Fixed end moments for Polygonal Loading 1

[JoeH78]

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?

 

[rowingengineer]

I tried to understand what your asking but I just can't follow.

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

 

[JoeH78]

At the attachment there is a rough sketch. Blue one is a rod element (say beam, girder), fuschia color is polygonal loading arrows are not loads they are intended to show just the loading direction.

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,

 

[prex]

Sorry, it isn't clear.
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

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[JoshPlumSE]

Approximate the loads as a series of partial length triangular loads (for which you should be able to find tabulated fixed end moments). Then use the principal of superposition to add up the fixed end moments for each partial length load.

 

[IDS]

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.

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/

 

[JoeH78]

Thank you all in advance, and sorry for not being verbose

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.

(I'm also not clear what you mean by a general form

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.

 

[prex]

If you have a single law of load change along the full span, then you can do that in a general form, but you had 4 regions with linear change over each in your sketch: it is not possible to generalize with any number of loading regions.
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

 

[ishvaaag]

About the year 2000 I explored a bit doing FEM work within a Mathcad environment. Unfortunately to your intent, it involves the problem solving abilities of Mathcad, that you will need to change for solving the corresponding equation by programming or whatever your environment provides.

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.

 

[ishvaaag]

Now the Fiexed-End Moment operative worksheets; later other work.

 

[Ron]

Hey Josh...feeding the competition?

 

[JoeH78]

So I see that there is not a simple solution to that, this topic in itself is well worth a chapter in a book.

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?)

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.

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.

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,

 

[ishvaaag]

The local stiffness matrices for line elements

 

[ishvaaag]

And the plane stress and plane strain elements as I took from texts for a 2D frame application.

 

[JoeH78]

Thanks isvaag,

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,

 

[RFreund]

I don't mean to distract from this thread but -

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

 

[BAretired]

Perhaps you could express the applied load in the form of a Fourier Series. The fixed end moment for the beam would be the integral of the fixed end moments for concentrated loads along the beam where P = w*dx.

BA

 

[ishvaaag]

Hi, RFreund

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.

 

[IDS]

I have just posted a spreadsheet with a UDF using Macaulay's Method to find fixed-end moments in a uniform or stepped beam with any number of trapezoidal loads or point loads and moments:

http://newtonexcelbach.wordpress.com/2011/11/13/beam-fixed-end-actions-by-macaulays-method/

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/

 

[JoeH78]

Thanks IDS's,

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?