How to add area moments of inertia for varying cross sections

In summary, the area moment of inertia for a shaft overhang can be calculated using a equation from papers or by assuming the entire shaft has a smaller diameter. It can be complicated, but often a more practical approach.
  • #1
friscoboya
6
1
Hello

I am doing a shaft design layout for a spindle to minimize deflection and I have to calculate the area moment of inertia for a shaft overhang.The shaft overhang is the distance from a bearing to the end of the shaft. Here is a picture detailing the location of the overhang. The black square would be the shaft overhang. Would I calculate the area moment of inertia for d1 and d2, add them together and average them? Or what is the correct method to add area moment of inertias? The equation for hollow cylinder area moment of inertia is.
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Thanks
 

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  • #2
It depends what you are going to do with the area moment of inertia. How are you going to use it?
 
  • #3
haruspex said:
It depends what you are going to do with the area moment of inertia. How are you going to use it?
So the shaft design would be for a CNC machine primarily used for milling. The area moment of inertia would be put into an equation I found from papers. Area moment of inertia is used for deflection calculation which is what I am focusing on. Yeah what's confusing me is the different diameters, it would be simple if the diameters were constant throughout the shaft overhang (black box).
 
  • #4
If you have access to Mechanical Engineering Design by Shigley-Mischke-Budynas, it shows you how the deflection should be calculated. You have to perform some integration. I find it quite complicated.
.
However, if you want to calculate a "limiting," or maximum possible deflection, you can assume the entire shaft has diameter d1 - the smaller diameter. This is often a more practical approach, but of course it depends on your application.

Another website is-
https://www.machinedesign.com/archive/calculate-deflection-stepped-shafts-easy-way

Still not easy in my opinion.
 
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Likes berkeman
  • #5
friscoboya said:
deflection calculation
Deflection under what load, applied where?
 

What is area moment of inertia and why is it important?

Area moment of inertia, also known as second moment of area, is a property of a cross section that describes its resistance to bending. It is important in structural engineering and mechanics because it helps determine the structural strength and stability of an object.

What is the formula for calculating area moment of inertia?

The formula for calculating area moment of inertia varies depending on the shape of the cross section. For example, for a rectangular cross section, the formula is 1/12*b*h^3, where b is the base and h is the height. For a circular cross section, the formula is π*r^4/4, where r is the radius. It is important to consult a reference table or use a software program to accurately calculate the area moment of inertia for a specific cross section.

How do you add area moments of inertia for varying cross sections?

To add area moments of inertia for varying cross sections, you can use the parallel axis theorem. This theorem states that the area moment of inertia of a shape can be calculated by adding the area moment of inertia of the shape at its centroid to the product of its area and the square of the distance between the centroid and the new axis. This method is useful for calculating the area moment of inertia for complex shapes made up of multiple smaller shapes.

What factors can affect the area moment of inertia of a cross section?

The area moment of inertia of a cross section can be affected by its shape, size, and material properties. For example, a larger cross section will have a larger area moment of inertia compared to a smaller one. Additionally, the material properties of the cross section, such as its density and modulus of elasticity, can also impact the area moment of inertia.

Why is it important to accurately calculate the area moment of inertia for a structural design?

Accurately calculating the area moment of inertia is crucial for ensuring the safety and stability of a structural design. An incorrect or underestimated value can lead to structural failure or collapse. Additionally, accurate area moment of inertia calculations can help optimize the design and reduce unnecessary material usage and costs.

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