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EVALUATION OF TIP BEHAVIOR OF DRILLED SHAFT IN FLORIDA LIMESTONE By JEONGSOO KO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 2010 Jeongsoo Ko To my lovely wife & personal secretary, Yookyeong Jang ACKNOWLEDGMENTS It is my great pleasure to give thanks to professors and friends who supported me to complete this dissertation. First and foremost, I would like to thank my great graduate supervisory committee, all of whom have offered invaluable support and encouragement throughout my whole academic career including this dissertation. Especially, I would like to express my sincere gratitude to Dr. Michael C. McVay for the opportunity to do this study and for his invaluable guidance during the research. I also wish to express my thanks to Dr. Jae Chung for teaching and helping in the Finite Element Analysis (FEA). Furthermore, I thank Dr. David Bloomquist, Dr. Dennis Hiltunen and Dr. Malisa Sarntinoranont for their time and effort they devoted to serving on my supervisory committee. I also wish to express my gratitude to Mr. Peter Lai of the FDOT for financial support. Also, I appreciate Dr. Klammler for his advice for my basic geostatistic concept. Furthermore, I appreciate Mr. Chuck Broward for his technical advice into my lab test. I would like to say thanks to all members in geotechnique group for their friendship. Furthermore, I would like to say thanks to all the members of Korean students in civil engineering for sharing my happiness. Finally, I thank my wife, Yookyeong for her patience, encouragement, and sacrifice for 7 years of my graduate study, and she is my perfect personal secretary and technical supporter. Also, I thank my parents and parents in law for their endless love and support. TABLE OF CONTENTS page A C KN O W LED G M E NTS .......... .......... .......... ......... .......................... ............... 4 LIST OF TABLES .............. ........................ ............. 8 LIS T O F F IG U R E S .................................................................. 9 A BSTRA CT ..................................................................................................... 16 CHAPTER 1 INTRODUCTION ............... .......... .......... ......... 18 1.1 Problem Statement .. ................................................. ............... 18 1.2 Objectives ...... ........................ .................. 20 1.3 S co pe .............................. .............. ...... 2 0 1.3.1 C entrifuge .............. .................... .. ...... ..................... 20 1.3.1.1. Loading (Axial and Lateral) ........... ................................... 21 1.3.1.2. Shaft diameter.............. ........ ....... .......... ............ 21 1.3.1.3. Length to diameter (L/D) Ratio ......................................... 21 1.3.1.4. Influence of Rock Voids or Limestone Recoveries .................... 22 1.3 .2 Laboratory T est ...................... ...... .......... .......................... 22 1.3.3 LRFD Assessment with FEA Analysis .......... ...... .............. 23 1.3.4 LRFD Assessment with Field Verification .......... .......... ............. 24 1.4 Overview of Dissertation ............... ..... ................................. 25 2 LITERATURE REVIEW ....................... ........ ................. 27 2.1 Designs for Drilled Shaft Subjected in Axial Load .................... ........ ....... 28 2.2 Relation Moment and Rotation .............. ......................... ............... 28 2.3 G eostatistics Background ........... ..... ................. .... ....... .... ............... 30 2.4 Implication of Geostatistics to Geotechnical Field Problems....................... ... 32 2.4.1 Application of Stochastic Simulation to Geotechnique Engineering ........ 34 2.4.2 Averaging Technique.................................................. 36 2.4.3 Characterization for Spatial Variability of Soil Properties...................... 37 3 CENTRIFUGE TEST SETUP AND RESULT................ ... ................... 41 3.1 C entrifuge Background ...................... .. .............. .................... ............... 4 1 3.2 Hardw are & Instrum entation ...................... .... ........................ ............... 43 3 .3 M ateria l ...... .... ........ ... ................................................ .. 44 3 .3 .1 S synthetic R ock ................ .......... ......... ....................... ............... 44 3.3.2 C concrete G rout.......................................... ................ ............... 46 3.3.2.1 M ix w ith silica sand ........... ........... .... ................. ................ 47 3.3.2.2 Mix with granite sand .......... ......... ............. .............. 47 3.4 Data Reduction .............. ............................... 47 3.4.1 Axial Loading .................................. ................. 47 3.4.1.1 Axial loading Side shear................. .. ..... .............. ... 49 3.4.1.2 Axial loading End bearing .............. ....................... .............. 49 3.4.2 Lateral Loading in Combination with Axial Loading ............................... 52 3.4.2.1 Lateral centrifuge data reduction.................................................. 52 3.5 Measured Versus Predicted Shaft Tip Response ................ ......... ......... 56 3.5.1 Axial Test Results............................................................. 56 3.5.2 Lateral Test Result Tip Shear Model ................................ ............... 57 3.5.3 Lateral Test Results MomentRotation Model ................................... 59 4 A TWODIMENSIONAL NUMERICAL ANALYSIS OF TIP RESISTANCE OF A DEEP FOUNDATION IN HETEROGENEOUS ROCK......................................... 94 4.1. Finite Element Analysis...................................................... ....... 95 4.1.1 Dimensions and Boundary Conditions............................ .................... 95 4.1.2 Constitutive Models for Materials ..................... .. .. ............ ... 96 4.1.2.1 Reinforced concrete .............. .................... ............ ............. 96 4.1.2.2 Lim estone rock ............. ... .. ....... ...... .......... ............ 96 4.1.3 R ockS haft Interface ................ ........... ....... ............... ........ .... .. 97 4.2 Simulating the Influence of Rock Variability in Florida Limestone on Tip S tiffn e s s ...................................................... ................ 1 0 0 5 USE OF GEOSTATISTICS IN LRFD ASSESSMENT WITH FIELD TEST R E S U L T .............................................................. ..................................... 1 1 6 5.1 Background of Usage with Spatial Correlation and Its Influence on LRFD resistance factors, ................. .... .... ................. 116 5.2 Development of LRFD resistance factors,4 for End Bearing and the 17th Street Bridge Case Study ................... ........ .. ............. ..... ............. 119 5.3 LRFD resistance factors,4 for End Bearing at the Fuller Warren Bridge........ 125 6 C O N C L U S IO N .............. .... ...................................................................... 14 4 APPENDIX A COMPRESSIVE FORCE (ON x aXIS, KIPS) ALONG DEPTH (ON Y aXIS, FT) FROM EACH PAIR OF STRAIN GAGE WITH TREND LINES OF LATERAL LO A D IN G S T E P ............ .................................................. ............. 149 B OUTPUT VOLTAGE (VOLT ON YAXIS) FROM EACH STRAIN GAGES VS. tESTING TIME (SEC ON XAXIS) WITH LOADINGS.................................. 153 C MOMENT CURVE WITH LOAD STEP ...... .................................. 157 D SHEAR CURVE WITH LOAD STEP................................ ............... 162 E RELATION Kv(VERTICAL STIFFNESS) VERSUS J(RIGIDITY PARAMETER)... 167 LIST O F REFERENCES ....... .................... ..................................... ............... 168 BIOGRAPHICAL SKETCH ................... ............................ 171 LIST OF TABLES Table page 11 Recommended Tests for End Bearing and Unit Tip Shear.............................. 26 31 Centrifuge Scaling Relationships (Chandrasekaran, 2001).............................. 65 32 Proportion of Component for Synthetic Limestone................ ......... .......... 65 33 Ratio of Em (with perlite) / Ei (no perlite) ..... ....................... .............. 65 34 Amount of Component for Mixing Concrete with Silica Sand ........................... 65 35 Result of Unconfined Compression Strength Test................ ......................... 65 36 Amount of Component for Mixing Concrete with Granite Sand ..................... 66 37 Result of Unconfined Compression Strength Test................ ......................... 66 38 Estimation of Em/Ei Based on RQD (Load Transfer for Drilled Shafts in Intermediate Geomaterials, 1996) ......... ......... ............ .. ............... 66 39 Comparison of Axial Tip Stresses................ .................................. ...... 66 41 Comparison of FEA results (CV6FEA) and analytical solution (CVsh from Eq. 14) at H/D=2.... .............................................................. 108 LIST OF FIGURES Figure page 21 A scatter plot and variance and covariance as a function of lag distance......... 39 22 Examples of randomly generated E for various correlation lengths.................. 40 31 New Cylindrical Sample Container ......... ................................. ............. 67 32 Instrumentation for Measuring Lateral & Vertical Forces and Deformation ........ 67 33 University of Florida 12.5 G ton Centrifuge.............. ..................... 68 34 Slotted Steel Pipe (Axial Reinforcement) & Strain Gages ............. ............... 68 35 Mixture of Synthetic Rock for Centrifuge Test ...... ........ .................... 69 36 Drilled Hole in Synthetic Limestone ...... .... ............................. 69 37 CastIn Model Shaft in Synthetic Limestone....... ...... ... ... .................. 70 38 S lip R ings and R otary U nion................................................... .... ................. 7 1 39 Schematic sketch of a centrifuge test setup ............ .. .... .................. 71 310 Grain Size Distribution for Limestone Aggregate................ ...... .......... .. 72 311 Strength Tests for Concrete and Rock ...... ................. ..... ........... 72 312 Synthetic Limestone with 5 mm Perlite Spheres Occupying 30 % Volume ........ 73 313 StressStrain Response Voided and nonVoided Limestone........................... 73 314 MohrCoulomb Envelop from Qu,Qt and Triaxial Tests on 10tsf Rock.............. 74 315 Strain G age Layout for L/D = 1 ...................... .... ...................... ............... 74 316 Compressive Forces (Q) along Depth ............................ ....... ................ 75 317 Mobilized Skin Frictions vs. Axial Displacement...................... ........ ............. 75 318 Measured Skin Friction Comparing Lila (2003) and McVay (1992) .................... 76 319 Measured Skin Friction Comparing Lila (2003) and McVay (1992) .................... 76 320 O'Neil Em/Ei vs. RQD and UF Em/Ei vs. Recovery .................... ............. 77 321 Two Layer System and Equivalent Modulus (Ueshita and Meyerhof, 1967)...... 77 322 Layered M oduli beneath a Drilled Shaft................................... ..................... 78 323 Output Voltage from Gages vs. Time with Loading; Axial Loading First and Combined Loading Following ............. ............................. ......... 78 324 Total Axial and Bending Strains along the Shaft CrossSection...................... 79 325 MomentCurvature Relationship for 9ft Diameter from a Pair of Strain Gages above Rock Surface. ............................. ............................ 79 326a Moment Distribution along a 9 ft Diameter, 50% Styrofoam at 1D below Tip of Shaft in 10 tsf Limestone from 2 tests ................................. ..... .......... 80 326b Moment Distribution along a 9 ft Diameter, 30% Styrofoam at 1D below Tip of Shaft in 45 tsf Limestone from 2 tests ............................ ..... .......... 80 326c Moment Distribution along 9 ft Shaft without Styrofoam below Tip in 35 tsf Rock from 2 tests.... ...................................... .......... 81 327 Increase in Tip Shear with Increasing Lateral Load, L/D = 1 ............. .............. 81 328 Increasing Tip Shear with Increasing Rock Strength ............. ................ 82 329 Shaft Shear at Failure with and without an Underlying Soft Layer.................... 82 330 Assessing Rotation from the MomentCurvature Relationship (a) Moment versus Depth (b) Moment Curvature Relationship................... .... .......... 83 331 Typical Lateral Deflections (y) vs. Depth ............... ......................................... 83 332a Tip Resistance (Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O'Neil(right) for 9 ft / 9 ft 10 tsf R ock .............. .... .......................................................... 84 332b Tip Resistance (Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O'Neil(right) for 9 ft / 27 ft 10 tsf R ock .............. .... .......................................................... 84 332c Tip Resistance (Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O'Neil(right) for 9 ft / 9 ft 30 tsf R ock .............. .... .......................................................... 84 332d Tip Resistance (Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O'Neil(right) for 9 ft / 27 ft 30 tsf R ock .............. .... .......................................................... 85 333a Tip Resistance (Model vs.O'Neill's Equation with Harmonic Average Method (left) and Geometric Average Method (right) for 10 tsf Rock with 50% Recovery at 1 D .......... ... .... ... ............................ .......... ........ 85 333b Tip Resistance (Model vs.O'Neill's Equation with Harmonic Average Method (left) and Geometric Average Method (right) for 30 tsf Rock with 50% R ecove ry at 1D ......... ... ....................................... ........................... 85 334 Tip Shear Stress vs. Tip Lateral Displacement.................... ................. 86 335 Tip Shear Stress vs. Tip Lateral Displacement.................... ................. 86 336 Shear in Shafts vs. Depth with and without Underlying Softer Layer.................. 87 337 Moment in Shafts vs. Depth with and without Underlying Softer Layer ........... 87 338 Tip Shear Minus Rock Strength Divided by Normal Tip Stress vs. Normalized Lateral Displacement.................................... ............. 88 339 Tip Shear minus Rock Strength Divided by Normal Tip Stress vs. Normalized Lateral Displacement.................................... ............. 88 340 MohrCoulomb Strength of Florida limestone......... ............ ............... 89 341 Tip Shear Model (a) Tip Shear Model Scheme (b) Tip Shear Model................. 89 342 Tip Moment Transfer as Function of Tip Rotation................... ............ 90 343 Tip Moment Transfer as Function of Tip Rotation................... ............ 90 344 Calculated vs. Measured Rotation using Bell (1991) Trend line is in red ........ 91 345 Calculated vs. Measured Rotation using Bell (1991) with Trend line in red........ 91 346 Stiffness Coefficient, K, from Doherty and Deeks (2006) ............... .............. 92 347 Calculated vs. Measured Rotation in 10 tsf Rock using Doherty and Deeks (2006) w ith Trend Line in Red .................................... ......................... ......... 92 348 Calculated vs. Measured Rotation in 30 tsf Rock using Doherty and Deeks (2006) w ith T rend Line in red ...................................... ....................... .......... 93 349 Recommended Tip Rotations vs. Moment Model from Bell (1991) (a) Tip Rotation Model Scheme (b) Tip Rotation Model.............................................. 93 41 Dimensions and boundary conditions of FEA Model ............. ... ............ 109 42 Nodal penetration phenomenon ............ .. ................................. ............ 109 43 Model of the rockshaft side interface boundary.............................................. 110 44 Centrifuge test and FEA results........................... ........................... ............ 111 45 Comparison of the tip stiffness among various CVE (a = 5 ft)........................ 112 46 Simulation results of the cases among various av (CVE =0.6 and L/D =3) ..... 113 47 Simulation results of the cases among various CVE (av=15ft and L/D=3) ....... 114 48 Comparison of tip stiffness (CVE=0.6) ............................... ............... ........ 115 51 Thick Solid Contour Lines Represent a = as2/a2 for a/D = [0, 20] and L/D = [0,10]. Thick Dashed Contour Lines Represent a = f (L/a) for D = 0. Thin Dotted Lines are Parabolas, Along Which A = const. (Klammler, 2010)........... 129 52 LRFD resistance factors,4,as a Function of Reliability Index (3 and COVR (Klam m ler, 2010) ............. ..... ..... ... ............. ............... 130 53 Secant vs. Tangent Young's Modulus on 17th Street Bridge Data from LTSO4 131 54 Tangent Mass Modulus of 17th Street Bridge (118 Values) .............................. 132 5.5 Secant Mass Modulus of 17th Street Bridge (118 values).............................. 132 56 Harmonic Mean Modulus, Eh, Assuming Correlation Length, a = 5 ft from 17th Street Bridge Data ........... ..................... .......... .. ............... .............. 133 57 Harmonic Mean Modulus, Eh, Assuming Correlation Length, a = 10 ft from 17th Street Bridge D ata ....................... ............... ............... .............. 133 58 Harmonic Mean Modulus, Eh, Assuming Correlation Length, a = 15 ft from 17th Street B ridge D ata ....................... ............... ............... .............. 134 59 Histogram/PDF of Contact Stress, qb, Using Eh with Correlation Length, a = 5ft from 17th Street Bridge Data .............................................. 134 510 Histogram/PDF of Contact Stress, qb, Using Eh with Correlation Length, a = 10 ft from 17th Street Bridge Data ............................ ..... .. ............... 135 511 Histogram/PDF of Contact Stress, qb, Using Eh with Correlation Length, a = 15 ft from 17th Street Bridge Data ............................ ..... .. ............... 135 512 Histogram/PDF of Displacement, Wt, Using Eh with Correlation Length, a = 5 ft from 17th Street Bridge Data ............................. .............................. 136 513 Histogram/PDF of Displacement, Wt, Using Eh with Correlation Length, a = 10 ft from 17th Street Bridge Data ............................ ..... .. ............... 136 514 Histogram/PDF of Displacement, Wt, Using Eh with Correlation Length, a = 15 ft from 17th Street Bridge Data ......... .................................... .. ................ 137 515 Secant Mass Modulus of Fuller Warren Bridge (96 values)............................ 138 516 Tangent Mass Modulus of Fuller Warren Bridge (96 values).......................... 138 517 Geometric Mean Modulus, Eh, Assuming a Correlation Length, a = 5 ft from Fuller W arren Bridge Data ........ ... ........ .. .......................... ..... .. ......... 139 518 Geometric Mean Modulus, Eh, Assuming a Correlation Length, a = 10 ft from Fuller W arren Bridge Data ....... ..... .......... .................. ... .. ......... 139 519 Geometric Mean Modulus, Eh, Assuming a Correlation Length, a = 15 ft from Fuller W arren Bridge Data ....... ..... .......... .................. ... .. ......... 140 520 Histogram/PDF of Contact Stress, qb, Using Eh with a Correlation Length, a = 5 ft from Fuller Warren Bridge Data................ ............... .. ............ 141 521 Histogram/PDF of Contact Stress, qb, Using Eh with a Correlation Length, a = 10 ft from Fuller Warren Bridge Data................ .............................. 141 522 Histogram/PDF of Contact Stress, qb, Using Eh with a Correlation Length, a = 15 ft from Fuller Warren Bridge Data................ .............................. 142 523 Histogram/PDF of Displacement, Wt, Using Eh with Correlation Length, a = 5 ft from Fuller W arren Bridge Data............................................... 142 524 Histogram/PDF of Displacement, Wt, Using Eh with Correlation Length, a = 10 ft from Fuller Warren Bridge Data................ .............................. 143 525 Histogram/PDF of Displacement, Wt, Using Eh with Correlation Length, a = 15 ft from Fuller Warren Bridge Data................ .............................. 143 A1 Result from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test N o 2 ........................... .................. .. ... .................... 14 9 A2 Result from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test N o 2 ........................... .................. .. ... .................... 14 9 A3 Result from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ....... ....... .............. .... ............... ...... ........... 150 A4 Result from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ....... ....... .............. .... ............... ...... ........... 150 A5 Result from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ....... ....... .............. .... ............... ...... ........... 150 A6 Result from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ............... ...... .... ................. .................. 151 A7 Result from 9 ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ................................................................... ..... ........ 151 A8 Result from 9 ft Diameter 27 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ................................................................... ..... ........ 151 A9 Result from 6 ft Diameter 18 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ................................................................... ..... ........ 152 A10 Result from 6 ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ................................................................... ..... ........ 152 B1 Output Voltage from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1 D below from Tip of Shaft a) Test No. 1 b)Test N o. 2 ....... ........ ............ .... ............... ..... ..... ......... 153 B2 Output Voltage from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1 D below from Tip of Shaft a) Test No. 1 b)Test N o. 2 ....... ........ ............ .... ............... ..... ..... ......... 153 B3 Output Voltage from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test N o. 1 b) Test N o. 2 ......... ................ .......... .. ............ .............. 154 B4 Output Voltage from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test N o. 1 b) Test N o. 2 ........................................ .............. .............. 154 B5 Output Voltage from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test N o. 1 b) Test N o. 2 ......... ................ .......... .. ............ .............. 154 B6 Output Voltage from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test N o. 1 b) Test N o. 2 ....... ................................ ............... .............. 155 B7 Output Voltage from ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test N o. 1 b) Test N o. 2 ......... ................ .......... .. ............ .............. 155 B8 Output Voltage from ft Diameter 27 ft Embedded Length in 30 tsf Rock a) Test N o. 1 b) Test N o. 2 ....... ................................ ............... .............. 155 B9 Output Voltage from ft Diameter 18 ft Embedded Length in 10 tsf Rock a) Test N o. 1 b) Test N o. 2 ......... ................ .......... .. ............ .............. 156 B10 Output Voltage from ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test N o. 1 b) Test N o. 2 ......... ................ .......... .. ............ .............. 156 C1 Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 ............... .................. 157 C2 Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 ................................ 157 C3 Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 .................... 158 C4 Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ....................... 158 C5 Moment Curve with Different Top Lateral Displacement from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 .................... 159 D1 Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 ................................ 162 D2 Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 ................................ 162 D3 Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ....................... 163 D4 Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 .......................... 163 D5 Shear Curve with Different Top Lateral Displacement from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ....................... 164 D6 Shear Curve with Different Top Lateral Displacement from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 .......................... 164 D7 Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ....................... 165 D8 Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 27 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 .......................... 165 D9 Shear Curve with Different Top Lateral Displacement from 6 ft Diameter 18 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ....................... 166 D10 Shear Curve with Different Top Lateral Displacement from 6 ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ....................... 166 E1 The relation J and Kv: a(v =0.2) and b(v =0.499).................. .................. 167 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EVALUATION OF TIP BEHAVIOR OF DRILLED SHAFT IN FLORIDA LIMESTONE By Jeongsoo Ko August 2010 Chair: Michael. C. McVay Major: Civil Engineering This study investigated the tip behavior of large diameter drilled shafts embedded short distances in Florida Limestone (i.e., L/D = 1 & 3). This dissertation focused on the modeling of axial, shear, and moment responses for tips of shafts subject to combined axial and lateral loading in homogeneous and heterogeneous limestone. For this study, laboratory (centrifuge) tests, Finite Element Analysis (FEA), and field investigation (from two sites: 17th Street and Fuller Warren Bridges) were conducted. The study from the centrifuge test found that the tip stiffness could be estimated within serviceability condition using the linear stiffness relation proposed by Misra and Doherty. Furthermore, O'Neil's method of assessing tip resistance versus displacement was accurate if the harmonic mass modulus of the heterogeneous rock (twolayer system) was assessed within two diameters (D) below the tip of the shaft. In the case of tip shear, it was found that a bilinear elastic plastic model with failure assessed using MohrCoulomb provided good results. For tip rotation, the model proposed by Bell (1991) gave very reasonable results even though it requires assessment of tip shear and lateral tip displacement. To evaluate the effect of spatial variability of rock material properties on the tip resistance of drilled shafts subjected to axial load, field investigation and two dimensional FEA were conducted. In addition, with regard to the vertical spatial variability of the rock (using harmonic mass modulus with the influence zone at 2D below the tip of a shaft), a relationship was derived between deterministic and probabilistic endbearing resistances in homogeneous and heterogeneous rock formations. The FEA was conducted at 1 in tip displacement, defined as the serviceability condition. An important finding from the field investigation was the variability of the LRFD resistance factors (4) from the spatial variability of the rock at the two sites. Using standard geostatistics measures (e.g., variogram, covariance, etc.), the variance of the harmonic mass modulus with the influence zone at 2D below the tip of a shaft was assessed, along with tip resistance and its associated variability. LRFD resistance factors based on FOSM were calculated with reliability values of 2.5 and 3.0. CHAPTER 1 INTRODUCTION 1.1 Problem Statement Recently, drilled shafts have become the deep foundation of choice for bridges and tall structures, as opposed to small diameter concrete shafts or driven piles. A major economic benefit occurs by replacing a large number of piles in a group with a single or several drilled shafts with diameter reduced footprint. Also, drilled shafts offer reduced construction noise and vibration versus driven piles. Shaft diameters have steadily increased due to the development of larger construction equipment. Because of their large diameters and concomitant large moments of inertia, they are able to resist lateral loads induced by hurricanes and ship impacts. In addition, installation involves a minimal foundation footprint, addressing rightofway issues. The size of drilled shafts has a significant effect on their load resistance behavior. For instance, under lateral loading, large diameter shafts develop a significant moment or couple from the shear transfer on the side of the shaft. McVay et al (2004) found the side shear influenced the backcalculated PY curves, representing soil structure horizontal interaction, i.e., horizontal resistance per length versus horizontal displacement. Furthermore, McVay et al (2004) found that the error in estimating the lateral resistance may be as high as 26% for a 12 ft diameter shaft. In addition to side shear, the tip behavior of large diameter shafts (i.e., 8 ft or greater) is greatly influenced by embedment length. For instance, at typical embedment depths of 20 ft or less, it was expected that these shafts will exhibit lateral tip displacement and rotation, especially for L/D < 5. For such rotations significant moments and shears at the shaft tip would develop in stiff/strong material, i.e. rock. These tip shears and moments would correspondingly change the shaft's internal shears and moments. The current design of drilled shafts does not consider the tip shear and tip moment's relationship to tip rotation because practice developed from small diameter shafts and large L/D embedments for which tip shear and moments are negligible. However, in the case of large diameter shafts or short L/D ratios, tip shears and moments must be accounted for in shaft designs (i.e., steel reinforcement requirements) Furthermore, due to the loss of foundation redundancy and the trend toward larger single shaft construction, field coring of rock near asbuilt, nonredundant shafts was now required (FDOT Structures Bulletin, 2005). In addition, end bearing of large diameter shafts was also strongly influenced by spatial variability within the tip zone (in 2D) because tip resistance is a function of tip compressibility. In current designs for tip resistance of drilled shafts in intermediate geomaterials by O'Neill (1997), the end bearing equation, including elastic modulus, was developed for simple homogeneousrock. In practice, however, there is no uniform rock. Variable heterogeneous rock, in the vicinity of the shaft tip was typically recovered in the field and laboratory strength tests (unconfined compression, and split tension) were performed to assess both strength and Young's Modulus. Generally, all the samples over the whole site have been averaged arithmetically. This arithmetic average may not properly characterize tip resistance of the shaft in heterogeneous rock because it is usually to high due to the general type of distribution, e.g. lognormal. Consequently, tip resistance from other averaging approaches, such as harmonic or geometric may be more appropriate. For current analysis/design (e.g. FBMultiPier) there is great need for a 1D simple tip stiffness model for a large diameterdeep foundation, i.e. drilled shaft) embedded short distances (L/D < 5) which considered spatial variability of the underlying rock. Furthermore, associated variability of the tip stiffness model should be quantified to evaluate the reduced shaft reliability due to layer variability. This dissertation focuses on the effect of spatial variability on the tip stiffness model with specific emphasis on the design aspects of the end bearing of a drilled shaft. 1.2 Objectives Objectives of this research include the following: * Predict the tip shear and moment associated with tip rotation, which should be accounted for in shaft design. * Develop a 1D simple tip model as spring model for easy assessment by engineers in designing shafts while considering spatial variability and the associated correlation in serviceability limit, i.e., 1 inch tip displacement. * Estimate the variability of tip stiffness due to spatial variability in used in assessing LRFD (Load and Resistance Factor Design) resistance factor (4). 1.3 Scope 1.3.1 Centrifuge In order to develop a tip shear and tip moment relationship with including tip rotation and end bearing for large diameter short shafts (L/D < 3), two different rock strengths (10 tsf and 30 tsf) representative of typical Florida limestone would be used. To insure repetitive results, synthetic limestone with same strength and compressibility characteristics would be used the following experimental conditions were varied in order to develop the analytical end bearing and tip shear for design. 1.3.1.1. Loading (Axial and Lateral) Initially, axial load tests were performed to assess existing lab/field unit tip resistance models as a function of tip displacement (e.g., Osterberg results). These tests were then used to identify the peak axial force for a given rock strength, Young's modulus, and L/D embedment. Next, the maximum lateral capacity of the test shaft was established using FBPier. In strong rock, the lateral capacity was controlled by its moment capacity for the longer shafts (L/D = 3); in shorter shafts (L/D = 1), rock strength was the controlling factor. The shafts' axial tip displacement, end bearing, tip translation, and shear were recorded. 1.3.1.2. Shaft diameter The effects of short shaft tip rotation was more pronounced for larger diameter shafts, e.g., D > 8 ft. Consequently, two different shaft diameters were tested to quantify its influence. Diameters of 6 and 9 ft were studied, since they are representative of medium and large diameter drilled shafts in Florida. 1.3.1.3. Length to diameter (L/D) Ratio For each shaft diameter, two different L/D ratios were tested to investigate the effect of embedment depth on the distribution of end bearing and tip shear. The embedment depth strongly influences the magnitude of displacement and rotation of the shaft, as well as its tip normal pressure and shear distribution. Long shafts (i.e., L/D > 5) are generally controlled by the shaft's moment capacity, whereas the soil/rock resistance controls shorter shaft response. L/D ratios between 1 and 3 were tested (see Table 1.1), since they represent typical Florida embedment depths. 1.3.1.4. Influence of Rock Voids or Limestone Recoveries Of particular interest was the influence of karst channels or rock voids usually expressed as Recovery Ratio: ratio length of core recovered to theoretical depth of rock cored < 100% on end bearing. To account for the effect of voids on a shaft's ultimate tip resistance and tip shear, perlite (Styrofoam) spheres, 30% to 50% by volume, were mixed with the synthetic limestone. Since voids are more of a concern for short shafts, the studies focused on shafts with a L/D ratio of 1 and were tested in weak and strong limestone, as shown in Table 1.1. In addition, since shafts may be embedded in various layers with different strength parameters, the extent and quality of rock below the shaft tip becomes extremely important. Current design practice assumed uniform rock properties to a depth below the tip of three diameters, which may or may not be conservative. To investigate this possibility, reduced modulus rock was located one and three diameters below the shaft tip (see Table 1.1), and was used to quantify its influence on end bearing capacities. Table 1.1 summarizes the 32 tests performed to quantify end bearing, tip shear, and tip moment for large diameter short shafts. Note, while there were 16 different scenarios (i.e., rock strength, embedment, voids etc.), each test was repeated twice to ensure accuracy. 1.3.2 Laboratory Test To obtain Young's modulus, cohesion, and friction angle of the synthetic limestone, laboratory tests were conducted. Unconfined compression and split tension tests were conducted with the same material used in the centrifuge test to obtain elastic modulus and cohesion. In addition, triaxial tests were performed to determine the rock's Mohr Coulomb friction angle with increasing confinement load and cohesion. The parameters obtained from the laboratory tests was used to characterize the limestone and as input for FE (Finite Element) analysis. 1.3.3 LRFD Assessment with FEA Analysis To evaluate the effect of spatial variability of rock material properties on tip resistance, finite element analysis (FEA) was conducted. A mathematical and computational model for simulating the multidimensional soilstructure interaction was developed and subsequently used for studying spatial variability effects on tip resistance of the drilled shaft. Using an axisymmetric boundary and loading conditions for the centrifuge testing, finite element models were developed using the ADINA finite element analysis program with a MohrCoulomb constitutive model to simulate the material behavior of the limestone. The input data for Florida limestone in FEA were obtained from unconfined compression, spilt tension, and triaxial tests. The experimental results obtained from the centrifuge tests were used to calibrate the FEA material model. Parametric sensitivity studies were then carried out to quantify the tip resistance of the drilled shaft under a service loading condition. Of interest was the expected variability of shaft tip resistance versus the variability of the elastic modulus of the limestone. To investigate the effect of spatial variability on the tip resistance, random of elastic moduli was generated satisfying site specific mean, coefficient of variation (CV), and correlation length (a). Due to the soil formation process, soil or rock variability was greater vertically than horizontally. Since full rock borings are generally far apart (horizontal correlation length = 0) and shaft diameter is in the 4 ft to 10 ft range, it was assumed that the rock moduli are randomly distributed variables in the vertical direction but of constant value in horizontal direction. Of interest was the 1 Dmegascopic tip stiffness models for such characterization. To develop tip stiffness, 500 simulations were performed with respect to three independent, verticallydistributed variables: CV a, and L/D. Using the FEA simulation results, geostatistical analysis was performed to derive mean tip stiffness and associated variability in terms of CVof displacement, which was a function of the CV of the elastic modulus and the spatial correlation length (i.e., covariance of the rock). The analysis also revealed a linear relationship between mean tip resistances and mean tip displacement, called a 1D simple tip stiffness model, which was based on the harmonic or geometric averages of the elastic modulus of the rock with respect to various depths (i.e., averaging lengths). The nominal resistance and the resistance factors of LRFD were then established from the CV of the tip resistance predicted by FEA 1.3.4 LRFD Assessment with Field Verification Current assessment of side and tip resistance of drilled shafts is performed using rock core samples recovered from the site, which are tested in the laboratory (e.g., unconfined compression and split tension tests) to obtain strength parameters. Generally, all the property measurements resulting from laboratory testing were averaged over the entire site, with an arithmetic mean assumed. Values one standard deviation above and below are discarded (FDOT Soil and Foundation Handbook, 2009). Unfortunately, current designs failed to consider spatial variability and associated correlation (i.e., covariance), when designing for end bearing. A probabilistic approach which can handle spatial variability and associated correlations at a specific pier or over the entire site, may be more appropriate. For instance, the designer should have the option of developing specific LRFD factors for a particular shaft/pier based on adjacent data (e.g., summary statistics, covariance, etc.) or by using the entire site's data for an assessment of general LRFD factors. However, this is offset by the higher cost of additional field testing. Examples of predicted tip response, as well as LRFD factors using spatial variability and correlation are presented for two FDOT (Florida Department of Transportation) sites: 17th Street and Fuller Warren Bridges. 1.4 Overview of Dissertation The dissertation includes the following Chapters: Chapter 2 is a literature review, and examines current shaft design for the tip resistance and relationship between moment and rotation, geostatistics theory and past research into geostatistics theory and geotechnical engineering, presented in the same order as the objectives. Chapter 3 provides the centrifuge test setup and results. In this chapter, the tip moment and tip shear are related to rotation, which can be then be applied to drilled shaft design. Chapter 4 presents the FEA analysis to evaluate the effect of spatial variability of rock material properties on the tip resistance of drilled shaft. Using the results of this chapter, 1D tip spring and associated variability of tip stiffness can be evaluated based on spatial variability and correlation. Chapter 5 presents the field data assessment for the LRFD factor. Field data were obtained from two FDOT sites: 17th Street and Fuller Warren Bridges. Unit Tip Shear Table 11. Recommended Tests for SDescription of D r Location & Recovery of D Rock Strength Limestone Di 2nd Layer system N/A 1 9ft Homogeneous N/A 3 System N/A 1 6ft N/A 3 10tst Two Layered System 1D Below from Tip of 1 Shaft 1 (Homogeneous Layer 9ft Shaft 1 above the % 3D Below from Tip of 1 Recovery Layer) Shaft 1 N/A 1 9ft Homogeneous N/A 3 System N/A 1 6ft N/A 3 30tsf Two Layered System 1D Below from Tip of 1 Shaft 1 (Homogeneous Layer 9fthaft 1 above the % 3D Below from Tip of 1 Recovery Layer) Shaft 1 End Bearing and CHAPTER 2 LITERATURE REVIEW The design of reinforced concrete drilled shafts subject to both axial and lateral loads, should include the assessment of moment, shear, and rotational force. The distributions of moment, shear, and rotation along the drilled shaft are major factors in design, as defined in ACI (America Concrete Institute) code. The distributions of these properties at tip of drilled shaft are not considered in current designs but are significant, especially for short L/D. In addition, for the last several decades, geotechnical design has been moving toward the Load and Resistance Factor Design (LRFD) methodologies as identified by American Association of State Highway and Transportation Officials (AASHTO). The LRFD factor (0) is based on known variability of the axially applied load (i.e., dead and live loads) and material properties used for the design at multiple limit states (service, impact, etc.). Furthermore, using the LRFD factor (0), a failure probability can be found similarly for each limit state factor. Limited knowledge of material properties was indicated as problem in obtained LRFD factor (0) as function of coefficient of variation of resistance. With heterogeneous soil particularly, the LRFD factor 0 can't be easily obtained due to the difficulty of characterizing its properties. This chapter reviews past studies about tip resistance and axial load, as well as the relationship between moment and rotation. It also reviews past studies that characterize heterogeneous soil and rock properties, such as spatial variability, in terms of geostatistics, which lead to a better understanding of spatial variability. Finally, it reviews past studies in order to derive the resistance with spatial variability of soil properties. 2.1 Designs for Drilled Shaft Subjected in Axial Load Hassan and O'Neill (1997) studied predicting loaddisplacement behavior of axially loaded drilled shafts in intermediate geomaterials (i.e., rocks). Drilled shafts are often located in intermediate geomaterials and derive resistance to axial load from a combination of side friction and end bearing. Hassan and O'Neill (1997) developed simple design equations with loadsettlement behavior of sockets considering the parameters which significantly affecting the socket's behavior. To verify Hassan and O'Neill's study (1997), field load test results were performed. The equations for tip resistance had the following parameters: L/D ratio, elastic modulus of drilled shaft, averaging rock modulus, and embedment length. Furthermore, Hassan and O'Neill (1997) suggested a service limit resistance based on settlements: 1 inch of tip displacement. Misra (2006) suggested tip soil stiffness, i.e., the relationship between tip force and tip displacement. The tip soil stiffness was related to shaft diameter and elastic properties of tip soil, Young's Moduls, E, and Poisson's ratio, pt. Their tip stiffness was based on an embedded rigid punch bearing upon elastic halfspace. 2.2 Relation Moment and Rotation Doherty's (2003) presented a study of the tip moment's relationship with rotation. The rotation at the tip can be calculated from the bending strains obtained from centrifuge testing. Doherty (2003) studied the load and displacement of rigid circular footing in non homogeneous elastic halfspaces using a semianalytical technique. The estimated load and displacement relation was presented in dimensionless elastic stiffness coefficients. Applied loadings in footing were vertical, horizontal, a moment, and a rotation. Four possible footing geometries were considered and the variation of soil shear modulus with depth was an important factor as identified by soil type, sand and clay. Circular footing at the bottom of an open trench, fully embedded footing, fully embedded footing with sidewall, and skirted foundation were modeled. The total displacements, vertical, horizontal, moment and tensional, were presented at middle of the tip side of the footing. Bell (1991) carried out a series of three dimensional finite element analyses to examine the effect of embedment (cases 1, 2 and 3) subjected to vertical, horizontal, and moment loads. The study showed that between horizontal and moment loads, a crosscoupling stiffness coefficient was needed. The study also obtained tensional stiffness coefficient. The author presented monographs or figures representing stiffness coefficient as function of Poisson's ratio, relevant footing geometry, and embedment ratio. Doherty (2006) addressed stiffness of flexible circular footing embedded in an elastic halfspace. Bending stiffness of the footing dictated the response for vertical and moment load cases. This study also presented graphical methods for estimating dimensionless elastic stiffness coefficients. Selvadurai (1979) found that the response of flexible footing in elastic medium subjected to vertical load as a function of rigidity parameter (J), which is perfectly flexible when J=0 and perfectly rigid when J=infinity. Rigidity parameter is a function of Poisson's ratio of soil and footing, thickness of the footing, radius of the footing, shear modulus of the soil, and elastic modulus of the footing. The crosscoupling stiffness coefficient between horizontal and moment was insignificant. 2.3 Geostatistics Background Geostatistics originated during the 1950s in the mining industry to estimate ore yield on mine excavated volumes. Since the 1980s, applications have expanded to include various engineering and science fields such as petroleum engineering (Deutsch, 1998), agriculture, meteorology, and hydrology (Goovaerts, 1997). In geotechnical engineering, geostatistics is a means of quantifying the variability of material properties spatially or from point to point. Three basic components of geostatistics are: semivariogram analysis (generally called variogram analysis), kriging, and stochastic simulation. One of the basic tools in geostatistics is the variogram, which quantifies the spatial correlation (or variance of local differences) of geotechnical properties. The spatial semivariance (y(h)) is expressed between two points by lag (separation) distance (h). To measure the spatial variability between two different variable sets with semivariance, two terminologies should be introduced: covariance (Cov) and correlation coefficient (p). Covariance is the measurement how two variables relate together, describing the variance of two variable sets. Correlation coefficient is defined as dividing the covariance of the two variables by the product of their standard deviation. Correlation coefficient ranges from 1 (increasing linear relationship) to 1 (decreasing linear relationship). If the variables are independent, the correlation coefficient is zero; this means covariance is equal to zero. Also, the scatter plot can be introduced to describe spatial variability. The scatter plot can be expressed with measured data pairs from the same variables and each data pair is separated by some distance (i.e., lag) from each other in Xdirection, Ydirection or XY direction. If the scatter plot is close to the 45 degree line (p =1) or 45 degree line (p =1), it indicates a high correlation coefficient. Semi variance is half variance of the increments between of two sets of variables. The variogram is described as relationship between semi variance and lag. Establishing the spatial correlation structure of a site having spatial variability in rock properties would require an extensive amount of sub surface exploration. This may not be feasible due to high cost. The variogram is a common method to develop the correlation length, which is an indicator of how much two values with distance are related. The longest distance shown between which two variables are related is correlation length. The variogram is a statistical tool that appraises the average decrease in similarity between two random variables as the distance between the variables increases. It describes spatial continuity as a function of distance and direction. The terminologies used to describe the variogram are "range," "still," and "nugget." With increasing lag, semivariance generally increases in the variogram. However, at some point, semivariance stops increasing and there is a plateau of semivariance with increasing lag. The distance at this point from zero lag is called the "range." The semi variance value at the plateau of variogram is known as the "still." When lag is zero, the semivariance used to be zero. If nonzero, the value of semivariance is known as the "nugget." To help explain the concept of a semivariogram and covariance function, Figures 2.1a and 2.1b show scatter plots that illustrate the variation of local values of E, which were sampled at a site in Florida at 1 ft and 5 ft vertical separation distances, respectively. It is evident that the scatter of the data cloud about the 450 line is less for the smaller separation distance of 1 ft, which is reflected by a smaller value of the semivariogram. Evaluating the semivariogram for a series of lag distances allows for construction of the variogram (the prefix "semi" is often omitted) drawn by the continuous line in Figure 2.1 c. In many cases, spatial correlation only persists over a range limited by the correlation length (denoted as av in the sequel when considered in the vertical direction) until the variogram reaches a plateau ("sill") at a value equal to the variance (o2) of the random function. As an alternative to the variogram, a spatial covariance function C(h) can be also used as a measure of spatial likeness (drawn by the dashed line in Figure 1c). The kriging technique is to linearly interpolate the value Z(x) of a random field Z(x) at an unobserved location x from observation z, = Z(x),i = 1,....,n of the random field at nearby locations, x1,..., x,, weighted according to spatial variogram or covariance values. This implies the kriging is possible with knowledge of the variogram or covariance. Stochastic simulation is a technique for generating multiple equal probable realizations. The most widely used techniques of stochastic simulation are Sequential Gaussian Simulation and sequential indicator simulation. Multiple realizations to fit data sets generated by variogram, kriging, and stochastic can be used in simulations to get the capacity variability of a foundation. 2.4 Implication of Geostatistics to Geotechnical Field Problems A number of commercial geostatistics software packages (e.g., GSLIB) are available for generating large numbers of random field realizations (stochastic simulation) of properties using either y(h) or C(h) in multidimensions. These two different realizations may be considered as two equally likely scenarios for an unknown situation in the real world. In Figure 2, example outcomes of 1D realizations at a 1 ft resolution are illustrated for two vertical correlation lengths of 5 ft (shown in Figures 2.2a and 2.2b) and 15 ft (shown in Figures 2.2c and 2.2d). Brighter areas indicated lower values of Young's Modulus, E, whereas darker areas indicated higher values of E. Visual inspection confirms the shorter scale variability of Figures 2.2a and 2.2b compared to Figures 2.2c and 2.2d, which was due to the contrast in two correlation lengths. The use of random input properties in Finite Element Analysis (FEA) of foundations was first reported by Fenton and Griffiths (2002 & 2005). Section 2.4.1 addresses this in further detail. Using a spatially correlated and lognormally distributed E, the authors numerically predicted settlements of the shallow foundation in a two dimensions. Results of their FEA simulations reveal that settlements of the shallow foundation can be estimated using an effective modulus calculated by the geometric average of E over the entire depth of their FEA model. They also modeled the uncertainty (variance) of settlement by applying a variance reduction factor to a volume averaging scheme in a general statistical context, (originally discussed by Vanmarcke (1984)). Sections 2.4.2 and 2.4.3 more closely address averaging and variance reduction factor of the mean (expected) settlements. In early numerical studies for design of deep foundations (Phoon, 2000), uncertainty was modeled by dividing the soil into multiple homogeneous layers of random yet uniform properties rather than considering more realistic spatial variations of properties. Recently, Fenton and Griffiths (2007) investigated the settlement and ultimate axial resistance of a shaft using onedimensional FEA in which spring stiffness was computed using a geometric average of a spatially correlated random variable for E. The only other study of the application of geostatistics to the analysis of deep foundations known to the authors is by Klammler et al. (2010). They reported the effect of spatial variability of ground properties on the ultimate skin friction of deep foundations. In their work, an analytical variance reduction factor was proposed as a function of a deep foundation dimensions and spatial covariance function. The author is not aware of any other investigation to determine end bearing resistance and uncertainty of deep foundations based on a geostatistical description of the soil/rock properties. 2.4.1 Application of Stochastic Simulation to Geotechnique Engineering Fenton and Griffiths (2002) estimated the uncertainty of shallow foundations based on settlement within the serviceability limit using twodimensional simulation. In case of single footing, the probability density function of the total settlement of the footing was considered as function of footing width for various statistics of the underlying soil. Fenton and Griffiths (2002) considered = isotropic soil with a Young's Modulus, E, as a random variable with a mean variance, and a correlation structure that was the same in the horizontal and vertical directions even though soils may show a stronger correlation in horizontal direction due to layering. Since elastic properties, E, are important elements of settlement, the calculation of immediate and consolidation should include it. With layered soil, the estimation of probabilities of settlement was associated with probabilities of E, chosen as the only spatially random soil property. Fenton and Griffiths (2002) assumed elastic modulus field was a lognormal distribution because of the non negative nature of material property. Fenton and Griffiths (2002) found that when correlation length increased, the larger variability in footing settlement was obtained.. Furthermore, Fenton and Griffiths (2002) investigated the effect of standard deviation of E and the effect of correlation length on settlement variability. A Monte Carlo approach is adopted to simulate realization of the elastic modulus field in the FEA mesh and run thousands of times to obtain footing settlements. Both the mean and variance of the settlements were found as function of CV of modulus and correlation length Using an analytical solution, estimated displacement using geometric averaging was obtained with deterministic displacement and E. Fenton and Griffiths (2005) estimated the uncertainty of shallow foundations based on settlement within serviceability limit using 3D simulation. To give support to the geometric averaging method, simulation with horizontally layered soil was conducted with elastic moduli varied from layer to layer but constant in each layer. Based on the displacement of the simulation, the effective averaging modulus in horizontally layered soil was the harmonic average. Alternatively, effective averaging modulus in vertically layered soil was the arithmetic average. As result, the geometric average of a random field was placed between arithmetic and harmonic averages. In 2007 study, Fenton and Griffiths examined reliabilitybased deep foundation design. The random behavior of piles subjected to vertical load and supported by spatially distributed soil was investigated for pile settlements. They assumed that the surrounding soil and pile were perfectly bonded, which means any displacement of pile corresponds to equivalent local displacement of the soil. Fenton and Griffiths (2007) suggested the soil's influence through a series of bilinear springs based on geometric averaging attached to the pile. Monte Carlo simulation of loading and spring stiffness was undertaken to assess the probabilistic behavior of deep foundation represented with a mean stiffness. Following the simulation, the statistical behavior of maximum loads was estimated. The authors concluded that arithmetic averaging of soil strength controlled the maximum of pile 2.4.2 Averaging Technique To estimate equivalent engineering properties considering the effect of spatial variability, an averaging technique for heterogeneous soil properties was needed. Deutsch (1989) suggested and empirical technique called power averaging. Power averaging is calculated with nonlinear regression using the results from 3D numerical simulations of flow conducted with different soil volumes and properties. Power averaging method is very simple to use, but the volumes and properties of soils were assumed to be uncorrelated with one another. A semiempirical technique was developed by King (1989) and Norris (1991). King (1989) suggested a renormalization technique for permeability, which generated the simulation grid and constant permeability was taken into each of its element. Four elements comprised one group and effective permeability, which was based on an analogy between water flow through soils of differing permeabilities, comprised another group. This was valid not only for uncorrelated permeability, but also for correlated. However, it was valid only for isotropic media. Norris (1991) suggested the representative elementary volume (REV) method for flow simulation with renormalization technique. REV defines specific averaging volume where all micro variations are averaged into macro values as a representative elementary property. With increasing volume, effective permeability was measured. The fluctuation in relation between increasing volume and effective permeability was shown. No significant fluctuation at a specific volume shows at the effective averaging volume. A rational approach was suggested by Deutsch (2002), with geometric mean as an estimation of effective permeability of heterogeneous soil. 2.4.3 Characterization for Spatial Variability of Soil Properties For the stochastic analysis, the main element of soil spatial variability should be assigned. Classical method, variogram, spatial correlation length, and variance reduction factor comprise the main elements. The classical method for characterizing soil properties involved statistics, such as mean, coefficient of variation (CV), and probability distribution. Another method for characterizing spatial variability was spatial correlation, which was able to show variation of soil properties from one point to another in the field. Deutsch (2002) expressed spatial correlation with using variograms, quantifying dissimilarity between two points in field. Vanmarcke (1984) also expressed spatial correlation with using a covariance function, quantifying similarity between two points in field. However, the two expressions (variogram and covariance function) were only suitable for stationary fields with constant means and standard deviations. Spatial correlation length, in terms of separation distance between two points with no (or insignificant) spatial correlation, is the maximum value of spatial continuity. Spatial correlation length is defined as the "sill" from a variogram, which was previously discussed. The variance reduction factor was the expression to obtain the final variance from averaging data within certain a volume. Vanmarcke (1984) found spatial average obtains a smaller variance due to a reduction in the probability distribution, which was smaller than the probability distribution of field data. Vanmarcke (1984) suggested the variance reduction factor which variance of spatial averages is correlated to the point variance. The variance reduction factor varies as a function of the averaging volume, the choice of covariance function and correlation length. His variance reduction factor was only given for 1D averaging process. However, a 3D variance reduction factor could also be calculated as the product of three 1D variance reduction factors by assumption of independent correlation structures. Vanarcke's variance reduction factor approached 1 with decreasing averaging volume to a very small value. Elkateb (2002) mentioned, with respect to geotechnical applications, the variance reduction factor was affected only by the averaging volume in vertical direction because the averaging volume in horizontal direction is usually small compared to spatial horizontal correlation length. N E 0 mu ii 4.E+06 3.5E+06  3.OE+06 S 2.5E+05  2 E+05  1_5E+06  10E+06 ScatterPlot 45 degree plot O.OE+00 0.OE+O0 5.0E+05 1.OE+06 1.5E+06 2.0E+06 2.5E+06 3.E+06 3.5E+06 Modulus, Z(ui) [al 4 OE+06 3.5E06 .C N2 5.E+06 S2 0E+065 1 OE06 O tr P 5E405 degree pl 0OE+OO O.OE+00 5.0E+05 1.OE+06 1.5E+06 2.0E+06 2.5E+06 3.OE+06 3.5E+06 Modulus, Z(ui) (b) Spatial Variance and Spatial Covariance Variance I I I c __% __ Variance 'C 0 1 2 3 4 5 6 7 9 10 11 12 SDistance h ovaan Figure 21. A scatter plot and variance and covariance as a function of lag distance Figure 21. A scatter plot and variance and covariance as a function of lag distance 15ft ~ I 30ft 35t 1  (a) (b) (c) (d) Figure 22. Examples of randomly generated E for various correlation lengths 10.668m 0  1.524m  3.048m  4.572m  6.096m S7.62m  9.144m 25ft 1 CHAPTER 3 CENTRIFUGE TEST SETUP AND RESULT This chapter discusses the development of tip behavior (i.e., tip axial resistance, tip shear, tip moment and tip rotation) model for drilled shafts in Florida limestone subjected in combined loading (axial and lateral), using centrifuge testing. Specifically, the effect of strength and stiffness of limestone, as well as voids (described with R.Q.D or Recovery Ratio) and layering (e.g., two layers with different R.Q.D or Recovery Ratio) on tip behavior was investigated in the case of heterogeneous Florida limestone. R.Q.D (Rock Quality Designation) is defined as the quotient as(Score/Tot). 100%, where "Score" is sum of core sticks longer than 4 inch measured along the center line of the core and "Tot" is total length of core run.). Also, the Recovery Ratio is defined as the percent of length of the recovered sample. The tests in Table 1.1 were selected to encompass both soft and medium strength rock for both short and intermediate embedment. In addition, since the character of Florida limestone is typically heterogeneous, it results in a variable modulus. To model this aspect, 1.5 mm diameter perlite spheres were used to capture the mass modulus, Em. A total of 32 centrifuge tests were conducted to characterize the tip behavior based on Table 1.1. A discussion of the centrifuge test setup and results follows. 3.1 Centrifuge Background To reduce the cost and time for testing in the field, laboratory tests have been developed to model field conditions. In geotechnical engineering, because the character of soil is dependent on stress for example strength and stiffness, the modeling of field stress as well as field displacements are very important. The centrifuge test is a laboratory test which is best able to reproduce field stress as well as simulating load transfer alongside and beneath drilled shafts and the surroundingFlorida limestone.Typical applications in geotechnical engineering where the centrifuge testing isuseful include: deep foundations for bridges and buildings, settlement of embankments, stability of slopes, earthretaining structures, tunnel stability, and seawalls. A centrifuge test develops a centrifugal force generated by angular velocity, w (=de/dt). Due to every particle in body rotated in same angular velocity, the speed of particle may be expressed with angular velocity (i.e., w = de/dt = (ds/dt)(1/r)), with r = radius of rotation axis. This means the speed of a particle (i.e., v(ds/dt) = w (de/dt) r) in a body increases linearly with distance, r. In addition, the centrifugal force is found as function of the revolutions per minute (i.e., angular velocity) and the distance from the center of rotation. The relation of centrifugal acceleration and gravitational acceleration can be used as scaling factor for converting between the centrifuge model and a prototype. When the centrifugal force is larger compared to earth's gravity, the scaling factor, N, is ratio the centrifugal acceleration to the normal gravitational acceleration. Using a scaling factor, the stress (1:1), length (1: N) and force (1: N2) can be scaled because the whole body, even each single particle in body, rotates with same angular velocity (see Table 3.1). For example, with 67 gravities, a 1.61 inch diameter model structure is scaled to 9 ft diameter in the field (i.e., size of field size = N size of model size). Since the scale ratio of stress is 1:1 between the centrifuge model and the field model, nonlinear material behavior (and strength) can be simulated. The centrifuge sample container has a depth of 12 inches and a diameter of 17 inches (shown in Figure 3.1), the length below bottom has 7 inches which can be minimized boundary effect. 3.2 Hardware & Instrumentation The tests were performed in the test container shown in Figure 3.1. The circular steel container holds the cast limestone specimen which represents homogeneous or heterogeneous field conditions. The aluminum superstructure provided the necessary reaction for both axial and lateral loading of the models. Mounted to the top of the container were two load cells for axial and lateral loading, two displacement transducers (LVDTs) for axial displacement, and one LVDT for lateral displacement. The test setup is shown Figure 3.2. Figure 3.3 shows the centrifuge at the University of Florida, which is rated at 12.5 Gtons with a maximum model size of 30 inches. To monitor load transfer, the moment, and shear for lateral loading, 4 to 6 pairs (depending on length) of 350 Ohm strain gages were attached to the longitudinal steel reinforcement in each specimen, as shown in Figure 3.4. For steel reinforcing, a 0.75 inch diameter steel pipe (for the 6 ft diameter shaft) and a 1.32 inch diameter steel pipe (for the 9 ft diameter shaft) were used. Both were slotted to characterize the steel ratio of 6.4% and 7.5%. This reinforcement was needed for the expected moments. The axial forces in the shaft were obtained from the average strain gage readings at discrete locations multiplied by the shaft modulus and crosssectional area. To assess bending (and thus flexure), the paired strain gages were monitored from which the curvature was computed and bending moments obtained. Note that the moment curvature relationship may be nonlinear for cracked crosssections. Once the bending moment distribution along a shaft is known, the shear distribution (i.e., derivative of moment with depth) is found. The shear and bending moments are determined along the shaft and the tip. Section 3.4 provides more detail. After placing the synthetic limestone (described in Section 3.3.1) in the centrifuge container as shown in Figure 3.5, it was cured for 14 days to the appropriate strength (from separate test samples). Then, holes were drilled into the synthetic limestone (see Figure 3.6). Next, the instrumented reinforcement cage was placed and concrete described in Section 3.3.2 of the shaft specimen was cast, as shown in Figure 3.7. The Bimba load actuator, LVDT vertical measurement devices, and vertical and horizontal load cell were subsequently placed on the experiment (Figure 3. 2) and all the instrumentation wiring was run through the slip rings (Figure 3.8) to the data acquisition unit outside of the centrifuge. Each slip ring had 24 channels and each channel was used to send and receive voltage from instrumentation. Furthermore, the pneumatic ports on the rotary union were used to send air pressure to air piston acting on the model as loading, axial, and lateral. Figure 3.9 was a schematic sketch of the experiment setup for a specimen of a lengthtodiameter ratio (L/D) of 1. 3.3 Material 3.3.1 Synthetic Rock The limestone used in the experiments, Figure 3.5, was reconstituted material (synthetic limestone), i.e., a mixture of crushed limestone, cement, and water (McVay et al. 2004). Mini Perlite spheres were added to characterize voids associated with typical recoveries from the field. Of primary interest was the influence of void size on rock properties, e.g., modulus and strength. The current practice (i.e., FHWA intermediate geomaterials) was to reduce modulus as a function of RQD or recovery values. For this research, crushed limestone was obtained from a pit in Newberry (Florida Limestone Products Inc.) by the Florida Department of Transportation's State Materials Office (SMO) and delivered to UF in bulk (approximately 2 tons). For volume considerations, (i.e., those necessitated by centrifuge testing) the limestone was subsequently sieved through a No. 10 standard sieve with a maximum particle diameter no larger than 0.00656 ft. From the grain size distribution shown in Figure 3.10, the aggregate was well graded. To minimize the affect of natural moisture in the material, it was dried for at least 24 hours. Next, the synthetic limestone was batched by mixing Portland cement and water in differing proportions. With crushed limestone, a mortar mixture with a waterto cement ratio (W/C) of 4 and an aggregatetocement (Agg/C) ratio of 15 is used (ASTM C270). Cylindrical samples were cast and cured at room temperature for 14 days before unconfined compressive tests were run. To obtain the desired strength, multiple trials of different proportions of limestone, cement, and water were used, as shown in Table 3.2. The unconfined compression strength testing was performed in accordance with ASTM D 2938. An ISTRON compression testing machine, was used to obtain both the unconfined compression strength, as well as the modulus in Figure 3.11. Once the desired strength was achieved, the tests were repeated at least three times to verify repeatability. Limestone specimens with and without perlite were also cast in 2" x 4" cylinders to identify the influence of voids on rock properties (e.g., strength and modulus) and are shown in Figure 3.12 and Figure 3.13. As discussed previously, perlite spheres cast with the synthetic limestone specimens attempted to mimic voids found in natural occurring rock (Figure 3.12). Figure 3.13 shows the stress versus strain response of intact (i.e., no perlite) versus 30% by volume perlite specimens. Both small (5 mm) and larger (10 mm) spheres were investigated. As is evident from the figure, a significant reduction (> 50 %) in strength and compressibility (i.e., the slope of stress versus strain) occurred with the addition of the perlite. Interestingly, the size of the perlite spheres was not a factor, suggesting that the karst nature of limestone could be characterized through sphere inclusion. Table 3.3 showed the comparison of voided modulus to nonvoided modulus based on volume or recovery. Experimental strength tests:Qt, Qu and triaxial tests (three triaxial and one spilt tension tests (Qt) were conducted according to ASTM D2850 and ASTM D3967, respectively). Subsequently, Mohr's stress circles were drawn in Figure 3.14 from which a strength envelope, Figure 3.14 was estimated. Using a curvefitted fourth order polynomial function, material strength characteristics such as cohesion (C=6ksf) and the angle of friction (0) were graphically estimated. The slope of the failure envelope (0) was estimated as 28 degrees at a maximum normal stress of 38 ksf (1.82 MPa) from the centrifuge test with 9 ft diameter and L/D =1 in 10 tsf rock. 3.3.2 Concrete Grout Since the rock strength used in the centrifuge tests are as high as 30 tsf (417 psi), the drilled shaft concrete strength had to be sufficient so as to ensure a limestone failure instead of a shaft failure. This was because the study focused on rotation of the drilled shaft models as well as translation (i.e., tip shear). To ensure the latter, a concrete strength of 360 tsf (5000 psi), which is typical for drilled shafts in Florida, was used. Due to the model size of the shafts, the concrete aggregate was limited in size to 2 mm and below. Several trial mixes were tested, incorporating five components; Portland cement type I, sand, silica fume, superplasticizer, and water. The silica fume and superplasticizer provided an increase in strength by reducing the water cement (w/c) ratio. The influence of sand particle size was investigated, and is presented in Section 3.3.2.1.2. 3.3.2.1 Mix with silica sand Several trial mixes using w/c ratios of 0.350.4 and 0.50.6, with final strength values of 34 ksi were obtained. After several attempts to obtain 5,000 psi, w/c ratio was reduced to 0.3 for higher strengths, as the target strength (i.e., 5 ksi) could not be obtained. Table 3.4 shows the amount of each component for 5 samples. Table 3.5 shows the dimensions of the samples and the maximum load applied to each. The samples were tested after 4 days of curing at room temperature. 3.3.2.2 Mix with granite sand Table 3.6 shows the amount of each component, by weight, required to make 4 samples. Table 3.7 shows the size of sample and the maximum load on each sample. The samples were tested 4 days after curing at room temperature. As shown in Table 3.6, the expected strength (i.e., 5 ksi) was obtained and was used to make the model shafts. Consequently, the mix design outlined in Tables 3.6 and 3.7 was used for drilled shaft construction, as well as the instrumentation and testing matrix outlined in Table 1.1 for a total of 32 instrumented centrifuge tests. 3.4 Data Reduction 3.4.1 Axial Loading The standard testing procedure consisted of first applying an axial load to each shaft until settlement reached 3% to 5% of the models' diameters. This was followed by lateral loading. The axial loading was representative of the vertical live and dead loads under service conditions. The data from the axial loading was used to develop the tip displacement vs. tip stress model. The reduction of the data was as follows. From several pairs of strain gage readings (Figure 3.15) with axial LVDT displacements and load cell output at the top of shaft, the transferred skin friction along the shaft and tip resistance were computed. Specifically, from each pair of strain gage, the axial strain or compressive strain for each applied axial load state was found. The compressive strain is given by: e = ft + snght (3.1) 2 Next, the transformed Young's modulus of the shaft was assessed from the pair of strain gages located above the rock surface along with the applied axial load. The Young's modulus of the shaft is given by: P E = 2 (3.2) where P = the applied axial load. Subsequently, from each axial strain, the compressive force (Q) with depth may be assessed: Q =Ex s xA (3.3) where A = gross shaft area. Figure 3.16 showed the typical axial force, Q, distribution within a shaft for various applied axial loads. Points are from strain gage measurements and the solid lines covering points at each top displacement are trend lines. 3.4.1.1 Axial loading Side shear Once the axial force distribution and set of strains along the shaft for a particular top displacement was determined, the shaftrock load transfer, T (fs) versus Z curve for the shaft can be computed by: AQ f, = (3.4) ,rxDxL where L is the interval length between adjacent pairs of strain gages. Figure 3.17 showed a typical fs versus Z curve for test shafts. For any applied axial load, a corresponding axial displacement and skin friction can be calculated at each elevation. However, due to the homogenous nature of the rock, the skin friction was expected to be uniform along the shaft as well as within range of the FDOT design values (McVay, 1992) and previous centrifuge tests (Lila, 2003). Figures 3.18 and 3.19 show all the TZ curves for the axial load portions of the centrifuge tests. Figure 3.18 shows the results for 10 tsf strength rock and Figure 3.19 shows results for 30 tsf strength rock. The results showed that the FDOT design criteria was quite good, however slightly conservative. The latter was attributed to the linear assumption of the strength envelope (Figure 3.14) in assessing cohesion from /2 sqrt (qu) sqrt (qt). The TZ curves shown in Figures 3.18 and 3.19 agreed very well with McVay (1992) and Lila (2003) results, which have subsequently been implemented into FB MultiPier. 3.4.1.2 Axial loading End bearing A major focus of this research involves end bearing and specifically mobilized tip resistance as a function of tip displacement. Currently, FBDEEP and FBMultiPIER employ the FHWA model formulated by O' Neill, which characterized tip resistance, qb as: qb = AWt067 (3.5) where A(Lambda) = Elastic compressibility parameter; Wt = Displacement at top of shaft (value assumed) The elastic compressibility parameters, A(Lambda), F(Gamma), and Q(Omega), may be computed as: 0L/5]le ]+0.13 (3.6) F= 0.37 0.15 1 logo 0 +0.13 (3.6) ILL E Q 1.14 f 0.05 L 1 logloE 0.44 (3.7) Leading to S05 10 67 A = 0.0134Em, (L+ L LF (3.8) L 7rLF (DJ The Young's modulus in Eqs. 3.6 to 3.8 was the mass modulus, Em, and was of significant importance. The latter was different from the Young's modulus of intact rock samples, Ei, measured in the laboratory (ASTM D3148). The rock mass Young's modulus, Em, represented the whole mass, including fissures, voids, slip planes, etc. O'Neill suggested a correlation (Table 3.8) between the Ei, and Em based on RQD. If RQD values were less than 20 percent, the 20 percent's RQD correlation was used (Load Transfer for Drilled Shafts in Intermediate Geomaterials, 1996). Of interest was the relationship between Florida limestone's mass modulus and intact Young's modulus, Ei, as reported in Table 3.3. Figure 3.20 shows the Em/Ei ratios as reported by O'Neil and UF. It showed that the UF data falls between O'Neil's open and closed joint data. Also, there existed a linear relationship between Em/Ei for recoveries above 50 %; below this value there was a sharp dropoff. Next in importance for estimating tip resistance was the influence of layered systems. The first researchers to address this were Ueshita and Meyerhof (1967), whose results were shown in Figure 3.21. Their theoretical solution for settlement or stress was found in terms of an equivalent modulus, Ee = K El, as shown in Figure 3.21. Others have suggested the harmonic mean, Eh (i.e. Menard), or: N harmomc 1 1 1 (3.9) (+ +..+) E1 E2 EN Where N is number of layers and EN is the modulus of each layer shown in Figure 3.22. Recently, the geometric averaging method, Eg, which accounts for spatial variability (i.e., covariance), provided an excellent correlation with an FEM analysis (Fenton and Griffiths, 2005). The latter may be expressed as: 1 4 Eg =exp( ZinE,) (3.10) n = Both the harmonic and geometric means were used to estimate tip resistance, qb versus displacement (Eq.3.5) for all axial centrifuge tests. The results are presented in Figure 3.32. 3.4.2 Lateral Loading in Combination with Axial Loading The first phase of the research was to conduct lateral load tests in combination with axial loading to obtain moment, shear, and tip rotation of the shafts. Variables that were used include: rock strength (10 tsf and 30 tsf), diameter (6 ft and 9 ft) and embedment ratio (L/D ratio of 1 and 3). Based on the latter tests, tip shear and rotation models were developed. A discussion on how the data reduction was performed is presented in the following section. 3.4.2.1 Lateral centrifuge data reduction As discussed previously, instrumentation was used to assess the lateral load and displacements at the top of shaft. The strain gage output is found by: 4xAV s= (3.11) GxE where, s was the strain (inches per inch) AVwas the change in bridge output voltage (mV) G was the gage factor (2.05) E was bridge excitation voltage (10 V) It was assumed that the relationship between strain gage resistance and voltage (i.e., Eq. 3.11) was linear. Figure 3.23 shows a typical output voltage from the strain gages located at the rock surface for the entire loading sequence (i.e., axial and subsequent axial with lateral). As expected, from 0 to 600 seconds, the individual gages on each side of the shaft indicated no bending since only an axial load was being applied. However after 600 seconds when the lateral load was applied the gage values diverged. A positive output voltage indicated tension and negative voltage indicated compression. Note that the voltage relationship was linear with the first application of the lateral load, but then became nonlinear due to concrete cracking. At each strain gage elevation, the bending strain may be computed from the gage values on each side of the shaft as: S lef ght (3.12) 2 Summing the axial strains (Eq. 3.1) and the bending strains would result in the measured strain on each side of the shaft (i.e., sa and Sb), as shown in Figure 3.24. Also of great interest was the momentcurvature relationship of the shaft's cross section. The latter was used in assessing the moments, shears, rotations, and lateral translations of the shaft below the rock surface. That relationship was: M= EIb = EI( (3.13) r where E = Young's Modulus of the shaft I = Moment of Inertia of crosssection r = Shaft's radius S= Curvature of the crosssection (i.e., change in rotation) The momentcurvature relationship may be simply expressed as M = BO where B is the nonlinear relationship of El. The latter may be obtained readily from the strain gage data in the shaft at the rock surface elevation where the moment is readily known (lateral load x height above rock). The curvature (see Eq. 3.13) was also obtained from strain gage data. A typical momentcurvature relation is shown in Figure 3.25. From the momentcurvature relationship, the moment distribution along the length of the shaft may be found from Eq. 3.13 by using the bending strains from the gage pairs along the shaft. Figure 3.26a, 3.26b, and 3.26c present a number of moment distribution curves along the length of the shafts under multiple lateral loading states. By examining Figure 3.26, it is apparent that the moment distribution along the shaft starts at zero at the top and increases linearly down to the rock surface (i.e., maximum moment) whereupon it starts to decrease due to the rock's resistance. The moment at the tip was a function of shaft length and rock strength. After the moment distribution along the shaft is found, the shear distribution within the shaft may be readily determined from: dMW V = (3.14) dz Figure 3.27 shows the increase in shear with increased lateral loading. The axial load is maintained at approximately 80% of the ultimate capacity. The rock had an unconfined compressive strength of 11 tsf, and the shaft was 9 ft in diameter and embedded 9 ft in the rock (L/D = 1). The shear at the tip of shaft is equal to the lateral force transferred to the rock at the bottom. From the figure it was evident that the more tip shears transferred with increasing lateral load due to the increasing lateral displacement, suggesting a model which mobilized tip shear based on tip lateral displacement. Increasing the rock strength (qu = 35 tsf versus 11 tsf) increased the available tip shear for comparable deformations. The results are shown in Figure 3.28. Both tests involved 9 ft diameter shafts embedded 9 ft into limestone under constant axial and varying lateral loads. The figure shows the shear distribution in the shafts under the same lateral top displacements. Note that failure occurred at 5% of the shaft's diameter, or 5.4 inches. Figure 3.29 shows the influence of a soft limestone layer (i.e., 50 % voids) one diameter below the tip of the shaft in 10 tsf strength rock. This was reflected by the shear crossing the axis (i.e., zero shear or lateral resistance), as well as the shear distribution along the length of the shaft as it reduces at the shaft's tip. In order to develop a tip shear model, the lateral translation of shaft tip was necessary. This was accomplished by evaluating the lateral displacement and rotation along the entire length of the shaft. The procedure used is as follows: * Choose a depth along shaft, as shown in Figure 3.30a; * From the moment trend line, assess the moment from the selected depth; * With the known moment, go to the momentcurvature relationship (Figure 3.30b) to obtain the curvature (i.e., change in slope); * Repeat steps 1 through 3 for several points along the shaft, making sure the tip is also selected; * Identify where there was zero lateral translation of the shaft (i.e., y = 0) by checking for zero lateral resistance (i.e., P = dV/dX = 0) in Figure 3.31; * From the curvature, the rotation at the top of shaft (0) is found; SAO 8= bt bb xL (3.15) 2r * where Sbt and Sbb are the bending strains at the top and bottom of each segment, respectively; r = shaft radius; and L = segment length; * Using the latter estimate of 0, the segment rotations, AO at each point were obtained along the shaft (i.e., 02= 01 A01, 03= 02 A02, etc.); * The computed Oi (see step 7) at each point represented the slope of the deflected shape at their respective positions (dy/dzi = Oi). Thus, the lateral displacement at each point can be calculated using dy = Oi* dzi; * Repeat steps 5 to 9 until the line passes through or was very close to the point of zero deflection and P = 0; * After shaft rotations and displacements were assessed for the top half of the shaft (i.e., above y = 0), the bottom half of the shaft values were determined, with special focus on the shaft tip. Figure 3.31 shows a typical displacement versus depth for a shaft using this method. The next section presents the shear versus tip displacement for all tests conducted, as well as the proposed shear tip model. 3.5 Measured Versus Predicted Shaft Tip Response The focus of this research was to validate or develop reliable methods to predict the axial, shear, and moment response of a drilled shaft tip founded in Florida limestone under combined axial and lateral loads. A number of models which characterize tip response were investigated (e.g., O'Neill, Bell, Dohetry, Ueshita, Misra). Some had been developed specifically for rock (O'Neill), while others were applicable for general elastic materials (e.g., Bell, & Dohetry). However, none had tested Florida limestone with its associated variability (see Chapters 4 and 5). For the lateral tip shear response, the work focused on extending the existing FDOT side friction model (McVay 1992) to account for normal stress and shaft diameter. A discussion of axial loading is presented next, followed by shear and then moment computations. 3.5.1 Axial Test Results Figure 3.32 shows the measured axial tip response for all shafts founded in 10 tsf and 30 tsf strength rock with different L/D ratios (1 & 3) in homogeneous limestone compared to O'Neil's prediction and the linear tip stiffness model proposed by Misra (2006) and Doherty (2006). Misra suggested tip stiffness, K,, relates to Pt (tip force) and ut (tip displacement). Divided by area, A, the tip resistance can be calculated as shown in Eq. 3.16. Qb =P,A (3.16) where, P,= Ku, andK, =0.3.ir.D.Es/(1/ 2) Doherty also suggested tip stiffness expressed in Eq. 3.17. K, = K G R (3.17) where, KJ is found in a relationship shown in Appendix E. Analysis of Figures 3.32a through 3.32d revealed that the tip resistance using tip stiffness showed good agreement with the experimental response upto 1" tip displacement. Furthermore, the estimation of tip resistance by O'Neil showed good agreement with the experimental response with two different moduli secantt and tangent). With secant modulus, the O'Neil prediction proved more conservative. Figure 3.33 also shows the predicted axial tip response using O'Neill's (Eqs. 3.5 3.8) model incorporating either the harmonic mean modulus, Eh, or geometric mean modulus, Eg, and accounting for moduli from two layers. Analysis of Figure 3.33 revealed that harmonic modulus in combination with the O'Neill's tip resistance model showed good agreement with the experimental response compared with geometric modulus. Fenton and Griffiths (2005) already discussed the geometric modulus was good in spatial variability. However, with spatial variability in only the vertical direction and the same condition in centrifuge test, the harmonic modulus gave the better result. 3.5.2 Lateral Test Result Tip Shear Model Figures 3.34 and 3.35 show the measured mobilized shear stress versus tip lateral displacement for all centrifuge tests. Figure 3.34 presents the lower strength (10 tsf) response and Figure 3.35 presents the higher (30 tsf) strength rock results. Also included in each figure (Red Lines/Diamonds Dots) were the results for the layered system (i.e., 50% recovery rock located one diameter below the tip of the shafts). As expected, the higher strength rock developed the higher tip shear resistance per unit of lateral tip displacement. Also, the response for the 9 ft shaft was similar to the 6 ft shaft. However, what was not expected was the significantly diminished resistance of shafts with 50 % recovery located one diameter below the shaft tip. Evidently, the more compressible voided zone resulted in smaller moments developing at the shaft tip, which in turn required larger moments in the overlying rock to resist the applied lateral load. Figures 3.36 and 3.37 show a onetoone comparison of the shears and moments with and without the underlying softer (voided) layer. The influence on tip compressibility at 1 inch axial tip displacement is shown in Table 3.9 for layered and nonlayered systems. The table also shows the influence of rock strength. To develop a tip shear model, the influence of rock strength and normal stress must be taken into account. Figures 3.38 and 3.39 showed the normalized shear stress on yaxis with the cohesion and the mobilized tip resistance versus the normalized lateral tip displacement on xaxis with the diameter of shaft. Figure 3.38 referred to the 10 tsf rock and Figure 3.39 referred to the 30 tsf rock. Each plot may be characterized by a bilinear representation (i.e., a linear increasing line) followed by a horizontal or constant line. The linear portion represented the mobilization of tip shear versus normalized lateral tip displacement. The horizontal or maximum value (i.e., 0.55 for 10 tsf rock, and 1.0 for 30 tsf rock) represented the failure state as shown in Figure 3.40. Consequently, the failure states represented the shear stresses within the Mohr Coulomb limit state. That is to say, they are represented by angles of internal friction of 28 and 45 degrees, respectively. Compared with Figure 3.14 from the triaxial test, a 28 degree friction angle matches with the 10 tsf rock. The generalized shear model for shaft tip is shown in Figures 3.41a and 3.41b. It was characteristic of an elasticplastic behavior model. To implement the model in a finite element code such as FBMultiPier, one would do the following: * Depending on the strength of the rock, the lateral spring stiffness (Figure 3.41a) was assessed from the initial slope of the curve (Figure 3.41b). If the rock strength was not 10 tsf or 30 tsf, the slope was interpolated (e.g., 20 tsf is halfway between 10 tsf and 30 tsf); * Next, the finite element solution was obtained for the axial tip displacements; * Knowing the axial tip displacement, the axial tip stress was computed from O'Neill's solution; * Subsequently, Plot 3.41b was analyzed with the known lateral tip displacement. The mobilized tip shear stress was assessed from the curve, along with the normal stress and cohesion of the rock 3.5.3 Lateral Test Results MomentRotation Model Similar to the tip shear distribution, the moment transfer from the shafts to the underlying rock was plotted versus tip rotation as a function of rock strength, shown in Figures 3.42 and 3.43. Figure 3.42 present the 6 and 9 ft shaft diameter results in 10 tsf rock while Figure 3.43 shows the same shafts in 30 tsf rock. As expected, the moments for the 9 ft diameter shafts were higher than the 6 ft shafts due to the greater contact area for similar bending stresses. As discussed earlier, with the lower tip resistance, the layered rock system developed smaller bending stresses due to the lower compressibility modulus. The first reported work on embedded piles in an elastic half space subjected to both lateral tip displacement and rotation was reported by Bell (1991). He performed hundreds of finite element analyses and developed a stiffness matrix [K] relating vertical displacement (u,), horizontal displacement (uH ) and rotation (0M) to vertical load (V), horizontal load (H), and moment (M), as follows: G x R 2) K, 0 0 uR HGx 2) 0 K2 K4 R (3.18) (G0 K K M( R2) K4 K3 OM I(GxR where G = shear modulus under the tip of shaft and R = radius of shaft As is evident from Eq. 3.18, there is cross coupling between the horizontal shear, H, and moment. Bell developed the moment and horizontal stiffness terms from footings located at the ground surface with: H K4 GRO2 4 (3.19) K3 GR3 For the case of embedded shafts, the stiffness terms were adjusted due to resistance provided by the shaft's sidewall. The embedded stiffness terms superscriptt, z) were expressed in terms of shaft embedment length ZD as follows: K1 = KI,K = K2,K4 = K4 K D R Z) Z Z +(3.20) Kz =K2(Z Z) Z R R \R Rj Using the metacentre concept, Bell uncoupled the horizontal and moment components by replacing K4z = 0 in Eq. 3.20, resulting in an equation for Zm: Z Z K Zm D 4 (3.21) R R K2 For the new K3m, with K4Z = 0, inserting Eq. 3.21 into Eq. 3.20 and solving for K3Z yields: Z Z K Z Z K K2 K3m K2 D D K4)) 2K4( D D 4))+K = K3 ,K (3.22) R R K2 R R K2 K2 H where, K2 = (3.23) GRuH Using K3m, 0, was calculated as: M OM = (3.24) GR3K3m The 0M term was dependent on the horizontal force, horizontal displacement, and moment. Also, Eqs. 3.18 to 3.24 were dependent on the shear modulus of the rock. It was assumed that the stressstrain behavior of the rock is linear to failure and Poisson's ratio, t, of the rock is 0.3. Using the Hook's law, the shear modulus, G, was calculated as: G (3.25) 2(1 + p) To calculate the shear modulus, G, in Eq. 3.25, Young's modulus, E, must be known for the underlying rock. As discussed in Section 3.4, either the harmonic mean modulus, Eh (Eq. 3.9), or geometric mean modulus, Eg (Eq. 3.10), may be used. The harmonic mean modulus for both the single and double layered systems was used and the measured versus calculated rotation was assessed for the various rock strengths. Figure 3.44 shows the results for the 10 tsf rock and Figure 3.45 shows the results for the 30 tsf rock. It is evident from Figures 3.44 and 3.45 that Bell's representation is noticeably below the strength of the rock, i.e., within the MohrCoulomb strength envelope. The latter occurred at approximately 0.03 radians (Figure 3.44), which wasn't reached in the case of the 30 tsf rock tests. It was important to note that even though the tip rotation, 0, was shown as its stiffness coefficient was not constant, but rather was a function of horizontal loads displacements, and moments, as shown in Eq. 3.18. o A measured Calculated d (3.26) GR3K3m A less complicated approach was provided by Dohrety and Deeks (2006) for an elastic plate embedded in a Gibson soil and rock medium. This model represented an increasing modulus with depth. As with Bell (1991), the moment and rotation may be expressed in an equation similar to Eq. 3.26. However, the stiffness coefficient, K, was determined using a monograph (Figure 3.46) as a function of the plate rigidity parameter, J. The plate rigidity parameter included Poisson's ratio (v), shear modulus (GR), Poisson's ratio (Vs) of the soil, Young's modulus (Es), radius (R), and thickness (t) of the footing. The rigidity parameter, J, is shown in Eq. 3.27: zi(3 4v) E t (3.27) J=4 (3vs)(1 27)v) G 24(1 v,)(1 v) GR R As identified earlier, the shear modulus in Eq. 3.27 varied with depth (z) according to: G(z)= GR( (3.28) R In Eq. 3.28, GR is the shear modulus of the rock located at a depth R below the rock surface, and a is a userdefined parameter. For instance, a was assumed to be equal to 0.5 for the 10 tsf rock (i.e., sensitive to overburden depth), whereas the with the 30 tsf rock, a is presumed to be equal to zero, or a constant G. Using Eq. 3.26, with K determined from Figure 3.46 and the rigidity parameter, J, defined from Eq.3.27, the measured and calculated rotations for different rock strengths were found. These are shown in Figures 3.47 and 3.48. The Dohrety and Deeks (2006) method also provided acceptable predictions between measured and predicted rotations for the developed moments. However, the Dohrety and Deeks (2006) trend lines are located in the middle or slightly above the measured results, suggesting that they were slightly less conservative. Consequently, even though Bell's moment versus rotation model also included tip shear and lateral translation and is thus more complicated than Dohrety and Deeks, its procedure is recommended for Florida limestone. Figures 3.49a and 3.49b show the model. To use this model, the following procedure should be followed: * Find the moment and lateral displacement at the tip from FBMultiPier; * Find the tip shear force from the tip shear model in Figure 3.41 with lateral tip displacement; * Find the stiffness coefficient, K3m, including an assumed tip rotation described in Eq. 3.24; * Repeat step 3 until the calculated rotation, K3 matches the measured tip rotation. Table 31. Centrifuge Scaling Relationships (Chandrasekaran, 2001) Quantity Prototype Model Acceleration 1 N Area Volume Length Mass Force Energy Stress Strain Velocity 1 1 Table 32. Proportion of Component for Synthetic Limestone Required Strength Percentage, % (by Weight) Qu (tsf) Crushed Limestone Cement Water 10 75 5 20 30 70 10 20 Table 33. Ratio of Em (with perlite) / Ei (no perlite) Recovery Modulus Ratio (Em/Ei) 70% 0.29 50% 0.065 Table 34. WC 0.3 Amount of Component for Mixing Concrete with Silica Sand Water(g) Cement(g) SP(oz) SF(g) C/A 213.8 712.7 0.45 60.6 0.5 Table 35. Result of Unconfined Compression Strength Test Sample # 1 2 Aggregate(g) 1425.5 3 Average Length (in) 4.06 4.063 4.055 Average Diameter (in) 2 2 2 Weight (g) 471 472 469 Unit Weight (pcf) 140.7 140.9 140.2 Max Load (Ibf) 15003 10996 13691 Strength (psi) 4776 3500 4358 Amount of Component for Mixing Concrete with Granite Sand Water(g) Cement(g) SP(oz) SF(g) C/A 213.8 611 0.5 52 0.5 Aggregate(g) 1221.9 Table 37. Result of Unconfined Compression Strength Test Sample # 1 2 3 Average Length (in) 3.965 4 3.97 Average Diameter (in) 2 2 2 Weight (g) 456 462 453 Unit Weight (pcf) 139.4 140 138.4 Max Load (Ibf) 15229 15866 16248 Strength (psi) 4847.5 5050.3 5171.9 Table 38. Estimation of Em/Ei Based on RQD (Load Intermediate Geomaterials, 1996) Transfer for Drilled Shafts in RQD (%) Em/Ei Closed Joint Em/Ei Open Joint 100% 1 0.6 70% 0.7 0.1 50% 0.15 0.1 20% 0.05 0.05 Table 39. Comparison of Axial Tip Stresses Description Translated Axial Stress(ksf) 10tsf Rock with intact rock Over with 50% recovery Layd 30tsf Rock with intact rock Over with 50% recovery NonLayered 10tsf Rock 1520 NonLayered 30tsf Rock 6080 Table 36. WC 0.35 Figure 31. New Cylindrical Sample Container Figure 32. Instrumentation for Measuring Lateral & Vertical Forces and Deformation Figure 33. University of Florida 12.5 G ton Centrifuge Figure 34. Slotted Steel Pipe (Axial Reinforcement) & Strain Gages Figure 35. Mixture of Synthetic Rock for Centrifuge Test Figure 36. Drilled Hole in Synthetic Limestone Figure 37. CastIn Model Shaft in Synthetic Limestone Figure 37. Castin Model Shaft in Synthetic Limestone U 11 Figure 38. Slip Rings and Rotary Union 0.8 ft / (0.244 m) Load cell plate / Steel plate Schematic sketch of a centrifuge test setup v E rin 00 0 7 LI.,I C < LVDT P LVDT  Strain gage f t Specimen 0.134 ft (0.04 m) Load frame Synthetic Limestone Figure 39. Figure 310. Grain Size Distribution for Limestone Aggregate Figure 311. Strength Tests for Concrete and Rock Sieve Analysis for LimeRock 100  S*Sieve Analysis #1 90 Sieve Analysis #2 Sieve Analysis #3 80    Sieve Analysis #4 Sleve Analysis $5 70 .060  S40 30 20 2E 0    ______ ______  10 0 10 1 0.1 0.01 Grain Size(mm) Figure 312. Synthetic Limestone with 5 mm Perlite Spheres Occupying 30 % Volume 250  Intact 10 tsf# 1 200 Irtact 10tsf# 2 30% LargeVoid 10 tsf # 1 S 30% LargeVoid 10 tsf # 2 S30% SrrmallVoid 10tsf#1 150 ""*30% SrrallVoid 10tsf #2 S100 50 0 0.01 0.02 0.03 0.04 0.05 0.06 Strain Figure 313. StressStrain Response Voided and nonVoided Limestone 20 10 0 10 20 30 40 50 Normal Stress (ksf) 60 70 80 90 100 Figure 314. MohrCoulomb Envelop from Qu,Qt and Triaxial Tests on 10tsf Rock FFrk SLacfe Figure 315. Strain Gage Layout for L/D = 1 Figure 316. Compressive Forces (Q) along Depth Skin Friction(10tsf) 2 3 Axial Displacement, z (in) Figure 317. Mobilized Skin Frictions vs. Axial Displacement Load Transfer along depth with different top displacement 0 2000 4000 6000 8000 10000 12000 D 5 eU p 10 t h 15 f t 20 25 Load(kips) Skin Friction(10tsf) S9/910tsf50%#1  A McVay  9/910tsf#1 6/61 tsf#1 9/271 Otsf#1 6/181 Ots f#1   9/910tsf50%#2 Lila 9/910tsf#2 6/61 Otsf#2 9/271 Otsf#2 6/181 Ots f#2 0 1 2 3 Axial Displacement, z (in) Figure 318. Measured Skin Friction Comparing Lila (2003) and McVay (1992) Skin Friction(30tsf) 160 140  120 S100 80  .2 60 9/930tsf50%1 9/930tsf50%#2 A McVay Lila 40 9/930tsf#1  9/930tsf#2 i 20 6/630tsf#1 6/630tsf#2 9/2730tsf#1 9/2730tsf#2 0 ,: 6/1830tsf#1 6/1830tsf#2 0 1 2 3 4 5 Axial Displacement,z (in) Figure 319. Measured Skin Friction Comparing Lila (2003) and McVay (1992) 100  90  80 70 = 60  0o f 50  O'NeilClosed o 40  O'NeilOpen oe 30 _ 30 UFLimestone 20 )< Trend 10  10 0 0.2 0.4 .. 0.6 0.8 1 Em/El Figure 320. O'Neil Em/Ei vs. RQD and UF Em/Ei vs. Recovery Figure 321. Two Layer System and Equivalent Modulus (Ueshita and Meyerhof, 1967) 2 fl.'Sti CI iriuaim 211w FWFI~cUy WoNdlV LtUW E. iEo m 9ta1r2 Figure 322. Layered Moduli beneath a Drilled Shaft 0.6 0.4 0 0.2 0. ) 200 00 600 800 1000 1200 .2 ^  0.4  0.6 0.8 _ Time (sec) Figure 323. Output Voltage from Gages vs. Time with Loading; Axial Loading First and Combined Loading Following SII < ~ ~ ~~* U __ m p L Centroidal Axis " StNeut Ax * Neutral Axis Figure 324. Total Axial and Bending Strains along the Shaft CrossSection Moment Curvature 80000 70000 60000 cl 50000 40000 I 30000 E 20000 10000 0 0 0.0002 0.0004 0.0006 Curvature 0.0008 0.001 Figure 325. MomentCurvature Relationship for 9ft Diameter from Gages above Rock Surface. a Pair of Strain Figure 326a. Moment Distribution along a 9 ft Diameter, 50% Styrofoam at 1D below Tip of Shaft in 10 tsf Limestone from 2 tests 15 10 5 5 20000 40000 80000 10000 10 15 Moment (kips ft) Figure 326b. Moment Distribution along a 9 ft Diameter, 30% Styrofoam at 1D below Tip of Shaft in 45 tsf Limestone from 2 tests 15 10 5 .4 i" 0 ) 10000 20 40(00 a 5 10 15 Moment (kips ft) ( 10000 20000 30000 40000 00 70( Moment (kips ft) Figure 326c. Moment Distribution along 9 ft Shaft without Styrofoam below Tip in 35 tsf Rock from 2 tests 6( a 6( Shear (kips) Figure 327. Increase in Tip Shear with Increasing Lateral Load, L/D = 1 Rock Surface 00 4000 20 ) 2000 4000 60 ip Shea Rock Surface 5 0 90 6 3000 6000 900 Shear (kips) Figure 328. Increasing Tip Shear with Increasing Rock Strength  9 f t/9 ft 10 tsf 1 D 50%  9 ft / 9 ft 10 tsf S10)00 5000 5000 S Shear (kips) Figure 329. Shaft Shear at Failure with and without an Underlying Soft Layer 1Mom4 t 200O 3t) 4 MomnuuAOdiqs.ti MomielUImak noo oor ^KO 1H 0DD  0 S10D?2 1mi4 0. OO6 UJ80D II Cwiaire Figure 330. Assessing Rotation from the MomentCurvature Relationship (a) Moment versus Depth (b) Moment Curvature Relationship Deflection (in) 0 Figure 331. Typical Lateral Deflections (y) vs. Depth __ NO J 60 AxialTip Displacement (in) \iilTi Dispbecincin i Figure 332a. Tip Resistance (Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O'Neil(right) for 9 ft / 9 ft 10 tsf Rock 50 50 AxialTip Displacement in) 5 6 1 AxialTip Displacement (in) Figure 332b. Tip Resistance (Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O'Neil(right) for 9 ft / 27 ft 10 tsf Rock 40 40 20 21" 101 II 0 rII [0 ./ 0 0 .5 1 1.5 2 2.5 3 3.5 I > 11. 1 2 2 5 3 3 Axial Tip Displacement (in) Axial Tip Displacement (in) Figure 332c.Tip Resistance (Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O'Neil(right) for 9 ft / 9 ft 30 tsf Rock 60 60 50 50 20 20 0 A Tip 0. P 0 1i2 3 4 5 6 0 1 2 3 4 56 Axial Tip Displacement (in) Axial Tip Diplacement (in) Figure 332b. Tip Resistance (Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O'Neil(right) for 9 ft / 27 ft 10 tsf Rock 140 140 120 120 e 100 100 o oC60 i60 40 T 40 20 I 20 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Axial Tip Displacement (in) Axial Tip Displacement (in) Figure 332c.Tip Resistance (Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O'Neil(right) for 9 ft / 9 ft 30 tsf Rock 84 250 25'' 250 ~oo 200 2 ', r trit,. T t I 50 5, p lll I I u UI'TI L y 1c y 0 1 3 4 54 Axial p Displacement (in) ial Diplacncicn in i Figure 332d. Tip Resistance (Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O'Neil(right) for 9 ft / 27 ft 30 tsf Rock Figure 333a. Tip Resistance (Model vs.O'Neill's Equation with Harmonic Average Method (left) and Geometric Average Method (right) for 10 tsf Rock with 50% Recovery at 1D Figure 333b. Tip Resistance (Model vs.O'Neill's Equation with Harmonic Average Method (left) and Geometric Average Method (right) for 30 tsf Rock with 50% Recovery at 1D 4.5 4 3.5 3 2.5 2 1.5 0.5 0 0 0.5 1 1.5 2 Axial lip Displacement (in) 6 5 C 2 1  0 0.5 1 1.5 2 Axial Tip Displacement (in) 20 18 16 14 C' 12 , 10 6 4 2 0 0 0.5 1 1.5 2 Axial Tip Displacement (in) 40 35 30 25 20 5 5  ,_, .. .. "' ,, . 0 0 0.5 1 1.5 2 Axial Tip Displacement (in) Rock Strength 10tsf . a 9ft 1LD 10tsf1D50%Aver 9 ft 1LD 10 tsfAver 6ft1LD10tsfAver , / 9 ft 3LD10tsfAver 6 ft 3LD 10 tsf Aer 1 1.5 2 Tip Lateral Displacement (in) Figure 334. Tip Shear Stress vs. Tip Lateral Displacement Rock Strength 30tsf S$9 ft 1LD 30 tsf 1D 50% Aver 9 ft 1LD 30 tsf Aer S 6ft1LD30tsfAver 9 ft 3LD 30 tsf Aer 6 ft 3LD 30 tsf Aver 0 0.2 0.4 0.6 0.8 1 1.2 Tip Lateral Displacement (in) Figure 335. Tip Shear Stress vs. Tip Lateral Displacement Shear (kips) Shear (kips) Figure 336. Shear in Shafts vs. Depth with and without Underlying Softer Layer 9 ft 9ft 30 tsf 9ft/19ft30tsf1D50% 20000 40000 O 80 Moment (kips ft) Moment (kips ft) Figure 337. Moment in Shafts vs. Depth with and without Underlying Softer Layer 9 ft/9 ft10 tsf 1D 50% 9 ft/ 9 ft10tsf 00 4000 2000 40C 49ft/9ft30tsf 19ft/9ft30tsf1D50% 100 .100) 5000 10 4A 09ft/9ft10tsf1D50% 9ft/9ft10tsf n10000 no 4000 Rock Strength 10 tsf 0 0.005 0.01 0.015 0.02 0.025 0.03 Tip Lateral Displacement/Diameter 0.035 Figure 338. Tip Shear Minus Rock Strength Divided by Normal Tip Stress vs. Normalized Lateral Displacement Rock Strength 30 tsf E0.8 .0.6 I 0.4 0.2 Figure 339. go 9 ft 1LD 30 tsf 1D 50% Aver 9 ft 1LD 30 tsf Aver 6ft1LD30tsfAver 9 ft 3LD 30 tsf Aver 6 ft 3LD 30 tsf Aver 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Tip Lateral Displacement/Diameter Tip Shear minus Rock Strength Divided by Normal Tip Stress vs. Normalized Lateral Displacement 0.7 0.6 0.5 0.4 z .0.3 1 0.2 0.1 4 (shear stress Cohesion. C) / or Shear StressCohesion, C Normal Stress Figure 340. MohrCoulomb Strength of Florida limestone Tip Shear Model ,,10 tsf ,30 tsf 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Lateral Displacement/Diameter Figure 341. Tip Shear Model (a) Tip Shear Model Scheme (b) Tip Shear Model Shaft Tip of Shaft Ti Shear Model Tip Shear Mlodel Rock Strength 10 tsf 30000 9/9 10tsf 1D 50% 9/9 10tsf 6/6 10tsf 25000 . 20000 C 15000 S10000 S5000 0 0.01 0.02 0.03 0.04 0.05 0.06 Tip Rotation (rad) Figure 342. Tip Moment Transfer as Function of Tip Rotation Rock Strength 30 tsf 25000 9/9 30 tsf 1D 50% 9/9 30 tsf A6/6 30 tsf  20000 E. S15000 E 10000 0 a. = 5000 0 0.005 0.01 0.015 0.02 0.025 0.03 Tip Rotation (rad) Figure 343. Tip Moment Transfer as Function of Tip Rotation a 0.02 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0.04 0.06 0.08 Measured Rotation (rad) Figure 344. Calculated vs. Measured Rotation using Bell (1991) Trend line is in red 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 9/9 30 tsf 1D 50% #1 9/9 30 tsf 1D 50% #2 9/9 30 tsf #1 9/9 30 tsf #2  trend , 0.005 0.01 0.015 0.02 Measured Rotation (rad) Figure 345. Calculated vs. Measured Rotation using Bell (1991) with Trend line in red 9/9 10 tsf 1D 50% #1 9/9 10 tsf 1D 50% #2  9/9 10 tsf #1 9/9 10 tsf #2  Trend "(i II 1 m 0 Y o(* II I LE42 I.O401 1.OE+00 1.B+0O1 LU.H2 1O12O03 J LOE02 L.EO I I.0+00 o IOE+ I_ LOE+02 IOE+03 .1 Fig. 14. KM for a=0: (a) v=0.2 and (b) v=0.499 35.0 300 25.0 20.0 150 I0.0 1.0402 1.Oe+0f 1.OE+02 Fig. 15. KMfora=1: (a) v=0.2 and (b) v=0.499 Figure 346. Stiffness Coefficient, K, from Doherty and Deeks (2006) 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0.01 0.02 Measured Rotation (rad) 0.03 0.04 Figure 347. Calculated vs. Measured Rotation in 10 tsf Rock using Doherty and Deeks (2006) with Trend Line in Red _  dAoI.J 6 dIRt  ad/R=4  1.IE+0t 9/9 10 tsf aver 6/6 10 tsf aver +9/9 10 tsf 1D 50%  9/9 10 tsf 1D 30% K 9/9 10 tsf 3D 50% 9/9 10 tsf 3D 30%  9/27 10 tsf  6/18 10 tsf *Trend ~13$~ ,, 1 m 0.03 0.025 0.02 S 0.015 0.01 0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 Measured Rotation (rad) Figure 348. Calculated vs. Measured Rotation in 30 tsf Rock using Doherty and Deeks (2006) with Trend Line in red 0I oi 2 U3 Iheas d FtAtion ~ (a) (b) Figure 349. Recommended Tip Rotations vs. Moment Model from Bell (1991) (a) Tip Rotation Model Scheme (b) Tip Rotation Model 9/9 30 tsf aver 16/6 30 tsf aver 9/9 30 tsf 1D 50%/ 9/9 30 tsf 1D 30%/ K 9/9 30 tsf 3D 50%/  9/9 30 tsf 3D 30%/  6/18 30 tsf  9/27 30 tsf Trend Tip ftationRodel TVIo lel   CHAPTER 4 A TWODIMENSIONAL NUMERICAL ANALYSIS OF TIP RESISTANCE OF A DEEP FOUNDATION IN HETEROGENEOUS ROCK In the case of deep foundations in rock, the current practice is to idealize the heterogeneous rock as an elastic homogeneous continuum. Arithmetic, harmonic or geometricaveraged value of individual elastic properties of the rock layers such as modulus of elasticity (E) is used in the computation of the bearing resistance. The appropriate averaging equations, as well as the averaging depth are unknown. In addition, the expected variance of the settlement about the mean is also unknown. The purpose of this chapter is to numerically validate the averaging as well as assess the variance of settlement using the same geostatistical principles presented earlier for bearing resistance of deep foundations subjected to axial loading. To conduct the study, a twodimensional finite element model has been developed which is capable of analyzing a load transfer mechanism under spatially heterogeneous rock conditions and predicting the loaddisplacement behavior accordingly. Although comprehensive calibration and validation against physical field test data was beyond the scope of this study, model validation of key material properties was carried out by laboratory and centrifuge testing (as delineated in Chapter 3), to confirm numerical components and aspects of the finite element model. Spatial heterogeneity was incorporated into the FEA model by assigning random values of E to each individual rock layer of 1 ft thickness (i.e., no horizontal variability considered here). Subsequently, FEA was repeated for a large number of rock property scenarios. Variability of E was hereby characterized by a coefficient of variation (CVE) and a vertical correlation length (av), which measured the degree and spatial scale of variability, respectively. Finally, results of the random FEA were used to develop a geostatistical spring model that was capable of predicting realistic tip loaddisplacement behavior and quantifying the uncertainty of rock conditions by the geostatistical properties of E. 4.1. Finite Element Analysis The numerical study described herein dealt with the development of a finite element soilstructure interaction model and subsequent use of that model in a parametric sensitivity study to evaluate the influence of heterogeneous limestone rock on the axial capacity of a rocksocketed drilled shaft. Primary focus was given to the task of determining whether the presence of the heterogeneity was shown to affect (based on simulation results) the axial bearing capacity of the deepfoundation and, if so, the extent to which point bearing response was altered by the spatial variation of material strength characteristics such as E. For modeling nonlinear soil response and soilstructure interaction, the nonlinear implicit/explicit finite element simulation code ADINA (Version 8.5) was employed. 4.1.1 Dimensions and Boundary Conditions Based on the centrifuge test setup as shown in Figure 3.9, a 9 ft diameter field model was selected for finite element model development. Using symmetry for the geometry and the loading conditions, a twodimensional (2D) axisymmetric finite element model was constructed (Figure 4.1). The dimensions of the FEA model were scaled from those of the centrifuge test model, which were 3 times shaft diameter (D) wide and of shaft length (L) plus three times diameter below the shaft tip. Element size of 0.5 ft by 0.5 ft was used in the discretization of a synthetic homogenous limestone used in the centrifuge tests, which was assumed to be a Representative Element Area (REA) on the axisymmetry plane at macroscopic scale. The degree of constraint provided by the boundaries of the system was modeled such that translational motion was allowed to expand along the boundaries of the system (Figure 4.1). The system was assumed to be in static equilibrium prior to any applied external loads. 4.1.2 Constitutive Models for Materials 4.1.2.1 Reinforced concrete Based upon the modified Hognestad's stressstrain relationship for concrete (MacGregor, 2003), comp =peak[20comp/ (comp/ )2 the material behavior of the concrete shaft under axial loading was approximated with an isotropic elastic model. The key assumptions made in the model are listed below. The compressive strength of reinforced concrete was obtained as 720,000 ksf using Eq. 3.2. E considered was for static loading rather than dynamic loading rates. 4.1.2.2 Limestone rock The ADINA soil model used in the simulation was a MohrCoulomb model that can predict pressuredependent failure for the rock material under consideration. Based upon the assumption of normality of plastic flow to the failure surface, the plastic strain rate vector had a component in the volumetric (hydrostatic) direction that resulted in increase of mean normal stresses. However, the linear postulate of the failure envelope can produce much greater shear failure stress (as minor normal stress (03) increases) than what was observed in the laboratory tests, as shown in Figure 3.14. Thus, the effect of increasing normal stress had on the shear strength of the rock material needed to be properly accounted for over a realistic range of maximum normal stresses. A tangent slope of the failure envelope (Figure 3.14) was estimated as 28 degrees at a maximum normal stress of 38 ksf. This slope was subsequently used as the angle of friction (0) in the simulation. The value of cohesion was also determined by a curvefit of the data obtained from the direct shear (Qt), unconfined compression (Qu), and three triaxial compression tests (Figure 3.14). It must be noted that any prediction of nonlinear MohrCoulomb failure envelope at normal stresses higher than those identified in Figure 3.14 was conjecture due to the confinement restrictions of the laboratory tests. Pressuredependent failure mechanisms associated with the nonlinear failure envelope should be considered in future research to quantitatively determine accurate stress fields. Nonetheless, the constitutive material model presented here offered an improved tool calibrated with experimental data for studying twodimensional stress state development near the tip of the rock finite elements that underwent a maximum tip displacement of 1 inch, i.e. the serviceability limit condition. 4.1.3 RockShaft Interface The deep foundation system consisted of a shaft structure formed by excavation of a cylindrical borehole into limestone rock where reinforcing steel and concrete was cast. When a shaft is subjected to axial loading, the shaft transfers the load by a combination of shear stresses developed along the cylindrical interface between the concrete and rock and normal stress emanating from the tip of the shaft. This inelastic load deformation mode was often seen as rock material failure along the interface based on the visual appearance that the shaft takes on after load transfer has occurred. When a shaft is designed to have sufficient axial capacity, the shaft remains structurally intact and the tip resistance contributes significant axial stiffness to the overall behavior of the rockshaft system, even if failure along the interface occurs. Considering the physical interface of the system constituents, the initial approach taken in modeling the interface involved the use of discrete contact edges of axisymmetric elements in which the interface represented the physical boundaries between the shaft and rock. An approximate means of accounting for the shear failure was attempted through the use of a contact model that simulates frictional resistance under Coulomb's Law of Friction. During the contact simulations performed using this modeling technique, this approach was found to be problematic. Nonphysical penetration at the corner node of the shaft into a rock element caused an artificial "gap" (i.e., the rock mesh was detached from the side of the shaft; see Figure 4.2), and produced zero contact force. This nonphysical penetration was a numerical instability that typically is associated with either the use of a coarse finite element mesh or too large incremental loading steps, or both (Bathe, 2000). Under certain conditions, elements formulated using numeric integration can undergo deformation modes in which strains sampled at the element integration points fail to capture all of the strain energy associated with the deformation. As a consequence, energy was numerically (rather than physically) dissipated, and the nonlinear solution process potentially becomes unstable (Bathe, 2000) leading to unreliable analysis results or termination of the simulation (i.e., nonconvergence). Problems associated with nonphysical nodal penetration can often be remedied by using a highresolution finite element mesh and small incremental loading steps. Reduction of numerical instabilities to an acceptable level (quantified by maintaining the pseudoenergy at less than 1% of total system energy) and prevention of nodal penetration were attempted by significantly increasing the resolution of the finite element mesh and using a small loading step, e.g. an increment load of 1 kip. Even with approximately 40,000 elements of the rock mesh, nodal penetration was not completely prevented. While stable solutions might be obtained, the very highresolution mesh with such a small loading step was deemed to be impractical from the standpoint of conducting a parametric study involving numerous, repeated simulations required for geostatistical realization of the field condition. In preliminary analysis, several hundred simulations were necessary for one geostatistical realization of a heterogeneous rock condition. An alternative, morenumericallyefficient solution was achieved by abandoning the approach of modeling the contact. In the physical shaftrock system, the circumference of the shaft was bonded to the surrounding rock. Approximate numerical modeling of this bond was accomplished using interface boundary elements (Figure 4.3a). This numerical approach simulated the mathematical link between two edges together at a common interface. Parts linked together in this manner may still deform and respond to load, as may the interface between them, but the edges of the two parts remained linked to each other on a pointbypoint (or nodebynode) basis (Figure 4.3b). That is, at no point on the interface boundary may the two tied parts separate from each other, even though the interface boundary element itself may deform. In contrast to the contact approach of the previous modeling technique, this approach led to a more controlled stress redistribution of internal stresses as the shaft rock interface yields, thus permitting stable solutions to be obtained at coarser mesh resolution. Determination of the yielding (i.e., shear failure at the interface) was carried out using a MohrCoulomb model with the elasticperfectly plastic yield condition such that the angle of friction (0) was very small, e.g., 0 =0. Thus, exceeding the failure shear stress at integration points in an interface element was avoided by limiting the maximum shear stress to the cohesion value. This yielding mechanism permitted experimentally observed amounts of volumetric deformation to be modeled and thus produced a better representation of both the shear failure along the interface and the load transfer to the tip of the shaft. For comparison to the centrifuge test results, the side resistance and end bearing predicted by FEA was shown in Figures 4.4a and 4.4b, respectively. While the load deflection curves of the side resistance were not identical, their general form was in good agreement. As is evident from Figure 4.4b, the load transfer by side shear was initially linear and becomes nonlinear as shear stress reaches the limiting shear stress (the strength of the synthetic rock). The tip stiffness, however, exhibited a linear relationship as the tip displacement reached a service limit condition of oneinch settlement defined in Commentary C10.6.2.6.1 of Section 10 of Foundations of AASHTO Bridge Design Specifications. Since the FEA material model has been developed using aforementioned Qt, Qu, and triaxial test data separate from centrifuge test results, the agreement shown in Figure 4.4 suggested a degree of validity in the load transfer both predicted by FEA and measured in the centrifuge tests. Due to the robustness and increased numerical efficiency of this modeling technique, it was used throughout the remainder of the study. 4.2 Simulating the Influence of Rock Variability in Florida Limestone on Tip Stiffness Having validated the FEA model for the homogeneous rock condition of the centrifuge tests, the influence of the vertical variability of E on the tip resistance of the 100 shaft for given settlements was investigated. Variability in E was represented through a lognormal distribution with mean (PE), coefficient of variation (CVE) and vertical correlation length (av). Generally, most soil/rock properties were assumed to be log normally distributed because of mathematical simplicity, their observance of positive skewness, and the requirement of nonnegativity. For instance, McVay (2008) reported CVE = 0.5 and av = 8 ft as a typical heterogeneous rock condition from boring sites of 17th Causeway Bridge over the Intercoastal Waterway located in Fort Lauderdale, Florida; Martino(2001) reported ranges of CVE ranging from 0.1 to 0.3. Based on the reports, three correlation lengths of 5 ft, 10 ft, and 15ft and three CVE's of 0.2, 0.4, and 0.6 were selected for the subsequent analyses of the cases for shafts with L/D of 1 and 3. The generation of vertically correlated E distributions for the FEA model (one E value per 1 ft rock layer thickness) may be undertaken by a number of different approaches. Most popular were Gaussian based methods such as Sequential Gaussian Simulation (SGS), LUdecomposition, etc. (Deutsh, 1998). These methods first generate Gaussian (standard normal) random fields with a prescribed spatial correlation structure, which were then transformed to a target distribution. In the present work, the LU method was applied to a spherical covariance model (Deutsh, 1998) for the underlying Gaussian field. In general, conversion from a standard normal variable N to a lognormal variable LN is achieved by LN = exp(PN + ONN), where PN and ON are the mean and standard deviation of the logtransformed variable. That is, In(LN). PN and ON may be obtained as functions of target values for PLN and CVLN being the mean and coefficient of variation of the lognormal variable. 101 Pu l In (4.1) 1T+, 7 V 2= ln(1+ CVN) (4.2) For later use, Eqs. 4.1 and 4.2 were inverted to express PLN and CVLN as functions of PN and ON as follows: /N = exp pY + (4.3) CVL = exp 1) (4.4) Figure 2.2 shows correlated variations of E with respect to various correlation lengths, av = 5 ft and 15 ft. For each combination of CVE, a,, and L/D, 1,000 ADINA simulations were performed for various realizations of local E values to compute a respective series of tip resistances versus tip displacements. Approximately 1,000 simulations per a parameter combination were found to be necessary in order to achieve a constant coefficient of variation of the tip resistance. Figure 4.5 presents the predicted tip resistances versus tip displacements for the homogeneous case (where PE is used as a constant E) and the respective results for the heterogeneous cases as a function of CVE, and av. Note that for the heterogeneous cases, there would be 1,000 simulation data lines (not shown in the current figure for visual clarity) in each graph. Only the respective mean lines are depicted. As is observable from Figure 4.5, the coefficient of variation of the modulus (CVE) had a significant effect on the mean tip resistance. Next, focus was shifted to derivation of the tip stiffness (resistance/displacement) in heterogeneous rock conditions as a function of the spatial properties of E. Fenton and 102 Griffiths (2005) showed that the settlement (5) of the shallow foundation in heterogeneous soil deposits can be predicted by a linear relationship of = det P (4.5) eff where 5det = deterministic settlement computed using a homogeneous modulus, PE and Eeff = effective modulus equal to the geometric average of moduli. ( HWfWf E, = exp WHJ J Jin E(x,y,z)dxdydzd (4.6) where Wf = width of a square footing and H = thickness of soil below the footing. In the case of horizontallylayered soil (variability only in vertical zdirection), Fenton and Griffiths (2005) suggested that the harmonic average (Eh) of modulus may be more appropriate for Eeff in Eq. 4.5: Eh FH dz 1 (4.7) H 0o E(z)) For the present situation of horizontally layered rock conditions, the validity of Eq. 4.5 (in combination with Eq. 4.7) had to be investigated for predicting tip resistance vs. tip displacements of the deep foundation. For this purpose, Eq. 4.7 was first used to express the expectation Ph and the coefficient of variation CVh of Eh in terms of the known parameters PE, CVE, and av. These known parameters were subsequently used to study the existence and optimal value of the vertical averaging length (H) below the shaft tip. Note that while Eh in Eq. 4.7 was a mean in the sense of a spatial average, Ph was a mean of either an ensemble average over many realizations or a statistical expectation. 103 Assuming that E in Eq. 4.7 was lognormal with expectation PE and coefficient of variation CVE, and since A = 1/E was also lognormal, the summary statistics of A are: (1) mean expectation (PA) equals to (1+ CV1)/ E and (2) coefficient of variation (CVA) equals to CVE. This can be proven by Eqs. 4.14.4 using the normal variable In(A) equals normal variable In(E). Introducing B= fAdzas the arithmetic average of A H over H, B can be approximated by setting pB equal to pA (both are lognormal distributions of expectation) and coefficient of variation CVB equal to a2CVA (Klammer, 2010), where a was a variance reduction factor for a vertical covariance function of A of the spherical type as: H H3 H a=1+ for 0<<1 2a, 20a3 a, Sa(4.8) 3a a2 H a = for 1< 4H 5H2 aV Similarly, as A was inferred from E, expectation and the coefficient of variation of the approximately lognormal Eh = 1/B can now be expressed as Ph = (1+CVg2)/ /B and CVh = CVB, respectively. Substitution of these relations into Eqs. 4.1 and 4.2 finally gave: 1+a CV2 /Ph =PE V (4.9) CVh = 4a CVE (4.10) Eqs. 4.9 and 4.10 demonstrated how both expectation and coefficient of variation of the harmonically averaged modulus increased proportionally to a as the relative averaging length (H/av) decreased. 104 Presented in the left columns of Figures 4.6 and 4.7 as functions of H/D were Ph from Eq. 4.9 and Phs obtained from applying Eq. 4.7 to each realization of the generated modulus values (used in FEA) and subsequently arithmetically averaging it over 1,000 simulations. The similarity between the analytical Ph and the stochastically simulated Phs showed the validity of the approximations of Eq. 4.9 and the LUdecomposition approach used for random field generation. Similarly, it is also shown in the plots the arithmetic mean pas should remain constant at the expectation of the lognormal modulus distribution [tE =1728 ksf. Fluctuations of this expected value can occur due to the finite number of realizations used. Thus it was recommended that the degree of fluctuations be observed as a measure to evaluate a minimum required number of realizations per graph. Since values of 5det and pE in Eq. 4.5 were known, each value of heterogeneous settlement 5 from FEA of individual E realizations can be converted into a respective value of Eeff. These values were used to construct the distribution of an FEA based effective modulus EFEA with expectation PFEA and coefficient of variation CVFEA. PFEA was independent of H/D and appeared as a horizontal line in the left columns of Figures 4.6 and 4.7. The graphs in the right columns of Figures 4.6 and 4.7 represented the relationships of respective coefficients of variation. Interestingly, all cases and values of H/D between 1.5 and 2, ph and CVh from Eqs.4.9 and 4.10 were able to closely predict values of PFEA and CVFEA in the FEA simulation, respectively. This justified the use of Eeff = Eh in Eq. 4.5 for prediction of a random (uncertain) settlement (5h) based on the harmonic averaging method. Since 6det and pE are deterministic constants and Eh is approximately lognormal (under the assumption of lognormal E), 6h approximately 105 maintained lognormality of expectation Ph = detlE (1+ Ch) / Ph and CVh = CVh. Using Eqs. 4.9 and 4.10, this lognormality of expectation was written as: sh = det (1+ CV) (4.11) CV = x CV, (4.12) Equation 4.11 stated that the geostatisticallypredicted mean settlement (/,h) at the tip of a drilled shaft in vertically heterogeneous rock was a function of the homogeneous tip settlement (5det) and the coefficient of variation of the rock mass modulus (CVE), but not a function of a. In contrast, the coefficient of variation of heterogeneous settlement (CV6h) depended on the degree of spatial averaging (a) and thus, the choice of the averaging length (H). Both /6h and CV6h were seen to increase with CVE. Equation 4.11 was validated in Figure 4.8, which shows the FEA results of the homogeneous tip settlement (5det) and the mean heterogeneous tip settlement (P6FEA) for CVE of 0.6 and the two spatial correlation lengths of 10 ft and 15 ft. Evident from Figure 4.5 are various values of L/D and CVE and from Figure 4.8 are various values of av. Analytical P/ and FEAsimulated /,FEA were in good agreement. Table 4.1 shows a collection of comparisons in respective coefficients of variation, CV6FEA and CV6h, for various dimensions (L/D), rock variabilities (CVE), correlation lengths (av), at H/D = 2. An average error was 2.5% in all the cases. Therefore, for current pile and shaft design practice where a spring model was used to represent the tip resistance (e.g., FBMultiPier, APile), the spring stiffness of the heterogeneous rock was estimated using Eq. 4.11: 106 K Homogeneous Kheterogeneous ( CV ) (4.13) where CVE = coefficient of variation of E of the rock mass. As expected, CVE = 0 resulted in the heterogeneous tip stiffness equal to the homogeneous value; however, in the case of high CVE (e.g., 0.61.0) the heterogeneous tip stiffness may be reduced by up to half of the homogeneous tip stiffness. It was important to realize that Eq. 4.13 led to a prediction of a mean (expected) value for settlement, which is associated with an uncertainty described by the coefficient of variation given by Eq. 4.12. A mean value of tip resistance may be obtained according to Eq. 4.13 by dividing the homogenous tip resistance by the 1+CV2. Assuming that the mean of skin friction along the side of the shaft was known, it can be summed with the expected tip resistance to obtain the expected total shaft resistance. Knowing the tip resistance was proportional to tip displacements, Eq. 4.12 gave the CV of tip resistance, which can then be converted to a tip resistance variance. Consequently, assuming independence between the side and tip resistances, the variance of the tip resistance may be summed with the variance of the side resistance to give the total variance of the pile/shaft resistance from which the resistance factor (4) of LRFD can be computed. The higher the variance (and the lower the 4 factor), the longer the shafts required for design. 107 Table 41. Comparison of FEA results (CV6FEA) and analytical solution (CVsh from Eq.14) at H/D=2 CVE = 0.2 CVE = 0.4 CVE = 0.6 CV6FEA CVsh CV6FEA CVsh CV6FEA CV5h av =5 0.10 0.09 0.20 0.18 0.30 0.26 L/D = 1 a = 10 0.13 0.12 0.25 0.24 0.37 0.35 a = 15 0.14 0.14 0.27 0.27 0.40 0.41 av =5 0.10 0.09 0.19 0.18 0.27 0.26 L/D = S a = 10 0.12 0.12 0.24 0.24 0.35 0.35 a = 15 0.15 0.14 0.29 0.27 0.44 0.41 108 Axis cf Symmetry 1l ,,'111 i 1/ a 00 a a a 0 6 r D b ad I/ I!'k 1 /] 4.5 ft (1.3716 rn Figure 41. Dimensions and 27ft (8.2296 m) boundary conditions of FEA Model Figure 42. Nodal penetration phenomenon 109 Interface Boundary Element (IBE) Applied nodal loads along the side of the shaft 0 0 0 0 Shaft Finite Element Rock Finite Element Shear deformation of IBE and Rock FE (a) Interface boundary element (IBE) (b) Load transfer of IBE Figure 43. Model of the rockshaft side interface boundary 110 2500 Figure 44. Centrifuge test and FEA results 111 2000 " n............ ........... S2000  From Centrifugel  * From Centrifuge2 8 1500 S 1000 500 0 0.00 0.04 0.08 0.12 0.16 0.20 Tip Displacement(ft) 2500 From FEA in Homogeneous Case 2000  From Centrifugel S From Centrifuge2 1500 C) 150^C  1000 500 0 0.00 0.04 0.08 0.12 0.16 0.20 Top Displacement(ft) ~ Lrrrrn LCn i~r Yir^irlrnnr~ll* Tirr 1400 1400 1200  1200 1000 L/D=1, CVE=0 2 I 1200 L/D=3, CVE=O 2 ooo1000 ooo1000 S800 800 600 ..,, 0+c ) ( 600 4 400 [A (omogeeous Case) 400 200 o t t .ll .r 200 1 0 002 004 006 008 01 012 014 0 002 004 006 .. 01 012 014 Tip Displacement (ft) Tip Displacement JI 1400  1400 __  1200 1000 L/D 3 VE0 4 ) 1 I 8uu _ __   800  ~_   s 600  j i    600 _ _* _    C400 400 200 200 0 002 004 006 008 0 1 012 0 14 0 002 ,,i 006 01 0 12 0 14 Tip Displacement (ft) .1' Displacement ,,, 1400  1400  1200 1200 i000 D=1, CVE= B1000 L/D=3, CVE=0 6 ,, 0 800 _ 800 _ 600 0 600 400 4 400 200 200 0 0 0 002 004 0 06 008 0 1 0 12 0 14 0 002 006 1 012 014 Tip Displacement (ft) I Displacement I, Figure 45. Comparison of the tip stiffness among various CVE (av = 5 ft) 112 112 1800 a,=5ft 1700 C ,1600 3 S 1400  1300 0 05 15 2 25 3 35 16UU  a,=1 0t 1700  ... .....  "4 1600 1500 1400  1300 0 05 1 15 2 25 3 35 H/D 1800 a,=l 5f a=15f ... .. ... .. ...... 4.. .... ..... 1700 '1600 1500 10iso___ 0 05 1 15 2 25 3 35 H/D a,=10fi MU r z  0 05 1 15 2 25 3 35 H/D a,=15fl J 0 05 1 15 2 25 3 35 0 05 1 15 2 25 3 35 H/D Figure 46. Simulation results of the cases among various av (CVE =0.6 and LID =3) 113 1800 06 17 CVE0.2 05 CVE=0.2 S04 . h' cp = cv.. 160003 1500 ___ 1300 I 1300    I 0        0 05 1 15 2 25 3 35 0 05 1 15 2 25 3 35 HD HD 1800 06 6    CVE=0.4 .............. ....... .. ...... CVE=0.4 1700 1300 _      04        S03 1500 02 1400 01 1300 0 0 05 1 15 2 25 3 35 0 05 1 15 2 25 3 35 H/D H/D 1700 ,1600 0 02 1400 1300 0 0 05 1 15 2 25 3 35 0 05 1 15 2 25 3 35 H/D H/D Figure 47. Simulation results of the cases among various CVE (av=15ft and LD=3) 114 g <, 04       1300 o              114 1400 1200 R.1000 8 800 .2 600 400 200 0 0.02 0.04 0.06 0.08 Tip Displacement (ft) 0.1 0.12 0.14 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Tip Displacement (ft) Figure 48. Comparison of tip stiffness (CVE=0.6) 115 CHAPTER 5 USE OF GEOSTATISTICS IN LRFD ASSESSMENT WITH FIELD TEST RESULT 5.1 Background of Usage with Spatial Correlation and Its Influence on LRFD resistance factors, 1 A deep foundation's resistance or capacity, Q, is the summation of side friction, Qs, and end bearing Qtip. In the case of cylindrical prismatic shafts of length L and diameter D [L], the pile/shaft's side resistance Qs may be determined from the product of the pile/shaft's surface area, As = rrDL [L2], and the mean unit skin friction fs [stress] over As or, mathematically, Qs = Asfs. Consequently, fs is related to point unit skin friction, qs, through: I= J.U d (5.1) s A, Since qs may be regarded as a spatially random ("regionalized") variable of a certain distribution (e.g., lognormal) and spatial correlation structure (variogram), fs is also a random variable in space and its properties are related to the properties of qs. The variable qs was defined through summary statistics (mean, m, and variance, o) and the covariance function, C(h) stresss2. C(h) is a measure of spatial correlation between values of qs, which are separated by a distance h [L], and is directly related to the variogram y(h) stresss] by y(h) = o2C(h). In practice, this assumption corresponded to situations where m, o2, and y(h) of a site were known with high confidence (e.g., from exhaustive core sample testing) and the shaft was located at a random location on the site. In the case of the variance, as2, of the pile/shaft side friction, it was expected that it would be less than the site's variance, c2 of qs as a result of the spatial averaging over the shaft surface or, Os2 < 02. In geostatistics, the problem of determining Os2 was 116 known as a "change of support", "scale up," or "regularization" problem (Isaaks and Srivastava, 1989), since the properties of a variable were averaged over different support sizes. In the present case, qs obtained from core samples may be thought of as a rather small support volume (i.e., points) as opposed to fs, which is defined on a support equal to the shaft's side surface As. In fact, a = as 2/2 dimensionlesss] can be introduced as the factor of variance reduction between qs and fs, and it was determined by the relationship (e.g., Journel and Huijbregts, 1978; Isaaks and Srivastava, 1989; Deutsch, 2002) as: a = C'(h) dAdA2 (5.2) s As As where C'(h) = C(h)/o2 dimensionlesss] was a covariance function normalized to unit variance. Equation 5.2 contained two integrals over the area As (in practice, it is a quadruple integral); however, it was nothing but the arithmetic average of the covariance values that correspond to all possible combinations of two points on As (i.e., the side of shaft). If the averaging domain As was not a continuous area but a discrete number of n dimensionlesss] points with statistically independent observations of some random variable, then Eq. 5.2 reduced to the well known formula of the standard error, s2 = o2/n, where a = 1/n. To assess the variance of side shear, os2, the covariance C'(h) in Eq. 5.2 was assumed to be spherical (Isaaks and Srivastava, 1989), with an isotropic correlation length, a, or: 117 C'(h) = 11.5h +0.5 (5.3) a (a) Note that C'(h) took on a value between 0 and 1 depending on the distance h. That is to say, when h 2 a, C'(h) = 0 (no correlation) and when h = 0, C'(0) = 1 (a one toone correlation). The correlation length (or range) "a" was established from the experimental variogram based on available data. Substituting Eq. 5.3 into Eq. 5.2 and integrating over the areas, a = as2/a2 is determined. The results are presented in a monograph as a function of L/D and a/D, as shown in Figure 5.1. Also shown in the graph is the case of D = 0 (thick dashed line), in which one averages the variability over a length or depth that would subsequently only be used for end bearing. Next, the LRFD spatial resistance factors (4) for shaft side shear may be assessed (Klammler, 2010) from COVRs = s/ms using either field or laboratory soil/rock strength assessments, or both: COVR = i=n =M COVq, (5.4) m^ m and: S+ (COIV[ QD]) + (COV[QL])2) (1+(COV[. ) (5.5) (E[AD I qD+ E[A r ..' .r Figure 5.2 shows the LRFD resistance factors (4) for known COVR and reliability index (p) values. Using COVRs (i.e., side shear) from Eq. 5.4 for COVR, typical P values (e.g., 2.5 3.0), and LRFD resistance factors, the value of 4 may be ascertained. 118 5.2 Development of LRFD resistance factors, for End Bearing and the 17th Street Bridge Case Study In the case of a shaft's tip resistance, Qtip, the shaft's unit tip stress, 9b, must be calculated and then multiplied by the tip's cross sectional area. However, the shaft's unit tip resistance is a function of the shaft's displacement, as was found in Eq. 3.5. The settlement of at the bottom of the shaft, Wb, was computed from the elastic shortening of the shaft as follows: [( 2Q+ Q,)L (5.6) zE, D2 where Ec = Young's Modulus of the concrete shaft Qtop = Force at Top of Shaft Qtip = Force at Bottom of Shaft (qb Ashaft) The shaft's tip resistance Qtip (i.e., qb), may be found by determining the shaft and rock's compressibility parameter, A, which is a function of Of, F, and Q as shown below: As is evident from Eqs. 3.6 3.8, the Young's modulus of the rock mass, Em, controlled the variability of the results. Of interest was the relationship between Florida limestone mass modulus and intact Young's modulus, Ei. Synthetic limestone specimens with various strengths were cast with different volume percentages of voids (perlite), as reported in Chapter 3. A comparison of no void Young's modulus, Ei, versus mass modulus values with different volume percentages (i.e., recovery) was completed. Shown in Figure 3.20 was the Em/Ei ratio as reported by O'Neil and UF. As shown, the UF data falls between O'Neil's open and closed joint data. Also, there exists a linear relationship between Em/Ei for recoveries above 50% and a sharp drop off below 50%. 119 Of interest was the variability in the field and its impact on the tip resistance of shafts. To study the latter, a series of drilled shaft load tests at a bridge site were investigated. Figure 5.3 shows a typical stressstrain plot from an unconfined test on Florida limestone recovered from the 17th Street Bridge near load test LTSO4 at pier 10. A total of 102 unconfined tests were performed by SMO personnel on rock cores recovered in six boreholes at 5 foot spacing near LTSO4. The analysis considered both the secant Young's modulus as well as the tangent Young's modulus. Generally, the secant modulus encompassed end effects as well as micro cracking which induces a reduction in modulus versus the tangent modulus (Figure 5.3). All observed loss in tangent modulus (i.e. micro cracking) was found to occur beyond 6 to 7% strain, which for shafts of interest (6 to 9 ft) equated to a vertical movement greater than two inches or the AASHTO service state. Also presented in each figure are summary statistics (median, mean, standard deviation, and CV) of the data, as well as a lognormal distribution fit to the data. Comparing Figures 5.4 and 5.5, the mean of the tangent modulus was approximately 1.8 times the secant mean modulus, but the variation, CV, of each was quite similar. From the boring logs, the recoveries varied from 65% to 95% with a mean value of 75%. Based on Figure 3.20, an Em/Ei ratio of 0.45 was selected, and the mass modulus was computed using both the secant and tangent Ei moduli. Figure 5.4 and 5.5 showed the probability density distribution for the tangent and secant mass moduli using both the recent LTSO4 data (102), as well as the original design data (16) set. Clearly it is apparent from Figures 5.4 and 5.5 that the mass modulus of the rock varied over the site as well as in three dimensions below the shaft, which must be 120 accounted for in Eqs. 3.6 through 3.8. As shown in Chapter 3 & 4 for the case of horizontally layered soil (variability only in the vertical direction), one should use the harmonic mean modulus, Eh (i.e., Eq. 3.9) or: Fenton (2005) showed an excellent correlation with FEM analysis using the geometric mean, Eg (i.e., Eq. 3.10) or: It should be recognized that the geometric mean generally lies between the arithmetic and harmonic mean and was used in this project to assess contact stresses (Eq. 3.5). Due to the soil formation process, soil or rock variability was usually greater vertically than horizontally. With full boring generally for apart and shaft diameter in 4ft to 10ft range, it is assumed that the rock in modulus, random variables, was distributed vertically and constant horizontally, i.e., horizontally layered soil. The next question was the influence of geospatial correlation on the harmonic mean modulus, Eh. Specifically, Eq. 4.7 required the sum or average of 1/Em over a distance (H=2D) below the shaft as resulted by FEA analysis in Chapter 4. If Y = 1/Em, then the term inside the parentheses in Eq. 4.7 became the simple arithmetic average, i.e., 1/n YY = F. Note, that because Y [1/Em] was a random variable, so is F and it would have the general summary statistics, (i.e., mF, and G2F), as well. As discussed for side friction in Section 5.1, the mean of F, mF, would be the same as my; however, the variance of F would be reduced by the averaging process (i.e., 1/nYY = F) or G2F = a Y2, where 2 y represents the variance of 1/Em over the site. As was found with side friction, Eq. 4.8 for a or the monograph, Figure 5.1, may be used to assess a for the case of D = 0 and an appropriate correlation length, a. Once the value of a has been 121 assessed, then the random function F (mF =m; and G2F = C2 2) was also known and may be substituted back into Eq. 4.7 to obtain Eh ( i.e., random function). In the case of the 17th Street Bridge, the 1/Em was calculated for all tangent data. The mean of 1/Em = my =0.0000919381 and the standard deviation of 1/Em = OF = 0.0000450687 were calculated. Next, a was obtained for H/D = 2, and a = 5, 10, and 15 from Figure 5.1 or Eq. 4.8 which were used to find G2F = a G2y. Subsequently, a Monte Carlo simulation was performed to generate typical F values, which were substituted into Eq. 4.7 to generate the distributions shown in Figures 5.6, 5.7, and 5.8. Of particular interest was a comparison of the harmonic mean modulus, Eh, with the original field data, Em, as well as the inverted lognormal distribution which is the dash curve in Figure 5.4. Its summary statistics showed a mean of 13,481 ksf, a standard deviation of 6569 ksf, and a CV equal to 0.487. The dash curve represented what the field samples should look like if enough samples were recovered and the mass modulus, Em, was lognormally distributed. As expected, the harmonic mean, Eh was reduced from both the field sample mean (13,481 ksf). Specifically, due to the spatial correlation represented in the covariance function as lengths of 5, 10, and 15 ft, the CV of Eh was reduced to 0.303, 0.383, and 0.426 respectively, compared to the CV of Em. The highest reduction in CV of Eh was due to the lack of correlation between Em over short distances (i.e., the data were more random), which resulted in lower variability in Eh when averaged harmonically. This was already proven in Chapter 4. Using the harmonic mean function, Eh (Figures 5.65.8), in Eqs. 3.6 3.8 and Eq. 3.5, the distribution of the contact stress, qb at the bottom of the drilled shaft was obtained for a top shaft movement of 1.6 inches. This is shown in Figures 5.9 ~ 5.11. 122 Note that the figures represented the expected distribution of end bearing on the east side of the site where all the data were collected (i.e., in the vicinity of LTSO4). Variograms developed for the data showed a typical vertical correlation length of approximately 8 ft, which from Figures 5.9 and 5.10 suggested a mean tip resistance of between 126130 ksf and a standard deviation above and below the mean of 3242 ksf. The actual recorded tip resistance was 130 ksf for LTSO4 and 100 ksf for LTSO3. Both were close to the mean and well within the one standard deviation (3242 ksf). Knowing the distribution of the end bearing, the LRFD resistance factors, 4, may be assessed for the shafts from Eq. 5.5 for multiple reliability index values, P (e.g., P = 2.5 and 3). The computed 4 values were also shown in each figure depending on the correlation length, a. It was apparent that there was a significant effect of a on the 4 factor for a specific reliability index, p. For example, in the case of P = 3.0, 4 varied from 0.58 to 0.44, and the design end bearing ranges from 0.58 (130 ksf) = 75.4 ksf to 0.44 (130 ksf) = 57.2 ksf. In addition, to estimate the variability of tip displacement with fixed load as conducted in Chapter 4, Eq. 5.7, as modified from Eq. 3.5, was used. The actual recorded tip resistance, 130 ksf for LTSO4, was used as the fixed load. (q A)1/0.67 = Wt (5.7) With a typical vertical correlation length of approximately 8 ft, which from Figure 5.13 and Figure 5.14 suggested a mean displacement of 0.156~0.158 ft (1.87~1.9 in) and a standard deviation of 0.06~0.08 ft (0.9~1 in). The actual recorded displacement was 1.9in for LTSO4 and 1.8in for LTSO3. Both were close to the mean and well within the one standard deviation. 123 Of interest was the estimation of variability of displacement based on variability of modulus. In Chapter 4, the variability of tip displacement was successfully estimated using Eq. 4.12, derived with no power of distribution of modulus. However, in Eq. 5.7, there was the power function of distribution of modulus, i.e., inside of A. The following step was used for obtaining a simple power function from Eqs 3.63.8. 1. Using the Eqs. 3.63.8 and A = (L/D)05 and B = A 0, a. F=0.37A0.15(A1)glogE +0.13+0.15(A1)glogEm b. B=0.14A+0.05(A1)glog1,E + 0.44 0.05(A1)glogE,, 0.10A3 B 2. Using 1.a, 1.b and Eq. 3.8, A15 = E15 S L(A2 +1)05 m 3. For simple F and B as function of Em, a. C=10037A0 15(A1)log,,Ec+0 13 b. D= 10(0 14A+0 05(A1)logEc+0 44) 1 c. F=0.15(A1)log0(C 15(A 1E) d. B=0.05(A1)log0,(DO 05(A1Em) 4. Knowing logoX = 0.43 InX, the constants of power function were found by E 0 43 a. log1,kE = logokm( )lglkm where, m was a chosen value of Em. m The function y = logio(kEm) in equations 3.c and 3.d may be approximated by the power function yp = aEmb by requiring y(Em=m) = yp(Em=m) and y'(Em=m) = yp'(Em=m), where the prime indicates the first derivative(y' = 1/Em and yp' = 124 abEmb1 ) with respect to Em and m is a chosen value of Em for which the approximation becomes exact. I I b. k=F=CO15(A 1) in 3.c and k=G=Do05(lA1) in 3.d and substituting into 2. 5. Finally, A15 rLA lg (logGm) 0 loglo(Gm) loglo(Fm) 15+0 43  1 A Em loglo(Gm) loglo(Fm)l =HE n 30L(A2+1)05 logio(Fm) a. H(0.07) and J(1.27) were obtained as function of D, L, Ec and m 1.5 b. W, bE as same form of simple power function, Y=Xk H To estimate the variation of power function ( i.e., CV of Y based on CV of X) take "In" on the both side in power function, such as InY=klnX. Using Eqs. 4.1~4.4, take mean and variance of the normal distribution and back to the lognormal mean and CV. CV = (1+ CV)k2 1 (5.8) With Eq. 5.8, the CV of displacement at each correlation length (i.e., 5 ft, 10 ft and 15 ft) were calculated as 0.391, 0.497, and 0.55, which were close to the CV of displacement from distribution. 5.3 LRFD resistance factors,4 for End Bearing at the Fuller Warren Bridge Like the 17th Street Bridge, the Fuller Warren Bridge site was revisited and three new borings along with sixtythree samples were recovered and tested by SMO personnel. Similar to the 17th Street Bridge data, the mean of the tangent modulus was approximately 1.7 times the secant mean modulus. However, the CV of each were quite different: 0.49 & 0.55 for 17th Street Bridge and 1.27 and 1.16for the Fuller Warren 125 Bridge. The latter will have a profound effect on the LRFD resistance factors, 4, for the two sites. From the boring logs, recoveries varied from 58% to 94% with a mean value of 77%. Based on Figure 3.20, an Em/Ei ratio of 0.5 was selected, and the mass modulus was computed using both the secant and tangent, Ei, moduli. Figures 5.15 and 5.16 showed the probability density distribution for the tangent and secant mass modulus using data between LTS3 and LT4 (63), as well as the original design data (33). In the case of Fuller Warren, the 1/Em was calculated for all the new and old tangent data and was shown Figure 5.16. The results showed a mean of 1/Em = my = 0.0004322,and a standard deviation of 1/Em = OF = 0.0004999. Next, a was obtained for L/D = 2 and ranges a = 5, 10, and 15 ft, from which G2F = a 2y was found. Subsequently, a Monte Carlo simulation was performed to generate typical F values which were then substituted into Eq. 3.10 to generate the distributions shown in Figures 5.17, 5.18 and 5.19. Of interest was a comparison of the harmonic mean modulus, Eh, with the original field data, Em, as well as with the inverted lognormal. It has a summary statistics of mean = 5468.129 ksf, standard deviation= 6338.638 ksf, and a CV = 1.16, as shown in Figure 5.16. The dash curve represents what the field samples should look like if enough samples were recovered and the mass modulus, Em, was lognormally distributed. Specifically, due to spatial correlation represented in the covariance function as correlation length, a, the CV of Eh was reduced to 0.7, 0.88, and 1.0, respectively, compared to the CV of Em. The largest reduction in the CV of Eh to 0.46 was due to the 126 lack of correlation between Em over short distances (i.e., more random), which resulted in lower variability in Eh when averaged harmonically. Using the harmonic mean function, Eh, and Eq. 3.5, the distribution of the contact stress, qb at the bottom of the drilled shaft was obtained for a top shaft movement of 3.2 inches (field data), as shown in Figures 5.20, 5.21, and 5.22. Note, the figures represent the expected distribution of end bearing at the site where the data were collected (LT2LT4). Variograms developed for the data showed typical vertical correlation lengths of approximately 14 ft, which from Figures 5.205.22 suggests a mean tip resistance of between 7087 ksf and one standard deviation above and below the mean of 40~70 ksf, respectively. The actual recorded tip resistances were 65 ksf for LT3, 108 ksf for LT4, and 160 ksf for LT2. The results were close to the mean and well within the one standard deviation (40 ksf to 70 ksf). The estimation of the variability of tip displacement was conducted with fixed load, i.e., the actual recorded tip resistance, 65ksf for average of LT3. With typical vertical correlation length of approximately 14ft, which from Figure 5.23 ~ Figure 5.25 suggested a mean displacement of 0.4~0.425ft(4.8in~5.1 in) and a standard deviation of 0.39~0.58(4.68~6.96in). The actual recorded displacement was 3.2in. The results were not close to the mean but within the one standard deviation. With Eq. 5.8, the CV of displacement at each correlation length, i.e., 5ft, 10ft and 15ft, were calculated as 0.96, 1.24 and 1.43 which were close the CV of displacement from distribution. A comparison of LRFD resistance factors from the 17th Street and Fuller Warren Bridges was quite interesting (see Figures 5.95.11 and Figures 5.205.22). The 127 variability of the resistance factors was controlled by the variability of Eh. For example, the data from Fuller Warren had much higher variability (0.70.99; shown in Figures 5.175.19) versus the lower variability (0.30.42; Figures. 5.65.8) seen in the 17th Street Bridge data The latter affected the variability in contact stresses which in turn affected 4 values. For instance, the 4 values ranged from 0.580.44 (3=3) and 0.67 0.52 (3=2.5) for the 17th Street Bridge data, whereas Fuller Warren's 4 values ranged from 0.230.127 (3=3) and 0.30.18 (3=2.5), as shown in Figures 5.20~5.22. Evidently, the design end bearing for both sites were quite different. In addition, if sufficient data was collected at any single pier/shaft, an individual LRFD resistance factor could be assessed and its value would most likely be higher than the value assigned to the entire site. 128 ii I /,, ,/ /.."7/ y,/ / /  0 2 4 6 8 10 12 14 16 18 20 a/D [] Figure 51. Thick Solid Contour Lines Represent a = Os2/o2 for a/D = [0, 20] and L/D = [0,10]. Thick Dashed Contour Lines Represent a = f (L/a) for D = 0. Thin Dotted Lines are Parabolas, Along Which A = const. (Klammler, 2010) 129 0.5 COV [] Figure 52. LRFD resistance factors,4,as a Function of Reliability Index 3 and COVR (Klammler, 2010) 130 Figure 53. Secant vs. Tangent Young's Modulus on 17th LTSO4 Street Bridge Data from 131 Stress vs. Strain Curve w/ 10tsf limestone 16000  14000 Etan 12000 /  10000 ir 8000 / ( 6000 4000 z 2000 0 0,005 0.01 0,015 0.02 Strain Figure 54. Tangent Mass Modulus of 17th Street Bridge (118 Values) 18 16 O riginalSecant 4 __M (kaf) 7618.32 p (kaf) 8125.55 12 rr (kaf) 4462.63 o c Cv 0 54921 1 1 O . N(" 6TR (Em ks G113f) s r 0 Range (Em,ksf) Figure 55. Secant Mass Modulus of 17th Street Bridge (118 values) 132 20 18 0 riginalTangent 16 \ M (kaf) 12167.79677 14 \ (kta) 1 48?1. 54557 o12 a ( ks) 6569.552861 3 8  0_ 6 = 4 2 0 I D .. . RangeEm,ks f) 300Distribution of Eh, 17th Street for a5ft 300 Eh a=5ft 250 M (ksf) 11402.56271 o20 w_ i (ksf) 11909.62945 0 0 a (ksf) 3614.105382 "c5 50 1 cv 0.303460775 50  Range (Eh,ksf) Figure 56. Harmonic Mean Modulus, Eh, Assuming Correlation Length, a = 5 ft from 17th Street Bridge Data Distribution of Eh, 17th Street for a=10ft 250 Eh a=1 Oft 200 M (ksf) 11668.57474 SM (ksf) 12472.8685 o50  (ksf) 4778.001941 0 l L cv 0.38307162 := 100  4  2 50  0 M Range (Eh,ksf). ..... Range (Eh,ksf) Figure 57. Harmonic Mean Modulus, Eh, Assuming Correlation Length, a = 10 ft from 17th Street Bridge Data 133 Figure 58. Harmonic Mean Modulus, Eh, Assuming Correlation Length, 17th Street Bridge Data Distribution of Tip Resistance with 6=1.6in, 17th Street for a=5ft with Eh W M (ksf) 122.267 ( p (ksf) 126.141 ._____ o (ksf) 32.2196 c cv 0.25543 N \2.5 0.67125 3 0.58581 * *l ll Bii i11 ~cc~_ii_ i I I I I I l II 1 1 1 1 1 1 1 I I I I I I I I I I I T oo 0 0 0 0.r a = 15 ft from Range (Qb,ksf) Figure 59. Histogram/PDF of Contact Stress, qb, Using Eh with Correlation 5ft from 17th Street Bridge Data Length, a = 134 Distribution of Eh, 17th Street for a=15ft 200 Eh a=15ft 10 S(ksf) 12805.67993 S50 O 0 "Rn f ( 0 E, f 0 Range (Eh,ksf) 300 250 ^200 150 0 100 0 O _ 50 0 250 200 0 150 100 0 O D 50 0 Distribution of Tip Resistance with 6=1.6in, 17th Street for a=10ft with Eh  M (ksf) 124.678  (ksf) 130.767 E oa (ksf) 41.9731 cv 0.32098 O B  N 2.5 0.57082 ._< __ 3 0.48396 S0 8 R 0 S 8 0 08 R 0 S Q O , , 0 CM 0C CO CO CO CO '^ , Range (Qb,ksf) Figure 510. Histogram/PDF of Contact Stress, qb, Using Eh with Correlation Length, a = 10 ft from 17th Street Bridge Data Distribution of Tip Resistance with 6=1.6in, 250 200 S150 100 _.Q _.Q 2 50 0_ 0 17th Street for a=15ft with Eh .) M (ksf) 125.328 = _o p (ksf) 133.464 E \ . o (ksf) 47.4915 l l cv 0.35584 1 N 2.5 0.52317 3 0.43682 LA I 088 0 0 0 0 0 0 0 0 0 Range (Qb,ksf) Figure 511. Histogram/PDF of Contact Stress, qb, Using Eh with Correlation 15 ft from 17th Street Bridge Data Length, a = 135 Figure 512. Histogram/PDF of Displacement, Wt, Using Eh with Correlation Length, a = 5 ft from 17th Street Bridge Data Distribution of Dsplacement with qb=130ksf, 17th Street for a=10ft with Eh 300 M (ft) 0.14192 250 250 (ft) 0.15853 7200  \ a (ft) 0.07838 150 l cv 0.49445 0 S50 0 .. ... 0 CDC CD CD c D CD CD CD CD 6 C6 Range (6,ft) Figure 513. Histogram/PDF of Displacement, Wt, Using Eh with Correlation 10 ft from 17th Street Bridge Data Length, a = 136 Distribution of Displacement with qb=130ksf, 17th Street for a=5ft with Eh 350 300 M (ft) 0.14611 250 j (ft) 0.15695 200 o (ft) 0.06133 50 cv 0.39076 150  100 1 1 O (0 CJ 0 t CQ (0 CJ O 't (0 (0 CJ 0O 't O d d d d d(ft) Range (6,ft) Distribution of Displacement with qb=130ksf, 17th Street for a=15ft with Eh O (CD CJ O CD C (CD CJ O D  O d Co L d o o o d d d Range (6,ft) (0 (0 CJ 0 t d6 d d d Figure 514. Histogram/PDF of Displacement, Wt, Using Eh with Correlation Length, a = 15 ft from 17th Street Bridge Data 137 300 250 c 200 150 (' 100 c l00 O _ 50 0 M (ft) 0.14082 S (ft) 0.15965  (ft) 0.0871 cv 0.54555 ', m h ii i i i i i n i n 25 O riginalSecant 20 M t(kf) IOJ 18 Sp (kaf) 3997.88 0 _5 (kaf) 5077.66 I c 1 27'09 o 5 0  P : ^ Ragek T) m 0 tl t03 Figure 515. Secant Mass Modulus of Fuller Warren Bridge (96 values) 20 18 L OriginalTangent 16 NM (kaf) 5.;1 .7461 65 14 L 1 (kaf) 5468.128667 0012 4__1__________ (kaf) 26; o 10 cV 1.159196995 4 2 0 Ra ge (m,'ks f) Figure 516. Tangent Mass Modulus of Fuller Warren Bridge (96 values) 138 I] I 600Distribution of Eh, Fuller Warren for a=5ft 600 Eh a=5ft 500 M (ksf) 2839.090212 400 (ksf) 3504.121119 S0 a (ksf) 2475.5851 300 cv 0.706478177 oo 0 100  0Range (h, Range (Eh,ksf) Figure 517. Geometric Mean Modulus, Eh, Assuming a Correlation Length, a = 5 ft from Fuller Warren Bridge Data Distribution of Eh, Fuller Warren for a=10ft 400 __Eh a=1 Oft 350 M (ksf) 3129.427258 o00 I (ksf) 4213.896554 50 o 0 (ksf) 3724.830951 Soo cv 0.883939818 100 50  Range (Eh,ksf) Figure 518. Geometric Mean Modulus, Eh, Assuming a Correlation Length, a = 10 ft from Fuller Warren Bridge Data 139 Figure 519. Geometric Mean Modulus, Eh, Assuming a Correlation Length, a = 15 ft from Fuller Warren Bridge Data 140 Distribution of Eh, Fuller Warren for a=15ft 450 400 Eh a=1 5ft 350 M (ksf) 3297.787915 300 wI\ (ksf) 4657.202235 o50  a (ksf) 4652.275009 I100  cv 0.99894202  50  c 00   50  Range (Eh,ksf) Distribution of Tip Resistance with 6=3.2in, Fuller Warren for a=5ft with Eh 300 250 o.200 150 100 c 0 o S50 0 Range SC(Qb,ksf) (Qbksf) 1 _M (ksf) 60.1666 oIJ (ksf) 69.9796 a (ksf) 40.4733 I KC\J cv 0.57836 2.5 0.30158 II I3 0.22938 Figure 520. Histogram/PDF of Contact Stress, qb, Using Eh with a Correlation Length, a = 5 ft from Fuller Warren Bridge Data Figure 521. Histogram/PDF of Contact Stress, qb, Using Eh with a Correlation Length, a = 10 ft from Fuller Warren Bridge Data 141 Distribution of Tip Resistance with 6=3.2in, Fuller Warren for a=10ft with Eh 250 0 M (ksf) 65.2963 200 0 o p (ksf) 80.8093 cE o a (ksf) 57.9344 150  150 N cv 0.71693 100  CO S 1  2.5 0.21767 0 O o 50 3 0.15708 Range (Qb,ksf) R O Figure 522. Histogram/PDF of Contact Stress, qb, Using Eh with a Correlation Length, a = 15 ft from Fuller Warren Bridge Data Figure 523. Histogram/PDF of Displacement, Wt, Using Eh with Correlation Length, a = 5 ft from Fuller Warren Bridge Data 142 Distribution of Tip Resistance with 6=3.2n, Fuller Warren for a=15ft with Eh 250 0 M (ksf) 68.2378 200 D o p (ksf) 87.3309 a (ksf) 69.7517 150 o Cl cv 0.79871 100 L P D S1 N 2.5 0.18114 m m 0 .l. .il l ll C O an O( 0O M Os Range (Qb,ksf) Distribution of Tip Resistance with qb=65ksf, Fuller Warren for a=5ft with Eh 500 M (ft) 0.29926 400 0 (ft) 0.40934 300 ao (ft) 0.38922 20 cv 0.95084 2100 100 o co C0 M o LO 0o M Co co 0 o CM Lo Range ( ,ft) cd cd ci ' Range (6,ft) Figure 524. Histogram/PDF of Displacement, Wt, Using Eh with Correlation Length, a = 10 ft from Fuller Warren Bridge Data Figure 525. Histogram/PDF of Displacement, Wt, Using Eh with Correlation Length, a = 15 ft from Fuller Warren Bridge Data 143 Distribution of Tip Resistance with qb=65ksf, Fuller Warren for a=10ft with Eh 500 M (ft) 0.26486 400 (ft) 0.41956 3 0 o (ft) 0.52591 300 cv 1 .25346 S200 O L 100 00 il O MO C, O CQJ M N O l MO OO Ma O CN Ln o d d 0O CN r o,] o,] oc, co cd o5 c Range (6,ft) Distribution of Tip Resistance with qb=65ksf, Fuller Warren for a=15ft with Eh 500 M (ft) 0.248 400 (ft) 0.42752 00 (ft) 0.58348 cv 1.3648 S200 S100 Range (6,ft) CHAPTER 6 CONCLUSION This dissertation encompasses the development of axial, lateral, and rotational tip models of short drilled shafts founded in heterogeneous limestone conditions. Focus has been made to characterize the tip behavior of largediameter but relatively short shafts under heterogeneous rock conditions. In this study, original contributions have been made by the author: * A onedimensional (1D) lateral spring model to simulate the tip lateral behavior of the drilledshaft with L/D ratio ranging one to three has been developed as a function of rock strength and normal stress. * A 1D axial spring model that includes spatial variability and correlation of rock modulus under serviceability limit has been developed to simulate the tip resistance of the drilled shaft in heterogeneous limestone. * An LRFD material resistance factor of heterogeneous limestone has been evaluated in consideration of the variability of axial tip stiffness. To study tip behavior of drilled shafts, the total of 16 centrifuge tests were performed, which varied shaft dimensions, rock strengths, rock recovery, Young's modulus, and layering. To ensure repeatability, each test was performed twice with synthetic limestone. After assessing the axial tip resistance during axial load, the shaft's tip shear resistance versus lateral tip displacement was found under combined load as shown in Figure 3.41b. Specifically, for a given lateral tip displacement, the measured tip shear was analyzed to obtain the normalized tip shear (i.e., shear stress minus cohesion divided by axial tip stress). Multiplying the normalized tip shear by axial tip stress and adding cohesion was found equal to the mobilized tip shear. The bilinear representation covered typical rock strengths from 10 to 30 tsf and was obtained from 16 different types of boundaries conditions (e.g., L/D = 1 and 3; homogeneous and layered scenarios). The bilinear characterization represented elastic behavior up to 144 failure, at which point no additional shear was mobilized. The values of the normalized shear failure were developed for rock angles of internal friction of 28 and 45 degrees with unconfined strengths, qu, of 10 and 30 tsf, respectively. The lateral tip shear model was mainly controlled by the strength of rock and the diameter of shaft. Once the axial and shear resistance of the tip has been determined, the moment versus rotation model suggested by Bell (1991) was recommended, as shown in Figure 3.49b. The relationship between moment and rotation was given by Eq. 3.24 and was function of tip shear and lateral tip displacement. This study also found that the tip lateral and rotational behaviors of drilled shaft was significantly affected by the tip axial resistance. The following conclusions on axial tip resistance were drawn from the centrifuge tests and the FEA results : * Florida limestone's mass modulus (i.e., Em) is sensitive to recoveries and is proportional to intact Young's modulus, Ei, from core samples as shown in Figure 3.20. The mass modulus, Em, should be assessed from the tangent Young's modulus, Ei,; use of secant Young's Modulus will generally be 1.8 to 2.0 times lower than tangent values (Fuller Warren and 17th Bridge sites). * The axial tip resistance was very sensitive to mass modulus of elasticity of rock, Em, within a zone of two diameters below the tip of the shaft in serviceability limit. * The numerical solution with spatial variability in heterogeneous rock has suggested that the use of either the harmonic mean or the geometric mean provides reasonable prediction of axial tip resistance. * For the multidirectional spatial variability, the geometric average was recommended. However, the harmonic average was found to agree favorably with the experimental results obtained from a condition simulated for 1D spatial variability, i.e., variability of rock modulus considered in the vertical direction only. Examples of predicting the axial tip resistance versus axial tip displacement using the FHWA method (O'Neill 1997) and linear stiffness relation by Misra (2006) and Doherty (2006) were shown in Figure 3.32 and 3.33 from centrifuge test. Both the harmonic and geometric mean were used for predicting the axial tip behavior in layered 145 system. These were assessed over a depth of 2D below the shaft tip. The following summaries of the axial tip behavior from centrifuge test were observed: * Initial tip stiffness, up to 1" axial tip displacement, can be estimated by Misra and Doherty's method with secant modulus. * The nonlinear axial tip resistance, above 1" axial tip displacement, can be estimated by O'Neil's Method with harmonic mean within 2D below the shaft tip of tangent modulus in spatial variability. To study the influence of the spatial variation of the modulus of elasticity on the axial tip resistance vs. axial tip displacement in serviceability, the heterogeneous rock conditions were considered in the FEA model that simulated a typical field condition in Florida with perfectlyhorizontallylayered limestone. The modulus of elasticity of the heterogeneous limestone was characterized as a lognormal random variable with a coefficient of variation (CVE), ranging from 0.20.6 and vertical correlation lengths (a,) ranging from 515 ft. For each random heterogeneous condition generated from a given set of rock characteristics (such as mean, CVE and av), 1,000 FEA simulations were performed on a given shaft L/D. Statistical analysis of the FEA results revealed below: * The mean axial tip response of the heterogeneous system obtained from the simulations can be linearly estimated from the axial tip response of homogeneous system with CVE as shown in Eq. 4.13. * The effective average depth of elastic modulus (2D below tip drilled shaft) should be used to predict the mean axial tip response. * The homogeneous axial tip response can be obtained from the Misra (2006) and Doherty (2006). * The variability of the axial tip displacement (Eq. 4.12) should be estimated as function of variability of elastic modulus and correlation length described in variance reduction factor (Klammler, 2010). 146 However, above one inch axial tip displacement, the variability of the axial tip displacement (Eq. 5.8) can be estimated as function of variability of elastic modulus and correlation length described in a variance reduction factor with a nonlinear relation. A significant finding of the present study suggests that smallerdiameter drilled shafts have more variance of tip resistance than largerdiameter drilled shafts in heterogeneous rock conditions. Therefore, the design resistance factor (4) of the rock in LRFD should be reduced to account for the size effect on the axial tip resistance. Moreover, the variance of the axial capacity of the deep foundation may be computed by summing the variance of side resistance (Klammler, 2010) with the variance of tip resistance. Finally, the effort studied the LRFD resistance factors (4) in light of the new harmonic modulus. Currently, one set of fixed resistance factors are used for drilled shafts in Florida. The spatial variability and associated correlation lengths (i.e., covariance function), suggest that LRFD resistance factors should vary from site to site. To study the effect of the spatial variability on LRFD resistance factors, two field investigations (i.e., Fuller Warren and 17th Bridge) were investigated. The analysis revealed: * The axial tip axial resistance is sensitive to Em assessed from the tangent Ei based on recovery and RQD. * As the correlation length increases with a high CVE, the variability of Qb of CVR increases, resulting in lower LRFD resistance factors. o CVEm and correlation length of 17th street (0.49 & 8ft) and Fuller Warren (1.17 & 14ft). o CVEh of 17th street (0.30.43) and Fuller Warren (0.71) due to variance reduction factor. o CVqb or CVR of 17th street (0.260.36) and Fuller Warren (0.580.8). 147 o LRFD resistance factor of 17th street (0.580.44 at 3=3) and Fuller Warren(0.23~0.13 at 3=3). * It is recommended that individual LRFD assessment of axial tip resistance be performed as a site by site. 148 APPENDIX A COMPRESSIVE FORCE (ON X AXIS, KIPS) ALONG DEPTH (ON Y AXIS, FT) FROM EACH PAIR OF STRAIN GAGE WITH TREND LINES OF LATERAL LOADING STEP 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 3000 0 1 2 3 4 5 6 7 8 9 10 Figure A1. Result from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 0 1000 2000 3000 4000 5000 6000 2000 4000 6000 8000 10000 12000 YJ Figure A2. Result from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 149 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 6000 0 1 2 3 4 5 6 7 8 9 10 (a) (b) Figure A3. Result from 9 ft Diameter 9 ft Embedded b) Test No. 2 0 2000 4000 6000 8000 10000 1200C O 0 1 / 1 2 2 3 3 4 4 5 5 6 6 7 / 7 8 8 9 9 10 10 Figure A4. Result from 9 ft Diameter 9 ft Embedded b) Test No. 2 500 1000 1500 2000 25001 Length in 10 tsf Rock a) Test No. 1 2000 4000 6000 8000 1000C Length in 30 tsf Rock a) Test No. 1 500 1000 1500 2000 Figure A5. Result from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 150 0 1000 2000 3000 4000 5000 600C 0 1000 2000 3000 4000 5000 6000 7000 0 0 2 2 3 3 4 4 5 5 6 6 7 7 (a) (b) Figure A6. Result from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 000 200000 4000 00 600000 800000 10000 OC 0 00 1000 00 2000 00 3000 00 4000 00 5000 00 6000 00 7000 00 000  000  500 * 500 10 10 00 15 0 1500 2000  20 00 2500 2500 30 00 30 00 35 00 35 on (a) (b) Figure A7. Result from 9 ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 000 500000 1000000 1500000 2000000 25000 00 000 500000 1000000 1500000 2000000 25000 OC 000 000 00 500 500 / / / 1000 10 00 1500 15 00 20 00 r 20 00 2500 25 00 30 00 30 00 (a) (b) Figure A8. Result from 9 ft Diameter 27 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 151 0 1000 2000 3000 4000 5000 6000 / , ee= Figure A9 Result from 6 ft Diameter 18 1 b) Test No. 2 0 5000 10000 15000 0 1000 2000 3000 4000 5000 6000 0 /1 /  2 71. 4 6 8 10 12 14 16 18 20 (b) Figure A10. Result from 6 ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 152 ft Embedded Length in 10 tsf Rock a) Test No. 2000( 0 2000 4000 6000 8000 10000 12000 1400C r 0 1  / *//X e*/ /11 1__ (f A( APPENDIX B OUTPUT VOLTAGE (VOLT ON YAXIS) FROM EACH STRAIN GAGES VS. TESTING TIME (SEC ON XAXIS) WITH LOADINGS 0,6 0,4 0,2 0 0.2 0,4 0,6 0,8 1 Figure B1. Output Voltage from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1 D below from Tip of Shaft a) Test No. 1 b)Test No. 2 Figure B2. Output Voltage from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1 D below from Tip of Shaft a) Test No. 1 b)Test No. 2 153 0,6 0,4 0,2 0 0,2 0,4 0.6 0,8 (a) (b) Figure B3. Output Voltage from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Figure B4. Output Voltage from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 0.4 0.2 0 0.2 0,4 0.6 0.8 0,2 0,1 0 0,1 0,2 0.3 0,4 0,5 0,6 0.7 0.8 Figure B5. Output Voltage from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 154 Figure B6. Output Voltage from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 I r, 1.5 0,5 " C 1,5 3 2 (a) (b) Figure B7. Output Voltage from ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 05 ~' C1 I.2 Figure B8. Output Voltage from ft Diameter 27 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 155 ~^ ,, ,o 0,5 Figure B9. Output Voltage from ft Diameter 18 ft Test No. 1 b) Test No. 2 32 i Embedded Length in 10 tsf Rock a) 2 4 Figure B10. Output Voltage from ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 156 APPENDIX C MOMENT CURVE WITH LOAD STEP Moment vs. Depth Moment vs. Depth Moment(kipsRt) M oment(kipsRt) Figure C1. Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 Moment vs. Depth Moment vs. Depth Moment(kipsRt) Moment(kipsit) Figure C2. Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 157 Morrnt vs. Depth Moment(kipsft) Moment(kipsft) Figure C3. Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Moment vs. Depth Moment vs. Depth Moment(kipsft) Moment(kipst) Figure C4. Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 158 Moment vs. Depth Moment vs. Depth 12 10 8 6 4 2  2 5n 3 4 8 Monent(kpsRt) Momentvs. Depth Moment(kipsRt) (a) (b) Figure C5. Moment Curve with Different Top Lateral Displacement from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Moment vs. Depth Moment vs. Depth Moment(kipsft) Moment(kipsRt) Figure C6. Moment Curve with Different Top Lateral Displacement from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 159 Moment vs. Depth M oment(kipsft) M oment(kipsft) Figure C7. Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Moment vs. Depth Moment(kipsft) Moment vs. Depth Moment(kipsft) Figure C8. Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 27 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 160 Moment vs. Depth 10.00 5.00 0.00 . 5.00 0 > 10.00 a) 15.00 20.00 25.00 Moment vs. Depth Moment(kipsft) (a) Moment(kipsft) (b) Figure C9. Moment Curve with Different Top Lateral Displacement from 6 ft Diameter 18 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Moment vs. Depth Moment vs. Depth )00 0 00 40000 60( 20.00 Moment(kipsft) (b) Figure C10. Moment Curve with Different Top Lateral Displacement from 6 ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 161 )00 " 30000 40000 50000 60000 70 Moment(kipsft) Moment vs. Depth APPENDIX D SHEAR CURVE WITH LOAD STEP Shear vs. Depth Shear(kips) Shear vs. Depth Shear(kips) Figure D1. Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 Shear vs. Depth i 5 10 15 Shear(kips) Shear vs. Depth (b) .000 100,0 v " ) 5000 10 15 Shear(kips) (b) Figure D2. Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 162 Shear vs. Depth 10 5  M AU 4000 2000 4000 6 _____ / .__________ ^Sk____ Shear(kips) Shear(kips) (a) (b) Figure D3. Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Shear vs. Depth Shear(kips) Shear vs. Depth 10 1000 5000 10 Sh Shear( kps) (a) (b) Figure D4. Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 163 Shear vs. Depth Shear vs. Depth Shear(kips) Shear(kips) Figure D5. Shear Curve with Different Top Lateral Displacement from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Shear vs. Depth Shear(kips) Shear vs. Depth 1 I Shear(kips) Figure D6. Shear Curve with Different Top Lateral Displacement from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 164 Shear vs. Depth Shear vs. Depth 1C00 5000 5000 1000 5 5 5000 10 00 I 0 (D > Shear(kips)Shear(kips) Shear(kips) (a) (b) Figure D7. Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Shear vs. Depth 10 100 10Q ) 1000 2000 3000 4C Shear(kips) Shear(kips) Shear vs. Depth 1O 00 00 2000 4000 6000 8000 1C S rI Shear(kips) Figure D8. Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 27 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 165 Shear vs. Depth Shear vs. Depth 00 1000 1000 2000 3000 4C S 15 )0 2( 0) LU Shear(kips) (a) Shear vs. Depth i1 00 1 1000 2000 3000 4( Shear(kips) (b) Figure D9. Shear Curve with Different Top Lateral Displacement from 6 ft Diameter 18 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Shear vs. Depth Shear vs. Depth Shear(kips) Shear(kips) Figure D10. Shear Curve with Different Top Lateral Displacement from 6 ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 166 APPENDIX E RELATION KV(VERTICAL STIFFNESS) VERSUS J(RIGIDITY PARAMETER) S( T(34v) Est3 J 24(1v,)(lv)GRR3 Where, v is Possion's ratio of limestone v, is Possion's ratio of footing E, is Elastic modulus of footing GR is Shear Modulus of limestone t is Thickness of footing R is Radius of footing .0. (b) 9.0 24.0 d/ 4.0 0,o ...  60 II d/RI.0 I 100 4A i 6C 1.OEOS I1E03 LOE01 l.CE401 t.OE 3 I.OE.05 1.0E43 I.OE l.OE+01 I.OEd3 J J Figure E1. The relation J and Kv: a(v =0.2) and b(v =0.499) 167 LIST OF REFERENCES ADINA R & D, Inc. (2006). ADINA user manual, Version 8.5, Watertown, MA. American Association of State Highway and Transportation Officials (AASHTO). (2009). LRFD Bridge Design Specifications, 4th ed., 2009 Interim Revisions. American Society for Testing and Materials (ASTM). (1997). C207, Standard specification for hydrated lime for masonry purpose, Philadelphia, PA. American Society for Testing and Materials (ASTM). (2002). D293895, Standard test method for unconfined compressive strength of intact rock core specimens, Philadelphia, PA. American Society for Testing and Materials (ASTM). (2004). D3967 95a, Standard test method for splitting tensile strength of intact rock core specimens, Philadelphia, PA. American Society for Testing and Materials (ASTM). (2007). D2850 03a, Standard test method for unconsolidatedundrained triaxial compression test on cohesive soils, Philadelphia, PA. Bathe, K. J. (1996). Finite Element Procedures, The Prentice Hall, Englewood Cliffs, NJ. Bell, R. W. (1991). The analysis of offshore foundations subjected to combined loading, M.S. Thesis, University of Oxford, Oxford, U.K. Bridge Software Institute (BSI). (2010). FBMultipier user's manual, Department of Civil and Coastal Engineering, University of Florida, Gainesville, FL. Chandrasekaran, V. S. (2001). "Numerical and centrifuge modeling in soil structure interaction." Indian Geotechnical Journal, 31(1), 159. Consolazio, G. R., Chung, J. H., and Gurley, K. R. (2003). "Impact simulation and full scale crash testing of a low profile concrete work zone barrier." Computers and Structures, 81(13), 13591374. Deutsch, C.V. (2002). Geostatistical reservoir modeling, The Oxford University Press, New York, NY. Deutsch, C.V., and Journel, A. G. (1998). GSLIB: Geostatistical software library and user's guide, 2nd ed., The Oxford University Press, New York, NY. Doherty, J. P., and Deeks, A. J. (2006). "Stiffness of a flexible circular footing embedded in an elastic half space." J. Geomechics., Vol(6), 4654. 168 Elkateb, T., Chalaturnyk, R., and Robertson, P. K. (2002). "An overview of soil heterogeneity: quantification and implication on geotechnical field problems." Canadian Geotechnical Journal, 40, 115. Fenton, G. A., and Griffiths, D. V. (2002). "Probabilistic foundation settlement on spatially random soil." J. Geotech. Geoenviron. Eng., 128(5), 381390. Fenton, G. A., and Griffiths, D. V. (2005). "Threedimensional probabilistic foundation settlement." J. Geotech. Geoenviron. Eng., 131(2), 232239. Fenton, G. A., and Griffiths, D. V. (2007). "Reliabilitybased deep foundation design." Probabilistic Applications in Geotechnical Engineering, Geotechnical Specialty Publication, Am. Soc. Civ. Eng., 170. Goovaerts, P. (1997). Geostatistics for natural resources evaluation, The Oxford University Press, New York, NY. Hassan, K. M., O'Neill, M. W., Sheikh, S. A., and Ealy, C.D. (1997). "Design method for drilled shafts in soft argillaceous rock." J. Geotech. Eng. Div., Am. Soc. Civ. Eng., 123(3), 272280. Isaaks, E. H., and Srivastava, R. M. (1989). An Introduction to applied geostatistics, The Oxford University Press, New York, NY. Jaeger, J. C., Cook, N. G., and Zimmerman,R. W. (1969). Fundamentals of rock mechanics, 4th ed., The Blackwell Publishing, London, UK. Kim, M. H. (2001). Analysis of osterberg and statnamic axial load testing and conventional lateral load testing, M.S. Thesis, University of Florida, Gainesville, FL. King, P. R. (1989). "The use of renormalization for calculating effective permeability." Transport in Porous Media, 4, 3758. Klammler, H., McVay, C. M., Lai, P., and Horhota, D.(2010). "Influence of spatially variable side friction on single drilled shaft resistance and LRFD resistance factors." J. Geotech. Geoenviron. Eng., In Press. MacGregor, J. G. (1992). Reinforced Concrete, 2nd ed., The Prentice Hall, Englewood Cliffs, NJ. Martino, S., Prestininzi, A., and Mugnozza, G. (2001). Mechanisms of deep seated gravitational deformations: parameters from laboratory testing for analogical and numerical modeling, in: S6kk1, Eloranta(Eds.), Rock Mechanics a Challenge for Society, Swets & Zeitinger Lisse, Netherlands,137142. 169 McVay, C. M., Bloomquist, D., Ko, J. S., Klammler, H., and Otero, J. (2008). "Distribution of end bearing and tip shear on drilled shafts founded in Florida limestone." Florida Department of Transportation, Report BD54559/UF 57247. McVay, M. C., Townsend, F. C., and Williams, R. C. (1992). "Design of socketed drilled shafts in limestone." J. Geotech. Eng. Div., Am. Soc. Civ. Eng., 118(10), 1626 1637. Misra, A., and Roberts, L. A. (2006). "Probabilistic analysis of drilled shaft service limit state using the "tz" method." Canadian Geotechnical Journal, 43, 13241332. National Instruments (NI). (1999). DAQ 6034E/6035E user manual, National Instruments Corporation, Austin, Texas. Niraula, L. D. (2004). Development of modified TZ curves for large diameter Piles/Drilled shafts in limestone for FBPIER, M.S. Thesis, University of Florida, Gainesville, FL. Norris, R. J., Lewis, J. M., and HeriotWatt, U. (1991). The geological modeling of effective permeability in complex heterolithic facess" In proceeding of the 66th Annual Technical Conference and Exhibition, SPE 22692, Dallas, TX, 359374. Phoon, K. K., Kulhawy, F. H., and Grigoriu, M. D. (2000). "Reliabilitybased design for transmission line structure foundations." Computers and Geotechnics, 26, 169 185. Reese, L. C., and O'Neil, M. W. (1999). "Drilled shafts: Construction methods and design procedures." FHWAIF99025, U.S. Department of Transportation, ADSC, Dallas, Texas. Styler, M. A. (2006). Development and implementation of the diggs format to perform LRFD resistance factor calibration of driven concrete piles in florida, M.S. Thesis, University of Florida, Gainesville, FL. Ueshita. K., and Meyerhof, G. G. (1967). "Deflection of multilayer soil system," J. Soil Mechs. Fndns. Div., Am. Soc. Civ. Eng., 93(SM5), 257282. Vanmarcke, E. H. (1984). Random fields: Analysis and synthesis, The MIT Press, Cambridge, Mass. 170 BIOGRAPHICAL SKETCH Jeongsoo Ko was born in Gwangju, South Korea. He spent his childhood in that beautiful city and completed primary, middle, high school, and university, leaving only to complete his military service. He was accepted to study civil engineering at Chosun University, Gwangju, South Korea in 1994. He earned his BSc in the Civil Engineering Department at Chosun University in March 2002. During his undergraduate studies, he spent 26 months performing military service at the border between South and North Korea. He realized that his knowledge was not yet adequate for dealing with realworld civil engineering problems, so he decided to go abroad for advanced education. He was accepted by the Civil and Coastal Engineering Department at the University of Florida and moved to the U.S. in July 2003. He had studied and worked on thick lift compaction with Dr. McVay for about one year. Including this period, he spent the 18 months acquiring a background in Geotechnical Engineering. He earned his master's degree in August 2005. In August 2005, he began his Ph.D. degree under the guidance of Dr. McVay in Civil Engineering at University of Florida. His research interests include the tip behavior of drilled shafts and spatial variability. 171 PAGE 1 1 E VALUATION OF T IP B EHAVIOR OF D RILLED S HAFT IN FLORIDA L IMESTONE By JEONGSOO KO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 PAGE 2 2 2010 J eongsoo K o PAGE 3 3 To m y lovely wife & personal secretary Yookyeong Jang PAGE 4 4 ACKNOWLEDGMENTS It is my great pleasure to give thanks to professors and friends who supported me to complete this dissertation. First and foremost, I would like to thank my great graduate supervisory committee, all of whom have offered invaluable support and encouragement throughout my whole academ ic career including this dissertation. Es pecially, I would like to express my s incere gratitude to Dr. Michael C. McVay for the opportunity to do this study and for his invaluable guidance during the research. I also wish to express my thanks to Dr. Jae Chung for teaching and helping in the F inite Element Analysis (FE A ) Furthermore I thank Dr. David Bloomquist, Dr. Dennis Hiltunen and Dr. Malisa Sar ntinoranont for their time and effort they devoted to serving on my supervisory committee. I also wish to express my gratitude to Mr. Peter Lai of the FDOT for financial support. A lso, I appreciate Dr. Klammler for his advice for my basic geostatistic concept. F urthermore, I appreciate Mr. Chuck Broward for his technical advice into my lab test. I would like to say thank s to all members in geotechnique group for their friendship. Furthe rmore I would like to say thank s to all the member s of Korean student s in civil engineering for sharing my happiness. Finally, I thank my wife, Yookyeong for her patience, encouragement, and sacrifice for 7 years of my graduate study, and she is my perfect personal secretary and technical supporter. Also, I thank my parents and parents in law for their endless love and support PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF TABLES ............................................................................................................ 8 LIST OF FIGURES .......................................................................................................... 9 ABSTRACT ................................................................................................................... 16 CHAPTER 1 INTRODUCTION .................................................................................................... 18 1.1 Problem Statement ........................................................................................... 18 1.2 Objectives ......................................................................................................... 20 1.3 Scope ................................................................................................................ 20 1.3 .1 Centrifuge ................................................................................................ 20 1.3.1.1. Loading (Axial and Lateral) ........................................................... 21 1.3.1.2. Shaft di ameter ............................................................................... 21 1.3.1.3. Length to diameter (L/D) Ratio ...................................................... 21 1. 3 .1.4. Influence of Rock Voids or Limestone Recoveries ........................ 22 1. 3 2 Laboratory Test ....................................................................................... 22 1. 3 3 LRFD Assessment with FEA Analysis ..................................................... 23 1.3.4 LRFD Assessment with Field Verification ................................................ 24 1.4 Overview of Dissertation ................................................................................... 25 2 LITERATURE REVIEW .......................................................................................... 27 2.1 Designs for Drilled Shaft Subjected in Axial Load ............................................. 28 2.2 Relation Moment and Rotation ......................................................................... 28 2.3 Geostatistics Background ................................................................................. 30 2.4 Implication of Geostatistics to Geotechnical Field Problems ............................. 32 2.4.1 Application of Stochastic Simulation to Geotechnique Engineering ........ 34 2.4.2 Averaging Technique ............................................................................... 36 2.4.3 Characterization for Spatial Variability of Soil Properties ......................... 37 3 CENTRIFUGE TEST SETUP AND RESULT .......................................................... 41 3.1 Centrifuge Background ..................................................................................... 41 3.2 Hardware & Instrumentat ion ............................................................................. 43 3.3 Material ............................................................................................................. 44 3.3.1 Synthetic Rock ........................................................................................ 44 3.3.2 Concrete Grout ........................................................................................ 46 3.3.2.1 Mix with silica sand ........................................................................ 47 3.3.2.2 Mix with granite sand ..................................................................... 47 PAGE 6 6 3.4 Data Reduction ................................................................................................. 47 3. 4. 1 Axial Loading ........................................................................................... 47 3.4.1.1 Axial loading Side shear ............................................................... 49 3.4.1.2 Axial loading End bearing ........................................................... 49 3.4.2 Lateral Loading in Combination with Axial Loading ................................. 52 3.4.2.1 Lateral centrifuge data reduction .................................................... 52 3.5 Measured Versus Predicted Shaft Tip Response ............................................. 56 3.5.1 Axial Test Results .................................................................................... 56 3.5.2 Lateral Test Result Tip Shear Model ..................................................... 57 3.5.3 Lateral Test Results Moment Rotation Model ....................................... 59 4 A T WO DIMENSIONAL NUMERICAL ANALYSIS OF TIP RESISTANCE OF A DEEP FOUNDATION IN HETEROGENEOUS ROCK ............................................ 94 4.1. Finite Element Analysis .................................................................................... 95 4.1.1 Dimensions and Boundary Conditions ..................................................... 95 4.1.2 Constitutive Models for Materials ............................................................ 96 4.1.2.1 Reinf orced concrete ....................................................................... 96 4.1.2.2 Limestone rock ............................................................................... 96 4.1.3 Rock Shaft Interface ................................................................................ 97 4.2 Simulating the Influence of Rock Variability in Florida Limestone on Tip Stiffness ............................................................................................................. 100 5 USE OF GEOSTATISTICS IN LRFD ASSESSEMENT WITH FIELD TEST RESULT ................................................................................................................ 116 5 .1 Background of Usage with Spatial Correlation and Its Influence on LRFD resistance factors, ........................................................................................... 116 5 .2 Development of LRFD resistance factors, for End Bearing and the 17th Street Bridge Case Study .................................................................................. 119 5 .3 LRFD resistance factors, for End Bearing at the Fuller Warren Bridge ........ 125 6 C ONCLUSION ...................................................................................................... 144 APPENDIX A COMPRESSIVE FORC E ( ON x aXIS, KIPS ) ALONG DEPTH ( ON Y aXIS, FT ) FROM EACH PAIR OF STRAIN GAGE WITH TREND LINE S OF LATERAL LOADING STEP ................................................................................................... 149 B OUTPUT VOLTAGE (VOLT ON Y AXIS) FROM EACH STRAIN GAGES VS. tESTING TIME (SEC ON X AXIS) WITH LOADINGS ........................................... 153 C MOMENT CURVE WITH LOAD STEP ................................................................. 157 D SHEAR CURVE WITH LOAD STE P ..................................................................... 162 E RELATION Kv(VERTICAL STIFFNESS) VERSUS J(RIGIDITY PARAMETER) ... 167 PAGE 7 7 LIST OF REFERENCES ............................................................................................. 168 BIOGRAPHICAL SKETCH .......................................................................................... 171 PAGE 8 8 LIST OF TABLE S Table page 1 1 Recommended Tests for End Bearing and Unit Tip Shear ................................. 26 3 1 Centrifuge Scaling Relationships (Chandras ekaran, 200 1 ) ................................ 65 3 2 Proportion of Component for Synthetic Limestone ............................................. 65 3 3 Ratio of Em (with perlite) / Ei (no perlite) ............................................................ 65 3 4 Amount of Component for Mixing Concrete with Silica Sand ............................. 65 3 5 Result of Unconfined Compression Strength Test .............................................. 65 3 6 Amount of Component for Mixing Concrete with Granite Sand .......................... 66 3 7 Result of Unconfined Compression Strength Test .............................................. 66 3 8 Estimation of Em/Ei Based on RQD (Load Transfer for Drilled Shafts in Intermediate Geomaterials, 1996) ...................................................................... 66 3 9 Comparison of Axial Tip Stresses ....................................................................... 66 4 1 Comparison of FEA results ( CV ) and analytical solution ( CVfrom Eq.14) at H/D=2 ................................................................................................ 108 PAGE 9 9 LIST OF FIGURES Figure page 2 1 A scatter plot and variance and covariance as a function of lag distance ........... 39 2 2 Examples of randomly generated E for various correlation lengths .................... 40 3 1 New Cylindrical Sample Container ..................................................................... 67 3 2 Instrumentation for Measuring Lateral & Vertical Forces and Deformation ........ 67 3 3 University of Florida 12.5 G ton Centrifuge ......................................................... 68 3 4 Slotted Steel Pipe (Axial Reinforcement) & Strain Gages .................................. 68 3 5 Mixture of Synthetic Rock for Centrifuge Test .................................................... 69 3 6 Drilled Hole in Synthetic Limestone .................................................................... 69 3 7 Cast In Model Shaft in Synthetic Limestone ....................................................... 70 3 8 Slip Rings and Rotary Union ............................................................................... 71 3 9 Schematic sketch of a centrif uge test setup ....................................................... 71 3 10 Grain Size Distribution for Limestone Aggregate ................................................ 72 3 11 Strength Tests for Concrete and Rock ............................................................... 72 3 12 Synthetic Limestone with 5 mm Perlite Spheres Occupying 30 % Volume ........ 73 3 13 Stress Strain Response Voided and nonVoided Limestone .............................. 73 3 14 Mohr Coulomb Envelop from Qu,Qt and Triaxial Tests on 10tsf Rock ............... 74 3 15 Strain Gage Layout for L/D = 1 ........................................................................... 74 3 16 Compressive Forces (Q) along Depth ................................................................ 75 3 17 Mobilized Skin Frictions vs Axial Displacement ................................................. 75 3 18 Measured Skin Friction Comparing Lila (2003) and McVay (1992) .................... 76 3 19 Measured Skin Friction Comparing Lila (2003) and McVay (1992) .................... 76 3 20 ONeil Em/Ei vs. RQD and UF Em/Ei vs. Recovery .............................................. 77 3 21 Two Layer System and Equivalent Modulus (Ueshita and Meyerhof, 1967) ...... 77 PAGE 10 10 3 22 Layered Moduli beneath a Drilled Shaft .............................................................. 78 3 23 Output Voltage from Gages vs. Time with Loading; Axial Loading First and Combined Loading Following ............................................................................. 78 3 24 Total Axial and Bending Strains along the Shaft Cross Section ......................... 79 3 25 Moment Curvature Relationship for 9ft Diameter from a Pair of Strain Gages above Rock Surface. .......................................................................................... 79 3 26a M oment Distribution along a 9 ft Diameter, 50% Styrofoam at 1D below Tip of Shaft in 10 tsf Limestone from 2 tests ................................................................ 80 3 26b M oment Distribution along a 9 ft Diameter, 30% Styrofoam at 1D below Tip of Shaft in 45 tsf Limestone from 2 tests ................................................................ 80 3 26c Moment Distribution along 9 ft Shaft without Styrofoam below Tip in 35 tsf Rock from 2 tests ................................................................................................ 81 3 27 Increase in Tip Shear with Increasing Lateral Load, L/D = 1 .............................. 81 3 28 Increasing Tip Shear with Increasing Rock Strength .......................................... 82 3 29 Shaft Shear at Failure with and without an Underlying Soft Layer ...................... 82 3 30 Assessing Rotation from the Moment Curvature Relationship ( a) Moment versus Depth ( b) Moment Curvature Relationship ............................................ 83 3 31 Typical Lateral Deflections (y) vs. Depth ............................................................ 83 3 32a Tip Resistance ( Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using ONeil(right) for 9 ft / 9 ft 10 tsf Rock ................................................................................................................... 84 3 32b Tip Resistance ( Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using ONeil(right) for 9 ft / 27 ft 10 tsf Rock ................................................................................................................... 84 3 32c Tip Resistance ( Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using ONeil(right) for 9 ft / 9 ft 3 0 tsf Rock ................................................................................................................... 84 3 32d Tip Resistance ( Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using ONeil(right) for 9 ft / 27 ft 3 0 tsf Rock ................................................................................................................... 85 3 3 3a Tip Resistance (Model vs.ONeills Equation with Harmonic Average Method (left) and Geometric Average Method (right) for 10 tsf Rock with 50% Recovery at 1D ................................................................................................... 85 PAGE 11 11 3 3 3b Tip Resistance (Model v s.ONeills Equation with Harmonic Average Method (left) and Geometric Average Method (right) for 30 tsf Rock with 50% Recovery at 1D ................................................................................................... 85 3 34 Tip Shear Stress vs. Tip Lateral Displacement ................................................... 86 3 35 Tip Shear Stress vs. Tip Lateral Displacement ................................................... 86 3 36 Shear in Shafts vs. Depth with and without Underlying Softer Layer .................. 87 3 37 Moment in Shafts vs. Depth with and without Underlying Softer Layer .............. 87 3 38 Tip Shear Minus Rock Strength Divided by Normal Tip Stress vs. Normalized Lateral Displacement .......................................................................................... 88 3 39 Tip Shear minus Rock Strength Divided by Normal Tip Stress vs. Normalized Lateral Displacement .......................................................................................... 88 3 40 Mohr Coulomb Strength of Florida limestone ..................................................... 89 3 41 Tip Shear Model (a) Tip Shear Model Scheme (b) Tip Shear Model .................. 89 3 42 Tip Moment Transfer as Function of Tip Rotation ............................................... 90 3 43 Tip Moment Transfer as Function of Tip Rotation ............................................... 90 3 44 Calculated vs. Measured Rotation using Bell (1991) Trend line is in red ......... 91 3 45 Calculated vs. Measured Rotation using Bell (1991) with Trend line in red ........ 91 3 46 Stiffness Coefficient, K, from Doherty and Deeks ( 2006) ................................... 92 3 47 Calculated vs. Measured Rotation in 10 tsf Rock using Doherty and Deeks (2006) with Trend Line in Red ............................................................................ 92 3 48 Calculated vs. Measured Rotation in 30 tsf Rock using Doherty and Deeks (2006) with Trend Line in red .............................................................................. 93 3 49 Recommended Tip Rotations vs. Moment Model from Bell (1991) (a) Tip Rotation Model Scheme (b) Tip Rotation Model ................................................. 93 4 1 Dimensions and boundary conditions of FEA Model ........................................ 109 4 2 Nodal penetration phenomenon ....................................................................... 109 4 3 Model of the rock shaft side interface boundary ............................................... 110 4 4 Centrifuge test and FEA results ........................................................................ 111 PAGE 12 12 4 5 Comparison of the tip stiffness among various CVE (av = 5 ft) .......................... 112 4 6 Simulation results of the cases among various av (CVE =0.6 and L/D =3) ..... 113 4 7 Simulation results of the cases among various CVE (av=15ft and L/D=3) ....... 114 4 8 Comparison of tip stiffness (CVE=0.6) .............................................................. 115 5 1 s 22 for a/D = [0, 20] and L/D = [0,10]. Thick Dashed Contour Lines Re (L/a) for D = 0. Thin Dotted Lines are Parabolas, Along Which A = const. (Klammler, 2010) ........... 129 5 2 LRFD resistance factors, ,as a Function of Reliability I R ( Klammle r 2 010) .............................................................................................. 130 5 3 Secant vs. Tangent Youngs Modulus on 17th Street Bridge Data f rom LTSO4 131 5 4 Tangent Mass Modulus of 17th S treet Bridge (118 Values) .............................. 132 5 .5 Secant Mass Modulus of 17th Street Bridge (118 values) ................................. 132 5 6 Harmonic Mean Modulus, Eh, Assu ming Correlation Length, a = 5 ft from 17th Street Bridge Data ............................................................................................ 133 5 7 Harmonic Mean Modulus, Eh, Assuming Correlation Length, a = 10 ft from 17th Street Bridge Data ..................................................................................... 133 5 8 Harmonic Mean Modulus, Eh, Assuming Correlation Length, a = 15 ft from 17th Street Bridge Data ..................................................................................... 134 5 9 Histogram/PDF of Contact Stress, qb, Using Eh with Correlation Length, a = 5ft from 17th Street Bridge Data ........................................................................ 134 5 10 Histogram/PDF of Contact Stress, qb, Using Eh with Correlation Length, a = 10 ft from 17th Street Bridge Data ..................................................................... 135 5 11 Histogram/PDF of Contact Stress, qb, Using Eh with Correlation Length, a = 15 ft from 17th Street Bridge Data ..................................................................... 135 5 1 2 Histogra m/PDF of Displacement Wt, Using Eh with Correlation Length, a = 5 ft from 17th Street Bridge Data .......................................................................... 1 36 5 1 3 Histogram/PDF of Displacement Wt, Using Eh with Correlation Length, a = 10 ft from 17th Street Bridge Data ..................................................................... 136 5 1 4 Histogram/PDF of Displacement Wt, Using Eh with Correlation Length, a = 1 5 ft from 17th Street Bridge Data ..................................................................... 137 5 15 Secant Mass Modulus of Fuller Warren Bridge (96 values) .............................. 138 PAGE 13 13 5 16 Tangent Mass Modulus of Fuller Warren Bridge (96 values) ............................ 138 5 1 7 Geometric Mean Modulus, Eh, Assuming a Correlation Length, a = 5 ft from Fuller Warren Bridge Data ................................................................................ 139 5 1 8 Geometric Mean Modulus, Eh, Assuming a Correlation Length, a = 10 ft from Fuller Warren Bridge Data ................................................................................ 139 5 19 Geometric Mean Modulus, Eh, Assuming a Correlation Length, a = 15 ft from Fuller Warren Bridge Data ................................................................................ 140 5 20 Histogram/PDF of Contact Stress, qb, Using Eh with a Correlation Length, a = 5 ft from Fuller Warren Bridge Data .................................................................. 141 5 21 Histogram/PDF of Contact Stress, qb, Us ing Eh with a Correlation Length, a = 10 ft from Fuller Warren Bridge Data ................................................................ 141 5 22 Histogram/PDF of Contact Stress, qb, Using Eh with a Correlation Length, a = 15 ft from Fuller Warren Bridge Data ................................................................ 142 5 23 Histogram/PDF of Displacement Wt, Using Eh with Correlation Length, a = 5 ft from Fuller Warren Bridge Data ..................................................................... 142 5 24 Histogram/PDF of Displacement Wt, Using Eh with Correlation Length, a = 10 ft from Fuller Warren Bridge Data ................................................................ 143 5 25 Histogram/PDF of Displacement Wt, Using Eh with Correlation Length, a = 1 5 ft from Fuller Warren Bridge Data ................................................................ 143 A 1 Result from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Tes t No. 2 ................................................................................................................. 149 A 2 Result from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 ................................................................................................................. 149 A 3 Result from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ..................................................................................................... 150 A 4 Result from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ..................................................................................................... 150 A 5 Result from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ..................................................................................................... 150 A 6 Result from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ..................................................................................................... 151 PAGE 14 14 A 7 Result from 9 ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ..................................................................................................... 151 A 8 Result from 9 ft Diameter 27 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ..................................................................................................... 151 A 9 Res ult from 6 ft Diameter 18 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ..................................................................................................... 152 A 10 Result from 6 ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ..................................................................................................... 152 B 1 Output Voltage from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 ...................................................................................................... 153 B 2 Output Voltage from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 ...................................................................................................... 153 B 3 Output Voltage from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ................................................................................... 154 B 4 Output Voltage from 9 ft Diameter 9 ft Embedded Length in 30 tsf Roc k a) Test No. 1 b) Test No. 2 ................................................................................... 154 B 5 Output Voltage from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ................................................................................... 154 B 6 Output Voltage from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ................................................................................... 155 B 7 Output Voltage from9 ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ................................................................................... 155 B 8 Output Voltage from9 ft Diameter 27 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ................................................................................... 155 B 9 Output Voltage from6 ft Diameter 18 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ................................................................................... 156 B 10 Output Voltage from6 ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ................................................................................... 156 C 1 Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 .......................................... 157 PAGE 15 15 C 2 Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 .......................................... 157 C 3 Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test N o. 2 ....................... 158 C 4 Moment Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ....................... 158 C 5 Moment Curve with Different Top Lateral Displacement from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ....................... 159 D 1 Shear Curve with Di fferent Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 .......................................... 162 D 2 Shear C urve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 .......................................... 162 D 3 Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 .......................... 163 D 4 Shear Curve with Different Top Lateral Displ acement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 .......................... 163 D 5 Shear Curve with Different Top Lateral Displacement from 6 ft Diameter 6 ft Embedded Length i n 10 tsf Rock a) Test No. 1 b) Test No. 2 .......................... 164 D 6 Shear Curve with Different Top Lateral Displacement from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 .......................... 164 D 7 Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 .......................... 165 D 8 Shear Curve with Different Top Lateral Displacement from 9 ft Diameter 27 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 .......................... 165 D 9 Shear Curve with Different Top Lateral Displacement from 6 ft Diameter 18 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 .......................... 166 D 10 Shear Curve with Different Top Lateral Displacement from 6 ft Diameter 18 ft E mbedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 .......................... 166 E 1 The relation J and Kv: a( =0.2) and b( =0.499) ............................................. 167 PAGE 16 16 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EVALUATION OF TIP BEHAVIOR OF DRILLE D SHAFT IN FLORIDA LIME STONE By Jeongsoo Ko Aug ust 2010 Chair: M ichael. C McVay Major: Civil Engineering This study investigated the tip behavior of large diameter drilled shafts embedded short distances in Florida Limestone (i.e. L/D = 1 & 3). This dissertation focused on the modeling of axial, shear, and moment responses for tip s of shafts subject to combined axial and lateral loading in homogeneous and heterogeneous limestone. For this study, laboratory (centrifuge) tests F inite E lement A nalysis (FEA) and field investig ation (from two sites : 17th Street and Fuller Warren Bridges) were conducted The study from the centrifuge test found that the tip stiffness could be estimated within serviceability condition using the linear stiffness relation proposed by Misra and Doher ty. Furthermore, ONeils method of assessing tip resistance versus displacement was accurate if the harmonic mass modulus of the heterogeneous rock (two layer system) wa s assessed within two diameters (D) below the tip of the shaft In the case of tip she ar, it was found that a bilinear elastic plastic model with failure assessed using Mohr Coulomb provided good results. For tip rotation, the model proposed by Bell (1991) gave very reasonable results even though it requires assessment of tip shear and lat eral tip displacement. PAGE 17 17 T o evaluate the effect of spatial variability of rock material properties on the tip resistance of drilled shaft s subjected to axial load, field investigation and two dimensional FE A were conducted. In addition, with regard to the v ertical spatial variability of the rock ( using harmonic mass modulus with the influence zone at 2D below the tip of a shaft) a relationship was derived between deterministic and probabilistic endbearing resistances in homogeneous and heterogeneous rock f ormations T he FEA was conducted at 1 in tip displacement defined as the serviceability condition. An important finding from the field investigation was the variability of the LRFD resistance factors ( ) from the spatial variability of the rock at the two sites. Using standard g eostatistics measures (e.g., v ariogram, covariance, etc.), the v ariance of the harmonic m ass m odulus with the influence zone at 2D below the tip of a shaft was assessed, along with tip resistance and its associated variability. LR FD resistance factors based on FOSM were calculated with reliability values of 2.5 and 3.0. PAGE 18 18 CHAPTER 1 INTRODUCTION 1.1 Problem Statement Recently drilled shaft s have become the deep foundation of choice for bridges and tall structures as opposed to sma ll diameter concrete shafts or driven piles. A major economic benefit occurs by replacing a large number of piles in a group with a single or several drilled shafts with diameter reduced footprint Also, drilled shafts offer reduc ed construction noise and vibration versus driven piles. Shaft diameters have steadily increased due to the development of larger construction equipment. Because of their large diameters and concomitant large moments of inertia, they are able to resist lateral loads induced by hur ricanes and ship impacts. In addition, installation involves a minimal foundation footprint, addressing right of way issues. The size of drilled shafts has a significant effect on their load resistance behavior. For instance, under lateral loading, large diameter shafts develop a significant moment or couple from the shear transfer on the side of the shaft. McVay et al (2004) found t he side shear influence d the back calculated P Y curves representing soil structure horizontal interaction, i.e., horizontal resistance per length v ersus horizontal displacement Furthermore, McVay et al (2004) found that the error in estimating the lateral resistance may be as high as 26% for a 12 ft diameter shaft. In addition to side shear, the tip behavior of large diameter shafts (i.e., 8 ft or greater) is greatly influenced by embedment length. For instance, at typical embedment depths of 20 ft or less, it wa s expected that these shafts will exhibit lateral tip displacement and rotation, especially for L/D < 5. For such rotations significant moments and shear s at the shaft tip w ould PAGE 19 19 develop in stiff/strong material, i.e. rock These tip shears and moments w ould correspondingly change the shafts internal shears and moments. The current design of drilled shaft s d oes n o t co nsider the tip shear and tip moment s relationship to tip rotation because practice developed from small diameter shafts and large L/D embedments for which tip shear and moments are negligible. However, in the case of large diameter shafts or short L/D r atios, tip shears and moments must be accounted for in shaft designs (i.e., steel reinforcement requirements) Furthermore, due to the loss of foundation redundancy and the trend toward larger single shaft construction, field coring of rock near as built non redundant shafts wa s now required (FDOT Structures Bulletin, 2005). In addition, end bearing of large diameter shafts wa s also strongly influenced by spatial variability within the tip zone (in 2D ) because tip resistance is a funct ion of tip compressi bility. I n current designs for tip resistance of drilled shafts in intermediate geom a t e ri al s by O Neill (199 7) the end bearing equation, including elastic modulus, wa s developed for simple homogeneousrock. In practice, however, there is no uniform rock Variable heterogeneous rock, in the vicinity of the shaft tip was typically recovered in the field and laboratory strength tests (unconfined compression, and split tension) wer e performed to assess both strength and Youngs Modulus G enerally, all the samples over the whole site have been averaged arithmetically. This arithmetic average may not properly characterize tip resistance of the shaft in heterogeneous rock because it is usually to high due to the general type of distribution, e.g. log normal Cons equently, tip resistance from other averaging approaches, such as harmonic or geometric may be more appropriate PAGE 20 20 For current analysis/design (e.g. FB MultiPier) there is great need for a 1D simple tip stiffness model for a large diameter deep foundation, i .e. drilled shaft) embedded short distances (L/D < 5) which consider ed spatial variability of the underlying rock Furthermore, associated variability of the tip stiffness model should be quantified to evaluate the reduced shaft reliability due to layer variability. This dissertation focuses on the effect of spatial variability on the tip stiffness model with specific emphasis on the design aspects of the end bearing of a drilled shaft. 1.2 Objectives Objectives of this research include the following: Pred ict the tip shear and moment associated with tip rotation, which should be accounted for in shaft design. Develop a 1D simple tip model as spring model for easy assessment by engineers in designing shafts while considering spatial variability and the assoc iated correlation in serviceability limit, i.e., 1 inch tip displacement. Estimate the variability of tip stiffness due to spatial variability in used in assessing LRFD ( Load and Resistance Factor Design) resistance factor ( ). 1.3 Scope 1. 3 .1 Centrifuge In order to develop a tip shear and tip moment relat ionship with includ ing tip rotation and end bearing for large diameter short shafts (L/D < 3 ), two different rock strengths (10 tsf and 30 tsf) represent ative of typical Flo rida limestone would be used. To insure repetitive results, synthetic limestone with same strength and compressibility characteristics would be used the following experimental conditions were varied in order to develop the analytical end bearing and tip s hear for design. PAGE 21 21 1. 3 .1. 1. Loading (Axial and Lateral) Initially axial load tests were performed to assess existing lab/fi eld unit tip resistance models as a function of tip displacement (e.g., Osterberg results). These tests were then used to identify the peak axial force for a given rock strength, Youngs m odul us and L/D embedment. Next, the maximum lateral capacity of the test shaft was established using FB Pier. In strong rock, t he lateral capacity was controlled by its moment capacity for the longer shafts ( L/D = 3); in shorter shafts (L/D = 1) rock strength was the controlling factor The shafts axial tip displacement end bearing, tip translation, and shear were recorded. 1. 3 .1.2 Shaft diameter The effects of short shaft tip rotation was more pr onounced for larger diameter shafts, e.g., D > 8 ft. Consequently two different shaft diameters were tested to quantify its influence. Diameters of 6 and 9 ft were studied, since they are representative of medium and large diameter drilled s hafts in Flor ida 1. 3 .1.3 Length to diameter (L/D) Ratio For each shaft diameter, two different L/D ratios were tested to investigate the effect of embedment depth on the distribution of end bearing and tip shear. The embedment depth strongly influences the magnitude of displace ment and rotation of the shaft, as well as its tip normal pressure and shear distribution. Long shafts (i.e., L/D > 5) are generally controlled by the shafts moment capacity, whereas the soil/rock resistance controls shorter shaft response. L/D ratios between 1 and 3 were tested (see Table 1.1) since they represent typical Florida embedment depths. PAGE 22 22 1. 3 .1.4 Influence of Rock Voids or Limestone Recoveries Of particular interest was the influence of karst channels or rock voids usually express ed as Recovery Ratio: ratio length of core recovered to theoretical depth of rock cored < 100% on end bearing. To account for the e ffect of voids on a shafts ultimate tip resistance and tip sh ear, perlite (Styrofoam) spheres 30% to 50% by volume, were mi xed with the sy nthetic limestone. Since voids are more of a concern for short shafts, the studies focused on shafts with a L/D ratio of 1 and were tested in weak and strong l imestone, as shown in Table 1.1. In addition, since shafts may be embedded in vari ous layers with different strength parameters, the extent and quality of rock below the shaft tip becomes extremely important. Current design practice assumed uniform rock properties to a depth below the tip of three diameters, which may or may not be cons ervative. To investigate this possibility, reduced modulus rock was located one and three diameters below the shaft tip (see Table 1.1), and was used to quantify its influence on end bearing capacities Table 1.1 summarizes the 32 tests performed to quanti fy end bearing, tip shear, and tip moment for large diameter short shafts. Note, while there were 16 different scenarios (i.e., rock strength, embedment, voids etc.), each test was repeated twice to ensure accuracy. 1. 3 2 Laboratory Test To obtain Y oung s modulus, cohesion and friction angle of the synthetic limestone laboratory tests were conducted. Unconfined compression and split tension tests were conducted with the same material used in the centrifuge test to obtain elastic modulus and cohesion. I n addition, triaxial tests were performed to determine the rocks Mohr Coulomb friction angle with increasing confinement load and cohesion. T he parameters PAGE 23 23 obtained from the laboratory test s was used to characterize the limestone and as input for FE (Finite E lement) analysis. 1. 3 3 LRFD Assessment with FEA Analysis To evaluate the effect of spatial variability of rock material properties on tip resistance finite element analysis (FEA) wa s conducted A mathematical and computational model for simulating the multidimensional soilstructure interaction was developed and subsequently used for studying spatial variability effects on tip resistance of the drilled shaft. Using an axisymmetric boundary and loading conditions for the centrifuge testing, finite element models were developed using the ADINA finite element analysis program with a Mohr Coulomb constitutive model to simulate the material behavior of the limestone. The input data for Florida limestone in FEA were obtained from unconfined compression spilt te nsion and triaxial tests The experimental results obtained from the centrifuge tests were used to calibrate the FEA material model Parametric sensitivity studies were then carried out to quantify the tip resistance of the drilled shaft under a service l oading condition. Of interest was the expected variability of shaft tip resistance v ersus the variability of the elastic modulus of the lime stone To investigate the effect of spatial variability on the tip resistance, random of elastic modul i was generat ed satisfying site specific mean, coefficient of variation ( CV ), and correlation length ( a). Due to the soil formation process, soil or rock variability was greater vertically than horizontally. Since full rock borings are generally far apart (horizontal c orrelation length = 0) and shaft diameter is in the 4 ft to 10 ft range, it was assumed that the rock moduli are randomly distributed variables in the vertical direction but of constant value in horizontal direction. Of interest was the 1D megascopic tip PAGE 24 24 stiffness models for such characterization. To develop tip stiffness, 500 simulations were performed with respect to three independent vertically distributed variables: CV, a and L/D. Using the FEA simulation results, g eostatistical analysis wa s performed to derive mean tip stiffness and associated variability in terms of CV of displacement which was a function of the CV of the elastic modulus and the spatial correlation length ( i.e., covariance of the rock ) The analysis also reveal ed a linear relationship between mean tip resistances and mean tip displacement called a 1D simple tip stiffness model, which was based on the harmonic or geometric averages of the elastic modulus of the rock with respect to various depths ( i.e., averaging lengths ) The nomin al resistance and the resistance factors of LRFD were then established from the CV of the tip resistance predicted by FEA 1. 3 4 LRFD Assessment with Field Verification Current assessment of side and tip resistance of drilled shaft s is performed using rock core samples recovered from the site, which are tested in the laboratory ( e.g., unco nfined compression and split tension tests) to obtain strength parameters Generally, all the property measurements resulting from laboratory testing were averaged over the entire site, with an arithmetic mean assumed. Values one standard deviation above and below are discarded (FDOT Soil and Foundation Handbook, 2009) Unfortunately, current designs fail ed to consider spatial variability and associated correlation (i.e., covariance), when designing for end bearing. A probabilistic approach which can handle spatial variability and associated correlations at a specific pier or over the entire site, may be more appropriate. For instance, the designer should have the option of developing specific LRFD factors for a particular shaft/pier based on adjacent data (e.g., summary statistics, covariance, etc.) or by using the entire sites data for an PAGE 25 25 assessment of general LRFD factors. However this is offset by the higher cost of additional field testing. Examples of predicted tip response, as well as LRFD factors using spatial variability and correlation are presented for two FDOT (Florida Department of Transportation) sites: 17th Street and Fuller Warren Bridges. 1. 4 Overview of Dis sertation The dissertation includes the following C hapters: Chapter 2 is a literature review, and examines current shaft design for the tip resistance and relationship between moment and rotation, geostatistics theory and past research into geostatistics t heory and geotechnical engineering, presented in the same order as the objectives. Chapter 3 provides the centrifuge test setup and results. In this chapter, the tip moment and tip shear are related to rotation, which can be then be applied to drilled shaf t design. Chapter 4 presents the FEA analysis t o evaluate the effect of spatial variability of rock material properties on the tip resistance of drilled shaft. Using the results of this chapter, 1D tip spring and associated variability of tip stiffness can be evaluated based on spatial variability and correlation. Chapter 5 presents the field data assessment for the LRFD factor. Field data were obtained from two FDOT sites: 17th Street and Fuller Warren Bridges. PAGE 26 26 Table 1 1 Recommended Tests for End Bearing and Unit Tip Shear Rock Strength Description of Limestone Diameter Location & Recovery of 2 nd Layer system L/D 10tst Homogeneous System 9ft N/A 1 N/A 3 6ft N/A 1 N/A 3 Two Layered System (Homogeneous Layer above the % Recovery Layer) 9ft 1D Below from Tip of Shaft 1 1 3D Below from Tip of Shaft 1 1 30tsf Homogeneous System 9ft N/A 1 N/A 3 6ft N/A 1 N/A 3 Two Layered System (Homogeneous Layer above the % Recovery Layer) 9ft 1D Below from Tip of Shaft 1 1 3D Below from Tip of Shaft 1 1 PAGE 27 27 CHAPTER 2 LITERATURE REVIEW The design of reinforced concrete drilled shaft s subject to both axial and lateral loads, should include the assessment of moment, shear and rotational force. T he distributions of moment, shear and rotation along the drilled shaft are major factors in design, as defined in ACI (America Concrete Institute) code. T he distributions of these properties at tip of drilled shaft are not considered in current designs but are significant, especial ly for short L/D. In addition, f or the last several decades, geotechnical design has been moving to ward the Load and Resistance Factor Design (LRFD) methodologies as identified by American Association of State Highway and Transportation Officials (AASHTO) The L RFD factor ( is based on known variability of the axially applied load ( i.e., dead and live load s) and material properties used for the design at multiple limit state s (service, impact, etc.) F urthermore, using the LRFD factor a failure prob ability can be found similarly for each limit state factor. L imit ed knowledge of material properties wa s indicated as problem in obtained LRFD factor as function of coefficient of variation of resistance. W ith heterogeneous soil particularly the LRFD factor can t be easily obtained due to the difficulty of characterizing its properties. T his chapter review s past stud ies about tip resistance and axial load, as well as the relationship between moment and rotation. It also review s past stu dies that char acterize heterogeneous soil and rock properties, such as spatial variability, in terms of geostatistics, which lead to a better understanding of spatial variability. Finally, it review s past studies in order to derive the resistance with spatial variabilit y of soil properties. PAGE 28 28 2. 1 Designs for Drilled Shaft Subjected in Axial Load Hassan and O Neill (1997) studied predicting loaddisplacement behavior of axially loaded drilled shafts in intermediate geomaterials ( i.e., rocks). D rilled shafts are often locate d in intermediate geomaterials and derive resistance to axial load from a combination of side friction and end bearing. Hassan and O Neill (1997) developed simple design equations with loadsettlement behavior of sockets considering the parameters which significantly affecting the socket s behavior. T o verif y Hassan and O Neill s study (1997), field load test result s were performed. T he equations for tip resistance had the following parameters : L/D ratio, elastic modulus of drilled shaft, averaging rock modulus and embedment length. F urthermore, Hassan and O Neill (1997) suggested a service limit resistance based on settlements: 1 inch of tip displacement Misra (2006) suggested tip soil stiffness, i.e., the relationship between tip force and tip displacem ent. The tip soil stiffness was related to shaft diameter and elastic properties of tip soil, Youngs Moduls, E, and Poissons ratio, Their tip stiffness was based on an embedded rigid punch bearing upon elastic half space. 2. 2 Relation Moment and Rotat ion Doherty s (2003) presented a study of the tip moments relationship with rotation. The rotation at the tip can be calculated from the bending strains obtained from centrifuge testing. Doherty (2003) studied the load and displacement of rigid circular footing in nonhomogeneous elastic half spaces using a semi analytical technique. T he estimated load and displacement relation was presented in dimensionless elastic stiffness coefficients. A pplied loadings in footing were vertical, horizontal, a moment a nd a rotation. Four PAGE 29 29 possible footing geometries were considered and the variation of soil shear modulus with depth was an important factor as identified by soil type, sand and clay. C ircular footing at the bottom of an open trench, fully embedded footing, fully embedded footing with sidewall and skirted foundation were modeled The total displacements, vertical, horizontal, moment and tensional were presented at middle of the tip side of the footing. B ell (1991) carried out a series of three dimensional finite element analyses to examine the effect of embedment (cases 1, 2 and 3) subjected to vertical, horizontal and moment loads. The study showed that between horizontal and moment loads a crosscoupling stiffness coefficient was needed. The study also obtained tensional stiffness coefficient. The author presented monographs or figures representing stiffness coefficient as function of Poisson s ratio relevant footing geometry and embedment ratio. Doherty (2006) addressed stiffness of flexible circular footing embedded in an elastic half space. B ending stiffness of the footing dictated the response for vertical and moment load cases. T his study also presented graphical methods for estimating dimensionless elastic stiffness coefficients. Selvadurai (1979) found that the response of flexible footing in elastic medium subjected to vertical load as a function of rigidity parameter (J), which is perfectly flexible when J=0 and perfectly rigid when J=infinity. R igidity parameter is a function of Poisson s rati o of soil and footing, thickness of the footing, radius of the footing, shear modulus of the soil and elastic modulus of the footing. The c ross coupling stiffness coefficient between horizontal and moment was insignificant. PAGE 30 30 2. 3 Geostatistics Background Ge ostatistics originated during the 1950s in the mining industry to estimate ore yield on mine excavated volumes. Since the 1980s, applications have expanded to include various engineering and science fields such as petroleum engineering (Deutsch, 1998), agr iculture, meteorology, and hy drology (Goovaerts, 1997). In g eotechnical engineering, geostatistics is a means of quantifying the variability of material properties spatially or from point to point. T hree basic components of geo statistic s are : semi variog ram analysis (generally called variogram analysis), kriging and stochastic simulation. One of the basic tools in geostatistics is the variogram which quantifies the spatial correlation (or variance of local differences) of geotechnical properties. The spatial semivariance ( () h ) is expressed between two points by lag (separation) distance ( h ). T o measure the spatial variability between two different variable sets with semivariance, two terminologies should be introduced : covariance ( Cov ) and correlation coefficient ( ) Covariance is the measurement how two variables relate together describing the variance of two variable sets. C orrelation coefficient is defined as dividing the covari ance of the two variables by the product of their standard deviation. Correlation coefficient ranges from 1 (increasing linear relationship) to 1 (decreasing linear relationship). If the variables are independent the correlation coefficient is zero; thi s mean s covariance is equal to zero. A lso the scatter plot can be introduced to describ e spatial variability. T he scatter plot can be expressed with measured data pairs from the same variables and each data pair is separated by some distance (i.e., lag ) f rom each other in X direction, Y direction or XY direction. I f the scatter plot is close to the 45 PAGE 31 31 degree line ( =1) or 45 degree line ( = 1), it indicates a high correlation coefficient. S emi variance is half variance of the increments between of two sets of variables. T he variogram is described as relationship between semi variance and lag. Establishing the spatial correlation structure of a site having spatial variability in rock properties would requ ire an extensive amount of sub surface exploration. T his may not be feasible due to high cost. T he variogram is a common method to develop the correlation length, which is an indicator of how much two values with distance are related. The longest distance shown between which two variables are related is correlation length. T he variogram is a statistic al tool that appraises the average decrease in similarity between two random variables as the distance between the variables increases. I t describes spatial c ontinuity as a function of distance and direction. T he terminologies used to describe the variogram are range, still, and nugget. W ith increasing lag, semi variance generally increases in the variogram. H owever, at some point, semi variance stops inc reasing and there is a plateau of semi variance with increasing lag. T he distance at this point from zero lag is called the range. T he semi variance value at the plateau of variogram is known as the still W hen lag is zero, the semi variance used to be zero. I f no n zero, the value of semi variance is known as the nugget. To help explain the concept of a semivariogram and covariance function, Figures 2 .1 a and 2.1 b show scatter plots that illustrate the variation of local values of E, which were sampled at a site in Florida at 1 ft and 5 ft vertical separation distances, respectively. It is evident that the scatter of the data cloud about the 45 line is less for PAGE 32 32 the smaller separation distance of 1 ft, which is reflected by a smaller value of the semiva riogram. Evaluating the semivariogram for a series of lag distances allows for construction of the variogram (the prefix semi is often omitted) drawn by the continuous line in Figure 2.1 c. In many cases spatial correlation only persists over a range lim ited by the correlation length (denoted as av in the sequel when considered in the vertical direction) until the variogram reaches a plateau ( sill ) at a value equal to the variance ( 2) of the random function. As an alternative to the variogram, a spatial covariance function C(h) can be also used as a measure of spatial likeness (drawn by the dashed line in Figure 1c). The kriging technique is to linearly interpolate the value ) (0x Z of a random field ) ( x Z at an unobserved location 0x from observation n i x Z zi i,....,1 ), ( of the random field at nearby locations, nx x ,...,1 weighted according to spatial variogram or covariance values. This implies the kriging is possible with knowledge of the variogram or covariance. Stochastic simulation is a technique for generating multiple equal probable realizations. The most widely used techniques of stochastic simulation are Sequential Gaussian Simulation and sequential indicator simulation. M ultiple realizations to fit data set s generated by variogram, kriging and stochastic can be used in simulations to get the capacity variability of a foundation. 2. 4 Implication of Geostatist ics to Geotechnical Field Problems A number of commercial geostatistics software packages (e.g. GSLIB) are available for generating large numbers of random field realizations (stochastic simulation) of properties u sing either or C(h) in multidimensions. These two PAGE 33 33 different realizations may be considered as two equally likely scenarios for an unknown situation in the real world. In Figure 2, example outcomes of 1D realizations at a 1 ft resolution are illustrated for two v ertical correlation lengths of 5 ft ( shown in Figures 2.2a and 2.2b) and 15 ft ( shown in Figures 2.2c and 2.2d) Brighter areas indicate d lower values of Youngs Modulus, E whereas darker areas indicate d higher values of E. Visual inspection confirms the shorter scale variability of Figures 2.2a and 2.2b compared to Figures 2.2c and 2.2d, which wa s due to the contrast in two correlation lengths. The use of random input properties in Finite Element Analysis ( FEA ) of foundations was first reported by Fenton and Griffiths (2002 & 2005). Section 2.4.1 addresses this in further detail. Using a spatially correlated and log normally distributed E, the authors numerically predict ed settlements of the shallow foundation in a two dimensions. Results of their FEA simu lations reveal that settlements of the shallow foundation can be estimated using an effective modulus calculated by the geometric average of E over the entire depth of their FEA model. They also model ed the uncertainty (variance) of settlement by applying a variance reduction factor to a volumeaveraging scheme in a general statistical context, ( originally discussed by Vanmarcke (1984 ) ) Sections 2.4.2 and 2.4.3 more closely address averaging and variance reduction factor of the mean (expected) settlements In early numerical studies for design of deep foundations (Phoon, 2000), uncertainty wa s modeled by dividing the soil into multiple homogeneous layers of random yet uniform properties rather than considering more realistic spatial variations of properties Recently, Fenton and Griffiths (2007) investigated the settlement and PAGE 34 34 ultimate axial resistance of a shaft using onedimensional FEA in which spring stiffness wa s computed using a geometric average of a spatially correlated random variable for E. The onl y other study of the application of geostatistics to the analysis of deep foundations known to the authors is by Klammler et al. (2010). They report ed the effect of spatial variability of ground properties on the ultimate skin friction of deep foundations. In their work, an analytical variance reduction factor wa s proposed as a function of a deep foundation dimensions and spatial covariance function. The author is not aware of any other investigation to determine end bearing resistance and uncertainty of deep foundations based on a geostatistical descript ion of the soil/rock properties 2.4.1 Application of S tochastic S imulation to G eotechnique E ngineering Fenton and Griffiths (2002) estimated the uncertainty of shallow foundations based on settlement within the serviceability limit using two dimensional simulation. I n case of single footing, the probability density function of the total settlement of the footing was considered as function of footing width for various statistics of the underlying soil. Fenton and Griffiths (2002) considered = isotropic soil with a Youngs Modulus, E, as a random variable with a mean variance, and a correlation structure that wa s the same in the horizontal and vertical directions even though soils may show a stronger correlation in horizontal direction due to layer ing S ince elastic properties, E are important elements of settlement, the calculation of immediate and consolidation should include it W ith layered soil, the estimation of probabilities of settlement wa s associated with probabilities of E, chosen as the only spatially random soil property. F enton and Griffiths (2002) assumed elastic modulus field was a lognormal distribution because of the non negative nature of material property F enton and Griffiths (2002) found t hat when PAGE 35 35 correlation length increased, the large r variability in footing settlement was obtained. F urthermore, F enton and Griffiths (2002) investigated the effect of standard deviation of E and the effect of correlation length on settlement variability. A Monte Carlo approach is adopted to simulate realization of the elastic modulus field in the FEA mesh and run thousands of times to obtain footing settlement s. Both the mean and variance of the settlements were found as function of CV of modulus and correlation length Using an analytical solution, estimated displacement using geometric averaging was obtained with deterministic displacement and E. Fenton and Griffiths (2005) estimated the uncertainty of shallow foundations based on settlement within serviceability limit using 3D simulation. To give support to the geometric averaging method, simulation with horizontally layered soil was conducted with elastic modul i varied from layer to layer but constant in each layer. Based on the displacement of the simula tion, the effective averaging modulus in horizontally layered soil wa s the harmonic average. A lternatively, effective averaging modulus in vertically layered soil was the arithmetic average. A s result, the geometric average of a random field wa s placed bet ween arithmetic and harmonic averages. In 2007 study, Fenton and Griffiths examined reliability based deep foundation design. The random behavior of piles subjected to vertical load and supported by spatially distributed soil was investigated for pile sett lements. They assumed that the surrounding soil and pile were perfectly bonded, which means any displacement of pile corresponds to equivalent local displacement of the soil. Fenton and Griffiths (2007) suggested the soil s influence through a series of bi linear springs based on geometric averaging attached to the pile Monte Carlo simulation of loading and spring stiffness PAGE 36 36 was undertaken to assess the probabilistic behavior of deep foundation represented with a mean stiffness. F ollowing the simulation, th e statistical behavior of maximum loads was estimated. The authors concluded that arithmetic averaging of soil strength controlled the maximum of pile 2.4.2 Averaging Technique To estimate equivalent engineering properties considering the effect of spatial variability, an averaging technique for heterogeneous soil properties was needed. Deutsch (1989) suggested and empirical technique called power averaging. Power averaging is calculated with nonlinear regression using the results from 3D numerical simulat ions of flow conducted with different soil volumes and properties. Power averaging method is very simple to use, but the volumes and properties of soils were assumed to be uncorrelated with one an other. A semi empirical technique was developed by King (1989) and Norris (1991). King (1989) suggested a renormalization technique for permeability, which generated the simulation grid and constant permeability was taken into each of its element. Four elements comprised one group and effective permeability, which was based on an analogy between water flow through soils of differing permeabilities, comprised another group. This was valid not only for uncorrelated permeability, but also for correlated. However, it was valid only for isotropic media. Norris (1991) sug gested the representative elementary volume (REV) method for flow simulation with renormalization technique. REV defines specific averaging volume where all micro variations are averaged into macro values as a representative elementary property. With incre asing volume, effective permeability was measured. The fluctuation in relation between increasing volume and effective permeability was shown. No significant PAGE 37 37 fluctuation at a specific volume shows at the effective averaging volume. A rational approach was suggested by Deutsch (2002), with geometric mean as an estimation of effective permeability of heterogeneous soil. 2.4.3 Characterization for Spatial Variability of Soil Properties For the stochastic analysis, the main element of soil spatial variability s hould be assigned. C lassical method, variogram, spatial correlation length, and variance reduction factor comprise the main elements. The classical method for characterizing soil properties involved statistics, such as mean, coefficient of variation (CV), and probability distribution. A nother method for characteriz ing spatial variability wa s spatial correlation, which wa s able to show variation of soil properties from one point to another in the field. Deutsch (2002) expressed spatial correlation with using variogram s, quantify ing dissimilarity between two points in field. Vanmarcke (19 84) also expressed spatial correlation with using a covariance function, quantify ing similarity between two points in field. H owever, the two expressions ( variogram and covari ance function) were only suitable for stationary fields with constant means and standard deviations. Spatial correlation length, in terms of separation distance between two points with no (or insignificant) spatial correlation, is the maximum value of spat ial continuity. S patial correlation length is defined as the sill from a variogram which was previously discussed The v ariance reduction factor wa s the expression to obtain the final variance from averaging data within certain a volume. Vanmarcke (1984 ) found spatial average obtains a smaller variance due to a reduction in the probability distribution, which wa s smaller than the probability distribution of field data. Vanmarcke (1984) suggested the variance reduction factor which variance of spatial averages is correlated to the point variance. The variance reduction factor varies as a function of the averaging volume, PAGE 38 38 the choice of covariance function and correlation length. His variance reduction factor wa s only given for 1D averaging process. H owever a 3D variance reduction factor could also be calculated as the product of three 1D variance reduction factors by assumption of independent correlation structures. Vanarckes variance reduction factor approached 1 with decreasing averaging volume to a ver y small value Elkateb (2002) mentioned, with respect to geotechnical applications, the variance reduction factor wa s affected only by the averaging volume in vertical direction because the averaging volume in horizontal direction is usually small compared to spatial horizontal correlation length. PAGE 39 39 Figure 2 1 A scatter plot and variance and covariance as a function of lag distance PAGE 40 40 (a) (b) (c) (d) Figure 2 2 Examples of randomly generated E for various correlation lengths PAGE 41 41 CHAPTER 3 CENTRIFUGE TEST SETUP AND RESULT Th is chapter discusses the development of tip behavior (i.e., tip axial resistance, tip shear, tip moment and tip rotation) model for drilled shafts in Florida l imeston e subjected in combined loading (axial and lateral), us ing centrifuge test ing Specifically, the effect of strength and stiffn ess of limestone, as well as voids (described with R.Q.D or Recovery Ratio) and layering (e.g., two layers with different R.Q.D or Recovery Ratio) on tip behavior was investigated in th e case of heterogeneous Florida limestone. R.Q.D (Rock Quality Designation) is defined as the quotient as (/)100% ScoreTot where Score is sum of core sticks longer than 4 inch measured along the center line of the core and Tot is t otal lengt h of core run.). Also the Recovery Ratio is defined as the percent of length of the recovered sample. The tests in Table 1.1 were selected to encompass both soft and medium strength rock for both short and intermediate embedment. In addition, since the ch aracter of Florida limestone is typically heterogeneous, it results in a variable modulus. To model this aspect, 1.5 mm diameter perlite spheres were used to capture the mass modulus, Em. A total of 32 centrifuge tests were conducted to characterize the ti p behavior based on Table 1.1. A discussion of the centrifuge test setup and results follows 3 .1 Centrifuge Background T o reduce the cost and time for test ing in the field, laboratory tests have been developed to model field conditions In geotechnical e ngineering, because the character of soil is dependent on stress for example strength and stiffness, the modeling of field stress as well as field displacements are very important. T he centrifuge test is a laboratory test which i s best able to reproduce fi eld stress as well as simulat ing load PAGE 42 42 transfer alongside and beneath drilled shaft s and the surrounding Florida limestone. Typical applications in geotechnical engineering where the centrifuge test ing is useful include: deep foundations for bridges and buildi ngs, settlement of embankments, stability of slopes, earthretaining structures, tunnel stability and seawalls. A centrifuge test develops a centrifugal force generated by angular velocity D ue to every particle in body rotated in same angular velocity, t he speed of particle may be expressed with angular velocity (i.e., radius of rotation axis. T his means the speed of a particle (i.e., v( ds/dt ) = r) in a body increases linearly with distance, r In addition, the centrifugal force is found as function of the revolutions per minute (i.e., angular velocity) and the distance from the center of rotation. The relation of centrifugal acceleration and gravitational acceleration can be used as scaling fact or for converting between the centrifuge mod el and a prototype. When the centrifugal force is larger compared to earths gravity, the scaling factor, N, is ratio the centrifugal acceleration to the normal gravitational acceleration. U sing a scal ing factor, the stress (1:1) length (1: N) and force (1: N2) can be scaled because the whole body, even each single particle in body, rotates with same angular velocity (see Table 3.1) For example, with 67 gravities, a 1.61 inch diameter model structure is scaled t o 9 ft diameter in the field (i.e., size of field size = N size of model size). Since the scale ratio of stress is 1: 1 between the centrifuge model and the field model, nonlinear material behavior (and strength) can be simulated PAGE 43 43 The centrifuge sample container has a depth of 12 inches and a diameter of 17 inches ( shown in Figure 3.1) the length below bottom has 7 inches which can be minimized boundary effect 3 .2 Hardware & Instrumentation The tests were performed in the test container shown in Figur e 3.1. The circular steel container holds the cast limestone specimen which represents homogeneous or heterogeneous field conditions The aluminum superstructure provided the necessary reaction for both axial and lateral loading of the models. Mounted to t he top of the container were two load cells for axial and lateral loading, two displacement transducers (LVDTs) for axial displacement, and one LVDT for lateral displacement. The test setup is shown Figure 3.2 Figure 3. 3 shows the centrifuge at the Univer sity of Florida, which is rate d at 12.5 G tons with a maximum model size of 30 inches To monitor load transfer the moment, and shear for lateral loading 4 to 6 pairs (depending on length) of 350 Ohm strain gages were attached to the longitudinal steel r einforcement in each specimen, as shown in Figure 3. 4 For steel reinforcing, a 0.75 inch diameter steel pipe (for the 6 ft diameter shaft) and a 1.32 inch diameter steel pipe (for the 9 ft diameter shaft) were used. Both were slotted to characterize the steel ratio of 6.4% and 7.5%. This reinforcement was needed for the expected moments. The axial forces in the shaft were obtained from the average strain gage readings at discrete locations multiplied by the shaft modulus and cross sectional area. To assess bending ( and thus flexure) the paired strain gages were monitored from which the curvature wa s compute d and bending moments obtained. Note that the moment curvature relationship may be nonlinear for cracked cross sections. Once the bending PAGE 44 44 moment distri bution along a shaft is known, the shear distribution (i.e. derivative of moment with depth) is found. T he shear and bending moments are determined along the shaft and the tip. Section 3.4 provides more detail After placing the synthetic limestone (descr ibed in S ection 3.3.1) in the centrifuge container as shown in Figure 3.5, it wa s cured for 14 days to the appropriate strength (from separate test samples). Then, holes were drilled into the synthetic limestone (see Figure 3. 6 ) Next, the instrumented rei nforcement cage wa s placed and concrete described in S ection 3.3.2 of the shaft specimen wa s cast as shown in Figure 3. 7 The Bimba load actuator, LVDT vertical measurement devices, and vertical and horizontal load cell were subsequently placed on the experiment (Figure 3. 2 ) and all the instrumentation wiring was run through the slip rings (Figure 3. 8 ) to the data acquisition unit outside of the centrifuge. Each slip ring had 24 channels and each channel was used to send and receive voltage from instrumentation. Furthermore, the pneumatic ports on the rotary union were used to send air pressure to air piston acting on the model as loading, axial and lateral. Figure 3. 9 wa s a schematic sketch of the experiment setup for a specimen of a lengthto diameter r atio (L/D) of 1. 3 .3 Material 3 .3.1 Synthetic Rock The limestone used in the experiments, Figure 3. 5 was reconstituted material (synthetic limestone), i.e. a mixture of crushed limestone, cement, and water (McVay et al. 2004). Mini Perlite spheres were a dded to characterize voids associated with typical recoveries from the field. Of primary interest wa s the influence of void size on rock properties, e.g modulus and strength. The current practice (i.e. FHWA intermediate geomaterials) wa s to reduce modulus as a function of RQD or r ecovery values PAGE 45 45 For this research crushed limestone was obtained from a pit in Newberry ( Florida Limestone Products Inc ) by the Florida Department of Transportations State Materials Office ( SMO ) and delivered to UF in bulk ( approx imate ly 2 tons) For volume considerations, ( i.e. those necessitated by centrifuge test ing ) the l imestone was subsequently sieved through a No 10 standard sieve with a maximum particle diameter no larger than 0.00656 ft. From the grain size distri bution shown in Figure 3. 10 the aggregate wa s well graded To minimize the affect of natural moisture in the material, it was dried for at least 24 hours. Next, the synthetic limestone was batched by mixing Portland cement and water in differing proportions With crushed limestone, a mortar mixture with a water to cement ratio (W/C) of 4 and an aggregateto cement (Agg/C) ratio of 15 is used (ASTM C270). C ylindrical samples were cast and cured at room temperature for 14 days before unconfined compressive t ests were run. T o obtain the desired strength, multiple trials of different proportions of limestone cement and water were used, as shown in Table 3. 2 The unconfined compression strength test ing was performed in accordance with ASTM D 2938. An ISTRON c ompression testing machine, was used to obtain both the unconfined compression strength, as well as the modulus in Figure 3. 11 O nce the desired strength was achieved, the test s w ere repeated at least three times to verify repeatability. Limestone specimens with and without perlite were also cast in 2 x 4 cylinders to identify the influence of voids on rock properties (e.g ., strength and modulus ) and are shown in Figure 3. 12 and Figure 3.13. As discussed previously, perlite spheres cast with the syntheti c limestone specimens attempted to mimic voids found in natural occurring rock (Figure 3. 12). PAGE 46 46 Figure 3. 13 show s the stress versus strain response of intact (i.e. no perlite) versus 30% by volume perlite specimens. Both small (5 mm) and larger (10 mm) sp heres were investigated. As is e vident from the f igure, a significant reduction (> 50 %) in strength and compressibilit y (i.e., the slope of stress versus strain) occurred with the addition of the perlite. Interestingly, the size of the perlite spheres was not a factor suggesting that the k arst nature of limestone could be characterized through sphere inclusion. Table 3. 3 show ed the comparison of voided modulus to nonvoided modulus based on volume or recovery E xperimental strength tests:Qt, Qu and triaxi al tests ( t hree triaxial and one spilt tension tests (Qt) were conducted according to ASTM D2850 and ASTM D3967, respectively ) Subsequently, Mohrs stress circles were drawn in Figure 3. 14 from which a strength envelope, Figure 3.14 was estimated. Using a curve fitted fourth order polynomial function, material strength characteristics such as cohesion (C =6ksf ) and the angle of friction ( ) were graphically estimated. The slope of the failure envelope ( ) w as estimated as 28 degrees at a maximum normal stress of 38 ksf (1.82 MPa) from the c entrifuge t est with 9 ft diameter and L/D =1 in 10 tsf rock 3 .3.2 Concrete Grout Since the rock strength used in the centrifuge test s are as high as 30 tsf ( 417 psi ), the drilled shaft concrete strength ha d to be sufficient so as to ensure a limestone failure instead of a shaft failure. This was because the study focused on rotation of the drilled shaf t models as well as translation ( i.e., tip shear ) To ensure the latter, a concrete strength of 360 tsf ( 5000 p si ) which is typical for drilled shafts in Florida, was used. Due to the model size of the shaft s, the concrete aggregate was limited in size to 2 mm and below Several trial mix es were tested, incorporating five co mponents; Portland PAGE 47 47 cement type I, sand, silica fume, super plasticizer, and water. The s ilica fume and super plasticizer provided an increas e in strength by reducing the water cement (w/c) ratio. The influence of sand particle size was investigated, and is presented in Section 3.3.2.1. 2 3 .3.2.1 Mix with s ilica s and Several trial mix es using w/c ratios of 0.350.4 and 0.5 0.6, with final strength values of 3 4 ksi were obtained. A fter several attempts to obtain 5,000 psi w/c ratio was reduced to 0.3 for h igh er strength s, as the target s trength ( i.e., 5 ksi ) could not be obtained. Table 3. 4 show s the amount of each component for 5 samples. Table 3. 5 show s the dimensions of the sample s and the maximum load applied to each. T he samples were tested after 4 day s of curing at room temperature. 3 .3.2.2 Mix with g ranite s and Table 3. 6 show s the amount of each component by weight, required to make 4 samples. Table 3. 7 show s the size of sample and the maximum load on each sample. T he samples were tested 4 days after curing at room temperature. As shown in Table 3.6, the expected strength ( i.e., 5 ksi ) was obtained and was used to make the model shafts Consequently, the mix design outlined in Tables 3. 6 and 3. 7 was used for drilled shaft construction, as well as the instrumentation and t esting m atrix outlined in Table 1.1 for a total of 32 instrumented centrifuge tests 3.4 Data Reduction 3. 4. 1 Axial Loading The standard testing procedure consisted of first applying an axial load to each shaft until settlement reached 3% to 5% of the models diameters. This was followed by PAGE 48 48 lateral loading. The axial loading was representative of the vertical live and dead loads under service conditions. The data from the axial loading was used to develop the tip displacement vs. tip st ress model. The reduction of the data wa s as follows. From several pairs of strain gage readings ( Figure 3. 15) with axial LVDT displacements and load cell output at the top of shaft, the transferred skin friction along the shaft and tip resistance were com puted. Specifically, from each pair of strain gage, the axial strain or compressive strain for each applied axial load state was found. The compressive strain is given by: 2 ) (right left a (3.1) Next, the transformed Youngs m odulus of the shaft was assessed from the pair of strain gages located above the rock surface along with the applied axial load. The Youngs m odulus of the shaft is given by : ar P E 2 (3.2) w here P = the applied axial load. Subsequently, from each axial strain, the compressive force (Q) with depth may be assessed : A E Qa (3.3) w here A = g ross shaft a rea. Figure 3. 1 6 show ed the typical axial force, Q, distribution within a shaft for various applied axial loads. P oints are from strain gage measurements and the solid lines cover ing points at each top displacement are trend lines PAGE 49 49 3. 4 1.1 Axial l oading Side s hear Once the axial force distribution and set of strains along the shaft for a particular top displacement wa s determined, the shaft rock load transfer, T (fs) versus Z curve for the shaft can be computed by : L D Q fs (3.4) w here L is the interval length between adjacent pairs of strain gages. Figure 3.17 show ed a typical fs v ersus Z curve for test shafts For any applied axial l oad, a corresponding axial displacement and skin friction can be calculated at each elevation. However, due to the homogenous nature of the rock, the skin friction was expected to be uniform along the shaft as well as within range of the FDOT design values (McVay, 1992) and previous centrifuge tests (Lila, 2003). Figures 3.18 and 3.19 show all the T Z curves for the axial load portions of the centrifuge tests. Figure 3.1 8 show s the results for 10 tsf strength rock and Figure 3.1 9 show s results for 30 tsf strength rock The results show ed that the FDOT design criteria wa s quite good, however slightly conservative. The latter wa s attributed to the linear assumption of the strength envelope (Figure 3.14) in assessing cohesion from sqrt (qu) sqrt (qt). The T Z curves shown in Figures 3.18 and 3.19 agreed very well with McVay (1992) and Lila (2003) results which have subsequently been implemented into FB MultiPier 3. 4.1 .2 Axial l oading End b earing A major focus of this research involves end bearing and spec ifically mobilized tip resistance as a function of tip displacement. Currently, FB DEEP and FBMultiPIER PAGE 50 50 employ the FHWA model formulated by O Neill, which characterized tip resistance, qb as : 0.67 btqW (3.5) w here (Lambda) = Elastic co mpressibility parameter; Wt = Displacement at top of shaft (value assumed) The elastic compressibility param eters, (Lambda), (Gamma), and (Omega), may be computed as: 0.5 0.5 c 10 mE LL 0.370.151log0.13 DDE (3.6) 0.5 0.5 c 10 mE LL 1.140.051log0.44 DDE (3.7) Leading to 0.67 0.5 mLL L 200 1 DD D 0.0134E L L 1 D (3.8) The Youngs m odulus in Eqs. 3. 6 to 3. 8 wa s the mass modulus, Em, and wa s of significant importance. The latter wa s different from the Youngs modulus of intact rock samples, Ei, measured in the laboratory (ASTM D3148). The rock mass Youngs m odulus, Em, represent ed the whole mass including fissures, voids, slip planes, etc. ONeill suggest ed a correlation (Table 3. 8 ) between the Ei, and Em based on RQD. If RQD values were less than 20 percent, the 20 percents RQD correlat ion was used (Load Transfer for Drilled Shafts in Intermediate Geomaterials, 1996) PAGE 51 51 Of interest was the relationship between Florida limestones m ass m odulus and intact Youngs m odulus, Ei, as reported in Table 3.3 Figure 3. 20 shows the Em/Ei ratios as r eported by ONeil and UF. It show ed that the UF data falls between ONeils open and closed joint data. Also, there exist ed a linear relationship between Em/Ei for r ecoveries above 50 % ; below this value there wa s a sharp dropoff Next in importance for estimating tip resistance wa s the influence of layered systems. The first researchers to address this were Ueshita and Meyerhof ( 1967) whose result s were shown in Figure 3. 21. Their theoretical solution for settlement or stress was found in terms of an e q uivalent m odulus, Ee = K E1, as shown in Figure 3. 21 Others have suggested the harmonic mean, Eh (i.e. Menard), or : ) 1 .. 1 1 (2 1 N harmonicE E E N E (3.9) Wher e N is number of layers and NE is the modulus of each layer shown in Figure 3.22 R ecently, the geometric averaging method, Eg, which account s for spatial variability (i.e. covariance) provided an excellent correlation with an FEM analysis (Fenton and Griffiths, 2005). The latter may be expressed as : 4 1) ln 1 exp(i i gE n E (3.10) Bot h the h armonic and g eometric means were used to estimate tip resistance, qb v ersus displacement (Eq.3.5) for all axial centrifuge tests. The results are presented in Figure 3.32. PAGE 52 52 3. 4.2 Lateral Loading in Combination with Axial Loading The first phase of t he research was to conduct lateral load tests in combination with axial loading to obtain moment, shear and tip rotation of the shafts. Variables that were used include: rock strength (10 tsf and 30 tsf), diameter (6 ft and 9 ft) and embedment ratio (L/D ratio of 1 and 3). Based on the latter tests, tip shear and rotation models were developed. A discussion on how the data reduction was performed is presented in the following section. 3.4.2.1 Lateral c entrifuge d ata r eduction As discussed previously ins trumentation was used to assess the lateral load and displacements at the top of shaft. The strain gage output is found by : E G V 4 (3.11) w here, wa s the strain (inches per inch) V wa s the change in bridge output voltage (mV) G wa s the gage factor (2.05) E wa s bridge excitation voltage (10 V) It was assumed that the relationship between strain gage resistance and voltage ( i.e. Eq. 3.11) was linear. Figure 3. 23 show s a typical output voltage from the strain gages located at the rock surface for the entire loading sequence (i.e. axial and subsequent axial with lateral). As expected, from 0 to 600 seconds, the individual gages on each side of the shaft indicated no be nding since only an axial load was being applied. However after 600 seconds when the lateral load was applied the gage values diverged. A positive output voltage indicated tension and negative voltage indicated PAGE 53 53 compression. Note that the voltage relations hip wa s linear with the first application of the lateral load, but then became non linear due to concrete cracking. At each strain gage elevation, the bending strain may be computed from the gage values on each side of the shaft as : 2right left b (3.12) Summing the axial strains ( Eq. 3.1 ) and the bending strains w ould result in the measured strain on each side of the shaft ( i.e. a and b), as shown in Figure 3. 24 Also of great interest was the moment curvature relationship of the shafts cross section. The latter was used in assessing the moments, shears, rotations and lateral translations of the shaft below the rock surface. That relationship wa s: EI r EI Mb (3.13) where E = Youngs Modulus of the shaft I = Moment o f Inertia of cross section r = Shafts radius = Curvature of the cross section (i.e. change in rotation) The moment curvature relationship may be simply expressed as M = B where B is the nonlinear relationship of EI. The latter may be obtained readily from the strain gage data in the shaft at the rock surface elevation where the moment is readily known (lateral load x height above rock). The curvature (see Eq. 3.13 ) wa s also obtained from strain gage data. A typical moment curvature relation is shown in Figure 3. 2 5 From the moment curvature relationship, the moment distribution along the length of the shaft may be found from Eq. 3.13 by using the bending strains from the gage PAGE 54 54 pairs along the shaft. Figure 3.26 a, 3.26b, and 3.26c present a number of moment distribution curves along the length of the shafts under multiple lateral loading states By examining Figure 3.26 it is apparent that the moment distribution along the shaft starts at zero at the top and increases linearly down to the rock surface (i.e. maximum moment) whereupon it starts to decrease due to the rocks resistance. The moment at the tip wa s a function of shaft length and rock strength. After the moment distribution along the shaft is found, the shear distribution within the shaft m a y be readily determined from : dz dM V (3.14) Figure 3.27 shows the increase in shear with increased lateral loading. The axial load is maintained at approximately 80% of the ultimate capacity. The rock had an unconfined compressive strength of 11 tsf, and the shaft was 9 ft in diameter and embedded 9 ft in the rock (L/D = 1). The shear at the tip of shaft is equal to the lateral force transferred to the rock at the bottom From the f igure it was evident that the more tip shears transfer red with increasing lateral load due to the increasing lateral displacement, suggesting a model which mobilized tip shear based on tip lateral d isplacement Increasing the rock strength (qu = 35 tsf v ersus 11 tsf) increase d the available tip shear for comparable deformations. The results are shown in Figure 3.28 Both tests involved 9 ft diameter shafts embedded 9 ft into limestone under constant axial and varying lateral loads. T he f igure shows the shear distribution in the shafts under the same lateral top displacements. Note that failure occurred at 5% of the shafts diameter or 5.4 in ches PAGE 55 55 Figure 3.29 shows the influence of a soft limestone layer (i.e. 50 % voids) one diameter below the tip of the shaft in 10 tsf strength rock. This wa s reflected by the shear crossing the axis (i.e. zero shear or lateral resistance) as well as the shear distribution along the length of the shaft as it reduces at the shafts tip. In order to develop a tip shear model, the lateral translation of shaft tip was necessary. This was accomplished by evaluating the lateral displacement and rotation along the entire length of the shaft. The procedure used is as follows: Choose a depth along shaft as shown in Figure 3. 30 a; From the m oment trend line, assess the moment from the selected depth; With the known m oment, go to the m oment curvature relationship (Figure 3. 30b ) to obtain the curvature (i.e. change in slope); Repeat steps 1 through 3 for several points along the shaft making sure the tip is also selected; Identify where there wa s zero lateral translation of the shaft ( i.e. y = 0 ) by checking for zero lateral resistance ( i.e. P = dV/dX = 0 ) in Figure 3. 31; From the curvature, the rotation at the top of shaft ( ) is found; L rbb bt 2 (3.15) where bt and bb are the bending strains at the top and bottom of each segment respectively ; r = shaft radius ; and L = segment length; Using the latter estimate of the segment rotations, at each point were obtained along the shaft (i.e., 2 = 11, 3 = 2 2, etc.) ; The computed i ( see step 7 ) at each point represent ed the slope of the deflected shape at their respective positions (dy/dzi = i). Thus, the lateral displacement at each point can be calculated using dy = i* dzi ; Repeat s teps 5 to 9 until the line passes through or was very close to the point of zero deflection and P = 0; PAGE 56 56 After shaft rotations and displacements were assesse d for the top half of the shaft ( i.e., above y = 0), the bottom half of the shaft values were determined, with special focus on the shaft tip. Figure 3. 31 shows a typical displacement v ersus depth for a shaft using t his method The next section presents t he shear versus tip displacement for all tests conducted as well as the proposed shear tip model 3.5 Measured Versus Predicted Shaft Tip Response The focus of this research was to validate or develop reliable methods to predict the axial, shear, and mome nt response of a drilled shaft tip founded in Florida limestone under combined axial and lateral loads. A number of models which characterize tip response were investigated (e.g., ONeill, Bell, Dohetry, Ueshita, Misra ). Some had been developed specifically for rock (ONeill), while others were applicable for general elastic materials (e.g., Bell, & Dohetry). However, none had tested Florida limestone with its associated variability (see Chapters 4 and 5). For the lateral tip shear response, the work focused on extending the existing FDOT side friction model (McVay 1992) to account for normal stress and shaft diameter. A discussion of axial loading is presented next, followed by shear and then moment computations 3.5 .1 Axial Test Results Figure 3. 32 sh ow s the measured axial tip response for all shafts founded in 10 tsf and 30 tsf strength rock with different L/D ratios (1 & 3) in homogeneous limestone compared to O Neil s prediction and the linear tip stiffness model proposed by Misra (2006) and Doherty (2006) Misra suggested tip stiffness ,tK, relates to Pt (tip force) and ut (tip displacement) D ivided by area, A, the tip resistance can be calculated as shown in Eq. 3.16. PAGE 57 57 /btQPA (3.16) where, tttPKu and 20.3/(1)t SsKDE Doherty also suggested tip stiffness expressed in Eq. 3.17. tvKKGR (3.1 7 ) where, vK is found in a relationship shown in Appendix E. Analysis of Figures 3. 3 2 a through 3. 3 2d reveal ed that the tip resistance using tip stiffness show ed good agreement w ith the experimental response upto 1 tip displacement Furthermore, the estimation of tip resistance by ONeil show ed good agreement w ith the experimental response with two different modul i (se cant and tangent). W ith secant modulus, the O Neil prediction proved more conservative. Figure 3. 3 3 also shows the predicted axial tip response using ONeills (Eqs. 3.53.8) model incorporating either the harmonic mean modulus, Eh, or geometric mean modul us, Eg, and accounting for moduli from two layers. Analysis of Figure 3. 3 3 reveal ed that harmonic modul us in combination with the ONeills tip resistance model show ed good agreement w ith the experimental response compared with geometric modulus. Fenton and Griffiths (2005) already discussed the geometric modulus wa s good in spatial variability. H owever, with spatial variability in only the vertical direction and the same condition in centrifuge test, the harmonic modulus g ave the better result 3.5 .2 Lateral Test Result Tip Shear Model Figure s 3. 34 and 3. 35 show the measured mobilized shear stress versus tip lateral displacement for all centrifuge tests. Figure 3. 34 presents the lower strength (10 tsf) response and Figure 3. 35 presents the higher (30 ts f) strength rock result s. Also PAGE 58 58 included in each f igure (Red Lines/Diamonds Dots) were the results for the layered system ( i.e. 50% recovery rock located one diameter below the tip of the shafts ) As expected, the higher strength rock developed the higher tip shear resistance per unit of lateral tip displacement. Also, the response for the 9 ft shaft was similar to the 6 ft shaft. However, what was not expected was the significantly diminished resistance of shafts with 50 % recovery located one diameter below the shaft tip. Evidently, the more compressible voided zone resulted in smaller moments developing at the shaft tip, which in turn required larger moments in the overlying rock to resist the applied lateral load. Figure s 3. 36 and 3.3 7 sho w a oneto o ne comparison of the shears and moments with and without the underlying softer (voided) layer. The influence on tip compressibility at 1 inch axial tip displacement is shown in Table 3. 9 for layered and nonlayered systems. T he table also shows the influe nce of rock strength To develop a tip shear model, the influence of rock strength and normal stress must be taken into account. Figures 3.38 and 3.39 show ed the normalized shear stress on y axis with the cohesion and the mobilized tip resistance versus t he normalized lateral tip displacement on x axis with the diameter of shaft Figure 3.38 refered to the 10 tsf rock and Figure 3.39 refered to the 30 tsf rock. Each plot may be characteriz ed by a bilinear representation ( i.e. a linear increasing line) fol lowed by a horizontal or constant line. The linear portion represent ed the mobilization of tip shear versus normalized lateral tip displacement. The horizontal or maximum value (i.e. 0.55 for 10 tsf rock, and 1.0 for 30 tsf rock ) represent ed the failure s tate as shown in Figure 3. 40 PAGE 59 59 Consequently, the failure states represent ed the shear stresses within the Mohr Coulomb limit state. That is to say, they are represented by angles of internal friction of 28 and 45 degrees respectively. Compared with Figure 3.14 from the triaxial test, a 28 degree friction angle matches with the 10 tsf rock. The generalized shear model for shaft tip is shown in Figures 3. 41 a and 3.41 b. It wa s characteristic of an elastic plastic behavior model. To implement the model in a f inite element code such as FB MultiPier, one would do the following: Depending on the strength of the rock, the lateral spring stiffness (Figure 3. 41 a) wa s assessed from the initial slope of the curve (Figure 3. 41 b). If the rock strength was not 10 tsf or 30 tsf, the slope wa s interpolated ( e.g., 20 tsf is halfway between 10 tsf and 30 tsf ) ; Next, the finite element solution wa s obtained for the axial tip displacements; Knowing the axial tip displacement, the axial tip stress wa s computed from ONeills so lution; Subsequently, Plot 3. 41 b wa s analyzed with the known lateral tip displacement T he mobilized tip shear stress wa s assessed from the curve, along with the normal stress and cohesion of the rock 3.5.3 Lateral Test Results Moment Rotation Model Similar to the tip shear distribution, the moment transfer from the shafts to the underlying rock was plotted versus tip rotation as a function of rock strength, shown in Figures 3. 42 and 3. 43. Figure 3. 42 present the 6 and 9 ft shaft diameter results in 10 ts f rock while Figure 3. 43 shows the same shafts in 30 tsf rock. As expected, the moments for the 9 ft diameter shafts were higher than the 6 ft shafts due to the greater contact area for similar bending stresses. As discussed earlier, with the lower tip resistance, the layered rock system developed smaller bending stresses due to the lower compressibility modulus PAGE 60 60 The first reported work on embedded piles in an elastic half space subjected to both lateral tip displacement and rotation was reported by Bell ( 1991). He performed hundreds of finite element analyses and developed a stiffness matrix [K] relating vertical displacement (Vu), horizontal displacement ( Hu ) and rotation (M) to vertical load (V), horizontal load (H) and moment (M) as follows : M H VR u R u K K K K K R G M R G H R G V3 4 4 2 1 2 2 20 0 0 0 ) ( ) ( ) ( (3.18) where G = s hear m odulus under the tip of shaft and R = radius of shaft As is evident from Eq. 3.18, there is cross coupling between the horizontal shear, H, and moment. Bell developed the moment and horizontal stiffness terms from footings located at the ground surface with: M MGR M K GR H K 3 3 2 4 (3.19) For the case of embedded shafts, the stiffness terms were adjusted due to resistance provided by the shafts sidewall. The embedded stiffness terms (superscript, z) were expressed in terms of shaft embedment lengthD Z as follows : 3 4 2 3 2 4 4 2 2 1 12 ) ( K R Z R Z K R Z R Z K K R Z R Z K K K K K K KD D Z D Z Z Z (3.20) PAGE 61 61 Using the metacentre concept, Bell uncoupled the horizontal and moment components by replacing 04ZK in Eq. 3. 20, resulting in an equation for mZ: 2 4K K R Z R ZD m (3.21) For the new mK3 w ith 04ZK inserting Eq. 3. 21 into Eq. 3. 20 and solving for ZK3 yields : 2 2 4 3 3 2 4 4 2 2 4 2 3)) ( ( 2 )) ( ( K K K K K K R Z R Z K K K R Z R Z K KD D D D m (3.22) w here, HGRu H K 2 (3.23) Using mK3 M wa s calculated as: m MK GR M 3 3 (3.24) The M term was dependent on the horizontal force, horizontal displacement and moment. Also Eqs. 3.1 8 to 3.2 4 were dependent on the shear modulus of the rock. It wa s assumed that the stress strain behavior of the rock is linear to failure and Poissons ratio, of the rock is 0. 3 Using the Hooks law the shear modulus, G, was calculated as : ) 1 ( 2 E G (3.25) To calculate the shear modulus, G, in Eq. 3.2 5 Youngs m odulus E, must be known for the underlying rock. As discussed in Section 3.4, either the harmonic mean mod ulus, Eh (Eq. 3.9) or geometric mean modulus, Eg (Eq. 3.10) may be used. The PAGE 62 62 harmonic mean modulus for both the single and double layered systems was used and the measured versus calculated rotation was assessed for the various rock strengths. Figure 3. 4 4 shows the results for the 10 tsf rock and Figure 3. 45 shows the results for the 30 tsf rock It is evident from Figures 3. 44 and 3. 45 that Bells representation is noticeably below the strength of the rock, i.e. within the Mohr Coulomb strength envelope. The latter occur r ed at approximately 0.03 radians (Figure 3. 44) which wasnt reached in the case of the 30 tsf rock tests. It wa s important to note that even though the tip rotation, was shown as its stiffness coefficient wa s not constant, but rathe r wa s a function of horizontal loads displacements and moments, as shown in Eq. 3.1 8 m measured CalculatedK GR M3 3 (3.26) A less complicated approach was provided by Dohrety and Deeks (2006) for an elastic plate embedded in a Gibson soil and rock medium. This model represent ed an increasing modulus with depth. As with Bell (1991), the moment and rotation may be expressed in an equation similar to Eq. 3.2 6 However, the stiffness coefficient, K, wa s determined using a monograph (Figure 3. 4 6 ) as a function of t he plate rigidity parameter, J. The plate rigidity parameter included P oissons ratio ( ), shear modulus (R G ) Poissons ratio ( S ) of the soil, Youngs modulus ( SE ), radi us (R ), and thickness (t) of the footing. The rigidity parameter, J, is shown in Eq. 3.27: R t G E JR S S) 1 )( 1 ( 24 ) 4 3 ( (3.27) PAGE 63 63 As identified earlier, the shear modulus in Eq 3.2 7 varie d with depth (z) according to: ) ( ) ( R Z G z GR (3.28) In Eq. 3. 28 GR is the shear modulus of the rock located at a depth R below the rock surface, and is a user defined parameter. For instance, was assumed to be equal to 0.5 for the 10 tsf rock (i.e. sensitive to overburden depth), whereas the with the 30 tsf rock is presumed to be equal to zero, or a constant G Using Eq. 3.2 6, with K determined from Figure 3. 46 and the rigidity parameter, J, defined from Eq.3.27, the measured and calculated rotations for different rock strengths were found. These are shown in Figures 3. 4 7 and 3. 48. T he Dohrety and Deeks (2006) method also provided acceptable predictions between measured and predicted rotations for the developed moments. However, the Dohrety and Deeks (2006) trend lines are located in the middle or slightly above the measured results, suggesting that they were slightly less conservative. Consequently, even though Bells m oment v ersus r otation model also included tip shear and lateral translation and is thus more complicated than Dohrety and Deeks, its proced ure is recommended for Florida limestone. Figures 3. 49a and 3. 49b show the model To use this model the following procedure should be followed: Find the moment and lateral displacement at the tip from FB MultiPier; Find the tip shear force from the tip s hear model in Figure 3.41 with lateral tip displacement; Find the stiffness coefficient, mK3 including an assumed tip rotation described in Eq. 3.24 ; PAGE 64 64 Repeat step 3 until the calculated rotation, mK3 matches the measured tip rotation PAGE 65 65 Table 3 1 Centrifuge Scaling Relationships (Chandrasekaran, 200 1 ) Quantity Prototype Model Acceleration 1 N Area N2 1 Volume N3 1 Length N 1 Mass N3 1 Force N2 1 Energy N3 1 Stress 1 1 Strain 1 1 Velocity 1 1 Table 3 2 P roportion of C omponent for Synthetic Limestone Required Strength Percentage, % (by Weight) Qu (tsf) Crushed Limestone Cement Water 10 75 5 20 30 70 10 20 Table 33 Ratio of Em (with perlite) / Ei (no perlite) Recovery Modulus Ratio (E m /E i ) 70% 0.29 50% 0.065 Table 3 4 Amount of Component for M ixing C oncrete with S ilica S and WC Water(g) Cement(g) SP(oz) SF(g) C/A Aggregate(g) 0.3 213.8 712.7 0.45 60.6 0.5 1425.5 Table 3 5 Result of U nconfined C ompression S trength T est Sample # 1 2 3 Average Length (in) 4.06 4.063 4.055 Average Diameter (in) 2 2 2 Weight (g) 471 472 469 Unit Weight (pcf) 140.7 140.9 140.2 Max Load (lbf) 15003 10996 13691 Strength (psi) 4776 3500 4358 PAGE 66 66 Table 3 6 Amount of Component for M ixing C oncrete with G ranite S and W C Water(g) Cement(g) SP(oz) SF(g) C/A Aggregate(g) 0.35 213.8 611 0.5 52 0.5 1221.9 Table 3 7 Result of U nconfined C ompression S trength T est Sample # 1 2 3 Average Length (in) 3.965 4 3.97 Average Diameter (in) 2 2 2 Weight (g) 456 462 453 Unit Wei ght (pcf) 139.4 140 138.4 Max Load (lbf) 15229 15866 16248 Strength (psi) 4847.5 5050.3 5171.9 Table 38 Estimation of Em/Ei Based on RQD (Load Transfer for Drilled Shafts in Intermediate Geomaterials, 1996) RQD (%) E m /E i Closed Joint E m /E i Open J oint 100% 1 0.6 70% 0.7 0.1 50% 0.15 0.1 20% 0.05 0.05 Table 39 Comparison of Axial Tip Stresses Description Translated Axial Stress(ksf) Layered 1 0tsf Rock with intact rock Over with 50% recovery 3 Layered 30tsf Rock with intact rock Over with 50% recovery 10 Non Layered 10tsf Rock 15~20 Non Layered 30tsf Rock 60~80 PAGE 67 67 Fi gure 31 New Cylindrical Sample Container Figure 3 2 Instrumentation for Measuring Lateral & Vertical Forces and Deformation PAGE 68 68 Figure 3 3 University of Florida 12.5 G ton Centrifuge Figure 3 4 Slotted Steel Pipe (Axial Reinforcement) & Strain Gages PAGE 69 69 Figure 3 5 Mixture of Synthetic R ock for Centrifuge Test Figure 3 6 Drilled Hole in Synthetic Limestone PAGE 70 70 Figure 3 7 Cast In Model Shaft in Synthetic Limestone PAGE 71 71 Figure 3 8 Slip Rings and Rotary Union 0.8 ft (0.244 m)3 @ 0.045 ft (0.0137 m) 0.134 ft (0.04 m) 1.0 ft (0.305 m) 0.134 ft (0.04 m) C L Strain gage LVDT Load cell Synthetic Limestone Specimen Load frame LVDT Steel plate Figure 3 9 Schematic sketch of a centrifuge test setup PAGE 72 72 0 10 20 30 40 50 60 70 80 90 100 0.01 0.1 1 10 Percent finer(by Weight) Grain Size(mm) Sieve Analysis for Lime Rock Sieve Analysis #1 Sieve Analysis #2 Sieve Analysis #3 Sieve Analysis #4 Sieve Analysis $5 Figure 3 10. Grain Size Distribution for L ime stone Aggregate Figure 3 11. Strength Test s for Concrete and Rock PAGE 73 73 Figure 312. Synthetic Limestone with 5 mm Perlite Spheres Occupying 30 % Volume Figure 313. Stress Strain Response Voided and nonVoided Limestone PAGE 74 74 0 10 20 30 40 50 60 20 10 0 10 20 30 40 50 60 70 80 90 100 Shear Stress (ksf) Normal Stress (ksf) Result from Qt Result from Qu Triaxial Test Result w/Confine 15.84ksf Triaxial Test Result w/Confine 13.68ksf Triaxial Test Result w/Confine 11.52ksf Nonlinear Failure Envelope Figure 314. Mohr Coulomb Envelop from Qu,Qt and Triaxial Tests on 10tsf Rock Figure 315. Strain Gage Layout for L/D = 1 PAGE 75 75 Figure 31 6 Compressive Forces (Q) along Depth Skin Friction(10tsf) 0 5 10 15 20 25 30 35 40 45 0 1 2 3 4 5 Axial Displacement, z (in) Skin Friction, fs (psi) Figure 31 7 Mobilized Skin Frictions vs Axial Displacement PAGE 76 76 Skin Friction(10tsf) 0 10 20 30 40 50 60 0 1 2 3 4 5 Axial Displacement, z (in) Skin Friction, fs (psi) 9/910tsf50%#1 9/910tsf50%#2 McVay Lila 9/910tsf#1 9/910tsf#2 6/610tsf#1 6/610tsf#2 9/2710tsf#1 9/2710tsf#2 6/1810tsf#1 6/1810tsf#2 Figure 31 8 Measured Skin Friction Comparing Lila (2003) and McVay (1992) Skin Friction(30tsf) 0 20 40 60 80 100 120 140 160 0 1 2 3 4 5 Axial Displacement,z (in) Skin Friction, fs (psi) 9/930tsf50%#1 9/930tsf50%#2 McVay Lila 9/930tsf#1 9/930tsf#2 6/630tsf#1 6/630tsf#2 9/2730tsf#1 9/2730tsf#2 6/1830tsf#1 6/1830tsf#2 Figure 31 9 Measured Skin Friction Comparing Lila (2003) and McVay (1992) PAGE 77 77 0 10 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 Recovery & RQD E m /Ei O'Neil Closed O'Neil Open UF Limestone Trend Figure 320. ONeil Em/Ei vs. RQD and UF Em/Ei vs. Recovery Figure 321. Two Layer System and Equivalent Modulus (Ueshita and Meyerhof, 1967) PAGE 78 78 Figure 322. Layered Moduli beneath a Drilled Shaft Figure 323. Output Voltage from Gages vs. Time wit h Loading; Axial Loading First and Combined Loading Following 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0 200 400 600 800 1000 1200 Time (sec) output voltage (volt) PAGE 79 79 Figure 324. Total Axial and Bending Strains along the Shaft Cross Section Moment Curvature 0 10000 20000 30000 40000 50000 60000 70000 80000 0 0.0002 0.0004 0.0006 0.0008 0.001 Curvature Moment (kips ft) Figure 32 5 Moment Curvature Relationship for 9ft Diameter from a Pair of Strain Gages above Rock Surface. PAGE 80 80 15 10 5 0 5 10 15 0 10000 20000 30000 40000 Depth (ft) Moment (kips ft) Figur e 3 2 6 a Moment Distribution along a 9 ft Diameter 5 0% Styrofoam at 1D below T ip of Shaft in 10 tsf Limestone from 2 tests 15 10 5 0 5 10 15 0 20000 40000 60000 80000 100000 Depth (ft) Moment (kips ft) Figure 3 2 6 b Moment Distribution along a 9 ft Diameter 30% Styrofoam at 1D below T ip of Shaft in 45 tsf Limestone from 2 tests PAGE 81 81 15 10 5 0 5 10 15 0 10000 20000 30000 40000 50000 60000 70000 Depth (ft) Moment (kips ft) Figure 3 2 6 c. Moment Distribution along 9 ft Shaft without Styrofoam below T ip in 35 tsf Rock from 2 tests 15 10 5 0 5 10 15 6000 4000 2000 0 2000 4000 6000 Shear (kips) Depth (ft) Rock Surface Tip Shear Figure 32 7 Increase in Tip Shear with Increasing Lateral Load, L/D = 1 PAGE 82 82 15 10 5 0 5 10 15 900 0 600 0 300 0 0 3000 6000 9000 Shear (kips) Depth (ft) Rock Surface Tip Shear Qu= 10 tsf Qu= 30 tsf Figure 32 8 Increasing Tip Shear with Increasing Rock Strength 15 10 5 0 5 10 15 10000 5000 0 5000 Shear (kips) Depth (ft) 9 f t/ 9 ft 10 tsf 1D 50% 9 ft / 9 ft 10 tsf Fi gure 32 9 Shaft Shear at Failure with and without an Underlying Soft Layer PAGE 83 83 (a) (b) Figure 330. Assessing Rotation from the Moment Curvature Relationship ( a) Moment versus Depth ( b) Mo ment Curvature Relationship 8 6 4 2 0 2 1 0 1 2 3 Depth (ft) Deflection (in) Figure 331. Typical Lateral Deflections (y) vs. Depth PAGE 84 84 0 10 20 30 40 50 60 0 0.5 1 1.5 2 2.5 3 3.5 Qb(ksf) Axial Tip Displacement (in) Centrifuge Test 1 Centrifuge Test 2 O'Nell(Secant) O'Neil(Tangent) 0 10 20 30 40 50 60 0 0.5 1 1.5 2 2.5 3 3.5 Qb(ksf) Axial Tip Displacement (in) Centrifuge Test 1 Centrifuge Test 2 Tip Stiffness from Misra Tip Stiffness from Doherty Figure 332a Tip Resistance ( Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O Neil(right) for 9 ft / 9 ft 10 tsf Rock 0 10 20 30 40 50 60 0 1 2 3 4 5 6 Qb(ksf) Axial Tip Displacement (in) Centrifuge Test 1 Centrifuge Test 2 O'Neil(Secant) O'Neil(Tangent) 0 10 20 30 40 50 60 0 1 2 3 4 5 6 Qb(ksf) Axial Tip Displacement (in) Centrifuge Test 1 Centrifuge Test 2 Tip Stiffness from Misra Tip Stiffness from Doherty Figure 332b Tip Resistance ( Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O Neil(right) for 9 ft / 27 ft 10 tsf Rock 0 20 40 60 80 100 120 140 0 0.5 1 1.5 2 Qb(ksf) Axial Tip Displacement (in) Centrifuge Test 1 Centrifuge Test 2 O'Neil(Secant) O'Neil(Tangent) 0 20 40 60 80 100 120 140 0 0.5 1 1.5 2 Qb(ksf) Axial Tip Displacement (in) Centrifuge Test 1 Centrifuge Test 2 Tip Stiffness from Misra Tip Stiffness from Doherty Figure 332c. Tip Resistance ( Tip Resistance prediction usi ng linear tip stiffness relation (left) and Tip Resistance prediction using O Neil(right) for 9 ft / 9 ft 3 0 tsf Rock PAGE 85 85 0 50 100 150 200 250 0 1 2 3 4 5 Qb(ksf) Axial Tip Displacement (in) Centrifuge Test 1 Centrifuge Test 2 O'Neil(Secant) O'Neil(Tangent) 0 50 100 150 200 250 0 1 2 3 4 5 Qb(ksf) Axial Tip Displacement (in) Centrifuge Test 1 Centrifuge Test 2 Tip Stiffness from Misra Tip Stiffness from Doherty Figure 332d Tip Resistance ( Tip Resistance prediction using linear tip stiffness relation (left) and Tip Resistance prediction using O Neil(right) for 9 ft / 27 ft 3 0 tsf Rock 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 Qb (ksf) Axial Tip Displacement (in) Centrifuge Test O'Neil using Harmonic Averaging 0 1 2 3 4 5 6 0 0.5 1 1.5 2 Qb (ksf) Axial Tip Displacement (in) Centrifuge Test O'Neil using Geometric Averaging Figure 33 3a. Tip Resistance (Model vs.ONeills Equation with Harmonic Average Method (left) and Geometric Average Method (right) for 10 tsf Rock with 50% Recovery at 1D 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 Qb (ksf) Axial Tip Displacement (in) Centrifuge Test O'Neil using Harmonic Averaging 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 Qb (ksf) Axial Tip Displacement (in) Centrifuge Test O'Neil using Geometric Averaging Figure 33 3b. Tip Resistance (M odel vs.ONeills Equation with Harmonic Average Method (left) and Geometric Average Method (right) for 30 tsf Rock with 50% Recovery at 1D PAGE 86 86 Rock Strength 10tsf 0 5 10 15 20 25 30 3 2.5 2 1.5 1 0.5 0 Tip Lateral Displacement (in) Tip Shear Stress (ksf) 9 ft 1LD 10 tsf 1D 50% Aver 9 ft 1LD 10 tsf Aver 6 ft 1LD 10 tsf Aver 9 ft 3LD 10 tsf Aver 6 ft 3LD 10 tsf Aver Figure 334. Tip Shear Stress vs. Tip Lateral Displacement Rock Strength 30tsf 0 20 40 60 80 100 120 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Tip Lateral Displacement (in) Tip Shear Stress (ksf) 9 ft 1LD 30 tsf 1D 50% Aver 9 ft 1LD 30 tsf Aver 6 ft 1LD 30 tsf Aver 9 ft 3LD 30 tsf Aver 6 ft 3LD 30 tsf Aver Figure 335. Tip Shear Stress vs. Tip Lateral Displac ement PAGE 87 87 15 10 5 0 5 10 15 6000 4000 2000 0 2000 4000 Shear (kips) Depth (ft) 9 ft / 9 ft 10 tsf 1D 50% 9 ft / 9 ft 10 tsf 15 10 5 0 5 10 15 15000 10000 5000 0 5000 10000 Shear (kips) Depth (ft) 9 ft / 9 ft 30 tsf 9 ft / 9 ft 30 tsf 1D 50% Figure 336. Shear in Shafts vs. Depth with and without Underlying Softer Layer 15 10 5 0 5 10 15 0 10000 20000 30000 40000 Moment (kips ft) Depth (ft) 9 ft / 9ft 10 tsf 1D 50% 9 ft / 9 ft 10 tsf 15 10 5 0 5 10 15 0 20000 40000 60000 80000 Moment (kips ft) Depth (ft) 9 ft / 9ft 30 tsf 9 ft / 9 ft 30 tsf 1D 50% Figure 337. Moment in Shafts vs. Depth with and without Underlying Softer Layer PAGE 88 88 Rock Strength 10 tsf 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 Tip Lateral Displacement/Diameter (Tip ShearCohesion)/ Tip Normal 9 ft 1LD 10 tsf 1D 50% Aver 9 ft 1LD 10 tsf Aver 6 ft 1LD 10 tsf Aver 9 ft 3LD 10 tsf Aver 6 ft 3LD 10 tsf Aver Figure 33 8 Tip Shear Minus Rock Strength Divided by Normal Tip Stress vs. Nor malized Lateral Displacement Rock Strength 30 tsf 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 Tip Lateral Displacement/Diameter (Tip ShearCohesion)/ Tip Normal 9 ft 1LD 30 tsf 1D 50% Aver 9 ft 1LD 30 tsf Aver 6 ft 1LD 30 tsf Aver 9 ft 3LD 30 tsf Aver 6 ft 3LD 30 tsf Aver Figure 33 9 Tip Shear minus Rock Strength Divided by Normal Tip Stress vs. Normalized Lateral Displacement PAGE 89 89 Figure 340. Mohr Coulomb Strength of Florida limestone Tip Shear Model 0 0.2 0.4 0.6 0.8 1 1.2 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 Lateral Displacement/Diameter (ShearCohesion)/Normal 10 tsf 30 tsf (a) (b) Figure 341. Tip Shear Model (a) Tip Shear Model Scheme (b) Tip Shear Model PAGE 90 90 Rock Strength 10 tsf 0 5000 10000 15000 20000 25000 30000 0 0.01 0.02 0.03 0.04 0.05 0.06 Tip Rotation (rad) Tip Moment (kip ft) 9/9 10 tsf 1D 50% 9/9 10 tsf 6/6 10 tsf Figure 342. Tip Moment Transfer as Function of Tip Rotation Rock Strength 30 tsf 0 5000 10000 15000 20000 25000 0 0.005 0.01 0.015 0.02 0.025 0.03 Tip Rotation (rad) Tip Moment (kip ft) 9/9 30 tsf 1D 50% 9/9 30 tsf 6/6 30 tsf Figure 343. Tip Moment Transfer as Function of Tip Rotation PAGE 91 91 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 0.02 0.04 0.06 0.08 Measured Rotation (rad) (rad) 9/9 10 tsf 1D 50% #1 9/9 10 tsf 1D 50% #2 9/9 10 tsf #1 9/9 10 tsf #2 Trend Figure 344. Calculated vs. Meas ured Rotation using Bell (1991) Trend line is in red 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0 0.005 0.01 0.015 0.02 Measured Rotation (rad) (rad) 9/9 30 tsf 1D 50% #1 9/9 30 tsf 1D 50% #2 9/9 30 tsf #1 9/9 30 tsf #2 trend Figure 345. Calculated vs. Measured Rotation using Bell (1991) with Trend line in red PAGE 92 92 Figure 34 6 Stiffness Coefficient, K, from Doherty and Deeks ( 2006 ) 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 0.01 0.02 0.03 0.04 Measured Rotation (rad) (rad) 9/9 10 tsf aver 6/6 10 tsf aver 9/9 10 tsf 1D 50% 9/9 10 tsf 1D 30% 9/9 10 tsf 3D 50% 9/9 10 tsf 3D 30% 9/27 10 tsf 6/18 10 tsf Trend Figure 347. Calculated vs. Measured Rotation in 10 tsf Rock using Doherty and Deeks (2006) with Trend Line in Red PAGE 93 93 0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.005 0.01 0.015 0.02 0.025 0.03 Measured Rotation (rad) (rad) 9/9 30 tsf aver 6/6 30 tsf aver 9/9 30 tsf 1D 50% 9/9 30 tsf 1D 30% 9/9 30 tsf 3D 50% 9/9 30 tsf 3D 30% 6/18 30 tsf 9/27 30 tsf Trend Figure 348. Calculated vs. Measured Rotation in 30 tsf Rock using Doherty and Deeks (2006) with Trend Line in red (a) (b) Figure 349. Recommended Tip Rotations vs. Moment Model from Bell (1991) (a) Tip Rotation Model Scheme (b) Tip Rotation Model PAGE 94 94 CHAPTER 4 A TWO DIMENSIONAL NUMERICA L A NALYSIS OF TIP RESISTANCE OF A DEEP FOUNDATION IN HETERO GENEOUS ROCK In th e case of deep foundations in rock, the current practice is to idealize the heterogeneous rock as an elastic homogeneous continuum. Arithmetic harmonic or geometric averaged value of individual elastic properties of the rock layers such as modulus of elas ticity (E) is used in the computation of the bearing resistance. Th e appropriate averaging equations, as well as the averaging depth are unknown. In addition, the expected variance of the settlement about the mean is also unknown. The purpose of this chap ter is to numerically validate the averaging as well as assess the variance of settlement using the same geostatistical principles presented earlier for bearing resistance of deep foundations subjected to axial loading. To conduct the study, a two dimensio nal finite element model has been developed which is capable of analyzing a load transfer mechanism under spatially heterogeneous rock conditions and predicting the loaddisplacement behavior accordingly. Although comprehensive calibration and validation against physical field test data wa s beyond the scope of this study, model validation of key material properties was carried out by laboratory and centrifuge testing (as delineated in C hapter 3) to confirm numerical components and aspects of the finite ele ment model. Spatial heterogeneity wa s incorporated into the FEA model by assigning random values of E to each individual rock layer of 1 ft thickness (i.e., no horizontal variability considered here). Subsequently, FEA wa s repeated for a large number of r ock property scenarios. Variability of E wa s hereby characterized by a coefficient of variation (CVE) and a vertical correlation length (av), which measure d the degree and spatial scale of variability, respectively. Finally, results of the random FEA were used to develop a geostatistical spring model that wa s capable PAGE 95 95 of predicting realistic tip loaddisplacement behavior and quantifying the uncertainty of rock conditions by the geostatistical properties of E 4 .1. Finite Element Analysis The numer ical study described herein dealt with the development of a finite element soil structure interaction model and subsequent use of that model in a parametric sensitivity study to evaluate the influence of heterogeneous limestone rock on the axial capacity of a rock socketed drilled shaft. Primary focus wa s given to the task of determining whether the presence of the heterogeneity wa s shown to affect (based on simulation results) the axial bearing capacity of the deepfoundation and, if so, the extent to which point bearing response wa s altered by the spatial variation of material strength characteristics such as E. For modeling nonlinear soil response and soilstructure interaction, the nonlinear implicit/explicit finite element simulation code ADINA (Version 8.5) was employed. 4 .1.1 Dimensions and Boundary Conditions Base d on the centrifuge test setup as shown in Figure 3.9 a 9 ft diameter field model wa s selected for finite element model development. Using symmetry for the geometry and the loading conditions, a twod imensional (2D) axisymmetric finite element model was constructed ( Figure 4 .1 ). The dimensions of the FEA model were scaled from those of the centrifuge test model, which were 3 times shaft diameter (D) wide and of shaft length (L) plus three tim es diameter below the shaft tip Element size of 0.5 ft by 0.5 ft wa s used in the discretization of a synthetic homogenous limestone used in the centrifuge tests, which wa s assumed to be a Representative Element Area (REA) on the axisymmetry plane at macroscopic scale. The degree of constraint provided by the boundaries of the system wa s modeled such that translational motion PAGE 96 96 wa s allowed to expand along the boundaries of the system (Figure 4 .1 ). The system wa s assumed to be in static equilibrium prior to any applied external loads 4 .1.2 Constitutive Models for Materials 4 .1.2.1 Reinforced c oncrete Based upon the modified Hognestads stress strain relationship for concrete ( MacGregor 2003) ] ) ( 2 [2 0 0 comp comp peak comp the material behavior of the concrete shaft under axial loading was approximated with an isotropic elastic model. The key assumptions made in the model are listed below. The compressive strength of reinforced concrete was obtained as 720 000 ksf using Eq. 3.2. E considered wa s for static loading rather t han dynamic loading rates 4 .1.2.2 Limestone r ock The ADINA soil model used in the simulation wa s a Mohr Coulomb model that can predict pressuredependent failure for the rock material under consideration. Based upon the assumption of normality of plastic flow to the failure surface, the plastic strain rate vector had a component in the volumetric (hydrostatic) direction that result ed in increase of mean normal stresses. However, the linear postulate of the failure envelope can produce much greater shear failure stress (as minor normal stress (3) increases) than what was observed in the laboratory tests, as shown in Figure 3.14 Thus, the effect of increasing normal stress ha d on the shear strength of the rock material needed to be properly accounted for over a realistic range of maximum normal stresses. A tangent slope of the failure envelope (Figure 3.14) wa s estimated as 28 degrees at a maximum normal stress of 38 ksf This slope wa s subsequently used as the angle of friction ( ) in the simulation. The value of cohesion wa s also determined by a curvefit PAGE 97 97 of the data obtained from the direct shear (Qt), unconfined compression (Qu), and three tri axial compression tests (Figure 3.14 ). It must be noted that a ny prediction of nonlinear Mohr Coulomb failure envelope at normal stresses higher than those identified in Figure 3.14 wa s conjecture due to the confinement restrictions of the laboratory tests. Pressuredependent failure mechanisms associated with the nonlinear failure envelope should be considered in future research to quantitatively determine accurate stress fields. Nonetheless, the constitutive material model presented here offer ed an improved tool calibrated with experimental data for studying twodi mensional stress state development near the tip of the rock finite elements that under went a maximum tip displacement of 1 inch i.e. the serviceability limit condition 4 .1.3 Rock Shaft Interface The deep foundation system consist ed of a shaft structure f ormed by excavation of a cylindrical borehole into limestone rock where reinforcing steel and concrete wa s cast. When a shaft is subjected to axial loading, the shaft transfer s the load by a combination of shear stresses developed along the cylindrical int erface between the concrete and rock and normal stress emanating from the tip of the shaft. This inelastic loaddeformation mode wa s often seen as rock material failure along the interface based on the visual appearance that the shaft takes on after load t ransfer has occurred. When a shaft is designed to have sufficient axial capacity, the shaft remains structurally intact and the tip resistance contributes significant axial stiffness to the overall behavior of the rock shaft system even if failure along t he interface occur s. Considering the physical interface of the system constituents, the initial approach taken in modeling the interface involved the use of discrete contact edges of PAGE 98 98 axisymmetric elements in which the interface represented the physical boundaries between the shaft and rock. An approximate means of accounting for the shear failure was attempted through the use of a contact model that simulates frictional resistance under Coulombs Law of Friction. During the contact simulations performed usi ng this modeling technique, th is approach was found to be problematic Nonphysical penetration at the corner node of the shaft into a rock element caused an artificial gap ( i.e., the rock mesh was detached from the side of the shaft ; see Figure 4 .2 ), and produced zero contact force. This nonphysical penetration wa s a numerical instability that typically is associated with either the use of a coarse finite element mesh or too large incremental loading st eps, or both (Bathe, 2000) Under certain conditions, elements formulated using numeric integration can undergo deformation modes in which strains sampled at the element integration points fail to capture all of the strain energy associated with the deformation. As a consequence, energy wa s numerically (rather than physically) dissipated, and the nonlinear solution process potentially becomes unstable (Bathe, 2000) leading to unreliable analysis results or termination of the simulation ( i.e. nonconvergence) Problems associated with nonphysical nodal penetration can often be remedied by using a highresolution finite element mesh and small incremental loading steps. Reduction of numerical instabilities to an acceptable level (quantified by maintaining the pseudoenergy at less than 1% of total system energy) and prevention of nodal penetration were attempted by significantly increasing the resolution of the finite element mesh and using a small loading step, e.g. an increment load of 1 kip. Even with approximately 40,000 elements of the rock mesh, nodal penetr ation was not completely PAGE 99 99 prevented. While stable solutions might be obtained, the very highresolution mesh with such a small loading step was deemed to be impractical from the standpoint of conducting a parametric study involving numerous, repeated simula tions required for geostatistical realization of the field condition. In preliminary analysis, several hundred simulations were necessary for one geostatistical realization of a heterogeneous rock condition. An alternative, morenumerically efficient solut ion wa s achieved by abandoning the approach of modeling the contact. In the physical shaft rock system, the circumference of the shaft wa s bonded to the surrounding rock. Approximate numerical modeling of this bond wa s accomplished using interface boundary elements (Figure 4 .3a ). This numerical approach simulated the mathematical link between two edges together at a common interface. Parts linked together in this manner may still deform and respond to load, as may the interface between them, but the edges o f the two parts remained linked to each other on a point by point (or node by node) basis ( Figure 4 .3b ) That is at no point on the interface boundary may the two tied parts separate from each other, even though the interface boundary element itself may d eform In contrast to the contact approach of the previous modeling technique, this approach led to a more controlled stress redistribution of internal stresses as the shaft rock interface yields, thus permitting stable solutions to be obtained at coarser mesh resolution. Determination of the yielding ( i.e., shear failure at the interface) wa s carried out using a Mohr Coulomb model with the elastic perfectly plastic yield condition such that the angle of friction ( ) wa s very small e.g., =0 Thus, exceeding the failure PAGE 100 100 shear stress at integration point s in an interface element was avoided by limiting the maximum shear stress to the cohesion value. This yielding mechanism permit ted experimentally observed amounts of volumetric deformation to be modeled and thus produced a better representation of both the shear failure along the interface and the load transfer to the tip of the shaft. For comparison to the centrifuge test results, the side resistance and en d bearing predicted by FEA wa s shown in Figures 4 .4 a and 4 .4 b, respectively. While the loaddeflection curves of the side resistance were not identical, their general form was in good agreement As is e vident from Figure 4 .4 b, the load transfer by side she ar wa s initially linear and becomes nonlinear as shear stress reaches the limiting shear stress (the strength of the synthetic rock). The tip stiffness, however, exhibited a linear relationship as the tip displacement reached a service limit condition of one inch settlement defined in Commentary C10.6.2.6.1 of Section 10 of Foundations of AASHTO Bridge Design Specifications Since the FEA material model has been developed using aforementioned Qt, Qu, and triaxial test data separate from centrifuge test res ult s, the agreement shown in Figure 4 .4 suggest ed a degree of validity in the load transfer both predicted by FEA and measured in the centrifuge tests. Due to the robustness and increased numerical efficiency of this modeling technique, it wa s used throughout the remainder of the study 4 .2 Simulating the Influence of Rock Variability in Florida Limestone on Tip Stiffness Having validated the FEA model for the homogeneous rock condition of the centrifuge tests the influence of the vertical variability of E on the tip resistance of the PAGE 101 101 shaft for given settlements wa s investigated. Variability in E wa s represented through a log E), coefficient of variation (CVE) and vertical correlation length (av). Generally, most soil/rock properties were assumed to be log normally distributed because of mathematical simplicity, their observ ance of positive skewness, and the requirement of nonnegativity. For instance, McVay (2008) report ed CVE = 0.5 and av = 8 ft as a typical heterogeneous rock condition from boring sites of 17th Causeway Bridge over the Interco a stal Waterway located i n Fort Lauderdale, Florida ; Martino(2001) report ed ranges of CVE ranging from 0.1 to 0.3. Based on the reports, three correlation lengths of 5 ft, 10 ft, and 15ft and three CVEs of 0.2, 0.4 and 0.6 were selected for the subsequent analyses of the cases f or shaft s with L/D of 1 and 3 The generation of vertically correlated E distributions for the FEA model (one E value per 1 ft rock layer thickness) may be undertaken by a number of different approaches. Most popular were Gaussian based methods such as Seq uential Gaussian Simulation (SGS), LU decomposition, etc. (Deutsh, 1998) These methods first generate Gaussian (standard normal) random fields with a prescribed spatial correlation structure, which were then transformed to a target distribution. In the pr esent work the LU method wa s applied to a spherical covariance model (Deutsh, 1998) for the underlying Gaussian field. In general, conversion from a standard normal variable N to a log N NN N are the mean and standard deviation of the log transformed variable. That is, ln N N LN and CVLN being the mean and coefficient of variation of the log normal variable. PAGE 102 102 21 lnLN LN NCV ( 4 .1 ) 2 21 lnLN NCV ( 4 .2 ) For later use, Eqs. 4 .1 and 4 .2 were LN and CVLN as functions N N as follows: 2 exp2 N N LN ( 4 .3 ) 1 exp2 N LNCV ( 4 .4 ) Figure 2.2 shows correlated variations of E with resp ect to various correlation lengths, av = 5 ft and 15 ft. For each combination of CVE, av, and L/D, 1,000 ADINA simulations were performed for various realizations of local E values to compute a respective series of tip resistances v ersus tip displacements Approximately 1, 000 simulations per a parameter combination were found to be necessary in order to achieve a constant coefficient of variation of the tip resistance. Figure 4 .5 presents the predicted tip resistances v ersus tip displacements for the homog E is used as a constant E) and the respective results for the heterogeneous cases as a function of CVE, and av. Note that for the heterogeneous cases, there would be 1,000 simulation data lines (not shown in the current figure for visua l clarity) in each graph. Only the respective mean lines are depicted. As is observable from Figure 4 .5 the coeffici ent of variation of the modulus ( CVE) had a significant effect on the mean tip resistance Next, focus was shifted to derivation of the tip stiffness ( resistance/displacement) in heterogeneous rock conditions as a function of the spatial properties of E. Fenton and PAGE 103 103 Griffiths (2005) show ed that the settlement ( ) of the shallow foundation in heterogeneous soil deposits can be predicted by a linear relationship o f det E effE ( 4 5 ) det = deterministic settlement computed using a homogeneous modulus E and Eeff = effective modulus equal to the geometric average of moduli H W W f gf fdz dy dx z y x E H W E0 0 0 2, ln 1 exp ( 4 6 ) where Wf = width of a square footing and H = thickness of soil below the footing. In the case of horizontally layered soil (variability only in vertical z direction), Fenton and Griffiths (2005) suggest ed that the harmonic average (Eh) of modulus may be more appropriate for Eeff in Eq. 4 .5 : 1 01 H hz E dz H E ( 4 7 ) For the present situation of horizontally layered rock conditions, the validity of Eq. 4 .5 ( in combination with Eq. 4 .7 ) had to b e investigated for predicting tip resi stance vs. tip displacements of the deep foundation. For this purpose, Eq. 4 .7 wa s first used to h and the coefficient of variation CVh of Eh in terms of the E, CVE, and av. T hese known parameters were subsequently used to study the existence and optimal value of the vertical averaging length ( H ) below the shaft tip. Note that while Eh in Eq. 4 .7 was h wa s a mean of either an ensemble average over many realizations or a statistical expectation PAGE 104 104 Assuming that E in Eq. 4 .7 wa s log E and coefficient of variation CVE, and since A = 1/E wa s also log normal the summary statistics of A are: (1) mean expectation ( A) e quals to 2(1)/EECV and (2) coefficient of variation ( CVA ) equals to CVE. This can be proven by E qs. 4 .1 4 .4 using the normal variable ln(A) equals normal variable ln(E). Introducing 1 BAdz H as the arithmetic averag e of A over H B can be approximated by setting B equal to A (both are log normal distributions of expectation) and coefficient of variation BCV equal to 1 2 ACV ( Klammer 2010 ) was a variance reduction factor for a vertical covariance function of A of the spherical type a s: v v v v v va H for H a H a a H for a H aH 1 5 4 3 1 0 20 2 12 2 3 3 ( 4 .8 ) Similarly as A wa s inferred from E expectation and the coefficient of variation of the approximately log normal Eh = 1/B can now be expressed as 2(1)/hBBCV and hBCVCV respectively. Substitution of these relations into Eqs. 4 .1 and 4 .2 finally g ave: 2 21 1E E E hCV CV ( 4 .9 ) E hCV CV ( 4 .10 ) Eqs. 4 .9 and 4 .10 demonstrated how both expectation and coefficient of variation of the harmonically averaged modulus increased proportionally to averaging length ( H/av) decreased. PAGE 105 105 Presented in the left columns of Figures 4 .6 and 4 .7 as functions of H/D were h from Eq. 4 .9 hs obtained from applying Eq. 4 .7 to each realization of the generated modulus values (used in FEA) and subsequently arithmetic ally averaging it over 1, 000 h hs showed the validity of the approximations of Eq. 4 .9 and the LU decomposition approach used for random field generation. Similarly it is also shown in the plots the as should remain constant at the expectation of the log normal modulus distribution E =17 28 ksf. Fluctuations of this expected value can occur due to the finite number of realizations used Thus it was recommended that t he degree of fluctuations be observed as a measure to evaluate a minimum required number of realizations per graph. 4 .5 were known, each value of heterogeneous individual E realizations can be converted into a respective value of Eeff. These values were used to construct the distribution of an FEA based effective modulus EFEA with expectation FEA and coefficient of variation CVFEA. FEA was independent of H/D and appeared as a horizontal line in the left columns of Figures 4 .6 and 4 .7 The graphs in the right columns of Figures 4 .6 and 4 .7 represented the relationships of respective coefficients of variation. Interestingly, all cases and values of H/D between 1 4 .9 and 4 .10 were able to closely predict values of FEA and CVFEA in the FEA simulation, respectively. This justified the use of Eeff = Eh in Eq. 4 .5 harmon approximately log normal (under the assumption of log PAGE 106 106 maintained log normality of expectation 2 det(1)/hEhhCV Using Eqs. 4 .9 and 4 .10 this log normality of expectation was written as: 2 det(1)hECV ( 4 .11 ) E hCV CV ( 4 .12 ) Equation 4 .11 stated that the geostatistically predicted mean settlement ( h ) at the tip of a drilled shaft in ve rtically heterogeneous rock was a function of the homogeneous tip settlement ( det) and the coefficient of var iation of the rock mass modulus ( CVE), but not a function of In contrast, the coefficient of variation of heterogeneous settlement ( CV) depended on the degree of spatial averaging ( ) and thus the choice of the averaging length ( H ) Both and CV were seen to increase with CVE. Equation 4 .11 wa s validated in Figure 4 .8 which shows the FEA results of the homogeneous tip settlement ( det) and the mean heterogeneous tip settlement ( ) for CVE of 0.6 and the two spatial correlation lengths of 10 ft and 15 ft Evident from Figure 4 .5 are various values of L/D and CVE and from Figure 4 .8 are various values of av. Analytical h and FEA simulate d FEA were in good agreement Table 4 .1 shows a collection of comparison s in respective coefficients of variation, CV and CV, for various dimensions (L/D), rock variabilities ( CVE), correlation lengths ( av), at H/D = 2. An av erage error was 2.5% in all the cases Therefore, f or current pile and shaft design practice where a spring model was used to represent the tip resistance ( e.g., FB MultiPier, APile ), the spring stiffness of the heterogeneous rock was estimated using Eq. 4 .11 : PAGE 107 107 ) 1 (2 hom E ogeneous ous heterogeneCV K K ( 4 .13 ) where CVE = coefficient of variation of E of the rock mass. As expected, CVE = 0 result ed in the heterogeneous tip stiffness equal to the homogeneous value; however, in the case of high CVE (e.g., 0.6 1.0) the heterogeneous tip stiffness may be reduced by up to half of the homogeneous tip stiffness It wa s important to realize that Eq. 4 .13 l ed to a prediction of a mean (expected) value for settlement, which is associated with an uncertainty described by the coefficient o f variation given by Eq. 4 .12 A mean value of tip resistance may be obtained according to Eq. 4 .13 by dividing the homogenous tip resistance by the 21ECV Assuming that the mean of skin friction along the side of the shaft was known, it can be summed with the expected tip resistance to obtain the expected total shaft resistance. Knowing the tip resistance was proportional to tip displacements, Eq. 4 .12 gave the CV of tip resistance, which can then be converted to a tip resistance var iance. Consequently, assuming independence between the side and tip resistances, the variance of the tip resistance may be summed with the variance of the side resistance to give the total variance of the pile/shaft resistance from which the resistance fa ctor ( ) of LRFD can be computed. The higher the variance (and the lower the factor) the longer the shafts required for design. PAGE 108 108 Table 4 1 Comparison of FEA results ( CV ) and analytical solution ( CVfrom Eq.14) at H/D=2 CVE = 0.2 CVE = 0.4 CVE = 0.6 CV CV CV CV CV CV L/D = 1 a v = 5 0.10 0.09 0.20 0.18 0.30 0.26 a v = 10 0.13 0.12 0.25 0.24 0.37 0.35 a v = 15 0.14 0.14 0.27 0.27 0.40 0.41 L/D = 3 a v = 5 0.10 0.09 0.19 0.18 0.27 0.26 a v = 10 0.12 0.12 0.24 0.24 0.35 0.35 a v = 15 0.15 0.14 0.29 0.27 0.44 0.41 PAGE 109 109 Figure 41 Dimensions and boundary conditions of FEA Model Gap Nodal Penentration Shaft Mesh Rock Mesh Figure 42 Nodal penetration phenomenon PAGE 110 110 Shaft Finite Element Rock Finite Element Interface Boundary Element (IBE) along the side of the shaft (a) Interface boundary element (IBE) Shear deformation of IBE and Rock FE Applied nodal loads (b) Load transfer of IBE Figure 4 3 Model of the rock sh aft side interface boundary PAGE 111 111 0 500 1000 1500 2000 2500 0.00 0.04 0.08 0.12 0.16 0.20 Tip Resistance (kips) Tip Displacement(ft) 0 500 1000 1500 2000 2500 0.00 0.04 0.08 0.12 0.16 0.20 Side Resistance (kips) Top Displacement(ft) Figure 4 4 Centrifuge test and FEA results PAGE 112 112 0 200 400 600 800 1000 1200 1400 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Tip Resistance (kips) Tip Displacement (ft) L/D=1, CV E =0.2 0 200 400 600 800 1000 1200 1400 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Tip Resistance (kips) Tip Displacement (ft) L/D=3, CVE=0.2 0 200 400 600 800 1000 1200 1400 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Tip Resistance (kips) Tip Displacement (ft) L/D=1, CVE=0.4 0 200 400 600 800 1000 1200 1400 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Tip Resistance (kips) Tip Displacement (ft) L/D=3, CVE=0.4 0 200 400 600 800 1000 1200 1400 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Tip Resistance (kips) Tip Displacement (ft) L/D=1, CVE=0.6 0 200 400 600 800 1000 1200 1400 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Tip Resistance (kips) Tip Displacement (ft) L/D=3, CVE=0.6 Figure 4 5 Comparison of the tip stiffness among various CVE (av = 5 ft) PAGE 113 113 1300 1400 1500 1600 1700 1800 0 0.5 1 1.5 2 2.5 3 3.5 Modulus (ksf) H/D av=5ft 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 2.5 3 3.5 CV Modulus H/D av=5ft 1300 1400 1500 1600 1700 1800 0 0.5 1 1.5 2 2.5 3 3.5 Modulus (ksf) H/D av=10ft 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 2.5 3 3.5 CV Modulus H/D av=10ft 1300 1400 1500 1600 1700 1800 0 0.5 1 1.5 2 2.5 3 3.5 Modulus (ksf) H/D av=15ft 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 2.5 3 3.5 CV Modulus H/D av=15ft Figure 4 6 Simulation results of the cases among various av (CVE =0.6 and L/D =3) PAGE 114 114 1300 1400 1500 1600 1700 1800 0 0.5 1 1.5 2 2.5 3 3.5 Modulus (ksf) H/D CVE=0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 2.5 3 3.5 CV Modulus H/D CVE=0.2 1300 1400 1500 1600 1700 1800 0 0.5 1 1.5 2 2.5 3 3.5 Modulus (ksf) H/D CVE=0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 2.5 3 3.5 CV Modulus H/D CVE=0.4 1300 1400 1500 1600 1700 1800 0 0.5 1 1.5 2 2.5 3 3.5 Modulus (ksf) H/D CVE=0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 2.5 3 3.5 CV Modulus H/D CVE=0.6 Figure 4 7 S imulation results of the cases among various CVE (av=15ft and L/D=3) PAGE 115 115 0 200 400 600 800 1000 1200 1400 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Tip Resistance (kips) Tip Displacement (ft) av=15ft 0 200 400 600 800 1000 1200 1400 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Tip Resistance (kips) Tip Displacement (ft) a v =10ft Analytical Solution Figure 4 8 Comparison of tip stiffness (CVE=0.6) PAGE 116 116 CHAPTER 5 U SE OF G EOSTATISTICS IN LRFD ASSESSEMENT WIT H FIELD TEST RESULT 5 .1 Background of Usage with Spatial Correlation and Its Influence on LRFD resistance factors, A deep foundations resistance or capacity, Q, is the summation of side friction, Qs, and end bearing Qtip. In the case of cylindrical prismatic shafts of length L and diameter D [L], the pile/shafts side resistance Qs may be determined from the product of the pile/shafts surface area As 2] and the mean unit skin friction fs [stress] over As or, mathematically, Qs = Asfs. Consequently, fs is related to point unit skin friction qs, through: sA s s sdA q A f 1 ( 5.1 ) Since qs may be regarded as a spatially random (regionalized) variable of a certain distribution (e.g., log normal) and spatial correlation structure (variogram), fs is also a random variable in space and its properties are related to the properties of qs. The variable qs wa s defined through summary statistics (mean, m and variance, ) and the covariance function, C(h) [stress2]. C(h) is a measure of spatial correlation between values of qs, which are separated by a distance h [L], and is directly related to the 22C(h). In practice, this assumption corresponded to 2, were known with high confidence (e.g., from exhaustive core sample testing) and the shaft wa s located at a random location on the site. s 2, of the pile/shaft side friction, it wa s expected that it would be less than the site s variance, 2 of qs as a result of the spatial averaging over the shaft surface or s 2 2s 2 wa s PAGE 117 117 known as a change of support, scale up, or regularization problem (Isaaks and Srivastava, 1989), since the properties of a variable were averaged over different support sizes. In the present cas e, qs obtained from core samples may be thought of as a rather small support volume (i.e., points) as opposed to fs, which is defined on a support equal to the shafts side surface Ass 22 [dimensionless] can be introduced as the factor of variance reduction between qs and fs, and it wa s determined by the relationship (e.g., Journel and Huijbregts, 1978; Isaaks and Srivastava, 1989; Deutsch, 2002) as: 2 1 2) ( 1 dA dA h C As sA A s ( 5.2 ) 2 [dimensionless] wa s a covariance func tion normalized to unit variance. Equation 5 .2 contain ed two integrals over the area As (in practice, it is a quadruple integral); however, it wa s nothing but the arithmetic average of the covariance values that correspond to all possible combinations of t wo points on As (i.e., the side of shaft). If the averaging domain As wa s not a continuous area but a discrete number of n [dimensionless] points with statistically independent observations of some random variable, then Eq. 5 2 reduced to the well known f ormula of the standard error s 2 = 2/n, where = 1/n. To assess the variance of side shear, s 2, the covariance C(h) in Eq. 5 .2 was assumed to be spherical (Isaaks and Srivastava, 1989) with an isotropic correlation length, a or: PAGE 118 118 35 0 5 1 1 )( a h a h h C ( 5.3 ) Note that C(h) t ook on a value between 0 and 1 depending on the distance h. That is to say, when h C(h) = 0 (no correlation) and when h = 0, C(0) = 1 (a oneto one correlation). The correlation length (or range) a wa s established from the experimental variogram based on available data. Substituting Eq. 5 .3 into Eq. 5 .2 and s 22 is determined. The results are presented in a monograph as a function of L/D and a/D as shown in Figure 5 .1. Also show n in the graph is the case of D = 0 (thick dashed line), in which one averages the var iability over a length or depth that wou l d subsequently only be used for end bearing. Next, the LRFD spatial resistance factors ( ) for shaft side shear may be assessed ( Klammler, 2010) from COVRs s/ms using either field or laboratory soil/rock strength assessment s, or both : sq s s RsCOV m m COV 2 ( 5.4 ) and: ( 5.5 ) Figure 5 .2 show s the LRFD resistance factors ( ) for known COVR and reliability index ( ) values. Using COVRs (i.e. side shear ) from Eq. 5 .4 for COVR, typical values (e.g. 2.5 3.0), and LRFD resistance factors, the value of may be ascertained. PAGE 119 119 5 .2 Development of LRFD resistance factors, for End Bearing and the 17th Street Bridge Case Study In the case of a shafts tip resistance, Qtip, the shafts unit tip stress, bq must be calculated and then multiplied by the tips cross section al area. However, the shafts unit tip resistance is a function of the shafts displacement, as was found in Eq. 3.5 The settlement of at the bottom of the shaft, Wb, wa s computed from the elastic short ening of the shaft as follows: 22 D E L Q Q W Wc tip top t b ( 5. 6 ) where Ec = Youngs Modulus of the concrete shaft Qtop = Force at Top of Shaft Qtip = Force at Bottom of Shaft (qb Ashaft) The shafts tip resistance Qtip (i.e. qb), may be found by determining the shaft and rocks compressibility parameter which is a function of f and as shown below : As is e vident from Eqs. 3.6 3.8 the Youngs m odulus of the rock mass, Em control led the variability of the results. Of interest was the relationship between Florida limestone m ass m odulus and intact Youngs m odulus, Ei. Synthetic l imestone specimens with various strengths were cast with different volume percentages of voids ( perlite ) as reported in Chapter 3. A comparison of no void Youngs m odulus, Ei, v ersus m ass m odulus values with different volume percentages (i.e. recovery) was completed. Shown in Figure 3.20 was the Em/Ei ratio as reported by ONeil and UF. As shown, the UF data falls between ONeils open and closed joint data. Also, there exist s a linear relationship between Em/Ei for r ecoveries above 50% and a sharp drop off below 50%. PAGE 120 120 Of interest was the variability in the fi eld and its impact on the tip resistance of shafts. To study the latter, a series of drilled shaft load tests at a bridge site were investigated. Figure 5 3 shows a typical stress strain plot from an unconfined test on Florida limestone recovered from the 17th Street Bridge near load test LTSO4 at pier 10. A total of 102 unconfined tests were performed by SMO personnel on rock cores recovered in six boreholes at 5 foot spacing near LTSO4. The analysis considered both the s ecant Youngs m odulus as well as the t angent Youngs m odulus. Generally, the secant modulus encompassed end effects as well as micro cracking which induces a reduction in modulus v ersus the tangent modulus (Figure 5.3 ). All observed loss in tangent modulus (i.e. micro cracking) was found to occur beyond 6 to 7% strain, which for shafts of interest (6 to 9 ft) equated to a vertical movement greater than two inches or the AASHTO service state. Also presented in each f igure are summary statistics (median, mean, standard deviation, and C V) of the data as well as a lognormal distribution fit to the data. C ompari ng Figures 5 4 and 5 5 the mean of the tangent modulus wa s approximately 1.8 times the secant mean modulus, but the variation, C V of each was quite similar. From the boring log s, the recoveries varied from 65% to 95% with a mean value of 75%. Based on Figure 3.20 an Em/Ei ratio of 0.45 was selected, and the m ass m odulus was computed using both the secant and tangent Ei moduli. Figure 5 4 and 5 5 show ed the probability density distribution for the t angent and secant m ass m odul i using both the recent LTSO4 data (102) as well as the original design data (16) set. Clearly it is apparent from Figures 5 4 and 5 5 that the mass modulus of the rock varie d over the site as well as in three dimensions below the shaft which must be PAGE 121 121 accounted for in Eqs. 3.6 through 3.8 As shown in Chapter 3 & 4 for the case of horizontally layered soil (variability only in the vertical direction) one should use the harmonic mean modulus, Eh (i.e., Eq 3.9) or: Fenton (2005) showed an excellent correlation with FEM analysis using the geometric mean, Eg (i.e., Eq. 3.10) or: It should be recognized that the geometric mean generally lies between the arithmetic and harmonic mean and was used in this projec t to assess contact stresses (Eq. 3.5 ). Due to the soil formation process, soil or rock variability wa s usually greater vertically than horizontally. With full boring generally for apart and shaft diameter in 4ft to 10ft range, it is assumed that the rock in modulus, random variables, wa s distributed vertically and constant horizontally i.e., horizontally layered soil The next question was the influence of geospatial correlation on the harmonic mean modulus, Eh. Specifically, Eq 4.7 require d the sum or average of 1/ Em over a distance ( H= 2 D) below the shaft as resulted by FEA analysis in Chapter 4. If Y = 1/ Em, the n the term inside the parentheses in Eq. 4 7 bec ame the simple arithmetic average, i.e. 1/n Y = F. Note, that because Y [ 1/ Em] wa s a random variable, so is F and it w ould have the general summary statistics, ( i.e. mF, and 2 F), as well. As discussed for side friction in Section 5.1, the mean of F, mF, w ould be the same as mY; however, the variance of F would be reduced by the averaging process (i.e ., 1/n Y = F) or 2 F = 2 Y, where 2 Y represent s the variance of 1/ Em over the site. As was found with side friction, Eq 4.8 for or the monograph, Figure 5 .1, may be used to assess for the case of D = 0 and an appropriate correlation length, a. Once the value of has been PAGE 122 122 assessed then the random function F (mF =mY; and 2 F = 2 Y) wa s also known and may be substituted back into Eq 4 7 to obtain Eh ( i.e., random function) In the case of the 17th Street Bridge, the 1/ Em was calculated for all tangent data. The mean of 1/ Em = mY = 0.0000919381 and the standard deviation of 1/ Em = F = 0.0000450687 were calculated Next, was obtained for H /D = 2 and a = 5, 10, and 15 from Figure 5 .1 or Eq. 4.8 which were used to find 2 F = 2 Y. Subsequently, a Mont e Carlo simulation was performed to generate typical F values which were substituted into Eq 4 7 to generate the distributions shown in Figures 5 6 5 7 and 5 8 Of particular interest wa s a comparison of the harmonic m ean m odulus, Eh, with the original field data, Em, as well as the inverted lognormal distribution which is the dash curve in Figure 5 4 Its summary statistics show ed a mean of 13, 481 ksf, a standard deviation of 6569 ksf, and a C V equal to 0. 487 The dash curve represent ed what the field samples should look like if enough samples were recovered and the mass modulus, Em, was lognormally distributed. As expected, the harmonic mean, Eh wa s reduced from both the field sample mean ( 13,481 ksf) Specifically, due to the spatial correlation re presented in the c ovariance function as lengths of 5 10, and 15 ft, the C V of Eh was reduced to 0.303 0. 383 and 0. 426 respectively, compared to the C V of Em. The highest reduction in C V of Eh wa s due to the lack of correlation between Em over short dist ances (i.e. the data were more random), which result ed in lower variability in Eh when averaged harmonically T his was already proven in Chapter 4. Using the harmonic m ean f unction, Eh (Figures 5 6 5 8 ), in Eqs 3.6 3.8 and Eq. 3.5 the distribution of the contact stress, qb at the bottom of the drilled shaft was obtained for a top shaft movement of 1.6 inches. This is shown in Figures 5 9 ~ 5 1 1 PAGE 123 123 Note that the f igures represent ed the expected distribution of end bearing on the east side of the site where all the data were collected (i.e. in the vicinity of LTSO4). Variograms developed for the data showed a typical vertical correlation length of approximately 8 ft, which from Figures 5 9 and 5 1 0 suggest ed a mean tip resistance of between 126 1 30 ksf an d a standard deviation above and below the mean of 32 42 ksf. The actual recorded tip resistance was 130 ksf for LTSO4 and 10 0 ksf for LTSO3. Both were close to the mean and well within the one standard deviation ( 3242 ksf). Knowing the distribution of t he end bearing, the LRFD resistance factors, may be assessed for the shafts from Eq 5. 5 for multiple reliability index values, (e.g., = 2.5 and 3). The computed values were also shown in each f igure depending on the correlation length, a. It wa s apparent that there wa s a significant effect of a on the factor for a specific reliability index For example, in the case of = 3.0, varie d from 0.5 8 to 0. 44 and the design end bearing ranges from 0.5 8 (1 30 ksf) = 75.4 ksf to 0. 44 (1 30 ksf) = 57.2 ksf. In addition, to estimate the variability of tip displacement with fixed load as conducted in Chapter 4, Eq. 5. 7 as modified from Eq. 3.5 was used T he actual recorded tip resistance 130 ksf for LTSO4 was used as the fixed load. 1/0.67 bt(q/)W ( 5. 7 ) With a typical vertical correlation length of approximately 8 ft, which from Figure 5. 1 3 and Figure 5. 1 4 suggested a mean displacement of 0.156~0.158 ft (1.87~1.9 in) and a standard deviation of 0.06~0.08 ft (0.9~1in). The actual recorded displacement was 1.9in for LTSO4 and 1.8in for LTSO3. B oth were close to the mean and well within the one standard deviation. PAGE 124 124 O f interest was the estimation of variability of displacement based on variability of modulus. In Chapter 4, the variability of tip displacement was successfully estimated using Eq. 4.12 derived with no power of distribution of modulus. H owever, in Eq. 5. 7 there was the power function of distribution of modulus, i.e., inside of The following step was used for obtaining a simple power function from Eqs 3.6 ~ 3.8 1 U sing the Eqs. 3.6 ~ 3.8 and A = (L/D)0.5 and B = A a 10 10=0.37A0.15(A1)log0.130.15(A1)log cmEE b 10 10=0.14A+0.05(A1)log0.440.05(A1)log cmBEE 2 Using 1.a, 1.b and Eq. 3.8 3 1.5 1.5 20.50.10 (1)mAB E LA 3 For simpl e and B as function of Em, a 100.37A0.15(A1)log0.13C=10cE b 10(0.14A+0.05(A1)log0.44)=10cED c. 1 0.15(1) 10=0.15(A1)log()A mCE d 1 0.05(A1) 10=0.05(A1)log()mBDE 4 K nowing 10log0.43ln XX the constants of power function were found by a 100.43 log 10 10loglog()km m mE kEkm m where, m wa s a chosen value of Em The function y = log10(kEm) in equations 3.c and 3.d may be approximated by the power function yp = aEm b by requiring y(Em=m) = yp(Em=m) and y(Em=m) = yp(Em=m), where the prime indicates the first derivative(y = 1/Em and yp = PAGE 125 125 abEm b 1 ) with respect to Em and m is a chosen value of Em for which the approximation becomes exact. b k=F= 1 0.15(1) AC in 3.c and k=G= 1 0.05(1) AD in 3.d and substituting into 2. 5 Finally, J m Fm Gm m Fm GmHE E m Fm Gm A L A ) ( log 1 ) ( log 1 43 0 5 1 ) ( log 1 ) ( log 1 43 0 10 10 5 0 23 5 110 10 1010) ( log ) ( log) 1 ( 30 a H(0.07) and J(1.27) were obtained as function of D, L, Ec and m b 1.5 J b tmq WE H as same form of simple power function, Y=Xk T o estimate the variation of power function ( i.e., CV of Y based on CV of X ) take ln on the both side in power function, such as lnY=klnX. U sing Eqs. 4.1~4.4, take mean and variance of the normal distribution and back to the log normal mean and CV. 22k YXCV(1CV)1 ( 5. 8 ) W ith Eq. 5. 8 the CV of displacement at each correlation length ( i.e., 5 ft, 10 ft and 15 ft ) were calculated as 0.391 0.497 and 0.55, which were close to the CV of displacement from distribution. 5 .3 LRFD resistance factors, for End Bearing at the Fuller Warren Bridge Like the 17th Street Bridge, the Fuller Warren Bridge site was revisited and three new borings along with sixty three samples were recovered and tested by SMO personnel Similar to the 17th Street Bridge data, the mean of the tangent modulus was approximately 1.7 times the secant mean modulus. However, the C V of each were quite different: 0.4 9 & 0.55 for 17th Street Bridge and 1.27 and 1.16 for the Fuller Warren PAGE 126 126 Bridge. The latter will have a profound effect on the LRFD resistance factors, for the two sites. From the boring logs, recoveries varied from 58% to 94% with a mean value of 77%. Based on Figure 3.20 an Em/Ei ratio of 0.5 was selected, and the m ass m odulus was computed using both the secant and tangent Ei, moduli. Figures 5 1 5 and 5. 1 6 show ed the probability density distribution for the t ang ent and secant m ass m odul us using data between LTS3 and LT4 (63), as well as the original design data (33). In the case of Fuller Warren, the 1/ Em was calculated for all the new and old tangent data and wa s shown Figure 5 1 6 T he results showed a mean of 1 / Em = mY = 0.0004322,and a standard deviation of 1/ Em = F = 0.0004999 Next, was obtained for L/D = 2 and ranges a = 5, 10, and 15 ft, from which 2 F = 2 Y was found. Subsequently, a Monte Carlo simulation was performed to generate typical F values which were then substituted into Eq 3.10 to generate the distributions shown in Figures 5 1 7 5 1 8 and 5 19. Of interest wa s a comparison of the harmonic m ean m odulus, Eh, with the original field data, Em, as well as with the inverted lognormal. It has a summary statistics of mean = 5468.129 ksf, standard deviation= 6338.638 ksf, and a CV = 1. 16 as shown in Figure 5. 1 6 The dash curve represents what the field samples should look like if enough samples were recovered and the mass modulus, Em, was lognormally distributed. Specifically, due to spatial correlation represented in the covariance function as correlation length, a, the C V of Eh was reduced to 0. 7 0.8 8 and 1. 0 respectively, compared to the C V of Em. The largest reduction in the C V of Eh to 0. 4 6 wa s due to the PAGE 127 127 lack of correlation between Em over short distances (i.e. more random), which result ed in lower variability in Eh when averaged harmonically Using the harmonic m ean f unction, Eh, and Eq. 3.5 the distribution of the contact stress, qb at the bottom of the drilled shaft was obtained for a top shaft movement of 3.2 inches (field data), as shown in Figures 5. 2 0 5. 2 1 and 5 2 2 Note, the f igures represent the expected distribution of end bearing at the site where the data were collected (LT2LT4). Variograms developed for the data showed typical vertical correlation lengths of approximately 14 ft, which from Figures 5. 2 0 5. 2 2 suggest s a mean tip resistance of between 7 0 8 7 ksf and one standard deviation above and below the mean of 40~70 ksf, respectively. The actual recorded tip resistances were 65 ksf for LT3, 108 ksf for LT 4 and 160 ksf for LT 2. T he results were close to the mean and well within the one standard deviation ( 40 ksf to 70 ksf). The estimation of the variability of tip displacement was conducted with fixed load, i.e., the actual recorded tip resistance 65 ksf for average of LT 3 With typical vertical correlation length of approximately 14ft, which from Figure 5. 2 3 ~ Figure 5. 2 5 suggested a mean displacement of 0. 4 ~0. 425 ft( 4.8 i n~ 5.1 in ) and a standard deviation of 0. 39~ 0.58( 4.68~6.96in). The actual recorded displacement was 3.2in. The results were not close to the mean but within the one standard deviation. W ith Eq. 5. 8 the CV of displacement at each correlation length, i.e., 5f t, 10ft and 15ft, were calculated as 0.96, 1.24 and 1.43 which were close the CV of displacement from distribution. A comparison of LRFD resistance factors from the 17th Street and Fuller Warren Bridges wa s quite interesting (see Figures 5 9 5 1 1 and Figures 5 2 0 5 2 2 ) The PAGE 128 128 variability of the resistance factors was controlled by the variability of Eh. For example, the data from Fuller Warren had much higher variability ( 0. 7 0.99 ; shown in Figures 5 1 7 5. 19) versus the lower variability ( 0.3 0. 42; Figures. 5 6 5 8 ) seen in the 17th Street Bridge data The latter affected the variability in contact stresses which in turn affected values. For instance, the values ranged from 0. 580. 44 ( =3) and 0. 67 0. 52 ( =2.5) for the 17th Street Bridge data, whereas Fuller Warrens values ranged from 0. 23 0.12 7 ( =3) and 0.30.1 8 ( =2.5) as shown in Figures 5. 2 0 ~5. 2 2 Evidently, the design end bearing for both sites were quite different. In a ddition, if sufficient data was collected at any single pier/shaft, an individual LRFD resistance factor could be assessed and its value would most likely be higher than the value assigned to the entire site. PAGE 129 129 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 5 6 7 8 9 10 a/D []L/D [] 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 = 0.1 Figure 5 1 Thick Solid Contour Lines Repr e s 22 for a/D = [0, 20] and L/D = (L/a) for D = 0. Thin Dotted Lines are Parabolas, Along Which A = const. (Klammler, 2010) PAGE 130 130 0.10.10.10.20.20.20.30.30.30.40.40.40.50.50.50.60.60.60.70.70.70.80.80.80.9 0.91 COVR [] [] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 Figure 5 2 LRFD resistance factors, ,as a Function of Reliability I n R ( Klammler 2 010) PAGE 131 131 Figure 5 3 Secant vs. Tangent Youngs Modulus on 17th Street Bridge Data f rom LTSO4 PAGE 132 132 0 2 4 6 8 10 12 14 16 18 20 Probability(%) Range(Em,ksf) Figure 5 4 Tangent Mass Modulus of 17th Street Bridge (118 Values) 0 2 4 6 8 10 12 14 16 18 Probability (%) Range (Em,ksf) Figure 5 5 Secant Mass Modulus of 17th Street Bridge (118 values) PAGE 133 133 0 50 100 150 200 250 300 Probability (%) Range (Eh,ksf) Distribution of Eh, 17th Street for a=5ft Eh a=5ftM (ksf)11402.5627111909.629453614.105382CV0.303460775 Figure 5 6 Harmonic Mean Modulus, Eh, Assuming Correlation Length, a = 5 ft from 17th Street Bridge Data 0 50 100 150 200 250 Probability (%) Range (Eh,ksf) Distribution of Eh, 17th Street for a=10ft Eh a=10ftM (ksf)11668.5747412472.86854778.001941CV0.38307162 Figure 5 7 Harmonic Mean Modulus, Eh, Assuming Correlation Length, a = 10 ft from 17th Street Bridge Data PAGE 134 134 0 50 100 150 200 Probability (%) Range (Eh,ksf) Distribution of Eh, 17th Street for a=15ft Eh a=15ftM (ksf)11740.4063912805.679935452.527151CV0.425789742 Figure 5 8 Harmonic Mean Modulus, Eh, Assuming Correlation Length, a = 15 ft from 17th Street Bridge Data 0 50 100 150 200 250 300 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 Probability (%) Range (Qb,ksf) Distribution of Tip Resistance with =1.6in, 17th Street for a=5ft with Eh M (ksf)122.267126.14132.2196CV0.25543 2.5 0.67125 3 0.58581 a=5ft w/ Influence zone, 2D below Figure 5 9 Histogram/PDF of Contact Stress, qb, Using Eh with Correlation Length, a = 5ft from 17th Street Bridge Data PAGE 135 135 0 50 100 150 200 250 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 Probability (%) Range (Qb,ksf) Distribution of Tip Resistance with =1.6in, 17th Street for a=10ft with Eh M (ksf)124.678130.76741.9731CV0.32098 2.5 0.57082 3 0.48396 a=10ft w/ Influence zone, 2D below Figure 5 1 0 Histogram/PDF of Contact Stress, qb, Using Eh with Correlation Length, a = 10 ft from 17th Street Bridge Data 0 50 100 150 200 250 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 Probability (%) Range (Qb,ksf) Distribution of Tip Resistance with =1.6in, 17th Street for a=15ft with Eh M (ksf)125.328133.46447.4915CV0.35584 2.5 0.52317 3 0.43682 a=15ft w/ Influence zone, 2D below Figure 5 1 1 Histogram/PDF of Contact Stress, qb, Using Eh with Correlation Length, a = 15 ft from 17th Street Bridge Data PAGE 136 136 0 50 100 150 200 250 300 350 0 0.06 0.12 0.18 0.24 0.3 0.36 0.42 0.48 0.54 0.6 0.66 0.72 0.78 0.84 0.9 Probability (%) Range ( ,ft) Distribution of Displacement with qb=130ksf, 17th Street for a=5ft with Eh M (ft)0.146110.156950.06133CV0.39076 Figure 5 1 2 Histogram/PDF of Displacement Wt, Using Eh with Correlation Length, a = 5 ft from 17th Street Bridge Data 0 50 100 150 200 250 300 0 0.06 0.12 0.18 0.24 0.3 0.36 0.42 0.48 0.54 0.6 0.66 0.72 0.78 0.84 0.9 Probability (%) Range ( ,ft) Distribution of Displacement with qb=130ksf, 17th Street for a=10ft with Eh M (ft)0.141920.158530.07838CV0.49445 Figure 5 1 3 Histogram/PDF of Displacement Wt, Using Eh with Correlation Length, a = 10 ft from 17th Street Bridge Data PAGE 137 137 0 50 100 150 200 250 300 0 0.06 0.12 0.18 0.24 0.3 0.36 0.42 0.48 0.54 0.6 0.66 0.72 0.78 0.84 0.9 Probability (%) Range ( ,ft) Distribution of Displacement with qb=130ksf, 17th Street for a=15ft with Eh M (ft)0.140820.159650.0871CV0.54555 Figure 5 1 4 Histogram/PDF of Displacement Wt, Using Eh with Correlati on Length, a = 1 5 ft from 17th Street Bridge Data PAGE 138 138 0 5 10 15 20 25 Probability (%) Range (Em,ksf) Figure 5 1 5 Secant Mass Modulus of Fuller Warren Bridge (96 values) 0 2 4 6 8 10 12 14 16 18 20 Probability (%) Range (Em,ksf) Figure 5 1 6 Tangent Mass Modulus of Fuller Warren Bridge (96 values) PAGE 139 139 0 100 200 300 400 500 600 Probability (%) Range (Eh,ksf) Distribution of Eh, Fuller Warren for a=5ft Eh a=5ftM (ksf)2839.0902123504.1211192475.5851CV0.706478177 Figure 5 1 7 Geometric Mean Modulus, Eh, Assuming a Correlat ion Length, a = 5 ft from Fuller Warren Bridge Data 0 50 100 150 200 250 300 350 400 450 Probability (%) Range (Eh,ksf) Distribution of Eh, Fuller Warren for a=10ft Eh a=10ftM (ksf)3129.4272584213.8965543724.830951CV0.883939818 Figure 5 1 8 Geometric Mean Modulus, Eh, Assuming a Correlation Length, a = 10 ft from Fuller Warren Bridge Data PAGE 140 140 0 50 100 150 200 250 300 350 400 450 Probability (%) Range (Eh,ksf) Distribution of Eh, Fuller Warren for a=15ft Eh a=15ftM (ksf)3297.7879154657.2022354652.275009CV0.99894202 Figure 5 19. Geometric Mean Modulus, Eh, Assuming a Correlation Length, a = 15 ft from Fuller Warren Bridge Data PAGE 141 141 0 50 100 150 200 250 300 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 Probability (%) Range (Qb,ksf) Distribution of Tip Resistance with =3.2in, Fuller Warren for a=5ft with Eh M (ksf)60.166669.979640.4733CV0.57836 2.5 0.30158 3 0.22938 a=5ft w/ Influence zone, 2D below Figure 5 2 0 Histogram/PDF of Contact Stress, qb, Using Eh with a Correlation Length, a = 5 ft from Fuller Warren Bridge Data 0 50 100 150 200 250 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 Probability (%) Range (Qb,ksf) Distribution of Tip Resistance with =3.2in, Fuller Warren for a=10ft with Eh M (ksf)65.296380.809357.9344CV0.71693 2.5 0.21767 3 0.15708 a=10ft w/ Influence zone, 2D below Figure 5 2 1 Histogram/PDF of Contact Stress, qb, Using Eh with a Correlation Length, a = 10 ft f rom Fuller Warren Bridge Data PAGE 142 142 0 50 100 150 200 250 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 Probability (%) Range (Qb,ksf) Distribution of Tip Resistance with =3.2in, Fuller Warren for a=15ft with Eh M (ksf)68.237887.330969.7517CV0.79871 2.5 0.18114 3 0.127 a=15ft w/ Influence zone, 2D below Figure 5 2 2 Histogram/PDF of Contact Stress, qb, Using Eh with a Correlation Length, a = 15 ft from Fuller Warren Bridge Data 0 100 200 300 400 500 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 Probability (%) Range ( ,ft) Distribution of Tip Resistance with qb=65ksf, Fuller Warren for a=5ft with Eh M (ft)0.299260.409340.38922CV0.95084 Figure 5 2 3 Histogram /PDF of Displacement Wt, Using Eh with Correlation Length, a = 5 ft from Fuller Warren Bridge Data PAGE 143 143 0 100 200 300 400 500 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 Probability (%) Range ( ,ft) Distribution of Tip Resistance with qb=65ksf, Fuller Warren for a=10ft with Eh M (ft)0.264860.419560.52591CV1.25346 Figure 5 2 4 Histogram/PDF of Displacement Wt, Using Eh with Correlation Length, a = 10 ft from Fuller Warren Bridge Data 0 100 200 300 400 500 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 Probability (%) Range ( ,ft) Distribution of Tip Resistance with qb=65ksf, Fuller Warren for a=15ft with Eh M (ft)0.2480.427520.58348CV1.3648 Figure 5 2 5 Histogram/PDF of Displacement Wt, Using Eh with Correlation Length, a = 1 5 ft from Fulle r Warren Bridge Data PAGE 144 144 CHAPTER 6 C ONCLUSION This dissertation encompasses the development of axial, lateral, and rotational tip models of short drilled shafts founded in heterogeneous limestone conditions Focus has been made to characterize the tip behavi or of large diameter but relatively short shafts under heterogeneous rock conditions. In this study, original contributions have been made by the author: A one dimensional (1D) lateral spring model to simulate the tip lateral behavior of the drilled shaft with L/D ratio ranging one to three has been developed as a function of rock strength and normal stress A 1 D axial spring model that includes spatial variability and correlation of rock modulus under serviceability limit has been developed to simula te the tip resistance of the drilled shaft in heterogeneous limestone. An LRFD material resistance factor of heterogeneous limestone has been evaluated in consideration of the v ariability of axial tip stiffness To study t ip behavior of drilled shafts the total of 16 centrifuge tests were performed, which varied shaft dimensions, rock strengths, rock recovery, Youngs modulus, and layering. To ensure repeatability, each test was performed twice with synthetic limestone. After assessing the axial tip resis tance during axial load, the shafts tip shear resistance versus lateral tip displacement was found under combined load as shown in Figure 3.41b. Specifically for a given lateral tip displacement the measured tip shear was analyzed to obtain the normali zed tip shear (i.e. shear stress minus cohesion divided by axial tip stress). Multiplying the normalized tip shear by axial tip stre ss and adding cohesion was found equal to the mobilized tip shear. The bilinear representation cover ed typical rock stren gths from 10 to 30 tsf and was obtained from 16 different types of boundaries conditions ( e.g. L/D = 1 and 3; homogeneous and layered scenarios ) The bilinear characterization represent ed elastic behavior up to PAGE 145 145 failure at which point no additional shear was mobilized. The values of the normalized shear failure were developed for rock ang les of internal friction of 28 and 45 degrees with unconfined strengths, qu, of 10 and 30 tsf, respectively. The lateral tip shear model was mainly c ontrolled by the str ength of rock and the diameter of shaft. Once the axial and shear resistance of the tip has been determined, the moment versus rotation model suggested by Bell (1991) was reco mmended, as shown in Figure 3. 49 b The relationship between moment and rotation w a s given by Eq. 3.24 and was function of tip she ar and lateral tip displacement Th is study also found that the tip lateral and rotational behavior s of drilled shaft was significantly affected by the tip axial resistance The following conclusions on axial tip resistance were drawn from the centrifuge test s and the FEA results : Florida limestones mass modulus (i.e., Em) is sensitive to recoveries and is proportional to intact Youngs modulus, Ei, from core samples as shown in Figure 3.20 The mass modulus, Em, should be assessed from the tangent Youngs modulus, Ei,; use of secant Youngs Modulus will generally be 1.8 to 2.0 times lower than tangent values (Fuller Warren and 17th Bridge sites) The axial tip resistance was very sensitive to mass modulus o f elasticity of rock Em, within a zone of two diameters below the tip of the shaft in serviceability limit The numerical solution with spatial variability in heterogeneous rock has suggested that the use of either the harmonic mean or the geometric mean provides reasonable prediction of axial tip resistance. For the multi directional spatial variability, the geometric average was recommended. H owever, the harmonic average was found to agree favorably with the experimental results obtained from a condition simulated for 1 D spatial variability i.e., variability of rock modulus considered in the vertical direction only Examples of predicting the axial tip resistance versus axial tip displacement using the FHWA method (ONeill 1997 ) and linear stiffness rel ation by Misra (2006) and Doherty (2006) were shown in Figure 3.32 and 3.33 from centrifuge test. B oth the harmonic and geometric mean were used for predicting the axial tip behavior in layered PAGE 146 146 system These were assessed over a depth of 2D below the shaft tip. T he following summaries of the axial tip behavior from centrifuge test were observed: Initial tip stiffness, up to 1 axial tip displacement, can be estimated by Misra and Doherty s method with secant modulus. T he nonlinear axial tip resistance, abov e 1 axial tip displacement, can be estimated by O Neil s Method with harmonic mean within 2D below the shaft tip of tangent modulus in spatial variability. T o study the influence of the spatial variation of the modulus of elasticity on the axial tip resi stance vs. axial tip displacement in serviceability the heterogeneous rock conditions were considered in the FEA model that simulate d a typical field condition in Florida with perfectly horizontally layered limestone. The modulus of elasticity of the heterogeneous limestone was characterized as a lognormal random variable with a coefficient of variation ( CVE), ranging from 0.20.6 and vertical correlation lengths (va) ranging from 5 15 ft. For each random heterogeneous condition g enerated from a giv en set of rock characteristics (such as mean, CVE and av), 1,000 FEA simulations were performed on a given shaft L/D. Statistical a nalysis of the FEA results revealed below: T he mean axial tip response of the heterogeneous system obtaine d from the simulations can be linearly estimated from the axial tip response of homogeneous system with CVE as shown in Eq. 4.13. The effective average depth of elastic modulus (2D below tip drilled shaft) should be used to predict the mean axial tip response. The homogeneous axial tip response can be obtained from the Misra (2006) and Doherty (2006). T he variability of the axial tip displacement (Eq. 4.12) should be estimated as function of variability of elastic modulus and correlation length described in variance reduction factor (Klammler, 2010). PAGE 147 147 However, above one in ch axial tip displacement, the variability of the axial tip displacement (Eq. 5.8) can be estimated as function of variability of elastic modulus and correlation length described in a varia nce reduction factor with a nonlinear relation. A s ignificant finding of the present study suggest s that smaller diameter drilled shafts ha ve more variance of tip resistance than larger diameter drilled shafts in heterogeneous rock conditions Therefore, the design resistance factor ( ) of the rock in LRFD should be reduced to account for the size effect on the axial tip resistance. Moreover, the variance of the axial capacity of the deep foundation may be computed by summing the variance of side resistanc e (Klammler, 2010) with the variance of tip resistance Finally, the effort studied the LRFD resistance factors ( ) in light of the new harmonic modulus Currently one set of fixed resistance factors are used for drilled shafts in Florida. The s patial v ariability and associated correlation lengths (i.e., covariance function), suggest that LRFD resistance factors should vary from site to site. T o study the effect of the spatial variability on LRFD resistance factors, two field investigations (i.e., Fuller Warren and 17th Bridge ) were investigated The analysis revealed: T he axial tip axial resistance is sensitive to Em assessed from the tangent Ei based on recovery and RQD. As the correlation length increas es with a high CVE, the variability of Qb of CVR i ncreases, resulting in lower LRFD resistance factors. o CVEm and correlation length of 17th street (0.49 & 8ft) and Fuller Warren (1.17 & 14ft). o CVE h of 17th street (0.3~0.43) and Fuller Warren (0.7~1) due to variance reduction factor o CVqb or CVR of 17th street (0.26~0.36) and Fuller Warren (0.58~0.8). PAGE 148 148 o LRFD resistance factor of 17th street (0.58~0.44 at =3) and Fuller Warren(0.23~0.13 at =3). It is recommended that individual LRFD assessment of axial tip resistance be performed as a site by site. PAGE 149 149 APPEND IX A C OMPRESSIVE FORC E ( ON X AXIS KIPS ) ALONG D EPTH ( ON Y AXIS FT ) FROM E ACH P AIR OF S TRAIN G AGE WITH T REND L INE S OF LATERAL LOADING STEP 10 9 8 7 6 5 4 3 2 1 0 0 500 1000 1500 2000 2500 10 9 8 7 6 5 4 3 2 1 0 0 500 1000 1500 2000 2500 3000 ( a ) (b) Figure A 1 Result from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 10 9 8 7 6 5 4 3 2 1 0 0 1000 2000 3000 4000 5000 6000 p() 10 9 8 7 6 5 4 3 2 1 0 0 2000 4000 6000 8000 10000 12000 p() ( a ) (b) Figure A 2 Result from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % St yrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 PAGE 150 150 10 9 8 7 6 5 4 3 2 1 0 0 1000 2000 3000 4000 5000 10 9 8 7 6 5 4 3 2 1 0 0 1000 2000 3000 4000 5000 6000 ( a ) (b) Figure A 3 Result from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 10 9 8 7 6 5 4 3 2 1 0 0 2000 4000 6000 8000 10000 12000 10 9 8 7 6 5 4 3 2 1 0 0 2000 4000 6000 8000 10000 p() ( a ) (b) Figure A 4 Result from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 7 6 5 4 3 2 1 0 0 500 1000 1500 2000 2500 Depth(ft) 7 6 5 4 3 2 1 0 0 500 1000 1500 2000 Depth(ft) ( a ) (b) Figure A 5 Result from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 151 151 7 6 5 4 3 2 1 0 0 1000 2000 3000 4000 5000 6000 p() 7 6 5 4 3 2 1 0 0 1000 2000 3000 4000 5000 6000 7000 ( a ) (b) Figure A 6 Result from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 35.00 30.00 25.00 20.00 15.00 10.00 5.00 0.00 0.00 2000.00 4000.00 6000.00 8000.00 10000.00 Tfd F(ki) 35.00 30.00 25.00 20.00 15.00 10.00 5.00 0.00 0.00 1000.00 2000.00 3000.00 4000.00 5000.00 6000.00 7000.00 ( a ) (b) Figure A 7 Result from 9 ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 30.00 25.00 20.00 15.00 10.00 5.00 0.00 0.00 5000.00 10000.00 15000.00 20000.00 25000.00 p() 30.00 25.00 20.0015.00 10.00 5.00 0.00 0.00 5000.00 10000.0015000.00 20000.00 25000.00 ( a ) (b) Figure A 8 Result from 9 ft Diamet er 27 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 152 152 20 18 16 14 12 10 8 6 4 2 0 0 1000 2000 3000 4000 5000 6000 20 18 16 14 12 10 8 6 4 2 0 0 1000 2000 3000 4000 5000 6000 ( a ) (b) Figure A 9 Result from 6 ft Diameter 18 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 20 18 16 14 12 10 8 6 4 2 0 0 5000 10000 15000 20000 Tfd F(ki) p() 25 20 15 10 5 0 0 2000 4000 6000 8000 10000 12000 14000 ( a ) (b) Figure A 10. Result from 6 ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 153 153 APPENDIX B O UTPUT V OLTAGE (V OLT ON Y A XIS ) FROM EACH STRAIN G AGES VS. TESTING T IME (S EC ON X A XIS ) WITH L OADING S ( a ) (b) Figure B 1 Output Voltage from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 (a) (b) Figure B 2 Output Voltage from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 PAGE 154 154 (a) (b) Figure B 3 Output Voltage from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ( a ) (b) Figure B 4 Output Voltage from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ( a ) (b) Figure B 5 Output Voltage from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 155 155 ( a ) (b) Figure B 6 Output Voltage from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 ( a ) (b) Figure B 7 Output Voltage from9 ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ( a ) (b) Figure B 8 Output Voltage from9 ft Diameter 27 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 156 156 ( a ) (b) Figure B 9 Output Voltage from6 ft Diameter 18 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 ( a ) (b) Figure B 10. Output Voltage from6 ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 157 157 APPENDIX C M OMENT C URVE WITH LOAD STEP Moment vs. Depth 15 10 5 0 5 10 15 0 10000 20000 30000 40000 50000 60000 Moment(kipsft) Elevation(ft) Moment vs. Depth 10 5 0 5 10 15 0 10000 20000 30000 40000 50000 60000 Moment(kipsft) Elevation(ft) ( a ) (b) Figure C 1 Moment Curve with D ifferent T op L ateral D isplacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of S haft a) Test No. 1 b)Test No. 2 Moment vs. Depth 15 10 5 0 5 10 15 0 20000 40000 60000 80000 100000 Moment(kipsft) Elevation(ft) Moment vs. Depth 15 10 5 0 5 10 15 0 20000 40000 60000 80000 100000 Moment(kipsft) Elevation(ft) ( a ) (b) Figure C 2 Moment Curve with D ifferent T op L ateral D isplacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 PAGE 158 158 Moment vs. Depth 15 10 5 0 5 10 15 0 10000 20000 30000 40000 50000 60000 Moment(kipsft) Elevation(ft) Moment vs. Depth 15 10 5 0 5 10 15 0 10000 20000 30000 40000 50000 Moment(kipsft) Elevation(ft) ( a ) (b) Figure C 3 Moment Curve with D ifferent T op L ateral D isplacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Moment vs. Depth 15 10 5 0 5 10 0 20000 40000 60000 80000 100000 120000 Moment(kipsft) Elevation(ft) Moment vs. Depth 15 10 5 0 5 10 15 0 20000 40000 60000 80000 100000 120000 Moment(kipsft) Elevation(ft) (a) (b) Figure C 4 Moment Curve with D ifferent T op L ateral D isplacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) T est No. 1 b) Test No. 2 PAGE 159 159 Moment vs. Depth 8 6 4 2 0 2 4 6 8 10 12 0 5000 10000 15000 20000 25000 30000 35000 Moment(kipsft) Elevation(ft) Moment vs. Depth 8 6 4 2 0 2 4 6 8 10 0 5000 10000 15000 20000 25000 30000 35000 Moment(kipsft) Elevation(ft) ( a ) (b) Figure C 5 Moment Curve with D ifferent T op L ateral D isplacement from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Moment vs. Depth 15 10 5 0 5 10 15 10000 0 10000 20000 30000 40000 50000 Moment(kipsft) Elevation(ft) Moment vs. Depth 8 6 4 2 0 2 4 6 8 10 10000 0 10000 20000 30000 40000 50000 60000 Moment(kipsft) Elevation(ft) (a) (b) Figure C 6 Moment Curve with D ifferent T op L ateral D isplacement from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 160 160 Moment vs. Depth 35.00 30.00 25.00 20.00 15.00 10.00 5.00 0.00 5.00 10.00 20000 0 20000 40000 60000 80000 Moment(kipsft) Elevation(ft) Moment vs. Depth 35.00 30.00 25.00 20.00 15.00 10.00 5.00 0.00 5.00 10.00 15.00 0 10000 20000 30000 40000 50000 60000 Moment(kipsft) Elevation(ft) (a) (b) Figure C 7 Moment Curve with D ifferent T op L ateral D isplacement from 9 ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Moment vs. Depth 30.00 25.00 20.00 15.00 10.00 5.00 0.00 5.00 10.00 15.00 5000 0 5000 10000 15000 20000 25000 30000 Moment(kipsft) Elevation(ft) Moment vs. Depth 30.00 25.00 20.00 15.00 10.00 5.00 0.00 5.00 10.00 15.00 20000 0 20000 40000 60000 80000 Moment(kipsft) Elevation(ft) (a) (b) Figure C 8 Moment Curve with D ifferent T op L ateral D isplacement from 9 ft Diameter 27 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 161 161 Moment vs. Depth 25.00 20.00 15.00 10.00 5.00 0.00 5.00 10.00 0 5000 10000 15000 20000 25000 Moment(kipsft) Elevation(ft) Moment vs. Depth 25.00 20.00 15.00 10.00 5.00 0.00 5.00 10.00 5000 0 5000 10000 15000 20000 25000 Moment(kipsft) Elevation(ft) (a) (b) Figure C 9 Moment Curve with D ifferent T op L ateral D isplacement from 6 ft Diameter 18 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Moment vs. Depth 20.00 15.00 10.00 5.00 0.00 5.00 10.00 15.00 10000 0 10000 20000 30000 40000 50000 60000 70000 Moment(kipsft) Elevation(ft) Moment vs. Depth 25.00 20.00 15.00 10.00 5.00 0.00 5.00 10.00 15.00 40000 20000 0 20000 40000 60000 Moment(kipsft) Elevation(ft) (a) (b) Figure C 10. Moment Curve with D ifferent T op L ateral D isplacement from 6 ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 162 162 APPENDIX D S HEAR CURVE WITH LOAD STEP Shear vs. Depth 15 10 5 0 5 10 15 8000 6000 4000 2000 0 2000 4000 6000 Shear(kips) Elevation(ft) Shear vs. Depth 10 5 0 5 10 15 10000 5000 0 5000 10000 Shear(kips) Elevation(ft) ( a ) (b) Figure D 1 Shear Curve with D ifferent T op L ateral D isplacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test N o. 2 Shear vs. Depth 15 10 5 0 5 10 15 15000 10000 5000 0 5000 10000 Shear(kips) Elevation(ft) Shear vs. Depth 15 10 5 0 5 10 15 15000 10000 5000 0 5000 10000 Shear(kips) Elevation(ft) ( a ) (b) Figure D 2 Shear Curve with D ifferent T op L ateral D isplacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock with 50 % Styrofoam Layering below 1D below from Tip of Shaft a) Test No. 1 b)Test No. 2 PAGE 163 163 Shear vs. Depth 15 10 5 0 5 10 15 6000 4000 2000 0 2000 4000 6000 Shear(kips) Elevation(ft) Shear vs. Depth 15 10 5 0 5 10 15 6000 4000 2000 0 2000 4000 6000 Shear(kips) Elevation(ft) ( a ) (b) Figure D 3 Shear Curve with D ifferent T op L ateral D isplacement from 9 ft Diameter 9 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Shear vs. Depth 15 10 5 0 5 10 15000 10000 5000 0 5000 10000 Shear(kips) Elevation(ft) Shear vs. Depth 15 10 5 0 5 10 15 10000 5000 0 5000 10000 Shear(kips) Elevation(ft) ( a ) (b) Figure D 4 Shear Curve with D ifferent T op L ateral D isplacement from 9 ft Diameter 9 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 164 164 Shear vs. Depth 8 6 4 2 0 2 4 6 8 10 12 6000 4000 2000 0 2000 4000 6000 Shear(kips) Elevation(ft) Shear vs. Depth 8 6 4 2 0 2 4 6 8 10 9000 6000 3000 0 3000 6000 9000 Shear(kips) Elevation(ft) ( a ) (b) Figure D 5 Shear Curve with D ifferent T op L ateral D isplacement from 6 ft Diameter 6 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Shear vs. Depth 8 6 4 2 0 2 4 6 8 10 12 16000 8000 0 8000 16000 Shear(kips) Elevation(ft) Shear vs. Depth 8 6 4 2 0 2 4 6 8 10 16000 8000 0 8000 16000 Shear(kips) Elevation(ft) ( a ) (b) Figure D 6 Sh ear Curve with D ifferent T op L ateral D isplacement from 6 ft Diameter 6 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 165 165 Shear vs. Depth 35 30 25 20 15 10 5 0 5 10 10000 5000 0 5000 10000 Shear(kips) Elevation(ft) Shear vs. Depth 35 30 25 20 15 10 5 0 5 10 15 5000 0 5000 10000 Shear(kips) Elevation(ft) ( a ) (b) Figure D 7 Shear Curve with D ifferent T op L ateral D isp lacement from 9 ft Diameter 27 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Shear vs. Depth 30 25 20 15 10 5 0 5 10 15 2000 1000 0 1000 2000 3000 4000 Shear(kips) Elevation(ft) Shear vs. Depth 30 25 20 15 10 5 0 5 10 15 4000 2000 0 2000 4000 6000 8000 10000 Shear(kips) Elevation(ft) ( a ) (b) Figure D 8 Shear Curve with D ifferent T op L ateral D isplacement from 9 ft Diameter 27 f t Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 166 166 Shear vs. Depth 25 20 15 10 5 0 5 10 2000 1000 0 1000 2000 3000 4000 Shear(kips) Elevation(ft) Shear vs. Depth 25 20 15 10 5 0 5 10 2000 1000 0 1000 2000 3000 4000 Shear(kips) Elevation(ft) ( a ) (b) Figure D 9 Shear Curve with D ifferent T op L ateral D isplacement from 6 ft Diameter 18 ft Embedded Length in 10 tsf Rock a) Test No. 1 b) Test No. 2 Shear vs. Depth 20 15 10 5 0 5 10 15 25000 20000 15000 10000 5000 0 5000 10000 Shear(kips) Elevation(ft) Shear vs. Depth 25 20 15 10 5 0 5 10 15 25000 20000 15000 10000 5000 0 5000 10000 Shear(kips) Elevation(ft) ( a ) (b) Figure D 10. Shear Curve with D ifferent T op L ateral D isplacement from 6 ft Diameter 18 ft Embedded Length in 30 tsf Rock a) Test No. 1 b) Test No. 2 PAGE 167 167 APPENDIX E R ELATION KV( VERTICAL STIFFNESS ) VERSUS J( RIGIDITY PARAMETER ) 3 3(34) 24(1)(1)s sREt GRJ Where, is Possion s ratio of limestone s is Possion s ratio of footing s E is Elastic modulus of footing RG is Shear Modulus of limestone t is Thickness of footing R is Radius of footing Figure E 1 The r elation J and Kv: a( =0.2) and b( =0.499) PAGE 168 168 LIST OF REFERENCES ADINA R & D, Inc. (2006). ADINA u ser m anual Version 8.5, Watertown, MA American Association of State Highway and Transportation Officials ( AASHTO ). (2009). LRFD Bridge Design Specifications, 4th ed. 2009 Interim Revisions American Society for Testing and Materials (ASTM). (1997). C207, Standard specification for h ydrated lime for m asonry p urpose, Philadelphia, PA American Society for Testing and Mat erials (ASTM). (2002). D293895, Standard t est m ethod for u nconfined compressive strength of i ntact r ock c ore specimens, Philadelphia, PA American Society for Testing and Materials (ASTM). (2004). D3967 95a, Standard t est m ethod for s plitting t ensile st rength of i ntact r ock core s pecimens, Philadelphia, PA. American Society for Testing and Materials (ASTM). (2007). D2850 03a, Standard t est m ethod for u nconsolidatedu ndrained t riaxial compression t est on c ohesive soils, Philadelphia, PA Bathe, K. J. (1 996). Finite Element Procedures, The Pren tice Hall, Englewood C liffs, NJ Bell, R. W. ( 1991) The a nalysis of o ffshore f oundations s ubjected to combined l oading, M S Thesis, Univ ersity of Oxford, Oxford, U.K. Bridge Software Institute (BSI). (2010) FB Mu ltip ier u sers m anual Department of Civil and Coastal Engineering, University of Florida, Gainesville, F L. Chandrasekaran, V. S. (2001). Numerical and centrifuge modeling in soil structure interaction. 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PAGE 171 171 BIOGRAPHICAL SKETCH Jeongsoo Ko was born in Gwangju, South Korea. He spent his childhood in that beautiful city and completed primary, middle, high school, and university, leaving only to complete his military service H e was accepted to study civil engineering at Chosun University, Gwangju, South Korea in 1994. He earned his BSc in the Civil Engineering Department at Chosun University in March 2002. During his undergraduate studies he spent 26 months performing military service at the border between South and North Korea. He realiz ed that his knowledge was not yet adequate for deal ing with real world civil engineering problem s, so he decided to go abroad for advanced education. H e was accepted by the Civil and Coastal Engineering Department at the University of Florida and moved to the U S in Ju l y 2003. He had studied and worked on thick lift compaction with Dr. McVay for about one year. I ncluding this period, he spent the 18 months acquiring a background in Geotechnical Engineering. H e earned his master s degree in August 2005. In August 2005, he began his Ph.D. degree under the guidance of Dr. McVay in C ivil E ngineering at University of Florida. H is research interests include the tip behavior of drilled shaft s and spatial variability 