Shear Strength and Stability of Highway Embankments in Ohio a Thesis Presented to the Faculty of the Russ College of Engineering

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Shear Strength and Stability of Highway Embankments in Ohio

A thesis presented to

the faculty of the Russ College of Engineering and Technology of Ohio University

In partial fulfillment

of the requirements for the degree

Master of Science

Xiao Han

March 2010

This thesis titled

Shear Strength and Stability of Highway Embankments in Ohio

XIAO HAN

has been approved for

the Department of Civil Engineering

and the Russ College of Engineering and Technology by

Teruhisa Masada

Professor of Civil Engineering

Dennis Irwin

Dean, Russ College of Engineering and Technology 3

ABSTRACT

HAN, XIAO, M.S., March 2010, Civil Engineering

Shear Strength and Stability of Highway Embankments in Ohio (182 pp.)

Director of Thesis: Teruhisa Masada

One of the primary factors that control the stability of earthen embankments is the

shear strength of embankment fill soils. Karl von Terzaghi and U.S. Navy found

empirical correlations between soil shear strength and lab or in-situ soil test results.

Subsequently, the Navy proposed typical effective friction angle values for different soils

types. However, it is not certain if these correlations and standard values are reliable for

Ohio soils. The main objective of this thesis is to establish reliable correlations between shear strength and index properties/in-situ test measurements for embankment fill soils existing in Ohio.

Relatively undisturbed soil samples were gathered from nine highway embankment sites spread throughout Ohio. Tri-axial compression tests were performed to determine the shear strength of these soil samples. During data analysis, statistical methods, such as regression analysis and T-tests, were utilized to assess the correlations.

Through the statistical analysis, optimized correlations were identified between shear strength and other characteristics of soil have been found.

Once the optimum correlations were established, soil shear strength properties were applied to sophisticated geotechnical software to perform a series of slope stability analysis for highway embankments. Lastly, results from the laboratory testing, field 4 testing, statistical data analysis, and slope stability analysis were all combined to suggest technical guidelines for highway embankment design/constructions in Ohio.

Approved: ______

Teruhisa Masada

Professor of Civil Engineering 5

ACKNOWLEDGMENTS

I would like to show my appreciation to my advisor, Dr. Teruhisa Masada, for assisting me through out my Master’s Program. I would also like to thank him for encouraging me to be diligence on this thesis and teaching me the skills that will be very helpful in my future career.

I would also like to thank Dr. Greg Springer, Dr. Munir Nazzal, and Dr. Omer

Tatari for being members of my thesis committee.

Finally, I would like express my appreciation to my parents who supported me in many ways through out the years I worked on my Master’s degree.

TABLE OF CONTENTS

Page

Abstract ...... Error! Bookmark not defined. Acknowledgments...... Error! Bookmark not defined. List of Tables ...... Error! Bookmark not defined. List of Figures ...... Error! Bookmark not defined. Chapter 1 : INTRODUCTION...... 15 1.1 Background ...... 15 1.2 Objectives ...... 16 1.3 Outline of Thesis ...... 16 Chapter 2 : Ohio Geology ...... 18 2.1 Geological Conditions ...... 18 2.2 Soil Classifications ...... 20 Chapter 3 : Research Methodologies ...... 22 3.1 Site Screening and Sample Collection ...... 22 3.1.1 Site Locations and Distribution ...... 22 3.1.2 Standard Penetration Test ...... 24 3.1.3 Soil Sampling Method ...... 26 3.2 Experiment ...... 28 3.2.1 Triaxial Compression Test - Introduction ...... 28 3.2.2 C-U Triaxial Test Procedure and Equipment ...... 30 Chapter 4 : Results and Data Analysis ...... 37 4.1 Introduction of Basic Theory ...... 37 4.2 Methodology to analyze the correlations ...... 40 4.3 Effective Internal Friction Angle vs. Plasticity Index (by Terzaghi) ...... 42 4.4 SPT-N vs. Unconfined Compression Strength (by Department of Navy) ...... 47 4.5 A Values and Level of Consolidation ...... 52 4.6 T-test for Three Major Soils ...... 56 4.7 Single Variable Regression Analysis ...... 69 4.7.1 Single Variable Regression Analysis for A-4a Soils ...... 71 7

4.7.2 Single Variable Regression Analysis for A-6a Soils ...... 79 4.7.3 Single Variable Regression Analysis for A-6b Soils ...... 89 4.7.4 Single Variable Regression Analysis for A-7-6 Soils ...... 102 4.8 Multiple Variables Linear Regression Analysis (Part I) ...... 112 4.9 Multiple Variables Linear Regression Analysis (Part II) ...... 117 4.10 Closing Remarks ...... 124 Chapter 5 : Highway Embankment Slope Stability ...... 127 5.1 Soil Slope Stability Theory ...... 127 5.2 Embankment Model Assumptions ...... 132 5.3 Results of Highway Embankment Slope Stability Analysis ...... 138 5.3.1 Homogeneous Embankments...... 138 5.3.2 Outcomes of Embankments Built on Two Soil Layers ...... 155 Chapter 6 : Summary and Conclusions ...... 174 6.1 Summary ...... 174 6.2 Conclusions ...... 175 6.2.1 Conclusions on Empirical Correlations ...... 175 6.2.2 Conclusions from T-Tests Performed ...... 176 6.2.3 Conclusions from Single-Variable Regression Analysis ...... 177 6.2.4 Conclusions from Multi-Variable Linear Regression Analysis ...... 178 6.2.5 Conclusions from Slope Stability Analysis ...... 179 References ...... 181 8

LIST OF TABLES Page

Table 2.1: AASHTO Classifications for Fine-Grained Soils ...... 20 Table 4.1: Effective Internal Friction Angle vs. Plasticity Index by Terzaghi ...... 42 Table 4.2: SPT-N vs. Unconfined Compression Strength (psi) by Department of Navy . 47 Table 4.3: The A value for A-4a soil ...... 53 Table 4.4: The A value for A-6a soil ...... 54 Table 4.5: The A value for A-6b soil ...... 54 Table 4.6: The A value for A-7-6 soil ...... 55 Table 4.7: The Critical Value of T-Distribution ...... 58 Table 4.8: Summary of T Tests Performed on A-4 Soil Subsets ...... 59 Table 4.9: Summary of T Tests Performed on A-6 Soil Subsets ...... 60 Table 4.10: The Calculation for T Value of A-4a in Various Locations (North and South) ...... 61 Table 4.11: The Calculation for T Value of A-4a in Various Locations (North and Central) ...... 62 Table 4.12: The Calculation for T Value of A-4a in Various Locations (Central and South) ...... 63 Table 4.13: The Calculation for T Value of A-6a in Various Locations (North and South) ...... 64 Table 4.14: The Calculation for T Value of A-6a in Various Locations (North and Central) ...... 65 Table 4.15: The Calculation for T Value of A-6a in Various Locations (Central and South) ...... 66 Table 4.16: The Calculation for T Value of A-6b in Various Locations (North and South) ...... 67 Table 4.17: The Calculation for T Value of A-7-6 in Various Locations (North and South) ...... 68 Table 4.18: Single Variable Linear Regression for A-4a for Corrected SPT Value ...... 71 Table 4.19: Single Variable Nonlinear Regression for A-4a for Corrected SPT Value .... 71 Table 4.20: Single Variable Linear Regression for A-4a for Unconfined Compression Strength ...... 72 Table 4.21: Single Variable Nonlinear Regression for A-4a for Unconfined Compression Strength ...... 73 Table 4.22: Single Variable Linear Regression for A-4a for Cohesion ...... 74 Table 4.23: Single Variable Nonlinear Regression for A-4a for Cohesion ...... 75 Table 4.24: Single Variable Linear Regression for A-4a for Effective Cohesion ...... 76 Table 4.25: Single Variable Nonlinear Regression for A-4a for Effective Cohesion ...... 76 Table 4.26: Single Variable Linear Regression for A-4a for Internal Friction Angle ...... 77 Table 4.27: Single Variable Nonlinear Regression for A-4a for Internal Friction Angle . 77 Table 4.28: Single Variable Linear Regression for A-4a for Effective Internal Friction Angle ...... 78 9

Table 4.29: Single Variable Nonlinear Regression for A-4a for Effective Internal Friction Angle ...... 78 Table 4.30: Single Variable Linear Regression for A-6a for Corrected SPT Value ...... 80 Table 4.31: Single Variable Nonlinear Regression for A-6a for Corrected SPT Value .... 80 Table 4.32: Single Variable Linear Regression for A-6a for Unconfined Compression Strength ...... 81 Table 4.33: Single Variable Nonlinear Regression for A-6a for Unconfined Compression Strength ...... 81 Table 4.34: Single Variable Linear Regression for A-6a for Cohesion ...... 82 Table 4.35: Single Variable Nonlinear Regression for A-6a for Cohesion ...... 83 Table 4.36: Single Variable Linear Regression for A-6a for Effective Cohesion ...... 84 Table 4.37: Single Variable Nonlinear Regression for A-6a for Effective Cohesion ...... 85 Table 4.38: Single Variable Linear Regression for A-6a for Internal Friction Angle ...... 87 Table 4.39: Single Variable Nonlinear Regression for A-6a for Internal Friction Angle . 87 Table 4.40: Single Variable Linear Regression for A-6a for Effective Internal Friction Angle ...... 88 Table 4.41: Single Variable Nonlinear Regression for A-6a for Effective Internal Friction Angle ...... 88 Table 4.42: Single Variable Linear Regression for A-6b for Corrected SPT Value ...... 90 Table 4.43: Single Variable Nonlinear Regression for A-6b for Corrected SPT Value .... 90 Table 4.44: Single Variable Linear Regression for A-6b for Unconfined Compression Strength ...... 91 Table 4.45: Single Variable Nonlinear Regression for A-6b for Unconfined Compression Strength ...... 92 Table 4.46: Single Variable Linear Regression for A-6b for Cohesion ...... 93 Table 4.47: Single Variable Nonlinear Regression for A-6b for Cohesion ...... 94 Table 4.48: Single Variable Linear Regression for A-6b for Effective Cohesion ...... 95 Table 4.49: Single Variable Nonlinear Regression for A-6b for Effective Cohesion ...... 96 Table 4.50: Single Variable Linear Regression for A-6b for Internal Friction Angle ...... 97 Table 4.51: Single Variable Nonlinear Regression for A-6b for Internal Friction Angle . 98 Table 4.52: Single Variable Linear Regression for A-6b for Effective Internal Friction Angle ...... 100 Table 4.53: Single Variable Nonlinear Regression for A-6b for Effective Internal Friction Angle ...... 101 Table 4.54: Single Variable Linear Regression for A-7-6 for Corrected SPT Value ...... 103 Table 4.55: Single Variable Nonlinear Regression for A-7-6 for Corrected SPT Value . 103 Table 4.56: Single Variable Linear regression for A-7-6 for Unconfined Compression Strength ...... 104 Table 4.57: Single Variable Nonlinear Regression for A-7-6 for Unconfined Compression Strength ...... 104 Table 4.58: Single Variable Linear Regression for A-7-6 for Cohesion ...... 105 Table 4.59: Single Variable Nonlinear Regression for A-7-6 for Cohesion ...... 105 Table 4.60: Single Variable Linear Regression for A-7-6 for Effective Cohesion ...... 106 Table 4.61: Single Variable Nonlinear Regression for A-7-6 for Effective Cohesion .... 107 10

Table 4.62: Single Variable Linear Regression for A-7-6 for Internal Friction Angle ... 109 Table 4.63: Single Variable Nonlinear Regression for A-7-6 for Internal Friction Angle ...... 109 Table 4.64: Single Variable Linear Regression for A-7-6 for Effective Internal Friction Angle ...... 110 Table 4.65: Single Variable Nonlinear Regression for A-7-6 for Effective Internal Friction Angle ...... 110 Table 4.66: Multiple Variables Linear Regression for A-4a Soils ...... 114 Table 4.67: Multiple Variables Linear Regression for A-6a Soils ...... 115 Table 4.68: Multiple Variables Linear Regression for A-6b Soils ...... 116 Table 4.69: Multiple Variables Linear Regression for A-7-6 Soils ...... 117 Table 4.70: Multiple Variables Linear Regression for A-4a Soils (Collinearity Largely Eliminated) ...... 122 Table 4.71: Multiple Variables Linear Regression for A-6a Soils (Collinearity Largely Eliminated) ...... 122 Table 4.72: Multiple Variables Linear Regression for A-6b Soils (Collinearity Largely Eliminated) ...... 123 Table 4.73: Multiple Variables Linear Regression for A-7-6 Soils (Collinearity Largely Eliminated) ...... 123 Table 4.74: Reliable Equations for A-4a Soil Shear Strength Parameters ...... 124 Table 4.75: Reliable Equations for A-6a Soil Shear Strength Parameters ...... 125 Table 4.76: Reliable Equations for A-6b Soil Shear Strength Parameters ...... 125 Table 4.77: Reliable Equations for A-7-6 Soil Shear Strength Parameters ...... 126 Table 5.1: Shear Strength Properties of Four Major Soils in Ohio ...... 133 Table 5.2: Unit Weights of Four Soil Types in Ohio (based on triaxial test data) ...... 136 Table 5.3: Unit Weights of Four Soil Types in Ohio (based on unconfined compression test data) ...... 137 Table 5.4: Factor of Safety in Short-Term Analysis (Homogeneous Embankment; Total Height 20 ft) ...... 138 Table 5.5: Factor of Safety in Short-Term Analysis (Homogeneous Embankments; Height 30 ft) ...... 141 Table 5.6: Factor of Safety in Short-Term Analysis (Homogeneous Embankments; Height 40 ft) ...... 144 Table 5.7: Factor of Safety in Long-Term Analysis (Homogeneous Embankments; Height 20 ft) ...... 146 Table 5.8: Factor of Safety in Long-Term Analysis (Homogeneous Embankments; Height 30 ft) ...... 149 Table 5.9: Factor of Safety in Long-Term Analysis (Homogeneous Embankments; Height 40 ft) ...... 152 Table 5.10: Factor of Safety in Short-Term Analysis (Two-Layer Embankments; Height 20 ft) ...... 155 Table 5.11: Factor of Safety in Short Term Analysis (Two-Layer Embankments; Height 30 ft) ...... 158 11

Table 5.12: Factor of Safety in Short-Term Analysis (Two-Layer Embankments; Height 40 ft) ...... 161 Table 5.13: Factor of Safety in Long-Term Analysis (Two-Layer Embankments; Height 20 ft) ...... 164 Table 5.14: Factor of Safety in Long-Term Analysis (Two-Layer Embankments; Height 30 ft) ...... 167 Table 5.15: Factor of Safety in Long-Term Analysis (Two-Layer Embankments; Height 40 ft) ...... 170

LIST OF FIGURES Page

Figure 2.1: Ohio’s Soil Regions ...... 19 Figure 2.2: Distribution of Three Major Soil types in Ohio ...... 21 Figure 3.1: Locations of Highway Embankment Sites ...... 23 Figure 3.2: Process of SPT...... 25 Figure 3.3: Plan View of Soil Sampling Plan ...... 26 Figure 3.4: Side View of Soil Sampling Plan ...... 27 Figure 3.5: Sealing of Shelby Tube Ends ...... 28 Figure 3.6: Pushing the Specimen Out by Oil Hydraulic Jack ...... 31 Figure 3.7: The Work for Restraining the Specimen ...... 32 Figure 3.8: The Course of Saturation ...... 33 Figure 3.9: The Facilities for Triaxial Test ...... 35 Figure 4.1: The Line of Soil Shear Strength Determined by Mohr-Coulomb Principle Stress Circle ...... 38 Figure 4.2: The Interactions between Each Type of Experimental Results ...... 41 Figure 4.3: Evaluation of Terzaghi’s Empirical PI-φ′ Relationship (for A-4a Soils in Ohio) ...... 43 Figure 4.4: Evaluation of Terzaghi’s Empirical PI-φ′ Relationship (for A-6a Soils in Ohio) ...... 44 Figure 4.5: Evaluation of Terzaghi’s Empirical PI-φ′ Relationship (for A-6b Soils in Ohio) ...... 45 Figure 4.6: Evaluation of Terzaghi’s Empirical PI-φ′ Relationship (for A-6 Soils in Ohio) ...... 46 Figure 4.7: Evaluation of Terzaghi’s Empirical PI-φ′ Relationship (for A-7-6 Soils in Ohio) ...... 46 Figure 4.8: Evaluation of Relationship built by Department of Navy (for A-4a Soils in Ohio) ...... 48 Figure 4.9: Evaluation of Relationship built by Department of Navy (for A-6a Soils in Ohio) ...... 49 Figure 4.10: Evaluation of Relationship built by Department of Navy (for A-6b Soils in Ohio) ...... 49 Figure 4.11: Evaluation of Relationship built by Department of Navy (for A-6 Soils in Ohio) ...... 50 Figure 4.12: Evaluation of Relationship built by Department of Navy (for A-7-6 Soils in Ohio) ...... 51 Figure 4.13: Geographical Distributions of Various Soil Types in Ohio ...... 57 Figure 4.14: Linearity Check between Two Independent Variables (A-4a Soils) ...... 119 Figure 4.15: Linearity Check Between Two Independent Variables (A-6a Soils) ...... 120 Figure 4.16: Linearity Check Between Two Independent Variables (A-6b Soils) ...... 120 Figure 4.17: Linearity Check Between Two Independent Variables (A-7-6 Soils) ...... 121 Figure 5.1: The Illustration of Different Failure Circle for Soil Slope ...... 128 Figure 5.2: Sketch for Mass Procedure Analysis ...... 129 13

Figure 5.3: Sketch for Method of Slices Analysis ...... 130 Figure 5.4: The Demonstration of Stresses on Each Slice ...... 131 Figure 5.5: Water Table Existing in Embankment ...... 134 Figure 5.6: Plots of Factor of Safety in Short-Term Analysis (Homogeneous Embankments; Height 20 ft) ...... 139 Figure 5.7: Most Critical Failure Arc in Short-Term Analysis (Homogeneous Embankment; Height 20 ft) ...... 140 Figure 5.8: Stresses Acting on of the Slice Shaded in Figure 5.7 ...... 141 Figure 5.9: Plots of Factor of Safety in Short-Term Analysis (Homogeneous Embankments; Height 30 ft) ...... 142 Figure 5.10: Most Critical Failure Arc in Short-Term Analysis (Homogeneous Embankment; Height 30 ft) ...... 143 Figure 5.11: Stresses Acting on of Slice Shaded in Figure 5.10 ...... 143 Figure 5.12: Plots of Factor of Safety in Short-Term Analysis (Homogeneous Embankments; Height 40 ft) ...... 144 Figure 5.13: Most Critical Failure Circle in Short- Term Analysis (Homogeneous Embankment; Height 40 ft) ...... 145 Figure 5.14: Stresses Acting on Slice Shaded in Figure 5.13 ...... 146 Figure 5.15: Plots of Factor of Safety in Long-Term Analysis (Homogeneous Embankments; Height 20 ft) ...... 147 Figure 5.16: Most Critical Failure Circle in Long- Term Analysis (Homogeneous Embankment; Height 20 ft) ...... 148 Figure 5.17: Stresses Acting on Slice Shaded in Figure 5.16 ...... 148 Figure 5.18: Plots of Factor of Safety in Long Term Analysis (Homogeneous Embankments; Height 30 ft) ...... 150 Figure 5.19: Most Critical Failure Circle in Long- Term Analysis (Homogeneous Embankment; Height 30 ft) ...... 151 Figure 5.20: Stresses Acting on Slice Shaded in Figure 5.19 ...... 151 Figure 5.21: Plots of Factor of Safety in Long-Term Analysis (Homogeneous Embankments; Height 40 ft) ...... 153 Figure 5.22: Most Critical Failure Circle in Long- Term Analysis (Homogeneous Embankment; Height 40 ft) ...... 154 Figure 5.23: Stresses Acting on Slice Shaded in Figure 5.22 ...... 154 Figure 5.24: Plots of Factor of Safety in Short Term Analysis (Two-Layer Embankments; Height 20 ft) ...... 156 Figure 5.25: Most Critical Failure Circle in Short Term Analysis (Two-Layer Embankment; Height 20 ft) ...... 157 Figure 5.26: Stresses Acting on Slice Shaded in Figure 5.25 ...... 157 Figure 5.27: Plots of Factor of Safety in Short Term Analysis (Two- Layer Embankments; Height 30 ft) ...... 159 Figure 5.28: Most Critical Failure Circle in Short Term Analysis (Two-Layer Embankment; Height 30 ft) ...... 160 Figure 5.29: Stresses Acting on Slice Shaded in Figure 5.28 ...... 160 14

Figure 5.30: Plots of Factor of Safety in Short Term Analysis (Two-Layer Embankments; Height 40 ft) ...... 162 Figure 5.31: Most Critical Failure Circle in Short Term Analysis (Two-Layer Embankment; Height 20 ft) ...... 163 Figure 5.32: Stresses Acting on Slice Shaded in Figure 5.31 ...... 163 Figure 5.33: Plots of Factor of Safety in Long-Term Analysis (Two- Layer Embankments; Height 20 ft) ...... 165 Figure 5.34: Most Critical Failure Circle in Long-Term Analysis (Two-Layer Embankment; Height 20 ft) ...... 166 Figure 5.35: Stresses Acting on Slice Shaded in Figure 5.34 ...... 167 Figure 5.36: Plots of Factor of Safety in Long-Term Analysis (Two-Layer Embankments; Height 30 ft) ...... 168 Figure 5.37: Most Critical Failure Circle in Long-Term Analysis (Two-Layer Embankment; Height 30 ft) ...... 169 Figure 5.38 Stresses Acting on Slice Shaded in Figure 5.37 ...... 169 Figure 5.39: Plots of Factor of Safety in Long-Term Analysis (Two-Layer Embankments; Height 40 ft) ...... 171 Figure 5.40: Most Critical Failure Circle in Long-Term Analysis (Two-Layer Embankment; Height 40 ft) ...... 172 Figure 5.41: Stresses Acting on Slice Shaded in Figure 5.40 ...... 172

CHAPTER 1 : INTRODUCTION

1.1 Background

Roadway construction projects often call for building embankments to level road

surfaces in uneven terrains. The embankment requires sufficient strength and stability for

reliable field performance. As the consolidation of embankment soil progresses, the

embankment’s compression capacity gradually decreases with remarkable increase of

compression strength. Because the sides of an embankment are unconfined, the factors

affecting its stability include angle of slope, height of embankment, pore pressure, moist

unit weight of soil fill, and soil shear strength.

The soil shear strength is the maximum shear stress that soil can endure before

plastic deformation takes place. It is determined by the soil’s fundamental shear strength

parameters -- cohesion and internal friction angle. The cohesion is related to ionic

attraction, covalent bond, and other chemical cementations caused by minerals. The

internal friction angle is influenced by the moisture content, and particle characteristics

(shape, arrangement, and size).

Due to the absence of detailed previous studies on soil shear strength,

geotechnical engineers in Ohio are forced to rely on conservative empirical relationships

established by Terzaghi and the Department of Navy for determining soil shear strength

properties. Furthermore, for major highway projects the cohesion and internal friction

angle need to be measured by performing triaxial compression tests, which can be quite

time consuming. There is a strong need in Ohio for establishing reliable correlations between shear strength properties and soil index properties for each type of soil typically 16

utilized for highway embankment construction, so that the embankment structures can be designed and built economically without compromising their serviceability.

1.2 Objectives

This thesis research was conducted to meet the following objectives:

• Evaluate the applicability of empirical correlations established by Terzaghi and

the US Department Navy to embankment fill soils in Ohio.

• Explore the correlations between soil shear strength properties and its

fundamental material properties through using a variety of statistical approaches.

• Based on the soil shear strength data derived from the research, perform a

parametric study of slope stability of embankment structures so that a set of

geotechnical guidelines can be recommended for highway embankments

constructed with soils typically found in Ohio.

1.3 Outline of Thesis

Chapter Two demonstrates the geological conditions in Ohio. Then the AASHTO

criterion of soil classification is introduced and the soil in Ohio’s embankment

construction is sorted in accordance with this criteria.

In Chapter Three, the research methodology is the main points. It contains the

requirements and approaches of collection of soil specimen. And the SPT test will be

described in this chapter. Furthermore, triaxial test, the way to determine the mechanical

properties of the soil specimen, is displayed in detail. 17

Chapter Four is the combination of the analysis methods and the results. Whether

the empirical relationships developed by Terzaghi and Department of Navy are suitable

for the soil in Ohio is tested in this chapter. To acquire exact mathematical correlations of the soil in Ohio, the exploration is conducted through the linear and nonlinear regression methods.

In Chapter Five, the basic theories of slope stability are presented. Then, the slope stabilities of the embankments constructed with the Ohio’s soil are calculated using computer software. According to the comparison of these results, the rules of the slope stability are revealed.

Chapter Six represents the conclusion and summary of entire results of the analyses in the former chapters. 18

CHAPTER 2 : OHIO GEOLOGY

In this chapter, geological conditions existing in Ohio are first presented to give the readers understanding of soil types commonly found in the state. Then, the Unified and AASHTO soil classification systems are described, so that each soil examined in

Ohio can be related to designations in the soil classification system.

2.1 Geological Conditions

Soils in Ohio formed over thousands of years, and the elements to build soil involve climate, parent material, organisms, slope/relief, and time. Ohio can be divided into unglaciated and glaciated regions. These regions are illustrated in Figure 2.1. The glaciated region resides in the north and the northwest. The soils in this region consist mainly of sands and gravels. Because of the enormous transport capacity of glaciers, many rock fragments found in the region can be traced to bedrock in Canada. Through their opposite properties, geologists depict drifts in glaciated region as either stratified or non-stratified. The former one, such as eskers and kames, shows layering. Non-stratified soils were deposited by glaciers and were typically graded. 19

Figure 2.1: Ohio’s Soil Regions (from Johnson 1975)

In Figure 2.1, the pink section in the north is high lime glacial lake sediments, and

the scarlet section in the west is high lime glacial drift of Wisconsin Age, and the red section in the east is low lime glacial drift of Wisconsin Age. The section located in southwest colored yellow is glacial drift of Illinois Age, and soils in the massive green area in the southeast formed from sandstone and shale.

2.2 Soil Classifications

Two soil classification systems are popular among geotechnical engineers, which

are the United Soil Classification System (USCS) and the American Association of State

Highway and Transportation Officials (AASHTO) Soil Classification system.

The USCS has two main steps. First of all, the soil is designated into either the

group of gravels and sands or the group of silts and clays. Then, particular methods are

employed to sort the soil into more specific groups. For gravels and sands, sieve analysis

results can land the soil into one of 36 granular soil types. For the group of silts and

clays, the Atterberg limit test results can lead the soil to one of 35 fine-grained soil types.

The AASHTO system also possesses two dominant steps. First, sieve analysis

can confirm if the soil is mostly coarse-grained or fine-grained. The first three groups,

A-1, A-2, and A-3, consist mainly of granular particles. The remaining four groups, A-4,

A-5, A-6 and A-7-6, have at least 35% of their particles belonging to silts and clays.

Secondly, liquid limit and plasticity index values can help engineers to classify the fine- grained soils further into one of the four groups.

Based on the AASHTO Soil Classification System, the fine-grained soils in Ohio are either A-4, or A-6, or A-7-6. Their index properties are listed in Table 2.1.

Table 2.1: AASHTO Classifications for Fine-Grained Soils Property Index A-4 A-6 A-7-6 Percentage Passing Sieve #200 (%) 36 min. 36 min. 36 min. Liquid Limit (%) 40 max. 40 max. 41 min. Plasticity Index (%) 10 max. 11 min. 11 min.

Ohio Department of Transportation (ODOT) has been allowing construction of major highway embankment structures, using A-4, A-6, and A-7-6 soils available locally.

In Ohio, A-4 soils often exist as lake deposits in the north along the shores of Lake Erie.

A-6 soils are contained in the unglaciated zone and thus are distributed from east to southeast. And, A-7-6 soils makes up the glaciated till, which occupies the western and central territory. Their locations are displayed in Figure 2.2, as discussed above.

The more detailed categories are available for A-4 and A-6 soil types in Ohio. By the percentage passing sieve No. 200, A-4 types can be separated further into A-4a and

A-4b subgroups. A-4a soils are A-4 soils that have between 36 and 49 percent of particles passing sieve no. 200. For A-4b soils the percent passing sieve no. 200 must be more than 50 percent. A-6 soils are broken down into subgroups by their plasticity index values. The range of plasticity index should be 11-15 for A-6a and at least 16 for A-6b.

A-4b and A-6b soils are known to be more frost susceptible than A-4a and A-6a soils.

Figure 2.2: Distribution of Three Major Soil types in Ohio (ref. Masada 2009) 22

CHAPTER 3 : RESEARCH METHODOLOGIES

The chapter presents methodologies employed in the current thesis research to meet its objectives. The first section describes the methodology used to test and sample soils encountered at representative highway embankment sites. The methodology used in the laboratory to measure shear strength of soil samples is provided in the second section.

And, the last section is devoted to discussing the general approaches taken to analyzing the test data.

3.1 Site Screening and Sample Collection

3.1.1 Site Locations and Distribution

The leading institution Ohio Research Institute for Transportation and

Environment (ORITE) consulted the funding agency Ohio Department of Transportation

(ODOT) to come up with a set of criteria for screening candidate highway embankment sites in Ohio. The criteria are listed below:

1. The sites should be located ideally near major highways, such as interstates and state routes.

2. The sites should be distributed across different regions of Ohio to represent major soil deposits (lake deposits, glaciated soils, and unglaciated soils) existing in the state.

3. The soils found at the sites should cover a wide range (A-4 to A-7-6) of cohesive materials that have been utilized in highway embankment construction.

4. The embankment fill should be mostly free of gravel particles and rock fragments. 23

5. It is preferred that the no guardrails exist at the sites.

6. The site should provide a large flat area where soil testing and sampling work

can be performed safely.

7. The height of embankment should be more than 25 feet.

The first three criteria ensure that the field/laboratory data will be comprehensive and tied to major road infrastructure in the state. The next three criteria will make the

subsurface exploration work at each site quite manageable. And, the last item is needed

to protect the embankment’s foundation from being disturbed during the site work.

In the end, a total of nine highway embankment sites were selected for the current

research project. Their locations are illustrated in Figure 3.1. Here, each site’s

identification consists of the county name (abbreviated in three letters) followed by the

route number of the roadway.

Figure 3.1: Locations of Highway Embankment Sites (ref. Masada 2009)

Figure 3.1 shows that the sites are quite evenly distributed across the state. There

are three sites located in the northern Ohio, which are LAK-2, ERI-2 and HAN-75. The

central region of Ohio contains three sites, which are FAY-35, MRW-71 and MUS-70.

And, three sites, HAM-275, ATH-33 and NOB-77, are found in the southern Ohio.

In terms of the state’s geological features, these nine sample sites can be divided into

three groups. In the lake deposit area, there are two sites labeled LAK-2 and ERI-2.

Around glaciate till distributed four sites which identified HAM-275, FAY-35, MRW-71 and HAN-75. The last three sites occupied unglaciated zone, they are ATH-33, MUS-70 and NOB-77.

3.1.2 Standard Penetration Test

SPT is the acronym for Standard Penetration Test, which is an in-situ test for measuring relative density/stiffness of soils.

During SPT, a split-spoon sampler (having inside diameter of 1.4 inches, outside diameter of 2 inches, and typical length of about 2 ft) is driven into the ground by dropping a 140-lb. hammer repeatedly over a height of 30 inches. In the current research, the number of hammer drops or blow count for each 6-inch penetration depth was recorded until the split-spoon was driven into 18 inches soil. Figure 3.5 shows a SPT in progress at one of the embankment sites. 25

Figure 3.2: Process of SPT

Different hammer types generate varied levels of energy transfer to the split- spoon. There are three types of SPT hammer used in the U.S. They are donut hammer

(about 45% energy delivery), safety hammer (about 60% energy delivery), and automatic

hammer (80-90% energy delivery). The last type, automatic hammer, is most widespread

because of its efficient energy delivery and prevention of human error. In the current

research, a dedicated mobile rig, equipped with an automatic hammer, was utilized to

conduct subsurface exploration work at all the embankment sites. The automatic hammer

was calibrated over a depth range of 1 to 19 ft, prior to the field work. According to the

results, the automatic hammer’s energy delivery ratio was 81.7%.

3.1.3 Soil Sampling Method

At each embankment site, soil samples were collected by pushing thin-wall steel tubes into soil. The type of steel tube used was Shelby Tube, which has length of 36 inches, inside diameter of 2.8 inches, and outside diameter of 3.0 inches. A minimum of four augered boreholes were placed close to the SPT hole. In these holes, soil sampling took place at three different depths to account for variations in the SPT blow count and/or changes in the soils. Figures 3.2 and 3.3 together illustrate the soil sampling plan executed at each project site.

Figure 3.3: Plan View of Soil Sampling Plan (ref. Masada 2009)

Figure 3.4: Side View of Soil Sampling Plan (ref. Masada 2009)

To avoid alteration of sample’s original condition, once the drilling operation was

accomplished, the ends of each steel tube were sealed by melted wax and then

capped/tapped. Figure 3.4 displays worker melting wax to seal the steel tube. All tubes

were secured to prevent them from moving and being disturbed during transportation to the laboratory.

Figure 3.5: Sealing of Shelby Tube Ends (ref. Masada 2009)

3.2 Experiment

3.2.1 Triaxial Compression Test - Introduction

Triaxial compression test is recognized as one of the reliable soil shear strength test methods applicable to highway embankment soil fills. After being extracted from the

Shelby tube, the soil specimen is ready for the triaxial test. While being confined inside the triaxial chamber, the soil sample can be saturated and then consolidated to simulate in-situ conditions. Depending on the actual conditioning and loading the test specimen receives, the triaxial test can be divided into three categories, which are C-D test, C-U test, and U-U test.

C-D (Consolidated-Drained) Test

This method is suitable for evaluating long term stability of the embankment fill slopes. In this test, the specimen is saturated and consolidated. The drainage valve should be kept open from the beginning of the saturation process to the end of the test.

For obtaining fully drained condition, application of the axial stress should be slow enough to squeeze water out of specimen without any increase in pore pressure. Hence, this test can be very time-consuming. That is the reason why it is not a very popular test in the lab.

C-U (Consolidated-Undrained) Test

This test is suitable for evaluating both long-term and short-term stability of embankment fill slopes. Up to the specimen consolidation stage, there are no differences between the C-D and C-U test procedures. In the C-U test, drainage valve stays open during the saturation and consolidation stages. Then, the valve is closed during the axial loading stage. During the load test, pore pressure sensor readings are recorded to present the test data in both total and effective stresses. This test can be started and completed typically within 3 to 4 days of time period.

U-U (Unconsolidated-Undrained) Test

This third triaxial test method is generally performed to evaluate undrained cohesion strength of existing silt and clay strata. After being saturated by back pressure, the specimen is loaded axially with the drainage valve closed. The specimen will not be consolidated between the saturation and loading stages. The apparent feature of this test is its speed and low cost. 30

3.2.2 C-U Triaxial Test Procedure and Equipment

In the current research project, relatively undisturbed embankment fill soil

samples were tested according to the C-U triaxial compression test procedure. In order to

supplement the triaxial test results, unconfined compression (UC) strength tests were also

conducted in parallel. The facility to conduct the C-U triaxial test was GEOTEC Sigma-

1 (provided by GEO-TAC, Inc. of Houston, Texas). The standard test procedure (ASTM

D-4767) was followed carefully to ensure high quality test results. The test steps outlined

below provide general guidelines. A manual of very detailed test procedure was

presented previously by Holko (2009).

Step 1: Extraction of Soil Sample

Since the Shelby tube is sealed by wax, cutting the tube into parts by electric saw

was an efficient approach to avoid disturbing the sample. The length of the dismembered

tube section was in the range of 5.6 to 7.0 inches. Then, with the help of a hydraulic jack,

the specimen was pushed out of tube carefully. Figure 3.6 shows this jacking operation.

As soon as the soil specimen was out of the tube, it was examined for the presence of

large rock stones or voids. When large stones or air voids were detected, the specimen

was rejected for testing. When the sizes of stones or voids are relatively small, some

patch work was done, as allowed by the ASTM test protocol, to improve the quality of

the specimen. Once the specimen was secured, its initial dimensions (diameter, height)

and weight were recorded, using a caliper and an electronic scale, for the determination of the initial unit weight. To acquire the natural moisture content, a thin layer of soil 31 sticking to the inner wall of the cut tube or a portion of the soil contained in the remaining sections of the original tube (located near the test specimen) was trimmed off and weighed. After oven-drying it for 24 hours, the loss in its weight was taken as the amount of water. The ratio of mass of water and that of dried soil provided the initial moisture content.

Figure 3.6: Pushing the Specimen Out by Oil Hydraulic Jack (ref. Holko 2009)

Step 2: Mounting of Test Specimen

The test specimen was positioned on top of the platen attached to the bottom assembly. The top platen was placed on top of the specimen. Each of these platens included a porous stone disk and ports for water inflow and drainage lines. Next, the specimen was wrapped around in a filter paper curtain (as an aid for the later consolidation stage). Once this was done, a thin latex rubber membrane encased the test 32 specimen, with o-rings sealing the membrane ends over the sides of the platens. Figure

3.7 shows one of the soil specimens going through the mounting steps.

Figure 3.7: The Work for Restraining the Specimen (ref. Holko 2009)

Step 3: Specimen Saturation

The chamber wall and the top assembly were both added to the bottom assembly complete the chamber construction. A set of flexible tubing was attached to connect the specimen ends and the chamber void space to the burettes on the control panel. The chamber void was filled with de-aired water until water overflowed from a port on the top assembly. Then, to initiate the saturation process water was rushed into the soil specimen by building a backpressure on top of the water contained in the de-airing water tank. During the initial stage of the saturation process, a vacuum line was hooked up to the upper end of the specimen to remove pore air from the specimen as much as possible. 33

Once water appeared in the line connected to the specimen’s top end, backpressure

pushed water into the upper portion of the test specimen as well. During the entire

saturation stage, the chamber pressure was maintained 2 psi higher than the backpressure

to ensure the specimen’s structural integrity. Figure 3.8 shows one of the test specimens

undergoing the saturation stage.

Figure 3.8: The Course of Saturation (ref. Holko 2009)

The accomplishment of saturation was judged by the Skempton’s B-value, which is the ratio between the increase of confining pressure and corresponding rise of pore pressure. It can be formulated as:

B = Δu / Δσ3 (3.1) where Δu = rise of pore pressure, and Δσ3 = increase of confining pressure.

The B-value was measured by closing the drainage valve, elevating the confining

pressure by 10.0 psi, and monitoring changes in pore pressure transducer reading over 34 duration of 2 minutes. If the B-value remains more than 0.95 for 2 minutes, the specimen was assumed to be sufficiently saturated.

Step 4: Specimen Consolidation

The specimen was consolidated for 24 hours as soon as it was saturated sufficiently. The difference between the cell pressure and backpressure constituted the effective consolidation pressure. To eliminate any overconsolidation behavior, the effective consolidation pressure was set slightly higher than the in-situ effective overburden pressure. The consolidation process was started by opening drainage valve after adjusting the chamber pressure and setting up suitable water columns in the control panel burettes. During the consolidation, water column readings were recorded frequently, especially during the first 3 hours, to record its progression. This operation continued until no changes in the water column readings were observed, which typically lasted for up to 24 hours.

Step 5: Axial Stress Application

Immediately after the consolidate stage, the triaxial chamber was positioned on the loading machine. The piston extending from the chamber’s top assembly was centered with respect to the load frame’s loading axis. Figure 3.9 displays the triaxial chamber resting in its correct position for the execution of the axial loading test. Closing of the drainage lines was the last important physical step to take prior to loading the soil specimen. 35

Figure 3.9: The Facilities for Triaxial Test (ref. Holko 2009)

Before loading the specimen, the loading rate had to be determined carefully using the data obtained during the consolidation stage. The time for 50% consolidation

(t50) was extracted from the plot of the changes in the water column readings vs. elapsed time. This value was then used to determine the specimen loading rate as per the ASTM protocol:

Loading rate (percent / minute) = 4 / (10 * t50) (3.2)

The above step for specifying the loading rate was needed to make sure that the

pore water reading would be in equilibrium during the entire loading test. The computer

data acquisition system connected to the physical system recorded the sensor (load cell,

axial displacement transducer, cell pressure transducer, and pore pressure transducer)

readings until the test specimen failed or reached the end of the test. The end of the

triaxial test was determined when the axial strain of specimen reached 15%. The 36

specimen failure was defined when the load cell registered a pronounced peak and

several constant readings consecutively.

At the end of each triaxial test, the experimental data was saved as an EXCEL data file. The chamber was unloaded and drained to access the test specimen. Once the rubber membrane was carefully removed, the specimen’s final physical condition was photographed and sketched. Whenever the specimen exhibited a distinct shear plane, the

orientation of the plane was measured using a protractor. Then, the specimen was cut

into pieces to check if it contained large rock fragments/stones inside. All the soil pieces were weighed and then dried in the oven to determine the specimen’s final moisture

content. 37

CHAPTER 4 : RESULTS AND DATA ANALYSIS

This chapter is devoted to the presentation of all the field and laboratory data

compiled in the current research, evaluation of empirical correlations established by

Terzaghi and the U.S. Navy. The relationships between each group of data are also

explored in this chapter, using several different regression methods.

4.1 Introduction of Basic Theory

The factors affecting soil shear strength are applied normal stress, cohesion, and

internal friction angle. This is shown below:

τ = c + σ*tan φ  (4.1)

where τ = shear strength, c = cohesion, σ = applied normal stress, and φ = internal friction angle.

In fact, the shear strength is derived from friction force between soil particles.

Compared to physical friction phenomenon, the normal stress generated by gravity and pressure is the fundamental element to trigger the shear strength of soil. The cohesion and internal friction angle figure the micro influence between soil particles. The former reveals bonding force, which is determined by ionic attraction, covalent bond, and chemical cementation. The latter is the degree of interlock between soil particles. This

factor is dictated by the soil particles’ geometry, relative location, mineral component,

and level of density.

The variables in shear strength function can be derived for each soil type from a

graphical plot constructed using C-U triaxial test data. For the plot, the abscissa is 38

assigned to normal stress and the ordinate to shear strength. At the point of shear failure of soil specimen in C-U triaxial test, the vertical pressure is equal to the major principal stress σ1, and the confining pressure is equal to the minor principal stress σ3. Marking σ3 on the abscissa, a half-circle can be drawn from the marked point with a diameter matching the difference between σ1 and σ3. When the data from two tests are processed

in this manner, two half-circles are established in the coordinate. A common tangent line

exists for the two half circles, which is the image of linear function of soil shear strength.

This graphical method is based on the Mohr-Coulomb Theory. The sketch of this plotting

process is shown is Figure 4.1.

Figure 4.1: The Line of Soil Shear Strength Determined by Mohr-Coulomb Principle Stress Circle

The method described above is the theoretical foundation for calculating shear

strength parameters c and φ. But, this method does not work well when the test results

are not perfectly aligned with each other. In such a case, which occurs often, another

graphical approach provides a more accurate and more convenient avenue for

determining the shear strength properties. First, the values of the stress path parameters p 39

and q are computed for the failure state observed in a few triaxial tests, according to the

following formula:

p = (σ1 + σ3) / 2 (4.2)

q = (σ1 - σ3) / 2 (4.3)

Here, the parameter p and q represents the mean stress and deviatoric stress,

respectively. The meanings of σ1 and σ3 are the same as those in the Mohr-Coulomb

theory.

Next, a p-q diagram is constructed with its abscissa assigned to p and ordinate to q. In the diagram, each test’s failure condition is represented by a point in the p-q diagram. Adopting the linear regression statistical method, a trend line can be established

through these points. Then, the c and φ values can be computed through the equations

below:

φ = arcsin (tan α) (4.4)

c = d / cos φ (4.5)

where α = slope of the trend line in the p-q diagram (degrees), and d = intercept of the trend line in the p-q diagram.

To calculate the effective stress based shear strength parameters c’ and φ’, the above approach can be repeated again, provided that total stresses are converted to effective stresses by using the pore pressure at failure. The plot is now called the p′-q′

diagram. Therefore, Equations 4.2 through 4.5 are changed to:

P’ = (σ’1 + σ’3) / 2 (4.6)

q’ = (σ’1 - σ’3) / 2 (4.7) 40

σ’1 = σ1 - uf , σ’3 = σ3 - uf (4.8)

φ’ = arcsin (tan α’) (4.9)

c’ = d’ / cos φ’ (4.10)

where uf = pore pressure at failure, α’ = slope of the trend line in the p′-q′ diagram, and d’ = intercept of the trend line in the p′-q′ diagram.

4.2 Methodology to analyze the correlations

After the completion of the entire field testing and the laboratory experiments, thousands of data are generated. In order to explore the correlations existing among the research data, the data set was divided into specific groups. First of all, the data can be

divided into field test and lab test by the location of experiment. The field test is

represented single -handedly by the SPT test. The lab test can be further separated into

soil Index properties test, unconfined compression strength test and C-U triaxial test. The

interactions between these test results guide the directions of the data analysis. Figure 4.2

illustrates these interactions. 41

Figure 4.2: The Interactions between Each Type of Experimental Results (ref. Masada, 2009)

In describing various correlations, it is convenient and efficient to use symbols to

narrate their interminable full name. This paragraph exhibits these symbols in the

parenthesis. The data of field test is corrected SPT-N value (SPT-(N60)1). The results of

unconfined compression test are Unconfined Compression Strength (qu) and Initial Dry

Unit Weight (γd-UC). Soil index properties include Specific Gravity (Gs), Percent of

Gravel (%G), Percent of Sand (%S), Percent of Silt (%M), Percent of Clay (%C), Plastic

Limit (PL), and Liquid Limit (LL). C-U triaxial test’s data contain Initial Dry Unit

Weight (γd-CU), Initial Moisture Content (w), Final Moisture Content (wf), Half time of

Consolidation (t50), Cohesion (c), Internal Friction Angle (φ), Effective Cohesion (c’),

Effective Internal Friction Angle (φ’) and the compact percent (%Cpt). The total number of data listed above was 1,400. 42

The approaches to analyze correlations were statistical methods, which included

single variable linear regression, multi-variable linear regression, single variable

nonlinear regression, and multi-variable nonlinear regression. The results of relationships

were expressed in terms of their equations and the coefficient of determination R2.

4.3 Effective Internal Friction Angle vs. Plasticity Index (by Terzaghi)

Among the same type of soil, the higher SPT-(N60)1 value represented higher

unconfined compression strength and stronger shear strength. In 1996, Terzaghi provided

a correlation between plasticity index and internal friction angle, which is the key factor

influencing the shear strength of cohesive soils. The empirical relationship is listed in

Table 4.1.

Table 4.1: Effective Internal Friction Angle vs. Plasticity Index by Terzaghi Plasticity Index (%) Effective Internal Friction Angle (degrees) 10 33.3 20 30.8 30 29.2 40 27.1 50 25.6 60 24.6 70 23.8 80 23.1

Reliability of the above relationship is supposed to be the listed friction angle

value three degrees. Terzaghi also provided reasonable range around the empirical

values, which are illustrated in Figure 4.3. The upper bond is two degrees above the

empirical value and lower bond is two degrees below the empirical value. According to 43

the distribution of the data obtained in the research, the correlation created by Terzaghi

can be evaluated for each cohesive soils found in Ohio. In the figures in this section, the

three dashed lines indicate the upper bound, the mean, and the lower bound of the

empirical relationship derived by Terzaghi’s. The red dots display the data collected from

the laboratory experiments.

Figure 4.3: Evaluation of Terzaghi’s Empirical PI-φ′ Relationship (for A-4a Soils in Ohio)

The plasticity index of A-4a soil was in the range of 7 to 13. There are nineteen points in Figure 4.3. Seven points (36.8%) are out of the band defined by the upper and lower bounds. In those seven points, three points (42.9%) are below the lower bound, and four points (57.1%) are above the upper bound. The fact that twelve out of nineteen points (63.2%) are distributed within the band shows that the Terzaghi’s correlation is mostly suitable for the A-4a soils in Ohio. 44

Because A-4b soil types were rarely encountered in the field phase of the

research, evaluation of the Terzagi’s empirical PI-φ′ correlation is not possible for the A-

4b soils found in Ohio.

Figure 4.4: Evaluation of Terzaghi’s Empirical PI-φ′ Relationship (for A-6a Soils in Ohio)

The plasticity index of A-6a soils varied between 11 and 17. In Figure 4.4, only two points (9.1%) are above the upper bound given by Terzaghi. No points are seen below the lower bound. This means that the empirical correlation of Terzaghi fits the A-

6a soils in Ohio very well. 45

Figure 4.5: Evaluation of Terzaghi’s Empirical PI-φ′ Relationship (for A-6b Soils in Ohio)

The A-6b’s plasticity index was in the range of 17 to 24 (see Figure 4.5). In

Figure 4.5, all nine points are located within the band specified by Terzaghi. However, the number of data points available for A-6b soil was not adequate to affirm the

Terzaghi’s correlation totally for this soil type. Thus, the data was combined for A-6a and

A-6b soils to carry out the empirical correlation’s evaluation, as shown in Figure 4.6 below. For the integrated A-6 soils, only two out of thirty one points (6.5%) stayed outside of the floating scope. Hence, the Terzaghi’s correlation is judged to be highly appropriate for the A-6 soils in Ohio. 46

Figure 4.6: Evaluation of Terzaghi’s Empirical PI-φ′ Relationship (for A-6 Soils in Ohio)

Figure 4.7: Evaluation of Terzaghi’s Empirical PI-φ′ Relationship (for A-7-6 Soils in Ohio)

The plasticity index of A-7-6 soils contained values ranging from 22 to 37. In

Figure 4.7, the points are centered on the lower bound of the Terzaghi’s relationship.

Nine out of twenty four data points (37.5%) are found below the lower bound. Therefore, the scope of Terzaghi’s correlation appears to be on the higher side for the A-7-6 soils in

Ohio. The correlation established by Terzaghi can be the guide for the A-7-6 soils in Ohio for embankment construction. To achieve a high level of accuracy, the Terzaghi’s 47 correlation should be moved down by 3 degrees uniformly for estimating φ′ of A-7-6 soils.

4.4 SPT-N vs. Unconfined Compression Strength (by Department of Navy)

The Department of Navy established the correlation between the corrected SPT-N value and unconfined compression strength in 1982. This correlation determine the range of unconfined compression strength of various SPT-(N60)1 value of clay types in different plasticity. Since this empirical correlation integrates the values derived from tremendous projects spread at different locations, so it is uncertain to determine this correlation is suitable for the embankment construction in Ohio. This can be testified by the comparison of results from SPT tests in Ohio to the correlation built by Department of Navy.

Table 4.2: SPT-N vs. Unconfined Compression Strength (psi) by Department of Navy SPT- qu of clays at low qu of clays at medium qu of clays at high (N60)1 plasticity (psi) plasticity (psi) plasticity (psi) 5 5.2 10.4 17.4 10 10.4 20.8 34.7 15 15.6 31.3 52.1 20 20.8 41.7 69.4 25 26.0 52.1 86.8 30 31.2 62.5 104.1

The clays at low plasticity means the liquid limit of clay is in the range of 10 to 30 and plastic index is in the range of 4 to 7. If the liquid limit of clay surpasses 50, the clay is treated as high plasticity. Since liquid limits of soils found in Ohio are lower than 50, 48

only low and medium boundary lines are demonstrated in the following tables. In Table

4.2, the unconfined compression strength of clays at low plasticity is treated as low limit,

and the one at medium plasticity is treated as high limit. The value from SPT tests in

Ohio is illustrated as points to show the distribution.

Figure 4.8: Evaluation of Relationship built by Department of Navy (for A-4a Soils in Ohio)

In Figure 4.8, there are ten points for the A-4a soil and only five points (50%) are in of the empirical limits. For the points out of empirical limits, two points (40%) falls the low limit and three points (60%) exceeds the high limit. Hence, the empirical limits built by Department of Navy are not suitable for the A-4a soil in Ohio.

Since the deficiency of effective data of the A-4b soil, the comparison between the empirical limits and A-4b soil in Ohio is neglected. 49

Figure 4.9: Evaluation of Relationship built by Department of Navy (for A-6a Soils in Ohio)

In Figure 4.9, eight points are listed in the figure above. There are five points

(62.5%) not existing inside the empirical limits. Four points (80%) are under the low empirical limit and only one point (20%) is above the high empirical limit. Hence the empirical correlation derived by Department of Navy is not suitable in Ohio.

Figure 4.10: Evaluation of Relationship built by Department of Navy (for A-6b Soils in Ohio)

In Figure 4.10, for A-6b soil in Ohio, there are only six values are available to be

adopt. However, only two points (33.3%) are not in the range of empirical limits. The distribution of points exceeding the empirical limits is average, one (50%) is above the high limit and one (50%) is below the low limit. The empirical correlation is suitable for the A-6b soil in Ohio, whereas the number of effective value is not sufficient to provide powerful proof. Therefore, the comparison of the whole A-6 soil will further testify the efficiency of empirical correlation.

Figure 4.11: Evaluation of Relationship built by Department of Navy (for A-6 Soils in Ohio)

In Figure 4.11, there fourteen values for the A-6 soil in Ohio, but seven points

(50%) are merely wrapped in the range of empirical limits. For the values jumped over

the limits, five values (71.4%) are less than the low limit and two values (28.6%) are

greater than the high limits. The range of empirical correlation is higher than A-6 soil in

Ohio. 51

Figure 4.12: Evaluation of Relationship built by Department of Navy (for A-7-6 Soils in Ohio)

The effective values for A-7-6 soil in Ohio are much more abundant than that of the other soils in Ohio. Nevertheless, in Figure 4.12, seven points (50%) located at the outside of empirical limits. Among the outside points, three points (42.9%) surpass the high limit and four points (57.1%) do not reach the low limit. The empirical correlation is also not suitable for the A-7-6 soil in Ohio.

According to the analysis above, the main soil in Ohio for embankment construction cannot be guided by the empirical correlation established by Department of

Navy at 1982. The unconfined compression strength of soil in Ohio should be tested by the specimens in the laboratory.

4.5 A Values and Level of Consolidation

For the consolidated drained (C-D) triaxial test, the volume of soil shrink under the external pressure, but the excess pore pressure keeps around zero. Oppositely, for the consolidated undrained triaxial test, the volume of soil remains the same as the original one and the excess pore pressure will increase above zero or decrease below zero. At the failure stage, if the soil specimen is consolidated into the normal level, the change of excess pore pressure will be positive. If the soil specimen is overconsolidated at the failure stage, the change of excess pore pressure will show negative. This trend can be described by the parameter named A value. The equation is listed below:

u −σ A = f 3 f (4.11) σ1 f −σ 3 f where uf = excess pore pressure, σ3f = confining pressure and σ1f = major principle stress.

The subscript “f” represents the failure stage. Tables 4.3 through 4.6 list the A values recorded during the current research project.

Table 4.3: The A value for A-4a soil Location Tube σ1f (psi) σ3f (psi) uf (psi) A value 9.2-9.7 A-2 149.6 45.1 5.5 -0.37895 9.2-9.7 D-2 167 52.7 9.7 -0.37620 FAY 9.2-9.7 E-2 205.3 60.2 1.9 -0.40179 14.7-15.2 B-3 104.8 47.9 24.7 -0.40773 15.4-15.8 B-3 141.8 54.1 20 -0.38883 4.1-4.6 A-2 77.9 37.8 24.9 -0.32170 4.0-4.5 D-2 96.6 44.9 27.9 -0.32882 4.7-5.2 D-2 153.5 59.9 9.5 -0.53846 LAK 14.7-15.2 C-3 104 48 21 -0.48214 14.6-15.1 A-3 116.4 54 24.2 -0.47756 14.6-15.1 D-3 173.2 59.8 3.7 -0.49471 13.3-13.8 D-2 112.6 45 18 -0.39941 13.8-14.3 C-2 129.5 52.4 19.7 -0.42412 13.3-13.7 C-2 127.7 60.2 31.4 -0.42667 MRW 17.9-18.4 B-3 114.9 50.1 24.7 -0.39198 18.2-18.6 D-3 122.2 60 39.2 -0.33441 17.6-18.1 C-3 110.7 64.8 44.3 -0.44662 Average -0.41295

Table 4.4: The A value for A-6a soil Location Tube σ1f (psi) σ3f (psi) uf (psi) A value 5.7-6.2 A-1 79.3 37.7 24.5 -0.31731 6.6-7.1 D-1 83.1 45.3 29.4 -0.42063 FAY 6.3-6.7 E-1 101.3 52.3 28.5 -0.48571 5.5-6.0 E-1 113.4 59.9 42 -0.33458 1.6-2.1 A-1 68.4 35.1 20.2 -0.44745 LAK 1.0-1.5 A-1 119.9 45.2 10.8 -0.46051 1.1-1.6 D-1 150.6 59.8 16.6 -0.47577 5.9-6.1 A-1 86.5 37.6 19.3 -0.37423 5.5-6.0 B-1 108.1 45.1 21.4 -0.37619 5.9-6.4 D-1 142.2 60.1 27.7 -0.39464 ATH 8.8-9.3 B-2 114.8 44.9 17.6 -0.39056 9.0-9.5 D-2 103.4 52.3 31.9 -0.39922 9.5-9.5 B-2 132.9 60.1 26.6 -0.46016 10.5-11.0 B-1 100.4 45.2 24 -0.38406 MRW 10.5-11.0 C-1 110.7 52.5 29.2 -0.40034 10.5-11.0 D-1 112.4 60 38.3 -0.41412 10.25 A-1 70.9 42.6 32.6 -0.35336 MUS 10.25 B-1 102.8 50.4 29.6 -0.39695 10.25 C-1 105.7 46.6 24.1 -0.38071 10.25 B-3 42.9 59.4 35.3 1.46061 NOB 10.25 C-3 50.2 84.2 35.2 1.44118 10.25 D-3 55.2 91.4 39.8 1.42541 Average -0.15179

Table 4.5: The A value for A-6b soil Location Tube σ1f (psi) σ3f (psi) uf (psi) A value 17.45 A-3 92.1 45.1 20.3 -0.52766 HAN 17.45 B-3 109.6 52.3 23.5 -0.50262 17.65 D-3 128.6 61.3 23.9 -0.55572 4.25 B-1 64 42 33.2 -0.40000 4.25 C-1 80 50 35.7 -0.47667 4.25 D-1 78.7 55.1 38.4 -0.70763 NOB 4.75 B-1 90 55.3 39 -0.46974 7.25 A-2 73.1 42.7 30.1 -0.41447 7.25 D-2 83.3 49.9 34.9 -0.44910 7.25 E-1 88 55.2 38.3 -0.51524 Average -0.50189

Table 4.6: The A value for A-7-6 soil Location Tube σ1f (psi) σ3f (psi) uf (psi) A value 2.5-3.0 A-1 51.6 35 27.1 -0.4759 3.1-3.6 A-1 65.6 45.1 33.6 -0.56098 2.5-3.0 D-1 90 60.1 40 -0.67224 5.1-5.6 A-2 63.4 39 26.4 -0.51639 HAM 4.9-5.4 C-2 64.7 44.8 33.5 -0.56784 4.6-5.1 D-2 86.3 59.9 41.2 -0.70833 10.3-10.8 A-3 66.3 42.6 27.8 -0.62447 10.2-10.6 D-3 78.5 46.4 26.9 -0.60748 20.0-20.5 A-3 96.3 51.6 25.3 -0.58837 ATH 20.0-20.5 B-3 101.6 59.9 32.1 -0.66667 20.0-20.5 D-3 125.3 64.3 28.9 -0.58033 2.95 B-1 95.9 59.5 37.2 -0.61264 3.25 B-1 65.4 45.2 32.6 -0.62376 3.50 D-1 48.6 35.2 29.5 -0.42537 6.50 D-2 73.2 50 34.7 -0.65948 ERI 7.05 D-2 55.5 40.2 31.6 -0.56209 7.15 B-2 90.3 59.9 39.8 -0.66118 11.75 B-3 80.8 45 29.1 -0.44413 11.75 C-3 82 52.3 34.6 -0.59596 11.75 D-3 87.6 57.2 38.8 -0.60526 6.55 D-1 90.2 55 32.8 -0.63068 6.75 C-1 80.5 47.1 27.7 -0.58084 HAN 7.00 A-1 71.5 40 22.3 -0.56190 10.95 A-2 70.3 41.9 26 -0.55986 10.95 B-2 75.9 48.9 32.1 -0.62222 Average -0.58858

The A value of normally consolidated clays is between 0.5 and 1.0, and the value

of overconsolidated clays located in the range of -0.5 to 0.0 (Braja M. Das, 2006).

According to the data listed above, all the specimens tested by the C-D triaxial test procedure exhibited A values of overconsolidated material, except for the A-6a soils collected from the Noble County location. 56

4.6 T-test for Three Major Soils

Figure 4.13 shows the locations of the ten embankment sites and the AASHTO

soil types found at each site. The figure clearly shows that each soil type was not

confined within one specific region of the state. For example, A-7-6 soil type was

encountered in northwestern, southwestern, and southeastern regions. Thus, a question can arise about the influence geographical location may have on the engineering properties of each soil type. Is the A-7-6 soil in the northwestern region basically the same as the A-7-6 soil found in the southwestern region? In addition, A-4 soil type included further subgroups of A-4a and A-4b. A-6 soil type was also subdivided into A-

6a and A-6b subgroups. This can lead to the second fundamental question – are there any significant differences between the subsets of A-4 soil type (or the subsets of A-6 soil type). In order to find satisfactory answers to these questions, a statistical technique (t- test) was employed.

Figure 4.13: Geographical Distributions of Various Soil Types in Ohio

The T-test is a statistical approach that can be used to evaluate the unity between two groups of data selected from different samples. First, let us assume that the two samples are the same. This can be stated mathematically through the following hypothesis:

H0: μ1 − μ2 = 0 (4.12)

where n1 and n2 = total number of data points in samples 1 and 2, and Sp = pooled variance (defined by the formula given below).

μ − μ t = 1 2 (4.13) 1 1 sp + n1 n2

where, the n1 and n2 = the total number of each sample. The Sp value indicates pooled

variance, its formula is:

2 2 2 s1 ()()n1 −1 + s2 n2 −1 sp = (4.14) n1 + n2 − 2

2 2 where s1 and s2 = variance in samples 1 and 2 (calculated by the following equations).

n1 n1 2 n2 n2 2 2 ⎛ ⎞ 2 ⎛ ⎞ n1∑∑x1i − ⎜ x1i ⎟ n2 ∑∑x2i − ⎜ x2i ⎟ 2 i==1 ⎝ i 1 ⎠ 2 i==1 ⎝ i 1 ⎠ s1 = and s2 = . (4.15) n1()n1 −1 n2 ()n2 −1

The criterion to judge the unity between the two samples is the critical t-values derived from the t-distribution, which are listed in Table 4.7. If the absolute value of t value is smaller than the critical t-value, the hypothesis H0 should be accepted. This means that the two samples can be treated as the same. The judgment process is expressed mathematically as:

If t < tα / 2,υ , then H0 is true. (4.16)

Generally, the level of significance is set at 0.05, which corresponds to a 95% reliability. Tables 4.8 and 4.9 below present the t-test results for the two soil subsets belonging to AASHTO A-4 and A-6 soil types.

Table 4.7: The Critical Value of T-Distribution

υ tα / 2,υ υ tα / 2,υ υ tα / 2,υ υ tα / 2,υ υ tα / 2,υ 1 3.078 7 1.415 13 1.350 19 1.328 25 1.316 2 1.886 8 1.397 14 1.345 20 1.325 26 1.315 3 1.638 9 1.383 15 1.341 21 1.323 27 1.314 4 1.533 10 1.372 16 1.337 22 1.321 28 1.313 5 1.476 11 1.363 17 1.333 23 1.319 29 1.311 6 1.440 12 1.356 18 1.330 24 1.318 + ∞ 1.282

Table 4.8: Summary of T Tests Performed on A-4 Soil Subsets Type # Gs LL PL PI %G %S %M %C Data

A-4a 17 2.68 26.2 16.4 9.8 8.7 25.1 40.2 25.9

A-4b 2 2.70 29.5 19.0 10.5 0.0 17.0 59.0 24.0

Sp / 0.026 3.76 2.25 2.24 4.70 1.87 4.14 5.75 / - -1.18 -1.54 - 2.48 5.79 -6.07 0.451 t value 0.086 0.438

Critical / 1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333 Value

Hypothesis / True True False True False False False True

Type γd (pcf) % Com. qu (psi) (N60)1 t50 (min) φ (deg.) φ′ (deg.) A-4a 121.2 101.0 39.3 32.1 4.5 24.4 33.4

A-4b 117.2 97.7 48.9 22.0 6.5 21.6 35.6

Sp 8.02 6.68 19.90 13.40 2.81 4.85 2.40 t value 0.670 0.670 -0.644 1.000 -0.962 0.791 -1.200

Critical 1.333 1.333 1.333 1.333 1.333 1.333 1.333 Value

Hypothesis True True True True True True True

Table 4.9: Summary of T Tests Performed on A-6 Soil Subsets # Type G LL PL PI %G %S %M %C Data s

A-6a 22 2.71 30.41 17.95 12.45 7.50 24.00 39.82 28.68

A-6b 9 2.71 38.33 20.67 17.67 7.33 14.44 43.11 35.44

Sp / 0.0387 4.944 2.635 3.154 13.04 13.78 25.52 45.79 - - - - - t value / 0.050 0.032 1.753 4.051 2.601 4.176 0.326 0.373

Critical / 1.311 1.311 1.311 1.311 1.311 1.311 1.311 1.311 Value

Hypothesis / True False False False True False True True

Type γd (pcf) % Com. qu (psi) (N60)1 t50 (min) φ (deg.) φ ′ (deg.) A-6a 119.80 108.91 37.20 32.27 7.30 20.05 33.48

A-6b 119.01 108.19 33.89 28.56 9.20 15.94 30.83

Sp 39.94 33.01 243.9 163.9 34.47 17.37 3.514 t value 0.050 0.055 0.034 0.057 -0.140 0.597 1.905

Critical Value 1.311 1.311 1.311 1.311 1.311 1.311 1.311

Hypothesis True True True True True True False

According to Table 4.8, the subsets of A-4 soil type are remarkably similar. Their properties are different only in terms of plastic limit and compositions. They are indistinguishable in terms of weight and shear strength properties. According to Table

4.9, the two subsets of A-6 soil type are more different from each other than the two subsets of the A-4 soil type.

Table 4.10: The Calculation for T Value of A-4a in Various Locations (North and South) # Type Gs LL PL PI %G %S %M %C Data North 6 2.37*10-31 3.47 0.167 2.70 3.47 2.97 1.07 3.90 South 5 7.50*10-04 0.00 0.00 0.00 0.30 0.30 0.300 0.300 Sp / 0.018 1.3880.304 1.225 1.435 1.335 0.8521.517 t value / 1.81 6.74 28.0 0.674 -10.4 -3.18 -4.39 15.1 Critical Value / 1.383 1.383 1.383 1.383 1.383 1.383 1.383 1.383 Hypothesis / False False False True False False False False

Type γd (pcf) % Com. qu (psi) (N60)1 t50 (min) φ (deg.) φ′ (deg.)

North 0.506 0.351 544 347 13.8 11.9 14.1

South 0.507 0.352 5.04 159 1.39 22.6 0.363

Sp 0.712 0.593 17.453 16.223 2.881 4.085 2.830 t value -10.9 -10.9 1.26 1.14 2.15 -2.33 -0.978

Critical 1.383 1.383 1.383 1.383 1.383 1.383 1.383 Value

Hypothesis False False True True False False True

Table 4.11: The Calculation for T Value of A-4a in Various Locations (North and Central)

Type # Gs LL PL PI %G %S %M %C Data 2.37*10- North 6 3.47 0.167 2.70 3.47 2.97 1.07 3.90 31 Central 6 0.00108 0.00 0.30 0.30 7.50 0.30 2.70 2.70 Sp / 0.026 1.472 0.540 1.369 2.618 1.429 1.5342.031

t value / 2.00 -3.92 2.14 -5.06 0.551 1.62 -10.3 5.97 Critical 1.372 1.372 1.372 1.372 1.372 1.372 1.3721.372 Value /

Hypothesis / False False False False True False False False

Type γd (pcf) % Com. qu (psi) (N60)1 t50 (min) φ (deg.) φ′ (deg.)

North 0.506 0.351 544 347 13.8 11.9 14.1

Central 0.588 0.408 0.867 67.5 1.57 14.8 3.20

Sp 0.827 0.689 18.458 16.0923.103 4.088 3.291 t value 29.6 29.6 3.38 0.807 0.642 0.713 -0.623

Critical Value 1.372 1.372 1.372 1.372 1.372 1.372 1.372

Hypothesis False False False True True True True

Table 4.12: The Calculation for T Value of A-4a in Various Locations (Central and South) Type # Data Gs LL PL PI %G %S %M %C

Central 5 0.00108 0.00 0.30 0.30 7.50 0.30 2.70 2.70 South 6 0.00075 0.00 0.00 0.00 0.30 0.30 0.300 0.300 Sp / 0.667 0.667 0.782 0.782 2.155 0.803 1.406 1.406

t value / -0.0409 36.8 15.7 15.7 -12.5 -13.2 13.4 13.4

Critical Value / 1.383 1.383 1.383 1.383 1.383 1.383 1.383 1.383

Hypothesis / True False False False False False False False

Type γd (pcf) % Com. qu (psi) (N60)1 t50 (min) φ (deg.) φ′ (deg.)

Central 0.588 0.408 0.867 67.5 1.57 14.8 3.20

South 0.507 0.352 5.04 159 1.39 22.6 0.363

Sp 0.910 0.843 1.219 7.455 1.214 3.344 1.505 t value -56.4 -50.7 -50.8 1.35 5.83 -6.07 -0.894

Critical Value 1.383 1.383 1.383 1.383 1.383 1.383 1.383

Hypothesis False False False True False False True

Table 4.13: The Calculation for T Value of A-6a in Various Locations (North and South) Type # Data Gs LL PL PI %G %S %M %C

North 3 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

South 16 0.00044 0.171 4.06 6.46 10.6 34.9 20.4 3.76

Sp / 0.020 3.879 1.893 2.388 3.058 5.551 4.239 1.822

t value / 4.03 -1.69 -0.892 -2.04 0.130 0.823 -1.99 1.91

Critical Value / 1.333 1.333 1.333 1.333 1.333 1.333 1.333 1.333

Hypothesis / False False True False True True False False

Type γd (pcf) % Com. qu (psi) (N60)1 t50 (min) φ (deg.) φ′ (deg.) North 0.00 0.00 7.57*10-29 0.00 1.58 18.7 0.303

South 32.9 27.2 64.6 213 59.6 20.4 5.19

Sp 5.390 4.890 7.552 13.7247.263 4.498 2.148 t value 1.59 1.59 6.02 -0.304 0.0638 1.87 -1.33

Critical Value 1.333 1.333 1.333 1.333 1.333 1.333 1.333

Hypothesis False False False True True False True

Table 4.14: The Calculation for T Value of A-6a in Various Locations (North and Central) Type # Data Gs LL PL PI %G %S %M %C

North 3 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Central 6 0.00048 1.20 4.80 1.20 1.20 1.20 1.20 10.8

Sp / 0.019 0.926 1.852 0.926 0.926 0.926 0.926 2.777

t value / 6.11 0.00 0.764 -1.53 -3.06 -7.64 -6.11 5.60

Critical Value / 1.415 1.415 1.415 1.415 1.415 1.415 1.415 1.415

Hypothesis / False True True False False False False False

Type γd (pcf) % Com. qu (psi) (N60)1 t50 (min) φ (deg.) φ′ (deg.) North 0.00 0.00 7.57*10-29 0.00 1.58 18.7 0.303

Central 13.9 11.5 118 86.7 5.36 10.0 0.815

Sp 3.148 2.862 9.166 7.869 2.069 3.535 0.818 t value -0.899 -0.899 2.99 1.89 2.08 1.62 -5.02

Critical Value 1.415 1.415 1.415 1.415 1.415 1.415 1.415

Hypothesis True True False False False False False

Table 4.15: The Calculation for T Value of A-6a in Various Locations (Central and South)

Type # Gs LL PL PI %G %S %M %C Data Central 6 0.00048 1.20 4.80 1.20 1.20 1.20 1.20 10.8

South 16 0.00044 0.171 4.06 6.46 10.6 34.9 20.4 3.76

Sp / 0.021 3.618 2.061 2.269 2.872 5.147 3.946 2.350

t value / -2.95 -2.38 -2.09 -1.90 1.64 3.20 -0.695 -7.83 Critical 1.325 1.325 1.325 1.325 1.325 1.325 1.325 1.325 Value /

Hypothesis / False False False False False False True False

Type γd (pcf) % Com. qu (psi) (N60)1 t50 (min) φ (deg.) φ′ (deg.)

Central 13.9 11.5 118 86.7 5.36 10.0 0.82 South 32.9 27.2 64.6 213 59.6 20.4 5.19

Sp 5.307 4.824 8.825 13.4826.784 4.222 2.023 t value 2.92 2.92 2.18 -2.03 -0.849 0.610 1.14

Critical Value 1.325 1.325 1.325 1.325 1.325 1.325 1.325

Hypothesis False False False False True True True

Table 4.16: The Calculation for T Value of A-6b in Various Locations (North and South) Type # Data Gs LL PL PI %G %S %M %C

North 3 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

South 6 0.00 1.20 0.30 0.30 15.9 8.97 2.27 5.47

Sp / 0.00 0.93 0.46 0.46 3.37 2.53 1.27 1.98

t value / Null 1.53 -7.64 10.7 -2.73 2.14 -15.2 12.4

Critical Value / 1.415 1.415 1.415 1.415 1.415 1.415 1.415 1.415

Hypothesis / True False False False False False False False

Type γd (pcf) % Com. qu (psi) (N60)1 t50 (min) φ (deg.) φ′ (deg.)

North 0.00 0.00 7.57*10-29 120.3 0.46 0.16 0.91 South 7.84 6.48 1.047 76.8 58.04 1.22 2.40

Sp 2.367 2.152 0.86 9.45 6.45 0.96 1.41

t value -6.91 -6.91 67.0 1.60 -0.143 10.0 -2.47

Critical Value 1.415 1.415 1.415 1.415 1.415 1.415 1.415

Hypothesis False False False False True False False

Table 4.17: The Calculation for T Value of A-7-6 in Various Locations (North and South)

Type # Gs LL PL PI %G %S %M %C Data North 14 0.000138 58.0 3.02 35.8 0.687 53.2 17.9 55.5

South 11 0.000785 26.1 1.07 26.3 14.4 26.4 5.82 10.3

Sp / 0.0205 6.64 1.47 5.63 2.58 6.45 3.56 5.99

t value / -1.65 2.15 3.05 1.74 -4.92 -2.82 1.85 4.05 Critical 1.319 1.319 1.319 1.319 1.319 1.319 1.319 1.319 Value /

Hypothesis / False False False False False False False False

Type γd (pcf) % Com. qu (psi) (N60)1 t50 (min) φ (deg.) φ′ (deg.)

North 27.0 22.4 71.9 89.7 806 4.58 6.13

South 10.8 8.97 137 24.6 178 8.64 3.32

Sp 4.47 4.07 10.0 7.83 23.08 2.52 2.22

t value -3.80 -3.80 -1.92 -2.26 2.06 -0.78 0.35

Critical Value 1.319 1.319 1.319 1.319 1.319 1.319 1.319 Hypothesis False False False False False True True

From the calculation results listed in tables from Table 4.10 to Table 4.17, the same soils in different location expose great distance. Although some chief factors of mechanical factors, such as unconfined compression strength and internal friction angle, appear coordination with the hypothesis, the advice is that conducting the unconfined compression test and triaxial test in the laboratory before the design process.

Since the size for each sample is not sufficient, so these comparisons are merely the recommendations instead of applicable guidelines in the project. The results from t- 69

test are reliable when each sample size surpasses hundreds. This project will cost lots of time and money.

4.7 Single Variable Regression Analysis

In the former chapters, the empirical limits established by Department of Navy have been proved to be inappropriate to the soils in Ohio embankment construction.

Though the Terzaghi’s correlation can provide the scope of the effective internal friction angle, the accuracy is not satisfying and the absence of other important mechanical properties is the defection for the design work. Hence, for the convenience of embankment design in Ohio, the relationship between the mechanical properties and experimental data is necessary.

To build the mathematical model explaining correlation between several factors, the best statistical method is regression. Since there are eighteen factors and total more than 1400 data, so only determine six chief factors to be the dependent variables. They are SPT-N corrected value SPT-(N60)1, unconfined compression strength qu, cohesion c,

effective cohesion c’, internal friction angle φ and effective internal friction angle φ’.

These factors are collected from complicated and time consuming experiments. The

other less important and more convenient to observed, for instance, physical properties and content are treated as independent factors. To provide consultation in independent variables selection in multiple variable regression analysis, the single variable regression analysis between dependent variable and each independent variable should be conducted 70

for determining which independent variable has distinct influence on dependent variables.

To predict the correlation in accuracy and entirety, the function types are divided into linear and nonlinear. Among the nonlinear function, it includes second degree polynomial, power, exponential, logarithmic, reciprocal and hyperbolic. Their mathematical formulae are listed below:

Linear: y = ax + b (4.17)

Second degree polynomial: y = ax2 + bx + c (4.18)

Power: y = axb (4.19)

Exponential: y = aebx (4.20)

Logarithmic: y = a + bLn(x) (4.21)

Reciprocal: y = a + b / x (4.22)

Hyperbolic: xy = a + bx (4.23) where y is the dependent variable, x is the independent variable, a and b are coefficients,

e is the natural logarithms.

The single variable regression analysis will cover every soil type in Ohio

embankment construction except A-4b soils, because the sufficiency of effective data for

this soil type is not available. For the other soil types illustrated in the following sections,

the single variable linear regression contains dependent variables to each independent

variable, but single variable nonlinear regression only display the equations with the

coefficient of determination R2 greater than 0.5. 71

4.7.1 Single Variable Regression Analysis for A-4a Soils

From Table 4.18 to Table 4.29, the regression results demonstrate the linear and nonlinear relationships among A-4a soil properties. The dependent variables include courted SPT value, unconfined compression strength, cohesion, effective cohesion, internal friction angle, and effective internal friction angle. The independent variables are consisted of percentage of soil components and soil mechanical properties.

Table 4.18: Single Variable Linear Regression for A-4a for Corrected SPT Value Y X R2 Equation SPT-(N60)1 PI 0.0432 y = 1.249x + 19.863 SPT-(N60)1 Gs 0.0204 y = 74.763x - 168.57 SPT-(N60)1 %G 0.0864 y = -0.8407x + 39.378 SPT-(N60)1 %C 0.2009 y = 2x - 25 SPT-(N60)1 %M 0.0717 y = -0.87x + 67.065 SPT-(N60)1 %S 0.0034 y = 0.4163x + 21.601 SPT-(N60)1 PL 0.1247 y = 2.1086x - 2.5471 SPT-(N60)1 LL 0.1122 y = 1.1969x + 0.7277 SPT-(N60)1 γd - UC 0.0035 y = -0.099x + 44.065 SPT-(N60)1 w - UC 0.0054 y = 0.3855x + 27.208 SPT-(N60)1 wf - CU 0.0911 y = 1.2109x + 15.128 SPT-(N60)1 qu 0.3535 y = 0.4021x + 16.243 SPT-(N60)1 t50 0.0026 y = -0.2556x + 33.203 SPT-(N60)1 φ 0.0069 y = -0.2344x + 37.782 SPT-(N60)1 φ’ 0.1155 y = 1.9178x - 31.983 SPT-(N60)1 %Compact 0.0033 y = -0.115x + 43.671 SPT-(N60)1 γd - CU 0.0328 y = -0.4007x + 83.492

Table 4.19: Single Variable Nonlinear Regression for A-4a for Corrected SPT Value Y X R2 Equation Function SPT-(N60)1 qu 0.8207 xy = 60.837x - 973.17 Hyperbolic 2 SPT-(N60)1 PI 0.6611 y = -3.2583x + 66.576x - 291.06 Polynomial SPT-(N60)1 t50 0.6159 xy = 34.998x - 14.99 Hyperbolic 2 SPT-(N60)1 qu 0.5972 y = 0.0179x - 1.3099x + 48.75 Polynomial 2 SPT-(N60)1 %C 0.5742 y = 0.8199x - 47.184x + 703.4 Polynomial 2 SPT-(N60)1 φ’ 0.5635 y = 1.3832x - 90.329x + 1498.2 Polynomial 72

Table 4.20: Single Variable Linear Regression for A-4a for Unconfined Compression Strength Y X R2 Equation qu PI 0.1491 y = -3.4313x + 72.841 qu Gs 0.1014 y = 246.68x - 622.63 qu %G 0.0017 y = 0.1728x + 37.831 qu %C 0.7007 y = 5.5233x - 118.24 qu %M 0.6573 y = -3.8942x + 196.02 qu %S 0.2678 y = 5.4853x - 98.443 qu PL 0.0002 y = 0.1167x + 37.419 qu LL 0.0492 y = -1.1719x + 70.011 qu γd - UC 0.3737 y = 1.5147x - 144.27 qu w - UC 0.2559 y = -3.9427x + 88.943 qu wf - CU 0.0696 y = -1.5653x + 61.222 qu t50 0.0147 y = -0.8996x + 43.362 qu φ 0.0069 y = -0.2344x + 37.782 qu φ’ 0.0436 y = 1.7428x - 18.863 qu %Compact 0.3754 y = 1.8218x - 144.7 qu γd - CU 0.0432 y = 0.6806x - 48.02

Table 4.21: Single Variable Nonlinear Regression for A-4a for Unconfined Compression Strength Y X R2 Equation Function -4.356 qu %M 0.8045 y = 3E+08x Power -0.105x qu %M 0.7936 y = 2411.6e Exponential qu %C 0.7930 xy = 213.2x - 4912. Hyperbolic 2 qu %M 0.7697 y = 0.5495x - 49.317x + 1124.7 Polynomial 2 qu %C 0.7008 y = 0.0182x + 4.4342x - 102.11 Polynomial qu %C 0.6969 y = 163.61ln(x) - 508.03 Logarithmic qu %M 0.6950 y = 6718/x - 129.3 Reciprocal qu %C 0.6880 y = -4775/x + 208.4 Reciprocal qu %M 0.6769 y = -162.4ln(x) + 638.4 Logarithmic 2 qu PI 0.6714 y = -4.4296x + 85.381x - 349.86 Polynomial 3.8426 qu %C 0.6346 y = 9E-05x Power 0.1288x qu %C 0.6287 y = 0.8844e Exponential qu %M 0.6050 xy = -121.1x + 6391. Hyperbolic 5.5843 qu %Compact 0.5901 y = 2E-10x Power 5.5754 qu γd - UC 0.5881 y = 9E-11x Power 0.0558x qu %Compact 0.5815 y = 0.1242e Exponential 2 qu %Compact 0.5795 y = -0.4065x + 82.758x - 4154.3 Polynomial 0.0464x qu γd - UC 0.5795 y = 0.1253e Exponential 2 qu γd - UC 0.5785 y = -0.2825x + 69.008x - 4156.4 Polynomial qu %G 0.559 xy = -1785.x + 632.5 Hyperbolic 2 qu w - UC 0.5502 y = -2.1136x + 48.26x - 219.52 Polynomial 2 qu φ' 0.5167 y = 2.1018x - 138.43x + 2306.3 Polynomial

Table 4.22: Single Variable Linear Regression for A-4a for Cohesion Y X R2 Equation c PI 0.0125 y = 0.211x + 10.056 c Gs None None c %G 0.0794 y = 0.2652x + 10.026 c %C 0.9486 y = -1.4685x + 55.38 c %M 0.5041 y = 0.6994x - 15.684 c %S 0.0977 y = -0.8821x + 34.04 c PL 0.2711 y = -1.0607x + 29.65 c LL 0.0611 y = -0.3109x + 20.17 c γd - UC 0.1095 y = -0.2157x + 38.479 c w - UC 0.0076 y = 0.1828x + 9.8131 c wf - CU 0.166 y = -0.8702x + 23.723 c qu 0.9159 y = -0.1982x + 20.584 c t50 0.0306 y = -0.4077x + 13.966 c φ 0.2097 y = 0.4728x - 0.1855 c φ’ 0.0938 y = -0.8395x + 39.849 c %Compact 0.1110 y = -0.2599x + 38.596 c γd - CU 0.1554 y = 0.3824x - 37.423

Table 4.23: Single Variable Nonlinear Regression for A-4a for Cohesion Y X R2 Equation Function c %C 0.9877 y = -0.1655x2 + 8.5955x - 96.136 Polynomial c qu 0.9529 y = -0.0023x2 + 0.0349x + 15.928 Polynomial c %G 0.9390 xy = 15.97x - 24.36 Hyperbolic c %C 0.9364 y = -44.15ln(x) + 161.28 Logarithmic c %C 0.9220 y = 1319/x - 33.05 Reciprocal c %C 0.9152 y = 1235.4e-0.16x Exponential c qu 0.9034 y = 28.061e-0.022x Exponential c %M 0.8987 y = -0.2562x2 + 22.054x - 454.72 Polynomial c %C 0.8983 y = 1E+08x-4.802 Power c %C 0.8840 xy = -34.46x + 1360. Hyperbolic c Phi 0.8737 y = -0.6192x2 + 35.375x - 477.91 Polynomial c PL 0.8178 y = -1.521x2 + 46.208x - 329.07 Polynomial c qu 0.7960 y = -7.839ln(x) + 40.61 Logarithmic c %S 0.7778 y = 2.1654x2 - 109.17x + 1382.8 Polynomial c qu 0.7525 y = 239.1x-0.846 Power c %M 0.7410 xy = 41.12x - 1139. Hyperbolic c w - UC 0.7217 y = 0.9874x2 - 23.157x + 143.28 Polynomial

c γd - UC 0.6740 y = 0.1229x2 - 29.595x + 1787.1 Polynomial c %Cpt 0.6695 y = 0.1758x2 - 35.261x + 1774.2 Polynomial c PI 0.6417 y = 0.9135x2 - 18.189x + 97.62 Polynomial c qu 0.6150 y = 237.3/x + 5.058 Reciprocal c Phi 0.5970 xy = 25.32x - 335.1 Hyperbolic c %M 0.5600 y = -1257/x + 44.11 Reciprocal c t50 0.5480 xy = 11.88x - 0.604 Hyperbolic c %M 0.5317 y = 29.755ln(x) - 97.28 Logarithmic

Table 4.24: Single Variable Linear Regression for A-4a for Effective Cohesion Y X R2 Equation c’ PI 0.107 y = 0.375x + 1.355 c’ Gs 4E-16 y = 1.891x - 0.183 c’ %G 9E-5 y = -0.005x + 4.964 c’ %C 0.140 y = 0.341x - 5.147 c’ %M 0.024 y = -0.093x + 8.631 c’ %S 0.339 y = 0.994x - 19.85 c’ PL 0.033 y = -0.223x + 8.632 c’ LL 0.011 y = 0.081x + 2.808 c’ γd - UC 0.013 y = 0.045x - 0.706 c’ w - UC 0.151 y = -0.491x + 10.96 c’ wf - CU 0.000 y = 0.038x + 4.411 c’ qu 0.461 y = 0.085x + 1.264 c’ t50 0.410 y = -0.903x + 9.146 c’ φ 0.014 y = 0.076x + 2.947 c’ φ’ 0.912 y = 1.583x - 47.47 c’ %Compact 0.014 y = 0.056x - 0.804 c’ γd - cu 0.022 y = 0.086x - 6.326

Table 4.25: Single Variable Nonlinear Regression for A-4a for Effective Cohesion Y X R2 Equation Function c’ φ’ 0.976 y = 1E-24x16.13 Power c’ φ’ 0.974 y = 3E-07e0.497x Exponential 2 c’ γd - CU 0.965 y = 0.545x - 143.6x + 9461. Polynomial c’ PI 0.955 y = -0.641x2 + 13.28x - 60.08 Polynomial c’ %C 0.951 y = 0.456x2 - 27.39x + 412.4 Polynomial c’ φ’ 0.926 y = 0.210x2 - 12.10x + 174.1 Polynomial c’ φ’ 0.910 xy = 55.50x - 1670. Hyperbolic c’ φ’ 0.909 y = 51.24ln(x) - 174.3 Logarithmic c’ φ’ 0.905 y = -1656/x + 55.07 Reciprocal c’ qu 0.877 xy = 10.38x - 197.6 Hyperbolic c’ φ 0.867 y = 0.424x2 - 23.83x + 330.2 Polynomial 2 c’ wf - UC 0.784 y = 1.004x - 25.15x + 157.5 Polynomial 2 c’ t50 0.738 y = -0.441x + 3.061x + 1.786 Polynomial c’ %G 0.666 xy = 5.808x - 6.904 Hyperbolic 2 c’ qu 0.561 y = 0.002x - 0.146x + 5.895 Polynomial -0.32x c’ t50 0.559 y = 17.32e Exponential

Table 4.26: Single Variable Linear Regression for A-4a for Internal Friction Angle Y X R2 Equation φ PI 0.1875 y = -0.9232x + 33.433 φ Gs 0.1956 y = 82.224x - 196.23 φ %G 0.2388 y = 0.4958x + 20.101 φ %C 6E-5 y = -0.0127x + 24.78 φ %M 0.0327 y = -0.2084x + 32.805 φ %S 0.1008 y = 0.8076x + 4.1332 φ PL 0.3857 y = -1.3159x + 46.014 φ LL 0.3959 y = -0.7978x + 45.302 φ γd - UC 0.2880 y = 0.3191x - 14.26 φ w – UC 0.4129 y = -1.2017x + 39.538 φ wf - CU 0.4840 y = -0.9906x + 38.269 φ qu 0.0156 y = 0.03x + 23.237 φ t50 0.0150 y = -0.2178x + 25.393 φ φ’ 0.0028 y = -0.1066x + 27.979 φ %Compact 0.2862 y = 0.3817x - 14.14 φ γd - CU 0.8366 y = 0.7183x - 67.789

Table 4.27: Single Variable Nonlinear Regression for A-4a for Internal Friction Angle Y X R2 Equation Function φ t50 0.9230 xy = 24.19x - 0.556 Hyperbolic φ γd - CU 0.8820 xy = 116.5x - 11800 Hyperbolic 3.8525 φ γd - CU 0.8581 y = 2E-07x Power φ qu 0.8550 xy = 23.26x + 57.19 Hyperbolic 0.0301x φ γd - CU 0.8545 y = 0.5039e Exponential 2 φ γd - CU 0.8381 y = 0.0037x - 0.2365x - 6.7467 Polynomial φ γd - CU 0.8333 y = 91.625ln(x) - 420.3 Logarithmic φ γd - CU 0.8280 y = -11634/x + 115.2 Reciprocal φ %G 0.6180 xy = -1260.x + 425.5 Hyperbolic φ wf - CU 0.5160 y = 198.7/x + 9.410 Reciprocal -0.602 φ wf - CU 0.5089 y = 115.39x Power 2 φ wf - CU 0.5068 y = 0.0612x - 2.8239x + 51.247 Polynomial φ wf - CU 0.5062 y = -14.5ln(x) + 62.26 Logarithmic φ wf - CU 0.5020 xy = 62.46x - 4591. Hyperbolic -0.042x φ wf - CU 0.5012 y = 42.951e Exponential φ %Cpt 0.5000 xy = 62.34x - 3814. Hyperbolic

Table 4.28: Single Variable Linear Regression for A-4a for Effective Internal Friction Angle Y X R2 Equation φ’ PI 0.0506 y = 0.2396x + 31.055 φ’ Gs 0.0380 y = -18.113x + 82 φ’ %G 0.0214 y = 0.0742x + 32.748 φ’ %C 0.0425 y = -0.163x + 38.046 φ’ %M 1E-6 y = -0.0006x + 33.419 φ’ %S 0.2932 y = 0.6879x + 16.116 φ’ PL 0.0623 y = -0.2643x + 37.731 φ’ LL 0.0002 y = -0.0099x + 33.653 φ’ γd - UC 0.0043 y = -0.0195x + 35.764 φ’ w - UC 0.0212 y = -0.136x + 35.105 φ’ wf - CU 0.0238 y = -0.1098x + 34.93 φ’ qu 0.0436 y = 0.025x + 32.409 φ’ t50 0.5591 y = -0.6649x + 36.371 φ’ φ 0.0028 y = -0.0266x + 34.044 φ’ %Compact 0.0039 y = -0.0224x + 35.655 φ’ γd - CU 0.0117 y = 0.0424x + 27.948

Table 4.29: Single Variable Nonlinear Regression for A-4a for Effective Internal Friction Angle Y X R2 Equation Function φ’ t50 0.988 xy = 28.95x + 15.10 Hyperbolic φ’ qu 0.964 xy = 35.47x - 72.07 Hyperbolic φ’ PI 0.923 xy = 35.13x - 15.82 Hyperbolic φ’ wf - CU 0.893 xy = 32.07x + 17.19 Hyperbolic φ’ φ 0.887 xy = 32.92x + 10.74 Hyperbolic φ’ w - UC 0.876 xy = 32.33x + 12.46 Hyperbolic φ’ %S 0.788 xy = 50.88x - 436.9 Hyperbolic φ’ LL 0.787 xy = 33.56x - 4.635 Hyperbolic φ’ %G 0.759 xy = -1487.x + 530.8 Hyperbolic φ’ PL 0.712 xy = 29.26x + 66.43 Hyperbolic φ’ %M 0.704 xy = 33.62x - 9.341 Hyperbolic -0.02x φ’ t50 0.576 y = 36.483e Exponential φ’ %C 0.572 xy = 28.20x + 146.6 Hyperbolic 2 φ’ t50 0.560 y = 0.0073x - 0.7469x + 36.539 Polynomial

In the linear regression for Corrected SPT value, each independent variable shows

extreme weak correlation. And in the nonlinear regression, the function displays most powerful correlation is hyperbolic. For the unconfined compression strength regression, it possesses strong correlation with the content of clay in linear regression and with content of silt in nonlinear regression. Among the regression for the cohesion, the factors of content of clay, content of silt and content of gravel express chief influences. To the effective cohesion, although effective internal friction angle shows remarkable effect, estimating one mechanical property based on another one is lack of practicability in the design project. The plasticity index and content of clay indicate strong correlation in nonlinear regression. The efficient nonlinear function type is polynomial. Through the nonlinear regressions for internal friction angle and effective internal friction angle, the meaningful and distinct independent variables are initial dry unit weight, initial moisture content and final moisture content. Hyperbolic is the best approach to display the nonlinear correlation.

4.7.2 Single Variable Regression Analysis for A-6a Soils

From Table 4.30 to Table 4.41, the regression consequences show the linear and nonlinear relationships among A-6a soil properties. The dependent variables contain courted SPT value, unconfined compression strength, cohesion, effective cohesion, internal friction angle, and effective internal friction angle. The independent variables are consisted of percentage of soil components and soil mechanical properties.

Table 4.30: Single Variable Linear Regression for A-6a for Corrected SPT Value Y X R2 Equation SPT-(N60)1 PI 0.067 y = -1.817x + 55.16 SPT-(N60)1 Gs 1E-5 y = 1.109x + 29.25 SPT-(N60)1 %G 0.244 y = -2.264x + 49.25 SPT-(N60)1 %C 0.202 y = 1.252x - 3.663 SPT-(N60)1 %M 0.293 y = -3.574x + 174.5 SPT-(N60)1 %S 0.009 y = 0.339x + 24.12 SPT-(N60)1 PL 0.050 y = 1.776x + 0.380 SPT-(N60)1 LL 0.003 y = 0.102x + 29.42 SPT-(N60)1 γd - UC 0.078 y = -0.590x + 103 SPT-(N60)1 w - UC 0.083 y = 2.910x - 6.184 SPT-(N60)1 wf - CU 0.123 y = 2.365x - 5.638 SPT-(N60)1 qu 0.002 y = -0.064x + 34.66 SPT-(N60)1 t50 0.024 y = 0.368x + 29.56 SPT-(N60)1 φ 0.091 y = 0.927x + 13.69 SPT-(N60)1 φ’ 0.037 y = 1.373x - 13.70 SPT-(N60)1 %Compact 0.078 y = -0.652x + 103.3 SPT-(N60)1 γd - CU 0.069 y = -0.680x + 115.7

Table 4.31: Single Variable Nonlinear Regression for A-6a for Corrected SPT Value Y X R2 Equation Function SPT-(N60)1 t50 0.845 xy = 35.80x - 14.37 Hyperbolic 2 SPT-(N60)1 γd - UC 0.584 y = -0.268x + 63.02x - 3661. Polynomial 2 SPT-(N60)1 %Cpt 0.583 y = -0.326x + 69.68x - 3680. Polynomial 2 SPT-(N60)1 %G 0.522 y = 0.724x - 14.97x + 97.85 Polynomial

Table 4.32: Single Variable Linear Regression for A-6a for Unconfined Compression Strength Y X R2 Equation qu PI 0.035 y = -1.078x + 50.77 qu Gs 0.013 y = 33.70x - 54.44 qu %G 0.075 y = -1.033x + 44.95 qu %C 0.095 y = 0.705x + 16.95 qu %M 0.451 y = -3.637x + 182.0 qu %S 0.030 y = 0.499x + 25.22 qu PL 0.285 y = -3.464x + 99.41 qu LL 0.027 y = 0.253x + 30.16 qu γd - UC 0.175 y = 0.727x - 49.89 qu w - UC 0.008 y = 0.740x + 27.41 qu wf - CU 0.331 y = -3.176x + 88.10 qu t50 0.046 y = 0.414x + 34.15 qu φ 0.188 y = 1.093x + 15.30 qu φ’ 0.042 y = -1.193x + 77.17 qu %Compact 0.173 y = 0.797x - 49.65 qu γd - CU 0.302 y = 1.160x - 105.2

Table 4.33: Single Variable Nonlinear Regression for A-6a for Unconfined Compression Strength Y X R2 Equation Function qu t50 0.890 xy = 39.27x - 2.136 Hyperbolic qu φ 0.548 xy = 55.63x - 348.9 Hyperbolic

Table 4.34: Single Variable Linear Regression for A-6a for Cohesion Y X R2 Equation c PI 0.195 y = 0.743x + 2.804 c Gs 0.618 y = -78x + 223.4 c %G 0.003 y = 0.107x + 11.23 c %C 0.557 y = -0.668x + 32.32 c %M 0.401 y = 1.160x - 33.15 c %S 0.577 y = 1.258x - 19.07 c PL 0.513 y = -1.653x + 40.31 c LL 0.009 y = -0.187x + 17.39 c γd - UC 0.015 y = 0.050x + 5.872 c w - UC 0.239 y = -0.935x + 24.11 c wf - CU 0.002 y = 0.070x + 10.74 c qu 0.166 y = -0.102x + 16.1 c t50 0.060 y = -0.188x + 13.33 c φ 0.315 y = -0.748x + 28.16 c φ’ 0.621 y = 1.821x - 49.05 c %Compact 0.016 y = 0.056x + 5.803 c γd - CU 0.001 y = 0.026x + 8.562

Table 4.35: Single Variable Nonlinear Regression for A-6a for Cohesion Y X R2 Equation Function c φ’ 0.860 y = -1.258x2 + 84.78x - 1414. Polynomial c Gs 0.823 y = -1846.x2 + 9975.x - 13459 Polynomial c t50 0.758 xy = 10.883x + 5.3591 Hyperbolic c %S 0.748 xy = 43.556x - 776.06 Hyperbolic c w - UC 0.736 y = 1.251x2 - 34.37x + 245.0 Polynomial c φ’ 0.709 y = 5E-09x6.162 Power c φ’ 0.698 y = 0.023e0.185x Exponential c φ’ 0.692 xy = 70.026x - 1942.3 Hyperbolic c Gs 0.688 y = 2E+10e-7.91x Exponential c Gs 0.684 y = 2E+10x-21.4 Power c φ’ 0.642 y = -2006/x + 71.935 Reciprocal c φ’ 0.631 y = 60.49ln(x) - 200.4 Logarithmic 2 c qu 0.616 y = -0.016x + 1.278x - 10.90 Polynomial c Gs 0.615 y = -211.ln(x) + 223.1 Logarithmic c Gs 0.612 y = 574.86/x - 200.12 Reciprocal c Gs 0.595 xy = -202.34x + 580.89 Hyperbolic c PI 0.590 xy = 21.025x - 109.74 Hyperbolic c %S 0.579 y = -776.97/x + 43.593 Reciprocal c %S 0.578 y = -0.059x2 + 4.221x - 55.81 Polynomial c %S 0.578 y = 31.30ln(x) - 88.31 Logarithmic c %C 0.572 y = -0.055x2 + 2.540x - 13.54 Polynomial c %C 0.551 y = -19.1ln(x) + 77.23 Logarithmic c %C 0.544 y = 543.31/x - 6.0759 Reciprocal c %M 0.522 xy = 56.207x - 1717.6 Hyperbolic c PI 0.516 y = -0.983x2 + 26.07x - 157.3 Polynomial c PL 0.516 y = -0.173x2 + 4.076x - 6.73 Polynomial c LL 0.516 y = 1.398x2 - 84.63x + 1288. Polynomial c φ 0.511 y = -0.452x2 + 18.54x - 175.6 Polynomial c PL 0.511 y = -27.2ln(x) + 89.20 Logarithmic c PL 0.510 y = 445.98/x - 14.185 Reciprocal c %S 0.504 y = 0.001x2.811 Power c %S 0.501 y = 0.718e0.112x Exponential

Table 4.36: Single Variable Linear Regression for A-6a for Effective Cohesion Y X R2 Equation c’ PI 0.540 y = 0.911x - 7.525 c’ Gs 0.881 y = -68.14x + 188.4 c’ %G 0.005 y = -0.14x + 4.185 c’ %C 0.834 y = -1.601x + 54.66 c’ %M 0.929 y = 1.38x - 49.71 c’ %S 0.040 y = -0.351x + 11.85 c’ PL 0.540 y = -3.646x + 68.14 c’ LL 0.540 y = 1.215x - 32.74 c’ γd - UC 0.749 y = -0.352x + 44.37 c’ w - UC 0.068 y = 0.445x - 2.605 c’ wf - CU 0.632 y = 1.096x - 14.78 c’ qu 0.511 y = -0.135x + 8.749 c’ t50 0.056 y = 0.137x + 2.274 c’ φ 0.778 y = -0.901x + 23.37 c’ φ’ 0.283 y = 0.887x - 26.18 c’ %Compact 0.748 y = -0.389x + 44.56 c’ γd - CU 0.759 y = -0.686x + 87.57

Table 4.37: Single Variable Nonlinear Regression for A-6a for Effective Cohesion Y X R2 Equation Function c’ w - UC 1.000 y = 1.819x2 - 49.27x + 334.1 Polynomial 2 c’ t50 0.979 y = 0.165x - 2.701x + 12.15 Polynomial c’ %C 0.977 y = -0.936x2 + 57.40x - 873.1 Polynomial c’ φ 0.965 y = -0.416x2 + 16.73x - 160.9 Polynomial c’ Gs 0.951 y = 4E+30x-69.5 Power c’ Gs 0.950 y = 4E+30e-25.5x Exponential c’ Gs 0.948 y = 785.2x2 - 4342x + 6003. Polynomial c’ %M 0.935 xy = 56.54x - 2042. Hyperbolic c’ %G 0.934 y = -2.07x2 + 22.63x - 55.84 Polynomial c’ %M 0.929 y = 53.10ln(x) - 190.4 Logarithmic c’ %M 0.929 y = -2042/x + 56.54 Reciprocal c’ Gs 0.885 y = 505.7/x - 182.8 Reciprocal c’ %M 0.884 y = 6E-30x18.71 Power c’ %M 0.884 y = 2E-08e0.486x Exponential c’ %Cpt 0.883 y = -0.052x2 + 10.61x - 534.1 Polynomial 2 c’ γd - UC 0.883 y = -0.042x + 9.505x - 526.1 Polynomial c’ Gs 0.883 y = -185.ln(x) + 188.8 Logarithmic c’ Gs 0.881 xy = -182.0x + 503.3 Hyperbolic -32 c’ γd - CU 0.834 y = 2E+67x Power -0.26x c’ γd - CU 0.834 y = 2E+14e Exponential c’ rd-uc 0.830 y = 2E+07e-0.13x Exponential c’ %Cpt 0.829 y = 2E+07e-0.14x Exponential c’ %C 0.827 y = -50.1ln(x) + 177.2 Logarithmic -15.4 c’ γd - UC 0.819 y = 2E+32x Power c’ %C 0.819 y = 1570/x - 45.73 Reciprocal c’ %Cpt 0.818 y = 6E+31x-15.4 Power 0.448x c’ wf - CU 0.809 y = 0.001e Exponential 7.145 c’ wf - CU 0.807 y = 5E-09x Power 2 c’ γd - CU 0.797 y = 0.060x - 15.69x + 1011. Polynomial c’ %C 0.791 xy = -46.70x + 1601. Hyperbolic c’ γd - CU 0.766 y = 10455/x - 81.86 Reciprocal c’ γd - CU 0.763 y = -84.7ln(x) + 410.8 Logarithmic c’ φ 0.760 y = -18.7ln(x) + 61.39 Logarithmic c’ PI 0.754 xy = 16.17x - 150.4 Hyperbolic c’ γd - CU 0.752 xy = -81.16x + 10370 Hyperbolic c’ φ 0.741 y = 387.7/x - 14.27 Reciprocal c’ γd - UC 0.739 y = -40.5ln(x) + 196.1 Logarithmic c’ %Cpt 0.738 y = -40.7ln(x) + 193.1 Logarithmic c’ γd - UC 0.729 y = 4657/x - 36.70 Reciprocal c’ %Cpt 0.728 y = 4253/x - 36.88 Reciprocal 86

Table 4.38: (continued) Y X R2 Equation Function c’ γd - UC 0.715 xy = -37.83x + 4788. Hyperbolic c’ %Cpt 0.714 xy = -38.02x + 4373. Hyperbolic -0.05x c’ qu 0.701 y = 25.06e Exponential c’ %C 0.692 y = 6E+07e-0.52x Exponential c’ %C 0.683 y = 2E+25x-16.4 Power c’ wf - CU 0.669 xy = 20.55x - 281.9 Hyperbolic -2.22 c’ qu 0.657 y = 8482.x Power c’ φ 0.637 y = 0.086x2 - 1.688x + 7.350 Polynomial c’ φ 0.630 y = 1720.e-0.29x Exponential c’ φ 0.627 xy = -15.24x + 409.1 Hyperbolic c’ wf - CU 0.627 y = 17.43ln(x) - 45.48 Logarithmic c’ wf - CU 0.619 y = -275.1/x + 20.14 Reciprocal c’ LL 0.619 xy = 41.40x - 1128 Hyperbolic c’ φ 0.611 y = 4E+08x-6.07 Power c’ t50 0.603 xy = 6.497x - 23.64 Hyperbolic c’ %S 0.581 y = -1.823x2 + 88.99x - 1079. Polynomial c’ LL 0.540 y = 37.04ln(x) - 122.2 Logarithmic c’ LL 0.540 y = -1128/x + 41.40 Reciprocal c’ PL 0.540 y = -63.8ln(x) + 186.9 Logarithmic c’ PL 0.540 y = 1115/x - 59.49 Reciprocal c’ PI 0.540 y = 11.75ln(x) - 25.69 Logarithmic c’ PI 0.540 y = -150.4/x + 16.17 Reciprocal 2 c’ qu 0.522 y = -0.002x + 0.032x + 5.465 Polynomial

Table 4.39: Single Variable Linear Regression for A-6a for Internal Friction Angle Y X R2 Equation φ PI 0.451 y = -1.536x + 39.38 φ Gs 0.076 y = -32.43x + 108.2 φ %G 0.500 y = -1.055x + 27.94 φ %C 0.133 y = 0.332x + 10.51 φ %M 0.461 y = -1.462x + 78.28 φ %S 0.190 y = 0.491x + 8.235 φ PL 0.171 y = -1.067x + 39.19 φ LL 0.020 y = -0.087x + 22.46 φ γd - UC 0.046 y = -0.148x + 37.83 φ w - UC 0.001 y = -0.142x + 21.92 φ wf - CU 0.005 y = 0.168x + 17.33 φ qu 0.188 y = 0.172x + 13.60 φ t50 0.000 y = -0.016x + 20.15 φ φ’ 0.005 y = 0.171x + 14.29 φ %Compact 0.047 y = -0.165x + 38.03 φ γd - CU 0.175 y = 0.351x - 23.10

Table 4.40: Single Variable Nonlinear Regression for A-6a for Internal Friction Angle Y X R2 Equation Function φ t50 0.930 xy = 18.85x + 8.170 Hyperbolic φ qu 0.828 xy = 27.17x - 245.7 Hyperbolic φ %C 0.599 xy = 29.67x - 269.2 Hyperbolic φ %S 0.586 xy = 27.79x - 179.0 Hyperbolic φ %G 0.564 y = 31.40e-0.06x Exponential φ %G 0.542 y = -0.091x2 + 0.554x + 21.79 Polynomial φ Gs 0.534 y = -2778.x2 + 15169x - 20678 Polynomial φ PI 0.504 y = -0.555x2 + 12.98x - 53.48 Polynomial

Table 4.41: Single Variable Linear Regression for A-6a for Effective Internal Friction Angle Y X R2 Equation φ’ PI 0.000 y = -0.013x + 33.65 φ’ Gs 0.273 y = -26.55x + 105.6 φ’ %G 0.048 y = -0.142x + 34.54 φ’ %C 0.063 y = -0.099x + 36.32 φ’ %M 0.007 y = 0.079x + 30.32 φ’ %S 0.188 y = 0.212x + 28.38 φ’ PL 0.052 y = -0.254x + 38.05 φ’ LL 0.083 y = -0.075x + 35.58 φ’ γd - UC 0.026 y = -0.048x + 39.32 φ’ w - UC 0.020 y = 0.203x + 30.79 φ’ wf - CU 0.022 y = 0.142x + 31.20 φ’ qu 0.042 y = -0.035x + 34.79 φ’ t50 0.114 y = -0.112x + 34.30 φ’ φ 0.005 y = 0.032x + 32.83 φ’ %Compact 0.026 y = -0.053x + 39.30 φ’ γd - CU 0.006 y = 0.029x + 29.86

Table 4.42: Single Variable Nonlinear Regression for A-6a for Effective Internal Friction Angle Y X R2 Equation Function φ’ t50 0.992 xy = 30.37x + 19.34 Hyperbolic φ’ %G 0.979 xy = 31.86x + 10.93 Hyperbolic φ’ qu 0.960 xy = 31.00x + 87.93 Hyperbolic φ’ LL 0.945 xy = 32.21x + 31.35 Hyperbolic φ’ φ 0.935 xy = 33.28x + 4.509 Hyperbolic φ’ %S 0.927 xy = 38.13x - 108.5 Hyperbolic φ’ %C 0.881 xy = 31.19x + 63.35 Hyperbolic φ’ PI 0.857 xy = 33.11x + 4.525 Hyperbolic φ’ wf - CU 0.844 xy = 35.56x - 32.81 Hyperbolic φ’ w - UC 0.765 xy = 36.57x - 40.53 Hyperbolic φ’ PL 0.686 xy = 28.51x + 88.43 Hyperbolic

For the correlation of Corrected SPT value and unconfined compression strength, only half time of consolidation expresses distinct influence. However this factor is determined in the triaxial test, the practical operation in design program is insignificant. 89

Specific gravity is the chief factor for the cohesion of A-6a soils. Through the regression

analysis of effective cohesion, it indicates obvious correlation with component and

physical properties. The former one contains the content of silt, the content of clay and

the content of gravel, and the latter one includes specific gravity, initial moisture content

and initial dry unit weight. The function type to express the nonlinear correlation for

effective cohesion is polynomial. The results of regression for internal friction angle is

equal to that of unconfined compression strength, the value of half time of consolidation

cannot be present in design stage. The characters affecting the effective internal friction

angle are similar to those influencing the effective cohesion. The factors of component are gravel’s, clay’s and sand’s content. The elements of physical properties are liquid limit, plasticity index and moisture content. Hyperbolic is the unique function model to illustrate the nonlinear correlation of effective internal friction angle.

4.7.3 Single Variable Regression Analysis for A-6b Soils

From Table 4.42 to Table 4.53, the regression consequences represent the linear and nonlinear relationships among A-6b soil properties. The dependent variables contain courted SPT value, unconfined compression strength, cohesion, effective cohesion, internal friction angle, and effective internal friction angle. The independent variables are consisted of percentage of soil components and soil mechanical properties. 90

Table 4.43: Single Variable Linear Regression for A-6b for Corrected SPT Value

Y X R2 Equation SPT-(N60)1 PI 0.079 y = 1.547x + 1.939 SPT-(N60)1 Gs 0.206 y = -175.7x + 505.9 SPT-(N60)1 %G 0.556 y = 1.432x + 10.86 SPT-(N60)1 %C 0.109 y = 0.354x + 16.48 SPT-(N60)1 %M 0.172 y = -0.572x + 53.67 SPT-(N60)1 %S 0.010 y = -0.295x + 33.39 SPT-(N60)1 PL 0.463 y = -5.268x + 137.8 SPT-(N60)1 LL 0.123 y = -3.339x + 156.6 SPT-(N60)1 γd - UC 0.063 y = -0.355x + 71.26 SPT-(N60)1 w - UC 0.150 y = 1.430x + 6.494 SPT-(N60)1 wf - CU 8E-6 y = 0.009x + 28.81 SPT-(N60)1 qu 0.218 y = 0.231x + 21.48 SPT-(N60)1 t50 0.064 y = 0.084x + 26.00 SPT-(N60)1 φ 0.087 y = 0.766x + 17.23 SPT-(N60)1 φ’ 0.097 y = -1.795x + 83.92 SPT-(N60)1 %Compact 0.163 y = -0.673x + 100.4 SPT-(N60)1 γd - CU 0.044 y = -0.292x + 61.91

Table 4.44: Single Variable Nonlinear Regression for A-6b for Corrected SPT Value Y X R2 Equation Function SPT-(N60)1 t50 0.988 xy = 33.32x - 48.37 Hyperbolic SPT-(N60)1 qu 0.840 xy = 40.25x - 291.9 Hyperbolic SPT-(N60)1 %G 0.826 xy = 41.74x - 130.2 Hyperbolic 0.580 SPT-(N60)1 %G 0.653 y = 6.651x Power 2 SPT-(N60)1 PL 0.649 y = -3.889x + 153.7x - 1482. Polynomial 2 SPT-(N60)1 %G 0.630 y = -0.125x + 4.332x - 3.002 Polynomial 0.056x SPT-(N60)1 %G 0.612 y = 13.47e Exponential SPT-(N60)1 %C 0.587 xy = 45.36x - 553.9 Hyperbolic SPT-(N60)1 %G 0.586 y = 14.66ln(x) - 6.872 Logarithmic SPT-(N60)1 φ 0.561 xy = 46.45x - 258.2 Hyperbolic 2 SPT-(N60)1 %C 0.560 y = 0.232x - 17.10x + 327.4 Polynomial SPT-(N60)1 %G 0.533 y = -114.3/x + 40.23 Reciprocal 2 SPT-(N60)1 PI 0.502 y = 3.677x - 131.2x + 1189. Polynomial

Table 4.45: Single Variable Linear Regression for A-6b for Unconfined Compression Strength Y X R2 Equation qu PI 0.938 y = 10.75x - 155.8 qu Gs 0.930 y = -752.6x + 2074. qu %G 0.472 y = 2.660x - 1.029 qu %C 0.877 y = 2.026x - 39.19 qu %M 0.902 y = -2.638x + 146.0 qu %S 0.384 y = 3.573x - 19.14 qu PL 0.864 y = -14.50x + 332.1 qu LL 0.281 y = 10.16x - 355.2 qu γd - UC 0.692 y = -2.362x + 313.0 qu w - UC 0.689 y = 6.163x - 64.56 qu wf - CU 0.027 y = -1.165x + 54.70 qu t50 0.064 y = 0.084x + 26.00 qu φ 0.857 y = 4.841x - 41.83 qu φ’ 0.097 y = -1.795x + 83.92 qu %Compact 0.690 y = -2.593x + 312.4 qu γd - CU 0.057 y = 0.674x - 43.44

Table 4.46: Single Variable Nonlinear Regression for A-6b for Unconfined Compression Strength Y X R2 Equation Function 2 qu Gs 0.998 y = 42764x - 23217x + 31513 Polynomial 2 qu PL 0.997 y = 6.632x - 285.7x + 3095. Polynomial 2 qu PI 0.985 y = 2.472x - 78.53x + 643 Polynomial 2 qu φ 0.979 y = 0.639x - 15.77x + 115.7 Polynomial 2 qu %C 0.974 y = 0.217x - 14.33x + 252.1 Polynomial -56.6 qu Gs 0.965 y = 1E+26x Power -20.9x qu Gs 0.964 y = 1E+26e Exponential -3.03 qu %M 0.958 y = 3E+06x Power 2 qu %M 0.953 y = 0.269x - 24.88x + 593.0 Polynomial -0.07x qu %M 0.950 y = 689.6e Exponential qu PI 0.946 xy = 234.8x - 3515. Hyperbolic 0.293x qu PI 0.933 y = 0.168e Exponential qu Gs 0.931 y = -2037ln(x) + 2065. Logarithmic qu Gs 0.931 y = 5512/x - 1999. Reciprocal qu Gs 0.929 xy = -1997.x + 5507 Hyperbolic qu PI 0.925 y = 192.0ln(x) - 516.4 Logarithmic qu %M 0.924 y = 4410/x - 72.55 Reciprocal 5.242 qu PI 0.923 y = 9E-06x Power qu %M 0.914 y = -108.ln(x) + 439.2 Logarithmic qu t50 0.913 xy = 19.06x + 106.1 Hyperbolic qu PI 0.911 y = -3408/x + 228.6 Reciprocal qu %C 0.909 xy = 115.1x - 2785. Hyperbolic 0.056x qu %C 0.905 y = 3.901e Exponential qu φ 0.899 xy = 119.5x - 1273. Hyperbolic qu PL 0.890 y = 6119/x - 264.6 Reciprocal -8.19 qu PL 0.887 y = 2E+12x Power 2.039 qu %C 0.880 y = 0.020x Power qu PL 0.878 y = -298.ln(x) + 935.4 Logarithmic -0.39x qu PL 0.875 y = 10839e Exponential qu %M 0.864 xy = -70.26x + 4313. Hyperbolic qu %C 0.851 y = 73.41ln(x) - 227.4 Logarithmic 0.130x qu φ 0.848 y = 3.799e Exponential qu PL 0.848 xy = -256.0x + 5941. Hyperbolic qu %C 0.822 y = -2595/x + 109.6 Reciprocal qu φ 0.783 y = 71.99ln(x) - 162.2 Logarithmic 2 qu w - UC 0.778 y = -1.806x + 65.78x - 545.1 Polynomial 1.939 qu φ 0.774 y = 0.148x Power qu w - UC 0.772 xy = 136.2x - 1597. Hyperbolic 2.887 qu w - UC 0.759 y = 0.010x Power 93

Table 4.45: (continued) Y X R2 Equation Function 0.175x qu w - UC 0.749 y = 1.796e Exponential -7.81 qu γd - UC 0.738 y = 5E+17x Power -0.06x qu γd - UC 0.737 y = 78058e Exponential 830.9x qu %Cpt 0.737 y = 0.012e Exponential -7.80 qu %Cpt 0.736 y = 2E+17x Power qu w - UC 0.713 y = -1655/x + 139.9 Reciprocal qu w - UC 0.703 y = 101.7ln(x) - 246.8 Logarithmic 2 qu γd - UC 0.692 y = -0.003x - 1.617x + 269.3 Polynomial qu γd - UC 0.692 y = -276.ln(x) + 1354. Logarithmic qu γd - UC 0.691 y = 32394/x - 241.1 Reciprocal 2 qu %Cpt 0.690 y = -0.005x - 1.325x + 244.9 Polynomial qu %Cpt 0.689 y = -276.ln(x) + 1325. Logarithmic qu φ 0.685 y = -1000/x + 101.1 Reciprocal qu %G 0.658 xy = 65.87x - 363.5 Hyperbolic qu γd - UC 0.651 xy = -241.6x + 32452 Hyperbolic qu %Cpt 0.649 xy = -241.1x + 29450 Hyperbolic 2 qu %G 0.587 y = 0.316x - 4.639x + 33.86 Polynomial qu %S 0.538 xy = 82.03x - 681.1 Hyperbolic 0.076x qu %G 0.523 y = 10.87e Exponential

Table 4.47: Single Variable Linear Regression for A-6b for Cohesion Y X R2 Equation c PI 0.134 y = 0.742x - 4.230 c Gs 0.017 y = -19.4x + 61.52 c %G 0.269 y = -0.392x + 13.59 c %C 0.086 y = 0.121x + 4.562 c %M 0.030 y = -0.093x + 12.90 c %S 0.621 y = 1.070x - 6.581 c PL 0.039 y = 0.547x - 2.424 c LL 0.855 y = 3.37x - 120.3 c γd - UC 0.133 y = -0.223x + 35.47 c w - UC 0.021 y = 0.239x + 5.170 c wf - CU 0.871 y = -4.691x + 97.70 c qu 0.018 y = 0.023x + 8.072 c t50 0.889 y = -0.308x + 13.78 c φ 0.133 y = 0.361x + 3.203 c φ’ 0.035 y = -0.560x + 26.17 c %Compact 0.134 y = -0.247x + 35.65 c γd - CU 0.999 y = 0.718x - 73.21 94

Table 4.48: Single Variable Nonlinear Regression for A-6b for Cohesion Y X R2 Equation Function c PI 1.000 y = -2.351x2 + 85.94x - 768.7 Polynomial c %G 1.000 y = 0.225x2 - 5.468x + 37.43 Polynomial c %C 1.000 y = -0.142x2 + 10.96x - 190.8 Polynomial c %M 1.000 y = -0.906x2 + 73.94x - 1457 Polynomial c %S 1.000 y = -0.640x2 + 18.78x - 124.7 Polynomial c PL 1.000 y = 3.636x2 - 148.0x + 1509. Polynomial 2 c γd - UC 1.000 y = -0.217x + 50.92x - 2962. Polynomial c w - UC 1.000 y = -26.63x2 + 866.8x - 6910. Polynomial 2 c wf - CU 1.000 y = 51.97x - 1997.x + 19180 Polynomial 2 c qu 1.000 y = -2.907x + 236.8x - 3592. Polynomial 2 c t50 1.000 y = 0.095x - 4.043x + 38.54 Polynomial c φ 1.000 y = -0.566x2 + 19.19x - 146.0 Polynomial c φ’ 1.000 y = -21.13x2 + 1285.x - 19514 Polynomial c %Cpt 1.000 y = -0.261x2 + 55.59x - 2940. Polynomial 2 c γd - UC 1.000 y = -0.006x + 2.207x - 157.8 Polynomial c γd - UC 1.000 y = -9277/x + 90.17 Reciprocal c γd - UC 1.000 xy = 90.16x - 9276. Hyperbolic c γd - UC 0.999 y = 81.67ln(x) - 378.0 Logarithmic 10.58 c γd - UC 0.980 y = 1E-21x Power 0.092x c γd - UC 0.976 y = 0.000e Exponential -0.72 c t50 0.974 y = 52.14x Power c t50 0.957 y = 81.56/x + 1.555 Reciprocal -0.04x c t50 0.954 y = 15.76e Exponential -12.2 c wf - CU 0.942 y = 3E+16x Power -0.63x c wf - CU 0.942 y = 1E+06e Exponential c LL 0.930 y = 2E-27x17.47 Power c LL 0.930 y = 2E-07e0.459x Exponential c t50 0.920 y = -5.39ln(x) + 22.71 Logarithmic c t50 0.909 xy = 1.837x + 78.06 Hyperbolic c qu 0.887 xy = 10.01x - 29.28 Hyperbolic c wf - CU 0.873 y = 1723/x - 82.26 Reciprocal c wf - CU 0.872 y = -89.9ln(x) + 273.3 Logarithmic c LL 0.863 xy = 135.8x - 4862. Hyperbolic c wf - CU 0.86 xy = -82.13x + 1721. Hyperbolic c LL 0.855 y = 128.0ln(x) - 457.9 Logarithmic c LL 0.855 y = -4862/x + 135.8 Reciprocal c %S 0.778 xy = 21.94x - 181.8 Hyperbolic c %S 0.774 y = 0.028x2.129 Power c %S 0.736 y = 0.895e0.152x Exponential 95

Table 4.47: (continued) Y X R2 Equation Function c %S 0.699 y = -207.2/x + 23.73 Reciprocal c %S 0.662 y = 15.06ln(x) - 31.08 Logarithmic c φ 0.560 xy = 13.73x - 71.78 Hyperbolic c %C 0.543 xy = 12.73x - 127.9 Hyperbolic

Table 4.49: Single Variable Linear Regression for A-6b for Effective Cohesion Y X R2 Equation c’ PI 0.020 y = -0.247x + 8.905 c’ Gs 0.141 y = 47x - 122.9 c’ %G 0.765 y = -0.566x + 11.33 c’ %C 0.048 y = -0.077x + 7.297 c’ %M 0.113 y = 0.153x - 2.090 c’ %S 0.143 y = 0.440x - 1.829 c’ PL 0.434 y = 1.555x - 27.6 c’ LL 0.377 y = 1.917x - 68.96 c’ γd - UC 0.021 y = 0.076x - 4.525 c’ w - UC 0.132 y = -0.508x + 12.43 c’ wf - CU 0.400 y = -2.724x + 56.12 c’ qu 0.140 y = -0.057x + 6.473 c’ t50 0.427 y = -0.183x + 7.450 c’ φ 0.020 y = -0.122x + 6.458 c’ φ’ 0.104 y = 0.823x - 20.83 c’ %Compact 0.020 y = 0.082x - 4.382 c’ γd - CU 0.778 y = 0.543x - 57.55

Table 4.50: Single Variable Nonlinear Regression for A-6b for Effective Cohesion Y X R2 Equation Function 2 c’ γd - CU 1.000 y = 0.09x - 19.95x + 1106. Polynomial c’ %Cpt 1.000 y = -0.238x2 + 50.99x - 2717. Polynomial c’ φ’ 1.000 y = -17.45x2 + 1062.x - 16154 Polynomial c’ φ 1.000 y = -0.516x2 + 17.03x - 129.4 Polynomial 2 c’ t50 1.000 y = 0.186x - 7.470x + 55.74 Polynomial 2 c’ qu 1.000 y = -2.330x + 189.7x - 2880. Polynomial 2 c’ wf - CU 1.000 y = 96.29x - 3695.x + 35410 Polynomial c’ PL 1.000 y = 2.391x2 - 96.16x + 966.6 Polynomial c’ %M 1.000 y = -0.742x2 + 60.82x - 1206. Polynomial c’ %S 1.000 y = -0.825x2 + 23.26x - 154.1 Polynomial c’ %C 1.000 y = -0.124x2 + 9.403x - 163.5 Polynomial c’ %G 1.000 y = 0.109x2 - 3.030x + 22.90 Polynomial c’ PI 1.000 y = -2.144x2 + 77.43x - 688.1 Polynomial c’ %G 0.915 y = 59.72/x - 1.483 Reciprocal 0.121x c’ γd - CU 0.876 y = 3E-06e Exponential 13.78 c’ γd - CU 0.867 y = 2E-28x Power c’ %G 0.856 y = -6.17ln(x) + 19.32 Logarithmic c’ γd - CU 0.783 xy = 66.99x - 7128. Hyperbolic c’ γd - CU 0.766 y = 61.28ln(x) - 285.8 Logarithmic c’ γd - CU 0.754 y = -6906/x + 65.05 Reciprocal c’ %G 0.753 y = 69.59x-1.22 Power c’ %S 0.662 y = 15.06ln(x) - 31.08 Logarithmic c’ %G 0.646 y = 13.89e-0.11x Exponential -0.79 c’ t50 0.610 y = 28.54x Power c’ t50 0.559 y = 53.39/x - 0.260 Logarithmic -0.04x c’ t50 0.558 y = 7.497e Exponential -12.7 c’ wf - CU 0.532 y = 7E+16x Power -0.66x c’ wf - CU 0.531 y = 1E+06e Exponential c’ LL 0.507 y = 2E-28x17.86 Power c’ LL 0.507 y = 5E-08e0.470x Exponential

Table 4.51: Single Variable Linear Regression for A-6b for Internal Friction Angle Y X R2 Equation φ PI 0.919 y = 2.042x - 20.37 φ Gs 0.865 y = -145.9x + 411.4 φ %G 0.258 y = 0.416x + 10.32 φ %C 0.922 y = 0.419x + 0.812 φ %M 0.831 y = -0.514x + 37.77 φ %S 0.502 y = 0.874x + 3.030 φ PL 0.624 y = -2.391x + 64.86 φ LL 0.483 y = 2.55x - 82.05 φ γd - UC 0.902 y = -0.590x + 85.99 φ w - UC 0.868 y = 1.598x - 9.209 φ wf - CU 0.041 y = -0.271x + 20.53 φ qu 0.857 y = 0.177x + 9.598 φ t50 0.064 y = 0.084x + 26.00 φ φ’ 0.097 y = -1.795x + 83.92 φ %Compact 0.901 y = -0.649x + 85.98 φ γd - CU 0.099 y = 0.168x - 3.666

Table 4.52: Single Variable Nonlinear Regression for A-6b for Internal Friction Angle Y X R2 Equation Function φ qu 0.995 xy = 24.05x - 220.0 Hyperbolic φ %C 0.988 xy = 32.42x - 563.5 Hyperbolic φ t50 0.983 xy = 9.685x + 49.67 Hyperbolic φ PI 0.966 xy = 53.46x - 660.9 Hyperbolic φ w - UC 0.955 xy = 42.59x - 411.5 Hyperbolic φ %C 0.925 y = -0.007x2 + 0.981x - 9.239 Polynomial φ %C 0.925 y = 15.51ln(x) - 39.27 Logarithmic φ %C 0.924 y = -560.0/x + 32.31 Reciprocal φ PI 0.919 y = 0.002x2 + 1.941x - 19.47 Polynomial φ PL 0.919 y = 2.038x2 - 85.63x + 911.5 Polynomial φ PI 0.919 y = 36.75ln(x) - 89.67 Logarithmic φ PI 0.917 y = -658.0/x + 53.29 Reciprocal 2 φ γd - UC 0.910 y = 0.024x - 6.269x + 419.8 Polynomial φ φ’ 0.910 y = 0.029x2 - 7.044x + 427.8 Polynomial φ γd - UC 0.905 y = 8197/x - 53.36 Reciprocal φ φ’ 0.904 y = 7453/x - 53.37 Reciprocal φ γd - UC 0.903 y = -69.6ln(x) + 348.3 Logarithmic φ φ’ 0.903 y = -69.6ln(x) + 341.7 Logarithmic φ w - UC 0.883 y = 0.273x2 - 7.229x + 60.71 Polynomial φ %M 0.876 y = 0.052x2 - 4.798x + 123.4 Polynomial 2 φ t50 0.873 y = 0.015x - 1.042x + 28.10 Polynomial φ %C 0.871 y = 0.494x0.968 Power φ PI 0.870 y = 0.021x2.298 Power φ PI 0.868 y = 1.606e0.127x Exponential φ Gs 0.865 y = -145.9x + 411.4 Polynomial φ Gs 0.865 y = -394.ln(x) + 409.5 Logarithmic φ Gs 0.865 y = 1067/x - 377.8 Reciprocal φ %C 0.864 y = 6.048e0.026x Exponential φ w - UC 0.861 y = 25.52ln(x) - 54.15 Logarithmic φ %G 0.860 y = 0.170x2 - 3.329x + 27.32 Polynomial -4.35 φ γd - UC 0.856 y = 2E+10x Power φ φ’ 0.856 y = 1E+10x-4.35 Power -0.03x φ γd - UC 0.856 y = 1245.e Exponential φ φ’ 0.855 y = 1245.e-0.04x Exponential φ w - UC 0.853 y = -403.7/x + 42.08 Reciprocal φ Gs 0.852 xy = -377.8x + 1067. Hyperbolic φ qu 0.848 y = 6.504ln(x) - 6.355 Logarithmic φ %M 0.847 y = 849.5/x - 4.608 Reciprocal 99

Table 4.51: (continued) Y X R2 Equation Function φ %M 0.840 y = -21.0ln(x) + 94.49 Logarithmic φ γd - UC 0.836 xy = -53.16x + 8172. Hyperbolic φ φ’ 0.835 xy = -53.15x + 7429. Hyperbolic φ qu 0.831 y = -213.8/x + 23.82 Reciprocal φ %G 0.806 xy = 21.80x - 69.90 Hyperbolic φ w - UC 0.799 y = 3.286e0.098x Exponential φ %S 0.795 xy = 27.52x - 164.0 Hyperbolic φ w - UC 0.793 y = 0.205x1.574 Power φ Gs 0.790 y = 5E+11x-24.2 Power φ Gs 0.790 y = 6E+11e-8.96x Exponential 0.010x φ qu 0.783 y = 10.49e Exponential 0.399 φ qu 0.774 y = 3.942x Power φ %M 0.769 y = 1940.x-1.29 Power φ %M 0.761 y = 59.34e-0.03x Exponential φ PL 0.656 y = 1016/x - 33.89 Reciprocal φ PL 0.640 y = -49.3ln(x) + 164.8 Logarithmic φ PL 0.555 y = 11485x-2.95 Power φ LL 0.544 xy = 111.7x - 3679. Hyperbolic φ PL 0.539 y = 287.6e-0.14x Exponential φ %S 0.528 y = -0.395x2 + 11.99x - 71.78 Polynomial φ LL 0.512 y = 1E-09x6.409 Power φ LL 0.512 y = 0.023e0.168x Exponential φ %S 0.511 y = -166.5/x + 27.69 Reciprocal φ %S 0.509 y = 1.913x0.783 Power φ %S 0.506 y = 12.16ln(x) - 16.55 Logarithmic φ %S 0.503 y = 6.764e0.056x Exponential

100

Table 4.53: Single Variable Linear Regression for A-6b for Effective Internal Friction Angle Y X R2 Equation φ’ PI 0.451 y = -0.669x + 42.65 φ’ Gs 0.464 y = 49x - 102.1 φ’ %G 0.321 y = -0.207x + 33.32 φ’ %C 0.387 y = -0.126x + 35.30 φ’ %M 0.546 y = 0.191x + 22.58 φ’ %S 0.410 y = -0.377x + 36.28 φ’ PL 0.398 y = 0.857x + 13.11 φ’ LL 0.141 y = -0.675x + 56.70 φ’ γd - UC 0.289 y = 0.156x + 12.26 φ’ w - UC 0.338 y = -0.457x + 37.93 φ’ wf - CU 0.043 y = 0.151x + 28.02 φ’ qu 0.485 y = -0.061x + 32.90 φ’ t50 0.030 y = 0.261x + 24.43 φ’ φ 0.422 y = -0.333x + 36.14 φ’ %Compact 0.287 y = 0.171x + 12.31 φ’ γd - CU 0.000 y = 0.003x + 30.46

101

Table 4.54: Single Variable Nonlinear Regression for A-6b for Effective Internal Friction Angle

Y X R2 Equation Function φ’ t50 0.998 xy = 29.75x + 6.659 Hyperbolic φ’ qu 0.995 xy = 27.98x + 73.62 Hyperbolic φ’ %G 0.980 xy = 28.48x + 23.77 Hyperbolic φ’ %C 0.956 xy = 25.56x + 178.1 Hyperbolic φ’ %M 0.956 xy = 38.48x - 321.6 Hyperbolic φ’ φ 0.946 xy = 24.88x + 91.21 Hyperbolic φ’ %S 0.938 xy = 25.55x + 73.14 Hyperbolic φ’ %Cpt 0.938 xy = -15.44x + 2159. Hyperbolic φ’ wf - CU 0.873 xy = 34.89x - 74.34 Hyperbolic φ’ w - UC 0.847 xy = 22.83x + 121.9 Hyperbolic φ’ PL 0.823 xy = 47.87x - 350.8 Hyperbolic φ’ γd - UC 0.736 xy = 48.72x - 2124. Hyperbolic φ’ PI 0.675 xy = 18.31x + 219.3 Hyperbolic φ’ w - UC 0.621 y = -0.527x2 + 16.58x - 96.99 Polynomial φ’ %M 0.620 y = 0.030x2 - 2.281x + 72.00 Polynomial 2 φ’ qu 0.620 y = 0.019x - 1.649x + 57.00 Polynomial φ’ Gs 0.564 xy = 162.9x - 358.5 Hyperbolic φ’ %M 0.555 y = 23.53e0.006x Exponential φ’ %M 0.544 y = 12.01x0.250 Power φ’ %M 0.535 y = 7.706ln(x) + 1.923 Logarithmic φ’ qu 0.526 y = 77.90/x + 27.83 Reciprocal φ’ %M 0.525 y = -307.0/x + 38.14 Reciprocal -0.07 φ’ qu 0.512 y = 39.61x Power φ’ %S 0.509 y = -0.357x2 + 9.680x - 31.53 Polynomial φ’ qu 0.502 y = -2.29ln(x) + 38.57 Logarithmic

The results of regression for A-6b are ample. This phenomenon reveals the

experiment data derived from A-6b soil correspond to statistical distribution. Whereas,

for the Corrected SPT value, only content of gravel can provides valuable information.

Oppositely, many factors show apparent impaction to the unconfined compression

strength. They are plasticity index, specific gravity, plastic limit, and the content of clay and that of silt. Among the linear regression of cohesion, it is mainly determined by 102 liquid limit, initial dry unit weight and final moisture content. But in accordance with the outcomes of nonlinear regression for cohesion and effective cohesion, almost every independent variable demonstrates perfect correlation without any flaw. These comprise contents of each component (gravel, silt, clay, and sand), plastic limit, plasticity index, initial dry unit weight and moisture content. The notable nonlinear function model is polynomial. Through the linear and nonlinear regression, the outstanding independent variables in the equations showing internal friction angle possess content of clay, content of silt, specific gravity, plasticity index, initial dry unit weight and initial moisture content. Yet the important elements only appear in nonlinear regression for effective internal friction angle. They are contents of each component, specific gravity, plasticity index, plastic limit, initial dry unit weight and moisture content. To simulate the nonlinear correlation of internal friction angle and its effective value, the most efficient function type is hyperbolic.

4.7.4 Single Variable Regression Analysis for A-7-6 Soils

From Table 4.54 to Table 4.65, the regression results express the linear and nonlinear relationships among A-6b soil properties. The dependent variables contain courted SPT value, unconfined compression strength, cohesion, effective cohesion, internal friction angle, and effective internal friction angle. The independent variables are consisted of percentage of soil components and soil mechanical properties. 103

Table 4.55: Single Variable Linear Regression for A-7-6 for Corrected SPT Value Y X R2 Equation SPT-(N60)1 PI 0.090 y = -0.427x + 32.68 SPT-(N60)1 Gs 0.001 y = -17.59x + 68.43 SPT-(N60)1 %G 0.092 y = 0.714x + 18.62 SPT-(N60)1 %C 0.324 y = -0.634x + 54.38 SPT-(N60)1 %M 0.391 y = -0.353x + 35.96 SPT-(N60)1 %S 0.410 y = 0.741x + 12.77 SPT-(N60)1 PL 0.317 y = -2.793x + 81.23 SPT-(N60)1 LL 0.275 y = -0.624x + 52.00 SPT-(N60)1 γd - UC 0.472 y = 1.043x - 88.05 SPT-(N60)1 w - UC 0.424 y = -1.357x + 50.47 SPT-(N60)1 wf - CU 0.487 y = -1.974x + 67.50 SPT-(N60)1 qu 0.153 y = 0.319x + 12.11 SPT-(N60)1 t50 0.077 y = -0.095x + 24.74 SPT-(N60)1 φ 0.274 y = 1.778x - 1.941 SPT-(N60)1 φ’ 0.021 y = -0.571x + 36.65 SPT-(N60)1 %Compact 0.450 y = 1.114x - 84.95 SPT-(N60)1 γd - CU 0.628 y = 0.960x - 84.21

Table 4.56: Single Variable Nonlinear Regression for A-7-6 for Corrected SPT Value Y X R2 Equation Function SPT-(N60)1 %G 0.885 xy = 21.51x + 7.240 Hyperbolic SPT-(N60)1 %S 0.853 xy = 27.75x - 36.66 Hyperbolic SPT-(N60)1 qu 0.724 xy = 33.16x - 304.8 Hyperbolic SPT-(N60)1 γd - CU 0.704 xy = 125.1x - 11367 Hyperbolic 5.680 SPT-(N60)1 γd - CU 0.662 y = 5E-11x Power 0.051x SPT-(N60)1 γd - CU 0.653 y = 0.067e Exponential 2 SPT-(N60)1 γd - CU 0.652 y = -0.026x + 6.869x - 407 Polynomial SPT-(N60)1 γd - CU 0.640 y = -11547/x + 126.8 Reciprocal SPT-(N60)1 γd - CU 0.635 y = 105.5ln(x) - 474.5 Logarithmic 2 SPT-(N60)1 %G 0.603 y = -0.630x + 7.197x + 11.96 Polynomial SPT-(N60)1 φ 0.595 xy = 45.07x - 299.8 Hyperbolic 0.370 SPT-(N60)1 %S 0.552 y = 8.858x Power SPT-(N60)1 γd - UC 0.545 xy = 128.3x - 11190 Hyperbolic SPT-(N60)1 %Cpt 0.524 xy = 125.0x - 9864. Hyperbolic SPT-(N60)1 %S 0.522 y = -37.12/x + 27.80 Reciprocal -0.10x SPT-(N60)1 wf - CU 0.512 y = 237.4e Exponential 2 SPT-(N60)1 γd - UC 0.501 y = -0.061x + 13.91x - 756.8 Polynomial

104

Table 4.57: Single Variable Linear regression for A-7-6 for Unconfined Compression Strength Y X R2 Equation qu PI 0.317 y = -0.979x + 54.53 qu Gs 0.070 y = -129.5x + 377.1 qu %G 0.000 y = 0.087x + 27.48 qu %C 0.319 y = -0.770x + 68.30 qu %M 0.407 y = -0.441x + 46.46 qu %S 0.458 y = 0.959x + 17.14 qu PL 0.034 y = -1.126x + 52.07 qu LL 0.347 y = -0.858x + 70.40 qu γd - UC 0.246 y = 0.921x - 68.56 qu w - UC 0.157 y = -1.012x + 49.75 qu wf - CU 0.167 y = -1.415x + 61.10 qu t50 0.016 y = -0.054x + 29.91 qu φ 0.408 y = 2.652x - 6.428 qu φ’ 0.016 y = -0.614x + 44.60 qu %Compact 0.252 y = 1.019x - 69.18 qu γd - CU 0.315 y = 0.831x - 63.31

Table 4.58: Single Variable Nonlinear Regression for A-7-6 for Unconfined Compression Strength Y X R2 Equation Function qu %S 0.864 xy = 39.35x - 78.89 Hyperbolic qu %G 0.835 xy = 26.49x + 5.36 Hyperbolic qu φ 0.699 xy = 66.62x - 485.0 Hyperbolic qu t50 0.635 xy = 20.31x + 260.0 Hyperbolic 0.034x qu %S 0.500 y = 17.80e Exponential

105

Table 4.59: Single Variable Linear Regression for A-7-6 for Cohesion Y X R2 Equation c PI 0.033 y = -0.198x + 11.05 c Gs 0.004 y = 13.29x - 30.10 c %G 0.094 y = 0.325x + 4.577 c %C 0.166 y = -0.269x + 19.47 c %M 0.234 y = -0.151x + 11.95 c %S 0.075 y = 0.149x + 4.012 c PL 0.479 y = -1.946x + 46.75 c LL 0.278 y = -0.457x + 28.09 c γd - UC 0.432 y = 0.549x - 51.79 c w - UC 0.330 y = -0.654x + 20.01 c wf - CU 0.336 y = -0.905x + 27.29 c qu 0.032 y = -0.094x + 8.275 c t50 0.158 y = -0.102x + 9.856 c φ 0.014 y = 0.201x + 3.198 c φ’ 0.018 y = -0.530x + 20.4 c %Compact 0.435 y = 0.605x - 51.85 c γd - CU 0.262 y = 0.317x - 28.87

Table 4.60: Single Variable Nonlinear Regression for A-7-6 for Cohesion YX R2 Equation Function c φ 0.895 y = -1.256x2 + 34.87x - 226.9 Polynomial c %G 0.827 xy = 6.293x + 2.951 Hyperbolic c %G 0.778 y = -8.495/x + 8.929 Reciprocal c PL 0.638 y = 1.405x2 - 62.17x + 688.8 Polynomial c %G 0.544 y = -0.291x2 + 3.412x + 1.539 Polynomial c %S 0.536 y = -0.059x2 + 1.564x - 0.971 Polynomial

106

Table 4.61: Single Variable Linear Regression for A-7-6 for Effective Cohesion Y X R2 Equation c’ PI 0.122 y = -0.178x + 8.150 c’ Gs 0.014 y = 10.81x - 25.88 c’ %G 0.002 y = -0.025x + 3.933 c’ %C 0.689 y = -0.281x + 17.99 c’ %M 0.567 y = -0.110x + 8.000 c’ %S 0.781 y = 0.286x + 0.557 c’ PL 0.640 y = -1.004x + 24.44 c’ LL 0.693 y = -0.345x + 20.43 c’ γd - UC 0.602 y = 0.289x - 27.02 c’ w - UC 0.434 y = -0.334x + 10.56 c’ wf - CU 0.753 y = -0.635x + 18.62 c’ qu 0.251 y = 0.242x - 2.368 c’ t50 0.200 y = -0.051x + 5.320 c’ φ 0.754 y = 1.037x - 9.051 c’ φ’ 0.091 y = -0.554x + 18.66 c’ %Compact 0.601 y = 0.317x - 26.94 c’ γd - CU 0.731 y = 0.256x - 24.44

107

Table 4.62: Single Variable Nonlinear Regression for A-7-6 for Effective Cohesion Y X R2 Equation Function -0.22x c’ wf - CU 0.899 y = 628.5e Exponential -5.48 c’ wf - CU 0.897 y = 1E+08x Power c’ φ 0.890 xy = 17.73x - 168.9 Hyperbolic c’ φ 0.882 y = -0.597x2 + 16.63x - 108.4 Polynomial 2 c’ qu 0.876 y = 0.145x - 6.767x + 79.38 Polynomial 9.810 c’ γd - CU 0.859 y = 3E-20x Power c’ %S 0.853 y = 1.058e0.097x Exponential c’ %S 0.851 y = 0.707x0.687 Power 0.090x c’ γd - CU 0.850 y = 0.000e Exponential c’ %C 0.837 y = 5E+09x-5.39 Power c’ %S 0.834 xy = 6.138x - 16.21 Hyperbolic c’ %C 0.830 y = 515.5e-0.10x Exponential c’ %S 0.794 y = 0.008x2 + 0.117x + 1.017 Polynomial c’ PL 0.792 y = 5E+10x-7.77 Power c’ PL 0.791 y = 5707.e-0.36x Exponential c’ φ 0.790 y = -175.8/x + 18.31 Reciprocal c’ w - UC 0.779 y = -0.164x2 + 6.952x - 67.80 Polynomial c’ %M 0.774 y = 0.052x2 - 5.847x + 165.1 Polynomial c’ %C 0.774 y = 0.052x2 - 5.847x + 165.1 Polynomial 2 c’ γd - CU 0.772 y = -0.013x + 3.114x - 178.7 Polynomial c’ φ 0.772 y = 13.57ln(x) - 30.20 Logarithmic 11.02 c’ γd - UC 0.767 y = 1E-22x Power c’ %Cpt 0.766 y = 5E-22x11.00 Power

c’ γd - CU 0.764 xy = 30.93x - 2974. Hyperbolic 0.106x c’ γd - UC 0.763 y = 4E-05e Exponential c’ %G 0.763 xy = 2.828x + 4.745 Hyperbolic c’ %Cpt 0.762 y = 4E-05e0.116x Exponential 2 c’ wf - CU 0.756 y = 0.026x - 1.920x + 34.12 Polynomial c’ wf - CU 0.755 y = -15.4ln(x) + 52.35 Logarithmic c’ wf - CU 0.754 y = 372.7/x - 12.35 Reciprocal c’ γd - CU 0.743 y = -3022/x + 31.38 Reciprocal c’ γd - CU 0.737 y = 27.90ln(x) - 127.2 Logarithmic c’ %M 0.735 y = 15.40e-0.04x Exponential c’ %S 0.721 y = 1.942ln(x) - 0.475 Logarithmic c’ LL 0.712 y = 0.013x2 - 1.624x + 51.64 Polynomial c’ %Cpt 0.709 y = -0.104x2 + 20.08x - 955.4 Polynomial 2 c’ γd - UC 0.709 y = -0.088x + 18.59x - 972.8 Polynomial c’ LL 0.709 y = 830.7/x - 13.62 Reciprocal c’ wf - CU 0.706 xy = -12.37x + 373.2 Hyperbolic 108

Table 4.61: (continued) Y X R2 Equation Function c’ %C 0.706 y = 805.6/x - 12.33 Reciprocal c’ LL 0.704 y = 749.7e-0.11x Exponential c’ LL 0.703 y = 7E+09x-5.55 Power c’ LL 0.702 y = -17.0ln(x) + 69.67 Logarithmic c’ %C 0.698 y = -15.1ln(x) + 62.93 Logarithmic c’ %M 0.697 y = 1664.x-1.73 Power c’ Gs 0.672 y = -4896.x2 + 26553x - 35994 Polynomial c’ %C 0.650 xy = -11.97x + 787.2 Hyperbolic c’ PL 0.645 y = 0.186x2 - 8.992x + 109.5 Polynomial c’ PL 0.642 y = 458.4/x - 18.59 Reciprocal c’ PL 0.641 y = -21.4ln(x) + 68.70 Logarithmic c’ w - UC 0.636 y = 47.21e-0.13x Exponential c’ %S 0.634 y = -9.023/x + 5.192 Reciprocal c’ γd - UC 0.623 xy = 32.87x - 3085. Hyperbolic c’ %Cpt 0.622 xy = 32.78x - 2798 Hyperbolic c’ φ 0.615 y = 0.000x3.947 Power c’ γd - UC 0.609 y = -3119/x + 33.18 Reciprocal c’ PL 0.609 xy = -18.53x + 457.2 Hyperbolic c’ %Cpt 0.608 y = -2828/x + 33.11 Reciprocal c’ γd - UC 0.605 y = 30.08ln(x) - 136.5 Logarithmic c’ %Cpt 0.604 y = 30.00ln(x) - 133.3 Logarithmic c’ w - UC 0.602 y = 14848x-2.81 Power c’ LL 0.597 xy = -13.37x + 818.5 Hyperbolic c’ φ 0.593 y = 0.075e0.299x Exponential c’ %M 0.527 y = -4.63ln(x) + 20.45 Logarithmic c’ φ’ 0.526 y = -1.215x2 + 67.78x - 939.8 Polynomial c’ qu 0.508 xy = 10.28x - 159.3 Hyperbolic

109

Table 4.63: Single Variable Linear Regression for A-7-6 for Internal Friction Angle Y X R2 Equation φ PI 0.141 y = -0.157x + 17.19 φ Gs 0.011 y = -12.44x + 46.44 φ %G 0.056 y = -0.163x + 13.43 φ %C 0.223 y = -0.155x + 21.06 φ %M 0.163 y = -0.067x + 15.74 φ %S 0.480 y = 0.236x + 10.27 φ PL 0.059 y = -0.357x + 20.59 φ LL 0.237 y = -0.170x + 21.38 φ γd - UC 0.088 y = 0.133x - 1.032 φ w - UC 0.031 y = -0.108x + 15.26 φ wf - CU 0.302 y = -0.458x + 23.69 φ qu 0.408 y = 0.153x + 8.620 φ t50 0.011 y = 0.010x + 12.47 φ φ’ 0.031 y = -0.204x + 18.48 φ %Compact 0.085 y = 0.142x - 0.695 φ γd - CU 0.266 y = 0.184x - 7.293

Table 4.64: Single Variable Nonlinear Regression for A-7-6 for Internal Friction Angle Y X R2 Equation Function φ %G 0.972 xy = 11.20x + 3.578 Hyperbolic φ %S 0.935 xy = 16.39x - 26.58 Hyperbolic φ qu 0.901 xy = 18.21x - 131.7 Hyperbolic φ t50 0.877 xy = 12.24x + 31.71 Hyperbolic φ %M 0.720 xy = 9.400x + 133.5 Hyperbolic φ %S 0.583 y = 0.017x2 - 0.170x + 11.70 Polynomial

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Table 4.65: Single Variable Linear Regression for A-7-6 for Effective Internal Friction Angle Y X R2 Equation φ’ PI 0.059 y = 0.088x + 24.96 φ’ Gs 0.002 y = 5.037x + 13.80 φ’ %G 0.003 y = 0.034x + 27.26 φ’ %C 0.017 y = 0.037x + 25.41 φ’ %M 0.011 y = 0.015x + 26.73 φ’ %S 0.010 y = -0.029x + 27.71 φ’ PL 0.016 y = 0.161x + 23.90 φ’ LL 0.040 y = 0.061x + 24.31 φ’ γd - UC 0.008 y = -0.035x + 31.09 φ’ w - UC 0.005 y = 0.040x + 26.50 φ’ wf - CU 0.035 y = 0.135x + 24.18 φ’ qu 0.016 y = -0.027x + 28.13 φ’ t50 0.054 y = -0.020x + 28.18 φ’ φ 0.031 y = -0.154x + 29.37 φ’ %Compact 0.009 y = -0.041x + 31.29 φ’ γd - CU 0.049 y = -0.069x + 34.94

Table 4.66: Single Variable Nonlinear Regression for A-7-6 for Effective Internal Friction Angle Y X R2 Equation Function φ’ t50 0.994 xy = 26.14x + 36.55 Hyperbolic φ’ %S 0.991 xy = 26.91x + 3.683 Hyperbolic φ’ %G 0.989 xy = 27.72x - 0.708 Hyperbolic φ’ qu 0.971 xy = 26.44x + 23.32 Hyperbolic φ’ %M 0.930 xy = 28.24x - 33.18 Hyperbolic φ’ φ 0.894 xy = 26.12x + 15.28 Hyperbolic φ’ PI 0.876 xy = 30.24x - 75.15 Hyperbolic φ’ LL 0.779 xy = 30.89x - 171.4 Hyperbolic φ’ %C 0.767 xy = 29.48x - 108.3 Hyperbolic φ’ wf - CU 0.736 xy = 32.16x - 111.3 Hyperbolic φ’ PL 0.547 xy = 31.33x - 84.79 Hyperbolic

None element display strong correlation to Corrected SPT value in linear regression, and only content of gravel and that of sand appear in the nonlinear regression consequence. All of them exist in the hyperbolic model. The homogenous situation 111

occurs in the regression analysis for unconfined compression strength. For the cohesion and effective cohesion, linear regression cannot explore the satisfying independent

variables to build reliable correlations. But some valuable factors are found in the

nonlinear regression. The content of gravel achieves well connection to the cohesion in

hyperbolic function type. And the effective cohesion acquires good relationship with the

content of clay, the content of sand, initial dry unit weight and final moisture content.

Linear regression is also not the effective selection for the internal friction angle and

effective internal friction angle. The content of gravel and that of sand exhibit in the

outcomes of nonlinear regression for these two dependent variables. Furthermore, the

results of effective internal friction angle contain two more valid elements, which are the

content of silt and plasticity index. Hyperbolic is the sole equation pattern to discover

those nonlinear correlations.

From the results in this section, single variable regression analysis is not suitable

to establish satisfying relationships between soil properties, especially to corrected SPT-

N value and unconfined compression strength. Among independent variables, the

percentages of components in soil notably influence each dependent variable. The

Atterberg limits and soil moisture content possess distinct effects on cohesion, effective

cohesion, internal friction angle and effective internal fiction angle. The efficient

equation types for cohesion and internal friction angle are respectively second degree

polynomial and hyperbolic functions.

112

4.8 Multiple Variables Linear Regression Analysis (Part I)

The soil structure is complicated combination, which indicates that the each mechanical property is impossible to connect with unique factor. Those equations containing several variables are more convincing to express the comprehensive correlations. However, there are twelve independent variables, so the number of

12 combination can reach ∑12Ci , whose result is 4083. For the six dependent variables, i=2 there are 24498 equations. It is redundant and unrealistic to measure the coefficient of determination R2 and then decide the equations in strong correlation. To prevent this complex problem, the approach is the acceptance and rejection of the independent variables. The criterion to adopt the independent variable is the coefficient of them in single variable regression analysis. The coefficient represents weight. The greater of its coefficient number, the independent variable possess more weight to influence the dependent variable. The results displayed in the following sections are the equations experiencing acceptance and rejection of independent variables. Furthermore, the possible function of nonlinear regression is countless. Therefore, for the multiple variables, only linear regression is in analysis.

The progress of multiple variables regression relies on SPSS software. SPSS is the abbreviation of Statistical Product and Service Solution, which can provide powerful statistical analysis. SPSS possess three methods to manipulate regression for a block of data. They are forward, backward and stepwise. Forward is the entry of variable. For the variable not in the equation, the one with smallest F value will be entered if the F value is smaller than PIN value. Backward is the elimination of variable. At each step, the 113

variable with largest F value will be removed from equation if the F value is larger than

POUT value. Stepwise is the selection of the variable. First, it processes backward

method until no independent variable can be removed. Then, it operates forward method

until none of independent variable can be added. This approach sustains until none of

independent variable satisfying the criteria of elimination and entrance. F is the value of

probability, and PIN and POUT are the limits of probability to determine the entry and

elimination.

Since the forward method picks up the independent variable for the equation at

the first step, the existence of none independent variable meeting the requirements is

possible. However, the backward and stepwise methods initially consider whole

independent variables, it is impossible to generate null results. Furthermore, the

calculation of stepwise method is more comprehensive. Hence, the statistical methods

for multiple variables linear regression are backward and stepwise. The default value for

PIN and POUT are both 0.05. The calculations conducted in this section adopt these

default values except specific declaration.

The multiple variables linear regression analysis will cover every soil type in

Ohio embankment construction except A-4b soils, because the sufficiency of effective

data for this soil type is not available. For the other soil types illustrated in the following

sections, the results of regression analysis only display the equations with the coefficient

of determination R2 greater than 0.5. For the results of same dependent variables listed below, the upper one is derived from backward method and the lower one is calculated through the stepwise method. 114

Table 4.67: Multiple Variables Linear Regression for A-4a Soils Y X R2 Equation (N ) = -2168.608+960.817(Gs)+15.822(%G) Gs, %G, %C, 60 (N ) 1.000 +16.132(%C)+6.539(%S) +5.813(PL)- 60 1 %S, PL, %Cpt 12.229(%Cpt) (N ) = 1370.435+28.454(PI)+129.616(Gs) - PI,Gs, %C, 60 (N60)1 1.000 13.655(%C)-20.890(%M) -22.391(%S)- %M, %S, γd 13.633(γd) Gs, %G, %C, q = -638.239+212.659(Gs)+4.197(%G) q 0.970 u u %S, %Cpt +10.411(%C)+6.955(%S)-3.973(%Cpt) qu %C, %S, LL 0.953 qu = -332.785+5.208(%C)+7.306(%S)+1.530(LL) c %C, %S, %Cpt 1.000 c = 62.494-1.496(%C)-1.1(%S)+0.207(%Cpt) c %C 0.949 c = 55.380-1.468(%C) c’ = -12.544+0.481(%C) +2.837(%S)- c’ %C, %S, %Cpt 1.000 0.660(%Cpt) c’ %S, %G, LL 1.000 c’ = -110.941+2.106(%S)+1.030(%G)+2.128(LL) φ Gs, %G 0.615 φ = -301.472+119.212(Gs)+0.687(%G) φ %S, w 0.585 φ = 104.009-1.959(%S)-2.415(w) φ’ = -57.281+32.890(Gs) +1.878(%S)- φ’ Gs, %S, %Cpt 0.809 0.443(%Cpt) φ’ Gs, %S, γd 0.810 φ’ = -57.709+33.074(Gs) +1.873(%S)-0.369(γd)

For the multiple variables linear regression in Table 4.66, the components of the

soil exert great influence to the A-4a soil mechanical properties, especially to the

Corrected SPT value and unconfined compression strength. To the internal friction angle, although the components appear in the block of independent variables, their coefficients are lack of weight to affect the dependent variable. The main factor is

specific gravity. In stepwise analysis, the PIN value is 0.42 for effective cohesion and

0.22 for Corrected SPT value.

115

Table 4.68: Multiple Variables Linear Regression for A-6a Soils Y X R2 Equation PI, %G, %M, (N ) = 84.221+12.917(PI)-7.897(%G)-7.592(%M) (N ) PL, LL, w, 1.000 60 60 1 +11.863(PL)-2.674(LL)-5.753(w)+0.774(%Cpt) %Cpt PI, Gs, %M, (N60) = 2107.777+0.097(PI)-857.641(Gs) - (N60)1 PL, LL, w, 1.000 9.418(%M)+18.956(PL) +1.247(LL)- %Cpt 1.320(w)+2.508(%Cpt) PI, %G, %M, q = -93.476-7.893(PI)+2.075(%G)+0.850(%M) - q PL, LL, w 1.000 u u 5.579(PL)+1.777(LL)+7.422(w)+1.224(%Cpt) %Cpt PI, Gs, %S, PL, q = -388.124-3.611(PI)+168.105(Gs)-1.020(%S) - q 1.000 u u LL, w, %Cpt 7.417(PL)+0.228(LL)+5.495(w)+0.847(%Cpt) PI, %G, γ , c = 9.948+1.918(PI) -1.041(%G)- c d 1.000 %Cpt 1.949(γd)+0.095(%Cpt) c = 232.891-81.412(Gs) +0.727(%S)- c Gs, %S, LL, w 1.000 0.633(LL)+0.037(w) c’ %S, w, %Cpt 1.000 c’ = 34.361+0.255(%S)+0.888(w)-0.464(%Cpt) c’ %M 0.930 c’ = -49.715+1.380(%M) PI, %G, %M, φ = 5.645-2.061(PI)-0.579(%G)+0.461(%M) - φ PL, LL, w, 0.642 0.586(PL)+0.242(LL) +0.934(w)+0.163(%Cpt) %Cpt φ PI, %G 0.660 φ = 38.582-1.021(PI)-0.760(%G)

For the results of A-6a soils listed in Table 4.67, Atterberg limits and components are the main factors for the Corrected SPT value and unconfined compression strength.

Whereas, the coefficients before Atterberg limits represent higher weight in the equations. To the cohesion and internal friction angle, no independent variable acquires remarkable effect in the relationships. The results for effective internal friction angle are invalid, since the coefficients of determination are no more than 0.3. The Atterberg limits show relative superiority to the components. The F entry value in stepwise for cohesion analysis is adjusted to 0.12.

116

Table 4.69: Multiple Variables Linear Regression for A-6b Soils Y X R2 Equation (N60)1 PL, LL 0.705 (N60)1 = 334.812-6.012(PL)-4.758(LL) (N60)1 %G 0.556 (N60)1 = 10.862+1.432(%G) %S, PL, LL, q = -38.999-0.039(%S) - q 1.000 u u %Cpt 15.330(PL)+8.615(LL)+0.555(%Cpt) PI, Gs, %C, q = -6.156+9.989(PI) +16.667(Gs)-0.700(%C)- q 1.000 u u PL 7.589(PL) c %G, %Cpt 1.000 c = 97.618-0.882(%G)-0.722(%Cpt) c PL, LL 1.000 c = -152.567+1.067(PL)+3.637(LL) c’ %G, %Cpt 1.000 c’= 52.112-0.804(%G)-0.351(%Cpt) c’ %G, w 1.000 c’ = -0.576-0.944(%G)+1.059(w) %G, %S, φ 0.929 φ= 67.712+0.090(%G)+0.252(%S)-0.524(%Cpt) %Cpt φ %C 0.922 φ = 0.813+0.419(%C)

To the Corrected SPT value and unconfined compression strength, the

relationships for A-6b soils are similar to those for A-6a soils. In Table 4.68, the

Atterberg limits display distinct influence to the dependent variables. However, the R2 value demonstrates that the relationships are not as strong as those in A-6a soil.

Oppositely to the cohesion and internal friction angle, the components replace the

Atterberg limits to develop important function. The F entry values in stepwise approach for cohesion and effective cohesion are respectively modulated to 0.25 and 0.33.

Because of the unqualified coefficient of determination, the equations for effective internal friction angle are absent in the table.

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Table 4.70: Multiple Variables Linear Regression for A-7-6 Soils Y X R2 Equation Gs, %C, %S, (N60) = 517.602-240.583(Gs)+1.291(%C) (N60)1 0.774 PL, LL, γd +1.255(%S)-4.955(PL)+0.940(LL)+1.247(γd) %G, %C, %M, qu = 87.779+0.523(%G)+0.440(%C) - qu %S, PL, LL, γd, 0.989 0.984(%M)+0.480(%S)+8.015(PL) - %Cpt 1.619(LL)+3.831(γd)-5.692(%Cpt) Gs, %M, PL, qu = -87.002+55.792(Gs)-1.042(%M) qu 0.980 LL, γd, %Cpt +8.878(PL)-1.524(LL)+4.459(γd)-6.029(%Cpt) PI, Gs, %C, %S, c = 304.328-0.074(PI)-192.832(Gs) +0.620(%C)- c 1.000 %Cpt 0.043(%S)+2.025(%Cpt) c %M, PL 0.804 c = 127.646-6.915(%M)+0.580(PL) PI, Gs, %S, c’= 158.752+0.026(PI) - c’ 1.000 %Cpt 73.936(Gs)+0.101(%S)+0.445(%Cpt) c’ %S 0.781 c’= 0.558+0.287(%S) %C, %S, LL, w, φ= 98.477+0.298(%C)+0.502(%S) -0.336(LL)- φ 0.729 %Cpt 0.558(w)-0.820(%Cpt) φ %G, %S 0.636 φ = 10.875 +0.266(%G)-0.279(%S)

In Table 4.69, the independent variables entered in the equation are abundant.

Though Atterberg limits and components emerged in the equations, their coefficients reduce notably. The weight of specific gravity shows absolute power to vary the dependent variables. Comparing to those in the other three soil types, the coefficients of compact percent increase a lot. The PIN value is upgraded to 0.13 in cohesion stepwise calculation. Similarly, the deletion of eligible relationships for effective internal friction angle occurred in A-7-6 soils.

4.9 Multiple Variables Linear Regression Analysis (Part II)

In the initial multi-variable linear regression analysis, the computer software used certain criteria to decide which independent variables can stay and which variables should be removed. The end results appeared to be very fruitful, as many multi-variable 118

models emerged with the R square value of 1.0 or very close to 1.0. However, it is rare

to observe perfect correlations being exhibited by several variables. Thus, the outcome of the initial analysis may be viewed abnormal. When linear relationships exist among independent variables, the R squared value of the model developed for the dependent variable tends to shoot up extremely high. This problem is known as the collinearity problem. It is possible that multiple collinearity existed within the data set and skewed the R squared values of the regression models.

In statistical method, presence of collinearity can be detected through several factors, which include tolerance, variance inflation factors (VIF), eigenvalue, and condition index. In the process of collinearity statistics, collinearity may appear and standard error of regression coefficients may be inflated when tolerances of the variables are close to zero. Regression equation is suspicious when a VIF value is greater than 2.0.

Through collinearity diagnostics, if several eigenvalues are nearly zero, the independent variables are confirmed to be highly correlated. This means insignificant changes in data of independent variables will endanger remarkable variations in their coefficients.

Condition indices, square roots of ratios of the highest eigenvalue to every other eigenvalues, represent the levels of difference among eigenvalues. The regression equation is problematic when condition index is larger than 15 and is affirmed to have collinearity when condition index is greater than 30.

In this second part of the multiple linear regression analysis, a series of collinearity analysis were carried out by developing correlation plots for combinations of any two independent variables available for each dependent variable of each major soil 119 type. Figures 4.14 through 4.17 show samples of the basic correlation plots developed using EXCEL. Here, the highest R square value case is reported for each soil type. A minimum R square value of 0.7 may be applied as a screening criterion to uncover possible collinearity cases. It appears that within the data set for A-4a soil data dry unit weight and moisture content possessed collinearity. Collinearity also appears to exist between % fines (= % silt + % clay) and liquid limit for the A-6a soil data set. The data accumulated for A-6b soils may be free of any collinearity problems. It appears that the data set established for A-7-6 soil data also have collinearity problem between the dry unit weight and moisture content.

Figure 4.14: Linearity Check between Two Independent Variables (A-4a Soils)

120

Figure 4.15: Linearity Check Between Two Independent Variables (A-6a Soils)

Figure 4.16: Linearity Check Between Two Independent Variables (A-6b Soils)

121

Figure 4.17: Linearity Check Between Two Independent Variables (A-7-6 Soils)

The next step in the second part of the multi-variable linear regression analysis

was to redo the analysis while minimizing the effects of collinearity problems existing

among independent variables. First, the outcome of the initial analysis was re-examined to identify ill-fated and/or unnecessary variables. Many of the regression equations with

R squared value of 1.0 contained specific gravity as one of the independent variables.

Also, common sense approach points out that the % compaction and dry unit weight are intimately correlated. Another common sense approach points out that only one set of dry unit weight and moisture content is needed in the analysis. Plasticity index (PI) is not needed, when the data already contain liquid limit and plastic limit. % silt and % clay can be consolidated into one variable, % fines. As the result, a total of seven (dry unit weight from triaxial test, final moisture content from triaxial test, plasticity index, specific gravity, % clay, % silt, and % compaction) independent variables were

eliminated before the second analysis. 122

To reduce the effect of collinearity, only stepwise method was adopted in the second SPSS analysis. A criterion for dropping any variable was set by POUT = 0.5.

The independent variable with largest VIF value was excluded after each regression analysis until the largest condition index was smaller than 30. However, the condition index may distinctly increase when some specific independent variable is removed from equation. Hence, the results listed in Tables 4.70 through 4.73 are the situations with lowest collinearity problem.

Table 4.71: Multiple Variables Linear Regression for A-4a Soils (Collinearity Largely Eliminated) Y X R2 Equation

(N60)1 %G, %S 0.279 (N60)1 = -70.409-2.332(%G)+4.888(%S)

qu %G, PL, w 0.799 qu = 41.104-0.906(%G)+9.737(PL)-12.214(w) c %F, PL 0.376 c = 0.181+0.504(%F)-1.388(PL) c’ %S, PL 0.398 c’ = -34.558+1.33(%S)+0.383(PL) φ %G, PL, w 0.464 φ = 55.62-0.309(%G)-1.067(PL)-0.875(w) φ’ %S, γd 0.726 φ’ = 28.457+1.557(%S)-0.282(γd)

Table 4.72: Multiple Variables Linear Regression for A-6a Soils (Collinearity Largely Eliminated) Y X R2 Equation

(N ) = -54.107-3.997(%G) +5.526(PL)- (N ) %G, PL, LL, w 0.628 60 1 60 1 0.061(LL)+1.425(w)

qu PL, γd 0.478 qu = 10.096-3.565(PL)+0.76(γd) c %G, w 0.240 c = 23.767+0.051(%G)-0.933(w) c’ LL 0.541 c’ = -32.748+1.216(LL) φ %G, LL, w 0.575 φ = 39.04-1.123(%G)-0.151(LL)-0.482(w) φ’ %G, PL, LL, w 0.171 φ’ = 37.379-0.105(%G)-0.078(LL)+0.153(w)

123

Table 4.73: Multiple Variables Linear Regression for A-6b Soils (Collinearity Largely Eliminated) Y X R2 Equation

(N60)1 %G, %S 0.682 (N60)1 = 23.515+1.656(%G)-1.066(%S)

qu %G, %S 0.654 qu = -31.761+2.116(%G)+2.589(%S) c %F 0.487 c = -82.867+1.166(%F) c’ %G 0.766 c’ = 11.336-0.567(%G) φ %G, %S 0.669 φ = 0.005+0.338(%G)+0.799(%S) φ’ %G, %S 0.618 φ’ = 37.604-0.17(%G)-0.327(%S) [Note] %F = % Fines = % Silt + %Clay.

Table 4.74: Multiple Variables Linear Regression for A-7-6 Soils (Collinearity Largely Eliminated) Y X R2 Equation

(N ) = 95.35-1.696(%G) - (N ) %G, %F, LL, w 0.633 60 1 60 1 0.532(%F)+0.543(LL)-2.315(w)

qu %F, PL 0.618 qu = 45.825-1.542(%F)+5.268(PL) c %G, PL 0.533 c = 60.948-0.338(%G)-2.562(PL) c’ %G, w 0.954 c’ = 28.097-0.742(%G)-0.999(w) φ %G, %S, w 0.647 φ = 14.537-0.385(%G)+0.235(%S)-0.136(w) φ’ = 21.34+0.109(%G) φ’ %G, %S, LL, w 0.064 +0.035(%S)+0.9(LL)+0.038(w) [Note] %F = % Fines = % Silt + %Clay.

The results varied significantly in the second multi-variable linear regression analysis. None of the regression equations acquired more than four independent variables. Furthermore, the R square values sharply decreased from those seen during the initial regression analysis. Liquid limit and initial dry unit weight rarely appeared in the final models.

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4.10 Closing Remarks

Multiple variables linear regression is a very effective method to exploit the

relationship existing between dependent variables and independent variables. However,

in the current study this analytical approach did not yield many fruitful results. This is

believed to be due to the relatively small data set available for each major soil type and collinearity problems affecting the outcome of the statistical analysis.

It is reminded that some results derived from the single variable regression analysis appeared to be much more reliable than the results of the multiple variable regression analysis. This can be seen in Tables 4.74 through 4.77 presented below.

These tables can provide simple approaches for estimating shear strength parameters of cohesive soils in Ohio for highway embankment projects.

Table 4.75: Reliable Equations for A-4a Soil Shear Strength Parameters Y X R2 Equation c %C 0.988 y = -0.1655x2 + 8.5955x - 96.136 c %G 0.939 xy = 15.97x - 24.36 c %M 0.899 y = -0.2562x2 + 22.054x - 454.72 c PL 0.818 y = -1.521x2 + 46.208x - 329.07 c’ PI 0.955 y = -0.641x2 + 13.28x - 60.08 c’ %C 0.951 y = 0.456x2 - 27.39x + 412.4 φ’ PI 0.923 xy = 35.13x - 15.82

125

Table 4.76: Reliable Equations for A-6a Soil Shear Strength Parameters Y X R2 Equation c Gs 0.823 y = -1846.x2 + 9975.x - 13459 c’ %C 0.977 y = -0.936x2 + 57.40x - 873.1 c’ Gs 0.951 y = 4+30x-69.5 c’ %M 0.935 xy = 56.54x - 2042. c’ %G 0.934 y = -2.07x2 + 22.63x - 55.84 φ’ %G 0.979 xy = 31.86x + 10.93 φ’ LL 0.945 xy = 32.21x + 31.35 φ’ %S 0.927 xy = 38.13x - 108.5 φ’ %C 0.881 xy = 31.19x + 63.35 φ’ PI 0.857 xy = 33.11x + 4.525

Table 4.77: Reliable Equations for A-6b Soil Shear Strength Parameters Y X R2 Equation c LL 0.855 y = 3.37x - 120.3 c PI 1.000 y = -2.351x2 + 85.94x - 768.7 c %G 1.000 y = 0.225x2 - 5.468x + 37.43 c %C 1.000 y = -0.142x2 + 10.96x - 190.8 c %M 1.000 y = -0.906x2 + 73.94x - 1457 c %S 1.000 y = -0.640x2 + 18.78x - 124.7 c PL 1.000 y = 3.636x2 - 148.0x + 1509. c’ PL 1.000 y = 2.391x2 - 96.16x + 966.6 c’ %M 1.000 y = -0.742x2 + 60.82x - 1206. c’ %S 1.000 y = -0.825x2 + 23.26x - 154.1 c’ %C 1.000 y = -0.124x2 + 9.403x - 163.5 c’ %G 1.000 y = 0.109x2 - 3.030x + 22.90 c’ PI 1.000 y = -2.144x2 + 77.43x - 688.1 φ Gs 0.865 y = -145.9x + 411.4 φ %C 0.988 xy = 32.42x - 563.5 φ PI 0.966 xy = 53.46x - 660.9 φ PL 0.919 y = 2.038x2 - 85.63x + 911.5 φ %M 0.876 y = 0.052x2 - 4.798x + 123.4 φ’ %G 0.980 xy = 28.48x + 23.77 φ’ %C 0.956 xy = 25.56x + 178.1 φ’ %M 0.956 xy = 38.48x - 321.6 φ’ %S 0.938 xy = 25.55x + 73.14 φ’ PL 0.823 xy = 47.87x - 350.8

126

Table 4.78: Reliable Equations for A-7-6 Soil Shear Strength Parameters Y X R2 Equation c %G 0.827 xy = 6.293x + 2.951 c’ %S 0.853 y = 1.058e0.097x c’ %C 0.837 y = 5E+09x-5.39 φ %G 0.972 xy = 11.20x + 3.578 φ %S 0.935 xy = 16.39x - 26.58 φ’ %S 0.991 xy = 26.91x + 3.683 φ’ %G 0.989 xy = 27.72x - 0.708 φ’ %M 0.930 xy = 28.24x - 33.18 φ’ PI 0.876 xy = 30.24x - 75.15

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CHAPTER 5 : HIGHWAY EMBANKMENT SLOPE STABILITY

In the final analytical phase of the current research work, a series of slope stability analysis are carried out based on the engineering properties measured previously. The soil stability is calculated through the ordinary method of slices proposed by Bishop. All the analyses are conducted with the use of powerful computer software called “GEO-

SLOPE,” and the embankment models analyzed refer to geometrical variations often seen among highway embankment structures in Ohio.

5.1 Soil Slope Stability Theory

The unrestrained soil slope keeps its stability if equilibrium can exist between the gravitational effect and soil’s shear resistance. If the gravity effect surpasses the soil’s shear strength consistently along some an interface, the slope will slide and fail. To indicate whether any given soil slope is capable of maintaining its stability, the factor of safety is calculated by:

τ r Fs = (5.1) τ d

where F = the factor of safety, τ = the average shear stress developed along a potential s d

failure surface, τ = the average shear resistance of the soil available along the potential r failure surface.

If the factor of safety is greater than 1.0, the soil can just resist the shearing action and avoid becoming unstable. In practice, the lowest acceptable safety factor value is set at 1.5 for making sure that the soil slope will most likely remain stable. 128

Among the all potential failure surfaces, the one with the smallest factor of safety

constitutes the most critical surface. For the infinite slope stability analysis, the critical

surface is straight line reaching the toe of the slope. For the finite slope stability analysis, the critical surface is often circular. If the circular failure arc develops above the toe of the slope, it is referred as the “slope circle.” The failure arc is called the “toe circle” if it goes through the toe of the slope. When the circular failure arc runs deep into the bearing soil layer below the embankment, it is called the “base failure circle.” These different failure modes are illustrated in Figure 5.1.

Figure 5.1: The Illustration of Different Failure Circle for Soil Slope

There are two approaches for calculating the factor of safety for the finite slopes.

One of the approaches is the mass procedure. In this case, the soil above the sliding surface is treated as two integrated blocks located on both sides of the center of the 129

potential circular failure arc. This method is convenient to analyze any embankment

consisting of homogeneous soil fill material. However, homogeneous soil embankments

may be rare in realistic projects. The other approach is called the method of slices. In this procedure, the soil above the potential failure surface is separated into several vertical slices. The stability of each part will be calculated by summing stability possessed by each slice. This method has built-in flexibility to solve the embankment structures under complicated conditions, characterized for example by multiple soil layers and the presence of groundwater table below the top of the embankment.

Figure 5.2 shows how the mass procedure can be applied to examine the stability of a homogeneous soil embankment.

Figure 5.2: Sketch for Mass Procedure Analysis

130

According to the Figure 5.2, the safety factor is calculated for the mass procedure as:

M r WL2 2−− WL 11 WL 2 2 WL 11 Fs == =2 (5.2) Mrdd⋅⋅ατ ⋅ r r ατ d

where M d = the moment causing the sliding, M r = the resisting moment, W1 = weight of soil block located on the higher side of the line going through the center of potential failure arc, W2 = weight of soil block located on the lower side of the line going through

the center of the failure circle, L1 = moment arm length of the soil block with weight W1,

L2 = moment arm length of soil block with weight W2, r = the radius of the failure arc,

and α = angle of the circular section that defines the entire arc length.

Next, Figure 5.3 shows how vertical slices are set up in the method of slices.

Figure 5.3: Sketch for Method of Slices Analysis

131

There are two common procedures currently in use for the method of slices. They

are the ordinary method of slices and the Bishop’s simplified method of slices. Figure

5.4 illustrates forces acting on a soil slice that controls its ultimate stability condition.

Figure 5.4: The Demonstration of Stresses on Each Slice

In this figure, Wn = the weight of the slice, Wnr = reaction to the weight of the

slice, Nnr and Tnr = the normal and tangential components of Wnr , Bn = width of the slice,

Ln = base length of the slice, θ = angle corresponding to the slope steepness, and Pn and

Tn = normal and friction forces applied by the adjacent slices. 132

The ordinary method of slices is conservative because it does not consider the

forces applied by the adjacent slices. The shear stress created by the gravitational self

loading is the factor causing the potential sliding action. It is expressed as:

τ dnr= W sinθ . The soil’s shear strength is determined by the Mohr-Coulomb theory as:

τ f =+c 'tan'σφ. Hence, the factor of safety is given by the following equation:

∑()cL'costan'nn+ W θ φ Fs = (5.3) ∑Wn sinθ where c′ = effective cohesion possessed by the soil mass in the slice, and φ′ = effective

angle of internal friction possessed by the soil mass in the slice.

Since the forces between the slices are difficult to determine, Bishop provided a

simplified approach in 1955. This method treats the forces between the slices as the

additional external forces ΔP and ΔT . Under the general circumstances, the difference

between the friction forces ΔT can be neglected. So the above equation can be simplified as:

∑()cB'tan'nn+ W φ Fs = (5.4) mWθ ∑ n sinθ

tanφ 'sinθ where m =+cosθ θ F s

5.2 Embankment Model Assumptions

In the slope stability analysis, the heights of the embankment are set at 20, 30 and

40 feet, and the depth of the foundation soil layer is specified to be 10 feet. The

embankment slope steepness is defined by the gradients of 2H:1V (2 horizontal to 1 133

vertical), 2.5H:1V and 3H:1V. The loadings due to live traffic and earthquake events are

neglected. This means that the embankment structure will support neither external

vertical force nor horizontal acceleration. The soil layers in the embankment are

considered isotropic. The shear strength properties derived from the triaxial tests are

listed in Table 5.1.

Table 5.1: Shear Strength Properties of Four Major Soils in Ohio c (psi) φ (degree) c’ (psi) φ’ (degree) A-4a 12.06 24.42 4.92 33.39 A-6a 11.87 20.03 3.42 33.48 A-6b 8.88 15.36 4.54 30.83 A-7-6 5.77 12.90 3.29 27.38

The failure arc is assumed to be always circular in the finite slope stability

analysis. The factor of safety should be calculated for both short-term and long-term

embankment conditions. In the short-term analysis, the density of the soil is represented

by the moisture density. Since the embankment is undrained, the internal friction angle is

zero. The cohesion equals to cu, which is half of the unconfined compression strength.

However, this value tends to be too large to reflect realistic assessment of undrained

cohesion. Thus, the c value is adopted in the simulation process.

In the long term analysis, since the action of evaporation and drainage, the

embankment is divided into two layers according to the position of the water table. The

soil above the water table is assumed to have the same moist unit weight used in the short

term analysis. The shear strength of the soil in this zone is determined by the soil’s cohesion and internal friction angle. The soil below the water table is saturated. Its unit 134 weight is represented by the saturated unit weight. The shear strength of the soil in the saturated zone is defined by the soil’s effective cohesion and effective internal friction angle. According to E. N. Bromhead (1986), the water table in the embankment may have the shape is illustrated in Figure 5.5. In this sketch, the horizontal and vertical distances between adjacent dots are all equal to one foot.

Figure 5.5: Water Table Existing in Embankment

In the test data assembled in the earlier phases of the research, only the initial dry unit weight values are is collected. The moist unit weight and saturated unit weight need to be calculated from the initial dry unit weight and other available data through the fundamental relationships. Furthermore, the initial dry unit weight value was taken from both unconfined compression test and triaxial test. Hence, there are two alternative data groups. The relationships for computing the unit weights are demonstrated below.

To calculate moist unit weight γ mos :

135

WWswmos+ γ W mos==+=+V 11 wmos ω (5.5) W mos γ dss W V γγω=+1 mos d() mos (5.6) where V = initial volume of specimen extracted from the tube, Ws = weight of soil solid,

Wwmos = the weight of water existed in the moist soil specimen, γ d = initial dry unit

weight, and ωmos = initial moisture content.

There are two methods for computing the saturated unit weight γ sat . The first method is based on the theory of three phases in soil, which assumes the solid volume to be constant. It has the following relationships:

G γ e =−sw 1 (5.7) γ d

Se= ω Gs (5.8) where e = void ratio, Gs = specific gravity of soil solid, γw = unit weight of water, S = degree of saturation, and w = soil moisture content.

e γ w 1 When the soil is saturated, the S value is equal to 1.0. So ωsat ==−. GGs γ ds

γγsat=+ d()1 ω sat (5.9) In this case, saturated moisture content has no relationship with final moisture content. Therefore, this result represents the unit weight of saturated soil without consolidation. This value may not be suitable for the long-term slope stability analysis.

The second method is deduced by the correlations that were identified during the laboratory experiments. Here, the volume of soil solid does not need to be equal to 1.

The following shows the derivation process for the second approach of computing the saturated unit weight:

γ s = Gswγ (5.10) 136

WVs = γ ss (5.11)

WWVGwfω f sωγ f s w s VVf =+=+=wf V s V s += V s +=+ VV s s()1 ω f G s (5.12) γγww γ w

WWWVGf s++ wf sswγω fssw VG γ Gsγ wf(1+ω ) γ f == = = (5.13) VVffVGsfs()1+ω 1+ G sfω

where γ s and Vs = unit weight and volume of soil solid without void, γ f and V f = unit

weight and volume of soil specimen after consolidation, Wwf = weight of water remaining

in the soil specimen after consolidation, and ω f = the final moisture content.

In this procedure, the consolidation is taken into consideration in calculating the

saturated unit weight. The main element in the equation is the final moisture content.

Only the specific gravity is related to the initial soil status. This method is not suitable to

the unconfined compression test since it does not run consolidation process. Table 5.2 lists the dry, moist, and saturated unit weights calculated for each major soil type based on the triaxial test data. Table 5.3 lists the dry and moist unit weights calculated for each

major soil type based on the unconfined compression strength test data.

Table 5.2: Unit Weights of Four Soil Types in Ohio (based on triaxial test data) γ (psf) γ (psf) Soil Type γ (psf) γ (psf) sat sat d mos (Theoretical) (Experimental) A-4a 128.36 143.70 142.93 139.13 A-6a 122.76 139.12 140.01 137.26 A-6b 112.70 132.80 133.58 133.11 A-7-6 109.56 132.15 131.32 127.29

137

Table 5.3: Unit Weights of Four Soil Types in Ohio (based on unconfined compression test data) γ (psf) Soil Type γ (psf) γ (psf) sat d mos (Theoretical Method) A-4a 121.22 136.28 138.45 A-6a 119.80 135.61 138.16 A-6b 119.56 137.82 137.94 A-7-6 104.52 127.00 128.15

In Table 5.2, the saturated unit weight is smaller than the moisture unit weight.

There are two reasons for this ridiculous outcome. In one hand, specific gravity is surveyed in unconfined compression test instead of being manipulated in triaxial test.

Therefore, the saturated moisture content based on UC test is reliable, and opposite situation happened in the calculation for CU test. The differences between moisture and saturated unit weight are no more than 1.5 pcf, these can occur when the error in specific gravity is only 0.1. In the other hand, after extracted from embankment and sealed in the tube, the soil kept heating by environment. Because of evaporation, the water molecule moved from the interior of specimen to the exterior. At last, they aggregated at the interface between specimen and internal wall of tube. In the triaxial test, the initial moisture content just observed from the soil stick to the internal wall of tube. Hence, the moisture contents collected in CU test are higher than actual moisture content in the specimens.

Finally, in the calculation of slope stability, the grope of unit weights coming from unconfined compression test (listed in Table 5.3) is selected.

138

5.3 Results of Highway Embankment Slope Stability Analysis

5.3.1 Homogeneous Embankments

This section presents the results of the short-term and long-term analyses performed on the embankments built with homogeneous soils. After the demonstration of each group of data, a figure is prepared to illustrate the failure arc with the smallest factor of safety in this group. And, the second figure reveals the stress of the heaviest slice in that failure circle.

Table 5.4 summarizes the results of the short-term analyses carried out on 20-ft high homogeneous embankments. Figure 5.6 plots the results provided in Table 5.4 graphically.

Table 5.4: Factor of Safety in Short-Term Analysis (Homogeneous Embankment; Total Height 20 ft) Soil Type 2H:1V 2.5H:1V 3H:1V A-4a 4.241 4.403 4.637 A-6a 4.195 4.355 4.586 A-6b 3.088 3.206 3.376 A-7-6 2.177 2.261 2.38

139

Figure 5.6: Plots of Factor of Safety in Short-Term Analysis (Homogeneous Embankments; Height 20 ft)

According to Table 5.4 and Figure 5.6, the short-term safety factor values are all above 2.0 for the homogeneous embankments with the height of 20 feet, regardless of the soil type and/or slope steepness considered. The smallest value is 2.177. This means the embankments built with uniform soils are reliable in the short-term conditions (or conditions existing up to the end of construction). The embankments constructed with A-

4a soils are the safest, and those built with A-6a soils are almost equally stable. Under the same slope steepness, the factors of safety of embankment slope composed of A-6b soils are about 1.0 lower than that given to the same embankment consisting of the A-4a soils. And, the factor of safety values assigned to the A-7-6 soil embankments are about

2.0 below those given to the A-4a soil embankments. 140

Figure 5.7 reveals the most critical failure arc identified during the short-term

analysis on 20-ft tall homogeneous embankments, which happens to be a base failure

circle. Figure 5.8 represents the stresses acting on the slice with largest gravity effect

(shaded slice), along the most critical arc shown in Figure 5.7.

Figure 5.7: Most Critical Failure Arc in Short-Term Analysis (Homogeneous Embankment; Height 20 ft)

141

Figure 5.8: Stresses Acting on of the Slice Shaded in Figure 5.7

The units of the stresses given in the above figure are all pound per square feet

(lb/ft2). The width of this slice is 1.92 feet, and the middle height is 21.62 feet. The base

length is 2.18 feet, and the base angle is 27.86 degree.

Table 5.5 summarizes the results of the short-term analyses carried out on 30-ft

high homogeneous embankments. Figure 5.9 plots the results provided in Table 5.5 graphically.

Table 5.5: Factor of Safety in Short-Term Analysis (Homogeneous Embankments; Height 30 ft)

Soil Type 2H:1V 2.5H:1V 3H:1V A-4a 2.995 3.158 3.383 A-6a 2.963 3.124 3.346 A-6b 2.181 2.299 2.463 A-7-6 1.538 1.621 1.737 142

Figure 5.9: Plots of Factor of Safety in Short-Term Analysis (Homogeneous Embankments; Height 30 ft)

According to Table 5.5 and Figure 5.9, the short-term safety factor values are all above 1.5 for the homogeneous embankments with the height of 30 feet, regardless of the soil type and/or slope steepness considered. The orders of factor of safety values are the same as that previously observed for 20-ft high homogeneous embankments. However, the separations between soil types are narrower.

Figure 5.10 reveals the most critical failure arc identified during the short-term analysis on 30-ft tall homogeneous embankments, which happens to be a base failure circle. Figure 5.11 represents the stresses acting on the slice with largest gravity effect

(shaded slice), along the most critical arc shown in Figure 5.10. 143

Figure 5.10: Most Critical Failure Arc in Short-Term Analysis (Homogeneous Embankment; Height 30 ft)

Figure 5.11: Stresses Acting on of Slice Shaded in Figure 5.10

144

The units of the stresses appearing in the above figure are all pound per square

feet. The width of this slice is 2.40 feet, and the middle height is 27.43 feet. The base

length is 2.68 feet and the base angle is 26.34 degree.

Table 5.6 summarizes the results of the short-term analyses carried out on 40-ft

high homogeneous embankments. Figure 5.12 plots the results provided in Table 5.6 graphically.

Table 5.6: Factor of Safety in Short-Term Analysis (Homogeneous Embankments; Height 40 ft)

Soil Type 2H:1V 2.5H:1V 3H:1V A-4a 2.344 2.512 2.729 A-6a 2.318 2.485 2.699 A-6b 1.707 1.829 1.987 A-7-6 1.203 1.29 1.401

Figure 5.12: Plots of Factor of Safety in Short-Term Analysis (Homogeneous Embankments; Height 40 ft) 145

According to Table 5.6 and Figure 5.12, the factor of safety values are acceptable

(above 1.5) for embankments built with A-4a and A-6 soils but not for embankments

consisting of A-7-6 soils. The orders are the same as those already observed previously.

And, the separations between the curves are even tighter than before.

Figure 5.13 reveals the most critical failure arc identified during the short-term

analysis on 40-ft tall homogeneous embankments, which happens to be a base failure

circle. Figure 5.14 represents the stresses acting on the slice with largest gravity effect

(shaded slice), along the most critical arc shown in Figure 5.13.

Figure 5.13: Most Critical Failure Circle in Short- Term Analysis (Homogeneous Embankment; Height 40 ft)

146

Figure 5.14: Stresses Acting on Slice Shaded in Figure 5.13

The units of the stresses appearing in the above figure all are all pound per square

feet. The width of this slice is 3.00 feet, and the middle height is 32.84 feet. The base

length is 3.34 feet and the base angle is 26.10 degree.

Table 5.7 summarizes the results of the long-term analyses carried out on 20-ft high homogeneous embankments. Figure 5.15 plots the results provided in Table 5.7 graphically.

Table 5.7: Factor of Safety in Long-Term Analysis (Homogeneous Embankments; Height 20 ft) Soil Type 2H:1V 2.5H:1V 3H:1V A-4a 3.235 3.531 3.837 A-6a 2.644 2.96 3.247 A-6b 2.923 3.224 3.503 A-7-6 2.348 2.595 2.824

147

Figure 5.15: Plots of Factor of Safety in Long-Term Analysis (Homogeneous Embankments; Height 20 ft)

From Table 5.7 and Figure 5.15, the distribution of the lines in the long term is more regular than those in the short term. The distances between the adjacent lines are all nearly 0.3. The location of A-4a and A-7-6 lines retains the same. The former one is on the top and the latter one stays at the bottom. Whereas, the A-6b embankment exhibits higher reliability than A-6a embankment does. This phenomenon does not appear in the short term homogeneous embankment analysis. The least factor of safety in this group is 2.348. Hence, the circle failure will not occur in the long term operation.

Figure 5.16 reveals the most critical failure arc identified during the long-term analysis on 20-ft tall homogeneous embankments, which happens to be a close to a toe circle. Figure 5.17 represents the stresses acting on the slice with largest gravity effect

(shaded slice), along the most critical arc shown in Figure 5.16. 148

Figure 5.16: Most Critical Failure Circle in Long- Term Analysis (Homogeneous Embankment; Height 20 ft)

Figure 5.17: Stresses Acting on Slice Shaded in Figure 5.16 149

Figure 5.16 shows the slices of long term single layer embankment in height of 20 feet. Figure 5.17 represents the stresses of slice with largest gravity in Figure 5.16. The units of the stresses in the above figure all are pound per square feet. The width of this slice is 1.74 feet, and the middle height is 15.86 feet. The base length is 1.96 feet and the base angle is 27.67 degree. Comparing to stresses of the short term embankment with same geometry, the shear stress increase only one pound per square feet, although the weight of slice abates 1752 psf.

Table 5.8 summarizes the results of the long-term analyses carried out on 30-ft high homogeneous embankments. Figure 5.18 plots the results provided in Table 5.8 graphically.

Table 5.8: Factor of Safety in Long-Term Analysis (Homogeneous Embankments; Height 30 ft)

Soil Type 2H:1V 2.5H:1V 3H:1V A-4a 2.485 2.794 3.076 A-6a 2.091 2.383 2.656 A-6b 2.27 2.551 2.806 A-7-6 1.822 2.052 2.262

150

Figure 5.18: Plots of Factor of Safety in Long Term Analysis (Homogeneous Embankments; Height 30 ft)

In Table 5.8 and Figure 5.18, the arrangement of the lines is almost the same to that of long term single layer embankment in the height of 20 feet. However, the range between the lines of A-6a and A-6b becomes short. The top point depresses 0.761 and the bottom point decrease 0.526. Since the values of the whole points are more than 1.5, the long term embankment slopes are reliable in the height of 30 feet.

Figure 5.19 reveals the most critical failure arc identified during the long-term analysis on 30-ft tall homogeneous embankments, which happens to be basically a toe circle. Figure 5.20 represents the stresses acting on the slice with largest gravity effect

(shaded slice), along the most critical arc shown in Figure 5.19. 151

Figure 5.19: Most Critical Failure Circle in Long- Term Analysis (Homogeneous Embankment; Height 30 ft)

Figure 5.20: Stresses Acting on Slice Shaded in Figure 5.19 152

Figure 5.19 shows the slices of long term single layer embankment in height of 30

feet. Figure 5.20 represents the stresses of slice with largest gravity in Figure 5.19. The

units of the stresses in the above figure all are pound per square feet. The width of this

slice is 2.31 feet, and the middle height is 20.24 feet. The base length is 2.55 feet and the

base angle is 25.36 degree.

Table 5.9 summarizes the results of the long-term analyses carried out on 40-ft high homogeneous embankments. Figure 5.21 plots the results provided in Table 5.9 graphically.

Table 5.9: Factor of Safety in Long-Term Analysis (Homogeneous Embankments; Height 40 ft) Soil Type 2H:1V 2.5H:1V 3H:1V A-4a 2.181 2.465 2.739 A-6a 1.868 2.146 2.403 A-6b 1.99 2.248 2.496 A-7-6 1.603 1.815 2.019

153

Figure 5.21: Plots of Factor of Safety in Long-Term Analysis (Homogeneous Embankments; Height 40 ft)

Compared to the short term results in the uniform geometry status, in Table 5.9 and Figure 5.21, the sequence of the lines does not change for the long term embankments. However, the lines of A-6a and A-6b get much nearer to each other. Their differences between the factors of safety of same slope degree are merely around 0.1. The bottom point belongs to the A-7-6 embankment with sharpest slope. Its value is 1.603, which is slightly higher than the number acceptable in the project.

Figure 5.22 reveals the most critical failure arc identified during the long-term analysis on 40-ft tall homogeneous embankments, which happens to be nearly a toe circle. Figure 5.23 represents the stresses acting on the slice with largest gravity effect

(shaded slice), along the most critical arc shown in Figure 5.22. 154

Figure 5.22: Most Critical Failure Circle in Long- Term Analysis (Homogeneous Embankment; Height 40 ft)

Figure 5.23: Stresses Acting on Slice Shaded in Figure 5.22 155

Figure 5.22 shows the slices of long term single layer embankment in height of 40 feet. Figure 5.23 represents the stresses of slice with largest gravity in Figure 5.22. The units of the stresses in the above figure all are pound per square feet. The width of this slice is 3.08 feet, and the middle height is 23.72 feet. The base length is 3.41 feet and the base angle is 25.50 degree.

5.3.2 Outcomes of Embankments Built on Two Soil Layers

The analyses in this part approach closer to the realistic situation. The spreads of the layers are in accordance with those in the embankments collected the soil specimen.

In the instruction of the soil layer, the soil type on the left of the plus sign is the top layer in the embankment. After the demonstration of each group of data, the following figure illustrates the failure circle of the embankment with the smallest factor of safety in this group. And the next figure reveals the stress of the heaviest slice in that failure circle.

Table 5.10 summarizes the results of the short-term analyses carried out on 20-ft high two-layer embankments. Figure 5.24 plots the results provided in Table 5.10 graphically.

Table 5.10: Factor of Safety in Short-Term Analysis (Two-Layer Embankments; Height 20 ft) Soil Type 2H:1V 2.5H:1V 3H:1V A-6a + A-4a 4.224 4.408 4.639 A-6a + A-7-6 2.468 2.487 2.606 A-7-6 + A-6b 3.076 3.199 3.372

156

Figure 5.24: Plots of Factor of Safety in Short Term Analysis (Two-Layer Embankments; Height 20 ft)

Since this is the combination of two soil layers, the trends of the factor of safety depending on the slope degree do not emerge the linear features. In Table 5.10 and Figure

5.24, the embankment containing A-4a possesses great reliability. However, the factors of safety of the embankments consisting of A-7-6 fall sharply, especially for the one’s foundation is A-7-6. All the data exceed 2.0, so the slope of the embankment is safe in the short term manipulation.

Figure 5.25 reveals the most critical failure arc identified during the short-term analysis on 20-ft tall two-layer embankments, which happens to be a base failure circle.

Figure 5.26 represents the stresses acting on the slice with largest gravity effect (shaded slice), along the most critical arc shown in Figure 5.25. 157

Figure 5.25: Most Critical Failure Circle in Short Term Analysis (Two-Layer Embankment; Height 20 ft)

Figure 5.26: Stresses Acting on Slice Shaded in Figure 5.25

158

Figure 5.25 represents the slices of short term double layers embankment in

height of 20 feet. Figure 5.26 illustrates the stresses of slice with largest gravity in Figure

5.26. The units of the stresses in the above figure all are pound per square feet. The width of this slice is 2.00 feet, and the middle height is 19.36 feet. The base length is

2.25 feet and the base angle is 27.37 degree.

Table 5.11 summarizes the results of the short-term analyses carried out on 30-ft high two-layer embankments. Figure 5.27 plots the results provided in Table 5.11 graphically.

Table 5.11: Factor of Safety in Short Term Analysis (Two-Layer Embankments; Height 30 ft)

Soil Type 2H:1V 2.5H:1V 3H:1V A-6a + A-4a 2.995 3.156 3.383 A-6a + A-7-6 1.707 1.754 1.875 A-7-6 + A-6b 2.157 2.277 2.433

159

Figure 5.27: Plots of Factor of Safety in Short Term Analysis (Two- Layer Embankments; Height 30 ft)

In Table 5.11 and Figure 5.27, the arrangement and nonlinear characteristic of the lines remains the same to the results listed in the Table 5.10 and Figure 5.24. The three lines get nearer. The points in the top line descend 1.25 averagely, and the points in the bottom line decline 0.93 averagely.

Figure 5.28 reveals the most critical failure arc identified during the short-term analysis on 30-ft high two-layer embankments, which happens to be a base failure circle.

Figure 5.29 represents the stresses acting on the slice with largest gravity effect (shaded slice), along the most critical arc shown in Figure 5.28. 160

Figure 5.28: Most Critical Failure Circle in Short Term Analysis (Two-Layer Embankment; Height 30 ft)

Figure 5.29: Stresses Acting on Slice Shaded in Figure 5.28

161

Figure 5.28 represents the slices of short term double layers embankment in

height of 30 feet. Figure 5.29 illustrates the stresses of slice with largest gravity in Figure

5.28. The units of the stresses in the above figure all are pound per square feet. The width of this slice is 2.50 feet, and the middle height is 25.93 feet. The base length is

2.80 feet and the base angle is 26.83 degree.

Table 5.12 summarizes the results of the short-term analyses carried out on 40-ft high two-layer embankments. Figure 5.30 plots the results provided in Table 5.12 graphically.

Table 5.12: Factor of Safety in Short-Term Analysis (Two-Layer Embankments; Height 40 ft)

Soil Type 2H:1V 2.5H:1V 3H:1V A-6a + A-4a 2.344 2.49 2.728 A-6a + A-7-6 1.322 1.369 1.495 A-7-6 + A-6b 1.686 1.793 1.967

162

Figure 5.30: Plots of Factor of Safety in Short Term Analysis (Two-Layer Embankments; Height 40 ft)

From Table 5.12 and Figure 5.30, the line showing the factor of safety of the embankment’s foundation consisting of A-4a locates at the top. The one representing the stability of the embankment’s foundation composed with A-7-6 seats at the bottom. With the change of height from 20 feet to 40 feet, the difference between the peak point and the foot point varies from 1.563 to 1.042. The stability for the bottom line is not reliable for the project. The gaps in the adjacent line become shorter with the rise of the height.

Figure 5.31 reveals the most critical failure arc identified during the short-term analysis of 20-ft tall two-layer embankments, which happens to be nearly a toe circle. Figure 5.32 represents the stresses acting on the slice with largest gravity effect (shaded slice), along the most critical arc shown in Figure 5.31. 163

Figure 5.31: Most Critical Failure Circle in Short Term Analysis (Two-Layer Embankment; Height 20 ft)

Figure 5.32: Stresses Acting on Slice Shaded in Figure 5.31

164

Figure 5.31 represents the slices of short term double layers embankment in

height of 40 feet. Figure 5.32 illustrates the stresses of slice with largest gravity in Figure

5.31. The units of the stresses in the above figure all are pound per square feet. The width of this slice is 3.16 feet, and the middle height is 29.53 feet. The base length is

3.49 feet and the base angle is 25.23 degree.

Table 5.13 summarizes the results of the long-term analyses carried out on 20-ft high two-layer embankments. Figure 5.33 plots the results provided in Table 5.13 graphically.

Table 5.13: Factor of Safety in Long-Term Analysis (Two-Layer Embankments; Height 20 ft)

Soil Type 2H:1V 2.5H:1V 3H:1V A-6a + A-4a 3.121 3.466 3.761 A-6a + A-7-6 2.304 2.536 2.758 A-7-6 + A-6b 2.933 3.236 3.529

165

Figure 5.33: Plots of Factor of Safety in Long-Term Analysis (Two- Layer Embankments; Height 20 ft)

In the long term, the factors of safety of the three statuses descend. For the consequences of slope degree of 2:1 listed in Table 5.13 and Figure 5.33, the values of three lines all drop. However, for the slope degree of 2.5:1 and 3:1, the stability of the lower two lines increase instead of decreasing. The raising values are no more than 1.6.

Hence, the three lines get closer, especially for the top one and the middle one. 166

Figure 5.34: Most Critical Failure Circle in Long-Term Analysis (Two-Layer Embankment; Height 20 ft)

167

Figure 5.35: Stresses Acting on Slice Shaded in Figure 5.34

Figure 5.34 represents the slices of long term double layers embankment in height of 20 feet. Figure 5.35 illustrates the stresses of slice with largest gravity in Figure 5.34.

The units of the stresses in the above figure all are pound per square feet. The width of this slice is 1.67 feet, and the middle height is 15.85 feet. The base length is 1.89 feet and the base angle is 28.24 degree.

Table 5.14 summarizes the results of the long-term analyses carried out on 30-ft high two-layer embankments. Figure 5.36 plots the results provided in Table 5.14 graphically.

Table 5.14: Factor of Safety in Long-Term Analysis (Two-Layer Embankments; Height 30 ft)

Soil Type 2H:1V 2.5H:1V 3H:1V A-6a + A-4a 2.434 2.735 3.027 A-6a + A-7-6 1.788 2.009 2.221 A-7-6 + A-6b 2.262 2.538 2.804 168

Figure 5.36: Plots of Factor of Safety in Long-Term Analysis (Two-Layer Embankments; Height 30 ft)

From Table 5.14 and Figure 5.36, the stability of the embankment composing of

A-6a and A-4a declines. Comparing the stability of the short term, the results rise in the second line and third line. For the 2:1 slope degree, the increasing value is 0.09. For the

2.5:1 slope degree, it ascends 0.25. For the 3:1 slope degree, the raising value is 0.35. 169

Figure 5.37: Most Critical Failure Circle in Long-Term Analysis (Two-Layer Embankment; Height 30 ft)

Figure 5.38 Stresses Acting on Slice Shaded in Figure 5.37

170

Figure 5.37 represents the slices of long term double layers embankment in height of 30 feet. Figure 5.38 illustrates the stresses of slice with largest gravity in Figure 5.37.

The units of the stresses in the above figure all are pound per square feet. The width of this slice is 2.35 feet, and the middle height is 19.96 feet. The base length is 2.52 feet and the base angle is 20.81 degree.

Table 5.15 summarizes the results of the long-term analyses carried out on 40-ft high two-layer embankments. Figure 5.39 plots the results provided in Table 5.15 graphically.

Table 5.15: Factor of Safety in Long-Term Analysis (Two-Layer Embankments; Height 40 ft)

Soil Type 2H:1V 2.5H:1V 3H:1V A-6a + A-4a 2.134 2.405 2.683 A-6a + A-7-6 1.571 1.77 1.977 A-7-6 + A-6b 1.969 2.217 2.471

171

Figure 5.39: Plots of Factor of Safety in Long-Term Analysis (Two-Layer Embankments; Height 40 ft)

From Table 5.15 and Figure 5.39, the factor of safety of the embankments composing of A-6a and A-4a reduces. Comparing the factor of safety of the short term, the consequences rise in the second line and third line. For the 2:1 slope degree, the increasing value is 0.26. For the 2.5:1 slope degree, it ascends 0.41. For the 3:1 slope degree, the raising value is 0.49. The bottom value of the third line is more than 1.5. In the height of 40 feet, these embankments in three slope degree are dependable in the design procedure. 172

Figure 5.40: Most Critical Failure Circle in Long-Term Analysis (Two-Layer Embankment; Height 40 ft)

Figure 5.41: Stresses Acting on Slice Shaded in Figure 5.40 173

Figure 5.40 represents the slices of long term double layers embankment in height of 40 feet. Figure 5.41 illustrates the stresses of slice with largest gravity in Figure 5.40.

The units of the stresses in the above figure all are pound per square feet. The width of this slice is 2.99 feet, and the middle height is 24.26 feet. The base length is 3.34 feet and the base angle is 26.77 degree. 174

CHAPTER 6 : SUMMARY AND CONCLUSIONS

6.1 Summary

Soil shear strength is the prerequisite factor in the design of embankment. However, this property needs to be determined by time consuming test. It will be convenient to utilize relationships to calculate soil shear strength with other soil properties. The specific objectives of this thesis are listed below:

• Obtain a comprehensive set of field and laboratory test data on cohesive soils that

are typically utilized in highway embankment construction work in Ohio;

• Evaluate the empirical shear strength correlations established by Terzaghi and the

US Department Navy are justified suitability in Ohio;

• Establish reliable relationships between soil shear strength and other more basic

soil properties through statistical regression methods; and

• Perform slope stability analysis, using soil properties acquired in the study, for

various configurations of highway embankments and make technical

recommendations on highway embankment construction projects.

The first objective was met by conducting standard penetration tests at highway embankment sites in Ohio and subjecting relatively undisturbed soil samples recovered from these sites to a variety of laboratory test procedures.

The second objective was achieved by applying the data collected in the current study to each of the empirical correlations previously proposed by Terzaghi and the US

Department Navy. 175

The third objective was met by fitting the test data compiled in the study to various

statistical regression methods, ranging from linear single-variable regression to nonlinear multi-variable regression models. This was done methodically using two computer software packages, EXCEL and SPSS.

The last objective was achieved by performing a series of soil embankment slope stability analysis through the method of slices, using a special computer software package

GEO-SLOPE. The slope stability analysis was performed with variations in the embankment height, slope steepness, soil compositions, and drainage conditions, to reflect a wide range of engineering characteristics found among highway embankments existing in Ohio.

6.2 Conclusions

6.2.1 Conclusions on Empirical Correlations

The empirical correlation, between the effective friction angle and plasticity index, established by Terzaghi is applicable to most of the cohesive soils found in Ohio.

The correlation fit the A-4a, A-6a and A-6b soil types well. The only exception was

Ohio’s A-7-6 soils, for which the empirical correlations were on the higher side. The empirical relationship, between the unconfined compression strength and the corrected

SPT-N values, provided by the U.S. Department Navy is not suitable for a majority of cohesive soils in Ohio. For most soils, the data points available from the current research plotted outside the limits specified by the Department of Navy. Only for A-6b soil type, 176

the empirical correlation faired better. However, the quantity of data came from A-6b

soil was not sufficient to really validate the empirical relationship.

6.2.2 Conclusions from T-Tests Performed

According to the Ohio Department of Transportation (ODOT) the A-4 soil type can be further differentiated into subgroups of A-4a and A-4b by the various percentages

of their components. Various engineering properties of the two subgroups were

examined by the t test method. The results indicated that the two subgroups of the A-4

soil type are essentially identical to each other. Similarly, ODOT separates the A-6 soil

type into two subgroups of A-6a and A-6b. According to the t test results, the A-6 soil

subgroups are slightly more different than the two subsets of the A-4 soil type. This was

particularly true for their shear strength properties. During the field phase of the current

research, some soil types were encountered repeatedly throughout different regions of the state. Additional t-tests were performed to determine if geographical location played any significant role in the soil’s engineering properties. According to these statistical analyses, geographical location did appear to influence engineering properties of cohesive soils existing in Ohio. All results calculated from t-test are only the recommendations since the sample sizes are not adequate.

177

6.2.3 Conclusions from Single-Variable Regression Analysis

Through single variable regression analyses, several general patterns appeared in the relationships between cohesive soil’s field/laboratory test results. They are listed below.

* Overall, the single variable regression analysis was not very successful in establishing many reliable shear strength correlations for cohesive soils studied in the current research.

* The corrected SPT-N value and unconfined compression strength have little connections with other soil mechanical properties.

* Soil compositional characteristics (such as % sand and % clay) have remarkable effect on the soil’s engineering properties, which include corrected SPT-N value, unconfined compression strength, cohesion, effective cohesion, internal friction angle, and effective internal friction angle.

* The Atterberg limits and soil moisture content have influences on the soil’s shear strength properties, which include cohesion, effective cohesion, internal friction angle, and effective internal friction angle.

* The second degree polynomial function is more effective in expressing the correlations for soil’s cohesion and effective cohesion. For soil’s internal friction angle and effective friction angle, the hyperbolic function is efficient in demonstrating the relationships between them and other independent variables.

178

6.2.4 Conclusions from Multi-Variable Linear Regression Analysis

Several different approaches were possible for conducting multi-variable regression analysis. However, in the current research only the linear regression model was employed to carry out the multi-variable analysis. This was because of the shear number of variables available in the study. The following summarizes observations made in this analytical phase of the thesis work.

* The soil composition characteristics (such as % sand and % clay) and the

Atterberg limits lay great effects on the corrected SPT-N value and unconfined compression strength.

* Although soil compositional characteristics and the Atterberg limits appear in many regression equations for soil shear strength factors, their overall influence on the soil’s engineering properties are relatively small.

* Whenever the specific gravity appears in the regression equations, its coefficient is large enough to influence the dependent variable.

* The multi-variable linear regression model is not very successful in establishing reliable correlation equations for the effective internal fiction angle of most of the cohesive soil types except for A-4a.

* In the multiple variables regression analysis, collinearity can cause negative influence on the reliabilities of equations. If the coefficient of determination between two independent variables is greater than 0.7, it will generate collinearity when they simultaneously appear in the same equation. 179

* The coefficient of determination decreases notably when the problem of collinearity is removed from the results. Hence, the efficient approaches to estimate soil shear strength parameters are the equations derived from single variable regression analysis.

6.2.5 Conclusions from Slope Stability Analysis

In the soil slope stability analysis, the models are divided into two general types in accordance with the number of soil layers making up the embankment structure.

According to the various locations of water table in the embankment, the models are separated into short-term and long-term analysis cases.

For the homogeneous embankment, the following conclusions can be drawn:

* The factor of safety for slope stability increases with the decrease in the embankment height and the rise in the slope steepness.

* The A-4a soil type shows the highest shear strength to resist the sliding shear force caused by its own weight, in both short-term and long-term conditions. In contrast, the A-7-6 soil type possesses the lowest soil shear strength to resist the sliding shear force, in both short-term and long-term conditions.

* From short term to long term, trend in the factor of safety of the same geometry conditions (controlled by height and slope steepness) observed for each soil type: A-4a results averagely decrease nearly 1.0, and A-6a data fall about 0.5, and A-6b consequences increase 0.15 averagely, and A-7-6 results gain 0.35. The factors of safety of embankments built with A-4a and A-6a will increase after years, and the factor of 180

safety of embankments constructed with A-6b and A-7-6 will decrease with the elapse of

time.

* In the short-term analysis, the factor of safety values are difference among the soil

types under the same embankment geometry conditions (which are controlled by the embankment height and slope steepness). In the long-term analysis, the safety factor values among the different soil types become more similar under the same embankment geometry conditions.

* The embankment constructed with A-7-6 soils is not safe at the height of 40 feet in the short term application, whatever in any slope degrees calculated in the analysis.

For embankments constructed from two soil layers, the following conclusions can be listed:

* In any geometry conditions considered in the slope stability analysis, the embankment becomes safer if its lower layer is built with A-4a soils.

* The slope stability of embankments built with A-7-6 soils increases with the elapse of time. This means that the embankments’ values of factor of safety in the long- term condition are higher than those in the short-term condition.

* If the embankment’s lower layer consists of A-7-6 soils, its height should not exceed 40 feet when the slope degree is steeper than 2H:1V. This is because the factors of safety do not reach the minimum value of 1.5 in the short-term condition. 181

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