Continuous beam deflection using method of superposition

[georgacus]

How can I get the final deflected shape using the method of superposition? My cantilever has 3 spans with 4 supports. See the pic attached which I've drawn using my new-found paint skills . At the fourth step, I only know how to get the deflected shape when either of the two middle supports are active. Not sure how to put it all together to get the final shape (step 5).

I've gotten the shapes so far by setting the deflection at the support points to be 0.

How do I go about doing this?

 

[BAretired]

The structure is indeterminate to the third degree.

1. Assume a unit load at each hinged support.
2. Calculate moment diagram for each unit load.
3. Calculate deflection at each support for each unit load.
4. Solve for R1, R2 and R3 such that Δ1, Δ2 and Δ3 = 0 under assumed load. There will be three equations to solve.

An easier way is to use a 2D frame program.

BA

 

[ChadV]

How are you solving the cantilevers with 1 and 2 supports? I'm not clear what you mean by "I've gotten the shapes so far by setting the deflection at the support points to be 0", and I can't think of a method I would use for a two-degree indeterminate structure that couldn't be easily applied to a three-degree indeterminate one (other than looking it up in a table or using a frame analysis program).

 

[rb1957]

you cannot superimpose the three displaced shapes (and get the right answer).

unit load method will do it, as described by BA.

from 1) (the cantilever solution) obtain the displacements at the 3 support positions, call them D1, D2, D3.

now apply a unit load at each loading point (one at a time) and obtain the displacements at the three loading points. eg, apply a unit load at point 1 (so you have a cantilever with a point load applied, yes?), displacements at the three loading points are d11, d12, and d13.
with the other unit loads you get d21, d22, d23, d31, d32, d33

then the idea is the three redundant support reactions (P1, P2, P3) drive the determinate solution (the loaded cantilever, 1), above) to have zero displacement at the supports, something like [d]*(P) = -(D).

it's easy to see this method work with one redundancy.

with mutiple redundancies, you might try moment distribution or "three moment" equation.


Quando Omni Flunkus Moritati

 

[IDS]

Probably the simplest way without being totally "black box" is get the reactions from a frame or continuous beam program, then apply all the loads (applied loads + reactions at pinned supports) to a cantilever analysis to get the deflections.

You could get the deflections from the computer as well of course, but calculating your own deflections you can verify that the computer reactions are consistent with zero deflection at the supports.

A free continuous beam spreadsheet can be found at:

http://newtonexcelbach.wordpress.com/2012/09/06/continuous-beam-spreadsheet-with-units/

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[msquared48]

Or spend $40.00 and get "Beampro". I use it every day for problems like this...

Mike McCann, PE, SE (WA)

 

[georgacus]

Thanks for the suggestions everyone.

I'm creating an excel spreadsheet that calculates the reactions, shear, moments, displacements. I took rb's idea to use moment distribution method and it's working out pretty well so far. Structural Analysis by Hibbeler has a nice section on it.

I'm getting my results verified using STAAD.Pro

 

[boyze]

perhaps this could help

Have Fun!

James A. Pike

http://www.xl4sim.com

http://www.erieztechnologies.com