## A Computerized Symbolic Integration Technique for Development of Triangular and Quadrilateral Composite Shallow-shell Finite Elements |

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Additions Multiplications analytic expressions APPENDIX arithmetic operations required bubble mode C-integrals coefficients of shell combinations of integrals computationally Computerized symbolic integration consistent load coefficients consistent mass corner nodes defined in equations degrees of freedom density parameters dihedral group eight nodes element domain element stiffness matrix elemental matrices elements developed extensional finite elements FINITE-ELEMENT floating-point arithmetic operations formulation FORTRAN freedom associated geometric stiffness group transformations group-theoretic techniques kth layer Langley Research Center linear combinations linear transformations MACSYMA nodal coordinates number of floating-point number of nodes Number of numerically Number of representative number of shape numerical evaluation numerical quadrature approach Numerically Integrated parallelogram elements phase of computation Poisson's ratio present study quadrature points quadrilateral elements ratios shape functions shell boundary shell element shell stiffnesses stiffness coefficients superscripts symbolic integration approach symbolic phase SYMINSE program symmetry group Taylor series tion trapezium Trapezoid triangular and quadrilateral triangular elements weighting coefficients