How can I accurately determine the stiffness of a joint using polynomial curves?

[RobHH]

Hi all,

I'm hoping someone can help me out with this problem. I am following an example of a previous report to analyse some test results, but am having trouble with this step. The tests plot moment-rotation curves of a horizontal member fixed into a vertical member. You run 5 tests, and for each test, fit a polynomial curve to the results. Then, it seems, you produce a single 'mean' curve and use this to derive the stiffness of the joint in rotation.

The problem i have is that using the results in the previous report, I cannot get an 'average' which matches that listed in the report. On the attached excel I have listed out the co-efficients for 2 lots of 5 tests, and showed the co-effecients which are listed in the report as the average. You'll see that I can't get them to match! I've tried mean, absolute and reciprocal averages all to no luck. The fact that a constant (x0) appears in the 'average' equation makes me thing I'm misunderstanding something. Also on the excel is a plot from one test which might help to illustrate.

Sorry if I've waffled, I just want to make sure I've explained properly! Any pointers would be greatly appreciated.

 

[Stephen Tashi]

The tests plot moment-rotation curves of a horizontal member fixed into a vertical member. You run 5 tests, and for each test, fit a polynomial curve to the results.

Is there a theoretical prediction for the curve if stiffness is known? For example, if stiffness is a constant $k$ then should the curve theoretically be $k = \frac{M}{\theta}$ ?

 

[Dr. Courtney]

Averaging polynomial fits is only meaningful if the range of input values is the same. This is because polynomial fits work well within the input range of the data, but quickly diverge outside the range of the data.

When we've done this in the past, we simply make a table of values on a small step size over the range of all the polynomials. The table for each best-fit polynomial might be a row in a spreadsheet. The average of all the polynomials (rows) can then be computed in an additional row. Finally, a polynomial fit can be done on the row that is the average, creating a best fit polynomial for the average.

 

[RobHH]

Stiffness is non-linear, so there is no formula I can use..

Dr Courtney, I think I see what you are saying... Tabulate the results of each equation (ie for rotation, M=) then average the actual results, plot this average and fit a polynomial to this? If I'm reading you right, I think that could work!