Understanding Calculus

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  Preface
  1. Why Study
  Calculus
  2. Numbers
  3. Functions
  4. The Derivative
  5. Differentiation
  6. Applications
  7. Free Falling
  Motion
  8. Understanding
  Derivative
  9. Derivative
  Approximations
  10. Integration
  Theory
  11. Understanding
  Integration
  12. Differentials

  Inverse Functions
  Exponents
  Exponential
  Functions
  Applications of
  Exponential
  Functions
  Sine and Cosine
  Function
  Sine Function
  Sine Function -
  Differentiation and
  Integration
  Oscillatory Motion
  Mean Value
  Theorem
  Taylor Series
  More Taylor Series
  Integration
  Techniques

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CHAPTER 11

Chapter 11 - Understanding Integration

Section 11.4 - Engineering Applications

Cantilever Beam

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. A picture is shown below:

If a load/force is applied at the end of the beam, the beam will bend downwards. Try this with a ruler in your hand to see how it bends. When a load is applied at the end the beam will experience the highest stress at the end where it is fixed. The stresses it experiences are proportional to how high the load is and how far the load is from the fixed end. In engineering, the term 'Bending Moment' is calculated from the product of the load multiplied by the distance . The greater the bending moment , the greater the chance it will break . Therefore:

Bending Moment = Load X Distance

In the example below of a single load at the end of the beam , the bending moment at the fixed end would be Load times the distance, d.

How would we find the bending moment for the case shown below?

Here a distributed load is increasing along the span of the beam with a triangular distribution. Triangular distributed loads are found commonly when a liquid is exerting pressure on a wall with the pressure increasing with the depth . An example would be the walls of your swimming pool. As the water goes deeper, they exert a linearly increasing pressure on the walls of the pool.

Since our distributed load is changing with the span of the beam, we need to apply our systematic approach to integration to solve the problem.

Step 1 - 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 as shown below:

We can write:

Load = some constant * distance

The constant is in units of load/distance and its value depends on the magnitude of the distributed load. Therefore the constant multiplied by distance equals the total load acting over that distance.

Step 2 - 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 or:

Step 3 - Write the differential dF, as a product of f(x) and an infinitely small change in the independent variable x, dx.

Substituting back into our equation for bending moment:

Step 4 - 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 or:

Questions

Determine the bending moment for this parabolic type of loading:

Next section -> Section 11.5 - Human Applications

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