# Formulas to Calculate deflection/bending of Free-Standing Pipe

Hello Everyone,

I have several questions regarding the deflection of a free-standing pipe wherein Force is applied on top.

Appreciate any feedback that I can get.

Supposing I have the following Pipe Specifications:

OD: 7in
ID: 6in
Yield Strength: 80,000psi
Modulus of Elasticity: 30000000psi
Poisson’s Ratio: 0.3

Here are my questions:

1. How do we calculate the maximum height that the above mentioned size of pipe will freely stand (without any support or braces)?

2. Assuming that the above mentioned pipe has a length of 15feet and for the sake of argument can be placed vertically without any support (free-standing) but fixed at the bottom end. If we apply 10,000lbs of Force at the top end, how do we calculate the:

A. Amount of maximum deflection relative to the center line of the pipe
B. Location of the maximum deflection – Is this going to be halfway through the length of the pipe (@7.5feet)?

3. Supposing the above mentioned pipe 7in OD, 6in ID is placed concentrically inside another pipe. Let’s assume that the ID of the outer pipe is 8in (and assuming that the outer pipe is fully rigid). And supposing that on the 7in OD inner pipe, 20,000lbs of Force is applied on top of it. For the sake of argument let’ say that the deflection caused by the 20,000lbs of Force applied will exceed the annular clearance between the OD of the Inner Pipe and Outer Pipe. How do we then calculate the amount of Force effectively transferred by the inner pipe onto the fully rigid outer pipe?

Thanks,

Soji

## Answers and Replies

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SteamKing
Staff Emeritus
Science Advisor
Homework Helper
Hello Everyone,

I have several questions regarding the deflection of a free-standing pipe wherein Force is applied on top.

Appreciate any feedback that I can get.

Supposing I have the following Pipe Specifications:

OD: 7in
ID: 6in
Yield Strength: 80,000psi
Modulus of Elasticity: 30000000psi
Poisson’s Ratio: 0.3

Here are my questions:

1. How do we calculate the maximum height that the above mentioned size of pipe will freely stand (without any support or braces)?
The answer to this question can be found in this article, under the "Self-buckling" section:

http://en.wikipedia.org/wiki/Buckling

2. Assuming that the above mentioned pipe has a length of 15feet and for the sake of argument can be placed vertically without any support (free-standing) but fixed at the bottom end. If we apply 10,000lbs of Force at the top end, how do we calculate the:

A. Amount of maximum deflection relative to the center line of the pipe
B. Location of the maximum deflection – Is this going to be halfway through the length of the pipe (@7.5feet)?
As long as the load is applied concentrically with the axis of the pipe and the applied load is less than the critical buckling load, there should be no deflection in the pipe. (Note: I have not calculated the critical buckling load for the case you are describing.)

3. Supposing the above mentioned pipe 7in OD, 6in ID is placed concentrically inside another pipe. Let’s assume that the ID of the outer pipe is 8in (and assuming that the outer pipe is fully rigid). And supposing that on the 7in OD inner pipe, 20,000lbs of Force is applied on top of it. For the sake of argument let’ say that the deflection caused by the 20,000lbs of Force applied will exceed the annular clearance between the OD of the Inner Pipe and Outer Pipe. How do we then calculate the amount of Force effectively transferred by the inner pipe onto the fully rigid outer pipe?
Thanks,

Soji
The answer to Quest. 2 above applies, unless the inner pipe buckles for some reason. What is the ID of the 8 in pipe?

To be sure, the lateral deflection of any vertical column is a sign that failure and collapse may be imminent. Run away.

Last edited:
As long as the load is applied concentrically with the axis of the pipe and the applied load is less than the critical buckling load, there should be no deflection in the pipe. (Note: I have not calculated the critical buckling load for the case you are describing.)
Well, given that they were given poisson's ratio, I would imagine that the OP is asking what the delta-y and delta-x dimensions would be given the load at the top.

that's just standard stress/strain calculations, though the lateral elongation of a tube due to axial compression can sometimes be tricky when trying to reason out the equations.

Steam King – Thanks for the inputs.

Let me modify my first question.

1. Supposing that the 7in OD, 6in ID pipe has a material density of 490lbm/ft3, if it were to be fixed at the bottom end, how tall/high can this pipe rise without swaying/bending due to its own weight? What formula can I apply?

Thank you,

Soji

SteamKing
Staff Emeritus
Science Advisor
Homework Helper
Steam King – Thanks for the inputs.

Let me modify my first question.

1. Supposing that the 7in OD, 6in ID pipe has a material density of 490lbm/ft3, if it were to be fixed at the bottom end, how tall/high can this pipe rise without swaying/bending due to its own weight? What formula can I apply?

Thank you,

Soji
This question was answered in Post #2. The formula is in the attached article mentioned there.

Steam King – Thanks for re-iterating where to find the formula for self-buckling. Appreciate it.

There’s a statement that followed the formula for Self-Buckling on the link you posted – “and B is the first zero of the Bessel function of the first kind of order -1/3, which is equal to 1.86635086...”

Just want to clarify on the formula:

h_critical=((9B^2)/4 EI/(ρgπr^2 ))^(1/3)

Can I simply apply the 1.86635086 value mentioned above as the for B on the above equation? Or it’s on a case-to-case basis. I’m not familiar with the Bessel function.

Thanks,

Shodji

SteamKing
Staff Emeritus
Science Advisor
Homework Helper
Steam King – Thanks for re-iterating where to find the formula for self-buckling. Appreciate it.

There’s a statement that followed the formula for Self-Buckling on the link you posted – “and B is the first zero of the Bessel function of the first kind of order -1/3, which is equal to 1.86635086...”

Just want to clarify on the formula:

h_critical=((9B^2)/4 EI/(ρgπr^2 ))^(1/3)

Can I simply apply the 1.86635086 value mentioned above as the for B on the above equation? Or it’s on a case-to-case basis. I’m not familiar with the Bessel function.

Thanks,

Shodji
For a free-standing circular pipe, B = 1.86635086 in the equation for the critical height of the pipe. If you have a different type of free-standing structure, this equation may not apply: you would probably have to refer to a text on elastic stability or buckling to find a suitable method of analysis.

The classic reference for this the "Buckling Strength of Metal Structures" (1952) by F. Bleich, but it's long been out of print and copies are hard to come by.

http://structurae.net/persons/data/index.cfm?id=d009664