Structural Method for Maculay method for engineering, Summaries of Mechanical Engineering

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Deflection of Flexural Members -

Macaulay’s Method

3rd Year

Structural Engineering

Dr. Colin Caprani

Contents

1. Introduction

1.1 General

Macaulay’s Method is a means to find the equation that describes the deflected shape of a beam. From this equation, any deflection of interest can be found.

Before Macaulay’s paper of 1919, the equation for the deflection of beams could not be found in closed form. Different equations for bending moment were used at different locations in the beam.

Macaulay’s Method enables us to write a single equation for bending moment for the full length of the beam. When coupled with the Euler-Bernoulli theory, we can then integrate the expression for bending moment to find the equation for deflection.

Before looking at the deflection of beams, there are some preliminary results needed and these are introduced here.

Some spreadsheet results are presented in these notes; the relevant spreadsheets are available from www.colincaprani.com.

1.2 Background

Euler-Bernoulli Bending Theory

Basic Behaviour Consider a portion of a bending member before and after the application of load:

We can see that the fibres of the material contract on the upper face, and so they must be in compression. Since they lengthen on the lower face they must then be in tension. Thus the stresses vary from compression to tension over the depth of the beam and so at some point through the cross section, there must therefore be material which is neither shortening nor lengthening, and is thus unstressed. This is the neutral axis of the section.

Geometry of Deformation Next we consider the above phenomenon in more detail. Consider a portion of the beam of length dx between planes AG and BH. We are particularly interest in the arbitrary fibre EF a distance y below the neutral axis, CD. Before loading, EF is the same distance as CD. After loading, C’D’ remains the same length as CD , since it is the neutral axis to give:

dx  Rd 

And so the strain in fibre EF is:

Change in length ' ' Original length

E F EF

 EF

But since EF  CD  dx  R d  we have:

 R^ y d ^ R d^ y

R d R

 ^ 

 ^  

Thus:

y

 R

And so strain is distributed linearly across the section. Note that since no constitutive law was used in this derivation, this relationship holds for any form of material behaviour (linearly elastic, plastic etc.).

Linear Elastic Behaviour Next we will consider a specific case of material behaviour linear elasticity for which we know:

E

^ 

And so we have:

y E R

And this gives:

E y  R

This is the equation of a straight line, and so the stress is linearly distributed across the cross section for a linear elastic material subject to bending.

Equilibrium with Applied Moment Lastly, we will consider how these stresses provide resistance to the applied moment and force. Consider the elemental area dA , a distance y from the neutral axis, as shown in the diagram. The force that this area offers is:

dF   dA

And the total longitudinal force on the cross section is:

M 1

EI^  R

Summary Combining the relationships found gives the fundamental expression, sometimes called the Engineers Theory of Bending :

M E

I y R

This expression links stress, moment and geometry of deformation and is thus extremely important.

General Deflection Equation

From the Euler-Bernoulli Theory of Bending, at a point along a beam, we know:

1 M

R^  EI

where:  R is the radius of curvature of the point, and 1 R is the curvature;  M is the bending moment at the point;  E is the elastic modulus;  I is the second moment of area at the point.

We also know that dx  R d  and so 1 R  d  dx. Further, for small displacements,

  tan  dy dx and so:

2 2

1 d y R^  dx

Where y is the deflection at the point, and x is the distance of the point along the beam. Hence, the fundamental equation in finding deflections is:

2 2^ xx

d y M dx^  EI

In which the subscripts show that both M and EI are functions of x and so may change along the length of the beam.

For the free-body diagram A to the cut X 1 (^)  X 1 , (^) M about X 1 (^)  X 1  0 gives:

M x x M x x

For the second cut (^) M about X (^) 2  X 2  0 gives:

M x x x M x x x

So the final equation for the bending moment is:

M  x   ^  40 x  4080^ x ^ x  4  04 ^ xx^  84  portionportion BCAB 

The equations differ by the  80  x  4 term, which only comes into play once we are

beyond B where the point load of 80 kN is.

Going back to our basic formula, to find the deflection we use:

2

d y M^ x^ y M^ x dx dx^ ^ EI ^ ^  EI

But since we have two equations for the bending moment, we will have two different integrations and four constants of integration.

1.3 Discontinuity Functions

Background

This section looks at the mathematics that lies behind Macaulay’s Method. The method relies upon special functions which are quite unlike usual mathematical functions. Whereas usual functions of variables are continuous, these functions have discontinuities. But it is these discontinuities that make them so useful for our purpose. However, because of the discontinuities these functions have to be treated carefully, and we will clearly define how we will use them. There are two types.

Notation

In mathematics, discontinuity functions are usually represented with angled brackets to distinguish them from other types of brackets:

 Usual ordinary brackets:    (^)   (^)   Usual discontinuity brackets: 

However (and this is a big one), we will use square brackets to represent our discontinuity functions. This is because in handwriting they are more easily distinguishable than the angled brackets which can look similar to numbers.

Therefore, we adopt the following convention here:

 Ordinary functions:    

 Discontinuity functions:   

Macaulay Functions

Macaulay functions represent quantities that begin at a point a. Before point a the function has zero value, after point a the function has a defined value. So, for example, point a might be the time at which a light was turned on, and the function then represents the brightness in the room: zero before a and bright after a.

Mathematically:

     

0 when when where 0,1,2,...

n n n

x a F x x a (^) x a x a n

^ 

When the exponent n  0 , we have:

F 0 (^)  x (^)   (^)  x  a (^) ^0  ^01 whenwhen^ xx^^  aa 

This is called the step function, because when it is plotted we have:

Singularity Functions

Singularity functions behave differently to Macaulay functions. They are defined to be zero everywhere except point a. So in the light switch example the singularity function could represent the action of switching on the light.

Mathematically:

   ^0 whenwhen where 1, 2, 3,...

n n F x x a x^ a x a n

   ^ 

^ 

The singularity arises since when n   1 , for example, we have:

F  1  x  ^ x^1  a   ^0 whenwhen^ xx^^  aa

Two singularity functions, very important for us, are:

1. When n   1 , the function represents a unit force at point a :

2. When n   2 , the function represents a unit moment located at point a :

Integration of Discontinuity Functions

These functions can be integrated almost like ordinary functions:

Macaulay functions ( n  0 ):

  ^ ^   ^ 

1 1 0 0 1 i.e. 1

x x (^) n^ n Fn x Fn^ x x a^ x^ a n n

  ^   

Singularity functions ( n  0 ):

  1      ^1 0 0

i.e.

x x (^) n n Fn x Fn x x a x a 

 ^   ^ ^ 