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I have been searching the web for hours now, and I have had some success, but I have not found everything I am looking for.

The best thing I have found was a similar arrangement in Hibbeler’s Mechanics of Materials (hence the name of the thread). This had a short rod of aluminium in a vice being crushed axially (z). This is a very common problem, and finding the equations for this is simple, many of the texts had these, and the result is a uniform change in diameter in both the x and y directions.

However, I am interested in a rod being crushed diametrically (x), that is, loaded laterally. I have found the equations for the stress in the two dimensions that correspond to the circular cross section, x and y. However, I am also interested in what is happening along the length of the rod, in the z direction.

What I have is for a Force (F) applied in the dimension x, the stress in x ig given by,

σ_{x}=-6F/πld

where l is the length of the cylinder, or length over which the force is being applied in my actual application (the gauge length for sensing purposes), and d is the diameter of the cylinder. In the y axis the stress is given by,

σ_{y}=2F/πld

That is, there is a compressive load from the crushing in the x direction, which results in an expansion in the perpendicular diameter (y), resulting in an elliptical cross section. Here is a link to an image I found

http://what-when-how.com/wp-content/uploads/2011/07/tmp1914_thumb.jpg

What I want is to know what is happening along the length. There must be some strain in this direction according to Poisson's ratio, but I could be wrong. The journal articles I have come across just assume that this is zero for simplicity. I assume they can do this because the length (l) is significantly greater than the diameter (d), but no justification is given for this. The interesting thing is that in their experimental results compared to their theoretical prediction, there is a slight difference, and I think this can be explained by what is happening in the length direction.

Any input would be greatly appreciated.

Kind Regards,

G

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# Mechanics of Solids Cylinders with Lateral Loads

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