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Stress, Strain and finding center.

Stress – Strain curve for metal.

This is the first of what will probably be three posts on the theory about the loading of beams and columns. This is why I am building the popsicle crane boom in the way I am doing it.

We have been tied up in details lately, so I think it is time to zoom out and remember why we are putting ourselves through this.  Way back there I had built a small seed starting greenhouse.  One of the problems with it was I did not put on a brace that was badly needed. The discussion of the brace lead to discussions about right triangles and that lead to calculations of statics and the forces directly along a beam.   Now we are working on a miniature crane and we are working through many of the same problems and principles.

I was wrong when I said I did not know if much is transferable from popsicle stick construction to larger full size construction.  The theory and thought process is very much transferable.  The only thing not transferable is the fastening method.

We will go through some math, but it is my goal in these sessions that the math is not the key factor, but that you have a keen understanding of the principles and it becomes “common sense”.   In other words, as the sign before the big drop on the roller coaster says: “Hold on to your hat”. It is going to be a long ride, but hopefully an enjoyable one.

All materials are the same, yet all are different. We will start with a non-ferrous metal for our reference point and then talk about the differences.  If we take a section of this metal, for example a rod, and subject it to a slowly increasing pull (stress) and measure the change in length (strain) of the sample and plot those we will end up with a curve very similar to that shown in the diagram.   Stress is defined as the force of the pull divided by the cross sectional area and here in USA it is given in units of pounds per square inch.  (I think elsewhere it is given in Newtons per square centimeter.)  Strain is defined as the elongation of the sample and that is more precisely defined by the ratio: change in length / original length.

The area of the chart labelled 1 to 2 is the area of most interest to us.  This is called the elastic deformation range and when the stress is removed the metal will return to the original length.  At stresses above this value the metal starts to deform permanently and this area is called plastic deformation.  As the metal deforms it “necks down” and becomes a smaller cross sectional area. If the stress continues to  increase it eventually reaches the point called ultimate failure and the sample breaks.

As said earlier all materials can be tested this way and produce a curve.  However, different materials act very differently.  A way of experimenting and developing a feel for what is happening is to actually pull samples of various materials with your hands and see what happens.  For example, stretch rubber bands and notice how they become thinner as they are stretched and notice if they return back to the original length once released.  The same can be done for thin pieces of plastic and even some small pieces of thin metals.

So far we have been talking about pulling the sample in tension.  The same curves exist in compression, but the curve may or may not be a mirror image for a given material.  For example, concrete is very strong in compression, but very weak in tension unless it is reinforced with steel inside it.  Steel is nearly the same in both compression and tension, but tends to buckle if exposed to compression.   Some materials have different curves depending upon the orientation of the material.  Wood is very different cross grain than it is along the grain.   However, the general concept still applies.

Basically in the elastic deformation range the material can be thought of as a spring.  Energy is being stored in the material as it is stretched and released as the metal returns back to it’s original length.   Now think of a weight being suspended by two rods of the same material.   If one rod is carrying more load than the other rod, the one carrying the most load will extend slightly and the other rod will start carrying more load until an equilibrium is reached.  Obviously, if the rods where very different lengths this change would not be enough to completely equalize the load but the concept is a helpful one to consider when thinking of the forces within a single rod of the material.

So, now where can this force be thought of as being transferred through the material?  The official answer is through the neutral axis.  So where is the neutral axis?  It is along a line through the cross section of the material called the centroid.   On symmetric shapes this is the exact center of the material. This is the reason I am trying to build my popsicle creations symmetric symmetric. This is the reason most structural material comes in symmetric shapes.  Think of I beams, pipes and tubing, etc.

In the very next post I will show how to find the centroid.  Unfortunately, I am out of time this morning and need to get ready for the pay-for job and this post is becoming long anyhow.

This concept of the spring will be very important once we start loading beams in the middle of the beam or if we have a beam supported on one end and have the load on the opposite end of the beam. (Think diving boards).



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