Fixed end moments for Polygonal Loading
Fixed end moments for Polygonal Loading
(OP)
hi all,
What's the fixed end moments for polygonal load on a simple FEM rod element, additionally what if the load is offseted from "I" end or "J" end of rod element?
What's the fixed end moments for polygonal load on a simple FEM rod element, additionally what if the load is offseted from "I" end or "J" end of rod element?






RE: Fixed end moments for Polygonal Loading
How could you do anything so vicious? It was easy my dear, don't forget I spent two years as a building contractor. - Priscilla Presley & Ricardo Montalban
RE: Fixed end moments for Polygonal Loading
So what's the equivalent nodal loads/fixed end moments for that polygonal loading type in general form?
Breakingdown the polygonal loading to simple primitive forms like, rectangle, triangle, trapeze is an option for solution, but I'm not looking for that, I need to write the fixed end moments in general form depending on polygonal load.
Hope that's clear enough,
RE: Fixed end moments for Polygonal Loading
To solve such a problem in general form, you was possibly thinking at replacing the actual load with some resultants. However, as a beam with two fixed ends is statically indeterminate, this is not possible and the end moments will inevitably depend on the actual detailed loading distribution.
The only thing you can do, IMHO, is to define the maximum number of parts into which the beam is divided with a defined load variation in each, then solve the problem parametrically, where the parameters define the entire load distribution.
prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
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RE: Fixed end moments for Polygonal Loading
RE: Fixed end moments for Polygonal Loading
I'm also not clear what you mean by a general form, but the procedure for calculating the fixed end moments, if you don't want to use the published equations for triangular or trapezoidal loads, is:
- Find the slope of the ends for the applied loads assuming simple supports.
- Find the end slopes due to a unit moment applied at the end.
- Solve the simultaneous equations to find the end moments that will give the calculated end slopes under your loads, with a simply supported span.
- The fixed end moments are the equal and opposite reactions to these moments.
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Fixed end moments for Polygonal Loading
Actually this is part of FEM software which I'm trying write and those polygonal loads are resulting from slab-to-beam load distrubituion mechanism (with yield line method, which still works lamely for now!) Basically I need all the applied loads in a more computerized manner which can be applied at the nodes.
With general form I meant, if I know(or extract) the equation of applied loads on members, so I thought that, I can write the fixed end moments as function of that applied loads.
For example :
if load acting upon members is f(x)= X'2+5*X then
fixed end moment is M(x) = X'4 + 5*x'3/6 + .....
something like that
Also, could you shed some lights on which is the best method of slab-to-beam load transfer mechanisms(computer friendly), as well?
Especially when slabs shapes are more complicated than quadrilaterals it's very hard to predict the collapsing/yielding lines.
RE: Fixed end moments for Polygonal Loading
If you can accept approximations, you could find a best fit polynomial law for a loading extending over the full span. However it is difficult to estimate the degree of approximation without defining an outline of the possible distributions (e.g.: are there regions of the span with zero loads?).
Series expansion of the load distribution is another possibility, you could write a procedure that automatically extends the expansion till a predetermined maximum error is met.
prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads
RE: Fixed end moments for Polygonal Loading
I was planning to try at least a general 2D Frame solver but my intent washed before I completed the task. However I completed some initial steps, such solving for fixed end moments for a variety of situations. Here I attach a printout of the mathcad worksheet that would solve your case, and later will upload the whole package of Mathcad 2000 professional worksheets on the matter, just in case it becomes useful to someone.
RE: Fixed end moments for Polygonal Loading
RE: Fixed end moments for Polygonal Loading
RE: Fixed end moments for Polygonal Loading
So everything is getting knotted at accurate estimation of load distrubition.
For load distrubition approximation, I'm using the internal angles bisectors of slab, alhtough that completely relies on geometrical formation of slabs, if slabs are complex I couldn't say that I'm very successful at that too.
Could you elaborate that a bit more? IIUC, incrementing the loads gradually over the slabs till the stress reaches the line where it's equal to yield strength of material, and so those lines govern the load distrubition? If my understanding is correct any entry-point how to implement it?
Thanks for your comments,
RE: Fixed end moments for Polygonal Loading
RE: Fixed end moments for Polygonal Loading
RE: Fixed end moments for Polygonal Loading
Valuable infos need time to study them,
By chance, do you know how the arched(curved shape) elements local matrix is composed and how does it been integrated into global stiffness matrix.
Do the solution of linear equations systems (i.e. gauss elimination, cholesky factorization etc..) differs when arched elements are involved in structure?
Thanks,
RE: Fixed end moments for Polygonal Loading
Ishvaaag - where did/do you work or what field, that your so involved in complex analysis, FEM and software? Or do you just choose to research these areas?
EIT
RE: Fixed end moments for Polygonal Loading
BA
RE: Fixed end moments for Polygonal Loading
I studied in the ETSAM, that is the Technical School of Architecture in Madrid. I am a chartered architect in private practice since 1977. About 1993 work started to weaken, so I decided to study how the things were being built in USA, obviously the leader at least technically in most of the fields. So from then on I have been exploring these matters in english language. I have already told elsewhere in this forum that I bought Mathcad simply as a better alternative to PCA-Column, since by 2000 I was interested in one sectional analysis program, and I decided to do it myself and retain a more general tool, that to me showed to be superior and easier, if not in total capacity, as suiting my intent, than Excel. With Mathcad in the hands and the building work weak to inexistent -as remains now- I decided to also explore the FEM thing. Very likely my personal characteristics, plus my environmental pressures and easements have facilitated what I have been able and wanted to do. Everything becomes easier if you have someone good as a teacher. I quite likely met only but a handful to the end of my career years, but I have found tons in the books then and later, to whom I am very grateful.
If I do not understand more is part as a combination of a life not entirely directed to building and technical things -I am too interested in too many things to properly focus in all pervasive manner in just one-, not having the best learning, nor much precise memory -at least to the scale the things I look at would require- ... well, if I have done something well at any time I realize that most work was done by others and I just joined some things with what I had.
RE: Fixed end moments for Polygonal Loading
h
The function uses the method described in my earlier post in this thread (9th November) and includes full open-source code.
The spreadsheet also includes a function for calculating shear forces, moments, slopes and deflections in a simply supported beam, and a similar function for a continuous beam (which will be the subject of an upcoming blog post).
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Fixed end moments for Polygonal Loading
I'm really so curious about that UDF using Macaulay's Method.
Is that really being inteded for Excel? I couldn't locate any workbook(with xls extension) in zipped file(Macaulay.zip), is that normal?
RE: Fixed end moments for Polygonal Loading
I will re-save it in a zipped file, and also include an xls (2003) version, so the contents of the new zip file should be:
Macaulay.xls
Macaulay.xlsb
When downloaded and unzipped there should be a module in the VBE project listing called mMacaulay (it was mMacauley in the original version, I have just corrected the spelling).
Please let me know if the new file downloads OK.
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Fixed end moments for Polygonal Loading
EIT
RE: Fixed end moments for Polygonal Loading
So as a result,
1. As long as the loading function remains first degree polyinomal then method described by Macauley can be used to calculate fixed-end moments. (With modifying the source code to accept more than 3 inputs for each two loading type)
2. If loading function is more complex, then curve estimation should be used by any means of numerical analyse methods (e.g. Fourier Series, finite difference, least square etcc...) to get the load function and to find fixed end moments mentioned method should be used.
What actually you mean with that P=w*dx?
(Centroid of concantrated loads)*(distance to end) = M ???
RE: Fixed end moments for Polygonal Loading
The fixed end moments for a concentrated load are known, so you simply sum the fixed end moments for each element of load along the beam. Those two sums will be the fixed end moments due to the variable load.
BA
RE: Fixed end moments for Polygonal Loading
For several trapezoidal loads, you would simply sum the results.
BA
RE: Fixed end moments for Polygonal Loading
The function will accept any number of trapezoidal or point loads (up to about 1 million!). You just select a bigger range.
If required it would be straightforward to use the same method for any degree of polynomial, but this would require some coding. Alternatively you can approximate any loading with a number of short trapezoidal loads, or point loads if you prefer.
That's exactly what the spreadsheet does.
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Fixed end moments for Polygonal Loading
BA
RE: Fixed end moments for Polygonal Loading
Not much use today, but it was in the pre-computer days, although I suppose it could still be useful in a spreadsheet
Michael.
Timing has a lot to do with the outcome of a rain dance.
RE: Fixed end moments for Polygonal Loading
BA