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How a beam resists bending. (Moment of Inertia)

Rubber Band simulation of a beam subjected to bending.

In the post “Stress, Strain and finding center“,  I talked about simple compression and tension and how beam or column elongates under tension and shrinks under compression.  In the elastic portion of the stress strain curve the beam is acting like a very very strong spring.   The goal of this post is to get a strong picture of the concepts involved in the moment of inertia and how a beam resists bending.  We will do some calculations, but our calculations will not be complete and somewhat rough.   To do the complete calculations would require the use of calculus and that is beyond where we need to go.  However, we will end up with a good understanding and that is a whole lot more important.

When a beam is subjected to bending forces the beam sees both compression and tension at the same time.  Imagine we somehow grip the ends of a beam and twist one end clockwise and the other end counter-clockwise.  We would be attempting to bend the beam and it would form a curve. Now also imagine we picture the beam as made up of fibers running from one end of the beam to the other end and all of these are glued together.  (An accurate picture of a piece of wood.)  The fibers on the outer part of the curve would be getting stretched (elongation — strain) and the fibers on the inner part of the curve would be getting compressed and both would be acting like springs attempting to straighten the beam.  Somewhere in the middle there would be neither stretching nor compression.   This point is called the neutral axis.

The same actions occur if a beam is supported by a support on either end and a weight is placed in the middle and also if a beam is supported at one end and the weight is close to the other end, for example a diving board.

So now we have two things to get pictured. First, where is the neutral axis? The neural axis is in the center of the cross section area of the beam and we just learned how to find that.  Question 1 down, now comes the interesting one.  How do we determine how much a given beam resists bending?

Rubber band simulation of a wider beam with bending forces.

As we think of the sections of fibers on the outer and inner part of the bend there are two things going on as the beam is bent.   Assume for a moment the beam is a solid rectangle and we are going to look at four sections of fibers.   One section is on the tension side on the outer edge of the beam.  Another section is also on the tension side of the beam but only one-half way from the neutral axis to the outer edge of the beam.  The other two sections are in the same relationship but on the compression side of the beam.  We will ignore the compression side for now.

As the beam is flexed the outer fibers will be stressed,(elongated) twice as much as the mid-way fibers and since strain (the spring pull force) is related to the elongation, the outer fibers will produce twice as much force.  Since the outer fibers are twice as far away, each pound of force there will have twice the effect as the force in the mid-way section because of the longer lever armThe resistance to bending is related to the distance from the neural axis squared!  The same thing happens on the compression side.

The first two pictures on this post is a simulation of that by placing rubber bands on a stick that is able to rotate around a pivot point.  As the rubber bands are moved from 1″ to 2″ from the pivot point it takes four times the force to rotate the the stick around the pivot point.  Said another way, It would take 8 rubber bands place at the 1″ mark to equal 2 rubber bands placed at the 2″ mark.  The fixture is relatively simple to construct so you can try the experiment.  My first fixture was popsicle sticks balanced on nails. It worked but would have been hard to photograph.

This explains why bicycles and drag racers are made with hollow tubing.  This provides resistance to bending while keeping the weight low.  It also explains why I beams are made the way they are.   The “meat” at the outer edges, the flanges, provides the strength to keep the beam from bending.   This also explains why bridges consist of superstructures to prevent bending of the bridge under load.  This puts the compressive beams as far above the roadway and tension beams as possible.

Another interesting thing to look at is aircraft and spacecraft structures.  These have been analyzed using computer programs and later tested to ensure they can meet the stresses necessary.  These normally have lots of holes in the structure to reduce the weight.  Most of those holes will be found in the center, web, part of the beams, because this has less effect on the resistance to bending.

This post is already becoming long with a lot of information in a small space so tomorrow I will post some real examples using popsicle sticks in the configurations I am using on the popsicle crane boom.  I think it will amaze you to see the increase in stiffness.

 

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