本研究透過修改WERAFT-S (Wave Equation Analysis for Raft Foundation - Statics) 分析開發動態荷載條件下的有限差分分析。 此三維模型包含二維筏基的變形和一維基樁的變形，在時域中進行分析得到各種穩態荷載下筏基的位移量，進而分析樁筏基礎位移，分析結果並與三維分析軟體Midas GTS NX進行比較。
在本研究中，彈簧和阻尼器之模型選取皆有頻率相依 (frequency-dependent) 及頻率不相依 (frequency-independent) 的情況，上述兩種皆應用於模擬土壤。 所建議的分析能夠得到與有限元素法分析相似的位移量。 本研究未考慮樁與樁的互制效應。 研究結果發現，目前的分析模型在基礎共振的模擬上仍有限制。

In this study a finite difference analysis toolkit is developed by modifying the WERAFT-S (Wave Equation Analysis for Raft Foundation - Statics) analysis for use in dynamic loading conditions. This three-dimensional model considers the two dimensional deformations of the raft and the one-dimensional deformations of the piles. The analysis is performed in the time domain to determine the displacement of a raft foundation and subsequently a piled raft foundation, under various steady state loads. Validation of the proposed analysis model is carried out by performing a three dimensional analysis using Midas GTS NX.
Frequency-dependent and frequency-independent spring and dashpot models have been used to model the soil in this study. The proposed analysis is able to determine the order of displacement of similar magnitude as the finite element analysis. Pile to pile interaction effects have not been considered in this study. It was found that the current model is limited in its capacity to reveal the resonance of the foundation.

Table of Contents
Abstract	I
Acknowledgement	III
List of Figures	VI
List of Tables	IX

Chapter 1 	Introduction	1
1.1 Research Motivation and Objective	1
1.2 Research Method	2
1.1 Research Contents	4

Chapter 2	Literature Review	5
2.1 Raft Foundation Analysis	5
2.2 Piled Raft Foundation Analysis	6
2.3 Foundation Subject to Dynamic Loading	9
2.4 Pile Stiffness	12
2.5 Kirchhoff-Love Hypothesis	13
2.6 Summary	13

Chapter 3	Theory and Method	15
3.1 Theory	15
3.2 Governing Equation	16
3.3Nodal Equations	17
3.4 Raft Displacement Analysis: WERAFT-D	25
3.5 Piled Raft Displacement Analysis: WEAPR-D	26
3.6 Finite Difference Model	29
3.6.1 Finite Difference Model- Raft Only	30
3.6.2 Finite Difference Model - Piled Raft 	32
3.7 Finite Element Model	33
3.7.1 Finite Element Model- Raft Only	33
3.7.2 Finite Difference Model- Piled Raft	36
3.8 Material Parameters	38

Chapter 4	Results	39
4.1 Raft Displacement	39
4.1.1 Raft Displacement: WERAFT-D	41
4.1.2 Raft Displacement- WERAFT-D vs. FEM Comparison	44
4.1.3 Raft Displacement- Finite Element on Linear Springs	49
4.2 Piled Raft Displacement: WEAPR-D	51

Chapter 5	 Conclusion 	61
5.1 Discussion and Conclusion	61
5.2 Limitations and Suggestions	62

References		63

Appendix A		67
Appendix B		68
Appendix C		70







List of Figures
Figure 1-1 Research Structure Diagram	3

Figure 2-1 Plate on Springs Model on Winkler Springs 	6
Figure 2-2 Plate-Beam-Springs Modelling of a Piled Raft Foundation 	8
Figure 2-2a Kitiyodom and Matsumoto (2002) 	8
Figure 2-2b Randolph (1994) 	8
Figure 2-3 Wolf (1995) Simplified Model	10
Figure 2-2a Disk on surface of half-space with truncated semi-infinite translational cone  	10
Figure 2-2a Discrete-element model (lumped-parameter) for translational cone	10

Figure 3-1 Discretized Raft Model on Linear Springs	15
Figure 3-2 Single Pile Equilibrium Model	26
Figure 3-3 Discretized Pile Model	29
Figure 3-4 WERAFT-D Raft Model	30
Figure 3-5 WERAFT-D Discretized Raft Model	31
Figure 3-6 WEAPR-D Piled Raft Model	32
Figure 3-7 Finite Element Meshing	34
Figure 3-7a Isometric View	34
Figure 3-7b Top View	34
Figure 3-8 Finite Element Model Boundary Constraints	36
Figure 3-9a 3-D Finite Element Pile Model	36
Figure 3-9b 3-D Finite Element Piled Raft Model	36
Figure 3-10 Piled Raft Foundation 3-D Finite Element Mesh	37

Figure 4-1 Surface Raft Foundation 	39
Figure 4-2 Raft Foundation Discretization 	41

Figure 4-3a Raft Foundation Displacement vs. Frequency- Lysmer (1965)	42
Figure 4-3b Raft Foundation Displacement vs. Frequency- Gazetas (1991) 	42
Figure 4-3c Raft Foundation Displacement vs. Frequency- Wolf (1995) 	43

Figure 4-4 Finite Element Model Dimension 	45
Figure 4-5 FEM Raft Foundation Displacement vs. Frequency 	45

Figure 4-6a WERAFT-D vs. FEM Comparison (Model R1) - Vs=120m/s	46
Figure 4-6b WERAFT-D vs. FEM Comparison (Model R1) - Vs=150m/s	47
Figure 4-6c WERAFT-D vs. FEM Comparison (Model R1) - Vs=180m/s	47

Figure 4-7a WERAFT-D vs. FEM Comparison (Model R2) - Vs=120m/s	48
Figure 4-7b WERAFT-D vs. FEM Comparison (Model R2) - Vs=150m/s	48
Figure 4-7c WERAFT-D vs. FEM Comparison (Model R2) - Vs=180m/s	49

Figure 4-8 FEM on Linear Springs and Dashpots Model	50
Figure 4-9 FEM on Linear Springs and Dashpots and Proposed Finite Difference Analysis 	50
Figure 4-10 Piled Raft Foundation Model	51
Figure 4-11 Piled Raft Foundation 3x3 Pile Configuration Layout 	53
Figure 4-12 Piled Raft Foundation 4x4 Pile Configuration Layout 	53

Figure 4-13a Piled Raft Foundation (PR1) vs. Raft Model (R1) Comparison - Vs=120m/s	54
Figure 4-13b Piled Raft Foundation (PR1) vs. Raft Model (R1) Comparison - Vs=150m/s	55
Figure 4-13c Piled Raft Foundation (PR1) vs. Raft Model (R1) Comparison - Vs=180m/s	55

Figure 4-14a Piled Raft Foundation (PR2) vs. Raft Model (R2) Comparison - Vs=120m/s	56
Figure 4-14b Piled Raft Foundation (PR2) vs. Raft Model (R2) Comparison - Vs=150m/s	56
Figure 4-14c Piled Raft Foundation (PR2) vs. Raft Model (R2) Comparison - Vs=180m/s	57

Figure 4-15a Piled Raft Foundation Model (PR2) vs. (PR1) Comparison - Vs=120m/s	58
Figure 4-15b Piled Raft Foundation Model (PR2) vs. (PR1) Comparison - Vs=150m/s	58
Figure 4-15c Piled Raft Foundation Model (PR2) vs. (PR1) Comparison - Vs=180m/s	59

Figure 4-16a Piled Raft Foundation (PR3) vs. Raft Model (R2) Comparison - Vs=150m/s	60
Figure 4-16a Piled Raft Foundation Model (PR3) vs. (PR2) Comparison - Vs=150m/s	60

Figure A-1 Graphs to determine dynamic stiffness and damping coefficients k ̃_z and c ̃_z using Gazetas, 1991 spring and dashpot model 	67

Figure C-1 Sample input load- 1.5 kPa amplitude	70
Figure C-2 Sample input load- 0.222 kPa amplitude	70
Figure C-3 Sample output displacement vs. time curve - WERAFT model	71
Figure C-4 Sample output displacement vs. time curve - FEM model	71




List of Tables

Table 3-1 Spring and Dashpot Models for a disk on Homogeneous Half-space 	25
Table 3-2 Pile Spring Models for a Homogeneous Half-space	28
Table 3-3 Finite Element Assigned Mesh Sizes 	33
Table 3-4 Empirical Formulas for Material Parameter Calculation 	38

Table 4-1 Raft Foundation Dimension and Loading 	39
Table 4-2 Material Parameters (Elastic Properties) 	40
Table 4-3 Finite Element Model Analysis Zone Dimension	44
Table 4-4 Piled Raft Foundation Model Parameters	52
Table 4-4 Pile Stiffness Parameters	52

Table 5-1 Analysis Time Comparison	62

Table B-1 Frequency independent soil spring and dashpot values	68
Table B-2 Frequency dependent soil spring and dashpot values	69

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