A DIRECT TECHNIQUE FOR THE HOMOGENIZATION OF PERIODIC BEAM-LIKE STRUCTURES BY TRANSFER MATRIX EIGEN-ANALYSIS

Abstract

A DIRECT APPROACH TO HOMOGENIZE LATTICE BEAM-LIKE STRUCTURES BY MEANS OF THE MATRIX EIGEN- AND PRINCIPAL VECTORS OF THE STATE TRANSFER MATRIX, IS PROPOSED AND DISCUSSED IN THE PRESENT PAPER. THE TIMO-SHENKO COUPLE-STRESS BEAM IS THE EQUIVALENT CONTINUUM MEDIUM ADOPTED FOR THE HOMOGENIZATION. THE UNIT CELL OF THE EXAMINED GIRD-ERS TRANSMITS TWO BENDING MOMENTS: ONE GENERATED BY THE COUPLE GIVEN BY THE AXIAL FORCES ACTING ON EACH NODAL SECTION, THE OTHER PRO-DUCED BY THE MOMENTS APPLIED AT THE NODE SECTIONS BY THE ADJACENT CELLS. THIS LATTER MOMENT IS MODELLED AS THE RESULTANT OF COUPLE-STRESS. THE MAIN ADVANTAGE OF THE METHOD IS TO OPERATE DIRECTLY ON THE SUB-PARTITIONS OF THE UNIT CELL STIFFNESS MATRIX. CLOSED FORM SOLU-TIONS FOR THE TRANSMISSION PRINCIPAL VECTORS OF THE PRATT AND X-BRACED GIRDERS ARE ALSO ATTAINED AND EMPLOYED TO DETERMINE THE STIFFNESSES OF THE RELATED EQUIVALENT BEAMS. UNIT CELLS HAVING MORE COMPLEX GEOMETRIES ARE ANALYSED NUMERICALLY AND, BY THE DIRECT APPROACH, IT IS SHOWN THAT THE PRINCIPAL VECTOR PROBLEM IS ALWAYS REDUCED TO THE INVERSION OF A WELL-CONDITIONED MATRIX. HENCE, ILL-CONDITIONING PROBLEMS, AFFECTING ALL THE KNOWN TRANSFER METHODS, ARE DO NOT SHOW UP IN OUR METHOD. FINALLY, COMPARING THE PREDICTIONS OF THE HOMOGE-NIZED MODELS WITH THE RESULTS OF A SERIES OF GIRDER ANALYSES CARRIED OUT WITH THE FINITE ELEMENT METHOD, A VALIDATION OF THE HOMOGENIZA-TION METHOD IS PERFORMED.