#Abscissa quadrature point full
#Abscissa quadrature point free

The method is validated through convergence study and comparison with analytical results for a prismatic cantilever circular column in full contact with water. Examples of stepped and parabolic cantilever circular columns are also shown. The method is illustrated with the example of a tapered cantilever circular column. A coordinate ascent hierarchical infinite element based on the wave equation is developed to describe the hydrodynamic pressure. A hierarchical finite element is developed to describe the displacements.

#Abscissa quadrature point free

Small amplitude free vibrations are assumed. The effects of water compressibility and gravity surface waves are considered. The three-dimensional elasticity theory is used to predict accurately local stresses in the flexible circular column. The boundary element mesh can be limited to the surface of the tunnel, significantly reducing the size of the boundary element mesh.ДЄ simple and highly accurate numerical method is presented for the free flexural vibration analysis of water-column interaction systems. The layer interfaces and the surface of the poroelastic half-space no longer have to be meshed, avoiding spurious reflections at mesh truncations. A case study of calculating vibrations from a semi-circular tunnel embedded in a layered poroelastic half-space is presented using the 2.5-D BEM along with the proposed 2.5-D Green's function. The proposed 2.5-D Green's function is verified by comparison with the existing solutions. The solutions presented in this paper are free of numerical instability for the high frequency and large layer thickness. Utilizing the boundary conditions of the free surface, interfaces and bottom half-space, as well as the general solutions, the complete 2.5-D Green's function for a layered poroelastic half-space is obtained using the transmission and reflection matrix (TRM) method. Based on Biot's theory, the general solutions are derived using the potential decomposition method and the Fourier transform. The complete dynamic two-and-a-half-dimensional (2.5-D) Green's function for an internal point load or fluid source buried in a layered poroelastic half-space is derived and applied to the 2.5-D boundary element method (BEM) in this paper. Furthermore, the single-phase elastic soil model would underestimate the vertical displacement responses caused by both the accelerating and decelerating trains at the speed 200В km/h. It is found that the deceleration of moving trains can cause a significant increase to the ground vibrations as well as the excess pore water pressure responses at the train speed 200В km/h. The time domain results are obtained by the fast Fourier transform (FFT). The acceleration or deceleration of the train is simulated by properly choosing the time history of the train speed using Fourier transforms combined with Fresnel integrals in the transformed domain. A detailed track system is considered incorporating rails, sleepers and embankment, which are modeled as EulerвЂ“Bernoulli beams, an anisotropic Kirchhoff plate, and an elastic layer, respectively. The ground is modeled as a saturated poroelastic half-space and Biot's theory is applied to characterize the soil medium, taking the coupling effects between the soil skeleton and the pore fluid into account. The dynamic response of a fully saturated poroelastic half-space due to accelerating or decelerating trains is investigated by a semi-analytical method.