^{1}School of Civil Engineering National Technical University of Athens (NTUA) Zografou Campus Athens 15773 Greece

^{2}Department of Mathematics, University of Patras, Rio

Abstract

The BEM is applied to the solution of the torsion problem of non-homogeneous anisotropic non-circular prismatic bars. The problem is formulated in terms of the warping function. This formulation leads to a second order partial differential equation with variable coefficients, subjected to a generalized Neumann type boundary condition. The problem is solved using the Analog Equation Method (AEM). According to this method, the governing equation is replaced by a Poisson’s equation subjected to a fictitious source under the same boundary condition. The fictitious load is established using the Boundary Element Method (BEM) after expanding it into a finite series of radial basis functions. The method has all the advantages of the pure BEM, since the discretization and integration are limited only on to the boundary. Numerical examples are presented which illustrate the efficiency and accuracy of the method.

[1] Lekhnitskii, S.G., Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day Inc., San Francisco, Calif., 1963.

[2] Muskhelishvili, N.I., Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff, Groningen, Holland, 1953.

[3] Sokolnikoff, I.S., Mathematical Theory of Elasticity, McGraw-Hill Book Co., Inc., New York, N.Y., 1956.

[4] Timoshenko, S. and Goodier, J.N., Theory of Elasticity, McGraw-Hill Book Co., Inc., New York, N.Y., 1970.

[5] Martynovich, B.T. and Martynovich T.L., “Use of Integral Equations in the Solution of Problems of Torsion of Rectilinear-Anisotropic Rods”, Izv. AN SSR, Mekhanika Tverdogo Tela, Vol. 19, pp. 112‑118, 1984.

[6] Shaw, F.S., “The Torsion of Solid and Hollow Prisms in the Elastic and Plastic Range by Relaxation Methods”, Report No. ACA-11, Australian Council for Aeronautics, 1944.

[7] Ely, J.F. and Zienkiewicz, O.C., “Torsion of Compound Bars-A Relaxation Solution”, International Journal of Mechanical Science, Vol. 1, pp. 356‑365, 1960.

[8] Zienkiewicz, O.C. and Cheung, Y.K., “Finite Elements in the Solution of Field Problems”, The Engineer, Vol. 220, pp. 507‑510, 1965.

[9] Hermann, L.R., “Elastic Torsional Analysis of Irregular Shapes”, Journal of the Engineering Mechanics Division, ASCE, Vol. 91, No. EM6, Proc. Paper 4562, pp. 11‑19, 1965.

[10] Krahula, J.L. and Lauterbach, G.F., “A Finite Element Solution for Saint-Venant Torsion”, American Institute of Aeronautics and Astronautics Journal, Vol. 7, pp. 2200‑2203, 1969.

[11] Valliappan, S. and Pulmano, V.A., “Torsion of Nonhomogeneous Anisotropic Bars”, Journal of the Structural Division, Proc. of the ASCE, Vol. 100, No. ST1, pp. 286‑295, 1974.

[12] Jaswon, M.A. and Ponter, A.R.S., “An Integral Equation Solution of the Torsion Problem”, Proceedings of the Royal Society (London) A, Vol. 273, pp. 237‑246, 1963.

[13] Ponter, A.R.S., “An Integral Equation Solution of the Inhomogeneous Torsion Problem”, SIAM Journal of Applied Mathematics, Vol. 14, pp. 819‑830, 1966.

[14] Katsikadelis, J.T. and Sapountzakis, E.J., “Torsion of Composite Bars by Boundary Element Method”, Journal of the Engineering Mechanics Division, ASCE, Vol. 111, pp. 1197‑1210, 1985.

[15] Chou, S.I. and Mohr, J.A., “Boundary Integral Method for Torsion of Composite Shafts”, Res. Mechanica, Vol. 29, pp. 41‑56, 1990.

[16] Hromadka II, T.V. and Pardoen, G., “Application of the CVBEM to nonuniform St. Venant Torsion”, Computer Methods in Applied Mechanics in Engineering, Vol. 53, pp. 149‑161, 1985.

[17] Chou S.I. and Shams-Ahmadi, M., “Complex Variable Boundary Element Method for Torsion of Hollow Shafts, Nuclear Engineering Design”, Vol. 136, pp. 255‑263, 1992.

[18] Shams-Ahmadi, M. and Chou, S.I., “Complex Variable Boundary Element Method for Torsion of Composite Shafts”, International Journal for Numerical Methods in Engineering, Vol. 40, pp. 1165‑1179, 1997.

[19] Dumir, P.C. and Kumar, R., “Complex Variable Boundary Element Method for Torsion of Anisotropic Bars”, Applied Mathematics Modelling, Vol. 17, pp. 80‑88, 1993.

[20] Sapountzakis, E.J., “Solution of non-uniform torsion of bars by an integral equation method”, Computers and Structures, Vol. 77, pp. 659‑667, 2000.

[21] Sapountzakis, E.J., “Nonuniform Torsion of multi-material composite bars by the boundary element Method”, Computers and Structures, Vol. 79, pp. 2805‑2816, 2001.

[22] Ecsedi, I., “Some analytical solutions for Saint-Venant torsion of non-homogeneous cylindrical bars”, European Journal of Mechanics A/Solids, Vol. 28, pp. 985–990, 2009.

[23] Rongqiao, X., Jiansheng, H. and Weiqiu, C., “Saint-Venant torsion of orthotropic bars with inhomogeneous rectangular cross section”, Composite Structures, Vol. 92, pp. 1449–1457, 2010.

[24] Ecsedi, I., “Some analytical solutions for Saint-Venant torsion of non-homogeneous anisotropic cylindrical bars”, Mechanics Research Communications, Vol. 52, pp. 95– 100, 2013.

[25] Katsikadelis, J.T., “The analog equation method. A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies”, Theoretical and Applied Mechanics, Vol. 27, pp. 13–38, 2002.

[26] Katsikadelis, J.T., The Boundary Element Method for Engineers and Scientists, 2nd Edition, Academic Press, Elsevier, UK, 2016.

[27] Tsiatas, G.C. and Katsikadelis, J.T., “Large deflection analysis of elastic space membranes”, International Journal for Numerical Methods in Engineering, Vol. 65, pp. 264‑294, 2006.

[28] Golberg, M.A., Chen, C.S. and Karur, S.P., “Improved multiquadric approximation for partial differential equations”, Engineering Analysis with Boundary Elements, Vol. 18, pp. 9‑17, 1996.