yeah, that's a bugger ! did you try the obvious (google "stress in curved pipe"). Some articles were behind paywalls (not doing that !). One clue ... what does "Ceasar" do ?

next obvious ... did you look in Roark ? there's a section on piping.

another day in paradise, or is paradise one day closer ?

can you hide this expression by working out a conservative factor, calc the factor then round up ?
or fake out the expression ...
x = r/R, k = (1+x/2*sin(alpha))/(1+x*sin(alpha))

another day in paradise, or is paradise one day closer ?

What are the definitions or R, r and alpha?

But I've been designing pipes for decades and never seen this before. Is it some version of the bourdon effect?

https://www.eng-tips.com/viewthread.cfm?qid=6924

Look it up but seems to be a similar equation.

Remember - More details = better answers
Also: If you get a response it's polite to respond to it.

R is bend radius, r is tube mean section radius, and θ is angle around tube section.

from the linked thread ...
"Internal pressure can affect the pipe bend or elbow in 2 ways. The Bourdon effect describes the tendency for the pipe bend to "open" under internal pressure and "close" under external pressure. The reason is as described by Electricpete. The other effect is the "stiffening" of pipe bends and elbows as internal pressure is increased. The stiffening results from the internal pressure resisting the "ovalization" of the cross section that normally occurs when a bending moment is applied to the bend. One or both of these effects can occur in large diameter pipe with large D/t ratios. Reference B31.3, Appendix D, Table D300 and note (7). If you have piping systems of moderate diameter but are designing to B31.3, Chapter IX (high pressure - Class 2500 or higher) you will want to consider these effects. Good pipe stress computer programs (e.g., Caesar II) will give you the option of "toggling" these effects off or on."

googling "bourdon effect" gets hits ... more threads on Eng-Tips !

I don't know what "Reference B31.3, Appendix D, Table D300 and note (7)." refers to.

another reference to Ceasar, so there may be something there ?

another day in paradise, or is paradise one day closer ?

I believe I somewhat found my own answer. From Roark, 6th Ed., Table 28, Case 5 (Table 13.1 in 7th Ed.). Toroidal shell with internal pressure gives max σ1=(pr)/(2t)*(2R-r)/(R-r) which is same as BDM equation at θ=-90deg. However, I would like to find what reference Roark used. I suspect Timoshenko but I haven't found it there yet.

• https://files.engineering.com/getfile.aspx?folder=d3b00222-908f-4377-bd90-b

ref 42 (from 7th Ed) ... ASME J. Applied Mechanics vol29 no 2

I suspect Ceasar user manual may define this too ?

another day in paradise, or is paradise one day closer ?

I found that I need AIAA Journal Vol.3, No.10, 1965 by Rossettos and Sanders, "Toroidal Shells Under Internal Pressure in the Transition Range". Anyone have a copy?