In the analysis of a slender column, we normally adopt the conventional newmark's procedure as it is simple and quite straight forward. However, this numerical model is dealing with the slender pinned-pinned ends columns. Has any modifIcation has been made with regards to this method to fixed-pinned conditions? Or is there any other theoretical model to cater this to suggestion?
I am not sure if understood well your question. If yes, for the theoretical model of a slender column one can use the analytical solution instead of the numerical Newmark's solver (approximation).
As you may find in any basic book of structural analysis, assuming that the load is increased over a perfect Bernoulli-Euler (i.e. slender, homogeneous, etc) column up to the first bifurcation, which is theoretically given by:
Pcr=pi^2*(E*I)/(Leq^2)
where the effective length Leq depends on the boundary conditions according to:
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Boundary conditions Theory In some Codes
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Both ends pin ended 1.0L 1.0L
Both ends fixed 0.5L 0.65L
One end fixed and the other end pinned 0.707L 0.8L
Dear Miguel, the theoretical model that im talking about is the model which can generated load-deflection relationships as per Newmark's procedure. In newmark's the load-deflection behaviour can be simulated for material which inherent nonlinearities such as rc columns, confned column etc. With all due respect, i dont think that analytical solution can solve the problem especially simulating the load -deflection behaviour of the column (which simply assumed the effective length say 0.7L based on the analysis of buckling of elastic slender column) disregarding the effects of nonlinearities stress-strain to the flexural rigidity of the column. Anyway, thanks for the opinion, really appreciate it.:)
In my opinion, you should distinguish the theoretical model (a differential equation) and a numerical method to solve it. So the newmark procedure to solve numerically a differential equation cannot be considered as a "theoretical model for slender column with pinned-fixed end restraints". I understand from your previous comment that you are interested in non linear stress-strain. A possible starting point would be to write an equilibrium equation, derive the total potential energy and study its stationary point. Under conservative loading this method is referred to Koiter; for some bem models, with linear elasticity you may look in italian to http://it.wikipedia.org/wiki/Carico_critico_euleriano; obvously there exists books and papers on this broad subject.
The attachment is in italian, any other related literature on this matter? Preferably the existing theoretical model which has been done or developed by other researchers. Thanks.
Theory of Beam-Columns, Volume 1: In-Plane Behavior and Design
by Wai-Fah Chen (Author), Toshio Atsuta (Author)
ISBN-13: 978-1932159769
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