Problem With Columns: Finding Load for Buckling

[BusyEarning]

Hi every body this is my first post on these forums.

I have a hollow column made from structural steel that has a Young's modulus of 200GN m^-2 and a yield stress of 140MN m^-2 it is 10m long. the larger radius R is 40mm the smaller r is 30mm

I am asked to find the load that will cause it to buckle which i have used the formula

stress = force/cross sectional area

and transposed it to

force = stress x cross sectional area

so

force = (140 x 10^6 ) x (2.199 x 10^-3) = 307.86 x 10^3 N

then the next question asks me to do the same but for a column of half the length ... but if i do this the answer will just be the same because the equation for stress does not take the length into account.

So i think i am doing something wrong in the first part of the question as well, otherwise the next part seems pointless.

I am not brilliant at mathematics so any help hints or advice would be appreciated. thanks in advance.

 

[SteamKing]

You are just manipulating the formula for axial stress, which is not the same as the formula which determines the critical load to cause the column to buckle. Do some more research in your notes and try again.

Hint: the column will buckle before the yield stress of the material is reached.

 

[BusyEarning]

Hi thank you for your help i think i have got it now , i was getting confused as in my textbook it shows this.

F_{c} = σ_{c} A = \pi^{2}EI/L^{2}_{e}

Which if someone can explain what this means because the formulas to me are not equal eg.
to me this is like A = B = C
thus A = C ? or am i missing something.

 

[SteamKing]

Hi thank you for your help i think i have got it now , i was getting confused as in my textbook it shows this.

F_{c} = σ_{c} A = \pi^{2}EI/L^{2}_{e}

Which if someone can explain what this means because the formulas to me are not equal eg.
to me this is like A = B = C
thus A = C ? or am i missing something.

You just need to know what each variable means:

I'll take a stab at it -

σ_{c} - critical buckling stress
A - area of the cross section of the column
F_{c} - critical load above which the column buckles

All the first part of the equation, F_{c} = σ_{c}A,
is saying is that the critical buckling load is equal to the critical buckling stress multiplied by the area of the column, which you already knew from the OP.

The meat of the equation is that F_{c} = \pi^{2}EI/L^{2}_{e}

E - modulus of elasticity of the material of the column
I - second moment of area of the column cross section
L_{e} - effective length of the column

A note here: L_{e}, the effective length of the column, depends on how the ends of the column are supported.

https://www.efunda.com/formulae/solid_mechanics/columns/columns.cfm

The table at the bottom of the link above gives values of effective length for different end conditions.

 

[BusyEarning]

Thank you again , I have thrown myself into the deep end i think here , I am Doing HND Electrical Engineering by distance learning and i have been out of education for 10yrs+ so i think i have forgotten all the rules of mathematics lol

 

[SteamKing]

Well, give your original problem another try now that you have better info to work with.