Wrote

Hello SteelCat,

Could you please tell me what are the equations you want to solve (as I sincerely don't know about the Jacobian procedure and the AlgLib solver).




Nint = ?;
Mxint = ?;
Myint = ?;

Hi Alvaro,

I want solve the system of the 3 equilibrium equations (internal stress resultants = external loads):
Nint = Ned ;
Mxint = Mxed;
My = Myed.

From the solution you obtain the 3 parameters that define the deformation (a plane) over the section.

Quote

I will give it a try, I understand that you are dealing with biaxial bending of a concrete column..

Yes, you are right.

Quote

By the way, I am proceeding with the shoelace Gauss formula to obtain the compression volume of concrete (a discrete approach, not a continuous double-integral approach) and also with the R diagram for concrete (not P-R).

I also was looking for a suitable discrete approach, but the compressed part of the section is unknown! Attached you can find my Excel-like discrete approach: It work,s but you have to manually change the guess values, because in Smath there isn't a "by changing" solver like the Excel's one... Forum discretization example.sm (123 KiB) downloaded 49 time(s).

From Google I see that R diagram is an interaction diagram: I do not know it, but consider that my spreadsheet want to be more general (if you change the costitutive equations for the materials), not only for ULS.

Quote

But I will try to find a solution for your equations (staying continuous)

Thanks.

Regards.

Wrote

as I sincerely don't know about the Jacobian procedure and the AlgLib solver).

Here is an example solving a system of 4 functions.
The other great class of applications is solving
for the 'n' parameters of a parametric function.
Plenty of examples for your next visit, if you wish so.
Cheers Alvaro ... Jean.

Solve al_nleqsolve CSTR.sm (38 KiB) downloaded 44 time(s).

Wrote

Here is an example solving a system of 4 functions.

Thank you Jean, very fast. I had to change Random() by random() but I think it is about the SMath versions.

Wrote

I also was looking for a suitable discrete approach

Hi SteelCat,

Last night I was deeply trying to solve that double integral of discontinuous functions whose conditions depend both on the integral dummy variables and the unknown variables...I never had such a hard system of equations to solve.
I followed the suggestions -thanks to Alvaro and Martin- of translating the boolean conditions into unit_step() functions, that Maxima handles..the problem is that, since your conditions depend on the unknowns..complicated equations are transferred to Maxima, and solving with mnewton() goes for about 7 min.

Here is an example (Maxima handles unit_step() and converts it into abs() and signum() functions).



I estimate that other non-linear solvers will also be time-consuming, that does not worth it for this problem.

Before my suggestion, I want to clearly understand:

1. I did not see strain limits in concrete (~3.5%%) nor in steel (~10%%), thus, I thought, you want to have a result of the equations, and then evaluate if this result is real? (I have seen in your Excel-like worksheet that you had for example concrete strains of about one percent, which is not possible).

2. Those observations lead me to ask...you have external forces acting in a defined concrete section, so, I understand you do not want to design, nor check the ultimate resistances..what you want is to describe the section's tensional state, right? This last being a problem of SLS...not ULS.

I have never dealt with this problem, but knowing that R and P-R diagrams are constitutive equations simplified for the design, I would use Sargin equations (EC2-3.1.5), which are a more realistic description of concrete stress-strain relations.


Now, I would suggest you to re-model your approach, in a manner that you will stay continuous..but not solving for a unique solution, rather trying all the possible solutions in a while loop (in this approach, there is a definite range to search the solution).




This approach simplifies:

1. You have as input data Mx and My..the neutral axis will have the same direction as M, no need to have curvatures in two directions as unknowns.

2. You can make an Area function (that green trapezoid) that you would use to compute differential forces (sigma*dA)..those forces being integrated (single integral, not double) in the orthogonal direction to the neutral axis.

3. You can make a while loop that stops when a tolerance for equilibrium is reached. The loop variable will be your N.A. position..will vary from A to B.

4. These tasks are rapidly computed...the while loop will run very fast, as you will define in each loop a tensional state for your PR


Now, the problem..

I did not see your strain limitations..and putting these limitations in the problem, is a very hard task, since you have to rotate your plane always remaining in the regions defined by EC2-6.1

This is a very nice problem, I am dealing with it in my concrete volume interaction (for rectangular sections).

But all these suggestions I am giving to you, work for me since I am applying them in my worksheet. I cannot share it because now it's a mess, but hopefully will end in June.

As a simple demonstration of the while loop varying neutral axis, here I attach a rectangular section beam designer according to EHE-08 (Spanish code).


F-S-R-SR-D.sm (128 KiB) downloaded 38 time(s).


You can "show inputs" in "acero requerido" to view the while loop.

I hope it can help you with your project and please let us know here of your advance.

Regards,

Alvaro.

Sorry for the delay, but I did not see the email notification..

Wrote

I estimate that other non-linear solvers will also be time-consuming, that does not worth it for this problem.

I agree with you.

Quote

1. I did not see strain limits in concrete (~3.5%%) nor in steel (~10%%), thus, I thought, you want to have a result of the equations, and then evaluate if this result is real? (I have seen in your Excel-like worksheet that you had for example concrete strains of about one percent, which is not possible).

2. Those observations lead me to ask...you have external forces acting in a defined concrete section, so, I understand you do not want to design, nor check the ultimate resistances..what you want is to describe the section's tensional state, right?

You understood well. In the professional practice, it is not always necessary to know the exact deformation of a section subjected to bending plus compression. It is sufficient to know the resistance limits, that you can calculate in correspondence with the limit deformations (at least for ULS). It is a "workaround".

Mine wants to be a general purpose sheet, to test the possibilities that you have with SMath: there are many tools written in VBA in the Excel environment. Them take advantage of its solver: you do not have to explicitly define a function to be solved. Smath it's different, it's a CAS. I compare the Excel solver to a multivariable while loop.

For uniaxial bending+compression Smath solved the equilibrium equations in few seconds with AlgLib NL solver. Now, little by little, I would like to increase the difficulty.

Quote

This approach simplifies:

1. You have as input data Mx and My..the neutral axis will have the same direction as M, no need to have curvatures in two directions as unknowns.

2. You can make an Area function (that green trapezoid) that you would use to compute differential forces (sigma*dA)..those forces being integrated (single integral, not double) in the orthogonal direction to the neutral axis.

3. You can make a while loop that stops when a tolerance for equilibrium is reached. The loop variable will be your N.A. position..will vary from A to B.

4. These tasks are rapidly computed...the while loop will run very fast, as you will define in each loop a tensional state for your PR

#1 it's valid only if Jx=Jy. The challenge of biaxial flexure lies in the fact that, in general, you do not know the inclination of the neutral axis. Otherwise I agree with you.

Quote

As a simple demonstration of the while loop varying neutral axis, here I attach a rectangular section beam designer according to EHE-08 (Spanish code).


F-S-R-SR-D.sm (129kb) downloaded 8 time(s).


You can "show inputs" in "acero requerido" to view the while loop.

I hope it can help you with your project and please let us know here of your advance.

Thanks Alvaro, I'll study it.

If all that "concrete stuff" exists as a standard, why do you want
to reverse Engineer its original creation ... especially from the
unpredictable Smath integrator.
Show some data from that standard ... what's your objective ?

Wrote

If all that "concrete stuff" exists as a standard, why do you want
to reverse Engineer its original creation ...
Show some data from that standard ... what's your objective ?

Hi Jean!
The question is not so simple: not everything is standardized, only basic assumptions are standardized. Everything else is in the textbooks of structural mechanics.

For the structural engineer, there are two ways to deal with the most complex problems: use software made by others or try to make your own.

As said, I would like to try using SMath for some of these. No reverse engineer.

It seems that also Alvaro is studying some similar engineering matter...

There is the main road and shortcuts...

Quote

especially from the unpredictable Smath integrator.

I can use only symbolic integrator to define the functions, it can't forget the unknowns. I know the theory of Simpson and Gauss: I know that you need points in the domain where to evaluate the integrand function. But do these algorithms maintain symbolic evaluation?

Thanks for your interest.

Here my advance:
I replaced the integrals with summations and Smath solved the system with the AlgLib solver in 17s !!! Forum example w-o units_7.sm (75 KiB) downloaded 37 time(s).

I also tried with FindRoot but I didn't get the solution: Forum example w-o units_6.sm (68 KiB) downloaded 28 time(s)..

I would be curious to know just why, if I try to input the Jacobian matrix to the solver, something goes wrong.

Above rainbow is doctored SS 6179.
If it works SS 7030: don't touch it to be ready for contribution.
FindRoots is setup, but does NOT stop, neither complains.

Forum example w-o units_7 FindRoots.sm (87 KiB) downloaded 34 time(s).

... as it looks: FindRoots fails simply.

Forum example w-o units_7 FindRoots.sm (93 KiB) downloaded 46 time(s).

Wrote

I replaced the integrals with summations and Smath solved the system with the AlgLib solver in 17s

Nicely improved! That is much better, for me it took 6 s.

Wrote

I replaced the integrals with summations and Smath solved the system with the AlgLib solver

... at this point: we should see these two functions in graphs,
and appreciate wrt the 3 parameters and manipulate, is it ?

It could be, but I don't think so ... due to the computation time

Hi. In the attached, a numerical procedure for double integrals using rkfixed ode solver.

With some improvements maybe this idea can be used instead sums. Also, it can be modified for accept units.

Forum example w-o units_4-BIS.sm (64 KiB) downloaded 55 time(s).

Best regards.
Alvaro.

Maybe you can try to beat the horse otherwise.

Forum discretization example OTHERWISE.sm (120 KiB) downloaded 51 time(s).

Wrote

Wrote

I replaced the integrals with summations and Smath solved the system with the AlgLib solver in 17s

Nicely improved! That is much better, for me it took 6 s.

Yes, my Smath is slower because I run it on a VM....

Wrote

Hi. In the attached, a numerical procedure for double integrals using rkfixed ode solver.

With some improvements maybe this idea can be used instead sums. Also, it can be modified for accept units.

Forum example w-o units_4-BIS.sm (64 KiB) downloaded 55 time(s).

Best regards.
Alvaro.

Wrote

Maybe you can try to beat the horse otherwise.

Forum discretization example OTHERWISE.sm (120 KiB) downloaded 51 time(s).

Thanks to both, I will study them in the next few days.

Wrote

Thanks to both, I will study them in the next few days.

BTW, if you have some typical graph(s) from your book,
make a clean image and attach. I will try to digitize.
If not too big, Tracer [freeware] can do manually,
then smooth a bit, spline ...
Cheers ... Jean

Wrote

Wrote

Thanks to both, I will study them in the next few days.

BTW, if you have some typical graph(s) from your book,
make a clean image and attach. I will try to digitize.
If not too big, Tracer [freeware] can do manually,
then smooth a bit, spline ...
Cheers ... Jean

Sorry Jean, no graph, only values...

Wrote

Sorry Jean, no graph, only values...

... from enough values: we can best approximate/graph !!!