×
INTELLIGENT WORK FORUMS
FOR ENGINEERING PROFESSIONALS

Log In

Come Join Us!

Are you an
Engineering professional?
Join Eng-Tips Forums!
  • Talk With Other Members
  • Be Notified Of Responses
    To Your Posts
  • Keyword Search
  • One-Click Access To Your
    Favorite Forums
  • Automated Signatures
    On Your Posts
  • Best Of All, It's Free!
  • Students Click Here

*Eng-Tips's functionality depends on members receiving e-mail. By joining you are opting in to receive e-mail.

Posting Guidelines

Promoting, selling, recruiting, coursework and thesis posting is forbidden.

Students Click Here

Jobs

Singularity Functions - Useful for beams with fixed ends?
2

Singularity Functions - Useful for beams with fixed ends?

Singularity Functions - Useful for beams with fixed ends?

(OP)
I use singularity functions all the time for determining shear and moment diagrams for beams in MathCAD.  My question is, can singularity functions be used when the beam has a fixed end(s)?  I was assuming for a fixed end, that θ'=0 would be a boundary condition for solving for c1 and c2 in my θ' and θ equations, but I am not getting correct answers.  I don't know if there is a mistake in my program somewhere, or if the mistake is my assumption of setting θ'=0 at the fixed ends.

Also, if anyone is familiar with both singularity functions and the moment distribution method, is there any advantage to the moment distribution method?  I am not too familiar with that method, but I would think the only advantage is that the method can be applied to 2-D applications where singularity functions are limited to a continuous beam.

Juston Fluckey, E.I.
Engineering Consultant

RE: Singularity Functions - Useful for beams with fixed ends?

Not sure to understand what you are exactly doing: can you outline your procedure?
Taking the example given here Singularity function - Wikipedia, you would be required to add a constant of integration  c  when calculating M : paraphrasing the example Because the moment is not zero at x=0, a constant of integration, c, is added (and of course end reactions need be left as unknowns and determined with  c  from u'=0 and u=0 at the second end and from equilibrium).
Does this make sense?

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: Singularity Functions - Useful for beams with fixed ends?

From what I see in this article

http://www.cgl.uwaterloo.ca/~tjlahey/sfunctions.pdf

the singularity functions are used mainly in the definition of the loading. Once that is made, they should be ready to stand any of the ordinary procedures of analysis (including any boundary conditions) analytical when feasible, and numerical where not. If I find any practical reference dealing with the amtter I will post it.

RE: Singularity Functions - Useful for beams with fixed ends?

P1ENG,

Check out "Machine Design" by Robert L. Norton, pg 116-131 discusses the use of singularity functions for beam loading including examples of beams with fixed ends. Roark also tackles beam loadings with singularity functions. As ishvaag states, once the loading function is developed and you have integrated it is a matter of proper boundary conditions.

I am a mechanical who now mainly practices structural engineering so I was exposed to singularity funtions as an ME however I have found that it is never discussed in civil, including structural engineering grad classes that I have taken.

It is a great method for programming and graphically displaying beam loadings.

Good Luck! If you have a specific case post a sketch and we can work through.
 

RE: Singularity Functions - Useful for beams with fixed ends?

Don't know MatchCad but what I programmed in Matlab was an if statememt when there are no degrees of freedom to skip the matrix imversion. You know the stiffness, load and displacememt so you can go strait to the post analysis.

RE: Singularity Functions - Useful for beams with fixed ends?

(OP)
I have some work to catch up on, but I will post a sample of my calculation soon.  I fully understand how to use singularity functions as I too am an ME doing structural work.  I don't want to look up conditions in Rourke's if I can do it all mathematically.  The great thing about Singularity Functions is you can solve it even if you have a statically indeterminate situation.  The thing is, once you add an unknown (moment due to fixed end condition), then you need to add an extra boundary condition in order to solve it.  The definition of my boundary condition is what I am questioning.

Juston Fluckey, E.I.
Engineering Consultant

RE: Singularity Functions - Useful for beams with fixed ends?

Both slope and deflection are zero at fixity ends for fixed beams; this, of course, is not true (except for some cases) for intermediate support points where you have elastic fixity, as in supports in continuous beams or nodes at frames, where the joint itself gets some rotation from the undeformed shape to equilibrium. In other words, you only will state slope=0 where there is true fixity to rotation.

RE: Singularity Functions - Useful for beams with fixed ends?

Just to remark, the slope of the tangent at true fixity points is zero, so you can set slope=0 there.

RE: Singularity Functions - Useful for beams with fixed ends?

... for horizontal beams, of course (or relative to the axis of the beam, if straight).

RE: Singularity Functions - Useful for beams with fixed ends?

(OP)
ishvaaag, that was what I was assuming.  I will have to go back and look at my program to see where I made the mistake.

Juston Fluckey, E.I.
Engineering Consultant

RE: Singularity Functions - Useful for beams with fixed ends?

are you looking at multi-span beams ?  i'm not sure that you can define the boundary conditions of an individual span without solving the entire beam first (using MDM or other).

RE: Singularity Functions - Useful for beams with fixed ends?

(OP)
O.K., some time freed up.  Take a look at my example (attached below) that will better explain what I am trying to do.  My results are incorrect per the RISA results of the second page.  At this point, I wouldn't mind if I made the dumbest, clearest mistake as long as someone can point it out.  Thanks!

Juston Fluckey, E.I.
Engineering Consultant

RE: Singularity Functions - Useful for beams with fixed ends?

(OP)
prex! Going over the responses so far, I think I now understand what you were saying.  I should introduce a constant, c, to my calculation of M(x).  I was also thinking of doing this, but it leaves me with an extra unknown.  I need an extra BC to be able to solve for it.  Would that extra BC come from M(20')=0?  I think so... I will redo my hand calculation and see what the outcome is!

Juston Fluckey, E.I.
Engineering Consultant

RE: Singularity Functions - Useful for beams with fixed ends?

(OP)
IGNORE ALL MY PREVIOUS POSTS AND ATTACHMENTS, unless you want to read about my struggles of checking my work and thinking through the method.

The previous attachment integrated the c1 term incorrectly in the δ(x) function.  The attached hand calculation shows correct values.  I don't know how RISA's solver handles the calculation of the reactions which are slightly different than mine, but I would think mine are 100% on.

Juston Fluckey, E.I.
Engineering Consultant

RE: Singularity Functions - Useful for beams with fixed ends?

I solve as you per RISA 3D (shear deformation influence disabled) and by the force method in Mathcad 2000 Professional using as sole unknowns M1, R1, R2 by your notation, and attach the results.

My E and I are different but for constant section irrelevant to moments and shears (not vertical deflections outside nodal restraints, or slopes anywhere). Both the Mathcad and RISA 3D solutions easily changeable to your E, I values.

In this case, I have not yet stated the problem solved in Mathcad by the force method with the singularity functions. All that is needed to do so is to include a previous definition of the singularity functions required in Mathcad, then modify the statement of the moment according to such function definition, to follow the same procedure to solution.

Since I don't know about the general use of the singularity functions for structural analysis, I can't see if there is the general convenience of retaining 7 unknowns where as here we can deal the matter (mostly) with 3. Since automatical solver, scarcely relevant at this scale.

A note for Mathcad users:
Note interestingly how the Mathcad vector was able to retain in Mathcad 2000 Professional solutions in different units (moments and vertical reactions). I have seen this kind of ability disabled in later releases.

For later relases it may turn necessary while solving to reduce to consistent (to be, implied) units and then recovery of the same, first by the corresponding division of the pertaining units of the component in the to be implied units and then multiplication of those corresponding after solution.


 

RE: Singularity Functions - Useful for beams with fixed ends?

(OP)
ishvaaag:  Thanks for the calculation sample.  I didn't think to use functions and the Find command in Mathcad.  My approach involved using vectors.  This will probably simplify my program very much and make it more versatile.

Also, my RISA results were for a W8x10.  Making it a Rigid member yields the same results as ishvaaag.  I wonder if the difference is due to shear deformation.

Next thought for everyone... what would happen if the right end was also fixed?  You would then loose your known BC of M(20')=0.

Juston Fluckey, E.I.
Engineering Consultant

RE: Singularity Functions - Useful for beams with fixed ends?

Quote:

You would then loose your known BC of M(20')=0.
But you replace it with y'(20')=0...

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: Singularity Functions - Useful for beams with fixed ends?

If the right end is also fixed we lose the known moment quantity at the right support but gain there the slope=o known, so it can also be solved within the same setup, on only the same 3 unknowns than before.

I have not checked this time against RISA 3D.

RE: Singularity Functions - Useful for beams with fixed ends?

(OP)
Sleep deprivation... thanks prex.  I am armed with a boundary condition arsenal to complete my program.  I'm glad this all worked out.

Juston Fluckey, E.I.
Engineering Consultant

RE: Singularity Functions - Useful for beams with fixed ends?

(OP)
Continuation of singularity function development: What if the beam doesn't have a continuous moment of inertia (e.g. a tapered beam).

I will do some thinking/research and get back to you all.  Feel free to post anything you know/find.

Juston Fluckey, E.I.
Engineering Consultant

RE: Singularity Functions - Useful for beams with fixed ends?

(OP)
Quick response. I was planning on doing a sample calculation with the moment of inertia being a function of location and comparing it to my RISA results, but you were quicker.  It seemed straight forward, but I wanted verification, which you gave me.  Thanks ishvaaag!  

Juston Fluckey, E.I.
Engineering Consultant

RE: Singularity Functions - Useful for beams with fixed ends?

(OP)
Additionally, it should be stated that the same could be done for varying modulus of elasticity.

Juston Fluckey, E.I.
Engineering Consultant

RE: Singularity Functions - Useful for beams with fixed ends?

(OP)

Quote (ishvaaag):

A note for Mathcad users:
Note interestingly how the Mathcad vector was able to retain in Mathcad 2000 Professional solutions in different units (moments and vertical reactions). I have seen this kind of ability disabled in later releases.

You are right.  I am using Mathcad 14 and it does not handle the units.

Juston Fluckey, E.I.
Engineering Consultant

RE: Singularity Functions - Useful for beams with fixed ends?

I was a civil major for undergrad and graduate. My emphasis was structural. We are taught so many different methods to solve for moments and reactions for determinate and indeterminate members. I remember my sophmore year we skipped the singularity functions. Four years ago at work there was a co-worker who was a mechanical engineer student and recall that he was studying singularity functions.  I usually choose moment distribution over the others since its quite simple for me. I have attached a copy of how simple moment distribution can be for beams.

Red Flag This Post

Please let us know here why this post is inappropriate. Reasons such as off-topic, duplicates, flames, illegal, vulgar, or students posting their homework.

Red Flag Submitted

Thank you for helping keep Eng-Tips Forums free from inappropriate posts.
The Eng-Tips staff will check this out and take appropriate action.

Reply To This Thread

Posting in the Eng-Tips forums is a member-only feature.

Click Here to join Eng-Tips and talk with other members!


Resources