Buckling check for secant shaft

[HanStrulo]

Hi,

I am new to secant shaft design and I am reading some of the previous calculations when I noticed a buckling check using this formula for the critical buckling load Tcritical=3*E*I/r^2.

If someone knows a reference I could read to be familiar with this formula? For more information, the shaft is only laterally loaded with earth pressure and water and no axial loading. I assume the buckling check is for the secant shaft giving in radially? Not sure.

Thanks alot!

 

[EireChch]

I tried chatGPT for this with no luck, but the below might be usefull for you

Sure! Here are some references to the equation you provided in the codes mentioned:

American Society of Civil Engineers (ASCE) 7 Standard:
In section C3.3 of ASCE 7-16 Minimum Design Loads and Associated Criteria for Buildings and Other Structures, the equation for the critical buckling load is provided in the form of the following expression:
Tcritical = π^2 * E * I / (K * L)^2

where K is a factor that accounts for the end conditions of the column and L is the unsupported length of the column. The equation you provided can be derived from this equation by assuming fixed-fixed end conditions and setting K=1.

American Institute of Steel Construction (AISC) Specification for Structural Steel Buildings:
In the AISC 360-16 Specification for Structural Steel Buildings, the equation you provided is given in Table B4.1 as the approximate critical buckling stress for an ideal pin-ended column, which is derived from the more general equation for the critical buckling load:
Pcr = π^2 * E * I / L^2

where Pcr is the critical buckling load and L is the unsupported length of the column. The equation you provided can be obtained from this equation by assuming a fixed-fixed end condition and substituting the radius of gyration r for L.

Eurocode EN 1993-1-1:
In Eurocode 3: Design of Steel Structures - Part 1-1: General Rules and Rules for Buildings, the equation for the critical buckling load is given in the form of the following expression:
Tcritical = π^2 * E * I / (L * λ)^2

where λ is the slenderness ratio of the column, defined as the ratio of the effective length of the column to the radius of gyration of its cross-section. The equation you provided can be derived from this equation by assuming a fixed-fixed end condition and setting λ=π.

Note that these references are not exhaustive and there may be variations in the way different codes and standards present the equation for critical buckling load.

 

[Settingsun]

Should it be R^3?

https://newtonexcelbach.com/2011/01/01/buckling-of-rings-and-arches/

 

[EireChch]

I might be driving beyond my headlights here being a geotech, but shouldnt buckling be linked to the shaft length? Why is r (presuming its radius) used?

 

[Settingsun]

EireChch, there's no axial load in this case so it isn't buckling like a building column. This check is for the circular cross-section buckling into an oval shape, while the pile remains vertical.

The page I linked to (and suspect is the situation that HanStrulo's calcs are meant to check) assumes uniform radial pressure. I'm picturing it's a simplification of the real situation, which would have unequal pressure on the active and passive sides, and no pressure along the length of the wall (shielded by the neighbouring secants).

 

[EireChch]

ah ok understood, thanks Steve.

 

[HanStrulo]

Thank you everyone for the responses.

I asked about the origin of the formula and my colleague directed me towards CIRIA 95 (the design and construction of sheet-piled cofferdams), this is an snapshot of the formula

It's from Timoshenko's beam theory apparently.

I tried to get into it in depth but i found it too time consuming so i just used the formula from CIRCA 95 as is to check for buckling under uneven earth pressure.

if anyone has a good book or a simplified version of the proof to the formula, I would really appreciate it.

Also, it should have been to the power of 3. I apologize for the mistake.