Help Deriving Equation Behind Kellogg C-9 Graph 2

[Nontrie]

Hello,

I’m a 3rd-year mechanical engineering student currently doing an internship in the piping discipline. Most of my work involves reviewing and building calculation sheets based on graph references, and I’ve been using the Design of Piping Systems by the M. W. Kellogg Company, particularly the graphs in Appendix C-7 and C-9.

So far, I’ve been able to derive the equation behind the C-7 graph, which represents anchor leg flexibility for in-plane displacement, using:

S = (3 * E * D * Δ) / L²

Where:

• S = Allowable anchor load
• E = Young’s modulus
• D = Pipe diameter
• Δ = Thermal expansion
• L = Required Leg length

However, I’m stuck on the C-9 graph, which corresponds to Length of leg required One support displacement normal to the plane of the member (i.e., out-of-plane or vertical movement of an elbow leg). The axes of both graphs are the same, so I initially assumed the difference lies in the displacement term. I tried using:

Δ' = √[(L)² + Δ²] − L

(where Δ is the normal displacement), but it doesn’t seem to match the shape of the graph.

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I extracted approximate data points from both graphs manually and plotted them in MATLAB alongside the theoretical equations.

My question is:​

Does anyone know the formula or method to derive the equation behind the C-9 graph? Or could anyone offer a hint on how the normal displacement is accounted for in the anchor leg stress equation?


Any insight from those who’ve worked with this reference before would be greatly appreciated!


Thanks in advance.

 

[Snickster]

The method in Kellogg appears to be the "Guided Cantilever Method". This is a very conservative method and could give results for stress over two time or more the actual values. In this method flexibility of piping fittings are ignored as well as stress concentration factors, etc. All that is considered is that each piping segment absorbs thermal expansion in the shape of a guided cantilever beam.

I used this method for simple piping systems at times but for anything more than simple I used a computer analysis such as Caesar II. Attached are the derivation of C-7 and C-9 based on the cantilever method and a copy of the book "Piping and Pipe Support Systems" by Paul R. Smith and Thomas Van Laan that discusses the guided cantilever method.

Correction: The derivation of C-9 appears in error as I assume "M" is constant but in reality "F" is constant in members. I will need to check and re-derive.

 

[Snickster]

Here is a good reference for the Guided Cantilever Method from Caesar II piping stress analysis seminar notes.