column buckling a serious matter

8/14/2019 Column Buckling a Serious Matter.

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where, k is the radius of gyration, and is defined as

(3)k =$%%%%%%%IA

= $%%%%%%%%%%%%%%smoaA

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.

ote: 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

m the mass moment of inertia, I will use the symbol, smoa, in my calculations. This is also useful in

athematica since there I is nominally defined to be!!!!!!!-1 .

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

load to the cross-sectional area,i.e.,

(4)Pcr

A=

p2E

Sr2

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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.

Table 1

Column End conditions =End Condition Theoretical

Value

AISC

Value

Conservative

Value

Round - Round Leff = L Leff = L Leff = L

Pinned - Pinned Leff = L Leff = L Leff = L

Fixed - Free Leff = 2L Leff = 2.1L Leff = 2.4L

Fixed - Pinned Leff = 0.707 L Leff = 0.80L Leff = L

Fixed - Fixed Leff = 0.5L Leff = 0.65L Leff = L

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

ormula.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

(5)SrD = p$%%%%%%%%%%2 ESy

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

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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 columns cross-sectional area, the likelihood of a

ailure 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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