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Solid Mechanics I
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Solid Mechanics I
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C9: Deflections of Beams and Shafts
9.1 Integration Method
- Theory - Example - Question 1 - Question 2
9.2 Discontinuity Functions (Macaulay’s Method)
- Theory - Example - Question 1 - Question 2 - Question 3

C9.2 Discontinuity Functions (Macaulay’s Method)

We saw in Question 2 of Chapter 9.1 that when a new load is introduced along the beam, the moment equation changes and we need a separate elastic curve equation to fully define the beam.

That’s pretty tedious ain’t it? Fortunately, we have what we call discontinuity functions that allow us to have an elastic curve that fully defines the beam, regardless of the number of loadings. How does this work?

We use what we call “Macaulay brackets”, that will only be “activated” when our distance x moves past where the load is applied:


Macaulay's bracket formula

C9.2 Discontinuity Functions (Macaulay’s Method)

We saw in Question 2 of Chapter 9.1 that when a new load is introduced along the beam, the moment equation changes and we need a separate elastic curve equation to fully define the beam.

That’s pretty tedious ain’t it? Fortunately, we have what we call discontinuity functions that allow us to have an elastic curve that fully defines the beam, regardless of the number of loadings. How does this work?

We use what we call “Macaulay brackets”, that will only be “activated” when our distance x moves past where the load is applied:


Macaulay's bracket formula

To illustrate how this works, we’ll show you a quick example. We’ll attempt to get the moment equation using the normal method and Macaulay’s method to show you the difference:


Example of using Macaulay's method and demonstrating its advantage

  • For x < 2m, <x-2> = 0 so M(x) = -Px which is equal to M1(x) of the normal method.
  • For x ≥ 2m, <x-2> = (x-2) so M(x) = -Px-P(x-2) which is equal to M2(x) of the normal method.

See! One moment equation that fully defines the beam! With this moment equation, we can then perform our usual double-integration to get our slope (dν/dx) and deflection (ν). Integration for <x-2>n is the same as how you would do for (x-2)n.

Let’s reinforce our understanding by looking at an example. But before that, here are some useful formulas for the Macaulay brackets.

Formula for different types of loading

Here, we give you the formula for the moment caused by different types of loading, expressed as a function of the Macaulay brackets:


Macaulay's bracket formula for various loadings: point moment; point load; uniform distributed load; triangular distributed load

Let’s look at our example now.

To illustrate how this works, we’ll show you a quick example. We’ll attempt to get the moment equation using the normal method and Macaulay’s method to show you the difference:


Example of using Macaulay's method and demonstrating its advantage

  • For x < 2m, <x-2> = 0 so M(x) = -Px which is equal to M1(x) of the normal method.
  • For x ≥ 2m, <x-2> = (x-2) so M(x) = -Px-P(x-2) which is equal to M2(x) of the normal method.

See! One moment equation that fully defines the beam! With this moment equation, we can then perform our usual double-integration to get our slope (dν/dx) and deflection (ν). Integration for <x-2>n is the same as how you would do for (x-2)n.

Let’s reinforce our understanding by looking at an example. But before that, here are some useful formulas for the Macaulay brackets.

Formula for different types of loading

Here, we give you the formula for the moment caused by different types of loading, expressed as a function of the Macaulay brackets:


Macaulay's bracket formula for various loadings: point moment; point load; uniform distributed load; triangular distributed load

Let’s look at our example now.

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