# Macaulays equation

1. Dec 27, 2003

### bracey

Dont know if this is the right place to post this, so let me know if it isnt!!

I have just started studying macaulays equation and for every question i have done i have always been given the values of E and I. I know what E is and i was just wondering what I is and how it can be calculated. Is it something to do with the area of the beam, is it a geometrical property or a material property???

2. Dec 27, 2003

### Integral

Staff Emeritus
I have never heard of this, you said beam, so perhaps the Mechanical engineers will know about it.

3. Dec 28, 2003

### enigma

Staff Emeritus
I'm not familiar with the specific method you've described (maybe I am, I just don't recognize the name).

I is the mass moment of inertia. It is a measurement of how far away from the axis of bending the mass is located.

If your beam runs in the 'Z' direction, then your mass moment is calculated like this:

$$\int{X^2 + Y^2}dm$$

This is a nasty calculation for anything other than simple shapes, and is typically looked up from a book or calculated by computer.

4. Jan 23, 2004

### rdt2

Macauley's Method (or the 'Method of Singularity Functions') is a neat way of doing manual calculations for beams. It's simple to perform but the underlying mathematical concept is quite sophisticated. It's been largely superseded by the finite element method now that computers are commonplace.

For static beam bending problems, I is the 'second moment of area':

integral (y^2 dA)

where y is the distance of the (infinitesimal) area dA from the neutral axis. This is analogous to the 'second moment of mass':

integral (y^2 dm)

used in dynamics. Unfortunately, both are given the symbol 'I' and both are sometimes called the 'moment of inertia'.

'I' can be calculated from first principles but the simple formulas for the most common cross-sections are usually picked from an enginering reference book.

5. Feb 7, 2008

### euan_cpl

I is the second moment of area for the beam - cross sectional beam Ixx = bd^3/12 where b= breadth d=depth when the cross section is symetrical for unsymetrical beams it a little harder to explain on here contact me with your email address and i will send you an attachment with my own working