Intro to Engineering Design, Sections 19/20

Beam Design Introduction

Situation:

I have three pieces of wood (Ľ"x2"x38") stacked up on each other that I am using to bridge a gap of 36 inches, so that my train which weighs 20 lb./ft can get across it.
The Toy Train Bridge Association (TTBA) codes tell me that I am only allowed to have a maximum deflection of 0.15 inches.

Problem Definition

Is there a problem? If so, what is it?

Collect data and analyze information.

I take a measurement with the train on the bridge, and I find that the bridge is sagging about 0.3 inches. Does it comply with regulations? No . . . so, I have a problem. The deflection I measured is greater than what is allowed by the TTBA. I need to fix it.

Generate Solutions

One thing I can do is tinker. I can prop the bridge with a book or something. But, tinkering is not engineering.

Tenet of
engineering

Engineering is not knowing information, but knowing where to find the information you need.

So, I go to the library and find a book about bridges. In the table of contents, I notice that there is a chapter entitled Deflections of Beams. I go to this chapter and find an example problem called "Simply-Supported Beam Under a Uniform Load." This looks a lot like my little bridge. In this section, I find something extremely useful - the following equation that predicts the deflection of such a beam:

where q is the force/length of the load, L is the length of the span, E is something called the modulus of elasticity, and I is something called the moment of inertia. I have no idea what these last two things are, but I can forget about them for a minute because I realize that this equation is close to what I need, but not exactly. This equation gives me the deflection of any point along the beam. What I really need is just one point - the maximum one. How do I find the equation for the maximum displacement?

Tinkering solution:	Eyeball the bridge and guess that it is about midway.
Better solution:	Graph the above equation and see where the deflection is greatest.
Best solution:	Use calculus to find the max. (i.e., find x value where .)

We find that the maximum occurs when x = L/2. So we plug this into the deflection equation to get the following expression for the maximum deflection:

But, we still have that problem of not knowing what E and I are. So, I go back to the bridge book and look them up.

I find that the elastic modulus (E) is a constant material property that relates the amount of stress in a material to how much it is being stretched. There is also the following table of E values for a few types of material.

Material	E (psi)
Wood	1,750,000
Aluminum	10,000,000
Titanium	15,500,000
Steel	30,000,000

I also find some information about the moment of inertia (I). It turns out that the moment of inertia does not depend upon material. Instead, it depends upon the shape of the beam cross-section. For a rectangular beam, the formula for the moment of inertia is

where b is the length of the base of the beam and h is the height of the beam, as shown in the following figure:

So, now I know something about all of the terms in my maximum deflection equation. I decide to plug the moment of inertia equation into my displacement equation so that I am only dealing with one equation. By doing this, I get

Let me check the equation by plugging in values for all the terms. By stacking my three pieces of wood, I essentially have a beam with a base of 2 inches and a height of 3 x 0.25 inches = 0.75 inches as shown in the following figure:

Now when I plug all of my values into the displacement equation I get

which is pretty close to what I measured. It looks like the equation is OK, but how can I solve my problem. I am obviously going to have to change something about the design of my bridge. But, what? Let’s examine our displacement equation to find out.

Notice that I can make vmax smaller by decreasing the numerator or increasing the denominator. So, now let’s generate some possible solutions in light of this finding.

Possible Solutions:

Mathematical Solution	Practical Solution
Decrease q.	Make the train lighter.
Decrease L.	Shorten the gap.
Increase E.	Choose a material with a higher E value.
Increase b.	Widen the base.
Increase h.	Increase the height.

Decide The Course Of Action

Let’s examine the solutions:

Make the train lighter.	I could shave off sections along the whole length of my train. But, my train is the whole purpose for having the bridge, and this seems a little drastic.
Shorten the gap.	I can’t - that’s why I need the bridge.
Choose a different material.	Right away I can rule out using titanium, because it is incredibly expensive and it doesn’t give me as high an E value as steel does (which I know is cheaper.) Perhaps I could try using pieces of aluminum or steel, though. Let’s see if they will work. Aluminum: Steel: Both materials will give me allowable deflections! So, the problem is not a hopeless one. I know that one viable solution is to purchase steel or aluminum pieces. But which one? The steel performs better than the aluminum does, but maybe it is more expensive? So, I call around to find some prices. I find a store that has exactly what I need; they carry Ľ"x2"x38" pieces of both steel and aluminum. And, guess what? It turns out that steel is twice as cheap as aluminum. So, I can toss out the aluminum idea. Then, I get a brainstorm. The steel performs so well that I may even be able to get by with less pieces. What if I tried using only one or two pieces? This means that h is now 0.25 or 0.5 inches. Will this work? Two Pieces: One Piece: Well, one piece sure won’t work. But, I could get by with two pieces. So, one solution to my problem would be to buy two pieces of steel to make my bridge. But, I still have further solutions to examine before I run out to the store.
Widen the base of the beam.	How could I do this? I can’t stretch the wood, but I could lay the pieces side-by-side, instead of stacking them up. This would make my base 6 inches wide, but would decrease the height to 0.25 inches Will this work? Yikes. That sure doesn’t work.
Increase the height of the beam.	How could I do this? I don’t have any more wood pieces I can stack up. But, what if I stacked them together then turned them on their side. This would make my base 0.75 inches wide and increase my height 2 inches. Would this work? This would work great. In fact, even if I used one piece (now my base would change to 0.25 inches) my deflection would be small enough. I’ll show you. But, there’s one problem. My train needs the bridge to be 2 inches wide. So, neither of the above solutions will work. What can I do? How about if I put one piece on its side and glue another one on top of it to make a T? This should give me the deflection I need. So, now I have two solutions: Buy two steel pieces, or Make a T out of two of my wood pieces I decide to choose the second, because there is no additional cost.

Implement the Solution

I glue my pieces together to make the T. But there is one problem. I can’t get the bridge to stand up. So, I cut away an inch of the ends of my T-stem (web), and now I can rest the bridge across the gap just fine.

Evaluate the Solution

Was my problem solved? Yes. In fact, I even have a piece of wood left over for another project. My supposed problem actually turned out to be a good opportunity.