Theoretical spring constant calculation for a tube of pipe

[ReliableSin]

Homework Statement

I require the a means to theoretically calculate the spring constant of a hollow tube.
Essentially, I need to find the diameter of the glass pipette which will have a spring constant of 100 Newtons/metre. If someone could point me towards relevant information on calculating, or using software to find the solution it would be greatly appreciated.

Homework Equations

This is what I need. I've been searching, but all I've been able to find are solutions based on spring geometry.

The Attempt at a Solution

No idea.

 

[tiny-tim]

Welcome to PF!

Hi ReliableSin! Welcome to PF!

I don't understand … what are you intending to do with this glass pipette?

Are you using it to support something, as a building column?

 

[ReliableSin]

I'm using it as a cantilever tip in some atomic force microscopy experiments. Usually we fabricate our own cantilevers using optical fiber with a spring constant of ~100 Newtons/metre. Now we want to use a pipette as the cantilever tip. As such, we need the pipette to have a spring constant of about the same value. We can adjust the spring constant of the pipette by tapering or etching. However, we also need to make a bend in this pipette, and the thinner we make the pipette before bending, the higher the chance that our bend will stop fluid from flowing through the pipette.
To sum up, I need to know how to get the spring constant of a tube so I calculate what is the largest pipette diameter I can use which will have an appropriate spring constant.

I've been reading through the pages on wikipedia on spring constants and deflection. I think I need to find the appropriate moment of inertia for a cylinder, which is simple, and sub this into the equations given for a normal cantilever. However, I'm having a hard time following their math, specifically how they eliminate their force term.

 

[berkeman]

Homework Statement

I require the a means to theoretically calculate the spring constant of a hollow tube.
Essentially, I need to find the diameter of the glass pipette which will have a spring constant of 100 Newtons/metre. If someone could point me towards relevant information on calculating, or using software to find the solution it would be greatly appreciated.

Homework Equations

This is what I need. I've been searching, but all I've been able to find are solutions based on spring geometry.

The Attempt at a Solution

No idea.

I'm using it as a cantilever tip in some atomic force microscopy experiments. Usually we fabricate our own cantilevers using optical fiber with a spring constant of ~100 Newtons/metre. Now we want to use a pipette as the cantilever tip. As such, we need the pipette to have a spring constant of about the same value. We can adjust the spring constant of the pipette by tapering or etching. However, we also need to make a bend in this pipette, and the thinner we make the pipette before bending, the higher the chance that our bend will stop fluid from flowing through the pipette.
To sum up, I need to know how to get the spring constant of a tube so I calculate what is the largest pipette diameter I can use which will have an appropriate spring constant.

I've been reading through the pages on wikipedia on spring constants and deflection. I think I need to find the appropriate moment of inertia for a cylinder, which is simple, and sub this into the equations given for a normal cantilever. However, I'm having a hard time following their math, specifically how they eliminate their force term.

It does sound like glass will be too brittle for this application. Have you considered making the bending pipette out of some other material?

 

[ReliableSin]

We use glass for these applications all the time. The issue is that the stiffness of a non-tapered glass pipette is too great to perform contact-mode measurements.

 

[Q_Goest]

I'm assuming you have a cantilevered tube made of glass, but then you imply that the tube is actually tapered. If it isn't tapered, you can use straight beam equations:

I = 3.14159 (D_o⁴-D_i⁴) / 64
Where:
D_o = OD
D_i = ID

I is moment of inertia in units of length raised to the 4'th power.

Deflection of this beam is:

d = F L³ / (3 E I)
Where:
d = deflection (units of length)
F = Force
L = beam length
E = bending modulus (units of force per length squared)

To get spring constant:

k = F / d

k = 1 / ( L³ / (3 E I) )

This assumes the beam is tubular and straight and of uniform cross section and the force is perpendicular to the axis of the tube.

If the tube is actually tapered, curved, nonuniform cross section, etc... it will get a bit more messy...

 

[ReliableSin]

Thank you so much! That's incredibly helpful. Could you point me towards your source for further reading on the topic?
And as you said, the tapering will modify these calculations. Will it essentially be a change of the moment of inertia, or is it more complex than that?

 

[Q_Goest]

The equations are provided in your typical mechanics of materials textbooks, but you can also look online. See if these can help:
http://www.engineersedge.com/beam_bending/beam_bending9.htm
http://www.engineersedge.com/beam_bending/tapered-snap-fit-beam.htm