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The motivation for this work is the development of load measurement techniques based on the velocity of propagating guided waves in structural members such as cable and rail. A finite element technique for modelling the dispersion characteristics of guided waves in a waveguide of arbitrary cross section subjected to axial load is presented. The results from the FE model are compared to results obtained from a simple Euler-Bernoulli beam model. A dimensionless measure of the sensitivity of phase and group velocity to load is defined as the fractional change in velocity divided by the applied strain. At frequency waveguide-characteristic-dimension products (fd) of greater than around 1 for phase velocity and 5 for group velocity the sensitivity to strain levels likely to be encountered in engineering materials is strain independent (indicating that the change in velocity is proportional to strain) and decreases with increasing frequency. In this fd range, phase velocity increases under tensile loading and group velocity decreases. For waveguides with simple cross sections, such as plates and circular rods, it is shown that the Euler-Bernoulli beam model provides acceptable results over the majority of the fd range where there is measurable sensitivity to load. However, for waveguides with more complex cross sections such as rail, the Euler-Bernoulli beam model is less satisfactory. In particular, it does not predict the subtleties of the sensitivity of certain modes at high frequencies, nor any sensitivity for the torsional fundamental mode.