楊永斌臺灣大學：土木工程學研究所許琳青Hsu, Lin-ChingLin-ChingHsu2007-11-252018-07-092007-11-252018-07-092006http://ntur.lib.ntu.edu.tw//handle/246246/50301在人口密集的區域，地下化的軌道運輸系統可有效紓解繁忙的地面交通，已經成為都會地區不可或缺的大眾運輸工具，但是隨著都市建築密度的提高以及人們對於生活環境品質的要求，軌道車輛行駛於地下隧道時所產生之振動往往經由周遭的土壤傳遞到鄰近的建築物，嚴重者則對附近居民的生活造成干擾，引起民怨，因此列車行駛於地下隧道時所引致之土壤振動問題，已經成為一個引起高度重視的環保議題，世界各先進國家均定有明確的規範以限制振動公害的大小，而相關研究也不斷的在進行中。 本論文將針對起因於列車行駛於地下隧道時所引致之土壤振動問題作一綜合性探討，首先就此種振動問題的特性、對環境的影響與相關振動評估規範以及近三十年來部分學者專家所得到的研究成果加以介紹，其中有關古典波傳理論在此種振動問題的應用亦有詳盡的討論；此外，本研究更以有限和無限元素混合法作為數值分析工具，對於各項影響土壤振動的參數作一系統性的分析，整個土壤與隧道結構系統將被分為近域與遠域兩部分，近域部分以傳統有限元素模擬，而遠域部分則以無限元素模擬邊界無限遠之特性。此數值模擬方法因可建立在傳統有限元素的架構中，在實際應用上容易被一般工程師所接受，更可有效模擬土壤的輻射阻尼效應。研究中曾針對列車行駛於明挖覆蓋的隧道系統所引致的地表振動加以分析，經由與其他數值分析方法所得到的結果之比較，此有限和無限元素混合法的正確性亦可得到證實。 研究中先以2維的有限與無限平面元素來模擬土壤-隧道互制作用系統，並將列車模擬為簡諧震盪的無限長線載重，進行各項土壤參數的分析，而另一方面，為了更深入探討列車的動力特性對於土壤振動的影響，本研究更將原本的2維元素擴展至2.5D的有限與無限元素，在此2.5D的模式下，列車可以移動點載重的方式來模擬；經由各項參數分析結果顯示，即使在一般地下化軌道運輸系統的營運速度下，當外力的振動頻率與土層的自然振動頻率接近時，土壤的動力反應即會產生明顯的共振現象，因此解決此種振動問題的根本方法，就是要避免車輛的振動頻率與土層的自然振動頻率過於接近，以減少共振反應發生的機會。Ground-borne vibrations resulting from the underground railway traffic have become an important environmental issue, which has received increasing attention from both engineers and researchers. In this dissertation, an integrated investigation is conducted of the ground-borne vibration problem induced by trains moving in underground tunnels. Starting from an extensive literature review, practical parameter analyses were carried out for a two-dimensional simulation of the soil-tunnel interaction system, which were then extended to a 2.5D simulation with account taken of the moving effect of the train loads. The numerical procedure adopted here is called the coupled finite/infinite element method. With this approach, a soil-tunnel system is divided into two regions, i.e., the near and far fields. The near field, including the loads and other geometric/material irregularities, is simulated by the finite elements as conventional, and the far field covering the soils with infinite boundary by the infinite elements. This hybrid approach can overcome the inherent drawback of the finite element method in simulating the radiation damping for waves traveling to infinity. On the other hand, it can be established within the framework of the finite element method, which is commonly used by most structure analysis programs and is likely to be favored by most practicing engineers. From the results of this research, it can be concluded that even in most underground railways systems with the normal operating velocities, the coincidence of the loading frequencies with the fundamental frequency of the soil layer can still result in an apparent increase of the vibration level. Accordingly, to relieve the vibration of this kind, a fundamental solution should be resorted to avoid the coincidence of the loading frequencies of train with the fundamental frequency of soil layer.TABLE OF CONTENTS Acknowledgement (Chinese) i Abstract (Chinese) iii Abstract v Table of Contents vii List of Tables xi List of Figures xii Chapter 1 Introduction 1.1 Background 1 1.2 Objectives 3 1.3 Arrangement of the Dissertation 6 Chapter 2 Literature Review 2.1 Introduction 9 2.2 Statement of Problem 10 2.3 Evaluation Criteria of Vibration 13 2.4 State of the Art Researches on Ground-Borne Vibrations 20 2.4.1 Analytical approach 20 2.4.2 Field measurement 25 2.4.3 Empirical prediction models 28 2.4.4 Numerical simulation 31 2.5 Methods of Reducing Ground-Borne Vibration 36 2.6 Concluding Remarks 38 Chapter 3 Fundamentals of Wave Propagation in Soil 3.1 Introduction 49 3.2 Types of Waves 50 3.3 One Dimensional Wave Equation 52 3.4 Wave Propagation Generated by Underground Dynamic Loads 56 3.4.1 Formulation of elastodynamic problems 56 3.4.2 Wave equation generated by body forces 57 3.4.3 Solution for a periodic point force acting in the interior of an elastic half-space 60 3.5 Surface Ground Vibration due to a Moving Train in a Tunnel 64 3.6 Concluding Remarks 72 Chapter 4 Coupled Finite/Infinite Element Method 4.1 Introduction 79 4.2 Formulation of the Method 81 4.3 Formulation of Infinite Element 85 4.3.1 Shape function 85 4.3.2 Equation of motion in frequency domain 88 4.3.3 Damping 90 4.3.4 Numerical integration 91 4.3.5 Selection of wave numbers 93 4.4 Requirements on Finite Element Mesh 94 4.5 Frequency-Independent Finite/Infinite Element Mesh 95 4.6 Numerical Example 98 4.6.1 Simulation of the system 99 4.6.2 Comparison of the results 101 4.6.3 Characteristics of wave propagation 103 4.7 Concluding Remarks 104 Chapter 5 Parametric Study of Ground Vibrations due to Trains Moving in Underground Tunnel 5.1 Introduction 119 5.2 Ground Vibrations due to an Underground Infinite Line Load 121 5.2.1 Homogeneous elastic half-space 122 5.2.2 A layered soil superposed on an elastic half space 125 5.2.3 Effect of depth of loading point 127 5.2.4 Effect of depth of soil stratum 127 5.2.5 Effect of Poisson’s ratio 128 5.2.6 Effect of material damping ratio 129 5.3 Ground Vibrations due to an Infinite Line Load Acting in a Tunnel 130 5.3.1 Effect of existence of tunnel structure 131 5.3.2 A layered soil superposed on an elastic half space 132 5.3.3 Effect of depth of bedrock 133 5.3.4 Effect of elastic modulus ratio of soil/tunnel 134 5.3.5 Effect of two layered soils overlying a bedrock 135 5.4 Concluding Remarks 138 Chapter 6 2.5 D Finite/Infinite Element Approach for Modeling Trains Moving in Underground Tunnels 6.1 Introduction 167 6.2 Formulation of the Method 169 6.3 Shape Functions of 2.5D Infinite Element 172 6.3.1 Modification of wave number k 173 6.3.2 Modification of displacement amplitude decay factor176 6.4 Ground Vibrations due to an Underground Moving Load Acting in a Tunnel 178 6.4.1 Tunnel embedded in a half-space 178 6.4.2 Tunnel embedded in a soil layer superposed on a bedrock 180 6.4.3 Effect of train velocity 181 6.4.4 Effect of shear speed of soil 182 6.4.5 Effect of Poisson’s ratio 183 6.4.6 Effect of damping ratio 183 6.5 Concluding Remarks 184 Chapter 7 Conclusions and Further Studies 7.1 Conclusions 201 7.2 Further Studies 205 References 209 Appendix 219 LIST OF TABLES Table 2.1 Standards related to evaluation of human exposure to vibration in buildings 40 Table 2.2 Multiplying factors given in ISO 2631-2:1989 to define vibration magnitudes below which the probability of adverse human reaction is low 40 Table 2.3 VDV suggested in BS 6472:1992 at which adverse reactions may be expected from residential building occupants 41 Table 2.4 Vibration criteria regulated by the vibration regulation law (1976) in Japan 41 Table 2.5 The effective frequency range corresponding to partial methods used to reduce vibration as suggested by Nelson (1996) and Wilson et al. (1983) 42 Table 4.1 Material properties for tunnel and soil 107 LIST OF FIGURES Figure 2.1 Conceptual representation of the frequency range for airborne sound, structure-borne sound and ground-borne vibration 43 Figure 2.2 Sectional elevation of podium block (Balendra et al. 1989) 44 Figure 2.3 Parameters affecting noise and vibrations in buildings as reported by Melke (1988) 45 Figure 2.4 Principal surfaces of the floor supporting the body adopted from ISO 2631-1:1997: (a) seated position, (b) standing position, (c) recumbent position 46 Figure 2.5 Building vibration combined direction (x-,y-,z-axis) acceleration base curve of ISO 2631-1:1997 47 Figure 3.1 Schematic of particle motions (a) P-wave (b) S-wave 74 Figure 3.1 Schematic of particle motions (c) Rayleigh wave 75 Figure 3.2 Rod with exponentially varying areas (infinitesimal element and layer) (Wolf 1985) 76 Figure 3.3 Homogeneous elastic half-space subjected to a general load on the surface 77 Figure 3.4 Periodic force normal to the boundary in the interior of a semi-infinite solid 77 Figure 3.5 The time harmonic Boussinesq problem 78 Figure 3.6 Model adopted by Metrikine and Vrouwenvelder (2000b) 78 Figure 4.1 Schematic of the finite/infinite element approach (soil-structure system) 108 Figure 4.2 Schematic of the finite/infinite element approach (soil- tunnel system) 108 Figure 4.3 Infinite element: (a) global coordinates, (b) local coordinates 109 Figure 4.4 One dimensional mapping: (a) global coordinates, (b) local coordinates 109 Figure 4.5 Schematic for determining the amplitude decay factor 110 Figure 4.6 Selection of wave numbers 110 Figure 4.7 Finite element mesh 111 Figure 4.8 Schematic of condensation to the boundary 111 Figure 4.9 Layout of a cut-and-cover tunnel 112 Figure 4.10 Finite/infinite element mesh 112 Figure 4.11 Flow chart of the Finite/infinite element mesh generation 113 Figure 4.12 Comparison of pseudo-resultant responses along the surface of the ground: (a) f = 10Hz, (b) f = 20Hz, (c) f = 40Hz, (d) f = 80Hz 114 Figure 4.13 X-direction and Y-direction responses along the surface of the ground (unsymmetrical load): (a) f = 10Hz, (b) f = 20Hz, (c) f = 40Hz, (d) f = 80Hz 115 Figure 4.14 X-direction and Y-direction responses along the surface of the ground (symmetrical load): (a) f = 10Hz, (b) f = 20Hz, (c) f = 40Hz, (d) f = 80Hz 116 Figure 4.15 Pseudo-resultant responses along the surface of the ground: (a) f = 10Hz, (b) f = 20Hz, (c) f = 40Hz, (d) f = 80Hz 117 Figure 5.1 Fundamental analytical models for half-space: (a) homogeneous, (b) soil layer overlying a bedrock 142 Figure 5.2 Finite and infinite element mesh: (a) with no tunnel, (b) with tunnel 143 Figure 5.3 Vertical displacements of ground surface: (a) for h = 0 m, (b) for h = 5m 144 Figure 5.3 Vertical displacements of ground surface: (c) for h = 10 m, (d) for h = 15 m 145 Figure 5.4 Maximum vertical displacements of surface points due to different loading depths 146 Figure 5.5 Vertical displacements of surface points due to different shear modulus ratios of soils (h = 15 m): (a) at origin, (b) at x = 15 m 147 Figure 5.6 Maximum vertical displacements of surface points due to different loading depths (with stratum thickness H = 30 m) 148 Figure 5.7 Maximum vertical displacements of surface points due to different thicknesses of soil stratum: (a) for H/h = 2, (b) for H/h = 3 149 Figure 5.7 Maximum vertical displacements of surface points due to different thicknesses of soil stratum: (c) for H/h = 4 150 Figure 5.8 Vertical displacements of surface points due to different thicknesses of soil stratum: (a) at origin, (b) at x = 15 m 151 Figure 5.9 Maximum vertical displacements of surface points due to different thicknesses of soil stratum 152 Figure 5.10 Vertical displacements of surface points due to different Poisson’s ratios: (a) at origin, (b) at x = 15 m 153 Figure 5.11 Vertical displacements of surface points due to different damping ratios: (a) at origin, (b) at x = 15 m 154 Figure 5.12 Half space with tunnel: (a) homogeneous half-space, (b) soil layer overlying a bedrock 155 Figure 5.13 Vertical displacements of surface points due to different simulation cases: (a) with no tunnel, (b) with tunnel 156 Figure 5.14 Vertical displacements of surface points due to different simulation cases: (a) at origin, (b) at x = 15 m 157 Figure 5.15 Vertical displacements of different points on the origin and inside the tunnel 158 Figure 5.16 Vertical displacements of different points due to different shear modulus ratios of soil: (a) for the origin, (b) for point C 159 Figure 5.17 Vertical displacements of different points due to different depths of bedrock: (a) for the origin, (b) for point C 160 Figure 5.18 Vertical displacements of different points due to different elastic modulus ratios (with H = 30 m): (a) for the origin, (b) for point C 161 Figure 5.19 A tunnel embedded in two layers of soil deposits lying over a bedrock: (a) H1 = 20 m, H2 = 10 m, (b) H1 = 10 m, H2 = 20 m 162 Figure 5.20 Vertical displacements of surface points due to different shear modulus ratios of soils (G1<G2): (a) at origin, (b) at x = 15 m 163 Figure 5.21 Vertical displacements of surface points due to different shear modulus ratios of soils (G1>G2): (a) at origin, (b) at x = 15 m 164 Figure 5.22 Vertical displacements of surface points due to different shear modulus ratios of soils (G1<G2): (a) at origin, (b) at x = 15 m 165 Figure 6.1 Schematic view of the soil-tunnel interaction system 186 Figure 6.2 Half space with tunnel: (a) homogeneous half-space, (b) soil layer overlying a bedrock 187 Figure 6.3 Vertical displacements for a moving point load with f0=4 Hz (half-space): (a) c=10 m/s, (b) c=20 m/s, (c) c=30 m/s, (d) c=40 m/s 188 Figure 6.4 Vertical displacements for a moving point load with f0=0 Hz (bedrock) (a) c=10 m/s, (b) c=20 m/s, (c) c=30 m/s, (d) c=40 m/s 189 Figure 6.5 Vertical displacements for a moving point load with f0=4 Hz (bedrock): (a) c=10 m/s, (b) c=20 m/s, (c) c=30 m/s, (d) c=40 m/s 190 Figure 6.6 Vertical displacements for a moving point load with f0=1 Hz (bedrock): (a) c=10 m/s, (b) c=20 m/s, (c) c=30 m/s, (d) c=40 m/s 191 Figure 6.7 Maximum vertical displacements for a moving point load with different excitation frequency: (a) at origin, (b) at x = 15 m 192 Figure 6.8 Effect of train velocity on the vibration attenuation induced by a moving load with f0=1 Hz: (a) displacement, (b) velocity, (c) acceleration 193 Figure 6.9 Vertical displacements at the origin due to different shear speeds of soil: (a) c=10 m/s, (b) c=20 m/s, (c) c=30 m/s 194 Figure 6.10 Vertical displacements at x = 15 m due to different shear speeds of soil: (a) c=10 m/s, (b) c=20 m/s, (c) c=30 m/s 195 Figure 6.11 Vertical displacements at the origin due to different Poisson’s ratios: (a) c=10 m/s, (b) c=20 m/s, (c) c=30 m/s 196 Figure 6.12 Vertical displacements at x = 15 m due to different Poisson’s ratios: (a) c=10 m/s, (b) c=20 m/s, (c) c=30 m/s 197 Figure 6.13 Vertical displacements at the origin due to different damping ratios: (a) c=10 m/s, (b) c=20 m/s, (c) c=30 m/s 198 Figure 6.14 Vertical displacements at x = 15 m due to different damping ratios: (a) c=10 m/s, (b) c=20 m/s, (c) c=30 m/s 1992420307 bytesapplication/pdfen-US列車地下隧道土壤振動無限元素trainunderground tunnelground-borne vibrationinfinite elementthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/50301/1/ntu-95-D90521020-1.pdf