Macaulay’s Method & Poisson’s Ratio

Author: Leicester CollegeDate created:Date revised: 2009

Abstract: This section examines a method of analysing the bending moment and shear force distribution along a uniformly loaded, simply supported beam. The method used is Macaulay’s and the stages involved in this method are described in detail. The method also determines the position of the maximum bending moment, the slope and deflection at this point.

Poisson’s ratio is the ratio between lateral and axial strain in a directly loaded system. The derivation of standard equations and their further development is shown.

ContentsMacaulay’s method for deriving bending moments............................................................2Lets look at a typical problem.............................................................................................3Determine R1 and R2.........................................................................................................4POISSON’S RATIO............................................................................................................8Credits.............................................................................................................................. 10

These files support the Edexcel HN unit – Mechanical Principles (NQF L4)

File name Unit outcome

Key words

Macaulay’s beams

Do not use this file.

Balancing 4.1 Beams, balancing, rotation, mass, stress, shafts

Flywheels 4.3,4.4 Kinetic, battery, flywheel, inertia, energy storage, energy

Macaulay’s method

2.1,2.2,2.3 Beams, stress, loading, UDL, slope, deflection, method

Poisson 1.1, 1.2 Poisson ratio, equations, axial, lateral, strain, stress

Selecting beams 2.2 Beams, columns, selection, slenderness, stress, section modulus

For further information regarding unit outcomes go to Edexcel.org.uk/ HN/ Engineering / Specifications

© Leicester College 2009 This work is licensed under a Creative Commons Attribution 2.0 License.

Macaulay’s Method

Macaulay’s method for deriving bending moments

For a beam with a number of concentrated loads, separate bending moment equations have had to be written for each part of the beam between adjacent loads.

Integration of each expression then gives the slope and deflection relationships for each part of the beam.

Each slope and deflection relationship includes constants of integration and these have to be determined for each part of the beam. The constants of integration are then found by equating slopes and deflections given by the expressions on each side of each load.

This can be very laborious if there are many loads. A much less laborious way of tackling the problem is to use the method known as Macaulay’s method.

Macaulay’s method involves writing just one equation for the bending moment of the entire beam and enables boundary conditions for any part of the beam to be used to obtain the constants of integration.

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Macaulay’s Method

Lets look at a typical problem

A simply supported beam of length 3.0m has concentrated loads of 40 kN at 1.0m from one end and 60kN at 2.0 m from the same end. (Shown below)

a. What is the deflection and slope of the beam under the 40kN load?

b. What is the maximum deflection and at what position along the beam does it occur?

The beam has a value of

40kN 60kN X

R1 R2

1m

2m

x X

3M

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Macaulay’s Method

Determine R1 and R2

Taking moments about R1 CWM = ACWM

(40 x 1) +( 60 x 2) = R2 x 3

and

Considering the bending moment about section XX due to the forces to the left of the section, then:

(40 and 60 are loads on the beam)

( )

Integrating this expression gives:

( Beam slope)

Integrating again gives:

- - (Beam deflection)

When x = 0 we have y = 0, B = 0. For this example the following figures are known

NOTE:

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=DeflectionSlope of the

graph

Macaulay’s Method

When we have . Substitute for

-

Hence This becomes:

And,

- -

The deflection at is:

Note; The 10 is included because all the forces have until now been in kN..

The term is zero because it has a negative value when The term is also zero. .

The slope of the beam at is given in a previous equation as:

Thus the slope is rad.

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The 2 comes from in this case therefore

A is a constant, which once calculated is carried through the rest of the calculations

Macaulay’s Method

Maximum deflection occurs when Using the Integrated expression:

By inspection of the problem it looks likely that the maximum deflection will occur between the two load positions. Hence the solution of the above equation will be for x having a value between 1 and 2m. With x having such a value the bracket will be zero and the bracket can be written as .

.

Every term has been multiplied by 2, to remove the fractions. Therefore to keep the equation balanced.

Solve the brackets first:

The roots of a quadratic equation of the form are given.

We ignore the negative value

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Re-introducing the rest of the equation

Macaulay’s Method

=1.527m

Now put the value of x back into the Macaulay expressions for the beam. We get;

1.527m (only the positive value has any significance).

The deflection at this value of is obtained using equation:

= -5

= -

=

Therefore

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Macaulay’s Method

Poisson’s Ratio

The ratio between LATERAL and AXIAL STRAIN

AXIAL STRAIN (X-X)

LATERAL STRAIN ( Y – Y)

Given the greek letter -

Can be found by the ratio lateral strain / axial strain – it is dimensionless

When a member is in tension - The lateral strain represents a decrease in width ( -ve lateral strain) The axial strain represents an elongation (+ve axial strain)

The reverse is true when the member is in compression

Strain due to stress acting along X-X direction = x / E

Strain due to stress acting along the Y-Y direction = - y / E

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Macaulay’s Method

Combining strain in the X-X direction we have;

1 x = x - y E E

Combining strain in the X-X direction we have;

2 y = y - x E E

WE can go further by using simultaneous equations and by substitution as shown below;

3 x = E (x + y )(1 - 2)

AND

4 y = E (y + x )(1 - 2)

Equations 1 to 4 can be used to solve problems involving 3 dimensional stress and strain

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Macaulay’s Method

Credits

This resource was created Leicester College and released as an open educational resource through the Open Engineering Resources project of the Higher Education Academy Engineering Subject Centre. The Open Engineering Resources project was funded by HEFCE and part of the JISC/HE Academy UKOER programme.

© 2009 Leicester College

This work is licensed under a Creative Commons Attribution 2.0 License.

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