2-buckling of columns

موضوع: 2-buckling of columns 2008-09-23, 10:53 pm

Columns & Buckling

When we speak of columns (and buckling) we are talking about members loaded in compression, often axially loaded, although columns may be loaded eccentrically. We also tend to think of columns as vertical members, however, the formulas we will utilize also apply to horizontal compression members, or to compression members in general. For instance, compression members of a truss may be considered to be columns pinned at each end point.

Columns may be divided into three general types: Short Columns, Intermediate Columns, and Long Columns. The distinction between types of columns is not well defined, but a generally accepted measure is based on the Slenderness Ratio. The Slenderness Ratio is the (effective) length of the column divided by its radius of gyration.

The radius of gyration is the distance from an axis which, if the entire cross sectional area of the object (beam) were located at that distance, it would result in the same moment of inertia that the object (beam) possesses. Or, it may be expressed as: Radius of Gyration: r_xx = (I_xx/A)^1/2 (radius of gyration about xx-axis)

So, Slenderness Ratio = L_e / r.

Notice that we have put a subscript "e" by the length of the column. This is to indicate that, depending on how the column is supported, we do not use the actual length but an ‘effective length’.

The effective length is given by: L_e = K L, where K could be called an effective length constant. The values for K depend on how the column is supported.

Pinned-Pinned: K= 1, Fixed-Pinned: K= .7, Fixed-Fixed: K = .5, Fixed-Free: K = 2 (See Diagram.)

A generally accepted relationship between the slenderness ratio and the type of Column is as follows:

Short Column: 0< L_e / r < 60
Intermediate Column: 60< L_e / r <120
Long Column: 120< L_e / r < 300

I. Short Columns: When the slenderness ratio is less than 60, a column will not fail due to buckling, as the ratio of the column length to the effective cross sectional area is too small. Rather a short, 'thick' column, axially loaded, will fail in simple compressive failure: that is when the load/area of the column exceeds the allowable stress, P/A > s _allow.

Eccentrically loaded short columns have a slightly more complicated result for compression failure, which we will look at later in this section.

We also will put off discussion of intermediate length columns until we have discussed long columns. Idealized buckling in long columns was first treated by the famous mathematician Leonard Euler.

II. Long Columns & Euler's Equation: In 1757, Leonard Euler (pronounced Oiler) developed a relationship for the critical column load which would produce buckling. A very brief derivation of Euler's equation goes as follows:

A loaded pinned-pinned column is shown in the diagram. A top section of the diagram is shown with the bending moment indicated, and in terms of the load P, and the deflection distance y, we can write:
1. M = - P y.
We also can write that for beams/columns the bending moment is proportional to the curvature of the beam, which, for small deflection can be expressed as:
2. (M /EI) = (d²y /dx²)
(See Strength of Material text chapters on beams and beam deformations.) Where E = Young's modulus, and I = moment of Inertial. Then substituting from EQ. 1 to EQ. 2, we obtain:
3. (d²y /dx²) = -(P/EI)y or (d²y /dx²) + (P/EI)y = 0
This is a second order differential equation, which has a general solution form of
4. :

We next apply boundary conditions: y = 0 at x = 0, and y = 0 at x = L. That is, the deflection of the column must be zero at each end since it is pinned at each end. Applying these conditions (putting these values into the equation) gives us the following results: For y to be zero at x =0, the value of B must be zero (since cos (0) = 1). While for y to be zero at x = L, then either A must be zero (which leaves us with no equation at all, if A and B are both zero), or
2-buckling of columns D91a4

. Which results in the fact that

2-buckling of columns D91a5

And we can now solve for P and find:

5.

, where P_cr stands for the critical load which can be applied before buckling is initiated.

6. By replacing L with the effective length, L_e, which was defined above, we can generalize the formula to:
2-buckling of columns D91a7

, which then applies to Pinned-Pinned, Fixed-Pinned, Fixed-Fixed, and Fixed-Free columns.

This equation is a form of Euler's Equation. Another form may be obtained by solving for the critical stress: 2-buckling of columns D91a8

and then remembering that the Radius of Gyration: r_xx = (I_xx/A)^1/2 , and substituting we can obtain:

which gives the critical stress in terms of Young's Modulus of the column material and the slenderness ratio.

Let us at this time also point out that Euler's formula applies only while the material is in the elastic/proportional region. That is, the critical stress must not exceed the proportional limit stress. If we now substitute the proportional limit stress for the critical stress, we can arrive at an equation for the minimum slenderness ratio such that Euler's equation will be valid.

As an example: For structural steel with a proportional limit stress of 35,000 psi., and a Young's Modulus of E = 30 x 10⁶ psi., we can obtain (L_e/r) =sqrt( 3.14² x 30 x 10⁶ psi / 35,000 psi) = 92 This is the minimum slenderness ratio for which we could use Euler's buckling equation with this column material.

Euler's Equation for columns while useful, is only reasonably accurate for long columns, or slenderness ratios of general range: 120 < L_e / r < 300, and in addition will work for axially loaded members with stress in the elastic region, but not with eccentrically loaded columns.

Some additional points need to be mentioned.

The slenderness ratio (L_e/r) depends on the radius of gyration, (r_xx = (I_xx/A)^1/2 & r_yy = (I_yy/A)^1/2 ), which in turns depends on the moment of inertia of the column cross section. The value of the moment of inertia depends on the axis about which it is calculated. That is, for a rectangular 2" x 4" cross section, the column will buckle in the 2" direction rather than the 4" direction, as the moment of inertia for the 2" direction is considerable smaller than the moment of inertia in the 4" direction. (I = (1/12) b d³ = (1/12) (4")(2")³ = 2.67 in⁴, as compared to I = (1/12) b d³ = (1/12) (2")(4")³ = 10.67 in⁴ ) It is important to point this out, so that when given a column cross section, the student must be sure to determine the minimum moment of inertia, as buckling will occur in the direction.

It should also be pointed out that while these formulas give reasonable values for critical loads causing buckling, it should not be assumed the values are completely accurate. Buckling is a complicated phenomena, and the buckling in any individual column may be influenced by misalignment in loading, variations in straightness of the member, presence of initial unknown stresses in the column, and defects in the material. Several examples of buckling calculations follow.

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