Canned Solutions for Triangular Loading

Does anyone have solutions for the attached triangular distributed loading conditions. I can easily work the second one out, but if there is a solution handy, I’d appreciate it.

Bending moment and Shear Force diagrams…


DikTriangular_Loads.pdf (13.7 KB)

The second one is just the algebraic sum of two simple span loading conditions-one uniform and the other triangular.

The first one is more complicated. I would just use Beampro to solve it. I assume you want all the equations though.

Thanks Mike… I can easily develop the formulae for the second one… I was hoping that someone had the ‘coefficients’ for the moments and shears of the first one… maybe including deflections. I was going to embed them in an SMath program so I didn’t have to check each time.

If I can find enough time to waste, I may solve some simultaneous equations based on data generated by my own FEM program… should be easy enough, just time consuming.


may be the sum of these 2 load conditions
triang load def.pdf (344.4 KB)

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@pisani49 Thanks… I’m in the process of solving it… I’ll post the SMath solution later.


@pisani49 @MSQUARED48 @KootK Many thanks to KootK for the formulae.

FEM_Trapazoidal_Load.pdf (271.2 KB) (239.8 KB)

I haven’t had a chance to check it out… was happy it worked for a UDL, right out of the gate… People should look into SMath, if they haven’t already.

Now to brush up on my 2-Cycle Moment Distribution… haven’t used that for 40 years…

trapezoidal… spelling not for sh*t…


I’ve got the simultaneous equation built in python already if you want an additional reference. You can see the the code for that here:

Trapezoidal/variable loading starts on line 847
Simultaneous Equation for interior supports starts on line 3567

Thanks… I’ll take a gander at the source… about 3000 lines or so… Python seems to be a pretty popular language.

I forgot to welcome you to the group… the site is just starting, but, has a lot of great people on board. Again, welcome and thanks.


Thank you…yeah so far have seen a few familiar user names here, appreciate the effort launching and maintaining a space for us all to discuss these various topics.

If you want to include shear deformations I’ve got some things in the works for that and can point you to those as well. I have SMath downloaded but haven’t pulled the trigger on installing as of yet.

Sweet! You are sooooo welcome. It’s great to see someone other than me getting some utility out of the equations as they took some doing to develop.

In my MathCAD application, I combined those equations with the summation function such that results were calculated for each service load prior to combination with very, very little effort. Well, very little effort for me, lots for MathCAD.

As you may or may not know, I’m in the process of possibly getting into the software development game. See the first post of this thread: Kootware. It’s become abundantly clear that the most highly prized feature in this space would be detailed output presented in a readable form. Ideally, I’d like users to be able to dump the output as MathCAD calcs. Unfortunately, that doesn’t appear to be technologically possible. I’ve been wondering if SMath might be able to what I’d want however. I shall investigate…

Do you have a copy of Roark’s textbook? Pilkey’s also has solved cases like this one.

I haven’t been able to follow the software thread since it grew beyond 100 posts.
The raw SMATH files are readable like XML files. I was once PO’d enough at SMATH not being able to delete a matrix row that I directly edited the file in NOTEPAD and got on with life.

I have a *.pdf version of an edition of some kind… somewhere. I was busy adding end moments to the system and adjusting shears… but got tied up on real work. I’ll hop back to it when I have some free time.

KootK’s formulae work very well.