Maximum Response of an Elasto-Plastic SDOF to Rectangular Load Pulses

[netoprax]

I have been working on a task in which we are to obtain the maximum response of an elasto-perfectly plastic SDOF.  The main reference is: Introduction to Structural Dynamics, by John M. Biggs, 1964. In the second chapter, they provide these tables with the maximum response of a SDOF to different types of load pulses (rectangular, triangular, etc Figs 2.23 to 2.26). See the figure for a rectangular load pulse:

as you can see from the graph, the maximum response is represented by the ductility factor Mu, ratio of the maximum displacement to the elastic displacement of the system. Mu depends only on the ratio between the duration of the pulse (td) and the natural period (T) as well as the maximum spring force (Rm) and the maximum load (F1).

In my case, since we are conducting a sensitivity analysis, it is not very practical for us to be retrieving the results from these graphs for every analysis we do.

Therefore, I proceeded with writing my mathcad file that obtain the complete response of a SDOF system so i could obtain the maximum response analytically.

However, my problem is, in trying to compare my results with this graph, I observed that the results dont match. So I took a step back and compared the results of the graph with that of an example in the Book itself:

other input parameters are:

This example is a constant force (it doesnt have a td like the graph/figure provided). I assume that for a large td/T, the results would be the same (if not, I would like to understand why).

In this case, I look at the table for the largest value of td/T and for Rm/F1 of 1.51, which should be somewhere close to 2.0 (marked in the first figure in this post).

However, the result in the book gives ym/yel as 0.806/0.543 = 1.48.

From 2 to 1.5 is a big difference, and using different values of td/T in my analytical solution of the complete response of an elasto-plastic SDOF problem under a constant pulse load, the results differ even more

My guess is that I am not using the right values either of displacement or something else when using these tables. Anyone have familiarity with this approach can explain why this difference in the results?

[Werner_E]

I definitely can't help with the interpretation of the books diagrams and comparison with your calculations as I have no experience in your field of work.

Just out of curiosity: Is this question somehow related to your unsolved thread here: Re: How to define a subroutine or an alternative... ?

[netoprax]

It's related yes, I forgot to accept the answer to that question, thanks for the reminder

and indeed, this forum might not be the best place to ask this question. I am just out of ideas/places to ask this question and discuss

I tried other forums but no luck yet. Apologies if the question is not suitable here.

[Werner_E]

I guess we have structural engineers here as well, so lets hope you have luck that somebody can shed light on this even though its not directly Mathcad or Prime related.

[vnamboodheri]

Hello @netoprax,

It looks like you have some responses from some community members. If any of these replies helped you solve your question please mark the appropriate reply as the Accepted Solution. 

Of course, if you have more to share on your issue, please let the Community know so other community members can continue to help you.

Thanks,
Vivek N.
Community Moderation Team.

[Fred_Kohlhepp]

It took a while to get Express to work it's way thru the equations, then it produced a graph fairly close to your example.

I suspect that the difference has to do with the return, most material would show some hysteresis as you came out of plastic deformation..

Place to start--I'll keep hacking.

[BillWadsworth · Accepted solution]

Just noticed this item. This happens to be an example I used when I wrote this sheet in 2009. I have enlarged the displacement graph so, using trace, you can read off values for ym and yel which agree with the book example. 

[Werner_E]

I still have no expertise in this field of work, but wouldn't the "max" function do a better job than using 'trace'?

But the OP seems to have lost interest anyway and we still don' know which version of Mathcad or Prime he is using.

[BillWadsworth]

Absolutely, Werner.  I suppose using the graph comes more naturally to the engineer's brain. It is easier to find the info already there in the diagram, rather than writing yet another formula, albeit a short one! 

[Werner_E]

Agreed on.

But from another thread of the same poster I remember that he is using Prime 9 and this means that he can't use the convenient trace facility we had in real Mathcad. Its one of the many features missing in Prime.

[netoprax]

Thank you very much for sharing this! I ended up being able to write a mathcad file for it that replicates the book's graph but only for pulse load with a duration so yours will be useful for any type of load. thanks again!

[BillWadsworth]

Glad you found it useful. 

I wrote another version which will handle an irregular loading regime which is not easily formulated. This takes a sequence of load measurements, fits a curve to them and then produces the continuous load/time input required. The example used comes from Boswell & D'Mello, Blackwell, 1993.

I can't remember now why I added the energy analysis! You might need something similar if you want to handle any type of load.

[netoprax]

I was able to write a code that replicates the graph, in case anyone comes across this issue and need guidance, feel free to message me and I'll be glad to help.