in a design problem in which the objective is to specify a size and a shape of a column cross section to carry a certain load.

Notice that the buckling load is dependent only on the geometry (length and cross section) of the column and the stiffness of the material represented by the modulus of elasticity. The strength of the material is not involved at all. For these reasons, it is often of no benefit to specify a high-strength material in a long column application. A lower-strength material having the same stiffness, E, would perform as well.

5. If KL/r is less than Cc, the column is short. Use the J. B. Johnson formula:

Use of the Euler formula in this range would predict a critical load greater than it really is. The J. B. Johnson formula is written as follows:

The critical load for a short column is affected by the strength of the material in addition to its stiffness, E. As shown in the preceding section, strength is not a factor for a long column when the Euler formula is used.

Eccentrically Loaded Columns


An eccentric load is one that is applied away from the centroidal axis of the cross section of the column, as shown in the graphic help entitled “Eccentric column”. Such a load exerts bending in addition to the column action that results in the deflected shape shown in the figure. The maximum stress in the deflected column occurs in the outermost fibers of the cross section at the midlength of the column where the maximum deflection, ymax occurs. Let's denote the stress at this point as sL/2. Then, for any applied load, P,

Note that this stress is not directly proportional to the load. When evaluating the secant in this formula, note that its argument in the parentheses is in radians. Also, because most calculators do not have the secant function, recall that the secant is equal to 1/cosine.

For design purposes, we would like to specify a design factor, N, that can be applied to the failure load similar to that defined for straight, centrally loaded columns. However, in this case, failure is predicted when the maximum stress in the column exceeds the yield strength of the material. Let's now define a new term, Py, to be the load applied to the eccentrically loaded column when the maximum stress is equal to the yield strength. The equation then becomes

Now, if we define the allowable load to be

or

this equation becomes

Required

This equation cannot be solved for either A or Pa, so an iterative solution is required.
Another critical factor may be the amount of deflection of the axis of the column due to the eccentric load:

More information with excel spreadsheet.

• Column Design (Part 1)
• Column Design (Part 2)
• Column Design (Part 3)
• Column Design (Part 4)
• Column Design (Part 5)
• Column Design (Part 6)