# Macaulay Method

Describes the Macaulay Method for calculating the deflection of Beams.

**Contents**

## Macaulay's Method - Introduction

**Definition****Macaulay'**

**s method**

**(The double integration method)**is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. Use of Macaulay's technique is very convenient for cases of discontinuous and/or discrete loading.

## Concentrated Loads

A

**beam**is a horizontal structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a**bending moment**.
Measuring from one end write down an expression for the

**Bending Moment**in the last section of the beam enclosing all less than in square brackets, i.e. Subject to the condition that all terms for which the quantities in the square brackets are negative are omitted ( i.e. given a value of zero), this equation may be said to represent the bending moment for all values of . If is less than then both the last two terms are omitted and so on.The brackets are integrated as a whole, i.e.
And,
By doing so it can be shown that the constants of integration are common to all sections of the beam, e.g. if
And,
And,
Now as the slope and deflection values must correspond (i.e.) at from which it can be seen that ' and '. The values of and are found as before ( Part 1).

## Uniformly Distributed Loads.

Supposing that a uniformly distributed load is applied from a distance to a distance measured from one end . Then in order to obtain an expression for the Bending Moment at a distance from the end, which will apply for all values of , it is necessary to continue the loading up to the section at , compensating this with an equal negative load from to (see diagram) Hence, Each length of the loading acts at its centre of gravity. The square brackets are interpreted as before. For but , omit and: Hence, This is clearly correct. The remaining steps of integration are the evaluation of the Constants, and proceeds as before.## Concentrated Bending Moment

It is possible to write:Example:

[imperial]

##### Example - Example 1

Problem

A simply supported beam of length carries a load at a distance from one end and from the one .
Find the

**position**and**magnitude**of the maximum deflection and**show**that the position is always approximately within of the centre.Workings

The maximum deflection (i.e.) zero slope will occur on the length a since
Taking the axes as shown in the diagram.
Integrating
And Integrating again gives:-
At and and therefore .
At ,
From which:
We need to find the value of when is zero. Using equation
(1) and omitting ( We can do this because at zero slope when )
Hence,
At the point of maximum deflection. To find the value of this deflection substitute into equation (2)
Re-writing Equation (3) to obtain the value of when gives:
This gives the distance from the centre of the beam to be :
Distance from Centre =
Which has a maximum value of

Solution

- The
**distance**from the centre of the beam is