VIBRATION ANALYSIS OF LAMINATED FUNCTIONALLY GRADED SHALLOW SHELLS WITH CLAMPED CUTOUT OF THE COMPLEX FORM BY THE RITZ METHOD AND THE R-FUNCTIONS THEORY

Abstract

THE R-FUNCTIONS THEORY AND RITZ APPROACH ARE APPLIED TO ANALYZE FREE VIBRATIONS OF LAMINATED SHALLOW SHELLS WITH DIFFERENT TYPES OF CURVATURES AND COMPLEX PLANFORMS. SHALLOW SHELLS ARE CONSIDERED AS SANDWICH SHELLS OF DIFFERENT TYPES: A) FACE SHEETS OF SHALLOW SHELLS ARE MADE OF A FUNCTIONALLY GRADED MATERIAL (FGM) WHILE CORES ARE MADE OF AN ISOTROPIC MATERIAL; B) FACE SHEETS OF SHALLOW SHELLS ARE ISOTROPIC, BUT THE CORE IS MADE OF FGM. IT IS ASSUMED THAT FGM LAYERS ARE MADE OF A MIXTURE OF METAL AND CERAMICS, AND THE EFFECTIVE MATERIAL PROPERTIES OF LAYERS ARE VARIED ACCORDING TO VOIGHT’S RULE. THE FORMULATION OF THE PROBLEM IS CARRIED OUT USING THE FIRST-ORDER (TIMOSHENKO’S TYPE) REFINED THEORY OF SHALLOW SHELLS. DIFFERENT TYPES OF BOUNDARY CONDITIONS, INCLUDING CLAMPED, SIMPLY SUPPORTED, FREE EDGE, AND THEIR COMBINATIONS, ARE STUDIED. THE PROPOSED METHOD AND THE DEVELOPED COMPUTER CODE WERE EXAMINED ON TEST PROBLEMS FOR SHALLOW SHELLS WITH RECTANGULAR PLANFORMS. IN ORDER TO DEMONSTRATE THE CAPABILITY OF THE DEVELOPED APPROACH, NOVEL RESULTS ARE PRESENTED FOR LAMINATED FGM SHALLOW SHELLS WITH A CUT OF A COMPLEX FORM. EFFECTS OF DIFFERENT MATERIAL DISTRIBUTIONS, MECHANICAL PROPERTIES OF THE CONSTITUENT MATERIALS, LAMINATION SCHEME, BOUNDARY CONDITIONS, AND GEOMETRICAL PARAMETERS ON NATURAL FREQUENCIES ARE SHOWN AND ANALYZED.