Simple Rigid Frame - Parallel Legs Have Multiple Members

[AWDMIKE]

Greetings!

I am interested, as a matter of an academic exercise (stemming from a design issue at work), how to perform an elastic analysis of a rigid frame (using analytical methods), with the following description:

Visualize a three-sided rectangle, similar to many of the stick figures in Roark's table for "Reaction and deflection for rigid frames" (for reference, I am using Case 5f, which in my 5th edition is a concentrated load on a vertical member with both ends fixed). All of the closed-form solutions I have found (obviously including what's in Roark's) has each of the three sides (two vertical sides and one horizontal side) consisting of only one type of structural shape per side. In other words, I've found no information considering, say for example, each of the vertical members consisting of two different structural shapes, with different lengths and different moments of inertia.

For simple frames consisting of a single structural shape for each of the three sides, I have had considerable success using Roark's, when comparing it to results from RISA. I have also had very good results when using MathCAD, as well as Excel. I am currently attempting to perform such an evaluation, for calculating the reaction forces at the base, for a frame in which there are two different structural shapes for each vertical side. In my evaluation, the frame is symmetric, so that each vertical side has the same structural shapes and lengths of each shape, as well as there being only one horizontal shape.

I hope the following sketch suffices, please note that I1 and I2 represent the different moments of inertias for each of the two shapes making up the vertical legs of the frame, and that each shape has an associated length, say, L1 and L2, where the frame is fixed at the base of I2.

I3
I1 I1
I2 I2

I'd like to assume rigid end connections, as well as elastic deformations. I experimented with various methods of equivalent moments of inertias, though my hand calculations did not match as well with RISA, and needless to say, I'm not a fan of guessing and checking anyway.

There is more to be understood on my part, and if anyone has any help/literature that they can offer/recommend, it would be appreciated. Thank you.

 

[ishvaaag]

You have the frames tabulated in the

Prontuario Ensidesa
tomo I
Manual para el cálculo de estructuras metálicas
Empresa Nacional Siderúrgica S.A.
from 1969 onwards, there are still editions ongoing, andt hink to remember have seen it posted (not sure on copyright status) somewhere in the web.
I have a seventies edition (3 tomes)

There Frames L1 L2 L3 I1 I2 I3 are represented for variegated support constraints.

Have also another fantastic reference for framed arches and frames mexican edition from an US one. However only symmetrical outfits that I remember. And in the closed cabinet I only access but seldom a year. Must be

PORTICOS Y ARCOS: Leontovich, Valerian - Editorial CECSA, MEXICO - Año Edición 1961

 

[miecz]

AWDMIKE

Are you familiar with the matrix method of analysis?

 

[AWDMIKE]

Hi ishvaaag, thank you for the reference. I will look into it.

Hi Miecz, I am only vaguely familiar with the matrix method of analysis. If I recall, I took a course in college which covered basic numerical analysis, and if this is what you are referring to, then I may have an introductory text on the subject. I will look into that as well.

Thanks for the help.

 

[miecz]

If you can come up to speed on matrix methods, I have a Mathcad routine that solves your frame, except it has pinned supports and uniform loads. It could be easily modified for fixed supports and a concentrated load.

 

[OCI]

How about moment distribution? Seems like this could easily be solved with that method.

 

[AWDMIKE]

I've had considerable success using the matrix method, which I took time to relearn this past weekend. I was also able to get it to work with Excel, of which I'm a big fan.

Other hand calculations seemed considerably tougher since I was using fixed supports and fixed connections, and I wanted each member to have various cross sectional properties.

Thanks again for the help!

 

[miecz]

Here's a PDF of a matrix solution for a pinned frame in Mathcad. Shear flexibility is ignored.