Calculation for multi-span continuous straight beam on supports with rotational restraint
Calculation for multi-span continuous straight beam on supports with rotational restraint
(OP)
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?
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?






RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
Mike McCann, PE, SE (WA)
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
another day in paradise, or is paradise one day closer ?
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
another day in paradise, or is paradise one day closer ?
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
So no, this problem is not a series of fixed beams...
Mike McCann, PE, SE (WA)
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
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.
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
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 ?
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
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.
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
My ConbeamU spreadsheet (download from https://newtonexcelbach.wordpress.com/2016/08/28/c...) 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/
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
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.
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
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.
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
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/2013/03/17/3...
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
Dik
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
The loads and support configuration are arbitary for programme testing/debugging purposes.
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
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.
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
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/
RE: Calculation for multi-span continuous straight beam on supports with rotational restraint
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.