Stress analysis of curved pipes with a hybrid formulation Luı ´sa Madureira a, * , Francisco Q. Melo b a Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Rua Roberto Frias, Porto 4200 465, Portugal b Department of Mechanical Engineering, University of Aveiro, Aveiro, Portugal Received 20 June 2003; revised 14 January 2004; accepted 14 January 2004 Abstract This work describes a hybrid formulation based on variational techniques, where unknown functions are combined with Fourier trigonometric expansions. The unknown functions refer to the toroidal shell distortion along the longitudinal direction, when the shell is submitted to generalised in-plane forces in the linear-elastic stress field. This set of functions is obtained by carrying out evaluations with a mathematical computer system to solve a system of differential equations, using eigenvalues and corresponding eigenvectors in the complex field. The analytical solution here developed is exact for each Fourier term used. The resulting stress distribution agrees very well with other methods obtained using a total expansion of trigonometric functions in the longitudinal and meridional directions. q 2004 Elsevier Ltd. All rights reserved. Keywords: Curved pipes; Hybrid formulation; Fourier series; System of ordinary differential equations; Eigenvalues 1. Introduction The stress analysis of curved pipes plays an important role in the design and integrity assessment of this type of structural component in piping engineering. The determi- nation of the stress distribution is a complex task [1] as it is not possible to achieve a solution defined with elementary mathematical functions. A popular procedure for numerical analysis consists in the use of finite element techniques [2–8]. Quite common doubly curved finite shell elements are based on assumed displacement fields where the problem solution consists of solving a system of equations where the unknown vector includes only nodal displace- ments. Case studies where the bending of curved pipes also includes association with tangent cylindrical elements face an important problem arising from the deformation discontinuity in the transition section between the curved element and the tangent part [9–12]. The present procedure proposes a hybrid approach to the problem where the vector of unknown variables includes shell forces and moments associated with displacements [13]. This technique overcomes the limitation of the classical totally assumed displacement solution, where deformation discontinuities may arise in the changes of the shell geometry [8]. The stress distribution in this type of shell structure is complex and it is only possible to develop approximate solutions. The hybrid solution here described proposes the combination of unknown functions with Fourier series, both for the forces and the displacement field of the toroidal shell. Here only one set of Fourier expansions is needed, being such set of functions defined along the circumferential direction of the curved pipe. These Fourier expansions are combined with analytical functions evaluated from a set of linear ordinary differential equations with constant coefficients. This method gave rise to a solution where only four Fourier terms were retained, improving the solutions with three terms published in Ref. [13]. Despite the low number of trigonometric terms, the solution showed remarkable precision when compared to other classical approaches. 2. Geometry and hybrid parameters A curved pipe is considered as a part of a toroidal shell, where the bend angle defines the extreme section distance. Fig. 1 shows the geometric parameters considered in the analysis. 0308-0161/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2004.01.001 International Journal of Pressure Vessels and Piping 81 (2004) 243–249 www.elsevier.com/locate/ijpvp * Corresponding author. Tel.: þ351-22-5081413; fax: þ 351-22-508- 1445. E-mail addresses: luisa.madureira@fe.up.pt (L. Madureira); fqm@ mec.ua.pt (F.Q. Melo).