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Symbolic Integration

ptc-4154079
1-Newbie

Symbolic Integration

Hi all,

I've been using Mathcad for a couple of years, but I've always had difficulty learning how to do simple symbolic manipulations with Mathcad and instead used a program like Maxima (which did things beautifully). However, I like to keep my stuff in one place and thus want to learn how to do this in Mathcad.

So here is what I'm trying to do. I'm trying to use Timoshenko's beam theory to analyse a statically inderminate beam. The process is very simple. I have two coupled differential equations and I need to successively integrate to find the displacement and the rotation in the beam. However, I'm having with one of the integrals. I've double checked and I'm pretty sure the result that mathcad gives is not correct. I'm sure it is something that I've done that has screwed it up, but I just can't spot what is wrong.

I've posted the file and highlighted the problem regions.

I was also wondering if someone could point out how I could get these equations in matrix form. If you look at my problem statement, I've got two beam segments. I've defined all the properties of the segments as vectors with two elements. I was hoping that I could set up the equilibrium equations (DEs) in a way that they are also vectors (along with the integration constants) like w(x):=[ ... + C.4.1 , .... + C.4.2] .

Any help is greatly appreciated.

Regards,

Ali

PS: I'm using Mathcad 15.0.0.436

15 REPLIES 15

Ali Nayeri wrote:

...I'm trying to use Timoshenko's beam theory to analyse a statically inderminate beam...

Hello!

See example "Statically indeterminate beams (propped cantilever and fixed-end beams)" from the book "Mechanics of Materials" by J.M. Gere, S.P. Timoshenko. I hope this example will be helpful.

Hi Vladimir,

Thank you for the message and the file. I had actually looked at Gere's book and some other resources about this. But your file and the reminder was very helpful. I was under pressure to deliver the assignment so I finally decided to just use Maxima and do everything in there (it is a bit ugly but it works). The main issue was that we had to use the DEs to solve the problem. That is we couldn't use superposition. So that is why I wanted to set up my calculations that way. I've attached the maxima file to this message (I hope that is allowed) if someone else needs to use it.

Bests,

Ali

You can attach a file by using the option "Use advanced editor".

http://communities.ptc.com/servlet/JiveServlet/showImage/2-171150-26656/Pic_1.jpg

I think you have some problems to fix, that don't really relate to Mathcad:

First, you are integrating across a discontinuity at the interior support, both in the load and the support reaction. At the interior support, note also that the beam slope change due to shear deformation is discontinuous.

Secondly, how is Mathcad to what values of x apply to what, the way its set up now, the loads and properties are a function of the range variable i, not x. Normally, the range variable is used this way to calculate for several, independent beams.

If you want to do symbolically, you need look at each span separately and apply the appropriate BC's at the interior support. (deflection and curvature(or moment), but not slope).

You can do numerically by writing everything in terms of x and considering interior reaction as a variable, and then solving for 0 deflection at the support.

Just for my own information, where did the shear correction factor come from?

RichardJ
19-Tanzanite
(To:ptc-4154079)

Ali Nayeri wrote:

However, I'm having with one of the integrals. I've double checked and I'm pretty sure the result that mathcad gives is not correct. I'm sure it is something that I've done that has screwed it up, but I just can't spot what is wrong.

The extra term is just a constant, so the result is somewhat unexpected, but it's not incorrect (solutions to an indefinite integral can of course differ by constants). It would be nice if you could force the symbolic processor to make constants of integration 0, but I don't know any way to do that.

wayne
3-Visitor
(To:RichardJ)

Did you just take off your fake glasses and mustache?

RichardJ
19-Tanzanite
(To:wayne)

Did you just take off your fake glasses and mustache?

??

Not sure what you mean.

wayne
3-Visitor
(To:RichardJ)

Your name changed while I was looking at you post.

MikeArmstrong
5-Regular Member
(To:wayne)

Richards name does seem to change each time I log in

Mike

When I look at this site without logging in, some names are replaced by a ptc number, Richard's is replaced by A.Non and the rest of us still have our names. After I've logged in, the ptc numbers and A.Non all disappear to be replaced by proper names (or at least, the names individuals have specified)!

Alan

I have the same observation.

RichardJ
19-Tanzanite
(To:AlanStevens)

Relatively recently several things seemed to change (for example, the advanced editor is no longer the default, which I find very irritating), including the way names are displayed. If you go to your profile page you will see an option on your right to change your Display Name. I changed this long ago, because of the e-mail privacy issues (although I forget the details). Change this to something. Next, from your profile page, click on "Edit Profile and Privacy" on the right, and look at the Privacy tab. What you set as your "Display Name" is listed as your username, and is visible to everyone. Your name will probably be set so that it's visible only to registered users. If someone is not logged in they will therefore see your "Display Name" if you have set one, and ptc???? otherwise. If they are logged in they will see your name, not your Display Name" (which, IMHO, is rather confusing). I think you can force it to always display your Display Name by increasing the privacy level on your name, but I haven't tried it.

My Display Name was actually set to something else (TheBogeyman) before these changes occurred, but once I realized how the new settings worked my sense of humor got the better of me

RichardJ
19-Tanzanite
(To:RichardJ)

I changed this long ago, because of the e-mail privacy issues (although I forget the details).

I remember now what it was. In the early days of these forums, if you didn't set your Display Name to something, your email address got sent out in email notifications from the forum.

RichardJ
19-Tanzanite
(To:wayne)

Your name changed while I was looking at you post.

What you see will change if you log in to the forums (see my recent post), but it shouldn't change otherwise. I certainly didn't change any of my settings.

Hi Richard and Wayne,

Thank you for getting back to me. I think your observations were correct. The truth is I'm not used to using mathcad for anything other than normal numerical work so when I started working with the symbolic features, things just fell apart. Richard, you are absolutely right about the extra term. I realized that the next morning after I had some sleep.

When I posted here I was under pressure to deliver the assignment. Even though I would have liked to use mathcad which I am familiar with, I finally decided to just use Maxima and do everything in there (it is a bit ugly but it works).

Wayne, you are right what I am trying to do is probably not quite right or at least efficient. I introduced the index "i" because the functions just didn't work without it. I was hoping that instead of setting up equations for each of the beams seperately, I could just create a matrix function and use that for integration/differentiation and evaluation of the results.

I've attached the Maxima file here if anyone wants to see it and work with it.

Bests,

Ali

PS: the file is not very big but it is very long due to the output from the program.

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