Fast and accurate methods for determining pipeline eigenfrequencies and associated bending stresses are essential to free span design, and hence of great interest to the pipeline industry. The Rayleigh-Ritz approach has been applied in combination with a displacement field taken as a Fourier sine series to obtain a novel, exact semi-analytical solution to an initially curved Euler-Bernoulli beam on a partial elastic foundation subject to axial force. The proposed methodology for calculation of free span eigenfrequencies determines both static and harmonic response with excellent accuracy, and includes nonlinear geometric effects and the pipe stiffening effect caused by static deformation due to gravity. This novel method is easily implemented into a computer program, and the calculation speed is greatly improved compared to traditional FE analyses. Compared to the widely applied semi-empirical formulae of Fyrileiv and Mørk (2002), accuracy is increased and the range of applicability extended. Detailed FE analyses have been carried out to validate the model, and results of the semi-analytical model have also been compared to results obtained using current state-of-the-art engineering practice as defined in Det Norske Veritas' recommended practice provisions DNV-RP-F105 of 2006.