| MadSci Network: Engineering |
An identical question was submitted to MadSci and is pending answer. The only difference in the question was that steel was the only material mentioned. Pending an answer to the other question, the following is offered, should you be looking for an immediate answer.
When the steel sheet is bent in the manner you describe, or plywood cut and configured in the manner you describe, at least two benefits accrue.
First, you have lateral stability, a steel plate standing on its side is not very stable as the slightest force would cause the plate to fall over. With sufficient weight applied to the sheet, the slightest lean could lead to bending and collapse.
Another aspect is the effective area the weight is supported by. For the later, you may want to wait for the answer to the other question, but basically, the weight is distributed over more area when supported by steel configured as a 'post' or box. With a sheet of steel on its side and the weight applied edgewise, and assuming a perfectly smooth edge to the steel, the weight is spread over an area equal to the width of the steel times the length of steel edge that is in contact with the weight (also assumed to be flat and perfectly smooth on the contact side). For the 'post' or box configuration, the effective contact area is dependent upon the surface beneath the steel.
A very important assumption made is that your sheet of steel, or same sheet of steel configured as a 'post' or box, is supported on something that doesn't give, such as a concrete floor. If the floor were soft enough that the floor allowed the plate or post to sink instead of collapsing and the sheet or post standing in a perfectly vertical manner, you would observe that both plate and post sink at about the same rate.
References: Halliday & Resnick, Physics, 1966; Kurt Geick's Engineering Formulas, Third Edition, section P6 Shear Strength; Lindeburg, EIT Reference Manual, 8th Edition, p. 40-18, Modes of Beam Failure.
Sidney Chivers, Admin MadSci Network
Additional answer from Justin Roux:
Hello Patrick, and thank you so much for your patience in waiting for my reply.
In your question, what you have hit on is probably the single most important concept in structural engineering - one that we call the 'second moment of area'. But let's go into that straight away since it represents a whole year of University study on its own. There are many ways in which a structural member can fail, be it a column, a beam, a wall, a cable, or whatever. Other than failing in pure tension, it may bend, snap, buckle as a whole, or buckle in a single place causing total collapse. All of these events are dependent upon not just the amount of material the member has in cross-section, but also its shape.
Imagine a plastic ruler. Paint it. Now bend it. On the outside of the bend the paint cracks and shows that the outside edge of the bend is in tension. On the inside of the bend the paint wrinkles to show that the inside edge is in compression. There must be, therefore, a point in the middle of the ruler that is in neither tension or compression - what we call the neutral axis - a point that is not loaded at all.
Now think of an 'I' beam - one of those that are commonly used for steel structured buildings. The 'web', or the vertical section of the 'I' is thinner than the 'flanges', which are the top and bottom sections. Either side of the beam is scooped out to give the 'I' shape since these areas do very little to affect the integrity of the beam. Imagine the beam is painted like our ruler when it bends - all of the work is done by the outside and inside edges and so the 'I' beam is the most effective shape for its job in comparison to its weight. Take the same amount of steel and make a circular rod of the same length and you will find it to be much less effective in the same situation. However, if you bend our 'I' beam sideways, it is easy to see how much less effective it will be because the flanges are no longer on the tension and compression faces. The neutral axis is flipped through 90 degrees - that's why we also have box sections, 'H' sections, channel sections, and the like. It's tough being a structural engineer.
Let's come back to your question then. take a piece of paper and notice how it flops around in your hand. This is because all of the material of the paper is very close to its neutral axis. Now fold it back and forth in a concertina fashion to make a long zig-zag and note that it stands freely and you should be able to stand a coffee cup on it. This is because much of the paper has been moved away from the imaginary neutral axis of the sheet giving it stiffness (the neutral axis is imaginary, a straight line which cannot be zig-zagged). The 'second moment of area' is the summation of all parts of a section with respect to their distance away from each other. It is the definitive property in designing against buckling, compression, bending, and all sorts of other engineering scenarios.
Here is an experiment to finish up with. Take a piece of stiff cardboard, fold it and tape it so that you have a square section beam. Support it at each end and put some weight in the middle. How much can you put on it before it buckles? Now make another from an identical piece of cardboard, but make it rectangular in section. How much does it take before it buckles? It depends which way up you put the rectangle doesn't it? Think about which section has the most material a long way from the neutral axis. The strongest one, yes?
Look all around you for examples of stiffeners that work simply because they provide move material away from the neutral axis. Think of the ribs inside a ship's hull, a piece of corrugated iron roofing, look at a piece of corrugated cardboard, the buttresses in a long brick wall... Notice that the bone in your jaw is taller than it is thick and think about how it is trying to bend when you bite....
Engineering - you can't escape it!
Enjoy,
Justin Roux.
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