Buckling

[thschrader]

I played around with my spreadsheet and discovered that if I applied the recommended design factor of 2.1 the Johnson formula produced a critical load of 127.36% or 40% utilisation - Is this mere coincidence once more???
...

Can you check the input in cell G15, critical buckling load Johnson-formula?
When using Sy=355 MPa in B17, Johnson gives 167 kN buckling load,
when using Sy=235 MPa, Johnson gives 197 kN
This is a little bit counterintuitive...

About slenderness-ratio:
The design-manuals account for this, you can get reduction-factors from tables (in german: "Knicklinie").
Therefore the design-load (with beam-imperfections) is significantly lower than the theoretical Euler-value.

[Laurie Hartley]

Can you check the input in cell G15, critical buckling load Johnson-formula?
When using Sy=355 MPa in B17, Johnson gives 167 kN buckling load,
when using Sy=235 MPa, Johnson gives 197 kN
This is a little bit counterintuitive...

Oh Oh! - I will and respond later today.

[Laurie Hartley]

i have looked into this Thomas and at first I was quite bamboozled so. I thought perhaps the formula image in cells I12:K16 was incorrect. I am not sure where I got this particular version of the formula from. I conducted the following checks:-

1. I changed the variables in my SMath sheet (attached) to match the spreadsheet at Mpa 235 - it came up with the same numbers - Johnson Pcr 197.34
2. I changed the variables in my SMath sheet (attached) to match the spreadsheet at Mpa 355 - it came up with the same numbers - Johnson Pcr 167.09
3. I then applied the same variables to the online calculator https://www.omnicalculator.com/physics/buckling
4. At 235 Mpa it produced 197.3 kN using the Johnson formula
5. At 355 Mpa it produced 198.5 kN using the Euler formula

So I concluded that there is a "variable based" switch made between the Johnson & Euler formulas but on the surface my spreadsheet Johnson formula is correct.
I went back into my spreadsheet to see if I could find this "switch" variable and found that when I change Sy variable the Critical slenderness ratio at Cell B22 also changes.
At 355 Mpa it is 108.06 i.e. smaller than the Actual slenderness ratio (S (or Sr)
At 235 Mpa it is 132.81 i.e. greater than the Actual slenderness ratio(S (or Sr)

IF the critical slenderness ratio formula image I am using in cells D20:E23 is correct this seems to be the reason for the apparent anomaly because the yield stress value is the divisor or in other words the lower the yield stress the higher the Critical slenderness ratio.

I have modified the cells H8 and H15 to suggest which result to adopt.

I realise as a non mathematician and non engineer my logic may be skewed so I welcome correction, but the more I look into this subject especially the different curves for the different buckling formulas the more intriguing I find it.

[thschrader]

...
I realise as a non mathematician and non engineer my logic may be skewed so I welcome correction, but the more I look into this subject especially the different curves for the different buckling formulas the more intriguing I find it.

Your logic is ok, mate.
When plotting euler/johnson (blue/red) you can see that in this case the johnson-formula
is "out of range" (invalid) for the second case with S355. Sh.. happens, coincidence.

Johnson is valid for high strut loading, the black curve, where Euler predicts wrong values.

Clue: your spreadsheet is ok. Sorry for the noise.

Here the variable (x-axis) is the beam-length in mm, y-axis is max loading in kN

euler_vs_johnson.JPG (79.17 KiB) Viewed 729 times

[thschrader]

Addendum:
Calculation with your latest spreadsheet.
I used these values for the euler/johnson curves above.
As I said above, johnson accounts for struts with low slenderness
and high loading ==> "plastic buckling"

When using S355 instead of S235 in your spreadsheet, the P_crit drops,
which seems counterintuitive. It is not, for the given values in the spreadsheet
the johnson formula is simply not longer valid for S355.

Thomas

your_spreadsheet_S235.JPG (271.39 KiB) Viewed 694 times

your_spreadsheet_S355.JPG (202.57 KiB) Viewed 694 times

[Laurie Hartley]

Addendum:
Calculation with your latest spreadsheet.
I used these values for the euler/johnson curves above.
As I said above, johnson accounts for struts with low slenderness
and high loading ==> "plastic buckling"

When using S355 instead of S235 in your spreadsheet, the P_crit drops,
which seems counterintuitive. It is not, for the given values in the spreadsheet
the johnson formula is simply not longer valid for S355.

Thank you again Thomas - that is what I thought.

I have also noted that there is an area on your graphs where the curves intercept within which the two formulas would produce results that won't vary much but either side of that the results are too conservative, or worse too optimistic.

I actually find it very reassuring that the three methods used on the models come up with identical results (SMath, Online Calculator & Spreadsheet)

This appears to indicate that the formulas used are identical but then leaves me wandering why FreeCAD produced a 3% different result on my original model. I know it's not much but shouldn't it also come up with the same result? I am going to have another look at that in the light of what I have learned in this thread.