Prestress estimation in elastic rods using dispersion-constrained inversion

Abstract

The theory of waveguides has been applied to various problems of engineering, including the evaluation of prestress in reinforced concrete members and non-destructive inspection or monitoring of long slender structures such as pipes and rails. In prestressed concrete beams or slabs, it is crucial to estimate the current level of prestress in the steel tendons for assessing the limit loads that these structural members can withstand. In this paper, we present a dispersion-constrained inversion for predicting the initial stress of circular-cross-section elastic rods on the basis of its harmonic torsional response. In the forward problem, the displacement is computed by implementing a hyper element representing the straight rod. The inversion problem is solved by optimizing the misfit function subjected to the dispersion constraints which are conveyed from the forward problem. The misfit function is defined as the L_2-norm of the difference between the measured and predicted responses and is minimized by means of gradient-based optimization. The gradient is derived in terms of derivatives with respect to the eigenvalues (wavenumbers), eigenvectors (mode shapes), Lagrange multipliers, and the prestress. We then estimate the prestress by solving the optimization problem through an iterative process. The proposed inversion is applied to several examples to examine its performance. Via these numerical examples, we demonstrate that our inverse algorithm can detect the prestress with satisfactory accuracy, even in the presence of noise in the measurements.