Indeterminate Beam Formulas

[odu0485]

Are commonly published indeterminate beam formulas for continuous beams only for use with a constant cross-section? I assume so. When you have varying cross-sections as in many steel bridge plate girders with flange transitions, you can't just simply apply these common equations, correct? You'd have to use something like stiffness method or computer software to determine moments and shears, correct?

 

[JAE]

Yes, correct.


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[IDS]

Correct.

The ConbeamU spreadsheet at the link below has a function to return fixed-end actions for a beam with any number of different segments, from which you can derive moment distributions for continuous beams. Or if you prefer, a function to analyse continuous beams with varying sections under any combination of trapezoidal and point loads.

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

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[STpipe]

You could also use the moment-distribution method. Simple enough and takes into account the changing stiffness of members.

 

[odu0485]

Great, thanks. For some reason, I always had a hard time remembering moment-distribution method. I always liked slope-deflection method. If you were to use slope-deflection, how would you handle a flange transition in mid-air (not at a support)? I know at supports, you'd break the beam up into sections between supports and determine fixed-end moments and go on from there. How do you apply this method at a cross-section change other than at a support?

 

[IDS]

How do you apply this method at a cross-section change other than at a support?

Just the same. You divide the beam into segments of constant EI and follow the same procedure.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[odu0485]

Got it. So I'd still calculate fixed-end moments on each side of the flange transition whether at a support or not and calculate rotations at each transition as well as at supports, then finally calculate moments at ends of each beam segment?

 

[IDS]

Got it. So I'd still calculate fixed-end moments on each side of the flange transition whether at a support or not and calculate rotations at each transition as well as at supports, then finally calculate moments at ends of each beam segment?

You don't need fixed-end moments at the flange transitions, only at supports.

Or you can do without fixed-end moments altogether. In my Conbeam spreadsheet the procedure used for a continuous beam with simple supports is:

Calculate end reactions and moments along the beam with end supports only.
Integrate M/EI twice to find slope and deflection at each support assuming no rotation at left hand end.
Rotate to bring right hand end back to zero deflection.
Apply unit load to each internal support, and calculate resulting deflections at each support location, assuming single span.
Solve simultaneous equations to find support reactions required for zero deflection at all internal supports.
Recalculate: total reactions, shear forces, moments, curvatures, slopes, and deflections along the beam.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/