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Simplified formulations of mass and geometric stiffness matrices in vibration and stability analyses of thin-walled structures

Ship hydroelastic analysis is a complex task of determining the interaction between coupled structure motion and vibrations with water. In the governing equation of motion the unified restoring and geometric stiffness plays an important role. This paper deals with simplified geometric stiffness formulation which has some advantages in hydroelastic analysis comparing to the consistent one used in stability analysis. From a mathematical point of view, buckling and natural vibrations are similar eigenvalue problems. However, due to the dependency of geometric stiffness on imposed load, buckling is more complicated. Vibration analysis of a thin-walled structure can be performed with a consistent mass matrix determined by the shape functions of all degrees of freedom (d.o.f.) used for construction of conventional stiffness matrix, or with a lumped mass matrix related to deflection d.o.f. In similar way stability of a structure can be analysed with consistent geometric stiffness matrix or geometric stiffness matrix with lumped buckling load, related only to the rotational d.o.f. In this paper, first, the simplified mass matrix for beam element is constructed employing shape functions of in-plane displacements for deflection, and then the same approach is used for construction of simplified geometric stiffness matrix for beam, and triangular and rectangular plate elements. Application of newly developed matrices is illustrated by analyzing natural vibrations of simply supported beam, as well as buckling of simply supported beam, and simply supported plate with different mesh densities. The results of direct calculations are compared with the analytical solution. Also, a comparison with commercial software results is provided. Finally, combinations of simplified and lumped matrices, called hybrid matrices, are analysed in order to increase accuracy of vibrations and stability analyses, respectively. The performed analyses show that the usage of simplified mass matrix in vibration analysis, as well as usage of simplified geometric stiffness matrix in buckling analysis leads to quite good results. In that sense, the application of developed geometric stiffness matrix in ship hydroelastic analysis is reasonable choice.

Mass matrix; Geometric stiffness; Buckling; Stability; Vibration; Numerical solution; Analytical solutions; FEM

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