Pipeline Elastic Bend Angle Calculation

<...divided the arc length by two times the radius...> appears from here to be incorrect.

yes, my doubt is to know whether i am lacking any understanding of pipeline elastic (natural ) bend. so it leads to 2R

...then challenge <...one of my colleague...> to show the calculation from first principles. The outcome is a learning exercise for one of the two participants in the challenge.

I guess Sb = subtended angle, but what are E and D?

If the angle θ is in radians, then arc length L = Rθ. (θ = L/R; R = L/θ.)

If the angle is in degrees, then 2R(pi) would be involved.

Sb is the bending stress, E being modulus of elasticity and D is the pipe diameter.
Using that formulae , you calculate the elastic bending radius.

Now for 30m length, using the elastic bending radius we are calculating the degree

Maybe you're bending one way and then back again....

https://www.victaulic.com/assets/uploads/literature/26.02.pdf

Not clear exactly as to what you mean by one way and then the other way.
This is a vertical bend whereby we are going down only

https://cr4.globalspec.com/thread/7318/Maximum-Cold-Bend-Angle

This was indeed an informative thread. I think the conclusion was for cold bends and it goes very well in the lines of Arc length divided by radius.

I was looking for a specific case of elastic bend

https://www.albinaco.com/calculators/bending-calculators

Cold bends are plastic. You are desiring "elastic". In other words no "permanent set" to the bend. Suggesting that If you release the load that imparts curvature, the pipe returns to its original shape (hence "elastic). If you use 2R then the angle suggested is reduced by 50% which would by inspection make sense if cold bends use simply R (or half the radius, twice the curvature)...having said that, simply reducing curvature by 50% doesn't necessarily demarcate the elastic and plastic regime...material behaviour of steel in the elasto-plasto regime is non-linear. Like other commenters suggested, do some more research. Lots of material mechanics literature resources out there to read thru.

I think you are confusing two completely different concepts.

The angle subtended by an arc is just geometry.

The elastic bend angle is a property of material given certain known parameters.

Hi Randall

based on the material properties, you can do approximations on the maximum allowable bending stress on the pipe. Then that allowable bending stress leads you to a minimum allowable bending radius to keep the stress within the calculated bending stress. Now from that radius, we are trying to find the maximum angle for a certain length

Of which none were given such as pipe diameter and material yield stress...not enough info to answer, though based on simple thin wall shell stability/collapse theory the equation does resolve somewhat to basic geometry.

But not enough info provided to justify assumptions needed to apply those boundary conditions.

Practically speaking, the radii in question for staying in elastic limits are on order of 100's to 1000's of feet. Typically used in laying pipelines.

Not a piping problem (noting phrase "vertical bend")...proceed with caution is my advice.

At best it is a dodgy rule of thumb taking no account of material or pipe diameter.
I'd guess a length of macaroni wouldn't bend to that rule
Del