Continuous Beam Deflections from Moment Diagram Non Prismatic

[Taimoor957]

I am building a spreadsheet for a continuous beam upto 4 spans and cantilevers (each span having three segments of different EI). I have done reactions via slope deflection method and then built shear diagram. For moment diagram I have used just numerical integration of shear diagram. I now want to find deflection at each of the points which i used for building shear and bending.

What I understand is i can use conjugate beam or moment area method for which i will require to get reactions of M/EI diagram and it will be quite of a work for irregular shaped bending moment diagram. Is there any simple way i can achieve it numerically if any of you have can think of simpler solution?

 

[IDS]

I am building a spreadsheet for a continuous beam upto 4 spans and cantilevers (each span having three segments of different EI). I have done reactions via slope deflection method and then built shear diagram. For moment diagram I have used just numerical integration of shear diagram. I now want to find deflection at each of the points which i used for building shear and bending.

What I understand is i can use conjugate beam or moment area method for which i will require to get reactions of M/EI diagram and it will be quite of a work for irregular shaped bending moment diagram. Is there any simple way i can achieve it numerically if any of you have can think of simpler solution?

If you are happy doing numerical integration to find the moment diagram, you can integrate that to get the slope, and integrate the slope to get the deflection.

You might like to have a look at my ConBeam spreadsheet that has functions that use that approach that will work for any number of spans and different section properties. You can download it from my blog (see link below), and there are VBA and Python versions. Also quite a few posts giving more details of the procedure (search for ConBeam and Macaulay's method).

 

[KootK]

each span having three segments of different EI

If the EI will step at discrete point rather than varying continuously, I would consider using the direct stiffness matrix method. Easy to program. You'd just put some nodes in at the EI steps programmatically.

 

[Taimoor957]

If you are happy doing numerical integration to find the moment diagram, you can integrate that to get the slope, and integrate the slope to get the deflection.

You might like to have a look at my ConBeam spreadsheet that has functions that use that approach that will work for any number of spans and different section properties. You can download it from my blog (see link below), and there are VBA and Python versions. Also quite a few posts giving more details of the procedure (search for ConBeam and Macaulay's method).

Thank you Doug for sharing that link, that's a treasure of knowledge and functions at no cost

I was trying to numerically double integrate moment diagram however couldn't figure out how to evaluate cantilever deflections and converge the deflections to zeros at the supports. I like the functions in your spreadsheet and will try to utilize them, new excel has now 'insert python' button in formulas tab so can be a possibility since the business i work with may not decide to buy or install pyxll

 

[Taimoor957]

If the EI will step at discrete point rather than varying continuously, I would consider using the direct stiffness matrix method. Easy to program. You'd just put some nodes in at the EI steps programmatically.

Thank you for the reply, yes they can have tapered segments and you are right will be simpler with direct stiffness method. I have done similar to what Doug mentioned except for using UDFs, i solved slope equations and fixed end moments for varying EI etc. in python symbolically and then created formulas to insert in excel and found joint moments, reactions, shear and moments.

With my original problem which is just a continuous beam so if internal supports can become hinges (M_conjugate = 0) and cantilever ends as fixed for conjugate beam, i may be able to figure out deflections by finding centroids + areas in the M/EI diagram drawn numerically. I will give it a go tomorrow.