Calculation for multi-span continuous straight beam on supports with rotational restraint

[faremusai]

I want to calculate the reaction force (vertical) and reaction moment on supports for a continuous straight beam with several unequal spans resting on supports with rotational restraint (i.e. reaction moment occur when the support rotates).

If rotational restraint of supports are not present, the reaction can be calculated by the Three Moment Equation. If rotational restraint of supports exist and known (in Nm/rad), how to calculate the reaction moment and beam slope at the supports? Is there a direct method for such kind of problem? If iteration is required, what kind of iteration scheme should be used?

For iteration method, I think an initial guess of support reaction moments or beam slope can be made and calculate some kind of correction factors for subsequent iteration until the convergence limit / equilibrium / compatibility is reached.

Are there any literature / journal paper discuss on this topic?

 

[msquared48]

You should be able to set up the appropriate model to get the results you want with the RISA2D or similar program.

Mike McCann, PE, SE (WA)

 

[rb1957]

or moment distribution handcalc

another day in paradise, or is paradise one day closer ?

 

[rb1957]

thinking a second longer, doesn't the moment restraint (I'm assuming this means fixed) break the continuous beam into a bunch of double fixed end beams ?

another day in paradise, or is paradise one day closer ?

 

[msquared48]

I take it that the beam is continuous over the support, but the support is pinned to the beam allowing rotation at the support. The moment seen is only in the beam, not the support. Easy to model in RISA.

So no, this problem is not a series of fixed beams...

Mike McCann, PE, SE (WA)

 

[jdgengineer]

Typically it's really hard to have a truly "fixed" support. If you don't know the stiffness of the rotational support restraint I'd envelope those reactions as either pinned (more moment in beam) or fixed (more moment in support). For the pinned scenario you could do as suggested by Mike. For the fixed condition you could model the supports as fixed in the same model. At first glance, I agree with RB that it could likely be simplified as fixed fixed beams. In reality, I've never really seen this type of support except for interior columns of moment frames. In these instances you would model the columns to get the rotational spring component.

 

[faremusai]

All supports are pin (i.e. rotatable, but not totally fixed nor totally free, with resisting moment depending on the angular displacement of the support like a coil spring). The rotational stiffness (unit in Nm/rad) of all supports are known.

When there is no load, all supports does not have rotational displacement. After loading is applied to the beam, the supports will be rotated due to the resultant bending moment inside the beam. This in turn will create reaction moment in the supports to resist the bending of beam. I want to find out how much the supports will be rotated, which is certainly have magnitudes smaller than the totally free-to-rotate case.

I want to make the numerical calculation in Matlab by solving the Three Moment Equation (may require some modifications), not FEA modelling. M script for solving the standand form Three Moment Equation for simple pin supports is ready available. I want to add some code to cater for the conditions of support rotational restraint.

 

[jdgengineer]

I think you would want to form full stiffness matrix if you are programming into MATLAB. You should be able to account for the rotational spring with the stiffness matrix. It will be much more complicated

 

[rb1957]

M^2 ... from OP "supports with rotational restraint (i.e. reaction moment occur when the support rotates)."

this is a little confusing (in that the beam moment resists rotation).

Agreed, there is not truly fixed joint, just as there are no spherical chickens, but it is a convenient analysis assumption.

Possibly the joints are partially fixed, with a finite rotational stiffness. There maybe be mdm tweeks to deal with this, but FEA is the easy (lazy?) way to solve.

"lazy" is not a snipe ... I'm lazy too; FEA is a wonderfully flexible tool for solving really difficult problems. These days whenever the handcalc gets away from really simple we tend to use the sledge to open the walnut ...

another day in paradise, or is paradise one day closer ?

 

[dhengr]

Faremusai:
You would do well the dig out your Engineering Mechanics, Strength of Materials and Structural Engineering text books and study them a bit on this general subject. The Three Moment Equation approach gives you one bound on the problem; assuming the reaction locations allow free beam rotation, and the moments, slopes and deflections are only dependent upon the relative stiffness of the three beam segments. The other bound on your problem is the Moment Distribution or matrix/FEA approach; which assumes the joints at the reactions are rigid and the moments, etc. in the beams and columns are dependent upon the three member’s relative stiffness at that joint.

You really haven’t explained your problem very well, and leave us wondering what you are trying to do and how you are going to do it. How are you going to apply this reaction point moment which has its magnitude dependent upon the slope of the one or two beams at the given reaction point? What does this look like, and why and how? As for your iteration process..., do the two bounding calcs. and determine the slopes and deflections for each. Make some engineering judgement as to which of the slopes is most appropriate (nearest the final condition) as a function of the stiffness of your torsion spring moment inducing mechanism; superimpose these reaction moments on the original bounding calcs; then adjust the new slopes and deflections accordingly, and repeat. Note that, most joints are not perfectly pinned or totally fixed, these are analysis conveniences of our making. And, the way we model them for computer software use, or the way we detail and fabricate them, we can really screw up the works. That is, the computer says one thing, and the structure doesn’t pay any attention to that output, it just doesn’t understand our complexification, and it acts the way we forced it to act, by the way we messed up the detailing.

 

[IDS]

The problem statement looks clear enough to me. He wants to analyse a continuous beam with rotational springs at the supports using a Matlab routine rather than setting it up in a frame analysis program. That seems a perfectly reasonable thing to do.

My ConbeamU spreadsheet (download from

https://newtonexcelbach.wordpress.com/2016/08/28/conbeamu-4-10/)

does the same thing with Excel and VBA.

The method I use is:
- Analyse the beam as simply supported at the end supports, and find the beam deflections and rotations at the supports
- Find the beam deflection and rotation for unit loads at each support. For rotational springs you need to apply both a unit rotation and a unit vertical deflection as two separate cases.
- Set up and solve the matrix equations for the forces at each support, knowing the deflections and rotations. This gives all the support reactions, which you can then treat as applied loads to the beam supported on the end supports only.

If you want to work through the code, it's in the Macaulay module, in the ConBeam function, starting at the comment ' Find deflections at support positions for unit load at each internal support. (Press Alt-F11 to open the Visual Basic editor).

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[faremusai]

Doug,

Thank you for the advice and spreadsheet. Your method sounds like the Unit Load Method.

I should have described the support conditions more clearly. It is a kind of support which can move vertically (with vertical stiffness like an elastic isolation pad, or end movement of free end of cantilever beam) and rotate about the support hinge (with rotational stiffness like a coil spring, or end rotational due to torsion of a cantilever beam). These two stiffness properties are assumed to be independent of each other.

 

[jdgengineer]

In hearing your description better, I think your best option is going to the stiffness method and using MATLAB to power through the linear algebra for you. This is essentially what RISA or SAP would do.

It's been awhile since I've gone through it but I did a fair amount in MATLAB in grad school and don't remember it being too bad.

 

[IDS]

It is a kind of support which can move vertically (with vertical stiffness like an elastic isolation pad, or end movement of free end of cantilever beam) and rotate about the support hinge (with rotational stiffness like a coil spring, or end rotational due to torsion of a cantilever beam). These two stiffness properties are assumed to be independent of each other.

That is exactly what my spreadsheet does. I know it is difficult ploughing through other peoples code, but if you should want to convert my code to Matlab, or even adapt it in VBA for your own purposes, I would be happy to help.

Alternatively, if you want to use the stiffness method, I have an open source frame analysis program that could be easily adapted. See:

https://newtonexcelbach.wordpress.com/2013/03/17/3dframe-ver-1-03-and-frame4-ver-3-07/

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[dik]

Any chance you can provide a sketch showing what you are intending to do?

Dik

 

[faremusai]

Attached is a sketch showing a sample layout of the beam and load.

The loads and support configuration are arbitary for programme testing/debugging purposes.

 

[IDS]

My spreadsheet results for the provided beam details. The assumed EI is for 30 GPa concrete and 1m x 0.25m deep slab.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[faremusai]

I have downloaded the ConBeamU.xlsb from the URL you provided and test run it.
But the layout in workbook "ConbeamU1" is a bit different from that shown in your screen capture.
Also, I have defined five supports but the result in "ConBeam Reactions" have result for 4 supports only (for x=0 1.5, 2.5 4.0). Result for support at x=4.5 is missing.

The spreadsheet was ran using Miscroft Excel version 2002. Does the version of Excel matter in running the VBA code?

In my example, the beam material is steel with E = 2.1e11 N/m2, I = 0.72e-06 m4.

 

[IDS]

faremusai - the attachment to my previous post is a copy of the spreadsheet as shown in the screen-shot.

The analysis uses user-defined functions, which means you can easily re-arrange the input and output to suit your particular problem.
Using the reaction output as an example, to display all 5 supports:
- select the output range you want (P22:T26)
- Press F2
- Press Ctrl-Shift-Enter

You can also move the output to a different range, or move any of the input ranges where you want, or use input from another sheet or another file.

Other changes I have made from the standard version are:
- Load and stiffness units changed to N and M. You can change the output as well if you want. Just change the units at the top of the column.
- I reset the output graph x ranges to automatic; previously it was set to 32 m.

There is information on the required input and output for each function on the Functions sheet, and details of changing the output ranges on the Using Array Funcs sheet.

Reading my own information, I'd forgotten I have automated the array formula entry procedure. Just select the top-left cell and press Ctrl-Shift-S, and the output will resize to the number of supports.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[faremusai]

Doug,

Thanks. All support reaction/result can be displayed now.

Your advice and spreadsheet and advice by jdgengineer is really helpful.

In the meantime, I will complete the Three-Moment-Equation vesrion first (the support rotational stiffness problem is the last obstacle in my program).

My target is to write a Matlab script for analysis of truss (mainly static load condition). It can be modelled as a 2D plane frame or 3D frame which will be subject to unsymmetrical load. The FE stiffnes method will be used.