Force method for more than one unknown displacement

[zaxen]

I have a steel column that i'm designing for significant thermal expansion. It is pinned between two cantilevering concrete slab elevations. The column is located towards the outside edge of the cantilevered slabs. It's being provided to control vertical deflection of these slabs. I am trying to use compatibility equations to determine the accurate compression in the column when it's subjected to high temperature. I am able to successfully apply the force method when accounting for one slab's flexibility, and the other end as fixed, so only one degree of freedom. I am seeking insights as to how to apply force method when there are two unknown displacements.

 

[271828]

Not sure how to explain such a thing in text. Get a decent textbook on basic structural analysis; problems with multiple redundant forces should be covered. In a nutshell, you'll release one of them and write a compatibility equation. Then go release the other and write a second. Two unknowns and two equations, with the redundant forces as the unknowns.

This problem could be solved with SAP2000. Perhaps your analysis program will allow you to apply the temperature change. You could do some easy problems to make sure it's doing what you want.

 

[271828]

See the following, starting on Page 6. It shows the general idea of writing multiple compatibility equations. I think you'll still want to get a book, though.

http://www.engineering.uiowa.edu/~design1/StructuralDesignII/Chapter5-ForceMethod.pdf

 

[paddingtongreen]

Calculate the upward and downward deflections of the slabs under unit load. You now have a spring rate for the slabs. Calculate the contraction of the column under unit load, you now have a spring rate for the column. Calculate the extended length of the column. That extension is the total of the sum of the slab deflections and the column contraction so that the force is the same. Equate the two spring rates to find the force that provides the total deflection.

Michael.
Timing has a lot to do with the outcome of a rain dance.

 

[rb1957]

with two redundancies you'll have two equations to solve (two unknowns).

with one unknown, you remove the redundancy, calc the displacement; then determine the displacement for a unit redundant load; and so the redundant load = displacement/(unit load displacement).

with two unknowns, you remove both, calc displacements at both locations d1 and d2. now apply a unit load at 1, calc displacements at 1 and 2, u11 and u12. similarly unit load at 2, calc displacements at 1 and 2, u21 and u22. then ...
R1*u11+R2*u12 = d1 and
R1*u21+R2*u22 = d2
(assuming of cource that the unit directions oppose the d1 and d2 displacements).

clear as mud ?