unsaturated soils and foundation design: theoretical

Unsaturated Soils and Foundation

Design: Theoretical Considerations

for an Effective Stress Framework

AN ABSTRACT OF THE THESIS OF

Josiah D. Baker for the degree of Master of Science in Civil Engineering presented on

June 2, 2016.

Title: Unsaturated Soils and Foundation Design: Theoretical Considerations for an Ef-

fective Stress Framework

Abstract approved:

T. Matthew Evans

This study presents the theoretical background necessary to model the bearing capacity

of shallow and deep foundations in partially saturated soils. The conventional bearing

capacity equations for shallow and deep foundations and the 𝛽-method for deep foun-

dation side resistance have been modified to include the effects of matric suction and

varying water contents according to the effective stress framework. A closed-form so-

lution has been proposed for the bearing capacity equation that modifies the overbur-

den, unit weight, and cohesion terms in the conventional equation. The 𝛽-method has

been modified to consider suction stresses along the deep foundation and a reduction

in K0 due to tension cracking near the surface. The modified bearing capacity equation

for shallow foundations shows good agreement to load-tests performed in partially sat-

urated soils. Monte Carlo simulations were performed on silt loams, sands, and clays

to characterize variance and distribution of bearing capacity. The results show that silts

have the largest variance while clays have the smallest variance in predicted bearing

capacity.

© Copyright by Josiah D. Baker

June 2, 2016

All Rights Reserved

Unsaturated Soils and Foundation Design: Theoretical Considerations for an

Effective Stress Framework

by

Josiah D. Baker

A THESIS

submitted to

Oregon State University

in partial fulfillment of

the requirements for the

degree of

Master of Science

Presented June 2, 2016

Commencement June 2017

Master of Science thesis of Josiah D. Baker presented on June 2, 2016.

APPROVED:

Major Professor, representing Civil Engineering

Head of the School of Civil and Construction Engineering

Dean of the Graduate School

I understand that my thesis will become part of the permanent collection of Oregon

State University libraries. My signature below authorizes release of my thesis to any

reader upon request.

Josiah D. Baker, Author

ACKNOWLEDGEMENTS

First, I would like to thank my advisor, Professor Matt Evans, for his dedication, support, and

insight in writing this thesis. During my undergraduate, his passion for geotechnical engineer-

ing spurred within me an interest to pursue this discipline. Since then, he has been instrumental

in the completion of my education and research both during my undergraduate and graduate

studies at Oregon State University.

I would also like to give my sincerest appreciation to my friends at Grant Avenue Baptist

Church for their continual love, making my time at Oregon State University a memorable ex-

perience. They have encouraged me to grow in my faith as a Christian.

I am grateful to my parents and siblings for their patience and love throughout college. They

helped me through all my life decisions. I could not find a more supportive family than this.

Finally, I would like to thank my fiancé Rebecca. She has been a wonderful friend, challenging

me each day to pursue excellence while still encouraging me to enjoy life’s important moments.

TABLE OF CONTENTS

Page

1. Introduction ............................................................................................................1

1.1. Statement of Problem ......................................................................................1

1.2. Purpose and Scope ..........................................................................................1

1.3. Outline .............................................................................................................1

1.4. Qualifications and Limitations ........................................................................2

2. Background ............................................................................................................3

2.1. Shallow Foundations .......................................................................................3

2.1.1. General Bearing Capacity Theory for Shallow Foundations ...................3

2.1.2. Various Improvements on Bearing Capacity Equation ...........................8

2.1.3. Recent Developments ............................................................................11

2.2. Deep Foundations .........................................................................................13

2.2.1. Analytical Theory ..................................................................................13

2.2.2. Recent Developments ............................................................................18

2.3. Mechanics of Unsaturated Soils ....................................................................19

2.3.1. Soil Water Characteristic Curve ............................................................19

2.3.2. Particle Level Principles ........................................................................20

2.3.3. Bishop’s Effective Stress Framework ...................................................23

2.3.4. Solutions for Bishop’s Effective Stress Parameter ................................24

2.3.5. Extended Mohr-Coulomb Failure Criterion ..........................................26

2.3.6. Matric Suction Profiles ..........................................................................27

2.3.7. At-Rest Earth Pressure Coefficient ........................................................29

2.3.8. Discussion of Unsaturated Soil Properties ............................................30

2.4. Summary .......................................................................................................32

3. Research Objectives and Methodology ...............................................................35

TABLE OF CONTENTS (CONTINUED)

Page

3.1. Objectives ......................................................................................................35

3.2. Shallow Foundations in Unsaturated Soils ...................................................37

3.2.1. Theoretical Development .......................................................................37

3.2.2. Considerations for Apparent Cohesion ..................................................42

3.2.3. Considerations for Unit Weight .............................................................44

3.2.4. Considerations for Overburden ..............................................................46

3.3. Deep Foundations in Unsaturated Soils ........................................................47

3.3.1. Theoretical Development .......................................................................47

3.3.2. Tension Cracking and K0 .......................................................................50

3.3.3. Unit Weight ...........................................................................................52

3.3.4. Suction Stresses .....................................................................................52

3.4. Summary .......................................................................................................53

4. Comparison to Measured Response of Shallow Foundations..............................54

4.1. Introduction ...................................................................................................54

4.2. Method for the Selection of Load Test Data .................................................54

4.3. Comparison of Predicted Bearing Capacity to Database ..............................56

4.3.1. Steensen-Bach et al. (1987) ...................................................................56

4.3.2. Briaud and Gibbens (1997) ....................................................................60

4.3.3. Larsson (1997) .......................................................................................65

4.3.4. Viana da Fonseca and Sousa (2002) ......................................................69

4.3.5. Rojas et al. (2007) ..................................................................................71

4.3.6. Vanapalli and Mohamed (2007) / Oh and Vanapalli (2008) .................75

4.3.7. Vanapalli and Mohamed (2013) ............................................................80

4.3.8. Wuttke et al. (2013) ...............................................................................84

TABLE OF CONTENTS (CONTINUED)

Page

4.4. Summary and Discussion ..............................................................................88

5. Parametric Studies ...............................................................................................90

5.1. Outline of Parametric Studies .......................................................................90

5.2. Soils Parameters Used in Parametric Study ..................................................90

5.3. Parametric Studies on Shallow Foundations .................................................93

5.3.1. Shallow Foundation Bearing Capacity Profiles .....................................93

5.3.2. Evaluation of van Genuchten’s 𝛼 and 𝑛 ..............................................103

5.3.3. Other Considerations for Shallow Foundation Bearing Capacity .......108

5.3.4. Vahedifard and Robinson (2015) .........................................................116

5.4. Parametric Study on the Modified 𝛽-method .............................................124

5.4.1. Development of Side Resistance Profiles ............................................124

5.4.2. Evaluation of Side Resistance Profiles ................................................129

5.5. Monte Carlo Simulations for Partially Saturated Soils ...............................145

5.5.1. Silt Loam Analysis ..............................................................................146

5.5.2. Sand Analysis ......................................................................................151

5.5.3. Clay Analysis .......................................................................................154

5.6. Discussion ...................................................................................................157

6. Conclusions and Future Work ...........................................................................160

6.1. Conclusions .................................................................................................160

6.2. Implications for Geotechnical Engineering Practice ..................................161

6.3. Future Work ................................................................................................162

References ..................................................................................................................165

LIST OF FIGURES

Figure Page

Figure 2-1. Definitions of ultimate bearing capacity (from Terzaghi 1943) .................4

Figure 2-2. Terzaghi (1943) failure surface (from Fellenius 2015, pp. 6-2) .................5

Figure 2-3. General shear failure for embedded shallow foundation (from Vesić

1973) ...........................................................................................................................6

Figure 2-4. (a) General, (b) local, and (c) punching shear failure (Vesić 1973) ...........6

Figure 2-5. Comparison of different Nγ factors. (Left: lin-lin ordinate, Right:

lin-log ordinate) ........................................................................................................10

Figure 2-6. (a) Water content vs. matric suction. (b) two grains in contact with

water between contacts. ............................................................................................21

Figure 2-7. Forces acting on an individual particle (after Lu and Likos 2004). ..........22

Figure 2-8. (a) SWCC for sand, silt and clays, (b) corresponding suction stress

profile (Lu et al. 2010) ..............................................................................................26

Figure 2-9. Matric suction profiles at various surface flux boundary conditions

for clay (Lu and Griffiths 2004). ...............................................................................28

Figure 3-1. Soil water characteristic curve for this example soil. ..............................40

Figure 3-2. Suction stress profile for this example soil. ..............................................41

Figure 3-3. Failure surface corresponding to ϕ' = 20°, B = 2 m, and D = 0.5 m. ........42

Figure 3-4. Saturation of the soil profile for the example. ...........................................43

Figure 3-5. Failure surface of the shallow foundation, colored by the saturation

profile. .......................................................................................................................43

Figure 3-6. Sketch of the conceptual deep foundation considered in this work. .........48

Figure 3-7. Variation of At-Rest earth pressure coefficient in partially saturated

soils. ..........................................................................................................................51

Figure 4-1. SWCC for Sollerod sand (Steensen-Bach et al. 1987). .............................56

Figure 4-2. Load displacement curves for Sollerod sand with varying

groundwater tables ....................................................................................................58

LIST OF FIGURES (CONTINUED)

Figure Page

Figure 4-3. Measured bearing capacity vs. calculated bearing capacity for

Steensen-Bach et al. (1987) ......................................................................................59

Figure 4-4. Calculated bearing capacity vs. GWT depth for Steensen-Bach et

al. (1987) ...................................................................................................................60

Figure 4-5. Load displacement curve from Briaud and Gibbens (1997) .....................62

Figure 4-6. Comparison of measured bearing capacity to the conventional and

modified approach for Briaud and Gibbens (1997) ..................................................63

Figure 4-7. Comparison of measured bearing capacity with respect to footing

width (B) plus embedded depth (D) for Briaud and Gibbens (1997) .......................64

Figure 4-8. Predicted bearing capacity with respect to footing with using the

modified approach for the soil data provided by Briaud and Gibbens (1997)..........65

Figure 4-9. Hyperbolic fits to load displacement curve at Vatthammar site

(Larsson 1997) ..........................................................................................................67

Figure 4-10. Fitted hyperbolic load displacement curve for Viana da Fonseca

and Sousa (2002) data ...............................................................................................70

Figure 4-11. Fitted SWCC for the Rojas et al. (2007) data. ........................................73

Figure 4-12. Linearly interpolated matric suction profile for Rojas et al. (2007)

data ............................................................................................................................74

Figure 4-13. Comparison of calculated qult for the Rojas et al. (2007) data using

the modified and unmodified bearing capacity equation. .........................................74

Figure 4-14. Fitted SWCC using van Genuchten (1980) (after Vanapalli and

Mohamed 2007) ........................................................................................................77

Figure 4-15. Comparison of actual bearing capacity to predictions from this

work and Vanapalli and Mohamed (2007) ...............................................................78

Figure 4-16. Bearing capacity vs. variation in average matric suction from this

work and Vanapalli and Mohamed (2007) ...............................................................79

Figure 4-17. Comparison of measured and predicted bearing capacity for 150

mm surface plate for Vanapalli and Mohamed (2013) .............................................80


Figure Page


work and Vanapalli and Mohamed (2013) for a 150×150 mm plate loaded on

the surface. ................................................................................................................81

Figure 4-19. SWCC and suction stress profile for Vanapalli and Mohamed

(2013) soil. ................................................................................................................82

Figure 4-20. Comparison of measured and calculated bearing capacity for 150

mm embedded plate for Vanapalli and Mohamed (2013) ........................................83


work and Vanapalli and Mohamed (2013) for a 150×150 mm plate embedded

150 mm. ....................................................................................................................84

Figure 4-22. Soil water characteristic curve for Hostun sand (after Wuttke et al.

2013) .........................................................................................................................86

Figure 4-23. Calculated and measured bearing capacities compared to the

average matric suction at D and D + B. ....................................................................87

Figure 4-24. Comparison of actual bearing capacity to predictions from the

conventional and modified approach for Wuttke et al. (2013) .................................88

Figure 4-25. Measured bearing capacity vs. predicted bearing capacity for

database of load tests in Chapter 4. ...........................................................................89

Figure 5-1. (a) Soil water characteristic curves for the 12 USDA Textural

Classes using van Genuchten parameters from Carsel and Parrish (1988).

Curves for sand, clay and silt loam are highlighted. (b) USDA Textural

Triangle (USDA 2016). ............................................................................................92

Figure 5-2. Shallow foundation bearing capacity profile of clay, silt, and sand

at varying friction angles. Note changing ordinate across figures. ...........................94

Figure 5-3. Shallow foundation bearing capacity vs. zgwt - D for clay, silt, and

sand at varying depths of embedment. Note changing ordinate across figures.

...................................................................................................................................95

Figure 5-4. Shallow foundation bearing capacity vs. groundwater table depth

for clay, silt, and sand at varying depths of embedment. Note changing

ordinate across figures. .............................................................................................96


with varying rates of flux. Note changing ordinate across figures. ..........................97


Figure Page


with varying θs. Note changing ordinate across figures............................................99


with varying θr. Note changing ordinate across figures..........................................100

Figure 5-8. Shallow foundation bearing capacity profile with varying α. .................101

Figure 5-9. Shallow foundation bearing capacity profile with varying n. Note

changing ordinate across figures. ............................................................................102

Figure 5-10. Varying van Genuchten’s α at various ground table depths for n =

1.1............................................................................................................................104


1.5............................................................................................................................104


3.0............................................................................................................................105

Figure 5-13. Varying van Genuchten’s n at various ground table depths for 𝛼 =

0.01 kPa-1. ...............................................................................................................106


0.1 kPa-1. .................................................................................................................106


1 kPa-1. ....................................................................................................................107

Figure 5-16. Comparison of the predicted bearing capacity for a sand using the

modified and conventional approach at various friction angles, D = 0 m. .............109

Figure 5-17. Comparison of the predicted bearing capacity for a sand using the

modified and conventional approach at various friction angles, D = 1.5 m. ..........110

Figure 5-18. Comparison of the predicted bearing capacity between the

modified and conventional approach at various friction angles for a material

with D = 0 m, n = 3, and α = 0.1 kPa-1. ...................................................................111

Figure 5-19. Comparison of the predicted bearing capacity between the

modified and conventional approach at various friction angles for a material

with D = 1.5 m, n = 3, and α = 0.1 kPa-1. ................................................................111


Figure Page

Figure 5-20. Comparison of the predicted bearing capacity for silt while varying

the footing width. This soil has zw = 4 m, 𝜙′ = 30˚ and D = 0.5 m. .......................112

Figure 5-21. Table of figures for qmod/qunmod. The x and y axis of the table

correspond to various ϕ' and zw/B ratios respectively. For each individual

figure, x and y axes are αzwγw and n, respectively. ..................................................114

Figure 5-22. Normailzation of the soil water characteristic curve.............................115

Figure 5-23. Calculated bearing capacity for hypothetical clay with D = 0 m

from Vahedifard and Robinson (2015) compared to modified approach in this

current work (left 𝜙′ = 25°, right 𝜙′ = 20°). .........................................................119

Figure 5-24. Calculated bearing capacity for hypothetical clay with D = 1.5 m

from Vahedifard and Robinson 2015 compared to modified approach in this

current work (left 𝜙′ = 25°, right 𝜙′ = 20°). Note changing ordinate across

figures. ....................................................................................................................119

Figure 5-25. Calculated bearing capacity for hypothetical sand with D = 0 from

Vahedifard and Robinson 2015 compared to modified approach in this current

work. .......................................................................................................................121

Figure 5-26. Calculated bearing capacity for hypothetical sand with D = 1.5 m

from Vahedifard and Robinson (2015) compared to modified approach in this

current work. ...........................................................................................................121

Figure 5-27. Comparison of calculated bearing capacity profiles using the

proposed approach, the Vesić solution, and Vahedifard and Robinson (2015)

for a surface foundation (left: 𝜙′ = 35° × 1.1, right: 𝜙′ = 30° × 1.1). .................122

Figure 5-28. Comparison of calculated bearing capacity profiles using the

proposed approach, the Vesić solution, and Vahedifard and Robinson (2015)

for an embedded foundation (left: 𝜙′ = 35°, right: 𝜙′ = 30°). ..............................123

Figure 5-29. Suction stress profile above the groundwater table for theoretical

sand, silt, and clay (θs = 0.4, θr = 0.06, q = 0 m/s for each figure). Note

changing abscissa across figures. ............................................................................125

Figure 5-30. Vertical effective stress as a function of depth from soil surface for

theoretical sand, silt, and clay (θs = 0.4, θr = 0.06, q = 0 m/s, Gs = 2.65 for

each figure). ............................................................................................................126


Figure Page

Figure 5-31. Modified β’ as a function of depth from soil surface for theoretical

sand, silt, and clay (θs = 0.4, θr = 0.06, q = 0 m/s, Gs = 2.65, ν = 0.3, δ = 30˚

for each figure). .......................................................................................................127

Figure 5-32. Side resistance as a function of depth from soil surface for

theoretical sand, silt, and clay (θs = 0.4, θr = 0.06, q = 0 m/s for each figure). ......128

Figure 5-33. Side resistance profiles for theoretical sand, silt and clay (𝜈 = 0.3,

𝐺𝑠 = 2.65, 𝜃𝑟 = 0.06, and 𝜃𝑠 = 0.4) (unit side resistance given in force/unit

perimeter) ................................................................................................................130

Figure 5-34. Side resistance profiles for theoretical silt with varying specific

gravity (𝑛 = 1.5, 𝛼 = 0.1 kPa-1, 𝜈 = 0.3, 𝜃𝑟 = 0.06, and 𝜃𝑠 = 0.4) (unit side

resistance given in force/unit perimeter) .................................................................131

Figure 5-35. Side resistance profiles for theoretical silt with varying residual

water content (𝑛 = 1.5, 𝛼 = 0.1 kPa-1, 𝜈 = 0.3, 𝐺𝑠 = 2.65, and 𝜃𝑠 = 0.4) (unit

side resistance given in force/unit perimeter) .........................................................132

Figure 5-36. Side resistance profiles for theoretical silt with varying saturated

water content (𝑛 = 1.5, 𝛼 = 0.1 kPa-1, 𝜈 = 0.3, 𝐺𝑠 = 2.65, and 𝜃𝑟 = 0.06) (unit

side resistance given in force/unit perimeter) .........................................................133

Figure 5-37. Suction stress profiles of theoretical clay for flowrates of q = -

0.2ks, 0, and 0.2ks. ...................................................................................................134

Figure 5-38. Side resistance profiles of theoretical clay for flowrates of q = -

0.2ks, 0, and 0.2ks (𝑛 = 1.5, 𝛼 = 0.1 kPa-1, 𝜈 = 0.3, 𝐺𝑠 = 2.65, and 𝜃𝑟 = 0.06)

(unit side resistance given in force/unit perimeter) .................................................135

Figure 5-39. Suction stress profiles of theoretical silt for flowrates of q = -0.2ks,

0, and 0.2ks. .............................................................................................................136

Figure 5-40. Side resistance profiles of theoretical silt for flowrates of q = -

0.2ks, 0, and 0.2ks (𝑛 = 1.5, 𝛼 = 0.1 kPa-1, 𝜈 = 0.3, 𝐺𝑠 = 2.65, and 𝜃𝑟 = 0.06)

(unit side resistance given in force/ unit perimeter) ................................................137

Figure 5-41. Side resistance profiles for 𝜈 = 0.2 at various 𝑛 and 𝛼 values. ............138




Figure Page

Figure 5-44. K0 as a function of depth from soil surface for α = 0.01 kPa-1, and

n = 3.0 for fixed groundwater table depths (θs = 0.4, θr = 0.06, q = 0 m/s, Gs

= 2.65, ν = 0.2, δ = 30˚) ..........................................................................................141

Figure 5-45. Unit weight profile for theoretical clay (θs = 0.4, θr = 0.06, q = 0

m/s, Gs = 2.65, α = 0.01 kPa-1, n = 1.1) ..................................................................143

Figure 5-46. Side resistance profiles of theoretical clay for 𝜈 = 0.2, 0.3, and 0.4

(𝑛 = 1.1, 𝛼 = 0.01 kPa-1, 𝐺𝑠 = 2.65, 𝜃𝑠 = 0.4 and 𝜃𝑟 = 0.06). Note changing

abscissa across figures. ...........................................................................................144

Figure 5-47. Probability histogram of silt loam properties from Carsel and

Parrish (1988)..........................................................................................................147

Figure 5-48. Probability histogram of silt loam properties used in this work

(after Carsel and Parrish 1988) ...............................................................................147

Figure 5-49. Cumulative distribution function of silt loam (after Carsel and

Parrish 1988) ...........................................................................................................148

Figure 5-50. 50,000 Monte Carlo realizations of silt loam, calculating bearing

capacity. From left to right (1) cumulative distribution of calculated bearing

capacities, (2) distribution of 𝛼 and 𝑛 input parameters, (3) probability

histogram of bearing capacities, and (4) summary of the plotted percentiles

and other data. .........................................................................................................149

Figure 5-51. Gamma, Weibull, and lognormal distribution functions fitted to

the CDF for silt loam. .............................................................................................150

Figure 5-52. Probability histogram of sand properties used in this work (after

Carsel and Parrish 1988) .........................................................................................151

Figure 5-53. 50,000 Monte Carlo realizations of sand, calculating bearing




and other data. .........................................................................................................152


the CDF for sand. ....................................................................................................153

Figure 5-55. Probability histogram of clay properties used in this work (after

Carsel and Parrish 1988) .........................................................................................154


Figure Page

Figure 5-56. 50,000 Monte Carlo realizations of clay, calculating bearing




and other data. .........................................................................................................155


the CDF for clay......................................................................................................156

LIST OF TABLES

Table Page

Table 2-1. Typical unsaturated soil properties by USDA textural class (Carsel

and Parrish 1988) ......................................................................................................31

Table 3-1. Soil properties for theoretical example of shallow foundation bearing

capacity in an unsaturated soil. .................................................................................39

Table 3-2. Foundation and groundwater properties for theoretical example in

Chapter 3. ..................................................................................................................41

Table 4-1. Properties of Sollerod sand and plate (Steensen-Bach et al. 1987) ............57

Table 4-2. Actual and Predicted results for the Sollerod load tests (Steensen-

Bach et al. 1987) .......................................................................................................59

Table 4-3. Soil properties at 3.0 m using hand auger (Briaud and Gibbens 1997)

..................................................................................................................................61

Table 4-4. Bearing capacity comparison for Briaud and Gibbens (1997) ...................62

Table 4-5. Soil properties from Larson (1997) ............................................................66

Table 4-6. Results from static load tests at Vatthammar (Larson 1997) .....................67

Table 4-7. Calculated bearing capacities at various GWT levels, for

Vatthammar (Larsson 1997) .....................................................................................68

Table 4-8. Calculated bearing capacities by varying q, for Vatthammar (Larsson

1997) .........................................................................................................................68

Table 4-9. Soil properties used for Viana da Fonseca and Sousa (2002) ....................70

Table 4-10. Soil properties used for Rojas et al. (2007) ..............................................72

Table 4-11. Matric suction from tests and maximum bearing capacity from

hyperbolic fit for Rojas et al. data (2007). ................................................................73

Table 4-12. Soil properties used in Vanapalli and Mohamed (2007) ..........................76

Table 4-13. Soil properties for Hostun sand (Wuttke et al. 2007) ...............................85

Table 5-1. Soil properties used in this parametric study ..............................................91

Table 5-2. Input parameters for clay used in Vahedifard and Robinson (2015). .......118

Table 5-3. Input parameters for sand used in Vahedifard and Robinson (2015). ......120

LIST OF TABLES (CONTINUED)

Table Page

Table 5-4. Soil properties used in this parametric study ............................................129

Table 5-5. R2, mean, standard deviation, and coefficient of variation predicted

from the Gamma, Weibull, and lognormal distribution functions fitted to silt

loam data. ................................................................................................................151


from the Gamma, Weibull, and lognormal distribution functions fitted to sand

data. .........................................................................................................................153


from the Gamma, Weibull, and lognormal distribution functions fitted to clay

data. .........................................................................................................................156

1

1. Introduction

1.1. Statement of Problem

Geotechnical engineering has long been established as a discipline that studies the in-

teraction of structures with soil. Soils inherently are complex material, being composed

of a solid phase (individual grains), a liquid phase (water), and a gaseous phase (air).

Analysis of foundation performance, by means of bearing capacity or settlement cal-

culations, requires many assumptions. These assumptions often include that soils either

exist in a dry state or a completely saturated state. Present studies, however, show that

suction stresses above the ground water table will have an impact on the performance

of a foundation. Thus, it is important, above the water table, to consider the effects of

partial saturation.

1.2. Purpose and Scope

This work seeks to incorporate the results of recent studies on partially saturated soils

– including soil water characteristic curves, matric suction profiles, assumed at-rest

earth pressure coefficients, and empirical/theoretical Bishop’s 𝜒 relationships (which

correlates matric suction to suction stress) – into the current framework for calculating

the bearing capacity for shallow and deep foundations. Currently, engineers often em-

ploy some form of Terzaghi’s bearing capacity equation for shallow foundations and

the 𝛽-method for an effective stress analysis of deep foundations (among many other

methods). These are the approaches that are considered herein.

1.3. Outline

This thesis first discusses the current state of shallow foundation design (Section 2.1)

and deep foundation design (Section 2.2) – equations and approaches that are used to

calculate and predict an ultimate bearing capacity. Then recent literature covering un-

saturated soil mechanics is discussed in Section 2.3. Section 3.1 presents research ob-

jectives. After these basic considerations are made, an explanation of the theoretical

development employed in this work is discussed in Section 3.2 and 3.3. These sections

will discuss how the current methods for calculating shallow and deep bearing capacity

2

can be modified to include the effects of partial saturation and suction. Chapter 4 covers

a comparative study between shallow foundation load tests from the literature and the

bearing capacity calculated by the modified approach discussed in Chapter 3. Chapter

5 is composed of two parts, a parametric study for both shallow and deep foundations,

and simple Monte Carlo simulations for shallow foundations in partially saturated soils

using the procedures proposed by Carsel and Parrish (1988) to vary unsaturated soil

parameters. This chapter will provide some insight on how variation in parameter space

affects bearing capacity. Finally, this thesis will be concluded in Chapter 6 with a sum-

mary of the results presented in this work, a discussion on the results, conclusions, and

a discussion on future work.

1.4. Qualifications and Limitations

The work presented in this thesis is based on well-recognized theories for effective

stress, suction stress, the water-content suction relationship for porous media, and bear-

ing capacity. These theories have been generally verified in the archival literature and

are broadly accepted in research and practice, but they have not previously been com-

bined in the manner presented herein. To the best of the author’s knowledge, the deri-

vations presented in this thesis are correct and consistent with the underlying theories.

However, no attempt has been made to verify or validate many of the presented results.

As such, some of the boundary cases considered (i.e., those at the extremes of possible

ranges of applicability) may result in predictions that are demonstrably outside of the

range of commonly accepted values.

This thesis seeks to lay the groundwork for the incorporation of the effects of partial

saturation in the practice of foundation design. The hope is that this seminal effort will

spur others to perform laboratory tests, develop physical models, and execute numeri-

cal simulations to further advance the understanding of the role that partial saturation

plays on the bearing capacity of foundations and how it should be considered in prac-

tice.

3

2. Background

2.1. Shallow Foundations

2.1.1. General Bearing Capacity Theory for Shallow Foundations

The calculation of shallow foundation bearing capacity has been a topic of research for

the past century and is still a topic of modern research. Prandtl (1920) studied the

punching resistance of metals, developing bearing capacity factors to assess the

strength of metals, which is still used today. Reissner (1924) subsequently developed

an addition bearing capacity factor, 𝑁𝑞. Terzaghi (1943) refined these works, creating

the foundation of modern geotechnical engineering and specifically a framework for

calculating the settlement and bearing capacity of shallow foundations. His work set

the precedence for future research in geotechnical engineering and many subsequent

researchers have modified his work, modifying bearing capacity equations, and bearing

capacity factors (Meyerhof 1951; 1963; De Beer 1970; Hansen 1970; Vesić 1973;

Kumbhojkar 1993).

The ultimate bearing capacity and failure of a shallow foundation has been defined in

a variety of ways. Terzaghi (1943) provided two ways bearing capacity can be deter-

mined from load settlement curves. The first method is the identification of a peak

strength value, which is indicated by the expression 𝑞𝐷 in Figure 2-1. This is defined

as the critical state at which the soil will deform plastically with no additional increase

in stress. Defining ultimate bearing capacity with this critical state seems the most rea-

sonable, however, this state is not often achieved as many soils will continue to increase

capacity while loading (strain hardening), or when the soil strength/foundation size is

large enough such that the critical state cannot be reached. In this case, a criterion for

peak strength must be set. In Figure 2-1, the method for calculating bearing 𝑞𝐷′ is the

intercept of two tangent lines, lines extending from the plastic and elastic region. Other

researchers have defined the ultimate state by defining a limiting settlement criterion

4

or by fitting a hyperbolic function (Kondner 1963). In this work, ultimate bearing ca-

pacity will be defined by either a peak strength, or the asymptote of a fitted hyperbolic

curve as proposed by Kondner (1963).

Figure 2-1. Definitions of ultimate bearing capacity (from Terzaghi 1943)

Terzaghi (1943) states that the ultimate bearing capacity is the load (applied over the

bearing area) that causes the failure above to occur. A shallow footing is a foundations

for which the width B is greater than or equal to the embedded depth of the footing D.

The length of the footing L is greater than the width B. General shear failure is the

assumed mode of failure used by Terzaghi (1943) in his ultimate bearing capacity equa-

tion and in subsequent bearing capacity solutions (Meyerhof 1951; De Beer 1970; Han-

sen 1970; Vesić 1973). General shear failure (Figure 2-4 (a)) describes failure where

soil slides across a failure surface that extends in one of two outward directions from

the edge of the foundation to the surface of the soil. This failure mode corresponds to

the critical state described previously.

5

Figure 2-2. Terzaghi (1943) failure surface (from Fellenius 2015, pp. 6-2)

There are three regions in the general shear failure surface: (1) the active wedge zone;

(2) the radial shear zone; and (3) the passive zone. The active zone is located immedi-

ately beneath the footing and declines at 45° + 𝜙/2 from the horizontal. The principal

stress in this zone is vertical. The radial shear zone extends the failure surface down

and away from the center of the footing and then extends into the passive zone. The

shape of the radial shear zone is defined by the logspiral equation, 𝑟 = 𝑟0𝑒θ tan𝜙. The

passive zone extends from the radial shear zone at an inclination of 45° − 𝜙/2 from

the horizontal. The active zone typically moves vertically down, pushing the passive

zones, while the other two zones slide horizontally. The underlying assumption is that

the soils lying above the failure surface exist in a state of plastic failure. Terzaghi (1943)

used limit equilibrium to assess the bearing capacity across the failure surface shown

in Figure 2-2. For an embedded foundation, Terzaghi (1943) suggested that shearing

resistance above the embedded depth of the footing be ignored, shown by the line b-c

in Figure 2-3, and instead considered as a uniform surcharge pressure 𝑞 = 𝛾𝐷, which

is also taken to be the effective stress at the depth of embedment.

6

Figure 2-3. General shear failure for embedded shallow foundation (from Vesić 1973)

Two other common modes of failure are punching shear failure (Figure 2-4 (c)), char-

acterized by the immediate compression of soil underneath the footing, and local shear

failure (Figure 2-4 (b)), characterized by failure patterns that only exist immediately

below the foundation (Terzaghi 1943; Vesić 1973). Local shear failure is similar to

general shear failure in that there is a wedge slip surface, but the failure surface does

not extend through to the surface (Vesić 1973).

Figure 2-4. (a) General, (b) local, and (c) punching shear failure (Vesić 1973)

The shearing resistance of the soil at the failure surface is determined with the Mohr-

Coulomb failure criterion:

𝜏𝑓 = 𝑐′ + 𝜎′ tan 𝜙′ 2-1

7

where 𝜏𝑓 is the shear stress at failure, c’ is the cohesion in the soil, 𝜎′ is the effective

stress and 𝜙′ is the friction angle. With the Mohr-Coloumb failure criterion, Terzaghi

(1943) proposed a bearing capacity for plane-strain failure of a strip (continuous) foot-

ing, and modified equations for square and circular footings. The ultimate bearing ca-

pacity equation is described in the equations below:

𝑞𝑢𝑙𝑡 = 𝑐′𝑁𝑐 + 𝜎𝑧𝐷′ 𝑁𝑞 + 0.5𝛾′𝐵𝑁𝛾 for continuous footings 2-2

𝑞𝑢𝑙𝑡 = 1.3𝑐′𝑁𝑐 + 𝜎𝑧𝐷′ 𝑁𝑞 + 0.4𝛾′𝐵𝑁𝛾 for square footings 2-3

𝑞𝑢𝑙𝑡 = 1.3𝑐′𝑁𝑐 + 𝜎𝑧𝐷′ 𝑁𝑞 + 0.3𝛾′𝐵𝑁𝛾 for circular footings 2-4

where 𝑐′ is the cohesion of a soil, 𝜎𝑧𝐷′ is the vertical effective stress at the depth of

footing embedment, 𝛾′ is the effective unit weight, 𝐵 is the footing width, and

𝑁𝑐, 𝑁𝑞 , and 𝑁𝛾 are bearing capacity factors. Theses equations were derived from limit

equilibrium, satisfying force and moment equilibrium. The active zone is loaded, push-

ing the logspiral and passive zone, which resist movement by passive and shear forces.

The bearing capacity factors from his derivation are in Eqs. 2-5, 2-7, and 2-8. Equations

for 𝑁𝑐 and 𝑁𝑞 had already been established by Prandtl (1920) and Reissner (1924) in

Eqs. 2-5 and 2-6, respectively. Note that as 𝜙′ → 0° 𝑁𝑐 = 5.14 for Terzaghi (1943)

and = 5.7 for Prandtl (1920).

Prandtl (1920)

𝑁𝑐 = (𝑁𝑞 − 1) cot 𝜙′ 2-5

Reissner (1924)

𝑁𝑞 = 𝑒𝜋 tan 𝜙′tan2(45° + 𝜙′/2) 2-6

Terzaghi (1943)

𝑁𝑞 =

𝑒(270°−𝜙′) tan 𝜙′

2 cos2 (45° +𝜙′2 )

2-7

8

𝑁𝛾 =

1

2tan 𝜙′ (

𝐾𝑃𝛾

cos2 𝜙′− 1) 2-8

where 𝐾𝑝𝛾 is the passive earth pressure coefficients and all other variables are as pre-

viously defined.

2.1.2. Various Improvements on Bearing Capacity Equation

Many researchers have modified the expressions within the general bearing capacity

equation proposed by Terzaghi in 1943. Some works have modified the bearing capac-

ity factors, 𝑁𝛾, 𝑁𝑐, and, 𝑁𝑞, while others have added additional terms, modifying the

equation for shape, inclination, and depth factors (Meyerhof 1961; Hansen 1970; De

Beer 1970; Vesić 1973; Kumbhojkar 1993). This section will only cover a few modi-

fications that have been proposed. Factors concerning inclination and slope inclination

will not be included in this literature review.

In 1961, Meyerhof introduced a revised 𝑁𝛾. The equation for this factor is:

𝑁𝛾 = (𝑁𝑞 − 1) tan(1.4𝜙) 2-9

His later work (1963) also included shape factors and depth factors, accounting for the

different shapes of rectangular footings and embedment depths.

Meyerhof Shape Factors:

𝑠𝑐 = 1 + 0.2𝑁𝜙𝐵/𝐿 2-10

𝑠𝑞 = 𝑠𝛾 = 1 if 𝜙 = 0° 2-11

𝑠𝑞 = 𝑠𝛾 = 1 + 0.1𝑁𝜙𝐵/𝐿 if 𝜙 > 10° 2-12

Meyerhof Depth Factors:

𝑑𝑐 = 1 + 0.2√𝑁𝜙𝐷/𝐵 2-13

9

𝑑𝑞 = 𝑑𝛾 = 1 if 𝜙 = 0° 2-14

𝑑𝑞 = 𝑑𝛾 = 1 + 0.1√𝑁𝜙𝐷/𝐵 if 𝜙 > 10° 2-15

where 𝑁𝜙 = tan2 (1

4𝜋 +

1

2𝜙), which is the friction angle bearing capacity factor. As

mentioned, the framework Terzaghi (1943) proposed neglecting shearing resistance

from soil above the depth of embedment (shown in Figure 2-3). The depth factors from

Eqs. 2-13, 2-14, and 2-15 allow for the consideration of additional shear strength from

the previously neglected soil. For these shape and depth factors, linear interpolation

must be used if the friction angle is between 0 and 10 degrees.

Eqs. 2-10 – 2-15 are intended to modify Terzaghi’s equation for continuous (strip)

footings, Eq. 2-2. Considering these additional factors, the predicted ultimate bearing

capacity for a rectangular shallow foundation becomes:

𝑞𝑢𝑙𝑡 = 𝑐′𝑁𝑐𝑠𝑐𝑑𝑐 + 𝜎𝑧𝐷′ 𝑁𝑞𝑠𝑞𝑑𝑞 + 0.5𝛾′𝐵𝑁𝛾𝑠𝛾𝑑𝛾 2-16

In 1970, Hansen proposed different factors to be used in Eq. 2-16. His work includes

modified shape and depth factors, and another expression for 𝑁𝛾 based entirely on em-

pirical data from previous researchers.

Hansen Shape Factors:

𝑠𝑐 = 1 + 0.2𝐵/𝐿 2-17

𝑠𝑞 = 1 + sin(𝜙) 𝐵/𝐿 2-18

𝑠𝛾 = 1 − 0.4𝐵/𝐿 2-19

Hansen Depth Factors:

𝑑𝑐 = 1 + 0.4𝑘 2-20

𝑑𝛾 = 1 2-21

𝑑𝑞 = 1 + 2 tan 𝜙(1 − sin 𝜙)2𝑘 2-22

10

𝑘 = [

𝐷

𝐵 if

𝐷

𝐵≤ 1

tan−1 (𝐷

𝐵) if

𝐷

𝐵> 1

2-23

Hansen 𝑵𝜸

𝑁𝛾 = 1.5(𝑁𝑞 − 1) tan 𝜙 2-24

Vesić (1973) also established new bearing capacity factors to be used in Terzaghi’s

bearing capacity equation. His considerations were made on the basis of experimental

load tests:

Vesić Shape Factors:

𝑠𝑐 = 1 + (𝐵/𝐿)(𝑁𝑞/𝑁𝑐) 2-25

𝑠𝑞 = 1 + tan(𝜙) 𝐵/𝐿 2-26

𝑠𝛾 = 1 − 0.4𝐵/𝐿 2-27

Vesić 𝑵𝜸

𝑁𝛾 = 2(𝑁𝑞 + 1) tan 𝜙 2-28

Figure 2-5. Comparison of different Nγ factors. (Left: lin-lin ordinate, Right: lin-log ordinate)

0 10 20 30 40

50

100

150

200

Meyerhof (1961)

Vesic (1973)

Hansen (1970)

Kumbhojkar (1993)

Friction Angle [deg]

Be

arin

g C

apac

ity F

acto

r

0 10 20 30 400.1

1

10

100

1 103

Meyerhof (1961)

Vesic (1973)

Hansen (1970)

Kumbhojkar (1993)

Friction Angle [deg]

11

Figure 2-1 compares the above three mentioned 𝑁𝛾 factors and the 𝑁𝛾 from Kumbho-

jkar (1993). Each method shows agreement for small friction angles (on a linear scale),

but the methods diverge for larger friction angles. Plotting on a logarithmic scale indi-

cates that the magnitude of difference between values at small friction angles is greater

than for large friction angles.

Meyerhof (1955) was the first to propose a solution for effective unit weight when the

groundwater table exists close to the base of the foundation. He proposed that soil unit

weight vary linearly between 𝛾𝑏 and 𝛾𝑚 for groundwater table depths of D (depth of

embedment) and D + B (footing width) respectively. This is summarized by Equation

2-29. Here 𝛾𝑏 is the buoyant unit weight of the soil and 𝛾𝑚 is the material unit weight

of the soil. This assumption is often still made in practice (Salgado 2008).

𝛾′ = [

𝛾𝑏 = 𝛾𝑠𝑎𝑡 − 𝛾𝑤 if 𝑧𝑤 < 𝐷

𝛾𝑏 +𝑧𝑤−𝐷

𝐵(𝛾𝑚 − 𝛾𝑏) if 𝐷 ≤ 𝑧𝑤 ≤ 𝐷 + 𝐵

𝛾𝑚 if 𝑧𝑤 > 𝐷 + 𝐵

2-29

2.1.3. Recent Developments

Recently, researchers have begun studying the effects of partial saturation and suction

stress in foundation performance through foundation load tests in partially saturated

soils (Steensen-Bach et al. 1987; Oloo 1997; Costa et al. 2003; Mohamed and Vanapalli

2006; Vanapalli and Mohamed 2013; Wuttke et al. 2013) and by continued modifica-

tion of the conventional bearing capacity equation (Vanapalli and Mohamed 2007; Oh

and Vanapalli 2008; Vahedifard and Robinson 2015) discussed in the previous section.

These researchers have shown that partially saturated soils, especially silts and clays,

often have bearing capacities greater than the predicted bearing capacity for a com-

pletely dry or completely saturated soil.

12

To account for partial saturation, the cohesion term is typically modified within the

bearing capacity equation to account for apparent cohesion caused by suction stresses

(Fredlund et al. 2012; Vanapalli and Mohamed 2007; Vahedifard and Robinson 2015).

Thus, unsaturated soil mechanics can be easily integrated into the convention bearing

capacity framework. With this consideration, Fredlund et al. (2012) suggest a stress

state variable approach, which uses a constant, 𝜙𝑏, (Fredlund 1978), which is a friction

angle that describes the contribution of strength due to partial saturation for the soil.

This contribution to strength is discussed more closely in Section 2.3.5.

Vanapalli and Mohamed (2007) derived a closed-form solution based on the Fredlund

(1978) 𝜙𝑏 expression. The final closed-form solution for the ultimate bearing capacity

of a shallow foundation in unsaturated soils is:

𝑞𝑢𝑙𝑡 = [𝑐′ + (𝑢𝑎 − 𝑢𝑤)𝑏(1 − 𝑆𝜓) tan 𝜙′ + (𝑢𝑎 −

𝑢𝑤)𝐴𝑉𝑅𝑆𝜓 tan 𝜙′] 𝑁𝑐 [1 + (𝑁𝑞

𝑁𝑐) (

𝐵

𝐿)] + 0.5𝐵𝛾𝑁𝛾 [1 − 0.4

𝐵

𝐿]

2-30

In this work, 𝑢𝑎 and 𝑢𝑤 are the air and water pressures within soil pores (the difference

is known as matric suction), (𝑢𝑎 − 𝑢𝑤)𝑏 is the air entry value (or the pressure differ-

ence at which air enters the soil pores at a significant rate), (𝑢𝑎 − 𝑢𝑤)𝐴𝑉𝑅 is the average

matric suction at the bottom of the foundation and stress bulb, 𝑆 is the average degree

of saturation, and finally 𝜓 is a bearing capacity fitting parameter. There are two things

to note about Equation 2-30: (1) two additional terms have been added to the cohesion

term c', (𝑢𝑎 − 𝑢𝑤)𝑏(1 − 𝑆𝜓) tan 𝜙′ captures the contribution of strength due to matric

suctions less than the air-entry value, while (𝑢𝑎 − 𝑢𝑤)𝐴𝑉𝑅𝑆𝜓 tan 𝜙′ captures the con-

tribution of strength due from matric suctions that are greater than the air-entry value;

and (2) additional strength due to matric suction is nonlinear, thus, tan 𝜙𝑏 is replaced

by 𝑆𝜓 and tan 𝜙′. In general, this equation agrees well with the laboratory results pre-

sented in the original paper.

13

Vanapalli and Mohamed (2013) and Vahedifard and Robinson (2015) have modified

Equation 2-30 to account for embedment depth and hydrostatic (steady) flow, which

manifests in the (𝑢𝑎 − 𝑢𝑤) expressions. This modified equation is:

𝑞𝑢𝑙𝑡 = {𝑐′ + (𝑢𝑎 − 𝑢𝑤)𝑏(1 − 𝑆𝑒,𝐴𝑉𝑅) tan 𝜙′ + [(𝑢𝑎 − 𝑢𝑤)𝑆𝑒]𝐴𝑉𝑅 tan 𝜙′}𝑁𝑐𝜉𝑐

+ 𝑞0𝑁𝑞𝜉𝑞 + 0.5𝛾𝐵𝑁𝛾𝜉𝛾 2-31

𝑆𝜓 has been replaced with 𝑆𝑒, the effective saturation, which is 𝑆𝑒 = (𝑆 − 𝑆𝑟)/(1 − 𝑆𝑟).

𝑆𝑟 is the residual saturation of the soil, which is the minimum amount of water the soil

pores will retain. Here, matric suction 𝑢𝑎 − 𝑢𝑤 and effective saturation 𝑆𝑒 are averaged

across a foundation stress bulb, which is considered to be from a depth D to D + 1.5B.

2.2. Deep Foundations

2.2.1. Analytical Theory

The strength and settlement of a deep foundation is a function of multiple factors, rang-

ing from the method of installation to the properties of the soil (Meyerhof 1976). The

ultimate bearing load is difficult to determine because it can only be achieved when the

foundation is in a plunging failure (Salgado 2008). While ultimate bearing load is used

in the literature, it is difficult to attain in practice as it requires very large loads and may

also not be possible due to soil hardening at the toe. Often, deep foundations capacities

are determined against serviceability requirements (limitations on total or differential

settlement or deflection). Ultimate resistance of a pile may be expressed in terms of the

toe bearing resistance, Qt, and the side resistance (skin friction or shaft), Qs. The ulti-

mate resistance of a pile is:

𝑄𝑢 = 𝑄𝑡 + 𝑄𝑠 = 𝑞𝑡𝐴𝑡 + ∑ 𝑓𝑠𝐴𝑠 2-32

where 𝑞𝑡 is the average unit bearing resistance across the area 𝐴𝑡 and 𝑓𝑠 is the unit side

resistance for a layer of soil with a surface area 𝐴𝑠 (Meyerhof 1976). Conceptually, this

14

equation implies that the deep foundation distributes the applied load 𝑄 first through

friction on the side of the foundation, with continual dissipation of load with depth

(O’Neill 1987). Eventually, the remaining force within the pile is applied at the toe.

Before toe bearing resistance can be fully mobilized, the limit of shaft resistance must

be achieved.

Side shear resistance requires horizontal stresses (or the stresses normal to the pile sur-

face) to develop friction (Burland 1973). While effective vertical stresses are relatively

easy to estimate and calculate, effective horizontal stresses are significantly more dif-

ficult, requiring the assumption of earth pressure coefficients. The earth pressure coef-

ficient 𝐾 is not simply just a function of soil type, but also time after installation, depth,

stress history, porewater conditions any many other conditions (O’Neill 2001). It is

difficult to characterize the effective stresses in partially saturated soils due to suction

stresses. Suction stresses may exist in partially saturated soil (clays and silts) causing

tension cracking to occur and lateral earth pressure coefficients to decrease (Lu and

Likos 2004). Once an estimation of 𝐾 is made, the maximum unit side resistance 𝑓𝑚𝑎𝑥

can be determined.

There are two popular method for calculating shaft friction of drilled shafts and piles,

the 𝛼 and 𝛽 method (Skempton 1959; Burland 1973). The 𝛼 method (Skempton 1959)

is a total stress solution used to evaluate the undrained shear strength in saturated clays.

This method is described by:

𝑓𝑠 = 𝛼𝑠𝑢 2-33

Although this is a popular method for calculating shaft resistance, there is little funda-

mental justification in its favor. Values for 𝛼 should not be extrapolated between pile

types and varying ground conditions (Burland 1973). The 𝛼 method has been criticized

because undrained shear strength is not a unique property, but is a function of loading

15

method. Another criticism is that side friction is an effective stress problem, relying on

normal stresses to achieve friction (O’Neill 2001; Burland 1973).

The 𝛽-method (Burland 1973) is an effective stress method for assessing the unit shear

resistance. The 𝛽-method incorporates the initial vertical shear stress in the soil. This

is described by:

𝑓𝑠 = 𝛽𝜎𝑣0′ 2-34

Numerous researchers have proposed empirical and theoretical equations for predicting

values of 𝛽. Typically these equations are a function of 𝜙′ and some earth pressure

coefficient, 𝐾, to relate vertical stresses with horizontal stresses. For a drilled shaft, 𝛽

is defined by Eq. 2-35:

𝛽 = 𝐾0 tan 𝛿 =𝜎ℎ

′

𝜎𝑣′tan 𝛿 = (1 − sin 𝜙′) tan 𝛿 2-35

The relationship between 𝜙′ and 𝛿 is a function of the roughness between the soil/foun-

dation interface. Generally, 𝛿 = 𝜙′ is assumed for a concrete drilled shaft (Kulhawy

1983). In this equation the normally-consolidated 𝐾0 (Jaky 1943) is shown (Meyerhof

1976; O’Neill 2001). This does not take into account over-consolidation, remolding,

elevated porewater pressures and other considerations during and after installation.

Stress history will not be considered as part of this research.

The basic premise of base or toe resistance is that resistance is a function of overburden

above the toe, fundamental soil properties such as soil density and friction angle, cohe-

sion, and the diameter of the foundation. Kulhawy (1983) uses the following equation

to predict toe bearing resistance:

16

𝑞𝑢𝑙𝑡 = 𝑐′𝑁𝑐𝜁𝑐𝑠𝜁𝑐𝑟𝜁𝑐𝑑 +

1

2𝐵𝛾𝑁𝛾𝜁𝛾𝑠 𝜁𝛾𝑟𝜁𝛾𝑑 + 𝑞𝑁𝑞𝜁𝑞𝑠𝜁𝑞𝑟𝜁𝑞𝑑 2-36

where 𝑐′ is cohesion, 𝛾 is the average soil unit weight, and 𝑞 is the effective vertical

stress at the toe. The bearing capacity factors for Eq. 2-36 are:

𝑁𝑞 = 𝑒𝜋 tan 𝜙 tan2(45° + 𝜙/2) 2-37

𝑁𝑐 = (𝑁𝑞 − 1) cot 𝜙 2-38

𝑁𝛾 = 2(𝑁𝑞 + 1) tan 𝜙 2-39

As with shallow foundation bearing capacity theory, as 𝜙 → 0°, 𝑁𝑐 → 5.14. For un-

drained loading in most clays, the bearing capacity equation reduces to approximately

𝑞𝑢𝑙𝑡 = 9𝑠𝑢. Correction factors used in Eq. 2-36 were originally proposed by Vesić

(1975) and Hansen (1970). The shape, depth, and rigidity factors are as follows:

Shape Factors:

𝜁𝑐𝑠 = 1 + 𝑁𝑞/𝑁𝑐 2-40

𝜁𝛾𝑠 = 0.6 2-41

𝜁𝑞𝑠 = 1 + tan 𝜙 2-42

Depth Factors:

𝜁𝑐𝑑 = 𝜁𝑞𝑑 − [

1 − 𝜁𝑞𝑑

𝑁𝑐 tan 𝜙] 2-43

𝜁𝛾𝑑 = 1 2-44

𝜁𝑞𝑑 = 1 + 2 tan 𝜙 (1 − sin 𝜙)2 [(𝜋

180) tan−1(𝐷/𝐵)] 2-45

Rigidity Factors:

𝜁𝑐𝑟 = 𝜁𝑞𝑟 − [

1 − 𝜁𝑞𝑟

𝑁𝑐 tan 𝜙] ≤ 1 2-46

𝜁𝛾𝑟 = 𝜁𝑞𝑟 2-47

17

𝜁𝑞𝑟 = exp{[−3.8 tan 𝜙]

+ [(3.07 sin 𝜙)(log10 2 𝐼𝑟𝑟)/(1 + sin 𝜙) ]} ≤ 1 2-48

The rigidity factors, originally proposed by Vesić (1975), require a solution for Irr, the

reduced rigidity index, Ir, the rigidity index, and Irc, the critical rigidity index:

𝐼𝑟𝑟 =

𝐼𝑟

1 + 𝐼𝑟Δ 2-49

𝐼𝑟 =

𝐸

2(1 + 𝜈𝑑)𝑞𝑎 tan 𝜙 2-50

𝐼𝑟𝑐 = 0.5 exp[2.85 cot(45° − 𝜙/2)] 2-51

In these equations, 𝑞𝑎 is the average stress between 𝐷 and 𝐷 + 𝐵, 𝐸 is the approximate

modulus of elasticity for the soil, 𝜈 is Poisson’s ratio, and Δ is the volumetric strain. Δ

and 𝜈𝑑 have been estimated by Trautmann and Kulhawy (1987) as:

Δ ≅ 0.005(1 − 𝜙𝑟𝑒𝑙) (𝑞𝑎

𝑝𝑎) 2-52

𝜈𝑑 = 0.1 + 0.3𝜙𝑟𝑒𝑙 2-53

𝜙𝑟𝑒𝑙 =

𝜙′ − 25°

45° − 25° 2-54

𝜙𝑟𝑒𝑙 is limited to the values of 0 and 1. Δ is limited to 10. One last check is required

for this method and that is to compare 𝐼𝑟𝑟 to the critical rigidity index, 𝐼𝑟𝑐. If 𝐼𝑟𝑟 > 𝐼𝑟𝑐

then 𝜁𝑐𝑟 = 𝜁𝛾𝑟 = 𝜁𝑞𝑟 = 1. Otherwise, 𝜁𝑐𝑟 = 𝜁𝛾𝑟 = 𝜁𝑞𝑟 < 1 as calculated, reducing the

toe resistance.

18

2.2.2. Recent Developments

Conventional foundation design typically considers the case where soil is completely

saturated beneath the groundwater table and either partially saturated (or wet), for fine-

grained soils, or dry, for coarse-grained soils, above the groundwater table (Vanapalli

and Taylan 2012). Suction stress is neglected in the calculation of effective stress. This

approach is conservative, as it does not consider the effects of partial saturation and

suction stress.

Vanapalli and Taylan (2011) proposed modifications to the 𝛼 and 𝛽 method such that

the effects of partial saturation and suction stresses were included. For a shaft embed-

ded in fine-grained soils, they proposed that:

𝑓𝑠 = 𝛼𝑠𝑢,𝑠𝑎𝑡 [1 +(𝑢𝑎 − 𝑢𝑤)

(𝑃𝑎

101.3 kPa)

𝑆𝜈

𝜇 ] 2-55

(𝑢𝑎 − 𝑢𝑤) is the matric suction, 𝑆 is the degree of saturation, 𝜈 and 𝜇 are fitting pa-

rameters, and 𝑃𝑎 is atmospheric pressure. The above equation implies that if either ma-

tric suction is zero (which occurs near the groundwater table) or saturation is zero

(which occurs at some distance above the groundwater table), there is no additional

frictional resistance in the unsaturated zone.

Vanapalli and Taylan (2011) proposed a modification to the 𝛽-method, capturing the

effects of unsaturated soil mechanics. The researchers suggested that the shaft re-

sistance be captured into two components: (1) frictional resistance due to horizontal

effective stress; and (2) apparent cohesion due to suction. That equation is:

𝑓𝑠 = 𝛽𝜎𝑧′ + (𝑢𝑎 − 𝑢𝑤)(𝑆𝜅)(tan 𝛿′) 2-56

19

where 𝜅 is a fitting parameter used for shear strength and the other terms are as previ-

ously described. In this equation, additional frictional resistance from suction stresses

are considered separately from the effective stress term. To avoid considering suction

stress twice it is important to note that the effective stress term does not include suction

stress.

2.3. Mechanics of Unsaturated Soils

2.3.1. Soil Water Characteristic Curve

Many phenomena present in soil can be explained through the lenses of unsaturated

soil mechanics. Unsaturated soil mechanics integrates the presence of an air phase, wa-

ter phase, and solid phase into one coherent framework. This section will begin with a

discussion on equations that describe the amount of water in soil pores, which also

indicates the amount of air in the pores.

Soil water characteristic curves (also known as soil water retention curves) describe the

relationship between soil suction and water content (Lu and Likos 2004). Several mod-

els have been proposed to fit discrete laboratory data and to continuously describe the

soil water characteristic curve. (Brooks and Corey 1964; van Genuchten 1980; Fred-

lund and Xing 1994). van Genuchten (1980) proposed Eq. 2-57 as a model for the soil

water characteristic curve (SWCC). His work allows for the prediction of effective sat-

uration and normalized water content as a function of matric suction. The equation is

as follows:

Θ = 𝑆𝑒 = [

1

1 + (𝛼𝜓)𝑛]

𝑚

2-57

In this equation, 𝛼, m and n are fitting parameters, however 𝛼 is often considered to be

(or related to) the inverse of the air entry value, or the pressure at which the air-phase

begins to fill the pores at a greater rate. To simplify the expression (which is useful

20

when fitting curves to data) van Genuchten defined 𝑚 = 1 −1

𝑛 to satisfy the Mualem

(1976) hydraulic conductivity model or 𝑚 = 1 −2

𝑛 to satisfy the Burdine (1953) hy-

draulic conductivity model. The units of 𝛼 are kPa-1, or m-1 when considering pressure

heads. Matric suction 𝜓 is defined as the difference between the pressure of a gas phase

and liquid phase, or 𝑢𝑎 − 𝑢𝑤. The normalized water content Θ and effective saturation

𝑆𝑒 are defined by Eq. 2-58.

Θ =

𝜃 − 𝜃𝑟

𝜃𝑠 − 𝜃𝑟= 𝑆𝑒 =

𝑆 − 𝑆𝑟

1 − 𝑆𝑟 2-58

where 𝜃 is the volumetric water content, 𝜃𝑠 is the saturated water content, 𝜃𝑟 is the

residual water content, 𝑆 is the soil saturation, and 𝑆𝑟 is the residual saturation. The

saturated water content is the maximum water content that the soil pores can contain

while the residual water content is the minimum water content that the pores retain.

The soil water characteristic curve, as defined by Eq. 2-57, is useful in defining the

quantity of water that exist in soil pores as a function of the matric suction. Further, if

the quantity of water contained in the pores are known, then suction stress (which dif-

fers from matric suction) can be predicted.

2.3.2. Particle Level Principles

Unsaturated soil mechanics is best understood when explained at a granular scale. Fig-

ure 2-6 below shows a soil water characteristic curve, as described in Section 2.3.1,

and a diagram for two particles in contact with water between contacts.

21

Figure 2-6. (a) Water content vs. matric suction. (b) two grains in contact with water between

contacts.

A common misconception is that suction stresses may increase indefinitely above the

groundwater table. While matric suction can increase to a thermodynamic limit of 106

kPa (Fredlund and Xing 1994), suction stresses acting on the soil particle do not

necessarily increase monotonically with matric suction. Figure 2-6 (a) plots a typical

soil water characteristic curve defined by van Genuchten (1980), which shows that

while matric suction increases, volumetric water content decreases, especially after

some air-entry pressure. As matric suction increases, the water content will reduce from

the saturated water and decrease to a residual water content, which describes volume

of water retained as a result of surface tension. As water content reduces, the area of

𝜃𝑟 𝜃𝑠

(a)

(b)

22

water contact between particles, 𝐴𝑠 =𝜋

4𝑑𝑚

2 , decreases. Consequently, while matric

suctions may be large, the water contact area is small, resulting in limited (or zero)

suction stresses (Lu and Likos 2006; Lu et al. 2009).

Figure 2-7. Forces acting on an individual particle (after Lu and Likos 2004).

Figure 2-7 above summarizes the forces that act on a particle in unsaturated conditions.

Generally, the air pressures acting on a particle are very small. On the contrary, the

water pressure acting on the particle can be very large. As mentioned, matric suction

𝑢𝑎 − 𝑢𝑤 can increase almost indefinitely, however, the suction stresses 𝜎𝑠 cannot. This

is directly related to the water contact area As. This can be summarized as:

𝜎𝑠 = (𝑢𝑎 − 𝑢𝑤)𝑓(𝐴𝑠) = 𝜓 𝑓(𝐴𝑠) = 𝜓𝜒 2-59

23

where 𝑢𝑎 − 𝑢𝑤 and 𝜓 are the matric suction (or the pressure difference between the air

and water phase), and 𝜒 is Bishop’s effective stress parameter – which is a function of

water contact area, and 𝜎𝑠 is the suction stress, or the net interparticle force generated

between of negative pore water pressure and surface tension (Bishop 1959; Lu and

Likos 2004; Lu and Likos 2006). A positive suction will result in a force that tends to

pull particles together, while a negative suction will tend to push particles away (this is

also positive porewater pressure). For fine-grained soil such as clays, suction stress is

also influenced by van der Waal attractions, electric double-layer repulsion, and chem-

ical cementation effects (Lu and Likos 2006). 𝜒 is considered over a range of zero (dry

conditions) to unity (saturated conditions).

Eq. 2-59 introduces the concept of Bishop’s effective stress parameter. This parameter

is intended to capture the variation in suction stress as a function of gradation, soil type,

particle packing – considerations that affect the water contact area, ultimately decreas-

ing suction stress. For sandy soils, the rate at which 𝜒 approaches zero is greater than

the rate at which 𝜓 increases, thus 𝜎𝑠 is generally small for high values of matric suc-

tion. For clayey soils, this is not always the case, causing 𝜎𝑠 to asymptotically approach

a residual value of suction stress (Lu and Likos 2004).

2.3.3. Bishop’s Effective Stress Framework

In 1959, Bishop began the work of characterizing the way partial saturation affects the

state of stress within soil. Conventionally, the effective stress is defined as (Terzaghi

1943):

𝜎′ = 𝜎 − 𝑢𝑤 2-60

where 𝜎′ is the effective stress within the soil, 𝜎 is the net normal stress, and 𝑢𝑤 is the

porewater pressure. This definition works well beneath the groundwater table, where

positive pore water pressures increase linearly with depth. This, however, is not the

case above the ground water table. Soils above the groundwater table are in a state of

24

partial saturation, where nonlinearity arises due to the existence of both a gas and liquid

phase in the pores. To consolidate these two ideas, Bishop introduced an effective stress

parameter 𝜒, which is a function of suction and soil physical properties, capturing non-

linearity in partially saturated soil behavior. Bishop proposed the following equation,

Eq. 2-61.

𝜎′ = 𝜎 − 𝑢𝑎 + 𝜒(𝑢𝑎 − 𝑢𝑤) 2-61

As explained earlier, 𝜒 is unity when saturated and zero when dry. When 𝜒 is equal to

one, the equation is equal to the Terzaghi (1943) definition of effective stress, which is

𝜎′ = 𝜎 − 𝑢𝑎 + 𝑢𝑎 − 𝑢𝑤 = 𝜎 − 𝑢𝑤.

2.3.4. Solutions for Bishop’s Effective Stress Parameter

Since the introduction of this effective stress method, several researchers have sought

to develop solutions/equations for the effective stress parameter 𝜒. Khalili and Khabbaz

(1998) proposed a form based entirely on empirical data. Their results have good cor-

relation to data. The form proposed uses the suction ratio, which is the ratio between

matric suction and the suction at air-entry. Air-entry suction 𝑢𝑒, which is also termed

expulsion pressure, is the suction at which water begins to drain significantly from the

pores, transitioning the soil from a saturated to unsaturated state. The equation for the

Khalili and Khabbaz (1998) effective stress parameter is:

𝜒 = {(

𝑢𝑎 − 𝑢𝑤

𝑢𝑒)

−0.55

for 𝑢𝑎 − 𝑢𝑤 > 𝑢𝑒

1 for 𝑢𝑎 − 𝑢𝑤 ≤ 𝑢𝑒

2-62

This equations implies that before the pore fluid is drained and while the matric suction

is small, the soil behaves as if under conventional soil mechanics principles. Above the

air-entry value, 𝜒 asymptotically approach zero as matric suction increases.

25

Lu and Likos (2004) suggested that 𝜒 is equal to effective saturation, 𝑆𝑒, and the nor-

malized water content, Θ. If this assumption is made, it can be used in conjunction with

the van Genucthen (1980) equation for predicting water content with respect to matric

suction. This theory has been validated experimentally through shear testing on par-

tially saturated soils (Lu and Likos 2006; Lu et al. 2009; Lu et al. 2010). Bishop’s

proposed form was:

𝜒 =

𝑆 − 𝑆𝑟

1 − 𝑆𝑟=

𝜃 − 𝜃𝑟

𝜃𝑠 − 𝜃𝑟 2-63

With the van Genuchten (1980) soil water characteristic curve (SWCC), this equation

becomes:

χ = [

1

1 + (𝛼𝜓)𝑛]

𝑚

2-64

This definition of Bishop’s 𝜒 allows suction stresses, 𝜎𝑠 = 𝜒𝜓, to be related to the soil

water characteristic curve and corresponding fitting parameters. Figure 2-8 summarizes

the concepts discussed to this point: (a) plot of the soil water characteristic curves ac-

cording to the van Genuchten (1980) model and (b) plot of the suction stress calculated

from Bishop’s 𝜒 defined in Equation 2-64. This figure shows typical behavior for

sands, silts, and clays. Suction stresses in sands typically peak at small matric suction

values and then decrease to zero. Silts may also have a peak suction stress value, but

will either approach an asymptotic value or decrease to zero. Clays will not have a peak

suction stress, but will increase to an asymptotic value of suction (Lu and Likos 2006;

Lu et al. 2010).

26

Figure 2-8. (a) SWCC for sand, silt and clays, (b) corresponding suction stress profile (Lu et

al. 2010)

2.3.5. Extended Mohr-Coulomb Failure Criterion

The Mohr-Coulomb failure criterion can be modified to include the effects of matric

suction and partial saturation by separating the effective stress into two terms, the nor-

mal stress and suction stress, using Bishop’s effective stress in Eq. 2-61. This method

is the ‘effective stress’ method used by Lu and Likos (2004). The new equation for the

failure plane is:

𝜏𝑓 = 𝑐′ + 𝜎′ tan 𝜙′ = 𝑐′ + (𝜎 + 𝜎𝑠) tan 𝜙′

= 𝑐′ + 𝜎 tan 𝜙′ + 𝜒 𝜓 tan 𝜙′ 2-65

Fredlund et al. (1978) originally proposed a differing modification to the Mohr-Cou-

lomb Failure Criterion. The form proposed by Fredlund et al. (1978) is a ‘total stress’

approach, incorporating a new variable, 𝜙𝑏, which is the friction angle with respect to

changes in 𝜓 at a constant 𝜎.

𝜏𝑓 = 𝑐′ + 𝜎 tan 𝜙′ + 𝜓 tan 𝜙𝑏 2-66

(a) (b)

27

Both equations may be recast as:

𝜏𝑓 = 𝑐′ + 𝑐′′ + 𝜎 tan 𝜙′ 2-67

This indicates that the additional strength associated with an increase in suction stresses

can be incorporated into a cohesion parameter, c’’, that is a function of suction. The

Fredlund et al. (1978) form fails to consider the non-linear suction stresses evident in

unsaturated soils. Generally, the form proposed by Lu and Likos (2004; 2006) is more

powerful as it considers non-linearity in the effective stress parameter, 𝜒, while adjust-

ing the effective stress to account for partial saturation. These two approaches, how-

ever, can be consolidated:

𝑐′′ = 𝜓 tan 𝜙𝑏 = 𝜒𝜓 tan 𝜙′ 2-68

where all terms are as previously defined.

2.3.6. Matric Suction Profiles

A matric suction profile must be assumed to implement the effects of unsaturated soils.

Generally, it is assumed that matric suction increases linearly with distance above the

groundwater table. That is, 𝜓 = 𝛾𝑤ℎ, where h is the distance above the groundwater

table. Effects from infiltration and evaporation can also be considered. Using Darcy’s

Law, Lu and Likos (2004) and Lu and Griffiths (2004) provide a derivation for matric

suction as a function of permeability, infiltration/evaporation rates, and height above

the groundwater table which is in Eq. 2-69.

𝜓 = −

1

𝛼[(1 +

𝑞

𝑘𝑠) 𝑒−𝛼𝛾𝑤𝑧 −

𝑞

𝑘𝑠] 2-69

where 𝑞 is the flux rate – evaporation is positive and infiltration negative, 𝑘𝑠 is the

saturated hydraulic conductivity, 𝑧 is the distance above the ground water table, and 𝛼

28

is the inverse of the air-entry pressure (kPa-1), which is related to (or can be assumed

as) the fitting parameter used in the van Genuchten equation.

Eq. 2-69 was derived using the Gardner (1958) one-parameter, exponential model for

hydraulic conductivity:

𝑘 = 𝑘𝑠𝑒−𝛼(𝑢𝑎−𝑢𝑤) 2-70

Figure 2-9. Matric suction profiles at various surface flux boundary conditions for clay (Lu

and Griffiths 2004).

Figure 2-9 shows how a matric suction profile varies with seasonal changes to the flux

conditions. This figure plots water pressure, 𝑢𝑤, which when negative, results in a pos-

itive matric suction 𝜓. It is generally assumed that the matric suction profile with a net

zero flux will behave linearly defined by Eq. 2-71.

𝜓 = 𝛾𝑤𝑧 2-71

where 𝛾𝑤 is the unit weight of water, and 𝑧 is distance above the groundwater table. In

Figure 2-9, when a soil profile is subjected to a positive flux, or evaporation, the matric

29

suction will increase above the groundwater table. Likewise, with a negative flux (in-

filtration/precipitation), more water is present in the soil profile decreasing the matric

suction in the pores (Lu and Griffiths 2004).

2.3.7. At-Rest Earth Pressure Coefficient

One direct consideration that is made from suction stresses is the availability of these

stresses to cause tension within the soil. This consequentially leads to surface cracking

if the tension is great enough. This phenomena can be understood through the at-rest

earth pressure coefficient established derived through Hooke’s law:

𝐾0 =

𝜎ℎ − 𝑢𝑎

𝜎𝑣 − 𝑢𝑎=

𝜈

1 − 𝜈−

1 − 2𝜈

1 − 𝜈

𝜒(𝑢𝑎 − 𝑢𝑤)

(𝜎𝑣 − 𝑢𝑎) 2-72

where 𝜈 is the Poisson’s ratio for the soil and the other terms are as previously defined.

This expression is extensively derived by Lu and Likos (2004). It is important to note

that this formulation is in terms of net total, not effective, stresses. While this expres-

sion has not been validated experimentally, it does indicate the importance unsaturated

soils in determining earth pressure coefficients. In deep foundation design, it is often

assumed that the first 1.5 m (or 1 foundation diameter) of soil provides no frictional

resistance. The depth of cracking, however, can be known through the relationship de-

scribed above. This expression can be further manipulated to include the unsaturated

soil parameters. Eq. 2-73 implements the van Genuchten equation for the SWCC to Eq.

2-72.

𝐾0 =𝜈

1 − 𝜈−

1 − 2𝜈

1 − 𝜈

ln[(1 + 𝑞/𝑘𝑠)𝑒−𝛾𝑤𝛼𝑧 − 𝑞/𝑘𝑠]

𝛼(𝜎𝑣 − 𝑢𝑎)(1 + {−ln[(1 + 𝑞/𝑘𝑠)𝑒−𝛾𝑤𝛼𝑧 − 𝑞/𝑘𝑠]}𝑛)(𝑛−1)/𝑛 2-73

The depth of cracking can be calculated as the value of z such that 𝐾0 = 0. Note that

for the case of 𝜓 = 0, the expression simplifies to the normally-consolidated solution

30

for 𝐾0 = 𝜈/(1 − 𝜈). If 𝜈𝑢 = 0.5 is used, the expression simplifies to 𝐾0 = 1, which is

expected in undrained, elastic loading conditions.

2.3.8. Discussion of Unsaturated Soil Properties

The van Genuchten (1980) equation describes the soil water characteristic curve, or a

soil’s water retention with respect to matric suction, as a function of five variables. The

first is 𝛼, which has the units of either m-1 or kPa-1. 𝛼 is related inversely to the air entry

pressure or head of a given soil. The second and third variables are 𝜃𝑠 and 𝜃𝑟, which

are the saturated and residual water contents. 𝜃𝑠 is limited to the porosity of a soil. The

last two variables are n and m, which are fitting parameters used to fit Eq. 2-57 to

laboratory data. Another important variable is the saturated hydraulic conductivity ks,

which has units of m/s.

Carsel and Parrish (1988) performed an extensive study on the variation of these un-

saturated soil parameters between the 12 different soil types identified in the USDA

Soil Classification System. This classification system simply describes soil as a func-

tion of percent clay, silts, and sands. Carsel and Parrish (1988) performed many statis-

tical analysis on over 15,000 soils including simple calculations of the mean and stand-

ard deviation of a soil type, and a thorough framework for performing Monte Carlo

simulations. Note, that the Mualem (1976) solution for m = 1 – 1/n was used by these

authors. Table 2-1 presents the average mean and standard deviations for unsaturated

soil parameters by soil texture according to Carsel and Parrish (1988).

31

Table 2-1. Typical unsaturated soil properties by USDA textural class (Carsel and Parrish

1988)

𝜽𝒔 𝜽𝒓 𝜶 (𝐜𝐦−𝟏) 𝑲𝒔 (𝐜𝐦/𝐡𝐫) 𝒏

�̅� 𝑠 �̅� 𝑠 �̅� 𝑠 �̅� 𝑠 �̅� 𝑠

Clay 0.38 0.09 0.068 0.034 0.008 0.012 0.20 0.42 1.09 0.09

Clay Loam 0.41 0.09 0.095 0.010 0.019 0.015 0.26 0.70 1.31 0.09

Loam 0.43 0.10 0.078 0.013 0.036 0.021 1.04 1.82 1.56 0.11

Loamy

Sand 0.41 0.09 0.057 0.015 0.124 0.043 14.59 11.36 2.28 0.27

Silt 0.46 0.11 0.034 0.010 0.016 0.007 0.25 0.33 1.37 0.05

Silt Loam 0.45 0.08 0.067 0.015 0.020 0.012 0.45 1.23 1.41 0.12

Silty Clay 0.36 0.07 0.070 0.023 0.005 0.005 0.02 0.11 1.09 0.06

Silty Clay

Loam 0.43 0.07 0.089 0.009 0.010 0.006 0.07 0.19 1.23 0.06

Sand 0.43 0.06 0.045 0.010 0.145 0.029 29.70 15.60 2.68 0.29

Sandy

Clay 0.38 0.05 0.100 0.013 0.027 0.017 0.12 0.28 1.23 0.10

Sandy

Clay Loam 0.39 0.07 0.100 0.006 0.059 0.038 1.31 2.74 1.48 0.13

Sandy

Loam 0.41 0.09 0.065 0.017 0.075 0.037 4.42 5.63 1.89 0.17

As mentioned, Carsel and Parrish (1988) developed framework for determining appro-

priate unsaturated parameters to be used in Monte Carlo simulations. It is important to

appropriately characterize these properties for several reasons: (1) most of these prop-

erties do not have a normal distribution, but may have lognormal, log ratio, or hyper-

bolic arcsine distributions (Carsel and Parrish 1988); and (2) many of the properties are

interrelated including n, 𝜃𝑟, 𝛼, and 𝑘𝑠. If any of these four variables change, it is more

than likely that one or more of the other three variables would change as a result. The

exception to this is the saturated water content, which is assumed to vary independently

32

from the other variables. This assumption is valid since typical variations in 𝜃𝑠 are

small (generally less than 25%).The procedures developed by Carsel and Parrish (1988)

produce suitable Monte Carlo realizations that considers relationships between each

property while still matching known distributions.

2.4. Summary

Section 2.1 and 2.2 covered the current practice of calculating bearing capacity for

shallow foundations and drilled shafts. Section 2.3 covered the current state of unsatu-

rated soil mechanics in the literature. For shallow foundations, bearing capacity is typ-

ically calculated through closed form equations that are based on some form of the

Terzaghi (1943) bearing capacity equation. Subsequent research have introduced vari-

ous shape and depth factors. For drilled shafts, the 𝛽-method is frequently used to cal-

culate side friction and total side resistance while toe bearing may be calculated through

bearing capacity equations that have a similar form to the shallow foundation bearing

capacity equations.

More recently, researchers have developed solutions to the shallow foundation bearing

capacity equation with respect to matric suctions that are present in a body of partially

saturated soils (Vanapalli and Mohamed 2007; Oh and Vanapalli 2008; Vahedifard and

Robinson 2015). These solutions are based on the Fredlund et al (1978) total stress

approach, but an effective stress approach (Lu and Likos 2004) may be more appropri-

ate. Another issue with these solutions is that average matric suction values are calcu-

lated across the stress bulb. Instead, average apparent cohesion should be calculated

across the failure surface since soils on this plane are being sheared. These solutions

also require an assumption for the air-entry value as well as bearing capacity fitting

parameters. It would be more appropriate to calculate bearing capacity (or apparent

cohesion) directly from the soil water characteristic curve. Vahedifard and Robinson

(2015) introduced a procedure for calculating bearing capacity for steady state flow.

This approach will be used and assessed in this current work for shallow and deep

33

foundations. Variation in overburden, or the effective stress at the depth of embedment,

and average soil unit weight within the failure surface are not considered in the litera-

ture.

Vanapalli and Taylan (2011) have proposed solutions for side resistance for both the

𝛼-method and 𝛽-method. For the 𝛽-method, Vanapalli and Taylan (2011) did not spe-

cifically vary the 𝛽 term in the expression, but instead introduced a separate term to

account for suction stresses. While this is a good assumption for the state of suction

stresses within a body of soil, the net normal horizontal stresses should also be affected

since suction will decrease the at-rest earth pressure coefficient, as discussed in Section

2.3.7. This means that 𝐾0 will decrease (or become zero when cracking), which implies

that 𝛽 should also be reduced.

The state of unsaturated soil mechanics, discussed in Section 2.3, is rapidly changing,

with many publications written within the last decade. Currently, there are two methods

for calculating shear strength with respect to suction: the total stress approach (Fred-

lund et al. 1978), and the effective stress approach (Lu and Likos 2004). For both these

methods, suction is calculated from the soil water characteristic curve (SWCC). These

methods have been validated through laboratory testing (Fredlund et al. 1978; Lu and

Likos 2006; Lu et al. 2009). Expressions for the earth pressure coefficients and steady-

state flow (or flux) in unsaturated soils have not yet been validated experimentally. The

solution for the steady-state flow (Lu and Griffith 2005) is based on the simple Gardner

(1958) solution to hydraulic conductivity, which has been replaced by more numeri-

cally complex solutions. New steady-state flow equations in unsaturated soils should

be developed for newer permeability models to improve the accuracy of assumed ma-

tric suction profiles.

There is a need in both shallow and deep foundations design for simple solutions that

account for the suction in partially saturated soils. This work will develop a simple

34

framework for modifying the conventional shallow foundation bearing capacity and

deep foundation side resistance to account for matric suction and partial saturation.

35

3. Research Objectives and Methodology

3.1. Objectives

The purpose of this study is to explore the theoretical implications of calculating the

bearing capacity of shallow and deep foundations in the presence of unsaturated soils.

This includes using the considerations listed in the Chapter 2: matric suction, infiltra-

tion/evaporation, effective stress parameters to produce suction stresses, surface ten-

sion/cracking, at-rest earth pressure (EP) coefficients, and the unsaturated failure crite-

rion.

For typical shallow foundation design, resistance is derived from three primary com-

ponents: cohesion, unit weight, and surcharge loads. These three components are di-

rectly influenced by the value of the friction angle through bearing capacity factors.

The shape of the failure surface is understood to be a function of friction angle. The

assumption in this work is that the shape of the failure surface does not change as a

function of varying suction stresses and water contents. It would be difficult to assume

changes in the failure surface according the variation in saturation and suction stresses.

To consolidate both unsaturated soil mechanics and shallow foundation design, it is

assumed that an apparent cohesion term from Eq. 2-68 can be directly implemented

into the bearing capacity equation. This is appropriate since the shallow foundation

framework was described according to the Mohr-Coulomb failure criterion. Apparent

cohesion is simply an extension of MC failure surface in the third dimension, with

matric suction as a third stress state variable. Furthermore, the unit weight can be ap-

proximated by finding the average unit weight within the failure surface. The saturation

profile (calculated from the SWCC) can be used to define the unit weight of water as a

function of distance above the groundwater table. The third term, the effective stress at

the depth of embedment, will be calculated by summing the suction stress 𝜎𝑠 and

weight of soil above the base of the shallow foundation when the groundwater table is

beneath the depth of embedment.

36

Considerations for the side resistance of deep foundations are simple in that deep foun-

dations fail vertically across the length of the deep foundation. Toe bearing resistance

is more difficult to approximate and is often near or below the groundwater table in a

dense soil layer. Contribution to strength from partial saturation will be calculated sep-

arately for shaft resistance and toe bearing.

Partial saturation will be considered in shaft resistance by integrating the suction stress

profile for implementation in the 𝛽-method. Furthermore, variation in vertical stresses

can be calculated from the unit weight profiles, which are a function of the saturation

profile derived from the van Genuchten soil water characteristic curve (Eq. 2-57). The

last consideration in shaft resistance is in the variation of earth pressure coefficients.

Suction stresses present in unsaturated soils will engender a decrease in soil contacts,

especially for near-surface soils. This phenomena will decrease horizontal earth pres-

sures and ultimately produce surface cracking. Eq. 2-73 will be used for K in the 𝛽-

method, allowing a single integrand across the surface of the shaft.

Many equations exist in an effort to predict toe bearing resistance. Toe bearing re-

sistance is usually defined based on settlement requirements since ultimate states are

difficult to achieve. Typically, toe bearing resistance is predicted from either CPT data

(or some similar field test), pile driving data, or bearing capacity equations. For the

consideration of partial saturation in toe bearing (which is only relevant if the toe is

above the groundwater table), bearing capacity equations must be used. In this work

the Vesić toe bearing capacity equations described in Kulhawy et al. (1983) will be

used. The same approach for considering partial saturation in shallow foundations will

be used for the deep foundation bearing capacity equation.

The remainder of this chapter is dedicated to the theoretical development of shallow

foundation bearing capacity and deep foundation side resistance in partially saturated

soils. A modified shallow foundation bearing capacity equation is developed in Section

37

3.2 and a modified 𝛽-method for deep foundation is developed in Section 3.3. Follow-

ing this chapter, the validity of the modified shallow foundation equation is assessed

by comparing measured load tests in Chapter 4. Parametric studies on both the modified

shallow foundation and 𝛽 equations are discussed in Chapter 5. Sensitivity to soil input

parameters, including van Genuchten fitting parameters, for these modified approaches

are discussed in this chapter. Design charts are developed and discussed in Chapter 5.

For consistency throughout this thesis, the Vesić bearing capacity equations will be

used. The modifications proposed to account for partially saturated soil can be applied

to any bearing capacity framework. Further, any definition of 𝜒, the SWCC, and matric

suction profile can be used as well. The Lu et al. (2006) definition of 𝜒, van Genuchten

(1980) SWCC, and Lu and Griffiths (2004) 𝜓-profile are used in this section and in the

remainder of this thesis.

3.2. Shallow Foundations in Unsaturated Soils

3.2.1. Theoretical Development

The general bearing capacity equation for shallow foundations can be modified to ac-

count for partial saturation in three ways: (1) modifying the effective stress at the depth

of embedment (or overburden) term; (2) modifying the soil unit weight term; and (3)

modifying the cohesion term. Overburden can be modified to account for the suction

stresses that exist at the depth of embedment for partially saturated soils. Soil unit

weight in partially saturated soils will vary above the groundwater table due to capil-

larity. An average unit weight can be calculated across the failure surface if the water

content throughout the soil profile is known. Water content is a function of distance

above the groundwater table and can be predicted from the soil water characteristic

curve discussed in Section 2.3.1. Cohesion can be modified to account for apparent

cohesion due to suction stresses across the failure surface. This is a valid assumption

because stresses along the failure surface can be expressed Mohr-Coulomb failure cri-

terion.

38

An assumption made in this work is that the shape of the failure surface is not affected

by suction stresses and variations in soil unit weight due to partial saturation above the

ground water table. Current solutions to the bearing capacity factors are based on the

friction angle and are not a function of variables such as cohesion and unit weight.

Therefore, it is assumed apparent cohesion and suction stresses play no role in varying

the shape of the failure surface. The general bearing capacity equation is in Eq. 3-1.

𝑞𝑢𝑙𝑡 = 𝑐′𝑁𝑐𝑠𝑐𝑑𝑐 + 𝜎𝑧𝐷′ 𝑁𝑞𝑠𝑞𝑑𝑞 + 0.5𝛾′𝐵𝑁𝛾𝑠𝛾𝑑𝛾 3-1

Any of the discussed bearing capacity factors could be selected, but for the sake of

using one framework the Vesić 𝑁𝛾 equation will be used. In the Vesić framework, the

Prandtl (1920) 𝑁𝑐 and Reissner (1924) 𝑁𝑞 equations are used in addition to the Vesić

(1973) 𝑁𝛾 and shape factors, and the Hansen (1970) shape factors. The Vesić depth

factor allows for the consideration of cohesion at the toe of a failure surface in embed-

ded foundations. Cohesion (and apparent cohesion) at the toe is considered in this work.

To account for partial saturation, the general bearing capacity equation is modified in

Eq. 3-2. From here, this equation will be called the modified bearing capacity equation

or the modified approach.

𝑞𝑢𝑙𝑡 = (𝑐′ + 𝑐′′̅̅ ̅)𝑁𝑐𝑠𝑐𝑑𝑐 + 𝑞𝑠𝑁𝑞𝑠𝑞𝑑𝑞 + 0.5𝛾′̅ 𝐵𝑁𝛾𝑠𝛾𝑑𝛾 3-2

where 𝑐′ is the soil cohesion, 𝑐′′̅̅ ̅ is the average apparent cohesion due to suction stresses

across the length of the failure surface, 𝑞𝑠 is the overburden (including the suction

stress), and 𝛾′̅ is the average effective soil unit weight across the area of the failure

surface. The other expressions are as defined in Section 2.1.1. Apparent cohesion, over-

burden, and average effective soil unit weight are defined by Eq. 3-3, 3-4, and 3-5.

39

𝑐′′̅̅ ̅ =

∫ 𝑐′′𝑑𝑙

𝑙

𝑙=

∫ 𝜎𝑠 tan 𝜙′ 𝑑𝑙

𝑙

𝑙=

∫ 𝜓𝜒 tan 𝜙′ 𝑑𝑙

𝑙

𝑙 3-3

𝑞𝑠 = 𝜎𝑧𝐷 + 𝜎𝑠,𝑧𝐷 = 𝜎𝑧𝐷 + (𝜓𝜒)𝑧𝐷 if 𝐷 < 𝑧𝑤

𝑞𝑠 = 𝜎𝑧𝐷 − 𝑢 if 𝐷 ≥ 𝑧𝑤 3-4

𝛾′̅ =

∫ 𝛾′(𝜓)𝑑𝐴

𝐴

𝐴 3-5

where 𝜎𝑠 is suction stress (Section 2.3.2 and 2.3.3), 𝜓 is matric suction (Section 2.3.6),

𝜒 is Bishop’s Chi (Section 2.3.4), 𝑙 is the length of the failure surface, 𝑞𝑠 is the net

normal stress at the depth of embedment, 𝜎𝑠,𝑧𝐷 is the suction stress at the depth of

embedment, 𝐷 is the depth of embedment, 𝑧𝑤 is the depth of the groundwater table, 𝑢

is porewater pressure, 𝐴 is the area contained by the failure surface, and 𝛾′ is the effec-

tive (or buoyant below the groundwater table) unit weight of the soil, which is a func-

tion of matric suction. Apparent cohesion is zero below the groundwater table and pos-

itive above. If the foundation is embedded beneath the groundwater table, pore water

pressure will be used instead of suction stress in Eq. 3-3. Likewise, if the failure surface

extends below the groundwater table, the moist unit weight is used above the ground-

water table and the buoyant unit weight below.

To demonstrate how this approach is implemented, a thorough example of a shallow

foundation failure in partially saturated soil will be considered. The following example

includes soil with the properties listed in Table 3-1.

Table 3-1. Soil properties for theoretical example of shallow foundation bearing capacity in

an unsaturated soil.

Variable 𝑮𝒔 𝜽𝒔 𝜽𝒓 𝜶 𝒏 𝝓′

Value 2.65 0.385 0.0385 0.175 kPa-1 2.5 20°

where 𝐺𝑠 is the specific gravity of the soil, 𝜃𝑠 is the saturated volumetric water content,

𝜃𝑟 is the residual water content, 𝛼 and 𝑛 are van Genuchten fitting parameters and 𝜙′

40

is the soil friction angle. 𝛼 and 𝑛 were selected such that the majority of suction stresses

would exist within 3 m of the groundwater table. 𝜙′ was selected such that the failure

surface would extend to nearly 3 m (for a 2 m wide footing embedded 0.5 m). This was

done to maximize the influence of matric suction which will distinguish the differences

between the two approaches.

The soil water characteristic curve above the groundwater table is presented in

Figure 3-1. This is calculated according to the van Genuchten equation (Eq. 3-6). The

underlying assumption in this figure is that water exists in a steady-state (i.e. no transi-

ent flow) and that water content/saturation is due to capillary rise alone. Eq. 3-7 is the

assumed matric suction profile where all the terms are as previously defined in Section

2.3.6. If there is no net flow, matric suction can be estimated from 𝜓 = 𝛾𝑤𝑧, where z

is distance above the groundwater table.

Θ = 𝑆𝑒 = [

1

1 + (𝛼𝜓)𝑛]

𝑚

3-6

𝜓 = −

1

𝛼[(1 +

𝑞


𝑞

𝑘𝑠] 3-7

Figure 3-1. Soil water characteristic curve for this example soil.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0

5

10

15

20

25

30

35

40

0 20 40 60 80 100

Volumetric Water Content

Dis

tan

ce A

bo

ve G

WT

[m]

Mat

ric

Suct

ion

[kP

a]

Saturation [%]

41

The suction stress profile can also be calculated for the soil in Table 3-1. In the effective

stress approach, suction stress is defined by 𝜎𝑠 = 𝜓𝜒. Using Eq. 3-8 (Lu et al. 2006;

Lu et al. 2009; Lu et al. 2010) to define Bishop’s 𝜒 leads to the suction stress profile in

Figure 3-2.

𝜒 =

𝑆 − 𝑆𝑟

1 − 𝑆𝑟=

𝜃 − 𝜃𝑟

𝜃𝑠 − 𝜃𝑟 3-8

Figure 3-2. Suction stress profile for this example soil.

The groundwater and foundation dimensions are described in Table 3-2. The founda-

tion in this example is a strip footing.

Table 3-2. Foundation and groundwater properties for theoretical example in Chapter 3.

Variable 𝒛𝒘 𝑩 𝑫

Value 3 m 2 m 0.5 m

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5

Dis

tan

ce A

bo

ve G

WT

[m]

Mat

ric

Suct

ion

[kP

a]

Suction Stress [kPa]

42

The Terzaghi failure surface with a friction angle of 20° would extend just above the

groundwater table, as shown in Figure 3-3. In this example, the failure surface extends

to a depth of 2.85 m. The groundwater table was set to a depth of 3 m such that it would

have maximum influence on the foundation performance. This foundation will be

loaded to the ultimate limit state, where continuous plastic flow occurs.

Figure 3-3. Failure surface corresponding to ϕ' = 20°, B = 2 m, and D = 0.5 m.

3.2.2. Considerations for Apparent Cohesion

From Figure 3-2, we can define the saturation of the soil based on the proximity of the

layer to the depth of the groundwater table. Figure 3-4 presents the assumed saturation

profile, colored based on saturation at a depth and superimposed on the failure surface.

This figure shows that while matric suction is assumed to increase linearly above the

groundwater table, saturation, and ultimately suction stress, vary nonlinearly.

43

Figure 3-4. Saturation of the soil profile for the example.

Using this approach, the saturation of the soil can be determined at any point along the

failure surface. This ultimately enables the calculation of average suction stress acting

along the failure surface. Figure 3-5 shows the saturation, as well as its corresponding

apparent cohesion 𝑐′′ = 𝜎𝑠 tan 𝜙′, across the failure surface according to its proximity

from the groundwater table.

Figure 3-5. Failure surface of the shallow foundation, colored by the saturation profile.

100

80

60

40

20

0.91 kPa

1.12 kPa

1.29 kPa

1.38 kPa 1.40 kPa

1.37 kPa 1.27 kPa

1.11 kPa

0.79 kPa

44

From Figure 3-5, suction stress and apparent cohesion can be calculated along the value

surface. Average apparent cohesion can be calculated along the failure surface by mul-

tiplying representative apparent cohesions by the length over which they act, summing

those values, and dividing by the total length (l). This process can be performed for

infinitesimally small increments using Eq. 3-3. Using this equation, 𝑐’’ ̅̅̅̅ is calculated

to be 1.11 kPa. Thus, the entire cohesion term (𝑐′ + 𝑐′′̅̅ ̅) is (0 kPa + 1.11 kPa).

The issue with simply calculating apparent cohesion from the average depth is that 𝜒

behaves extremely nonlinearly (Section 2.3.4), even while 𝜓 behaves linearly above

the groundwater table for 𝑞 = 0 m/s (Section 2.3.6). Other issues are that the ground-

water table may exist within the failure surface, and 𝜒 may be limited due to cavitation

in pores. Consider this: for the previous example, the failure surface extended to a depth

of 2.85 m. Assuming the average failure surface depth is, say, 1.4 m, would the average

apparent cohesion be 0 kPa if the groundwater table was at this depth? No, while there

is no apparent cohesion below the groundwater table, apparent cohesion still exist at

the beginning and end of the failure surface, which will contribute to strength. There-

fore, the apparent cohesion must be averaged across the entire failure surface length,

and cannot be assumed from the average failure surface depth. If the groundwater table

depth was 1.4 m, the average apparent cohesion would be 0.41 kPa.

3.2.3. Considerations for Unit Weight

Similar assumptions can be made for this example in regards to the saturation profile.

Using this profile, the unit weight of water can be estimated throughout the entire fail-

ure surface. The averaged unit weight can be used directly within the bearing capacity

framework.

45

Figure 3-5. Variation in saturation within the failure surface for this example.

The average unit weight can be calculated in a similar way to the average apparent

cohesion. Using the saturation profile in Figure 3-5 (or the SWCC in

Figure 3-1), the moist unit weight profile can be calculated according from Eq. 3-9.

𝛾𝑚 = (𝐺𝑠(1 − 𝜃𝑠) + 𝜃)𝛾𝑤 = (𝐺𝑠(1 − 𝜃𝑠) + 𝜃𝑠𝑆)𝛾𝑤 3-9

where 𝛾𝑚 is the moist unit weight, 𝐺𝑠 is the specific gravity, 𝜃𝑠 is the saturated volu-

metric water content (equal to porosity), 𝜃 is the volumetric water content, 𝑆 is the

saturation, and 𝛾𝑤 is the unit weight of water. Once the unit weight profile is known,

the average unit weight can be calculated by taking a surface integral of unit weight

across the area of the failure surface then dividing by the total area.

𝛾′̅ =

∫ 𝛾′(𝜓)𝑑𝐴

𝐴

𝐴 3-10

𝛾′ is the moist unit weight, defined by Eq. 3-9, above the groundwater table, and the

buoyant unit weight below the groundwater table (which is the saturated unit weight

minus the unit weight of water). From Eq. 3-10, the average unit weight was calculated

80

60

80

40

80

100

80

20

80

46

as 𝛾 = 17.3 kN/m3. It is important to note that the soils above the base of the footing

are not included in this average, but are considered in the overburden term. The Vesić

equation accounts for shearing resistance above the base of the footing, which is con-

sidered in the apparent cohesion expression and implemented through depth factors.

The conventional approach assumes that the average unit weight varies between the

buoyant unit weight when the groundwater table is above the depth of embedment and

a dry/moist unit weight when the groundwater table is greater than the depth of embed-

ment plus the footing width (D + B). For this particular soil, the unit weight using the

conventional approach is estimated to be 16.4 kN/m3, which is a 0.9 kN/m3 difference.

3.2.4. Considerations for Overburden

The third term in Vesić’s bearing capacity equation is overburden. Overburden is the

effective stress at the base of the footing. In the presence of unsaturated soils, this ef-

fective stress will include the effects of suction stress. Overburden is the simplest con-

sideration in the Vesić equation, requiring only knowledge of the soil unit weight above

the footing and the suction stress at the embedment depth. From Figure 3-2, the suction

stress at the depth of embedment is 2.7 kPa. The net normal (total) stress can be calcu-

lated as the integral of the soil unit weight from the surface to the depth of embedment.

This is shown by Eq. 3-11.

𝑞𝑠 = 𝜒𝜓 + ∫ 𝛾(𝑧)𝑑𝑧

𝐷

0

= 𝜎𝑠 + 𝜎𝑡 3-11

The calculated surcharge in this example is 𝑞𝑠 = 11.1 kPa. The conventional approach

would use the estimate 16.4 kN/m3 from the previous section, multiplied by the depth

of embedment (0.5 m), resulting in an 8.2 kPa overburden.

Putting all these considerations together, the modified inputs can be used with the Vesić

equation:

47

𝑞𝑢𝑙𝑡 = (𝑐′ + 𝑐′′̅̅ ̅)𝑁𝑐𝑠𝑐𝑑𝑐 + 𝑞𝑠𝑁𝑞𝑠𝑞𝑑𝑞 + 0.5𝛾′̅ 𝐵𝑁𝛾𝑠𝛾𝑑𝛾

𝑞𝑢𝑙𝑡 = ( 1.12 kPa)𝑁𝑐𝑠𝑐𝑑𝑐 + (11.1 kPa)𝑁𝑞𝑠𝑞𝑑𝑞 + 0.5(17.3 kN/m3)𝐵𝑁𝛾𝑠𝛾𝑑𝛾

The bearing capacity is calculated to be 187 kPa. If the unmodified approach was used,

the bearing capacity would be calculated as 144 kPa. Thus, the modified approach pre-

dicts a 30% increase in bearing capacity relative to the conventional approach.

3.3. Deep Foundations in Unsaturated Soils

3.3.1. Theoretical Development

As discussed in Section 3.1, the effects of partial saturation on side resistance and toe

bearing are different, requiring two separate calculations. For side resistance, suction

stresses will increase strength along the length of the deep foundation since failure oc-

curs across this surface. This failure, described by the 𝛽-method, can simply be reduced

to the extended Mohr-Coulomb failure criterion (Section 2.3.5). Suction stresses, how-

ever, can work detrimentally against the performance of side resistance since the at-

rest earth pressure coefficient will decrease due to suction above the groundwater table

(Section 2.3.7). Considerations for toe bearing in partially saturated soils can be treated

like shallow foundations, requiring an assumption for apparent cohesion, surcharge and

unit weight. Figure 3-6 visualizes the different modes of resistance – side resistance

and toe bearing resistance – used in this work.

48

Figure 3-6. Sketch of the conceptual deep foundation considered in this work.

As mentioned, resistance for deep foundations accumulates along the length of the shaft

and at the toe. The shear resistance at failure in partially saturated soils can be described

by the extended Mohr-Coulomb failure criterion in Eq. 3-12.

𝜏𝑓 = 𝑐′ + 𝜎ℎ′ tan 𝛿 = 𝑐′ + 𝜎ℎ tan 𝛿 + 𝜎𝑠 tan 𝛿 3-12

where 𝜏𝑓 is the shear stress at failure, 𝑐′ is the cohesion, 𝜎ℎ is the horizontal stress, 𝛿′

is the foundation-soil interface friction angle, and 𝜎𝑠 is the suction stress. For the pur-

poses of the work presented herein, 𝛿 = 𝜙′ is assumed. This assumption is typical for

drilled shafts, however, changing 𝛿 to some fraction of 𝜙′ would serve only to scale

the results slightly, but would not change the conclusions. Suction stress is omnidirec-

tional, and thus 𝜎𝑠 = 𝜒𝜓 is not multiplied by 𝐾0. As described in Section 2.3.7, the

49

horizontal stress can be calculated as a function of matric suction according to Eq. 3-13

(Lu and Likos 2004).

𝐾0 =

𝜎ℎ

𝜎𝑣=

𝜈

1 − 𝜈−

1 − 2𝜈

1 − 𝜈

𝜎𝑠

𝜎𝑣 3-13

where 𝜈 is Poisson’s ratio, 𝐾0 is the at-rest earth pressure coefficient, and 𝜎𝑣 is the

vertical total stress. Implementing 𝐾0 into Eq. 3-23 leads to the development of the

modified 𝛽-method which is presented in Eqs. 3-14 through 3-17.

𝜏𝑓 = 𝑓𝑠 = 𝐾0𝜎𝑣 tan 𝛿 + 𝜎𝑠 tan 𝛿 3-14

𝑓𝑠 = 𝛽𝜎𝑣 + 𝜎𝑠 tan 𝛿 3-15

𝑓𝑠 = (𝛽 +𝜎𝑠

𝜎𝑣tan 𝛿) 𝜎𝑣 = 𝛽′𝜎𝑣 3-16

𝛽′ = 𝐾0 tan 𝛿 +𝜎𝑠

𝜎𝑣tan 𝛿 3-17

In this series of equations, 𝛽′ is similar to the unmodified 𝛽, except that the 𝐾0 derived

by Lu and Likos (2004) is used for partially saturated soils. In saturated, normally con-

solidated soils, the 𝐾0 in Eq. 3-18 is used. The purpose of developing 𝛽′ is to compare

conventional 𝛽 solutions in the literature that only consider vertical total stress in par-

tially saturated soils.

𝐾0 =𝜈

1 − 𝜈 3-18

For simplicity, this modified 𝛽-method can be summarized to the expression 𝛽′𝜎𝑣′,

where 𝛽′ is defined by 3-17 above the groundwater table and conventionally below the

groundwater table, and 𝜎𝑣′ is the total stress (not including suction stress) above the

groundwater table and the effective stress below the groundwater table.

50

Toe bearing resistance in unsaturated soils can be calculated from Eq. 3-19. Similar to

shallow foundations, cohesion, soil unit weight, and overburden will be modified to

account for partially saturated soils. Unlike shallow foundations, the failure surface is

more difficult to characterize, thus the averaged values for 𝛾′ ̅̅ ̅ and 𝑐′′̅̅ ̅ will be calculated

from D (depth to toe) to D + B (plus foundation diameter).

𝑞𝑢𝑙𝑡 = (𝑐′ + 𝑐′′̅̅ ̅)𝑁𝑐𝜁𝑐𝑠𝜁𝑐𝑟𝜁𝑐𝑑 +

1

2𝐵𝛾′ ̅̅ ̅𝑁𝛾𝜁𝛾𝑠 𝜁𝛾𝑟𝜁𝛾𝑑

+ 𝑞𝑠𝑁𝑞𝜁𝑞𝑠𝜁𝑞𝑟𝜁𝑞𝑑

3-19

Equations for the modified 𝑐′′̅̅ ̅, 𝛾′ ̅̅ ̅, and 𝑞𝑠 are presented in Eqs. 3-20 to 3-22. The deri-

vation and significance of Eqs. 3-20 through 3-22 will not be discussed here as it is

similar to the modifications used in the shallow bearing capacity equation.

𝑐′′̅̅ ̅ = tan 𝜙′ ∫ 𝜎𝑠(𝑧)𝑑𝑧

𝐷+𝐵

𝐷

3-20

𝑞𝑠 = 𝜒𝜓 + 𝜎𝑡 = 𝜎𝑠 + 𝜎𝑡 if 𝐷 < 𝑧𝑤

𝑞𝑠 = 𝜎𝑡 − 𝑢 if 𝐷 ≥ 𝑧𝑤 3-21

𝛾′ ̅̅ ̅ = ∫ 𝛾′(𝑧)𝑑𝑧

𝐷+𝐵

𝐷

3-22

where all terms are as previous defined. Here 𝛾′ is the moist unit weight above the

groundwater table and the buoyant unit weight below the groundwater table.

3.3.2. Tension Cracking and K0

Eq. 3-13 indicates that the at-rest earth pressure coefficient above the ground water

table is a function of both the total vertical stress and the suction stress above the

groundwater table.

51

Figure 3-7. Variation of At-Rest earth pressure coefficient in partially saturated soils.

Variation in suction stresses and surcharge loading will affect the profile of K0 in un-

saturated soils above the ground water table. Figure 3-7 presents an example profile,

comparing the suction profile with the K0 profile. Below the groundwater table, the at-

rest earth pressure remains constant. For fine-grained soils, suction stress will readily

increase above the groundwater table, resulting in tension cracks (K0 = 0). Above crack-

ing, K0 is set equal to 0 since no frictional strength is present when cracking occurs.

Further, although suction stress increases above the depth of cracking, these suctions

are ignored since there is a loss of contact between the soil and foundation. For coarse-

grained materials, decreases in K0 are small since suction stresses are generally smaller;

tension cracking does not typically occur in coarse-grained soils. The K0 and 𝜎𝑠 pre-

sented are implemented directly into the modified 𝛽-method. This modified approach

will also include the effects of tension cracking.

52

3.3.3. Unit Weight

The saturation for any matric suction can be defined above the groundwater table, thus

soil total unit weight is understood with assumptions of a few material parameters such

as Gs, 𝜃𝑟, and 𝜃𝑠. The total stress profile above the groundwater table can be defined

by the following equations:

𝜎𝑡 = ∫ 𝛾𝑡𝑑𝑧

𝑧

0

= ∫ 𝛾𝑑(1 + 𝑤)𝑑𝑧𝑧

0

= ∫ 𝛾𝑑 (1 + 𝜃(𝜓)𝛾𝑤

𝛾𝑑) 𝑑𝑧

𝑧

0

3-23

𝜃(𝜓) = 𝜃𝑟 + (𝜃𝑠 − 𝜃𝑟)[1 + (𝛼𝜓)𝑛]−𝑚 3-24

𝛾𝑑 = 𝐺𝑠(1 − 𝜃𝑠)𝛾𝑤 3-25

where all variables are as previously defined. Note that 𝜃𝑠 = 𝑉𝑉/𝑉𝑇 is the same as the

porosity of the soil. Integrating 𝜎𝑡 to the depth of embedment, yields the total stress at

the depth of embedment which can be implemented into Eq. 3-21. This expression for

𝜎𝑡 can also be used to calculate the vertical total stress for the modified 𝛽-method in

Eq. 3-17.

3.3.4. Suction Stresses

Suction stress, which is shown in Figure 3-7, is defined by Eq. 3-26. 𝜓 is the matric

suction, and 𝜒 is Bishop’s effective stress parameter. These two variables have been

defined in section 2.3.6 and 2.3.4. To reiterate, in this work 𝜓 and 𝜒 will be defined by

Eqs. 3-27 and 3-28. In the case of zero flux, 𝜓 will reduce to 𝜓 = 𝛾𝑤𝑧.

𝜎𝑠 = 𝜓𝜒 3-26

𝜓 = −

1

𝛼[(1 +

𝑞


𝑞

𝑘𝑠] 3-27

χ =

𝜃 − 𝜃𝑟

𝜃𝑠 − 𝜃𝑟=

𝑆 − 𝑆𝑟

1 − 𝑆𝑟= [

1

1 + (𝛼𝜓)𝑛]

𝑚

3-28

53

Above the groundwater table, 𝜎𝑠 can be used to calculate 𝛽′ in Eq. 3-17 and to calculate

suction stresses at the toe in Eq. 3-21 if the toe of the drilled shaft is above the ground-

water table.

3.4. Summary

In this section the conventional foundation bearing capacity equations and the 𝛽-

method for side resistance in drilled shafts have been modified to account for matric

suction and suction stresses. In Section 3.2, an apparent cohesion term has been added

to the shallow foundation bearing capacity equation to account for suction stresses

along the failure surface. The unit weight term has been modified to vary as a function

of the average moisture content within the failure surface as defined by the soil water

characteristic curve. The surcharge term has been modified to include suction stress at

the depth of footing embedment. These same changes have also been considered for

the toe resistance in deep foundations.

In Section 3.3, the 𝛽-method has been modified to account for suction stresses applied

to the soil-foundation interface. A modified normally-consolidated 𝐾0 will be used in

this work as it accounts for suction stresses pulling soil grains apart, decreasing contacts

and ultimately decreasing 𝐾0.

Implementation of these procedures as outlined in Chapter 3 allows for the considera-

tion of partially saturated soils in shallow and deep foundation design. The validity of

this approach is assessed for shallow foundations in Chapter 4. Parametric studies on

both modified shallow bearing capacity equation and 𝛽-method are performed in Chap-

ter 5.

54

4. Comparison to Measured Response of Shallow Foundations

4.1. Introduction

Chapter 4 is a comparative study between the measured responses of shallow founda-

tion load tests from the literature and the calculated bearing capacity from the modified

approach proposed in this work. This section is important in assessing the ability of the

proposed approach to reasonably predict bearing capacity for shallow foundations in

partially saturated soils.

Section 4.2 describes the method by which eight different sets of load tests were se-

lected from the literature for use in this analysis. Section 4.3 is composed of several

parts: (1) description of soil properties given in the literature or assumed in this work;

(2) description of the load-displacement curve and method of calculating ultimate bear-

ing capacity from the literature; (3) calculation of the ultimate bearing capacity using

the modified equation proposed in this work; and (4) a comparison between the meas-

ured and calculated bearing capacity. This analysis is done for eight sets of load tests,

which comprise the entire Section 4.3. Finally, a summary of the results are in Section

4.4. This section will also include a discussion outlining the significance of the work

and results of Chapter 4.

4.2. Method for the Selection of Load Test Data

There are many works in the literature concerning the bearing capacity of shallow foun-

dation, but there are very few that include soil water characteristic curves (SWCC) and

permeability. This information is crucial for this work so the soil water characteristic

curve must be predicted or provided.

An understanding of the grain size distribution/soil classification is required to predict

unsaturated soil properties as the distribution will control the behavior of the pore fluid.

In this work, soil water characteristic curves will be selected in one of two ways: (1)

55

from the Carsel and Parrish (1988) USDA soil classifications; and (2) by use of an

unsaturated soil database, SoilVision (Fredlund 2011), if the grain size distribution

curve is provided. Carsel and Parrish (1988) collected unsaturated properties for over

15,000 soil samples and calculated expected values of van Genuchten (1980) parame-

ters according to the USDA textural classification. Average values were reported for

each classification, organizing 𝛼, 𝑛, 𝜃𝑠, 𝜃𝑟, and 𝑘𝑠 based on percent clays, silts, and

sands. The SoilVision database software (Fredlund 2011) can also categorize unsatu-

rated soil based on soil type, and more specifically by the grain size distribution. By

supplying a grain size distribution curve and mass/volume information, SoilVision is

able to calculate a corresponding SWCC either by pedotransfer functions or by com-

parison to its 6,000+ soil database. Through this process, unsaturated parameters can

be predicted for implementation in this work.

In summary, average values from Carsel and Parrish (1988) will be selected when a

grain size distribution curve is not provided but soil classification is. If, however, a

grain size distribution is provided with the load test, the SoilVision software (Fredlund

2011) will be used instead to find a suitable SWCC.

Another requirement for the literature used in this comparative study is that either an

ultimate bearing capacity was achieved in the load test (i.e. the soil settles without any

additional load) or the bearing capacity can be calculated from the load displacement

curve. If the ultimate state was not achieved, the Kondner (1963) hyperbolic equation

will be fitted to the load displacement curve and 𝑞𝑢𝑙𝑡 will be taken as the hyperbolic

asymptote.

Some shallow foundation load test literature that meets these requirements are:

Steensen-Bach et al. (1987) – complete SWCC

Briaud and Gibbens (1997/1999) – grain size distribution

Larsson (1997) – soil classification

Viana da Fonseca and Sousa (2002) – soil classification

56

Rojas et al. (2007) – complete SWCC

Vanapalli and Mohamed (2007) – complete SWCC

Vanapalli and Mohamed (2013) – complete SWCC

Wuttke et al. (2013) – complete SWCC

Analysis of these manuscripts will reported by publication year.

4.3. Comparison of Predicted Bearing Capacity to Database

4.3.1. Steensen-Bach et al. (1987)

Steensen-Bach et al. (1987) performed scaled plate load tests on Sollerod sand, varying

the groundwater table for each test. Wetting and drying curves were provided for the

Sollerod sand. For this work, both wetting (imbibition) and drying (drained) curves will

be used to predict bearing capacity. The measured wetting and drying data were fitted

to the van Genuchten (1980) Equation in Figure 4-1.

Figure 4-1. SWCC for Sollerod sand (Steensen-Bach et al. 1987).

Table 4-1 presents properties of Sollerod sand, soil water characteristic curve proper-

ties, and the dimensions of the plate.

57

Table 4-1. Properties of Sollerod sand and plate (Steensen-Bach et al. 1987)

van Genuchten Drying SWCC Parameters‡

Value van Genuchten Wetting SWCC Parameters‡

Value Foundation Parameters

Value

van Genuchten’s 𝛼 (kPa-1)

0.136 van Genuchten’s

𝛼 (kPa-1) 0.292

Embedment Depth, 𝐷 (m)

0

van Genuchten’s 𝑛


𝑛 3.826

Footing Width, 𝐵 (m)

0.022

van Genuchten’s 𝑚


𝑚 0.739

Footing Length, 𝐿 (m)

0.022

Residual Water Content, 𝜃𝑟

0.012 Residual Water

Content, 𝜃𝑟 0.00

Groundwater Table Depth, 𝑧𝑤 (m)

-

Saturated Water Content, 𝜃𝑠

0.358 Saturated Water

Content, 𝜃𝑠 0.340

Volume/Mass Parameters

Value Strength

Parameters† Value

Specific Gravity, 𝐺𝑠

2.704 Friction Angle,

𝜙′ (˚) 36

Porosity, 𝑛 0.365 Cohesion,

𝑐′ (kPa) 4.4

† Determined from drained triaxial compression tests.

‡ Fitted to the wetting and drying data from Steensen-Bach et al. (1987) using the van Genuchten (1980)

equation.

One issue that can occur with model tests is a scale effect wherein the size of the footing

is not sufficiently large compared to the particle size and it affects the results of the

load test. Sollerod sand has a 𝐷50 = 0.14 mm. Kusakabe (1995) and Herle and Tejch-

man (1997) recommend 𝐵/𝐷50 > 100 to minimize scaling effects. For this soil and a

foundation width of 22 mm, 𝐵/𝐷50 = 157, thus, the recommendation suggests that there

is little influence due to scale.

Steensen-Bach et al. (1987) performed load tests on Sollerod sand with five different

groundwater depths, 0 mm, 100 mm (×2), 200 mm, 400 mm, and 820 mm. The 820

mm, 200 mm, and one of the two 100 mm tests were loaded to failure, which is indi-

cated by the dashed black line in Figure 4-2. For the other load tests, the hyperbolic

Kondner equation was used to fit the load-displacement data and determine the bearing

58

capacity. Figure 4-2 presents the load displacement curves from Steensen-Bach et al.

(1987) for each load test.

Figure 4-2. Load displacement curves for Sollerod sand with varying groundwater tables

Table 4-2 below compares the results of the load tests in Figure 4-2 with the analytical

solution determined from the modified bearing capacity equation (for both the wetting

and drying curves) developed in this work and also the unmodified bearing capacity

equation.

0 2 4 6 8 10

250

500

750

1000

Displacement [mm]

Str

ess

[kP

a]

h = 820 mm

h = 100 mm

h = 0 mm

59

Table 4-2. Actual and Predicted results for the Sollerod load tests (Steensen-Bach et al. 1987)

Depth of GWT (mm)

𝒒𝒖𝒍𝒕 (kPa)

Measured Bearing Capacity

Calculated from Wetting

Curve

Calculated from Drying

Curve

Calculated from Conventional

Approach

0 110 393 393 393 100 679 507 507 395

100† 222 507 507 395 200† 549 618 634 395 400 1051 646 884 395

820† 770 489 832 395

† These tests were loaded to failure

The unmodified equation is unable to capture the effects of suction stress that are pre-

sent in unsaturated soils. To compare the data in Table 4-2, the actual bearing capacity

is plotted against the calculated bearing capacity in Figure 4-3 and the bearing capacity

vs. depth of the ground water table is plotted in Figure 4-4.

Figure 4-3. Measured bearing capacity vs. calculated bearing capacity for Steensen-Bach et

al. (1987)

0

500

1000

1500

0 500 1000 1500

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]

Measured Bearing Capacity [kPa]

Wetting SWCC

Drying SWCC

Conventional Approach

60

Figure 4-4. Calculated bearing capacity vs. GWT depth for Steensen-Bach et al. (1987)

Figure 4-3 and Figure 4-4 show that the drying soil water characteristic curve per-

formed the best in predicting the bearing capacity, but wetting curves are notoriously

difficult to measure due to air entrapped in the measuring system. Results for bearing

capacity could vary dramatically depending on if the water phase in the soil has drained

to equilibrium or has imbibed via capillary rise. Steensen-Bach et al. (1987) have indi-

cated that the sands were fully saturated and then drained to a desired groundwater table

height. This implies that the drying SWCC is more appropriate to use, which is con-

firmed by the results.

4.3.2. Briaud and Gibbens (1997)

Briaud and Gibbens (1997) performed five full scale load tests on shallow foundations

on medium dense, uniform, silty silica sand. Footing dimensions ranged from

1 m × 1 m to 3 m × 3 m. Two soil samples were taken for examination at 0.6 m and 3.0

m. Since the theoretical failure surface extends far below 0.6 m, the sample from 3.0 m

is used for analysis. This sample is described in Table 4-3.

0

500

1000

1500

0 200 400 600 800 1000

Bea

rin

g C

apac

ity

[kP

a]

GWT Depth [mm]

Wetting SWCCDrying SWCCConventional ApproachOriginal Data

61

Table 4-3. Soil properties at 3.0 m using hand auger (Briaud and Gibbens 1997)

van Genuchten SWCC

Parameters‡ Value

Foundation Parameters

Value

Van Genuchten’s 𝛼 (kPa-1)

1.08 Embedment Depth,

𝐷 (m) 0.711– 0.889

Van Genuchten’s 𝑛

3 Footing Width, 𝐵

(m) 1.0 – 3.0

Van Genuchten’s 𝑚

0.125 Footing Length, 𝐿

(m) 1.0 – 3.0


0.0286 Groundwater Table

Depth, 𝑧𝑤 (m) 4.9

Hydraulic Conductivity,

𝑘𝑠 (m/s) -


Value Strength

Parameters† Value



𝜙′ (˚) 32


𝑐′ (kPa) -

† Determined from mean borehole shear tests, provided by original authors

‡ Determined using the Fredlund and Wilson PTF in SoilVision to fit grain size distribution

The load displacement curves for the five different load tests and the corresponding

hyperbolic fits are shown in Figure 4-5. The bearing capacity calculated from the curve

fits are reported in Table 4-4. Predicted bearing capacities using both the conventional

and modified approaches were calculated using the information given on Table 4-3.

62

Table 4-4. Bearing capacity comparison for Briaud and Gibbens (1997)

Load Test Depth

[m] Measured Qult [MN]

Measure qult [kPa]


[kPa]

Modified Approach

[kPa]

3 m x 3 m (N) 0.762 12.0 1327 878 1504 3 m x 3 m (S) 0.889 9.7 1061 962 1585 2.5 m x 2.5 m 0.762 8.0 1284 816 1494 1.5 m x 1.5 m 0.762 3.9 1720 703 1510

1 m x 1 m 0.711 2.0 2046 623 1517

Figure 4-5. Load displacement curve from Briaud and Gibbens (1997)

Comparing the measured and calculated bearing capacities on Figure 4-6 clearly indi-

cates that the modified approach captures the general behavior better than the conven-

tional approach. Interestingly, the measured data presented in Table 4-4 suggests that

the bearing capacity actually increases for smaller footings. The conventional approach

to calculating bearing capacity predicts a linear increase in bearing capacity as footing

width increases; however, the modified unsaturated approach takes into account the

predicted failure surface. To reiterate, for small footings, like the 1 m footing, the fail-

ure surface is not expected to extend past the groundwater table (at 4.9 m). Also, suction

0 50 100 150 200

3

6

9

12

Displacement [mm]

Lo

ad [

MN

] 3m × 3m (S)

3m × 3m (N)

1m × 1m

1.5m × 1.5m

2.5m × 2.5m

63

stresses in this region are prevalent. The failure surface of larger footings, like the 3 m

footings, are more likely to extend into groundwater table and into regimes where suc-

tion stresses are small. Thus, average apparent cohesion is smaller for such footings. A

comparison is made in Figure 4-7, showing that this modified approach can indeed

account for the extension of the failure surface above and below the groundwater table

and large suction stress regimes.

Figure 4-6. Comparison of measured bearing capacity to the conventional and modified

approach for Briaud and Gibbens (1997)

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

Cal

cula

ted

Be

arin

g C

apac

ity

[kP

a]


ModifiedApproach

ConventionalApproach

64

Figure 4-7. Comparison of measured bearing capacity with respect to footing width (B) plus

embedded depth (D) for Briaud and Gibbens (1997)

The modified approach was used to calculate the bearing capacity using a range of

footing widths as in Figure 4-8. For the unsaturated parameters selected in Table 4-3,

the minimum bearing capacity predicted to be around 1,494 kPa with a footing width

of 2.5 m. This lines up with the results of the full scale load tests, which indicated that

the largest footings had the smallest bearing capacity. Figure 4-8 below also suggests

that as footing width increases after 2.5 m, the bearing capacity would then increase at

an approximately linear rate, which then corresponds to the behavior expected from the

bearing capacity equation.

0

500

1000

1500

2000

2500

0 1 2 3 4 5

Bea

rin

g C

apac

ity

[MP

a]

D + B [m]

Modified ApproachConventional ApproachField Results

65

Figure 4-8. Predicted bearing capacity with respect to footing with using the modified

approach for the soil data provided by Briaud and Gibbens (1997).

4.3.3. Larsson (1997)

Three load tests at the Vatthammar site were reported by Larsson (1997): 0.5 m, 1.0 m,

and 2.0 m square footings. In the upper 5 m, the soil was composed of 13% clay. Below

6m, there was approximately 10% clay. Table 4-5 summarizes soil and foundation

properties for the Vatthammar site.

0 2 4 6 81400

1600

1800

2000

2200

Footing Width [m]

Be

arin

g C

apa

city

[k

Pa]

66

Table 4-5. Soil properties from Larson (1997)

van Genuchten SWCC

Parameters‡ Value


Value



𝐷 (m) 0


1.37 Footing Width, 𝐵

(m) 0.5 – 2



(m) 0.5 – 2


0.034 Groundwater Table

Depth, 𝑧𝑤 (m) -


𝑘𝑠 (m/s) 7×10-7


Value Strength

Parameters† Value



𝜙′ (˚) 35


𝑐′ (kPa) 0

† Determined from drained triaxial compression tests

‡ Assumed from the average silt values from Carsel and Parrish (1988)

The groundwater table was not reported in this work, however at Vatthammar the

groundwater table location varies dramatically with evaporation and precipitation. Lar-

son (1997) suggests that 30 mm (1.2 in) of rainfall could raise the groundwater table

by 1 m. Larson reported that a significant amount of rainfall occurred between the test-

ing of the 0.5 m footing and the 1.0 m footing, which for the analysis in this work is

assumed to be 15 mm (infiltration), raising the groundwater table by 0.5 m.

The load displacement curves for the three load tests are presented in Figure 4-9. The

resulting bearing capacities are shown in Table 4-6.

67

Table 4-6. Results from static load tests at Vatthammar (Larson 1997)

Field Test Fitted Pult (kN) Fitted qult (kPa)

0.5×0.5 476 1900 1.0×1.0 920 920 2.0×2.0 1670 417

Figure 4-9. Hyperbolic fits to load displacement curve at Vatthammar site (Larsson 1997)

From these load tests, bearing capacity decreased for the larger footing sizes. This can

be explained either by the rainfall or changes in the failure surface shape that would be

detrimental to strength (like extending the failure surface into the saturated region of

the soil). Again, heavy rainfall was reported to occur between the 0.5 m footing and the

1.0 m and 2.0 m footings.

Larsson describes the soil at this site to be silty; thus, we can use Carsel and Parrish

(1988) to selected typical van Genuchten parameters as listed in Table 4-5. Table 4-7

presents calculated bearing capacities for various groundwater table locations since this

information was not given. The bolded values are the values that most accurately match

the bearing capacity in Table 4-6.

0 50 100 150 200

250

500

750

1 103

Displacement [mm]

Lo

ad [

kN

]

0 1 2 3 4

250

500

750

1 103

Displacement [mm]

Lo

ad [

kN

]

1.0 m

68

Table 4-7. Calculated bearing capacities at various GWT levels, for Vatthammar (Larsson

1997)

zgwt

Calculated qult (kPa)

B = 0.5 m, D = 0 m, qult = 1900 kPa

B = 1.0 m, D = 0 m, qult = 920 kPa

B = 2.0 m, D = 0 m, qult = 417 kPa

Modified Conventional Modified Conventional Modified Unmodified

0.5 m 402 103 450 166 573 292

1.0 m 745 103 730 207 829 333

2.0 m 1342 103 1284 207 1280 414

3.0 m 1797 103 1786 207 1711 414

3.25 m 1899 103 1894 207 1819 414

Table 4-7 clearly shows that the conventional bearing capacity equation cannot fully

capture the effects of suction stress, severely under-predicting the true capacity. The

values in this table do not account for the influence of rainfall, as previously discussed.

The assumptions made in the next paragraphs are theoretical, but are included to

demonstrate the value of including flux into the bearing capacity equation. Since no

rainfall occurred during the testing of the 0.5 m footing, a groundwater table depth of

3.25 m is the most probable. If it rained approximately at a rate of 15 mm/day for the

1 m and 2 m footings with and without moving the groundwater table, two more pre-

dictions for bearing capacity can be made. The next analysis follows the same assump-

tions used to calculate Table 4-7, but with varying flux and groundwater table depths.

Table 4-8. Calculated bearing capacities by varying q, for Vatthammar (Larsson 1997)

qult (kPa)

B = 0.5 m, D = 0 m B = 1.0 m, D = 0 m B = 2.0 m, D = 0m

q = 0 mm/day, z = 3.25 m q = 0 mm/day, z = 3.25 m q = 0 mm/day, z = 3.25 m

1899 1894 1819


1899 966 1071


1899 933 1006

The influence of q has a significant impact on the bearing capacity of shallow founda-

tions (more than varying the value for zw). The inclusion of infiltration decreased the

69

total bearing capacity by 928 kPa and 748 kPa for the 1 m and 2 m foundations. This

example shows the importance of considering rainfall (or evaporation) as it affects

bearing capacity. In this example, rainfall potentially decreased the bearing capacity by

50%.

4.3.4. Viana da Fonseca and Sousa (2002)

To validate a hyperbolic model, Viana da Fonseca and Sousa performed a load test

using a circular foundation on Portuguese residual soil. This soil was classified as either

a silty sand (SM) or a silty-clayey sand (SM-SC). These materials generally have

greater than 12% fines (silt and clay), thus we could predict the soil to be a sandy loam,

sandy clay loam, or loam in the USDA textural classification scheme. Table 4-9 sum-

marizes the given soil properties and the inferred soil water characteristic curve param-

eters from average values presented by Carsel and Parrish (1988) for sandy loam and

loam.

70

Table 4-9. Soil properties used for Viana da Fonseca and Sousa (2002)

Inferred SWCC Properties‡

Sandy Loam

Sandy Clay Loam

Loam Foundation Parameters

Value


0.765 0.602 0.367 Embedment Depth,

𝐷 (m) 0


1.89 1.48 1.56 Footing Width, 𝐵

(m) 1.2


0.471 0.324 0.359 Footing Length, 𝐿

(m) -


0.065 0.1 0.078 Groundwater Table

Depth, 𝑧𝑤 (m) 1.0


𝑘𝑠 (m/s) - - -


Value Strength

Parameters† Value



𝜙′ (˚) 37-38

Porosity, 𝑛 0.375-0.459

Cohesion, 𝑐′ (kPa)

9-12

† Determined from triaxial compression tests

‡ Inferred from the average sandy loam, sandy clay loam, and loam values from Carsel and Parrish

Figure 4-10. Fitted hyperbolic load displacement curve for Viana da Fonseca and Sousa

(2002) data

0 20 40 60 80 100

300

600

900

1200

Displacement [mm]

Aver

age

Fo

oti

ng P

ress

ure

[k

Pa]

71

To determine the bearing capacity for this loading condition, the Kondner (1963) hy-

perbolic equation was fitted to the results. From Figure 4-10, the predicted bearing

capacity was determined to be 𝑞𝑢𝑙𝑡 = 1865 kPa.

For the three different soil types described in Table 4-9, the predicted bearing capacities

are 1573 kPa for the sandy loam, 1824 kPa for the sandy clay loam, and 1875 kPa for

the loam. The unmodified Vesić bearing capacity equation calculates 1427 kPa. All

three inferred soil types calculate greater bearing capacities than the unmodified Vesić

solution. The loam (1875 kPa) is most similar to the bearing capacity from the fitted

hyperbolic equation (1865 kPa).

4.3.5. Rojas et al. (2007)

Rojas et al. (2007) performed full-scale plate load tests on circular foundations in un-

saturated lean clay. Seven tests were performed on variably saturated clay, including at

full saturation. Table 4-10 below describes the in-situ soil conditions and van Genuch-

ten parameters fitted to measured data in Figure 4-11.

72

Table 4-10. Soil properties used for Rojas et al. (2007)

van Genuchten SWCC

Parameters‡ Value


Value



𝐷 (m) 0



(m) 0.31



(m) -


0 Groundwater Table



𝑘𝑠 (m/s) -


Value Strength

Parameters† Value



𝜙′ (˚) 26


𝑐′ (kPa) 3

† Method for obtaining strength parameters was not reported.

‡ Fitted to the van Genuchten (1980) equation.

Since Rojas et al. (2007) did not provide a depth for the groundwater table, the matric

suction profile was linearly interpolated based on the matric suction data in Table 4-11.

Figure 4-12 presents the linearly interpolated matric suction profiles that are used in

the modified bearing capacity equation. In this figure, depth is measured from the sur-

face.

Table 4-11 describes the seven different loading scenarios. Rojas et al. (2007) did not

provide estimates for the location of the groundwater table, but instead provided matric

suction values at different depths, which are included on Table 4-11. Figure 4-11 in-

cludes a fit for the soil water characteristic curve using the van Genuchten (1980) form.

73

Figure 4-11. Fitted SWCC for the Rojas et al. (2007) data.

Table 4-11. Matric suction from tests and maximum bearing capacity from hyperbolic fit for

Rojas et al. data (2007).

Test No.

Matric Suction 𝝍 [kPa] at Depth [m] 𝒒𝒖𝒍𝒕

[kPa] 0.1 0.3 0.6 0.9

S1 0 0 0 0 450

S2 0 3 0 0 376

U1 10 4 13 0 552

U2 48 38 15 0 671

U3 56 46 4 0 961

U4 60 53 6 4 980

U5 63 57 11 0 1112

0.01 0.1 1 10 100 1 103

1 104

0

0.1

0.2

0.3

0.4

Data from Pressure Plates

Data from Tensionmeter

Fitted VG Equation

Suction [kPa]

Vo

lum

etri

c W

ate

r C

on

ten

t [%

]

74

Figure 4-12. Linearly interpolated matric suction profile for Rojas et al. (2007) data

The modified bearing capacity calculations accounting for partial saturation through

these matric suction profiles results in the values presented in Figure 4-13.

Figure 4-13. Comparison of calculated qult for the Rojas et al. (2007) data using the modified

and unmodified bearing capacity equation.

0 0.2 0.4 0.6 0.8 1

20

40

60

80

Depth [m]

Mat

ric

Su

ctio

n [

kP

a]

0

300

600

900

1200

1500

0 300 600 900 1200 1500

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]


ModifiedApproach

ConventionalApproach

S1

U1

S2

U2

U1

U3

U1

U4

U1

U5

U1

75

Figure 4-13 implies that the modified approach suggested in this work can reasonably

predict bearing capacity for these tests; however, the modified approach over predicts

the bearing capacity, which is most significant in loading scenario U2. Calculations for

Scenarios U3, U4, and U5 also overpredicted, but much less than for U2. These over

predictions may potentially be attributed to poor characterization of the SWCC or of

the matric suction profile.

4.3.6. Vanapalli and Mohamed (2007) / Oh and Vanapalli (2008)

Vanapalli and Mohamed (2007) performed scaled plate load tests on partially saturated

sands to quantify the bearing capacity when suction is present. Similar measurements

were made by Vanapalli and Mohamed to develop a soil water characteristic curve.

The van Genuchten equation was fitted in Figure 4-14 to the existing SWCC data from

Vanapalli and Mohamed (2007) for use in this work. Four different tests were per-

formed at average matric suctions of 0 kPa, 2 kPa, 4 kPa, and 6 kPa. The USCS soil

classification is poorly graded sand (SP). The soil properties are presented in Table

4-12. Soil properties used in Vanapalli and Mohamed (2007)

76

Table 4-12. Soil properties used in Vanapalli and Mohamed (2007)

van Genuchten SWCC

Parameters‡ Value


Value



𝐷 (m) 0



(m) 0.1



(m) 0.1


0 Groundwater Table



𝑘𝑠 (m/s) -


Value Strength

Parameters† Value



𝜙′ (˚) 35.3 or

39


𝑐′ (kPa) 0.6

† Determined from direct shear tests. The authors recommend 39° to account for dilatancy

‡ Determined by fitting the van Genuchten (1980) equation to the measured SWCC

D60 and D30 were reported as 0.22 mm and 0.18 mm respectively. Since the footing

width is 100 mm, B/D60 = 454 and B/D30 = 555, and therefore B/D50 > 100. It is not

expected that scale effects had an influence on the results since B/D50 is greater than

the Kusakabe (1995) and Herle and Tejchman (1997) threshold.

77

Figure 4-14. Fitted SWCC using van Genuchten (1980) (after Vanapalli and Mohamed 2007)

In this work, measurements of matric suction were made at the bottom of the foundation

and at a depth of 1.5B from the bottom of the foundation. The reported suction was

then the average of these two values. An equivalent ground water could be determined

from Eq. 4-1 assuming matric suction increases linearly above the groundwater table.

This equation was developed to match the provided average matric suctions since the

framework presented in this work uses groundwater table depths instead.

𝜓 =

1

2(𝑧𝑤𝛾𝑤 + (𝑧𝑤 − 1.5𝐵)𝛾𝑤) 4-1

where 𝜓 is the matric suction, 𝑧𝑤 is the depth of the groundwater table, 𝐵 is the footing

width, and 𝛾𝑤 is the unit weight of water. Solving for depth of the ground water table:

𝑧𝑤 =𝜓

𝛾𝑤+ 0.75𝐵 if 𝜓 > 0.75𝐵𝛾𝑤 4-2

𝑧𝑤 =2𝜓

𝛾𝑤 if 𝜓 < 0.75𝐵𝛾𝑤 4-3

For average matric suctions of 0 kPa, 2 kPa, 4 kPa, and 6 kPa the equivalent ground-

water table depth is 0 m, 0.279 m, 0.483 m, and 0.687 m.

0 2 4 6 8 10

20

40

60

80

100

Suction [kPa]

Satu

rati

on

[%

]

78

Figure 4-15 compares the predicted bearing capacity to the measured bearing capacity

using a friction angle of 𝜙′ = 39° and the groundwater table depths described previ-

ously. The bearing capacity was calculated using both the modified approach presented

in this work and the conventional approach. Vanapalli and Mohamed (2007) proposed

a closed-formed solution to the bearing capacity of shallow foundations in Eq. 4-4

which is also compared in this figure. This equation is discussed in Section 2.1.3.


𝑢𝑤)𝐴𝑉𝑅𝑆𝜓 tan 𝜙′] 𝑁𝑐 [1 + (𝑁𝑞

𝑁𝑐) (

𝐵

𝐿)] + 0.5𝐵𝛾𝑁𝛾 [1 − 0.4

𝐵

𝐿]

4-4

The modified approach and the equation proposed by Vanapalli and Mohamed (2007)

show good agreement with the measured bearing capacity.

Figure 4-15. Comparison of actual bearing capacity to predictions from this work and

Vanapalli and Mohamed (2007)

Figure 4-16 compares the bearing capacity of the model shallow foundation to the av-

erage suction applied at D and D + B. Interestingly, the peak bearing capacity calcu-

lated using the modified approach proposed herein and the measured bearing capacities

0

200

400

600

800

1000

0 200 400 600 800 1000

Cal

cula

ted

Be

arin

g C

apac

ity

[kP

a]


This Work

Vanapalli andMohamed (2007)ConventionalApproach

79

from Vanapalli and Mohamed (2007) are essentially the same. The bearing capacity

predicted in this work quickly reduces after 5 kPa. This is quite different to the work

of Vanapalli and Mohamed, who predict a slower decline in strength with an increase

in suction.

Figure 4-16. Bearing capacity vs. variation in average matric suction from this work and


The bearing capacity profile calculated from the proposed modified approach shows a

peak at approximately 5 kPa followed by a dramatic decrease in bearing capacity after

6 kPa, converging to the conventional approach at higher suctions. The bearing capac-

ity profile calculated from the Vanapalli and Mohamed (2007) equation shows an in-

crease in bearing capacity until 6 kPa followed by a shallow decrease in bearing capac-

ity at higher suctions. The SWCC in Figure 4-14 shows an air-entry suction of approx-

imately 4 kPa followed by rapid desaturation, therefore the profile predicted by this

current work is more consistent with the SWCC than that predicted by Vanapalli and

Mohamed (2007).

0

300

600

900

1200

1500

0 2 4 6 8 10

Be

arin

g C

apac

ity

[kP

a]

Matric Suction [kPa]

This Work




80

4.3.7. Vanapalli and Mohamed (2013)

Vanapalli and Mohamed (2013) used the same soil as in Vanapalli and Mohamed

(2007); refer to Table 4-12 for the soil properties. The difference between this and the

previous work was the use of a larger 150 mm plate embedded at depths of 0 mm and

150 mm. The surface plate was loaded at average suction stresses of 0 kPa, 2 kPa, 4

kPa, and 6 kPa, which can be represented with groundwater table depths of 0 m, 0.317

m, 0.521 m, and 0.725 m as calculated with Eqs. 4-1, 4-2 and 4-3. The embedded plate

was tested at 0 kPa, 2 kPa, and 6 kPa, which can be represented with groundwater table

depths of 0 m, 0.317 m, and 0.725 m. The authors recommend that when the footing is

embedded, a friction angle of 35.3° should be used instead of 39° to account for a re-

duction in dilation.

Figure 4-17. Comparison of measured and predicted bearing capacity for 150 mm surface

plate for Vanapalli and Mohamed (2013)

Figure 4-17 compares measured bearing capacity to the bearing capacity calculated by

the modified approach, the Vanapalli and Mohamed approach, and the conventional

approach. Both the modified and the Vanapalli and Mohamed approach show close

0

200

400

600

800

1000

0 200 400 600 800 1000

Cal

cula

ted

Be

arin

g C

apac

ity

[kP

a]

Measured Capacity [kPa]

This Work

Vanapalli andMohamed (2013)ConventionalApproach

81

agreement to the actual bearing capacity, while the conventional approach significantly

underpredicts the bearing capacity.

Figure 4-18 shows the variation in bearing capacity with suction. This work predicts a

dramatic decrease in strength after 4 kPa of suction, whereas Vanapalli and Mohamed

(2013) predict a slower decline in bearing capacity, which begins at 6 kPa. A dramatic

decline in bearing capacity is expected due to soil being a poorly graded sand. It is

expected that sands have low air-entry suctions (i.e. high α values) and a higher value

for n. The air-entry suction defines the suction where air begins to rapidly enter into

the pores. van Genuchten’s n describes the rate of desaturation with increasing suction.

A higher n means a steeper SWCC and thus, suction stress is expected to only have an

influence over a narrow range of suctions (as opposed to clays, which typically have

higher n values).


Vanapalli and Mohamed (2013) for a 150×150 mm plate loaded on the surface.

To emphasize the short range over which matric suction will influence behavior, Figure

4-19 shows the SWCC and suction stress for the Vanapalli and Mohamed soil with

0

300

600

900

1200

1500

0 2 4 6 8 10

Bea

rin

g C

apac

ity

[kP

a]


This Work




82

respect to distance above the groundwater table. As stated previously, water content

decreases rapidly above the groundwater table, resulting in negligible suction stresses

at suctions greater than 4 or 5 kPa. If the groundwater table was at a depth of 1 m

(approximately 10 kPa matric suction) there should be little suction stress acting on the

failure surface.

Figure 4-19. SWCC and suction stress profile for Vanapalli and Mohamed (2013) soil.

The second set of testing Vanapalli and Mohamed (2013) performed was on plates

embedded at a depth of 150 mm. Since the plates were embedded, the authors argued

that effects due to dilation could be ignored. However, the confining stress associated

with 150 mm of overburden is quite small; thus, in this work, both the original value

of 𝜙′ and the recommended scaled value of 1.1𝜙′ are considered to account for the

dilation of the densely packed sand. Figure 4-20 compares the measured bearing ca-

pacity for the embedded plate against the modified approach for friction angles of 35.3˚

and 39˚ and against the Vanapalli and Mohamed approach. Figure 4-21 presents the

bearing capacity profile from these two approaches and the measured bearing capacity.

0 0.1 0.2 0.3

0.5

1

1.5

2

Gravimetric Water Content

Dis

tan

ce A

bo

ve G

WT

[m

]

0 1 2 3 4 5

0.5

1

1.5

2

Suction Stress [kPa]

Dis

tan

ce A

bo

ve G

WT

[m

]

83

Figure 4-20. Comparison of measured and calculated bearing capacity for 150 mm embedded

plate for Vanapalli and Mohamed (2013)

This work shows that friction angles of 39° and 35.3° bracket the measured bearing

capacity. This may imply that the assumption that dilation can be ignored for embedded

foundations is not valid, but rather, that dilation is suppressed, which is consistent with

conventional shear strength theory. The equation proposed by Vanapalli and Mohamed

show good agreement to the measured data.

0

300

600

900

1200

1500

0 300 600 900 1200 1500

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]


This Work (Friction = 35.3)

This Work (Friction = 39.0)


84


Vanapalli and Mohamed (2013) for a 150×150 mm plate embedded 150 mm.

4.3.8. Wuttke et al. (2013)

Wuttke et al. (2013) performed scaled load tests on shallow strip foundations. In this

work, the soil was desaturated via suction in order to replicate the suction stress that

occurs in-situ due to the presence of a groundwater table. The soil used in this work is

Hostun sand, which has a USCS classification of poorly graded sand (SP). The SWCC

was determined experimentally, so no prediction is required. The fitted SWCC is

shown in Figure 4-22. In this work, the plate is loaded over partially saturated soil and

suctions is measured at various depths. The sand specimen is saturated from the bottom.

The properties of Hostun sand are presented in Table 4-13.

0

500

1000

1500

2000

0 2 4 6 8 10

Bea

rin

g C

apac

ity

[kP

a]


This Work (Friction = 35.3)This Work (Friction = 39.0)Vanapalli and Mohamed (2013)Measured Bearing Capacity

85

Table 4-13. Soil properties for Hostun sand (Wuttke et al. 2007)

van Genuchten Drying SWCC Parameters‡

Value van Genuchten Wetting SWCC

Parameters Value


Value



𝛼 (kPa-1) 0.906

Embedment Depth, 𝐷 (m)

0



𝑛 3.411

Footing Width, 𝐵 (m)

0.079



𝑚 0.701

Footing Length, 𝐿 (m)

0.477


0.017 Residual Water

Content, 𝜃𝑟 0.017

Groundwater Table Depth, 𝑧𝑤 (m)

-


𝑘𝑠 (m/s) -


𝑘𝑠 (m/s) -


Value Strength

Parameters† Value



𝜙′ (˚) 46.7


𝑐′ (kPa) -

† Determined under plane-strain loading conditions using a double-wall cell

‡ Determined by fitting the van Genuchten (1980) equation to the measured SWCC

Kusakabe (1995) and Herle and Tejchman (1997) recommend a threshold of 𝐵/𝐷50 ≥

100 to avoid particle size affects. Hostun sand has 𝐷50 = 0.35 mm while the scaled

footing has a width of 𝐵 = 79 mm. 𝐵/𝐷50 = 226 which is over twice the recom-

mended value, implying that scaled effects should be small.

86

Figure 4-22. Soil water characteristic curve for Hostun sand (after Wuttke et al. 2013)

In this work, measurements of matric suction were made at the bottom of the foundation

and at a depth of 1.5B from the bottom of the foundation. The reported suction was

then the average of these two values. This is similar to the work done by Vanapalli and

Mohamed (2007). Wuttke et al. (2013) made measurements at 2 kPa, 3 kPa, 4 kPa,

completely dry, and completely saturated. This corresponds to groundwater tables

depths of 263mm, 365mm, 467mm, very deep (approximated as 1000mm), and 0mm.

The bearing capacity profile was calculated using the modified approach and the con-

ventional bearing capacity equation in Figure 4-23. In Figure 4-24, the measured bear-

ing capacity was compared to values calculated from the modified and conventional

approach. For the modified approach, two separate calculations were made using van

Genuchten properties for the wetting and drying curves shown in Figure 4-22.

0.1 1 10 1000

10

20

30

40

Wetting Curve

Drying Curve

Suction [kPa]

Vo

lum

etri

c W

ater

Co

nte

nt

[%]

87

Figure 4-23. Calculated and measured bearing capacities compared to the average matric suc-

tion at D and D + B.

Figure 4-23 shows that the measured bearing capacity was generally between both the

wetting and drying curve. The calculated bearing capacity using both the wetting and

drying soil water characteristic curves performed well compared to the measured bear-

ing capacity and compared to the conventional bearing capacity equation, as in Figure

4-24.

0

300

600

900

1200

0 2 4 6

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]

Matric suction [kPa]

Wetting SWCC

Drying SWCC



88

Figure 4-24. Comparison of actual bearing capacity to predictions from the conventional and

modified approach for Wuttke et al. (2013)

4.4. Summary and Discussion

This chapter provided validation for the modified bearing capacity equation proposed

in this work. The comparative study in Section 4.3 included several load tests con-

ducted on both model and full-scaled shallow foundations. Bearing capacities were cal-

culated using the modified bearing capacity equation and the conventional equation.

Figure 4-25 summarizes the comparative study in Section 4.3, comparing the measure

bearing capacity for the load tests in this section against the predicted bearing capacity

using the modified and conventional approach. Trendlines were fitted to the data for

comparison against the 1:1 line. Data from Larsson (1997) was not included in this

figure since the groundwater table depth was not provided in this work.

0

200

400

600

800

0 200 400 600 800

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]


WettingSWCCDrainedSWCCConventionalApproach

89

Figure 4-25. Measured bearing capacity vs. predicted bearing capacity for database of load

tests in Chapter 4.

The linear trendline fitted to the modified approach shows closer agreement to the 1:1

line than the conventional approach. The conventional bearing capacity equation will

generally underpredict bearing capacity. The slope of the trendline for modified ap-

proach is 0.93 while the slope for the conventional approach is 0.43. Using linear re-

gression against the 1:1 line gives a coefficient of determination, R2, of 0.81 for the

modified approach and -2 for the conventional. The closer the coefficient of determi-

nation is to 1, the better the fit. A negative R2 implies that the conventional approach

does not at all correlate with the measured bearing capacity.

In summary, this chapter shows that the conventional bearing capacity equation will

generally underpredict the bearing capacity except at lower bearing capacities (which

the majority are saturated load tests). The modified bearing capacity equation shows

close agreement to the measured bearing capacity data presented in this section, vali-

dating the approach used in this work. Suction stresses (not matric suction) in partially

saturated soils will have the tendency to increase soil strength, increasing the bearing

capacity.

0

600

1200

1800

2400

0 600 1200 1800 2400

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]


Modified

Conventional

Linear (Modified)

Linear (Conventional)

90

5. Parametric Studies

5.1. Outline of Parametric Studies

In this section, shallow and deep foundation performance will be evaluated with respect

to varying soil parameters such as 𝜙′, 𝜃𝑟, 𝜃𝑠, 𝛼, 𝑛, 𝑚, and 𝑘𝑠 where all terms are as

previously defined. Further considerations will also be made to foundation shape – such

as footing width 𝐵 and embedment depth 𝐷, groundwater table depth 𝑧𝑤, and variation

to water infiltration 𝑞. In this study, considerations will not be made to the cohesion 𝑐′,

specific gravity 𝐺𝑠, and footing length 𝐿 since these parameters will not affect the foun-

dation performance differently in unsaturated soils than in saturated soils. The soil

types that will be selected in this study are sand, silt, and clay. These three soil types

serve as the extremes (as in the sand and clay) and average (silt) for typical soil water

characteristic curves.

Section 5.2 will present the soil properties that correspond to the theoretical sand, silt,

and clay that are used in this parametric study. Section 5.3 will cover parametric studies

on shallow foundations. Section 5.4 will cover parametric studies on deep foundations,

presenting only a discussion on shaft resistance. Section 5.5 will cover Monte Carlo

analyses for shallow foundation bearing capacity using Carsel and Parrish (1988) to

calculate appropriate realizations for unsaturated soil properties. Chapter 5 will be

concluded with a discussion in Section 5.6.

5.2. Soils Parameters Used in Parametric Study

The soil types that will be used in this study are sand, silt, and clay. 𝛼 generally varies

by a magnitude between sand, silt and clay. The selection of 𝛼 = 1.0, 0.1, and 0.01

kPa-1 for sand, silt, and clay are simple values that describe the general ranges of 𝛼

between these three soil types. van Genuchten’s fitting parameter 𝑛 is mathematically

limited to being greater than 1.0, so for clays, a value of 1.1 was selected. For theoret-

ical silt and sand, 𝑛 values of 1.5 and 3.0 will be used. These value have been selected

91

to describe the range at which 𝛼 and 𝑛 vary between all three soil types. Hydraulic

conductivity between sand, silt, and clay can vary by a magnitude or more, thus, 𝑘𝑠 =

10-5, 10-6, and 10-7 m/s respectively are selected as typical values. 𝜃𝑠 = 0.4 and 𝜃𝑟 =

0.06 are used to describe the saturated and residual water contents, respectively. The

below table summarizes the hydraulic properties used in the following parametric stud-

ies for each of theoretical sand, silt, and clay.

Table 5-1. Soil properties used in this parametric study

𝜶 (𝐤𝐏𝐚−𝟏) 𝒏 𝒌𝒔 (𝐦/𝐬) 𝜽𝒔 𝜽𝒓

Clay 0.01 1.1 10-7 0.4 0.06

Silt 0.10 1.5 10-6 0.4 0.06

Sand 1.00 3.0 10-5 0.4 0.06

In this study, the bearing capacity of a shallow foundation and the 𝛽-method for deep

foundations will be evaluated with the ranges of variables proposed in Table 5-1. Fur-

ther, the typical 𝜃𝑠 value will be taken as 0.4, but will be studied at 0.35 and 0.45.

Likewise, the typical 𝜃𝑟 value will be taken as 0.06, but will be studied at 0.02 and 0.1.

Van Genuchten’s 𝑚 will be set equal to 𝑚 = 1 − 1/𝑛 (Mualem 1976; van Genuchten

1980). For all three soil types 𝑐′ = 0 kPa, and 𝐺𝑠 = 2.65. 𝜙′ will be studied at 25°,

30°, and 35°. When the effects of fluid flux are studied, values ranging from 𝑞 =

−3.14 × 10−8 m/s to 1.15 × 10−8 m/s will be used. This range is recommended by

Lu and Likos (2004) as representative flux values.

Carsel and Parrish (1988) determined average values for unsaturated soil properties

including 𝜃𝑠, 𝜃𝑟, 𝛼 and 𝑛, as discussed in Section 2.3.8. Typical soil water characteristic

curves have been plotted from these mean values listed in Table 2-1 using the van

Genuchten (1980) equation:

92

𝜃 = 𝜃𝑟 +

𝜃𝑠 − 𝜃𝑟

(1 + (𝛼𝜓)𝑛)1−1/𝑛 5-1

Figure 5-1 presents (a) soil water characteristic curves for the 12 USDA textural classes

from Carsel and Parrish (1988) (gray and blue lines) and for the theoretical sand, silt,

and clay (red dashed lines) used in this study and (b) the USDA textural triangle, de-

scribing the means by which each soil is classified. The theoretical sand and clay, listed

in Table 5-1, serve as the extremes for unsaturated soil response. The air-entry value

for sands is significantly smaller than clays and the fitting parameter n is generally

larger, resulting in a more abrupt decrease in water content with increasing suction. The

theoretical silt serves to describe SWCC that are in between these two extremes, as

presented in Figure 5-1 (a). It is intended that the selection of the previously described

parameters would sufficiently describe the three extreme soil textures presented in Fig-

ure 5-1 (b).

Figure 5-1. (a) Soil water characteristic curves for the 12 USDA Textural Classes using van

Genuchten parameters from Carsel and Parrish (1988). Curves for sand, clay and silt loam are

highlighted. (b) USDA Textural Triangle (USDA 2016).

C & P Silt Loam

Theoretical Sand

Theoretical Silt

93

The soil water characteristic curves for sand, silt loam, and clay noted Carsel and Par-

rish (1988) as they compare relatively well with the values that are used for this study.

The theoretical soils used in this study, as shown with the dashed lines, bracket the

ranges of SWCCs described in Carsel and Parrish (1988).

5.3. Parametric Studies on Shallow Foundations

The shallow foundation portion of the parametric study will be composed of founda-

tions that have a footing width 𝐵 = 1 m. All other footing shape parameters and water

depth will be normalized to this value with the exception of the footing length, 𝐿, which

is considered to be very long (i.e. a strip footing). The depth of embedment will be

studied at four different depths: 𝐷 = 0, 𝐵/2, 𝐵, and 3𝐵/2. The variation of the ground-

water table will be set to 𝑧𝑤 = 𝐷 to 𝑧𝑤 = 𝐷 + 4𝐵.

5.3.1. Shallow Foundation Bearing Capacity Profiles

The first set of analyses in the shallow foundation parametric study calculate bearing

capacity as a function of the groundwater table depth. The deeper the groundwater ta-

ble, the higher the matric suction is near the soil surface. A plot comparing bearing

capacity vs. depth of the groundwater table will subsequently be called the bearing

capacity profile. Evaluating the influence of matric suction on bearing capacity is the

focus of this research, so beginning the parametric study with these bearing capacity

profiles is important. Bearing capacity will be calculated according to the methodology

described for shallow foundations in Section 3.2. For each figure, one input parameter

will be varied to assess its effect on bearing capacity. For these figures, zw will be eval-

uated to a depth of 4 m, which is deeper than the foundation stress bulb D + B.

The first figure, Figure 5-2, presents the influence of friction angle on partially satu-

rated soils. Three different friction angles, 𝜙′ = 25°, 30°, and 35° are assessed for the

theoretical sand, silt, and clay with an embedment depth D = 0 m. Note that in Figure

5-2 and in the subsequent figures of bearing capacity profiles, the y-axis will not be

94

plotted to the same scale. This is because the theoretical suction stresses in clays are

large, resulting in high bearing capacities.

Figure 5-2. Shallow foundation bearing capacity profile of clay, silt, and sand at varying

friction angles. Note changing ordinate across figures.

Figure 5-2 demonstrates the large influence suction stress have on fine-grained soils.

The more fine a soil is (lower 𝑛 and 𝛼), the greater the influence generally is. With the

same friction angle, clays see a bearing capacities that are significantly greater than for

sands. Since the value predicted for clays and silts are large, it is important to note that

these are theoretical predictions, uncalibrated to physical measurements. As expected,

with an increase in friction angle, the strength increases as well, although it is a non-

linear increase. In subsequent figures, 𝜙′ will be held at 30°.

0

500

1000

1500

2000

2500

3000

0 1 2 3 4

Bea

rin

g C

apac

ity

[kP

a]

GWT Depth [m]

Clay, n = 1.1α = 0.01 kPa-1

0

400

800

1200

1600

2000

0 1 2 3 4

Bea

rin

g C

apac

ity

[kP

a]GWT Depth [m]

Silt, n = 1.5α = 0.1 kPa-1

0

100

200

300

400

500

600

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

GWT Depth [m]

Sand, n = 3.0α = 1.0 kPa-1

050010001500200025003000

-2 3

φ' = 25°

φ' = 30°

φ' = 35°

95

Figure 5-3 presents the influence of foundation embedment depth on the bearing ca-

pacity profile in partially saturated soils. The 1 m footing is embedded D = 0, B/2, B,

and 3B/2 and is assessed for theoretical sand, silt, and clay.

Figure 5-3. Shallow foundation bearing capacity vs. zgwt - D for clay, silt, and sand at varying

depths of embedment. Note changing ordinate across figures.

Figure 5-3 compares bearing capacity profile to embedment depth. The embedment

depth is subtracted from the groundwater table depth so that the plots would all begin

at 0. This figure shows that increasing the depth of embedment of a shallow foundation

in unsaturated soils will only increase the bearing capacity by a constant value for all

ranges of zgwt – D. The increase in bearing capacity can be attributed to the overburden

0

500

1000

1500

2000

2500

3000

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

zgwt - D [m]

Clay, n = 1.1α = 0.01 kPa-1

0

400

800

1200

1600

2000

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

zgwt - D [m]

Silt, n = 1.5α = 0.1 kPa-1

0

300

600

900

1200

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

zgwt - D [m]

Sand, n = 3.0α = 1.0 kPa-1

050010001500200025003000

-2 3

D = 0

D = B/2

D = B

D = 3B/2

96

term in the bearing capacity equation, which accounts for suction and normal stresses,

and the depth factors introduced by Hansen (1970).

Similarly, the data in Figure 5-3 can be plotted against groundwater depth only. This is

done in Figure 5-4.

Figure 5-4. Shallow foundation bearing capacity vs. groundwater table depth for clay, silt,

and sand at varying depths of embedment. Note changing ordinate across figures.

Figure 5-4 shows another interesting trend for embedded foundations in partially satu-

rated soils. While the previous figure shows that increasing depth of embedment in-

creases bearing capacity by a constant, this does not prevent shallower foundations to

0

500

1000

1500

2000

2500

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

zgwt [m]

Clay, n = 1.1α = 0.01 kPa-1

0

400

800

1200

1600

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

zgwt [m]

Silt, n = 1.5α = 0.1 kPa-1

0

200

400

600

800

1000

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

zgwt [m]

Sand, n = 3.0α = 1.0 kPa-1

05001000150020002500

-2 3

D = 0

D = B/2

D = B

D = 3B/2

97

have a greater bearing capacity. This is observed in a silt, where D = 0 exceeds D = B/2

for a portion of the figure.

Another component of this research is to assess the influence of infiltration and evap-

oration on bearing capacity. The matric suction profile and its implementation in cal-

culating bearing capacity are discussed in Sections 2.3.6 and 3.2.1. Figure 5-5 plots the

bearing capacity profile at three different rates of flux, q = -3.14E-8 m/s (which is in-

filtration), q = 0 (which is no flow), and q = 1.15E-8 m/s (which is evaporation). This

is done for the theoretical sand, silt, and clay.

Figure 5-5. Shallow foundation bearing capacity profile of clay, silt, and sand with varying

rates of flux. Note changing ordinate across figures.

0

500

1000

1500

2000

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

GWT Depth [m]

Clay, ks = 10-7 m/s

0

400

800

1200

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

GWT Depth [m]

Silt, ks = 10-6 m/s

0

50

100

150

200

250

0 1 2 3 4

Bea

rin

g C

apac

ity

[kP

a]

GWT Depth [m]

Sand, ks = 10-5 m/s

0500100015002000

-2 3

q = -3.14E-8 m/s

q = 0 m/s

q = 1.15E-8 m/s

98

Two behaviors can be distinguished from these plots: (1) in the theoretical silt and clay,

bearing capacity increases with evaporation; and (2) in sand, bearing capacity increases

with infiltration. The explanation for this discrepancy is that in sands suction stresses

are very small (e.g., Lu et al. 2010) and at net zero flow or evaporation the pores are

likely at or near residual saturation, implying that suction stresses are low. When infil-

tration occurs in sands, the pores become filled with water increasing the average unit

weight and suction stress, thus increasing the bearing capacity. Alternatively, in silts

and clays the suction stresses are much higher, allowing for water to be retained in the

pores. Infiltration in silts and clays would cause the suction stress to decrease and there-

fore the bearing capacity as well.

Figure 5-6 and Figure 5-7 address the influence of 𝜃𝑠 and 𝜃𝑟 on the shallow foundation

bearing capacity profile in partially saturated soils. 𝜃𝑠 is evaluated in Figure 5-6 with

values of 0.35, 0.40, and 0.45. 𝜃𝑟 is evaluated in Figure 5-7 with values of 0.02, 0.06,

and 0.10. These two soil properties ultimately control the quantity of waters retained

within the soil pores.

99

Figure 5-6. Shallow foundation bearing capacity profile of clay, silt, and sand with varying

θs. Note changing ordinate across figures.

Figure 5-6 presents the influence of varying the saturated volumetric water content.

Essentially this is the same as varying the porosity. Varying the porosity will ultimately

affect the unit weight term in the bearing capacity equation; a lower porosity will mean

a higher dry unit weight. While the figure above shows that sands are most influenced

by this parameter, silts and clays are also affected, but due to the magnitude of bearing

capacities cannot be visualized. Other than changing the dry unit weight of the soil,

varying 𝜃𝑠 does little in regards to influencing behavior.

0

250

500

750

1000

1250

1500

1750

0 1 2 3 4

Bea

rin

g C

apac

ity

[kP

a]

GWT Depth [m]

Clay, n = 1.1α = 0.01 kPa-1

0

150

300

450

600

750

900

0 1 2 3 4

Bea

rin

g C

apac

ity

[kP

a]

GWT Depth [m]

Silt, n = 1.5α = 0.1 kPa-1

0

50

100

150

200

250

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

GWT Depth [m]

Sand, n = 3.0α = 1.0 kPa-1

0500100015002000

-2 3

θs = 0.35

θs = 0.40

θs = 0.45

100

Figure 5-7. Shallow foundation bearing capacity profile of clay, silt, and sand with varying θr.

Note changing ordinate across figures.

Figure 5-7 plots the influence of the residual volumetric water content on the bearing

capacity profile. The residual water content has little influence on the behavior of silt

and clay. On the other hand, sands are affected by the value of the residual water con-

tent. In sands, the pores are more likely to be drained and at a residual state. Thus,

varying the residual water content will directly influence the volume of water retained

in the pores. When the volume of water is varied, the moist unit weight of the soil will

be influenced in the same manner, either increasing or decreasing the bearing capacity.

For silts and clays, if the groundwater table was sufficiently deep and the pores at a

residual state, then the residual volumetric water content would influence the bearing

capacity as well.

0

500

1000

1500

2000

0 1 2 3 4

Bea

rin

g C

apac

ity

[kP

a]

GWT Depth [m]

Clay, n = 1.1α = 0.01 kPa-1

0

150

300

450

600

750

900

0 1 2 3 4

Bea

rin

g C

apac

ity

[kP

a]

GWT Depth [m]

Silt, n = 1.5α = 0.1 kPa-1

0

35

70

105

140

175

210

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

GWT Depth [m]

Sand, n =3.0α = 1.0 kPa-1

0500100015002000

-2 3

θr = 0.02

θr = 0.06

θr = 0.10

101

The last consideration that will be made in these series of plots is how varying 𝛼 (Figure

5-8) and 𝑛 (Figure 5-9) will affect the bearing capacity profile. In varying these prop-

erties, the soil cannot be simply called a ‘sand’, ‘silt’, or ‘clay’ since the variation in 𝛼

and 𝑛 exceeds what is typical for each material. 𝛼 will be assessed at 0.01, 0.1 and 1.0

kPa-1. 𝑛 will be assessed at 1.1, 1.5, and 3.0. The purpose for this exercise is to deter-

mine the sensitivity of bearing capacity to these soil water characteristic curve fitting

parameters.

Figure 5-8. Shallow foundation bearing capacity profile with varying α.

In Figure 5-8, van Genuchten’s n is held constant at 1.1, 1.5 and 3.0 by subplot. 𝛼 is

varied at 0.01, 0.1, and 1 kPa-1. This figure shows that a lower 𝛼 will result in higher

0

400

800

1200

1600

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

GWT Depth [m]

n = 1.1

0

400

800

1200

1600

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

GWT Depth [m]

n = 1.5

0

400

800

1200

1600

0 1 2 3 4

Bea

rin

g C

apac

ity

[kP

a]

GWT Depth [m]

n = 3.0

0500100015002000

-2 3

α = 0.01 1/kPa

α = 0.10 1/kPa

α = 1.00 1/kPa

102

bearing capacity. 𝛼 varies the shape of the soil water characteristic curve. Increasing 𝛼

decreases the air-entry suction, while decreasing 𝛼 increases the air-entry suction. For

lower 𝛼, air-entry occurs at a higher matric suction, which implies that the groundwater

table depth must be greater in order for air to begin rapidly entering the pores, and

thereby inducing suction stresses. For n = 3.0, bearing capacity profile shows a defini-

tive peak, and then begins to decrease for 𝛼 = 0.1 and 1.0 kPa-1. This is typical for

coarse-grained soils.

Figure 5-9. Shallow foundation bearing capacity profile with varying n. Note changing

ordinate across figures.

Figure 5-9 is similar to Figure 5-8 except that 𝛼 is held constant at 0.01 kPa−1,

0.1 kPa−1, and 1 kPa−1 for each subplot. For 𝛼 = 0.1 and 1 kPa-1, smaller values for

0

400

800

1200

1600

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

GWT Depth [m]

α = 0.01 kPa-1

0

400

800

1200

1600

0 1 2 3 4

Be

arin

g C

apac

ity

[kP

a]

GWT Depth [m]

α = 0.10 kPa-1

0

400

800

1200

1600

0 1 2 3 4

Bea

rin

g C

apac

ity

[kP

a]

GWT Depth [m]

α = 1.0 kPa-1

0500100015002000

-2 3

n = 1.1

n = 1.5

n = 3.0

103

n result in larger bearing capacity. For 𝛼 = 0.01 kPa-1, n = 1.1 and n = 3.0 have higher

bearing capacities than n = 1.5, although the difference is small. Again, this can also be

explained due to suction stresses. With n = 1.5, the average suction will be smaller

throughout the entire soil profile and at the depth of embedment (which is D = 0). Thus,

the average apparent cohesion cʹʹ and the effective stress at the depth of embedment

will be smaller, decreasing bearing capacity.

To summarize the findings in this section, Figure 5-8 and Figure 5-9, show that the van

Genuchten fitting parameters, 𝛼 and n, have the greatest influence on the behavior of

bearing capacity. An accurate determination of these parameters, and thus the soil water

characteristic curve, is thus crucial in calculating reasonable bearing capacity values. It

has been shown that other input parameters influence the bearing capacity of shallow

foundations in partially saturated soils, but these changes are small (or constant) as in

𝜃𝑠 and 𝜃𝑟, or only varying the unmodified portion of the bearing capacity equation, like

𝜙′. Flux 𝑞 also has a considerable influence on the bearing capacity, especially for clays

and silts.

5.3.2. Evaluation of van Genuchten’s 𝛼 and 𝑛

As mentioned, van Genuchten’s fitting parameters have the most significant influence

on bearing capacity performance for foundations in partially saturated soils. With that

in mind, these variables should be assessed more closely. The results presented in the

following figures will look more closely at these two parameters, varying them along

the x-axis (instead of zw) while holding the groundwater table depth fixed. The bearing

capacity will be calculated at depths of the groundwater table equal to 0 m, 1 m, 2 m,

3 m, 5 m, and 10 m while either varying 𝛼 or n. In this example, 𝜙′ = 30°, 𝜃𝑠 = 0.4,

𝜃𝑠 = 0.06, and 𝐷 = 0 m.

Figure 5-10 through Figure 5-12 plot bearing capacity with respect to van Genuchten’s

𝛼 fitting parameter while n is fixed at 1.1, 1.5, and 3.0. As mentioned, this is done at

various depths of groundwater tables.

104

Figure 5-10. Varying van Genuchten’s α at various ground table depths for n = 1.1.


105


Figure 5-10 and Figure 5-11 exhibit different behavior than Figure 5-12. In these two

figures, as the depth of the groundwater table increases, the bearing capacity will al-

ways increase. This is the tendency of fine-grained soils (i.e., those with relatively low

n-values), where suction stress increases with matric suction. Figure 5-12 shows dif-

ferent behavior in that the groundwater table depth does not necessarily increase bear-

ing capacity. For a groundwater table at a depth of z = 2 m and higher 𝛼 values (typical

for silts and sands), the bearing capacity is greater than at any other groundwater table

depth. When 𝛼 decreases to values more typical of clays, the bearing capacity for z = 1

m is less than the other depths. This behavior is typical for coarse-grained soils, which

have limited suction stresses. In coarse-grained soils, the bearing capacity will initially

increase as the groundwater table depth increases due to suction stresses, but will then

decrease to the conventional Vesić solution as suction stress dissipates. Another notable

trend is that as n increases, the influence of 𝛼 on bearing capacity decreases for 𝛼 values

typical of silts and sands.

106

Figure 5-13. Varying van Genuchten’s n at various ground table depths for 𝛼 = 0.01 kPa-1.

Figure 5-14. Varying van Genuchten’s n at various ground table depths for 𝛼 = 0.1 kPa-1.

107

Figure 5-15. Varying van Genuchten’s n at various ground table depths for 𝛼 = 1 kPa-1.

Figure 5-13 to Figure 5-15 plot bearing capacity with respect to van Genuchten’s n

fitting parameter while 𝛼 is fixed at 0.01, 0.1, and 1 kPa-1. Figure 5-13 exhibits similar

behavior to Figure 5-10 and Figure 5-11, which shows that as the groundwater table

depth increases, bearing capacity also increases. This indicates that 𝛼 = 0.01 kPa−1

behaves predominately as a fine-grained soil. Figure 5-14 and Figure 5-15 behave the

same as Figure 5-12, where the bearing capacity does not necessarily increase with

groundwater table depth. These two figures show a transitory regime, labeled by sands,

where smaller groundwater table depths will have greater bearing capacities. Bearing

capacity decreases/drops significantly as n increases, especially for higher values of 𝛼.

The 5 m groundwater table in Figure 5-13 shows that the bearing capacity initially

decreases with increasing n, but then decreases. This indicates that the average apparent

cohesion across the failure surface does not necessarily decrease with an increase in n.

The results show that for low 𝛼, average suction stress may increase with n.

108

Figure 5-10 through Figure 5-15 have demonstrated the different behaviors of fine-

grained versus coarse-grained soils. In fine-grained soils (low 𝛼 and n), bearing capac-

ity is expected to increase with the groundwater table depth. This is due to the suction

stresses present in the soil, which will continue to increase with matric suction in fine-

grained soils. In coarse-grained soils (higher 𝛼 and n), bearing capacity does not nec-

essarily follow this pattern depending on the values of 𝛼 and n. As these values in-

crease, the influence of the suction decreases, and the calculated bearing capacity ap-

proaches the conventional solution, which is equal to the bearing capacity calculated in

dry conditions.

5.3.3. Other Considerations for Shallow Foundation Bearing Capacity

With these basic trends established, it is important to assess other factors concerning

shallow foundation bearing capacity. In this section the following will be addressed:

(1) a brief comparison between the modified approach and the conventional approach;

(2) discussion on phenomena where bearing capacity decreases and increases with foot-

ing width; and (3) the presentation of design charts for shallow foundations in partially

saturated soils.

Figure 5-16 show the bearing capacity profile for a theoretical sand (𝛼 = 1 kPa-1 and n

= 3) with an embedment depth D = 0 m. The bearing capacity profile is made for soils

with 𝜙′ = 25˚, 30˚, and 35˚. This figure actually shows that the conventional bearing

capacity equation performs well against the modified equation. There are portions

where the conventional approach will predict a bearing capacity greater than the mod-

ified approach. This is because the modified approach calculates a smaller average unit

weight, due to drainage in the pores, than the conventional method. Overall, there is

agreement between the two methods.

109

Figure 5-16. Comparison of the predicted bearing capacity for a sand using the modified and

conventional approach at various friction angles, D = 0 m.

Figure 5-17 plots the same bearing capacity profile, except that the foundation is em-

bedded at depth D = 1.5 m. As with the previous figure, the calculated bearing capacity

is the same between the conventional and modified equations for saturated (z = 0 m)

and dry conditions (the groundwater table is deep). For the modified approach, suction

will increase the effective stress at the depth of embedment 𝜎𝑧𝐷′ and also the average

apparent cohesion 𝑐′′. In the conventional approach, the soil weight is assumed to vary

linearly between D and D + B, which is why the conventional method does not predict

an increase in bearing capacity until z = D. The modified approach, however, indicates

that a shallow increase in bearing capacity should occur between z = 0 to D due to

suction. Apart from this exception, both solutions decently agree.

0

100

200

300

400

500

0 0.5 1 1.5 2 2.5 3

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]

Depth of GWT [m]

Modified Approach

Conventional Approach 𝜙′ = 35°

𝜙′ = 30°

𝜙′ = 25°

110

Figure 5-17. Comparison of the predicted bearing capacity for a sand using the modified and

conventional approach at various friction angles, D = 1.5 m.

Figure 5-18 and Figure 5-19 show how varying unsaturated properties affect the bear-

ing capacity profile. In these two example, the unsaturated soil parameters selected are

n = 3 and 𝛼 = 0.1 kPa-1. n was not decreased since as this soil would exhibit the fine-

grained behavior discussed in the previous section, Section 5.3.2, where ultimate bear-

ing capacity increases with groundwater table depth and does not return to the conven-

tional bearing capacity solution. Figure 5-18 and Figure 5-19 have embedment depths

of D = 0 and D = 1.5 m. In these two figures, a clear peak is achieved. This peak is

attributed to suction which is manifested in the effective vertical stress at the depth of

embedment 𝜎𝑧𝐷′ and apparent cohesion 𝑐′′. At higher depths, suction stresses will dis-

sipate due to smaller water contact area between particles.

0

500

1000

1500

2000

0 1 2 3 4

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]

Depth of GWT [m]

Modified Approach


𝜙′ = 35°

𝜙′ = 30°

𝜙′ = 25°

111

Figure 5-18. Comparison of the predicted bearing capacity between the modified and

conventional approach at various friction angles for a material with D = 0 m, n = 3, and α =

0.1 kPa-1.

Figure 5-19. Comparison of the predicted bearing capacity between the modified and

conventional approach at various friction angles for a material with D = 1.5 m, n = 3, and α =

0.1 kPa-1.

Figure 5-20 shows an interesting phenomena that was discussed in the Briaud and Gib-

bens (1997) analysis. Many load tests have shown that the bearing capacity does not

0

200

400

600

800

1000

0 1 2 3 4 5 6

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]

Depth of GWT [m]

Modified Approach


𝜙′ = 35°

𝜙′ = 30°

𝜙′ = 25°

0

500

1000

1500

2000

0 2 4 6 8 10

Cal

cula

ted

Be

arin

g C

apac

ity

[kP

a]

Depth of GWT [m]

Modified Approach


𝜙′ = 35°

𝜙′ = 30°

𝜙′ = 25°

112

always increase with footing with. The conventional bearing capacity equation does

not predict this phenomena since the footing width term in the equation, 0.5𝐵𝛾′, sug-

gests that increases in bearing capacity are linearly proportional to increases in footing

width. While this term is still used in the modified approach, variation in effective unit

weight 𝛾′, apparent cohesion 𝑐′′, and overburden 𝜎𝑧𝐷′ are dictated by the size and shape

of the failure surface. That being said, as the footing width increases, depth of the fail-

ure surface also increases. For this example, the groundwater table depth is 4 m. For

very small footing widths (B = 0.5 m), the failure surface will not extend below the

groundwater table and apparent cohesion 𝑐′′ and 𝛾′ will be greater. If the footing width

is large enough (B > 1.5 m), the failure surface will extend into the groundwater table,

decreasing average apparent cohesion and unit weight.

Figure 5-20. Comparison of the predicted bearing capacity for silt while varying the footing

width. This soil has zw = 4 m, 𝜙′ = 30˚ and D = 0.5 m.

As previously discussed, the unsaturated parameters 𝜃𝑟 and 𝜃𝑠 have little influence on

the behavior or performance of a shallow foundation (with the exception of clean sands)

but 𝛼 and n have been shown to contribute significantly to the calculated modified

bearing capacity. Figure 5-21 is a table of figures comparing the calculated bearing

0

250

500

750

1000

1250

1500

1750

0 1 2 3 4 5 6

Cal

cula

ted

Be

arin

g C

apac

ity

[kP

a]

Footing Width [m]

Modified Approach


113

capacity to five input parameters: friction angle 𝜙′, groundwater table depth 𝑧𝑤, footing

width 𝐵, van Genuchten’s 𝛼, and 𝑛. Cohesion c’, depth of embedment 𝐷, and specific

gravity 𝐺𝑠 have been shown to only incrementally affect bearing capacity for all values

of 𝜙′, 𝑧𝑤, 𝐵, 𝛼, and 𝑛. The consideration of c’, 𝐷, and 𝐺𝑠 would not produce interest-

ing results. In Figure 5-21 𝐺𝑠 = 2.65, 𝜃𝑠 = 0.4, 𝜃𝑟 = 0.06, and is considered to be a strip

footing.

In the development of Figure 5-21 it is important to note that a few variables can be

normalized: 𝑧𝑤 with 𝐵, 𝑞𝑢𝑙𝑡,𝑚𝑜𝑑𝑖𝑓𝑖𝑒𝑑 with 𝑞𝑢𝑙𝑡,𝑢𝑛𝑚𝑜𝑑𝑖𝑓𝑖𝑒𝑑, and 𝛼 with 𝑧𝑤. The depth of

the groundwater table and footing width can be normalized since the size and shape of

the failure surface will remain proportional. The modified bearing capacity and the un-

modified bearing capacity can be normalized by dividing the two since variation in

apparent cohesion, unit weight, and overburden will only increase (or decrease) accord-

ing the bearing, shape, and depth factors which are the same for the two equations. That

is, an increase in, say, apparent cohesion will always increase the bearing capacity by

a factor 𝑁𝑐𝜁𝑠𝑐𝜁𝑑𝑐 . Lastly, 𝛼𝜓 or 𝛼𝑧𝑤𝛾𝑤 will normalize the soil water characteristic

curve, as shown in Figure 5-22. In normalizing these variables, contours of

𝑞𝑢𝑙𝑡,𝑚𝑜𝑑𝑖𝑓𝑖𝑒𝑑/𝑞𝑢𝑙𝑡,𝑢𝑛𝑚𝑜𝑑𝑖𝑓𝑖𝑒𝑑 are plotted in Figure 5-21. These charts can be used in

design as a tool for predicting the bearing capacity of a shallow foundation in partially

saturated soils. If 𝐺𝑠, 𝜃𝑟, 𝜃𝑠, 𝑐′, or 𝐷 were to change, the bearing capacity would vary

by a fixed value, which can be calculated from the conventional bearing capacity equa-

tion.

114

𝜙′ = 40° 𝜙′ = 30° 𝜙′ = 20°

𝑧 𝑤/𝐵

=0

.5

𝑧 𝑤/𝐵

=1

𝑧 𝑤

/𝐵=

2

𝑧 𝑤/𝐵

=5

Figure 5-21. Table of figures for qmod/qunmod. The x and y axes of the table correspond to various ϕ' and zw/B ratios

respectively. For each individual figure, x and y axes are αzwγw and n, respectively.

van

Gen

uch

ten’s

𝑛

𝛼𝑧𝑤𝛾𝑤

115

Figure 5-22. Normailzation of the soil water characteristic curve.

Figure 5-21 summarizes the most prominent behavior shallow foundations in partially

saturated soils. Soils with low friction angle will have the tendency to be more influ-

enced by suction, which is indicated by contour lines with higher values. This does not

imply that calculated bearing capacities are higher. This figure also indicates that higher

values of 𝛼 and 𝑛 will calculate higher modified bearing capacities, especially as the

0.01 0.1 1 10 100 1 103

0

0.1

0.2

0.3

1/kPa

0.1/kPa

0.01/kPa

= z [kPa]

0.01 0.1 1 10 1000

0.1

0.2

0.3

1/kPa

0.1/kPa

0.01/kPa

= z

116

depth of the groundwater table increases. Regimes where the modified bearing capacity

is less than the unmodified bearing capacity exist where the contour is 1.0 or less. This

figure is fairly simple to use and implementation in practice would be easy when the

SWCC or soil type are known.

This section has covered other considerations for shallow foundation bearing capacity

in partially saturated soils. For sands, there is close agreement between the modified

approach and the conventional approach. Therefore, the use of the conventional bearing

capacity equation is suitable coarse-grained soils, but not fine-grained soils, where suc-

tion stresses are high. Footing width is an important consideration as this will control

the depth of the failure surface. Larger footings have larger failure surface, which may

be detrimental to the ultimate loading stress depending on the location of the failure

surface. Design charts for shallow foundations in partially saturated soils have been

introduced. For implementation of unsaturated soil in practice, ease and accessibility is

an important consideration, and design charts allow for quick estimates of the potential

increase (or decrease) in bearing capacity when compared to conventional method.

5.3.4. Vahedifard and Robinson (2015)

Vahedifard and Robinson (2015) built off of the work by Lu and Likos (2004), who

first developed a closed-form solution to the matric suction profile in soil with varying

rates of infiltration, and Vanapalli and Mohamed (2007), who have modified the Ter-

zaghi bearing capacity equation to account for partially saturated soils. Again, the

equation that was derived originally by Vanapalli and Mohamed (2007) is:


𝑢𝑤)𝐴𝑉𝑅𝑆𝜓 tan 𝜙′] × 𝑁𝑐𝜉𝑐 + 0.5𝐵𝛾𝑁𝛾𝜉𝛾 5-2

The modified equation proposed by Vahedifard and Robinson (2015) is:

117

𝑞𝑢𝑙𝑡 = {𝑐′ + 𝑎(1 + 𝑆𝑒,AVR) tan 𝜙′ + [(𝑢𝑎 − 𝑢𝑤)𝑆𝑒]AVR tan 𝜙′}𝑁𝑐ξc

+ 𝑞0𝑁𝑞𝜉𝑞 + 0.5�̅�𝐵𝑁𝛾𝜉𝛾 5-3

Essentially the above equation was modified to account for three things: (1) inclusion

of a surcharge term denoted by 𝑞0; (2) modification to the cohesion term which is done

to include the apparent cohesion due to matric suction; and (3) averaged unit weight

values. The surcharge 𝑞0 is calculated as the effective stress, not including suction

stress, at the embedment depth 𝐷. Variation in soil unit weight �̅� and matric suction

(𝑢𝑎 − 𝑢𝑤) are calculated as the average across the stress bulb, 𝐷 to 𝐷 + 1.5𝐵. 𝑆𝑒,AVR

is the average saturation value across the stress bulb. 𝑎 is the air entry suction. Vahe-

difard and Robinson (2015) have segregated the effects of suction stress into two terms:

𝑎(1 + 𝑆𝑒,AVR) tan 𝜙′ and [(𝑢𝑎 − 𝑢𝑤)𝑆𝑒]AVR tan 𝜙′ where (𝑢𝑎 − 𝑢𝑤)𝑆𝑒 is suction

stress 𝜎𝑠. These two expressions account for suction up to the air-entry value and suc-

tions greater than the air-entry value respectively.

This approach is similar to the method used in this work with the following exceptions:

(1) no dry unit weight or saturated unit weight is assumed, this is controlled by the

variables 𝜃𝑟, 𝜃𝑠, and 𝐺𝑠; (2) the apparent cohesion due to suction stress is calculated

across the failure surface, as opposed to the average from D to D + B, and is not sepa-

rated into two different expressions; and (3) the surcharge 𝑞0 is inclusive to any suction

stress applied by unsaturated soils at the depth of embedment. Again, the apparent co-

hesion term used in this work is:

𝑐′′ = 𝜒𝜓 tan 𝜙′ 5-4

Which is integrated across the failure surface:

𝑐′′̅̅ ̅ =

∫ 𝑙 ∙ 𝑐′′𝑑𝑙

∫ 𝑙 𝑑𝑙 5-5

118

Equation 5-5 is directly implement into the conventional bearing capacity equation.

Since the approach provided by Vahedifard and Robinson (2015) is similar to the ap-

proach developed in this work, it is insightful to make comparisons between the two.

The following figures compare the parametric studies conducted by Vahedifard and

Robinson (2015) and a comparison made by the modified approach used in this thesis.

The comparison in Figure 5-23 and Figure 5-24 compares strip footings with a width

of 1 m over clay soil. The soil has the following properties:

Table 5-2. Input parameters for clay used in Vahedifard and Robinson (2015).

Properties

Value SWCC

Properties Values

𝝓′ 20 − 25° 𝛼 0.005 kPa−1 𝒄′ 10 kPa 𝑛 1.8 𝑮𝒔 2.65 𝜃𝑠 0.369

𝜃𝑟 0.0 𝑘𝑠 5 ∙ 10−8 m/s

Vahedifard and Robinson (2015) use an 18 kN/m3 soil for all groundwater table depths.

𝜃𝑠 was selected to fix the saturated unit weight to 20 kN/m3. The dry unit weight cor-

responding to this porosity is 16.4 kN/m3. This range brackets the value used by Vahe-

difard and Robinson (2015).

119

Figure 5-23. Calculated bearing capacity for hypothetical clay with D = 0 from Vahedifard

and Robinson (2015) compared to modified approach in this current work (left 𝜙′ = 25°,

right 𝜙′ = 20°).

Figure 5-24. Calculated bearing capacity for hypothetical clay with D = 1.5 m from

Vahedifard and Robinson 2015 compared to modified approach in this current work (left

𝜙′ = 25°, right 𝜙′ = 20°). Note changing ordinate across figures.

0

1000

2000

3000

4000

0 3 6 9 12

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]

Depth of GWT [m]

q = 1.15E-8 m/s

q = 0 m/s

q = -3.14E-8 m/s

V & H (2015)

0

1000

2000

3000

4000

0 3 6 9 12

Cal

cula

ted

Be

arin

g C

apac

ity

[kP

a]

Depth of GWT [m]

q = 1.15E-8 m/s

q = 0 m/s

q = -3.14E-8 m/s

V & R (2015)

0

1000

2000

3000

4000

5000

0 3 6 9 12

Cal

cula

ted

Be

arin

g C

apac

ity

[kP

a]

Depth of GWT [m]

q = 1.15E-8 m/sq = 0 m/sq = -3.14E-8 m/sV & R (2015)

0

1000

2000

3000

0 3 6 9 12

Cal

cula

ted

Be

arin

g C

apac

ity

[kP

a]

Depth of GWT [m]

q = 1.15E-8 m/sq = 0 m/sq = -3.14E-8 m/sV & R (2015)

120

Figure 5-23 and Figure 5-24 show very close agreement between Vahedifard and Rob-

inson (2015) and the modified approach developed in this current work. The two fig-

ures show agreement both in magnitude of bearing capacities and in consideration to

variation in infiltration values. Having a negative flux value (precipitation) will result

in a decrease in bearing capacity. This is due to a decrease in suction stresses. Alterna-

tively, a positive flux value (evaporation) will result in an increase in bearing capacity

due to higher suction stresses. There is some difference between the calculated bearing

capacities for the embedded foundation. The approach proposed in this work predicts

smaller bearing capacities for embedded foundations.

The soil properties used for theoretical sand in Figure 5-25 and Figure 5-26 below are

described in Table 5-3. Vahedifard and Robinson (2015) use 1.1𝜙′ for when 𝐷 = 0 m

to account for dilation in sands. Note, since flow does not appreciably affect the bearing

capacity in sands, only one solution is plotted for Vahedifard and Robinson (2015) at

𝑞 = 0 m/s.

Table 5-3. Input parameters for sand used in Vahedifard and Robinson (2015).

Properties

Value SWCC

Properties Values

𝝓′ 30 − 35° 𝛼 0.1 kPa−1 𝒄′ 0 kPa 𝑛 4.0 𝑮𝒔 2.65 𝜃𝑠 0.369

𝜃𝑟 0.0 𝑘𝑠 3 ∙ 10−5 m/s

121

Figure 5-25. Calculated bearing capacity for hypothetical sand with D = 0 m from Vahedifard

and Robinson 2015 compared to modified approach in this current work.

Figure 5-26. Calculated bearing capacity for hypothetical sand with D = 1.5 m from

Vahedifard and Robinson (2015) compared to modified approach in this current work.

Figure 5-25 and Figure 5-26 show that the approach proposed in this work and by

Vahedifard and Robinson (2015) do not have good agreement in calculated bearing

capacities in sands. These two figures, however, show agreement in that sandy soils are

0

1000

2000

3000

0 3 6 9 12

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]

Depth of GWT [m]

q = 1.15E-8 m/s

q = 0 m/s

q = -3.14E-8 m/s

V & R (2015)

𝜙′ = 35° × 1.1

𝜙′ = 35° × 1.1

𝜙′ = 30° × 1.1

𝜙′ = 30° × 1.1

0

2000

4000

6000

0 3 6 9 12

Cal

cula

ted

Be

arin

g C

apac

ity

[kP

a]

Depth of GWT [m]

q = 1.15E-8 m/s

q = 0 m/s

q = -3.14E-8 m/s

V & H (2015)

𝜙′ = 35°

𝜙′ = 30°

𝜙′ = 35°

𝜙′ = 30°

122

largely unaffected by variation in flux. There is no noticeable difference between infil-

tration, evaporation, and no flow. To assess the difference more closely, solutions to

bearing capacity for the current proposed method, the Vesić approach, and Vahedifard

and Robinson (2015) are compared in Figure 5-27 and Figure 5-28. For the Vesić so-

lution, the assumed moist unit weight is 𝛾 = 18 kN/m3, which does not fluctuate as

the groundwater table moves. In Figure 5-27, solutions for the surface foundation with

1.1𝜙′ are presented. In Figure 5-28, solutions for the embedded foundation are pre-

sented. Only solutions for 𝑞 = 0 m/s are presented in these figures.

Figure 5-27. Comparison of calculated bearing capacity profiles using the proposed approach,

the Vesić solution, and Vahedifard and Robinson (2015) for a surface foundation (left: 𝜙′ =35° × 1.1, right: 𝜙′ = 30° × 1.1).

0

1000

2000

3000

0 3 6 9 12

Cal

cula

ted

Be

arin

g C

apac

ity

[kP

a]

Depth of GWT [m]

This Work

Vesic Approach

V & R (2015)

0

300

600

900

1200

0 3 6 9 12

Cal

cula

ted

Be

arin

g C

apac

ity

[kP

a]

Depth of GWT [m]

This Work

Vesic Approach

V & R (2015)

123

Figure 5-28. Comparison of calculated bearing capacity profiles using the proposed approach,

the Vesić solution, and Vahedifard and Robinson (2015) for an embedded foundation (left:

𝜙′ = 35°, right: 𝜙′ = 30°).

Figure 5-27 and Figure 5-28 show that the approach proposed in this work generally

has good agreement with the Vesić solution. Vahedifard and Robinson (2015) calcu-

lated around twice the amount predicted by the Vesić equation for each scenario. Since

sands are being modeled, the bearing capacity solutions should be similar when zw = 0

m and when zw is deep. This is the case for the approach proposed in this work. There

are some discrepancies between the current proposed solution and the Vesić solution

for the embedded foundation at deep groundwater tables. This is attributed to a differ-

ence in definition for soil unit weight. The unit weight in the proposed approach is

allowed to vary between 20 kN/m3 and 16.4 kN/m3, while 18 kN/m3 is fixed for the

Vesić solution. The air-entry suction for sands has the tendency to be relatively low,

therefore, the predicted unit will quickly tend towards 16.4 kN/m3 as the groundwater

table increases in depth. Differences between the approach proposed by Vahedifard and

Robinson (2015) and in this work can be attributed to either different bearing capacities

(although the differences should not be this large) or human error.

0

1000

2000

3000

4000

5000

0 3 6 9 12

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]

Depth of GWT [m]

This Work

Vesic Approach

V & H (2015)

0

500

1000

1500

2000

2500

0 3 6 9 12

Cal

cula

ted

Bea

rin

g C

apac

ity

[kP

a]

Depth of GWT [m]

This WorkVesic ApproachV & R (2015)

124

5.4. Parametric Study on the Modified 𝛽-method

5.4.1. Development of Side Resistance Profiles

In this section sensitivity of the modified beta method will be assessed with various

soil properties. This study will use the same soils described at the beginning of this

chapter in Section 5.2, theoretical sand, silt, and clay. For each of these soil types, the

frictional resistance will be calculated as a function of groundwater table depth. A plot

comparing the groundwater table depth to total unit side resistance 𝑞𝑠 will subsequently

referred to as the side resistance profile. The total unit side resistance is defined by Eq.

5-6.

𝑞𝑠 = ∫ 𝑓𝑠

𝐷

0

𝑑𝑧 → 𝐹𝑜𝑟𝑐𝑒/𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 5-6

where 𝑓𝑠 is the side resistance of the pile or drilled shaft, and 𝐷 is the pile depth. 𝑓𝑠 is

measured in units of pressure (kPa) and describes the frictional resistance at a single

point along the deep foundation. Integrating by depth yields 𝑞𝑠 which has units of kN/m

or MN/m. To calculate the total side resistance 𝑄𝑠, the total unit side resistance can be

multiplied by the circumference of the pile or drilled shaft for units of kN or MN. The

shape of the pile or drilled shaft is not important in this discussion, therefore, there is

no selection of pile diameter. Eq. 5-7 describes the side resistance above the ground-

water table, which was developed in Section 3.3.1.

𝑓𝑠 = (𝐾0 tan 𝛿 +

𝜎𝑠

𝜎𝑣tan 𝛿 +

𝑐′

𝜎𝑣) 𝜎𝑣 = 𝛽′𝜎𝑣 5-7

where all variables are as previously described. In this formulation of the 𝛽′-method,

matric suction is considered in two ways: (1) variation of the at-rest earth pressure co-

efficient; and (2) variation in suction stress. Equations for suction stress 𝜎𝑠, at-rest earth

pressure coefficient 𝐾0, and vertical effective stress 𝜎𝑣 are discussed in Section 3.3.

Figure 5-29 plots the suction stress profile for the theoretical sand, silt, and clay.

125

Figure 5-29. Suction stress profile above the groundwater table for theoretical sand, silt, and

clay (θs = 0.4, θr = 0.06, q = 0 m/s for each figure). Note changing abscissa across figures.

These suction stress profiles will serve as the basis for comparing the difference in

performance of sand, silt, and clay in partially saturated soils. As a general note, suction

stresses for sands are significantly smaller than silts and clays, and also peak shortly

above the groundwater table. Clays have the largest suction stresses. Using the suction

stress profile, 𝐾0 (Section 3.3.1) can be calculated as a function of depth from the sur-

face. This requires an assumption for the groundwater table depth. Finally, with these

considerations side resistance 𝑓𝑠 can be calculated according to the modified 𝛽-method

in Eq. 5-7. Figure 5-30 plot the vertical effective stress profile as a function of depth

from the soil surface for groundwater table depths of 0 m, 5 m, 10 m, 20 m, and 30 m.

The vertical effective stress is calculated from van Genuchten (1980) SWCC and is

discussed in Section 3.3.3. In this figure the specific gravity is 2.65.

0

1

2

3

4

5

0 0.25 0.5 0.75

Dis

tan

ce A

bo

ve G

WT

[m]

σs [kPa]

San

d, n

= 3

.0α

= 1

1/k

Pa

0

5

10

15

20

25

30

0 25 50 75

σs [kPa]Si

lt,n

= 1

.5α

= 0

.1 1

/kP

a

0

5

10

15

20

25

30

0 100 200 300

σs [kPa]

Cla

y, n

= 1

.1α

= 0

.01

1/k

Pa

126

Figure 5-30. Vertical effective stress as a function of depth from soil surface for theoretical

sand, silt, and clay (θs = 0.4, θr = 0.06, q = 0 m/s, Gs = 2.65 for each figure).

The only significant trend in this figure is that clay and silts with the same porosity and

specific gravity will generally have greater vertical stress than a similar sand. This in-

dicates that silts and clays retain a greater volume of water in pores than do sands,

increasing the soil unit weight. Once the soil profile and suction stresses are known,

the modified beta, 𝛽′, can be calculated. Figure 5-31 plot the modified beta as a function

of depth from the soil surface for groundwater table depths of 0 m, 5 m, 10 m, 20 m,

and 30 m. In this figure Poisson’s ratio 𝜈 and the interface friction angle 𝛿 are 0.3 and

30° respectively.

0

5

10

15

20

25

30

0 250 500 750

De

pth

fro

m S

urf

ace

[m]

0

5

10

15

20

25

30

0 250 500 750

σ' [kPa]

0

5

10

15

20

25

30

0 250 500 750

0

00.20.40.6GWT = 0 m GWT = 5 m GWT = 10 m GWT = 20 m GWT = 30 m

San

d, n

= 3

.0

α =

1 1

/kP

a

Silt

, n =

1.5

α

= 0

.1 1

/kP

a

Cla

y, n

= 1

.1

α =

0.0

1 1

/kP

a

127

Figure 5-31. Modified β’ as a function of depth from soil surface for theoretical sand, silt, and

clay (θs = 0.4, θr = 0.06, q = 0 m/s, Gs = 2.65, ν = 0.3, δ = 30˚ for each figure).

Figure 5-31 shows the most important trends in the understanding of deep foundations

in partially saturated soils. First, sands predict little influence from partially saturated

soils; there is little to no deviation from the conventional 𝛽, which is calculated as

𝜈 tan 𝛿 /(1 − 𝜈) = 0.24. This is also the same value that would be predicted for silts

and clays if the conventional approach was used and δ = 30˚ and ν = 0.3. Silts and

clays, however, have a different trend, especially near the surface. When 𝛽′ is equal to

zero, tension cracking is predicted near and no shear strength between the soil and pile

is achieved. Clays will typically have deeper tension cracks than silts. At depths below

tension cracking, silts and clays are highly influenced by suction stress, which is indi-

cated by the 𝛽′ values that are greater than 0.24. Once this 𝛽′ profile is known, side

resistance can be calculated. Figure 5-32 plot the side resistance as a function of depth

from the soil surface for groundwater table depths of 0 m, 5 m, 10 m, 20 m, and 30 m.

In this figure Poisson’s ratio 𝜈 and the interface friction angle 𝛿 are 0.3 and 30° respec-

tively.

0

5

10

15

20

25

30

0 0.2 0.4 0.6

De

pth

fro

m S

urf

ace

[m]

0

5

10

15

20

25

30

0 0.2 0.4 0.6

β'

0

5

10

15

20

25

30

0 0.2 0.4 0.6

0


San

d, n

= 3

.0

α =

1 1

/kP

a

Silt

, n =

1.5

α

= 0

.1 1

/kP

a

Cla

y, n

= 1

.1

α =

0.0

1 1

/kP

a

128

Figure 5-32. Side resistance as a function of depth from soil surface for theoretical sand, silt,

and clay (θs = 0.4, θr = 0.06, q = 0 m/s for each figure).

Figure 5-32 show the general trends in the behavior of theoretical sand, silt, and clay.

Silts and clays achieve cracking (when fs = 0) as the groundwater table increases in

depth. Tension cracking is not observed in sands since suction stress are low. Although

suction stresses cause cracking in silts and clays, it also results in large increases for

side resistance. For all three soil types, side resistance increases with depth due to an

increase in the vertical stress with depth. This vertical effective stress causes 𝑓𝑠 to in-

crease seemingly linearly with depth since the large majority of strength comes from

the vertical stress term (𝛽𝜎𝑣), however, suction stresses are present (especially in silts

and clays) through the expression 𝜎𝑠 tan 𝛿 that causes non-linearity in these plots. In-

tegrating any line in Figure 5-32 from 0 to the pile length L calculates a single value of

𝑞𝑠 for a specific groundwater table depth. If this same integral is conducted for many

groundwater table depths, 𝑞𝑠 can be plotted as a function of 𝑧𝑤, which is subsequently

called the side resistance profile.

0

5

10

15

20

25

30

0 100 200

De

pth

fro

m S

urf

ace

[m]

0

5

10

15

20

25

30

0 100 200

fs = βσv + σstanδ [kPa]

0

5

10

15

20

25

30

0 100 200

0


San

d, n

= 3

.0

α =

1 1

/kP

a

Silt

, n =

1.5

α

= 0

.1 1

/kP

a

Cla

y, n

= 1

.1

α =

0.0

1 1

/kP

a

129

For this parametric study, the side resistance profile will be considered for deep foun-

dation with lengths of 10 m, 15 m, 20 m, 25 m, and 30 m and groundwater table depths

ranging from 0 m to 30 m. The influence of Poisson’s ratio ν on the modified 𝛽-method

in partially saturated soils will be assessed by varying ν at 0.2, 0.3, and 0.4. It is im-

portant to note that interface friction angle 𝛿 will not be varied since this expression

has not been modified. 𝛿 will be taken as 30° for the remainder of this section. Values

for 𝑞𝑠 can be adjusted to a different interface friction angle by multiplying 𝑞𝑠 by

tan 𝛿 / tan 30°. As mentioned previously, the three soil types being considered are the

theoretical sand, silt, and clay. Table 5-4 summarizes the three soil types and

Table 5-4. Soil properties used in this parametric study

Property 𝜶 (𝐤𝐏𝐚−𝟏) 𝒏 𝒌𝒔 (𝐦/𝐬) 𝜽𝒔 𝜽𝒓

Clay 0.01 1.1 10-7 0.4 0.06

Silt 0.10 1.5 10-6 0.4 0.06

Sand 1.00 3.0 10-5 0.4 0.06

5.4.2. Evaluation of Side Resistance Profiles

Figure 5-33 presents the side resistance profile for theoretical sand, silt, and clay. This

profile was calculated according to Eq. 5-7 and is the integral of the three plots in

Figure 5-32 from 0 m to the length of the pile L. In this figure 𝜈 = 0.3, 𝐺𝑠 = 2.65, 𝜃𝑟 =

0.06, and 𝜃𝑠 = 0.4. As mentioned, the groundwater table depth varies from 0 m to 30

m. Pile lengths are assessed at 10 m, 15 m, 20 m, 25 m, and 30 m.

130

Figure 5-33. Side resistance profiles for theoretical sand, silt and clay (𝜈 = 0.3, 𝐺𝑠 = 2.65, 𝜃𝑟

= 0.06, and 𝜃𝑠 = 0.4) (unit side resistance given in force/unit perimeter)

Figure 5-33 shows several interesting trends between the three different soil types. The

first thing to note is that for each pile length, unit side resistance will generally either

increase to an asymptotic limit or achieve a peak resistance and begin to decrease (as

shown in the clay for L = 10 m). Generally, as the depth of the groundwater table in-

creases side resistance increases as well. This is mostly due to the increase in vertical

effective stress as porewater pressures dissipate. Clays and silts have higher peak unit

side resistance than do sands, which is attributed to the high suction stresses in these

soils. The last notable trend is that when the groundwater table extends below 10 m for

the 10 m pile embedded in clay, effects of tension cracking begin to manifest in a de-

creased side resistance. Eventually, side resistance is zero when the groundwater table

is 25 m. This implies that the tension cracks extend to a depth of 10 m.

San

d, n

= 3

.0

α =

1 1

/kP

a

Silt

, n =

1.5

α

= 0

.1 1

/kP

a

Cla

y, n

= 1

.1

α =

0.0

1 1

/kP

a

131

Figure 5-34 through Figure 5-36 study the effects volume/mass properties on theoreti-

cal silt. Silt was selected for these comparisons as silts typically have an intermediate

soil water characteristic curve relative to sands and clays. This will serve to generalize

the effects of varying specific gravity, porosity, and residual water content. In these

figures, all other variables are held constant, so that effects from 𝜃𝑟, 𝜃𝑠, or 𝐺𝑠 can be

individually characterized. Figure 5-34 sets the specific gravity at 2.60, 2.65, and 2.70

from left to right.

Figure 5-34. Side resistance profiles for theoretical silt with varying specific gravity (𝑛 = 1.5,

𝛼 = 0.1 kPa-1, 𝜈 = 0.3, 𝜃𝑟 = 0.06, and 𝜃𝑠 = 0.4) (unit side resistance given in force/unit perim-

eter)

Other than increasing the unit weight of the soil (and thus 𝜎𝑣), varying specific gravity

does not significantly change the behavior of the side resistance curves. It can be con-

cluded that the specific gravity has little influence on the mean modified 𝛽 profile.

Figure 5-35 sets 𝜃𝑟 equal to 0.02, 0.04, and 0.06 from left to right. Figure 5-36 sets 𝜃𝑠

equal to 0.35, 0.40, and 0.45 from left to right.

132

Figure 5-35. Side resistance profiles for theoretical silt with varying residual water content (𝑛

= 1.5, 𝛼 = 0.1 kPa-1, 𝜈 = 0.3, 𝐺𝑠 = 2.65, and 𝜃𝑠 = 0.4) (unit side resistance given in force/unit

perimeter)

San

d, n

= 3

.0

α =

1 1

/kP

a

Silt

, n =

1.5

α

= 0

.1 1

/kP

a

Cla

y, n

= 1

.1

α =

0.0

1 1

/kP

a

133

Figure 5-36. Side resistance profiles for theoretical silt with varying saturated water content

(𝑛 = 1.5, 𝛼 = 0.1 kPa-1, 𝜈 = 0.3, 𝐺𝑠 = 2.65, and 𝜃𝑟 = 0.06) (unit side resistance given in

force/unit perimeter)

These two figures shows that varying the saturated and residual water contents over a

reasonable range of values does not significantly change the performance or behavior

of a deep foundation. Increasing the residual water content will cause more water to be

retained in soil pores at high suctions, which will slightly increase the side resistance.

Decreasing the porosity increases the unit weight of the soil, which will also increase

side resistance. It is concluded that properties pertaining to soil volume and mass (𝜃𝑟,

𝜃𝑠, and 𝐺𝑠) do appreciably affect the predicted 𝑞𝑠.

Another consideration that can be made is the influence of varying water flow rates to

account for infiltration and evaporation. Fluctuation in flow will be studied for both

clay and silts (sands will not be included since suction stresses are low and negligible)

which have an assumed permeability of 𝑘𝑠 = 10−7 m/s and 𝑘𝑠 = 10−6 m/s respec-

tively. Figure 5-37 plot different 𝜎𝑠 for clays using a flow rate of 𝑞 = 0.2𝑘𝑠 (evapora-

tion), 𝑞 = 0 m/s, and 𝑞 = −0.2𝑘𝑠 (infiltration).

San

d, n

= 3

.0

α =

1 1

/kP

a

Silt

, n =

1.5

α

= 0

.1 1

/kP

a

Cla

y, n

= 1

.1

α =

0.0

1 1

/kP

a

134

Figure 5-37. Suction stress profiles of theoretical clay for flowrates of q = -0.2ks, 0, and 0.2ks.

The suction stress profile shows that positive flux (evaporation) will increase the suc-

tion stress. In this example, suction is thermodynamically limited to 106 kPa. Negative

flux (infiltration) will decrease the suction stress.

Figure 5-38 calculates the side resistance profile for theoretical clay exposed to

flowrates of q = -0.2ks (infiltration), 0, and 0.2ks (evaporation). During evaporation,

side resistance will decrease when compared to q = 0. During infiltration, the matric

suction throughout the profile is decreased, but this will also decrease the depth of ten-

sion cracking which increases the length along the pile where frictional resistance can

develop. This leads to an increased side resistance for clays exposed to infiltration.

0

5

10

15

20

25

30

0 100 200 300 400

Dis

tan

ce A

bo

ve G

WT

[m]

0

5

10

15

20

25

30

0 100 200 300 400

σs [kPa]

0

5

10

15

20

25

30

0 100 200 300 400

Cla

y q

= -

0.2

k s

Cla

y q

= 0

Cla

y q

= 0

.2k s

135

Figure 5-38. Side resistance profiles of theoretical clay for flowrates of q = -0.2ks, 0, and 0.2ks



The same example is done for silt which has 𝑘𝑠 = 10−6 m/s. Figure 5-39 plots 𝜎𝑠 for

the theoretical silt exposed to flowrates of q = -0.2ks (infiltration), 0, and 0.2ks (evapo-

ration). Note that this figure is plotted as distance above the groundwater table so that

it is applicable to all groundwater table locations.

136

Figure 5-39. Suction stress profiles of theoretical silt for flowrates of q = -0.2ks, 0, and 0.2ks.

Figure 5-39 shows that the selected van Genuchten fitting parameters (𝛼 and 𝑛) result

in a greater sensitivity to flux 𝑞 than in clay. Therefore, 𝜎𝑠 approaches an asymptotic

value for infiltration (𝑞 < 0) and increases to the thermodynamic limit 106 (kPa) at a

lower height above the groundwater table than clay. This also implies that the location

of tension cracking will be more significantly altered than for clays. With evaporation,

tension cracking is more abrupt and occurs at a greater depth for this silt depending on

the depth of the groundwater table. Alternatively with infiltration, tension cracking will

occur at essentially the same location for any depth of the groundwater table and is

shallower than in clays. It is important to note that this is a theoretical calculation for

the suction stress and has not been validated in real soil. For q = 0.2ks, cracking will

occurs at a distance of less than 5 m according to Figure 5-39. This, of course is not

reasonable at all groundwater table depths.

Side resistance profiles of theoretical silt for various flowrates of q = -0.2ks (infiltra-

tion), 0, and 0.2ks (evaporation) are shown in Figure 5-40. Like the clay, when evapo-

ration occurs the suction stresses are large, increases the depth of tension cracking

which results in side resistance profiles that decrease to zero. Despite the reduction in

0

5

10

15

20

25

30

0 25 50 75

Dis

tan

ce A

bo

ve G

WT

[m]

0

5

10

15

20

25

30

0 25 50 75

σs [kPa]

0

5

10

15

20

25

30

0 25 50 75

Silt

q

= -

0.2

k s

Silt

q

= 0

Silt

q

= 0

.2k s

137

suction stress, infiltration produces greater side resistance than when q = 0 m/s. This is

attributed to the reduction in tension cracking.

Figure 5-40. Side resistance profiles of theoretical silt for flowrates of q = -0.2ks, 0, and 0.2ks



As mentioned in the shallow foundation parametric study, 𝛼 and 𝑛 are the most im-

portant variables in determine the influence of suction stress on foundation perfor-

mance. Figure 5-41 through Figure 5-43 plot the side resistance profiles for various

values of 𝛼 (0.01 kPa-1, 0.1 kPa-1, and 1 kPa-1) and n (1.1, 1.5, and 3.0). Figure 5-41,

Figure 5-42, and Figure 5-43 have a fixed Poisson’s ratio of 0.2, 0.3, and 0.4 respec-

tively. These figures were intended to show the extent in variation between values of

total unit side resistance calculated in this work while varying unsaturated soil proper-

ties. Since 𝜃𝑟, 𝜃𝑠, and 𝐺𝑠 do not vary the results significantly, these charts can be used

in a design scenario for the selection of unit side resistance for normally consolidated

soils, especially when the soil water characteristic curve is known. In all three figures,

𝜃𝑟 = 0.06, 𝜃𝑠 = 0.40, and 𝐺𝑠 = 2.65.

138

Figure 5-41. Side resistance profiles for 𝜈 = 0.2 at various 𝑛 and 𝛼 values.

139


140


141

Figure 5-41 through Figure 5-43 summarizes the results of the parametric study for the

side resistance of deep foundations in normally-consolidated soils. First, as the Pois-

son’s ratio increases, the influence of tension cracking decreases. Soils with large ν are

will generally have a greater horizontal strain, and thus be prone to less cracking. Con-

sequently, a smaller Poisson’s ratio will cause smaller horizontal strain and thus hori-

zontal stress in soils, which then reduces the required suction to produce tension crack-

ing. The subplot in Figure 5-41 exhibits interesting behavior for 𝛼 = 0.01 kPa-1 and n

= 3.0. In this figure, total unit side resistance initially decreases as the groundwater

table depth increases, but then begins to increase again. This can be explained through

the K0 profile, which is calculated for five different groundwater depths (0 m, 5 m, 10

m, 20 m, and 30 m) in Figure 5-44.

Figure 5-44. K0 as a function of depth from soil surface for α = 0.01 kPa-1, and n = 3.0 for

fixed groundwater table depths (θs = 0.4, θr = 0.06, q = 0 m/s, Gs = 2.65, ν = 0.2, δ = 30˚)

Figure 5-44 indicates that for a soil with 𝛼 = 0.01 kPa-1 and 𝑛 = 3.0, as the groundwater

table depth increases, the depth of cracking (which occurs when K0 = 0) initially in-

creases, but then decreases at zw = 30 m. What this implies is that when the groundwater

0

5

10

15

20

25

30

0 0.2 0.4 0.6

D [

m]

K0

0


142

table is deep (30 m), the suction stress is sufficiently small and the vertical stress is

sufficiently large enough to limit cracking.

Figure 5-41 through Figure 5-43 generally show that the depth of tension cracking will

increase for smaller Poisson’s ratios. This also implies that the length along which side

resistance can develop is shorter. Across all Poisson’s ratios, the matric suction and

suction stress profile remains the same. Therefore, soils with a higher Poisson’s ratio

will be influenced less by tension cracking. That is why the side resistance profiles for

ν = 0.4 do not decrease to zero. On the other hand, for ν = 0.2, the side resistance often

decreases with groundwater table depth due to tension cracks.

Another trend from Figure 5-41 through Figure 5-43 is the distinction between fine and

coarse grained material. Generally, for large 𝛼 and 𝑛 values (i.e. sands) the soil will

behave similar to the soil will be largely uninfluenced by matric suction. The depth of

tension cracking is also smaller (or non-existent) for larger values of 𝛼 and 𝑛. Tension

cracks are generally larger for smaller 𝛼 and 𝑛 (i.e. clays) values. In general, larger

Poisson’s ratios produces a greater transfer in vertical to horizontal stresses, resulting

stronger soils that are less influenced by tension cracking. Soils with larger Poisson’s

ratios will also have larger side resistance due to suction stresses. Since less cracking

occurs, there is more pile length to accumulate resistance due to suction and vertical

stresses.

In this last example, the modified 𝛽-method will be compared to the conventional ap-

proach. In the conventional approach, variation in K0 due to suction stress will be ne-

glected. That is K0 will be fixed to 𝜈/(1 − 𝜈), which for 𝜈 = 0.2, 0.3, and 0.4 is 0.25,

0.429, and 0.667. The conventional 𝛽-method also ignored additional strength from

suction stress, which is manifested in the expression 𝜎𝑠 tan 𝛿 = 𝜒𝜓 tan 𝛿. For this ex-

ample, the same 𝛾 profile will be used for both the conventional and modified ap-

proaches. The 𝛾 is based off of the van Genuchten (1980) SWCC, but it is assumed that

from exploratory drilling, similar unit weight values could be discovered. The unit

143

weight profile is plotted as a function of distance above the groundwater table, where

0 is at the groundwater table. This is presented in Figure 5-45.

Figure 5-45. Unit weight profile for theoretical clay (θs = 0.4, θr = 0.06, q = 0 m/s, Gs = 2.65,

α = 0.01 kPa-1, n = 1.1)

In Figure 5-45, the unit weight profile is plotted for theoretical clay. Clay was selected

because of its sensitivity to matric suction, resulting in larger suction stresses and po-

tentially more tension cracking. At the groundwater table, the pores are fully saturated

and the unit weight is equal to 19.5 kN/m3, which is calculated from 𝛾𝑤(𝐺𝑠(1 − 𝜃𝑠) +

𝜃𝑠). As the distance above the groundwater table increases, the unit weight will de-

crease to an asymptotic value of 16.2 kN/m3, which is calculated from

𝛾𝑤(𝐺𝑠(1 − 𝜃𝑠) + 𝜃𝑠). Below the groundwater table, the soil unit weight is calculated

as the buoyant unit weight, which in this case is 9.7 kN/m3. Using this soil profile, the

side resistance profile can be calculated for theoretical sands for both the modified and

conventional approach. This profile is calculated for 𝜈 = 0.2, 0.3, and 0.4, which is

presented in Figure 5-46.

0

5

10

15

20

25

30

15 16 17 18 19 20

Dis

tan

ce A

bo

ve G

WT

[m]

Unit Weight γ, [kN/m3]

144

Figure 5-46. Side resistance profiles of theoretical clay for 𝜈 = 0.2, 0.3, and 0.4 (𝑛 = 1.1, 𝛼 =

0.01 kPa-1, 𝐺𝑠 = 2.65, 𝜃𝑠 = 0.4 and 𝜃𝑟 = 0.06). Note changing abscissa across figures.

In Figure 5-46, the conventional 𝛽-method is indicated by thin, gray lines. For 𝜈 = 0.2,

the modified approach predicted less side resistance than in the conventional approach.

As the groundwater table depth increased, this difference becomes more obvious. Ten-

sion cracking will control the behavior of the deep foundation for lower values of 𝜈.

For 𝜈 = 0.3, the conventional and modified approach showed good agreement, espe-

cially for smaller groundwater table depths. At lower groundwater table depths, the

conventional approach predicted greater side resistance for shorter foundations, and

smaller side resistance in longer foundations. Finally, for 𝜈 = 0.4, the modified ap-

proach calculated more side friction than the conventional approach. For this Poisson’s

ratio, tension cracking is less of an issue, and is able to accumulate more strength due

to suction stress along the length of the deep foundation.

In general, the modified approach is able represent phenomena that is expected in soil.

Soils with a lower Poisson’s ratio will exhibit more tension cracking due to its inability

to transfer vertical strains to horizontal strains, thus limiting horizontal stresses. Soils

145

with a higher Poisson’s ratio can transfer these stresses more readily, reducing the mag-

nitude of tension cracking. These soils are inevitably stronger, especially when suction

stresses are considered. As mentioned, fine-grained soils are more sensitive to matric

suction, resulting in both higher suction stresses and deeper tension cracking. Coarse-

grained soils are largely unaffected by matric suction and the side resistance profile

calculated from the modified approach is essentially the same as the conventional ap-

proach. Finally, evaporation will cause suction stresses to increase within the soil pro-

file, which results in deeper tension cracking. Soils with 𝜈 = 0.3, will ultimately see a

reduction in strength as opposed to infiltration and no flow.

5.5. Monte Carlo Simulations for Partially Saturated Soils

In this section, basic Monte Carlo simulations will be performed on three soil types,

clay, silt loam, and sand, which are provided by Carsel and Parrish (1988). Clays and

sands were selected to capture the extremes of unsaturated soil behavior. Silt loams

were selected as a soil that serves to approximate a median between the soil water char-

acteristic curve between clays and sands. A more detailed discussion on various un-

saturated soil properties is in 2.3.9.

The most significant work conducted by Carsel and Parrish (1988) is the statistical

correlations between the distribution of four unsaturated properties, 𝜃𝑟, 𝑛, 𝑘𝑠, and 𝛼.

That is, when one of these four properties varies it is statistically likely the other three

will vary to some degree as well. The complexity of determining suitable variables to

input into a Monte Carlo simulation was simplified by the transformation matrices and

procedures provided by Carsel and Parrish (1988). Carsel and Parrish (1988) suggest

that 𝜃𝑠 has a normal distribution and for can be considered independent from the other

four variables. In this Monte Carlo analysis, 𝜃𝑠 will not be varied to simplify the com-

parison. The selected average 𝜃𝑠 for sand, silt loam, and clay are 0.43, 0.45, and 0.38

respectively. To compare between soil types, a friction angle of 𝜙′ = 30°, specific

146

gravity of 𝐺𝑠 = 2.65, and cohesion of 𝑐′ = 0 kPa have been selected. The foundation

is a 1 m wide surface strip footing with a groundwater table of 𝑧𝑤 = 1 m.

In order to develop suitable parameters to be implemented into Monte Carlo simula-

tions, Carsel and Parrish (1988) presented procedures by which mean values for 𝑘𝑠, 𝜃𝑟,

𝛼, and 𝑛 can be transformed by a transformation matrix to create a new set of variables

that comply with known distributions for each variables. The procedures are as follows:

(1) generate a 4×1 vector of random numbers that are normally distributed. This vector

will then be transformed to the corresponding distribution for each variable; (2) multi-

ply the transformation matrix by the randomly generated vector then add this to a vector

of means; (3) perform a check that verifies this new vector falls within truncation limits,

if not repeat to step 1. These are numerical or physical truncations; (4) apply an inverse

transformation to account for distribution type; and (5) verify that these final values

comply with limits on variation, if not repeat to step 1. Following these procedures will

create one realization. Carsel and Parrish provided an example of 1000 realizations of

silt loam. This soil will be compared and analyzed first.

5.5.1. Silt Loam Analysis

The reported average properties for silt loam are 𝜃𝑠 = 0.45, 𝜃𝑟 = 0.067, 𝑘𝑠 = 12.5 ×

10−7 m/s, 𝛼 = 0.20 kPa−1, and 𝑛 = 1.41. The following distributions are the results

of the transformation procedures provided by Carsel and Parrish (1988) for hydraulic

conductivity ks, residual water content 𝜃𝑟, van Genuchten’s 𝛼, and van Genuchten’s n.

Figure 5-47 is from the original paper, while Figure 5-48 is the distributions used in

this current work. Note that the distributions in Figure 5-47 used only 1,000 realizations

while those in Figure 5-48 are based on 50,000 realizations, which is why they are

smoother than those presented by the original authors. Figure 5-49 is the continuous

cumulative distribution function compared to reported discrete percentiles reported by

Carsel and Parrish (1988).

147

Figure 5-47. Probability histogram of silt loam properties from Carsel and Parrish (1988)

Figure 5-48. Probability histogram of silt loam properties used in this work (after Carsel and

Parrish 1988)

148

Figure 5-49. Cumulative distribution function of silt loam (after Carsel and Parrish 1988)

This comparison shows that the distribution of soil water characteristic curves can be

accurately estimated. The cumulative distribution function shows that the distribution

used in this work is the same as the distribution provided by Carsel and Parrish (1988),

giving confidence that the property distribution and correlation are the same as in the

original work.

For further analysis, the shallow foundation bearing capacity was calculated for 50,000

realizations according to the properties listed at the beginning of Section 5.5 and the

distribution of unsaturated properties in Figure 5-49. 50,000 iterations were selected to

1) reduce noise in the distribution of bearing capacities, and 2) to limit error in the

calculated mean bearing capacity. The results are presented in Figure 5-50.

149

Figure 5-50. 50,000 Monte Carlo realizations of silt loam, calculating bearing capacity. From

left to right (1) cumulative distribution of calculated bearing capacities, (2) distribution of 𝛼

and 𝑛 input parameters, (3) probability histogram of bearing capacities, and (4) summary of

the plotted percentiles and other data.

There are three subplot within Figure 5-50. The first two on the left are a cumulative

distribution and relative frequency plot of the calculated shallow foundation bearing

capacity using the modified approach developed in this work. The distribution of bear-

ing capacity is skewed to the left, meaning the mean predicts under the median. The

span of predicted values ranges from a minimum of 205.1 kPa (-40.2% from the mean)

to 408.7 kPa (+19.1% from the mean), which corresponds to a range of 59.4% of the

mean. The third subplot shows the range of 𝛼 and 𝑛, where each point represents one

realization. Contours of bearing capacity are included on this subplot.

Various percentiles are plotted on all three figures. Red corresponds to the 5th and 95th

percentiles, green to the 25th and 75th percentile, blue with the median, and magenta

150

with the mean. By visual inspection, the mean and median split the range of 𝛼 and 𝑛

by 1/3.

This figure shows a clear distinction between the unmodified Vesić solution, which

predicts a bearing capacity of 167.3 kPa, to the mean calculated from the modified

approach proposed in this work, which predicts 343.2 kPa. Partially saturated silt loam

with the same foundation dimension and groundwater table depth can have bearing

capacities greater than two to three times the unmodified Vesić solution.

The gamma, Weibull, and lognormal distribution functions were fitted to the cumula-

tive distribution function in Figure 5-50. The fitted distributions are presented in Figure

5-51. Table 5-5 below presents the R2 for each fit, the mean and standard deviation

predicted from each function, and the coefficient of variance.

Figure 5-51. Gamma, Weibull, and lognormal distribution functions fitted to the CDF for silt

loam.

151

Table 5-5. R2, mean, standard deviation, and coefficient of variation predicted from the

Gamma, Weibull, and lognormal distribution functions fitted to silt loam data.

𝑅2 𝜇 (kPa) 𝜎 (kPa) COV (%) Gamma 0.9896 347.3 34.6 10.0 Weibull 0.9988 343.8 35.9 10.4

Lognormal 0.9881 347.7 34.7 9.9

The Weibull distribution function does the best in terms of fitting the measured cumu-

lative distribution function for silt, however, each of the three distribution functions

have reasonable good fits. The predicted mean from the Weibull distribution function

is the closest to the true mean from the Monte Carlo simulations. It can be concluded

that silt loam can be adequately described with a Weibull distribution.

5.5.2. Sand Analysis

Similarly for a sand, the reported average properties for sand are 𝜃𝑠 = 0.43, 𝜃𝑟 =

0.045, 𝑘𝑠 = 8.25 × 10−5 m/s, 𝛼 = 1.48 kPa−1, and 𝑛 = 2.68. The corresponding

distributions are below:

Figure 5-52. Probability histogram of sand properties used in this work (after Carsel and

Parrish 1988)

152

Calculating the shallow foundation bearing capacity for 50,000 realizations of sand

produces Figure 5-53 below:

Figure 5-53. 50,000 Monte Carlo realizations of sand, calculating bearing capacity. From left

to right (1) cumulative distribution of calculated bearing capacities, (2) distribution of 𝛼 and

𝑛 input parameters, (3) probability histogram of bearing capacities, and (4) summary of the

plotted percentiles and other data.

The distribution for sands are skewed to the right, meaning the mean predicts over the

median. The minimum and maximum are 158.2 kPa (-5.2% from mean) and 211.3 kPa

(+26.7% from mean) respectively, which corresponds to a range of 31.9% of the mean.

The most important conclusion for sands is that the unmodified Vesić solution to the

bearing capacity is greater than (but not significantly greater than) the mean and median

from the modified approach proposed in this work. The Vesić equation fairly accurately

calculates the bearing capacity of partially saturated sand.

153

The gamma, Weibull, and lognormal distribution functions were also fitted to the CDF

for sand. The fitted distributions are presented in Figure 5-54. Table 5-6 below presents

the R2 for each fit, the mean and standard deviation predicted from each function, and

the coefficient of variance.

Figure 5-54. Gamma, Weibull, and lognormal distribution functions fitted to the CDF for

sand.


Gamma, Weibull, and lognormal distribution functions fitted to sand data.


Lognormal 0.9874 166.1 4.2 2.5

For sands, both the gamma and lognormal distribution do well in describing the behav-

ior of sands. The predicted mean for the lognormal distribution function is the closest

to the true mean (166.8 kPa) from the Monte Carlo simulations. On this basis, sands

can be most adequately described by the lognormal distribution. It is important to note

that the coefficient of variation is smaller for sands than in silt loams. That implies that

there is greater uncertainty in silt loam unsaturated parameters.

154

5.5.3. Clay Analysis

The reported average properties for clay are 𝜃𝑠 = 0.38, 𝜃𝑟 = 0.068, 𝑘𝑠 = 5.56 ×

10−7 m/s, 𝛼 = 0.082 kPa−1, and 𝑛 = 1.09. The distributions for 𝑘𝑠 and 𝑛 are trun-

cated due to the restriction on these variables. Physically, permeability cannot be less

than 0, and numerically van Genuchten’s n cannot be less than or equal to 1. The fol-

lowing distributions are below:

Figure 5-55. Probability histogram of clay properties used in this work (after Carsel and

Parrish 1988)

155

Figure 5-56. 50,000 Monte Carlo realizations of clay, calculating bearing capacity. From left

to right (1) cumulative distribution of calculated bearing capacities, (2) distribution of 𝛼 and

𝑛 input parameters, (3) probability histogram of bearing capacities, and (4) summary of the

plotted percentiles and other data.

The distribution of the calculated bearing capacity for clays is unlike both silt loam and

sand. Here the bearing capacity distribution is truncated. This is due numerical limita-

tions on the fitting parameter 𝑛, which cannot be less than or equal to 1. The minimum

and maximum calculated bearing capacity is 371.0 kPa (-11.8% from the mean) and

424.1 kPa (+0.8% from the mean), which corresponds to a range of 12.8% from the

mean.

The unmodified Vesić solution calculates a value of 187.7 kPa, which is significantly

lower than the mean and median solution calculated by the modified approach. The

Monte Carlo simulations indicate that this modified method predicts significant, possi-

bly unrealistic, increases in bearing capacity due to partial saturation in clay soils. This

156

significant increase in bearing capacity, relative to sands with the same footing and

friction angle, needs to be assessed physically through model and full-scale footing

tests.

The fitted distributions are presented in Figure 5-57 presents fitted distribution func-

tions for clays. Table 5-7 below presents the R2 for each fit, the mean and standard

deviation predicted from each function, and the coefficient of variance.

Figure 5-57. Gamma, Weibull, and lognormal distribution functions fitted to the CDF for

clay.


Gamma, Weibull, and lognormal distribution functions fitted to clay data.


Lognormal 0.9608 421.6 2.4 0.6

From Figure 5-57 it is clear that while each distribution function decently describes the

behavior of clay, none of the distribution functions can capture the truncated behavior

at the end of the CDF that is caused by the truncation of 𝑛. It can be concluded from

the 𝑅2 value that Weibull distribution function most adequately describes the variation

of bearing capacity in clays with respect to unsaturated soil properties. Clays had the

157

lowest variation in predicted bearing capacity, followed by sands and silts. This implies

that the unsaturated soils properties for clays had the least variation.

5.6. Discussion

The parametric studies included in this chapter have revealed many interesting behav-

iors for foundations in partially saturated soils. For the shallow foundations, van

Genuchten’s 𝛼 and 𝑛 fitting parameter had the greatest influence on foundation perfor-

mance, varying the magnitude of bearing capacity as well as controlling the shape of

the bearing capacity profile. For 𝛼 and 𝑛 values typical of fine-grained soils, bearing

capacity increased with depth. This was not the case for coarse-grained soils, which

often had a peak bearing capacity at some groundwater table depth. Generally, bearing

capacity increased as 𝛼 and 𝑛 decreased. Another interesting behavior is that the ratio

between the conventional bearing capacity equation and unmodified bearing capacity

equation increased with lower 𝛼 and 𝑛 values. For higher friction angles, this ratio de-

creased.

The 𝛽-method was modified by incorporating suction stresses and by implementing a

new K0 that accounts for tensile stresses in the soil. Tension cracking, which occurs

more predominantly near the surface in fine-grained soils (i.e. small 𝛼 and 𝑛 values),

is a direct by-product of this varied K0. Suction stresses throughout the soil profile,

leads to an increase in side resistance, but is countered by a reduced K0 and tension

cracking near the soil surface. The parametric study on the modified 𝛽-method has

resulted in many interesting conclusions: (1) the selection of low Poisson’s ratio (𝜈 =

0.2) results in more tension cracking, while higher Poisson’s ratios (𝜈 = 0.4) develops

less tension cracking and therefore greater total unit side resistance; (2) tension crack-

ing occurs more frequently in fine-grained soils (low 𝛼 and 𝑛) and will therefore peak

in side resistance as the groundwater table lowers; (3) coarse-grained soils are not con-

trolled by tension cracking and have low suction stresses, therefore side resistance will

continue to increase (to an asymptotic limit) as the groundwater table is lowered; (4)

158

while evaporation increases the suction stress within the soil, it is detrimental to

strength as it increases the depth of cracking; (5) infiltration reduces tension cracking

but also decreases suction stress – soils exposed to infiltration saw a slight increase in

strength from no net flow.

For sands, both the conventional bearing capacity and 𝛽-method generally agreed with

the modified approaches. The inclusion of unsaturated soil mechanics is only signifi-

cant in fine-grained soils such as clays and silts. This behavior is also shown in the

Monte Carlo simulation for sands.

Monte Carlo simulations are an important tool that provide a robust determination of

the confidence bounds for the range of bearing capacity relative to the mean bearing

capacity. These simulations can be further advanced if variation in other soil properties

are well understood (like friction angle, cohesion, and seasonal variation in flux and

groundwater table depth).

The purpose of Monte Carlo simulations in Section 5.5 was to determine typical distri-

butions of unsaturated soil properties and how they might affect the calculated bearing

capacity using the modified approach. In terms of distribution, silts and sands are sim-

ilar but skewed in different directions. The bearing capacity for clays is truncated due

to the numerical limitation placed on the fitting parameter 𝑛. In general, the unmodified

Vesić solution compares well with modified approach used in this work in predicting

the bearing capacity of partially saturated sands.

It has been shown that the Weibull distribution, for silt loam and clays, most adequately

characterizes the variation of bearing capacity with respect to differing unsaturated

properties. For sands, the lognormal distribution sufficiently describes the variation in

bearing capacity. In general, however, the gamma, Weibull, and lognormal distribution

all do well in fitting the results of the Monte Carlo simulation.

159

Another interesting result of these Monte Carlo simulation is the differences in the

range of values. Sands have a range of 31.9% (COV = 2.5%), silt loams 59.4% (COV

= 9.9%), and clays 12.8% (COV = 0.6%) of the mean. Clays have the smallest variation

which indicates that the distribution of unsaturated soil properties does not significantly

affect or change the calculated bearing capacity. Silt loams, which have soil water char-

acteristic curves between sands and clays, have the greatest variation in bearing capac-

ity.

160

6. Conclusions and Future Work

6.1. Conclusions

The purpose of this work was to develop a framework for calculating ultimate bearing

capacity in shallow and deep foundations in partially saturated soils. This theoretical

development included modifying the conventional shallow foundation bearing capacity

equations and the 𝛽-method to incorporate recent literature on unsaturated soil mechan-

ics. Implementation of unsaturated soil mechanics into the conventional bearing capac-

ity equations includes considering apparent cohesion as it varies above the groundwater

table, soil unit weight according to matric suction and the soil water characteristic

curve, and the inclusion of suction stress on overburden. The modified 𝛽-method, has

been designed to include the consideration of suction stress along the drilled shaft and

a reduction of K0 due to tension cracking near the surface. Implementation of these

procedures are discussed in Chapter 3.

The proposed theoretical equation for calculating shallow foundation bearing capacity

has been evaluated with a comparative study of load tests performed on plates and

shallow foundations in partially saturated soils. The modified bearing capacity equa-

tions shows close agreement to measured bearing capacities, having an R2 = 0.81, while

the conventional bearing capacity has very little agreement with the measured data.

This implies that the proposed model for bearing capacity is suitable for use in shallow

foundation design. Based on the load tests considered, the conventional method se-

verely underpredicts the bearing capacity in partially saturated soils and its use may

result in overdesign.

Parametric studies were performed on the modified shallow bearing capacity equation

and modified 𝛽-method for side resistance. For shallow foundations, variation in bear-

ing capacity is most sensitive to the selection of van Genuchten’s 𝛼 and 𝑛 parameters.

In comparing the modified bearing capacity to the convention equational, low values

for 𝛼 and 𝑛 resulted in the highest contrast. Finally, sensitivity to these van Genuchten

161

fitting parameters decreased for larger friction angles. The parametric study for the

modified 𝛽-method showed that side resistance was greatly influenced by tension

cracking, especially for shorter foundations embedded in fine-grained soils. If the

groundwater table was sufficiently deep and the foundation short, tension cracking sig-

nificantly reduced performance. Infiltration generally reduced the depth of tension

cracking, but also reduced the suction stresses in the soil. Evaporation increased the

depth of cracking while increasing suction stresses in the soil. The amount of variation

in suction and cracking was ultimately dependent on material type. Implementation of

Monte Carlo simulations for shallow foundations in partially saturated soils showed

that bearing capacity calculated from silts had the greatest variability, while clays had

the smallest. Finally, for sands, both the conventional bearing capacity equation and

conventional 𝛽-method performed well against the modified approaches.

6.2. Implications for Geotechnical Engineering Practice

The modified shallow foundation bearing capacity equation and 𝛽-method for deep

foundations serve to describe many phenomena in partially saturated soils, especially

in fine-grained soils. As such, it is important to discuss the implication of these pro-

posed methods for use in geotechnical engineering practice.

The proposed shallow foundation bearing capacity equation has shown good agreement

with measured bearing capacity values presented in the literature. This gives some con-

fidence that calculating bearing capacity for partially saturated soils using this approach

may yield suitable bearing capacity values. The database of load tests used in this work

should be expanded in subsequent work to confirm the validity of this approach.

Figure 5-21 is a table of figures that plots contours of the ratio between the bearing

capacity calculated from the modified approach and the conventional Vesić approach.

Within this figure, considerations are made to varying friction angles, ground water

162

table depths, and van Genuchten parameters n and 𝛼. This figure allows for quick esti-

mates of the potential increase (or decrease) of bearing capacity due to partial satura-

tion, especially if the soil texture, grain size distribution, or SWCC are well understood.

The value selected from this figure can be multiplied by the conventional Vesić bearing

capacity equation. For a more robust analysis, Monte Carlo simulations can be con-

ducted, as described in this work, to assess the variability in bearing capacity with re-

spect to unsaturated soil properties. These simulations can also be modified to include

variation in friction angle, steady-state flow, cohesion, saturated water content, and

seasonal variation in the groundwater table depth.

The modified 𝛽-method should be used with caution as no comparison has been made

with measured side resistances of deep foundations. Further, predicted tension cracking

depths from the Lu and Likos (2004) K0 equation and the influence of suction stress on

the soil/foundation interface have not been validated in the literature. Figure 5-41 to

Figure 5-43 plot unit side resistance as a function of Poisson’s ratio, groundwater table

depth, and van Genuchten parameters n and 𝛼. These figures can be used to quickly

estimate the unit side resistance of a deep foundation embedded in partially saturated

soils. While the modified 𝛽-method has not been validated through load testing, it is

useful in assessing whether tension cracking may be an issue or if suction stresses may

contribute to side resistance.

6.3. Future Work

The effects of partial are not typically considered geotechnical engineering applications

such as in foundation design. As a result, there is still a significant amount of research

to be completed on this topic. Shallow and deep foundations design considerations in

partially saturated soils is still lacking in many areas such as settlement. The author has

outlined the following as potential future work for research that pertains to foundation

design in partially saturated soil:

163

- Finite element modeling using the modified Mohr-Coulomb failure criterion or

another unsaturated constitutive model (like the Barcelona Basic Model) in par-

tially saturated soils to assess the impacts of partial saturation on load-displace-

ment behavior.

- Comparison of the modified shallow and deep foundation equations with dif-

ferent definitions for Bishop’s effective stress parameter and different fitting

equations for the soil water characteristic curve.

- Characterization of in-situ matric suction profiles in partially saturated soils,

especially stratified soils. The matric suction profile equation used in this work

is based on the Gardner (1958) hydraulic conductivity model, which has not

been validated in real soils.

- Development of matric suction profile equations in stratified, partially saturated

soils. This work assumes a uniform soil profile since matric suction models in

stratified soils have not been developed.

- Creation of a new matric suction profile equation based off of more recent hy-

draulic conductivity models like Mualem (1976). The matric suction profile

used in this work is based on the Gardner (1958) model.

- Development and application of transient flow equations in partially saturated

soils for shallow and deep foundation bearing capacity. This work assumes

steady-state flow conditions, but this is not reasonable since weather conditions

and water flow are irregular.

- Development of settlement framework/equation for shallow and deep founda-

tions in partially saturated soils. This work only addressed the need for devel-

oping bearing capacity equations for shallow and deep foundations.

- Validation of the modified 𝛽-method through a comparative study on reported

𝛽 coefficients in the literature for normally consolidated, partially saturated

soils.

- The consideration of stress history in the modified 𝛽-method. OCR can be im-

plemented into the modified 𝛽-method to remove the limitation of normally

consolidated soils.

164

- Full-scale shallow foundation load tests in partially saturated soils with a well-

defined soil water characteristic curve, especially in clays. Much of the litera-

ture used in the shallow foundation comparative study was for model footings

loaded on fine sands.

- Development of procedures/standards to measure matric suction during full-

scale load testing. Matric suction measurements should become standard in in-

dustry as it accounts for a considerable amount of strength in fine-grained soils.

The consideration of unsaturated soil mechanics in geotechnical engineering is an im-

portant task. Geotechnical engineering has been plagued with uncertainty, uncertainty

that is continually being reduced. The inclusion of partially saturated soils allows for

more robust designs and a greater understanding of the behavior of soils as a whole.

165

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