3D Analysis of Right Angle Tunnels under Wave Propagation Effect with BEM

Tunnel is integral part of the infrastructure of modern society and is used for a wide range of applications, including highway, sewage and water transport. This structure built in areas subject to earthquake activity and must withstand safely seismic waves. Seismic waves are characterized in terms of the deformations and strains imposed on the structure by the surrounding ground, often due to the interaction between two. Dynamic analysis of lined tunnel with bending in longitudinal axis is not a common case and is not exist in literature as rich as long lined tunnel with straight axis. In this paper the case was analyzed with BEM in full space time domain. Tunnel with bending in longitudinal axis and large backfilling is dynamically analyzed in three dimensional contexts by assuming linear or viscoelastic material behavior, full contact between soil and structure at their interface without modeling free surface of ground. An analytical boundary integral equation is developed for the efficient time domain dynamic response analysis of a bending cavity with arbitrary cross-section geometry. The numerical solution of boundary integral equation is obtained after discretizing both space and time variation. First curvilinear coordinates, which Z axis is along main longitudinal line, is introduced. Geometry of bend, displacement and traction is expressed in this coordinates. Complex shape of the cross section is mapping into a circular shape with a unit radius by trigonometric shape functions. This mapping enables us to discretize the boundary only along the longitudinal axis of tunnel. Therefore boundary is divided into N circular element this is because of advantages of circular element. Harmonic substantial behavior of deformation and traction, which expressed in cylindrical coordinates, results their variation with central angle expressed in Fourier series. This helps us to decrease the number of equation very much. Observation time is divided into M time step. Appropriate time step is chose because it is very important for numerical stability of transient dynamic problems. Weak and strong singularity occurs during integration process. Direct approach for the rigorous treatment and numerical evaluation is used. For this purpose circular element divided into subdivision elements and integration is done in each element. Singular points exist in one subdivision. Polar coordinates and CPV sense is used for evaluating strong singularity and three coordinates transformation is used for eliminating weak singularity. After all, equation should be solved for the total displacement and tractions at inner and outer surface of tunnel. This is accomplished by especial out of core block equation solver of gauss type for non-symmetric matrices. Numerical results are presented for the case of lined curve tunnel of circular cross section with some values in bending radius, tunnel radius and thickness of tunnel. Plain strain theory results are compared with the straight portion of the model.