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### Mathematics in the Applied Sciences

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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.
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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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