J Mar Sci Technol (2005) 10:96–102 DOI 10.1007/s00773-005-0195-0 Numerical weight function method for the structural analysis of ships: a speedy direct calculation with condensed structural information Yoichi Sumi, Takuji Yano, and Anowarul T.M.M. Bashar Department of Systems Design for Ocean-Space, Yokohama National University, Yokohama 240-8501, Japan to arbitrary loading conditions. Recently, Sumi 4 has developed a simple numerical method to calculate a weight function using the finite-element method, by which one can obtain stress intensity factors of arbi- trarily shaped three-dimensional cracks in a compli- cated three-dimensional structure, such as those in weld details in a ship structure. 5 In this article, a numerical weight function method has been further extended to the structural response analyses of two-dimensional elasticity, plate bending, and general three-dimensional shell structures by apply- ing the Maxwell–Betti reciprocal theorem to the origi- nal and properly defined auxiliary problems. The present numerical weight function may be considered as a finite-element version of a Green’s function in an integral equation solution scheme. Although ship structures are certainly analysed by the finite-element method, the weight function approach has not yet been realized. A numerical weight function of a displacement component is derived from the displacement field calcu- lated from the auxiliary problem by using the finite- element method, in which a certain concentrated force is applied at the evaluation point under the same dis- placement boundary condition of the original problem. Similarly, those corresponding to stress components are derived by the displacement field subjected to certain force-couples applied in the vicinity of the evaluation point of stresses. It should be noted that a weight func- tion is a universal function of a structure, which depends only upon the geometry and displacement boundary condition of the structure. Having investigated the accuracy of the method for the two-dimensional elastic problem and the plate- bending problem by considering the effects of the mesh size and the distance of the load application points of the force couple, it is confirmed that the present method may offer a very efficient and accurate numerical tool to analyse a structure subjected to a vast range of loading conditions, because structural responses can simply be Abstract The weight function method was originally derived for crack problems to calculate stress intensity factors for arbitrary loading conditions. In this article, a numerical weight function method has been extended to formulate the struc- tural response analyses of two-dimensional elasticity, plate- bending, and three-dimensional plate-structures by using the finite-element method. The solution procedure is based on the well-known Maxwell–Betti reciprocal theorem, which is ap- plied to the original and properly defined auxiliary problems. The present numerical weight function may be considered as a finite-element version of a Green’s function in an integral equation solution scheme. Although ship structures are cer- tainly analysed by the finite-element method in a practical design procedure, the weight function approach has not yet been realized. The method is very useful for the analysis of structures subjected to a vast range of loading conditions, because structural responses can simply be calculated by the inner product of the universal weight function and load vec- tors. The validity and convergence characteristics of the present method are investigated by two-dimensional elastic and plate-bending problems, respectively. Finally, the method is applied to the calculation of the response amplitude opera- tor of a stress component at a critical structural detail of a double-hull tanker, and the speed and efficiency of the method are quantitatively discussed based on the practical results. Key words Weight function · Reciprocal theorem · Formula- tion of structural analysis · Ship structures Introduction The weight function method was originally derived by Bueckner 1,2 and Rice 3 to calculate the stress intensity factors of two- and three-dimensional cracks subjected Address correspondence to: Y. Sumi (sumi@structlab.shp.ynu.ac.jp) Received: November 22, 2004 / Accepted: February 24, 2005