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Buckling of column


Column Buckling a serious matter.
? Normally when structural members are in compression it’s a good thing. They will not fail except by crushing (exceeding their compressive yield strength), and fatigue does not occur for elements in compression. However if the geometry of the member is such that it is a “column” then buckling can occur. Buckling is particularly dangerous because it is a catastrophic failure that gives no warning. That is, the structural system collapses often resulting in total destruction of the system and unlike yielding failures, there may be no signs that the collapse is about to occur. Thus design engineers must be constantly on vigil against buckling failure.

? The classic analysis for buckling comes to us from Euler.The fundamental analysis is for a pinned-pinned joint.For this case,the critical load,i.e.,the load at which the column will collapse is given by

p2 E I p2 E A Pcr = ????????????????? = ???????????????????? L2 Sr 2
? where, E is the modulus of Elasticity, I is the second moment of area, L is the length of the column, A is the cross - sectional area of the column, and Sr is the slenderness ratio Han expression of the aspect ratio of the columnL.
L Sr = ???? k

(1)

? The slenderness ratio is defined as

(2)

2

L09Buckling.nb

? where, k is the radius of gyration, and is defined as
I smoa k = $%%%%%%% = $%%%%%%%%%?%%%%% ?????? ??????????? ???? ? A A (3)

? where, again I is the second moment of area, and A is the cross-sectional area of the column. This comes from the definition of radius of gyration, I = k2 A . Note: because of the confusion that sometimes occurs between I, the second moment of area (which is esentially an expression of the distribution of the cross-sectional area relative to the neurtral axis) and Im the mass moment of inertia, I will use the symbol, smoa, in my calculations. This is also useful in è!!!!!!! Mathematica since there I is nominally defined to be - 1 . load to the cross-sectional area,i.e.,
Pcr p2 E ?????????? = ????????2 ?? ???? A Sr (4)

? The critical load can also be represented as a critical load unit by presenting it as a ratio of the critical

L09Buckling.nb

3

? Don't confuse this with a stress, it is not a stress!

? With a little imagination, we can see that different methods of attaching the ends of the column will lead to different likehoods of collapse. These different end conditions have been analyzes and are presented as a set of effective lengths, i.e., by how much and in which direction (longer or shorter) do these different end conditions change the behavior of the column.

To put it another way, how long would a pinned-pinned column have to be to exhibit the same tendency to buckle? Table 4-3, page 204, presents several different end conditions and their corresponding “effective” length.
Column End conditions = Theoretical AISC Conservative Value Value Value Round - Round Leff = L Leff = L Leff = L Pinned - Pinned Leff = L Leff = L Leff = L Fixed - Free Leff = 2 L Leff = 2.1 L Leff = 2.4 L Fixed - Pinned Leff = 0.707 L Leff = 0.80 L Leff = L Fixed - Fixed Leff = 0.5 L Leff = 0.65 L Leff = L End Condition Table 1

? Another interesting and important aspect of this analysis is that as beams get shorter (into a region we call intermediate) they exhibit the tendency to fail at loads less than are predicted using the Euler formula.This has lead to the development of a companion expression (using a parabolic curve fit) to properly account for failures in this intermediate region.This parabolic equation,first suggested by J.B.Johnson now bears his name.The point at which we shift from the Euler formula to the J.B.Johnson formula is usually taken as half the yield strength of the material,i.e.,Sy/2. This point can be calculated from
2E SrD = p $%%%%%%%%%% ?????????? ? Sy (5)

? Thus in doing a column buckling analysis, once one has recognized that buckling failure mode exists, is to compute the Slenderness ratio of the column and the Slenderness ratio at the dividing line between J.B.Johnson parabola,and Euler and determine which expression is appropriate for calculating critical load.

4

? Thus in doing a column buckling analysis, once one has recognized that buckling failure mode exists, is to compute the Slenderness ratio of the column and the Slenderness ratio at the dividing line between J.B.Johnson parabola,and Euler and determine which expression is appropriate for calculating critical load.
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? See Example 4-10 for a design that requires this analysis.

? Note that the above analysis assumes that the loading is centered on the column. If the load is not centered, and thus is not equally distributed over the column’s cross-sectional area, the likelihood of a failure will increase. The approach that is used for these “eccentrically” loaded beams requires what is called the secant column formula. This approach requires an iterative solution since the ratio of P/A occurs on both sides of the equation. Such solutions are greatly enabled by the use of computer software.

This is the most important aspect of buckling analysis!!! since as you will see, it greatly increases the likelihood of failure.
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? See an example I brewed up here.PopsicleStickColumn.nb

? In summary,the design engineer should keep the following issues in mind when dealing with machine elements in compression (these are in order of importance,most to least). 1. Is the loading eccentric? 2. What are the end conditions? 3. What is the slenderness ratio? 4. Is the column Euler or J.B.Johnson,or is it “short”? 5. What is the critical load? All of these questions must be answered as part of the analysis.

Stresses in Cylinders
? Many instances occur in mechanical design that involve cylindrical shaped structures with different pressure distributions on the inside and outside.Specific formulas exist for calculating the stress distributions for these cylinders.Note that the equations for thin walled cylinders are simply approximations for the thick walled cylinder solutions.
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