IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 _________________________________________________________________________________________ Volume: 03 Issue: 03 | Mar-2014, Available @ http://www.ijret.org 233 BUCKLING ANALYSIS OF LINE CONTINUUM WITH NEW MATRICES OF STIFFNESS AND GEOMETRY C.N Okoli 1 , J.C. Ezeh 2 , O.M. Ibearugbulem 3 1, 2, 3 Civil Engineering Department, Federal University of Technology, Owerri, Nigeria Abstract This research work present buckling analysis of line continuum with new matrices of elastic stiffness and geometric stiffness. The stiffness matrices were developed using energy variational principle. Two deformable nodes were considered at the centre and at the two ends of the continuum which brings the number of deformable node to six. The six term Taylor McLaurin’s shape function was substituted into strain energy equation and the result functional was minimized, resulting in a 6 x 6 stiffness matrix used herein. The six term shape function is also substituted into the geometric work equation and minimized to obtain 6 x 6 geometric stiffness matrix for buckling analysis. The two matrices were employed, as well as traditional 4 x 4 matrices in classical buckling analysis of four line continua. The results from the new 6 x 6 matrices of stiffness and geometry were very close to exact results, with average percentage difference of 2.33% from exact result. Whereas those from the traditional 4 x 4 matrices and 5 x 5 matrices differed from exact results, with average percentage difference of 23.73% and 2.55% respectively. Thus the newly developed 6 x 6 matrices of stiffness and geometry are suitable for classical buckling analysis of line continuum. Keywords: 6x6 stiffness system; buckling; geometry; line continuum; variational principle; deformable node; shape function; classical; numerical; analysis; beam ---------------------------------------------------------------------***--------------------------------------------------------------------- 1. INTRODUCTION The design of multi-storey building frames requires that a structural engineer is familiar with structural instability that can occur in such a building. Hence, thorough analysis and calculations is required. However, the classical method of analysis using the traditional 4 x 4 stiffness matrix system tends to find solution that are close to exact solution. Unfortunately, as observed by Ibearugbulem et al (2013), the traditional 4 x 4 stiffness matrix and its load vector cannot classically analyze flexural line continua except using them numerically. This difficult in using the traditional classical approach is evident in the work of Iyengar (1988), Chopra (1995) and Yoo and Lee (2011). The traditional 4 x 4 numerical method of analysis (more than one element in one continuum) becomes a good alternative to the tradition 4 x 4 classical method. The problem with numerical method is that, it is tedious (Melosh, 1963; Long, 1973, 1992, 2009. Cook et al, 1989; Huebner et al, 1995; Bath, 1996; Zienkiewicz and Taylor, 2000; Ibearugbulem et al, 2013) and frequently give results that differ greatly with exact classical results. Hence there is need for classical matrix approach that would be less cumbersome and at the same time give results that are close to exact results. Ibearugbulem et al (2013) developed 5 x 5 stiffness matrices capable of classically analyzing stability and dynamic line continuum, but some of their solutions are not yet exact solution. This work present buckling analysis of line continuum with new matrices of stiffness and geometry. The governing differential equation of line continuum is first integrated to obtain a general solution of the continuum that has specific number of terms. The general solution obtained is used in energy variational principle to get a new and more reliable 6 x 6 stiffness matrices for classical buckling analysis of line continuum. 2. DIRECT INTEGRATION OF GOVERNING EQUATION The line continuum governing equation is: The solution of the equation (1) is assumed thus: Equation (2) can be written as: