Material Analysis

"End Posts", "Top Chords", "Vertical Hips"...  All these pieces of a bridge are collectively called "Bridge Members".  There are three ways a bridge member can fail.  It can be crushed, it can be torn apart or it can buckle.  This page talks about the first two, and of those, talks about crushing first. 

Maximum Compressive Force

Suppose you took a 1" x 1" x 1" cube of balsa wood, put a pressure plate on top of it and then put on successively heavier weights.  At some point, the walls of the internal cells of the balsa wood would collapse and the cube would be crushed.    If you recorded the total weight as you did this experiment, you would find that there was a borderline weight.  If you stayed on the light side of the borderline, the cube would be fine.  If you crossed over to the heavier side of this borderline, the cube would be crushed.  This borderline is called the balsa wood's maximum compressive strength, and the force is called the maximum compressive force.

You would find that this borderline changed based on how dense the balsa wood was.  If it was low density, the border line would be around 680 pounds.  If it was medium density, the borderline would be around 1,750 pounds.  If it was high density, the borderline would be around 2,830 pounds.

Intuitively, this seems a little bit weird.  Clearly, if you tried to balance a 680 pound walrus on the top of a 1" x 1" x 100" piece of balsa wood, it would "snap". (the balsa wood, not the walrus.  The walrus would go "plop").  But that kind of failure is called "buckling", not compressive failure.  Here we assume the piece of wood is so short that no buckling occurs.

"Compressive Force Before Failure" scales with surface area.  If we put weights on a 1/2" x 1/2" x 1" piece of balsa wood, it would hold 1/2 x 1/2 or 1/4 as much weight.  So a 1" x 1" piece of wood that could hold a 680 pound walrus would now only hold a 170 pound walrus (170 = 680/4) if we cut away 3/4 of it.

So, what possible use is this?  Well, recall from our internal force analysis that if we put 50 pounds on each corner, that results in a force of 78.11 pounds on the "End Post".  So, for every 78.11 pounds of load on the "End Post" we have 200 pounds we can put on the bridge (4 corners x 50 pounds).  Stated differently, for every pound of load on the "End Post" load, we can support 200/78.11 = 2.56 pounds of load.

Since the "End Post" cross section is 1/2 x 1/2 or 1/4 inches squared, then if it was made of light balsa wood, it could tolerate at most a 170 pound load passing through it.  Or, 170 x 2.56 = 435 pounds on top. Now, here's the important part. This is more weight than Katherine's bridge held, even though she used heavy balsa wood for the "End Post".  Heavy balsa wood is 4 times as strong as light balsa wood and so the bridge could have held over 1,700 pounds if "End Post" compressive failure was the bridge's weak point.  Clearly it wasn't.  (In the buckling section, we will discover that if Katherine had known about algebra, trig, Maximum Compressive Force and buckling, she could have made her "End Post" a little lighter and used the weight savings to put in more cross bracing.  She didn't, but now you do!)

Maximum Tensile Force

As balsa wood can be crushed, it can also be torn.  The equations for tearing balsa wood are similar to those for crushing balsa wood and depends on force per unit area.  In particular, suppose we had another hapless walrus, weighing 1,100 pounds, suspended from a rope connected to a 1" x 1" x 1" piece of low density balsa wood.  He would be fine until he ate anything, and then he would cross the "Tensile Force before Failure" borderline and the balsa wood would tear apart.

Relationship between compressive and tensile strength

The table below uses the compressive an tensile strengths listed here and divides one by the other for each density of balsa wood.

Balsa Wood Tensile/Compressive Strength Ratios