A friend of mine posted this on their LJ and I have no idea how one would go about solving this. And no, this isn’t homework for either one of us. I think she saw the question in some magazine that had a “tough quiz” or something.
Anyway, I figured that not only might I get it answered, but that someone might explain in English what “modulus of elasticity” and all the rest mean, and why I should care in my everyday life. (That isn’t being facetious, I really am curious).
OK, on to the question:
"There is a steel beam 50 feet long. The entire construction crew (total weight: 6000 pounds) was standing at the center of the beam. Assume the modulus of elasticity is 29,000,000 pounds per square inch, the moment of inertia is 850 inches^4, the beam has simple pin connections at either end, and all loads other than the weight of the crew are disregarded.
How much will the beam deflect? Please round to the nearest tenth of an inch."
This isn’t really a “math geek” question… This is a second-semester mechanical engineering question, although to derive the necessary equations, one needs a pretty strong math background. I wonder what kind of magazine this question was in…
“Real” mechanical engineers, like me, just look up the formula in beam deflection tables in a handbook.
“Modulus of elasticity” aka Young’s Modulus, is the ratio of strain to stress in a linear material. This is a fancy way to say “how much a given material stretches under a given load”. The number 29,000,000 pounds per in[sup]2[/sup] is the typical number for most steels.
Of course, you had to ask on Sunday, when I’m not at the office.
In a set-up like this, the maximum deflection is at the center, and is equal to:
(load * length[sup]3[/sup]) / (48 * Modulus * Moment of Inertia).
For a simply supported beam (meaning that the ends are constrained in translation, but that there is no resistance to an applied moment):
y max = -§(L)^3/(48)(E)(I)
where P is the applied load, L is the beam length, E is the modulus of elasticity, and I is the cross-sectional moment of inertia of the beam.
More generally, if you want to find the deflection of a point not at the center:
y = [-§(x)/48(E)(I)](3(L)^2 - 4(x)^2), where x is the distance from the end.
Incidentally, you mention that the entire construction crew is at the center of the beam, and that this load is 6000 lbs. Assuming an average weight of 200 lbs per man (construction workers being big guys), this means that 30 guys are standing on each other’s shoulders at the center of the beam. Impressive.
Since this isn’t a homework problem, I’ll be studying this thread in some detail. One can never be too well prepared. If the fellow across the street quizes me about modulus of elasticity, I’ll be ready!
Trust me, I’m 28, I work for a blood agency, and I am not an engineer. If this was homework, the GPA I had in university would probably have been negative.
Here is a really neat page all about beam stiffness (with cool pictures and videos) for those who’d like to do more reading on the subject. And here’s a little bit all about Young’s modulus (i.e. the modulus of elasticity). It’s a British page so all the units are metric tho’. (hmph, wusses. Here in Texas we do ALL our calcs using Standard units…)
In a standard 2-D Cartesian coordinate system (i.e. X and Y axes), X increases to the right, while Y increases upwards. A beam with weight on it deflects downward (i.e. in the negative Y direction), so the negative sign is necessary to return a positive value for deflection.
As far as the +/- signs goes, we call deflection downward, “positive” deflection and upward deflection is “negative” deflection. (A good way to remember this is that positive deflection is a smiley face and negative deflection is a frowny face. :mad: ) Mostly 'cause you get downward deflection way more than you do negative deflection. But units don’t really matter as long as you know which way is up and down. I tend to always do my calcs the opposite.
