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# Statically indeterminate continuous beam using double integration

## Statically indeterminate continuous beam using double integration

(OP)
Hi Everyone,

I'm trying to find the reaction load at point A using the double integration method. (I know there are simple and less time consuming method than double integration).

The problem I have is, I'm getting different reaction load when I compare it with the reaction load I get from SAP model program. (difference by almost 4kN)

Can anyone find any problem or error with my hand calculation? Does my calculation make sense? or am I making any mistake?

Thank you

### RE: Statically indeterminate continuous beam using double integration

In equation 4, shouldnt the 0.25Ra x^3 term actually be 0.5/3 instead of 0.5/2?
I stopped going further.

### RE: Statically indeterminate continuous beam using double integration

Be aware that Sap automatically considers shear deformation effects which you don’t consider in your calculations.

Jason McKee
proud R&D Manager of
Cross Section Analysis & Design
Software for the structural design of cross sections
Moment Curvature Analysis
Interaction Diagrams
Reinforcement Design etc.

### RE: Statically indeterminate continuous beam using double integration

I'd imagine the shear deformations would be fairly minor in comparison to the flexural deformations given the length of spans involved compared with the implied beam depths. So percentage error would be fairly small between a analysis accounting & not accounting for shear deformations, certainly enough to get fairly close agreement to confirm the hand or computer analysis was of the correct order. Suspect the rounding off of significant figures involved in some terms in the hand analysis would have more of an effect on the 'exact' answer than the exclusion of the shear deformation.

Don't discount the fact that there might be something wrong in the SAP2000 model also! (You never know, I've certainly seen my fair share of people including myself on occasion accidentally entering loads or stiffnesses and being an order (or two) of magnitude out)

### RE: Statically indeterminate continuous beam using double integration

I would approach this a different way:

Remove the reaction at B and replace the cantilever with a shear and moment applied at C. compute the deflection at B for the new reduced simply structure then compute the restoring force ie the load at B that would produce the inverse of that deflection on a single simple span beam. Once you know the reaction at B then it is just simple statics to get the reaction at A.

Looks like you swapped your sign convention on the moment at section D for Md and section E for Me.
EIY at section D 0.5/2 Ra x^3 should be 0.5/3 Ra x^3

Open Source Structural Applications: https://github.com/buddyd16/Structural-Engineering

### RE: Statically indeterminate continuous beam using double integration

Hello

I have the beam program in ACAD Mech 2018. If you post your cross section dimensions (for I) I could give you the reactions. It can be done quite quickly.

### RE: Statically indeterminate continuous beam using double integration

I get 4.2182 kN for reaction A.

Try your beam model on Optimal Beam (online beam calculator), it solves statically indeterminate continuous beams using Direct/Matrix Stiffness Method. The free version will give you the reactions.

https://optimalbeam.com/

You can specify unlimited support types and loading conditions on your beam model. It gives you support reactions, shear force, and bending moment diagrams in the free version. The paid gives additional perks such as deflection and stress diagrams, reports, and ability to save models.

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