harm“ ‘ ‘i', "a ‘ mu. .‘m- A L. MU," ' 1571‘”. W t ' ‘ v a “"v [- -:_~ A x,~~_ r-"vm‘ . ‘ t g. ”4 w.. "“3 ESIS J Illllllillllllllllllllllllflllllllllllllll 31293 01019 8442 This is to certify that the dissertation entitled APPLICATIONS OF GEOSTATISTICS TO SETTLEMENT PROBLEMS presented by Mostafa Kamal Ashoor has been accepted towards fulfillment of the requirements for Doctor of PhilosophX degreein Civil Engineering Major rofessor Date February 23, 1994 M5u;.nnnrn.mm.‘..1 4 .-_' In" - r .- .- 0.12771 LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOlD FINES return on or before date due. DATE DUE DATE DUE DATE DUE MSU IsAn Affirmative Action/Equal Opportunity Institution 7 , , W , czlcirc para—9.1 APPLICATIONS OF GEOSTATISTICS TO SETTLEMENT PROBLEMS By Mostafa Kamal Ashoor A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil and Environmental Engineering 1994 it ABSTRACT APPLICATIONS or GEOSTATISTICS To SETTLEMENT PROBLEMS By Mostafa Kamal Ashoor This research investigates improved techniques for estimating and modeling the settlement and differential settlement of shallow foundations on noncohesive soils using standard penetration test (SPT) values. Two geostatistical methods (trend surfaces and Kriging) are employed to model the spatial variability of the soil standard penetration resistance values (commonly known as N values) in a three-dimensional field. Lack of homogeneity in the N-value data in the vertical direction: 1. is analyzed geotechnically considering such factors as the soil’s relative density, overburden pressure, stiffness. . .etc, and; 2. is accounted for by a nonconstant—mean assumption, and; 3. is tested by statistical multiple comparison techniques. ‘ Aside from the difficulties of measurement bias; which is accounted for by the above-mentioned modeling, the N values vary from point to point, and the question remains as to how to combine the varying measurements of N values within the depth 0f influence under the foundation footing into one design N value to be used for the deterministic models for settlement estimation. This question is tackled in this research by using the two geostatistical methods to generate a two—point estimate for the design N value. The two point estimates are in W_gggfl_ * fl _, Mostafa Kamal Ashoor tum weighted to obtain a single value. It is hoped that this proposed estimate will be favoured by foundation designers on the grounds of simplicity. Moreover, this two-point estimate is shown to be: Comparable to the current procedures of estimating the design N value in the 1. sense that the N values at different depths are given similar weights in both methods. 2. Easy to be used by the spatial models in a way that can help in transforming the spatial N (x,y,z) models into a planar N (x, y) or S(x, y) model. This latter model enables one to do such things as contouring analysis or planar settlement compari— sons and such like. The developed method is verified using simulated data of an assumed field. The practical reliability is then tested by conducting the modeling on available case histories. The modeling of N values and settlements proposed herein provides a capability to study the practicality of obtaining more N data in order to take advantage of the resulting lower degree of uncertainty. ACKNOWLEDGNIEN TS Researching and writing a dissertation is a job that is shared by more than the author. I had terrific support from my committee chairman and academic advisor, Professor Thomas Wolff. I wish to acknowledge my appreciation to his scientific talent, self denial and modesty. No words can express my gratefulness to him. As aformer engineer for the US Corps of Engineers, the greatest engineering agency in the world, his practical suggestions and criticisms were invaluable in alerting me to problem areas in the research and helping me to sort them out and improve the clarity of the deveIOped models. I shall remain ever grateful for his academic guidance and support. I wish to give special thanks and express my acknowledgment and gratitude to Professor Mark Snyder, who kindly, together with Professor Wolff, volunteered his time and effort to help me to keep going when things seemed to stop. His unceasing encouragement and scientific guidance helped make this research a reality. I sincerely wish to express my appreciation and gratitude to the chairperson of Civil and Environmental Engineering. Professor William Saul, for his assistance, support and his talented leadership. I also would like to extend my sincere appreciation and gratitude to the other members of my guidance committee, Professors William Taylor, Dennis Gilliland and Ronald Harichandran for their valuable suggestions and constructive criticisms. Finally, thanks to my family members; my wife Magda and my wonderful Children Radwa, Kamal and Sara. They helped me keep going and keep on schedule ("What chapter are you writing now, Dad?"). he TABLE OF CONTENTS LIST OF TABLES. LIST OF FIGURES. CHAPTER 1. BACKGROUND, OBJECTIVES, AND SCOPE. 1 . 1 Introduction. 1.2 The Framework Of Settlement Prediction From N Values. 1.3 Research Objectives. 1.4 Research Scope. 2. CURRENT PRACTICE IN SETTLEMENT PREDICTION USING THE SPT TEST 2.1 General. 2.2 SPT Test. 2.3 2.4 2.5 2.2.1 SPT Test Precision And Bias. 2.2.2 Practical Advantages Of The SPT Test. Measuring Settlements. Settlement Prediction Methods (Group 1). Settlement Prediction Methods (Group 2). iv Page 11 ll l2 l3 14 15 16 21 b!— TABLE OF CONTENTS, Continued CHAPTER VARIABILITY OF SOIL DATA AND SUMNIARY OF 3 . GEOSTATISTICAL MODELING METHODS. 3.1 Variability Of N Values. 3.1.1 Sources Of Variability Of N Values. 3.1.2 N—Value Depth Effect And Correction For Overburden Pressure. 3.2 Summary Of Geostatistical Modeling Methods. 3.2.1 The Random Field Theory. 3.2.2 Trend Surface Analysis. 3.2.2.1 The First Degree Polynomial Model. 3.2.2.2 The Second Degree Polynomial Model. 3.2.2.3 The Four—Dirnensional Trend Surface. 3.2.2.4 Measuring Goodness-Of—fit Of A Trend Surface 3.2.3 Interpolation Schemes For N Value Modeling. 3.2.3.1 Triangulation. 3.2.3.2 Kriging. 3.2.3.2.1 Underlying Concepts. 3.2.3.2.2 The Lagrange Parameter. 3.2.3.2.3 The Ordinary Kriging System. Page 25 25 26 27 30 30 35 38 39 4O 47 48 52 53 TABLE OF CONTENTS, Continued CHAPTER Page 3.2.3.2.4 The Covariance Matrix. 55 4. MODELING THE N—FUNCTION FOR SETTLEMENT PREDICTION. . 58 4.1 General. 58 4.2 The "Two—Point" Estimate For Settlement Prediction. .5 9 4.3 Application Of "Trend Surface" Theory To Obtain The "Two—Point" Estimate And Settlement Prediction. 61 4.3.1 Using The "Trend Surface" N Function To Predict The Settlements And Differential Settlements. _ 69 4.3.2 Constructing The Prediction Confidence Intervals For A Given Data Set. 75 4.4 Application Of The Kriging Technique To SPT Data. 78 4.4.1 Preparation Of N Data Before Modeling. 79 4.4.2 Kriging The N Regression Coefficients. - 80 4.4.3 Using The Estimated N Function To Predict The Settlements And Differential Settlements. 83 4.4.4 Constructing The Prediction Confidence Intervals For A Given Data Set. 84 5 . VERIFICATION OF THE DEVELOPED MODELS. 87 5.1 Verification Using Simulated Data Of An Assumed Field. 87 vi CHAPTER 5.1.1 5.1.2 5.1.3 TABLE OF CONTENTS, Continued Verification Of The Deve10ped Models By The Trend Surface Analysis. 5.1.1.1 Generating The Simulated "True Unobserved" And "Observed" N Values. 5.1.1.2 Using Trend Surface Analysis To Predict The N Values. 5.1.1.3 Predicting N Values At Eight Given Locations. 5.1.1.4 Analysis Of Variance. 5.1.1.5 Comments On Prediction Using Trend Surface Analysis. Verification Of The Deve10ped Models By The Kriging Technique. 5.1.2.1 Generating The N Values. 5.1.2.2 Using The Kriging Technique To Predict The N Values. 5.1.2.3 Analysis Of Variance. 5.1.2.4 Comments On The Prediction By Kriging. Verification Of The Developed "Two—Point" Estimate. 5.1.3.1 The "Eight-Point Estimate" Design. 5.1.3.2 The "Two-Point Estimate" Design. vii Page 88 88 92 98 99 100 101 101 107 116 116 118 120 121 TABLE OF CONTENTS, Continued HAPTER Page 5.2 Verification Using Case Histories. 123 5.2.1 Applying The Trend Surface alysis To Case History No.1 123 5.2.1.1 Project Description. 123 5.2.1.2 Footing Details. 124 5.2.1.3 SPT Locations. 124 5.2.1.4 Applying The Trend Surface Analysis Procedure. . 126 5.2.2 Applying The Kriging Technique To Case History No.1 144 5.2.2.1 Applying The Kriging Technique To The N Values. 144 5.2.2.2 Using The Estimated N Function To Obtain The Design N Value. 157 5.2.2.3 Using The Design N Value For Settlement Prediction. 15 7 5.2.2.4 Constructing The Prediction Intervals For The Estimated Values Of Both N And S. 159 viii .Iu. TABLE OF CONTENTS, Continued PTER Page 5.2.3 Summary And Analysis Of The Predicted Settlements For Six Case Histories. 162 5.2.3.1 The Case Histories Selection Criteria And Summary. 162 5.2.3.2 Analyzing The Results. 167 SUIVIIVIARY AND CONCLUSIONS. 171 6.1 Summary. . 171 6.2 Detailed Summary Of Recommended Procedures. 175 6.3 Conclusions. 180 6.4 Recommendations. 185 iNDIX A. 187 'NDIX B. 197 NDIX C. 209 OF REFERENCES. 275 LIST OF TABLES iLE Page Classification Of Relative Density In Terms Of N Values. 64 The Simulated "True Unobserved" N Values For Trend Surface. 90 The Simulated "Observed" N Values For Trend Surface. 91 The "Estimated Unobseved" N Values By Trend Surface VS. The "True Unobserved" N Values. 98 The Coordinates Of The location Of The Simulated "True Unobserved" Boring For Kriging. 102 The N Values Of The Simulated "T rue Unobserved" Boring "o". 103 The Simulated "Observed" Boring Locations For Kriging. 104 The N Values Of The Simulated "Observed" Borings. ' 106 The N Values Predicted By Kriging Vs. The Simulated "True Unobserved" N Values. 114 The "Eight-Point" Estimate Calculations. 120 The Predicted Settlements At Points Spaced At 20 ft For Case History No.1. 141 The Design N Values For The Six Case Histories. 165 The Estimated N Function And The Design N Values For The Six Case Histories. 166 Summary Of The Predicted Settlements Vs. The Measured Values For The Six Case Histories. 166 LIST OF TABLES, Continued BLE 1 Settlement Ratios For The Six Case Histories. Summary Table Of The Two Considered Methods. Correction Of N Values Of Boring No. (B-102) For Overburden Pressure. Correction Of N Values Of Boring No. (B-105) For Overburden Pressure. Correction Of N Values Of Boring No. (B—106) For Overburden Pressure. Correction Of N Values Of Boring No. (B-109) For Overburden Pressure. Correction Of N Values Of Boring No. (B-2) For Overburden Pressure. Correction Of N Values Of Boring No. (B—103) For Overburden Pressure. The N Data With The Spatial Locations For Case History No.1. The N Data With The Spatial Locations For The Upper layer. The N Data With The Spatial Locations For The Lower Layer. The Correction Factors For Overburden Pressure Of N Values Of Case History No.2. The Correction Of N Values Of Borings NO.(I) And N012)- The Correction Of N Values Of Borings NO.(3) And NO-(4)- Page 1 67 184 187 188 189 189 190 191 192 197 204 211 212 213 LIST OF TABLES, Continued E i Page The Correction Of N Values Of Borings NO.(5) And NO.(6). 214 The Correction Factors For Overburden Pressure Of N Values Of Case History No.3. 229 The Correction Of N Values Of Boring B1. 230 The Correction Of N Values Of Boring B2. 231 The Correction Of N Values Of Boring B3. '232 The Correction Of N Values Of Boring (D—5). 244 The Correction Of N Values Of Boring (D—8). 245 The Correction Of N Values Of Boring (9). 257 The Correction Of N Values Of Boring (34). 25 8 The Correction Of N Values Of Boring (416). 266 The Correction Of N Values Of Boring (417). 267 LIST OF FIGURES JRE Page Typical N (2) Functions. 6 N Value Versus Relative Density "Dr" And Vertical Effective Stress "pv’", (After D’Appolonia). 29 Parameters Of Homogeneous Randomly Varying Soil Properties. 31 Parameters Of Homogeneous Randomly Varying Soil Properties In 3—Dimensions. 32 The Decay Of The "Variance Function" As The Averaging Interval Increases. 33 Three Nearby Samples N1, N2 And N3 Surrounding The Point In Question "o". 45 Components Of Random Function In Kriging Model. 49 Weights Used By Parry. 59 The Weifhts Suggested By This Research. 59 Schmertmann’s Strain Influence Factor Vs. Depth. 60 The 3 Geometric Co-ordinates (X, Y & Z). 61 The Trend Surface Is Decomposed Into A Trend And A Random Error. 63 The Different N Functions For The Different Subsoil Layers. 70 78 The Boring Locations In The (X,Y) Plane. Representing The N Values By A Linear Regression Function. 80 LIST OF FIGURES, Continued. URE The Assumed Triangular Distribution Of N Value. Trend Surface Results For First Model. Trend Surface Results For First Model, Continued. The Variation Of N Value At A Given X Value As Described By The First Model. The Variation Of N Value At A Given Z Value As Described By The Second Model. Trend Surface Results For Second Model. Trend Surface Results For Second Model, Continued. Analysis Of Variance Of The N Values Predicted In The First Trial Vs. The "True Unobserved" N Values. Analysis Of Variance Of The N Values Predicted In The Second Trial Vs. The "True Unobserved" N Values. Top View Of The Locations Of The Simulated "Observed" Borings And The Location In- Question "0". The Linear Regression Function Of Boring (4). The Linear Regression Function Of Boring (8)- The Linear Regression Function Of Boring (14)- The Linear Regression Of The Simulated "True Unobserved"" N Values And The Predicted N Function By Kriging At The Locatron o . xiv Page 89 92 92 94 94 96 97 99 99 105 109 110 111 115 LIST OF FIGURES, Continued. IRE . Analysis Of Variance Of The N Values Predicted By Kriging Vs. The Simulated "True Unobserved" N Values. Schrnertmann’s Strain Influence Factor Vs. Depth. The Chosen Footing For The Settlement Prediction Study. Top Plan Of The Boring Locations For Case History No.1. (After Borden and Lien, 1988). Generalized Subsurface Profile For Case History No.1. (After Borden and Lien, 1988). Subsoil Stratification And The Representative N Values. Plotting Of The N Function Given By Equation 5.23 Together With The N Values Of The Six Borings At The Depth Of B/2: Z= 877.25 ’ Plotting Of The N Function Given By Equation 5.24 Together With The N Values Of The Six Borings At The Depth Of 3B/2: 2: 865.7 The Settlement Contours. The Linear Regression Function Of Boring (3402)- The Linear Regression Function Of Boring (B405)- The Linear Regression Function Of Boring (3406)- The Linear Regression Function Of Boring (B409)- The Linear Regression Function Of Boring (B-102) Plotted Over Data Without Restricting The Boring Depth. XV Page 116 119 124 125 125 134 138 139 143 146 147 148 149 150 Ib- LIST OF FIGURES, Continued. URE Page The Predicted N Function Vs. The Regression Line Of The Observed N Values Of Boring (B-103). 156 t-Test For The Paired Difference Between The Settlements Predicted By Trend Surface Analysis And The Measured Values. 168 t—Test For The Paired Difference Between The Settlements Predicted By Kriging And The Measured Values. ‘ 169 t-Test For The Paired Difference Between The Settlements Predicted By The Designers And The Measured Values. 170 Flowchart Of The Trend Surface Method. 177 Flowchart Of The Kriging Method. 179 Oneway ANOVA Analysis For Case History No.1. 195 Modeling The N Function For The Upper layer. 199 Modeling The N Function For The Lower layer. 205 Site Plan And Boring Locations Of Case History No.2. 210 The Computer Output Of Case History No.2. 218 Site Plan And Boring Locations Of Case History No.3. 228 236 The Computer Output Of Case History No.3. xvi LIST OF FIGURES, Continued. URE Site Plan And Boring Locations Of Case History No.4. The Computer Output Of Case History No.4. Site Plan And Boring Locations Of Case History No.5. The Computer Output Of Case History No.5. Site Plan And Boring Locations Of Case History No.6. The Computer Output Of Case History No.6. xvii Page 243 250 256 262 265 271 BACKGROUND , OBJECTIVES , AND SCOPE INTRODUCTION Soil deformations under the action Of external loads are a major consideration he design of structure foundations. Nevertheless, economic feasibility dictates that astimation of such deformations must be made on little and widely scattered test data may have considerable variability. In this study an investigation is made as to how information can be best "mathematically" represented, and whether improved irmance predictions can be made using such representation. The loads that can be safely applied to a foundation soil should fulfill two .tions: The shear stresses deve10ped in the soil mass should not exceed some tolerable fraction of the shear strength. The settlement and differential settlements of foundations should not exceed certain tolerable values. hesive soils, both of these criteria require careful evaluation. Cohesionless soils, other hand, have good bearing capacity in most cases and settlement usually 5 the design of shallow foundations. Except for relatively narrow shallow footings se materials where the water table is high, the allowable pressure which may 'ed to a cohesionless foundation will be governed by settlement considerations ther than shear strength. In other words, for dimensioning shallow foundations upOI‘ted by a sandy soil, the failure parameter for verifying stability with respect shear failure is less critical and the design of this type of foundations will be dictated the deformation parameter for estimating the settlements, (See ,e. g. , Jeyapalan, .6; Moussa, 1982; Simons and Menzies, 1976). Meyerhof (1956) and D’Appolonia I-rissette (1968) quantify these recommendations noting that, except in cases where footing width is less than about 3 ft or 4 ft, the allowable footing settlement is ally exceeded before bearing capacity considerations become important. The state-of-the-practice for shallow foundation design is that penetration stance measurements, commonly called "N values", are obtained from the standard etration test (SPT) and used to estimate settlements , either directly or as predictors lastic parameters. The focus of this study is to develop a statistically rational approach for loying the SPT N values to estimate the settlements and differential settlements thallow foundations on sandy soils. The approach would be of particular benefit re the number of footings to be designed is greater than the number of borings. lIS dissertation the SPT test results will be referred to as N values, and the ion which represents the N value as a function of the testing location will be ed to as the N function. Although the current procedures of estimating the settlements in sands have ent views regarding the depth of influence Of a footing (e.g. Meyerhof 1965, rtmann 1970, Burland 1985) or the correction of the N values (e. g. Skempton Liao 1986, Seed et al. 1976, Peck et al. 1974, Bazaraa 1967), most of them adOpt 1b er a simple average or a weighted average of the N values. In contrast, the lication Of geostatistics to the N data - as suggested by this research - will employ available data to formulate a model for the N values. Two modeling techniques are proposed to model an N function on the assumption the N values which are obtained from a limited number of borings are Lidered as a sample and thatthe N function has estimated parameters and associated :rtainty. The function developed from this sample will be used to estimate the penetration resistance function of the whole site, and in particular to generate "two- t estimates" under footing locations. It is stated (DeGroot and Baecher, 1993) that , stronger inferences are ble by dealing with "data statistically than by relying solely on intuitive data )retation. Geostatistical spatial models are a relatively recent addition to the tics literature and their applications are being used with increasing frequency. ie, (1991) stated that, any discipline that works with data collected from different 1 locations needs to develop models that indicate when there is dependence en measurements at different locations. These models are used to summarize bserved data or to predict unobserved ones. The strength of geostatistics ron, 1963) over more classical approaches is that it recognizes spatial variability h the large scale and the small scale, or in statistical notation it models atial trends and spatial correlation. This research considered two applications of geostatistics, namely the "trend analysis" and the"interpolation modeling techniques" such as Kriging, to te the spatial trends or to model the spatial correlation of the N values. 4 Modeling the N data using a geostatistical approach will make it possible to ssign a degree of uncertainty to the resulting model. This concept is used to lustrate a technique for assessing the trade off between the estimation precision and 1e SPT samng costs. Such a technique will help the engineer designer decide on a anng plan which is reasonable for any foundation exploration scheme. The planar (x, y) settlement function obtained from the methods developed :rmits one to draw contours of the settlements, which can be used to help the signer to adjust sizes as necessary to minimize expected settlement or differential ttlement over the site. Such contours could be used also to predict the expected ape of the settlement surface under buildings with many similarly - located )tings over broad areas such as warehouses and parking ramps. The accuracy of the proposed models is tested two ways. First they are verified ng simulated data Of an assumed field. Second, their practical reliability is checked conducting the suggested modeling on a number of available case histories. :omparison is made between the predicted settlements using these approaches and the res which were reported by the previous investigators. THE FRAMEWORK OF SETTLEMENT PREDICTION FROM N VALUES The estimation of the correlation between a series of a scattered N values been studied by a number of investigators, for example (Vanmarcke, 1977), ikula, 1983). However, none of them developed a model to help in selecting a design 1e from scattered data for settlement prediction. Most of the current procedures for predicting footing settlements in sands from SPT test adopt either a simple average or a weighted average of the N values. The of an average N value implies the prediction of the average settlement. It also lies some statistical relationship between the data. A prudent designer would desire :r an accurate prediction or at least a likely maximum predicted value for the ement. The problem is how to develop an accurate settlement prediction as data are m to be scattered and variable. The settlement problem is best posed in a statistical ework. The problem lends itself to subdivision in the following way: The SPT data occur in the form of a number of N values which are measured .' each boring at suitable vertical spacings. The boring locations are represented by ° ates in the (x,y) plane, so the N values are collected from different Spatial ns and can be represented by a set of discrete functions as: N(z) (xi : yi) z : is the testing depth. (thi) : are the horizontal coordinates of the boring i. 6 such functions is illustrated in Figure 1.1. N(z) (X11 1 Ya) _ N(z) (X1 I Y1) fi— N(z) (XOIYO) ; N(z)(X21Y2) fi- 21, Figure 1.1 : Typical N(z) functions. ring the N values may Show considerable scatter. sidering the three coordinates x, y & 2, it can be said that the N values are (three-dimensional field. It is required tO estimate the N values at different 16311] the location (xo,yo), where no data were available, which can be by the function: N(z)(xo : Yo) , it iS required to compute the confidence interval on the estimate to some nfidence to be used for evaluating the quality of prediction. As was noted earlier in the introductory section, many researchers have related settlement to the N -value. It was mentioned that these procedures employed the N 6 either explicitly or implicitly as predictors of elastic parameters. Therefore, it is ired to use. the estimated N function N (2) (x0 , yo) to formulate a design N value to sed for the deterministic models of settlement estimations. Here also, it is required aluate the quality of the settlement prediction. RESEARCH OBJECTIVES This research is proposed to add a useful contribution regarding how to treat :r of N value data by modeling it as a spatial'function , and to use points on this ion to estimate the settlements and differential settlements. patial data analysis and modeling will be done by the use of suitable geostatistical iques (specifically trend surface analysis and interpolation modeling). recise research objectives are as follows: Setting forward guidelines for employing the geostatistical modeling for obtaining better settlement predictions. Proposing geostatistical methods to treat the data scatter of N values to provide a rational estimate of N value at locations where no data are available. Assessing how to use the resulting N functions in conjunction with selected existing settlement functions to predict the settlements and differential settlements. Testing the two prOposed geostatistical methods in combination with a settlement prediction method on a number of previously—published case histories and comparing the results with the measured values to assess the accuracy of the proposed model. RESEARCH SCOPE To accomplish the required objectives, the research chapters are organized ows: Chapter 2 reviews the SPT test procedure as well as the techniques used for ing settlements of shallow foundations. These are followed by reviewing the procedures for predicting settlements in sands using the SPT test results directly using SPT results to estimate deformation parameters to be used with elastic chapter 3, the variability of N values is analyzed in a geotechnical framework ' g such factors as the soil relative density, overburden pressure, ,etc. The sources Of the variability of N values are also reviewed. Special 5 is made on the tendency for the N value to increase with depth and its 11 for overburden pressure. tical modeling methods from the published literature are reviewed, emphasizing - adaptability of the geostatistical approaches which are suitable for characterizing the tter of the soil properties. These include the random field theory, the trend surface : ysis theory, and interpolation schemes such as triangulation and Kriging techniques. In chapter 4, a two-point estimate technique is deve10ped to combine the ing measurements of N values within the depth of influence under the footing into design N value to be used in deterministic models of settlement estimation. This is o wed by development of the suggested approaches for the application of both the (1 surface analysis theory and the Kriging technique to obtain the SPT function and nately the two point estimate. Recommendations are made as to the class of problems which each is preferred. lications of the models to practice are also discussed. These include the evaluation 1e quality of prediction versus the quantity Of the data and their monetary costs and the contours of the expected settlements. In chapter 5, the two suggested methods for settlement prediction, trend surface Isis and Kriging, are each tested two different ways. First they are verified using lated data of an assumed field. These are followed by the verification of the Oped "two-point" settlement estimate by using the assumed field and an assumed g. Second, the reliability of the proposed methods for practical applications is ed by conducting the suggested modeling on a number of the available .case 'es. Results are compared with the measured values. Based on these studies, 'sons are made to find out the advantages and disadvantages of using each of the didate procedures for estimating N (x,y,z). 10 In the final chapter, summary, critical discussion, conclusions and recommenda- ns are presented. CHAPTERZ, CURRENT PRACTICE IN SETTLEMENT PREDICTION USING THE SPT TEST GENERAL The standard penetration test continues to be widely pOpular among foundation ers because it is economical and easy to perform. It has been studied and discussed rge number of investigators and its correlations with basic soil parameters are well shed. A considerable number of methods for the estimation of settlements in onless soils rely on SPT data. These include: (Terzaghi and Peck, 1950; hof, 1956; Peck and Bazaraa, 1967; Alpan, 1964; Schmertmann, 1970; and 1971). The published methods may be divided in two main groups poulos, 1992): Those which give a direct estimation of settlement from , the results of in— situ tests. Those based on elastic theory which depend on estimated material deformation parameters from the interpretation Of in-situ test results. The next four sections will summarize the current practice in the SPT test re, the measurement of settlements , and prediction methods related to groups two respectively. 11 l2 SPT TEST The standard penetration test is currently the most popular and economical s to obtain subsurface information regarding cohesiOnless soils. The test has been ardized by ASTM D 1586 as "Standard Method for Penetration Test and Split — :1 Sampling of Soils". ‘ This test involves using a 140 lb (63.5 kg) driving maSs falling free from a t of 30 in (0.76 m) to drive the standard split-barrel samwer which has an inside :ter of 1.5 in, an outside diameter of 2 in and a length of 18 in. The sampler is n a distance of 18 in (0.45 m) into the soil at the bottom of the boring. - ampler is first driven a distance of 6 in (0.15 In) to seat it on undisturbed soil and Imber of blows recorded. The sum of the blow counts for each of the next two six icrements is taken as the penetration resistance (N value) in blows per foot unless Ist increment cannot be completed (either from encountering rock or because the count exceeds 100). In this case the blow count for the last 12 in (0.3 m) is ted and taken as the N value. The ASTM standard (ASTM D 1586, 84) - states that the boring is advanced entally to permit intermittent or continous sampling. Typically the test intervals :d are 5 ft (1.5 m) or less in homogeneous strata with test and sampling locations y change of strata. According to many investigators (see e. g. Gibbs & Holtz, 1957 ; Seed et.al., Peck et.al., 1974 ; Bazaraa, 1967 ; Liao, 1986 ; Skempton, 1986), the N value ular soils is affected by the effective overburden pressure. For that reason, the : Obtained from field exploration under different effective overburden pressures l3 :hould be changed to correspond to a standard value of overburden pressure when used 0 estimate the relative density (D,). .2.1 SPT TEST PRECISION AND BIAS The SPT test has been studied and reported by many investigators. These include [eyerhof (1957), Gibbs and Holtz (1957), D’Appolonia (1968), Peck and Bazaraa 967), Schmertmann (1970), and Kovacs and Salomone (1982). The studies by Schmertmann, De Mello, and Bazaraa as summarized by Bowles 988), noted that the SPT test is difficult to reproduce. Some of the factors which affect e reproducibility include variations and interference in the free fall of the drive weight, 3 use of worn or damaged drive shoe, failure to properly seat the sampler in the bottom the boring, the inadequate cleaning of loosened material from the bottom of the ring, failure to maintain sufficient hydrostatic pressure in the borehole so that the test e becomes "quick", and driving a stone ahead of the sampler. In another field measurement study, Kovacs and Salomone concluded that the rgy delivered to the drill stern varies with the number of turns of rope around the ead, the fall height, drill rig type, hammer type, and operator characteristics. The ASTM standard (D 1586, 84) noted that variations in N values of 100 % or e have been observed when using different standard penetration test apparatus and lers for adjacent borings in the same soil formation. When using the same apparatus driller, the current opinion (ASTM D 15 86, 84) indicates that N values in the same at the same overburden stress can be reproduced with a coefficient of variation of t 10%. It is noted also (D 1586, 84) that the use of faulty equipment can 14 rificantly contribute to differences in N values obtained between Operator — drill rig BIDS. 2 PRACTICAL ADVANTAGES OF THE SPT TEST It is stated by the ASTM standard (D 1586, 84) that the SPT test is used Jsively in a great variety of geotechnical explOration projects. . Many local :Iations and widely published correlations which relate SPT blow counts, or N value, :he engineering behavior of earthworks and foundations are available. Regardless of the variability of the SPT test results, the SPT is not likely to be ioned for several reasons. Bowles (1988) cites the following: I The test is too economical in terms of cost per unit information. If performed every 2.5 ft of depth a tube recovery length of 18 in, including seating length, produces a visual profile of around 60% of the visually examined. The test results in recovery Of very disturbed samples, but they can still be tested for index properties, and with appropriate conservatism, tested for strength properties. Long service life Of the enormous amount of equipment in use. The accumulation of a large SPT data baselwhich is continually expanding. The fact that other methods can be readily used to supplement the SPT when the borings indicate more refinement in sample/ data collection. 15 1.3 MEASURING SETTLEMENTS As the focus of this research involves prediction of settlements, which are verified y field measurements, it is relevant to briefly review the methods of measuring the :ttlements. It has been stated (Sze’chy and Varga, 1978) that the measuring of settlements m not really require any sophisticated apparatus or theoretical training, only me organized thinking and accuracy. The measurement techniques used rely mainly on mventional surveying. Hanna (1973) stated that in most projects simplicity is sential because of the pressures of finance and time limitations. For the great majority foundations, most if not all of the required measurements can be obtained by simple rveying techniques. The most commonly obtained information is the determination of :vation and change in elevation by offset measurement from a line of sight. For determination of absolute movement, it is essential that the datum benchmark located well away from the zone of ground movement; otherwise it may also be ected by the ground movements, ( a distance of 60 In from the building is usually ugh to be clear of any effects of settlements of the building ). A permanent chmark is used if available. Where this is not available, a benchmark or a number enchmarks, depending on the size of the project, are constructed, (Burland et. al., 2). The reference points on the foundations are either rigidly attached to the structure olting or welding, or special demountable points are employed. The reference pin socket commonly used for survey of foundation settlements is comprised of a steel rass socket grouted into a hole in the side of the foundation. The reference pin 16 rews into the socket. When not in use the socket is protected with a cover plate. The elling staff is placed on top of the reference pin. On steel structures such as oil tanks, 3 may be welded to the outside of the tank about 1 ft above ground level. Each lug, ich has a protective cover, has a steel ball bearing about 1 in diameter welded to it ich forms the reference mark for the survey, (Hanna, 1973). Settlements of strata lying deep under a building can be measured by a simple suring rod bored or driven down to the layer in question with an enlarged tip at the om of the rod. The bored rod is usually protected against corrosion by asphalt ting over the full length of the rod and cloth wrapping impregnated with oil. iriven, the rod should be protected by a casing which is slightly retracted in the end, :e’chy and Varga, 1978). Sze’chy and Varga (1978) state that even if the observed data have some :ertainty, they still can be used for settlement evaluation and noted that, it is always :er to know that settlements between the limits of, say, 5 cm to 15 cm are to be cipated than to be in doubt whether settlements will be of a few millimeters only or le to reach magnitudes of tens of centimeters. In other words, even with some rtainty a reasonable estimate of the magnitude is important. SETTLEMENT PREDICTION METHODS (GROUP 1) Some of the published methods related to the first group which employ N values tly for settlement prediction - as. cited by many authors including Jeyapalan, 1986; nS and Menzies, 1976 ; Oweis, 1979 ; Haji, 1990 ; and Bowles, 1988 - are as S: Meyerhof (1965): recommended predicting the settlement as the ratio of the net undation pressure q to an empirically determined pressure q’ which is expected to ult in a settlement of one inch: (2.1) S=q/q’ ere S = settlement in inches. q = net foundation pressure, in (ton/ftz) q’== (N/3) [(B+l)/2B]2 ; for B > 4 ft. = N/8 ; for B < 4 ft. B = footing width in feet. Peck and Bazaraa (1969): developed a refinement of Meyerhof’s :edure by introducing correction terms and give the settlement as: S=(2/3) (q/q’) (pd/pw). (2.2) q’= (N ’/3) [(B+1)/2B]2. N’ = N value corrected for overburden pressure. pd: effective overburden pressure at a depth B/2 below the base of the footing for the dry condition. pw= pressure at the same level (B/2 below the base) with water table present. == pd, if there is no water table present. _.,__,..__h ._, _ .. Parry (1971): proposed an empirical equation giving the settlement as: S=(l/N) (ZOOqB) CD. CW.CT. (2.3) are = settlement in (mm). N = weighted average N value using the weights of (3,2 &l) for N values between the depths of (0 to 2B/3) , 1(2B/3 to 4B/3) and (4B/ 3 to 2B) respectively. q = applied pressure in (MN / m2). B = footing width in (In). CD,CW,CT= factors for the influence of excavation, water table and the thickness of the compressible layer. V’s equation was developed by assuming that the settlement is a function of the width e loaded area, the magnitude of the bearing pressure and the deformation modulus 3 soil, which is implied by the weighted N values. It is of note that the weighting rs permit incorporation of the spatial trend of the N values. This method could .bly be included in Group 2 (Section 2.5), but it is included here because it does not y the modulus E explicitly. Alpan (1964): recommended predicting the settlement as: S=(O.14q/25) (N’) '1°8[ZB/(B+l)12 (2.4) S = settlement in inches. q = applied pressure (t/ftz). 19 N’ = N corrected for overburden pressure. B = footing width in feet. pan’s equation was based on predicting the settlement of a plate one square foot at undation level using measured N values corrected for effective overburden pressure and an extrapolating this predicted settlement up to the settlement of the full scale structure ing the Terzaghi and Peck correlation. This correlation formulates the relationship tween the settlement of a footing of width B and the settlement of a one square foot t plate loaded to the same loading intensity as: s,,/S,=(ZB/(B+:L))2 (2.5) Schultze and Sherif(1973): estimated the settlement by : s=q,Bf/[1.71N0-87(B/B,)°-5(i+o.4D/B)] (2.6) ere S = settlement in (cm). qt = total foundation pressure (kg/emf). B = footing width (cm) B1 = 1 cm. f = parameter depending on width to length ratio and thickness of compressible stratum. D = depth of embedment (cm). 1126 & Sherif’s equation was based on a statistical correlation using linear elastic ry as a model. m.rfi_._._ _ __ .1 2O neral Cements On The Methods Of " Group 1" To summarize the inherent assumptions of the methods of "Group 1" it can be lthat the Meyerhof’s method is an analytical expression of Terzaghi and Peck’ s I known settlement design chart that allows estimating allowable foundation ssure such that settlements would not exceed 1 inch. In this method, the ition of the water table is ignored. Peck and Bazaraa’s method is a refinement of Meyerhof’s method. They Immended that the predicted settlements based on the Terzaghi and Peck design It be reduced by one-third but still proposed that the settlement estimate is to be eased when the depth to water table below the foundation base is less than B/ 2. as recommended also that the N values are to be corrected for the effect of the rburden pressure. The results of a study by Schmertmann (1970) suggested that the Terzaghi and r design Chart is quite unconservative for large foundations and it may be in error. rther recommended that the Meyerhof method in its present form Should be ded. Schmertmann’s approach is described in the next section. Parry’s methodiassumes that the settlement is a function of both the footing width he bearing pressure and inversely proportional to the N value. This is formulated ilding an empirical model using measured settlements. Alpan’s method is based on predicting the settlement of a one square. foot plate Indation level using measured N values corrected for effective overburden pressure 'rren extrapolating this predicted settlement up to the settlement at full scale footing . the Terzaghi and Peck correlation. 21 Schultze and Sherif’ 3 method is based on statistical correlation using linear astic theory as a model. Several coefficients were employed to ensure a high degree 7" correlation. ,5 SETTLEMENT PREDICTION METHODS (GROUP 2) The second group of methods are those which estimate the deformation meters, in particular the modulus E, from the interpretation of in situ tests or boratory tests, then apply elastic theory to estimate strains, and then integrate vertical rains to obtain settlements. It has been stated (Bazaraa 1982) that the use of the theory of asticity for estimating stresses or settlements for foundations in sand is not eoretically correct. However, with reasonable correlation of E with standard netration N value ( for a certain depth under the footing), this method can be nsidered as a reasonable empirical method for estimating the settlement (Bazaraa, £2, pp. 68). me of the published methods related to this group — as cited by many authors including apalan, 1986 ; Oweis, 1979 ; Schmertmann, 1970 ; Webb, 1969 ; are as follows: Schmertmann (1970): gives the following equation for calculating the 1 settlement: ZB s=Cl.C2.pE [(Izi/E) .dzi]. (2.7) z=0 . 22 p = increase in effective overburden pressure at foundation level. C1 is a depth embedment factor. C2 is an empirical creep factor. I2 is the strain influence factor. B is the deformation modulus = 4N for silts or slightly cohesive silt - sand to 12N for sandy gravel and gravel. Webb (1969): estimated the settlement by: S=2 [(pzi/E)dz,-]. (2-8) i=1 , p2, = vertical stress in soil layer 1 produced by footing load. dzi = thickness of layer 1. ral Comment On The Methods Of "Group 2" To summarize the methods of "Group 2", it can be said that the methods by rtmann and Webb are quasi-elastic (Oweis, 1979). Both methods predict settlem- roportional to the width of foundation (B) and the foundation pressure (p) nversely proportional to the modulus of deformation (E). In both methods the .us (E) depends solely on the N value irrespective Of foundation width and re, but the modulus is implicitly weighted in Schmertmann’s method. 23 Webb’s method implies that maximum strains occur immediately beneath the use of the footing where vertical stresses are at maximum values. This is contrary 0 results from tests on small plates that indicate maximum Strains occur at depths f 0.5 B to 0.75 B below the base of footing (Oweis, 1979). Schmertmann recognized this by assuming a maximum strain at a depth of 1.5 B. The Schmertmann’s method accounts for the observed rapid attenuation of ettlements with depth by considering only a thickness of compressible layer equal to B when estimating settlements. Although the models of "'Group 1" assume a single value of "N", the models if "Group 2" assume precise information regarding "N", i.e. each layer is represented- ~y different N and E values. or N = N(z) and E = E(z). Schmertmann made the point that the distribution of vertical strain under the enter of a footing on a uniform sand is not qualitatively similar to the distribution of the Icrease in vertical stress; rather the greatest strain occurs at a depth of about B/2. He reposed an influence factor whose value increases with the depth "z" according to the inction: Iz=[0.l+(z/B)] (2.9) aching a maximum value at the depth of (B/2) then decreasing according to the nction: Iz=(0.4/B) (219—2) (2.10) til it reaches a value of (0.0) at the depth of (2B). 24 The conclusion here is that the N value is. a main input in most settlement xiiction methods. In chapter 3, the approaches of treating the data scatter of N values : reviewed. For the prediction of settlements at many points in the (x, y) plane, it is tired to deveIOp a method which has the simplicity of single N value techniques but :serves the information regarding the spatial structure of the N values. This will be lressed in chapter 4 by developing a method employing the variability of N values as ,,(x,y) together With a selected settlement prediction method. VARIABILITY OF SOIL DATA AND SUMlVIARY OF GEOSTATISTICAL MODELING lVlETHODS .l VARIABILITY OF N VALUES Although the use of SPT data to estimate design parameters is well established in literature, little guidance is given on how to treat data scatter (Wolff, 1989). rrelations provide estimates of soil properties as a function of a single N value, and rious designers may enter correlation equations with values anywhere from the average lue down to values below the minimum measured. Wolff (1989) showed that the signer’s assumptions and judgements may significantly influence the final design :ommendations. Haji (1990) noted that the interpretation of the soil boring data and the ection of a representative N value depends to a great extent on the experience and the )wledge of the foundation engineer rather than on a Specific procedure or method. The design N value is used to estimate the angle of internal friction, which in. will be used to assess the soil bearing capacity; consequently the decision whether se shallow foundations or deep foundations is dictated by the design N value. hermore, if shallow footings are selected the dimensioning of the footings and the 'cted settlement will all be a function of this N value. 25 1.1 Sources Of Variability Of N Values Soil by its nature has an inherent variability. This variability may result from ilot & El-Ramli, 1982): The variation of soil layer thicknesses. The heterogeneities within the soil layers in terms of different relative densities, different moisture contents, different overburden pressures, and different stress histories. From the viewpoint of mathematical modeling , Baecher (1978) attributed the 'ation of soil properties to 3 sources: Drift in average properties. Random fluctuations about the average. Inhomogeneities. The modeling problem is to represent mathematically the spatial variation ch is attributed to the first and second sources. The variation due to inhomogeneities not. be tackled by mathematical modeling; however, it is suggested herein that ndom field of the soil properties can be divided into a number of subfields, each vhich satisfies the homogeneity condition. Thus, a soil field could be modeled a set of mathematical functions. This concept will be developed further in section The above-mentioned inherent variability, combined with the variability resulting the testing repeatability or reproducability errors, commonly leads to considerable r in SPT values over a construction site. .1.2 N-Value Depth Effect And Correction For Overburden Pressure For "consistent" materials (i.e. a thick deposit of soil having uniform composition d relative density) the variation of N in the vertical direction is minated by the soil stiffness which increases with depth due to the overburden essure and the corresponding increase in confining pressure. But the question of how soil modulus E, a measure of stiffness, would increase with depth in such a material 3 been the subject of different and varying opinions: The earliest literature on stiffness of a semi-infinite mass started with Boussinesq (1885) who developed solutions for stress and strain in an elastic homogeneous half space, but did not take any account of the systematic increase of stiffness with depth. Feda (1963) presented a model of an elastic half-space whose modulus increased parabolically with depth. Gibson (1967) considered the soil as an elastic half-space whose modulus increases linearly with depth. Many authors (e.g. Skempton, 1986; Liao and Whitman, 1986; Seed et al., 6; Peck et al., 1974; Bazaraa 1967) recommended that field N values should be ected for the vertical effective overburden pressure before using them for acterizing the sand. The justification for such correction is explained as follows: In sand, the settlement (under a given stress change) is strongly correlated to its relative density. The N value reflects both the relative density (D,) and the effective stress. This correction makes it possible for the N value to be correlated directly to the relative density; consequently this corrected N value can replace the relative density in the correlation which was mentioned in ( 1) above. That will lead to a more accurate prediction of settlement from N value. Regarding item 1 above, D’Appolonia et al.. (1968) confirmed the correlation tween settlement and relative density by noting that the principle variables ntrolling settlement for a particular granular soil under a given static loading nfiguration are the initial density and the initial state of stress in the deposit. erefore, analytical and empirical methods of estimating settlement of footings on sand uire a direct or indirect measurement of in situ density and stress state. Regarding item 2 above, D’Appolonia ( 1968) presented a relationship between the relative density and the effective stress, similar to that shown in Figure 3.1. seen in the figure, the N value cannot uniquely define both relative density and ass; there are an infinite number of combinations of stress level and relative density t will result in the same measured N value, (D’Appolonia 1968). Regarding item 3 above, the correction of N value was suggested to make it sible for an adjusted value N’ to reflect only the effect of the relative density while ng the effective stress value. D’Appolonia (1968) reported that "Empirical tionships have been developed to correct the blow- count for in situ vertical tive stress. Thus, the SPT can be correlated to relative density". He also added "When the SPT resistance was corrected for overburden pressure, Meyerhof’s 0d accurately predicts the measured settlements". Based on the above considerations, the N values will be corrected, using the Liao Whitman correction method, before incorporating them into the proposed models. 29. /\ Drl Dr2 Dr3 Figure 3.1 : N Value Versus Relative Density "Dr" And Vertical Effective Stress "Pv’", (After D’Appolonia). 30 5.2 . SUlVIMARY OF GEOSTATISTICAL MODELING METHODS Having described the inherent variability of N value data, it is desired to represent .uch data with one or more functions that capture overall trend and variability around hat trend. Sections 3.2.1 through 3.2.3 below summarize some geostatistical methods vhich are suitable for characterizing the scatter of soil properties: i.2.1 The Random Field Theory This theory was employed by Vanmarcke (1977) to study the implications of .tochastic variability of soil properties. He quantified the variability of the soil >rofile using the "variance function". The variance function was defined as the ratio >f the variance of the moving average function to the variance of the random 'unction of the soil property, before smoothing, related to one spatial coordinate "z", s illustrated in Figure 3.2. As a smoothing procedure, the moving average implies the removal of local laxima and minima of the random variable function resulting in reducing the data ariability and decreasing the variance. For the SPT data, the random variable is the N alue, consequently the variance of the moving average over an area is smaller than the niance of the N values, which can be expressed as: 02[1_\-7] 11(2) 7T Scale of fluctuation L Moving ave. function. Random function Depth : Z v / SD. of the moving ave. function i '6 SD. of the random function. Figure 3.2 : Parameters Of Homogeneous Randomly Varying Soil Properties. 32 ------------.-‘D---------- --———o-——— 8..” W T Scale of fluctuation Figure 3.3 : Parameters Of Homogeneous RandOmly Varying Soil Properties In 3- Dimensions. 33 Variance function A 1.0 v —- .— 0 EA ,3 O . / Averaging interval Correlation distance = V * A Figure 3.4 : The Decay Of The "Variance Function." As The Averaging Interval Increases. 34 acreases is used to estimate the " correlation distance" or the " scale of fluctuation" if the variable values, as illustrated in Figures (3.2, 3 & 4). The " scale of fluctuation" 3 obtained by multiplying the asymptotic value of the variance function by the tveraging distance corresponding to this value Figure 3.3. Physically the "scale of fluctuation" is a measure of the distance within which :he soil property shows relatively strong correlation from point to point or the iistance over which the soil can be treated as statistically homogeneous. What is often needed within this "statistically homogeneous" soil mass is (V anmarcke , 1977) the probability density function of some " spatially averaged " soil property. Consequently, :he expected value and the variance of the soil property within this soil mass are )btained in conformity with the distributional characteristics of the observed zalues. However , Vanmarcke (1978) later made the comment that the scarcity of .ubsurface data makes it difficult to validate complex stochastic models. For :xample , it is seldom possible to distinguish between competing functional forms 1f the correlation function. At best, the data permit one to estimate the decay arameter (such as the correlation distance) of any chosen functional form for the )rrelation or variance function. Therefore, in this research, the random field theory is )t recommended for the development of the proposed models. ‘A.’—-. -w - - —— .- 35 3.2.2 Trend Surface Analysis The trend surface analysis technique was suggested by Krumbein and Graybill (1965), summarized by Davis (1973 and 1986) and extended by Baecher, (1978). In this procedure, a trend surface describing a soil property N = N (x,y,z) is estimated from data by taking a least squares fit. Precisely, the trend surface analysis is an adaptation of the statistical field of multiple regression, and the techniques have been borrowed directly from the discipline. In some cases (Davis, 1986) one can even use the powerful tests of hypotheses of multiple regression on geologic problems. Fhus, one approach for consideration in a Spatial approach to the settlement prediction problem is to estimate a trend surface from data by taking a least squares fit with confidence limits determined as in regression analysis. Lancaster and Salkauskas (1986) noted that, if there is a sizeable error that enters into the data in a random way that should be smoothed out, then judgement must be exercised in assessing the degree of smoothing to be applied and the choice if smoothing procedure. In this instance smoothing is referred to in the statistical ense which implies the removal of extraneous local maxima and minima and the ientification of the underlying trend. Here, several different least squares fitting _ :chniques may be considered. There are several items which must be considered in attempting to fit a surface rough the data scatter of N values. These are: The trend surface geologic definition. The trend surface operational definition. The functional forms of the trend surface. I M;~_a.4—__x—————_- 36 Regarding the first item, Cressie (1991) reported that geostatistics recognizes atial variability as being the sum of two components, the large scale component or atial trend and the small scale component or spatial deviation. Trend surface analysis nsiders only large scale variation, assuming independent errors. However , what considered to be "large scale" and "small scale" is largely subjective. The question then, is how to objectively separate the data into two components the distinction between the components is entirely subjective. To remedy this estion, Davis (1986) suggested that this be done using an operational definition which ecifies the way in which the data are to be treated instead of a geologic definition of nd and deviation. In accordance with the second item, the operational definition, a trend may be med as a function of the coordinates of a set of observations so constructed that : squared deviations about it are minimized. The expansion of this definition will yield t: The trend is a function of the coordinates, meaning that an observation is considered to be in part a function of the location (x , y , z) of the observation. This function has the form of an equation whose terms are added together. Each term is the product of a coefficient and some combination of the coordi- nates. The sum of the squared deviations from the mean defines the variance of the sample. So, it can be seen that the trend can be regarded as a function having the smallest variance about it. ' Regarding the third item, the functional form of the trend of the N function (x,y,z), one seldom would have any prior knowledge about what the functional form the trend should be. Instead, one does the next best thing, and approximates the r c own function with one of arbitrary nature. Commonly , a polynomial expansion used which uses the powers and cross-products of the coordinates. Polynomials : extremely flexible, and if expanded to sufficiently high orders , can conform very complex surfaces. There is however, some mathematical basis to using the plest (lowest order) model in the abscence of information to justify a more complex del. It is noted also that the 2 term should be different than the x and y terms, due a epth effect. It is important to note that polynomial functions are used for trend analysis arily as a matter of convenience. For polynomials, the equations necessary to find coefficients of the trend may easily be established and solved by computer programs vis, 1986, pp. 411). The use of polynomials does not mean that the N functions are 'nomial functions; their unknown nature is only approximated by a polynomial nsion. Other approximations may be more appropriate in specific instances, but in ral are less convenient. The trend surface can be fit to different models (Box, Hunter and Hunter, 1978, 13-521; Davis, 1986, pp. 430). These models are initially considered. These are: The first degree polynomial model. The second degree polynomial model. The four-dimensional trend surface. ese models are mentioned just for setting the stage. The fitted model to any particular se should be based on the variation of the data and prior knowledge, historical data, d theory. These models are discussed in sections (3.2.2.1) through (3.2.2.3). .2.1 The First Degree Polynomial Model The first degree polynomial model is expressed as: N=bo+b1X+b2Z+e (3 . 2) b’s are the model constants, and; e is the. error term. This model: Allows the design to be efficiently fitted. This is done by estimating the model constants by using the least squares fit. Allows checks to .be made to determine whether this planer model is representationally adequate. This is done by conducting the interaction checks. The planar model implies that the effects of the variables are additive. Interaction between the variables X , Z would be measured by the coefficient b12 of an added cross-product term X Z in the model. If the value of the b12 is not significant, then the planer model is adequate. Provides some estimate of experimental error by estimating the error variance of the replicated observations. 2.2.2 The Second Degree Polynomial Model The second degree polynomial model is expressed as: N=b0 +le+b2Z+bllX2+b22Z2+b12XZ+e (3 . 3) To check the adequacy of the second degree model, the combinations of d-order terms have to be checked. For example the coefficients b111 and b122 are tfficients of X3 and of XZz, respectively, in a third-degree polynomial. Both of these 3 coefficients would be zero if the surface were described by a second-degree model, M & Hunter, 1978). .2.3 The Four-Dimensional Trend Surface A logical extension of polynomial trend surface analysis is the inclusion of three dimensions X, Y and Z (and if significant, their powers) as independent ables (Davis, 1986). The use of a four-dimensional model is preferable when the data more erratic because it is more general than the other models. In this technique, the dependent variable is regressed upon X, Y and Z. In the . of N more emphasis is put on the inclusion of the Z term. Its inclusion is not only :al but necessary as it is known that the depth effect is the main factor in the bility of the N values. In three dimensions contour lines become contour envelopes. A completed sis can be represented as a solid containing nested contour envelopes. The lume between two successive contour envelopes is occupied by points which have - same range of predicted value of dependent variable. The dependent variables in - case histories reported by previous geostatistical investigators commonly were asures such as the percentage composition of some constituent or mineral. In this earch, the dependent variable is taken as the N value. .2.4 Measuring Goodness-of-fit Of A Trend Surface Higher values for the coefficient of determination R2 should be expected in a four ensional model. The R2 value is frequently used empirically as a measure of the ree to which the trend fits the data (Lancaster and Salkauskas, 1986). R2=SSR/SST ' (3 . 4) :re SSR = regression sum of squares. =Zfi2_(2fi)2/n (3.5) SST = sum of squared deviations from mean. =ZN2-(2NV/n ”-9 The value of the R2 lies in the range of (O — 1). When R=1, all of the data n the fitted trend and there is no residual or deviation; when R=O, the observed 3 failed to show any trend. Values of R2 between 0.8 and l are often considered to indicate a significant nd in the data, and values of R2 between 0 and 0.2 suggest that the trend is not ll established (Lancaster and Salkauskas, 1986). In practice, R2 is not likely to at the limits of the range (0 - 0.2) or (0.8 -l), but rather somewhere in between these its. The closer it is to one, the greater is said to be the degree of association . een the dependent and the independent variables (Neter and Wasserman, 1974). For a model to be accepted, it should pass a test of significance; to select among els which are known to fit with significance, R2 and sometimes the standard error .stimate are used as criteria. The significance of a trend surface may be tested statistically, by comparing the .ance due to deviations from the mean to the variance due to deviations from the d. This comparison is conducted by using the F test, which is valid only if the . satisfy the following conditions (Davis, 1986): The population of data is normally distributed about the regression. The population has a constant variance and does not change with changes in the independent variable (i.e. the variance is constant for all x, y & 2 locations). The samples are drawn without bias from this population. One does not really know if these conditions are true or not, but, in the abscence idence to the contrary, it is assumed so in order to go forward. The significance e trend may then be tested by performing the F test as follows: otal variation of N is divided into two components: the trend and the deviations. the F ratio is calculated: (3.7) MSR = regression mean square. = (regression sum of squares) / degrees of freedom. MS D = (sum of squared deviations from the mean - sum of squared deviations from the trend) / d.f. z (33, - ssR) / (d.fT - d.fR). d.fT = total degrees of freedom. —- number of data points - 1 d.fR = trend degrees of freedom. = number of coefficients (b’s) in trend surface not counting b0. resulting F value is then compared to the critical value of F probability distribution 3 selected level of significance. A significance level of 5 % is commonly used. The critical F value is a measure of the variation that might be expected solely ue to randomness in sampling the data and the les‘ser the calculated P value the better he data are fitted to the trend and the lesser the chance that the fit is a coincidence. e F- test result is a test of the hypothesis (Davis, 1986): H02B1=B2=..=Bm=0 (3.8) B’s are the true population regression coefficients. Lis null hypothesis implies that there is no trend. Section (3.2.3) summarizes the interpolation schemes including the Kriging :hnique, which is an alternative technique for spatial modeling. In chapter (5), :emative models from both techniques ( the trend surface and the Kriging) will be used make settlement estimates for the same case histories. Based on the comparison .ween the estimates by each of them, an evaluation of the aptness of each will be .de. 44 3.2.3 _ Interpolation Schemes For .N Value Modeling It could be argued that the N value at a given point should not influence the ature of the fitted surface at distant points. If this is the case, then no constraint is mposed on the choice of any local modeling technique. On the other hand, (Lancaster d Salkauskas, 1986), if there is sufficient confidence in the data to demand that he fitted surface must contain every data point - in the influence zone — then e interpolation modeling techniques may be considered. In other words, regression mphasizes trend over local fit; if one wishes to emphasize local fit over trend, then the terpolation modeling techniques are preferred. Interpolation modeling techniques take one of two forms. The first is the simple oint estimation scheme for interpolating two—dimensional data, such as the geometric chnique known as "triangulation". The second is the estimation method that is designed » give the best estimate for one of the statistical criteria. This latter technique is lown as "Kriging", after its developer, D. G. Krige, a South African mining engineer rd pioneer in the application of statistical techniques to mine evaluation. 2.3.1 Triangulation Triangulation is a method of interpolating a value f(x,y) and is done by fitting a me through three samples that surround the point being estimated (Isaaks and .vastava, 1989). X & Y are not necessarily situated at the ground surface but rather, :y are situated in the plane of the three data points which can take any orientation. e points need not be regularly spaced, but they have to surround the point being imated "nicely" (i.e. in relatively different directions and at nearly equal distances). 45 e equation of a plane can be expressed generally as: z=ax+by+c (3 . 9) the SPT data, where it is desired to estimate N values using coordinate rmation, z is the N value. Given the coordinates and the N values of three nearby samples N1 , N2 N3 that nicely surround the point being estimated "0" as shown in Figure 3.5 , one calculate the coefficients a , b and c by solving the following system of equations: Nl=axl+byl+c (3.10) N2=axz+by2+c (3.1.1) N3=ax3+by3+c (3.12) N1 N2 + (Xrayo + (X205) + 0 N3 + (Kai/3) Figure 3.5 : Three Nearby Samples N l,Nz And N3 Surrounding ‘ The Point In Question "0". If the three points are located in a straight line — i.e. not surrounding the point '0" nicely - then the three equations cannot be solved for all given values of N1 , N2 (1 N3. This method of estimation depends on which three nearby samples are used define the plane. There are several ways one could choose to triangulate the ple data set. Triangulation is typically not recommended for extrapolation purposes. In fact ris violates the "surrounding" requirement. If the point "0" at which an estimate is quired is contained within the triangle 123. , one can directly calculate the triangula- m estimate at "0" without simultaneous equations as: lvo= (l/A123) (A023N1+A013N2 +A012N3) ' (3 ' 13) The A’s represent the areas of the triangles given in their subscripts. The anulation estimate, therefore, is a weighted linear combination in which each value weighted according to the area of the opposite triangle (Isaaks and Srivastava, 89, pp. Equation 11.6). Other interpolation schemes involving four or more points or higher order faces have been developed and are commonly applied in finite element analysis; ever, they do not have the statistical advantages of Kriging as discussed in section .3.2. .3.2 Kriging Kriging is a probabilistic method used for fitting a surface to irregularly scattered 'nts in space, (Krige, 1966). This technique has found increased application in recent s,(Lancaster and Salkauskas, 1986), for example by Spikula, (1983) and Baecher, 81). What distinguishes Kriging from the regression or trend surface technique is t attempts are made to localize the computation by excluding distant points the calculations of the interpolant at any fixed point. What distinguishes Kriging simple polynomial interpolation such as triangulation is that attempts are made to the best estimate for one statistical criteria. Triangulation, on the other hand, gives estimated value based on an entirely geometric criteria. Isaaks and Srivastava (1989) note that Kriging is often associated with the )nym "BLUE" for "best linear unbiased estimator": Kriging ,is "linear" because its estimates are weighted linear combinations of the available data. It is "unbiased" since it attempts to set the mean error equal to zero. It is "best" because it aims at minimizing the variance of the errors. This is the main distinguishing feature of the Kriging method. An important aspect of Kriging is that one never knows the mean error and fore cannot guarantee that it is exactly zero. Nor does one know the variance of the s; it cannot be minimized. The best one can do is to build a model of the data being and work with the average error and the error variance for the model. In Kriging a probability model (a random function model) is used in which ias and the error variance can both be calculated and weights chosen for the earby samples that ensure that the average error for the model is exactly zero and at the modeled error variance is minimized. .2.3.2.1 Underlying Concepts It is assumed (Lancaster and Salkauskas, 1986) that the data is a samme from random function v(p), which is the sum of a " slowly " varying random polynomial \ V of degree m, called the drift, and a "rapidly" varying random component r(p) , 'ch is assumed to have ‘ zero mean or expected value E [r] = 0 and which is sponsible for the noise-like nature of v(p), Figure 3.6. ms v(p) = d(p) + r(p) , E [r] = 0 -( 3.14) It is assumed further that the covariance structure of r(p) can be obtained and that : covariance between values of r(p) at points p and q depends only on the distance tWeen p and q. en Cov [r(p) , r(q)] = f(distance between p & q) ( 3.15 ) Now it is desired to estimate the unknown true value v(po) at a point po re no sample is available by a linear combination of the available samples: v(po) =2 w,v(p,) . (3.16) i=1 re wi is the weight associated with the random variable at the location (i). It. v(p) A P1 Pn random . ---._.-._.-.---...._ . A function r(P) 2"Rapidly” varying (W random component_ Drift d(p): J ”Slowly” varying d(p) random polynomial. \ Spatial distance. Figure 3.6 : Components Of Random Function In'Kriging Model. 50 This set of weights is allowed to change as unknown values are estimated at ‘ferent locations, in such a way that the variance of the error "r" is minimized. This 'or is given by: r(po) =2 WiV(pl-) —Vtrue(po) . (3 .17) i=1 The probabilistic solution to this problem assumes that for any point at which 3 desired to estimate the unknown value, the model is a stationary random :tion that consists of several random variables, one for the value at each of the (ple locations, V(p1), . . , V(p,,), in the subset used to predict V(po), and one for the nown value at the point where the estimate is desired V(po). Each of these random iables has the same probability law at all locations; the expected value of the lom variable is E[V], (Isaaks and Srivastava, 1989). This means that all the random variables V(p,),. . V(p,,) and V(po) in the subset ‘aken to have the same expected value EM and the same variance. For the SPT this means that E[N(x,y,z)] and Var[N(x,y,z)] are the same everywhere in any at and the measured values are deviations from E[N]. Thus the estimation error R at the point po is also a random variable and Men by: Error = Estimated value— True value R(po) =2 WiV(pl-) -Vtrue(po) (3.18) i=1 51 r the unbiasedness condition states that E [ R(po) ] = O :efore: E[R(po)]=0=E[V]ZWi—E[V] (3.19) i=1 sequently: Z wi=1 (3 .20) i=1 Now to get the best estimate of V(po) it is desired to minimize the error ane Var [ R(po)] , (Isaaks and Srivastava, 1989, pp. 278). The error variance )btained as the variance of the difference of the estimated value and the true 3 of the variable which is given by: 0122:02[I7(P0)-V(P0)] (3.21) Lnding and manipulating some terms leads to: 9V[I7(Po) , v(po) ] -COV[I7(Po) ,V(PO) ] -COV[V(PO) ,l7(Po)] DV[V(P ), V(P )] O O >V[l7'(Po) , 17(Po)] —2C0V[I7’(PO) , V(Po) ] +COV[V(P0) , V N1 (Z'XI Y) V N At'A Given (x,y). N2 ( zlx.y) .__ gure 4.5 : The Trend Surface Is Decomposed Into A. Trend And A Random Error. I. Ta the 12 hon) Sllbst filler flucu 64 4. If more than one layer is justified, a separate three-dimensional model for the N « values as N = f(x,y,z) is constructed for each layer as shown in Figure 4.5. Regarding item "1" above, the relative density Dr values are estimated by using the correlation between the relative density and the N values. Terzaghi and Peck (1948) gave the first classification of relative density in terms of the N values. Values of Dr were assigned to this classification by Gibbs and Holtz (1957). The combined results are shown in Table 4.1. Table 4.1 : Classification Of Relative Density In Terms Of N values. Dr N Relative density 0.15 0 - 4 very loose 0.15 - 0.35 I 4 - 10 loose 0.35 — 0.65 10 — 30 medium 0.65 — 0.85 30 — 50 dense 0.85+ 50+! I very dense As far as the layer depth is concerned, Vanmarcke’s (1977, pp. 1229) noted that e layer depth in a real profile may vary more or less erratically as a function of the )rizontal dimension; the type of modeling he proposed is meant to supplement , not to bstitute for the conventional soil profile and the concept of "local homogeneity" is tended to cover not only averages, but the nature and the overall appearance of the .ctuations as well. As such, it suffices for this research to stratify the soil into layers b by rat _._.._.,_._...._.. _... .— .—_. _.. ._ 65 y using geotechnical judgment which considers the average of N values as well as the ture and the overall appearance of the fluctuations. Regarding item "2" above, the "one-way ANOVA analysis" will provide a tistical measure that the stratification is reflecting real different layers with erent relative densities and thus verify the judgmental stratification. The measured ues of N within each layer are considered as one treatment, so the number of trnents in the ANOVA analysis will be the same as the number of layers adOpted. 's kind of analysis is suggested because it distinguishes the amount of variation within between treatments. So in order to test the dependence of the N -mean values on depth, the difference between the layer means has to be large compared with the 'ations within layers. In other words, to test the null hypothesis that the ferent layers means are equal, the F ratio - which is the quotient of the "between ers mean square" by the "within layers mean square" - is compared to the F (babilty distribution value at the desired significance level (Davis, 1986; Box, Hunter .Hunter, 1978, pp. 187)“. If the difference between the layers means is proven to be significant, then the gested stratification is accepted; otherwise no stratification is justified and the whole :urface soil is considered as one layer, (e.g., although differences may exist to depth effects, the differences in average properties are not significant enough to cut representing the whole subsurface soil by a single spatial model). Regarding item "3" above, the difference between any two consecutive layers has 3 significant, otherwise they should be combined into one layer. This may then :ase layer variance, but the estimation uncertainty of the model will be reduced inst betv that UllC mm 0011 AN D1 311 66 because one will have the advantage of estimating the parameters of only one model instead of two by using the same amount of data. Therefore a trade-off is being made between estimation uncertainty and random spatial variation. Baecher (1978) stated that exploration data have a finite number of degrees of freedom : the more parameters estimated, the more uncertainty in each. To reduce overall predictive uncertainty requires both more data and a more spatially disaggregate stochastic model. The comparison between layers has to be made by using a suitable multiple comparison technique. The difference between the multiple comparison test and the ANOVA test is explained as follows: The ANOVA test’s objective is to test whether the stratification is justified irrespective of which number of layers was used to conduct the test. The multiple comparison test’s objective is testing whether the suggested number of layers is aCcepted; otherwise a lesser number - by combining some of them together — is considered. here are many. multiple comparison tests available in the literature, (Levine, 1991; mm, 1961; Dunnett, 1964 and Tukey, 1949). Three such tests will be used herein; they 6 briefly summarized below. D (Least Significant Difference) Test: This test (levine, 1991) is done to test the significance of the differences between airs of means. It is simply a convenient form of the t test. It allows the use of ifferent alpha levels for different comparisons. The LSD test is not done if every ossible paired comparison is to be tested. This test involves computing a single value or "critc criterion statistic; to be ze Tukey 1 value : compan testing when 1 be used This te,‘ the crin DImn'. the ma Selene) The 1 made. to 0111) rec01m 67 r "criterion value" against which each desired paired comparison is tested, i.e. this riterion value is compared to each difference between means that is to be tested for tistical significance or to check whether the corre8ponding'true difference is not likely be zero. key HSD (Honestly Significant Difference) Test: This test (Tukey, 1949) provides a convenient way of computing a single ue against which all of the possible differences between pairs of means can be mpared for statistical significance. It also uses some confidence limits for ting the significance of the differences between pairs of means, but it is used hen the intention is to maintain one "family-wise" Type I (alpha) error level to :used for all comparisons. For "n" layers, there are n(n-1)/2 possible pairs of layers. 118 test compares every one of these possible n(n-1)/2 difference between means to 3 criterion value to test it for statistical significance. Inn Test: Whereas the Tukey test runs all of the possible pairs of comparisons among :means, the Dunn test (Dunn, 1961) is used when the intention is to test only some ected number of pairs. At the same time this test maintains an overall "family-wise" e I error probability that does not change with the number of comparisons to be e. The Dunn test is the alternative to the Tukey when the decision has been made nly test a subset of all possible paired comparisons. Consequently, this test is not mmended unless the number of layers is relatively large. C011 the N ) nec be ten bet Th his uh 68 Regarding item "4" above, a three-dimensional model for the N values as N =f(x,y,z) is built for the entire foundation or for each layer separately. Some considerations about the building of this model are as follows: 1. The model for each layer is built by conducting a multiple regression analysis on the relevant N values. The resulting regression surfaces can be used to predict the soil N value at any depth beneath any location on site. The three parameters x, y and z are necessarily independent; consequently their effects are additive and no interactions should be expected between any of them. This leads to the exclusion of all the cross—product terms , of any order, from the model. As a result, the general form of the model becomes: N(X, y, z) =bo+blx+b2y+b3z+b4xz+b5y2+b622+. . . (4 . 1) The degree of the model and the terms that have to be included are assessed for six case histories in chapter five. However, certain relevant information are given herein to be rtilized to develop guidelines prior to such analyses. The most important term that should be included is the constant term b0, because t is the basis from which the N value varies in the x , y and 2 directions. Philosophi- ally satisfactory model would have a b, value that is also a reasonable value for N. The variability of the N value in the z direction will be controlled by the oil stiffness which increases with depth due to overburden pressure. The correlation etween N and E as a measure of stiffness is well established (Bowles, 1988; Webb, 969 ; Anagnostopoulos,1990), but no consensus is evident among the researchers as > how "B" would increase with depth. As a result of this uncertainty, the orders of the Ztt 1681] level unde N va Thes sehh 44h] T0 1)} 4.6 69 :rms is an issue to be examined using the regression analysis by examining the rlting values of their coefficients. The F test approach can be used to test the aptness of this model. If it is not epted, a different case of this general form of the model should be employed and ed again. The predictive power or " goodness of fit" of the resulting model can be ntified by calculating the R2 value, which represents the percent of total variation in :xplained by the formulated model. The geometric location of the footing in addition to the determined foundation :1 will identify the three coordinates of the footing as well as the depth of influence .er its loading, consequently the resulting model can be used to determine the design 'alue for this footing. * >se design N value(s) are used in the next stage which is the estimation of lements and differential settlements. 1 USING THE "TREND SURFACE" N FUNCTION TO PREDICT THE SETTLEMENTS AND DIFFERENTIAL SETTLEMENTS lredict the settlement under a specific footing of breadth "B" as shown in Figure l the following N functions are formulated using corrected N values: N1=f(x,y,z); z Kriged in two dimensions. This form is suggested by this research based on a study by Gibson (1967) which .ered the subsurface soil as an elastic half-space whose modulus E increases y with depth. The relationship between the soil modulus E in turn and the N is realized by the examination of the deterministic equations for settlement tion which were developed by Meyerhof, Peck, Bazaraa and Parry and were irized earlier in Section 2.4. The inclusion of the N values in the denominators e strain determination equations implies the assumption that there is a linear ship between N and E. Consequently , the representation of the N values by linear regression mentioned bon' Figure 4.8 4.4.2 Kriging The fol Where an eStin The ho the bofing 10c; 0f h Values wi HSE>Ct10n 3'2 experimeHlal t prOCedul-e‘ Ct 80 r regression functions in the depth Z is considered as reasonable. Thus the above ioned borings will be represented as shown in Figure 4.8. N2: 8.2+ bzz 1%: ar+lmz + + boring(Z) Nu: an+ bnz boring(l) + ' boring(n) No: ao+ boz + the point where an estimate is desired boring + boring . + igure 4.8 : Representing The N Values By A Linear Regression Function. Kriging The N Regression Coefficients: The following procedure is suggested to estimate the N function at any point an estimation for it is desired: The horizontal distance (h) between every pair of locations is calculated including ring locations and the locations where estimation is required. As such, the number tlues will equal to (n+1)n/2 for each "unknown" location. The covariances which describe the. spatial continuity of the data are calculated. ion 3.2.3.2.4 reference was made to Davis (1986) who noted that in principle, the nental covariance values could be used directly to provide values for the estimation are. Consequently, if the available data were enough to calculate the covariance values I the cov discrete require modele case th the (hit Baeche expouc by: Where 81 between every pair of locations, then these values could be used directly to build variance matrix required for Kriging. However, the covariance is known only at te points representing the sampled locations. In practice, covariance may be ed for any distance. For this reason, the discrete experimental covariances may be ed by a continuous function that can be evaluated for any desired distance. In this e employed covariance model should be chosen in a way that is consistent with . Based on the discussion in Section 3.2.3.2.4, there is an agreement between er et. al. (1993) and Soulie’ et al.(1990) on the suitability of the squared :ntial model for describing the variability of soil properties. This model is given C(h) :oze(-h2/h:) - (4.19) the distance between two points for which the covariance is desired. autocovariance distance. the distance at which C(h) decays to (l/e) C(O), where e is the base of the natural logarithms. ing the value of the autocovariance distance (ho), reference is made to Baecher ho noted that considerable empirical work has verified the theoretical contention properties are spatially correlated, and he suggested a distance of 50 m - 100 m finite correlation. Then the covariance of every pair of locations is calculated this model. This will total to (n+1)n/2 covariances. where in order Or equiv: COHSQqu and Calculating the weight matrix from: WeC"1hD (4.20) ere C = (n+1) * (n+1) matrix D = (n+1) * 1 vector and the matrix elements are as given earlier. Kriging the N values - whiCh are now represented as functions in z - order to estimate the unknown N value at the point (xo , y) as follows: I) fi(xo,yo) =2 wiN(x,-,y,-> (4.21) i=1 quivalently: ao+boz=2 W1 (are-z) (4 .22) i=1 :2 Wrai+2 Wibiz (4 .23) i=1 i=1 sequently: n 620:2 wia, (4 .24) i=1 (4.25) was sugg 1.1“; Serum Section 83 Then, the estimated value N(xo , y,) is equal to (a0 + boz), which is a function in 2. Using The Estimated N Function To Predict The Settlements And Differential Settlements Having obtained the N value as a function in z, the "two-point" estimate — which uggested in Section 4.2— can be used to estimate the design N value as follows: The N values at the depths of B/2 & 3B/2 are given by: n(B/,,=ao+bO1/2,N Thed Which B01111; Thed 5.2.2. Where 157 It is observed in the Figure that the two lines differ, but they are reflective of the " local average" versus a single location average. 5.2.2.2 USING THE ESTIMATED N FUNCTION TO OBTAIN THE DESIGN N VALUE The "two-point" estimate can now be used to estimate the design N value as follows: The N values at the depths of (B/2 = 5.75 ft) and (3B/2 = 17.25 ft) are given by: 1703/2)=a0+bo(B/2+embedment) =23.51—0.495 (5.75+8.5) =16 .46 and Kim/2) =ao+bo(3B/2+embedment) =23 . 51—0.495 (17 .25+8 . 5) =10 .76 The design N value at this location "0" is then given by the weighted average: N=(l/3) [217(3/2,+1§‘I(3B/2)] =(1/3) [2(16.46)+(10.76)]=14.56 which compare to the measured values of (18, 17, 67, 64) in this general vicinity in Boring (B-103). The design N value obtained by the trend surface analysis was 36.1. 5.2.2.3 USING THE DESIGN N VALUE FOR SETTLEMENT PREDICTION: The predicted settlement "S" in inch is given by: S = (2/N) [q * (2B / B+1)2] ; (Peck and Bazaraa, 1969). where B = wid q = net = 650 Therefore S = (2 / = 0.58 The settlement The previous dimensional cc pressuremeter The measured Therefore the 1 settlement by z 0.24 in is less 158 B = width of footing = 11.5 ft. q = net foundation pressure in (tsf). = 650 Kips / (2000 * 11.5’ * 22.5’) = 1.256 (tsf). Therefore S = (2/ 14.56) [1.256 * (2 *11.5/ 11.5 +1)2] = 0.58 inch. The settlement predicted by the trend surface analysis was 0.24 inch. The previous investigators give a predicted settlement as 0.53 inch using the one- dimensional compression laboratory test and a predicted settlement as 0.12 inch using the pressuremeter test. The measured settlement as reported by the previous investigator = 0.3 inch. Therefore the predicted settlement (by Kriging) of 0.58 in is greater than the measured settlement by approximately 93 % and the predicted settlement (by trend surface) of 0.24 in is less than the measured value by approximately 20 %. 5.2.2.4 ‘ The pre med prediction van' Te; Term (2 Term (3 Therefore, thI The p; emulate to an N Value is gi‘ 1J59 5.2.2.4 CONSTRUCTING THE PREDICTION INTERVALS FOR THE ESTIlVIATED VALUES OF BOTH N AND S The prediction variance is given by: n n n 2 2 OR=E:Z:Wl"m'7j'C1'j_22I'VJL'C.1'0+0N (5-35) i=1 j=l i=1 From the previous results of the weight and covariances matrices the terms of this prediction variance are calculated as follows: II II Term(1) =2 2 w, . wj. Cij=18 . 89 1=1 ]=1 n Term(2) =22 Wi. Cio i=1 = 2 (w2 C2° + w5 C50 +w6 C6o + w9 C90) = 99.44 rerm(3)=o§=133.27 Therefore, the prediction variance is given by: o§=52.7 The prediction variance can be used to construct the prediction interval on the estimate to any desired degree of confidence. Thus, the 90 % prediction interval for the N value is given by: Nat(M1U,,Om¢o§=(14.5642.353¢52.7)=14.56417.1 Howe\ standard pene accepting or n role in the acc be used - as I For these rea testing. In this regard 2:) This says the: Confidence int Variance of t} surface analys After I value by Kn'g trend surface. Which is large the effect of t the data of t] predlClion by —+ —r-—.-v-I—'-- __ _ _‘_ .. A 7 I ww_~_~_h~—- __._ _ ~._._._.— __.‘ . ._r -. ._ _.. ~. A 160 However, in practice, the 90 % confidence interval is pretty broad. For the standard penetration test, the confidence interval is not intended to be a criteria for accepting or rejecting an N observation. The experience of the driller plays an important role in the acceptance of an observation. Furthermore, the upper confidence limit could be used — as recommended by some researchers e.g. Baecher,]981 - as a design value. For these reasons, the 50 % confidence interval is considered reasonable for the SPT testing. In this regard, the 50% prediction interval for the N value is given by: A N:t((4_1) film/oi: (14 . 5650 .765752 .7") =14 . 56:5 . 55 This says the 50 % confidence limits are: (9.01 and 20.11), which is a relatively broad confidence interval for the predicted N value, obviously because of the high value of the variance of the N values. For comparison, the 50 % confidence limits from the trend surface analysis were (32.8 and 39.4). After the examination of these results, it is observed that the predicted design N value by Kriging was 14.56 which is lower than the predicted design N value of 36.1 by trend surface. On the other hand, the 50% confidence interval by Kriging was 11.1 which is larger than the 50% confidence interval of 6.6 by trend surface. This reflects the effect of testing a "hard spot" at the location of Boring (B-103) and the exclusion of the data of this boring from the prediction by Kriging while including them in the prediction by trend surface. The 5( follows: The 50% coni were: (0.22 1 Here a; Boring (B-103 0.24 in. On- reflects the efi 161 The 50 % confidence limits of the settlement prediction (by Kriging) are as follows: S=(2/(1\7¥t((4_1),0.25)\/0E)) tq*(2B/B+1>ZI = (2/2011) [1.256 * (2*11.5/11.5+1)2] and (2/9.01)[1.256*(2*11.5/11.5+1)2] = (0.41 and 0.95) in. The 50 % confidence limits of the settlement prediction (by the trend surface analysis) were: ( 0.22 and 0.26) in. Here again, the effect of including the data of the tough spot, at the location of Boring (B-103), in the prediction by trend surface resulted in a predicted settlement of 0.24 in. On the other hand, the predicted settlement by Kriging was 0.58 in, which reflects the effect of using the local average instead of the N value at the " hard spot". ‘ 5.2.3 SUM] SIX( Fiver Carolina - we and the settle six case histo 1. Split] 2. A twe 3« Large 4. . A loa< 5' A gen 6. A III: 5.2.3.1 The These histories. Tm Well defined meant that eaI 1' Every boring 2‘ The f0 3' The 10 4' The ac “. r.<:-V'.'_ “"71-‘n 162 5.2.3 SUMTVIARY AND ANALYSIS OF THE PREDICTED SETTLEMENTS FOR SIX CASE HISTORIES Five more case histories - in addition to the split level office building in North Carolina - were analyzed using both the trend surface analysis and the Kriging technique and the settlements are predicted for each of them using both techniques. A list of the six case histories together with the references are as follows: 1. Split level office building in North Carolina ; (Borden and Lien,l988). 2. A twenty - story block in Nigeria ; (Grimes and Cantly, 1965). 3. Large tanks in Kansas City ; (Davisson and Salley, 1972). 4. .A load test in northern Spain ; (Picomell and Del-Monte, 1988). 5. A generator in Pennsylvania ; (Fischer et. a1. , 1972). 6. A lift bridge in Delaware ; (Seymour et. al. , 1972). 5.2.3.1 The Case Histories Selection Criteria And Summary: These six case histories were selected from among tens of published case histories. The selection was controlled by the criteria of accepting the case only if it has well defined data from the viewpoint of the spatial and geotechnical modeling. This meant that each case history should satisfy the following requirements: 1. Every N value should have a defined location as N (x,y,z). This means that the boring locations in the (X , Y) plane and the depth of each N value are reported. 2. The footing dimensions and locations are available: 3. The loading to the base of the footing is available. 4. The actual settlement is measured and reported. The selected 3 were found to The an: in details earlie of this analysis summarized as The tre 1. RefereI histon'es, the 1 field, hence 5 for each case ; selected as prI Case history I N: 163 The selected six case histories satisfied these four requirements. Case histories which were found to violate one or more of these requirements were not selected. The analysis of these case histories - except for the first one which was analyzed in details earlier in this chapter - is summarized in Appendix "C". However, the results of this analysis are presented together here for the sake of comparison. These results are summarized as follows: The trend surface analyses yielded the following results: 1. Reference is made to what was mentioned in Section 5.2 that, for the actual case histories, the best fit of trend surface models cannot be easily judged as in the assumed field, hence 5 to 15 models were tried for each case. The best model was determined for each case according to the criteria explained in Section 5.2. The models which were selected as providing the best fit to the data of the different case histories were: Case history No.1: N=-3008929 .7+10.058Xo'5+78.1351/0'5+2'74208.9Z0'5 -0.612X—3.794Y—7027.8Z+1.36722 (5.37) R2: 0.528 Substituting the coordinates (104.17, 92.7) of the predicted location in this model yields the N(z)funtion at this location as follows: N = 1.367 22 + 4590.3 Z + 2742089 (-Z + 891.5)” — 81872892 Case history No.2: N=35.66-16 .7X0'5+1.38X+0 . 0019X2+4 . 56Y0'5-0 . 37Y +1.73Zo'5-0.107Z--0.001Z2 (5.38) R2 The Na, func Case history Case history The Na) fun Case history The N(z) fill] 164 R2: 0.458 The Na, function at the predicted location, (115.8, 58.42), is given by: N = -0.001 22 — 0.107 Z + 1.73 205 + 54.47 Case history No.3: 1\I'=—1254—0.0082X+1476z°-5-624Z+114.321-5 —7.68Z2 (5.39) R2: 0.65 The NZ) function at the predicted location, (315, 178.38), is given by: N = -7.68 Z2 + 114.3 Z15 — 624 Z + 1476 20'5 — 1256.58 Case history No.4: N=—15.26+0.216X+5.67zo-5—0.417Z+0.001Z2 +7.9ZZ3 (5.40) R2 = 0.62 The N (Z, function at the predicted location, (32.5, 25), is given by: N = 7.92 E—7 Z3+ 0.001 22 — 0.417 Z + 5.67 20‘5 — 8.24 Case history No.5 : N=15197—o.02X—4302ZO-5+36oz—0.74z2 (5.41) R2: 0.53 The N (2) function at the predicted location, (260.6, 0.0), is given by: N = —0.7442 22 + 360.89 Z — 4302.4 Z0'5+ 15191.79 Case history The NZ) fun 2. The d are 'sl. Table Case The l The esfihmt: 811111111me 165 Case history No.6: 1\7=27836—1.62X—807220-5+675.2'—1.622 (5.42) R2: 0.503 The N(z) function at the predicted location, (760, 0.0), is given by: N = -1.6 Z2 + 675 Z - 8072 Z05 + 26604.8 2. The design N values were as shown in Table 5.11. The rounded integer values are ‘shown between brackets. Table 5.11 : The Design N Vaues For The Six Case Histories. Case History No. Design N Value 1 36.1 (36) 2 6.91 (7 ) 3 3.16 (3 ) 4 12.29 (12) 5 . 65.6 (66) 6 39.18 (39) The prediction by Kriging yielded the following results: The estimated N functions and the design N values for the six case histories are summarized in Table 5.12. Table 5.12 : Case History l 2 Table 5.13 x Case History No. \ 1 2 7 ,1, remis- =..-- “rv- -- - ' 166 Table 5.12 : The Estimated N Function And The Design N Values For The Six Case Histories. Case History No. . The Estimated N Function. The Design N Value. 1 N = 23.51 - 0.495 Z 14.56 (15) 2 N = 10.745 - 0.019 Z 10.18 (10) 3 N = 7.5 + 0.368 Z 4.13 (4) 4 N = 6.572 + 0.139 Z 12.38 (12) 5 N = 0.747 + 33.593 Z 74.6 (75) 6 N = - 123.5 + 0.97 Z 37.23 (37) Table 5 .13 : Summary Of The Predicted Settlements Versus The Measured Values For The Six Case Histories. Settlement (inch) Case - Predicted By Measured History ‘ - The Designer Trend Kriging No. (method ref.) Surface 1 0.12 0.24 0.58 0.3 2 1.5 . 2.5 2.29 0.97 3 2.1: (Peck et. al.) 3.9: (Schmertmann) 4.03 3.1 3.3 4 1.4: (Peck et. al.) 1.2: (Meyerhof) 1.93 1.92 1.56 5 1.3: (Terzaghi & Peck) 0.76: (Meyerhof) 0.48 0.43 0.5 6 0.3 0.36 0.38 0.4 Tat hist Tal Wt 110) 167 The predicted settlements by both methods are summarized as follows: Table 5 . 13 shows the predicted settlements versus the measured values for the six case histories. Table 5 . 14 shows the ratios of the predicted settlements to the measured values. Table 5.14 : Settlement Ratios For The Six Case Histories. Case Settlement Ratio: (Predicted / Measured) History No. By Designer Using Trend Surface By Kriging 1 0.4 0.8 1.93 2 1.55 2.57 2.36 3 0.64 , 1.18 1.22 0.94 4 0.9 , 0.77 1.24 1.23 5 2.6 , 1.52 0.96 0.86 6 0.75 0.9 0.95 5.2.3.2 Analyzing The Results: The predicted settlements which appear in Table 5.13 were compared to the measured values. Investigation of the predicted settlements by both techniques as well as by designers gives an impression that both designer and statistical methods significantly overpredicted the settlement of the second case history suggesting that it could be a potential outlier that can be dropped out from the comparison. The comparison was made by running a simple t—test of zero mean on the paired differences. The hypothesis to be tested is that the difference between the predicted settlement (8,) and the measured settlement (Sm) is zero. The t—statistic is computed as: Thef betwe Thec GroI Groi Va '—J The: bar 010. insig 168 t(nl+nz-2) , (oz/2) = [ (Eh—Em) — 01/ USS/.711) +(Sj/n2) 10's (5 '43) where Sp = standard deviation of the predicted settlements. S.m = standard deviation of the measured settlements. n = number of cases in each group. The first t-test was conducted using the computer program "SPSS" to test the differences between the settlement predictions by the trend surface analysis and the measured values. The computer output of this test appears in Figure 5 .30. Number of Cases Mean Standard Deviation Standard Error Group 1 5 1.4080 1.618 .724 Group 2 5 1.2120 1.273 .569 Pooled Variance Estimate Separate Variance Estimate F 2 -Tail 1: Degrees of 2 -Tail t Degrees of 2 ~Tail Value Prob . Value Freedom Prob . Value Freedom Prob . 1.62 .653 .21 8 .837 .21 7.58 .837 Figure 5.30 : t-Test For The Paired Difference Between The Settlements Predicted By The Trend Surface Analysis And The Measured Values. The second t—test was conducted to test the differences between the settlement predictions by Kriging and the measured values. The output of this test appears in Figure 5.31. From the results of the t-test: the predictions by the trend surface yielded a t value of 0.21. The critical t value is 2.306 (at the level of 0.05 and the degrees of freedom of 8) Since 0.21 is well below 2.306 then, the null hypothesis that the differences are insignificant is accepted. Gro Gro Va 2.30 cone. trenc' by It yield did 1 001le isin 169 Number of Cases 'Mean Standard.Deviation Standard Error Group 1 5 1.2820 1.198 .536 Group 2 5 1.2120 1.273 .569 Pooled variance Estimate Separate Variance Estimate F 2—Tail t Degrees of 2-Tai1 t Degrees of 2-Tail Value Prob. Value Freedom Prob. Value Freedom Prob. 1.13 .910 .09 8 .931 .09 7.97 .931 Figure 5.31 : t—Test For The Paired Difference Between The Settlements Predicted By Kriging And The Measured Values. The predictions by Kriging yielded a t value of 0.09. Since 0.09 is well below 2.306, then the predictions by Kriging are also consistent with the null hypothesis. The conclusion here is that the differences - either between the settlement predicted by the trend surface analysis and the measured settlements or between the settlements predicted by Kriging and the measured settlements — are insignificant. Although both of them yielded acceptable settlement predictions at the level of 0.05, the predictions by Kriging did better. This is reflected by the t value of 0.09 in the Kriging case being less than the t value of 0.21 in the trend analysis case. The general conclusion here is that the trend surface analysis is preferred to Kriging as long as the trend is fitted with an R2 value of at least 0.8, otherwise the Kriging is recommended. Kriging more strongly reflects "local" effects which also strongly affect indvidual footing settlement. The designer’s predictions were also compared to the measured settlements. The computer output of the t—test appears in Figure 5.32. The results show that the difference is insignificant with a t value of 0.25 which is less than the critical value. 170 Number of Cases Mean Standard Deviation Standard Error Group 1 8 1.3850 1.198 .424 Group 2 5 1.2120 1.273 .569 Pooled variance Estimate Separate variance Estimate F 2-Tail t Degrees of 2-Tai1 t Degrees of 2-Tai1 Value Prob. value Freedom Prob. Value Freedom Prob. 1.13 .831 .25 11 .809 .24 8.22 .813 Figure 5.32 : t—Test For The Paired Difference Between The Settlements Predicted By The Designers And The Measured Values. The examination of the results of the t—test between the designers predictions and the measured values shows that the t value was 0.25 which is more than the t values from both the trend surface and Kriging. These values were 0.21 and 0.09 respectively. This result shows the capability of the techniques proposed herein of yielding settlement predictions which are (on the average) more accurate. The conclusion here is that the proposed techniques have the following advantages over the current procedures: 1. They yield (on the average) more accurate results due to consistency in selecting the data and determining the design N value as the weighted average of the two N value estimates, obtained at the depths of B/ 2 and 3B/ 2 under the footing. This conveys information regarding both the average of N values within the zone of influence as well as the rate of increase with depth. This is tested in this section. 2. They are stronger in the sense that they provide confidence limits at the desired level of significance. This can help the foundation designer make stronger decisions about the structures supported by these foundations. 3. The upper limits of the estimated N values produced by these techniques can be used as design values to produce less conservative designs with lower costs and yet based on a rationalized criteria. CHAPTER6 SUlVIlVIARY AND CONCLUSIONS 6.1 SUMlWARY Cohesionless soils generally provide good bearing capacity, and settlement usually controls the design of shallow foundations. The state-of-the-practice for design of shallow foundations on cohesionless soils is that N values obtained from the standard penetration test (SPT) are used to estimate settlements, either directly or as predictors of elastic parameters such as the soil modulus "E". A difficulty of working with N values is their inherent Spatial variability. It is therefore important to be able to select an appropiate design N value that can be used with confidence with these settlement equations and parameter correlations. Regardless of the variability of the results, the standard penetration test is not likely to be abandoned because it has remained the most convenient and economical means to obtain subsurface information in cohesionless soils. The availability of computers and software increasingly permits application of relatively sophisticated analysis tools to practical problems; accordingly it is important to develop a technique to treat the data scatter accurately and consistently. It was undertaken by this research to investigate the use of geostatistical techniques for spatial data analysis and modeling of continuous N -value functions. Geostatistical modeling was chosen because it can account for spatial variability at both the large scale (spatial trend) and the small scale (spatial correlation). Two geostatistical techniques were considered, namely trend surface analysis and Kriging. Trend surface analysis, a geostatistical version of nonlinear 171 it» 172 regression, fits data only to large scale variations, whereas the Kriging technique ignores the trend and fits data to more localized small scale variations only. Based on a review of factors which affect the N value (soil relative density, overburden pressure, stiffness. . etc.) as well as the intended use of the model (settlement analysis) it was determined that a correction of the N values for the overburden pressure should be made before applying the suggested modeling techniques. In keeping with current practice, the N values were accordingly corrected based on the [correction equation recommended by Liao and Whitman (1986). Some of the geostatistical approaches potentially suitable for characterizing the scatter of the soil properties were summarized and two were selected for modeling the N value function. The considered two approaches were presented and their adaptability to the modeling of the soil properties was discussed. As a single N value would still ultimately need to be extracted from such a function, the "two—point" estimate approxima- tion was developed. This technique combines the N values within the depth of influence under the footing into one design N value based on the weighted average of two N value estimates obtained at depths B/2 and 3B/2. The results were shown to be comparable to the current procedures of estimating the design N value. The weighted combination derived from the "two-point" estimate permits transforming the spatial N(x,y,z) models into a planar N (x,y) model. This latter model form was used in the analysis of the case histories to do contouring analysis and the planar settlement comparisons. In the context of the adaptability of the trend surface analysis, it was suggested by this research to introduce an addition to it to fit the reality of soil stratification more closely. This addition was the multiple layers concept. Lack of homogeneity in the N 173 data in the vertical direction was accounted for by a nonconstant—mean assumption, and this was tested (later in the analysis of the case histories) by statistical multiple comparison techniques. The functional forms of the trend surfaces were approximated- with polynomial expansions which use the powers of the coordinates. Five to fifteen candidate polynomial forms were tried for each case history. The polynomial providing the best fit was then determined by comparing both the (R2) coefficients values and the standard errors of estimate and by judging the numerical value of each term in the model within the limits of the site. The polynomial approximation was suggested to take advantage of its flexibility which can conform to very complex surfaces if expanded to sufficiently high orders; however, in most case histories, simpler lower orders polynomials were found to provide the best fit for the N values. In the context of adapting the Kriging technique to the prediction of N values, it was decided to represent the N values of each boring by a linear regression function and perform the Kriging on the regression coefficients rather than on the N values themselves. This remedies the inconvenience that results from the lack of homogeneity in the vertical direction and transforms a 3-dimensional Kriging problem into a more manageable 2-dimensional one. The weights obtained from Kriging the regression coefficients were used to predict the regression coefficients at the point in question and ultimately the N value. The different forms of the covariance functions which describe the spatial continuity of the data were reviewed. The squared exponential model was used later in the analysis of the case histories to build the covariance matrix which was used for Kriging. 174 The determination of the model precision was discussed and an evaluation of the quality of prediction versus the quantity of the data and their monetary costs were consid- ered. This was achieved by considering the trade off between the number and location of data points and the confidence intervals of the resulting model. It was Shown that the confidence band of the estimated N value is proportional to (1/ n)":5 in which "n" is the number of N values. Finally, the developed models for settlement prediction were evaluated two ways. The first evaluation was made using simulated data of an assumed field. Given an assumed N function, the models were used to fit some data sampled from this function and to estimate the N values at group of unsampled points. The " estimated unobserved" values were then compared to the "true unobserved" ones. The results of this evaluation were consistent and showed that the developed methods can be used systematically with confidence conforming with the quality of the available data. Remaining in question was the effect of real and scattered data of a real site. These results ascertained also what was assumed initially regarding the superiority of using the trend surface analysis method when the data follow an underlying trend. The power of using the Kriging technique to handle the data with high degrees of randomness was also ascertained. The second method of evaluation was to test the practical reliability of the developed models by conducting the suggested modeling on a set of actual case histories. The variability of the real N values in the practical application to a real site may not be as ideal as that of a theoritical field. However, the variability of N values of the six tried case histories was found to be representable by equations whose X, Y and Z terms are not exceeding the second order. Consequently, it was easy to try out all the possible combinations and to 175 reach an acceptable goodness of fit. The resulting settlement predictions were compared with the measured settlements using t-test of zero mean on the paired differences. The results of these verifications confirmed the reliability of the developed techniques. The outputs included stronger inferences regarding the settlement predictions in the sense that they estimate the settlements and provide confidence limits at the desired level of confidence. This can help the foundation designer make stronger decisions about the structures supported by these foundations. 6.2 DETAILED SUMNIARY OF RECOMMENDED PROCEDURES Two statistical methods were developed for settlement predictions from results of standard penetration test, one using the trend surface analysis and the other using the Kriging technique. A summary outline of these methods follows: The outline of the trend surface analysis method: 1. Correct N values for overburden pressure. 2. Stratify soil into layers reflecting the homogeneity (or lack thereof) in N in the vertical direction. 3. Test the stratification by statistical multiple comparison techniques (LSD, Tukey, and Dunn tests). 4. Define a coordinate system. 5. Propose a number of candidate nonlinear three dimensional models to the N values within each layer. Evaluate their (R2) and standard error values to ascertain 176 that the models have acceptable prediction capabilities. Investigate the physical shape to ensure acceptable representation. Use the "two—point" estimate to transform the selected N(x,y,z) model into a planar N (x,y) model for the design N value, and assess the confidence limits of this model using a (50%) confidence level. Use the model N(x,y) in conjunction with Bazaraa’s settlement equation to produce a planar S(x,y) model. Use the S(x,y) model for contouring expected settlements of replicate foundations in (x,y) as well as for assessing the quality of prediction. The above procedure is summarized in Figure 6.1. 177 Correct N values for overburden pressure. Modify Stratification l V L Stratify soil into layers. Reject 7 V 1 Test stratification f V l— ‘ statistically accept v I Propose a number of candidate models as N(x,y,z), for each layer l V I Determine the model providing the best fit l V J Use the "two9point" estimate to transform the selected N(x,y,z) model into a planar N (x,y) model. Bazaraa’s model 8: f(N, P, B) l V l_ V A Assess the 50% confidence limits for the model N (x,y) V J Determine planar S (x,y) model V l Predict settlement. Plot “S" contours. .Assess the 50% confidence limits for settlements Figure 6.1: Flowchart Of The Trend Surface Method. 10. _,_,._.__, . _.~. .4.— 178 The outline of the Kriging method: Correct N values for overburden pressure. Stratify soil into layers. The Kriging method is only recommend for single strata. Fit N values of each boring to a linear equation as (N = a+ bz) using regression. Compute the covariance matrix (C) describing the spatial continuity of the data using either direct calculations (if enough data are available) or a squared exponential model as : cm) :oze ”‘Z/hil Compute the covariance vector (D). Calculate the weight matrix (W) as (W = C'1.D). Determine the estimated N function at the point in question as a linear regression function with parameters obtained as the weighted sum of the regression parameters of the considered borings. Use the "two-point" estimate with‘the N (2),,y to predict N (x,y). Use the predicted design N value in conjunction with Bazaraa’s model to estimate the settlement. Assess the quality of prediction by constructing the confidence interval to the level of (50 %) using the t—statistic and the prediction variance which is given by: 11 n n 2-2 Z _ 2 2 ' i=1 The above procedure is summarized in Figure 6.2. 179 Correct N values for overburden pressure. T V I Stratify soil into layers 1] v I Fit N values of each boring to a linear equation N= a+bz T Y 41 Compute covariance matrix (C). r V L, Compute covariance vector (D). T Y I Calculate the weight matrix (W). l V I Determine the N function at the predicted location as N (z). V L Use the "two-point" estimate with N(z) to predict N (x,y) V l Bazaraa’s model Assess the 50% confidence limits S= f(N, P, B) for the predicted N value. I L V 1 Estimate settlement "S" at predicted location Figure 6.2: Flowchart Of The Kriging Method. 1 80 6 .3 CONCLUSIONS The following Conclusions can be made concerning the findings in this study: 1. Applicability Of Geostatistics: A general conclusion can be drawn from this research that geostatistics can be a powerful tool to extract a representative function from among spatially scattered N values and to provide a rational estimate of N value where no measurements were taken. 2. Use Of Trend Surface: The trend surface analysis does not create a trend by itself and its use can be justified only if a large scale trend in the N data is in fact present. In this case the trend surface analysis is employed to smooth out the random variation that enters into the data and to identify the underlying trend in order to use it for prediction. A fitted surface to a data scatter with an (R2) value of less than 0.2 would lead only to misleading results. A value of > 0.8 would imply confidence in the fitted model. For practical problems, however, it was possible to obtain R2 values of about 0.5 — 0.6. 3. Candidate Models For Trend Surface: Where the trend surface analysis is justified (e. g., R2 > 0.2), then it is important to investigate the variability of the data carefully, by trying out some preliminary models, and select a model which can adequately represent this variability. 181 Use Of Kriging: The Kriging technique was found to be superior to the trend surface analysis if a certain degree of randomness is present. In this case, the N value at a given location is assumed to have no influence on the N values at distant locations. If the data follow a trend, then Kriging can still be used to find a point on the trend provided that the employed data are located in the vicinity of that point in order to justify ignoring the trend. Quality Of Data: The precision of the settlement prediction depends on both the predicting model qualifications and the data qualifications. If the data qualifications are inferior (e.g. a subsoil investigation based on data from a single boring), then neither the trend surface nor the Kriging technique can produce reliable predictions. The inferior quality of data could also result from using an unreliable drilling equipments or inexperienced crew. Prediction Confidence Intervals: In constructing the prediction confidence intervals for practical design problems, it might be considered reasonable to adopt a degree of confidence of 50 % rather than the conventional levels used in statistical inference of 90 or 95 %. The level of 50% inference is more practical with the N value data due to the higher variability encountered. Where 90% or greater is used, the variance of N values resulted in a relatively broad confidence intervals even though the predicted 182 values were close to the measured values. This suggests that the construction of broad confidence intervals is meaningless. Deviation Of The Predicted Settlements From The Measured Values: Using trend surface analysis, the average of the ratio of predicted to measured settlements of the considered six case histories was found to be 1.28. Using the Kriging technique, this ratio was found to be 1.37. This deviation is believed to result from employing Bazaraa’s settlement equation in conjunction with the estimated N values to predict the settlements. Bazaraa’ 5 equation (1969) is a refinement of Meyerhof’s equation (1965) and this in turn is an analytical expression of Terzaghi and Peck’s well—known settlement design chart. These settlement equations have an empirical nature and hence have the common disadvantage of all empirical models of being applicable ideally only to the conditions where they were developed from. These conditions include the sand gradation and top size, the particle shapes and angularities, the relative densities . . etc. In any other case, it is unlikely to have these conditions of soil properties identical to the properties of the soils used to develop these settlement functions in the first place, the thing which result in. these deviations upon the use of these settlement equations. One more reason for the deviation of the predicted settlement from the measured value is that the design loads are often not realized. 183 8. The Depth Of Influence Under A Footing: Investigating the results of the case histories has also raised an important issue regarding the depth of influence within which the settlements are computed and below which the soil compression is left out of consideration. If the subsoil is in the form of a shallow loose layer laying on a much firmer thick deposit, and if the footing width is relatively large, then the assumption that the depth of influence is (2B) may give unj ustifiedly low settlement predictions because the upper layer which contributes most of the compressibility will not be given a realistic weight during the settlement computations. Using the Schmertmann’ 5 method, the strain influence factor implies the weights that are given to the N values at different depths. The distribution of the strain influence factor with depth was based on a theoretical work assuming that the modulus E is constant with depth. Consequently, the depth of influence of 2B as well as the values of the strain influence factor at different depths will all be confused if the modulus E increased rapidly in the vertical direction. In other foundation cases where a large number of footings are closely located, they will have some additional "overlap" effect extending the depth of influence to greater than (2B). 9. Summary Table Of The Two Methods: The advantages, disadvantages and uses of the two considered methods are summarized in Table 6.1. -_‘m ch. 184 Table 6.1: Summary Table Of The Two Considered Methods. Method Trend Surface Analysis. Kriging. Advanta- 1. Most useful if a sign- 1. Used where the data ges ificant trend is pres— are erratic with a ent in the data. Certain degree of 2. It is a geostatistical randomness. version of the nonlin- 2. Applicable to almost ear regression which any N value data. has been studied thor- 3. Predicted better than oughly and its theori- trend surface when tical background is modeling the N value well established in data to predict sett- the literature. lements. Disadv- 1. The process of select- 1. Computational effort antages ing the model provid— is relatively lengthy. ing the best fit is 2. Constructing the time consuming. covariances between 2.May lead in the absence the different locat- of a significant trend ions is laborious. 'to misleading results. Required 1. Where large scale var- 1. If there is enough co- iations occur with an nfidence in data to underlying trend. wish to emphasize 2. Where it is desired to local fit over trend. emphasize the trend 2. For fitting data to over local fit. small scale variations 6.4 — ..r_ ”fin-ff ~~“‘. ‘- '~." ' _ . 185 RECONIMENDATIONS In improving the settlement prediction techniques many aspects are involved. This research has considered the treatment of the scatter of N data and assessing the quality of prediction. As far as a comprehensive improved settlement prediction technique is concerned, it is important to consider the following recommendations: 1. There is no point in using sophisticated statistical modeling for settlement ' prediction while employing data of inferior quality. Therefore, it is recommended to base the settlement analysis on the results of at least four borings. Additionally, unreliable drilling equipment and inexperienced crew are obviously to be avoided. The data base regarding the covariance functions representing the variability of N values, which are used for Kriging, is not yet well established. Improvement is therefore required regarding the suitable functional forms of covariance functions to be used for cohesionless soils with varying prOperties. This can be done by testing different soils and establishing correlations including the N values, the distances between the tested locations and the best fits of the covariance functions. In preparation of N data before Kriging, the N values could, in some cases, be representable by a nonlinear regression function. In these cases, it is recommend- ed to represent the N values of each boring by a regression function of the form: N=a,+a,Z°-5+a,z+a,zz+... The same functional form is used for all borings. The parameters a0, a1, a2 . . are then Kriged in two dimensions. The estimated N function at the point in question is then obtained as a nonlinear regression function in the same form with 186 parameters obtained as the weighted sum of the regression parameters of the considered borings. Only by comparing the observed settlements with predicted values and by evaluating measurements can a development of new settlement functions and refinement of their accuracy be achieved. It is, therefore, urged to set up a settlement-recording program right from the begining for any structure of major importance or having unusual foundation conditions. Using trend surface analysis, the average of the ratio of predicted to measured settlements of the considered six case histories was found to be 1.28. Using the Kriging, this ratio was found to be 1.37. The reasons of this deviation were analyzed in Section 6. 3. These two ratios imply the overestimation of the predicted settlements in both cases. To make up for this overestimation, the following recommendation is suggested. If a level of confidence of 50% is adopted, then it is considered reasonable to use the upper confidence limit of the N value as a design value if need be. Using the upper confidence limit of N value means adopting higher bearing capacity and greater allowable pressures. This leads to smaller footing sizes with reduced costs. In other words, less conserva— tive designs are produced for lower costs and yet based on a rationalized criteria. Taking the depth of influence as (2B) under a footing of width (B) could be incorrect (e. g. where the subsoil is in the form of shallow loose layer laying on a much firmer thick deposit and the footing width is relatively big). The depth of influence of (2B) should therefore be questioned if need be to include only the layer that contributes most of the compressibility. APPENDICES APPENDIX A ' APPENDIX A A. SUBSURFACE SOIL STRATIFICATION OF CASE HISTORY No.1 A.1 Correction of N values for the overburden pressure: The corrected N value is given by: Nl =01 *N Where C1 = (l/Pv’)"'5 ; (Liao, 1986). pv’ = effective overburden pressure (tsf) = depth in (ft) * unit weight (lb/ft3)/2000 The correction results for the different borings are shown in Tables A.1 to A6. Table A.1 : Correction Of N Values Of Boring No. (B-102) For Overburden Pressure. Depth Unit wt. Effective Correction (ft) (lb/cf) Pressure factor N N1=C1*N (lb/sf) (C1) 2.0 125.0 250.0 2.828 8 22.627 4.0 125.0 500.0 . 2.000 11 22.000 7.5 125.0 937.5 1.460 13 18.987 10.0 125.0 1250.0 1.264 13 16.443 17.0 125.0 2125.0 0.970 17 16.492 21.0 125.0 2625.0 0.872 15 13.093 26.0 125.0 3250.0 0.784 22 17.258 31.0 125.0 3875.0 0.718 35 25.144 187 l? 188 Table A.1 : Continued. 36.0 62.6 4187.8 0.691 47 32.480 41.0 62.6 4500.8 0.666 79 52.661 Table A.2 : Correction Of N Values Of Boring No. (B-105) For Overburden Pnsmne. Depth Unit wt. Effective Correction (ft) (lb/cf) Pressure factor N' N1=C1*N (lb/sf) (Cl) 2.5 125.0 312.5 2.529 13 32.887 5.0 125.0 625.0 1.788 17 30.410 8.0 125.0 1000.0 1.414 18 25.455 10.0 125.0 1250.0 1.264 13 16.443 15.0 125.0 1875.0 1.032 7 7.229 20.0 125.0 2500.0 0.894 5 4.472 25.0 125.0 3125.0 0.800 7 5.600 30.0 125.0 3750.0 0.730 6 4.381 35.0 62.6 4250.2 0.685 7 4.801 40.0 62.6 4563.2 0.662 32 21.185 45.0 62.6 4876.2 0.640 48 30.740 50.0 62.6 5189.2 0.620 37 22.970 189 Table A.3 : Correction Of N Values Of Boring No. 03-106) For Overburden Pnrmue. Depth Unit wt. Effective Correction (ft) (lb/cf) Pressure factor N N1=C1*N (lb/sf) (C1) 2.0 125.0 250.0 2.828 8 22.627 5.0 125.0 625.0 1.788 29 51.876 8.0 125.0 1000.0 1.414 32 45.254 10:0 125.0 1250.0 1.264 14 17.708 15.0 125.0 1875.0 1.032 15 15.491 19.5 125.0 2437.5 0.905 18 16.304 25.0 62.6 3000.2 0.816 42 34.291 30.0 62.6 3313.2 0.776 38 29.523 Table A.4 : Correction Of N Values Of Boring No. (B-109) For Overburden Premuna , Depth Unit wt. Effective Correction (ft) (lb/cf) Pressure factor N N1=C1*N (lb/sf) (C1) 3.0 125.0 375.0 2.309 10 23.094 4.0 125.0 500.0 2.000 13 26.000 6.5 125.0 812.5 ‘ 1.568 12 18.827 9.0 125.0 1125.0 1.333 12 16.000 14.0 125.0 1750.0 1.069 13 13.897 19.0 125.0 2375.0 0.917 14 12.847 190 Table A.4 : Continued. 23.5 125.0 2937.5 0.825 10 8.251 28.5 62.6 3406.5 0.766 16 12.259 33.0 62.6 3688.2 0.736 38 27.982 Table A.5 : Correction Of N Values Of Boring No. (B-2) For Overburden inmne. Depth Unit wt. Effective Correction (ft) (lb/cf) Pressure factor N N1=C1*N (lb/sf) (C1) 2.0 125.0 250.0 2.828 8 18.475 5.0 125.0 625.0 1.788 11 19.677 8.0 125.0 1000.0 1.414 12 16.971 10.0 125.0 1250.0 1.264 15 18.974 15.0 125.0 1875.0 1.032 12 12.829 19.5 125.0 2437.5 0.905 14 12.522 25.0 62.6 3000.2 0.816 20 16.330 30.0 62.6 3313.2 0.776 25 19.424 Table A.6 : 191 Correction Of N Values Of Boring No. (B-103) For Overburden Pnsmue. Depth Unit wt. Effective Correction (ft) (lb/cf) Pressure factor N N1=C1*N (lb/sf) (C1) 2. 125.0 250. 2.828 15 42.426 5. 125.0 625. 1.788 18 32.199 8. 125.0 1000. 1.414 17 24.042 10. 125.0 1375. 1.206 18 21.709 15. 125.0 1875. 1.032 17 17.558 20. 125.0 2500. 0.894 67 15.198 25. 125.0 3250. 0.784 64 50-206 30. 62.6 3563. 0.749 69 51.696 35. 62.6 3844. 0.721 150 108.187 40. 62.6 4189. 0.691 100 69.097 45. 62.6 4439. 0.671 150 100.680 50. 62.6 4752. 0.648 52 33.734 55. 62.6 5034. 0.630 100 63.031 60. 62.6 5409. 0.608 400 243.214 65. 62.6 5691. 0.593 120 71.136 70. 62.6 6004. 0.577 600 346.283 75. 62.6 6192. 0.568 300 170.496 192 A.2 Oneway ANOVA For The Corrected N Values: Table A.7 :The N data with the spatial locations For Case History No.1. LAYER CORRECTED N x Y z 1 22.627 11.66 136.52 897 1 22.000 11.66 136.52 895 1 18.987 11.66 136.52 891.5 1 22.627 19.17 84.27 888 1 51.877 19.17 84.27 885 1 42.426 104.17 92.7 891 2 16.444 11.66 136.52 889 2 16.492 11.66 136.52 882 2 13.093 11.66 136.52 878 2 17.258 11.66 136.52 873 2 25.145 11.66 136.52 868 2 32.480 11.66 136.52 863 2 52.662 11.66 136.52 858 2 45.255 19.17 84.27 882 2 17.709 19.17 84.27 880 2 15.492 19.17 84.27 875 2 16.305 19.17 84.27 870. 2 34.292 19.17 84.27 865 2 29.524 19.17 84.27 860 2 32.199 104.17 92.7 888 2 24.042 104.17 92.7 885 Table A.7 : Continued. 193 21.709 17.558 59.927 50.206 51.696 18.475 19.677 16.971 18.974 12.829 12.522 16.330 19.424 23.094 26.000 18.827 16.000 13.898 12.847 8.251 12.260 27.983 32.888 30.411 25.456 104.17 104.17 104.17 104.17 104.17 149.16 149.16 149.16 149.16 149.16 149.16 149.16 149.16 194.17 194.17 194.17 194.17 194.17 194.17 194.17 194.17 194.17 201.67 201.67 201.67 92.7 92. 92. 92. 92. 69 69 69 69. 69 69 69. 69 84 84 84. 84. 84 84 84. 84. 84. 141. 141. 141. .94 .94 .94 94 .94 .94 94 .94 .27 .27 27 27 .27 .27 27 27 27 57 57 57 882 878 873 867 862 888 886 883 881 877 871 867 862 893 892 889. 887 882 877 872. 867. 863 897. 895 892 Table A.7 : Continued. 194 m-V. .' -~ -_. ......._~ 2 16.444 7.230 4.472 5.600 4.382 4.802 21.185 30.741 22.970 73.721 75.412 60.987 143.075 108.187 69.097 100.680 33.734 63.031 243.214 71.136 346.283 170.496 59.950 117.607 57.128 201.67 201.67 201.67 201.67 201.67 201.67 201.67 201.67 201.67 11.66 11.66 11.66 11.66 104.17 104.17 104.17 104.17 104.17 104.17 104.17 104.17 104.17 201.67 201.67 201.67 141.57 141.57 141.57 141.57 141.57 141.57 141.57 141.57 141.57 136.52 136.52 136.52 136.52 92.7 92.7 92.7 92.7 92.7 92.7 92.7 92.7 92.7 141.57 141.57 141.57 890 885 880 875 870 865 860 855 850 853. 849 844 840 857. 852 848 843 838. 832. 828 823 820 844 840. 835 195 ONEWAY ANOVA FOR THE CORRECTED N VALUES ----------0NEWAY---------- variable N By variable LAYER Analysis of variance Sum of MEan F F Source D.F. Squares Squares Ratio Prob. Between Groups 2 98504.0575 49252.0287 30.8483 .0000 Within Groups 68 108567.8161 1596.5855 Totalt 70 207071-8735 ONEWAY ANOVA FOR THE CORRECTED N VALUES ----------ONEWAY-----—---- Standard Standard Group Count mean Deviation Error 95 Pct Conf Int for Mean Grp 1 6 30.0907 13.6159 5.5587 15.8018 To 44.3795 Grp 2 49 22.2537 12.7585 1.8226 18.5890 To 25.9183 Grp 3 16 112.1086 81.5792 20.3948 68.6381 To 155.5791 Total 71 43.1650 54.3891 6.4548 30.2913 To 56.0387 Fixed Effects Mbdel 39.9573 4.7421 33.7023 To 52.6276 Random Effects Model 39.5239 -126.8945 To 213.2245 Random Effects Mbdel - Estimate of Between Component variance 2882.0583 ------—- -———--——-——-—- ————-----———-——---——————-—--—--———- ——--———-— ONEWAY ANOVA FOR THE CORRECTED N VALUES Group Minimum Maximum Grp 1 18.9870 51.8770 Grp 2 4.3820 59.9270 Grp 3 33.7340 346.2830 Total 4-3820 346.2830 Figure.A.1 - Oneway ANOVA Analysis For Case History No.1 196 Tests for Homogeneity of variances Cochrans C = Max. Variance/Sum(Variances) = .9503, P _ .000 (Approx.) Bartlett-Box F = 49.704 , P = .000 Maximum variance / Minimum variance 40.885 ONEWAY ANOVA FOR THE CORRECTED N VALUES - - - - - - - - - - O N E W A Y - - - - - - — - - - variable N By variable LAYER MUltiple Range Test Tukey-HSD Procedure Ranges for the .050 level - 3.39 3.39 The ranges above are table ranges. The value actually compared with Nean(J)-Mean(I) is.. 28.2541 * Range * Sqrt(1/N(I) + 1/N(J)) (*) Denotes pairs of groups significantly different at the .050 level NTJH (C. 8) 2. The estimated N function is given by: N=7.504835+0.3681199Z (c.9) 3. The "two—point" estimate of N values are: N(B/2)=27 .75,N(3B/2)=68.24 (c.10) 4. The design N value is given by the weighted average: N=(l/3) [247(3/2)+1V<33/2)] =41.24 (c.11) 5. The predicted settlement is 3.3 in. L___¥ 235 Therefore the predicted settlement of 3.1 in is within about 6 % of the measured value of 3.3 in. 6. The 90 % confidence limits 0f the settlement prediction are: (2.2 and 5.13) in. The 50% confidence limits are: (2.76 and 3.48) in. C.2.5 APPLYING THE TREND SURFACE ANALYSIS TECHNIQUE The trend surface analysis results are summarized as follows: 1. The model which is fitted to the data is given by: N=—1254-0.0082X+1476Z°'5—624Z+114.3Z1'5—7.68Z2 (C.12) (R2: 0.65). 2. ~ The design N value is N: 3.16. 3. The predicted settlement is 4.03 in. Therefore the predicted settlement of 4.03 in is within about 22% of the measured value of 3.3 in. 4. The 90% confidence limits of the settlement prediction are: (0.77 and 7.29) in. The 50% confidence limits are: (1.52 and 6.54) in. 5. The areal distribution of settlement in inches is given by the equation: S=12.744/(1.63-0.0082X) ((3.13) The computer Output is shown in Figure C .4. I; 236 SUBSURFACE SOIL STRATIFICATION LAYER N 1 X(ft) Y(ft) Z(ft) 1 10.73313 252 159 5 1 17.70875 252 159 10 1 12.39355 252 159 15 1 12.52198 252 159 20 1 9.07604 252 159 25 2 15.68736 252 159 30 2 12.73524 252 159 35 2 15.08334 252 159 40 2 19.34534 252 159 45 1 8.94427 378 163 5 1 13.91402 378 163 10 1 15.49193 378 163 15 1 16.09969 378 163 20 2 18.15209 378 163 25 2 12.54989 378 163 30 1 7.15542 690 206 5 1 17.70875 690 206 10 1 4.13118 690 206 15 2 21.46625 690 206 20 2 29.70342 690 206 25 2 13 .33426 690 206 30 3 66.64594 252 159 50 3 25 .47049 378 163 35 3 20.11112 378 163 40 3 46.29064 378 163 45 Figure C.4 : The Computer Output Of Case History No. 3 H a; 237 3 79.97512 378 163 50 3 23.97222 690 206 35 3 71 .82544 690 206 40 3 9 .67267 690 206 45 3 10.66335 690 206 50 3 13.53319 690 206 55 3 74.93563 690 206 60 3 36.37433 690 206 65 3 44.21414 690 206 70 3 13 .77878 690 206 75 3 18.47529 690 206 80 LARGE TANKS FOUNDED ON A SANDY SITE IN KANSAS CITY ---------- ONEWAY---------- Variable N By Variable LAYER Analysis of Variance Sum of Mean F F Source D.F. Squares Squares Ratio Prob. Between Groups 2 4615 .2291 2307.6145 8.0817 .0014 Within Groups 33 9422.7228 285.5371 Total 35 14037.9519 Figure C.4 : Continued. "‘_J‘-.'~ ago—.-2. '- 238 Standard Standard Group Count Mean Deviation Error 95 Pct Conf Int for Mean _ Grp 1 12 12.1566 4.2944 1.2397 9.4280 To 14.8851 Grp2 9 17.5619 5.5016 1.8339 13.3330 To 21.7908 Grp3 15 37.0626 25.3232 6.5384 23.0390 To 51.0861 Total 36 23.8854 20.0271 3.3378 17.1092 To 30.6616 Fixed Effects Model 16.8978 2.8163 18.1556 To 29.6152 Random Effects Model 8.2271 -11.5135 To 59.2843 Random Effects Model - Estimate of Between Component Variance 172.0917 Group Minimum Maximum Grp 1 4.1312 17.7087 Grp 2 12.5499 29.7034 Grp 3 9.6727 79.9751 Total 4.1312 79.9751 Tests for Homogeneity of Variances Cochrans C = Max. Variance/Sum(Variances) = .9294, P = .000 (Approx.) Bartlett-Box F = 17.901 , P = .000 Maximum Variance / Minimum Variance 34.773 ---------- ONEWAY------—--- Variable N By Variable LAYER Multiple Range Test Tukey-HSD Procedure Ranges for the .050 level - 3 .46 3.46 The ranges above are table ranges. . The value actually compared with MeanO)—Mean(1) is.. 11.9486 * Range * Sqrt(1/N(I) + 1/N(J)) Figure C.4 : Continued. 239 (*) Denotes pairs of groups significantly different at the .050 level Mean 12.1566 17.5619 37.0626 Group Grpl Grp2 prs GGG PPP 123 ** MODELING THE N FUNCTIONS FOR THE SUBSURFACE SOIL LAYER N1 X(ft) Y(ft) Z(ft) 1 10.73313 252 159 5 l 17.70875 252 159 10 1 12.39355 252 159 15 1 12.52198 252 159 20 1 9.07604 252 159 25 2 15.68736 252 159 30 2 12.73524 252 159 35 2 15.08334 252 159 40 2 19.34534 252 159 45 1 8.94427 378 163 5 1 13.91402 378 163 10 1 15.49193 378 163 15 l 16.09969 378 163 20 2 18.15209 378 163 25 2 12.54989 378 163 30 1 7.15542 690 206 5 Figure C.4 : Continued. 240 l 17.70875 690 206 10 1 4.13118 690 206 15 2 21.46625 690 206 20 2 29.70342 690 206 25 2 13.33426 690 206 30 THE FITTED MODEL: N = BO+B1*X+B2*Z**05+B3*Z+B4*Z**15+B5*Z**2. All the derivatives will be calculated numerically. Run stopped after 6 model evaluations and 3 derivative evaluations. Iterations have been stopped because the relative reduction between successive residual sums of squares is at most SSCON = 1.000E-08 Nonlinear Regression Summary Statistics Dependent Variable N Source DF Sum of Squares Mean Square Regression 6 1906.13 828 317.68971 Residual 6 70. 10254 1 1 .68376 Uncorrected Total 12 1976.24081 (Corrected Total) 11 202.85765 R squared = l - Residual SS / Corrected SS = .65442 Asymptotic 95 % Asymptotic Confidence Interval Parameter Estimate Std. Error Lower Upper BO -1254.598035 608.99558842 -2744.756558 235.56048759 B1 -.008244547 .006109621 -.023194251 .006705158 B2 1476.5232332 72555692499 -298.8506053 3251.8970716 B3 -624.0239724 314.13350566 -1392.680970 144.63302546 B4 114.28725122 58.781601694 —29.54614660 258.12064903 B5 -7.683474492 4.023853769 -17.52948997 2.162540983 Figure C.4 : Continued. 241 Asymptotic Correlation Matrix of the Parameter Estimates B0 B1 B2 B3 B4 B5 B0 1.0000 -.1133 -.9995 .9979 -.9953 .9916 B1 -.1133 1.0000 .1102 -.1111 .1113 -.1104 B2 -.9995 .1102 1.0000 -.9995 .9978 -.9951 B3 .9979 -.1111 -.9995 1.0000 -.9994 .9978 3B4 -.9953 .1113 .9978 ..9994 1.0000 -.9995 B5 .9916 -.1104 -.9951 .9978 -.9995 1.0000 Figure C.4 : Continued. 242 C.3 THE SETTLEMENT PREDICTION OF CASE HISTORY No.4 C.3.1 PROJECT GENERAL DESCRIPTION The site investigation for a steel mill factory expansion in Lesaka, Spain , revealed the presence of a loose to medium dense silty sand stratum. The settlements estimated for this stratum were large, and were thought to be critical for the normal operation of the equipment. This justified the implementation of a field load test to verify the expected settlements. The settlements of this load test were previously studied and reported by Picomell, M. et. al. (1988). The load teSt was implemented by stockpiling steel sheet coils over an area of 65 feet by 50 feet , this led to an average contact pressure of 3.1 st (1.55 tsf). The load test was terminated when the settlement plates had stopped settling. C.3.2 SUBSOIL INVESTIGATION The reported subsoil investigation included 2 borings , (D5 & D8). These borings were drilled to depths of about 125 ft each. The boring locations are shown in Figure C5. The subsurface soil can be discribed as follows: The subsurface soils , at this site , are predominantly ' granular and can be grouped into two main strata. The surficial stratum consist of greenish gray silty gravel with some sand, and frequent boulders. At the site of the load test, this stratum is about 23 ft thick. In other areas of the expansion, irregular lenses of medium stiff to stiff silty clay have been encountered embedded in this stratum. The deeper silt— sand stratum has 243 Building 0 _ 0633550 U *1 Ll Ll Ba 82 6 a 7 s 08 3 . s :3 0'5 . KEY I08 Location of Boring D—8 8 Location of Settlemen: \\\\‘Boundary of Loaded Area Plate No. 6 10 Figure C.5 : Site Plan And Boring Locations Of Case History No. 4 .- ll; 244 Table C.9 : The Correction Of N Values Of Boring (D-S). Boring (D-5) Depth Correction N N1 =C1*N (ft) factor(C1) 5 2.527 65 164.307 10 1.787 93 166.230 ' 15 1.459 105 153.239 20 1.263 98 123.862 40 0.893 105 93.839 46 0.833 56 46.669 57 0.748 44 32.941 62 0.717 30 21.535 82 0.624 34 21.222 85 0.613 54 33.106 89 0.599 60 , 35.948 94 0.582 58 33.813 245 Table C.10 : The Correction Of N Values Of Boring (D-8). Boring (D-8) Depth Correction -N N1 =C1*N (ft) factor (Cl) 6.0 2.307 36 83.072 12.0 1.631 22 35.897 16.0 1.413 14 19.783 18.0 1.332 17 22.648 23.8 1.158 10 11.586 26.2 1.104 9 9.938 31.2 1.011 9 9.107 36.1 0.940 14 13.170 41.3 0.879 7 6.156 45.9 0.834 17 14.183 50.8 0.793 10 7.930 55.8 0.756 23 17.403 69.0 0.680. 18 12.248 81.0 0.628 34 21.353 89.0 0.599 53 31.754 117.0 0.522 25 13.063 130.0 0.495 25.778 52 #77574“ .._..._,.. it Aim-77‘ I“ ’ __ ' 246 a characteristic maroon color and consists of successive layers of silty sand and sandy silt with variable amounts of gravel, occasionally, the gravel size particles predominate. This stratum contains what appears to be large boulders of limestone 10 ft or more in thickness. The density of this stratum ranges from loose to medium dense. The thickness of this stratum is eXtremely variable throughout the site, and at the location of the field load test exceeds the 100 ft investigated. This silt-sand stratum rests on limestone bedrock. The corrected N values for the overburden pressure are shown in Tables C.9 and C.10. C.3.3 FOUNDATIONS AND SETTLEMENT MEASUREMENTS The load test was implemented to verify the compressibility of the silty sand layer. For this purpose, the load test was located in the area where this layer appeared closer to the ground surface. The load was applied on the soil by stockpiling steel sheet coils directly on the ground surface. During the installation of the load, a record was kept of the individual weights of the coils and an attempt was made to distribute them in such a manner as to achieve a nearly uniform load throughout the loaded area. The dimensions of the loaded area were 65 ft by 50 ft and the average load amounted to, 3.1 st (1.55 tsf). The settlements were monitored with twelve settlement plates installed throughout the loaded area and in its immediate vicinity. The locations of the settlement plates are shown in Figure C5. The settlement plates consisted of a riser welded to a one square foot plate. The plates were installed in an excavation at a depth of 2 ft. The settlements were monitored by surveying the elevation of the t0p of the risers. The load was left 247 in place for ten days and the settlement plates were surveyed on a daily basis. The average settlement was 1.56 in. From the settlement record it was obvious that the settlement occured rapidly coinciding with the application of the load. C.3.4 APPLYING THE KRIGING TECHNIQUE Considering that a settlement prediction is required at the center point of the loading area foundation. The Kriging results are summarized as follows: 1. The calculated covariance function is given by the equation: C(h) =128.6339‘4'44E’5(h2) ((3.14) 2. The estimated N function is given by: 8E6.571761+0.13946832 (c.15) 3. The "two-point" estimate of N values are: N(B/Z) le'OS'N(BB/2)=l7'03 (C.1-6) 4. The design N value is given by the weighted average: N=(1/3) [2N(B/2)+1’\‘7(3B/2,] =12.38 (c.17) 5. The predicted settlement is 1.92 in. 4‘ C.3.5 248 Therefore the predicted settlement of 1.92 in is within about 20 % of the measured value of 1.56 in. The 90 % confidence limits of the settlement prediction are: (1.16 and 5.75) in. The 50% confidence limits are: (1.74 and 2.15) in. APPLYING THE TREND SURFACE ANALYSIS TECHNIQUE The trend surface analysis results are summarized as follows: The model which is fitted to the data is as follows: N=~15.26+0.216X+5.67Z°'5-0.417Z+0.001Z2 +7.92z3, (R2=0.62) (c.18) The "two-point" estimate of N values are: N(B/2)=10.39,N(3B/2)=16.09 (c.19) The design N value is: N=(1/3) [2N(B/2)+N(3B/2)] =12.29 (c.20) The predicted settlement is 1.93 in. Therefore the predicted settlement of 1.93 in is within about 20 % of the measured value of 1.56 in. 249 5. The 90 % confidence limits of the settlement prediction are: (1.1 and 13.3) in. The 50% confidence limits are: (1.45 and 2.91) in. 6. The areal distribution of settlement in inches is given by the equation: S=23.84/(0.216X¥5.267) (C.21) The computer output is shown in Figure C.6. SUBSURFACE SOIL STRATIFICATION LAYER N1 X(ft) Y(ft) 2(8) 1 83.0720 33.15 9.78 6.0 1 35.8971 33.15 9.78 12.0 1 19.7831 33.15 9.78 16.0 1 22.6485 33.15 9.78 18.0 1 165.3070 107.39 25.75 5.0 1 166.2305 107.39 25.75 10.0 1 153.2398 107.39 25.75 15.0 1 123.8622 107.39 25.75 20.0 1 93.8398 107.39 25.75 40.0 2 11.5861 33.15 9.78 23.8 2 9.9384 33.15 9.78 26.2 2 9.1073 33.15 9.78 31.2 2 13.1705 33.15 9.78 36.1 2 6.1567 33.15 9.78 41.3 2 14.1831 33.15 9.78 45.9 2 719304 33.15 9.78 50.8 2 17.4036 33.15 9.78 55.8 2 12.2483 33.15 9.78 69.0 2 21.3532 33.15 9.78 81.0 2 46.6699 107.39 25.75 46.0 2 32.9415 107.39 25.75 57.0 2 21.5354 107.39 25.75 62.0 2 21.2227 107.39 25.75 82.0 2 33.1064 107.39 25.75 85.0 3 31.7547 33.15 9.78 89.0 Figure C.6 : The Computer Output Of Case History No. 4 2 5 1 3 13.0639 33.15 9.78 117.0 3 25.7786 33.15 9.78 130.0 3 35.9487 107.39 25.75 89.0 3 33.8136 107.39 25.75 94.0 ---------- ONEWAY--——------ Analysis of Variance Sum of Mean F F Source D.F. Squares Squares Ratio Prob. Between Groups 2 352905477 17645.2739 14.8462 .0001 Within Groups 26 30902.0337 11885398 Total 28 661925814 ---------- ONEWAY----—-—--- Standard Standard Group Count Mean Deviation Error 95 Pct Conf Int for Mean Grp l 9 95.9867 59.9605 19.9868 49.8970 To 142.0763 Grp 2 15 18.5702 11.3417 2.9284 12.2894 To 24.8510 Grp 3 5 28.0719 9.2070 4.1175 16.6401 To 39.5037 Total 29 44.2342 48.6212 9.0287 25.7397 To 62.7288 Fixed Effects Model 34.4752 6.4019 31.0750 To 57.3935 Random Effects Model 27.8852 -75 .7476 To 164.2161 18715502 Random Effects Model - Estimate of Between Component Variance Figure C.6 : Continued. 2 5 2 Group Minimum Maximum Grp 1 19.7831 166.2305 Grp 2 6.1567 46.6699 Grp 3 13.0639 35.9487 Total 6.1567 166.2305 Tests for Homogeneity of Variances Cochrans C = Max. Variance/SumCVariances) = .9440, P = .000 (Approx.) Bartlett-Box F = 15.904 , P = .000 Maximum Variance / Minimum Variance 42.413 Multiple Range Test Tukey-HSD Procedure Ranges for the .050 level - 3.51 3.51 The ranges above are table ranges. The value actually compared with Mean(J)-Mean(l) is.. 24.3777 * Range * Sqrt(1/N(I) + 1/N(J)) (*) Denotes pairs of groups significantly different at the .050 level GGG rrr PPP Mean Group 2 3 1 18.5702 Grp 2 28.0719 Grp3 95.9867 Grpl ** Figure C.6 : Continued. re. MODELING THE N FUNCTION .' ._v- ___ .-_l_._- __ _. 253 LAYER N1 X(ft) Y(ft) 2(8) 2 11.5861 33.15 9.78 23.8 2 9.9384 33.15 9.78 26.2 2 9.1073 1 33.15 9.78 31.2 2 13.1705 33.15 9.78 36.1 2 6.1567 33.15 9.78 41.3 2 14.1831 33.15 9.78 45.9 2 7.9304 33.15 9.78 50.8 2 17.4036 33.15 9.78 55.8 2 12.2483 33.15 9.78 69.0 2 21.3532 33.15 9.78 81.0 2 46.6699 107.39 25.75 46.0 2 32.9415 107.39 25.75 57.0 2 21.5354 107.39 25.75 62.0 2 21.2227 107.39 25.75 82.0 2 33.1064 107.39 25.75 85.0 3 31.7547 33.15 9.78 89.0 3 13.0639 33.15 9.78 117.0 3 25.7786 33.15 9.78 130.0 3 35.9487 107.39 25.75 89.0 3 33.8136 107.39 25.75 94.0 THE FITTED MODEL: N = B0+B1*X+B2*Z**05 +B3 *Z +B4*Z**2+B5*Z**3 . 20 cases are written to the compressed active file. Figure C.6 : Continued. 254 All the derivatives will be calculated numerically. Run stopped alter 6 model evaluations and 3 derivative evaluations. Iterations have been stopped because the relative reduction between successive residual sums of squares is at most SSCON = 1.000E-08 Nonlinear Regression Summary Statistics Dependent Variable N Source DF Sum of Squares Mean Square Regression 6 10315.18500 1719.19750 Residual 14 937.71472 66.97962 Uncorrected Total 20 11252.89972 (Corrected Total) 19 2478 .49464 R squared = 1 - Residual SS / Corrected SS = .62166 Asymptotic 95 % Asymptotic Confidence Interval Parameter Estimate Std. Error Lower Upper B0 4526345780 413.72188570 -902.6086507 872.08173514 B1 .216191623 .060909113 .085554569 .346828678 B2 5.673904168 177.22526971 3744364951 385.78430340 B3 -.417516829 21.645162943 46.84177417 46.006740509 B4 .001097525 .113220251 -.241735761 .243930811 B5 7.927580E-07 .000323570 -.000693195 .000694781 Asymptotic Correlation Matrix of the Parameter Estimates B0 B1 B2 B3 B4 B5 B0 1 .0000 -.0805 ~.9985 .9943 -.9810 .9639 B1 -.0805 1 .0000 .0801 —.0822 . 0725 -.05 28 B2 -.9985 .0801 1.0000 -.9987 .9899 -.9762 B3 .9943 -.0822 -.9987 1 .0000 -.995 8 .9855 B4 -.9810 .0725 .9899 -.9958 1.0000 -.9968 B5 .9639 -.0528 -.9762 .9855 -.9968 1.0000 Figure C.6 : Continued. 255 C.4 THE SETTLEMENT PREDICTION OF CASE HISTORY No. 5 C.4.1 PROJECT GENERAL DESCRIPTION This project consists of a 30 ft thick mat which supports several nuclear, electrical and associated facilities with loads ranging from 8,000 to 10,000 Kips/sf. The mat is founded upon the partially cemented silty sands of the Vincentown Formation. The settlements of this project were previously studied and reported by the investigators of "Dames and Moore, Granford, New Jersey" , (1972). Suitable foundation soils are some 70 ft below grade. The site was backfilled to foundation grade with a lean concrete up to 30 ft in thickness. C.4.2 SUBSOH. INVESTIGATION The site was investigated by drilling 35 borings to varying depths in the generating station area. The locations of the borings, with respect to the proposed structures, are shown in Figure C.7. Most of the borings were terminated at depths on the order of 100 feet below ground surface. The borings at the site indicate the top 25 to 35 feet consist of interbedded mixtures of clay , silt and sand , generally hydraulic fill or loose alluvium. The soils are generally soft in consistency and occasionally contain some organic material. Below these upper soils, the Kirkwood Formation was encountered to depths varying from 65 to 70 ft below the ground surface. This Formation consists of moderately firm to firm clayey soils. The Vincetown Formation of Miocene Age underlies the Kirkwood and consists of sand and silty sand layers, some well cemented. 256 J 7 o---~-~ : Er -l ’0' )3- in“ L‘f‘." / '-'+--‘ .T-I I . ' ’ (9' “ | a M'.‘ I I, “ xi rear“ wm av to! u. ' '7 i c“; 3 i E. 3 ' manual" sun Items. x‘ g I I H .. ' $.73; “Hun I I :0 _43 UNIT I ' ¥ I l I _ ‘ ‘2 ‘ $1 "5 : é. &.-l I. cos: IUNI T t I ‘ ’5’5' ' ‘ '? I. / I I ‘ ' - As 2 - r I I-\ e 1‘ ‘5 ’l ‘ I- v.o._ .. umr—w-— 0- i mu! 2 ‘\ ’ ' \. ’IG-m‘ ( a 9 : é 7‘ Ill -o I ‘ 7 ‘ a. ,.-....--- . . e . ~ ~------= I | I ' . ”8 I l . . ' ,. g. I ”’— ‘ a 7: a .7 I u‘ u' ' .F- - . : gm .9 ; . pg 0 W'S') ' I . ' a ‘ "' 3") I I pa ”All ' l . ' : .-_I,- -*~"-':-"-.9 I i I 1‘ '£:L-:°:':-..°_'-°:_".- l.’ H - - . _ i unit In «on i " r- ”l ' '" ' '1 5 I I I i l I I : v‘.“-_- I . .,._1 1 . U ’0 . ~ .I 615 i e , 5 the who” - _ I in . i .9— . -- -t. . . .' a —- - .J . 9A FIG. Figure C.7 : Site Plan And Boring Locations Of Case History No. 5 ‘ Table C.11 : The Correction Of N Values Of Boring (9). 257 Boring (9) Depth Correction N N1 =C1*N (ft) factor (C1) 2 3.162 2.14 6.767 5 2.000 4.28 8.560 31 0.803 94.30 75 .743 36 0.744 23.60 17.578 41 0.695 19.30 13.430 46 0.655 27.90 18.287 51 0.621 34.30 21.312 54 0.603 19.30 11.643 56 0.592 79.30 46.952 58 0.581 25.70 14.944 61 0.566 23.60 13.371 67 0.538 30.00 16.151 71 0.519 32.14 16.708 76 0.499 51.45 25.684 81 0.480 77.14 37.092 87 0.461 64.30 29.659 91 0.449 90.00 40.452 Table C.12 : The Correction Of N Values Of Boring (34). 258 Boring (34) Depth Correction N N1 = C1*N (ft) factor (C1) 3 2.582 3 7.746 7 1.690 4 6.761 21 0.975 2 1.951 27 0.860 3 2.582 31 0.803 27 21.687 36 0.744 6 4.469 41 0.695 9 6.263 47 0.648 19 12.315 51 0.621 15 9.320 53 0.609 16 9.746 57 0.586 10 5.867 58 0.581 23 13.374 61 0.566 30 16.997 67 0.538 44 23.688 71 0.519 37 19.235 77 0.495 26 12.879 81 0.480 28 13.463 259 Table C.12 : Continued. 86 0.464 94 43.650 91 0.449 52 23.372 96 0.435 62 27.026 100 0.425 44 18.740 At about 135 ft below the ground surface the cementation grades out. Sandy and silty sand soil continue to the depths penetrated by the borings. Bedrock in the area is in excess of 1800 ft below the ground surface and therefore will have no appreciable influ- ence on the foundation settlement of the proposed facilities. The corrected N values for the overburden pressure are shown in Tables CH and C. 12. C.4.3 FOUNDATIONS AND SETTLEMENT MEASUREMENTS The 30 ft thick mat supports several facilities including unit 2 which is selected for settlement anlysis. The diameter of the mat equals to 150 ft and the load which is carried by this mat has an ultimate pressure of 8 kips/sf (4 tsf). The settlements of the mat were monitored at regular intervals from the initial placement onwards. The average settlement which was reported by the previous investigator was 0.5 in. 2 6 0 C.4.4 APPLYING THE KRIGIN G TECHNIQUE Considering that a settlement prediction is required at the center point of unit 2. The Kriging results are summarized as follows: 1. The calculated covariance function is given by the equation: C(h) =107.056e‘4-44E-s(h2) (C.22) 2. The estimated N function is given by: NE0.7466782—33.59394 (0.23) 3. The design N value is: N = 74.6 4. The predicted settlement is 0.43 in. Therefore the predicted settlement of 0.43 in is within about 14 % of the measured value of 0.5 in. 5. The 90% confidence limits of the settlement prediction are: (0.35 and 0.55) in. The 50% confidence limits are: (0.41 and 0.44) in. C.4.5 APPLYING THE TREND SURFACE ANALYSIS TECHNIQUE The trend surface analysis results are summarized as follows: 1. The model which is fitted to the data is as follows: Ne15197—0.02X>4302205+3602—o.7422,(R2=0.53). (C-24) 2. The design N value = 65 .6 261 3. The predicted settlement is 0.48 in. Therefore the predicted settlement of 0:48 in is within about 4 % of the measured value of 0.5 in. 4. The 90 % confidence limits of the settlement prediction are: (0.41 and 0.60) in. The 50% confidence limits are; (0.45 and 0.53) in. 5. The areal distribution of settlement in inches is given by the equation: S=32/(72.11-0.025X) (c.25) The computer output is shown in Figure C.8. 262 MODELING THE N FUNCTIONS FOR THE SUBSURFACE SOIL LAYER N1 X(ft) Y(ft) Z(ft) 1 16.7088 130.3 78.78 71 1 25.6849 130.3 78.78 76 1 37.0925 130.3 78.78 81 1 29.6594 130.3 78.78 87 1 40.4520 130.3 78.78 91 1 19.2354 421.21 230.3 71 1 12.8798 421.21 230.3 77 1 13.4637 421.21 230.3 81 1 43.6502 421.21 230.3 86 1 23.3723 421.21 230.3 91 1 27.0269 421.21 230.3 96 1 18.7404 421 .21 230.3 100 THE FITTED MODEL: N = B0+B1*X+B2*Z**0.5 +B3*Z+B4*Z**2. 12 cases are written to the compressed active file. Run stopped after 7 model evaluations and 4 derivative evaluations. Iterations have been stOpped because the magnitude of the largest correlation between the residuals and any derivative column is at most RCON = 1.000E-08 Nonlinear Regression Summary Statistics Dependent Variable N Source DF Sum of Squares Mean Square Regression 5 8530.63765 1706. 12753 Residual 7 550.57966 78.65424 Uncorrected Total 12 9081 .21732 (Corrected Total) 1 l 1 177.6 13 82 R squared = l - Residual SS / Corrected SS = .53246 Figure C.8 : The Computer Output Of Case History No. 5 263 Asymptotic 95 % Asymptotic Confidence Interval Parameter Estimate Std. Error Lower Upper BO 15197.58 15018.059720 -21106.82770 49917.308755 B1 -.024978 1 83 .019002146 -.06991 1 1 1 8 .019954752 B2 4302.410445 4372.3288091 -14641.32518 6036.5042933 B3 360.89524397 357.43892798 4843135136 1206.1040015 B4 -.744228194 .704496432 -2.410097543 .921641155 Asymptotic Correlation Matrix of the Parameter Estimates B0 B1 B2 B3 B4 B0 1 .0000 . 1051 -.9999 .9995 -.9980 B1 .1051 1.0000 -.1082 .1112 -.1177 B2 -.9999 -. 1082 1.0000 -.9999 .99 89 B3 .9995 .1112 -.9999 1.0000 -.9995 B4 -.9980 -.1177 .9989 -.9995 1.0000 Figure C.8 : Continued. 264 C.5 THE SETTLEMENT PREDICTION OF CASE HISTORY No. 6 C .5 .1 PROJECT GENERAL DESCRIPTION Settlement studies were made in conjunction with the design for a relatively large railroad lift bridge over the Chesapeake and Delaware Canal near Summit, Delaware. The bridge design and plans were prepared by the firm of Howard, Needles, Tammen & Bergendoff for the US Army Corps of Engineers. Because of the size of the structure and the critical problems that could result from differential movements it was recommended that settlement measurements be made. The bridge foundations included two tower piers each of which is supported by a footing of 60 ft'width. The footings were founded on piles to minimize any tilting which could be critical to the operation of the bridge lift span and to minimize settlements. C.5.2 SUBSOH. INVESTIGATION The reported subsoil investigation included 2 borings , one at each tower pier. The N values corrected for the overburden pressure are shown in Tables C. 13 and C. 14 The boring locations with respect to the tower piers are shown in Figure C.9. These borings were to determine the soil types as well as the relative density of the granular soils using the standard penetration test. The site consists of sedimentary deposits of approximately 3000 ft thickness overlaying a pre—Cambrian crystalline bedrock. The sedimentary deposits range from Pleistocene down to Cretaceous. The geological profile is relatively consistent at the bridge site. These foundation soils are highly over - consolidated , having been subjected to loads considerably greater than the present overburden pressure. 265 Seal i nice! ”4" 1”“: r I" - N34 F3 1L9\-‘{§ {I ‘1'»: (ram: “IT The A If $754—— ‘I ‘LProposaedChml Figure C.9 : Site Plan And Boring Locations Of Case History No. 6 266 Table C.13 : The Correction Of N Values Of Boring (416). Boring (416) Depth Correction N N1 = C1*N (ft) factor (C1) 182 0.296 242 71.752 187 0.292 135 39.488 191 0.289 240 69.463 196 0.285 202 57.714 200 0.282 120 33.941 204 0.280 240 67.213 208 0.277 188 52.141 212 0.274 137 37.636 216 0.272 606 164.932 218 0.270 600 162.548 222 0.268 400 107.384 228 0.264' 400 105.962 232 0.262 400 105.045 238 0.259 400 103.712 242 0.257 240 61.711 267 Table C.14 : The Correction Of N Values Of Boring (417). Boring (417) Depth Correction N N1 = C1*N (ft) factor (C1) 62 0.5080 154 78.2321 66 0.492 78 38.404 70 0.478 67 32.032 73 0.468 400 187.265 78 0.452 600 271.746 82 0.441 600 265.035 84 0.436 167 72.884 90 0.421 300 126.491 93 0.414 300 124.434 98 0.404 64 25.859 268 C5 .3 FOUNDATIONS AND SETTLEMENT MEASUREIVIENTS The bottom of the pier footing seals were placed at an elevation of 42 ft below ground surface. The estimated pile tip elevation were elevation of 115 ft below ground surface. Therefore the depth of the compressible layer which caused the settlement is (180 -115 = 65 ft). Measurements of the vertical movement of the tower piers were made at the four corners of the pier and were taken over a period of about 70 months. These measurements showed a little heaving during the first 400 days and was followed by a small settlement below the original level and then further heave. It is believed that the heave resulted from the release of overburden “pressure by excavation through a highly over — consolidated soil and the following settlement was the result of the compression of the compressible sands due to the construction of the bridge foundations and superstructure. So the movement which is attributable to the sand settlement below pile tips is 0.4 inch. The pressure due to the construction at the elevation of 115 ft amounted to 0.9 tsf. C.5 .4 APPLYING THE KRIGING TECHNIQUE Considering that a settlement prediction is required at the center point of the south tower pier foundation. The Kriging results are summarized as follows: 1. The calculated covariance function is given by the equation: 269 C(h)=29l.66e‘4-°E‘5(h2) (C.26) The estimated N function is given by: fi=0.97z—123.5 (c.27) The "two-point" estimate of N values are: N(B/2)=l7.73,N(3B/2)=76.21 ((2.28) The design N value is given by the weighted average: N= (1/3) [2fi(B/2,+1§‘7(3B/2)] =37 .23 (c.29) The predicted settlement is 0.38 in. Therefore the predicted settlement of 0.38 in is within about 5 % of the measured value of 0.4 in. The 90% confidence limits of the settlement prediction are: (0.33 and 0.64) in. The 50% confidence limits are: (0.37 and 0.39) in. 270 C.5.5 APPLYING THE TREND SURFACE ANALYSIS TECHNIQUE The trend surface analysis results are summarized as follows: 1. The model which is fitted to the data is as follows: N=27836-1.62X—807220-5+6752—1.622, (122:0.503). (C.30) 2. The "two—point" estimate of N values: . N(B/2)=4.55,N(33/2)=108.44 (c.31) 3. The design N value is given by: N=(l/3) [2N(B/2)+N(3B/2)] =39.18 (c.32) 4. The predicted settlement is 0. 36 in. Therefore the predicted settlement of 0. 36 in is within about 9 % of the measured value. of 0.4 in. 5. The 90 % confidence limits of the settlement prediction are: (0.22 and 1.04) in. The 50% confidence limits are: (0.29 and 0.48) in. 6. . The areal distribution of settlement in inches is given by the equation: 5:14.34/(1275.758-l.627X) (C.33) The computer output is shown in Figure C. 10. 271 LAYER N1 X(ft) Y(ft) Z(ft) 1 71.7529 760 20.2 182 1 39.4887 760 20.2 187 1 69.4632 760 20.2 191 1 57.7143 760 20.2 196 1 33.9411 760 20.2 200 1 67.2134 760 20.2 204 1 52.1418 760 20.2 208 1 37.6368 760 20.2 212 1 ' 78.2321 220 22.5 62 1 38.4045 220 22.5 66 1 32.0321 220 22.5 70 2 164.9323 760 20.2 216 2 162.5485 760 20.2 218 2 107.3849 760 20.2 222 2 105.9626 760 20.2 228 2 105.0451 760 20.2 232 2 103.7126 760 20.2 238 2 61.7111 760 20.2 242 2 187.2658 220 22.5 73 2 271.7465 220 22.5 I 78 2 265.0357 220 22.5 82 2 72.8848 220 22.5 84 2 126.4911 220 22.5 90 2 124.4342 220 22.5 93 2 25.8599 220 22.5 98 Figure C.10 : The Computer Output Of Case History No. 6 272 ---------- ONEWAY---------- VariableN By Variable LAYER Analysis of Variance Sum of Mean F F Source D.F. Squares Squares Ratio Prob. Between Groups 1 41517.4676 41517.4676 14.1013 .0010 ' Within Groups 23 67717.4130 2944.2353 Total 24 109234.8806 T-TEST/ VARIABLE N. Independent samples of LAYER Group 1: LAYER EQ 1.00 Group 2: LAYER EQ 2.00 t-test for: N Number Standard Standard of Cases Mean Deviation Error Group 1 11 52.5474 17.078 5.149 Group 2 l4 134.6439 70.602 18.869 Pooled variance Estimate Separate variance Estimate ' - ' Degrees of 2-Tail F 2-Tail t Degrees of 2 Tail t Value Prob. value Freedom Prob. value Freedom Prob. 17.09 000 -3.76 23 001 -4.20 14.90---- .001 Figure C.10 : Continued. 273 MODELING THE N FUNCTIONS FOR THE SUBSURFACE SOIL LAYER N 1 X(ft) Y(ft) Z(ft) 1 71.7529 760 20.2 182 1 39.4887 760 20.2 187 1 69.4632 760 20.2 191 1 57.7143 760 20.2 196 1 33.9411 760 20.2 200 1 67.2134 760 20.2 204 l 52. 1418 760 20.2 208 1 37.6368 760 20.2 212 1 78.2321 220 22.5 62 1 38.4045 220 22.5 66 1 32.0321 220 22.5 70 THE FITTED MODEL: N = B0+B1*X+B2*Z**0.5 +B3*Z+B4*Z**2+B5*Z**3. There are 11 cases. There is enough memory for them all. Run st0pped after 11 model evaluations and 4 derivative evaluations. Iterations have been stopped because the relative reduction between successive residual sums of squares is at most SSCON = 1.000E—08 Nonlinear Regression Summary Statistics Dependent Variable N Source DF Sum of Squares Mean Square Regression 6 31841.69154 5306.94859 Residual 5 1448.37907 289.67581 Uncorrected Total 11 33290.07062 (Corrected Total) 10 2916.60145 R squared = 1 — Residual SS / Corrected SS = .50340 Figure C.10 : Continued. 274. Asymptotic 95 % Asymptotic Confidence Interval Parameter Estimate Std. Error Lower Upper B0 B 1 B2 B3 B4 B5 27836.718891 52453.781314 «107060.0186 162613.45635 -1.627071610 4.198324875 -12.41920927 9.165066053 «8072968442 15557.148607 48063.89207 31917.955181 675.68407215 1328.4194959 -2739.126954 4090.4950985 -1.603972812 3.233142627 -9.915030521 6707084896 .002087879 .004293028 -.008947700 .013123458 Asymptotic Correlation Matrix of the Parameter Estimates B0 B 1 B2 B3 B4 B5 B0 B1 B2 B3 B4 B5 1 .0000 «9833 -.9999 .9997 -.9983 .9952 -.9833 1.0000 .9817 -.9800 .9732 -.9634 -.9999 .9817 1 .0000 -.9999 .9989 -.9963 .9997 - .9800 -.9999 1 .0000 -.9994 .9972 -.9983 .9732 .9989 -.9994 1.0000 -.9992 .9952 -.9634 -.9963 .9972 -.9992 l .0000 Figure C.10 : Continued. 10. 11. 12. LIST OF REFERENCES Alpan I. , "Estimating Settlements 0f Foundations on Sands", Civil Engineering and Public Works Review , Vol. 59, No. 700, pp 1415—1418, 1964. Annual Book of ASTM Standards, Philadelphia, PA, 1993. Anagnotopoulos, A. G. , "The Compressibility of Cohesionless Soils", Geotech- nique, No.3, 1990. Baecher, G.B. , " Analyzing Exploration Strategies", Site Characterization & Exploration, AS CE, Proceedings, Specialty Workshop, Northwestern University, Evanston, Illinois 1978. Baecher, G.B. , " Optimal Estimators For Soil Properties", J. Geotech. Engng Div., ASCE, 107(5), pp. 649—653, May, 1981. Baines, A.H.J. , "Methods of Detecting Non— Randomness in a Given Series of Observations", Technical Report - Series "R" , No.Q.C./R/12 , British Ministry of Supply , As Reproduced in the"Statistics Manual", Research Dept, US. Naval Ordnance Test Station, Dover Edition, New York, 1960. Bazaraa , A.R., "Standard Penetration Test", Foundation Engng, Vol. 1: Soil Properties, Found. Design And Construction", Published by: "Presses de l’e’cole nationale des Ponts et chausse’es, Paris, 1982. Berry , DA. and Lindgren, B.W., "Statistics Theory and Methods", Brooks/Cole, Publishing Company, Pacific Grove, California,1990. Borden, R.H. , and Lien,W. , "Settlement Predictions In Residual Soils By Dilatometer, Pressuremeter And One-Dimensional Compression Test: Comparison With Measured Field Response", Proc. 2nd. Int. Conf. On Case, Histories In Geot. Engng, St. Louis, Mo.,' Paper No. 6.83, pp. 1449-1454, June, 1988. Bowles, J. E. , "Foundation Analysis And Design", 4th Ed. , Mc-Graw Hill, 1988. Box, G.E., Hunter , W.G. and Hunter, J.S., "Statistics For Experimenters, An Introduction to Design, Data Analysis, and Model Building", John Wiley & Sons, New York, 1978. Burland , J .B., and Burbidge , M. C., "Settlement of Foundations On Sand And Gravel", Proc. Instn Civ. Engrs, Part 1, pp 1325-1381, Dec., 1985. 275 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 276 Burland, J .B., and Moore, J .F.A., and Smith, P.D.K., "A Simple And Precise Borehole Extensometer", Geotechnique, Vol. 22, No.1,pp. 174—177,1972. Clayton, C.R., & Hababa, M.B., and Simons, N.E., "Dynamic Penetration Resistance and the Prediction of the Compressibility of a Fine- Grained Sand- A Laboratory Study", Geotechnique, 35, No.1, 19-31, 1985. Cressie, N.A.C., "Statistics For Spatial Data", John Wiley & Sons New York, 1991. D’Appolonia, D.J. & D’Appolonia, E. and Brissette, R.F., "Settlement of Spread Footings on Sand, "Jour.of Soil Mech. and Found. Div. , ASCE, Vol. 94, No SM3, Proc. Paper 5959, pp 735-760, May, 1968. D’Appolonia, D.J. & D’Appolonia, E. and Brissette, R.F., "Settlement Of Spread Footings On Sand", (Discussion),Jour. of Soil Mech. and Found. Div., ASCE, Vol.96, No SM2, pp 754-762, 1970. Das, B.M., "Principles Of Foundation Engng", Brooks, Cole Engng Div., Monterey, California 93940, 1984. Davis, J.C., "Statistics and Data Analysis in Geology", Wiley, lst ed., 1973, 2nd ed., 1986. Davisson, M.T. , and Salley, J .R. , "Settlement Histories Of Four large Tanks On Sand. 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