Passive Earth Pressures: Design Parameters for Common
Force-Displacement Approaches
Reynold David Meyer II
A project submitted to the faculty of
Brigham Young University
in partial fulfillment of the requirements for the degree of
Master of Science
Kyle M. Rollins, Chair
Norman L. Jones
Fernando S. Fonseca
Department of Civil & Environmental Engineering
Brigham Young University
April 2012
Copyright © 2012 Reynold David Meyer II
All Rights Reserved
ABSTRACT
Passive Earth Pressures: Design Parameters for Common
Force-Displacement Approaches
Reynold David Meyer II
Department of Civil & Environmental Engineering, BYU
Master of Science
Prior large scale lateral load testing was performed on pile caps with varying geometries
and backfill configurations. The testing simulated the lateral response of backfill against bridge
abutments. The backfill soil included clean sand, silty sand, fine gravel, and coarse gravel. In this
report, the backfills were considered either loose or dense depending on their relative densities.
Several of the tests were performed using wingwalls in an effort to assess the contribution of
plane strain effects. The purpose of this testing was to determine the lateral passive resistance of
several types of backfills against bridge abutments.
The purpose of this project was to first collect and tabulate the data from previous studies
and then to compare the results. The measured force-displacement relationships for each backfill
were then compared with common force-displacement approaches. The common forcedisplacement methods used were: PYCAP, ABUTMENT, and Shamsabadi’s general hyperbolic
force-displacement relationship. The analytical models were fit with the measured data and
appropriate design parameters were back-calculated. The design parameters can be used to
model the force-displacement relationship and to determine the total lateral passive resistance of
various backfills against bridge abutments.
Keywords: passive resistance, ABUTMENT, PYCAP, backfill
TABLE OF CONTENTS
LIST OF TABLES ....................................................................................................................... ix
LIST OF FIGURES ................................................................................................................... xiii
1
Introduction ........................................................................................................................... 1
1.1
2
3
4
Objectives ....................................................................................................................... 2
Literature Review ................................................................................................................. 3
2.1.1
Caltrans Seismic Design Approach............................................................................. 3
2.1.2
PYCAP ........................................................................................................................ 5
2.1.3
ABUTMENT .............................................................................................................. 8
2.1.4
Closed-Form Modified Hyperbolic Force-Displacement (HFD) Equation .............. 10
Testing .................................................................................................................................. 13
3.1
Rollins & Cole (2006)................................................................................................... 13
3.2
Kwon (2007) ................................................................................................................. 15
3.3
Valentine (2007) ........................................................................................................... 16
3.4
Pruett (2009) ................................................................................................................. 16
3.5
Cummins (2009) ........................................................................................................... 18
3.6
Nasr (2010) ................................................................................................................... 19
3.7
Heiner (2010) ................................................................................................................ 19
3.8
Strassburg (2010) .......................................................................................................... 20
3.9
Bingham (2012) ............................................................................................................ 21
3.10
Jessee (2012) ................................................................................................................. 22
3.11
UCLA (Stewart et al., 2007) ......................................................................................... 23
Results .................................................................................................................................. 25
4.1
Loose Clean Sand (2D) ................................................................................................. 25
v
5
6
4.2
Loose Clean Sand (3D) .................................................................................................27
4.3
Dense Clean Sand (2D) .................................................................................................30
4.4
Dense Clean Sand (3D) .................................................................................................34
4.5
Loose Silty Sand (3D) ...................................................................................................37
4.6
Dense Silty Sand (2D) ...................................................................................................39
4.7
Dense Silty Sand (3D) ...................................................................................................42
4.8
Loose Fine Gravel (3D) .................................................................................................45
4.9
Dense Fine Gravel (3D) ................................................................................................47
4.10
Loose Coarse Gravel (3D) .............................................................................................50
4.11
Dense Coarse Gravel (3D) ............................................................................................52
Comparison of Results .........................................................................................................55
5.1
Friction Angle, φ ...........................................................................................................55
5.2
Cohesion ........................................................................................................................65
5.3
Initial Soil Modulus, E i .................................................................................................70
5.4
Strain at 50% of Ultimate Strength, ε 50 .........................................................................77
5.5
Modified HFD: F ult /B eff ................................................................................................81
5.6
Modified HFD: yave .......................................................................................................84
5.7
Passive Earth Pressure Coefficient, K p .........................................................................87
5.8
Caltrans ..........................................................................................................................96
Conclusion ...........................................................................................................................103
6.1
Conclusions .................................................................................................................103
6.1.1 Friction Angle ............................................................................................................104
6.1.2 Cohesion
................................................................................................................104
6.1.3 Initial Soil Modulus ...................................................................................................105
6.1.4 Strain at 50% of Ultimate Strength, ε 50 .....................................................................106
vi
6.1.5
Modified HFD: F ult /B eff .......................................................................................... 106
6.1.6
Modified HFD: yave ................................................................................................. 107
6.1.7
Passive Earth Pressure Coefficients ........................................................................ 107
6.1.8
Caltrans ................................................................................................................... 108
6.2
Recommendations for Future Research ...................................................................... 108
References .................................................................................................................................. 109
vii
viii
LIST OF TABLES
Table 2.1: Stiffness Ranges for Various Soil Densities (Duncan and Mokwa, 2001) .....................8
Table 2.2: Typical Adhesion Factors (NAVFAC, 1982) .................................................................8
Table 2.3: Ranges of ε 50 for Various Soil Types (Shamsabadi et al., 2007) .................................10
Table 2.4: Suggested HFD Parameters for Abutment Backfills (Shamsabadi et al., 2007) ..........12
Table 3.1: Summary of Backfill Soil Parameters (Rollins and Cole, 2006) ..................................14
Table 3.2: Unit Weights and Shear Strength Parameters for Backfill Materials (Rollins
and Cole, 2006) ..................................................................................................................14
Table 3.3: Summary of Backfill Soil Parameters (Kwon, 2007) ...................................................15
Table 3.4: Gradation Properties of the Fine Gravel (Pruett, 2009) ................................................17
Table 3.5: Results of Direct Shear Testing for Fine Gravel (Pruett, 2009) ...................................17
Table 3.6: Gradation Properties for the Coarse Gravel (Pruett, 2009) ..........................................18
Table 3.7: Results of Direct Shear Testing for Coarse Gravel (Pruett, 2009) ...............................18
Table 3.8: Direct Shear Results for Loose Clean Sand (Cummins, 2009) ....................................19
Table 3.9: Gradation Parameters for Loose Well-Graded Sand (Strassburg, 2010)......................21
Table 3.10: Soil Properties for Dense Clean Sand (Jessee, 2012) .................................................23
Table 4.1: PYCAP Parameters for Loose Clean Sand (2D) ..........................................................26
Table 4.2: ABUTMENT Parameters for Loose Clean Sand (2D) .................................................27
Table 4.3: Modified HFD parameters for Loose Clean Sand (2D)................................................27
Table 4.4: PYCAP Parameters for Loose Clean Sand (3D) ..........................................................29
Table 4.5: ABUTMENT Parameters for Loose Clean Sand (3D) .................................................29
Table 4.6: Modified HFD Parameters for Loose Clean Sand (3D) ...............................................29
Table 4.7: PYCAP Parameters for Dense Clean Sand (2D) ..........................................................33
Table 4.8: ABUTMENT Parameters for Dense Clean Sand (2D) .................................................33
Table 4.9: Modified HFD Parameters for Dense Clean Sand (2D) ...............................................34
ix
Table 4.10: PYCAP Parameters for Dense Clean Sand (3D) ........................................................36
Table 4.11: ABUTMENT Parameters for Dense Clean Sand (3D) ...............................................36
Table 4.12: Modified HFD Parameters for Dense Clean Sand (3D) .............................................37
Table 4.13: PYCAP Parameters for Loose Silty Sand (3D) ..........................................................38
Table 4.14: ABUTMENT Parameters for Loose Silty Sand (3D) .................................................38
Table 4.15: Modified HFD Parameters for Loose Silty Sand (3D) ...............................................39
Table 4.16: PYCAP Parameters for Dense Silty Sand (2D) ..........................................................41
Table 4.17: ABUTMENT Parameters for Dense Silty Sand (2D).................................................41
Table 4.18: Modified HFD Parameters for Dense Silty Sand (2D) ...............................................42
Table 4.19: PYCAP Parameters for Dense Silty Sand (3D) ..........................................................44
Table 4.20: ABUTMENT Parameters for Dense Silty Sand (3D).................................................44
Table 4.21: Modified HFD Parameters for Dense Silty Sand (3D) ...............................................45
Table 4.22: PYCAP Parameters for Loose Fine Gravel (3D) ........................................................46
Table 4.23: ABUTMENT Parameters for Loose Fine Gravel (3D) ..............................................46
Table 4.24: Modified HFD Parameters for Loose Fine Gravel (3D) .............................................47
Table 4.25: PYCAP Parameters for Dense Fine Gravel (3D) .......................................................49
Table 4.26: ABUTMENT Parameters for Dense Fine Gravel (3D) ..............................................49
Table 4.27: Modified HFD Parameters for Dense Fine Gravel (3D) ............................................50
Table 4.28: PYCAP Parameters for Loose Coarse Gravel (3D) ....................................................51
Table 4.29: ABUTMENT Parameters for Loose Coarse Gravel (3D) ..........................................51
Table 4.30: Modified HFD Parameters for Loose Coarse Gravel (3D) .........................................52
Table 4.31: PYCAP Parameters for Dense Coarse Gravel (3D) ...................................................54
Table 4.32: ABUTMENT Parameters for Dense Coarse Gravel (3D) ..........................................54
Table 4.33: Modified HFD Parameters for Dense Coarse Gravel (3D) ........................................54
Table 5.1: Ranges for the PYCAP Friction Angles .......................................................................57
x
Table 5.2: Ranges for the ABUTMENT Friction Angles ..............................................................57
Table 5.3: Ranges for the PYCAP Cohesion Values .....................................................................67
Table 5.4: Ranges for the ABUTMENT Cohesion Values............................................................67
Table 5.5: Stiffness Ranges for Various Soil Densities (Duncan and Mokwa, 2001) ...................71
Table 5.6: Ranges for the Initial Soil Modulus ..............................................................................72
Table 5.7: A Comparison of Relative Density with the Initial Soil Modulus of the Silty
Sand....................................................................................................................................73
Table 5.8: A Comparison of Relative Density with the Initial Soil Modulus of the Clean
Sand....................................................................................................................................73
Table 5.9: A Comparison of Relative Density with the Initial Soil Modulus of the Loose
Gravel .................................................................................................................................74
Table 5.10: A Comparison of Relative Density with the Initial Soil Modulus of the
Coarse Gravel.....................................................................................................................74
Table 5.11: Ranges of Initial Soil Modulus for Different Relative Densities ...............................74
Table 5.12: Ranges for the Strain at 50% of Ultimate Strength ....................................................78
Table 5.13: Ranges for F ult /B eff for the Modified HFD Equation .................................................82
Table 5.14: Ranges of yave for the Modified HFD Equation .........................................................85
Table 5.15: Ranges for the Measured Passive Earth Pressure Coefficient ....................................90
Table 5.16: Comparisons of the Measured K p Value with the Rankine and Coulomb
Methods for the Silty Sands ...............................................................................................91
Table 5.17: Comparisons of the Measured K p Value with the Rankine and Coulomb
Methods for the Clean Sands .............................................................................................91
Table 5.18: Comparisons of the Measured K p Value with the Rankine and Coulomb
Methods for the Fine Gravels ............................................................................................92
Table 5.19: Comparisons of the Measured K p Value with the Rankine and Coulomb
Methods for the Coarse Gravels ........................................................................................92
xi
xii
LIST OF FIGURES
Figure 2.1: Passive Force-Deflection Relationship for Caltrans Seismic Design Model
(Adapted from Caltrans Seismic Design Manual (2004)) ...................................................4
Figure 2.2: Hyperbolic Passive Force-Displacement Curve (Duncan and Mokwa, 2001) ..............7
Figure 2.3: Mobilization of the Passive Resistance of a Backfill Material (Shamsabadi et al.,
2007) ....................................................................................................................................9
Figure 2.4: Hyperbolic Force-Displacement Curve (Shamsabadi et al., 2008) .............................11
Figure 3.1: Grain Size Distribution for Clean Sand (Heiner, 2010) ..............................................20
Figure 3.2: The Particle-Size Distribution for a Clean Sand. (Bingham, 2012) ............................22
Figure 4.1: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Loose Clean Sand (2D) (Strassburg, 2010). .........26
Figure 4.2: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Loose Clean Sand (3D) (Cummins, 2009). ..........28
Figure 4.3: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Loose Clean Sand (3D) (Strassburg, 2010). .........28
Figure 4.4: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (2D) (Bingham, 2012). ...........30
Figure 4.5: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (2D) (Jessee #1, 2012). ..........31
Figure 4.6: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (2D) (Jessee #2, 2012) ...........31
Figure 4.7: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (2D) (Jessee #3, 2012) ...........32
Figure 4.8: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (3D) (Rollins & Cole,
2006). .................................................................................................................................34
Figure 4.9: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (3D) (Heiner, 2010). ..............35
Figure 4.10: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (3D) (Bingham, 2012). ...........35
xiii
Figure 4.11: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Loose Silty Sand (3D) (Kwon, 2007).................. 37
Figure 4.12: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Silty Sand (2D) (Stewart et al., 2007) ...... 40
Figure 4.13: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Silty Sand (2D) using the Plane
Strain Friction Angle (Stewart et al., 2007) ...................................................................... 40
Figure 4.14: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Silty Sand (3D) (Rollins & Cole,
2006) ................................................................................................................................. 42
Figure 4.15: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Silty Sand (3D) (Valentine, 2007)............ 43
Figure 4.16: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Loose Fine Gravel (3D) (Pruett, 2009) ............... 45
Figure 4.17: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Fine Gravel (3D) (Rollins & Cole,
2006) ................................................................................................................................. 47
Figure 4.18: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Fine Gravel (3D) (Kwon, 2007) ............... 48
Figure 4.19: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Fine Gravel (3D) (Pruett, 2009) ............... 48
Figure 4.20: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Loose Coarse Gravel (3D) (Pruett, 2009) ........... 50
Figure 4.21: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Coarse Gravel (3D) (Rollins & Cole,
2006) ................................................................................................................................. 52
Figure 4.22: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Coarse Gravel (3D) (Pruett, 2009) ........... 53
Figure 5.1: Comparison of the PYCAP Friction Angles for All of the Soil Types. ..................... 56
Figure 5.2: Comparison of the ABUTMENT Friction Angles for All of the Soil Types. ............ 56
Figure 5.3: A Comparison of the PYCAP Friction Angles with Relative Density for the
Unconfined Backfills. ....................................................................................................... 59
xiv
Figure 5.4: A Comparison of the PYCAP Friction Angles with Relative Density for the
Unconfined, Non-Cohesive Backfills. .............................................................................. 60
Figure 5.5: A Comparison of the ABUTMENT Friction Angles with Relative Density for
the Unconfined Backfills. ................................................................................................. 61
Figure 5.6: A Comparison of the ABUTMENT Friction Angles with Relative Density for
the Unconfined, Non-Cohesive Backfills. ........................................................................ 62
Figure 5.7: A Comparison of the PYCAP Friction Angles for the Unconfined (3D)
Backfills. ........................................................................................................................... 63
Figure 5.8: A Comparison of the PYCAP Friction Angles for the Dense Backfills. ................... 63
Figure 5.9: A Comparison of the ABUTMENT Friction Angles for the Unconfined (3D)
Backfills. ........................................................................................................................... 64
Figure 5.10: A Comparison of the ABUTMENT Friction Angles for the Dense Backfills. ........ 65
Figure 5.11: Comparison of the Cohesion Values used in PYCAP for all of the Soil Types....... 66
Figure 5.12: Comparison of the Cohesion Values used in ABUTMENT for all of the Soil
Types. ................................................................................................................................ 66
Figure 5.13: A Comparison of the PYCAP Cohesion Values for the Unconfined (3D)
Backfills. ........................................................................................................................... 68
Figure 5.14: A Comparison of the PYCAP Cohesion Values for the Dense Backfills. ............... 69
Figure 5.15: A Comparison of the ABUTMENT Cohesion Values for the Unconfined
(3D) Backfills.................................................................................................................... 69
Figure 5.16: A Comparison of the ABUTMENT Cohesion Values for the Dense Backfills. ...... 70
Figure 5.17: Comparison of the Initial Soil Modulus for All of the Soil Types. .......................... 71
Figure 5.18: A Comparison of Initial Soil Modulus, E i , against Relative Density. ..................... 75
Figure 5.19: A Comparison of the Initial Soil Modulus Values for the Unconfined (3D)
Backfills. ........................................................................................................................... 76
Figure 5.20: A Comparison of the Initial Soil Modulus Values for the Dense Backfills. ............ 77
Figure 5.21: Comparison of the Strain at 50% of Ultimate Strength for All of the Soil
Types. ................................................................................................................................ 78
Figure 5.22: A Comparison of the Strain at 50% of Ultimate Strength against Relative
Density for the Unconfined Backfills. .............................................................................. 79
xv
Figure 5.23: A Comparison of the Strain at 50% of Ultimate Strength for the Unconfined
(3D) Materials. ...................................................................................................................80
Figure 5.24: A Comparison of the Strain at 50% of Ultimate Strength for the Dense
Materials. ...........................................................................................................................81
Figure 5.25: Comparison of F ult /B eff for All of the Soil Types. ....................................................82
Figure 5.26: A Comparison of the F ult /B eff Values for the Unconfined (3D) Backfills. ...............83
Figure 5.27: A Comparison of the F ult /B eff Values for the Dense Backfills..................................84
Figure 5.28: Comparison of yave for all of the Soil Types. ............................................................85
Figure 5.29: A Comparison of yave for the Unconfined (3D) Backfills. ........................................86
Figure 5.30: A Comparison of yave for the Dense Backfills. .........................................................87
Figure 5.31: Comparison of the Passive Earth Pressure Coefficient for All of the Soil Types. ....90
Figure 5.32: A Comparison of the Passive Earth Pressure Coefficients against Relative
Density for the Unconfined Backfills. ...............................................................................93
Figure 5.33: A Comparison of the Ultimate Passive Resistance per Effective Area against
Relative Density for the Unconfined Backfills. .................................................................94
Figure 5.34: A Comparison of the Passive Earth Pressure Coefficients for the Unconfined
(3D) Backfills.....................................................................................................................95
Figure 5.35: A Comparison of the Passive Earth Pressure Coefficients for the Dense
Backfills. ............................................................................................................................95
Figure 5.36: A Comparison of P ult for the Caltrans Method and the Measured Results for a
Wall Height of 3.67 ft. .......................................................................................................97
Figure 5.37: A Comparison of K for the Caltrans Method and the Measured Results for a
Wall Height of 3.67 ft. .......................................................................................................98
Figure 5.38: A Comparison of P ult for the Caltrans Method and the Measured Results for a
Wall Height of 5.5 ft. .........................................................................................................99
Figure 5.39: A Comparison of K for the Caltrans Method and the Measured Results of the
Dense Materials with a Wall Height of 5.5 ft. .................................................................100
Figure 5.40: A Comparison of P ult for the Caltrans Method and the Measured Results for
Loose Materials with a Wall Height of 5.5 ft. .................................................................101
Figure 5.41: A Comparison of K for the Caltrans Method and the Measured Results for
Loose Materials with a Wall Height of 5.5 ft. .................................................................102
xvi
1
INTRODUCTION
Pile foundations and abutment walls are critical in the lateral stability of a bridge under
seismic and wind loadings. The lateral stability comes through the interactions of the piles and
soil as well as the abutment wall and the backfill soil. As the abutment wall is subjected to lateral
loading and displaces, passive resistance within the backfill develops. This passive resistance can
provide much of the lateral stability during seismic and wind loadings. The passive resistance of
a backfill material depends on several factors including the magnitude and the direction of
movement, the strength and stiffness of the soil, the friction between the abutment and the soil,
and the geometry of the abutment wall (Duncan and Mokwa, 2001).
Engineers use the passive resistance of a backfill to determine the lateral strength and
performance of bridge foundations. Several methods, such as Rankine, Coulomb, and log-spiral,
are used to estimate the ultimate passive resistance under static loading. However, using the
ultimate passive resistance in design is not always an appropriate approach. There are cases in
which the magnitude of the abutment wall deflection needed to develop ultimate passive
resistance may exceed the limit specified for a particular bridge. There are also displacementbased design approaches in which the structure is designed for a specific amount of movement,
which may not allow the full development of passive resistance.
Due to displacement limitations, engineers can use several methods to develop forcedisplacement relationships for a given abutment backfill. These methods include simple linear
1
elastic as well as hyperbolic models that will predict the passive force at any given abutment wall
displacement. Duncan and Mokwa (2001), Shamsabadi (2007), Shamsabadi (2008), and
CALTRANS (2004) each provide engineers with models to predict the force-displacement
relationship for an abutment backfill.
1.1
Objectives
Developing the force-displacement backbone curve for a particular abutment-backfill
system can often be complex and uncertain. Although easier methods have been developed to
assist engineers in their design (e.g. Caltrans, PYCAP, ABUTMENT, and a general hyperbolic
force-displacement relationship), determining the soil parameters to be used in these methods
can also be difficult. Engineers need a quick and accurate method for producing soil parameters
to be used in the development of force-displacement backbone curves.
There have been several tests performed by Dr. Kyle Rollins and his students at Brigham
Young University (BYU) to determine passive force-deflection curves for various soils and soil
geometries against bridge abutments. However the results from these tests have generally been
analyzed one test at a time without much consideration of how one relates to other test results. In
this study, the results of all these tests were collected and tabulated, and then the results were
compared with one another. Using the collected data and common methods for approximating
the force-displacement relationships of various soils, appropriate soil parameters were backcalculated for the various design approaches. Knowing the type of backfill material, this data set
gives engineers access to parameters for use in developing force-displacement curves without
extensive geotechnical testing.
2
2
LITERATURE REVIEW
This chapter provides a summary of the analytical methods used in developing forcedisplacement relationships for soil against a bridge abutment or pile cap. These methods include
Caltrans, PYCAP, ABUTMENT, and a closed-form modified Hyperbolic Force-Displacement
(HFD) equation (Shamsabadi, 2008). In this report, the Caltrans approach was used only as a
comparison to the measured data. PYCAP, ABUTMENT, and the modified HFD equation were
used to back-calculate soil parameters that would provide a best fit to the measured data.
2.1.1
Caltrans Seismic Design Approach
The Caltrans seismic design approach was developed from research performed by
Maroney (1995) and Romstad et al. (1996) at the University of California, Davis. The testing
was performed on large scale abutments against (1) a well-graded silty sand and (2) a silty clay.
The Caltrans approach uses equations for the abutment stiffness and ultimate passive force to
form a bi-linear force-displacement relationship. Figure 2.1 demonstrates this bi-linear
relationship. Equations for the abutment stiffness and ultimate passive force are shown in
Equations (2-1) and (2-2).
3
Figure 2.1: Passive Force-Deflection Relationship for Caltrans Seismic Design Model
(Adapted from Caltrans Seismic Design Manual (2004))
𝑘𝑎𝑏𝑢𝑡 = (20 𝑘𝑖𝑝/𝑖𝑛./𝑓𝑡) ∗ 𝑤 ∗ (
𝑃𝑢𝑙𝑡 = (5.0 𝑘𝑠𝑓) ∗ 𝐴𝑒𝑓𝑓 ∗ (
𝐻
)
5.5 𝑓𝑡
(2-1)
𝐻
)
5.5 𝑓𝑡
(2-2)
where
k abut = stiffness of the backfill material, kip/in.
w = effective width of the abutment, ft
H = height of the abutment, ft
P ult = ultimate passive force, kips
A eff = effective area of the abutment
In 2011, Caltrans revised Equation (2-1) to include a value of 50 kip/in./ft instead of 20
kip/in./ft. Both the 2004 and 2011 versions of the equation are used in the Comparison of Results
section. Equations (2-1) and (2-2) include a height proportionality factor (H/5.5 ft) based on the
4
abutment height used for the UC Davis testing (5.5 ft). The Caltrans approach is meant to be
used for dense materials only and does not account for the type of backfill material used; thus the
equations are only a function of the dimensions of the abutment. This method was later adopted
in the Caltrans Seismic Design Criteria manual (2004).
2.1.2
PYCAP
Duncan and Mokwa (2001) concluded that the Log Spiral Theory, when corrected for 3D
effects, provides an accurate method for determining the ultimate passive pressure of a soil.
PYCAP is an excel spreadsheet that is based on the Log Spiral Theory and provides an efficient
method for determining the ultimate passive resistance of a soil. Using the ultimate passive
resistance along with an estimated value for soil stiffness and a hyperbolic expression, the
passive force-displacement relationship can be estimated for a particular backfill material.
Duncan and Mokwa (2001) also concluded that the passive resistance of a soil depends on the
magnitude and direction of the abutment movement, the strength and stiffness of the soil, the
frictional resistance between the soil and the abutment, and the shape of the abutment.
Duncan and Mokwa (2001) developed a Microsoft Excel spreadsheet, known as PYCAP,
which incorporates the factors and methods mentioned previously. PYCAP provides users with a
quick and effective method for determining the passive force-displacement relationship for a
backfill material. PYCAP is limited to cases in which the wall is vertical, the ground surface is
horizontal, and any surcharge is uniform (Duncan and Mokwa, 2001). The spreadsheet includes
the Brinch-Hansen (1966) correction factor for three-dimensional (3D) end effects. Equation (23) shows the Brinch-Hansen correction factor for 3D end effects.
5
where
𝑅
1.6𝐵
0.4𝑅𝑜 𝐴3 𝐵 2
2
= 1 + 𝑅𝑜 �3 �1.1𝐴4 +
+
�
𝑅𝑜
1 + 5 𝑙 ⁄ℎ 1 + 0.05 𝑙 ⁄ℎ
(2-3)
R = resistance factor
R o = resistance factor in the basic case (h = H, l = L)
A = 1 - h/H
B = 1 – (l/L)2
l = actual length of the anchor slab or pile cap
h = actual height of the anchor slab or pile cap
L = distance between the centers of two consecutive slabs
H = distance from the lower edge of the slab to the ground surface.
The passive force-displacement relationship is approximated in PYCAP by the
hyperbolic relationship shown in Equation (2-4).
𝑃=
where
�𝐾
1
𝑚𝑎𝑥
𝑦
(2-4)
𝑦
+ 𝑅𝑓 𝑃 �
𝑢𝑙𝑡
P = passive resistance, kips
y = deflection, in.
K max = initial stiffness, kip/in.
P ult = ultimate passive resistance, kips
R f = failure ratio, P ult /hyperbolic asymptote
6
Figure 2.2 displays the passive force-displacement relationship as approximated by
Equation (2-4).
Figure 2.2: Hyperbolic Passive Force-Displacement Curve (Duncan and Mokwa, 2001)
The backfill soil and abutment properties required for PYCAP include: cap width (b), cap
height (H), embedment depth (z), surcharge (q s ), cohesion (c), soil friction angle (φ), wall
friction (δ), soil modulus (E i ), Poisson’s ratio (υ), soil moist unit weight (γ m ), adhesion factor
(α), and the maximum deflection normalized by the wall height (Δ max /H). Poisson’s ratio can be
calculated using Equation (2-5). Table 2.1 provides a range of stiffness values for different soils
densities as described in Duncan and Mokwa (2001). Typical values for the adhesion factor are
shown in Table 2.2. Duncan and Mokwa (2001) recommends a value of 0.04 for the maximum
deflection normalized by the wall height, which is the estimated amount of movement necessary
to fully develop passive pressures.
𝜐=
1 − 𝑠𝑖𝑛 𝜑
2 − 𝑠𝑖𝑛 𝜑
(2-5)
7
Table 2.1: Stiffness Ranges for Various Soil Densities (Duncan and Mokwa, 2001)
Table 2.2: Typical Adhesion Factors (NAVFAC, 1982)
2.1.3
ABUTMENT
Shamsabadi et al. (2007) developed the computer program, ABUTMENT, which uses a
mobilized logarithmic-spiral failure surface and modified hyperbolic stress-strain behavior to
estimate the passive force-displacement relationship of a backfill material. In order to develop
the force-displacement relationship, ABUTMENT assumes that mobilized passive wedges are
formed for each level of wall displacement, and as a result, intermediate passive forces are
determined using force-based, limit-equilibrium equations. The ultimate passive resistance is
developed when the displacement is large enough to fully mobilize the shear strength of the soil.
Figure 2.3 shows the mobilization of intermediate passive wedges and how they relate to the
passive force-displacement curve.
8
Figure 2.3: Mobilization of the Passive Resistance of a Backfill Material (Shamsabadi et al.,
2007)
The force-displacement curve is a function of the backfill soil properties and the
movement of the abutment. The backfill soil and abutment properties used in ABUTMENT
include: abutment height, abutment effective width, soil friction angle (φ), wall friction angle (δ),
soil cohesion, abutment adhesion, soil density, strain at 50 percent of the failure stress (ε 50 ),
Poisson’s ratio (υ), failure ratio (R f ), and surcharge. The effective width of the abutment is
calculated by multiplying the Brinch-Hansen (1966) 3D correction factor with the actual width
of the abutment. Recommended ranges of ε 50 for various soil types are displayed in Table 2.3
9
Table 2.3: Ranges of ε 50 for Various Soil Types (Shamsabadi et al., 2007)
2.1.4
Closed-Form Modified Hyperbolic Force-Displacement (HFD) Equation
The modified HFD equation was developed by Shamsabadi et al. (2008) to provide a
simpler method for estimating the passive force-displacement curve of a given backfill material.
The closed-form equation was developed from testing performed at the University of California,
Davis (Romstad et al., 1995), the University of California, Los Angeles (Stewart et al., 2007),
and BYU (Rollins & Cole, 2006). The passive force-displacement relationship can be described
by the general hyperbolic form shown in Equation (2-6). A typical hyperbolic forcedisplacement curve is shown in Figure 2.4.
𝐹(𝑦) =
𝐶𝑦
1 + 𝐷𝑦
(2-6)
10
Figure 2.4: Hyperbolic Force-Displacement Curve (Shamsabadi et al., 2008)
The variables C and D are functions of the following soil parameters: average soil
stiffness (K), ultimate passive resistance (F ult ), maximum displacement (y max ), and the
displacement corresponding to half of the ultimate passive resistance (y ave ). For a granular
backfill, Shamsabadi et al. (2008) used 90.84 for C and 2.70 for D, and for a cohesive backfill,
45.42 was used for C and 1.35 for D. Equations (2-7), (2-8), and (2-9) show how the values of C
and D are calculated. Suggested HFD parameters for pressure, average soil stiffness, and
maximum displacement are shown in Table 2.4.
𝐶 = �2𝐾 −
𝐷 = 2�
𝐹𝑢𝑙𝑡
�
𝑦𝑚𝑎𝑥
(2-7)
𝐾
1
−
�
𝐹𝑢𝑙𝑡 𝑦𝑚𝑎𝑥
(2-8)
1
𝐹𝑢𝑙𝑡
𝐾=2
𝑦𝑎𝑣𝑒
(2-9)
11
Table 2.4: Suggested HFD Parameters for Abutment Backfills (Shamsabadi et al., 2007)
Shamsabadi et al. (2008) developed a height adjustment factor that can be applied to
Equation (2-6) to account for abutments in which the height is not 5.5 ft. The height adjustment
factor for granular materials is shown in Equation (2-10).
𝐻 1.5
𝑓𝑠 = �
�
5.5 𝑓𝑡
(2-10)
The height adjustment factor was applied to the modified HFD equation to compare with
test data on an abutment with a height of 3.67 ft at BYU (Rollins and Cole, 2006). Shamsabadi et
al. (2008) showed that the force-displacement curve calculated from the adjusted modified HFD
equation agreed reasonably well with the measured data from BYU.
12
3
TESTING
The measured data used for this project came from a variety of tests performed by Dr.
Kyle Rollins and several graduate students at BYU. Testing data from UCLA (Stewart et al.,
2007) is also included. Both full-scale and small-scale tests were performed. For the full-scale
tests at BYU, steel pipe piles were driven and reinforced pile caps were built on top of the piles.
The dimensions of the pile caps varied for each of the tests. For the small-scale tests, a 4-in. thick
concrete panel was pushed against compacted backfill material. In some of the tests the width of
the backfill was confined to the width of the pile cap in order to simulate wingwalls and
eliminate 3D end effects. For the UCLA test, a hydraulic actuator provided the vertical force to
keep the abutment from moving upward during lateral loading.
The tests were performed using different backfill materials at varying densities. The pile
caps were loaded laterally using hydraulic actuators to develop the passive force-displacement
curves. The passive resistance curve of each backfill material was calculated by subtracting the
force against the pile cap without backfill from the force against the pile cap with backfill.
3.1
Rollins & Cole (2006)
Rollins and Cole (2006) performed large-scale testing on four different backfill materials:
clean sand, silty sand, fine gravel, and coarse gravel. The height of the pile cap was 3.67 ft and
the width was 17 ft. The soil was not confined by wingwalls and extended 5 ft beyond the width
13
of the pile cap on each side to capture 3D end effects. The results of a mechanical sieve analysis
as well as the plasticity index of each backfill material are shown in Table 3.1.
Table 3.1: Summary of Backfill Soil Parameters (Rollins and Cole, 2006)
The specific gravity was 2.66, 2.68, 2.70, and 2.80 for the clean sand, silty sand, fine
gravel and coarse gravel, respectively. In-situ direct shear tests were performed on all of the
backfill materials except the clean sand, which was performed in the laboratory. The in-situ
moisture contents (w), dry unit weights (γ d ), relative densities (D r ), friction angles (φ), cohesion
(c), and interface friction ratio (δ/φ) are shown in Table 3.2.
Table 3.2: Unit Weights and Shear Strength Parameters for Backfill
Materials (Rollins and Cole, 2006)
14
3.2
Kwon (2007)
Kwon (2007) performed large-scale testing on two different backfill materials: loose silty
sand and dense fine gravel. The silty sand and fine gravel were the same material tested in
Rollins and Cole (2006). The height of the pile cap was 3.67 ft and the width was 17 ft. The soil
was not confined by wingwalls and extended 5 ft beyond the width of the pile cap on each side to
capture 3D end effects. The results of a mechanical sieve analysis as well as the plasticity index
of each backfill material are shown in Table 3.3.
Table 3.3: Summary of Backfill Soil Parameters (Kwon, 2007)
The specific gravity of the silty sand was measured to be 2.68, the average dry unit
weight was 99.9 pcf, and the average moisture content was 11.1%. The specific gravity of the
fine gravel was measured as 2.70, the average dry unit weight was 132.6 pcf, and the average
moisture content was 6.1%.
A laboratory direct shear test was performed on the silty sand. The friction angle was
determined to be 32.4° and the cohesion was 230 psf. Due to the difficulty of performing a direct
shear test on the fine gravel, no test was performed. However, a direct shear test was performed
on a similar material and the friction angle and cohesion of that material were determined to be
44° and 410 psf, respectively.
15
3.3
Valentine (2007)
Valentine (2007) performed large-scale testing on a dense silty sand. The height of the
pile cap was 3.67 ft and the width was 17 ft. The soil was not confined by wingwalls and
extended 5 ft beyond the width of the pile cap on each side to capture 3D end effects.
The silty sand used in Valentine (2007) was the same material used in Rollins and Cole
(2006). A mechanical sieve analysis determined the silty sand to be composed of 5.6% gravel,
53.6% sand, and 40.8% fines. The coefficient of uniformity was 14.8 and the coefficient of
gradation was 2.8. The average dry density and moisture content of the silty sand were 110.8 pcf
and 10.7%, respectively. Direct shear tests were performed both in the laboratory and in-situ.
The laboratory friction angle was 28.8° and the cohesion was 363 psf and the in-situ friction
angle was determined to be 29.1° with a cohesion of 148 psf.
3.4
Pruett (2009)
Pruett (2009) performed large-scale testing on two different backfill materials: fine gravel
and coarse gravel. Each material was tested in both a loose state and a dense state. The height of
the pile cap was 5.5 ft and the width was 11 ft. The soil was not confined by wingwalls and
extended 6 ft beyond the width of the pile cap on each side to capture 3D end effects.
The results of a mechanical sieve analysis on the fine gravel are displayed in Table 3.4.
The relative densities of the dense fine gravel and loose fine gravel were 74% and 35%,
respectively. These values were estimated from the relative compaction using the correlation
developed by Lee and Singh (1971). For the dense fine gravel, the average in-situ dry density
16
was 125.4 pcf and the average moisture content was 9.7%. For the loose fine gravel, the average
in-situ dry density was 114.6 pcf and the average moisture content was 6.6%.
Table 3.4: Gradation Properties of the Fine Gravel (Pruett, 2009)
Direct shear tests for the loose and the dense fine gravel were performed both in the
laboratory and in-situ. The results of this testing are shown in Table 3.5. Modified direct shear
testing was performed to determine the interface friction angle (δ) for the backfill material
against concrete. The interface friction angle for the dense fine gravel was determined to be 30.5
degrees, resulting in a δ/φ ratio of 0.61.
Table 3.5: Results of Direct Shear Testing for Fine Gravel (Pruett, 2009)
The results of a mechanical sieve analysis on the coarse gravel are displayed in Table 3.6.
The relative densities of the dense coarse gravel and loose coarse gravel were 82% and 48%,
respectively. These values were estimated from the relative compaction using the correlation
developed by Lee and Singh (1971). For the dense coarse gravel, the average in-situ dry density
17
was 135.0 pcf and the average moisture content was 2.9%. For the loose coarse gravel, the
average in-situ dry density was 125.4 pcf and the average moisture content was 1.9%.
Table 3.6: Gradation Properties for the Coarse Gravel (Pruett, 2009)
An in-situ direct shear test was performed for both the dense coarse gravel and the loose
coarse gravel. The results of this testing are shown in Table 3.7. Also displayed in Table 3.7 is
the friction angle estimated using a correlation with relative compaction as developed by Duncan
(2004). The same δ/φ ratio developed for the fine gravel (0.61) was used to determine the
interface friction angles of the coarse gravel. For the dense coarse gravel, the interface friction
angle was between 32° and 33°. For the loose coarse gravel, the interface friction angle was
30.5°.
Table 3.7: Results of Direct Shear Testing for Coarse Gravel (Pruett, 2009)
3.5
Cummins (2009)
Cummins (2009) performed large scale testing on loose clean sand. The height of the pile
cap was 5.5 ft and the width was 11 ft. The backfill soil was not confined by wingwalls and
extended 2.4 ft beyond the width of the pile cap on each side to capture 3D end effects.
18
From a mechanical sieve analysis, the clean sand was determined to contain 6% gravel,
92% sand, and 2% fines. The coefficients of uniformity and curvature were 8.7 and 1.2,
respectively. The average dry density of the sand was determined to be 98.6 pcf and the moisture
content was 8%. Using the Lee and Singh (1971) correlation, the relative density was estimated
as 44 %. Laboratory direct shear tests were performed on the sand to determine the soil friction
angle, wall friction angle, and cohesion. These values are shown in Table 3.8.
Table 3.8: Direct Shear Results for Loose Clean Sand (Cummins, 2009)
3.6
Nasr (2010)
Nasr (2010) used results from previous testing to determine the effects on the total
mobilized passive resistance of a backfill due to plane strain stress effects and 3D geometric end
effects. The data was taken from research performed by Kwon (2007), Pruett (2009), and Cummins
(2009). Nasr (2010) back-calculated soil parameters from the passive force-displacement curves
created using the PYCAP and ABUTMENT programs.
3.7
Heiner (2010)
Heiner (2010) performed large-scale testing on a dense clean sand. The height of the pile
cap was 5.5 ft and the width was 11 ft. The soil was not confined by wingwalls and extended 5.5
ft beyond the width of the pile cap on each side to capture 3D end effects.
19
The clean sand used in Heiner (2010) contained less than 5% fines and could be generally
classified as concrete sand. The grain size distribution curve is shown in Figure 3.1. The average
dry density and moisture content of the clean sand were 106.5 pcf and 9.0%, respectively. The
minimum void ratio was 0.48 and the maximum void ratio was 0.96. The relative density of the
clean sand was calculated to be 80% using the minimum and maximum void ratios. Several
direct shear tests were performed in the laboratory. The friction angle of the dense clean sand
was determined to be 40.5° with a cohesion value of zero.
Figure 3.1: Grain Size Distribution for Clean Sand (Heiner, 2010)
3.8
Strassburg (2010)
Strassburg (2010) performed large-scale testing on a loose well-graded sand. The height
of the pile cap was 5.5 ft and the width was 11 ft. Two tests were performed on the loose sand:
an unconfined test in which the soil extended 5.5 ft beyond the width of the pile cap on each side
to capture 3D end effects and a confined test with slip planes to simulate plane strain conditions.
Table 3.9 contains the results of a mechanical sieve analysis performed on the sand. The
in-situ dry unit weights for the loose sand unconfined (3D) and loose sand slip plane (2D) were
20
107.0 pcf, and 109.5 pcf, respectively. The average moisture content of the loose sand
unconfined (3D) was 5.6% and the loose sand slip plane (2D) was 7.9%.
Table 3.9: Gradation Parameters for Loose Well-Graded Sand (Strassburg, 2010)
D60
D50
D30
D10
Cu
Cc
Gravel Sand Fines
Backfill Soil
%
%
%
in.
in.
in.
in.
Well-graded Sand
12
87
1
0.056 0.042 0.021 0.006 8.91
1.26
Direct shear and triaxial tests were performed in the laboratory. The friction angle of the
sand was determined to be 36.0° and there was no cohesion. Using the Lee and Singh (1971)
correlation, the relative densities of the loose sand were calculated. For the loose sand
unconfined (3D) the relative density was 31% and for the loose sand slip plane (2D) it was 41%.
3.9
Bingham (2012)
Bingham (2012) performed large-scale testing on a dense well-graded sand. The same
sand was used for testing performed by Strassburg (2010). The height of the pile cap was 5.5 ft
and the width was 11 ft. Two tests were performed on the dense sand: an unconfined test in
which the soil extended 6 ft beyond the width of the pile cap on each side to capture 3D end
effects and a confined test with slip planes to simulate plane strain conditions.
Figure 3.2 shows an average particle-size distribution curve obtained from multiple
mechanical sieve analyses performed during testing. Table 3.9 shows some results from the
mechanical sieve analysis. The in-situ dry unit weights for the dense sand unconfined (3D) and
dense sand slip plane (2D) were 120.3 pcf, and 118.9 pcf, respectively. The average moisture
content of the dense sand unconfined (3D) was 7.7% and for the dense sand slip plane (2D) the
average moisture content was 8.8%.
21
100%
90%
Percent Finer
80%
70%
60%
50%
40%
30%
20%
10%
0%
100
10
1
0.1
0.01
Particle diameter (mm)
Figure 3.2: The Particle-Size Distribution for a Clean Sand. (Bingham, 2012)
Direct shear and triaxial shear tests were performed in the laboratory. The friction angle
of the sand was determined to be 43.1° and the corresponding cohesion was 823 psf. Using the
Lee and Singh (1971) correlation, the relative densities of the dense sand were calculated. For
the dense sand unconfined (3D) the relative density was 84% and for the dense sand slip plane
(2D) it was 79%.
3.10 Jessee (2012)
Jessee (2012) performed small-scale testing on a dense clean sand. A 4 in. thick concrete
panel, with a height of 2 ft and width of 4.125 ft, was pushed against the compacted backfill
material. The width of the backfill was limited to the width of the concrete panel using slip
planes. The test was performed three times with the same backfill conditions.
The sand classified as a clean poorly-graded material. The coefficients of uniformity and
curvature were 3.7 and 0.7, respectively. The average dry unit weight for the clean sand was
22
determined to be 111 pcf and the average moisture content during testing was 8.0%. Direct shear
tests were performed in the lab to determine the shear strength parameters of the clean sand. The
drained friction angle (φ) was found to be 49° and the cohesion was 90 psf. The interface friction
angle (δ) between the sand and concrete was measured to be 33°. This provided an interface
friction angle to soil friction angle ratio (δ/φ) of 0.68. Other soil parameters for the dense clean
sand can be found in Table 3.10.
Table 3.10: Soil Properties for Dense Clean Sand (Jessee, 2012)
Backfill Soil Properties
USCS Classification
Cu
SP
3.7
Cc
0.7
Gs
e
ϕ (°)
δ (°)
2.65
0.49
49.0
33.2
Modified Proctor
γd(max)
115.4
wopt
16.0
Avg. γd
111.0
Avg. w (%)
(during compaction)
Avg. w (%)
(during testing)
Avg. S (%)
Avg. ψ (kPa)
Avg. ca (kPa)
11.3
8.0
43.5
9.6
3.8
3.11 UCLA (Stewart et al., 2007)
The UCLA testing was performed on a seat-type abutment with a height of 5.5 ft and a
width of 15 ft. Plywood was placed on each side of the abutment to simulate wingwalls and
confine the backfill width to 16 ft. The backfill material was a well-graded sand with silt. The
fines content was 10% and the D 50 ranged from 0.7-0.85 mm.
23
Sand cone testing was used to determine the unit weight and moisture content of the sand.
The dry unit weight was 118.3 pcf and the median moisture content was 6.5%. A specific gravity
of 2.7 was assumed to calculate the median in-situ void ratio as 0.38 and from this value the
relative density was calculated as 92%. A triaxial test determined the friction angle to be 40° and
the cohesion range from 300-500 psf. The interface friction angle between the soil and the
relatively smooth concrete wall was determined to be 14°.
24
4
RESULTS
Testing was performed on 11 categories of backfill material, each with varying soil types
and configurations. For each soil, the measured passive force-displacement curve was used to
back-calculate parameters that could be used in common approaches for developing the passive
force-displacement relationship. The common approaches include PYCAP, ABUTMENT, and
the modified hyperbolic force-displacement equation. For each test on a dense material, the
measured passive force-displacement curve is compared with the bi-linear force-displacement
model formulated using the Caltrans Seismic Design approach. For the cases in which end
effects were accounted for (3D), the effective width computed using the Brinch-Hansen equation
was used in the ABUTMENT program. For the modified HFD equation, the maximum
displacement was assumed to be 5% of the pile cap height.
Many of the parameters used herein were determined previously or discussed in Chapter
3. ABUTMENT parameters for the Rollins & Cole (2006) testing were taken from Shamsabadi
(2007). Parameters for PYCAP and ABUTMENT on the Kwon (2007) testing were taken from
Nasr (2010).
4.1
Loose Clean Sand (2D)
Strassburg (2010) performed testing on loose clean sand confined to the width of the wall
by slip planes in order to negate 3D end effects. The measured results of this testing was fit to
25
curves from typical methods for determining the passive force-displacement relationship. The
force-displacement curves are found in Figure 4.1.
350
300
Force (kips)
250
200
150
Measured
100
PYCAP
ABUTMENT
50
Modified HFD
0
0
0.5
1
1.5
2
Displacement (in.)
2.5
3
3.5
Figure 4.1: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Loose Clean Sand (2D) (Strassburg, 2010).
The back-calculated parameters for PYCAP, ABUTMENT, and the modified HFD
equation are summarized in Table 4.1, Table 4.2, and Table 4.3, respectively.
Table 4.1: PYCAP Parameters for Loose Clean Sand (2D)
Strassburg
Parameter
Symbol
Unit
(2010)
Cap width
b
11
ft
Cap height
H
5.5
ft
Cohesion
c
60
psf
Soil friction angle
φ
42.0
degrees
Wall friction angle
Initial soil modulus
δ
Ei
31.5
400
degrees
ksf
Poisson's ratio
Moist unit weight
υ
γm
0.25
118.2
pcf
26
Table 4.2: ABUTMENT Parameters for Loose Clean Sand (2D)
Strassburg
Parameter
Symbol
Unit
(2010)
Cap width
b
11
ft
Cap height
H
5.5
ft
Soil cohesion
c
0.13
ksf
Soil friction angle
φ
42.0
degrees
Wall friction angle
δ
31.5
degrees
Soil density
Strain at 50% of ultimate strength
γ
ε50
0.1182
0.01
kcf
-
Poisson's ratio
Failure ratio
υ
Rf
0.25
0.9
-
Table 4.3: Modified HFD parameters for Loose Clean Sand (2D)
Strassburg
Parameter
Symbol
Unit
(2010)
Ultimate passive resistance/effective width
Fult /beff
30
kip/ft
Maximum displacement/Height
Displacement at half of the ultimate passive resistance
4.2
ymax /H
0.05
in/in
yave
0.8
in
Loose Clean Sand (3D)
Cummins (2009) and Strassburg (2010) performed testing on unconfined loose clean
sand. The measured results of these tests were fit to curves from typical methods for determining
the passive force-displacement relationship. The force-displacement curves are shown in Figure
4.2 and Figure 4.3.
27
100
Force (kips)
80
60
Measured
PYCAP
ABUTMENT
Modified HFD
40
20
0
0
0.5
1
1.5
Displacement (in.)
2
2.5
Figure 4.2: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Loose Clean Sand (3D) (Cummins, 2009).
350
300
Force (kips)
250
200
150
Measured
100
PYCAP
ABUTMENT
50
Modified HFD
0
0
1
2
Displacement (in.)
3
4
Figure 4.3: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Loose Clean Sand (3D) (Strassburg, 2010).
28
The back-calculated parameters for PYCAP, ABUTMENT, and the modified HFD
equation are summarized in Table 4.4, Table 4.5, and Table 4.6, respectively.
Table 4.4: PYCAP Parameters for Loose Clean Sand (3D)
Cummins Strassburg
Parameter
Symbol
Unit
(2009)
(2010)
Cap width
b
11
11
ft
Cap height
H
5.5
5.5
ft
Cohesion
c
10
60
psf
Soil friction angle
φ
27.7
36.0
degrees
Wall friction angle
Initial soil modulus
δ
Ei
20.8
200.5
27
210
degrees
ksf
Poisson's ratio
Moist unit weight
υ
γm
0.35
110
0.25
113
pcf
Table 4.5: ABUTMENT Parameters for Loose Clean Sand (3D)
Cummins Strassburg
Parameter
Symbol (2009)
Unit
(2010)
Cap width
b
11
11
ft
Cap height
H
5.5
5.5
ft
Soil cohesion
c
0.01
0.01
ksf
Soil friction angle
φ
27.7
36.0
degrees
Wall friction angle
δ
20.8
27
degrees
Soil density
γ
0.11
0.113
kcf
ε50
0.003
0.006
Strain at 50% of ultimate strength
Poisson's ratio
υ
0.35
0.25
0.97
0.94
Rf
Failure ratio
Table 4.6: Modified HFD Parameters for Loose Clean Sand (3D)
Cummins Strassburg
Parameter
Symbol
(2010)
(2009)
Ultimate passive resistance/effective width
Fult /beff
8.7
16.3
Unit
kip/ft
Maximum displacement/Height
Displacement at half of the ultimate passive resistance
29
ymax /H
0.05
0.05
in/in
yave
1.0
0.8
in
4.3
Dense Clean Sand (2D)
Bingham (2012) and Jessee (2012) performed testing on dense clean sand confined to the
width of the wall by slip planes in order to negate 3D end effects. Jessee (2012) performed three
different tests on small-scale abutments. The measured results of both the Bingham (2012) and
Jessee (2012) tests were fit to curves from typical methods for determining the passive forcedisplacement relationship. The passive force-displacement curves are shown in Figure 4.4,
Figure 4.5, Figure 4.6, and Figure 4.7.
800
700
Force (kips)
600
500
400
300
200
100
Measured
ABUTMENT
Caltrans
PYCAP
Modified HFD
1.5
2
2.5
Displacement (in.)
3
0
0
0.5
1
3.5
Figure 4.4: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (2D) (Bingham, 2012).
30
Force (kips)
50
45
40
35
30
25
20
15
10
5
0
Measured
ABUTMENT
Caltrans
0
0.5
1
Displacement (in.)
PYCAP
Modified HFD
1.5
2
Force (kips)
Figure 4.5: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (2D) (Jessee #1, 2012).
50
45
40
35
30
25
20
15
10
5
0
Measured
ABUTMENT
Caltrans
0
0.5
1
Displacement (in.)
PYCAP
Modified HFD
1.5
2
Figure 4.6: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (2D) (Jessee #2, 2012)
31
50
45
40
Force (kips)
35
30
25
20
15
Measured
ABUTMENT
Caltrans
10
5
PYCAP
Modified HFD
0
0
0.5
1
1.5
Displacement (in.)
2
2.5
Figure 4.7: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (2D) (Jessee #3, 2012)
These results show that the Caltrans method underestimates the passive forcedisplacement relationship for dense sands confined by slip planes. The back-calculated
parameters for PYCAP, ABUTMENT, and the modified HFD equation are summarized in Table
4.7, Table 4.8, and Table 4.9, respectively. The same parameters were used for all three Jessee
(2012) tests except in the modified HFD method.
32
Table 4.7: PYCAP Parameters for Dense Clean Sand (2D)
Bingham Jessee
Parameter
Symbol (2012)
Unit
(2012)
Cap width
b
11
4.125
ft
Cap height
H
5.5
2
ft
Cohesion
c
60
90
psf
Soil friction angle
φ
48.3
50.0
degrees
Wall friction angle
Initial soil modulus
δ
Ei
34.8
680
33.5
1000
degrees
ksf
Poisson's ratio
Moist unit weight
υ
γm
0.23
129.3
0.20
120.0
pcf
Table 4.8: ABUTMENT Parameters for Dense Clean Sand (2D)
Bingham Jessee
Unit
Parameter
Symbol (2012)
(2012)
Cap width
b
11
4.125
ft
Cap height
H
5.5
2
ft
Soil cohesion
c
0.06
0.13
ksf
Soil friction angle
φ
49.9
50.0
degrees
Wall friction angle
δ
35.9
33.5
degrees
Soil density
γ
0.1293
0.12
kcf
ε50
0.008
0.004
Strain at 50% of ultimate strength
Poisson's ratio
υ
0.23
0.20
Rf
0.92
0.95
Failure ratio
33
Table 4.9: Modified HFD Parameters for Dense Clean Sand (2D)
Parameter
Ultimate passive resistance/effective width
Bingham Jessee
Jessee
Jessee
Symbol (2012) (2012) #1 (2012) #2 (2012) #3 Unit
Fult /beff
58.9
50.7
48.5
47.8
kip/ft
Maximum displacement/Height
ymax /H
0.05
0.05
0.05
0.05
in/in
yave
0.68
0.20
0.14
0.15
in
Displacement at half of the ultimate passive resistance
4.4
Dense Clean Sand (3D)
Rollins & Cole (2006), Heiner (2010), and Bingham (2012) performed testing on
unconfined dense clean sand. The measured results of these tests were fit to curves from typical
methods for determining the passive force-displacement relationship. The passive forcedisplacement curves are shown in Figure 4.8, Figure 4.9, and Figure 4.10.
300
Force (kips)
250
200
150
Measured
PYCAP
ABUTMENT
Modified HFD
Caltrans
100
50
0
0
0.5
1
1.5
2
Displacement (in.)
2.5
3
Figure 4.8: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (3D) (Rollins & Cole, 2006).
34
600
Force (kips)
500
400
300
Measured
200
PYCAP
ABUTMENT
Modified HFD
100
Caltrans
0
0
0.5
1
1.5
2
Displacement (in.)
2.5
3
Figure 4.9: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (3D) (Heiner, 2010).
900
800
Force (kips)
700
600
500
400
Measured
300
PYCAP
ABUTMENT
200
Modified HFD
100
Caltrans
0
0
0.5
1
1.5
2
Displacement (in.)
2.5
3
Figure 4.10: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Clean Sand (3D) (Bingham, 2012).
35
The back-calculated parameters for PYCAP, ABUTMENT, and the modified HFD
equation are summarized in Table 4.10, Table 4.11, and Table 4.12, respectively. For Rollins &
Cole (2006) and Heiner (2010), the effective widths were used in the ABUTMENT program.
The Brinch-Hansen 3D factor for Rollins & Cole (2006) was 1.360 and for Heiner (2010) the 3D
factor was 1.636.
Table 4.10: PYCAP Parameters for Dense Clean Sand (3D)
Rollins & Heiner Bingham
Parameter
Symbol Cole (2006) (2010)
Unit
(2012)
Cap width
b
17
11
11
ft
Cap height
H
3.67
5.5
5.5
ft
Cohesion
c
0
0
60
psf
Soil friction angle
φ
40.7
40.5
43.5
degrees
Wall friction angle
Initial soil modulus
δ
Ei
31.3
775
31.2
800
31.3
450
degrees
ksf
Poisson's ratio
Moist unit weight
υ
γm
0.26
116.8
0.26
105
0.23
129.5
pcf
Table 4.11: ABUTMENT Parameters for Dense Clean Sand (3D)
Rollins & Heiner Bingham
Parameter
Symbol Cole (2006) (2010)
Unit
(2012)
Cap width
b
23.1
18
11
ft
Cap height
H
3.67
5.5
5.5
ft
Soil cohesion
c
0.08
0.1
0
ksf
Soil friction angle
φ
39.3
41.5
43.3
degrees
Wall friction angle
δ
30.3
32
31.2
degrees
Soil density
γ
0.1168
0.1161
0.1295
kcf
ε50
0.002
0.0065
0.007
Strain at 50% of ultimate strength
Poisson's ratio
υ
0.27
0.29
0.23
Rf
0.98
0.91
0.92
Failure ratio
36
Table 4.12: Modified HFD Parameters for Dense Clean Sand (3D)
Rollins & Heiner Bingham
Parameter
Symbol Cole (2006) (2010) (2012) Unit
Fult /beff
Ultimate passive resistance/effective width
19.5
27.0
37.7
kip/ft
Maximum displacement/Height
Displacement at half of the ultimate passive resistance
4.5
ymax /H
0.05
0.05
0.05
in/in
yave
0.16
0.58
0.85
in
Loose Silty Sand (3D)
Kwon (2007) performed testing on unconfined loose silty sand. The measured results of
these tests were fit to curves from typical methods for determining the passive forcedisplacement relationship. The force-displacement curves are shown in Figure 4.11.
100
Force (kips)
80
60
Measured
40
PYCAP
ABUTMENT
20
Modified HFD
0
0
0.5
1
1.5
Displacement (in.)
2
2.5
Figure 4.11: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Loose Silty Sand (3D) (Kwon, 2007).
37
The back-calculated parameters for PYCAP, ABUTMENT, and the modified HFD
equation are summarized in Table 4.13, Table 4.14, and Table 4.15, respectively. The BrinchHansen correction factor used for this material was 1.177.
Table 4.13: PYCAP Parameters for Loose Silty Sand (3D)
Kwon
Parameter
Symbol
Unit
(2007)
Cap width
b
17
ft
Cap height
H
3.67
ft
Cohesion
c
60
psf
Soil friction angle
φ
27.7
degrees
Wall friction angle
Initial soil modulus
δ
Ei
20.8
150
degrees
ksf
Poisson's ratio
Moist unit weight
υ
γm
0.35
110
pcf
Table 4.14: ABUTMENT Parameters for Loose Silty Sand (3D)
Kwon
Parameter
Symbol (2007)
Unit
Cap width
b
20.0
ft
Cap height
H
3.67
ft
Soil cohesion
c
0.06
ksf
Soil friction angle
φ
27.7
degrees
Wall friction angle
δ
20.8
degrees
Soil density
γ
0.11
kcf
0.003
ε50
Strain at 50% of ultimate strength
Poisson's ratio
υ
0.35
Rf
0.97
Failure ratio
38
Table 4.15: Modified HFD Parameters for Loose Silty Sand (3D)
Kwon
Parameter
Symbol
Unit
(2007)
Ultimate passive resistance/effective width
Fult /beff
8.3
kip/ft
Maximum displacement/Height
Displacement at half of the ultimate passive resistance
4.6
ymax /H
0.05
in/in
yave
0.7
in
Dense Silty Sand (2D)
UCLA (Stewart et al., 2007) performed testing on confined dense silty sand. The
measured results of this testing were fit to curves from typical methods for determining the
passive force-displacement relationship. The force-displacement curves are shown in Figure 4.12
and Figure 4.13. Two scenarios were used in comparing the measured data with the common
methods for the UCLA dense silty sand (2D): the triaxial friction angle and the plane strain
friction angle. According to Kulhawy and Mayne (1990), the plane strain friction angle is about
10/9 of the triaxial friction angle. The use of the plane strain friction angle allowed for lower
values of cohesion to be used.
39
600
Force (kips)
500
400
300
Measured
PYCAP
200
ABUTMENT
Modified HFD
100
Caltrans
0
0
1
2
3
4
Displacement (in.)
5
6
Figure 4.12: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Silty Sand (2D) (Stewart et al., 2007)
600
Force (kips)
500
400
300
Measured
200
PYCAP
ABUTMENT
100
Modified HFD
Caltrans
0
0
1
2
3
4
Displacement (in.)
5
6
Figure 4.13: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Silty Sand (2D) using the Plane Strain
Friction Angle (Stewart et al., 2007)
40
The back-calculated parameters for PYCAP, ABUTMENT, and the modified HFD
equation are summarized in Table 4.16, Table 4.17, and Table 4.18, respectively. The use of the
plane strain friction angle allowed for the cohesion to be reduced in both PYCAP and
ABUTMENT.
Table 4.16: PYCAP Parameters for Dense Silty Sand (2D)
UCLA (Stewart UCLA Plane
Parameter
Symbol et al., 2007)
Unit
Strain
Cap width
b
15
15
ft
Cap height
H
5.5
5.5
ft
Cohesion
c
380
190
psf
Soil friction angle
φ
40.0
44.4
degrees
Wall friction angle
Initial soil modulus
δ
Ei
14.0
1000
15.6
1000
degrees
ksf
Poisson's ratio
Moist unit weight
υ
γm
0.26
126.0
0.23
126.0
pcf
Table 4.17: ABUTMENT Parameters for Dense Silty Sand (2D)
UCLA (Stewart UCLA Plane
Parameter
Symbol et al., 2007)
Unit
Strain
Cap width
b
15
15
ft
Cap height
H
5.5
5.5
ft
Soil cohesion
c
0.3
0.05
ksf
Soil friction angle
φ
39
44.4
degrees
Wall friction angle
δ
29.25
33.3
degrees
Soil density
γ
0.126
0.126
kcf
ε50
0.0036
0.0036
Strain at 50% of ultimate strength
Poisson's ratio
υ
0.27
0.27
0.98
0.98
Rf
Failure ratio
41
Table 4.18: Modified HFD Parameters for Dense Silty Sand (2D)
UCLA (Stewart
Parameter
Symbol et al., 2007) Unit
Ultimate passive resistance/effective width
Fult /beff
30.3
kip/ft
Maximum displacement/Height
Displacement at half of the ultimate passive resistance
4.7
ymax /H
0.05
in/in
yave
0.30
in
Dense Silty Sand (3D)
Rollins & Cole (2006) and Valentine (2007) performed testing on unconfined dense silty
sand. The measured results of these tests were fit to curves from typical methods for determining
the passive force-displacement relationship. The passive force-displacement curves are shown in
Figure 4.14 and Figure 4.15.
350
300
Force (kips)
250
200
Measured
150
PYCAP
100
ABUTMENT
Modified HFD
50
Caltrans
0
0
0.5
1
1.5
2
Displacement (in.)
2.5
3
Figure 4.14: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Silty Sand (3D) (Rollins & Cole, 2006)
42
350
300
Force (kips)
250
200
Measured
150
PYCAP
100
ABUTMENT
Modified HFD
50
Caltrans
0
0
0.5
1
1.5
2
Displacement (in.)
2.5
3
Figure 4.15: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Silty Sand (3D) (Valentine, 2007)
The back-calculated parameters for PYCAP, ABUTMENT, and the modified HFD
equation are summarized in Table 4.19, Table 4.20, and Table 4.21, respectively. For Rollins &
Cole (2006) and Valentine (2007), the effective widths were used in the ABUTMENT program.
The Brinch-Hansen 3D factor for Rollins & Cole (2006) was 1.187 and for Valentine (2007) the
3D factor was 1.271.
43
Table 4.19: PYCAP Parameters for Dense Silty Sand (3D)
Rollins & Valentine
Parameter
Symbol Cole (2006) (2007)
Unit
Cap width
b
17
17
ft
Cap height
H
3.67
3.67
ft
Cohesion
c
570.2
650
psf
Soil friction angle
φ
27.9
30
degrees
Wall friction angle
Initial soil modulus
δ
Ei
20.9
800
26.3
800
degrees
ksf
Poisson's ratio
Moist unit weight
υ
γm
0.35
120.9
0.30
122.6
pcf
Table 4.20: ABUTMENT Parameters for Dense Silty Sand (3D)
Rollins & Valentine
Parameter
Symbol Cole (2006) (2007)
Unit
Cap width
b
20.4
21.6
ft
Cap height
H
3.67
3.67
ft
Soil cohesion
c
0.647
0.5
ksf
Soil friction angle
φ
27
30.5
degrees
Wall friction angle
δ
21
22.9
degrees
Soil density
γ
0.1209
0.1226
kcf
ε50
0.003
0.003
Strain at 50% of ultimate strength
Poisson's ratio
υ
0.35
0.33
Rf
0.97
0.97
Failure ratio
44
Table 4.21: Modified HFD Parameters for Dense Silty Sand (3D)
Rollins & Valentine
Parameter
Symbol Cole (2006) (2007) Unit
Ultimate passive resistance/effective width
Fult /beff
28.5
27.5
kip/ft
Maximum displacement/Height
Displacement at half of the ultimate passive resistance
4.8
ymax /H
0.05
0.05
in/in
yave
0.19
0.3
in
Loose Fine Gravel (3D)
Pruett (2009) performed large scale testing on unconfined loose fine gravel. The
measured results of this testing were fit to curves from typical methods for determining the
passive force-displacement relationship. The passive force-displacement curves are shown in
Figure 4.16.
200
180
160
Force (kips)
140
120
100
80
Measured
60
PYCAP
40
ABUTMENT
20
Modified HFD
0
0
0.5
1
1.5
2
Displacement (in.)
2.5
3
3.5
Figure 4.16: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Loose Fine Gravel (3D) (Pruett, 2009)
45
The parameters for PYCAP and ABUTMENT were taken from Pruett (2009). The backcalculated parameters for PYCAP, ABUTMENT, and the modified HFD equation are
summarized in Table 4.22, Table 4.23, and Table 4.24, respectively. For PYCAP the BrinchHansen 3D factor used was 1.54 and for ABUTMENT program, the 3D factor was 1.48.
Table 4.22: PYCAP Parameters for Loose Fine Gravel (3D)
Pruett
(2009)
Parameter
Symbol
Unit
Cap width
b
11
ft
Cap height
H
5.5
ft
Cohesion
c
0
psf
Soil friction angle
φ
31.0
degrees
Wall friction angle
Initial soil modulus
δ
Ei
31.0
491
degrees
ksf
Poisson's ratio
Moist unit weight
υ
γm
0.30
122.1
pcf
Table 4.23: ABUTMENT Parameters for Loose Fine Gravel (3D)
Pruett
(2009)
Parameter
Symbol
Unit
Cap width
b
16.28
ft
Cap height
H
5.5
ft
Soil cohesion
c
0
ksf
Soil friction angle
φ
33.0
degrees
Wall friction angle
δ
25.0
degrees
Soil density
Strain at 50% of ultimate strength
γ
ε50
0.1221
0.005
kcf
-
Poisson's ratio
Failure ratio
υ
Rf
0.30
0.94
-
46
Table 4.24: Modified HFD Parameters for Loose Fine Gravel (3D)
Pruett
Unit
Parameter
Symbol
(2009)
11.5
kip/ft
Ultimate passive resistance/effective width
Fult /beff
Maximum displacement/Height
Displacement at half of the ultimate passive resistance
4.9
ymax /H
0.05
in/in
yave
1.3
in
Dense Fine Gravel (3D)
Rollins & Cole (2006), Kwon (2007), and Pruett (2009) performed testing on unconfined
dense fine gravel. The measured results of these tests were fit to curves from typical methods for
determining the passive force-displacement relationship. The passive force-displacement curves
are shown in Figure 4.17, Figure 4.19, and Figure 4.19.
300
Force (kips)
250
200
150
Measured
PYCAP
ABUTMENT
Modified HFD
Caltrans
100
50
0
0
0.2
0.4
0.6
0.8
1
Displacement (in.)
1.2
1.4
1.6
Figure 4.17: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Fine Gravel (3D) (Rollins & Cole, 2006)
47
350
300
Force (kips)
250
200
Measured
150
PYCAP
100
ABUTMENT
Modified HFD
50
Caltrans
0
0
0.5
1
1.5
Displacement (in.)
2
2.5
Figure 4.18: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Fine Gravel (3D) (Kwon, 2007)
700
600
Force (kips)
500
400
300
Measured
PYCAP
200
ABUTMENT
Modified HFD
100
Caltrans
0
0
0.5
1
1.5
2
2.5
Displacement (in.)
3
3.5
Figure 4.19: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Fine Gravel (3D) (Pruett, 2009)
48
4
The back-calculated parameters for PYCAP, ABUTMENT, and the modified HFD
equation are summarized in Table 4.25, Table 4.26, and Table 4.27, respectively. For Rollins &
Cole (2006) and Pruett (2009), the effective widths were used in the ABUTMENT program. The
Brinch-Hansen 3D factor for Rollins & Cole (2006) was 1.260, for Kwon (2007) the 3D factor
was 1.403, and for Pruett (2009) the 3D factor was 1.950. The cohesion in PYCAP for Pruett
(2009) was changed from 84 psf to 0 psf to attain better agreement with the measured data.
Table 4.25: PYCAP Parameters for Dense Fine Gravel (3D)
Rollins &
Kwon
Pruett
Parameter
Symbol Cole (2006) (2007)
Unit
(2009)
Cap width
b
17
17
11
ft
Cap height
H
3.67
3.67
5.5
ft
Cohesion
c
79.4
0
0*
psf
Soil friction angle
φ
34.0
41.0
44.0
degrees
Wall friction angle
Initial soil modulus
δ
Ei
25.5
706
30.75
600
27
670
degrees
ksf
Poisson's ratio
Moist unit weight
υ
γm
0.31
132.2
0.26
141.0
0.30
137.8
pcf
*Pruett (2009) used c = 84 psf
Table 4.26: ABUTMENT Parameters for Dense Fine Gravel (3D)
Rollins &
Pruett
Kwon
Parameter
Symbol Cole (2006)
Unit
(2009)
(2007)
Cap width
b
21.4
24.4
21.5
ft
Cap height
H
3.67
3.67
5.5
ft
Soil cohesion
c
0.084
0.015
0.084
ksf
Soil friction angle
φ
34.0
42.0
44.0
degrees
Wall friction angle
δ
26.0
31.5
27.0
degrees
Soil density
γ
0.1322
0.141
0.1378
kcf
ε50
0.0015
0.004
0.004
Strain at 50% of ultimate strength
Poisson's ratio
υ
0.31
0.25
0.30
0.98
0.95
0.98
Rf
Failure ratio
49
Table 4.27: Modified HFD Parameters for Dense Fine Gravel (3D)
Rollins & Kwon Pruett
Parameter
Symbol Cole (2006) (2007) (2009)
Ultimate passive resistance/effective width
Fult /beff
15.3
23.1
31.0
Maximum displacement/Height
ymax /H
0.05
0.05
0.05
Displacement at half of the ultimate passive resistance
yave
0.095
0.24
0.66
Unit
kip/ft
in/in
in
4.10 Loose Coarse Gravel (3D)
Pruett (2009) performed large scale testing on unconfined loose coarse gravel. The
measured results of this test were fit to curves from typical methods for determining the passive
force-displacement relationship. The passive force-displacement curves are shown in Figure
4.20. The parameters for PYCAP and ABUTMENT were taken from Pruett (2009).
350
300
Force (kips)
250
200
150
Measured
PYCAP
100
ABUTMENT
Modified HFD
50
0
0
0.5
1
1.5
2
2.5
Displacement (in.)
3
3.5
4
Figure 4.20: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Loose Coarse Gravel (3D) (Pruett, 2009)
50
The back-calculated parameters for PYCAP, ABUTMENT, and the modified HFD
equation are summarized in Table 4.28, Table 4.29, and Table 4.30, respectively. For both the
ABUTMENT program and PYCAP, the Brinch-Hansen 3D factor used was 1.72.
Table 4.28: PYCAP Parameters for Loose Coarse Gravel (3D)
Pruett
Parameter
Symbol
Unit
(2009)
Cap width
b
11
ft
Cap height
H
5.5
ft
Cohesion
c
0
psf
Soil friction angle
φ
37.4
degrees
Wall friction angle
Initial soil modulus
δ
Ei
28.1
200
degrees
ksf
Poisson's ratio
Moist unit weight
υ
γm
0.28
128.7
pcf
Table 4.29: ABUTMENT Parameters for Loose Coarse Gravel (3D)
Pruett
Parameter
Symbol
Unit
(2009)
Cap width
b
18.6
ft
Cap height
H
5.5
ft
Soil cohesion
c
0
ksf
Soil friction angle
φ
40.0
degrees
Wall friction angle
δ
24
degrees
Soil density
Strain at 50% of ultimate strength
γ
ε50
0.1278
0.0074
kcf
-
Poisson's ratio
Failure ratio
υ
Rf
0.30
0.98
-
51
Table 4.30: Modified HFD Parameters for Loose Coarse Gravel (3D)
Pruett
Parameter
Symbol
Unit
(2009)
Fult /beff
Ultimate passive resistance/effective width
16.4
kip/ft
Maximum displacement/Height
Displacement at half of the ultimate passive resistance
ymax /H
0.05
in/in
yave
1.08
in
4.11 Dense Coarse Gravel (3D)
Rollins & Cole (2006) and Pruett (2009) performed testing on unconfined dense coarse
gravel. The measured results of these tests were fit to curves from typical methods for
determining the passive force-displacement relationship. The passive force-displacement curves
are shown in Figure 4.21 and Figure 4.22.
500
450
400
Force (kips)
350
300
250
200
Measured
PYCAP
ABUTMENT
Modified HFD
Caltrans
150
100
50
0
0
0.5
1
1.5
2
Displacement (in.)
2.5
3
Figure 4.21: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Coarse Gravel (3D) (Rollins & Cole, 2006)
52
900
800
700
Force (kips)
600
500
400
Measured
PYCAP
ABUTMENT
Modified HFD
Caltrans
300
200
100
0
0
0.5
1
1.5
2
Displacement (in.)
2.5
3
3.5
Figure 4.22: Comparison of Measured Passive Force-Deflection Curves with Curves
Computed using Various Methods for Dense Coarse Gravel (3D) (Pruett, 2009)
The back-calculated parameters for PYCAP, ABUTMENT, and the modified HFD
equation are summarized in Table 4.31, Table 4.32, and Table 4.33, respectively. For each of the
coarse gravel tests, effective widths were used for the ABUTMENT program using the BrinchHansen 3D factor. For Rollins & Cole (2006) the 3D factor was 1.4 and for Pruett (2009) the 3D
factor was 1.89.
53
Table 4.31: PYCAP Parameters for Dense Coarse Gravel (3D)
Rollins &
Pruett
Parameter
Symbol Cole (2006) (2009)
Unit
Cap width
b
17
11
ft
Cap height
H
3.67
5.5
ft
Cohesion
c
150.4
286
psf
Soil friction angle
φ
41.3
41
degrees
Wall friction angle
Initial soil modulus
δ
Ei
31.0
600
26.65
830
degrees
ksf
Poisson's ratio
Moist unit weight
υ
γm
0.25
147.2
0.30
138.4
pcf
Table 4.32: ABUTMENT Parameters for Dense Coarse Gravel (3D)
Pruett
Rollins &
Parameter
Symbol Cole (2006) (2009)
Unit
Cap width
b
23.8
20.8
ft
Cap height
H
3.67
5.5
ft
Soil cohesion
c
0.251
0.286
ksf
Soil friction angle
φ
40
41.0
degrees
Wall friction angle
δ
30
30.75
degrees
Soil density
γ
0.1472
0.1384
kcf
ε50
0.005
0.0037
Strain at 50% of ultimate strength
Poisson's ratio
υ
0.30
0.30
Rf
0.95
0.98
Failure ratio
Table 4.33: Modified HFD Parameters for Dense Coarse Gravel (3D)
Rollins & Pruett
Parameter
Symbol Cole (2006) (2009)
Ultimate passive resistance/effective width
Fult /beff
35.4
39.7
Unit
kip/ft
Maximum displacement/Height
Displacement at half of the ultimate passive resistance
54
ymax /H
0.05
0.05
in/in
yave
0.34
0.72
in
5
COMPARISON OF RESULTS
The back-calculated parameters for PYCAP, ABUTMENT, and the modified HFD
method are summarized in this section of the report. The parameters summarized include the
friction angle, cohesion, initial soil modulus, the strain at 50% of ultimate strength, the ultimate
passive force normalized by the effective width, and the displacement at 50% of the ultimate
strength. The passive earth pressure coefficient of each test is also summarized and compared to
the Rankine and Coulomb theories and the ultimate passive pressure and stiffness of each test are
compared with the Caltrans 2004 and 2010 methods.
5.1
Friction Angle, φ
The soil friction angle is one of the more important parameters in developing the passive
force-displacement relationship in PYCAP and ABUTMENT. The friction angles backcalculated using PYCAP and ABUTMENT for all of the tests are shown in Figure 5.1 and Figure
5.2 respectively. The values shown are the minimum, 25th quartile, median, 75th quartile, and the
maximum. Values shown as small boxes, e.g. Loose Silty Sand 3D, contained no variation and
represent a single point. The ranges of the friction angles for all of the soil types are summarized
in Table 5.1 and Table 5.2.
55
PYCAP Friction Angle (degrees)
55
50
45
40
35
30
25
Loose Dense Dense Loose Loose Dense Dense Loose Dense Loose Dense
Fine Coarse Coarse
Silty Silty Silty Clean Clean Clean Clean Fine
Sand Sand Sand Sand Sand Sand Sand Gravel Gravel Gravel Gravel
3D
2D
3D
2D
3D
2D
3D
3D
3D
3D
3D
Figure 5.1: Comparison of the PYCAP Friction Angles for All of the Soil Types.
ABUTMENT Friction Angle (degrees)
55
50
45
40
35
30
25
Loose Dense Dense Loose Loose Dense Dense Loose Dense Loose Dense
Silty Silty Silty Clean Clean Clean Clean Fine
Fine Coarse Coarse
Sand Sand Sand Sand Sand Sand Sand Gravel Gravel Gravel Gravel
3D
2D
3D
2D
3D
2D
3D
3D
3D
3D
3D
Figure 5.2: Comparison of the ABUTMENT Friction Angles for All of the Soil Types.
56
Table 5.1: Ranges for the PYCAP Friction Angles
PYCAP Friction Angle (degrees)
Soil Type
2D
3D
Loose Clean Sand
27.7 - 36.0
42.0
Dense Clean Sand
48.3 - 50.0
40.5 - 43.7
Loose Silty Sand
n/a
27.7
Dense Silty Sand
40.0 - 44.4
27.9 - 30.0
Loose Fine Gravel
n/a
31.0
Dense Fine Gravel
34.0 - 44.0
n/a
Loose Coarse Gravel
n/a
37.4
Dense Coarse Gravel
41.0 - 41.3
n/a
Table 5.2: Ranges for the ABUTMENT Friction Angles
ABUTMENT Friction Angle (degrees)
Soil Type
2D
3D
42.0
Loose Clean Sand
27.7 - 36.0
Dense Clean Sand
49.9 - 50.0
39.3 - 43.3
n/a
27.7
Loose Silty Sand
Dense Silty Sand
39.0 - 44.4
27.0 - 30.5
n/a
33.0
Loose Fine Gravel
n/a
Dense Fine Gravel
34.0 - 44.0
n/a
40.0
Loose Coarse Gravel
n/a
Dense Coarse Gravel
40.0 - 41.0
The results from the friction angle comparisons are consistent with anticipated results.
The tests in which the backfill was confined to the width of the wall (2D) resulted in higher
friction angles than the tests on the same material that were unconfined (3D). Soils under plane
strain conditions typically provide higher strength than those in an unconfined condition.
Kulhawy and Mayne (1990) observed the plane strain friction angle to be about 10/9 of the
triaxial friction angle.
In most cases, the friction angles of the backfill materials in the dense state were higher
than those of the loose state. The higher compactive effort provided a higher relative density as
57
well as a higher strength for the backfill material. The friction angles for the dense clean sand
were higher than those of the dense coarse gravel, which was unexpected. This may be a result of
using higher values for cohesion instead of increasing the friction angles of the dense coarse
gravels in PYCAP and ABUTMENT. For the dense clean sand (3D) cases, very little to no
cohesion was used in developing the passive force-displacement relationships whereas in the
dense coarse gravel (3D) cases, 150 psf to 286 psf were used to match the passive-force
displacement curves with the measured results.
The friction angles back-calculated from PYCAP are shown as varying with the relative
density in Figure 5.3 and Figure 5.4. The plots include only the friction angles of the unconfined
(3D) backfill materials to prevent plane strain conditions from influencing the results. Figure 5.4
includes only the non-cohesive materials, since the non-cohesive materials typically had lower
friction angles due to the use of cohesion in PYCAP. In most cases the friction angles for both
methods are the same but for there is a slight difference. The plots show that the friction angles
for the soils increase as the relative density also increases, which is consistent with the expected
results.
58
PYCAP Friction Angle (degrees)
45.0
φ = 0.1773Dr + 25.119
R² = 0.3155
41.0
37.0
33.0
29.0
25.0
30
40
50
60
70
Relative Density, Dr (%)
80
90
Figure 5.3: A Comparison of the PYCAP Friction Angles with Relative Density for the
Unconfined Backfills.
59
PYCAP Friction Angle (degrees)
45.0
φ = 0.235Dr + 23.051
R² = 0.6391
41.0
37.0
33.0
29.0
25.0
30
40
50
60
70
Relative Density, Dr (%)
80
90
Figure 5.4: A Comparison of the PYCAP Friction Angles with Relative Density for the
Unconfined, Non-Cohesive Backfills.
The friction angles back-calculated from ABUTMENT are shown as varying with the
relative density in Figure 5.5 and Figure 5.6. The plots include only the friction angles of the
unconfined (3D) backfill materials to prevent plane strain conditions from influencing the
results. Figure 5.6 includes only the non-cohesive materials, since the non-cohesive materials
typically had lower friction angles due to the use of cohesion in ABUTMENT. In most cases the
friction angles for both methods are the same but for some there is a slight difference. The
comparisons show that the friction angles of the soils increase as the relative density also
increases, which is consistent with the expected results.
60
ABUTMENT Friction Angle (degrees)
45.0
φ = 0.1703Dr + 25.905
R² = 0.2819
41.0
37.0
33.0
29.0
25.0
30
40
50
60
70
Relative Density, Dr (%)
80
90
Figure 5.5: A Comparison of the ABUTMENT Friction Angles with Relative Density for
the Unconfined Backfills.
61
ABUTMENT Friction Angle (degrees)
45.0
φ = 0.2216Dr + 24.279
R² = 0.6158
41.0
37.0
33.0
29.0
25.0
30
40
50
60
70
Relative Density, Dr (%)
80
90
Figure 5.6: A Comparison of the ABUTMENT Friction Angles with Relative Density for
the Unconfined, Non-Cohesive Backfills.
The average friction angle of each soil type is shown in Figure 5.7 and Figure 5.8. Figure
5.7 shows the average PYCAP friction angles of the dense and loose unconfined (3D) soils.
Figure 5.8 shows the PYCAP friction angles of the confined and unconfined dense materials.
The back-calculated values for the friction angles are consistent with the expected results. The
dense materials exhibited higher friction angles than the loose materials. Because of plane strain
conditions, the confined material friction angles are higher than those of the unconfined
materials.
62
PYCAP Friction Angle (degrees)
43
41
39
37
35
33
31
29
Dense Mean
Loose Mean
27
25
Silty Sand
Clean Sand
Fine Gravel Coarse Gravel
Figure 5.7: A Comparison of the PYCAP Friction Angles for the Unconfined (3D) Backfills.
PYCAP Friction Angle (degrees)
55
50
45
40
35
2D Mean
30
3D Mean
25
Dense Silty
Sand
Dense Clean
Sand
Dense Fine
Gravel
Dense Coarse
Gravel
Figure 5.8: A Comparison of the PYCAP Friction Angles for the Dense Backfills.
The average friction angle of each soil type is shown in Figure 5.9 and Figure 5.10.
Figure 5.9 shows the average ABUTMENT friction angles of the dense and loose unconfined
(3D) soils. Figure 5.10 shows the ABUTMENT friction angles of the confined and unconfined
63
dense materials. As with the PYCAP friction angles, the back-calculated values for the friction
angles are consistent with the expected results. The dense materials exhibited higher friction
angles than the loose materials, although the difference between the dense and loose coarse
gravel average friction angles is much closer than those of PYCAP. Because of plane strain
conditions, the confined material friction angles are higher than those of the unconfined
materials.
ABUTMENT Friction Angle (degrees)
43
41
39
37
35
33
31
Dense Mean
29
Loose Mean
27
25
Silty Sand
Clean Sand
Fine Gravel Coarse Gravel
Figure 5.9: A Comparison of the ABUTMENT Friction Angles for the Unconfined (3D)
Backfills.
64
ABUTMENT Friction Angles (degrees)
55
50
45
40
35
2D Mean
30
3D Mean
25
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.10: A Comparison of the ABUTMENT Friction Angles for the Dense Backfills.
5.2
Cohesion
The cohesion value is an important parameter in developing the passive force-
displacement relationship in PYCAP and ABUTMENT. The cohesion values back-calculated
using PYCAP and ABUTMENT are shown in Figure 5.11 and Figure 5.12, respectively. The
ranges of the cohesion values for all of the soil types are summarized in Table 5.3 and Table 5.4.
65
700
PYCAP Cohesion (psf)
600
500
400
300
200
100
0
Loose Dense Dense Loose Loose Dense Dense Loose Dense Loose Dense
Fine Coarse Coarse
Silty
Silty
Silty Clean Clean Clean Clean Fine
Sand Sand Sand Sand Sand Sand Sand Gravel Gravel Gravel Gravel
3D
2D
3D
2D
3D
2D
3D
3D
3D
3D
3D
Figure 5.11: Comparison of the Cohesion Values used in PYCAP for all of the Soil Types.
700
Abutment Cohesion (psf)
600
500
400
300
200
100
0
Loose Dense Dense Loose Loose Dense Dense Loose Dense Loose Dense
Silty Silty Silty Clean Clean Clean Clean Fine Fine Coarse Coarse
Sand Sand Sand Sand Sand Sand Sand Gravel Gravel Gravel Gravel
3D
2D
3D
2D
3D
2D
3D
3D
3D
3D
3D
Figure 5.12: Comparison of the Cohesion Values used in ABUTMENT for all of the Soil
Types.
66
Table 5.3: Ranges for the PYCAP Cohesion Values
PYCAP Cohesion (psf)
Soil Type
2D
3D
Loose Clean Sand
10 - 60
60
Dense Clean Sand
60 - 90
0 - 60
Loose Silty Sand
n/a
60
Dense Silty Sand
190 - 380
570 - 650
Loose Fine Gravel
n/a
0
Dense Fine Gravel
0 - 79.4
n/a
Loose Coarse Gravel
n/a
0
Dense Coarse Gravel
150 - 286
n/a
Table 5.4: Ranges for the ABUTMENT Cohesion Values
ABUTMENT Cohesion (psf)
2D
3D
Soil Type
10
130
Loose Clean Sand
Dense Clean Sand
60 - 130
0 - 100
n/a
60
Loose Silty Sand
Dense Silty Sand
50 - 300
500 - 647
n/a
0
Loose Fine Gravel
n/a
Dense Fine Gravel
15 - 84
n/a
0
Loose Coarse Gravel
n/a
Dense Coarse Gravel
251 - 286
As expected, the dense silty sands had the highest cohesion, ranging from 300-647 psf.
However, the loose silty sand needed a much smaller value for cohesion, 60 psf, in order to
develop a passive force-displacement curve that matched the measured data. Most of the clean
sands needed cohesion values between 60 and 90 psf to obtain good fits. Some cohesion was
calculated from the direct shear and triaxial tests. The dense coarse gravel needed higher
cohesion than would be expected, 150-286 psf. The reason for these cohesion values is because
the dense coarse gravel contained 10-12% of silty fines. The parameters used in PYCAP and
ABUTMENT for both dense coarse gravel (3D) tests correspond to the measured soil parameters
67
obtained from testing in the laboratory. Both the measured parameters and the PYCAP and
ABUTMENT parameters were determined in Rollins and Cole (2006), Shamsabadi (2007), and
Pruett (2009).
The average cohesion of each soil type is shown in Figure 5.13, Figure 5.14, Figure 5.15,
and Figure 5.16. Figure 5.13 and Figure 5.15 show the average PYCAP and ABUTMENT
cohesion values, respectively, of the dense and loose unconfined (3D) soils. Figure 5.14 and
Figure 5.16 show the PYCAP and ABUTMENT cohesion values, respectively, of the confined
and unconfined dense materials.
700
Dense Mean
PYCAP Cohesion (psf)
600
Loose Mean
500
400
300
200
100
0
Silty Sand
Clean Sand
Fine Gravel Coarse Gravel
Figure 5.13: A Comparison of the PYCAP Cohesion Values for the Unconfined (3D)
Backfills.
68
700
2D Mean
PYCAP Cohesion (psf)
600
3D Mean
500
400
300
200
100
0
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.14: A Comparison of the PYCAP Cohesion Values for the Dense Backfills.
ABUTMENT Cohesion (psf)
700
Dense Mean
600
Loose Mean
500
400
300
200
100
0
Silty Sand
Clean Sand
Fine Gravel Coarse Gravel
Figure 5.15: A Comparison of the ABUTMENT Cohesion Values for the Unconfined (3D)
Backfills.
69
ABUTMENT Cohesion (psf)
700
2D Mean
600
3D Mean
500
400
300
200
100
0
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.16: A Comparison of the ABUTMENT Cohesion Values for the Dense Backfills.
5.3
Initial Soil Modulus, E i
The initial soil modulus is a parameter used in PYCAP to develop the passive force-
displacement relationship of a backfill material. Duncan and Mokwa (2001) provides ranges of
initial soil modulus corresponding to density which are summarized in Table 5.5. The values for
initial soil modulus of the various soil types are shown in Figure 5.17 and summarized in Table
5.6.
70
Table 5.5: Stiffness Ranges for Various Soil Densities (Duncan and Mokwa, 2001)
1200
Initial Soil Modulus, Ei (ksf)
1000
800
600
400
200
0
Loose Dense Dense Loose Loose Dense Dense Loose Dense Loose Dense
Silty Silty Silty Clean Clean Clean Clean Fine Fine Coarse Coarse
Sand Sand Sand Sand Sand Sand Sand Gravel Gravel Gravel Gravel
3D
2D
3D
2D
3D
2D
3D
3D
3D
3D
3D
Figure 5.17: Comparison of the Initial Soil Modulus for All of the Soil Types.
71
Table 5.6: Ranges for the Initial Soil Modulus
Initial Soil Modulus, Ei (ksf)
2D
Soil Type
3D
Loose Clean Sand
201 - 210
400
Dense Clean Sand
680 - 1000
450 - 800
Loose Silty Sand
n/a
150
Dense Silty Sand
1000
800
Loose Fine Gravel
491
n/a
Dense Fine Gravel
n/a
600 - 706
200
Loose Coarse Gravel
n/a
Dense Coarse Gravel
n/a
600 - 830
The results from the initial soil modulus are consistent with those anticipated. The tests in
which the backfill was confined to the width of the wall (2D) resulted in a higher initial soil
modulus than the tests on the same material that were unconfined (3D). Soils under plane strain
conditions typically provide higher stiffness than those in an unconfined condition. In most
cases; the initial soil modulus of the backfill materials in the dense state was higher than those of
the loose state. The higher compactive effort provided a higher relative density as well as a
higher stiffness for the backfill material.
Table 5.7, Table 5.8, Table 5.9, and Table 5.10 summarize the relative density of each
backfill with the initial soil modulus and the corresponding Duncan and Mokwa (2001) ranges.
Table 5.11 shows the range of back-calculated initial soil modulus values and the values
recommended by Duncan and Mokwa (2001). Based on the field load tests, the initial soil
modulus, E i , in the Duncan and Mokwa hyperbolic equation typically ranges from 450 to 1000
ksf with a mean of 819 ksf for the dense materials, which is a typical fill for bridge approaches.
The range is 150 to 491 ksf with a mean of 275 ksf for the loose sands and gravels, which is
typical for naturally deposited soil bridge abutments.
72
Table 5.7: A Comparison of Relative Density with the Initial Soil Modulus of the
Silty Sand
Duncan & Mokwa
Dr
Ei
Range
Loose Silty Sand 3D
Kwon (2007)
40
150
400 - 800
Dense Silty Sand 2D
UCLA (Stewart et al., 2007)
92
1000
600 - 1200
UCLA Plane Strain
92
1000
600 - 1200
Dense Silty Sand 3D
Rollins and Cole (2006)
67
800
500 - 1000
Valentine (2007)
80
800
600 - 1200
Table 5.8: A Comparison of Relative Density with the Initial Soil Modulus of the
Clean Sand
Duncan & Mokwa
Dr (%) Ei (ksf)
Range (ksf)
Loose Clean Sand 2D
Strassburg (2010)
41
400
400 - 800
Loose Clean Sand 3D
Cummins (2009)
44
201
400 - 800
Strassburg (2010)
31
210
400 - 800
Dense Clean Sand 2D
Bingham (2012)
79
680
600 - 1200
Jessee (2012) #1
80
1000
600 - 1200
Jessee (2012) #2
80
1000
600 - 1200
Jessee (2012) #3
80
1000
600 - 1200
Dense Clean Sand 3D
Rollins and Cole (2006)
63
775
500 - 1000
Heiner (2010)
80
800
600 - 1200
Bingham (2012)
84
450
600 - 1200
73
Table 5.9: A Comparison of Relative Density with the Initial Soil Modulus of the
Loose Gravel
Duncan & Mokwa
Dr
Ei
Range
Loose Fine Gravel 3D
Pruett (2009)
35
491
400 - 800
Dense Fine Gravel 3D
Rollins and Cole (2006)
54
706
500 - 1000
Kwon (2007)
85
600
600 - 1200
Pruett (2009)
74
670
600 - 1200
Table 5.10: A Comparison of Relative Density with the Initial Soil Modulus of the
Coarse Gravel
Duncan & Mokwa
Ei
Dr
Range
Loose Coarse Gravel 3D
Pruett (2009)
48
200
400 - 800
Dense Coarse Gravel 3D
Rollins and Cole (2006)
69
600
500 - 1000
Pruett (2009)
82
830
600 - 1200
Table 5.11: Ranges of Initial Soil Modulus for Different Relative Densities
Mean Duncan & Mokwa
Density
Dr
Compacted
Loose
40%
Ei = 150 - 491
275
Ei = 400 - 800
Medium
60%
Ei = 600 - 800
720
Ei = 500 -1000
Dense
80%
Ei = 450 - 1000
819
Ei = 600 - 1200
The values for the initial soil modulus are consistent with the ranges provided by Duncan
and Mokwa (2001). In the preloaded or compacted category, the range for the loose state is 400800 ksf and for the dense state the range is 600-1200 ksf. All of the initial soil modulus values of
the dense backfill materials fall within the range given, except dense clean sand (3D) (Bingham,
2012). Three of the five loose materials (loose clean sand (3D), loose silty sand (3D), and loose
74
coarse gravel (3D)) do not fall within the range given by Duncan and Mokwa (2001). For these
materials, the values for initial soil modulus are between 150 and 200 ksf which is about 200 ksf
below the range for compacted material. Due to the majority of the loose materials falling below
the range, it may be necessary to modify the range to be 200-800 ksf.
The plot of the initial soil modulus, E i , against relative density is shown in Figure 5.18.
The plot includes all of the unconfined (3D) tests performed and shows an upward trend for the
soil modulus as relative density increases. Increasing the density of a soil also increases the
stiffness.
900
Initial Soil Modulus, Ei (psf)
800
700
600
500
400
300
Ei = 9.4682Dr - 38.648
R² = 0.505
200
100
0
30
40
50
60
70
Relative Density, Dr (%)
80
90
Figure 5.18: A Comparison of Initial Soil Modulus, E i , against Relative Density.
The average initial soil modulus, E i , of each soil type is shown in Figure 5.19 and Figure
5.20. Figure 5.19 shows the average initial soil modulus of the dense and loose unconfined (3D)
75
soils. Figure 5.20 shows the initial soil modulus of the confined and unconfined dense materials.
The dense materials exhibited higher stiffness than the loose materials and the confined materials
showed higher stiffness than the unconfined values due to plane strain conditions.
900
Initial Soil Modulus, Ei (ksf)
800
700
600
500
Dense Mean
Loose Mean
400
300
200
100
0
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.19: A Comparison of the Initial Soil Modulus Values for the Unconfined (3D)
Backfills.
76
Initial Soil Modulus, Ei (ksf)
1200
1000
800
600
400
200
2D Mean
3D Mean
0
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.20: A Comparison of the Initial Soil Modulus Values for the Dense Backfills.
5.4
Strain at 50% of Ultimate Strength, ε 50
The strain at 50% of ultimate strength, ε 50 , is a parameter used in ABUTMENT. The
values for ε 50 are shown in Figure 5.21 and the ranges for these values are given in Table 5.12.
The loose materials were expected to exhibit higher strain at 50% of the ultimate strength than
the dense materials because they have lower stiffness. The unconfined silty sands have the same
value for ε 50 while the strain of the dense unconfined clean sand is larger than the strain for the
loose clean sand. Heiner (2010) and Bingham (2012) force-displacement curves for the dense
clean sand (3D) category show low stiffness, which might account for the relatively high strain
values.
77
Strain at 50% of Ultimate Strength, ε50
0.012
0.01
0.008
0.006
0.004
0.002
0
Loose Dense Dense Loose Loose Dense Dense Loose Dense Loose Dense
Silty Silty Silty Clean Clean Clean Clean Fine
Fine Coarse Coarse
Sand Sand Sand Sand Sand Sand Sand Gravel Gravel Gravel Gravel
3D
2D
3D
2D
3D
2D
3D
3D
3D
3D
3D
Figure 5.21: Comparison of the Strain at 50% of Ultimate Strength for All of the Soil
Types.
Table 5.12: Ranges for the Strain at 50% of Ultimate Strength
Strain at 50% of Ultimate Strength, ε50
Soil Type
2D
3D
Loose Clean Sand
0.010
0.003 - 0.006
Dense Clean Sand
0.004 - 0.008
0.002 - 0.007
Loose Silty Sand
n/a
0.003
Dense Silty Sand
0.0036
0.003
Loose Fine Gravel
n/a
0.005
Dense Fine Gravel
n/a
0.0015 - 0.004
Loose Coarse Gravel
n/a
0.0074
Dense Coarse Gravel
n/a
0.0037 - 0.005
The strain at 50% of ultimate strength, as used in ABUTMENT, as a function of the
relative density of the backfill materials and is plotted in Figure 5.22. Only the results of the
unconfined (3D) backfill materials are plotted to remove the influence of plane strain conditions.
The loose materials should have lower stiffness that would allow for more displacement to occur
78
and therefore a higher strain. This may have been a result of the dense sands that exhibited low
stiffness.
Strain at 50% of Ultimate Strength, ε50
0.0080
0.0070
0.0060
0.0050
0.0040
0.0030
0.0020
ε50 = 2E-06Dr + 0.0042
R² = 0.0003
0.0010
0.0000
30
40
50
60
70
Relative Density, Dr (%)
80
90
Figure 5.22: A Comparison of the Strain at 50% of Ultimate Strength against Relative
Density for the Unconfined Backfills.
The average strain at 50% of the Ultimate Strength, ε 50 , of each soil type is shown in
Figure 5.23 and Figure 5.24. Figure 5.23 shows the average initial soil modulus of the dense and
loose unconfined (3D) soils. Figure 5.24 shows the initial soil modulus of the confined and
unconfined dense materials. Due to an insufficient amount of test data, other comparisons were
not included. These results are not entirely consistent with the expected results. The dense clean
sand test results showed higher values of strain than the loose sand, which is inaccurate. Also the
confined silty sand exhibited higher strain than the unconfined silty sand. As previously
79
mentioned, the possible reasons for these differences include the low stiffness values of the
dense sand (3D).
Strain at 50% of Ultimate Strength
0.008
0.007
0.006
0.005
0.004
0.003
0.002
Dense Mean
Loose Mean
0.001
0
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.23: A Comparison of the Strain at 50% of Ultimate Strength for the Unconfined
(3D) Materials.
80
Strain at 50% of Ultimate Strength
0.006
0.005
0.004
0.003
0.002
2D Mean
0.001
3D Mean
0
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.24: A Comparison of the Strain at 50% of Ultimate Strength for the Dense
Materials.
5.5
Modified HFD: F ult /B eff
The ratio of the ultimate passive resistance over the effective width of the wall or
abutment is used to calculate the variables C and D according with Equation (2.6). The effective
width of the structure is the actual width multiplied by the Brinch-Hansen 3D correction factor.
The values of F ult /B eff for the soils used during testing are shown in Figure 5.25 and the range of
these values is summarized in Table 5.13. The confined dense clean sand had the highest
measured passive resistance per foot of wall. This is consistent with each of the back-calculated
parameters. The values for F ult /B eff generally increase for the loose condition as soil type moves
from silty sand to the sand and gravels. For the dense conditions, the values of F ult /B eff for all of
the materials are similar except for the coarse gravel, which is higher than the others. As
expected, the dense materials provide higher values of F ult /B eff than the loose materials and the
values for the confined (2D) materials are also higher than those for the unconfined (3D)
materials.
81
70
60
Fult/Beff (kip/ft)
50
40
30
20
10
0
Loose Dense Dense Loose Loose Dense Dense Loose Dense Loose Dense
Silty Silty Silty Clean Clean Clean Clean Fine
Fine Coarse Coarse
Sand Sand Sand Sand Sand Sand Sand Gravel Gravel Gravel Gravel
3D
2D
3D
2D
3D
2D
3D
3D
3D
3D
3D
Figure 5.25: Comparison of F ult /B eff for All of the Soil Types.
Table 5.13: Ranges for F ult /B eff for the Modified HFD Equation
Fult /Beff (kip/ft)
Soil Type
2D
3D
Loose Clean Sand
30.0
8.7 - 16.3
Dense Clean Sand
47.8 - 58.9
19.5 - 37.7
Loose Silty Sand
n/a
8.3
Dense Silty Sand
30.3
27.5 - 28.5
Loose Fine Gravel
n/a
11.5
Dense Fine Gravel
n/a
15.3 - 31.0
Loose Coarse Gravel
n/a
16.4
Dense Coarse Gravel
n/a
35.4 - 39.7
The average ultimate resistance normalized by the effective width, F ult /B eff , of each soil
type is shown in Figure 5.26 and Figure 5.27. Figure 5.26 shows the average F ult /B eff of the
dense and loose unconfined (3D) soils. Figure 5.27 shows the F ult /B eff of the confined and
82
unconfined dense materials. The value F ult /B eff is the passive resistance of the backfill and
increases with increasing density. Aside from the fine gravel, the F ult /B eff values increase as soil
type moves from silty sand to sand and then the gravels. The confined conditions also show a
greater value for F ult /B eff over the unconfined conditions.
40
35
Fult/Beff (kip/ft)
30
25
20
15
10
Dense Mean
5
Loose Mean
0
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.26: A Comparison of the F ult /B eff Values for the Unconfined (3D) Backfills.
83
60
Fult/Beff (kip/ft)
50
40
30
20
2D Mean
3D Mean
10
0
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.27: A Comparison of the F ult /B eff Values for the Dense Backfills.
5.6
Modified HFD: y ave
The displacement associated with 50% of the ultimate passive resistance, y ave , is used to
calculate the variables C and D according with Equation (2.6). The yave values for the various
soil types are shown in Figure 5.28 and the range of these values is summarized in Table 5.14.
The value of y ave for a soil is related to the stiffness of that soil, therefore higher values for y ave
are expected for the loose materials than for the dense materials. The results for all of the
material types are consistent with the expected results. The loose gravels had the highest
displacement at 50% of ultimate strength. The confined materials should also have lower
displacement at 50% of ultimate strength than the unconfined materials due to the increase in
strength and stiffness from plane strain conditions. However, the dense silty sand (2D) had a
higher y ave than the dense silty sand (3D). This unanticipated result may be a consequence of the
small sample size used in this research.
84
1.4
1.2
yave (in.)
1
0.8
0.6
0.4
0.2
0
Loose Dense Dense Loose Loose Dense Dense Loose Dense Loose Dense
Silty Silty Silty Clean Clean Clean Clean Fine
Fine Coarse Coarse
Sand Sand Sand Sand Sand Sand Sand Gravel Gravel Gravel Gravel
3D
2D
3D
3D
3D
2D
3D
2D
3D
3D
3D
Figure 5.28: Comparison of y ave for all of the Soil Types.
Table 5.14: Ranges of y ave for the Modified HFD Equation
Yave (in.)
Soil Type
3D
2D
Loose Clean Sand
0.8
0.8 - 1.0
Dense Clean Sand
0.14 - 0.68
0.16 - 0.85
Loose Silty Sand
n/a
0.7
Dense Silty Sand
0.3
0.19 - 0.30
Loose Fine Gravel
n/a
1.3
Dense Fine Gravel
n/a
0.095 - 0.66
Loose Coarse Gravel
n/a
1.08
Dense Coarse Gravel
0.34 - 0.72
n/a
The average displacement at 50% of the ultimate strength, yave , of each soil type is shown
in Figure 5.29 and Figure 5.30. Figure 5.29 shows the average y ave of the dense and loose
unconfined (3D) soils. Figure 5.30 shows the y ave of the confined and unconfined dense
materials. The value yave is the displacement at 50% of the ultimate strength and should decrease
with increasing density and confinement. The results are consistent with the anticipated results
85
except in the confined silty sand, which has a higher yave than the unconfined silty sand. This is
consistent with the ε 50 results for the confined dense silty sand. (See the Strain at 50% of
Ultimate Strength, ε 50 section.) As previously discussed, a possible reason for these
discrepancies is the low stiffness measured in the dense unconfined sand tests.
Displacement at 50% of Ultimate
Strength, yave (in.)
1.40
1.20
1.00
0.80
0.60
0.40
Dense Mean
0.20
Loose Mean
0.00
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.29: A Comparison of y ave for the Unconfined (3D) Backfills.
86
Displacement at 50% of Ultimate
Strength, yave (in.)
0.60
0.50
0.40
0.30
0.20
2D Mean
3D Mean
0.10
0.00
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.30: A Comparison of y ave for the Dense Backfills.
5.7
Passive Earth Pressure Coefficient, K p
The passive pressure of a backfill material is the resistance of the material to compressive
forces caused by the movement of a structure into the soil. Several methods have been developed
to estimate the ultimate passive pressure of a soil. Some of these methods include the Rankine
theory and the Coulomb theory. The passive earth pressure coefficient, K p , is calculated using
Equation (5-1) for the Rankine method and Equation (5-2) for the Coulomb method.
∅
𝐾𝑝 = 𝑡𝑎𝑛2 �45 + �
2
𝐾𝑝 =
(5-1)
𝑠𝑖𝑛2 (𝛽 − ∅)
(5-2)
2
𝑠𝑖𝑛(∅ + 𝛿) 𝑠𝑖𝑛(∅ + 𝛼)
𝑠𝑖𝑛2 𝛽 𝑠𝑖𝑛(𝛽 + 𝛿) �1 − �
�
𝑠𝑖𝑛(𝛽 + 𝛿) 𝑠𝑖𝑛(𝛽 + 𝛼)
87
where
β = wall angle
φ = soil friction angle
δ = soil-wall interface friction angle
α = angle of the backfill material
For each of the tests, the passive earth pressure coefficients were back-calculated using
Equation (5-3), which accounts for the unit weight and cohesion of the backfill material. The
cohesion was assumed to be negligible for the granular backfill materials; therefore the second
term in Equation (5-3) was not used to calculate K p , except for the tests on silty sand.
where
1
𝑃𝑝 = 𝛾𝐻 2 𝐾𝑝 + 2𝑐𝐻 �𝐾𝑝
2
(5-3)
P p = passive force per unit length of structure
γ = unit weight of the backfill material
H = height of the structure
K p = passive earth pressure coefficient
c = cohesion
The passive earth pressure coefficients as back-calculated are shown in Figure 5.31 and
the ranges are summarized in Table 5.15. The results from the comparisons below were
consistent with those anticipated. The tests in which the backfill was confined to the width of the
wall (2D) resulted in a higher value for K p than the tests on the same material that were
88
unconfined (3D). The soils under plane strain conditions were able to provide more passive
resistance than those in an unconfined condition. For all tests, the passive earth pressure
coefficients of the backfill materials in the dense state were higher than those of the loose state.
The higher compactive effort provided a higher relative density as well as a higher passive
resistance for the backfill material.
The passive resistance of the dense clean sand (2D) was higher than expected. The dense
clean sand (2D) had more than two times the passive resistance of the dense coarse gravel (3D).
Gravels are typically associated with higher resistance to loading than sands. However, the
confinement of the dense clean sand may be the reason for the higher passive resistance. As
shown in the previous sections, the confined (2D) tests resulted in higher strengths and stiffness.
Confined testing on a dense coarse gravel is needed in order to compare the passive resistance of
these tests.
89
50
Passive Earth Pressure Coefficient, Kp
45
40
35
30
25
20
15
10
5
0
Loose Dense Dense Loose Loose Dense Dense Loose Dense Loose Dense
Silty Silty Silty Clean Clean Clean Clean Fine Fine Coarse Coarse
Sand Sand Sand Sand Sand Sand Sand Gravel Gravel Gravel Gravel
3D
2D
3D
2D
3D
2D
3D
3D
3D
3D
3D
Figure 5.31: Comparison of the Passive Earth Pressure Coefficient for All of the Soil Types.
Table 5.15: Ranges for the Measured Passive Earth Pressure Coefficient
Passive Earth Pressure Coefficient, K p
Soil Type
2D
3D
Loose Clean Sand
17.0
3.8 - 9.5
Dense Clean Sand
30.1 - 46.6
13.5 - 18.1
Loose Silty Sand
n/a
2.5
Dense Silty Sand
11.3
6.5 - 9.3
Loose Fine Gravel
n/a
5.9
Dense Fine Gravel
n/a
9.1 - 15.0
Loose Coarse Gravel
n/a
8.3
Dense Coarse Gravel
n/a
16.5 - 19.8
The passive earth pressure coefficient for each test with the calculated Rankine and
Coulomb passive earth pressure coefficients are summarized in Table 5.16 through Table 5.19.
The differences from the measured values for both the Rankine and Coulomb methods are also
90
shown. A negative value for the difference denotes an underestimation of the actual value while
a positive value shows an overestimation.
Table 5.16: Comparisons of the Measured K p Value with the Rankine and Coulomb
Methods for the Silty Sands
Kp
Difference
Measured Rankine Coulomb Rankine Coulomb
Loose Silty Sand 3D
Kwon (2007)
2.5
3.3
8.7
33%
253%
Dense Silty Sand 2D
UCLA (Stewart et al., 2007)
11.3
4.6
24.9
-59%
121%
UCLA Plane Strain
11.3
5.7
66.8
-50%
492%
Dense Silty Sand 3D
Rollins and Cole (2006)
6.5
2.7
5.2
-59%
-20%
Valentine (2007)
9.3
2.9
6.1
-69%
-35%
Table 5.17: Comparisons of the Measured K p Value with the Rankine and Coulomb
Methods for the Clean Sands
Kp
Difference
Measured Rankine Coulomb Rankine Coulomb
Loose Clean Sand 2D
Strassburg (2010)
17.0
3.9
13.5
-77%
-21%
Loose Clean Sand 3D
Cummins (2009)
3.8
4.0
13.6
5%
254%
Strassburg (2010)
9.5
3.9
13.5
-60%
42%
Dense Clean Sand 2D
Bingham (2012)
30.1
5.3
36.5
-82%
21%
Jessee (2012) #1
46.6
7.2
164
-85%
251%
Jessee (2012) #2
46.1
7.2
164
-84%
255%
Jessee (2012) #3
46.2
7.2
164
-85%
254%
Dense Clean Sand 3D
Rollins and Cole (2006)
13.5
4.4
22.5
-67%
67%
Heiner (2010)
15.7
4.7
27.3
-70%
74%
Bingham (2012)
18.1
5.3
47.4
-71%
162%
91
Table 5.18: Comparisons of the Measured K p Value with the Rankine and Coulomb
Methods for the Fine Gravels
Kp
Difference
Measured Rankine Coulomb Rankine Coulomb
Loose Fine Gravel 3D
Pruett (2009)
5.9
4.0
24.2
-31%
314%
Dense Fine Gravel 3D
Rollins and Cole (2006)
9.1
3.5
10.5
-61%
15%
Kwon (2007)
13.2
5.6
59.7
-58%
352%
Pruett (2009)
15.0
5.6
30.1
-62%
101%
Table 5.19: Comparisons of the Measured K p Value with the Rankine and Coulomb
Methods for the Coarse Gravels
Kp
Difference
Measured Rankine Coulomb Rankine Coulomb
Loose Coarse Gravel 3D
Pruett (2009)
8.3
4.5
23.7
-45%
185%
Dense Coarse Gravel 3D
Rollins and Cole (2006)
19.8
4.6
24.9
-77%
26%
Pruett (2009)
16.5
4.7
27.8
-71%
69%
On average, the Rankine method underestimated the measured K p of the silty sands by
41% and the Coulomb method overestimated by 162%. For the clean sands, the Rankine method
underestimated the measured K p by 67% while the Coulomb method overestimated by 153%.
The Rankine method underestimated the measured K p of the fine gravels by 53% and the
Coulomb method overestimated by 195%. The Rankine method underestimated the measured K p
of the coarse gravels by 64% and the Coulomb method overestimated by 93%.
These results indicate that the Rankine method typically underestimates while the
Coulomb method typically overestimates the passive earth pressure coefficient. In many of the
tests the Coulomb method overestimates the value of K p by a factor of 2 or 3. The Rankine
method only accounts for the friction angle of the soil and not the geometry whereas the
92
Coulomb method accounts for soil properties as well as geometry. This may explain the
significant differences between the measured and calculated values.
The passive earth pressure coefficients as a function of the relative density is plotted in
Figure 5.32. While the ultimate passive resistance as a function of the relative density is plotted
in Figure 5.33. Only the unconfined (3D) data was used in these comparisons. The ultimate
passive resistance per effective wall area was used and assumes the distribution of P ult to be
uniform across the entire wall. The results show an increase in both K p and P ult with increasing
relative density for the backfill materials.
Passive Earth Pressure Coefficient, Kp
25.0
Kp = 0.2013Dr - 1.4545
R² = 0.5194
20.0
15.0
10.0
5.0
0.0
30
40
50
60
70
Relative Density, Dr (%)
80
90
Figure 5.32: A Comparison of the Passive Earth Pressure Coefficients against Relative
Density for the Unconfined Backfills.
93
Ultimate Passive Resistance (ksf)
8.0
7.0
6.0
5.0
4.0
3.0
2.0
Pult/Aeff = 0.0746Dr - 0.8999
R² = 0.6089
1.0
0.0
30
40
50
60
70
Relative Density, Dr (%)
80
90
Figure 5.33: A Comparison of the Ultimate Passive Resistance per Effective Area against
Relative Density for the Unconfined Backfills.
The average passive earth pressure coefficients, K p , of each soil type are shown in Figure
5.34 and Figure 5.35. Figure 5.34 shows the average K p of the dense and loose unconfined (3D)
soils. Figure 5.35 shows the K p of the confined and unconfined dense materials. Due to an
insufficient amount of test data, other comparisons were not included. The passive earth pressure
coefficient, K p, should increase with increasing density and confinement. The results of the backcalculated parameters are consistent with this expected result. Also, the passive earth pressure
coefficients generally increase as the soil type moves from silty sand to sand and then to the
gravels.
94
Passive Earth Pressure Coefficient, Kp
20
18
16
14
12
10
8
6
4
Dense Mean
2
Loose Mean
0
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.34: A Comparison of the Passive Earth Pressure Coefficients for the Unconfined
(3D) Backfills.
Passive Earth Pressure Coefficient, Kp
45
40
35
30
25
20
15
10
2D Mean
5
3D Mean
0
Silty Sand
Clean Sand
Fine Gravel
Coarse Gravel
Figure 5.35: A Comparison of the Passive Earth Pressure Coefficients for the Dense
Backfills.
95
5.8
Caltrans
The results of the dense backfill tests were compared with the bi-linear Caltrans seismic
design approach, as described in the Literature Review. This section also provides comparisons
of the ultimate passive resistance, P ult , and backfill stiffness, K, of the measured data with the
values calculated by the Caltrans approach. The backfill stiffness of the measured data was taken
as the slope of the force-displacement relationship from the first data point until 50% of the
ultimate strength.
Comparisons were made with the Caltrans method for two different wall heights, 3.67 ft
and 5.5 ft. The values of P ult as well as K for a wall height of 3.67 ft are shown in Figure 5.36
and Figure 5.37, respectively. No confined (2D) tests were performed on the pile caps with a
height of 3.67 ft. The Caltrans method was developed for dense materials but the results of tests
on the loose material were included on the plots. The Caltrans approach approximates the
passive resistance as well as the stiffness of the dense materials reasonably well.
96
Ultimate Passive Resistance, Pult (kips)
500
Caltrans
3D Measured Dense
3D Measured Loose
400
300
200
100
0
15
17
19
21
23
Effective Width (ft)
25
27
Figure 5.36: A Comparison of P ult for the Caltrans Method and the Measured Results for a
Wall Height of 3.67 ft.
97
1200
3D Measured Dense
3D Measured Loose
Caltrans 2004
Caltrans 2011
Soil Stiffness, K (kips/in.)
1000
800
600
400
200
0
15
17
19
21
23
Effective Width (ft)
25
27
Figure 5.37: A Comparison of K for the Caltrans Method and the Measured Results for a
Wall Height of 3.67 ft.
The dense material values of P ult as well as K for a wall height of 5.5 ft are shown in
Figure 5.38 and Figure 5.39, respectively. The Caltrans 2004 method provides an accurate
estimation of P ult and K for the unconfined (3D) dense materials with a wall height of 5.5 ft.
However, the estimations for the measured values of P ult for the confined (2D) materials are not
as good. The value for K as calculated by Caltrans 2010 provides a good fit with the confined
data but the passive resistance is underestimated for the confined cases.
98
Ultimate Passive Resistance, Pult (kips)
900
800
700
600
500
400
300
Caltrans
2D Measured Dense
3D Measured Dense
200
100
0
10
12
14
16
18
Effective Width (ft)
20
22
24
Figure 5.38: A Comparison of P ult for the Caltrans Method and the Measured Results for a
Wall Height of 5.5 ft.
99
1600
2D Measured Dense
3D Measured Dense
Caltrans 2004
Caltrans 2011
Soil Stiffness, K (kips/in.)
1400
1200
1000
800
600
400
200
0
10
12
14
16
18
20
Effective Width (ft)
22
24
Figure 5.39: A Comparison of K for the Caltrans Method and the Measured Results of the
Dense Materials with a Wall Height of 5.5 ft.
The values of P ult as well as K for the loose materials with a wall height of 5.5 ft are
shown in Figure 5.40 and Figure 5.41, respectively. The data for the loose backfill materials are
shown despite the Caltrans method being typically used for dense materials. As expected, the
Caltrans method overestimates both P ult and K for the unconfined (3D) loose materials,
especially the the value for K where the Caltrans 2010 method is used. However, the confined
(2D) loose material fits well with the values calculated using the Caltrans 2004 method. More
testing is needed for this case but if the measured point is representative of other confined (2D)
loose backfill material, the Caltrans method may be appropriate for estimating P ult and K for
loose confined materials.
100
Ultimate Passive Resistance, Pult (kips)
800
700
600
500
400
300
200
Caltrans
2D Measured Loose
3D Measured Loose
100
0
10
12
14
16
18
Effective Width (ft)
20
22
24
Figure 5.40: A Comparison of P ult for the Caltrans Method and the Measured Results for
Loose Materials with a Wall Height of 5.5 ft.
101
1600
Caltrans 2004
2D Measured Loose
3D Measured Loose
Caltrans 2011
Soil Stiffness, K (kips/in.)
1400
1200
1000
800
600
400
200
0
10
12
14
16
18
20
Effective Width (ft)
22
24
Figure 5.41: A Comparison of K for the Caltrans Method and the Measured Results for
Loose Materials with a Wall Height of 5.5 ft.
102
6
CONCLUSION
Large-scale as well as small-scale testing has been performed by Dr. Kyle Rollins and
several graduate students at BYU in order to determine the passive force-displacement
relationship of various backfill materials under different conditions. The four backfill materials
used during testing included clean sand, silty sand, fine gravel, and coarse gravel. The backfill
conditions varied from loose to dense and confined (2D) to unconfined (3D).
The results of these tests were collected and compared with common methods used in
developing force-displacement relationships of backfill materials. These methods include
PYCAP (Duncan & Mokwa, 2001), ABUTMENT (Shamsabadi, 2007), and the modified
hyperbolic force-displacement equation as presented by Shamsabadi (2008). Using the common
force-displacement approaches and the measured data, parameters necessary to develop the
passive force-displacement curves without extensive geotechnical testing were back-calculated.
6.1
Conclusions
Based upon the data and analyses presented in this project, the following conclusions and
recommendations are made:
103
6.1.1
Friction Angle
The results from the friction angle comparisons were consistent with those anticipated.
The tests in which the backfill was confined to the width of the wall (2D) resulted in higher
friction angles than the tests on the same material that were unconfined (3D). The 2D failure
geometries produce higher passive force, and therefore higher friction angles, because the soil
fails in a plane strain. In most cases, the friction angles of the backfill materials in the dense state
were higher than those of the loose state. The higher compactive effort provided a higher relative
density as well as a higher strength for the backfill material.
The friction angles of the materials generally increased as soil type moved from the silty
sand up to the sand and gravel. The friction angles of the dense clean sands were higher than the
friction angles of the dense coarse gravels. This may be due to the sand being confined to the
width of the wall and the gravel being unconfined. No testing has been performed on gravels in
confined conditions (2D); therefore further testing on fine and coarse gravels in the confined
condition is needed.
6.1.2
Cohesion
The silty sand materials were expected to have the highest values of cohesion. This was
true for the dense silty sands; however, the loose silty sand needed a much smaller value for
cohesion in order to develop a passive force-displacement curve that matched the measured data.
Despite being a cohesionless soil, most of the clean sands needed some cohesion to obtain good
fits with the measured data. In many cases, this was done to keep the soil friction angle of the
dense sands equal to or less than 50 degrees. The dense coarse gravels contained 10-12% silty
fines and therefore had higher cohesion values than expected for a coarse gravel. The parameters
used in PYCAP and ABUTMENT for both tests correspond to the measured soil parameters
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obtained from laboratory testing. The measured parameters as well as the PYCAP and
ABUTMENT parameters were determined in Rollins and Cole (2006), Shamsabadi (2007), and
Pruett (2009). Testing on additional cohesive materials, such as a clay backfill, is needed.
6.1.3
Initial Soil Modulus
The results from the initial soil modulus comparisons were consistent with what was
expected. Similar to the friction angle comparisons, tests in which the backfill was confined to
the width of the wall (2D) resulted in a higher initial soil modulus than the tests on the same
material that were unconfined (3D). As expected, soils under higher compactive effort provide
higher relative densities as well as higher values of stiffness.
The values for the initial soil modulus compare reasonably well with the ranges provided
by Duncan and Mokwa (2001). The stiffness of the materials generally increased as soil type
moved from the silty sand up to the sand and gravel except in the dense conditions. For the dense
materials, the stiffness was lowest for the gravels and then increased as soil type moved to silty
sand and then clean sand. All of the values of initial soil modulus for the dense backfill materials
fall within the range given except for one while three of the five loose materials do not fall
within the range provided. For these materials, the values for initial soil modulus are about 200
ksf below the range for pre-loaded or compacted material. Due to the number of loose materials
falling below the range, it is suggested to modify the range to 200-800 ksf. Further testing is
recommended to obtain more data in an effort to determine trends for the various backfill
materials.
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6.1.4
Strain at 50% of Ultimate Strength, ε 50
The strain at 50% of ultimate strength, ε 50 , is a parameter used in ABUTMENT to
estimate the passive force-displacement relationship of a backfill material. The loose materials
were expected to exhibit higher strain at 50% of the ultimate strength than the dense materials
because they have lower stiffness. This was observed in all of the materials except for the clean
sands, in which the dense sand had higher strains than the loose sand. There may be several
reasons for the higher than expected strains including adjusting other parameters, such as friction
angle or cohesion, to achieve higher stiffness in the force-displacement relationship or the low
stiffness of the measured force-displacement curves for the dense clean sand (3D) category as
measured by Heiner (2010) and Bingham (2012).
6.1.5
Modified HFD: F ult /B eff
The ratio of the ultimate passive resistance over the effective width of the wall or
abutment is used to calculate the variables C and D according with Equation (2.6). The confined
dense clean sand had the highest measured passive resistance per foot of wall. This is consistent
with each of the back-calculated parameters. The values for F ult /B eff generally increase for the
loose condition as soil type moves from silty sand to the sand and gravels. For the dense
conditions, the values of F ult /B eff for all of the materials are close to the same except the coarse
gravel, which is higher than the others. As expected, the dense materials provide higher values of
F ult /B eff than the loose materials and the confined (2D) materials are higher than the unconfined
(3D) materials, due to plane strain conditions.
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6.1.6
Modified HFD: y ave
The displacement associated with 50% of the ultimate passive resistance, y ave , is used to
calculate the variables C and D according with Equation (2.6). The value of y ave for a soil is
related to the stiffness of that soil, therefore higher values for y ave are expected for the loose
materials than for the dense materials. The loose gravels had the highest displacement at 50% of
ultimate strength. The confined materials should also have lower displacement at 50% of
ultimate strength than the unconfined materials due to the increase in strength and stiffness from
plane strain conditions. However, the dense silty sand (2D) had a higher y ave than the dense silty
sand (3D). This unanticipated result may be a consequence of the small sample size used in this
research, therefore more testing with these variables is needed.
6.1.7
Passive Earth Pressure Coefficients
The soils under plane strain conditions were able to provide more passive resistance than
those in an unconfined condition, which led to higher values of K p . Also, the passive earth
pressure coefficients of the backfill material in the dense state were higher than those of the loose
state. The higher compactive effort provided a higher relative density as well as a higher passive
resistance for the backfill material.
On average for all backfill materials, the Rankine method underestimated the passive
earth pressure coefficient by 56% while the Coulomb theory overestimated by 150%. In many of
the tests the Coulomb method overestimated the value of K p by a factor of 2 or 3. The
differences in the values of K p are most likely due to the underlying assumptions associated with
each of the methods.
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6.1.8
Caltrans
The Caltrans approach provides a bi-linear estimation of the passive force-displacement
relationship for dense backfill materials. Caltrans provided a good approximation for P ult and K
of the unconfined (3D) dense measured data but underestimated the confined (2D) dense
materials. The Caltrans method also overestimated the passive resistance and soil stiffness for the
unconfined loose material. However, the Caltrans method provided a good estimation of P ult and
K for the confined loose material. More testing is needed, but the Caltrans method can possibly
be used in approximating the passive resistance of a loose material confined to the width of the
wall. The Caltrans 2010 method provides a good approximation for the stiffness, K, of the
unconfined materials with a wall height of 3.67 ft.
6.2
Recommendations for Future Research
Given the limited amount of data on several of the soil conditions, there is a need for
additional tests to be performed. These test results will provide parameters that are a better
representation of the backfill material populations. In addition, a wider sample of backfill
material conditions, such as including clayey materials or confined (2D) gravels, would be
beneficial.
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