Keywords: dynamic stiffness matrix, shell of revolution, continuous element, harmonic response, distributed load, dynamic transfer matrix.

This paper shows how distributed loads could be taken into account in the dynamic stiffness method [1,2] also known as the continuous element method [3]. These kinds of methods are particularly suitable to deal with dynamic problems for a large frequency range [4]. They are based on the construction of the Dynamic stiffness matrix that gives an accurate relation between force and displacement fields defined on the boundaries of structural elements. Discretization of the structure is cut down to a minimum that is only defined by its topology.

Several kinds of elements have been developed: straight beams, curved beams, plates [5], axisymmetric shells [6]. The lack of discretization implies that loadings have to be defined on the boundaries. Concentrated or linearly distributed loads are taken into account with the definition of the necessary number of elements in such a way that they are applied on a given boundary. On the other hand, distributed loads such as pressure and surface forces were not taken into account in these works. This paper discards this limitation.

The described procedure is based on force and displacement solutions of a non-homogeneous first-order differential system. This procedure is implemented in the case of an axisymetric shell continuous element. The non-homogeneous term of the equation is defined from distributed loads [7]. It gives rise to a particular solution that complements the dynamic transfer relation between the unknowns defined on the edges of the element. Inversion of this relation allows us to obtain a new dynamic stiffness relation. This relation involves the dynamic stiffness matrix and a complementary force vector that is applied on the edges. This vector can be seen as an equivalent concentrated load that gives equivalent displacements of the edges to those given by distributed loads. Solution of the dynamic stiffness relation allows us to obtain the unknown boundary displacements for a given distributed load. Post-processing consists in computing displacements of any point of the shell. This procedure is achieved by a numerical integration of the state vector solution between the two edges.

The results obtained by this procedure are compared with finite element results. Harmonic responses of a cylinder submitted to an internal pressure and a radial distributed load are computed and are compared with finite element responses. A good agreement and a significative saving of computing time are observed.

1

R.W. Clough, J. Penzien, "Dynamic of structures", Mc Graw-Hill, New-York, 1975.

2

A.Y.T. Leung, "Dynamic stiffness and substructures", Springer, London, 1993.

3

P.H. Kulla, "Continuous elements, some practical examples", ESTEC Workshop Proceedings : modal representation of flexible structures by continuum methods, 1989.

4

J.B. Casimir, C.Duforet, T.Vinh, "Dynamic Behaviour of Structures in Large Frequency Range by Continuous Element Methods", Journal of Sound and Vibration, 267, 1085-1106, 2003. doi:10.1016/S0022-460X(02)01533-X

5

J.B. Casimir, S.Kevorkian, T.Vinh, "The Dynamic Stiffness Matrix of two-dimensional elements : application to Kirchhoff's plate continuous element.", Journal of Sound and Vibration, 287, 571-589, 2005.

6

J.B. Casimir, M.C. Nguyen, I. Tawfiq, "Thick shells of revolution : Derivation of the dynamic stiffness matrix of continuous elements and application to a tested cylinder". Computers and Structures, 85, 1845-1857, 2007. doi:10.1016/j.compstruc.2007.03.002

7

J.L. Batoz, G. Dhatt, "Modélisation des structures par éléments finis Vol. 3", Hermes, Paris, France, 1990.

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