# Length of the column where buckling is likely to occur

## Homework Statement

i am struggling with a question i have.. i cant find the right equation to use to find what the minimum length of the column at which buckling is likely to occur??
can anyone offer any help with this as my lesson books show me nothing on how you find length?
i have D=80mm d=60mm
youngs modulus 200GNm-2
yield stress 140MNm-2

## The Attempt at a Solution

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PhanthomJay
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The minimum length at which the column will buckle depends on the applied axial compressive load which is not given...

The question seems to be a tube. IN that case there are at least two modes of buckling. First, there is the overall buckling of the member as if it were a long thin rod and Euler's formula might be said to apply. But this could be preceded by a local buckling of the tube, especially if it is thin in relation to the diameter. I think you are probably looking to apply the Euler equation of buckling, in which you will be interested in the EFFECTIVE length.

The question seems to be a tube. IN that case there are at least two modes of buckling. First, there is the overall buckling of the member as if it were a long thin rod and Euler's formula might be said to apply. But this could be preceded by a local buckling of the tube, especially if it is thin in relation to the diameter. I think you are probably looking to apply the Euler equation of buckling, in which you will be interested in the EFFECTIVE length.

well at present i am trying to find the length so been looking at using these...

to find I... I = (D^4-d^4)*pi / 64
and for area,,,, A = (D^2-d^2)*pi / 4

E.S.R = (sq) (pi^2*E)/oc(critical stress)
and then
L = E.S.R * (sq) I/A

am i going down the right route with these as someone has told me the answer for length 5.94 but i need to find it my self as i cant just put that lol

PhanthomJay
Homework Helper
Gold Member
The minimum length at which the column will buckle depends on the applied axial compressive load which is not given...
Since P_cr = pi^2(EI)/(kL)^2, you cannot solve for the effective length unless you know the value of the compressive load. Are you sure you have stated the problem correctly as worded?

Edit: Maybe the problem is looking for the max length before buckling occurs prior to the material reaching its yield stress?? If so, P_cr = (yield stress)*A, then solve for L using the buckling formula for a fixed-fixed column. I can't do the math...too many zeros.

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my question i have says... what is the minimum length of the column at which buckling is likely to occur ... and the values i have put down are all i have on the sheet..
then i have what will mode of failure be and at what load..

i have included an image of my workings out so far for length,, which i have seen on the internet that 5.94 is the length just needed to find it myself... could anyone confirm this???

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PhanthomJay
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I painstakingly did the math and i get about 6 meters length when the column buckles at yield stress $\sigma_{cr} = \sigma_y$ , hope the math is Ok but in any event the problem statement is poorly worded.

That all pans out. I get the same.

Isn't the diagram in post 1 a annulus? which would mean that second moment of area formula is pi (R^4 - r^4) / 4?

PhanthomJay
Homework Helper
Gold Member
Isn't the diagram in post 1 a annulus? which would mean that second moment of area formula is pi (R^4 - r^4) / 4?
Hi there newbie, at this time we wish to welcome you to these Forums!

Yes, sir, pi (R^4 - r^4) / 4 is the same as pi(D^4-d^4)/ 64

my question i have says... what is the minimum length of the column at which buckling is likely to occur ... and the values i have put down are all i have on the sheet..
then i have what will mode of failure be and at what load..

i have included an image of my workings out so far for length,, which i have seen on the internet that 5.94 is the length just needed to find it myself... could anyone confirm this???

Hi: What does E.S.R stand for in your calculations?

Hi,

Can anyone please help me with this question, it is driving me mad. I don't have the compressive load value, and so Eulers formula is useless :-( I have looked at MATYYH3's answer and don't understand what E.S.R is. :-(

Please can someone just nudge me in the right direction?

Thanks

PhanthomJay
Homework Helper
Gold Member
Hi,

Can anyone please help me with this question, it is driving me mad. I don't have the compressive load value, and so Eulers formula is useless :-( I have looked at MATYYH3's answer and don't understand what E.S.R is. :-(

Please can someone just nudge me in the right direction?

Thanks
ESR, a dimensionless number, stands for Effective Slenderness Ratio, and is equal to kL/r, where L is the length of the column, r is its radius of gyration, and k is a function of the end boundary conditions (k = 0.5 for a fixed-fixed column). The radius of gyration, r, is sq rt of (I/A).

For this problem, not clearly worded, in calculating the minimum length for buckling, the problem is looking for the max length before buckling occurs prior to the material reaching its yield stress. Thus critical buckling stress is set equal to yield stress. So the critical compressive buckling load to use is (yield stress*crossection area).

When dealing with buckling stresses, it is sometimes preferred to re-write the critical buckling stress formula , PI^2EI/(A(kL)^2) , as PI^2E/(kL/r)^2, or PI^2E/(ESR)^2, although more often than not it can sometimes confuse the issues.