Buckling And Vibration Analyses Of Line Continuum Using 6 X 6 Stiffness Matrices

ABSTRACT 

This research work presents buckling and vibration analyses of line continuum using 6 x 6 stiffness matrices. The study used six term Taylor McLaurin’s polynomial series shape function and the stiffness matrices were developed using energy variational principle. Two deformable nodes were considered at the centre of the continuum which brings the number of deformable nodes to six. The six term Taylor McLaurin’s shape function was substituted into strain energy equation and the resulting functional was minimized, resulting in a 6 x 6 stiffness matrix. The six term shape function was also substituted into the geometric work and inertia work (kinetic energy) equations respectively and minimized to obtain 6 x 6 geometric and inertia stiffness matrices for buckling and vibration analyses respectively. They were used together with the traditional 4 x 4 matrices, in classical and numerical studies for buckling and vibration analyses of four line continua and a portal frame. The results of the critical buckling load, from classical analysis (exact solution) for P – R, C – C, C – R and C – F columns were 9.87KN, 39.45KN, 20.19KN and 2.47KN respectively.  Results of the classical buckling analysis of the present study (6 x 6 stiffness matrices) for P – R, C – C, C – R and C – F columns were 9.875KN, 42KN, 20.286KN and 2.47KN respectively. The results of natural frequency from classical analysis (exact solution) for P – R, C – C, C – R and C – F beams were 9.87Hz, 22.37Hz, 15.42Hz and 3.52Hz respectively, while the results of natural frequency from classical analysis of the present study (6 x 6 stiffness system) for P – R, C – C, C – R and C – F beams were 9.87Hz, 22.45Hz, 15.43Hz and 3.52Hz respectively. Results of fixed base portal frame analysis using 4 x 4 stiffness matrices were 0.7172KN and 2.4228Hz for critical buckling load and natural frequency respectively, while the results of fixed base portal frame analysis using the present study (6 x 6 stiffness matrix) were 0.9795KN and 2.5291Hz for critical buckling load and natural frequency respectively. The above results showed that Taylor MaClaurin’s series (polynomial) truncated at six term, gave better approximate solution. 

Keywords: Buckling; Classical; Deformable Node; Geometry; Inertia; Line Continuum; Numerical; Shape Function; 6 x 6 Stiffness Matrix; Vibration.