Mathematics in the Applied Sciences
Date: 02/24/97 at 09:53:24 From: eric huggard Subject: Civil Engineering I would like to know some specific concepts of math and physics that are commonly used by civil engineers.
Date: 03/02/97 at 23:50:52 From: Doctor Barney Subject: Re: Civil Engineering Arithmetic, algebra, geometry, calculus, differential equations, complex analysis, probability and statistics. Civil engineers use all of these forms of math and many, many other forms as well. This is a great question! I could write about this for days, but that would probably give you more information than you need. Actually, there are some common misconceptions about the use of mathematics in the applied sciences, like engineering: most engineers actually spend a very small portion of their time carrying out mathematical calculation. But that does not mean that mathematics is not important to engineering. In fact, mathematics is indispensable. Let me expand my answer into four different areas, because I have thought of four different ways in which civil engineers (and other engineers) use mathematics. I'm sure there are many others. These are just the first four that come to mind: 1. To help them understand the chemistry and physics fundamental to the construction of civil engineering projects; 2. To carry out the technical calculations necessary to plan a construction project; 3. To help them with modeling and simulations to predict the behavior of structures before they are actually built; and 4. To help them with business decisions and other 'non-technical' aspects of their jobs. 1. To understand chemistry and physics. Civil engineers are frequently concerned with two fundamental technical questions: The first question, "How strong is this material?" can be answered through material science, which is a branch of chemistry. The second, "How strong will this part need to be?" is usually answered by statics and dynamics, which are branches of physics. Now, long before the engineer ever enrolled in his or her first engineering class, he or she probably had a good understanding of the basic principals of chemistry and physics (from a high school science class, perhaps) and virtually all of these fundamental scientific principals are described, analyzed, proven, and predicted through the use of one or more form of mathematics. And let me stress that the math is not only needed to pass those classes in the first place, but even as the engineers (and scientists) apply scientific principals in their respective fields, they continue to use the mathematics which define these principals. For example, suppose you were designing a concrete freeway over-pass. One of the things you would need to understand is how the concrete cures as a function of time. How long does one 'batch' need to cure before construction can continue? When does it need to be tested? How long until the bridge can be opened to regular traffic? (Concrete appears to 'dry' in a day or two, but actually does not reach its full strength for over a month!) Well, it turns out that the strength of the concrete as a function of time is described by an equation of the form S=c(1-e^-kt) where S is the strength at time t and c and k are constants specific to the type of concrete you are using. (How strong is the concrete at time t=0? How strong is it as t approaches infinity? How long will it take to get to half of its final strength?) Now, the important point here is that the engineer is not the person who calculates when the concrete will reach a point when it can be used without concern to its cure being complete; that is a job for a chemist or a material scientist at the factory where the concrete is made, who will make this calculation based on the type and amount of chemicals used in each particular type of concrete. But nevertheless, the engineers need to understand the properties of the materials they are working with, and in this case those properties are described by an exponential equation. 2. To carry out the technical calculations necessary to plan a construction project. This is the more obvious aspect of the engineers' jobs that everyone tends to think about: "If a circular concrete support column for a particular bridge needs to be able to hold up 28 tons and the concrete being used can support 4,575 pounds per square foot, what diameter does the column need to be? If the column is 22 feet tall, how many cubic yards of concrete will be needed to make the column?" ...that sort of thing. Engineers really do carry out some of these types of calculations, but the mathematics textbooks tend to simplify the problems quite a bit (which is not necessarily a bad thing; this type of problem really does illustrate how specific mathematical concepts are applied to other fields.) For engineers in the world, the support column mentioned above might also need to hold 28 tons after it is eighty years old, has been crashed into by three cars and a truck, and has survived four earthquakes and one flood. Now how would you calculate the diameter? By the way, don't forget that the concrete at the bottom of the column also needs to hold the weight of the column itself in addition to the 28 ton 'load'. See what I mean? In the real world the problems are always harder. 3. To help them with modeling and simulations. These days, before anyone builds anything that costs very much money, they usually develop some type of mathematical model, and analyze it using a computer. A mathematical model is a set of equations that describe what we think would happen to something if we really built it the way that it is described in the model. You may have seen a computerized 'stick drawing' of the space shuttle on the TV news, where they show the shuttle turning from side-to-side and it looks like it's three-dimensional. Well, that 'stick-drawing' is a graphical representation of a mathematical model. And NASA did not make that computer model just so they could see what the shuttle would look like, they made it so they could learn as much as they could about it before it was built. Things like how it would fly, how strong certain parts would have to be, and how hot it would get on re-entry. The equations that describe what happens to any one part are not very difficult to write down, but solving the equations for all of these parts at the same time would be extremely time-consuming. That is why this type of modeling is almost always done using a computer. The computer solves the mathematical equations, the computer scientists programmed the equations into the computer, but (pay attention here) the engineers had to write down the equations in the first place. For that, they needed to know a lot of mathematics, especially calculus and differential equations. Many, many other engineering projects are modeled using mathematics, although the model may not be as complicated as the one for the space shuttle. Engineering projects like bridges and buildings, which you may not hear about on the TV news but which are also very, very important, are usually simulated or modeled before they are built. 4. To help them with business decisions and other 'non-technical' aspects of their jobs. I know that this is probably not the answer you were looking for, but I feel obliged to discuss it, for completeness. The fact is, very little engineering ever occurs without someone spending some money, and in most cases, a whole lot of money. And because of that, many engineers spend a significant portion of their time (almost half?) concerned with the business aspects of their projects, as opposed to the technical aspects. For example, before you build a bridge, you had better know how much it will cost, how long it will take, and at what point in the project you will need the money. What type of equipment you will need and for how long. Exactly what materials you will need, and when, and all that sort of thing. You would also need to figure out the cheapest kind of bridge you could build. If one bridge is cheaper but takes longer to build, would that be better or worse? What is the environmental impact of building a certain type of bridge in a certain location? All of these 'business' questions can be very complicated issues, and different forms of mathematics can help people answer them: mathematics like algebra, probability, statistics, even calculus. This brings up the point that engineers are not the only professionals who need a strong mathematical background. Many professionals are routinely faced with this same type of 'business' problem that can be solved using mathematics. Professionals such as business executives, lawyers, accountants, business analysts, government officials, financial planners, etc. The point is, although civil engineers do have technical aspects of their jobs which require specialized mathematical skills, they also have many other aspects of their jobs which require the same mathematical skills that most other professionals need, which should not be underestimated. -Doctor Barney, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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