Macaulay's Equation: Calculate I & Learn E,I Properties

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In summary, I is the mass moment of inertia. It is a measurement of how far away from the axis of bending the mass is located. Macauley's Method (or the 'Method of Singularity Functions') is a neat way of doing manual calculations for beams.
  • #1
bracey
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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?
 
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  • #2
I have never heard of this, you said beam, so perhaps the Mechanical engineers will know about it.
 
  • #3
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:

[tex]\int{X^2 + Y^2}dm[/tex]

This is a nasty calculation for anything other than simple shapes, and is typically looked up from a book or calculated by computer.
 
  • #4
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 It's been largely superseded by the finite element method now that computers are commonplace.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 engineering reference book.
 
  • #5
bracey said:
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?


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
 

1. What is Macaulay's equation and how is it used in science?

Macaulay's equation is a mathematical formula used to calculate the moment of inertia (I) and other properties of a beam or structure. It takes into account the shape, size, and distribution of mass of the object to determine its resistance to bending. It is commonly used in structural engineering and mechanics to analyze and design structures.

2. What are the different parts of Macaulay's equation?

Macaulay's equation consists of two main parts: the bending moment (M) and the bending moment diagram (BMI). The bending moment is the external force applied to the beam, while the bending moment diagram is a graph showing the variation of bending moment along the length of the beam. The equation also includes the distance (x) from the point of interest to the point of zero bending moment and the length (L) of the beam.

3. How do you calculate the moment of inertia using Macaulay's equation?

The moment of inertia (I) is calculated by breaking the beam into smaller segments and integrating each segment using Macaulay's equation. The formula for calculating the moment of inertia is I = ∫Mx dx, where M is the bending moment and x is the distance from the point of interest to the point of zero bending moment. This integral is then evaluated for each segment and summed up to get the total moment of inertia for the entire beam.

4. Can Macaulay's equation be used for all types of beams?

Yes, Macaulay's equation can be used for all types of beams, including straight beams, curved beams, and beams with varying cross-sections. It is a universal equation that takes into account the shape and size of the beam, making it applicable for all types of structures.

5. How accurate is Macaulay's equation in real-world applications?

Macaulay's equation is a widely accepted and accurate method for calculating the moment of inertia and other properties of beams. However, its accuracy may be affected by factors such as material properties, boundary conditions, and simplifications made in the analysis. In real-world applications, it is recommended to use computer software or perform physical testing for more precise results.

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