# 3 Strain Points on a Cylinder

I have an infinitely long thick walled cylinder (steel casing). I have three strain gauges measuring strain along the length axis(z axis); the gauges are equally spaced. The gauges are on the outside of the cylinder.

How can I calculate the strain in the xy plane (cylinder cross section)?

My first notion was to use the following Poisson's formula:
Δd = -dv(ΔL/L) where d is the diameter of a rod. I would then assume my cylinder is three rods (the cylinder is placed in solid rock). I can calculate the axial strain in each rod; but not the direction. I was thinking that I could add the strains to get the strain in the xy plane but I am having a problem visualizing the vector direction.

Am I completely wrong in my assumptions? Is there a better way to quantify this problem?

I have never posted in this forum. This is not a homework question despite my phrasing it that way. If it belongs in the homework forum please let me know.

Thanks, n

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Is it possible for the 3 strain gauges to give zero readings (i.e. no strain in the z-direction) but still there is strain in the x-y plain? If so, then your task is impossible.

I think the 3 strain gauges setup can deduce bending, but cannot sense horizontal forces that produces no bending.

I am no expert in mechanical engineering. Just some ideas here.

I don't understand your scenario, as stated. So, I'll make some presumptions:

If I were to look at the cylinder on axis, it might look like a clock face (circle) coming out the page. So, place the gauges at 12-oclock, 3-oclock, and 6-oclock postions.

If the cylinder was in pure axial strain, then all gauges would give the same non-trivial value.

If the cylinder was in pure bending, then the top (12-oclock) gauge would have a value equal but opposite in sign to the bottom gauge. The 3-oclock gauge would have zero value.

If the cylinder was in pure shear, then the top and bottom gauges would have a zero value, whereas the 3-oclock gauge would have a non-zero value. This is true only if the direction of shear is oriented top-to-bottom. Alternative values would exist for different direction of shear force.

Variation of those combinations would indicate the cylinder is under both axial and bending forces. This can be resolved with a system of linear equations.

Now that the above is clear, it can be noted that some orientations of the force with respect to the strain gauge locations may lead to insufficient equations. This can be easily resolved by repositioning the three gauges instead at 0-deg, 120-deg, and 240-deg. Because of the symmetry, all orientations of force vector can be resolved by the gauges. Of course, the equations change some what, but this should be trivial.

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My strain gauges are located at 0, 120 and 240 degrees. The strain gauges are only measuring strain in one direction (the pipe axis). Is it possible to calculate vector forces in a direction other than the pipe axis? If so how?

Background: I have a long steel pipe casing 1km long installed vertically in bedrock. I am measuring strain with fiber optic fiber strain gauges which can measure a strain at any given point along the fibers length. So at z depth I have three strain measurements at each location (1, 120, and 240 degrees). I am trying to make sense of the hypothetical data.

More details: I have a steel casing about 1km long installed in bed rock vertically. I have three fiber optic wires attached to the pipe at 0, 120, and 240 degrees. The fiber gauges will only measure strain in one direction (the length of the pipe). due to the nature of the fiber, I can measure strain at any depth. So at z depth can I identify a vector force? How?

My strain gauges are located at 0, 120 and 240 degrees. The strain gauges are only measuring strain in one direction (the pipe axis). Is it possible to calculate vector forces in a direction other than the pipe axis? If so how?
Yes, however only some but not all stresses and their strains.

A structural material has 6 degrees of freedom:
3 forces: axial x-x, shear y-y, and shear z-z.
3 moments: torque x-x, bending y-y, and bending z-z.

In this nomenclature, your pipe would have its length aligned on the x-x axis.

One strain gauge can detect any one of the forces.
Two strain gauges can detect any one of the moments. Two are required because the nature of a moment is a coupling of two forces (note "coupling" as in "couple", or two.)

So, what should you do? Well, you need to reveal more of what you are trying to achieve so that it can be suggested how the gauges should be arranged. On the other hand, if the study and its data is already done, then:

Only the following forces can be determined:
1 force: axial x-x.

Only the following moments can be determined:
2 moments: bending y-y, and bending z-z.

I can how my pipe is moving by comparing the location of the neutral plane to the location of the original neutral plane using the following equation:

(M(z)/EI)y1=e1(z)
(M(z)/EI)y2=e2(z)

for sensor 1 and sensor 2 keeping in mind my z direction is the length of the pipe.

But how does one calculate the moment?