# CRITICAL BUCKLING LOAD BASED ON MODAL ANALYSIS BY RAYLEIGH METHOD AND FINITE ELEMENT METHOD OF A NONPRISMATIC SLENDER REINFORCED CONCRETE COLUMN

## Palavras-chave:

Buckling load, Modal analysis, Rayleigh method, Shape functions, Analytical solution, Finite element method## Resumo

In this study, an analytical and computational procedure were developed for determining the

critical buckling load. The analytical solution was based on the Rayleigh method and the computational

one on the finite element method (FEM). Rayleigh method preconize that one equation named as shape

(or trial) function should be defined to represent the vibrational movement of the system. Therefore, the

result obtained by this method is entirely conditioned to the correct choice of this equation. Different

equations even respecting the boundary conditions of the problem can lead to different results. Four

mathematical expressions as shape function were used in the present study: a trigonometric, two

polynomials and a potential equation. All these functions obey to the boundary conditions of the problem

and were valid in the whole domain. Therefore, the integrals obtained by the Rayleigh method were

solved considering the structural geometry. With comparative purpose the results obtained on the

analytical procedure were compared with those yielded by computational modelling using a finite

element modal analysis. The structure analyzed was a 46-m-high reinforced concrete pole, including its

foundation, which has geometry and reinforcement arrangement varying along its length. For both

solutions, three important items were considered: the geometric nonlinearity, due to the slenderness of

the system; the material nonlinearity and the creep of the concrete. The last one aspect was introduced

into the analysis by means of Eurocode criteria. Significant differences on the absolute value of the

critical load were found in comparison with the adopted procedures, being possible to observe that the

potential equation led to results too distant from the other equations. Analysis considering an elapsed

time of 4000 days revealed an average decreasing of 22% on the intensity of the critical buckling load.

The FEM presented the biggest percentual difference, 28%.