Closed form backcalculation algorithms for pavement analysis

CLOSED-FORM BACKCALCULATION ALGORITHMS FOR PAVEMENT ANALYSIS BAGUS HARIO SETIADJI NATIONAL UNIVERSITY OF SINGAPORE 2009 CLOSED-FORM BACKCALCULATION ALGORITHMS FOR PAVEMENT ANALYSIS BAGUS HARIO SETIADJI (B.Eng. (Hons.), ITB, Indonesia) (M.Eng., ITB, Indonesia) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2009 ACKNOWLEDGEMENTS In the name of Allah, the Most Gracious, the Most Merciful. All praises and thanks be to Allah who has provided the knowledge and guidance to the author in finishing this research work. A deepest appreciation is expressed to the author’s thesis advisor Professor Fwa Tien Fang for his invaluable assistance, supervision, and advice throughout the duration of the research. The author would also like to express his gratitude to National University of Singapore (NUS) for providing him the Research Scholarship and the opportunity to pursue the Doctoral degree program in Department of Civil Engineering. The author would like to thank all my friends, Dr. Ong Ghim Ping Raymond, Dr. Lee Yang Pin Kelvin, Mr. Joselito Guevarra, Mr. Hendi Bowoputro, Mr. Kumar Anupam, Mr. Srirangam Santosh Kumar, Mr. Farhan Javed, Mr. Wang Xinchang, Mr. Qu Xiaobo, Ms. Yuan Pu, Ms. Ju Fenghua, Mr. Hadunneththi Rannulu Pasindu, Mr. Cao Changyong, and Mr. Yang Jiasheng for the support and friendship. Gratitude is also extended to Mr. Goh Joon Kiat, Mr. Mohammed Farouk, Mr. Foo Chee Kiong, Mrs. Yap-Chong Wei Leng and Mrs. Yu-Ng Chin Hoe of the Transportation Engineering Laboratory. A special appreciation is expressed to the author’s parents, lovely wife, Amelia Kusuma Indriastuti, and sons, Bagus Jati Pramono and Bagus Dwisatyo Nugroho, for their patience, devotion and understanding given when the author was finishing the study in National University of Singapore (NUS). i TABLE OF CONTENTS ACKNOWLEDGEMENTS i TABLE OF CONTENTS ii SUMMARY vi LIST OF TABLES .vii LIST OF FIGURES viii NOMENCLATURE x CHAPTER INTRODUCTION 1.1 1.2 1.3 1.4 1.5 Definition of Pavement Systems Rigid Pavement System . 1.2.1 Background 1.2.2 Significance of k Values in Design and Evaluation of Rigid Pavements Flexible Pavement System . 1.3.1 Background 1.3.2 Multi-layered System in Design and Evaluation of Flexible Pavements Objectives and Scope of Work . Organization of Thesis . CHAPTER LITERATURE REVIEW 2.1 2.2 2.3 2.4 Introduction . Determination of Layer Moduli 11 2.2.1 Direct Test Methods . 11 2.2.1.1 k and Composite k Value of Rigid Pavement System 11 2.2.1.2 Elastic Layer Moduli of Flexible Pavement System 13 2.2.2 Correlation with Other Engineering Properties . 14 2.2.3 Non-destructive Test (NDT) Methods . 15 Backcalculation Algorithms for Layer Moduli . 17 2.3.1 Closed-form Algorithms . 19 2.3.1.1 ILLI-BACK . 19 2.3.1.2 NUS-BACK . 21 2.3.1.3 2L-BACK 23 2.3.2 Trial-and-Error Best Fit Algorithms 25 2.3.2.1 ERESBACK 26 2.3.2.2 MICHBACK 27 2.3.2.3 EVERCALC 29 2.3.3 Regression Method . 31 2.3.4 Database Search Algorithm (DSA) Method 32 2.3.5 Summary 33 Research Issues in Determination of Layer Moduli 34 ii CHAPTER EVALUATION OF BACKCALCULATION ALGORITHM FOR RIGID PAVEMENT SYSTEM 45 3.1 3.2 3.3 Introduction . 45 Selection of Backcalculation Algorithm for Rigid Pavements . 45 3.2.1 Background 45 3.2.2 Evaluation Procedure of Backcalculation Algorithms . 46 3.2.3 Long-Term Pavement Performance (LTPP) Program 49 3.2.4 Input Parameter and Assumptions Used in Analysis . 50 3.2.5 Comparison of Backcalculation Algorithms 51 3.2.5.1 Basis of Comparison 51 3.2.5.2 Results of Comparative Analysis 52 3.2.6 Summary 59 Consideration of Finite Slab Size in Backcalculation Analysis of Rigid Pavements . 61 3.3.1 Background 61 3.3.2 Methods of Backcalculation . 62 3.3.2.1 Backcalculation Procedure for One-slab and Nine-slab Algorithm (ONE-BACK and NINE-BACK) 63 3.3.2.2 Backcalculation Using Crovetti’s Corrections for Finite Slab Size 68 3.3.2.3 Backcalculation Using Korenev’s Corrections for Finite Slab Size 70 3.3.3 LTPP Database and Input Parameter Used in Evaluation 70 3.3.4 Analysis of Effect of Finite Slab Size . 71 3.3.4.1 Results of Backcalculation Analysis . 71 3.3.4.2 Basis of Evaluation 71 3.3.4.3 Results of Evaluation Analysis 72 3.3.5 Summary 78 CHAPTER DEVELOPING k-Es RELATIONSHIP OF RIGID PAVEMENT SYSTEM USING BACKCALCULATION APPROACH 105 4.1 4.2 Introduction . 105 Examining k-Es Relationship of Pavement Subgrade Based on LoadDeflection Consideration 105 4.2.1 Background 105 4.2.2 Review of k-Es Relationship by Past Researchers . 107 4.2.2.1 k-Es Relationship by AASHTO 107 4.2.2.2 k-Es Relationship by Khazanovich et al. . 109 4.2.2.3 k-Es Relationship by Vesic and Saxena . 110 4.2.3.4 k-Es Relationship by Ullidtz . 111 4.2.3 Proposed Procedure for Deriving k-Es Relationship 112 4.2.3.1 Main Considerations 112 4.2.3.2 Backcalculation of Equivalent k-Model and Es-Model 113 4.2.4 Derivation of k-Es Relationship Using LTPP Data 114 4.2.4.1 LTPP Database 115 4.2.4.2 Comparing of Equivalent k-Model and Equivalent Es-Model 115 iii 4.3 4.2.4.3 Proposed Methods of Estimating k from Es based on Equivalent k-Model and Es-Model 117 4.2.5 Comparison of Different k-Es Relationships 118 4.2.5.1 Comparison with Measured k Values . 118 4.2.5.2 Choice of Method to Estimate k from Es . 120 4.2.6 Summary 122 Examining k-Es Relationship of Rigid Pavement System by Considering Presence of Subbase Layer . 123 4.3.1 Background 123 4.3.2 Determination of Composite k Value by Existing Method . 125 4.3.2.1 Determination of Composite k by AASHTO 125 4.3.2.2 Determination of Composite k by PCA 127 4.3.2.3 Determination of Composite k by FAA 127 4.3.3 Proposed Procedure to Determine Composite k Value 128 4.3.3.1 Main Consideration 128 4.3.3.2 Backcalculation of Equivalent k-Model, Es-Model and Es/sb-Model . 129 4.3.3.3 Derivation of k- Es/sb relationship . 131 4.3.3.4 Relationship between lk and l Es / sb . 133 4.3.3.5 Proposed Method of Estimating Composite k from Esb and Es Based on Equivalent k-model and Es-model 134 4.3.4 Comparison of Composite k Values by Proposed Method and Existing Design Methods . 134 4.3.4.1 Comparison based on under- and over-estimation of k values 135 4.3.4.2 Comparison based on RMSE and RMSPE . 136 4.3.4.3 Summary Remarks on Method to Estimate Composite k from Es and Esb . 137 4.3.5 Summary 138 CHAPTER DEVELOPMENT OF FORWARD CALCULATION SOLUTIONS FOR THREE- AND FOUR-LAYER FLEXIBLE PAVEMENT SYSTEMS . 152 5.1 5.2 5.3 5.4 Introduction . 152 Solution for Surface Deflection 153 5.2.1 Determination of Surface Deflection Equation 153 5.2.1.1 Boundary Conditions for Three-layer Flexible System . 153 5.2.1.2 Determination of Three-layer System Coefficients . 155 5.2.1.3 Boundary Conditions for Four-layer Flexible System . 162 5.2.1.4 Determination of Four-layer System Coefficients . 164 5.2.2 Comparison of Solutions with Other Methods 169 Comment on the Effect of Temperature on Asphalt Layer 171 Summary . 171 CHAPTER DEVELOPMENT OF CLOSED-FORM BACKCALCULATION ALGORITHM FOR MULTI-LAYER FLEXIBLE PAVEMENT SYSTEM . 174 6.1 Introduction . 174 iv 6.2 6.3 6.4 Development of Backcalculation Procedure . 174 6.2.1 Proposed Procedure 174 6.2.2 Nelder-Mead Optimization Method 176 6.2.3 Determination of Unique Solution 180 Comparison of the Backcalculated Moduli with Other Backcalculation Programs 181 6.3.1 Comparison Using Exact Deflections . 183 6.3.2 Comparison Using Deflection with Random Measurement Errors 184 Summary . 186 CHAPTER CONCLUSIONS AND RECOMMENDATIONS . 199 7.1 7.2 7.3 7.4 7.5 Introduction . 199 Backcalculation of Layer Moduli of Rigid Pavement . 199 7.2.1 The Use of Infinite-Slab Backcalculation Algorithm to Evaluate Layer Moduli 199 7.2.2 The Use of Finite-Slab Backcalculation Algorithm to Evaluate Layer Moduli 200 Development of k-Es Relationship on Rigid Pavement System . 201 7.3.1 k-Es Relationship on Two-layer Rigid Pavement System 201 7.3.2 k-Es Relationship on Three-layer Rigid Pavement System with Consideration of Subbase Layer . 202 Closed-form Backcalculation of Layer Moduli of Flexible Pavement . 203 Recommendation for Further Research 204 LIST OF REFERENCES 206 APPENDIX A FINAL TERMS OF CONSTANTS C1 AND D1 . 218 APPENDIX B LIST OF PAPERS RELATED WITH THIS STUDY . 246 v SUMMARY Many backcalculation algorithms based on multi-layer elastic theory and plate theory were developed to backcalculate the layer moduli of a flexible and rigid pavement system, respectively. Unfortunately, they not always give the unique answer due to the use of iterative trial and error approach in developing the algorithms. In this study, a development and evaluation of closed-form backcalculation algorithms was proposed. The aims of this research were to examine the merits of currently available closed-form backcalculation algorithms, and develop a procedure to derive the composite modulus of subgrade reaction (composite k value) for a rigid pavement with a subbase layer using a suitable closed-form backcalculation algorithm; and to develop a closed-form backcalculation algorithm for multi-layer flexible pavement system. The results showed that the closed-form backcalculation algorithm, NUS-BACK, was suitable to evaluate the layer moduli of an infinite- and finite-slab rigid pavement system. The next result produced was the relationship of two radius of relative stiffness of different foundation model, namely lk-lEs and lk-lEs/sb relationship, was suitable to determine k and composite k values from their respective layer moduli Es; and Es and Esb. Another important achievement was the proposed closed-form backcalculation algorithms for three- and four-layer flexible pavement developed in this study, 3L-BACK and 4L-BACK, could produce slightly more accurate backcalculated moduli than those of other iterative-based backcalculation programs. vi LIST OF TABLES Table 2.1 Table 2.2 Table 2.3 Table 2.4 Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 3.5 Table 3.6 Table 3.7 Table 3.8 Table 3.9 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 5.1 Table 5.2 Table 6.1 Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 6.6 Effect of Untreated Subbase on k Values . 37 Design k Values for Cement Treated Subbases 37 Values for coefficient A, B, C and D in Equation (2-8) 38 Values for coefficient x, y and z in Equation (2-10) . 38 Measured Properties of 26 JCP Sections for Analyzing k . 80 Root-Mean-Square Percent Errors for k and Ec Backcalculated Using NUS-BACK (Load Level = 71.1 kN) 81 Measured Properties of 50 JCP Sections for Analyzing Ec 82 Measured Properties of 76 CRCP Sections for Analyzing Ec . 83 RMSPE of Backcalculated Pavement Properties and Coefficient of Correlation with Measured Values from Four Different Methods 84 RMSPE of Backcalculated Pavement Properties with Temperature Consideration 85 RMSPE of Backcalculated Pavement Properties from Five Different Methods 86 Percentages of Over-Estimation and Under-Estimation Cases 87 Statistical Tests on Pairwise Differences between Backcalculated and Measured Pavement Properties . 89 Properties of 50 JCP sections . 139 Properties of 75 CRCP sections . 140 RSME of Estimated k Values with Respect to Measured k Values . 141 RSME of Estimated k Values with Respect to Backcalculated k Values . 141 RSME and RMSPE of Estimated Composite k Values with Respect to Measured k Values from Different Methods . 141 Comparison of Computed Surface Deflections on Three-layer Flexible System 172 Comparison of Computed Surface Deflections on Four-layer Flexible System . 172 Comparison of Backcalculated Layer Moduli for Three-layer Flexible Pavement System by Different Methods 188 Comparison of Backcalculated Layer Moduli for Four-layer Flexible Pavement System by Different Methods 188 Deflections with Random Measurement Errors for Three-layer Flexible Pavement System . 189 Deflections with Random Measurement Errors for Four-layer Flexible Pavement System . 190 Summary of Statistics of Backcalculated Layer Moduli from Different Methods for Three-layer Flexible Pavement System . 191 Summary of Statistics of Backcalculated Layer Moduli from Different Methods for Four-layer Flexible Pavement System . 192 vii LIST OF FIGURES Figure 2.1 Figure 2.2 Representation of Dense Liquid Foundation 39 Chart for Estimating Composite k value Based on 1972 AASHTO Interim Guide . 40 Figure 2.3 Chart for Estimating Composite k value Based on 1993 AASHTO Guide . 41 Figure 2.4 Approximate Relationship between k values and Other Soil Properties . 42 Figure 2.5 Approximate Relationship between MR values and Other Soil Properties 43 Figure 2.6 Representation of Multi-Layer Pavement Structure as Equivalent Two-Layer System . 44 Figure 3.1 Comparison between Measured and Backcalculated k values of JCP (Load Level = 71.1 kN) from Four Different Methods . 89 Figure 3.2 Comparison between Measured and Backcalculated Ec values of JCP (Load Level = 71.1 kN) from Four Different Methods . 90 Figure 3.3 Comparison between Measured and Backcalculated Ec values of CRCP (Load Level = 71.1 kN) from Four Different Methods 91 Figure 3.4 Absolute Percent Errors of Backcalculated k values (Load Level = 71.1 kN) . 92 Figure 3.5 Absolute Percent Errors of Backcalculated Ec values of JCP (Load Level = 71.1 kN) . 93 Figure 3.6 Absolute Percent Errors of Backcalculated Ec values of CRCP (Load Level = 71.1 kN) . 94 Figure 3.7 Comparison between Backcalculated and Measured of k and Ec From Different Methods 95 Figure 3.8 Cumulative Frequency Plots for Backcalculated k and Ec . 99 Figure 3.9 Frequency Distributions of Percent Errors of Backcalculated Value of k and Ec 101 Figure 4.1 Equivalent k-model and Equivalent E-model . 142 Figure 4.2 Proposed Approach for Deriving Relationship between k and Es 143 Figure 4.3 k-Es Relationship Derived from Equivalent k-model and Equivalent Es-model . 144 Figure 4.4 lk-lEs Relationship Derived from Equivalent k-model and Equivalent Es-model . 145 Figure 4.5 Comparison of Different lk-lEs Relationship 146 Figure 4.6 Estimating k from Es by Different Methods . 147 Figure 4.7 Equivalent k-model and Equivalent Es-model 148 Figure 4.8 Equivalent k-model, Es-model and Es/sb-model . 149 Figure 4.9 Comparison between Predicted and Measured k Values . 150 Figure 4.10 Frequency Distributions of Percent Errors of Predicted k Values . 151 Figure 5.1 Schematic of Three-layer Flexible Pavement under Concentrated Load 173 Figure 5.2 Schematic of Four-layer Flexible Pavement under Concentrated Load 173 Figure 6.1 Geometries of Nelder-Mead Method . 193 Figure 6.2 Procedures of Nelder-Mead Algorithm 194 viii (Denom C ) ( ) ( ) = − NSTX NZ + − N STXY − KNSTY NZ + L − KN STXYZmh2 ( NSTVZ (NZ 15 ) + L )mh ( ) ( + 16 K NT X NZ + L m h1 h23 − K NSTX NZ + L m h22 − K NS 2VX NZ + L ) ( ) − 16 N T X Zm h h + NT X (NZ + L )m h h + N STVX m h h + N T XY m h h + 16 N T X m h h − 8K NT Z (NZ + L )m h h − KN T YZ m h h + N T XZ m h h − NT Z (NZ + L )mh − N STVX Zmh − N T Y Zmh − N T X Zm h h + N STX Zmh − 4K + K NSTVX NZ + L m h1 h2 + 16 KN T XYZm h h 2 2 2 2 − N S VX − N STX m h + 16 K LNT Xm h h + K LNSTVXm h1 h2 + KLNT Ym h1 h2 2 ( ) − K LNSTXm h22 − K LNS 2VX − KLNSTY − KN STVXYZmh1 − K NSTZ NZ + L mh2 * e mh1 e − mH (Denom C1 )16 = (Denom C1 )17 LN ST X m h1 h2 − LN ST XZmh1 − LN S 2TX e mh1 e −3mH ( ) ( ) = − NSTXY − NT 2VXY + 4m h12 − KNSTXZmh2 + KNST X NZ + L m h1 h2 ( ) ( + N ST X 2Ym h1 h2 − KNST Z NZ + L mh1 − N ST XYZmh1 − KNS 2TX NZ + L ( ) ( ) ) − N S 2TX 2Y − KNSTVXZmh1 − KT 2V ( NZ + L ) + 4m h12 − KST NZ + L − KNT Z m h1 h2 ( ) − KNT 2VXZmh1 + 4m h12 e − mh1 e mh2 e mH 231 ( (Denom C1 )18 = ) ( ) ( ) − NTY NZ + L − KLNSVX − KNTZ NZ + L mh2 − K NTY NZ + L m h22 ( ) + N 2TXYZmh2 − N 2TVXYZmh1 − K NSVY NZ + L − N 2TY − N SVX 2Y − 12 N 2TX 2Ym h22 − 10 KN TY mh2 − 24 KN TX Zm h + 16 KN TXZ m h22 − KLNTXm h22 ( ) − 16 K NTZ NZ + L m h23 − 16 K N 2TYZ m h22 − N S 2TX 2Y + 12 K LNST 2Ym h1 h2 ( )( ) Y (1 + 4m h ) − 16 K ( ) ( − K LNS 2TY − KNTV X NZ + L + 4m h12 − KNSTZ NZ + L mh1 − LNT 3Y + 4m h12 ( ) − N T YZ + 4m h − N TV X N TVYZ m h1 h2 ( ) + 16 K N ST YZ m h1 h2 − N ST XYZmh1 − K NSV Z NZ + L mh1 − K N 2TVYZ m h1h2 ( NSVZ (NZ ) + L )mh − K NTVZm h1 h22 − KN 2TV XZ m h12 − KNTVZ NZ + L mh1 − KN 2TVX Zm h1 h22 − KN TVY Zmh1 − KN SV X Zmh1 − K ( ) − KN SVX Zmh2 ( + KNST X NZ + L m h1 h2 + KLNST Xm h1 h2 + KLNST Xm h1 h2 − KNS 2TX NZ + L ( ) − N TVXYZmh1 − LNTY − KN SVXZ − KLNT Xm h + 4m h + KLNST Xm h1 h2 2 ( − KLNTVXm h1 h2 − KLNS TX − K LNSVY − K LNTV Y + 4m h12 ( ) ( ) − KN 2TV X Zmh2 + 4m h12 − 16 K NTVZ NZ + L m h1h22 ( )( ) ( ) )( − K NTV Zmh2 NZ + L + 4m h − K NT Zm h23 NZ + L + 4m h12 ( ) ( ) ) ( + KN T XZ m h + 4m h − K N T YZ m h + 4m h − KN T Y Zmh + 4m h12 2 2 ( + 16 K NST Zm h1 h22 − K N S 2TZ mh2 − KN 2T X Zm h23 + 4m h12 ( )( ) ) − KNT Zmh2 NZ + L + 4m h − KN ST XZ m h1h2 − 16 KN TVX Zm h1 h22 + KNTVXZm h1 h2 + 16 KN ST X Zm h1 h22 − KNS 2TX Zmh2 − KN 2TY Zmh2 ( )( ) ( )( (1 + 4m h ) − K LNS 2TZmh2 − K NTV 2Y NZ + L + 4m h12 − K NT 3Ym h22 NZ + L + 4m h12 ( ) ( ) + N T XYZmh2 + 4m h − KN T Y Zmh2 + 4m h − N T Y ( ) − K N S 2TYZ − N 2T X 2Ym h22 + 4m h12 + KN ST 2Y Zmh1 − KN 2TVY Zmh1 − N 2TVX 2Ym h1h2 − 8K LNTVYm h1 h2 + N ST X 2Ym h1 h2 e − mh1 e mh2 e − mH (Denom C1 )19 = ( ( ) 16 KN 2T X NZ + L m h1 h23 + 16 N 3T 2VX 2YZm h12 h2 − LN STXY ) − LN 2T 2VXY + 4m h12 − N 3T 2VXYZ m h12 + 12 KLN STVX m h1 h2 − KN V X Z m h − 16 KN 3T 2VXZ m h12 h2 − N STVX 2YZmh1 − KLN S 2VX − N 3V X 2YZmh1 − KN 2V X Z m h1 h2 − KN STVX Z m h1 h2 + LN 2T XYm h1 h2 − LN 2T 2YZmh1 − KLN STVXZmh1 − KLN 2T XZm h1 h22 − KLN STX m h22 ( − KLN 2V XZmh1 − KLN 2VXZmh2 − LN 2VXY − KLNT 2VXm h22 + 4m h12 ( ) − KLN 2V X + 4m h12 − 8KLN 2V X m h1 h2 − KLN 2VX m h22 ( ) ) ) − KLN 2T 2VXZmh2 + 4m h12 − KLN STXZmh2 + 16 KN 3T 2VX Z m h12 h22 * e − mh1 e mh2 e −3mH 232 ) ) ) (Denom C1)20 = ( ) ( ) − KLNSTY − KNSTY NZ + L − K NSTZ NZ + L mh2 ( ) − KN STXYZmh2 − N STXY − K NSTVZ NZ + L mh1 − KN STVXYZmh1 ( ) − KN T VXYZmh2 (1 + 4m h ) − K NT VZ NZ + L mh2 (1 + 4m h12 ) ( ) + K N T Z m h h − N T VXY (1 + 4m h ) − KNT VY NZ + L (1 + 4m h12 ) 2 + KN 2T 2YZ m h1h2 − KLNT 2VY (1 + 4m h12 ) e − mh1 e mh2 e − mH (Denom C1 )21 = ( − KLN S 2TXY − KLN 2TXYm h22 − KLN SVXY + 12 KLN ST XYm h1 h2 ) − K N 2TVXZ NZ + L m h1h22 − 16 K LN 2TXZm h23 − 16 KN 3TVXYZ m h1 h2 − N TVXY Zmh1 − KN 3TV XYZ m h12 − 16 K N 3TV XZ m h12 h2 − KLN 2TVXYm h1 h2 ( ) − KLN 2TVXY + 4m h12 − KLN 2TVYZmh1 − K LN SV XZmh1 − KLN 2TYZmh2 ( ) − K LN SVXZmh2 − LN 2TY − KLN ST 2YZmh1 − 8K LN 2T XZm h23 + 4m h12 ( ) ( − K LN TV XZmh2 + 4m h + 16 K LN ST XZm h h − KLN T YZmh2 + 4m h12 2 − K N TVXZ m h h − 16 K LN TVXZm h h − K LN S TZmh2 2 ( ) 2 ( ) ( − KLN 2T XYm h22 + 4m h12 − KLN 2TV XY + 4m h12 − LN 2T 3Y + 4m h12 ) ) * e − mh1 e mh2 e −3mH (Denom C1 )22 = − KLN STY − K LN STYZmh2 − K LN STVYZmh1 ( ) ( ) − K LN 2T 2VYZmh2 + 4m h12 − KLN 2T 2VY + 4m h12 e − mh1 e mh2 e −3mH ( (Denom C1 )23 = ) ( KV NZ + L − NSTXY − KNSTVXZmh1 − KST NZ + L ) ( − KNSTX m h22 − KNSTVXZmh1 − NVXY − KNS 2VX − KNVY − K S 2V NZ + L ( ) − KNS VX + 12 KNSTVX m h1h2 − KNSTY + K STV NZ + L m h1 h2 ) − NT 2VXY (1 + 4m h12 ) − 24 KNT XZm h1 h22 − K NVYZmh2 − K NV 2YZmh1 − KNT 2VX m h22 (1 + 4m h12 ) − KNV X (1 + 4m h12 ) − 12 KNVX m h22 ( ) ( ) + KT (NZ + L )m h h + 32 KNT X m h h + KNT Y m h h − K NSTYZmh − K ST (NZ + L )m h + KNSTXZmh + KNSTVX m h h + NT XYm h h − NT YZmh − KT V (NZ + L )(1 + 4m h ) + KNT VXZm h h − K NV X m h h − K 3V NZ + L m h22 + 16 K NT 2YZmh2 + 16 K 3T NZ + L m h1 h23 + KNT 2VXZm h12 h2 2 2 + KNSTVX m h1 h2 − K NT VYZmh2 (1 + 4m h ) − K NSTYZmh2 − KNT VY (1 + 4m h12 ) ( ) ( ) ( ) − K 3V NZ + L m h1 h2 − K 3V NZ + L (1 + 4m h12 ) − K 3T 2V NZ + L m h22 (1 + 4m h12 ) + KNT VXZmh2 (1 + 4m h ) − KNT VY (1 + 4m h ) − KNT VX m h (1 + 4m h12 ) ( ) 2 − KT 2V NZ + L (1 + 4m h12 ) + K NSTVYZmh1 + KNVXZmh2 − KNV X m h1 h2 * e − mh1 e − mh2 e mH 233 (Denom C1 )24 = − KSVX − 2TY + 12 KST Xm h1h2 − KS 2TX − KTXm h22 ( ) ( ) ( ) − KTV X + 4m 2h12 − KT Xm h22 + 4m 2h12 − T 3Y + 4m 2h12 − KTVXm h1h2 * e − mh1 e − mh2 e 3mH (Denom C1 )25 = { { } X {NZ 32 KN 2TV X Zm h12 h2 − KNSVX NZ + L − KLNSVX − N SVX 2Y } + KNTVX NZ + L m h1 h2 − KN SV X Zmh1 − KNTV { } }( + L + 4m h12 ( ) − KNTV X NZ + L m h12 − KLNTV X + N 2TVX 2Ym h1 h2 − N 2TV X 2Y + 4m h12 { } − KNTVZ NZ + L mh1 − N TVXYZmh1 * e (Denom C1 )26 = − mh1 e ( − mh2 e ) − mH ) − 3KNSVXY − KNTV XY + 4m h12 + 8KNTVXYm h12 + 16 K NTV XZm h12 h2 { } { }( ) + K 2TV NZ + L m h1 h2 − KNTVYZmh1 − K 2TV NZ + L + 4m h12 − K NSV XZmh1 { } − K SV NZ + L * e − mh1 e −4 mh2 e mH (Denom C1 )27 ( ) = − KSTY − KVY − KT 2VY + 4m h12 + 12 K STVXm h1 h2 − K S 2VX ( ) ( + KT 2Ym h1 h2 + 16 K 2T Xm h1h23 − K STXm h22 − K 2T 2VXm h22 + 4m h12 − K 2V X + 4m h12 − K 2V Xm h1 h2 − K 2VXm h22 * e − mh1 e − mh2 e 3mH ( ) (Denom C1 )28 = − K SVY + K 2TVYm h1 h2 − K 2TV 2Y + 4m h12 * e −2 mh1 e −6 mh2 e 3mH (Denom C1 )29 = − 2T NZ + L − 12 NTX m h22 − NTY − K SV NZ + L − KNSVXY { } { {NZ { } } { } + L}m h h − KNTV XY (1 + 4m h ) ) − NT Y (1 + 4m h ) − NSVX − NS 2TX − KNS 2TXY − K S 2T NZ + L − KNTYZmh2 − K 2T NZ + L m h22 + NTXZmh2 + 12KNST XYm h1 h2 + 12K ST { }( − K NTVZ m h1 h2 − K 2TV NZ + L + 4m h12 − NST XZmh1 + 12 NST X m h1 h2 − 8KNTXYm h22 − 24 K NTVXZm h1 h22 ( ) − K NTV XZmh2 + 4m h12 − K NSV XZmh1 − KNTVYZmh1 − K NSVXZmh2 ( ) ( − KNTYZmh2 − 16 K NTXZm h23 − KNT 3YZmh2 + 4m h12 − NTV X + 4m h12 − NTVX m h1 h2 − KNTVYZmh1 − KNTYZmh2 − 8K NT XZm h ( ) ) + 16K NST XZ − K NS 2TXZ − KNT XYm h22 + 4m h12 − 8KNTVXYm h1 h2 ( ) { ) − T {NZ } ( ) + L}(1 + 4m h ) + KNST ( − NTV X + 4m h − K T NZ + L m h + 4m h + NT XZmh2 + 4m h12 ( − NT X m h22 + 4m h12 2 ) YZmh1 − 8K LTVm h1 h2 * e −2 mh1 e mH (Denom C1 )30 = ( ) − 2STX − T 2VX + 4m h12 * e −2 mh1 e mH 234 ) (Denom C1 )31 = { } { } − NSTX NZ + L − LNSTX − NVX NZ + L − LNVX − N S 2VX { } { } − N STXY − KNSTY NZ + L − KLNSTY − K NS 2VX NZ + L − K LNS 2VX { } − N VXY + K NSTVX NZ + L m h1 h2 − K N STVXZ m h1 h2 + K LNSTVXm h1 h2 ( ) { } − KN T VXYZmh2 + 4m h − K NSTX NZ + L m h22 − K LNSTXm h22 − KN STVXYZmh1 + 32 K N T VXZ m h h + 16 KN 2T 2VXYZm3 h12 h2 { } 2 { } ( ) + 16 K NT X NZ + L m h1h23 + 16 K LNT Xm h1 h23 − K NT 2VX NZ + L m h22 + 4m h12 ( ) { } − K LNT 2VXm h22 + 4m h12 − K NVX NZ + L m h22 − K LNVXm h1 h2 { } T VXY (1 + 4m h ) − K − 12 K NV X NZ + L m h1 h2 − K LNV Xm h1 h2 − KN 2V XYZmh1 + 16 N 2T 2VX Zm h12 h2 − 2N { } { − LNT VX − KN VXYZmh2 − KN V XYZmh1 + 16 KN T XYZm h h − 16 N T X Zm h h { } } N 2V XZ m h12 − NT 2VX NZ + L m h12 − NT 2VX NZ + L 2 2 + NT X NZ + L m h1 h2 + 16 N 2T X m h1 h23 + N 2T XY m h1h2 + N STVX m h1 h2 { } { } − K NSTVX NZ + L mh1 − KN 2T 2YZ m h1 h2 − K NT Z NZ + L m h1 h22 { } − N T VXZ m h + N T XZ m h1 h2 − NT Z NZ + L mh1 − N T X Zm h1 h22 − N T Y Zmh1 − N STVX mh1 − KN STXYZmh2 − N STVX Zmh1 + N STXZmh2 { } − N STX m h22 + KLNT 2Ym h1 h2 − K NV Z NZ + L mh1 − KN 2V XYZmh1 { } { } ( − K NVZ NZ + L mh2 − K N 2V XZ mh2 − KNVY NZ + L − K LNV X + 4m h12 ( ) { } ( + K LNSTVXm h1 h2 − KLNT VY + 4m h − K NT VZ NZ + L mh2 + 4m h { } { }( − K NSTZ NZ + L mh2 − KN STXYZmh2 − KNT 2VY NZ + L + 4m h12 ( − KN STVXYZmh1 − N STVX Zmh1 − N V X + 4m h { }( ) ( − K NV X NZ + L + 4m h + N T VX Z + 4m h ( ) ) ) ) ) ) − N T VX m h + 4m h + K N STVXZ mh1 − KN 2VXYZmh2 2 + N VX Zmh2 − N STVXZmh1 − N 2V X m h1 h2 − N 2VX m h22 + N STVX m h1 h2 e −2 mh1 e −mH (Denom C1 )32 { } = − LN SVX − N 2TV X NZ + L m h12 − N SV X Zmh1 + 16 N 3TV X Zm h12 h2 + LN 2TVX m h1 h22 − N 3TV X Z m h12 − LN 2TVXZmh1 * e − mh1 e −3 mH { } (Denom C1 )33 = NST XY + KST NZ + L + KNST XZmh2 * e −4 mh1 e mh2 e mH (Denom C1 )34 = KNSTVX NZ + L + NT 2Y NZ + L + K NSTVY NZ + L { } { } { } + N 2T 2VXYZmh1 + KN 2T 2VXZ m h1 h2 + N STVX 2Y − KN 2T XZ m h22 { } + KLNT Xm h22 + K NSTVZ NZ + L mh2 + K N 2T 2YZ m h22 { } + K NT Z NZ + L m h + KN T Z mh2 + 8KN 2T X Zm h23 + KN 2T 2Y Zmh1 { } + KN STVX Zmh2 + KN 2T 2Y Zmh2 + K NT 2Y NZ + L m h22 − N 2T XYZmh2 + N T X Ym h + N T Y + LNT Y + KLNSTVX e 2 − mh1 e mh2 e − mH 235 (Denom C1 )35 = LN 2TVXY + N 3TV X 2YZmh1 + KN 3TV X Z 2m h1 h2 + KLN SV X + KLN 2TVX m h22 + KLN 2TVXZmh2 e − mh1 e mh2 e −3mH { (Denom C1 )36 } = KNST 2Y NZ + L + KLNST 2Y + K N ST Z mh2 + KN ST XYZmh2 + K LNST Zmh2 + N ST XY (Denom C1 )37 e − mh1 e mh2 e − mH = KLN STVXY + KN 3T 2VXYZ m h1 h2 + K N 3T 2VXZ m h1 h22 + KLN 2T 2YZmh2 + K LN STVXZmh2 + K LN 2T XZmh2 + N 3T 2VXY Zmh1 + LN 2T 2Y + KLN 2T XYm h22 e − mh1 e mh2 e −3mH (Denom C1 )38 = K LN ST 2YZmh2 + KLN ST 2Y e −4 mh1 e mh2 e −3mH { (Denom C1 )39 } {NZ = KNTVY + KTV NZ + L + NTVXY + KNSV X + KNTV XZmh1 + K NTVYZmh2 + KNTVX m h22 + K SV } { } + L + K 3TV NZ + L m h22 − KNTVXZmh * e − mh1 e − mh2 e mH (Denom C1 )40 = KSTVX + T 2Y + KT Xm h22 e −4 mh1 e −2 mh2 e 3mH (Denom C1 )41 = KN 2V X Zmh1 + KNV X NZ + L + N 2V X 2Y + KLNV X e −4 mh1 e −2 mh2 e − mH (Denom C1 )42 = KNV XY + K 2V NZ + L + K NV XZmh1 e −4 mh1 e −4 mh2 e mH (Denom C1 )43 = KTVY + K SV X + K 2TVXm h22 e −4 mh1 e −4 mh2 e 3mH (Denom C1 )44 = K 2V 2Y e −4 mh1 e −6 mh2 e 3mH (Denom C1 )45 = KNSTVXY + NSTVX + NT 2Y + K STV NZ + L + KNT YZmh2 { { } } { { } } + K T NZ + L m h + NT VXZmh1 − NT XZmh2 + T NZ + L + NT X m h22 2 + KNSTVXZmh2 + 8K NT XZm h23 + KNT XYm h22 e − mh1 e mH (Denom C1 )46 = (Denom C1 )47 ST X e −4 mh1 e 3mH { } { } = NTVX NZ + L + LNTVX + N 2TVXY + KLNTVY + K NSV X NZ + L + K LNSV X + K N 2TV XZ m h1 h2 + N TV X Zmh1 + KN 2TV XYZmh1 { } + KN 2TVXYZmh2 + K NTVX NZ + L m h22 + K LNTVXm h22 − N TVX Zmh2 { } + N TVX m h22 + N SV X + K NTVZ NZ + L mh2 + KN 2TVYZ e − mh1 e − mH (Denom C1 )48 = N 3V X Zmh1 + LN 2V X e −4 mh1 e −3mH 236 [ ( ) { } + KTV {NZ + L}(1 + 4m h ) + KS {NZ + L}− KNTZ m h h + NSXY + KNTV XZmh (1 + 4m h ) + KNSVXZmh + KNSXZmh e e (Denom C1 )49 ( ) = NTVXY + KNTVXZ + 2m h12 + KTV NZ + L + NTVXY + 4m h12 mh2 (Denom C1 )50 ( ) mH ( ) ( = N 2T 2Y + 2m h12 + N 2V X 2Y + 2m h12 − N 2T XYZmh2 + 2m h12 ( ) ( ) ) + N 2T X 2Ym h22 + 2m h12 + N STVX 2Y + 2m h12 + N S X 2Y + K LNS 2Y ( ) − 12 K LNSTYm h1 h2 + LNT Y + 2m h + KLNT Xm h22 + 16 KLNT Xm h12 h22 { } + KN T Y Zmh2 − KNSTX NZ + L m h1 h2 − KLNSTXm h1 h2 + KN STXZ m h1 h2 { } ( + 24 K NVZ NZ + L m h h + KN T 2Y Zmh2 − 16 KN 2T XZ m h22 + 2m h12 2 ( + KN STVX Zmh2 − 16 KN STVX Zm h h2 + 16 K N T YZ m h + 2m h ( ) 2 ) ) + KN T Y Zmh2 + 2m h + KN T Y Zmh2 + KN VY Zmh1 + 24 K N VYZ m h1 h2 { } + KN Y Zmh2 + 12 K N YZ m h22 − 16 K N STYZ m h1 h2 + K NT Z NZ + L m h23 { } { } + L}+ NT {NZ + K LNT Zm h + K NSTVZ NZ + L mh2 + KNT Zmh2 NZ + L mh2 { + KN T X Zm h + K NT YZ m h + K NSTVY NZ { 2 } } +L ( + K LNTYm h + KNV X NZ + L + KLNSTVX + K LNV Y − N T 2VX Ym h1 h2 + 4m h12 { 2 } ( ) ) + 2KNT VZ {NZ − KNT VX NZ + L m h1 h2 + 4m h + 16 KN T XZ m h h − N STX Ym h1 h2 ( + N 2T 2VXYZmh1 + 4m h12 2 } ( ( 2 ) + KNSTVX {NZ ) VYm h h (1 + 4m h ) + K }( + KLNS X + KLNXm h − K LNT ( 2 ) ( LNSTVY + 4m h12 { ) + L + 4m h12 + KN S XZ + KLNS X + KLNV X + 4m h + N VXYZmh1 + KLNVXm h1 h2 } ( ) ) + KN V X Zmh1 + 4m h + 16 K N V Z m h h + K NV Z NZ + L mh1 + 4m h12 { } ( ) 2 ( + K NT VZ NZ + L m h h + 4m h − KN T VXZ m h1 h2 + 4m h ( 2 ) ( ) ) + K N T VYZ m h1 h2 + 4m h + KN T VY Zmh1 + 4m h − K N STVZ m h12 h2 ( ) ( + L}mh − K N S VZ mh1 + KN T VX Zm h h + 4m h + KNT VZ {NZ + L}m h2 h1 + 4m h12 2 { − K N STVYZ m h + K N V YZ m h − KN VXZ m h1 h2 + KNVZ NZ ) + L mh2 + 4m h12 + KN STZ mh1 − KN T Z m h h + N STXYZmh1 + KLNSTZmh1 ) + KN STVXZ m h + 16 KN V X Zm h h + 16 KN VX Zm h h + 16 K LNV Z m h12 h2 2 2 ( + KNS 2VX Zmh1 − 16 K LNSTVZmh1 + K LNS 2VZmh1 + KN 2V X Zmh2 + 4m h12 { } ( ) { (1 + 4m h ) } + 16 K N VZ m h h + K NV Z NZ + L mh2 + 4m h + K NT Z NZ + L m h 2 − 16 K N STZ m h1 h22 + K N S Z mh2 + KN 2T X Zm h22 { } ( ) ) (1 + 4m h ) { } + KNT Z NZ + L mh2 + 4m h12 + K N Z m h23 − KN XZ m h22 + KNZ NZ + L mh2 + 16 KN VX Zm h h + 16 KN X Zm h + K LNZm h − 16 KN STX Zm h h 2 2 { }( + L}(1 + 4m h ) + KN S X Zmh2 − 16 K LNSTZm h1 h22 + K LNS Zmh2 + K NV 2Y NZ + L + 4m h12 { } ( ) { + K NT Y NZ + L m h + 4m h + K N S YZ + NT Y NZ 2 ) − KN STY Zmh1 − N XYZmh2 + N YZ + LNY + N STXYZmh1 + N 2VX 2Ym h1 h2 + N X 2Ym h22 − N 2VXYZmh1 + 8K LNVYm h1 h2 + K LNYm h22 − N STX 2Ym h1 h2 + N 2Y e mh2 e − mH 237 (Denom C1 )51 = { ( ) { } LN 2TVXY + 2m h12 − 16 KN 2TV X NZ + L m h12 h22 − KLN 2TV X m h1 h2 } { } − 24 KN TVX NZ + L m h h − KLN TVX m h h − KN SV X NZ + L m h12 2 { } 2 − KLN SV X − KN SVX NZ + L m h1 h2 − KLN SVX m h1 h2 − KN 2TVXZm h12 h2 − KLN TVXZm h h − 16 KN TV X Z m h13 h2 − 16 N 3TVX 2YZm h12 h2 − LN ST XYm h12 2 ( + KLN 2TVX m h22 + KLN SV X + KLN ST XZmh2 − 16 KLN 2T X m h1 h23 + 4m h12 ( ) + 32 KLN ST X m h h − LN T XYm h1 h2 + 4m h − KLN STX m h1 h2 2 ( ) { ) } − 16 KLN TX m h h − KLN T XZm h h + 4m h − KN 2TV XZ NZ + L m h13 2 ( − KLN TV XZmh1 + 16 KLN ST XZm h h − LN T YZmh1 + 4m h 2 ( ) ) − KLN S TXZmh1 − KLN TXZm h h − KLN ST X m h + 4m h + KLN S 2TX m h1 h2 2 { 2 } − KLN S X − KLN SX m h + KN TV XZ NZ + L m h + KLN 2TV XZmh1 2 ( + KLN SVXZmh1 + KN SV X Z m h + KN TVXZ m h12 h2 + KLN 2TVXZmh2 + 4m h12 + KLN SXZmh2 + KN SVX Z m h1 h2 + LN SXY e mh2 e ) − mH { }( ) ( ) + N TVXY (1 + 2m h ) + K NTVZ {NZ + L}mh (1 + 2m h ) + KN TVXYZmh (1 + 2m h ) + KN TV XYZmh (1 + 4m h ) + K NTV Z {NZ + L}mh (1 + 4m h ) + K NSVZmh {NZ + L} − K N TVZ m h h + KN SVXYZmh + K NSZ {NZ + L}mh − 8K N TZ m h h (Denom C1 )52 = KLNSY + KNTVY NZ + L + 2m h12 + KLNTVY + 2m h12 2 + KN SXYZmh2 + KN SYZ − KN 2TYZ m h1 h2 + N SXY e mh2 e − mH (Denom C1 )53 = ( { ) + KLN STVXY + 4m h + KLN T XYm h } ( (1 + 2m h ) + 2K ) KLN S XY + KLN 2T 2YZ NZ + L mh2 + 4m h12 − KLN STXYm h1 h2 2 LN STVXZmh2 − 16 K LN STVXZm h h + 16 K LN STVXZm h h + 16 K LN VXZm h1 h22 + K LNS 2VXZmh1 2 { } 2 + K LN S XZmh2 + K N 2VXZ NZ + L m h1 h22 + KN 3V XYZ m h12 + 16 K N 3V XZ m h12 h2 ( ) ( ) XZ {NZ ( + KLN T VYZmh1 + 4m h + 16 K LN T XZm h22 + 2m h12 + KLN 2V XY + 2m h12 ( ) + KLN T Y + K LN T VXZm h h + 4m h + K N V 2 ( ) } +Lm h + K LN V XZmh1 + K LN V XZmh2 + 4m h − 16 K LN STXZm h1 h22 ( + KN VXYZ m h1 h2 + K LN XZm h − KLN STXYm h1 h22 + LN 2T 2Y + 4m h12 + N VXY Zmh1 + KN VXYZ m h1 h2 + KLN XYm h + KLN VXYm h1 h2 ( ) 2 ) ) − KLN 2T 2VXY + 4m h12 + KLN STYZmh1 * e mh2 e −3mH (Denom C1 )54 = ( ) ( ) K LN 2TVYZmh2 + 2m h12 + KLN 2TVY + 2m h12 + KLN SVYZmh1 ( ) + K LN 2TV 2YZmh1 + 4m h12 + K LN SYZmh2 + KLN SY e mh2 e −3mH 238 (Denom C1 )55 = ( ) KNSY + NTVXY + 2m h12 + NSXY + 12 KNSX m h22 + KNS X ( ) ( ) ( ) + NST XY + 2m h12 + KNSV X + 2m h12 + KNSV X + KNTVY + 2m h12 + 16 KNSVX m h1 h2 − 16 KNS TX m h1 h2 + 16 KNTVXZm h h2 + 64 KNST X m h h 2 ( ) − 16 KNT X m h h (1 + 2m h ) − 16 KNT X m h (1 + 4m h ) + 24 KNT XZm h h (1 + 4m h ) + 24 KNTXZm h h + KNS TXZmh + K NTVYZmh (1 + 4m h ) − 16 K NTVYZm h h + K NSVYZmh + K NSYZmh + KNST X m h (1 + 4m h ) + K NST YZmh + KNST Y 2(1 + 2m h ) + K TV {NZ + L}m h + K NTVYZmh + K SV {NZ + L} − KNTVXZmh + KTV {NZ + L} + KNTVX m h + KST {NZ + L} + KNTVX m h + KTV {NZ + L}(1 + 4m h ) + KS {NZ + L} − NT XYm h h (1 + 4m h ) − 32 KNTVX m h h − NTXYm h h + NT YZmh (1 + 4m h ) + KNTV XZmh (1 + 4m h ) − 16 KNSTXZm h h + NTYZmh + K NTV YZmh (1 + 4m h ) − 32 K TV {NZ + L}m h h − K TV m h h {NZ + L}(1 + 4m h ) − 16 K T {NZ + L}m h h (1 + 4m h ) − 16 K NT YZm h h (1 + 4m h ) + 32 K ST {NZ + L}m h h − K NS TZ − KT m h h {NZ + L}(1 + 4m h ) + 16 K NST YZm h h − 16 K NTYZm h h − 16 K T {NZ + L}m h h − KT {NZ + L}m h h − 16 KNST XZm h h − 32 KNTVX m h h − KNTY m h h − K LS Tm h h + KNSV X (1 + 4m h ) + K NS TZ m h h + K SV {NZ + L}(1 + 4m h ) + K ST {NZ + L}m h (1 + 4m h ) − KNST XZmh (1 + 4m h ) − 8K NS TZ m h h + K S {NZ + L} + KST {NZ + L}(1 + 4m h ) − K NS TYZmh + K NSZ m h − KNSXZmh + KS {NZ + L} + K LSVm h h + K LSm h h − 8K LS Tm h h e e − 8KNS 2TX m h1 h2 − 48KNTX m h1 h23 − KNTV X m h1 h2 + 4m h12 2 2 2 2 + KT Xm h (Denom C1 )57 2 2 2 2 2 − mh2 ( ) (1 + 2m h ) + 8KVXm h h (Denom C1 )56 = 2 2 2 2 2 2 2 mH ( ) VXm h h (1 + 4m h ) e KV X + 2m h12 + KS X + KSTVX + 2T Y + 2m h12 − 12 KSTXm h1 h2 + KXm h + Y − KT ( 2 ) { }( − mh2 e 3mH ) = KLNSTVX + 2m h12 + KNSTVX NZ + L + 2m h12 − 16 KN STVX Zm h12 h2 + KN S VX Zmh1 + KN STVXZ m h − 16 KN STVX Zm h h2 + 64 KN 2T 2VX Zm h13 h22 { } − KNSTX NZ + L m h1 h2 − KLNSTXm h1 h2 − 32 KN 2T 2VXZ m h13 h2 − N STX Ym h1 h2 { }( ) ( − N STX Ym h − KNT 2VXm h1 h2 NZ + L + 4m h12 − N 2T 2VX Ym h1 h2 + 4m h12 { }( ) + L}− N ( ) + KNT VZmh1 NZ + L + 4m h + KN STZ mh1 + KLNSTZmh1 + N T VXYZmh1 + 4m h12 { + N STXYZmh1 + KNS X NZ ( ) S X 2Y − KLNT 2VXm h1 h2 + 4m h12 + KLNS X * e − mh2 e − mH (Denom C1 )58 = ( ) KNS XY + KNSTVXY + 4m h12 + KNSTVXY − 8KNSTXYm h1 h2 ( ) { }( − KNT VXYm h1 h2 + 4m h − 16 K NSTVXZm h h2 + K STV NZ + L + 2m h12 { }( S {NZ ) { } ) − K T Vm h1 h2 NZ + L + 4m h − K ST NZ + L m h1 h2 + 32 K NT VXZm h h + K NS VXZmh1 + K } ( ) 2 + L + KNT VYZmh1 + 4m h12 + KNSTYZmh1 e − mh2 e mH 239 ) ( ( ) + KST Y (1 + 2m h ) + K TVXm h − K TV Xm h h (1 + 4m h ) + 32 K ST Xm h h + K ST Xm h (1 + 4m h ) − 16 K T Xm h h (1 + 4m h ) − KT Ym h h (1 + 4m h ) (Denom C1 )59 = ) KSY + KTVY + 2m h12 + K SV X + 2m h12 − 12 K S 2TXm h1 h2 2 2 2 − 32 K 2TVXm h12 h22 − 16 K 2TXm h1 h23 − KTYm h1 h2 + 8K SVXm h1 h2 + K SXm h22 + K S X * e − mh2 e 3mH ( (Denom C1 )60 = ) ( ) ( ) K STVY + 4m h12 − K 2T 2VYm h1 h2 + 4m h12 − K STYm h1 h2 + K S 2Y * e − mh2 e 3mH { (Denom C1 )61 = } { } K S NZ + L + KNS XY + NT 2Y + 2m h12 + NY − K ST NZ + L m h1 h2 ( ) ( ) ( − K NSTZ m h1 h2 + 12 KNT YZmh2 + 2m h − NT XZmh2 + 2m h + KNV XY + 2m h12 ( ) { } ( + KNVYZmh1 + KNSTVXY + 2m h12 − 12 KNSTXYm h1 h2 + K 2T NZ + L m h22 + 2m h12 ( ) + 16 K NT Z m h h + 16 K NT XZm h + 2m h + NSTXZmh1 + NS X ( 2 ) ( ) ) ) + KNT XZm h + 2m h − KNT VXYm h1 h2 + 4m h12 + KNYZmh2 + K NSTVXZmh2 2 { }( ) { }( T V {NZ − 16 K NSTVXZm h h2 + K V NZ + L + 2m h + K STV NZ + L + 2m h12 { }( ) + 2T NZ + L + 2m h + NSTV + NT X m h + NV X − K ( ) ( 2 ) } ) ( ) XZmh (1 + 4m h ) + K NT VXZm h h + 4m h + KNT VYZmh1 + 4m h + K NV XZmh1 + 4m h 2 + 16 K NV XZm h h + K NVXZm h h + K NS VXZmh1 + K NV 2 2 ( + L m h1 h2 + 4m h12 + K NXZm h23 − 16 K NSTXZm h1 h22 + 16 K NVXZm h1 h22 + K NS XZmh2 + KNVXYm h1 h2 ( ) ( + KNXYm h22 − NT 2VX m h1 h2 + 4m h12 − NSTX m h1 h2 + NT XZmh1 + 4m h12 ( ) ( + L}m h ) ( ) + NSTVX + 4m h + NV X + 4m h + 8K NVZ m h1 h2 + NT X m h + 4m h12 { − KNSTYZmh1 + K NZ 2 { } − NXZmh2 + NZ + L + NVX m h1 h2 2 ) + NX m h22 − NVXZmh1 + K LVm h1 h2 − NSTX m h1 h2 e mH (Denom C1 )62 = ( ) 2TVX + 2m h12 + SX e 3mH 240 ) (Denom C1 )63 = { }( − 16 K N TV XZm h13 h2 − 16K N TVXZm h12 h22 + K NSV X NZ + L + 4m h12 { } X {NZ X {NZ X {NZ { } + K LNSV X − K NS TX NZ + L m h1 h2 − 8K LNS TXm h1 h2 + KNST Y NZ + L ( )+ K } { } + K LNSVXm h h − 32 K NTV + L}m h h − K NTV Xm h h {NZ + L} − K LNTV Xm h h − 16 K NT + L}m h h (1 + 4m h ) − 16 K LNT Xm h h (1 + 4m h ) + 32 K NST X {NZ + L}m h h + K LNST Xm h (1 + 4m h ) + NSX {NZ + L} + LNSX + 32 N TX Zm h h − 32 K LNTXm h h + 16 N TVX Zm h h + NTVX {NZ + L}(1 + 4m h ) + LNTVX − N TVXZ m h − NST X {NZ + L}(1 + 4m h ) + LNST X − N T XZ mh (1 + 4m h ) − NT X {NZ + L}m h h (1 + 4m h ) + K NTVX {NZ + L}m h + K LNTVXm h + KLNTVY + KNTVY {NZ + L}(1 + 4m h ) + 8KN TVYZ m h + N TVXY (1 + 2m h ) − 16 N ST X Zm h h − N ST X Z (1 + 4m h ) + N ST XY (1 + 2m h ) + KN SVXYZmh + K NS X {NZ + L} + K LNS X − 8KN TVXYZm h h + KN TV XYZmh (1 + 4m h ) + K NSX {NZ + L}m h + K N SV XZ m h + K NSV X {NZ + L}(1 + 4m h ) + K NSVZ {NZ + L}mh + 8K LNSVXm h h + K NTVZmh {NZ + L}(1 + 4m h ) + 16 K NTVZ {NZ + L}m h h − 32 K NTVX {NZ + L}m h h − 8K N TVXZ m h h + 16 N T X Zm h h (1 + 4m h ) + N T X Zm h h (1 + 4m h ) + KN TVXYZmh − N TVX mh + N SV X + N TVX m h + K NST Z {NZ + L}mh + KN ST XYZmh − KLNT Ym h h (1 + 4m h ) + K NTV Zmh {NZ + L}(1 + 4m h ) + K NSZ {NZ + L}mh + KN TVXYZmh (1 + 4m h ) + KN SXYZmh + KN SVXYZmh + KNSY {NZ + L} − 8KN TVXYm h h + N SXY − N TV X m h h (1 + 4m h ) − 16KN T XYZm h h (1 + 4m h ) − N T XY m h h (1 + 4m h ) + KLNST Y + 4m h NSV 2 2 2 2 2 2 1 2 − 16 N 2T X 3m4h1h23 (1 + 4m2h1) + 16KN 2ST XYZm3h12 h2 − 16KN 2TVXYZm3h12 h2 2 2 2 2 + L + K LNSV X + K NSVX NZ + L m h1 h2 − 16KN 2TXYZm3h1h22 − 16 K N 2TXZ 2m4h1h23 − NTX {NZ + L}m2h1h2 − 16 N 2ST X 2Zm3h12 h2 − 32 N 2TVX 3m4h12 h22 − 32 N 2TX 3m4h1h23 − 32 K LNTVXm4h12 h22 + 32 N 2STX 3m4h12 h22 − N S TX m h1 h2 − N TXY m h1 h2 + 32K LNST Xm h12 h22 − K LNS TXm h1 h2 ( ) { }( ) + 8K NT Z {NZ + L}m h h (1 + 4m h ) + 8KN T YZ m h h (1 + 4m h ) + N T Y Zmh (1 + 4m h ) − 16 K NST Z {NZ + L}m h h + K NS TZ {NZ + L}mh − NT Z {NZ + L}mh (1 + 4m h ) − 8KN ST YZ m h + 8KN TYZ m h h + 8K NTZ {NZ + L}m h h − N TXZ 2m h h + NTZ {NZ + L}mh + N ST XZ m h + N S TX Zmh + N TY Zmh + N SV X (1 + 4m h ) + 8K NSVXZ m h h + K NST X {NZ + L}m h (1 + 4m h ) + KNST XYZmh (1 + 4m h ) − 8K NS TX {NZ + L}m h h + N ST X m h (1 + 4m h ) − KN S TXYZmh + N TV X Zmh1 + 4m h11 + K NTV Zmh1 NZ + L + 4m h12 2 2 2 + KN SVXYZmh1 + KN SXYZmh2 − N SX Zmh2 + N S TX Zmh1 + N SVX m h1 h2 + N SX m h22 + K LNSXm h1 h2 − N S TX m h1 h2 + N S X e 3mH 241 ) (Denom C1 )64 = ( ) LN STVX + 2m h12 − 32 N 3T 2VX Z m h13 h2 − 16 N STVX Zm h12 h2 ( ) − LN T VX m h1 h2 + 4m h − LN STX m h1 h2 + 32 N 3T 2VX Zm h13 h22 + N 3T 2VXZ m h13 ( ) + LN 2T 2VXZmh1 + 4m h12 + LN STXZ + N STVX Z m h12 + N STVX Z m h12 + LN S X + N S VX Zmh1 e −3mH A.2 Final terms of constant D1   18   −  ∑ ( Num D1 )i   Numerator D1   D1m = =   i =1  27 Denominator D1  ∑ (Denom D1 )i   i =1   A.2.1 The list of Numerator D1 (Num D1 )1 = − KNTYZmh2 − K NTZ m h22 − K NSVZ − K LSV + NTXZmh2 − NTZ − K LTm h22 − NSVX − LT − NTY e mh1 (Num D1 )2 = − KNVZ − NVXY − KLV e mh1 e mh2 (Num D1 )3 = − KLNSVX − KLNTXm h22 − LNTY e mh1 e mh2 e −2 mH (Num D1 )4 = − KLNVY e mh1 e mh2 e −2 mH (Num D1 )5 = − KNSTZ − KLST − NSTXY e mh1 e −2 mh2 (Num D1 )6 = − KTXm h22 − KSVX − TY e mh1 e −2 mh2 e mH (Num D1 )7 = − KSTY e mh1 e −4 mh2 e mH (Num D1 )8 = − VX e mh1 e mH (Num D1 )9 = − LNSTX e mh1 e −2 mH (Num D1 )10 = NT X m h22 (1 − 2mh1 ) − NT XZmh2 (1 − 2mh1 ) + T NZ + L (1 − 2mh1 ) ( ) ( ) ( ) + K 2V NZ + L (1 − 2mh1 ) + K 2T NZ + L (1 − 2mh1 ) + KNT 2YZmh2 (1 − 2mh1 ) + KNSTYZ + NT 2Y (1 − 2mh1 ) + K NSTZ mh2 − KNVYZ − K NVZ mh2 + NV X (1 − 2mh1 ) − NVX mh2 + NVXZ − K LVmh2 + NSTX mh2 + NSTXZ + K LSTmh2 e − mh1 242 ( ) (Num D1 )11 = KTV NZ + L (1 − 2mh1 ) + NTVXY (1 − 2mh1 ) e − mh1 e mh2 (Num D1 )12 = KLNT Xm h22 (1 − 2mh1 ) + KLNV X (1 − 2mh1 ) + LNT 2Y (1 − 2mh1 ) + KLNSTXmh2 − KLNVXmh2 e − mh1 e mh2 e − mH (Num D1 )13 = KLNTVY (1 − 2mh1 ) e − mh1 e mh2 e −2 mH (Num D1 )14 = KTV NZ + L (1 − 2mh1 ) + NTVXY (1 − 2mh1 ) e − mh1 e −2 mh2 (Num D1 )15 = KT Xm h22 (1 − 2mh1 ) + KV X (1 − 2mh1 ) + T 2Y (1 − 2mh1 ) − KVXmh2 ( ) + KSTXmh2 e − mh1 e − mh2 e mH (Num D1 )16 = (Num D1 )17 KTVY (1 − 2mh1 ) e − mh1 e −4 mh2 e mH = TVX (1 − 2mh1 ) e − mh1 e mH (Num D1 )18 = LNTVX (1 − 2mh1 ) e − mh1 e −2 mH A.2.2 The list of Denominator D1 (Denom D1 )1 = ( K NTZ m h22 + KNTYZmh2 + K NSVZ + K LSV − NTXZmh2 + T NZ + L + NSVX + NTY + NTX m h22 + K LTm h22 e mh1 ( ) (Denom D1 )2 = KV NZ + L + NVXY e mh1 e mh2 (Denom D1 )3 = KLNTXm h22 + KLNSVX + LNTY e mh1 e mh2 e −2 mH (Denom D1 )4 = KLNVY e mh1 e mh2 e −2 mH (Denom D1 )5 = KST NZ + L + NSTXY e mh1 e −2 mh2 (Denom D1 )6 = KSVX + TY + KTXm h22 e mh1 e −2 mh2 e mH (Denom D1 )7 = KSTY e mh1 e −4 mh2 e mH (Denom D1 )8 = VX e mh1 e mH (Denom D1 )9 = LNSTX e mh1 e −2 mH ( ) 243 ) (Denom D1 )10 = ( ) ( ) K SV NZ + L + KNTYZmh2 + K 2T NZ + L m h22 + NTY + NSVX ( ) − NTXZmh2 + T NZ + L + NTX m h22 e −2 mh1 ( ) (Denom D1 )11 = KST NZ + L + NSTXY e −2 mh1 e mh2 (Denom D1 )12 = KN 2TVXZ m h1h2 + N 2TVXYZmh1 + LNTY + KLNSVX + KLNTXm h22 * e −2 mh1 e mh2 e −2 mH (Denom D1 )13 = KLNSTY e −2 mh1 e mh2 e −2 mH (Denom D1 )14 = KV NZ + L + KNV XZmh1 + NVXY e −2 mh1 e −2 mh2 (Denom D1 )15 = KSVX + TY + KTXm h22 e −2 mh1 e −2 mh2 e mH (Denom D1 )16 = KVY e −2 mh1 e −4 mh2 e mH (Denom D1 )17 = STX e −2 mh1 e mH (Denom D1 )18 = N 2V X Zmh1 + LNVX e −2 mh1 e −2 mH (Denom D1 )19 = − NTVXY + 4m h12 − KTV NZ + L + 4m h12 − KS NZ + L ( ) ( ) ( )( ) ( ) + KNTZ m h1h2 − NSXY e mh2 (Denom D1 )20 = ( ) ( − KLNT Xm h22 + 4m h12 − KN 2V XZ m h12 − KLNV X + 4m h12 ( ) ) + KLNSTXm h1h2 − LNT 2Y + 4m h12 − N 2VXYZmh1 − KN 2VXZ m h1h2 − KLNVXm h1h2 − KLNS X − KLNXm h22 e mh2 e −2 mH ( ) (Denom D1 )21 = − KLNSY − KLNTVY + 4m h12 e mh2 e −2 mH (Denom D1 )22 = − KTV NZ + L + 4m h12 − KNSVXZmh1 − KS NZ + L ( )( ( ) ) ( ) + KNTVXZm h12 h2 − NTVXY + 4m h12 − NSXY e −2 mh2 (Denom D1 )23 = ( ) ( ) ( − KT Xm h22 + 4m h12 − T 2Y + 4m h12 − KV X + 4m h12 ) − KVXm h1h2 − KXm h22 + KSTXm h1h2 − KS X − Y e −2 mh2 e mH ( ) (Denom D1 )24 = − KTVY + 4m h12 − KSY e −4 mh2 e mH (Denom D1 )25 = − TVX + 4m h12 − SX e mH ( ) 244 ( (Denom D1 )26 = ) − N 2TVXZ m h12 − LNTVX + 4m h12 − LNSX + N 2TVX Zm h12 h2 − N SVX Zmh1 e −2 mH ( (Denom D1 )27 = ) ( ) ( )( − NV X + 4m h12 − K 2V NZ + L m h1h2 − K 2V NZ + L + 4m h12 ( ) ( ) ( ) ( ) − NT Y (1 + 4m h ) + K ST (NZ + L )m h h (1 + 4m h ) − K S (NZ + L ) − NT X m h (1 + 4m h ) − T (NZ + L )(1 + 4m h ) + KNSTYZmh − KNVYZmh − KNYZmh − K (NZ + L )m h + NXZmh − (NZ + L ) − NSTXZmh − NVX m h h ) − K 2T NZ + L m h22 + 4m h12 + NT XZmh2 + 4m h12 − KNT 2YZmh2 + 4m h12 2 2 2 − NX m h + NVXZmh1 + NSTX m h1h2 − NS X − NY 245 APPENDIX B LIST OF PAPERS RELATED WITH THIS STUDY B.1 S/N Publication in Journal Paper Title Year Name of Journal Evaluation of Backcalculation Methods for Nondestructive Determination of Concrete Properties 2006 Transportation Research Record Volume 1949, pp. 83-97 Consideration of Finite Slab Size in Backcalculation Analysis of Jointed Concrete Pavements 2007 Transportation Research Record Volume 2005, pp. 124-142 Examining k -E Relationship of Pavement Subgrade Based on Load-Deflection Consideration 2009 Journal of Transportation Engineering Volume 135, Issue 3, pp. 140-148 Year 2007 Name of Conference Special International Conference on Pavement Technology (ICPT) on Road Construction and Maintenance Technology Beijing, China. B.2 Presented in Conference S/N Paper Title Statistical Evaluation for Backcalculation of Concrete Pavement Properties th 246 Backcalculation Analysis of Rigid Pavement Properties Considering Presence of Subbase Layer 2008 The 87 Annual Meeting, Transportation Research Board Washington D.C. Influence of Slab Thickness Variation on Backcalculation of Concrete Pavement Properties 2008 the International Conference on Transportation and Infrastructure (ICTI) Beijing, China Estimating Modulus of Subgrade Reaction for Rigid Pavement Design 2009 The Eastern Asia Society for Transportation Studies (EASTS) Surabaya, Indonesia st th [...]... of currently available closed- form backcalculation algorithms, and develop a backcalculationbased procedure to derive the composite k value for a rigid pavement with a subbase layer using a suitable closed- form backcalculation algorithm; and (b) to develop a closed- form backcalculation algorithm for a three-layer flexible pavement system, and another for a four-layer flexible pavement system The scope... multi-layer flexible pavements, and the issues involved Chapter 3 presents comparisons of several closed- form backcalculation computer programs of concrete pavement using measured deflections from the database of the USA Long Term Pavement Performance (LTPP) Project (Elkin et al., 2003) The effect of finite slab size in backcalculation analysis of concrete pavement using the selected closed- form backcalculation. .. used as an input for the backcalculation of the moduli of the overlying layers Brief descriptions of backcalculation algorithms for both rigid and flexible pavements are given in the following sub-sections 2.3.1 Closed- form Algorithms 2.3.1.1 ILLI-BACK ILLI-BACK is a closed- form algorithm proposed by Hoffman and Thompson (1981) for calculation of pavement properties of an infinite rigid pavement slab... To evaluate the available existing closed- form and non -closed- form backcalculation algorithms for rigid pavements and assess their suitability for nondestructive determination of composite k value, addressing the issues of slab size, the choice of seed modulus values, and the choice of the forward deflection computation method 2 To propose a procedure based on the backcalculation approach to determine... linear backcalculation currently available for the purpose of backcalculation analysis One approach makes use of theoretical closed- form solutions to directly compute the elastic modulus of each layer by using layer thickness and deflections from one or more sensors (Li et al., 1996; Fwa et al., 2000) Another approach of backcalculation applies some form of iterative process that varies the various pavement. .. using hypothetical data 6 5 To develop closed- form backcalculation methods of three- and four-layer flexible pavement systems respectively 6 To perform verification of the proposed backcalculation algorithms of three- and four-layer flexible pavements using hypothetical data 1.5 Organization of Thesis Chapter 1 presents the background of the study highlighting the need for a rational analytical procedure... function FWD Fallingweight Deflectometer h Layer thickness k modulus of subgrade reaction (for rigid pavement system) or ratio of layer moduli (ratio of E2 to E1 for flexible pavement system) LTPP Long term pavement performance MR Resilient modulus n Ratio of layer moduli (ratio of E4 to E3 for four-layer flexible pavement system) P load r Distance of FWD sensor from the center of load RMSE Root mean... determine the composite k value of a rigid pavement by means of deflection matching of equivalent pavement systems 3 To perform a validation of the computed composite k value by the proposed procedure against actual measured field data reported in the literature 4 To develop a forward calculation program for three- and four-layer flexible pavements respectively and perform a verification to examine the robustness... experience for specific locations and for certain material types For rehabilitation and overlay design, the use of nondestructive test to determine the composite k value is more popular than destructive tests, because destructive tests are not practical for this type of design In this type of design, the responses of the pavement under a test load will be employed as input to backcalculation analysis for. .. method to determine E in a nondestructive manner Many backcalculation algorithms based on multi-layer elastic 5 theory have been used to backcalculate the layer moduli Unfortunately, similar to the case of backcalculation analysis for rigid pavements, they do not always give the same answer due to the use of different approaches in developing the algorithms 1.4 Objectives and Scope of Work The main . suitable closed- form backcalculation algorithm; and to develop a closed- form backcalculation algorithm for multi-layer flexible pavement system. The results showed that the closed- form backcalculation. CLOSED- FORM BACKCALCULATION ALGORITHMS FOR PAVEMENT ANALYSIS BAGUS HARIO SETIADJI NATIONAL UNIVERSITY OF SINGAPORE 2009 CLOSED- FORM BACKCALCULATION. available closed- form backcalculation algorithms, and develop a backcalculation- based procedure to derive the composite k value for a rigid pavement with a subbase layer using a suitable closed- form