# How to calculate the stiffness of an arch?

1. Sep 27, 2010

### DyslexicHobo

Hi all,

I'm trying to design an arch for a project (will be about 4" long). This arch will be experiencing a load on its vertex, and I need to make sure that the arch will not deform very much.

What is a good way to model the stiffness of an arch like this? As far as I know, there is no fundamental solid mechanics equations that I can use for this, and I don't know how to use any finite elements software very well.

Any tips on how to model this?

Thanks.

Edit: Link to sketch of arch. I'm looking for the simplest first-order estimate that would be more accurate than modeling it as just a square simply supported beam.

Last edited: Sep 27, 2010
2. Sep 27, 2010

### Studiot

The strength and stiffness of an arch depends upon the strength and stiffness of its abutments.

Arch analysis does not have a closed form solution, like beam analysis.

So you will need to provide more information for more help - there is nothing like a good sketch.

3. Sep 27, 2010

### DyslexicHobo

Here is a general sketch of how I was planning on designing the beam.

The dimensions are not set in stone, and I realize the current geometry is far too complicated to model mathematically. I'm trying to learn how to use COMSOL, but until then I'd like at least a first-order estimate.

Thank you.

http://yfrog.com/nabeamsketchrev01ap

Picture: http://yfrog.com/nabeamsketchrev01ap

4. Sep 27, 2010

### Studiot

Is this a beam with a curved soffit or is it an arch?

In other words how is is supported?

On the 0.5 x 0.25 pads on the bottom at the ends?

If it is really just a curved beam there are some references for curved beams here

You could apply the Wilson and Quereau correction factors to a standard beam analysis.

Last edited: Sep 27, 2010
5. Sep 27, 2010

### DyslexicHobo

I'd like to model it as a 2-dimensional analysis if possible. If that's possible, one end is going to be constrained in all axes (x and y, as well as moments about x and y). The other end will be constrained in only the y-direction, but the moments will be constrained in both directions simply due to the geometry I chose (flat surface instead of a "pin").

All of the analysis in my solid mechanics book that I found shows "simply supported" beams (one end is a "pin" connection that constrains movement translationally, but does not provide reactionary moments. the other end is a "roller" which constrains movement translationally in only the y-direction).

Also, would estimating the second moment of area be useful here?

Bottom line: I'm looking for a simple first-order approximation of beam stiffness. How would I calculate the stiffness of an arch with a rectangular cross-section? I feel like that would be the place to start.

Last edited: Sep 27, 2010
6. Sep 27, 2010

### DyslexicHobo

So what I think I'm going to do is try to find the second moment of area and plug it into the equation: P/delta = (48*E*I)/L^3

Where P/delta would be the stiffness. Now I just need to figure out how to find the second moment of area easily.

7. Sep 28, 2010

### Studiot

An arch is a compression structure that does not carry moment.
It converts a horizontal load to horizontal and vertical force reactions at its abutments. Because there are both reactions at both abutments it is statically indeterminate.

But, from what you have said, you are considering a restrained or constrained (curved) beam.
This has vertical reactions and moments at its abutments, but not horizontal ones.

The appropriate formula for such a beam with a single central load is

$$\frac{P}{\delta } = \frac{{192EI}}{{{L^3}}}$$

You should look this up if you do not know it.

Your beam is quite close to flat so the correction factor, K for the curvature will be close to unity.