Calculating Minimum Column Length for Buckling: Stress and Variables

[Niall11]

1. The problem statement, all variables and given/known
I have been asked to calculate the minimum length of this column (attatched) at which buckling is likely to occur

Homework Equations

E.S.R = sqrt(π^2*E / σ)

2nd moment of Area I = AK^2

E.S.R = L/K

I= π/32*(D^4-d^4)

3. The Attempt at a Solution
so E.S.R = sqrt(π^2*200*10^9 / 140*10^6) = 118.74

Since I = AK^2 and E.S.R = L/K
L= (E.S.R)*K = (E.S.R)*sqrt(I/A)

so I = π/32 * (0.08^4-0.06^4) = 2.75*10-6Now L = 188.7 * sqrt(2.75*10-6 / Area)

ive got that far but don't know a suitable equation for area, I'm not looking for answers just for someone who can tell me if I'm on the right track, any help would be appreciated!

 

[SteamKing]

1. The problem statement, all variables and given/known
I have been asked to calculate the minimum length of this column (attatched) at which buckling is likely to occur

Homework Equations

E.S.R = sqrt(π^2*E / σ)

2nd moment of Area I = AK^2

E.S.R = L/K

I= π/32*(D^4-d^4)

3. The Attempt at a Solution
so E.S.R = sqrt(π^2*200*10^9 / 140*10^6) = 118.74

Since I = AK^2 and E.S.R = L/K
L= (E.S.R)*K = (E.S.R)*sqrt(I/A)

so I = π/32 * (0.08^4-0.06^4) = 2.75*10-6Now L = 188.7 * sqrt(2.75*10-6 / Area)

ive got that far but don't know a suitable equation for area, I'm not looking for answers just for someone who can tell me if I'm on the right track, any help would be appreciated!

They're talking about the area of the cross section of the column. You are given the outer diameter and the inner diameter of the column.

What else do you need? Do you know how to calculate the area of a circle?

 

[Niall11]

thanks for the fast reply, I was thinking I needed the area of the whole column for some reason.

yes area of circle = π*r^2 so area of outer circle - area of inner circle = cross sectional area of column;

π (0.04)^2 - π(0.03)^2 = 2.199*10^-3 m^2

meaning L = 118.7 * sqrt(2.75*10^-6 / 2.199*10^-3) = 4.19m

effective length is 1/2 of the column length for a column which both ends fixed therefore length = 4.19m * 2 = 8.38m

 

[SteamKing]

1. The problem statement, all variables and given/known
I have been asked to calculate the minimum length of this column (attatched) at which buckling is likely to occur

Homework Equations

E.S.R = sqrt(π^2*E / σ)

2nd moment of Area I = AK^2

E.S.R = L/K

I= π/32*(D^4-d^4)

This formula for the second moment of area of the column is incorrect. You used the polar moment, which is I_x + I_y

You are supposed to use the least value of the second moment of area, which for a circle is I = (π/64)D⁴

Since this is a hollow cylinder, I = (π/64)*(D_o⁴ - D_i⁴)

You'll have to re-do your calculations of the length of the column accordingly.

 

[Niall11]

oh right I have on my notes that least 2nd moment of area is π/32(D^4-d^4)!

so least 2nd moment of area = π/64 (D^4-d^4) = π/64 (0.08^4-0.06^4) = 1.374 * 10^-6

now effective L comes out at 2.94m and length 5.94m!
I'm hoping that is correct if I've used all the right formulae

 

[SteamKing]

oh right I have on my notes that least 2nd moment of area is π/32(D^4-d^4)!

so least 2nd moment of area = π/64 (D^4-d^4) = π/64 (0.08^4-0.06^4) = 1.374 * 10^-6

now effective L comes out at 2.94m and length 5.94m!
I'm hoping that is correct if I've used all the right formulae

Yes, your length calculation looks correct now.

FWIW, this same problem has cropped up on PF several times before.

 

[Niall11]

Thanks a lot that was a big help, oh right I didn't realize ill look next time!

 

[Marv K]

If the effective length is half of the actual length, are those figures not incorrect? I only ask cause I'm doing the same course and wanted to check. Is the effective length not 2.968m?

 

[SteamKing]

If the effective length is half of the actual length, are those figures not incorrect? I only ask cause I'm doing the same course and wanted to check. Is the effective length not 2.968m?

The effective length is approx. 2.94 m, but the OP made a slight error in doubling this figure to find the actual length, which should be approx. 5.88 m.

 

[Marv K]

Ok, sorry to be annoying, but worried I've done it wrong now.

If E.S.R. = 118.74

I = 1.374*10^-6

A = 2.199*10^-3

Then the equation is Le = 118.74*sqrt(1.374*10^-6)/(2.199*10^-3)

Which equals 2.968m?

Also, sorry if I've written that equation terribly.

 

[SteamKing]

Ok, sorry to be annoying, but worried I've done it wrong now.

If E.S.R. = 118.74

I = 1.374*10^-6

A = 2.199*10^-3

Then the equation is Le = 118.74*sqrt(1.374*10^-6)/(2.199*10^-3)

Which equals 2.968m?

Also, sorry if I've written that equation terribly.

I'm not sure what you point is here. Le = 2.968 m or Le = 2.94 m is the same number for all intents, given that this is a buckling problem.

Depending on how you round the intermediate computations, there will be some variation in the fractional portion of the final result.