2D Numerical Modeling of Pipeline Structures with Non-Axisymmetric Loads and Spectral Elements for Identifying Reliable Sensor Spacing

Abstract

The stable leak detection in gas pipelines using a continuous monitoring system can be used as an early diagnostic tool to prevent catastrophic failures. Acoustic Emission (AE) method has advantages among other Structural Health Monitoring systems through detecting leaks in real time with long range sensors as well as locating leak positions as multi-dimensional space while sensor spacing is a key to detect the target leak rate reliably. The leak amplitude depending on the pipe condition (e.g. internal pressure, buried or on ground, leak rate) and the attenuation curves are needed to define the reliable AE sensor spacing. In this paper, the attenuation profiles of various dimensions (thickness and diameter) pipelines and frequencies are obtained through efficient 2D finite element modeling. The attenuation profiles obtained from numerical studies are combined with the laboratory scale pipe testing that simulates a range of leak rates and pipe conditions. The classical finite element method along with Fourier expansion and superposition technique is studied in literature in order to analyze axisymmetric solids subjected to non-axisymmetric dynamic loading. The superposition method conjoint with traditional formulation is not beneficial as compared to three-dimensional analyses when the Fourier expansion requires many harmonics to represent the non-axisymmetric load, which occurs in case of concentrated load (i.e. leak). This paper presents an efficient formulation of axisymmetric pipelines with non-axisymmetric loading, specifically concentrated load representing leak in pipelines, using the spectral element method which is a special form of finite element based on the nodal coordinates of Lagrange shape functions obtained from the solutions of the orthogonal polynomials such as Legendre, Chebyshev. The algorithm using spectral element method is developed to model a 2D axisymmetric metal pipe. An efficient mathematical approach for the calculation of stiffness matrix in the case of ring element with rectangular cross section is developed in order to minimize the computational time for each harmonic of tangential Fourier expansion and corresponding deformation field. The paper provides the AE sensor spacing needed for the target leak rates for a range of pipe dimensions and wave frequencies, which can be utilized to define the sensor spacing for a particular pipeline and sensor frequency.