Civil Engineering

**Calculus in Civil Engineering**

In all aspects of engineering, when confronted with a problem, one usually defines the problem with a model using mathematical equations describing the relationships of the different aspects of the problem, usually through calculus.

How Calculus is Used in Engineering

Basic things that occur all the time in engineering are rates of change with respect to time, or space of such variables as heat, wave, gas, electric current, electromagnetic fields, conductivity, vibrations in solids like bridges and buildings, and many others.

Calculus in Engineering Cont.

The basic problems seek to maximize or minimize a quantity (such cost or profit, or a surface area of some object, or the distance a projectile can achieve). You’ll also find calculus in probability and that in turn is used in engineering when you can't get a model for your problem but instead you have many data items from which you extract relationships. These problems use calculus (derivatives and integrals) to be formulated and then solved

Basic Use of Calculus in Engineering

The simple beam formula to calculate the stress in a beam with various forms of end attachment from fixed (buried in concrete for example) to pinned like the attachment points on many bridge supports and with various loads from distributed loads to point loads. The derivation of each comes from a combination of algebra and calculus. You can derive the shear stress distribution from algebra and get the moment distribution by integrating the shear stress

EXAMPLE

Identify which dimension is changing with respect to another dimension and determine the independent variable.In our triangular distributed loading case the constant changes linearly with distance

Write the differential dF, as a product of f(x) and an infinitely small change in the independent variable x, dx.

A structural beam in Civil Engineering is designed to support load over a span. A specific type of beam is a cantilever beam which is beam with one end completely fixed so that it can not move.

You must determine the form of the functional relationship between the interacting conditions. This is synonymous with writing the equation, where every dimension is assumed to be a constant. In Civil Engineering, a distributed load is expressed as a constant in units of load per unit distance. For the case where the loading is a uniform rectangular distributed load over the span

Integrate both sides of the function from some value x=a to x=b to calculate the net change in the dependent dimension F.In the triangular loading case, c(d) just equals some constant, c multiplied by d. We could certainly have parabolic and even exponential distributed loading functions. But for triangular loading, we just need to replace c(d) with constant times d