RESULTANT FORCE
OF LATERAL EARTH PRESSURE
IN UNSTABLE SLOPES
by
Weiqiong Huang
A thesis submitted to the Faculty of the University of Delaware in
partial fulfillment of the requirements for the degree of Master of Civil
Engineering
Spring 2010
Copyright 2010 Weiqiong Huang
All Rights Reserved
RESULTANT FORCE
OF LATERAL EARTH PRESSURE
IN UNSTABLE SLOPES
by
Weiqiong Huang
Approved: __________________________________________________________
Dov Leshchinsky, Ph.D
Professor in charge of thesis on behalf of the Advisory Committee
Approved: __________________________________________________________
Harry W. Shenton III, Ph.D.
Chair of the Department of Civil and Environmental Engineering
Approved: __________________________________________________________
Michael J. Chajes, Ph.D
Dean of the College of Engineering
Approved: __________________________________________________________
Debra Hess Norris, M.S.
Vice Provost for Graduate and Professional Education
iii
ACKNOWLEDGMENTS
I would like to express my deepest appreciation to my advisor, Professor
Dov Leshchinsky for providing me with this invaluable study opportunity in University
of Delaware. Thanks very much for his guidance, instruction, special support and
infinite patience to me throughout my graduate studies in geotechnical engineering.
I would like to give special thanks to all my fellow graduate classmates in
the graduate student office. Thanks very much for their help on my study. I benefit a lot.
It‟s such a nice experience for me to study engineering with these wonderful
classmates.
Finally, I would like to thank all my families for their support and
encouragement on my study.
iv
TABLE OF CONTENTS
LIST OF TABLES .......................................................................................................... v
LIST OF FIGURES ....................................................................................................... xi
ABSTRACT ................................................................................................................. xiv
CHAPTER 1 INTRODUCTION ............................................................................ 1
CHAPTER 2 FORMULATION ............................................................................. 5
CHAPTER 3 RESULTS ....................................................................................... 12
CHAPTER 4 CONCLUSIONS AND RECOMMENDATIONS ......................... 55
APPENDIX A TABLES OF Ka_h FOR DESIGN CHARTS
WHEN =20°, 25°, 30°, 35°, 40° AND 45°. .................................. 57
APPENDIX B COMPARISON OF Ka_h FROM LOG SPIRAL
EQUIVALENT COULOMB AND Ka_h FROM
CLASSICAL COULOMB ............................................................ 106
APPENDIX C COMPUTER CODING SUBROUTINES .................................... 155
REFERENCES ........................................................................................................... 169
v
LIST OF TABLES
Table A.1.1 Ka_h for Design Charts when =20° =0 D=1/3 ............................. 58
Table A.1.3 Ka_h for Design Charts when =20° =1/3 D=1/3 ........................... 60
Table A.1.4 Ka_h for Design Charts when =20° /=1/3 D=1/2 ........................... 61
Table A.1.5 Ka_h for Design Charts when =20° =2/3 D=1/3 ........................... 62
Table A.1.6 Ka_h for Design Charts when =20° =2/3 D=1/2 ............................ 63
Table A.1.7 Ka_h for Design Charts when =20° =1 D=1/3 .............................. 64
Table A.1.8 Ka_h for Design Charts when =20° =1 D=1/2 .............................. 65
Table A.2.1 Ka_h for Design Charts when =25° /=0 D=1/3 .............................. 66
Table A.2.2 Ka_h for Design Charts when =25° =0 D=1/2 .............................. 67
Table A.2.3 Ka_h for Design Charts when =25° =1/3 D=1/3 ........................... 68
Table A.2.4 Ka_h for Design Charts when =25° =1/3 D=1/2 ........................... 69
Table A.2.5 Ka_h for Design Charts when =25° =2/3 D=1/3 ........................... 70
Table A.2.6 Ka_h for Design Charts when =25° =2/3 D=1/2 ........................... 71
Table A.2.7 Ka_h for Design Charts when =25° /=1 D=1/3 .............................. 72
Table A.2.8 Ka_h for Design Charts when =25° /=1 D=1/2 .............................. 73
Table A.3.1 Ka_h for Design Charts when =30° =0 D=1/3 .............................. 74
Table A.3.2 Ka_h for Design Charts when =30° =0 D=1/2 .............................. 75
Table A.3.3 Ka_h for Design Charts when =30° =1/3 D=1/3 ........................... 76
Table A.3.4 Ka_h for Design Charts when =30° =1/3 D=1/2 ........................... 77
Table A.3.5 Ka_h for Design Charts when =30° =2/3 D=1/3 ........................... 78
vi
Table A.3.6 Ka_h for Design Charts when =30° =2/3 D=1/2 ........................... 79
Table A.3.7 Ka_h for Design Charts when =30° =1 D=1/3 .............................. 80
Table A.3.8 Ka_h for Design Charts when =30° =1 D=1/2 .............................. 81
Table A.4.1 Ka_h for Design Charts when =35° =0 D=1/3 .............................. 82
Table A.4.2 Ka_h for Design Charts when =35° =0 D=1/2 .............................. 83
Table A.4.3 Ka_h for Design Charts when =35° =1/3 D=1/3 ........................... 84
Table A.4.4 Ka_h for Design Charts when =35° =1/3 D=1/2 ........................... 85
Table A.4.5 Ka_h for Design Charts when =35° =2/3 D=1/3 ........................... 86
Table A. 4.6 Ka_h for Design Charts when =35° =2/3 D=1/2 ........................... 87
Table A.4.7 Ka_h for Design Charts when =35° =1 D=1/3 .............................. 88
Table A.4.8 Ka_h for Design Charts when =35° =1 D=1/2 ............................. 89
Table A.5.1 Ka_h for Design Charts when =40° =0 D=1/3 .............................. 90
Table A.5.2 Ka_h for Design Charts when =40° =0 D=1/2 .............................. 91
Table A.5.3 Ka_h for Design Charts when =40° =1/3 D=1/3 ........................... 92
Table A.5.4 Ka_h for Design Charts when =40° =1/3 D=1/2 ........................... 93
Table A.5.5 Ka_h for Design Charts when =40° =2/3 D=1/3 ........................... 94
Table A.5.6 Ka_h for Design Charts when =40° =2/3 D=1/2 .......................... 95
Table A.5.7 Ka_h for Design Charts when =40° =1 D=1/3 .............................. 96
Table A.5.8 Ka_h for Design Charts when =40° =1 D=1/2 .............................. 97
Table A.6.1 Ka_h for Design Charts when =45° =0 D=1/3 .............................. 98
Table A.6.2 Ka_h for Design Charts when =45° =0 D=1/2 .............................. 99
Table A.6.3 Ka_h for Design Charts when =45° =1/3 D=1/3 ......................... 100
Table A.6.4 Ka_h for Design Charts when =45° =1/3 D=1/2 ......................... 101
Table A.6.5 Ka_h for Design Charts when =45° =2/3 D=1/3 ......................... 102
vii
Table A.6.6 Ka_h for Design Charts when =45° =2/3 D=1/2 ......................... 103
Table A.6.7 Ka_h for Design Charts when =45° =1 D=1/3 ............................ 104
Table A.6.8 Ka_h for Design Charts when =45° /=1 D=1/2 ............................ 105
Table B.1.1 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=20° /=0 D=1/3) ................... 107
Table B.1.2 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=20° /=1/3 D=1/3) ............. 108
Table B.1.3 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=20° /=2/3 D=1/3) ............. 109
Table B.1.4 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=20° /=1 D=1/3) ............... 110
Table B.1.5 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=20° /=0 D=1/2) ................ 111
Table B.1.6 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=20° /=1/3 D=1/2) ............. 112
Table B.1.7 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=20° /=2/3 D=1/2) ............. 113
Table B.1.8 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=20° /=1 D=1/2) ................ 114
Table B.2.1 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=25° /=0 D=1/3) ................ 115
Table B.2.2 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=25° /=1/3 D=1/3) ............. 116
Table B.2.3 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=25° /=2/3 D=1/3) ............. 117
Table B.2.4 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=25° /=1 D=1/3) ................ 118
Table B.2.5 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=25° /=0 D=1/2) ................ 119
viii
Table B.2.6 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=25° /=1/3 D=1/2) ............. 120
Table B.2.7 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=25° /=2/3 D=1/2) ............. 121
Table B.2.8 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=25° /=1 D=1/2) ................ 122
Table B.3.1 Comparison of Ka_h from log spiral Equivalent Coulomb a
nd Ka_h from classical Coulomb (=30° /=0 D=1/3) .................. 123
Table B.3.2 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=30° /=1/3 D=1/3) ............. 124
Table B.3.3 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=30° /=2/3 D=1/3) ............. 125
Table B.3.4 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=30° /=1 D=1/3) ................ 126
Table B.3.5 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=30° /=0 D=1/2) ................ 127
Table B.3.6 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=30° /=1/3 D=1/2) ............. 128
Table B.3.7 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=30° /=2/3 D=1/2) ............. 129
Table B.3.8 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=30° /=1 D=1/2) ................ 130
Table B.4.1 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=35° /=0 D=1/3) ................ 131
Table B.4.2 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=35° /=1/3 D=1/3) ............. 132
Table B.4.3 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=35° /=2/3 D=1/3) ............. 133
Table B.4.4 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=35° /=1 D=1/3) ................ 134
ix
Table B.4.5 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=35° /=0 D=1/2) ................... 135
Table B.4.6 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=35° /=1/3 D=1/2) ............. 136
Table B.4.7 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=35° /=2/3 D=1/2) ............. 137
Table B.4.8 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=35° /=1 D=1/2) ................ 138
Table B.5.1 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=40° /=0 D=1/3) ................ 139
Table B.5.2 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=40° /=1/3 D=1/3) ............. 140
Table B.5.3 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=40° /=2/3 D=1/3) ............. 141
Table B.5.4 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=40° /=1 D=1/3) ................ 142
Table B.5.5 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=40° /=0 D=1/2) ................ 143
Table B.5.6 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=40° /=1/3 D=1/2) ............. 144
Table B.5.7 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=40° /=2/3 D=1/2) ............. 145
Table B.5.8 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_hfrom classical Coulomb (=40° /=1 D=1/2) ................. 146
Table B.6.1 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=45° /=0 D=1/3) ................ 147
Table B.6.2 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=45° /=1/3 D=1/3) ............. 148
Table B.6.3 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=45° /=2/3 D=1/3) ............. 149
x
Table B.6.4 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=45° /=1 D=1/3) ............... 150
Table B.6.5 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=45° /=0 D=1/2) ................... 151
Table B.6.6 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=45° /=1/3 D=1/2)................ 152
Table B.6.7 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=45° /=2/3 D=1/2) ............. 153
Table B.6.8 Comparison of Ka_h from log spiral Equivalent Coulomb
and Ka_h from classical Coulomb (=45° /=1 D=1/2) ................ 154
xi
LIST OF FIGURES
Figure 1. Equivalent face AB used to calculate Coulomb‟s resultant in
NCMA (1997) ........................................................................................... 4
Figure 2. Notation, convention, and assumed direction of interface friction ........... 6
Figure 3. Resultant force components considered in classical Coulomb‟s
analysis .................................................................................................... 10
Figure 4. Coefficient for horizontal resultant as function of batter (=20):
(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1 ......................... 13
Figure 4. Coefficient for horizontal resultant as function of batter (=20):
(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1 ......................... 14
Figure 5. Coefficient for horizontal resultant as function of batter (=30):
(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1 ......................... 16
Figure 5. Coefficient for horizontal resultant as function of batter (=30):
(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1 ......................... 17
Figure 6. Coefficient for horizontal resultant as function of batter (=40):
(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1 ......................... 18
Figure 6. Coefficient for horizontal resultant as function of batter (=40):
(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1 ......................... 19
Figure 7. Coefficient for horizontal resultant as function of batter when
D=1/2 (=20°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1 ................. 21
Figure 7. Coefficient for horizontal resultant as function of batter when
D=1/2 (=20°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1 ................. 22
Figure 8. Coefficient for horizontal resultant as function of batter when
D=1/2 (=30°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1 ................. 23
Figure 8. Coefficient for horizontal resultant as function of batter when
D=1/2 (=30°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1 ................. 24
xii
Figure 9. Coefficient for horizontal resultant as function of batter when
D=1/2 (=40°) :(a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1 ................. 25
Figure 9. Coefficient for horizontal resultant as function of batter when
D=1/2 (=40°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1 ................. 26
Figure 10. Coefficient for horizontal resultant as function of batter (=25°)
when /=0: (a) D=1/3 (b) D=1/2 ........................................................... 27
Figure 11. Coefficient for horizontal resultant as function of batter (=25°)
when /=1/3: (a) D=1/3 (b) D=1/2 ........................................................ 28
Figure 12. Coefficient for horizontal resultant as function of batter (=25°)
when /=2/3: (a) D=1/3 (b) D=1/2 ........................................................ 29
Figure 13. Coefficient for horizontal resultant as function of batter (=25°)
when /=1: (a) D=1/3 (b) D=1/2 ........................................................... 30
Figure 14. Coefficient for horizontal resultant as function of batter (=35°)
when /=0: (a) D=1/3 (b) D=1/2 ........................................................... 31
Figure 15 Coefficient for horizontal resultant as function of batter (=35°)
when /=1/3: (a) D=1/3 (b) D=1/2 ........................................................ 32
Figure 16. Coefficient for horizontal resultant as function of batter (=35°)
when /=2/3: (a) D=1/3 (b) D=1/2 ........................................................ 33
Figure 17. Coefficient for horizontal resultant as function of batter (=35°)
when /=1: (a) D=1/3 (b) D=1/2 ........................................................... 34
Figure 18. Coefficient for horizontal resultant as function of batter (=45°)
when /=0:(a) D=1/3 (b) D=1/2 ............................................................ 35
Figure 19. Coefficient for horizontal resultant as function of batter (=45°)
when /=1/3: (a) D=1/3 (b) D=1/2 ........................................................ 36
Figure 20 Coefficient for horizontal resultant as function of batter (=45°)
when /=2/3: (a) D=1/3 (b) D=1/2 ........................................................ 37
Figure 21. Coefficient for horizontal resultant as function of batter (=45°)
when /=1: (a) D=1/3 (b) D=1/2 ........................................................... 38
Figure 22. Comparison of horizontal thrust coefficient using Coulomb‟s
resultant inclination (Eq. 6 and Fig. 3): (a) =20 (b) =30
(c) =40 ................................................................................................. 40
xiii
Figure 22. Comparison of horizontal thrust coefficient using Coulomb‟s
resultant inclination (Eq. 6 and Fig. 3): (a) =20; and (b) =30
(c) =40 ................................................................................................. 41
Figure 22. Comparison of horizontal thrust coefficient using Coulomb‟s
resultant inclination (Eq. 6 and Fig. 3): (d) =20; and (e) =30
(f) =40 .................................................................................................. 42
Figure 22. Comparison of horizontal thrust coefficient using Coulomb‟s
resultant inclination (Eq. 6 and Fig. 3): (d) =20; and (e) =30
(f) =40 .................................................................................................. 43
Figure 23. Comparison of horizontal thrust coefficient using Coulomb‟s
inclination when D=1/2: (a) =20° (b) =30° (c) =40° ........................ 44
Figure 23. Comparison of horizontal thrust coefficient using Coulomb‟s
inclination when D=1/2: (a) =20° (b) =30° (c) =40° ........................ 45
Figure 24. Comparison of horizontal thrust coefficient using Coulomb‟s
resultant inclination when D=1/2: (a) =20° (b) =30° (c) =40° ......... 46
Figure 24. Comparison of horizontal thrust coefficient using Coulomb‟s
resultant inclination when D=1/2: (a) =20° (b) =30° (c) =40° ......... 47
Figure 25. Slip surfaces comparison for horizontal crest: (13a) =20; and
(13b) =30 ............................................................................................. 49
Figure 26. Slip surfaces comparison for inclined crest: (14a) =20; (14b)
=30; (14c) =40 ................................................................................. 51
Figure 27. Slip surfaces comparison for horizontal crest when D=1/2: (a)
=20° (b) =30° ...................................................................................... 52
Figure 28 Slip surfaces comparison for inclined crest when D=1/2: (a) =20°
(b) =30° (c) =40°................................................................................. 54
xiv
ABSTRACT
Traditionally, resultant force of lateral earth pressure serves as the basis for
design nearly vertical walls. Conversely, slopes are designed to be stable using a
factor of safety approach. However, with the availability of heavy facing elements such
as gabions or with soil reinforcement combined with some facing system, steep slopes
are increasingly being constructed. Steep slopes are considered to be unstable unless
supported; that is, such slopes require facings to resist lateral earth pressure.
Extending Coulomb‟s formulation to such slopes may not be conservative as a planar
slip surface may not be critical. Presented are the results of a formulation to find the
resultant lateral force which utilizes the log-spiral failure mechanism. The friction at
the interface soil-facing is assumed to act on vertical surface only thus replicating the
geometry of stacked rectangular facing units. Given the batter, the backslope, the
height, the unit weight and design friction angle of the backfill, and the interface
friction, one can quickly determine the corresponding lateral earth pressure coefficient.
Formulation equivalent to Coulomb‟s is also presented. Its results show that for batters
up to 20°, the common approach of using Coulomb method, including the assumed
direction to coincide with the batter, yield results that are quite close to those stemming
from the log-spiral analysis. Hence, use of Coulomb‟s analysis for such small batters
is a reasonable as its formulation is simple.
1
Chapter 1
INTRODUCTION
Design of earth structures often relies on the resultant force of lateral earth
pressure distribution. This is common in analysis of earth retention systems where
Coulomb method is utilized. As the face inclination slope surface gets shallower, the
planar slip surface used in Coulomb‟s force equilibrium may not be as critical as a
curved surface and therefore, it may render unconservative results. An alternative
approach to Coulomb‟s then is to combine an adequate limit equilibrium approach with
a curved slip surface. While the principles of the alternative approach remain the same
as Coulomb‟s, its use is not as straightforward as it requires computerized optimization
(i.e., maximization). The formulation in this thesis provides an algorithm solving the
moment equilibrium equation for a log-spiral slip surface. Such a surface degenerates
to a Coulomb‟s planar surface when the slope face is near vertical. Hence, it provides
a seamless extension to Coulomb method to deal with unstable slopes. That is, to
provide the resultant force, carried by a retention system, needed to maintain such a
slope stable. Implementing this algorithm in a computer code is simple as it represents a
closed-from solution. This thesis also provides charts which constitute the critical
solution to the log-spiral analysis, all in the familiar format of Coulomb‟s Ka.
2
One may question whether the solution in this thesis is of academic value
only. It is argued that such a solution also has practical significance. Two such
examples are provided:
Occasionally unreinforced steep slopes are retained or stabilized by heavy
facing such as large gabions, large concrete blocks, or large rocks (rockery).
Such facings serve, in essence, as aesthetic gravity retaining walls. It is not
unusual to observe large movements of such facings, especially after rainfall.
These precipitations do not need to saturate the retained soil; it increases the
moisture content to a point where the apparent cohesion due to soil matrix
suction diminishes thus making a „stable‟ slope unstable requiring the support
of the facing. While these facings are initially stable mainly since there is
very little lateral earth pressure to resist, the loss of apparent cohesion makes
these seemingly dormant elements necessary for stability. Proper slope
stability analysis can be used in design; alternatively, the results produced in
this thesis can be used in the contexts of resultant of lateral earth pressures
combined with common design of gravity retaining walls to produce
adequately stable facing support. That is, simple assessment of facing to
resist sliding, overturning, and bearing capacity considering eccentric loading
can be conducted to ensure long term stability of the retention system.
A second example has to do with geosynthetic-reinforced masonry block
walls. NCMA‟s (1997) design manual for segmental retaining walls (SRW)
3
utilizes Coulomb‟s equation to calculate the required force in the
reinforcement while limiting the batter to 20 (or slope of 70). This
limitation is customary when using Coulomb‟s, recognizing that with larger
batter Coulomb‟s may be unconservative. However, one may question
whether using Coulomb‟s equation without any adjustment to the actual
geometry of the stacked facing units is appropriate. Refer to Fig. 1. NCMA
uses Coulomb‟s equation considering the average face batter inclination ω.
Such batter assumes that interaction soil-blocks occur along an interface
defined line AB. However, it is doubtful that such interaction can
physically be achieved on non-vertical interfaces such as CD. That is,
considering a typical construction process, block interaction with soil will
happens (if at all), cumulatively, primarily along vertical segments, such as
AC, of the blocks. This thesis also examines the NCMA‟s approximation.
The formulation and results are limited to cohesionless soil. Extension to
include cohesion is simple but not advisable.
4
Figure 1. Equivalent face AB used to calculate Coulomb‟s resultant in
NCMA (1997)
5
Chapter 2
FORMULATION
Fig. 2 shows the notation and convention used in formulating the problem.
Rather than considering reinforcement at the slip surface as done in the formulation
presented by Leshchinsky et al. (2010), a resultant force at the face of the slope is
introduced to render the soil mass stable – see Fig. 2. Adequate modification of the
moment limit equilibrium equation presented by Leshchinsky et al. (2010), considering
a resultant force at a prescribed elevation and inclination at the face of the slope, is
straightforward and is not shown here. Details for deriving the moment limit
equilibrium equation for a log-spiral mechanism are provided by Baker (1981) and
Leshchinsky and San (1994). It is noted that the formulation here is limited to
cohesionless soils. Inclusion of cohesion is straightforward but may not be prudent in
the context of design.
6
Figure 2. Notation, convention, and assumed direction of interface friction
For the assumed direction of the resultant force due to lateral earth
pressures (Fig. 2), the classical expression representing the horizontal component of
this resultant is:
(1) 𝑃 = 𝑃ℎ = 1
2𝛾𝐻2𝐾𝑎 cos 𝛿 =
1
2𝛾𝐻2𝐾𝑎_ℎ
7
where Ph = the horizontal component of the resultant [i.e., P= Ph =Pa cos()] – see Fig.
2; is the interface soil-facing friction angle; H = the height of the slope; = the unit
weight of the soil; Ka = the lateral earth pressure coefficient assuming that interface
friction acts along vertical surfaces only; and Ka_h is a convenient parameter directly
rendering the horizontal component of the resultant.
Writing the moment equilibrium about the pole of the log-spiral, (xc, yc),
and rearranging the terms to match the format of Eq. 1, one gets:
𝐾𝑎_ℎ = 𝐾𝑎 cos 𝛿 =
2
H2 Ae−βcosβ − Ae−β2 cosβ
2 Ae−βsinβ Ae−β cosβ − sinβ dβ
β2
β1
− tanω H
3tanω + Ae−β1 sinβ
1
− 2
H tanω Ae−β1 cosβ
1− Ae−β2 cosβ
2− H Ae−β1 sinβ
1+
H
2tanω
−1
H2 Ae−β2 sinβ
2− Ae−β1 sinβ
1− H tanω Ae−β1 cosβ
1− Ae−β2 cosβ
2− H
× Ae−β1 sinβ1
+ H tanω +1
3 Ae−β2 sinβ
2− Ae−β1 sinβ
1− H tanω
/ Ae−β1 cosβ1− D + tanδ Ae−β1 sinβ
1+ Dtanω (2)
where 1 and 2 = the polar coordinates of Point 1 and 2 (see Fig. 2; Point 1 is at the
origin of the Cartesian coordinates where the slip surface emerge and Point 2 is the
point where this surface starts); A = log-spiral constant (analogous to radius in a circle);
= tan() and is the design internal angle of friction; ω = batter [slope face
8
inclination is (90 - ω)]; and D = assumed height, measured from (0,0), where the
resultant acts.
Eq. 2 yields non-dimensional results. Adopting the common assumption
in assessing the resultant force of lateral earth pressure, the height at which Pa (or P)
acts is taken at D=H/3.
It can be verified that the trace of the log-spiral in Cartesian coordinates,
Fig. 2, can be expressed by the following parametric equations:
x = 𝑥𝑐 + Ae−β sin β (3a)
y = 𝑦𝑐 − Ae−β cos β (3b)
where xc, yc = the location of the pole of the log-spiral relative to the Cartesian
coordinate system. Considering that Point 1 is at (0,0) and that Point 2 must be on the
crest (see Fig. 2), manipulation of Eqs. 3a and 3b yields the following expression:
A = H 1−𝑡𝑎𝑛 𝑡𝑎𝑛
e−β1 cos β1+sin β1 tan α −e−β2 cos β2+sin β2 tan α (4)
where = backslope angle – Fig. 2.
9
At this stage, Eq. 2 can be solved via a simple maximization process
(Leshchinsky et al. 2010):
1. Assume values for 1 and 2
2. Solve Eq. 4 to obtain the constant A of the log-spiral
3. For an assumed D (=H/3), solve Eq. 2 to calculate Ka_h
4. Considering all calculated values, is max (Ka_h) rendered? If yes, the complete
critical solution – Ka_h and its active wedge defined by the associated
log-spiral (A, xc, yc) – was found. If not, change 1 and 2 and go to Step 2.
The process is repeated for all feasible values of 1 and 2 to ascertain that
max (Ka_h) was indeed captured.
5. Now Ph (Eq. 1) can be calculated
This numerical iterative process is analogous to finding the factor of safety
in slope stability analysis that is associated with a circular slip surface. That is, in such
analysis minimization of the safety factor is done by changing three parameters
defining a circle: center (xc, yc) and radius (R). When circles that emerge only at the toe
are considered, the minimization is done with respect to two parameters – analogous to
the log-spiral case here.
To realize a formulation that is equivalent to Coulomb‟s, refer first to Fig.
3. It is seen that the interface friction between the facing and the soil is along the slope
(see also line AB in Fig. 1; in effect it represents the average slope angle).
10
Figure 3. Resultant force components considered in classical Coulomb‟s analysis
Hence, the horizontal component of the resultant force would be:
𝑃ℎ = 1
2𝛾𝐻2𝐾𝑎 cos(𝛿 − 𝜔) =
1
2𝛾𝐻2𝐾𝑎_ℎ (5)
Similar to the manipulation used to derive Eq. 2, one can assemble the following
equation to yield the horizontal component, Ph, of the resultant Pa:
11
𝐾𝑎_ℎ = 𝐾𝑎 cos(𝛿 − 𝜔) =
cos(𝛿 − 𝜔) 2
H2 Ae−βcosβ − Ae−β2 cosβ2 Ae−βsinβ Ae−β cosβ −
β2
β1
sinβdβ
− tanω H
3tanω + Ae−β1 sinβ
1
− 4
H tanω Ae−β1 cosβ
1− Ae−β2 cosβ
2− H Ae−β1 sinβ
1+
H
2tanω
−2
H2 Ae−β2 sinβ
2− Ae−β1 sinβ
1− H tanω Ae−β1 cosβ
1− Ae−β2 cosβ
2− H
× Ae−β1 sinβ1
+ H tanω +1
3 Ae−β2 sinβ
2− Ae−β1 sinβ
1− H tanω
/ Ae−ψm β1 cosβ1− D cosω − Ae−ψm β1 sinβ
1+ D tanω sinω +
tanδAe−ψmβ1cosβ1−Dsinω+tanδAe−ψmβ1sinβ1+D tanωcosω (6)
The right hand side of Eq. 6 is different from Eq. 2 only in the denominator. That is, the
denominator represents the resisting moment generated by the resultant force that is
needed to stabilize the mass. Its value depends on the inclination of the resultant Pa.
The solution process of Eq. 6 is identical to that of Eq. 2
12
Chapter 3
RESULTS
All results are presented considering the horizontal component of the
resultant force, Ph, that is needed to render a stable retention system. Such systems may
be reinforced or unreinforced soil with facing at the sloping face. Knowledge of the
horizontal coefficient needed to find this resultant, Ka_h, one can easily assess the
coefficient that renders the actual, inclined resultant force. That is, for the modified
approach, which motivated this work, Ka=Ka_h/cos() – see Eq. 2 or Fig. 2. For a
log-spiral case that is equivalent to Coulomb‟s, in a sense that it uses the same direction
of resultant, Ka=Ka_h/cos(-ω) – see Eq. 6 or Fig. 3. Typically, Ka_h is a major
parameter in assessing the stability of the facing or the reactive force in reinforcement
(e.g., NCMA 1997).
Figs. 4a-4d show Ka_h versus the batter for design friction angle =20,
various backslopes, from horizontal to 18.4 (1:3), and interface frictions / varying
from zero to one. Note that the batter, ω, commonly used in defining retaining walls is
adopted for the slope retention system. In common notation of slopes a batter of 20 is
equivalent to a slope at an angle of (90-20)=70.
13
4a
4b
Figure 4. Coefficient for horizontal resultant as function of batter (=20):
(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40 50 60 70
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=20°
=0
D=13
Backslope [V:H]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40 50 60 70
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=20°
=13
D=13
Backslope [V:H]
14
4c
4d
Figure 4. Coefficient for horizontal resultant as function of batter (=20):
(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40 50 60 70
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=20°
=23
D=13
Backslope [V:H]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40 50 60 70
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=20°
=1
D=13
Backslope [V:H]
15
Similar to Fig. 4, Figs. 5a-5d and 6a-6d are for =30 and =40,
respectively. However, the maximum backslope angle extends to 26.6 as backfill
with design strength >26.6 enable such inclinations. One can readily calculate Ph
when combining Ka_h from these charts with Eq. 1. Hence, these charts can be
considered as design chart.
16
5a
5b
Figure 5. Coefficient for horizontal resultant as function of batter (=30):
(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50 60
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=30°
=0
D=13
Backslope [V:H]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50 60
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=30°
=13
D=13
Backslope [V:H]
17
5c
5d
Figure 5. Coefficient for horizontal resultant as function of batter (=30):
(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50 60
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=30°
=23
D=13
Backslope [V:H]
0.0
0.1
0.2
0.3
0.4
0.5
0 10 20 30 40 50 60
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=30°
=1
D=13
Backslope [V:H]
18
6a
6b
Figure 6. Coefficient for horizontal resultant as function of batter (=40):
(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 10 20 30 40 50
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=40°
=0
D=13
Backslope [V:H]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 10 20 30 40 50
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=40°
=13
D=13
Backslope [V:H]
19
6c
6d
Figure 6. Coefficient for horizontal resultant as function of batter (=40):
(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 10 20 30 40 50
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=40°
=23
D=13
Backslope [V:H]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 10 20 30 40 50
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=40°
=1
D=13
Backslope [V:H]
20
As expected from simple slope stability analysis, for =20 lateral
support is needed up to ω=70; for =30, up to ω=60; and for =40, up to ω=50.
At ω=0, the log-spiral results (Eq. 2) degenerate to Coulomb‟s and the log-spiral
practically turns into a planar surface. Not surprisingly, increase in batter rapidly
decreases the value of Ka_h.
If the location of the resultant force is assumed at D=H/2, the resultant is
larger than obtained for D=H/3. As can be seen in Figures 7, Ka_h decreases with the
increase of the internal friction angle . For example, consider the case of zero batter
and zero interface friction. For =20°, the value of Ka_h is 0.725; for =30°, the value
of Ka_h is 0.54 and for =40°, the value of Ka_h is less than 0.30. Ka_h decreases
when the interface soil friction angle δ goes up. All Ka_h values decrease with the
increase of the batter ω. Design charts for =25°, 35° and 45° are also provided in
Figures 10 -21.
21
7a
7b
Figure 7. Coefficient for horizontal resultant as function of batter when D=1/2
(=20°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40 50 60 70
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=20°
=0
D=12
Backslope [V:H]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40 50 60 70
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=20°
=13
D=12
Backslope [V:H]
22
7c
7d
Figure 7. Coefficient for horizontal resultant as function of batter when D=1/2
(=20°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40 50 60 70
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=20°
=23
D=12
Backslope [V:H]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40 50 60 70
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=20°
=1
D=12
Backslope [V:H]
23
8a
8b
Figure 8. Coefficient for horizontal resultant as function of batter when D=1/2
(=30°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50 60
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=30°
=0
D=12
Backslope [V:H]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50 60
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=30°
=13
D=12
Backslope [V:H]
24
8c
8d
Figure 8. Coefficient for horizontal resultant as function of batter when D=1/2
(=30°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50 60
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=30°
=23
D=12
Backslope [V:H]
0.0
0.1
0.2
0.3
0.4
0.5
0 10 20 30 40 50 60
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=30°
=1
D=12
Backslope [V:H]
25
9a
9b
Figure 9. Coefficient for horizontal resultant as function of batter when D=1/2
(=40°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 10 20 30 40 50
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=40°
=0
D=12
Backslope [V:H]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 10 20 30 40 50
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=40°
=13
D=12
Backslope [V:H]
26
9c
9d
Figure 9. Coefficient for horizontal resultant as function of batter when D=1/2
(=40°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 10 20 30 40 50
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=40°
=23
D=12
Backslope [V:H]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 10 20 30 40 50
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=40°
=1
D=12
Backslope [V:H]
27
10a
10b
Figure 10. Coefficient for horizontal resultant as function of batter (=25°)
when /=0: (a) D=1/3 (b) D=1/2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=25°
=0
D=13
Backslope [V:H]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=25°
=0
D=12
Backslope [V:H]
28
11a
11b
Figure 11. Coefficient for horizontal resultant as function of batter (=25°)
when /=1/3: (a) D=1/3 (b) D=1/2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=25°
=13
D=13
Backslope [V:H]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=25°
=13
D=12
Backslope [V:H]
29
12a
12b
Figure 12. Coefficient for horizontal resultant as function of batter (=25°)
when /=2/3: (a) D=1/3 (b) D=1/2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=25°
=23
D=13
Backslope [V:H]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=25°
=23
D=12
Backslope [V:H]
30
13a
13b
Figure 13. Coefficient for horizontal resultant as function of batter (=25°)
when /=1: (a) D=1/3 (b) D=1/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=25°
=1
D=13
Backslope
[V:H]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
=25°
=1
D=12
Backslope
[V:H]
31
14a
14b
Figure 14. Coefficient for horizontal resultant as function of batter (=35°)
when /=0: (a) D=1/3 (b) D=1/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0 5 10 15 20 25 30 35 40 45 50 55
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=35°
=0
D=13
Backslope
[V:H]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0 5 10 15 20 25 30 35 40 45 50 55
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=35°
=0
D=12
Backslope
[V:H]
32
15a
15b
Figure 15 Coefficient for horizontal resultant as function of batter (=35°) when
/=1/3: (a) D=1/3 (b) D=1/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 5 10 15 20 25 30 35 40 45 50 55
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=35°
=13
D=13
Backslope
[V:H]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 5 10 15 20 25 30 35 40 45 50 55
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=35°
=13
D=12
Backslope
[V:H]
33
16a
16b
Figure 16. Coefficient for horizontal resultant as function of batter (=35°) when
/=2/3: (a) D=1/3 (b) D=1/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 5 10 15 20 25 30 35 40 45 50 55
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=35°
=23
D=13
Backslope
[V:H]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 5 10 15 20 25 30 35 40 45 50 55
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=35°
=23
D=12
Backslope
[V:H]
34
17a
17b
Figure 17. Coefficient for horizontal resultant as function of batter (=35°)
when /=1: (a) D=1/3 (b) D=1/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 5 10 15 20 25 30 35 40 45 50 55
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=35°
=1
D=13
Backslope
[V:H]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 5 10 15 20 25 30 35 40 45 50 55
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=35°
=1
D=12
Backslope
[V:H]
35
18a
18b
Figure 18. Coefficient for horizontal resultant as function of batter (=45°)
when /=0:(a) D=1/3 (b) D=1/2
0.00
0.05
0.10
0.15
0.20
0.25
0 5 10 15 20 25 30 35 40 45
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=45°
=0
D=13
Backslope
[V:H]
0.00
0.05
0.10
0.15
0.20
0.25
0 5 10 15 20 25 30 35 40 45
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=45°
=0
D=12
Backslope
[V:H]
36
19a
19b
Figure 19. Coefficient for horizontal resultant as function of batter (=45°)
when /=1/3: (a) D=1/3 (b) D=1/2
0.00
0.05
0.10
0.15
0.20
0.25
0 5 10 15 20 25 30 35 40 45
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=45°
=13
D=13
Backslope
[V:H]
0.00
0.05
0.10
0.15
0.20
0.25
0 5 10 15 20 25 30 35 40 45
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=45°
=13
D=12
Backslope
[V:H]
37
20a
20b
Figure 20 Coefficient for horizontal resultant as function of batter (=45°)
when /=2/3: (a) D=1/3 (b) D=1/2
0.00
0.05
0.10
0.15
0.20
0.25
0 5 10 15 20 25 30 35 40 45
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=45°
=23
D=13
Backslope
[V:H]
0.00
0.05
0.10
0.15
0.20
0.25
0 5 10 15 20 25 30 35 40 45
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=45°
=23
D=12
Backslope
[V:H]
38
21a
21b
Figure 21. Coefficient for horizontal resultant as function of batter (=45°)
when /=1: (a) D=1/3 (b) D=1/2
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0 5 10 15 20 25 30 35 40 45
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=45°
=1
D=13
Backslope
[V:H]
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0 5 10 15 20 25 30 35 40 45
Batter, (degrees)
Ka_h
1:∞
1:10
1:5
1:3
1:2
=45°
=1
D=12
Backslope
[V:H]
39
Differences between Coulomb‟s equation and the equivalent log-spiral
approach (Eq. 6) can be seen in Figs. 22a-22f. For zero batter the results are, as
expected, practically identical. The differences are not very large when considering a
batter of ω=20, especially when the backslope is small. The 20 batter is
Coulomb‟s traditional limit as its planar slip surface is considered to become
significantly uncritical or unconservative beyond this limit.
40
22a
22b
Figure 22. Comparison of horizontal thrust coefficient using Coulomb‟s resultant
inclination (Eq. 6 and Fig. 3): (a) =20 (b) =30 (c) =40
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0 1/3 2/3 1
/
Ka
_h
Backslope V:H=1:∞, Logspiral
Backslope V:H=1:∞, Coulomb
=20°, =0°
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0 1/3 2/3 1
Ka_h
Backslope V:H=1:∞, Logspiral
Backslope V:H=1:∞, Coulomb
Backslope V:H=1:2, Logspiral
Backslope V:H=1:2, Coulomb
=30°, =0°
41
22c
Figure 22. Comparison of horizontal thrust coefficient using Coulomb‟s resultant
inclination (Eq. 6 and Fig. 3): (a) =20; and (b) =30 (c) =40
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 1/3 2/3 1
Ka
_h
Backslope V:H=1:∞, Logspiral
Backslope V:H=1:∞, Coulomb
Backslope V:H=1:2, Logspiral
Backslope V:H=1:2, Coulomb
=40°, =0°
42
22d
22e
Figure 22. Comparison of horizontal thrust coefficient using Coulomb‟s resultant
inclination (Eq. 6 and Fig. 3): (d) =20; and (e) =30 (f) =40
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 1/3 2/3 1
/
Ka
_h
Backslope V:H=1:∞, Logspiral
Backslope V:H=1:∞, Coulomb
=20°, =20°
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 1/3 2/3 1
Ka
_h
Backslope V:H=1:∞, Logspiral
Backslope V:H=1:∞, Coulomb
Backslope V:H=1:2, Logspiral
Backslope V:H=1:2, Coulomb
=30°, =20°
43
22f
Figure 22. Comparison of horizontal thrust coefficient using Coulomb‟s resultant
inclination (Eq. 6 and Fig. 3): (d) =20; and (e) =30 (f) =40
However, if the location of the resultant force is assumed at D=H/2, the
values of Coulomb‟s equation and the values of equivalent log-spiral approach are
different. Refer to Figures 23-24, one can see that the values of Ka_h are different for
=20 regardless of ω. The values of Ka_h for both methods are similar when =30,
especially for large backslope. The values of Ka_h for both methods become close to
each other when backslope turns larger.
0.00
0.05
0.10
0.15
0 1/3 2/3 1
Ka
_h
Backslope V:H=1:∞, Logspiral
Backslope V:H=1:∞, Coulomb
Backslope V:H=1:2, Logspiral
Backslope V:H=1:2, Coulomb
=40°, =20°
44
23a
23b
Figure 23. Comparison of horizontal thrust coefficient using Coulomb‟s inclination
when D=1/2: (a) =20° (b) =30° (c) =40°
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0 1/3 2/3 1
Ka
_h
Backslope V:H=1:∞, Logspiral
Backslope V:H=1:∞, Coulomb
=20°, =0°, D=1/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0 1/3 2/3 1
Ka
_h
Backslope V:H=1:∞, Logspiral
Backslope V:H=1:∞, Coulomb
Backslope V:H=1:2, Logspiral
Backslope V:H=1:2, Coulomb
=30°, =0°, D=1/2
45
23c
Figure 23. Comparison of horizontal thrust coefficient using Coulomb‟s
inclination when D=1/2: (a) =20° (b) =30° (c) =40°
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 1/3 2/3 1
Ka
_h
Backslope V:H=1:∞, Logspiral
Backslope V:H=1:∞, Coulomb
Backslope V:H=1:2, Logspiral
Backslope V:H=1:2, Coulomb
=40°, =0°, D=1/2
46
24a
24b
Figure 24. Comparison of horizontal thrust coefficient using Coulomb‟s resultant
inclination when D=1/2: (a) =20° (b) =30° (c) =40°
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0 1/3 2/3 1
Ka
_h
Backslope V:H=1:∞, Logspiral
Backslope V:H=1:∞, Coulomb
=20°, =20°, D=1/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 1/3 2/3 1
Ka
_h
Backslope V:H=1:∞, Logspiral
Backslope V:H=1:∞, Coulomb
Backslope V:H=1:2, Logspiral
Backslope V:H=1:2, Coulomb
=30°,=20°,D=1/2
47
24c
Figure 24. Comparison of horizontal thrust coefficient using Coulomb‟s resultant
inclination when D=1/2: (a) =20° (b) =30° (c) =40°
Comparing the results in Fig. 22d with those in Figs. 5a-5d, one realizes
that Ka_h stemming from Eq. 2 and 6 for the log-spiral are not significantly different for
ω=20. However, this is not the case for large values; i.e., large interface friction ,
especially when combined with large , render the Ka_h for the case in Fig. 2 that is
larger than the conventional assumption in Fig. 3. For ω≤20 it appears that the
differences are not very significant. Hence it confirms that current practice of using
Coulomb‟s assumption regarding the inclination of interface friction (Fig. 3) or even
0.00
0.05
0.10
0.15
0 1/3 2/3 1
Ka
_h
Backslope V:H=1:∞, Logspiral
Backslope V:H=1:∞, Coulomb
Backslope V:H=1:2, Logspiral
Backslope V:H=1:2, Coulomb
=40°, =20°, D=1/2
48
Coulomb formulation utilizing planar slip surface (e.g., NCMA 1997) is a reasonable
approximation.
Refer to Figs. 25-26. One can see the traces of the critical slip surfaces
using Coulomb‟s and the equivalent log-spiral formulation (Eq. 6). In fact, these
figures complement Figs. 22a-22f. One sees that as and the backslope angle go up,
the differences between Coulomb‟s and the log-spiral traces (for ω=20) get smaller,
thus justifying the small differences in Ka_h observed in Fig. 22.
49
25a
25b
Figure 25. Slip surfaces comparison for horizontal crest:
(13a) =20; and (13b) =30
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
X=x/H
Y=
y/H
Log Spiral,δ/Φ=0 (Ka_h=0.3687)
Coulomb, δ/Φ=0 (Ka_h=0.3572)
Log Spiral, δ/Φ=2/3 (Ka_h=0.3310)
Coulomb, δ/Φ=2/3 (Ka_h=0.3166)
Φ=20°,ω=20°
backslope V:H=1:∞
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
X=x/H
Y=
y/H
Log Spiral,δ/Φ=0 (Ka_h=0.2105)
Coulomb, δ/Φ=0 (Ka_h=0.1993)
Log Spiral, δ/Φ=2/3 (Ka_h=0.1813)
Coulomb, δ/Φ=2/3 (Ka_h=0.1743)
Φ=30°,ω=20°
backslope V:H=1:∞
50
26a
26b
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
X=x/H
Y=
y/H
Log Spiral,δ/Φ=0 (Ka_h=0.4252)
Coulomb, δ/Φ=0 (Ka_h=0.4228)
Log Spiral, δ/Φ=2/3 (Ka_h=0.3859)
Coulomb, δ/Φ=2/3 (Ka_h=0.3856)
Φ=20°,ω=20°
backslope V:H=1:5
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
X=x/H
Y=
y/H
Log Spiral,δ/Φ=0 (Ka_h=0.2304)
Coulomb, δ/Φ=0 (Ka_h=0.2236)
Log Spiral, δ/Φ=2/3 (Ka_h=0.1995)
Coulomb, δ/Φ=2/3 (Ka_h=0.1983)
Φ=30°,ω=20°
backslope V:H=1:5
51
26c
Figure 26. Slip surfaces comparison for inclined crest:
(14a) =20; (14b) =30; (14c) =40
Figure 27-28 illustrate comparisons of critical slip surfaces- for D=H/2.
With the increase of and the backslope angle, the difference between Coulomb‟s and
the log-spiral traces get smaller.
Generally, Coulomb‟s surfaces are deeper at the crest and the impact of
on surfaces location is small.
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0.2 0 0.2 0.4 0.6 0.8 1
X=x/H
Y=
y/H
Log Spiral,δ/Φ=0 (Ka_h=0.1158)
Coulomb, δ/Φ=0 (Ka_h=0.1079)
Log Spiral, δ/Φ=2/3 (Ka_h=0.0960)
Coulomb, δ/Φ=2/3 (Ka_h=0.0956)
Φ=40°,ω=20°
backslope V:H=1:5
52
27a
27b
Figure 27. Slip surfaces comparison for horizontal crest when D=1/2:
(a) =20° (b) =30°
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
X=x/H
Y=
y/H
Log Spiral,δ/Φ=0 (Ka_h=0.4488)
Coulomb, δ/Φ=0 (Ka_h=0.3572)
Log Spiral, δ/Φ=2/3 (Ka_h=0.3961)
Coulomb, δ/Φ=2/3 (Ka_h=0.3166)
Φ=20°,ω=20°, D=1/2
backslope V:H=1:∞
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
X=x/H
Y=
y/H
Log Spiral,δ/Φ=0 (Ka_h=0.2466)
Coulomb, δ/Φ=0 (Ka_h=0.1993)
Log Spiral, δ/Φ=2/3 (Ka_h=0.2095)
Coulomb, δ/Φ=2/3 (Ka_h=0.1743)
Φ=30°,ω=20°, D=1/2
backslope V:H=1:∞
53
28a
28b
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
X=x/H
Y=
y/H
Log Spiral,δ/Φ=0 (Ka_h=0.4675)
Coulomb, δ/Φ=0 (Ka_h=0.4228)
Log Spiral, δ/Φ=2/3 (Ka_h=0.4230)
Coulomb, δ/Φ=2/3 (Ka_h=0.3856)
Φ=20°,ω=20°, D=1/2
backslope V:H=1:5
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
X=x/H
Y=
y/H
Log Spiral,δ/Φ=0 (Ka_h=0.2561)
Coulomb, δ/Φ=0 (Ka_h=0.2236)
Log Spiral, δ/Φ=2/3 (Ka_h=0.2209)
Coulomb, δ/Φ=2/3 (Ka_h=0.1983)
Φ=30°,ω=20°, D=1/2
backslope V:H=1:5
54
28c
Figure 28 Slip surfaces comparison for inclined crest when D=1/2: (a) =20° (b)
=30° (c) =40°
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0.2 0 0.2 0.4 0.6 0.8 1
X=x/H
Y=
y/H
Log Spiral,δ/Φ=0 (Ka_h=0.1297)
Coulomb, δ/Φ=0 (Ka_h=0.1079)
Log Spiral, δ/Φ=2/3 (Ka_h=0.1063)
Coulomb, δ/Φ=2/3 (Ka_h=0.0956)
Φ=40°,ω=20°, D=1/2
backslope V:H=1:5
55
Chapter 4
CONCLUSIONS AND RECOMMENDATIONS
Presented is a limit equilibrium formulation which uses log-spiral
mechanism to find the resultant lateral force needed to stabilize an unstable slopes.
This force can be reacted by facing units (e.g., rockery) or reinforcement (e.g., NCMA).
Similar to Coulomb‟s analysis, in calculating the resultant one needs to assume the
inclination of the resultant considering the interface friction. In very steep slopes (i.e.,
slope angle larger than 70 ), it is customary to utilize Coulomb‟s equation using the
average angle of the batter as the interface along which maximum shear force occurs.
Such an average inclination is not physically feasible when considering stacked facing
units (e.g., large blocks, gabions). The formulation presented enables one to consider
the interface friction correctly. Also, it is not limited to very steep slopes; rather, it can
produce values for any unstable homogenous, simple slope.
Design charts that can be readily used were generated. Given the batter,
the backslope, the internal angle of friction, and the soil-facing interface friction, one
can obtain the lateral earth pressure coefficient to render the magnitude of the resultant
force.
It was observed that up to a batter of 20 the customary current
approximation, which utilizes Coulomb‟s equation, is practically accurate. It becomes
56
somewhat unconservative when the interface friction angle is large, especially when the
internal angle of friction is high. This work supplements Coulomb‟s results, mainly for
slopes shallower than 70.
Earthquakes are characterized by strong vibrations occurring in the
sub-surface due to the release of large amounts of energy within a short time period
through sudden disturbances in the earth‟s crust or in the upper part of the mantle.
Many soil retaining walls can be severely damaged during the major earthquakes. Seed
and Whitman (1970) pointed out that soil retaining walls designed for static condition
normally withstand earthquake ground motion of substantial magnitude and special
seismic design measures is not required in many cases. However, in earthquake-active
areas, seismic force must be input in the design process to fulfill safety requirements
(Huang & Chen 2004).
In the current formulation, a static model is developed for the resultant of
lateral earth pressures for slopes with batters ranging from zero to very large angles
(>45°). Seismicity has not been included in this model. It is recommended
reformulating the problem and including a pseudo-seismic coefficient to consider.
57
APPENDIX A
TABLES OF 𝐊𝐚_𝐡 FOR DESIGN CHARTS
WHEN =20°, 25°, 30°, 35°, 40°AND 45°.
58
Table A.1.1 Ka_h for Design Charts when =20° =0 D=1/3
Back-
slope
(v):(h)
=20 /=0 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Log-spiral: Ka_h
1: ∞ 0.4903 0.4365 0.3944 0.3603 0.3318 0.3072 0.2849 0.2635 0.2414 0.2169 0.1881 0.1523 0.1072 0.0513 0
1:25 0.5047 0.4491 0.4051 0.3694 0.3395 0.3137 0.2903 0.2679 0.2450 0.2199 0.1904 0.1540 0.1081 0.0516 0
1:15 0.5152 0.4584 0.4131 0.3762 0.3453 0.3186 0.2944 0.2713 0.2478 0.2221 0.1921 0.1552 0.1088 0.0518 0
1:10 0.5296 0.4714 0.4243 0.3859 0.3536 0.3255 0.3002 0.2762 0.2518 0.2253 0.1945 0.1570 0.1099 0.0521 0
1:7.5 0.5455 0.4861 0.4372 0.3970 0.3631 0.3336 0.3070 0.2818 0.2564 0.2290 0.1974 0.1591 0.1111 0.0525 0
1:5 0.5840 0.5230 0.4701 0.4259 0.3881 0.3549 0.3250 0.2967 0.2687 0.2389 0.2052 0.1647 0.1145 0.0537 0
1:4 0.6221 0.5602 0.5047 0.4568 0.4152 0.3784 0.3449 0.3134 0.2824 0.2501 0.2140 0.1712 0.1186 0.0552 0
1:3 0.7305 0.6689 0.6112 0.5566 0.5062 0.4592 0.4148 0.3723 0.3309 0.2891 0.2447 0.1945 0.1344 0.0622 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
59
Table A.1.2 Ka_h for Design Charts when =20° =0 D=1/2
Back-
slope
(v):(h)
=20 /=0 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Log-spiral: Ka _h
1: ∞ 0.5508 0.5006 0.4598 0.4255 0.3958 0.3688 0.3434 0.3178 0.2901 0.2566 0.2163 0.1702 0.1169 0.0544 0
1:25 0.5586 0.5076 0.4661 0.4311 0.4007 0.3731 0.3470 0.3207 0.2925 0.2586 0.2181 0.1717 0.1177 0.0547 0
1:15 0.5641 0.5126 0.4706 0.4352 0.4042 0.3762 0.3496 0.3229 0.2942 0.2600 0.2195 0.1728 0.1184 0.0549 0
1:10 0.5715 0.5193 0.4767 0.4406 0.4091 0.3805 0.3532 0.3259 0.2966 0.2621 0.2215 0.1744 0.1194 0.0552 0
1:7.5 0.5793 0.5265 0.4833 0.4466 0.4145 0.3852 0.3573 0.3293 0.2993 0.2646 0.2238 0.1763 0.1206 0.0556 0
1:5 0.5969 0.5430 0.4985 0.4605 0.4270 0.3963 0.3670 0.3375 0.3059 0.2708 0.2299 0.1813 0.1238 0.0567 0
1:4 0.6241 0.5671 0.5189 0.4774 0.4410 0.4080 0.3769 0.3460 0.3136 0.2778 0.2365 0.1870 0.1276 0.0581 0
1:3 0.7305 0.6689 0.6117 0.5596 0.5115 0.4668 0.4247 0.3845 0.3449 0.3040 0.2588 0.2057 0.1414 0.0647 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
60
Table A.1.3 Ka_h for Design Charts when =20° =1/3 D=1/3
Back-
slope
(v):(h)
=20 /=1/3 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Log-spiral: Ka _h
1: ∞ 0.4560 0.4131 0.3786 0.3501 0.3259 0.3046 0.2848 0.2651 0.2442 0.2204 0.1915 0.1550 0.1090 0.0520 0
1:25 0.4703 0.4254 0.3892 0.3591 0.3336 0.3111 0.2902 0.2696 0.2479 0.2234 0.1939 0.1567 0.1100 0.0523 0
1:15 0.4809 0.4346 0.3971 0.3659 0.3394 0.3159 0.2943 0.2730 0.2508 0.2257 0.1957 0.1580 0.1107 0.0526 0
1:10 0.4956 0.4475 0.4082 0.3755 0.3476 0.3229 0.3001 0.2779 0.2548 0.2289 0.1982 0.1599 0.1118 0.0529 0
1:7.5 0.5122 0.4621 0.4210 0.3866 0.3571 0.3310 0.3069 0.2836 0.2595 0.2327 0.2012 0.1621 0.1131 0.0533 0
1:5 0.5528 0.4991 0.4539 0.4154 0.3820 0.3522 0.3249 0.2986 0.2719 0.2429 0.2092 0.1681 0.1167 0.0545 0
1:4 0.5933 0.5375 0.4887 0.4464 0.4091 0.3756 0.3448 0.3153 0.2857 0.2541 0.2181 0.1748 0.1210 0.0561 0
1:3 0.7112 0.6532 0.5985 0.5479 0.5009 0.4567 0.4146 0.3741 0.3341 0.2933 0.2492 0.1986 0.1372 0.0633 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
61
Table A.1.4 Ka_h for Design Charts when =20° /=1/3 D=1/2
Back-
slope
(v):(h)
=20 /=1/3 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Log-spiral: Ka_h
1: ∞ 0.5163 0.4763 0.4427 0.4136 0.3878 0.3638 0.3406 0.3166 0.2902 0.2563 0.2159 0.1699 0.1168 0.0544 0
1:25 0.5249 0.4840 0.4496 0.4198 0.3932 0.3685 0.3446 0.3199 0.2928 0.2585 0.2179 0.1715 0.1177 0.0547 0
1:15 0.5311 0.4895 0.4545 0.4242 0.3971 0.3719 0.3475 0.3224 0.2947 0.2602 0.2195 0.1727 0.1184 0.0549 0
1:10 0.5393 0.4970 0.4613 0.4303 0.4025 0.3767 0.3515 0.3258 0.2975 0.2627 0.2217 0.1745 0.1195 0.0552 0
1:7.5 0.5480 0.5050 0.4686 0.4369 0.4084 0.3819 0.3561 0.3295 0.3005 0.2654 0.2243 0.1766 0.1207 0.0556 0
1:5 0.5684 0.5236 0.4856 0.4524 0.4224 0.3943 0.3669 0.3387 0.3079 0.2727 0.2311 0.1821 0.1242 0.0568 0
1:4 0.5981 0.5488 0.5066 0.4697 0.4368 0.4066 0.3774 0.3478 0.3160 0.2803 0.2385 0.1883 0.1283 0.0583 0
1:3 0.7112 0.6535 0.6008 0.5524 0.5074 0.4652 0.4253 0.3867 0.3482 0.3077 0.2623 0.2085 0.1431 0.0652 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
62
Table A.1.5 Ka_h for Design Charts when =20° =2/3 D=1/3
Back-
slope
(v):(h)
=20 /=2/3 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Log-spiral: Ka _h
1: ∞ 0.4300 0.3950 0.3665 0.3427 0.3222 0.3037 0.2861 0.2681 0.2483 0.2248 0.1953 0.1578 0.1109 0.0528 0
1:25 0.4440 0.4072 0.3769 0.3516 0.3298 0.3101 0.2915 0.2727 0.2520 0.2279 0.1978 0.1596 0.1119 0.0531 0
1:15 0.4544 0.4162 0.3848 0.3583 0.3355 0.3150 0.2956 0.2761 0.2549 0.2302 0.1997 0.1610 0.1127 0.0534 0
1:10 0.4690 0.4290 0.3958 0.3679 0.3437 0.3219 0.3015 0.2810 0.2589 0.2336 0.2024 0.1630 0.1139 0.0537 0
1:7.5 0.4854 0.4435 0.4085 0.3789 0.3531 0.3300 0.3083 0.2867 0.2637 0.2374 0.2056 0.1654 0.1153 0.0541 0
1:5 0.5267 0.4804 0.4413 0.4075 0.3779 0.3512 0.3262 0.3017 0.2762 0.2477 0.2139 0.1718 0.1191 0.0554 0
1:4 0.5689 0.5191 0.4762 0.4386 0.4050 0.3745 0.3461 0.3184 0.2901 0.2592 0.2231 0.1789 0.1236 0.0570 0
1:3 0.6943 0.6392 0.5884 0.5413 0.4972 0.4555 0.4156 0.3769 0.3383 0.2984 0.2545 0.2032 0.1404 0.0645 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
63
Table A.1.6 Ka_h for Design Charts when =20° =2/3 D=1/2
Back-
slope
(v):(h)
=20 /=2/3 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Log-spiral: Ka_h
1: ∞ 0.4917 0.4587 0.4303 0.4052 0.3824 0.3608 0.3394 0.3168 0.2912 0.2562 0.2157 0.1698 0.1167 0.0544 0
1:25 0.5010 0.4670 0.4377 0.4119 0.3883 0.3660 0.3438 0.3205 0.2942 0.2588 0.2180 0.1715 0.1177 0.0547 0
1:15 0.5077 0.4730 0.4432 0.4167 0.3926 0.3697 0.3470 0.3232 0.2963 0.2608 0.2197 0.1728 0.1185 0.0549 0
1:10 0.5166 0.4811 0.4505 0.4234 0.3985 0.3749 0.3515 0.3270 0.2994 0.2635 0.2222 0.1748 0.1196 0.0553 0
1:7.5 0.5262 0.4899 0.4586 0.4306 0.4050 0.3807 0.3565 0.3312 0.3028 0.2667 0.2251 0.1770 0.1210 0.0557 0
1:5 0.5485 0.5103 0.4773 0.4477 0.4205 0.3945 0.3685 0.3414 0.3110 0.2749 0.2326 0.1831 0.1248 0.0569 0
1:4 0.5782 0.5352 0.4979 0.4650 0.4353 0.4074 0.3799 0.3512 0.3197 0.2836 0.2408 0.1898 0.1291 0.0585 0
1:3 0.6945 0.6411 0.5923 0.5471 0.5049 0.4651 0.4271 0.3901 0.3525 0.3123 0.2664 0.2117 0.1449 0.0658 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
64
Table A.1.7 Ka_h for Design Charts when =20° =1 D=1/3
Back-
slope
(v):(h)
=20 /=1 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Log-spiral: Ka_h
1: ∞ 0.4093 0.3808 0.3574 0.3377 0.3206 0.3049 0.2894 0.2729 0.2539 0.2302 0.1994 0.1609 0.1130 0.0537 0
1:25 0.4231 0.3928 0.3677 0.3465 0.3281 0.3113 0.2948 0.2775 0.2578 0.2335 0.2022 0.1629 0.1141 0.0540 0
1:15 0.4334 0.4017 0.3754 0.3532 0.3338 0.3161 0.2989 0.2809 0.2606 0.2359 0.2042 0.1644 0.1150 0.0543 0
1:10 0.4478 0.4143 0.3864 0.3626 0.3419 0.3230 0.3047 0.2859 0.2648 0.2395 0.2071 0.1665 0.1162 0.0546 0
1:7.5 0.4641 0.4287 0.3990 0.3736 0.3512 0.3309 0.3115 0.2916 0.2696 0.2435 0.2106 0.1691 0.1177 0.0551 0
1:5 0.5053 0.4656 0.4316 0.4020 0.3759 0.3520 0.3294 0.3066 0.2822 0.2541 0.2195 0.1759 0.1218 0.0564 0
1:4 0.5480 0.5045 0.4665 0.4330 0.4029 0.3752 0.3491 0.3233 0.2962 0.2657 0.2292 0.1835 0.1266 0.0581 0
1:3 0.6789 0.6276 0.5804 0.5365 0.4951 0.4559 0.4181 0.3811 0.3440 0.3049 0.2610 0.2088 0.1440 0.0658 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
65
Table A.1.8 Ka_h for Design Charts when =20° =1 D=1/2
Back-
slope
(v):(h)
=20 /=1 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Log-spiral: Ka_h
1: ∞ 0.4737 0.4459 0.4216 0.3998 0.3795 0.3599 0.3400 0.3184 0.2931 0.2565 0.2157 0.1697 0.1167 0.0544 0
1:25 0.4836 0.4548 0.4296 0.4070 0.3859 0.3655 0.3448 0.3225 0.2964 0.2595 0.2182 0.1716 0.1178 0.0547 0
1:15 0.4907 0.4613 0.4355 0.4122 0.3906 0.3696 0.3484 0.3255 0.2988 0.2617 0.2201 0.1730 0.1186 0.0550 0
1:10 0.5004 0.4701 0.4435 0.4195 0.3970 0.3753 0.3533 0.3297 0.3021 0.2648 0.2229 0.1751 0.1198 0.0553 0
1:7.5 0.5108 0.4797 0.4523 0.4274 0.4042 0.3817 0.3589 0.3344 0.3058 0.2684 0.2261 0.1776 0.1213 0.0558 0
1:5 0.5350 0.5020 0.4728 0.4462 0.4212 0.3969 0.3722 0.3458 0.3151 0.2778 0.2345 0.1842 0.1254 0.0571 0
1:4 0.5633 0.5257 0.4929 0.4638 0.4371 0.4112 0.3849 0.3568 0.3249 0.2876 0.2436 0.1916 0.1301 0.0588 0
1:3 0.6807 0.6314 0.5860 0.5438 0.5042 0.4667 0.4306 0.3950 0.3583 0.3180 0.2714 0.2154 0.1470 0.0664 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
66
Table A.2.1 Ka_h for Design Charts when =25° /=0 D=1/3
Back-
slope
(v):(h)
=25 /=0 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Log-spiral: Ka_h
1: ∞ 0.4059 0.3563 0.3163 0.2831 0.2547 0.2295 0.2060 0.1830 0.1593 0.1335 0.1042 0.0702 0.0319 0
1:25 0.4166 0.3653 0.3237 0.2892 0.2596 0.2334 0.2091 0.1854 0.1611 0.1348 0.1051 0.0707 0.0320 0
1:15 0.4243 0.3718 0.3291 0.2937 0.2633 0.2363 0.2114 0.1872 0.1625 0.1358 0.1057 0.0710 0.0321 0
1:10 0.4347 0.3808 0.3366 0.2998 0.2683 0.2404 0.2146 0.1898 0.1644 0.1372 0.1066 0.0715 0.0322 0
1:7.5 0.4459 0.3906 0.3448 0.3066 0.2739 0.2449 0.2182 0.1926 0.1665 0.1387 0.1076 0.0720 0.0323 0
1:5 0.4718 0.4140 0.3646 0.3232 0.2875 0.2560 0.2271 0.1995 0.1718 0.1425 0.1101 0.0734 0.0327 0
1:4 0.4952 0.4356 0.3833 0.3391 0.3008 0.2668 0.2358 0.2063 0.1770 0.1463 0.1127 0.0748 0.0332 0
1:3 0.5466 0.4839 0.4269 0.3768 0.3327 0.2933 0.2573 0.2233 0.1900 0.1559 0.1192 0.0785 0.0344 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
67
Table A.2.2 Ka_h for Design Charts when =25° =0 D=1/2
Back-
slope
(v):(h)
=25 /=0 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Log-spiral: Ka_h
1: ∞ 0.4468 0.4007 0.3620 0.3285 0.2985 0.2706 0.2436 0.2160 0.1861 0.1535 0.1174 0.0771 0.0339 0
1:25 0.4530 0.4060 0.3665 0.3323 0.3016 0.2732 0.2457 0.2175 0.1873 0.1545 0.1181 0.0775 0.0340 0
1:15 0.4573 0.4097 0.3697 0.3350 0.3039 0.2751 0.2471 0.2186 0.1882 0.1552 0.1186 0.0778 0.0341 0
1:10 0.4630 0.4146 0.3739 0.3386 0.3070 0.2776 0.2491 0.2201 0.1894 0.1562 0.1193 0.0782 0.0342 0
1:7.5 0.4690 0.4199 0.3785 0.3425 0.3102 0.2803 0.2513 0.2218 0.1908 0.1573 0.1201 0.0787 0.0344 0
1:5 0.4834 0.4324 0.3892 0.3515 0.3178 0.2866 0.2564 0.2261 0.1944 0.1601 0.1222 0.0799 0.0348 0
1:4 0.5003 0.4466 0.4009 0.3612 0.3257 0.2930 0.2617 0.2305 0.1980 0.1630 0.1243 0.0811 0.0352 0
1:3 0.5468 0.4862 0.4341 0.3885 0.3478 0.3107 0.2759 0.2419 0.2072 0.1702 0.1296 0.0844 0.0363 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
68
Table A.2.3 Ka_h for Design Charts when =25° =1/3 D=1/3
Back-slope
(v):(h)
=25 /=1/3 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Log-spiral: Ka_h
1: ∞ 0.3736 0.3339 0.3010 0.2731 0.2486 0.2262 0.2049 0.1834 0.1607 0.1353 0.1059 0.0714 0.0323 0
1:25 0.3841 0.3427 0.3083 0.2791 0.2535 0.2302 0.2080 0.1859 0.1625 0.1366 0.1068 0.0719 0.0324 0
1:15 0.3917 0.3490 0.3136 0.2835 0.2571 0.2331 0.2104 0.1877 0.1639 0.1376 0.1074 0.0722 0.0325 0
1:10 0.4021 0.3578 0.3209 0.2895 0.2620 0.2371 0.2136 0.1903 0.1659 0.1390 0.1083 0.0727 0.0327 0
1:7.5 0.4135 0.3674 0.3290 0.2963 0.2676 0.2417 0.2172 0.1931 0.1680 0.1406 0.1094 0.0732 0.0328 0
1:5 0.4401 0.3904 0.3486 0.3127 0.2812 0.2527 0.2261 0.2001 0.1734 0.1445 0.1120 0.0746 0.0332 0
1:4 0.4644 0.4120 0.3672 0.3285 0.2943 0.2636 0.2349 0.2070 0.1787 0.1484 0.1146 0.0761 0.0337 0
1:3 0.5183 0.4616 0.4108 0.3661 0.3263 0.2900 0.2564 0.2241 0.1919 0.1582 0.1214 0.0799 0.0349 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
69
Table A.2.4 Ka_h for Design Charts when =25° =1/3 D=1/2
Back-slope
(v):(h)
=25 /=1/3 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Log-spiral: Ka_h
1: ∞ 0.4152 0.3778 0.3455 0.3167 0.2903 0.2652 0.2403 0.2141 0.1852 0.1530 0.1171 0.0770 0.0339 0
1:25 0.4218 0.3835 0.3504 0.3209 0.2938 0.2681 0.2425 0.2158 0.1865 0.1540 0.1179 0.0774 0.0340 0
1:15 0.4265 0.3876 0.3539 0.3239 0.2963 0.2701 0.2441 0.2171 0.1874 0.1548 0.1184 0.0777 0.0341 0
1:10 0.4327 0.3930 0.3585 0.3278 0.2997 0.2729 0.2464 0.2188 0.1888 0.1559 0.1192 0.0782 0.0342 0
1:7.5 0.4392 0.3987 0.3635 0.3321 0.3033 0.2759 0.2488 0.2206 0.1903 0.1572 0.1201 0.0787 0.0344 0
1:5 0.4551 0.4122 0.3750 0.3419 0.3116 0.2828 0.2544 0.2252 0.1942 0.1602 0.1223 0.0800 0.0348 0
1:4 0.4728 0.4269 0.3871 0.3518 0.3197 0.2895 0.2599 0.2299 0.1981 0.1633 0.1246 0.0813 0.0352 0
1:3 0.5197 0.4669 0.4206 0.3795 0.3423 0.3078 0.2747 0.2419 0.2078 0.1711 0.1303 0.0848 0.0364 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
70
Table A.2.5 Ka_h for Design Charts when =25° =2/3 D=1/3
Back-
slope
(v):(h)
=25 /=2/3 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Log-spiral: Ka_h
1: ∞ 0.3489 0.3164 0.2891 0.2654 0.2441 0.2243 0.2049 0.1848 0.1627 0.1376 0.1079 0.0727 0.0328 0
1:25 0.3591 0.3250 0.2962 0.2713 0.2490 0.2283 0.2081 0.1873 0.1647 0.1390 0.1088 0.0732 0.0329 0
1:15 0.3665 0.3312 0.3014 0.2756 0.2526 0.2312 0.2104 0.1891 0.1661 0.1400 0.1095 0.0736 0.0330 0
1:10 0.3766 0.3398 0.3086 0.2816 0.2575 0.2352 0.2137 0.1917 0.1680 0.1414 0.1104 0.0741 0.0332 0
1:7.5 0.3877 0.3493 0.3166 0.2883 0.2630 0.2397 0.2173 0.1946 0.1703 0.1431 0.1115 0.0746 0.0333 0
1:5 0.4140 0.3719 0.3359 0.3045 0.2765 0.2508 0.2262 0.2017 0.1758 0.1471 0.1142 0.0761 0.0338 0
1:4 0.4385 0.3933 0.3544 0.3202 0.2897 0.2616 0.2350 0.2086 0.1812 0.1511 0.1170 0.0776 0.0342 0
1:3 0.4941 0.4429 0.3979 0.3578 0.3215 0.2881 0.2565 0.2258 0.1945 0.1611 0.1239 0.0816 0.0355 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
71
Table A.2.6 Ka_h for Design Charts when =25° =2/3 D=1/2
Back-
slope
(v):(h)
=25 /=2/3 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Log-spiral: Ka_h
1: ∞ 0.3920 0.3608 0.3331 0.3080 0.2843 0.2614 0.2381 0.2131 0.1849 0.1529 0.1170 0.0769 0.0338 0
1:25 0.3991 0.3669 0.3384 0.3124 0.2881 0.2645 0.2406 0.2151 0.1863 0.1540 0.1178 0.0774 0.0340 0
1:15 0.4040 0.3712 0.3421 0.3156 0.2908 0.2667 0.2424 0.2164 0.1873 0.1548 0.1184 0.0777 0.0341 0
1:10 0.4107 0.3770 0.3472 0.3200 0.2945 0.2698 0.2448 0.2183 0.1888 0.1560 0.1193 0.0782 0.0342 0
1:7.5 0.4177 0.3832 0.3526 0.3246 0.2984 0.2731 0.2475 0.2204 0.1905 0.1574 0.1203 0.0787 0.0344 0
1:5 0.4341 0.3973 0.3648 0.3353 0.3076 0.2807 0.2537 0.2253 0.1946 0.1607 0.1227 0.0801 0.0348 0
1:4 0.4517 0.4119 0.3769 0.3452 0.3159 0.2877 0.2595 0.2302 0.1988 0.1640 0.1251 0.0815 0.0352 0
1:3 0.4985 0.4518 0.4103 0.3729 0.3386 0.3062 0.2747 0.2428 0.2092 0.1724 0.1313 0.0853 0.0365 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
72
Table A.2.7 Ka_h for Design Charts when =25° /=1 D=1/3
Back-
slope
(v):(h)
=25 /=1 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Log-spiral: Ka_h
1: ∞ 0.3289 0.3023 0.2796 0.2597 0.2414 0.2240 0.2063 0.1873 0.1659 0.1407 0.1105 0.0744 0.0334 0
1:25 0.3388 0.3107 0.2866 0.2655 0.2463 0.2279 0.2095 0.1899 0.1678 0.1421 0.1114 0.0749 0.0335 0
1:15 0.3460 0.3168 0.2918 0.2698 0.2498 0.2308 0.2118 0.1917 0.1693 0.1432 0.1121 0.0752 0.0336 0
1:10 0.3559 0.3252 0.2988 0.2757 0.2547 0.2349 0.2151 0.1943 0.1713 0.1447 0.1131 0.0758 0.0338 0
1:7.5 0.3668 0.3345 0.3067 0.2823 0.2602 0.2394 0.2188 0.1973 0.1736 0.1464 0.1143 0.0764 0.0339 0
1:5 0.3926 0.3568 0.3258 0.2984 0.2736 0.2504 0.2277 0.2045 0.1792 0.1506 0.1171 0.0779 0.0344 0
1:4 0.4169 0.3780 0.3441 0.3140 0.2867 0.2612 0.2365 0.2115 0.1847 0.1547 0.1199 0.0795 0.0349 0
1:3 0.4727 0.4276 0.3875 0.3514 0.3184 0.2876 0.2581 0.2288 0.1983 0.1650 0.1272 0.0837 0.0362 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
73
Table A.2.8 Ka_h for Design Charts when =25° /=1 D=1/2
Back-
slope
(v):(h)
=25 /=1 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Log-spiral: Ka_h
1: ∞ 0.3743 0.3478 0.3238 0.3015 0.2802 0.2590 0.2371 0.2131 0.1853 0.1531 0.1171 0.0769 0.0338 0
1:25 0.3817 0.3543 0.3294 0.3064 0.2843 0.2625 0.2398 0.2152 0.1869 0.1544 0.1180 0.0774 0.0339 0
1:15 0.3870 0.3589 0.3334 0.3098 0.2873 0.2649 0.2418 0.2168 0.1880 0.1553 0.1187 0.0778 0.0340 0
1:10 0.3941 0.3651 0.3388 0.3145 0.2913 0.2683 0.2446 0.2189 0.1896 0.1566 0.1196 0.0783 0.0342 0
1:7.5 0.4016 0.3717 0.3447 0.3196 0.2956 0.2719 0.2475 0.2212 0.1915 0.1581 0.1206 0.0788 0.0344 0
1:5 0.4186 0.3868 0.3580 0.3312 0.3056 0.2804 0.2545 0.2267 0.1959 0.1617 0.1233 0.0803 0.0348 0
1:4 0.4355 0.4009 0.3699 0.3415 0.3145 0.2879 0.2607 0.2318 0.2004 0.1653 0.1259 0.0819 0.0353 0
1:3 0.4815 0.4399 0.4026 0.3686 0.3369 0.3065 0.2763 0.2450 0.2114 0.1743 0.1326 0.0859 0.0366 0
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 0
74
Table A.3.1 Ka_h for Design Charts when =30° =0 D=1/3
Back-slope
(v):(h)
=30 /=0 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60
Log-spiral: Ka_h
1: ∞ 0.3333 0.2881 0.2511 0.2199 0.1929 0.1686 0.1457 0.1234 0.1004 0.0759 0.0494 0.0215 0
1:25 0.3413 0.2945 0.2561 0.2239 0.1960 0.1709 0.1475 0.1246 0.1012 0.0764 0.0496 0.0216 0
1:15 0.3469 0.2991 0.2598 0.2268 0.1982 0.1726 0.1488 0.1255 0.1018 0.0768 0.0498 0.0216 0
1:10 0.3544 0.3052 0.2647 0.2307 0.2013 0.1749 0.1505 0.1267 0.1027 0.0773 0.0501 0.0217 0
1:7.5 0.3624 0.3119 0.2701 0.2349 0.2046 0.1775 0.1524 0.1281 0.1036 0.0779 0.0503 0.0217 0
1:5 0.3803 0.3272 0.2825 0.2448 0.2123 0.1834 0.1568 0.1313 0.1058 0.0792 0.0510 0.0219 0
1:4 0.3958 0.3408 0.2936 0.2538 0.2194 0.1889 0.1609 0.1343 0.1078 0.0805 0.0516 0.0221 0
1:3 0.4271 0.3687 0.3171 0.2730 0.2347 0.2008 0.1699 0.1408 0.1123 0.0833 0.0530 0.0225 0
1:2 0.5359 0.4689 0.4073 0.3502 0.2988 0.2522 0.2096 0.1702 0.1328 0.0965 0.0602 0.0248 0
75
Table A.3.2 Ka_h for Design Charts when =30° =0 D=1/2
Back-slope
(v):(h)
=30 /=0 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60
Log-spiral: Ka_h
1: ∞ 0.3613 0.3192 0.2833 0.2517 0.2230 0.1960 0.1695 0.1426 0.1148 0.0854 0.0544 0.0230 0
1:25 0.3660 0.3231 0.2865 0.2542 0.2250 0.1975 0.1706 0.1435 0.1154 0.0858 0.0546 0.0230 0
1:15 0.3693 0.3259 0.2887 0.2560 0.2264 0.1986 0.1714 0.1441 0.1159 0.0861 0.0547 0.0231 0
1:10 0.3737 0.3294 0.2917 0.2584 0.2283 0.2000 0.1725 0.1450 0.1165 0.0865 0.0549 0.0231 0
1:7.5 0.3783 0.3333 0.2948 0.2609 0.2303 0.2016 0.1738 0.1459 0.1172 0.0870 0.0552 0.0232 0
1:5 0.3900 0.3428 0.3024 0.2670 0.2351 0.2054 0.1767 0.1482 0.1189 0.0881 0.0558 0.0234 0
1:4 0.4017 0.3522 0.3099 0.2729 0.2397 0.2090 0.1795 0.1503 0.1204 0.0891 0.0563 0.0235 0
1:3 0.4285 0.3739 0.3273 0.2867 0.2505 0.2173 0.1859 0.1551 0.1239 0.0915 0.0576 0.0239 0
1:2 0.5359 0.4690 0.4078 0.3529 0.3035 0.2584 0.2169 0.1779 0.1401 0.1024 0.0639 0.0261 0
76
Table A.3.3 Ka_h for Design Charts when =30° =1/3 D=1/3
Back-slope
(v):(h)
=30 /=1/3 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60
Log-spiral: Ka_h
1: ∞ 0.3045 0.2681 0.2374 0.2110 0.1874 0.1656 0.1446 0.1234 0.1011 0.0769 0.0501 0.0218 0
1:25 0.3122 0.2743 0.2424 0.2149 0.1905 0.1680 0.1464 0.1247 0.1020 0.0774 0.0504 0.0219 0
1:15 0.3177 0.2787 0.2460 0.2177 0.1927 0.1697 0.1477 0.1256 0.1026 0.0778 0.0506 0.0219 0
1:10 0.3250 0.2847 0.2508 0.2216 0.1957 0.1720 0.1494 0.1269 0.1035 0.0783 0.0508 0.0220 0
1:7.5 0.3329 0.2912 0.2560 0.2258 0.1990 0.1745 0.1513 0.1283 0.1044 0.0789 0.0511 0.0220 0
1:5 0.3510 0.3061 0.2682 0.2355 0.2067 0.1805 0.1558 0.1315 0.1067 0.0802 0.0518 0.0222 0
1:4 0.3667 0.3193 0.2791 0.2444 0.2137 0.1860 0.1599 0.1345 0.1087 0.0815 0.0524 0.0224 0
1:3 0.3987 0.3471 0.3022 0.2633 0.2289 0.1979 0.1690 0.1412 0.1133 0.0845 0.0539 0.0228 0
1:2 0.5131 0.4509 0.3932 0.3408 0.2931 0.2493 0.2088 0.1707 0.1342 0.0980 0.0613 0.0252 0
77
Table A.3.4 Ka_h for Design Charts when =30° =1/3 D=1/2
Back-slope
(v):(h)
=30 /=1/3 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60
Log-spiral: Ka_h
1: ∞ 0.3334 0.2990 0.2687 0.2413 0.2158 0.1912 0.1665 0.1408 0.1138 0.0849 0.0541 0.0229 0
1:25 0.3384 0.3031 0.2721 0.2440 0.2180 0.1929 0.1677 0.1418 0.1144 0.0853 0.0544 0.0230 0
1:15 0.3419 0.3061 0.2745 0.2460 0.2195 0.1941 0.1686 0.1424 0.1150 0.0857 0.0545 0.0230 0
1:10 0.3465 0.3099 0.2777 0.2486 0.2216 0.1957 0.1698 0.1434 0.1156 0.0861 0.0548 0.0231 0
1:7.5 0.3514 0.3140 0.2811 0.2514 0.2238 0.1974 0.1711 0.1444 0.1164 0.0866 0.0550 0.0232 0
1:5 0.3636 0.3238 0.2890 0.2577 0.2288 0.2014 0.1743 0.1468 0.1182 0.0878 0.0556 0.0233 0
1:4 0.3755 0.3334 0.2966 0.2638 0.2337 0.2052 0.1773 0.1491 0.1199 0.0889 0.0562 0.0235 0
1:3 0.4023 0.3553 0.3142 0.2778 0.2447 0.2138 0.1840 0.1542 0.1236 0.0914 0.0576 0.0239 0
1:2 0.5131 0.4511 0.3952 0.3446 0.2984 0.2557 0.2159 0.1780 0.1408 0.1031 0.0643 0.0262 0
78
Table A.3.5 Ka_h for Design Charts when =30° =2/3 D=1/3
Back-slope
(v):(h)
=30 /=2/3 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60
Log-spiral: Ka_h
1: ∞ 0.2821 0.2522 0.2265 0.2039 0.1833 0.1637 0.1442 0.1241 0.1023 0.0780 0.0510 0.0221 0
1:25 0.2894 0.2582 0.2314 0.2078 0.1863 0.1660 0.1460 0.1253 0.1031 0.0786 0.0512 0.0222 0
1:15 0.2947 0.2625 0.2349 0.2105 0.1885 0.1677 0.1473 0.1263 0.1038 0.0790 0.0514 0.0223 0
1:10 0.3018 0.2683 0.2396 0.2143 0.1915 0.1700 0.1490 0.1276 0.1047 0.0795 0.0517 0.0223 0
1:7.5 0.3094 0.2746 0.2447 0.2185 0.1948 0.1726 0.1510 0.1290 0.1056 0.0801 0.0520 0.0224 0
1:5 0.3269 0.2891 0.2566 0.2281 0.2024 0.1785 0.1555 0.1323 0.1079 0.0816 0.0527 0.0226 0
1:4 0.3424 0.3021 0.2674 0.2368 0.2094 0.1840 0.1597 0.1354 0.1101 0.0829 0.0534 0.0228 0
1:3 0.3745 0.3294 0.2902 0.2556 0.2245 0.1959 0.1688 0.1421 0.1148 0.0859 0.0549 0.0232 0
1:2 0.4929 0.4346 0.3815 0.3331 0.2886 0.2474 0.2087 0.1719 0.1360 0.0998 0.0626 0.0257 0
79
Table A.3.6 Ka_h for Design Charts when =30° =2/3 D=1/2
Back-slope
(v):(h)
=30 /=2/3 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60
Log-spiral: Ka_h
1: ∞ 0.3125 0.2835 0.2573 0.2332 0.2102 0.1875 0.1643 0.1395 0.1130 0.0845 0.0540 0.0229 0
1:25 0.3177 0.2879 0.2610 0.2361 0.2125 0.1894 0.1657 0.1405 0.1138 0.0850 0.0542 0.0229 0
1:15 0.3214 0.2909 0.2635 0.2382 0.2142 0.1907 0.1666 0.1413 0.1143 0.0853 0.0544 0.0230 0
1:10 0.3263 0.2950 0.2669 0.2410 0.2165 0.1924 0.1679 0.1423 0.1150 0.0858 0.0546 0.0231 0
1:7.5 0.3314 0.2994 0.2706 0.2440 0.2189 0.1943 0.1694 0.1434 0.1159 0.0864 0.0549 0.0231 0
1:5 0.3435 0.3092 0.2787 0.2507 0.2243 0.1985 0.1727 0.1460 0.1178 0.0876 0.0556 0.0233 0
1:4 0.3552 0.3187 0.2863 0.2568 0.2292 0.2025 0.1758 0.1485 0.1196 0.0888 0.0562 0.0235 0
1:3 0.3816 0.3404 0.3039 0.2709 0.2404 0.2114 0.1828 0.1539 0.1236 0.0915 0.0577 0.0239 0
1:2 0.4931 0.4363 0.3849 0.3379 0.2945 0.2540 0.2157 0.1786 0.1418 0.1040 0.0649 0.0263 0
80
Table A.3.7 Ka_h for Design Charts when =30° =1 D=1/3
Back-slope
(v):(h)
=30 /=1 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60
Log-spiral: Ka_h
1: ∞ 0.2633 0.2389 0.2175 0.1983 0.1804 0.1628 0.1448 0.1255 0.1041 0.0797 0.0521 0.0226 0
1:25 0.2704 0.2447 0.2222 0.2021 0.1834 0.1652 0.1466 0.1269 0.1050 0.0803 0.0524 0.0226 0
1:15 0.2755 0.2489 0.2256 0.2048 0.1855 0.1669 0.1479 0.1278 0.1057 0.0807 0.0526 0.0227 0
1:10 0.2823 0.2545 0.2303 0.2086 0.1885 0.1692 0.1497 0.1291 0.1066 0.0812 0.0529 0.0227 0
1:7.5 0.2897 0.2607 0.2353 0.2127 0.1918 0.1717 0.1516 0.1306 0.1076 0.0819 0.0532 0.0228 0
1:5 0.3067 0.2748 0.2470 0.2222 0.1994 0.1777 0.1562 0.1340 0.1100 0.0834 0.0539 0.0230 0
1:4 0.3219 0.2876 0.2576 0.2308 0.2063 0.1831 0.1604 0.1371 0.1122 0.0848 0.0546 0.0232 0
1:3 0.3534 0.3145 0.2802 0.2494 0.2213 0.1950 0.1696 0.1439 0.1170 0.0879 0.0563 0.0237 0
1:2 0.4736 0.4202 0.3716 0.3269 0.2854 0.2465 0.2096 0.1740 0.1386 0.1023 0.0643 0.0262 0
81
Table A.3.8 Ka_h for Design Charts when =30° =1 D=1/2
Back-slope
(v):(h)
=30 /=1 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55 60
Log-spiral: Ka_h
1: ∞ 0.2958 0.2710 0.2482 0.2267 0.2058 0.1849 0.1629 0.1388 0.1126 0.0842 0.0538 0.0228 0
1:25 0.3012 0.2756 0.2521 0.2299 0.2084 0.1869 0.1644 0.1399 0.1134 0.0847 0.0541 0.0229 0
1:15 0.3051 0.2789 0.2548 0.2322 0.2102 0.1883 0.1655 0.1407 0.1140 0.0851 0.0543 0.0229 0
1:10 0.3102 0.2832 0.2585 0.2352 0.2127 0.1902 0.1669 0.1418 0.1148 0.0857 0.0545 0.0230 0
1:7.5 0.3156 0.2878 0.2624 0.2384 0.2153 0.1923 0.1685 0.1430 0.1157 0.0862 0.0548 0.0231 0
1:5 0.3277 0.2981 0.2710 0.2457 0.2212 0.1970 0.1720 0.1458 0.1178 0.0877 0.0556 0.0233 0
1:4 0.3389 0.3072 0.2786 0.2519 0.2263 0.2011 0.1753 0.1484 0.1198 0.0890 0.0563 0.0235 0
1:3 0.3646 0.3284 0.2958 0.2659 0.2376 0.2102 0.1827 0.1542 0.1241 0.0919 0.0578 0.0239 0
1:2 0.4756 0.4238 0.3764 0.3326 0.2918 0.2533 0.2163 0.1801 0.1434 0.1053 0.0656 0.0265 0
82
Table A.4.1 Ka_h for Design Charts when =35° =0 D=1/3
Back-slope
(v):(h)
=35 /=0 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55
Log-spiral: Ka_h
1: ∞ 0.2710 0.2300 0.1963 0.1678 0.1430 0.1205 0.0994 0.0788 0.0580 0.0366 0.0154 0
1:25 0.2768 0.2345 0.1998 0.1704 0.1449 0.1218 0.1003 0.0794 0.0583 0.0367 0.0154 0
1:15 0.2809 0.2377 0.2022 0.1722 0.1462 0.1228 0.1010 0.0798 0.0585 0.0368 0.0155 0
1:10 0.2862 0.2420 0.2054 0.1747 0.1480 0.1241 0.1018 0.0803 0.0588 0.0370 0.0155 0
1:7.5 0.2919 0.2465 0.2089 0.1773 0.1499 0.1255 0.1028 0.0809 0.0592 0.0371 0.0155 0
1:5 0.3043 0.2566 0.2167 0.1832 0.1543 0.1286 0.1049 0.0823 0.0600 0.0375 0.0156 0
1:4 0.3148 0.2653 0.2235 0.1884 0.1582 0.1314 0.1069 0.0836 0.0607 0.0378 0.0157 0
1:3 0.3350 0.2824 0.2370 0.1988 0.1660 0.1370 0.1108 0.0861 0.0622 0.0385 0.0159 0
1:2 0.3929 0.3330 0.2791 0.2321 0.1915 0.1559 0.1240 0.0948 0.0674 0.0410 0.0165 0
83
Table A.4.2 Ka_h for Design Charts when =35° =0 D=1/2
Back-slope
(v):(h)
=35 /=0 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55
Log-spiral: Ka_h
1: ∞ 0.2901 0.2521 0.2193 0.1901 0.1635 0.1384 0.1140 0.0897 0.0652 0.0404 0.0165 0
1:25 0.2937 0.2549 0.2214 0.1918 0.1647 0.1393 0.1147 0.0902 0.0655 0.0405 0.0166 0
1:15 0.2961 0.2568 0.2230 0.1930 0.1656 0.1399 0.1151 0.0905 0.0656 0.0406 0.0166 0
1:10 0.2995 0.2595 0.2250 0.1945 0.1668 0.1408 0.1157 0.0909 0.0659 0.0407 0.0166 0
1:7.5 0.3033 0.2624 0.2272 0.1962 0.1681 0.1418 0.1164 0.0914 0.0662 0.0409 0.0167 0
1:5 0.3122 0.2693 0.2325 0.2002 0.1711 0.1440 0.1180 0.0925 0.0668 0.0412 0.0167 0
1:4 0.3203 0.2756 0.2373 0.2038 0.1737 0.1459 0.1194 0.0934 0.0674 0.0415 0.0168 0
1:3 0.3373 0.2888 0.2474 0.2114 0.1794 0.1501 0.1223 0.0954 0.0687 0.0421 0.0170 0
1:2 0.3930 0.3336 0.2827 0.2384 0.1995 0.1646 0.1326 0.1023 0.0730 0.0443 0.0176 0
84
Table A.4.3 Ka_h for Design Charts when =35° =1/3 D=1/3
Back-slope
(v):(h)
=35 /=1/3 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55
Log-spiral: Ka_h
1: ∞ 0.2462 0.2131 0.1850 0.1606 0.1387 0.1183 0.0986 0.0788 0.0584 0.0371 0.0156 0
1:25 0.2518 0.2174 0.1883 0.1631 0.1405 0.1196 0.0995 0.0794 0.0587 0.0372 0.0157 0
1:15 0.2557 0.2204 0.1907 0.1649 0.1418 0.1206 0.1002 0.0799 0.0590 0.0373 0.0157 0
1:10 0.2609 0.2245 0.1938 0.1673 0.1436 0.1219 0.1011 0.0804 0.0593 0.0375 0.0157 0
1:7.5 0.2664 0.2289 0.1972 0.1698 0.1455 0.1232 0.1020 0.0811 0.0597 0.0376 0.0160 0
1:5 0.2787 0.2386 0.2048 0.1756 0.1499 0.1264 0.1042 0.0825 0.0605 0.0380 0.0158 0
1:4 0.2892 0.2470 0.2114 0.1807 0.1537 0.1292 0.1061 0.0837 0.0612 0.0383 0.0159 0
1:3 0.3095 0.2637 0.2246 0.1910 0.1615 0.1348 0.1101 0.0863 0.0628 0.0391 0.0161 0
1:2 0.3687 0.3145 0.2661 0.2240 0.1868 0.1536 0.1234 0.0952 0.0681 0.0417 0.0168 0
85
Table A.4.4 Ka_h for Design Charts when =35° =1/3 D=1/2
Back-slope
(v):(h)
=35 /=1/3 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55
Log-spiral: Ka_h
1: ∞ 0.2664 0.2350 0.2071 0.1816 0.1578 0.1347 0.1117 0.0884 0.0645 0.0401 0.0165 0
1:25 0.2701 0.2380 0.2094 0.1834 0.1591 0.1357 0.1124 0.0889 0.0648 0.0403 0.0165 0
1:15 0.2727 0.2400 0.2110 0.1847 0.1601 0.1363 0.1129 0.0892 0.0650 0.0404 0.0165 0
1:10 0.2762 0.2428 0.2132 0.1863 0.1613 0.1373 0.1136 0.0897 0.0653 0.0405 0.0166 0
1:7.5 0.2800 0.2457 0.2155 0.1881 0.1626 0.1383 0.1143 0.0902 0.0656 0.0406 0.0166 0
1:5 0.2890 0.2527 0.2208 0.1922 0.1657 0.1406 0.1160 0.0914 0.0663 0.0410 0.0167 0
1:4 0.2971 0.2590 0.2257 0.1959 0.1685 0.1427 0.1175 0.0924 0.0670 0.0413 0.0168 0
1:3 0.3139 0.2722 0.2359 0.2037 0.1744 0.1470 0.1206 0.0945 0.0683 0.0419 0.0169 0
1:2 0.3690 0.3168 0.2712 0.2309 0.1949 0.1620 0.1313 0.1018 0.0729 0.0443 0.0176 0
86
Table A.4.5 Ka_h for Design Charts when =35° =2/3 D=1/3
Back-slope
(v):(h)
=35 /=2/3 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55
Log-spiral: Ka_h
1: ∞ 0.2266 0.1993 0.1756 0.1546 0.1352 0.1167 0.0983 0.0792 0.0591 0.0376 0.0159 0
1:25 0.2318 0.2034 0.1789 0.1571 0.1371 0.1180 0.0992 0.0798 0.0594 0.0378 0.0159 0
1:15 0.2356 0.2064 0.1812 0.1588 0.1384 0.1190 0.0999 0.0803 0.0597 0.0379 0.0159 0
1:10 0.2405 0.2103 0.1842 0.1612 0.1401 0.1203 0.1007 0.0809 0.0600 0.0380 0.0160 0
1:7.5 0.2458 0.2145 0.1875 0.1637 0.1420 0.1216 0.1017 0.0815 0.0604 0.0382 0.0160 0
1:5 0.2576 0.2239 0.1949 0.1694 0.1463 0.1248 0.1039 0.0829 0.0612 0.0386 0.0161 0
1:4 0.2677 0.2321 0.2014 0.1744 0.1501 0.1276 0.1059 0.0842 0.0620 0.0390 0.0162 0
1:3 0.2876 0.2483 0.2144 0.1846 0.1578 0.1333 0.1099 0.0869 0.0636 0.0397 0.0164 0
1:2 0.3474 0.2984 0.2553 0.2172 0.1831 0.1521 0.1233 0.0959 0.0691 0.0424 0.0171 0
87
Table A. 4.6 Ka_h for Design Charts when =35° =2/3 D=1/2
Back-slope
(v):(h)
=35 /=2/3 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55
Log-spiral: Ka_h
1: ∞ 0.2481 0.2215 0.1973 0.1747 0.1531 0.1317 0.1098 0.0873 0.0639 0.0399 0.0164 0
1:25 0.2520 0.2246 0.1997 0.1766 0.1545 0.1327 0.1106 0.0878 0.0643 0.0400 0.0165 0
1:15 0.2547 0.2268 0.2015 0.1780 0.1555 0.1334 0.1111 0.0882 0.0645 0.0401 0.0165 0
1:10 0.2582 0.2296 0.2037 0.1797 0.1569 0.1344 0.1118 0.0887 0.0648 0.0403 0.0165 0
1:7.5 0.2619 0.2326 0.2061 0.1816 0.1583 0.1355 0.1126 0.0893 0.0651 0.0404 0.0166 0
1:5 0.2707 0.2395 0.2115 0.1858 0.1615 0.1379 0.1144 0.0905 0.0659 0.0408 0.0167 0
1:4 0.2786 0.2457 0.2164 0.1895 0.1644 0.1401 0.1160 0.0916 0.0666 0.0411 0.0167 0
1:3 0.2951 0.2587 0.2265 0.1974 0.1703 0.1446 0.1193 0.0939 0.0680 0.0418 0.0169 0
1:2 0.3492 0.3027 0.2616 0.2247 0.1912 0.1601 0.1305 0.1016 0.0729 0.0444 0.0176 0
88
Table A.4.7 Ka_h for Design Charts when =35° =1 D=1/3
Back-slope
(v):(h)
=35 /=1 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55
Log-spiral: Ka_h
1: ∞ 0.2094 0.1872 0.1675 0.1496 0.1326 0.1158 0.0986 0.0802 0.0601 0.0384 0.0162 0
1:25 0.2145 0.1912 0.1706 0.1520 0.1344 0.1172 0.0995 0.0808 0.0605 0.0386 0.0162 0
1:15 0.2180 0.1940 0.1729 0.1537 0.1357 0.1181 0.1002 0.0812 0.0608 0.0387 0.0162 0
1:10 0.2228 0.1978 0.1759 0.1560 0.1375 0.1194 0.1011 0.0818 0.0611 0.0389 0.0163 0
1:7.5 0.2278 0.2019 0.1791 0.1585 0.1394 0.1208 0.1021 0.0825 0.0615 0.0390 0.0163 0
1:5 0.2392 0.2110 0.1863 0.1642 0.1436 0.1240 0.1043 0.0840 0.0624 0.0395 0.0164 0
1:4 0.2489 0.2189 0.1926 0.1691 0.1474 0.1268 0.1063 0.0853 0.0632 0.0398 0.0165 0
1:3 0.2682 0.2347 0.2053 0.1791 0.1551 0.1324 0.1104 0.0881 0.0648 0.0406 0.0167 0
1:2 0.3271 0.2841 0.2458 0.2114 0.1802 0.1513 0.1240 0.0973 0.0705 0.0435 0.0175 0
89
Table A.4.8 Ka_h for Design Charts when =35° =1 D=1/2
Back-slope
(v):(h)
=35 /=1 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50 55
Log-spiral: Ka_h
1: ∞ 0.2328 0.2100 0.1889 0.1688 0.1491 0.1292 0.1084 0.0864 0.0635 0.0397 0.0164 0
1:25 0.2368 0.2133 0.1915 0.1709 0.1507 0.1304 0.1092 0.0870 0.0638 0.0398 0.0164 0
1:15 0.2395 0.2156 0.1933 0.1723 0.1518 0.1312 0.1097 0.0874 0.0640 0.0399 0.0164 0
1:10 0.2432 0.2186 0.1958 0.1742 0.1533 0.1322 0.1105 0.0879 0.0644 0.0401 0.0165 0
1:7.5 0.2471 0.2217 0.1983 0.1762 0.1548 0.1334 0.1114 0.0885 0.0647 0.0403 0.0165 0
1:5 0.2555 0.2286 0.2039 0.1807 0.1582 0.1359 0.1133 0.0899 0.0656 0.0407 0.0166 0
1:4 0.2631 0.2346 0.2087 0.1845 0.1612 0.1382 0.1150 0.0911 0.0663 0.0410 0.0167 0
1:3 0.2789 0.2473 0.2187 0.1923 0.1673 0.1428 0.1184 0.0935 0.0679 0.0418 0.0169 0
1:2 0.3317 0.2905 0.2534 0.2196 0.1883 0.1588 0.1302 0.1018 0.0732 0.0446 0.0177 0
90
Table A.5.1 Ka_h for Design Charts when =40° =0 D=1/3
Back-slope
(v):(h)
=40 /=0 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50
Log-spiral: Ka_h
1: ∞ 0.2174 0.1808 0.1506 0.1250 0.1028 0.0827 0.0639 0.0459 0.0282 0.0116 0
1:25 0.2216 0.1838 0.1528 0.1266 0.1039 0.0834 0.0644 0.0461 0.0283 0.0116 0
1:15 0.2245 0.1860 0.1544 0.1277 0.1046 0.0839 0.0647 0.0463 0.0284 0.0116 0
1:10 0.2283 0.1888 0.1564 0.1292 0.1056 0.0846 0.0651 0.0465 0.0285 0.0116 0
1:7.5 0.2322 0.1919 0.1586 0.1308 0.1067 0.0853 0.0655 0.0467 0.0286 0.0116 0
1:5 0.2408 0.1985 0.1635 0.1342 0.1091 0.0868 0.0665 0.0472 0.0288 0.0117 0
1:4 0.2479 0.2040 0.1676 0.1372 0.1111 0.0882 0.0673 0.0477 0.0290 0.0117 0
1:3 0.2611 0.2146 0.1755 0.1429 0.1151 0.0908 0.0689 0.0486 0.0294 0.0118 0
1:2 0.2957 0.2433 0.1975 0.1592 0.1266 0.0985 0.0737 0.0513 0.0306 0.0121 0
91
Table A.5.2 Ka_h for Design Charts when =40° =0 D=1/2
Back-slope
(v):(h)
=40 /=0 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50
Log-spiral: Ka_h
1: ∞ 0.2305 0.1964 0.1669 0.1406 0.1166 0.0942 0.0726 0.0516 0.0312 0.0124 0
1:25 0.2332 0.1984 0.1683 0.1417 0.1174 0.0947 0.0729 0.0518 0.0313 0.0125 0
1:15 0.2352 0.1999 0.1694 0.1424 0.1179 0.0950 0.0732 0.0519 0.0314 0.0125 0
1:10 0.2378 0.2018 0.1708 0.1435 0.1186 0.0955 0.0735 0.0521 0.0314 0.0125 0
1:7.5 0.2406 0.2039 0.1723 0.1445 0.1194 0.0960 0.0738 0.0523 0.0315 0.0125 0
1:5 0.2471 0.2087 0.1758 0.1470 0.1211 0.0972 0.0745 0.0527 0.0317 0.0126 0
1:4 0.2527 0.2128 0.1788 0.1492 0.1226 0.0982 0.0752 0.0531 0.0319 0.0126 0
1:3 0.2638 0.2210 0.1848 0.1534 0.1256 0.1002 0.0765 0.0538 0.0322 0.0127 0
1:2 0.2959 0.2454 0.2028 0.1663 0.1346 0.1062 0.0803 0.0560 0.0333 0.0129 0
92
Table A.5.3 Ka_h for Design Charts when =40° =1/3 D=1/3
Back-slope
(v):(h)
=40 /=1/3 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50
Log-spiral: Ka_h
1: ∞ 0.1970 0.1670 0.1416 0.1196 0.0997 0.0813 0.0635 0.0460 0.0285 0.0117 0
1:25 0.2009 0.1700 0.1438 0.1211 0.1008 0.0820 0.0640 0.0462 0.0286 0.0117 0
1:15 0.2037 0.1720 0.1453 0.1222 0.1015 0.0825 0.0643 0.0464 0.0287 0.0117 0
1:10 0.2073 0.1748 0.1473 0.1236 0.1025 0.0831 0.0647 0.0466 0.0288 0.0118 0
1:7.5 0.2111 0.1776 0.1495 0.1252 0.1036 0.0839 0.0651 0.0469 0.0289 0.0118 0
1:5 0.2195 0.1840 0.1542 0.1286 0.1060 0.0854 0.0661 0.0474 0.0291 0.0118 0
1:4 0.2264 0.1893 0.1582 0.1315 0.1080 0.0868 0.0669 0.0479 0.0293 0.0119 0
1:3 0.2395 0.1995 0.1658 0.1371 0.1120 0.0894 0.0686 0.0488 0.0297 0.0120 0
1:2 0.2743 0.2275 0.1874 0.1531 0.1234 0.0972 0.0735 0.0515 0.0309 0.0123 0
93
Table A.5.4 Ka_h for Design Charts when =40° =1/3 D=1/2
Back-slope
(v):(h)
=40 /=1/3 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50
Log-spiral: Ka_h
1: ∞ 0.2110 0.1826 0.1573 0.1341 0.1124 0.0915 0.0711 0.0509 0.0309 0.0124 0
1:25 0.2138 0.1847 0.1588 0.1352 0.1132 0.0921 0.0715 0.0511 0.0310 0.0124 0
1:15 0.2157 0.1861 0.1599 0.1360 0.1137 0.0925 0.0717 0.0512 0.0311 0.0124 0
1:10 0.2184 0.1881 0.1613 0.1371 0.1145 0.0930 0.0720 0.0514 0.0312 0.0124 0
1:7.5 0.2212 0.1902 0.1629 0.1382 0.1153 0.0935 0.0724 0.0516 0.0313 0.0125 0
1:5 0.2276 0.1950 0.1664 0.1407 0.1171 0.0948 0.0732 0.0521 0.0315 0.0125 0
1:4 0.2331 0.1991 0.1694 0.1429 0.1186 0.0958 0.0739 0.0525 0.0317 0.0126 0
1:3 0.2439 0.2073 0.1755 0.1473 0.1217 0.0979 0.0752 0.0532 0.0320 0.0126 0
1:2 0.2753 0.2313 0.1934 0.1603 0.1309 0.1042 0.0793 0.0556 0.0331 0.0129 0
94
Table A.5.5 Ka_h for Design Charts when =40° =2/3 D=1/3
Back-slope
(v):(h)
=40 /=2/3 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50
Log-spiral: Ka_h
1: ∞ 0.1802 0.1555 0.1340 0.1149 0.0972 0.0802 0.0634 0.0463 0.0289 0.0119 0
1:25 0.1840 0.1583 0.1361 0.1164 0.0982 0.0809 0.0638 0.0465 0.0290 0.0119 0
1:15 0.1866 0.1603 0.1376 0.1175 0.0990 0.0814 0.0641 0.0467 0.0290 0.0119 0
1:10 0.1900 0.1629 0.1396 0.1189 0.1000 0.0821 0.0646 0.0469 0.0291 0.0119 0
1:7.5 0.1936 0.1657 0.1416 0.1204 0.1010 0.0828 0.0650 0.0472 0.0292 0.0120 0
1:5 0.2016 0.1718 0.1462 0.1237 0.1034 0.0844 0.0660 0.0477 0.0295 0.0120 0
1:4 0.2082 0.1769 0.1501 0.1266 0.1054 0.0857 0.0669 0.0482 0.0297 0.0121 0
1:3 0.2208 0.1868 0.1576 0.1321 0.1093 0.0884 0.0685 0.0492 0.0301 0.0122 0
1:2 0.2551 0.2140 0.1787 0.1479 0.1207 0.0961 0.0735 0.0520 0.0314 0.0125 0
95
Table A.5.6 Ka_h for Design Charts when =40° =2/3 D=1/2
Back-slope
(v):(h)
=40 /=2/3 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50
Log-spiral: Ka_h
1: ∞ 0.1955 0.1713 0.1492 0.1286 0.1088 0.0893 0.0698 0.0502 0.0307 0.0123 0
1:25 0.1983 0.1735 0.1509 0.1298 0.1096 0.0898 0.0702 0.0504 0.0308 0.0124 0
1:15 0.2002 0.1750 0.1520 0.1306 0.1102 0.0903 0.0704 0.0506 0.0308 0.0124 0
1:10 0.2028 0.1769 0.1534 0.1317 0.1110 0.0908 0.0708 0.0508 0.0309 0.0124 0
1:7.5 0.2055 0.1790 0.1550 0.1328 0.1118 0.0914 0.0712 0.0510 0.0310 0.0124 0
1:5 0.2117 0.1836 0.1585 0.1354 0.1137 0.0927 0.0720 0.0515 0.0312 0.0125 0
1:4 0.2170 0.1877 0.1615 0.1376 0.1153 0.0938 0.0727 0.0519 0.0314 0.0125 0
1:3 0.2276 0.1957 0.1675 0.1420 0.1184 0.0960 0.0742 0.0528 0.0318 0.0126 0
1:2 0.2581 0.2193 0.1854 0.1552 0.1278 0.1025 0.0785 0.0553 0.0330 0.0129 0
96
Table A.5.7 Ka_h for Design Charts when =40° =1 D=1/3
Back-slope
(v):(h)
=40 /=1 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45 50
Log-spiral: Ka_h
1: ∞ 0.1649 0.1449 0.1271 0.1107 0.0951 0.0796 0.0637 0.0469 0.0294 0.0121 0
1:25 0.1684 0.1476 0.1291 0.1122 0.0961 0.0803 0.0641 0.0472 0.0295 0.0121 0
1:15 0.1709 0.1495 0.1305 0.1132 0.0969 0.0808 0.0644 0.0474 0.0296 0.0122 0
1:10 0.1741 0.1520 0.1324 0.1146 0.0979 0.0815 0.0649 0.0476 0.0297 0.0122 0
1:7.5 0.1776 0.1546 0.1344 0.1161 0.0989 0.0822 0.0653 0.0479 0.0298 0.0122 0
1:5 0.1851 0.1605 0.1389 0.1194 0.1013 0.0838 0.0663 0.0484 0.0301 0.0123 0
1:4 0.1915 0.1654 0.1426 0.1222 0.1033 0.0852 0.0672 0.0489 0.0303 0.0123 0
1:3 0.2036 0.1749 0.1499 0.1276 0.1072 0.0878 0.0689 0.0499 0.0307 0.0124 0
1:2 0.2367 0.2015 0.1706 0.1432 0.1185 0.0956 0.0740 0.0528 0.0321 0.0127 0
97
Table A.5.8 Ka_h for Design Charts when =40° =1 D=1/2
Back-slope
(v):(h)
=40 /=1 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45 50
Log-spiral: Ka_h
1: ∞ 0.1817 0.1611 0.1419 0.1235 0.1055 0.0872 0.0686 0.0495 0.0304 0.0123 0
1:25 0.1846 0.1634 0.1436 0.1248 0.1064 0.0879 0.0690 0.0498 0.0305 0.0123 0
1:15 0.1866 0.1649 0.1448 0.1257 0.1070 0.0883 0.0693 0.0499 0.0306 0.0123 0
1:10 0.1891 0.1670 0.1464 0.1269 0.1079 0.0889 0.0696 0.0502 0.0307 0.0123 0
1:7.5 0.1918 0.1691 0.1480 0.1281 0.1087 0.0895 0.0701 0.0504 0.0308 0.0124 0
1:5 0.1977 0.1737 0.1516 0.1307 0.1107 0.0908 0.0710 0.0509 0.0310 0.0124 0
1:4 0.2028 0.1776 0.1545 0.1330 0.1123 0.0920 0.0718 0.0514 0.0312 0.0125 0
1:3 0.2128 0.1853 0.1605 0.1374 0.1156 0.0943 0.0733 0.0523 0.0317 0.0126 0
1:2 0.2422 0.2083 0.1781 0.1506 0.1252 0.1012 0.0779 0.0551 0.0330 0.0129 0
98
Table A.6.1 Ka_h for Design Charts when =45° =0 D=1/3
Back-slope
(v):(h)
=45 /=0 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45
Log-spiral: Ka_h
1: ∞ 0.1716 0.1391 0.1125 0.0901 0.0707 0.0534 0.0374 0.0224 0.0090 0
1:25 0.1745 0.1411 0.1139 0.0910 0.0713 0.0537 0.0375 0.0225 0.0090 0
1:15 0.1765 0.1425 0.1148 0.0917 0.0717 0.0539 0.0377 0.0226 0.0090 0
1:10 0.1791 0.1444 0.1161 0.0925 0.0722 0.0542 0.0378 0.0226 0.0090 0
1:7.5 0.1818 0.1464 0.1175 0.0934 0.0728 0.0546 0.0380 0.0227 0.0090 0
1:5 0.1876 0.1506 0.1204 0.0953 0.0740 0.0553 0.0384 0.0228 0.0090 0
1:4 0.1923 0.1541 0.1228 0.0970 0.0750 0.0559 0.0387 0.0230 0.0091 0
1:3 0.2009 0.1605 0.1273 0.1000 0.0770 0.0570 0.0393 0.0232 0.0091 0
1:2 0.2222 0.1771 0.1391 0.1080 0.0821 0.0601 0.0409 0.0239 0.0093 0
99
Table A.6.2 Ka_h for Design Charts when =45° =0 D=1/2
Back-slope
(v):(h)
=45 /=0 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45
Log-spiral: Ka_h
1: ∞ 0.1805 0.1503 0.1241 0.1009 0.0798 0.0604 0.0420 0.0249 0.0097 0
1:25 0.1826 0.1517 0.1250 0.1015 0.0803 0.0606 0.0422 0.0250 0.0097 0
1:15 0.1840 0.1527 0.1257 0.1020 0.0806 0.0608 0.0423 0.0250 0.0097 0
1:10 0.1860 0.1540 0.1267 0.1026 0.0810 0.0610 0.0424 0.0250 0.0097 0
1:7.5 0.1880 0.1555 0.1276 0.1033 0.0814 0.0613 0.0425 0.0251 0.0097 0
1:5 0.1925 0.1586 0.1298 0.1047 0.0823 0.0618 0.0428 0.0252 0.0098 0
1:4 0.1963 0.1613 0.1316 0.1059 0.0831 0.0623 0.0431 0.0253 0.0098 0
1:3 0.2035 0.1664 0.1351 0.1082 0.0846 0.0632 0.0436 0.0255 0.0098 0
1:2 0.2228 0.1801 0.1446 0.1146 0.0886 0.0656 0.0449 0.0261 0.0100 0
100
Table A.6.3 Ka_h for Design Charts when =45° =1/3 D=1/3
Back-slope
(v):(h)
=45 /=1/3 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45
Log-spiral: Ka_h
1: ∞ 0.1553 0.1285 0.1059 0.0863 0.0688 0.0526 0.0373 0.0226 0.0091 0
1:25 0.1580 0.1304 0.1072 0.0872 0.0694 0.0530 0.0375 0.0227 0.0091 0
1:15 0.1599 0.1318 0.1082 0.0878 0.0698 0.0532 0.0376 0.0227 0.0091 0
1:10 0.1624 0.1336 0.1094 0.0887 0.0703 0.0535 0.0377 0.0228 0.0091 0
1:7.5 0.1650 0.1354 0.1107 0.0895 0.0708 0.0538 0.0379 0.0228 0.0091 0
1:5 0.1706 0.1395 0.1136 0.0914 0.0721 0.0545 0.0383 0.0230 0.0092 0
1:4 0.1752 0.1428 0.1159 0.0930 0.0731 0.0551 0.0386 0.0231 0.0092 0
1:3 0.1836 0.1490 0.1203 0.0960 0.0750 0.0563 0.0392 0.0234 0.0092 0
1:2 0.2047 0.1650 0.1318 0.1039 0.0801 0.0594 0.0408 0.0241 0.0094 0
101
Table A.6.4 Ka_h for Design Charts when =45° =1/3 D=1/2
Back-slope
(v):(h)
=45 /=1/3 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45
Log-spiral: Ka_h
1: ∞ 0.1651 0.1396 0.1169 0.0962 0.0770 0.0588 0.0413 0.0246 0.0096 0
1:25 0.1671 0.1410 0.1179 0.0969 0.0774 0.0590 0.0414 0.0247 0.0097 0
1:15 0.1685 0.1420 0.1186 0.0974 0.0778 0.0592 0.0415 0.0247 0.0097 0
1:10 0.1704 0.1434 0.1195 0.0980 0.0782 0.0595 0.0417 0.0248 0.0097 0
1:7.5 0.1724 0.1448 0.1205 0.0987 0.0786 0.0597 0.0418 0.0248 0.0097 0
1:5 0.1768 0.1479 0.1227 0.1002 0.0796 0.0603 0.0421 0.0250 0.0097 0
1:4 0.1805 0.1506 0.1245 0.1014 0.0804 0.0608 0.0424 0.0251 0.0097 0
1:3 0.1876 0.1556 0.1280 0.1038 0.0819 0.0618 0.0429 0.0253 0.0098 0
1:2 0.2062 0.1696 0.1376 0.1102 0.0861 0.0643 0.0443 0.0259 0.0099 0
102
Table A.6.5 Ka_h for Design Charts when =45° =2/3 D=1/3
Back-slope
(v):(h)
=45 /=2/3 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45
Log-spiral: Ka_h
1: ∞ 0.1634 0.1378 0.1156 0.0957 0.0775 0.0600 0.0431 0.0263 0.0106 0
1:25 0.1664 0.1398 0.1171 0.0968 0.0782 0.0605 0.0433 0.0264 0.0106 0
1:15 0.1684 0.1413 0.1181 0.0975 0.0786 0.0607 0.0434 0.0264 0.0106 0
1:10 0.1711 0.1433 0.1195 0.0984 0.0792 0.0611 0.0436 0.0266 0.0106 0
1:7.5 0.1739 0.1454 0.1209 0.0994 0.0799 0.0615 0.0438 0.0267 0.0107 0
1:5 0.1800 0.1499 0.1241 0.1016 0.0813 0.0624 0.0442 0.0268 0.0107 0
1:4 0.1850 0.1536 0.1268 0.1033 0.0824 0.0630 0.0447 0.0270 0.0107 0
1:3 0.1943 0.1605 0.1316 0.1068 0.0846 0.0644 0.0454 0.0273 0.0109 0
1:2 0.2179 0.1782 0.1446 0.1157 0.0905 0.0680 0.0472 0.0281 0.0110 0
103
Table A.6.6 Ka_h for Design Charts when =45° =2/3 D=1/2
Back-slope
(v):(h)
=45 /=2/3 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45
Log-spiral: Ka_h
1: ∞ 0.1759 0.1507 0.1277 0.1063 0.0859 0.0662 0.0468 0.0281 0.0111 0
1:25 0.1781 0.1523 0.1290 0.1072 0.0865 0.0665 0.0470 0.0282 0.0111 0
1:15 0.1797 0.1535 0.1298 0.1077 0.0868 0.0667 0.0471 0.0282 0.0111 0
1:10 0.1817 0.1550 0.1308 0.1084 0.0873 0.0670 0.0472 0.0283 0.0111 0
1:7.5 0.1841 0.1566 0.1320 0.1092 0.0879 0.0673 0.0475 0.0284 0.0111 0
1:5 0.1889 0.1602 0.1345 0.1110 0.0890 0.0680 0.0478 0.0285 0.0112 0
1:4 0.1931 0.1632 0.1366 0.1125 0.0900 0.0687 0.0482 0.0286 0.0112 0
1:3 0.2008 0.1688 0.1406 0.1152 0.0918 0.0697 0.0488 0.0290 0.0113 0
1:2 0.2216 0.1842 0.1515 0.1227 0.0968 0.0729 0.0506 0.0297 0.0114 0
104
Table A.6.7 Ka_h for Design Charts when =45° =1 D=1/3
Back-slope
(v):(h)
=45 /=1 D=1/3
[degrees]
0 5 10 15 20 25 30 35 40 45
Log-spiral: Ka_h
1: ∞ 0.1280 0.1102 0.0943 0.0797 0.0657 0.0517 0.0376 0.0232 0.0094 0
1:25 0.1305 0.1120 0.0956 0.0806 0.0662 0.0521 0.0378 0.0233 0.0094 0
1:15 0.1321 0.1132 0.0965 0.0812 0.0666 0.0523 0.0379 0.0233 0.0094 0
1:10 0.1343 0.1148 0.0976 0.0820 0.0672 0.0527 0.0381 0.0234 0.0094 0
1:7.5 0.1366 0.1165 0.0989 0.0828 0.0677 0.0530 0.0383 0.0235 0.0095 0
1:5 0.1416 0.1202 0.1015 0.0846 0.0689 0.0537 0.0387 0.0236 0.0095 0
1:4 0.1457 0.1233 0.1037 0.0862 0.0699 0.0543 0.0390 0.0238 0.0095 0
1:3 0.1533 0.1290 0.1079 0.0891 0.0718 0.0555 0.0396 0.0241 0.0096 0
1:2 0.1727 0.1438 0.1187 0.0967 0.0769 0.0587 0.0414 0.0248 0.0097 0
105
Table A.6.8 Ka_h for Design Charts when =45° /=1 D=1/2
Back-slope
(v):(h)
=45 /=1 D=1/2
[degrees]
0 5 10 15 20 25 30 35 40 45
Log-spiral: Ka_h
1: ∞ 0.1402 0.1217 0.1045 0.0881 0.0719 0.0558 0.0397 0.0240 0.0095 0
1:25 0.1421 0.1232 0.1056 0.0888 0.0724 0.0561 0.0399 0.0241 0.0095 0
1:15 0.1435 0.1242 0.1063 0.0893 0.0727 0.0563 0.0400 0.0241 0.0095 0
1:10 0.1453 0.1256 0.1073 0.0900 0.0732 0.0566 0.0402 0.0242 0.0096 0
1:7.5 0.1471 0.1269 0.1083 0.0907 0.0736 0.0569 0.0404 0.0243 0.0096 0
1:5 0.1511 0.1299 0.1105 0.0922 0.0747 0.0576 0.0407 0.0244 0.0096 0
1:4 0.1544 0.1324 0.1123 0.0935 0.0755 0.0581 0.0410 0.0246 0.0096 0
1:3 0.1609 0.1372 0.1157 0.0959 0.0772 0.0592 0.0416 0.0248 0.0097 0
1:2 0.1780 0.1500 0.1251 0.1025 0.0817 0.0620 0.0433 0.0256 0.0099 0
106
APPENDIX B
COMPARISON OF 𝐊𝐚_𝐡 FROM LOG SPIRAL EQUIVALENT COULOMB
AND 𝐊𝐚_𝐡 FROM CLASSICAL COULOMB
107
Table B.1.1 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=20° /=0 D=1/3)
=20 /=0 D=1/3
Back-
slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.4903 0.4671 0.4462 0.4273 0.4101 0.3944 0.3799 0.3666 0.3542 0.3427 0.3318
(0.4903) (0.4777) (0.4651) (0.4522) (0.4392) (0.4260) (0.4127) (0.3991) (0.3854) (0.3714) (0.3572)
1:25 0.5047 0.4808 0.4591 0.4394 0.4215 0.4051 0.3899 0.3760 0.3630 0.3509 0.3395
(0.5047) (0.4917) (0.4784) (0.4651) (0.4517) (0.4379) (0.4241) (0.4100) (0.3958) (0.3815) (0.3669)
1:15 0.5152 0.4910 0.4687 0.4485 0.4300 0.4131 0.3975 0.3830 0.3696 0.3571 0.3453
(0.5152) (0.5018) (0.4882) (0.4745) (0.4607) (0.4467) (0.4325) (0.4182) (0.4036) (0.3889) (0.3740)
1:10 0.5296 0.5050 0.4821 0.4611 0.4419 0.4243 0.4081 0.3930 0.3790 0.3659 0.3536
(0.5296) (0.5157) (0.5016) (0.4874) (0.4732) (0.4587) (0.4441) (0.4294) (0.4144) (0.3993) (0.3839)
1:7.5 0.5455 0.5206 0.4972 0.4755 0.4556 0.4372 0.4202 0.4045 0.3898 0.3761 0.3631
(0.5455) (0.5311) (0.5165) (0.5018) (0.4870) (0.4721) (0.4571) (0.4418) (0.4264) (0.4108) (0.3950)
1:5 0.5840 0.5587 0.5346 0.5117 0.4901 0.4701 0.4515 0.4341 0.4179 0.4025 0.3881
(0.5841) (0.5685) (0.5528) (0.5369) (0.5211) (0.5051) (0.4890) (0.4727) (0.4563) (0.4397) (0.4228)
1:4 0.6221 0.5965 0.5721 0.5486 0.5261 0.5047 0.4846 0.4658 0.4480 0.4312 0.4152
(0.6222) (0.6054) (0.5888) (0.5719) (0.5551) (0.5382) (0.5212) (0.5039) (0.4865) (0.4690) (0.4512)
1:3 0.7305 0.7052 0.6808 0.6571 0.6339 0.6112 0.5889 0.5672 0.5462 0.5259 0.5062
(0.7306) (0.7113) (0.6921) (0.6729) (0.6537) (0.6344) (0.6152) (0.5958) (0.5762) (0.5566) (0.5367)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
108
Table B.1.2 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=20° /=1/3 D=1/3)
=20 /=1/3 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.4560 0.4377 0.4209 0.4056 0.3915 0.3785 0.3666 0.3554 0.3449 0.3352 0.3259
(0.4550) (0.4438) (0.4325) (0.4210) (0.4093) (0.3973) (0.3851) (0.3728) (0.3602) (0.3474) (0.3344)
1:25 0.4703 0.4511 0.4336 0.4176 0.4028 0.3891 0.3765 0.3647 0.3538 0.3434 0.3335
(0.4699) (0.4582) (0.4463) (0.4343) (0.4220) (0.4095) (0.3969) (0.3840) (0.3710) (0.3578) (0.3443)
1:15 0.4809 0.4611 0.4431 0.4265 0.4112 0.3971 0.3840 0.3717 0.3603 0.3495 0.3394
(0.4807) (0.4686) (0.4564) (0.4440) (0.4314) (0.4186) (0.4055) (0.3924) (0.3790) (0.3653) (0.3516)
1:10 0.4955 0.4751 0.4563 0.4390 0.4230 0.4082 0.3945 0.3816 0.3696 0.3583 0.3476
(0.4956) (0.4830) (0.4703) (0.4574) (0.4443) (0.4310) (0.4175) (0.4039) (0.3901) (0.3760) (0.3617)
1:7.5 0.5122 0.4910 0.4713 0.4533 0.4365 0.4210 0.4066 0.3931 0.3804 0.3684 0.3571
(0.5122) (0.4990) (0.4858) (0.4723) (0.4587) (0.4449) (0.4309) (0.4169) (0.4025) (0.3880) (0.3732)
1:5 0.5527 0.5304 0.5092 0.4896 0.4711 0.4539 0.4378 0.4226 0.4083 0.3948 0.3820
(0.5528) (0.5384) (0.5238) (0.5092) (0.4944) (0.4795) (0.4644) (0.4491) (0.4336) (0.4180) (0.4021)
1:4 0.5934 0.4091 0.5483 0.5271 0.5072 0.4887 0.4711 0.4544 0.4386 0.4235 0.5704
(0.5934) (0.5778) (0.5621) (0.5463) (0.5305) (0.5144) (0.4982) (0.4819) (0.4655) (0.4488) (0.4319)
1:3 0.7113 0.6875 0.6646 0.6420 0.6201 0.5984 0.5778 0.5578 0.5383 0.5194 0.5009
(0.7113) (0.6927) (0.6741) (0.6556) (0.6369) (0.6184) (0.5996) (0.5807) (0.5618) (0.5426) (0.5232)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
109
Table B.1.3 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=20° /=2/3 D=1/3)
=20 /=2/3 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.4300 0.4151 0.4014 0.3889 0.3773 0.3665 0.3565 0.3471 0.3384 0.3301 0.3222
(0.4262) (0.4163) (0.4062) (0.3958) (0.3852) (0.3744) (0.3633) (0.3520) (0.3404) (0.3287) (0.3166)
1:25 0.4440 0.4283 0.4139 0.4006 0.3884 0.3769 0.3663 0.3564 0.3470 0.3382 0.3298
(0.4413) (0.4308) (0.4202) (0.4092) (0.3982) (0.3868) (0.3752) (0.3634) (0.3514) (0.3391) (0.3266)
1:15 0.4544 0.4382 0.4233 0.4095 0.3967 0.3848 0.3737 0.3633 0.3535 0.3443 0.3355
(0.4523) (0.4414) (0.4304) (0.4191) (0.4076) (0.3959) (0.3840) (0.3719) (0.3595) (0.3468) (0.3340)
1:10 0.4690 0.4520 0.4363 0.4218 0.4084 0.3958 0.3841 0.3731 0.3628 0.3530 0.3437
(0.4676) (0.4561) (0.4446) (0.4328) (0.4209) (0.4086) (0.3962) (0.3836) (0.3707) (0.3576) (0.3444)
1:7.5 0.4854 0.4677 0.4512 0.4360 0.4218 0.4085 0.3961 0.3845 0.3735 0.3630 0.3531
(0.4846) (0.4726) (0.4605) (0.4482) (0.4356) (0.4229) (0.4100) (0.3968) (0.3835) (0.3699) (0.3560)
1:5 0.5267 0.5072 0.4891 0.4721 0.4562 0.4413 0.4272 0.4139 0.4013 0.3893 0.3779
(0.5265) (0.5133) (0.4998) (0.4862) (0.4724) (0.4584) (0.4443) (0.4300) (0.4153) (0.4007) (0.3856)
1:4 0.5689 0.5481 0.5285 0.5101 0.4927 0.4762 0.4606 0.4457 0.4316 0.4180 0.4050
(0.5689) (0.5544) (0.5397) (0.5249) (0.5099) (0.4948) (0.4795) (0.4641) (0.4484) (0.4326) (0.4165)
1:3 0.6943 0.6719 0.6500 0.6287 0.6082 0.5884 0.5691 0.5505 0.5323 0.5145 0.4972
(0.6944) (0.6765) (0.6586) (0.6406) (0.6227) (0.6046) (0.5864) (0.5682) (0.5497) (0.5310) (0.5122)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
110
Table B.1.4 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=20° /=1 D=1/3)
=20 /=1 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.4093 0.3972 0.3861 0.3758 0.3663 0.3574 0.3492 0.3414 0.3341 0.3272 0.3206
(0.4012) (0.3924) (0.3834) (0.3742) (0.3648) (0.3549) (0.3448) (0.3345) (0.3239) (0.3130) (0.3019)
1:25 0.4231 0.4103 0.3984 0.3874 0.3772 0.3677 0.3588 0.3505 0.3427 0.3352 0.3281
(0.4164) (0.4071) (0.3976) (0.3878) (0.3778) (0.3674) (0.3568) (0.3460) (0.3349) (0.3235) (0.3119)
1:15 0.4334 0.4200 0.4076 0.3961 0.3854 0.3754 0.3661 0.3574 0.3491 0.3413 0.3338
(0.4275) (0.4179) (0.4080) (0.3977) (0.3873) (0.3766) (0.3657) (0.3544) (0.3431) (0.3313) (0.3193)
1:10 0.4478 0.4336 0.4205 0.4083 0.3970 0.3864 0.3765 0.3671 0.3583 0.3499 0.3419
(0.4430) (0.4327) (0.4223) (0.4116) (0.4006) (0.3895) (0.3780) (0.3664) (0.3544) (0.3422) (0.3297)
1:7.5 0.4641 0.4492 0.4353 0.4224 0.4103 0.3990 0.3884 0.3784 0.3689 0.3599 0.3512
(0.4603) (0.4495) (0.4384) (0.4272) (0.4157) (0.4040) (0.3919) (0.3797) (0.3673) (0.3546) (0.3416)
1:5 0.5053 0.4886 0.4730 0.4584 0.4446 0.4316 0.4193 0.4076 0.3966 0.3860 0.3759
(0.5032) (0.4910) (0.4787) (0.4660) (0.4533) (0.4402) (0.4270) (0.4135) (0.3998) (0.3859) (0.3717)
1:4 0.5480 0.5298 0.5127 0.4965 0.4811 0.4665 0.4527 0.4394 0.4267 0.4146 0.4029
(0.5470) (0.5334) (0.5198) (0.5059) (0.4919) (0.4776) (0.4632) (0.4485) (0.4336) (0.4185) (0.4032)
1:3 0.6789 0.6578 0.6375 0.6178 0.5988 0.5804 0.5625 0.5450 0.5280 0.5114 0.4951
(0.6788) (0.6617) (0.6444) (0.6272) (0.6098) (0.5924) (0.5748) (0.5570) (0.5391) (0.5209) (0.5026)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
111
Table B.1.5 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=20° /=0 D=1/2)
=20 /=0 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.5508 0.5293 0.5097 0.4918 0.4752 0.4598 0.4454 0.4320 0.4193 0.4073 0.3958
(0.4903) (0.4777) (0.4651) (0.4522) (0.4392) (0.4260) (0.4127) (0.3991) (0.3854) (0.3714) (0.3572)
1:25 0.5586 0.5368 0.5169 0.4986 0.4817 0.4661 0.4514 0.4377 0.4247 0.4124 0.4007
(0.5047) (0.4917) (0.4784) (0.4651) (0.4517) (0.4379) (0.4241) (0.4100) (0.3958) (0.3815) (0.3669)
1:15 0.5641 0.5421 0.5220 0.5035 0.4864 0.4706 0.4558 0.4418 0.4287 0.4162 0.4042
(0.5152) (0.5018) (0.4882) (0.4745) (0.4607) (0.4467) (0.4325) (0.4182) (0.4036) (0.3889) (0.3740)
1:10 0.5715 0.5492 0.5288 0.5101 0.4928 0.4767 0.4616 0.4474 0.4340 0.4213 0.4091
(0.5296) (0.5157) (0.5016) (0.4874) (0.4732) (0.4587) (0.4441) (0.4294) (0.4144) (0.3993) (0.3839)
1:7.5 0.5793 0.5568 0.5362 0.5172 0.4996 0.4833 0.4680 0.4535 0.4399 0.4269 0.4145
(0.5455) (0.5311) (0.5165) (0.5018) (0.4870) (0.4721) (0.4571) (0.4418) (0.4264) (0.4108) (0.3950)
1:5 0.5969 0.5739 0.5529 0.5335 0.5154 0.4985 0.4827 0.4677 0.4535 0.4400 0.4270
(0.5841) (0.5685) (0.5528) (0.5369) (0.5211) (0.5051) (0.4890) (0.4727) (0.4563) (0.4397) (0.4228)
1:4 0.6241 0.6000 0.5777 0.5568 0.5373 0.5189 0.5016 0.4853 0.4698 0.4551 0.4410
(0.6222) (0.6054) (0.5888) (0.5719) (0.5551) (0.5382) (0.5212) (0.5039) (0.4865) (0.4690) (0.4512)
1:3 0.7305 0.7053 0.6809 0.6571 0.6340 0.6117 0.5903 0.5696 0.5497 0.5303 0.5115
(0.7306) (0.7113) (0.6921) (0.6729) (0.6537) (0.6344) (0.6152) (0.5958) (0.5762) (0.5566) (0.5367)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
112
Table B.1.6 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=20° /=1/3 D=1/2)
=20 /=1/3 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.5163 0.4994 0.4837 0.4691 0.4555 0.4427 0.4306 0.4192 0.4083 0.3978 0.3878
(0.4550) (0.4438) (0.4325) (0.4210) (0.4093) (0.3973) (0.3851) (0.3728) (0.3602) (0.3474) (0.3344)
1:25 0.5249 0.5076 0.4916 0.4766 0.4626 0.4496 0.4372 0.4254 0.4142 0.4035 0.3932
(0.4699) (0.4582) (0.4463) (0.4343) (0.4220) (0.4095) (0.3969) (0.3840) (0.3710) (0.3578) (0.3443)
1:15 0.5311 0.5135 0.4972 0.4821 0.4679 0.4545 0.4419 0.4300 0.4185 0.4077 0.3971
(0.4807) (0.4686) (0.4564) (0.4440) (0.4314) (0.4186) (0.4055) (0.3924) (0.3790) (0.3653) (0.3516)
1:10 0.5393 0.5213 0.5048 0.4894 0.4749 0.4613 0.4484 0.4362 0.4245 0.4133 0.4025
(0.4956) (0.4830) (0.4703) (0.4574) (0.4443) (0.4310) (0.4175) (0.4039) (0.3901) (0.3760) (0.3617)
1:7.5 0.5480 0.5299 0.5130 0.4973 0.4825 0.4686 0.4555 0.4430 0.4310 0.4195 0.4084
(0.5122) (0.4990) (0.4858) (0.4723) (0.4587) (0.4449) (0.4309) (0.4169) (0.4025) (0.3880) (0.3732)
1:5 0.5684 0.5495 0.5320 0.5155 0.5001 0.4856 0.4719 0.4588 0.4462 0.4341 0.4224
(0.5528) (0.5384) (0.5238) (0.5092) (0.4944) (0.4795) (0.4644) (0.4491) (0.4336) (0.4180) (0.4021)
1:4 0.5981 0.5774 0.5581 0.5399 0.5227 0.5065 0.4913 0.4767 0.4629 0.4496 0.4368
(0.5934) (0.5778) (0.5621) (0.5463) (0.5305) (0.5144) (0.4982) (0.4819) (0.4655) (0.4488) (0.4319)
1:3 0.7112 0.6876 0.6646 0.6425 0.6213 0.6008 0.5809 0.5617 0.5431 0.5250 0.5074
(0.7113) (0.6927) (0.6741) (0.6556) (0.6369) (0.6184) (0.5996) (0.5807) (0.5618) (0.5426) (0.5232)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
113
Table B.1.7 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=20° /=2/3 D=1/2)
=20 /=2/3 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.4917 0.4778 0.4648 0.4526 0.4411 0.4303 0.4199 0.4100 0.4005 0.3913 0.3824
(0.4262) (0.4163) (0.4062) (0.3958) (0.3852) (0.3744) (0.3633) (0.3520) (0.3404) (0.3287) (0.3166)
1:25 0.5010 0.4867 0.4734 0.4608 0.4490 0.4377 0.4271 0.4168 0.4070 0.3975 0.3883
(0.4413) (0.4308) (0.4202) (0.4092) (0.3982) (0.3868) (0.3752) (0.3634) (0.3514) (0.3391) (0.3266)
1:15 0.5077 0.4931 0.4795 0.4667 0.4546 0.4432 0.4322 0.4218 0.4118 0.4021 0.3926
(0.4523) (0.4414) (0.4304) (0.4191) (0.4076) (0.3959) (0.3840) (0.3719) (0.3595) (0.3468) (0.3340)
1:10 0.5166 0.5017 0.4878 0.4747 0.4623 0.4505 0.4393 0.4286 0.4182 0.4082 0.3985
(0.4676) (0.4561) (0.4446) (0.4328) (0.4209) (0.4086) (0.3962) (0.3836) (0.3707) (0.3576) (0.3444)
1:7.5 0.5262 0.5110 0.4968 0.4833 0.4706 0.4586 0.4470 0.4360 0.4254 0.4151 0.4050
(0.4846) (0.4726) (0.4605) (0.4482) (0.4356) (0.4229) (0.4100) (0.3968) (0.3835) (0.3699) (0.3560)
1:5 0.5485 0.5324 0.5174 0.5034 0.4900 0.4773 0.4651 0.4534 0.4421 0.4312 0.4205
(0.5265) (0.5133) (0.4998) (0.4862) (0.4724) (0.4584) (0.4443) (0.4300) (0.4153) (0.4007) (0.3856)
1:4 0.5782 0.5602 0.5433 0.5273 0.5122 0.4979 0.4843 0.4713 0.4589 0.4469 0.4353
(0.5689) (0.5544) (0.5397) (0.5249) (0.5099) (0.4948) (0.4795) (0.4641) (0.4484) (0.4326) (0.4165)
1:3 0.6945 0.6725 0.6514 0.6310 0.6113 0.5923 0.5738 0.5559 0.5385 0.5215 0.5049
(0.6944) (0.6765) (0.6586) (0.6406) (0.6227) (0.6046) (0.5864) (0.5682) (0.5497) (0.5310) (0.5122)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
114
Table B.1.8 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=20° /=1 D=1/2)
=20 /=1 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.4737 0.4621 0.4511 0.4408 0.4309 0.4216 0.4126 0.4040 0.3956 0.3875 0.3795
(0.4012) (0.3924) (0.3834) (0.3742) (0.3648) (0.3549) (0.3448) (0.3345) (0.3239) (0.3130) (0.3019)
1:25 0.4836 0.4716 0.4603 0.4495 0.4394 0.4296 0.4203 0.4113 0.4027 0.3942 0.3859
(0.4164) (0.4071) (0.3976) (0.3878) (0.3778) (0.3674) (0.3568) (0.3460) (0.3349) (0.3235) (0.3119)
1:15 0.4907 0.4785 0.4669 0.4559 0.4455 0.4355 0.4259 0.4167 0.4078 0.3991 0.3906
(0.4275) (0.4179) (0.4080) (0.3977) (0.3873) (0.3766) (0.3657) (0.3544) (0.3431) (0.3313) (0.3193)
1:10 0.5004 0.4878 0.4758 0.4645 0.4538 0.4435 0.4336 0.4241 0.4149 0.4059 0.3970
(0.4430) (0.4327) (0.4223) (0.4116) (0.4006) (0.3895) (0.3780) (0.3664) (0.3544) (0.3422) (0.3297)
1:7.5 0.5108 0.4979 0.4856 0.4740 0.4629 0.4523 0.4421 0.4322 0.4227 0.4133 0.4042
(0.4603) (0.4495) (0.4384) (0.4272) (0.4157) (0.4040) (0.3919) (0.3797) (0.3673) (0.3546) (0.3416)
1:5 0.5350 0.5212 0.5083 0.4959 0.4841 0.4728 0.4619 0.4514 0.4411 0.4311 0.4212
(0.5032) (0.4910) (0.4787) (0.4660) (0.4533) (0.4402) (0.4270) (0.4135) (0.3998) (0.3859) (0.3717)
1:4 0.5633 0.5476 0.5328 0.5188 0.5055 0.4929 0.4809 0.4694 0.4583 0.4476 0.4371
(0.5470) (0.5334) (0.5198) (0.5059) (0.4919) (0.4776) (0.4632) (0.4485) (0.4336) (0.4185) (0.4032)
1:3 0.6807 0.6604 0.6409 0.6220 0.6037 0.5860 0.5688 0.5520 0.5357 0.5198 0.5042
(0.6788) (0.6617) (0.6444) (0.6272) (0.6098) (0.5924) (0.5748) (0.5570) (0.5391) (0.5209) (0.5026)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
115
Table B.2.1 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=25° /=0 D=1/3)
=25 /=0 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.4059 0.3847 0.3653 0.3476 0.3314 0.3163 0.3024 0.2893 0.2771 0.2656 0.2547
(0.4059) (0.3926) (0.3793) (0.3659) (0.3523) (0.3388) (0.3250) (0.3113) (0.2974) (0.2834) (0.2694)
1:25 0.4166 0.3947 0.3747 0.3563 0.3394 0.3237 0.3092 0.2957 0.2830 0.2710 0.2596
(0.4166) (0.4029) (0.3890) (0.3751) (0.3612) (0.3471) (0.3330) (0.3188) (0.3045) (0.2902) (0.2757)
1:15 0.4243 0.4020 0.3814 0.3626 0.3452 0.3291 0.3142 0.3003 0.2872 0.2749 0.2633
(0.4243) (0.4102) (0.3960) (0.3818) (0.3675) (0.3532) (0.3387) (0.3242) (0.3096) (0.2949) (0.2802)
1:10 0.4347 0.4120 0.3907 0.3712 0.3532 0.3366 0.3211 0.3067 0.2931 0.2804 0.2683
(0.4347) (0.4201) (0.4055) (0.3908) (0.3761) (0.3613) (0.3465) (0.3316) (0.3165) (0.3015) (0.2864)
1:7.5 0.4459 0.4228 0.4009 0.3807 0.3621 0.3448 0.3288 0.3138 0.2997 0.2864 0.2739
(0.4459) (0.4308) (0.4158) (0.4006) (0.3855) (0.3702) (0.3550) (0.3396) (0.3242) (0.3087) (0.2932)
1:5 0.4718 0.4478 0.4250 0.4033 0.3833 0.3646 0.3472 0.3309 0.3157 0.3012 0.2875
(0.4719) (0.4557) (0.4395) (0.4234) (0.4072) (0.3911) (0.3748) (0.3585) (0.3422) (0.3257) (0.3093)
1:4 0.4952 0.4705 0.4470 0.4245 0.4032 0.3833 0.3648 0.3474 0.3310 0.3155 0.3008
(0.4953) (0.4782) (0.4612) (0.4442) (0.4271) (0.4101) (0.3930) (0.3759) (0.3588) (0.3416) (0.3244)
1:3 0.5466 0.5207 0.4959 0.4721 0.4491 0.4269 0.4060 0.3863 0.3675 0.3497 0.3327
(0.5467) (0.5278) (0.5090) (0.4901) (0.4714) (0.4527) (0.4339) (0.4152) (0.3964) (0.3777) (0.3588)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
116
Table B.2.2 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=25° /=1/3 D=1/3)
=25 /=1/3 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.3736 0.3568 0.3412 0.3269 0.3135 0.3010 0.2893 0.2784 0.2679 0.2580 0.2486
(0.3727) (0.3612) (0.3494) (0.3376) (0.3256) (0.3135) (0.3012) (0.2888) (0.2763) (0.2636) (0.2509)
1:25 0.3841 0.3665 0.3503 0.3353 0.3213 0.3083 0.2961 0.2846 0.2737 0.2633 0.2535
(0.3836) (0.3715) (0.3594) (0.3470) (0.3345) (0.3219) (0.3092) (0.2963) (0.2834) (0.2703) (0.2572)
1:15 0.3917 0.3736 0.3569 0.3415 0.3271 0.3136 0.3010 0.2891 0.2779 0.2673 0.2571
(0.3914) (0.3790) (0.3665) (0.3537) (0.3409) (0.3281) (0.3150) (0.3018) (0.2886) (0.2752) (0.2618)
1:10 0.4021 0.3833 0.3660 0.3499 0.3349 0.3209 0.3078 0.2955 0.2838 0.2726 0.2620
(0.4020) (0.3891) (0.3760) (0.3629) (0.3497) (0.3363) (0.3228) (0.3093) (0.2956) (0.2818) (0.2680)
1:7.5 0.4135 0.3940 0.3760 0.3592 0.3436 0.3290 0.3154 0.3025 0.2903 0.2787 0.2676
(0.4135) (0.4000) (0.3865) (0.3729) (0.3592) (0.3454) (0.3314) (0.3174) (0.3034) (0.2891) (0.2749)
1:5 0.4401 0.4192 0.3997 0.3815 0.3645 0.3486 0.3336 0.3195 0.3061 0.2933 0.2812
(0.4401) (0.4256) (0.4109) (0.3963) (0.3815) (0.3666) (0.3518) (0.3367) (0.3217) (0.3065) (0.2913)
1:4 0.4644 0.4426 0.4219 0.4025 0.3843 0.3672 0.3510 0.3358 0.3213 0.3075 0.2943
(0.4644) (0.4488) (0.4333) (0.4177) (0.4020) (0.3863) (0.3705) (0.3547) (0.3387) (0.3227) (0.3067)
1:3 0.5183 0.4950 0.4725 0.4508 0.4303 0.4108 0.3922 0.3746 0.3578 0.3417 0.3263
(0.5184) (0.5008) (0.4833) (0.4658) (0.4483) (0.4307) (0.4132) (0.3955) (0.3778) (0.3601) (0.3423)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
117
Table B.2.3 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=25° /=2/3 D=1/3)
=25 /=2/3 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.3489 0.3352 0.3225 0.3106 0.2995 0.2891 0.2792 0.2699 0.2609 0.2524 0.2441
(0.3456) (0.3356) (0.3253) (0.3148) (0.3041) (0.2933) (0.2823) (0.2710) (0.2597) (0.2481) (0.2365)
1:25 0.3591 0.3447 0.3313 0.3189 0.3072 0.2962 0.2858 0.2760 0.2666 0.2577 0.2490
(0.3566) (0.3459) (0.3351) (0.3242) (0.3131) (0.3017) (0.2902) (0.2786) (0.2668) (0.2548) (0.2428)
1:15 0.3665 0.3516 0.3378 0.3249 0.3128 0.3014 0.2907 0.2805 0.2708 0.2615 0.2526
(0.3644) (0.3534) (0.3423) (0.3310) (0.3195) (0.3078) (0.2960) (0.2841) (0.2720) (0.2597) (0.2474)
1:10 0.3766 0.3611 0.3467 0.3332 0.3205 0.3086 0.2974 0.2867 0.2766 0.2669 0.2575
(0.3750) (0.3635) (0.3519) (0.3402) (0.3282) (0.3161) (0.3040) (0.2916) (0.2791) (0.2664) (0.2537)
1:7.5 0.3877 0.3715 0.3565 0.3423 0.3291 0.3166 0.3049 0.2937 0.2830 0.2728 0.2630
(0.3865) (0.3746) (0.3625) (0.3502) (0.3379) (0.3253) (0.3127) (0.2998) (0.2869) (0.2737) (0.2606)
1:5 0.4140 0.3964 0.3798 0.3643 0.3497 0.3359 0.3229 0.3105 0.2987 0.2874 0.2765
(0.4136) (0.4005) (0.3872) (0.3739) (0.3604) (0.3468) (0.3331) (0.3193) (0.3054) (0.2913) (0.2771)
1:4 0.4385 0.4196 0.4018 0.3851 0.3693 0.3544 0.3402 0.3267 0.3138 0.3015 0.2897
(0.4384) (0.4243) (0.4101) (0.3957) (0.3814) (0.3669) (0.3522) (0.3375) (0.3227) (0.3077) (0.2927)
1:3 0.4941 0.4728 0.4526 0.4335 0.4153 0.3979 0.3813 0.3655 0.3503 0.3356 0.3215
(0.4941) (0.4779) (0.4616) (0.4453) (0.4289) (0.4125) (0.3960) (0.3794) (0.3627) (0.3459) (0.3290)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
118
Table B.2.4 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=25° /=1 D=1/3)
=25 /=1 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.3289 0.3177 0.3073 0.2975 0.2883 0.2796 0.2714 0.2635 0.2559 0.2486 0.2414
(0.3218) (0.3132) (0.3043) (0.2950) (0.2856) (0.2760) (0.2661) (0.2560) (0.2457) (0.2352) (0.2245)
1:25 0.3388 0.3270 0.3159 0.3056 0.2958 0.2866 0.2779 0.2695 0.2615 0.2538 0.2463
(0.3326) (0.3235) (0.3141) (0.3044) (0.2944) (0.2844) (0.2741) (0.2635) (0.2527) (0.2419) (0.2307)
1:15 0.3460 0.3337 0.3223 0.3115 0.3013 0.2918 0.2827 0.2740 0.2656 0.2576 0.2498
(0.3404) (0.3309) (0.3212) (0.3112) (0.3009) (0.2905) (0.2798) (0.2690) (0.2580) (0.2467) (0.2353)
1:10 0.3559 0.3430 0.3309 0.3196 0.3089 0.2988 0.2893 0.2801 0.2714 0.2629 0.2547
(0.3510) (0.3410) (0.3308) (0.3203) (0.3097) (0.2988) (0.2877) (0.2764) (0.2650) (0.2534) (0.2416)
1:7.5 0.3668 0.3532 0.3405 0.3286 0.3174 0.3067 0.2966 0.2870 0.2777 0.2688 0.2602
(0.3626) (0.3521) (0.3413) (0.3305) (0.3193) (0.3079) (0.2964) (0.2847) (0.2728) (0.2607) (0.2485)
1:5 0.3926 0.3777 0.3636 0.3503 0.3377 0.3258 0.3145 0.3037 0.2933 0.2833 0.2736
(0.3898) (0.3781) (0.3662) (0.3542) (0.3420) (0.3296) (0.3171) (0.3043) (0.2914) (0.2783) (0.2651)
1:4 0.4169 0.4007 0.3854 0.3709 0.3571 0.3441 0.3316 0.3197 0.3083 0.2973 0.2867
(0.4149) (0.4023) (0.3894) (0.3763) (0.3632) (0.3499) (0.3364) (0.3227) (0.3089) (0.2949) (0.2808)
1:3 0.4727 0.4540 0.4362 0.4192 0.4030 0.3875 0.3727 0.3584 0.3446 0.3313 0.3184
(0.4719) (0.4570) (0.4421) (0.4269) (0.4117) (0.3963) (0.3809) (0.3653) (0.3495) (0.3337) (0.3177)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
119
Table B.2.5 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=25° /=0 D=1/2)
=25 /=0 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.4468 0.4273 0.4093 0.3924 0.3767 0.3620 0.3481 0.3348 0.3222 0.3101 0.2985
(0.4059) (0.3926) (0.3793) (0.3659) (0.3523) (0.3388) (0.3250) (0.3113) (0.2974) (0.2834) (0.2694)
1:25 0.4530 0.4331 0.4147 0.3976 0.3815 0.3665 0.3523 0.3388 0.3259 0.3136 0.3016
(0.4166) (0.4029) (0.3890) (0.3751) (0.3612) (0.3471) (0.3330) (0.3188) (0.3045) (0.2902) (0.2757)
1:15 0.4573 0.4372 0.4185 0.4012 0.3849 0.3697 0.3553 0.3416 0.3285 0.3160 0.3039
(0.4243) (0.4102) (0.3960) (0.3818) (0.3675) (0.3532) (0.3387) (0.3242) (0.3096) (0.2949) (0.2802)
1:10 0.4630 0.4426 0.4236 0.4060 0.3895 0.3739 0.3593 0.3453 0.3320 0.3193 0.3070
(0.4347) (0.4201) (0.4055) (0.3908) (0.3761) (0.3613) (0.3465) (0.3316) (0.3165) (0.3015) (0.2864)
1:7.5 0.4690 0.4483 0.4290 0.4111 0.3943 0.3785 0.3636 0.3494 0.3358 0.3228 0.3102
(0.4459) (0.4308) (0.4158) (0.4006) (0.3855) (0.3702) (0.3550) (0.3396) (0.3242) (0.3087) (0.2932)
1:5 0.4834 0.4619 0.4419 0.4232 0.4057 0.3892 0.3735 0.3587 0.3445 0.3310 0.3178
(0.4719) (0.4557) (0.4395) (0.4234) (0.4072) (0.3911) (0.3748) (0.3585) (0.3422) (0.3257) (0.3093)
1:4 0.5003 0.4777 0.4566 0.4369 0.4184 0.4009 0.3844 0.3687 0.3538 0.3395 0.3257
(0.4953) (0.4782) (0.4612) (0.4442) (0.4271) (0.4101) (0.3930) (0.3759) (0.3588) (0.3416) (0.3244)
1:3 0.5468 0.5214 0.4976 0.4752 0.4540 0.4341 0.4151 0.3972 0.3800 0.3636 0.3478
(0.5467) (0.5278) (0.5090) (0.4901) (0.4714) (0.4527) (0.4339) (0.4152) (0.3964) (0.3777) (0.3588)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
120
Table B.2.6 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=25° /=1/3 D=1/2)
=25 /=1/3 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.4152 0.3995 0.3848 0.3710 0.3579 0.3455 0.3337 0.3223 0.3113 0.3007 0.2903
(0.3727) (0.3612) (0.3494) (0.3376) (0.3256) (0.3135) (0.3012) (0.2888) (0.2763) (0.2636) (0.2509)
1:25 0.4218 0.4058 0.3907 0.3766 0.3631 0.3504 0.3382 0.3266 0.3153 0.3044 0.2938
(0.3836) (0.3715) (0.3594) (0.3470) (0.3345) (0.3219) (0.3092) (0.2963) (0.2834) (0.2703) (0.2572)
1:15 0.4265 0.4102 0.3949 0.3805 0.3668 0.3539 0.3415 0.3296 0.3182 0.3071 0.2963
(0.3914) (0.3790) (0.3665) (0.3537) (0.3409) (0.3281) (0.3150) (0.3018) (0.2886) (0.2752) (0.2618)
1:10 0.4327 0.4160 0.4004 0.3857 0.3718 0.3585 0.3459 0.3337 0.3220 0.3107 0.2997
(0.4020) (0.3891) (0.3760) (0.3629) (0.3497) (0.3363) (0.3228) (0.3093) (0.2956) (0.2818) (0.2680)
1:7.5 0.4392 0.4223 0.4063 0.3913 0.3771 0.3635 0.3506 0.3382 0.3262 0.3146 0.3033
(0.4135) (0.4000) (0.3865) (0.3729) (0.3592) (0.3454) (0.3314) (0.3174) (0.3034) (0.2891) (0.2749)
1:5 0.4551 0.4371 0.4202 0.4043 0.3893 0.3750 0.3613 0.3483 0.3357 0.3235 0.3116
(0.4401) (0.4256) (0.4109) (0.3963) (0.3815) (0.3666) (0.3518) (0.3367) (0.3217) (0.3065) (0.2913)
1:4 0.4728 0.4536 0.4355 0.4185 0.4024 0.3871 0.3725 0.3586 0.3452 0.3323 0.3197
(0.4644) (0.4488) (0.4333) (0.4177) (0.4020) (0.3863) (0.3705) (0.3547) (0.3387) (0.3227) (0.3067)
1:3 0.5197 0.4977 0.4769 0.4572 0.4385 0.4206 0.4036 0.3874 0.3718 0.3568 0.3423
(0.5184) (0.5008) (0.4833) (0.4658) (0.4483) (0.4307) (0.4132) (0.3955) (0.3778) (0.3601) (0.3423)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
121
Table B.2.7 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=25° /=2/3 D=1/2)
=25 /=2/3 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.3920 0.3790 0.3667 0.3550 0.3438 0.3331 0.3228 0.3128 0.3031 0.2937 0.2843
(0.3456) (0.3356) (0.3253) (0.3148) (0.3041) (0.2933) (0.2823) (0.2710) (0.2597) (0.2481) (0.2365)
1:25 0.3991 0.3857 0.3730 0.3609 0.3494 0.3384 0.3277 0.3175 0.3075 0.2977 0.2881
(0.3566) (0.3459) (0.3351) (0.3242) (0.3131) (0.3017) (0.2902) (0.2786) (0.2668) (0.2548) (0.2428)
1:15 0.4040 0.3904 0.3774 0.3651 0.3534 0.3421 0.3313 0.3208 0.3106 0.3006 0.2908
(0.3644) (0.3534) (0.3423) (0.3310) (0.3195) (0.3078) (0.2960) (0.2841) (0.2720) (0.2597) (0.2474)
1:10 0.4107 0.3967 0.3834 0.3708 0.3587 0.3472 0.3360 0.3252 0.3148 0.3045 0.2945
(0.3750) (0.3635) (0.3519) (0.3402) (0.3282) (0.3161) (0.3040) (0.2916) (0.2791) (0.2664) (0.2537)
1:7.5 0.4177 0.4034 0.3898 0.3768 0.3644 0.3526 0.3411 0.3300 0.3193 0.3088 0.2984
(0.3865) (0.3746) (0.3625) (0.3502) (0.3379) (0.3253) (0.3127) (0.2998) (0.2869) (0.2737) (0.2606)
1:5 0.4341 0.4188 0.4043 0.3905 0.3774 0.3648 0.3527 0.3410 0.3296 0.3185 0.3076
(0.4136) (0.4005) (0.3872) (0.3739) (0.3604) (0.3468) (0.3331) (0.3193) (0.3054) (0.2913) (0.2771)
1:4 0.4517 0.4351 0.4195 0.4046 0.3904 0.3769 0.3639 0.3513 0.3392 0.3274 0.3159
(0.4384) (0.4243) (0.4101) (0.3957) (0.3814) (0.3669) (0.3522) (0.3375) (0.3227) (0.3077) (0.2927)
1:3 0.4985 0.4791 0.4606 0.4431 0.4263 0.4103 0.3949 0.3801 0.3658 0.3520 0.3386
(0.4941) (0.4779) (0.4616) (0.4453) (0.4289) (0.4125) (0.3960) (0.3794) (0.3627) (0.3459) (0.3290)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
122
Table B.2.8 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=25° /=1 D=1/2)
=25 /=1 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.3743 0.3633 0.3528 0.3428 0.3331 0.3238 0.3147 0.3059 0.2972 0.2887 0.2802
(0.3218) (0.3132) (0.3043) (0.2950) (0.2856) (0.2760) (0.2661) (0.2560) (0.2457) (0.2352) (0.2245)
1:25 0.3817 0.3704 0.3595 0.3491 0.3391 0.3294 0.3200 0.3109 0.3019 0.2931 0.2843
(0.3326) (0.3235) (0.3141) (0.3044) (0.2944) (0.2844) (0.2741) (0.2635) (0.2527) (0.2419) (0.2307)
1:15 0.3870 0.3754 0.3643 0.3536 0.3433 0.3334 0.3238 0.3144 0.3053 0.2962 0.2873
(0.3404) (0.3309) (0.3212) (0.3112) (0.3009) (0.2905) (0.2798) (0.2690) (0.2580) (0.2467) (0.2353)
1:10 0.3941 0.3821 0.3706 0.3596 0.3491 0.3388 0.3289 0.3193 0.3098 0.3005 0.2913
(0.3510) (0.3410) (0.3308) (0.3203) (0.3097) (0.2988) (0.2877) (0.2764) (0.2650) (0.2534) (0.2416)
1:7.5 0.4016 0.3893 0.3775 0.3661 0.3552 0.3447 0.3345 0.3245 0.3147 0.3051 0.2956
(0.3626) (0.3521) (0.3413) (0.3305) (0.3193) (0.3079) (0.2964) (0.2847) (0.2728) (0.2607) (0.2485)
1:5 0.4186 0.4055 0.3929 0.3808 0.3692 0.3580 0.3471 0.3364 0.3260 0.3157 0.3056
(0.3898) (0.3781) (0.3662) (0.3542) (0.3420) (0.3296) (0.3171) (0.3043) (0.2914) (0.2783) (0.2651)
1:4 0.4355 0.4212 0.4075 0.3944 0.3819 0.3699 0.3583 0.3470 0.3360 0.3252 0.3145
(0.4149) (0.4023) (0.3894) (0.3763) (0.3632) (0.3499) (0.3364) (0.3227) (0.3089) (0.2949) (0.2808)
1:3 0.4815 0.4643 0.4478 0.4321 0.4170 0.4026 0.3886 0.3751 0.3621 0.3493 0.3369
(0.4719) (0.4570) (0.4421) (0.4269) (0.4117) (0.3963) (0.3809) (0.3653) (0.3495) (0.3337) (0.3177)
1:2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
123
Table B.3.1 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=30° /=0 D=1/3)
=30 /=0 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.3333 0.3141 0.2964 0.2801 0.2650 0.2511 0.2380 0.2258 0.2142 0.2033 0.1929
(0.3333) (0.3199) (0.3065) (0.2930) (0.2796) (0.2662) (0.2528) (0.2394) (0.2260) (0.2127) (0.1993)
1:25 0.3413 0.3214 0.3031 0.2862 0.2706 0.2561 0.2426 0.2300 0.2180 0.2067 0.1960
(0.3413) (0.3274) (0.3135) (0.2996) (0.2858) (0.2720) (0.2581) (0.2444) (0.2306) (0.2169) (0.2033)
1:15 0.3469 0.3266 0.3079 0.2906 0.2746 0.2598 0.2459 0.2330 0.2208 0.2092 0.1982
(0.3469) (0.3327) (0.3185) (0.3043) (0.2902) (0.2761) (0.2620) (0.2480) (0.2340) (0.2200) (0.2062)
1:10 0.3544 0.3337 0.3143 0.2965 0.2800 0.2647 0.2504 0.2371 0.2245 0.2126 0.2013
(0.3544) (0.3398) (0.3252) (0.3106) (0.2961) (0.2817) (0.2672) (0.2528) (0.2385) (0.2242) (0.2099)
1:7.5 0.3624 0.3412 0.3213 0.3029 0.2859 0.2701 0.2553 0.2415 0.2285 0.2162 0.2046
(0.3624) (0.3474) (0.3323) (0.3174) (0.3024) (0.2876) (0.2727) (0.2580) (0.2433) (0.2286) (0.2141)
1:5 0.3803 0.3582 0.3373 0.3176 0.2994 0.2825 0.2667 0.2519 0.2380 0.2248 0.2123
(0.3804) (0.3643) (0.3483) (0.3325) (0.3167) (0.3010) (0.2853) (0.2697) (0.2543) (0.2389) (0.2236)
1:4 0.3958 0.3730 0.3513 0.3307 0.3115 0.2936 0.2769 0.2613 0.2465 0.2326 0.2194
(0.3959) (0.3790) (0.3623) (0.3457) (0.3292) (0.3128) (0.2964) (0.2801) (0.2640) (0.2479) (0.2320)
1:3 0.4271 0.4029 0.3799 0.3579 0.3368 0.3171 0.2986 0.2813 0.2649 0.2494 0.2347
(0.4271) (0.4088) (0.3905) (0.3725) (0.3546) (0.3368) (0.3192) (0.3016) (0.2841) (0.2669) (0.2497)
1:2 0.5359 0.5084 0.4819 0.4562 0.4314 0.4073 0.3838 0.3612 0.3395 0.3187 0.2988
(0.5360) (0.5131) (0.4905) (0.4681) (0.4460) (0.4241) (0.4023) (0.3807) (0.3593) (0.3380) (0.3169)
124
Table B.3.2 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=30° /=1/3 D=1/3)
=30 /=1/3 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.3045 0.2891 0.2748 0.2615 0.2491 0.2374 0.2264 0.2160 0.2061 0.1966 0.1874
(0.3038) (0.2921) (0.2805) (0.2687) (0.2569) (0.2451) (0.2332) (0.2213) (0.2092) (0.1973) (0.1852)
1:25 0.3122 0.2962 0.2813 0.2675 0.2545 0.2424 0.2309 0.2201 0.2098 0.2000 0.1905
(0.3117) (0.2997) (0.2875) (0.2753) (0.2631) (0.2508) (0.2386) (0.2262) (0.2139) (0.2015) (0.1892)
1:15 0.3177 0.3012 0.2859 0.2717 0.2584 0.2460 0.2342 0.2231 0.2125 0.2024 0.1927
(0.3173) (0.3050) (0.2925) (0.2800) (0.2675) (0.2550) (0.2424) (0.2298) (0.2172) (0.2046) (0.1920)
1:10 0.3250 0.3080 0.2922 0.2774 0.2637 0.2508 0.2386 0.2271 0.2162 0.2057 0.1957
(0.3249) (0.3120) (0.2992) (0.2863) (0.2734) (0.2605) (0.2475) (0.2346) (0.2217) (0.2087) (0.1958)
1:7.5 0.3329 0.3153 0.2989 0.2837 0.2694 0.2560 0.2434 0.2315 0.2202 0.2094 0.1990
(0.3329) (0.3197) (0.3063) (0.2931) (0.2797) (0.2664) (0.2531) (0.2398) (0.2265) (0.2132) (0.1999)
1:5 0.3510 0.3320 0.3144 0.2980 0.2826 0.2682 0.2546 0.2417 0.2295 0.2179 0.2067
(0.3510) (0.3367) (0.3225) (0.3083) (0.2941) (0.2799) (0.2657) (0.2516) (0.2375) (0.2234) (0.2094)
1:4 0.3667 0.3468 0.3282 0.3108 0.2944 0.2791 0.2646 0.2509 0.2380 0.2256 0.2137
(0.3667) (0.3516) (0.3366) (0.3217) (0.3067) (0.2918) (0.2769) (0.2622) (0.2403) (0.2326) (0.2179)
1:3 0.3987 0.3774 0.3569 0.3375 0.3194 0.3022 0.2861 0.2707 0.2561 0.2422 0.2289
(0.3987) (0.3821) (0.3656) (0.3491) (0.3327) (0.3164) (0.3001) (0.2839) (0.2678) (0.2517) (0.2358)
1:2 0.5131 0.4876 0.4629 0.4390 0.4157 0.3932 0.3716 0.3508 0.3309 0.3116 0.2931
(0.5132) (0.4916) (0.4702) (0.4491) (0.4281) (0.4073) (0.3867) (0.3661) (0.3457) (0.3253) (0.3051)
125
Table B.3.3 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=30° /=2/3 D=1/3)
=30 /=2/3 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2821 0.2695 0.2578 0.2468 0.2364 0.2265 0.2172 0.2082 0.1996 0.1913 0.1832
(0.2794) (0.2694) (0.2593) (0.2491) (0.2387) (0.2282) (0.2176) (0.2069) (0.1961) (0.1852) (0.1743)
1:25 0.2894 0.2763 0.2640 0.2525 0.2416 0.2314 0.2216 0.2123 0.2033 0.1947 0.1863
(0.2873) (0.2769) (0.2663) (0.2556) (0.2448) (0.2339) (0.2229) (0.2118) (0.2006) (0.1895) (0.1782)
1:15 0.2947 0.2812 0.2685 0.2566 0.2454 0.2348 0.2248 0.2152 0.2060 0.1971 0.1885
(0.2928) (0.2821) (0.2712) (0.2602) (0.2491) (0.2379) (0.2267) (0.2153) (0.2039) (0.1925) (0.1810)
1:10 0.3018 0.2877 0.2746 0.2622 0.2506 0.2396 0.2291 0.2191 0.2096 0.2004 0.1915
(0.3002) (0.2891) (0.2778) (0.2664) (0.2550) (0.2434) (0.2318) (0.2201) (0.2084) (0.1966) (0.1848)
1:7.5 0.3094 0.2948 0.2811 0.2683 0.2562 0.2447 0.2338 0.2235 0.2135 0.2040 0.1948
(0.3082) (0.2966) (0.2849) (0.2731) (0.2613) (0.2494) (0.2374) (0.2253) (0.2132) (0.2010) (0.1889)
1:5 0.3269 0.3111 0.2962 0.2822 0.2691 0.2566 0.2448 0.2335 0.2228 0.2124 0.2024
(0.3263) (0.3138) (0.3011) (0.2884) (0.2756) (0.2628) (0.2499) (0.2371) (0.2242) (0.2113) (0.1983)
1:4 0.3424 0.3255 0.3097 0.2948 0.2807 0.2673 0.2547 0.2426 0.2311 0.2200 0.2094
(0.3420) (0.3287) (0.3153) (0.3019) (0.2884) (0.2748) (0.2612) (0.2476) (0.2340) (0.2204) (0.2068)
1:3 0.3745 0.3557 0.3379 0.3211 0.3053 0.2902 0.2759 0.2622 0.2491 0.2366 0.2245
(0.3745) (0.3595) (0.3445) (0.3295) (0.3146) (0.2996) (0.2846) (0.2696) (0.2546) (0.2397) (0.2248)
1:2 0.4925 0.4687 0.4458 0.4235 0.4021 0.3815 0.3616 0.3424 0.3239 0.3060 0.2886
(0.4929) (0.4727) (0.4526) (0.4327) (0.4128) (0.3930) (0.3734) (0.3539) (0.3344) (0.3149) (0.2956)
126
Table B.3.4 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=30° /=1 D=1/3)
=30 /=1 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2633 0.2531 0.2435 0.2344 0.2257 0.2175 0.2096 0.2020 0.1946 0.1874 0.1804
(0.2574) (0.2490) (0.2404) (0.2316) (0.2225) (0.2133) (0.2040) (0.1945) (0.1847) (0.1749) (0.1650)
1:25 0.2704 0.2596 0.2495 0.2400 0.2309 0.2222 0.2140 0.2060 0.1983 0.1907 0.1834
(0.2650) (0.2562) (0.2473) (0.2380) (0.2286) (0.2189) (0.2092) (0.1993) (0.1892) (0.1791) (0.1688)
1:15 0.2755 0.2644 0.2539 0.2440 0.2346 0.2256 0.2171 0.2089 0.2009 0.1931 0.1855
(0.2705) (0.2614) (0.2521) (0.2425) (0.2329) (0.2230) (0.2129) (0.2028) (0.1925) (0.1821) (0.1716)
1:10 0.2823 0.2707 0.2598 0.2494 0.2396 0.2303 0.2213 0.2128 0.2045 0.1964 0.1885
(0.2778) (0.2683) (0.2587) (0.2488) (0.2387) (0.2284) (0.2181) (0.2075) (0.1969) (0.1861) (0.1753)
1:7.5 0.2897 0.2776 0.2661 0.2553 0.2451 0.2353 0.2260 0.2170 0.2084 0.2000 0.1918
(0.2857) (0.2758) (0.2657) (0.2553) (0.2449) (0.2343) (0.2235) (0.2126) (0.2016) (0.1905) (0.1794)
1:5 0.3067 0.2934 0.2809 0.2690 0.2577 0.2470 0.2368 0.2269 0.2175 0.2083 0.1994
(0.3036) (0.2928) (0.2817) (0.2705) (0.2591) (0.2477) (0.2361) (0.2244) (0.2126) (0.2007) (0.1888)
1:4 0.3219 0.3076 0.2941 0.2813 0.2691 0.2576 0.2465 0.2359 0.2257 0.2159 0.2063
(0.3194) (0.3077) (0.2959) (0.2839) (0.2718) (0.2595) (0.2473) (0.2348) (0.2224) (0.2098) (0.1973)
1:3 0.3534 0.3372 0.3219 0.3073 0.2934 0.2802 0.2675 0.2553 0.2436 0.2323 0.2213
(0.3520) (0.3386) (0.3253) (0.3117) (0.2981) (0.2844) (0.2707) (0.2568) (0.2431) (0.2292) (0.2152)
1:2 0.4736 0.4516 0.4305 0.4101 0.3905 0.3716 0.3533 0.3355 0.3183 0.3016 0.2854
(0.4735) (0.4546) (0.4359) (0.4172) (0.3985) (0.3799) (0.3613) (0.3427) (0.3242) (0.3056) (0.2871)
127
Table B.3.5 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=30° /=0 D=1/2)
=30 /=0 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.3613 0.3436 0.3271 0.3116 0.2971 0.2833 0.2702 0.2577 0.2457 0.2342 0.2230
(0.3333) (0.3199) (0.3065) (0.2930) (0.2796) (0.2662) (0.2528) (0.2394) (0.2260) (0.2127) (0.1993)
1:25 0.3660 0.3480 0.3311 0.3154 0.3005 0.2865 0.2731 0.2604 0.2482 0.2364 0.2250
(0.3413) (0.3274) (0.3135) (0.2996) (0.2858) (0.2720) (0.2581) (0.2444) (0.2306) (0.2169) (0.2033)
1:15 0.3693 0.3510 0.3340 0.3180 0.3029 0.2887 0.2752 0.2623 0.2499 0.2380 0.2264
(0.3469) (0.3327) (0.3185) (0.3043) (0.2902) (0.2761) (0.2620) (0.2480) (0.2340) (0.2200) (0.2062)
1:10 0.3737 0.3551 0.3377 0.3214 0.3061 0.2917 0.2779 0.2648 0.2522 0.2400 0.2283
(0.3544) (0.3398) (0.3252) (0.3106) (0.2961) (0.2817) (0.2672) (0.2528) (0.2385) (0.2242) (0.2099)
1:7.5 0.3783
(0.3624)
0.3594
(0.3474)
0.3417
(0.3323)
0.3251
(0.3174)
0.3095
(0.3024)
0.2948
(0.2876)
0.2808
(0.2727)
0.2674
(0.2580)
0.2546
(0.2433)
0.2422
(0.2286)
0.2303
(0.2141)
1:5 0.3900
(0.3804)
0.3702
(0.3643)
0.3516
(0.3483)
0.3342
(0.3325)
0.3179
(0.3167)
0.3024
(0.3010)
0.2878 0.2738
(0.2697)
0.2604
(0.2543)
0.2475
(0.2389)
0.2351
(0.2236) (0.2853)
1:4 0.4017 0.3809
(0.3790)
0.3614
(0.3623)
0.3432
(0.3457)
0.3261
(0.3292)
0.3099
(0.3128)
0.2946
(0.2964)
0.2800
(0.2801)
0.2660
(0.2640)
0.2526
(0.2479)
0.2397
(0.2320) (0.3959)
1:3 0.4285
(0.4271)
0.4056
(0.4088)
0.3841
(0.3905)
0.3640
(0.3725)
0.3452
(0.3546)
0.3273
(0.3368)
0.3104
(0.3192)
0.2944
(0.3016)
0.2791
(0.2841)
0.2645
(0.2669)
0.2505
(0.2497)
1:2 0.5359
(0.5360)
0.5084
(0.5131)
0.4819
(0.4905)
0.4563
(0.4681)
0.4315
(0.4460)
0.4078
(0.4241)
0.3851
(0.4023)
0.3634
(0.3807)
0.3426
(0.3593)
0.3227
(0.3380)
0.3035
(0.3169)
128
Table B.3.6 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=30° /=1/3 D=1/2)
=30 /=1/3 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.3334 0.3190 0.3055 0.2926 0.2804 0.2687 0.2574 0.2466 0.2361 0.2258 0.2158
(0.3038) (0.2921) (0.2805) (0.2687) (0.2569) (0.2451) (0.2332) (0.2213) (0.2092) (0.1973) (0.1852)
1:25 0.3384 0.3237 0.3098 0.2966 0.2841 0.2721 0.2606 0.2495 0.2387 0.2282 0.2180
(0.3117) (0.2997) (0.2875) (0.2753) (0.2631) (0.2508) (0.2386) (0.2262) (0.2139) (0.2015) (0.1892)
1:15 0.3419 0.3270 0.3128 0.2994 0.2867 0.2745 0.2628 0.2515 0.2406 0.2299 0.2195
(0.3173) (0.3050) (0.2925) (0.2800) (0.2675) (0.2550) (0.2424) (0.2298) (0.2172) (0.2046) (0.1920)
1:10 0.3465 0.3313 0.3168 0.3032 0.2901 0.2777 0.2657 0.2542 0.2431 0.2322 0.2216
(0.3249) (0.3120) (0.2992) (0.2863) (0.2734) (0.2605) (0.2475) (0.2346) (0.2217) (0.2087) (0.1958)
1:7.5 0.3514 0.3358 0.3211 0.3071 0.2938 0.2811 0.2689 0.2571 0.2457 0.2346 0.2238
(0.3329) (0.3197) (0.3063) (0.2931) (0.2797) (0.2664) (0.2531) (0.2398) (0.2265) (0.2132) (0.1999)
1:5 0.3636 0.3470 0.3313 0.3165 0.3024 0.2890 0.2761 0.2637 0.2518 0.2402 0.2288
(0.3510) (0.3367) (0.3225) (0.3083) (0.2941) (0.2799) (0.2657) (0.2516) (0.2375) (0.2234) (0.2094)
1:4 0.3755 0.3579 0.3413 0.3256 0.3108 0.2966 0.2831 0.2701 0.2576 0.2455 0.2337
(0.3667) (0.3516) (0.3366) (0.3217) (0.3067) (0.2918) (0.2769) (0.2622) (0.2403) (0.2326) (0.2179)
1:3 0.4023 0.3826 0.3641 0.3466 0.3300 0.3142 0.2992 0.2848 0.2710 0.2576 0.2447
(0.3987) (0.3821) (0.3656) (0.3491) (0.3327) (0.3164) (0.3001) (0.2839) (0.2678) (0.2517) (0.2358)
1:2 0.5131 0.4876 0.4630 0.4394 0.4169 0.3952 0.3744 0.3543 0.3350 0.3164 0.2984
(0.5132) (0.4916) (0.4702) (0.4491) (0.4281) (0.4073) (0.3867) (0.3661) (0.3457) (0.3253) (0.3051)
129
Table B.3.7 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=30° /=2/3 D=1/2)
=30 /=2/3 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.3125 0.3004 0.2890 0.2780 0.2675 0.2573 0.2475 0.2379 0.2285 0.2193 0.2102
(0.2794) (0.2694) (0.2593) (0.2491) (0.2387) (0.2282) (0.2176) (0.2069) (0.1961) (0.1852) (0.1743)
1:25 0.3177 0.3053 0.2936 0.2823 0.2714 0.2610 0.2508 0.2410 0.2313 0.2219 0.2125
(0.2873) (0.2769) (0.2663) (0.2556) (0.2448) (0.2339) (0.2229) (0.2118) (0.2006) (0.1895) (0.1782)
1:15 0.3214 0.3088 0.2968 0.2853 0.2742 0.2635 0.2532 0.2432 0.2333 0.2237 0.2142
(0.2928) (0.2821) (0.2712) (0.2602) (0.2491) (0.2379) (0.2267) (0.2153) (0.2039) (0.1925) (0.1810)
1:10 0.3263 0.3133 0.3010 0.2892 0.2779 0.2669 0.2564 0.2461 0.2360 0.2262 0.2165
(0.3002) (0.2891) (0.2778) (0.2664) (0.2550) (0.2434) (0.2318) (0.2201) (0.2084) (0.1966) (0.1848)
1:7.5 0.3314 0.3181 0.3055 0.2934 0.2818 0.2706 0.2597 0.2492 0.2389 0.2288 0.2189
(0.3082) (0.2966) (0.2849) (0.2731) (0.2613) (0.2494) (0.2374) (0.2253) (0.2132) (0.2010) (0.1889)
1:5 0.3435 0.3293 0.3158 0.3029 0.2905 0.2787 0.2672 0.2561 0.2453 0.2347 0.2243
(0.3263) (0.3138) (0.3011) (0.2884) (0.2756) (0.2628) (0.2499) (0.2371) (0.2242) (0.2113) (0.1983)
1:4 0.3552 0.3400 0.3256 0.3120 0.2989 0.2863 0.2742 0.2625 0.2512 0.2401 0.2292
(0.3420) (0.3287) (0.3153) (0.3019) (0.2884) (0.2748) (0.2612) (0.2476) (0.2340) (0.2204) (0.2068)
1:3 0.3816 0.3645 0.3482 0.3328 0.3180 0.3039 0.2903 0.2773 0.2647 0.2524 0.2404
(0.3745) (0.3595) (0.3445) (0.3295) (0.3146) (0.2996) (0.2846) (0.2696) (0.2546) (0.2397) (0.2248)
1:2 0.4931 0.4697 0.4472 0.4256 0.4049 0.3849 0.3656 0.3470 0.3289 0.3114 0.2945
(0.4929) (0.4727) (0.4526) (0.4327) (0.4128) (0.3930) (0.3734) (0.3539) (0.3344) (0.3149) (0.2956)
130
Table B.3.8 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=30° /=1 D=1/2)
=30 /=1 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2958 0.2856 0.2758 0.2663 0.2571 0.2482 0.2395 0.2309 0.2225 0.2142 0.2058
(0.2574) (0.2490) (0.2404) (0.2316) (0.2225) (0.2133) (0.2040) (0.1945) (0.1847) (0.1749) (0.1650)
1:25 0.3012 0.2907 0.2806 0.2708 0.2613 0.2521 0.2431 0.2343 0.2256 0.2170 0.2084
(0.2650) (0.2562) (0.2473) (0.2380) (0.2286) (0.2189) (0.2092) (0.1993) (0.1892) (0.1791) (0.1688)
1:15 0.3051 0.2943 0.2839 0.2739 0.2642 0.2548 0.2456 0.2366 0.2277 0.2190 0.2102
(0.2705) (0.2614) (0.2521) (0.2425) (0.2329) (0.2230) (0.2129) (0.2028) (0.1925) (0.1821) (0.1716)
1:10 0.3102 0.2991 0.2884 0.2781 0.2682 0.2585 0.2490 0.2398 0.2306 0.2216 0.2127
(0.2778) (0.2683) (0.2587) (0.2488) (0.2387) (0.2284) (0.2181) (0.2075) (0.1969) (0.1861) (0.1753)
1:7.5 0.3156 0.3042 0.2932 0.2826 0.2723 0.2624 0.2526 0.2431 0.2337 0.2245 0.2153
(0.2857) (0.2758) (0.2657) (0.2553) (0.2449) (0.2343) (0.2235) (0.2126) (0.2016) (0.1905) (0.1794)
1:5 0.3277 0.3154 0.3038 0.2925 0.2816 0.2710 0.2607 0.2506 0.2407 0.2309 0.2212
(0.3036) (0.2928) (0.2817) (0.2705) (0.2591) (0.2477) (0.2361) (0.2244) (0.2126) (0.2007) (0.1888)
1:4 0.3389 0.3258 0.3133 0.3013 0.2897 0.2786 0.2677 0.2571 0.2467 0.2365 0.2263
(0.3194) (0.3077) (0.2959) (0.2839) (0.2718) (0.2595) (0.2473) (0.2348) (0.2224) (0.2098) (0.1973)
1:3 0.3646 0.3496 0.3353 0.3216 0.3085 0.2958 0.2836 0.2717 0.2601 0.2488 0.2376
(0.3520) (0.3386) (0.3253) (0.3117) (0.2981) (0.2844) (0.2707) (0.2568) (0.2431) (0.2292) (0.2152)
1:2 0.4756 0.4543 0.4338 0.4140 0.3949 0.3764 0.3585 0.3411 0.3242 0.3078 0.2918
(0.4735) (0.4546) (0.4359) (0.4172) (0.3985) (0.3799) (0.3613) (0.3427) (0.3242) (0.3056) (0.2871)
131
Table B.4.1 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=35° /=0 D=1/3)
=35 /=0 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2710 0.2536 0.2376 0.2228 0.2091 0.1963 0.1844 0.1732 0.1626 0.1526 0.1430
(0.2710) (0.2577) (0.2447) (0.2316) (0.2188) (0.2059) (0.1932) (0.1806) (0.1681) (0.1558) (0.1437)
1:25 0.2768 0.2588 0.2423 0.2271 0.2129 0.1998 0.1875 0.1759 0.1650 0.1547 0.1449
(0.2768) (0.2631) (0.2497) (0.2363) (0.2230) (0.2098) (0.1968) (0.1839) (0.1711) (0.1585) (0.1460)
1:15 0.2809 0.2626 0.2457 0.2301 0.2156 0.2022 0.1896 0.1779 0.1667 0.1562 0.1462
(0.2809) (0.2669) (0.2532) (0.2395) (0.2260) (0.2126) (0.1993) (0.1862) (0.1732) (0.1604) (0.1478)
1:10 0.2862 0.2675 0.2501 0.2341 0.2192 0.2054 0.1925 0.1804 0.1690 0.1583 0.1480
(0.2863) (0.2719) (0.2579) (0.2439) (0.2299) (0.2163) (0.2027) (0.1892) (0.1760) (0.1629) (0.1501)
1:7.5 0.2919 0.2728 0.2549 0.2384 0.2231 0.2089 0.1956 0.1832 0.1715 0.1604 0.1499
(0.2919) (0.2772) (0.2628) (0.2484) (0.2342) (0.2201) (0.2063) (0.1925) (0.1790) (0.1657) (0.1525)
1:5 0.3043 0.2844 0.2655 0.2480 0.2318 0.2167 0.2027 0.1895 0.1771 0.1654 0.1543
(0.3044) (0.2889) (0.2736) (0.2585) (0.2435) (0.2288) (0.2142) (0.1998) (0.1856) (0.1717) (0.1580)
1:4 0.3148 0.2942 0.2746 0.2563 0.2393 0.2235 0.2088 0.1950 0.1820 0.1698 0.1582
(0.3148) (0.2987) (0.2827) (0.2670) (0.2514) (0.2362) (0.2210) (0.2061) (0.1914) (0.1769) (0.1627)
1:3 0.3350 0.3132 0.2924 0.2727 0.2542 0.2370 0.2210 0.2060 0.1919 0.1786 0.1660
(0.3350) (0.3176) (0.3005) (0.2835) (0.2669) (0.2505) (0.2343) (0.2184) (0.2026) (0.1873) (0.1721)
1:2 0.3929 0.3681 0.3444 0.3217 0.3000 0.2791 0.2595 0.2410 0.2236 0.2071 0.1915
(0.3930) (0.3724) (0.3521) (0.3322) (0.3125) (0.2932) (0.2742) (0.2555) (0.2370) (0.2189) (0.2012)
132
Table B.4.2 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=35° /=1/3 D=1/3)
=35 /=1/3 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2462 0.2323 0.2192 0.2071 0.1957 0.1850 0.1748 0.1652 0.1560 0.1472 0.1387
(0.2456) (0.2343) (0.2230) (0.2117) (0.2004) (0.1891) (0.1779) (0.1667) (0.1556) (0.1445) (0.1336)
1:25 0.2518 0.2373 0.2238 0.2112 0.1994 0.1883 0.1778 0.1679 0.1584 0.1493 0.1405
(0.2514) (0.2396) (0.2279) (0.2162) (0.2046) (0.1930) (0.1814) (0.1700) (0.1585) (0.1472) (0.1360)
1:15 0.2557 0.2408 0.2270 0.2141 0.2020 0.1907 0.1799 0.1698 0.1601 0.1508 0.1418
(0.2554) (0.2434) (0.2314) (0.2195) (0.2076) (0.1957) (0.1839) (0.1723) (0.1606) (0.1491) (0.1376)
1:10 0.2609 0.2455 0.2313 0.2180 0.2055 0.1938 0.1828 0.1723 0.1623 0.1528 0.1436
(0.2607) (0.2483) (0.2360) (0.2237) (0.2115) (0.1993) (0.1873) (0.1753) (0.1633) (0.1516) (0.1399)
1:7.5 0.2664 0.2506 0.2358 0.2221 0.2093 0.1972 0.1858 0.1750 0.1648 0.1550 0.1455
(0.2663) (0.2535) (0.2409) (0.2283) (0.2157) (0.2032) (0.1908) (0.1785) (0.1663) (0.1543) (0.1423)
1:5 0.2787 0.2618 0.2461 0.2314 0.2177 0.2048 0.1927 0.1812 0.1703 0.1599 0.1499
(0.2787) (0.2652) (0.2516) (0.2382) (0.2249) (0.2118) (0.1987) (0.1857) (0.1729) (0.1602) (0.1477)
1:4 0.2892 0.2714 0.2549 0.2394 0.2250 0.2114 0.1986 0.1865 0.1751 0.1642 0.1537
(0.2892) (0.2749) (0.2608) (0.2468) (0.2329) (0.2191) (0.2055) (0.1919) (0.1786) (0.1655) (0.1525)
1:3 0.3095 0.2903 0.2723 0.2553 0.2395 0.2246 0.2106 0.1974 0.1848 0.1729 0.1615
(0.3095) (0.2940) (0.2786) (0.2634) (0.2484) (0.2335) (0.2188) (0.2043) (0.1900) (0.1757) (0.1619)
1:2 0.3687 0.3464 0.3249 0.3043 0.2847 0.2661 0.2486 0.2320 0.2162 0.2012 0.1868
(0.3688) (0.3499) (0.3313) (0.3130) (0.2948) (0.2769) (0.2593) (0.2419) (0.2248) (0.2078) (0.1912)
133
Table B.4.3 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=35° /=2/3 D=1/3)
=35 /=2/3 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2266 0.2151 0.2044 0.1943 0.1847 0.1756 0.1670 0.1586 0.1506 0.1428 0.1352
(0.2244) (0.2148) (0.2051) (0.1953) (0.1855) (0.1755) (0.1656) (0.1556) (0.1456) (0.1357) (0.1257)
1:25 0.2318 0.2199 0.2088 0.1982 0.1883 0.1789 0.1699 0.1612 0.1529 0.1449 0.1371
(0.2300) (0.2200) (0.2100) (0.1998) (0.1896) (0.1793) (0.1690) (0.1588) (0.1485) (0.1383) (0.1281)
1:15 0.2356 0.2233 0.2118 0.2010 0.1908 0.1812 0.1719 0.1631 0.1546 0.1464 0.1384
(0.2340) (0.2237) (0.2134) (0.2029) (0.1925) (0.1821) (0.1715) (0.1610) (0.1506) (0.1401) (0.1297)
1:10 0.2405 0.2278 0.2160 0.2048 0.1942 0.1842 0.1747 0.1656 0.1568 0.1484 0.1401
(0.2391) (0.2285) (0.2179) (0.2071) (0.1964) (0.1856) (0.1748) (0.1640) (0.1532) (0.1426) (0.1320)
1:7.5 0.2458 0.2327 0.2204 0.2088 0.1979 0.1875 0.1777 0.1682 0.1592 0.1505 0.1420
(0.2446) (0.2337) (0.2226) (0.2115) (0.2005) (0.1894) (0.1783) (0.1673) (0.1562) (0.1452) (0.1343)
1:5 0.2576 0.2435 0.2302 0.2178 0.2061 0.1949 0.1844 0.1743 0.1646 0.1553 0.1463
(0.2568) (0.2451) (0.2333) (0.2215) (0.2097) (0.1978) (0.1861) (0.1744) (0.1628) (0.1511) (0.1397)
1:4 0.2677 0.2528 0.2388 0.2256 0.2132 0.2014 0.1902 0.1796 0.1694 0.1596 0.1501
(0.2672) (0.2548) (0.2423) (0.2299) (0.2175) (0.2051) (0.1928) (0.1806) (0.1684) (0.1563) (0.1444)
1:3 0.2876 0.2711 0.2557 0.2411 0.2274 0.2144 0.2020 0.1902 0.1790 0.1682 0.1578
(0.2875) (0.2738) (0.2601) (0.2465) (0.2330) (0.2195) (0.2061) (0.1928) (0.1796) (0.1666) (0.1536)
1:2 0.3474 0.3270 0.3077 0.2893 0.2719 0.2553 0.2396 0.2245 0.2101 0.1963 0.1831
(0.3474) (0.3303) (0.3133) (0.2964) (0.2798) (0.2632) (0.2468) (0.2306) (0.2145) (0.1987) (0.1831)
134
Table B.4.4 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=35° /=1 D=1/3)
=35 /=1 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2094 0.2002 0.1914 0.1831 0.1751 0.1675 0.1602 0.1531 0.1461 0.1393 0.1326
(0.2045) (0.1967) (0.1886) (0.1803) (0.1719) (0.1632) (0.1546) (0.1457) (0.1368) (0.1279) (0.1188)
1:25 0.2145 0.2047 0.1956 0.1869 0.1786 0.1706 0.1630 0.1556 0.1484 0.1414 0.1344
(0.2099) (0.2017) (0.1933) (0.1846) (0.1759) (0.1669) (0.1580) (0.1488) (0.1397) (0.1303) (0.1211)
1:15 0.2180 0.2080 0.1986 0.1896 0.1810 0.1729 0.1650 0.1574 0.1501 0.1428 0.1357
(0.2138) (0.2053) (0.1966) (0.1877) (0.1787) (0.1696) (0.1604) (0.1511) (0.1416) (0.1322) (0.1228)
1:10 0.2228 0.2124 0.2025 0.1932 0.1843 0.1759 0.1677 0.1599 0.1522 0.1448 0.1375
(0.2189) (0.2100) (0.2010) (0.1918) (0.1825) (0.1731) (0.1636) (0.1539) (0.1443) (0.1346) (0.1250)
1:7.5 0.2278 0.2170 0.2068 0.1971 0.1879 0.1791 0.1706 0.1625 0.1546 0.1469 0.1394
(0.2242) (0.2150) (0.2057) (0.1962) (0.1866) (0.1768) (0.1670) (0.1571) (0.1471) (0.1372) (0.1273)
1:5 0.2392 0.2274 0.2163 0.2058 0.1958 0.1863 0.1772 0.1684 0.1600 0.1517 0.1436
(0.2362) (0.2262) (0.2161) (0.2059) (0.1956) (0.1852) (0.1747) (0.1641) (0.1536) (0.1431) (0.1326)
1:4 0.2489 0.2364 0.2246 0.2134 0.2028 0.1926 0.1829 0.1736 0.1646 0.1559 0.1474
(0.2463) (0.2358) (0.2250) (0.2142) (0.2033) (0.1923) (0.1813) (0.1703) (0.1592) (0.1481) (0.1372)
1:3 0.2682 0.2542 0.2411 0.2285 0.2167 0.2053 0.1945 0.1841 0.1741 0.1645 0.1551
(0.2664) (0.2545) (0.2427) (0.2306) (0.2187) (0.2065) (0.1945) (0.1824) (0.1704) (0.1584) (0.1464)
1:2 0.3271 0.3093 0.2923 0.2761 0.2606 0.2458 0.2316 0.2180 0.2050 0.1924 0.1802
(0.3265) (0.3112) (0.2960) (0.2807) (0.2655) (0.2503) (0.2353) (0.2202) (0.2054) (0.1905) (0.1759)
135
Table B.4.5 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=35° /=0 D=1/2)
=35 /=0 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2901 0.2741 0.2592 0.2451 0.2319 0.2193 0.2072 0.1957 0.1847 0.1739 0.1635
(0.2710) (0.2577) (0.2447) (0.2316) (0.2188) (0.2059) (0.1932) (0.1806) (0.1681) (0.1558) (0.1437)
1:25 0.2937 0.2774 0.2622 0.2478 0.2343 0.2214 0.2092 0.1975 0.1862 0.1753 0.1647
(0.2768) (0.2631) (0.2497) (0.2363) (0.2230) (0.2098) (0.1968) (0.1839) (0.1711) (0.1585) (0.1460)
1:15 0.2961 0.2797 0.2642 0.2497 0.2360 0.2230 0.2106 0.1987 0.1873 0.1763 0.1656
(0.2809) (0.2669) (0.2532) (0.2395) (0.2260) (0.2126) (0.1993) (0.1862) (0.1732) (0.1604) (0.1478)
1:10 0.2995 0.2827 0.2670 0.2522 0.2382 0.2250 0.2124 0.2004 0.1888 0.1776 0.1668
(0.2863) (0.2719) (0.2579) (0.2439) (0.2299) (0.2163) (0.2027) (0.1892) (0.1760) (0.1629) (0.1501)
1:7.5 0.3033 0.2861 0.2701 0.2550 0.2407 0.2272 0.2144 0.2022 0.1904 0.1791 0.1681
(0.2919) (0.2772) (0.2628) (0.2484) (0.2342) (0.2201) (0.2063) (0.1925) (0.1790) (0.1657) (0.1525)
1:5 0.3122 0.2942 0.2773 0.2615 0.2466 0.2325 0.2191 0.2064 0.1942 0.1824 0.1711
(0.3044) (0.2889) (0.2736) (0.2585) (0.2435) (0.2288) (0.2142) (0.1998) (0.1856) (0.1717) (0.1580)
1:4 0.3203 0.3015 0.2839 0.2674 0.2519 0.2373 0.2234 0.2102 0.1976 0.1854 0.1737
(0.3148) (0.2987) (0.2827) (0.2670) (0.2514) (0.2362) (0.2210) (0.2061) (0.1914) (0.1769) (0.1627)
1:3 0.3373 0.3169 0.2978 0.2800 0.2632 0.2474 0.2325 0.2183 0.2048 0.1918 0.1794
(0.3350) (0.3176) (0.3005) (0.2835) (0.2669) (0.2505) (0.2343) (0.2184) (0.2026) (0.1873) (0.1721)
1:2 0.3930 0.3682 0.3448 0.3228 0.3022 0.2827 0.2642 0.2468 0.2303 0.2145 0.1995
(0.3930) (0.3724) (0.3521) (0.3322) (0.3125) (0.2932) (0.2742) (0.2555) (0.2370) (0.2189) (0.2012)
136
Table B.4.6 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=35° /=1/3 D=1/2)
=35 /=1/3 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2664 0.2533 0.2410 0.2292 0.2179 0.2071 0.1967 0.1866 0.1768 0.1672 0.1578
(0.2456) (0.2343) (0.2230) (0.2117) (0.2004) (0.1891) (0.1779) (0.1667) (0.1556) (0.1445) (0.1336)
1:25 0.2701 0.2567 0.2441 0.2320 0.2205 0.2094 0.1988 0.1885 0.1785 0.1687 0.1591
(0.2514) (0.2396) (0.2279) (0.2162) (0.2046) (0.1930) (0.1814) (0.1700) (0.1585) (0.1472) (0.1360)
1:15 0.2727 0.2591 0.2462 0.2340 0.2223 0.2110 0.2002 0.1898 0.1796 0.1697 0.1601
(0.2554) (0.2434) (0.2314) (0.2195) (0.2076) (0.1957) (0.1839) (0.1723) (0.1606) (0.1491) (0.1376)
1:10 0.2762 0.2623 0.2491 0.2366 0.2246 0.2132 0.2022 0.1915 0.1812 0.1711 0.1613
(0.2607) (0.2483) (0.2360) (0.2237) (0.2115) (0.1993) (0.1873) (0.1753) (0.1633) (0.1516) (0.1399)
1:7.5 0.2800 0.2657 0.2522 0.2394 0.2272 0.2155 0.2042 0.1934 0.1829 0.1726 0.1626
(0.2663) (0.2535) (0.2409) (0.2283) (0.2157) (0.2032) (0.1908) (0.1785) (0.1663) (0.1543) (0.1423)
1:5 0.2890 0.2738 0.2595 0.2460 0.2331 0.2208 0.2091 0.1977 0.1868 0.1761 0.1657
(0.2787) (0.2652) (0.2516) (0.2382) (0.2249) (0.2118) (0.1987) (0.1857) (0.1729) (0.1602) (0.1477)
1:4 0.2971 0.2812 0.2662 0.2520 0.2385 0.2257 0.2134 0.2016 0.1903 0.1792 0.1685
(0.2892) (0.2749) (0.2608) (0.2468) (0.2329) (0.2191) (0.2055) (0.1919) (0.1786) (0.1655) (0.1525)
1:3 0.3139 0.2965 0.2801 0.2646 0.2499 0.2359 0.2226 0.2099 0.1976 0.1858 0.1744
(0.3095) (0.2940) (0.2786) (0.2634) (0.2484) (0.2335) (0.2188) (0.2043) (0.1900) (0.1757) (0.1619)
1:2 0.3690 0.3472 0.3266 0.3072 0.2887 0.2712 0.2545 0.2386 0.2234 0.2089 0.1949
(0.3688) (0.3499) (0.3313) (0.3130) (0.2948) (0.2769) (0.2593) (0.2419) (0.2248) (0.2078) (0.1912)
137
Table B.4.7 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=35° /=2/3 D=1/2)
=35 /=2/3 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2481 0.2371 0.2266 0.2165 0.2067 0.1973 0.1881 0.1791 0.1703 0.1617 0.1531
(0.2244) (0.2148) (0.2051) (0.1953) (0.1855) (0.1755) (0.1656) (0.1556) (0.1456) (0.1357) (0.1257)
1:25 0.2520 0.2407 0.2298 0.2194 0.2094 0.1997 0.1903 0.1811 0.1721 0.1633 0.1545
(0.2300) (0.2200) (0.2100) (0.1998) (0.1896) (0.1793) (0.1690) (0.1588) (0.1485) (0.1383) (0.1281)
1:15 0.2547 0.2431 0.2321 0.2215 0.2113 0.2015 0.1919 0.1825 0.1734 0.1644 0.1555
(0.2340) (0.2237) (0.2134) (0.2029) (0.1925) (0.1821) (0.1715) (0.1610) (0.1506) (0.1401) (0.1297)
1:10 0.2582 0.2464 0.2351 0.2242 0.2138 0.2037 0.1939 0.1844 0.1751 0.1659 0.1569
(0.2391) (0.2285) (0.2179) (0.2071) (0.1964) (0.1856) (0.1748) (0.1640) (0.1532) (0.1426) (0.1320)
1:7.5 0.2619 0.2498 0.2382 0.2271 0.2164 0.2061 0.1961 0.1864 0.1769 0.1675 0.1583
(0.2446) (0.2337) (0.2226) (0.2115) (0.2005) (0.1894) (0.1783) (0.1673) (0.1562) (0.1452) (0.1343)
1:5 0.2707 0.2578 0.2454 0.2337 0.2224 0.2115 0.2010 0.1908 0.1808 0.1711 0.1615
(0.2568) (0.2451) (0.2333) (0.2215) (0.2097) (0.1978) (0.1861) (0.1744) (0.1628) (0.1511) (0.1397)
1:4 0.2786 0.2649 0.2520 0.2396 0.2277 0.2164 0.2054 0.1947 0.1844 0.1743 0.1644
(0.2672) (0.2548) (0.2423) (0.2299) (0.2175) (0.2051) (0.1928) (0.1806) (0.1684) (0.1563) (0.1444)
1:3 0.2951 0.2800 0.2656 0.2520 0.2390 0.2265 0.2146 0.2030 0.1918 0.1810 0.1703
(0.2875) (0.2738) (0.2601) (0.2465) (0.2330) (0.2195) (0.2061) (0.1928) (0.1796) (0.1666) (0.1536)
1:2 0.3492 0.3299 0.3116 0.2941 0.2775 0.2616 0.2464 0.2318 0.2178 0.2043 0.1912
(0.3474) (0.3303) (0.3133) (0.2964) (0.2798) (0.2632) (0.2468) (0.2306) (0.2145) (0.1987) (0.1831)
138
Table B.4.8 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=35° /=1 D=1/2)
=35 /=1 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2328 0.2234 0.2144 0.2057 0.1972 0.1889 0.1808 0.1728 0.1649 0.1570 0.1491
(0.2045) (0.1967) (0.1886) (0.1803) (0.1719) (0.1632) (0.1546) (0.1457) (0.1368) (0.1279) (0.1188)
1:25 0.2368 0.2271 0.2178 0.2088 0.2001 0.1915 0.1832 0.1749 0.1668 0.1587 0.1507
(0.2099) (0.2017) (0.1933) (0.1846) (0.1759) (0.1669) (0.1580) (0.1488) (0.1397) (0.1303) (0.1211)
1:15 0.2395 0.2297 0.2202 0.2110 0.2021 0.1933 0.1848 0.1764 0.1682 0.1600 0.1518
(0.2138) (0.2053) (0.1966) (0.1877) (0.1787) (0.1696) (0.1604) (0.1511) (0.1416) (0.1322) (0.1228)
1:10 0.2432 0.2331 0.2233 0.2139 0.2047 0.1958 0.1870 0.1784 0.1700 0.1616 0.1533
(0.2189) (0.2100) (0.2010) (0.1918) (0.1825) (0.1731) (0.1636) (0.1539) (0.1443) (0.1346) (0.1250)
1:7.5 0.2471 0.2366 0.2266 0.2169 0.2075 0.1983 0.1894 0.1806 0.1719 0.1633 0.1548
(0.2242) (0.2150) (0.2057) (0.1962) (0.1866) (0.1768) (0.1670) (0.1571) (0.1471) (0.1372) (0.1273)
1:5 0.2555 0.2444 0.2338 0.2235 0.2136 0.2039 0.1945 0.1852 0.1761 0.1672 0.1582
(0.2362) (0.2262) (0.2161) (0.2059) (0.1956) (0.1852) (0.1747) (0.1641) (0.1536) (0.1431) (0.1326)
1:4 0.2631 0.2513 0.2401 0.2293 0.2188 0.2087 0.1988 0.1892 0.1797 0.1704 0.1612
(0.2463) (0.2358) (0.2250) (0.2142) (0.2033) (0.1923) (0.1813) (0.1703) (0.1592) (0.1481) (0.1372)
1:3 0.2789 0.2658 0.2533 0.2414 0.2298 0.2187 0.2079 0.1975 0.1872 0.1772 0.1673
(0.2664) (0.2545) (0.2427) (0.2306) (0.2187) (0.2065) (0.1945) (0.1824) (0.1704) (0.1584) (0.1464)
1:2 0.3317 0.3147 0.2983 0.2827 0.2678 0.2534 0.2395 0.2261 0.2132 0.2006 0.1883
(0.3265) (0.3112) (0.2960) (0.2807) (0.2655) (0.2503) (0.2353) (0.2202) (0.2054) (0.1905) (0.1759)
139
Table B.5.1 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=40° /=0 D=1/3)
=40 /=0 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2174 0.2018 0.1875 0.1743 0.1620 0.1506 0.1399 0.1298 0.1204 0.1114 0.1028
(0.2174) (0.2048) (0.1923) (0.1800) (0.1679) (0.1560) (0.1443) (0.1327) (0.1215) (0.1105) (0.0998)
1:25 0.2216 0.2056 0.1908 0.1772 0.1645 0.1528 0.1418 0.1315 0.1218 0.1126 0.1039
(0.2216) (0.2086) (0.1958) (0.1832) (0.1707) (0.1586) (0.1465) (0.1349) (0.1233) (0.1121) (0.1012)
1:15 0.2245 0.2081 0.1931 0.1792 0.1663 0.1544 0.1432 0.1327 0.1229 0.1135 0.1046
(0.2245) (0.2113) (0.1982) (0.1854) (0.1727) (0.1603) (0.1482) (0.1362) (0.1247) (0.1133) (0.1022)
1:10 0.2283 0.2116 0.1961 0.1819 0.1687 0.1564 0.1450 0.1343 0.1242 0.1147 0.1056
(0.2283) (0.2148) (0.2014) (0.1883) (0.1754) (0.1627) (0.1503) (0.1382) (0.1263) (0.1148) (0.1035)
1:7.5 0.2322 0.2152 0.1993 0.1847 0.1712 0.1586 0.1469 0.1360 0.1257 0.1160 0.1067
(0.2323) (0.2184) (0.2047) (0.1913) (0.1781) (0.1653) (0.1526) (0.1402) (0.1281) (0.1163) (0.1049)
1:5 0.2408 0.2230 0.2063 0.1909 0.1767 0.1635 0.1512 0.1397 0.1289 0.1187 0.1091
(0.2408) (0.2263) (0.2120) (0.1979) (0.1842) (0.1707) (0.1575) (0.1446) (0.1320) (0.1197) (0.1079)
1:4 0.2479 0.2295 0.2122 0.1962 0.1813 0.1676 0.1548 0.1429 0.1317 0.1211 0.1111
(0.2479) (0.2328) (0.2180) (0.2034) (0.1891) (0.1752) (0.1616) (0.1483) (0.1353) (0.1227) (0.1104)
1:3 0.2611 0.2417 0.2234 0.2062 0.1903 0.1755 0.1618 0.1490 0.1370 0.1258 0.1151
(0.2611) (0.2450) (0.2292) (0.2137) (0.1986) (0.1839) (0.1694) (0.1553) (0.1416) (0.1283) (0.1154)
1:2 0.2957 0.2739 0.2532 0.2336 0.2149 0.1975 0.1814 0.1663 0.1522 0.1390 0.1266
(0.2958) (0.2771) (0.2590) (0.2413) (0.2239) (0.2070) (0.1905) (0.1746) (0.1589) (0.1438) (0.1291)
140
Table B.5.2 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=40° /=1/3 D=1/3)
=40 /=1/3 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1970 0.1843 0.1726 0.1616 0.1513 0.1416 0.1325 0.1238 0.1154 0.1074 0.0997
(0.1966) (0.1857) (0.1751) (0.1643) (0.1537) (0.1434) (0.1330) (0.1228) (0.1127) (0.1029) (0.0932)
1:25 0.2009 0.1879 0.1757 0.1644 0.1538 0.1438 0.1344 0.1254 0.1169 0.1087 0.1008
(0.2006) (0.1894) (0.1784) (0.1674) (0.1566) (0.1459) (0.1353) (0.1248) (0.1145) (0.1045) (0.0946)
1:15 0.2037 0.1903 0.1779 0.1663 0.1555 0.1453 0.1357 0.1266 0.1179 0.1096 0.1015
(0.2035) (0.1921) (0.1808) (0.1696) (0.1585) (0.1476) (0.1368) (0.1262) (0.1158) (0.1055) (0.0954)
1:10 0.2073 0.1936 0.1808 0.1689 0.1578 0.1473 0.1375 0.1281 0.1192 0.1107 0.1025
(0.2072) (0.1954) (0.1838) (0.1724) (0.1611) (0.1499) (0.1389) (0.1281) (0.1174) (0.1069) (0.0967)
1:7.5 0.2111 0.1970 0.1839 0.1716 0.1602 0.1495 0.1393 0.1298 0.1207 0.1120 0.1036
(0.2111) (0.1990) (0.1872) (0.1754) (0.1638) (0.1523) (0.1411) (0.1300) (0.1192) (0.1085) (0.0981)
1:5 0.2195 0.2045 0.1906 0.1776 0.1655 0.1542 0.1435 0.1334 0.1238 0.1147 0.1060
(0.2194) (0.2068) (0.1943) (0.1819) (0.1698) (0.1577) (0.1460) (0.1344) (0.1230) (0.1119) (0.1011)
1:4 0.2264 0.2107 0.1962 0.1827 0.1700 0.1582 0.1470 0.1365 0.1265 0.1171 0.1080
(0.2264) (0.2132) (0.2002) (0.1874) (0.1746) (0.1622) (0.1500) (0.1380) (0.1263) (0.1148) (0.1036)
1:3 0.2395 0.2227 0.2070 0.1923 0.1786 0.1658 0.1538 0.1425 0.1318 0.1216 0.1120
(0.2396) (0.2253) (0.2114) (0.1976) (0.1840) (0.1707) (0.1578) (0.1450) (0.1325) (0.1203) (0.1085)
1:2 0.2743 0.2549 0.2364 0.2190 0.2027 0.1874 0.1730 0.1595 0.1468 0.1348 0.1234
(0.2743) (0.2576) (0.2412) (0.2251) (0.2094) (0.1940) (0.1789) (0.1641) (0.1497) (0.1357) (0.1222)
141
Table B.5.3 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=40° /=2/3 D=1/3)
=40 /=2/3 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1802 0.1699 0.1602 0.1510 0.1423 0.1340 0.1262 0.1186 0.1112 0.1041 0.0972
(0.1785) (0.1695) (0.1604) (0.1512) (0.1420) (0.1328) (0.1236) (0.1145) (0.1055) (0.0967) (0.0878)
1:25 0.1840 0.1732 0.1631 0.1537 0.1447 0.1361 0.1280 0.1202 0.1127 0.1053 0.0982
(0.1825) (0.1731) (0.1636) (0.1542) (0.1448) (0.1353) (0.1259) (0.1166) (0.1073) (0.0982) (0.0892)
1:15 0.1866 0.1756 0.1652 0.1555 0.1463 0.1376 0.1293 0.1213 0.1137 0.1062 0.0990
(0.1852) (0.1757) (0.1659) (0.1563) (0.1467) (0.1370) (0.1274) (0.1180) (0.1085) (0.0993) (0.0901)
1:10 0.1900 0.1787 0.1680 0.1580 0.1485 0.1396 0.1310 0.1228 0.1150 0.1074 0.1000
(0.1888) (0.1789) (0.1690) (0.1591) (0.1491) (0.1393) (0.1294) (0.1197) (0.1102) (0.1006) (0.0914)
1:7.5 0.1936 0.1819 0.1709 0.1606 0.1509 0.1416 0.1329 0.1245 0.1164 0.1086 0.1010
(0.1927) (0.1824) (0.1722) (0.1620) (0.1518) (0.1417) (0.1317) (0.1217) (0.1118) (0.1022) (0.0927)
1:5 0.2016 0.1891 0.1774 0.1664 0.1560 0.1462 0.1369 0.1280 0.1195 0.1113 0.1034
(0.2009) (0.1900) (0.1791) (0.1683) (0.1575) (0.1470) (0.1364) (0.1260) (0.1157) (0.1056) (0.0956)
1:4 0.2082 0.1951 0.1828 0.1712 0.1604 0.1501 0.1403 0.1311 0.1222 0.1136 0.1054
(0.2077) (0.1963) (0.1849) (0.1737) (0.1625) (0.1514) (0.1404) (0.1296) (0.1188) (0.1083) (0.0981)
1:3 0.2208 0.2066 0.1932 0.1806 0.1688 0.1576 0.1470 0.1369 0.1274 0.1182 0.1093
(0.2206) (0.2082) (0.1959) (0.1838) (0.1717) (0.1598) (0.1480) (0.1364) (0.1250) (0.1138) (0.1029)
1:2 0.2551 0.2379 0.2218 0.2066 0.1922 0.1787 0.1659 0.1537 0.1422 0.1312 0.1207
(0.2551) (0.2403) (0.2256) (0.2112) (0.1969) (0.1828) (0.1690) (0.1554) (0.1422) (0.1292) (0.1165)
142
Table B.5.4 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=40° /=1 D=1/3)
=40 /=1 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1649 0.1566 0.1487 0.1412 0.1340 0.1271 0.1204 0.1139 0.1075 0.1013 0.0951
(0.1610) (0.1537) (0.1463) (0.1386) (0.1309) (0.1230) (0.1150) (0.1070) (0.0989) (0.0910) (0.0830)
1:25 0.1684 0.1597 0.1515 0.1437 0.1363 0.1291 0.1222 0.1155 0.1089 0.1025 0.0961
(0.1648) (0.1572) (0.1494) (0.1415) (0.1335) (0.1253) (0.1172) (0.1089) (0.1007) (0.0924) (0.0843)
1:15 0.1709 0.1620 0.1535 0.1455 0.1379 0.1305 0.1234 0.1166 0.1099 0.1033 0.0969
(0.1674) (0.1597) (0.1516) (0.1435) (0.1353) (0.1270) (0.1187) (0.1103) (0.1019) (0.0936) (0.0852)
1:10 0.1741 0.1649 0.1562 0.1479 0.1400 0.1324 0.1251 0.1181 0.1112 0.1045 0.0979
(0.1709) (0.1628) (0.1546) (0.1462) (0.1377) (0.1292) (0.1206) (0.1121) (0.1035) (0.0949) (0.0865)
1:7.5 0.1776 0.1680 0.1590 0.1504 0.1422 0.1344 0.1269 0.1196 0.1126 0.1057 0.0989
(0.1745) (0.1662) (0.1577) (0.1491) (0.1404) (0.1315) (0.1227) (0.1140) (0.1051) (0.0964) (0.0878)
1:5 0.1851 0.1749 0.1651 0.1560 0.1472 0.1389 0.1309 0.1231 0.1157 0.1084 0.1013
(0.1825) (0.1735) (0.1644) (0.1552) (0.1459) (0.1367) (0.1274) (0.1181) (0.1089) (0.0997) (0.0907)
1:4 0.1915 0.1806 0.1704 0.1606 0.1514 0.1426 0.1342 0.1261 0.1183 0.1107 0.1033
(0.1891) (0.1797) (0.1701) (0.1604) (0.1507) (0.1410) (0.1313) (0.1216) (0.1120) (0.1025) (0.0931)
1:3 0.2036 0.1916 0.1803 0.1697 0.1596 0.1499 0.1407 0.1319 0.1234 0.1152 0.1072
(0.2017) (0.1913) (0.1808) (0.1703) (0.1598) (0.1492) (0.1388) (0.1283) (0.1180) (0.1078) (0.0978)
1:2 0.2367 0.2220 0.2081 0.1950 0.1825 0.1706 0.1592 0.1484 0.1380 0.1281 0.1185
(0.2357) (0.2229) (0.2101) (0.1974) (0.1847) (0.1721) (0.1595) (0.1472) (0.1350) (0.1230) (0.1113)
143
Table B.5.5 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=40° /=0 D=1/2)
=40 /=0 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2305 0.2162 0.2028 0.1902 0.1782 0.1669 0.1560 0.1456 0.1357 0.1260 0.1166
(0.2174) (0.2048) (0.1923) (0.1800) (0.1679) (0.1560) (0.1443) (0.1327) (0.1215) (0.1105) (0.0998)
1:25 0.2332 0.2186 0.2049 0.1921 0.1799 0.1683 0.1573 0.1468 0.1367 0.1269 0.1174
(0.2216) (0.2086) (0.1958) (0.1832) (0.1707) (0.1586) (0.1465) (0.1349) (0.1233) (0.1121) (0.1012)
1:15 0.2352 0.2204 0.2065 0.1934 0.1811 0.1694 0.1583 0.1476 0.1374 0.1275 0.1179
(0.2245) (0.2113) (0.1982) (0.1854) (0.1727) (0.1603) (0.1482) (0.1362) (0.1247) (0.1133) (0.1022)
1:10 0.2378 0.2227 0.2086 0.1953 0.1827 0.1708 0.1595 0.1487 0.1383 0.1283 0.1186
(0.2283) (0.2148) (0.2014) (0.1883) (0.1754) (0.1627) (0.1503) (0.1382) (0.1263) (0.1148) (0.1035)
1:7.5 0.2406 0.2252 0.2108 0.1972 0.1844 0.1723 0.1608 0.1499 0.1393 0.1292 0.1194
(0.2323) (0.2184) (0.2047) (0.1913) (0.1781) (0.1653) (0.1526) (0.1402) (0.1281) (0.1163) (0.1049)
1:5 0.2471 0.2309 0.2158 0.2017 0.1884 0.1758 0.1639 0.1525 0.1416 0.1312 0.1211
(0.2408) (0.2263) (0.2120) (0.1979) (0.1842) (0.1707) (0.1575) (0.1446) (0.1320) (0.1197) (0.1079)
1:4 0.2527 0.2359 0.2202 0.2056 0.1918 0.1788 0.1665 0.1548 0.1436 0.1329 0.1226
(0.2479) (0.2328) (0.2180) (0.2034) (0.1891) (0.1752) (0.1616) (0.1483) (0.1353) (0.1227) (0.1104)
1:3 0.2638 0.2458 0.2290 0.2133 0.1986 0.1848 0.1718 0.1594 0.1476 0.1364 0.1256
(0.2611) (0.2450) (0.2292) (0.2137) (0.1986) (0.1839) (0.1694) (0.1553) (0.1416) (0.1283) (0.1154)
1:2 0.2959 0.2746 0.2548 0.2363 0.2190 0.2028 0.1876 0.1732 0.1596 0.1468 0.1346
(0.2958) (0.2771) (0.2590) (0.2413) (0.2239) (0.2070) (0.1905) (0.1746) (0.1589) (0.1438) (0.1291)
144
Table B.5.6 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=40° /=1/3 D=1/2)
=40 /=1/3 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.2110 0.1992 0.1880 0.1773 0.1671 0.1573 0.1478 0.1386 0.1297 0.1210 0.1124
(0.1966) (0.1857) (0.1751) (0.1643) (0.1537) (0.1434) (0.1330) (0.1228) (0.1127) (0.1029) (0.0932)
1:25 0.2138 0.2017 0.1902 0.1793 0.1688 0.1588 0.1491 0.1398 0.1307 0.1219 0.1132
(0.2006) (0.1894) (0.1784) (0.1674) (0.1566) (0.1459) (0.1353) (0.1248) (0.1145) (0.1045) (0.0946)
1:15 0.2157 0.2034 0.1918 0.1807 0.1701 0.1599 0.1501 0.1406 0.1314 0.1225 0.1137
(0.2035) (0.1921) (0.1808) (0.1696) (0.1585) (0.1476) (0.1368) (0.1262) (0.1158) (0.1055) (0.0954)
1:10 0.2184 0.2058 0.1938 0.1825 0.1717 0.1613 0.1514 0.1418 0.1324 0.1234 0.1145
(0.2072) (0.1954) (0.1838) (0.1724) (0.1611) (0.1499) (0.1389) (0.1281) (0.1174) (0.1069) (0.0967)
1:7.5 0.2212 0.2083 0.1961 0.1845 0.1734 0.1629 0.1527 0.1430 0.1335 0.1243 0.1153
(0.2111) (0.1990) (0.1872) (0.1754) (0.1638) (0.1523) (0.1411) (0.1300) (0.1192) (0.1085) (0.0981)
1:5 0.2276 0.2140 0.2011 0.1890 0.1774 0.1664 0.1558 0.1457 0.1359 0.1263 0.1171
(0.2194) (0.2068) (0.1943) (0.1819) (0.1698) (0.1577) (0.1460) (0.1344) (0.1230) (0.1119) (0.1011)
1:4 0.2331 0.2189 0.2055 0.1928 0.1808 0.1694 0.1585 0.1480 0.1379 0.1281 0.1186
(0.2264) (0.2132) (0.2002) (0.1874) (0.1746) (0.1622) (0.1500) (0.1380) (0.1263) (0.1148) (0.1036)
1:3 0.2439 0.2286 0.2142 0.2006 0.1877 0.1755 0.1638 0.1527 0.1420 0.1317 0.1217
(0.2396) (0.2253) (0.2114) (0.1976) (0.1840) (0.1707) (0.1578) (0.1450) (0.1325) (0.1203) (0.1085)
1:2 0.2753 0.2569 0.2396 0.2233 0.2079 0.1934 0.1797 0.1666 0.1542 0.1423 0.1309
(0.2743) (0.2576) (0.2412) (0.2251) (0.2094) (0.1940) (0.1789) (0.1641) (0.1497) (0.1357) (0.1222)
145
Table B.5.7 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=40° /=2/3 D=1/2)
=40 /=2/3 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1955 0.1856 0.1760 0.1668 0.1579 0.1492 0.1408 0.1326 0.1246 0.1167 0.1088
(0.1785) (0.1695) (0.1604) (0.1512) (0.1420) (0.1328) (0.1236) (0.1145) (0.1055) (0.0967) (0.0878)
1:25 0.1983 0.1881 0.1782 0.1688 0.1597 0.1509 0.1423 0.1339 0.1257 0.1176 0.1096
(0.1825) (0.1731) (0.1636) (0.1542) (0.1448) (0.1353) (0.1259) (0.1166) (0.1073) (0.0982) (0.0892)
1:15 0.2002 0.1898 0.1798 0.1702 0.1609 0.1520 0.1433 0.1348 0.1265 0.1183 0.1102
(0.1852) (0.1757) (0.1659) (0.1563) (0.1467) (0.1370) (0.1274) (0.1180) (0.1085) (0.0993) (0.0901)
1:10 0.2028 0.1921 0.1819 0.1720 0.1626 0.1534 0.1446 0.1359 0.1275 0.1192 0.1110
(0.1888) (0.1789) (0.1690) (0.1591) (0.1491) (0.1393) (0.1294) (0.1197) (0.1102) (0.1006) (0.0914)
1:7.5 0.2055 0.1945 0.1840 0.1740 0.1643 0.1550 0.1460 0.1371 0.1285 0.1201 0.1118
(0.1927) (0.1824) (0.1722) (0.1620) (0.1518) (0.1417) (0.1317) (0.1217) (0.1118) (0.1022) (0.0927)
1:5 0.2117 0.2001 0.1890 0.1784 0.1683 0.1585 0.1491 0.1399 0.1310 0.1222 0.1137
(0.2009) (0.1900) (0.1791) (0.1683) (0.1575) (0.1470) (0.1364) (0.1260) (0.1157) (0.1056) (0.0956)
1:4 0.2170 0.2049 0.1933 0.1822 0.1717 0.1615 0.1517 0.1423 0.1331 0.1241 0.1153
(0.2077) (0.1963) (0.1849) (0.1737) (0.1625) (0.1514) (0.1404) (0.1296) (0.1188) (0.1083) (0.0981)
1:3 0.2276 0.2143 0.2018 0.1898 0.1784 0.1675 0.1571 0.1470 0.1372 0.1277 0.1184
(0.2206) (0.2082) (0.1959) (0.1838) (0.1717) (0.1598) (0.1480) (0.1364) (0.1250) (0.1138) (0.1029)
1:2 0.2581 0.2419 0.2267 0.2122 0.1985 0.1854 0.1729 0.1610 0.1495 0.1385 0.1278
(0.2551) (0.2403) (0.2256) (0.2112) (0.1969) (0.1828) (0.1690) (0.1554) (0.1422) (0.1292) (0.1165)
146
Table B.5.8 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=40° /=1 D=1/2)
=40 /=1 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1817 0.1733 0.1651 0.1572 0.1495 0.1419 0.1345 0.1272 0.1199 0.1127 0.1055
(0.1610) (0.1537) (0.1463) (0.1386) (0.1309) (0.1230) (0.1150) (0.1070) (0.0989) (0.0910) (0.0830)
1:25 0.1846 0.1759 0.1675 0.1593 0.1514 0.1436 0.1360 0.1285 0.1211 0.1138 0.1064
(0.1648) (0.1572) (0.1494) (0.1415) (0.1335) (0.1253) (0.1172) (0.1089) (0.1007) (0.0924) (0.0843)
1:15 0.1866 0.1777 0.1691 0.1608 0.1527 0.1448 0.1371 0.1295 0.1219 0.1145 0.1070
(0.1674) (0.1597) (0.1516) (0.1435) (0.1353) (0.1270) (0.1187) (0.1103) (0.1019) (0.0936) (0.0852)
1:10 0.1891 0.1800 0.1712 0.1627 0.1545 0.1464 0.1385 0.1307 0.1230 0.1154 0.1079
(0.1709) (0.1628) (0.1546) (0.1462) (0.1377) (0.1292) (0.1206) (0.1121) (0.1035) (0.0949) (0.0865)
1:7.5 0.1918 0.1825 0.1735 0.1647 0.1563 0.1480 0.1399 0.1320 0.1242 0.1164 0.1087
(0.1745) (0.1662) (0.1577) (0.1491) (0.1404) (0.1315) (0.1227) (0.1140) (0.1051) (0.0964) (0.0878)
1:5 0.1977 0.1878 0.1783 0.1691 0.1602 0.1516 0.1431 0.1348 0.1267 0.1186 0.1107
(0.1825) (0.1735) (0.1644) (0.1552) (0.1459) (0.1367) (0.1274) (0.1181) (0.1089) (0.0997) (0.0907)
1:4 0.2028 0.1924 0.1824 0.1728 0.1635 0.1545 0.1458 0.1372 0.1288 0.1205 0.1123
(0.1891) (0.1797) (0.1701) (0.1604) (0.1507) (0.1410) (0.1313) (0.1216) (0.1120) (0.1025) (0.0931)
1:3 0.2128 0.2015 0.1906 0.1802 0.1701 0.1605 0.1511 0.1419 0.1330 0.1242 0.1156
(0.2017) (0.1913) (0.1808) (0.1703) (0.1598) (0.1492) (0.1388) (0.1283) (0.1180) (0.1078) (0.0978)
1:2 0.2422 0.2281 0.2148 0.2020 0.1898 0.1781 0.1668 0.1559 0.1454 0.1352 0.1252
(0.2357) (0.2229) (0.2101) (0.1974) (0.1847) (0.1721) (0.1595) (0.1472) (0.1350) (0.1230) (0.1113)
147
Table B.6.1 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=45° /=0 D=1/3)
=45 /=0 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1716 0.1577 0.1450 0.1333 0.1225 0.1125 0.1031 0.0943 0.0860 0.0782 0.0707
(0.1716) (0.1597) (0.1481) (0.1367) (0.1257) (0.1149) (0.1045) (0.0943) (0.0844) (0.0749) (0.0659)
1:25 0.1745 0.1603 0.1472 0.1352 0.1241 0.1139 0.1043 0.0953 0.0869 0.0789 0.0713
(0.1745) (0.1623) (0.1504) (0.1388) (0.1275) (0.1165) (0.1058) (0.0955) (0.0855) (0.0759) (0.0666)
1:15 0.1765 0.1620 0.1488 0.1366 0.1253 0.1148 0.1051 0.0960 0.0875 0.0794 0.0717
(0.1765) (0.1641) (0.1521) (0.1403) (0.1288) (0.1177) (0.1068) (0.0964) (0.0862) (0.0765) (0.0671)
1:10 0.1791 0.1643 0.1508 0.1383 0.1268 0.1161 0.1062 0.0969 0.0882 0.0800 0.0722
(0.1791) (0.1665) (0.1542) (0.1422) (0.1305) (0.1192) (0.1082) (0.0975) (0.0872) (0.0773) (0.0678)
1:7.5 0.1818 0.1668 0.1529 0.1401 0.1283 0.1175 0.1073 0.0979 0.0890 0.0807 0.0728
(0.1818) (0.1689) (0.1564) (0.1442) (0.1323) (0.1207) (0.1095) (0.0987) (0.0881) (0.0782) (0.0685)
1:5 0.1876 0.1720 0.1574 0.1440 0.1317 0.1204 0.1098 0.1000 0.0908 0.0822 0.0740
(0.1876) (0.1742) (0.1611) (0.1484) (0.1361) (0.1240) (0.1124) (0.1012) (0.0904) (0.0800) (0.0701)
1:4 0.1923 0.1762 0.1611 0.1473 0.1346 0.1228 0.1119 0.1018 0.0923 0.0834 0.0750
(0.1923) (0.1785) (0.1650) (0.1519) (0.1391) (0.1267) (0.1148) (0.1032) (0.0922) (0.0815) (0.0713)
1:3 0.2009 0.1840 0.1681 0.1533 0.1398 0.1273 0.1158 0.1051 0.0951 0.0858 0.0770
(0.2010) (0.1863) (0.1720) (0.1582) (0.1448) (0.1318) (0.1192) (0.1071) (0.0955) (0.0845) (0.0738)
1:2 0.2222 0.2034 0.1856 0.1689 0.1534 0.1391 0.1260 0.1138 0.1025 0.0920 0.0821
(0.2222) (0.2057) (0.1896) (0.1741) (0.1591) (0.1446) (0.1306) (0.1172) (0.1043) (0.0920) (0.0802)
148
Table B.6.2 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=45° /=1/3 D=1/3)
=45 /=1/3 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1553 0.1440 0.1335 0.1237 0.1145 0.1059 0.0977 0.0900 0.0827 0.0756 0.0688
(0.1549) (0.1449) (0.1349) (0.1250) (0.1153) (0.1059) (0.0966) (0.0875) (0.0787) (0.0701) (0.0619)
1:25 0.1580 0.1464 0.1356 0.1255 0.1161 0.1072 0.0989 0.0910 0.0835 0.0763 0.0694
(0.1577) (0.1474) (0.1371) (0.1270) (0.1171) (0.1075) (0.0980) (0.0887) (0.0797) (0.0710) (0.0626)
1:15 0.1599 0.1480 0.1370 0.1268 0.1172 0.1082 0.0997 0.0917 0.0841 0.0768 0.0698
(0.1598) (0.1492) (0.1387) (0.1285) (0.1184) (0.1086) (0.0989) (0.0895) (0.0804) (0.0716) (0.0631)
1:10 0.1624 0.1502 0.1389 0.1284 0.1186 0.1094 0.1008 0.0926 0.0848 0.0774 0.0703
(0.1623) (0.1514) (0.1408) (0.1303) (0.1200) (0.1100) (0.1002) (0.0906) (0.0814) (0.0724) (0.0638)
1:7.5 0.1650 0.1525 0.1409 0.1302 0.1201 0.1107 0.1019 0.0935 0.0856 0.0781 0.0708
(0.1649) (0.1539) (0.1429) (0.1323) (0.1217) (0.1115) (0.1015) (0.0918) (0.0823) (0.0732) (0.0645)
1:5 0.1706 0.1574 0.1452 0.1339 0.1234 0.1136 0.1043 0.0956 0.0874 0.0795 0.0721
(0.1706) (0.1589) (0.1475) (0.1363) (0.1254) (0.1148) (0.1044) (0.0943) (0.0845) (0.0750) (0.0659)
1:4 0.1752 0.1615 0.1488 0.1370 0.1261 0.1159 0.1063 0.0973 0.0888 0.0808 0.0731
(0.1751) (0.1631) (0.1513) (0.1398) (0.1284) (0.1175) (0.1067) (0.0963) (0.0863) (0.0765) (0.0672)
1:3 0.1836 0.1690 0.1554 0.1429 0.1312 0.1203 0.1101 0.1006 0.0916 0.0831 0.0750
(0.1836) (0.1708) (0.1582) (0.1460) (0.1340) (0.1223) (0.1110) (0.1001) (0.0896) (0.0794) (0.0696)
1:2 0.2047 0.1881 0.1724 0.1578 0.1443 0.1318 0.1201 0.1091 0.0989 0.0892 0.0801
(0.2047) (0.1900) (0.1756) (0.1617) (0.1482) (0.1350) (0.1223) (0.1100) (0.0982) (0.0868) (0.0759)
149
Table B.6.3 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=45° /=2/3 D=1/3)
=45 /=2/3 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1415 0.1322 0.1235 0.1152 0.1075 0.1001 0.0930 0.0862 0.0797 0.0733 0.0671
(0.1403) (0.1318) (0.1233) (0.1148) (0.1064) (0.0981) (0.0899) (0.0818) (0.0738) (0.0661) (0.0585)
1:25 0.1441 0.1344 0.1254 0.1170 0.1090 0.1014 0.0941 0.0872 0.0805 0.0740 0.0677
(0.1430) (0.1342) (0.1255) (0.1168) (0.1082) (0.0997) (0.0913) (0.0830) (0.0749) (0.0669) (0.0592)
1:15 0.1458 0.1360 0.1268 0.1182 0.1100 0.1023 0.0949 0.0879 0.0811 0.0745 0.0681
(0.1448) (0.1359) (0.1270) (0.1182) (0.1094) (0.1007) (0.0922) (0.0838) (0.0756) (0.0675) (0.0597)
1:10 0.1482 0.1381 0.1286 0.1198 0.1114 0.1035 0.0959 0.0887 0.0818 0.0751 0.0686
(0.1473) (0.1381) (0.1291) (0.1199) (0.1110) (0.1021) (0.0934) (0.0849) (0.0765) (0.0683) (0.0604)
1:7.5 0.1506 0.1403 0.1305 0.1214 0.1129 0.1047 0.0970 0.0897 0.0826 0.0758 0.0692
(0.1498) (0.1405) (0.1311) (0.1219) (0.1127) (0.1036) (0.0947) (0.0859) (0.0774) (0.0692) (0.0611)
1:5 0.1559 0.1449 0.1347 0.1251 0.1160 0.1075 0.0994 0.0917 0.0843 0.0772 0.0704
(0.1553) (0.1454) (0.1355) (0.1259) (0.1163) (0.1067) (0.0975) (0.0884) (0.0796) (0.0709) (0.0625)
1:4 0.1602 0.1488 0.1381 0.1281 0.1186 0.1098 0.1014 0.0934 0.0858 0.0784 0.0714
(0.1598) (0.1495) (0.1392) (0.1292) (0.1191) (0.1094) (0.0998) (0.0905) (0.0813) (0.0724) (0.0638)
1:3 0.1683 0.1559 0.1444 0.1337 0.1236 0.1140 0.1051 0.0966 0.0885 0.0807 0.0733
(0.1680) (0.1569) (0.1460) (0.1352) (0.1246) (0.1143) (0.1040) (0.0942) (0.0845) (0.0752) (0.0662)
1:2 0.1887 0.1742 0.1608 0.1481 0.1363 0.1252 0.1148 0.1050 0.0957 0.0868 0.0784
(0.1887) (0.1758) (0.1631) (0.1507) (0.1386) (0.1268) (0.1152) (0.1039) (0.0931) (0.0825) (0.0724)
150
Table B.6.4 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=45° /=1 D=1/3)
=45 /=1 D=1/3
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1280 0.1206 0.1136 0.1069 0.1005 0.0943 0.0884 0.0825 0.0768 0.0712 0.0657
(0.1250) (0.1183) (0.1115) (0.1046) (0.0976) (0.0905) (0.0834) (0.0763) (0.0692) (0.0622) (0.0554)
1:25 0.1305 0.1228 0.1155 0.1086 0.1019 0.0956 0.0895 0.0835 0.0776 0.0719 0.0662
(0.1276) (0.1207) (0.1137) (0.1065) (0.0993) (0.0920) (0.0847) (0.0774) (0.0701) (0.0631) (0.0561)
1:15 0.1321 0.1243 0.1168 0.1097 0.1030 0.0965 0.0902 0.0841 0.0782 0.0724 0.0666
(0.1293) (0.1223) (0.1151) (0.1078) (0.1004) (0.0930) (0.0855) (0.0782) (0.0708) (0.0636) (0.0566)
1:10 0.1343 0.1262 0.1185 0.1112 0.1043 0.0976 0.0912 0.0850 0.0789 0.0730 0.0672
(0.1317) (0.1244) (0.1170) (0.1095) (0.1019) (0.0944) (0.0868) (0.0792) (0.0717) (0.0644) (0.0572)
1:7.5 0.1366 0.1282 0.1203 0.1128 0.1057 0.0989 0.0923 0.0859 0.0797 0.0737 0.0677
(0.1341) (0.1266) (0.1190) (0.1113) (0.1035) (0.0958) (0.0880) (0.0803) (0.0727) (0.0652) (0.0578)
1:5 0.1416 0.1327 0.1243 0.1163 0.1087 0.1015 0.0946 0.0879 0.0814 0.0751 0.0689
(0.1393) (0.1314) (0.1233) (0.1152) (0.1070) (0.0989) (0.0907) (0.0827) (0.0748) (0.0669) (0.0594)
1:4 0.1457 0.1363 0.1275 0.1192 0.1112 0.1037 0.0965 0.0896 0.0828 0.0763 0.0699
(0.1436) (0.1352) (0.1268) (0.1184) (0.1098) (0.1014) (0.0929) (0.0846) (0.0764) (0.0684) (0.0605)
1:3 0.1533 0.1431 0.1336 0.1245 0.1160 0.1079 0.1001 0.0927 0.0855 0.0786 0.0718
(0.1515) (0.1425) (0.1334) (0.1243) (0.1152) (0.1061) (0.0971) (0.0883) (0.0796) (0.0711) (0.0629)
1:2 0.1727 0.1606 0.1492 0.1385 0.1283 0.1187 0.1096 0.1009 0.0926 0.0846 0.0769
(0.1716) (0.1608) (0.1501) (0.1394) (0.1288) (0.1183) (0.1080) (0.0979) (0.0880) (0.0783) (0.0690)
151
Table B.6.5 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=45° /=0 D=1/2)
=45 /=0 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1805 0.1679 0.1559 0.1447 0.1341 0.1241 0.1145 0.1053 0.0965 0.0880 0.0798
(0.1716) (0.1597) (0.1481) (0.1367) (0.1257) (0.1149) (0.1045) (0.0943) (0.0844) (0.0749) (0.0659)
1:25 0.1826 0.1696 0.1575 0.1461 0.1353 0.1250 0.1153 0.1060 0.0971 0.0886 0.0803
(0.1745) (0.1623) (0.1504) (0.1388) (0.1275) (0.1165) (0.1058) (0.0955) (0.0855) (0.0759) (0.0666)
1:15 0.1840 0.1709 0.1586 0.1470 0.1361 0.1257 0.1159 0.1065 0.0976 0.0889 0.0806
(0.1765) (0.1641) (0.1521) (0.1403) (0.1288) (0.1177) (0.1068) (0.0964) (0.0862) (0.0765) (0.0671)
1:10 0.1860 0.1726 0.1600 0.1482 0.1372 0.1267 0.1167 0.1072 0.0981 0.0894 0.0810
(0.1791) (0.1665) (0.1542) (0.1422) (0.1305) (0.1192) (0.1082) (0.0975) (0.0872) (0.0773) (0.0678)
1:7.5 0.1880 0.1743 0.1616 0.1496 0.1383 0.1276 0.1175 0.1079 0.0987 0.0899 0.0814
(0.1818) (0.1689) (0.1564) (0.1442) (0.1323) (0.1207) (0.1095) (0.0987) (0.0881) (0.0782) (0.0685)
1:5 0.1925 0.1782 0.1649 0.1525 0.1408 0.1298 0.1194 0.1095 0.1000 0.0910 0.0823
(0.1876) (0.1742) (0.1611) (0.1484) (0.1361) (0.1240) (0.1124) (0.1012) (0.0904) (0.0800) (0.0701)
1:4 0.1963 0.1815 0.1678 0.1550 0.1429 0.1316 0.1209 0.1108 0.1011 0.0919 0.0831
(0.1923) (0.1785) (0.1650) (0.1519) (0.1391) (0.1267) (0.1148) (0.1032) (0.0922) (0.0815) (0.0713)
1:3 0.2035 0.1878 0.1733 0.1597 0.1470 0.1351 0.1239 0.1133 0.1033 0.0937 0.0846
(0.2010) (0.1863) (0.1720) (0.1582) (0.1448) (0.1318) (0.1192) (0.1071) (0.0955) (0.0845) (0.0738)
1:2 0.2228 0.2047 0.1880 0.1725 0.1581 0.1446 0.1320 0.1202 0.1091 0.0986 0.0886
(0.2222) (0.2057) (0.1896) (0.1741) (0.1591) (0.1446) (0.1306) (0.1172) (0.1043) (0.0920) (0.0802)
152
Table B.6.6 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=45° /=1/3 D=1/2)
=45 /=1/3 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1651 0.1545 0.1444 0.1349 0.1257 0.1169 0.1084 0.1002 0.0923 0.0846 0.0770
(0.1549) (0.1449) (0.1349) (0.1250) (0.1153) (0.1059) (0.0966) (0.0875) (0.0787) (0.0701) (0.0619)
1:25 0.1671 0.1563 0.1460 0.1362 0.1268 0.1179 0.1093 0.1010 0.0929 0.0851 0.0774
(0.1577) (0.1474) (0.1371) (0.1270) (0.1171) (0.1075) (0.0980) (0.0887) (0.0797) (0.0710) (0.0626)
1:15 0.1685 0.1575 0.1471 0.1371 0.1277 0.1186 0.1099 0.1015 0.0934 0.0855 0.0778
(0.1598) (0.1492) (0.1387) (0.1285) (0.1184) (0.1086) (0.0989) (0.0895) (0.0804) (0.0716) (0.0631)
1:10 0.1704 0.1592 0.1485 0.1384 0.1287 0.1195 0.1107 0.1022 0.0939 0.0859 0.0782
(0.1623) (0.1514) (0.1408) (0.1303) (0.1200) (0.1100) (0.1002) (0.0906) (0.0814) (0.0724) (0.0638)
1:7.5 0.1724 0.1609 0.1500 0.1397 0.1299 0.1205 0.1115 0.1029 0.0946 0.0865 0.0786
(0.1649) (0.1539) (0.1429) (0.1323) (0.1217) (0.1115) (0.1015) (0.0918) (0.0823) (0.0732) (0.0645)
1:5 0.1768 0.1648 0.1534 0.1426 0.1324 0.1227 0.1134 0.1045 0.0959 0.0876 0.0796
(0.1706) (0.1589) (0.1475) (0.1363) (0.1254) (0.1148) (0.1044) (0.0943) (0.0845) (0.0750) (0.0659)
1:4 0.1805 0.1680 0.1562 0.1451 0.1345 0.1245 0.1150 0.1058 0.0970 0.0886 0.0804
(0.1751) (0.1631) (0.1513) (0.1398) (0.1284) (0.1175) (0.1067) (0.0963) (0.0863) (0.0765) (0.0672)
1:3 0.1876 0.1742 0.1616 0.1498 0.1386 0.1280 0.1180 0.1084 0.0992 0.0904 0.0819
(0.1836) (0.1708) (0.1582) (0.1460) (0.1340) (0.1223) (0.1110) (0.1001) (0.0896) (0.0794) (0.0696)
1:2 0.2062 0.1906 0.1761 0.1624 0.1496 0.1376 0.1262 0.1154 0.1052 0.0954 0.0861
(0.2047) (0.1900) (0.1756) (0.1617) (0.1482) (0.1350) (0.1223) (0.1100) (0.0982) (0.0868) (0.0759)
153
Table B.6.7 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=45° /=2/3 D=1/2)
=45 /=2/3 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1523 0.1433 0.1347 0.1264 0.1184 0.1106 0.1031 0.0957 0.0885 0.0814 0.0744
(0.1403) (0.1318) (0.1233) (0.1148) (0.1064) (0.0981) (0.0899) (0.0818) (0.0738) (0.0661) (0.0585)
1:25 0.1542 0.1450 0.1362 0.1277 0.1196 0.1117 0.1040 0.0965 0.0892 0.0820 0.0749
(0.1430) (0.1342) (0.1255) (0.1168) (0.1082) (0.0997) (0.0913) (0.0830) (0.0749) (0.0669) (0.0592)
1:15 0.1556 0.1462 0.1373 0.1287 0.1204 0.1124 0.1046 0.0970 0.0896 0.0823 0.0752
(0.1448) (0.1359) (0.1270) (0.1182) (0.1094) (0.1007) (0.0922) (0.0838) (0.0756) (0.0675) (0.0597)
1:10 0.1574 0.1478 0.1387 0.1299 0.1214 0.1133 0.1054 0.0977 0.0902 0.0829 0.0756
(0.1473) (0.1381) (0.1291) (0.1199) (0.1110) (0.1021) (0.0934) (0.0849) (0.0765) (0.0683) (0.0604)
1:7.5 0.1594 0.1495 0.1402 0.1312 0.1226 0.1143 0.1062 0.0984 0.0908 0.0834 0.0761
(0.1498) (0.1405) (0.1311) (0.1219) (0.1127) (0.1036) (0.0947) (0.0859) (0.0774) (0.0692) (0.0611)
1:5 0.1636 0.1533 0.1434 0.1341 0.1251 0.1165 0.1081 0.1001 0.0922 0.0846 0.0771
(0.1553) (0.1454) (0.1355) (0.1259) (0.1163) (0.1067) (0.0975) (0.0884) (0.0796) (0.0709) (0.0625)
1:4 0.1672 0.1564 0.1462 0.1365 0.1272 0.1183 0.1097 0.1014 0.0934 0.0856 0.0779
(0.1598) (0.1495) (0.1392) (0.1292) (0.1191) (0.1094) (0.0998) (0.0905) (0.0813) (0.0724) (0.0638)
1:3 0.1739 0.1624 0.1515 0.1411 0.1312 0.1218 0.1127 0.1040 0.0956 0.0875 0.0795
(0.1680) (0.1569) (0.1460) (0.1352) (0.1246) (0.1143) (0.1040) (0.0942) (0.0845) (0.0752) (0.0662)
1:2 0.1919 0.1784 0.1656 0.1535 0.1421 0.1312 0.1209 0.1111 0.1016 0.0926 0.0838
(0.1887) (0.1758) (0.1631) (0.1507) (0.1386) (0.1268) (0.1152) (0.1039) (0.0931) (0.0825) (0.0724)
154
Table B.6.8 Comparison of Ka_h from log spiral Equivalent Coulomb and Ka_h from classical Coulomb
(=45° /=1 D=1/2)
=45 /=1 D=1/2
Back-slope [degrees]
(v):(h) 0 2 4 6 8 10 12 14 16 18 20
Log-spiral: Ka_h (Coulomb: Ka_h)
1:∞ 0.1402 0.1326 0.1253 0.1182 0.1113 0.1045 0.0979 0.0913 0.0848 0.0783 0.0719
(0.1250) (0.1183) (0.1115) (0.1046) (0.0976) (0.0905) (0.0834) (0.0763) (0.0692) (0.0622) (0.0554)
1:25 0.1421 0.1344 0.1269 0.1196 0.1125 0.1056 0.0988 0.0921 0.0855 0.0789 0.0724
(0.1276) (0.1207) (0.1137) (0.1065) (0.0993) (0.0920) (0.0847) (0.0774) (0.0701) (0.0631) (0.0561)
1:15 0.1435 0.1356 0.1280 0.1206 0.1134 0.1063 0.0994 0.0927 0.0860 0.0793 0.0727
(0.1293) (0.1223) (0.1151) (0.1078) (0.1004) (0.0930) (0.0855) (0.0782) (0.0708) (0.0636) (0.0566)
1:10 0.1453 0.1372 0.1294 0.1218 0.1145 0.1073 0.1003 0.0934 0.0866 0.0799 0.0732
(0.1317) (0.1244) (0.1170) (0.1095) (0.1019) (0.0944) (0.0868) (0.0792) (0.0717) (0.0644) (0.0572)
1:7.5 0.1471 0.1388 0.1308 0.1231 0.1156 0.1083 0.1012 0.0942 0.0872 0.0804 0.0736
(0.1341) (0.1266) (0.1190) (0.1113) (0.1035) (0.0958) (0.0880) (0.0803) (0.0727) (0.0652) (0.0578)
1:5 0.1511 0.1424 0.1340 0.1259 0.1181 0.1105 0.1031 0.0958 0.0887 0.0816 0.0747
(0.1393) (0.1314) (0.1233) (0.1152) (0.1070) (0.0989) (0.0907) (0.0827) (0.0748) (0.0669) (0.0594)
1:4 0.1544 0.1453 0.1366 0.1282 0.1201 0.1123 0.1046 0.0972 0.0898 0.0826 0.0755
(0.1436) (0.1352) (0.1268) (0.1184) (0.1098) (0.1014) (0.0929) (0.0846) (0.0764) (0.0684) (0.0605)
1:3 0.1609 0.1511 0.1417 0.1327 0.1241 0.1157 0.1076 0.0998 0.0921 0.0846 0.0772
(0.1515) (0.1425) (0.1334) (0.1243) (0.1152) (0.1061) (0.0971) (0.0883) (0.0796) (0.0711) (0.0629)
1:2 0.1780 0.1664 0.1553 0.1448 0.1347 0.1251 0.1158 0.1069 0.0982 0.0898 0.0817
(0.1716) (0.1608) (0.1501) (0.1394) (0.1288) (0.1183) (0.1080) (0.0979) (0.0880) (0.0783) (0.0690)
155
APPENDIX C
COMPUTER CODING SUBROUTINES
This appendix lists all the computer coding subroutines that were used to
generate data for the Ka . Fan Zhu (2008) wrote these programs and modified them
to match the formulations in this thesis. All subroutines are written for MATLAB©
edition (version 7.0) and could be run only using MATLAB.
Program C-1 is the computer coding used for generating Ka for design
charts as well as the traces of critical slip surfaces. Five subroutines are needed: Ka.m,
Sur.m, calpah.m, calSur.m and ressal.m. Ka.m is the main subroutine. Sur.m is used
to generate slip surfaces.
Similarly, Program C-2 is the computer coding used to generate equivalent
log spiral Kaand its corresponding slip surfaces. Kan.m is the main subroutine for
Ka while Surn.m is used for slip surfaces. Four subroutines are needed in all: Surn.m,
calPahn.m, Kan.m and calSurn.m. All these subroutines should be installed in the
working directory of MATLAB before running.
156
C1.1 CalPah.m
function calPah=Pah
(b1,b2,A,D,sai,omega,delta,Kh)
calPah=(-1/24*(-27*exp(-b2*sai)*exp(-2*b2*sai)*cos(3*b2)*sai^2+27*exp
(-b2*sai)*exp(-2*b1*sai)*cos(2*b1+b2)*sai^2-54*exp(-b2*sai)*cos(b2)*e
xp(-b1*sai)^2*sai^2+54*exp(-b2*sai)^3*cos(b2)*sai^2+27*exp(-b2*sai)*e
xp(-2*b1*sai)*cos(-2*b1+b2)*sai^2-18*exp(-3*b2*sai)*cos(b2)*sai^2+18*
exp(-3*b2*sai)*cos(3*b2)*sai^2-27*exp(-b2*sai)*exp(-2*b2*sai)*cos(b2)
*sai^2+18*exp(-3*b1*sai)*cos(b1)*sai^2-18*exp(-3*b1*sai)*cos(3*b1)*sa
i^2-24*sai*exp(-3*b1*sai)*sin(b1)+24*sai*exp(-3*b2*sai)*sin(b2)+3*exp
(-b2*sai)*exp(-2*b1*sai)*cos(-2*b1+b2)+3*exp(-b2*sai)*exp(-2*b1*sai)*
cos(2*b1+b2)-6*exp(-b2*sai)*cos(b2)*exp(-b1*sai)^2-2*exp(-3*b1*sai)*c
os(3*b1)-3*exp(-b2*sai)*exp(-2*b2*sai)*cos(b2)-3*exp(-b2*sai)*exp(-2*
b2*sai)*cos(3*b2)+2*cos(3*b2)*exp(-3*b2*sai)+6*exp(-b2*sai)^3*cos(b2)
+6*cos(b2)*exp(-3*b2*sai)-6*exp(-3*b1*sai)*cos(b1))*A^3/(9*sai^2+1)-1
/2*tan(omega)*(1/3*tan(omega)+A*exp(-b1*sai)*sin(b1))-(A*exp(-b1*sai)
*cos(b1)-A*exp(-b2*sai)*cos(b2)-1)*tan(omega)*(A*exp(-b1*sai)*sin(b1)
+1/2*tan(omega))-(1/2*A*exp(-b2*sai)*sin(b2)-1/2*A*exp(-b1*sai)*sin(b
1)-1/2*tan(omega))*(A*exp(-b1*sai)*cos(b1)-A*exp(-b2*sai)*cos(b2)-1)*
(2/3*A*exp(-b1*sai)*sin(b1)+2/3*tan(omega)+1/3*A*exp(-b2*sai)*sin(b2)
)-.41666666666666666666666666666667e-1*(9.*exp(-3.*b1*sai)*sin(b1)*sa
i^2+9.*exp(-3.*b1*sai)*sin(3.*b1)*sai^2-54.*exp(-1.*b2*sai)^2*exp(-1.
*b1*sai)*sin(b1)*sai^2+27.*exp(-1.*b2*sai)^2*exp(-1.*b1*sai)*sin(-1.*
b1+2.*b2)*sai^2-27.*exp(-1.*b2*sai)^2*exp(-1.*b1*sai)*sin(b1+2.*b2)*s
ai^2+27.*exp(-1.*b2*sai)^3*sin(3.*b2)*sai^2-9.*exp(-3.*b2*sai)*sin(b2
)*sai^2-9.*exp(-3.*b2*sai)*sin(3.*b2)*sai^2+27.*exp(-1.*b2*sai)^3*sin
(b2)*sai^2+24.*sai*exp(-3.*b2*sai)*cos(b2)-24.*sai*exp(-3.*b1*sai)*co
s(b1)-3.*exp(-1.*b2*sai)^2*exp(-1.*b1*sai)*sin(b1+2.*b2)-6.*exp(-1.*b
2*sai)^2*exp(-1.*b1*sai)*sin(b1)+9.*exp(-3.*b1*sai)*sin(b1)-9.*sin(b2
)*exp(-3.*b2*sai)+exp(-3.*b1*sai)*sin(3.*b1)+3.*exp(-1.*b2*sai)^2*exp
(-1.*b1*sai)*sin(-1.*b1+2.*b2)+3.*exp(-1.*b2*sai)^3*sin(b2)+3.*exp(-1
.*b2*sai)^3*sin(3.*b2)-1.*sin(3.*b2)*exp(-3.*b2*sai))*Kh*A^3/(9.*sai^
2+1.)-1/2*Kh*tan(omega)*(A*exp(-b1*sai)*cos(b1)-2/3)-Kh*tan(omega)*(A
*exp(-b1*sai)*cos(b1)-A*exp(-b2*sai)*cos(b2)-1)*(1/2*A*exp(-b2*sai)*c
os(b2)+1/2*A*exp(-b1*sai)*cos(b1)-1/2)-1/2*Kh*(A*exp(-b1*sai)*cos(b1)
-A*exp(-b2*sai)*cos(b2)-1)*(A*exp(-b2*sai)*sin(b2)-A*exp(-b1*sai)*sin
(b1)-tan(omega))*(2/3*A*exp(-b2*sai)*cos(b2)+1/3*A*exp(-b1*sai)*cos(b
1)-1/3))/(A*exp(-b1*sai)*cos(b1)-D+tan(delta)*(A*exp(-b1*sai)*sin(b1)
+D*tan(omega)));
157
C1.2 calSur.m
function calSur=Sur(b1,b2,A,D,sai,omega,delta,Kh)
calSur=(-1/24*(-27*exp(-b2*sai)*exp(-2*b2*sai)*cos(3*b2)*sai^2+27*exp
(-b2*sai)*exp(-2*b1*sai)*cos(2*b1+b2)*sai^2-54*exp(-b2*sai)*cos(b2)*e
xp(-b1*sai)^2*sai^2+54*exp(-b2*sai)^3*cos(b2)*sai^2+27*exp(-b2*sai)*e
xp(-2*b1*sai)*cos(-2*b1+b2)*sai^2-18*exp(-3*b2*sai)*cos(b2)*sai^2+18*
exp(-3*b2*sai)*cos(3*b2)*sai^2-27*exp(-b2*sai)*exp(-2*b2*sai)*cos(b2)
*sai^2+18*exp(-3*b1*sai)*cos(b1)*sai^2-18*exp(-3*b1*sai)*cos(3*b1)*sa
i^2-24*sai*exp(-3*b1*sai)*sin(b1)+24*sai*exp(-3*b2*sai)*sin(b2)+3*exp
(-b2*sai)*exp(-2*b1*sai)*cos(-2*b1+b2)+3*exp(-b2*sai)*exp(-2*b1*sai)*
cos(2*b1+b2)-6*exp(-b2*sai)*cos(b2)*exp(-b1*sai)^2-2*exp(-3*b1*sai)*c
os(3*b1)-3*exp(-b2*sai)*exp(-2*b2*sai)*cos(b2)-3*exp(-b2*sai)*exp(-2*
b2*sai)*cos(3*b2)+2*cos(3*b2)*exp(-3*b2*sai)+6*exp(-b2*sai)^3*cos(b2)
+6*cos(b2)*exp(-3*b2*sai)-6*exp(-3*b1*sai)*cos(b1))*A^3/(9*sai^2+1)-1
/2*tan(omega)*(1/3*tan(omega)+A*exp(-b1*sai)*sin(b1))-(A*exp(-b1*sai)
*cos(b1)-A*exp(-b2*sai)*cos(b2)-1)*tan(omega)*(A*exp(-b1*sai)*sin(b1)
+1/2*tan(omega))-(1/2*A*exp(-b2*sai)*sin(b2)-1/2*A*exp(-b1*sai)*sin(b
1)-1/2*tan(omega))*(A*exp(-b1*sai)*cos(b1)-A*exp(-b2*sai)*cos(b2)-1)*
(2/3*A*exp(-b1*sai)*sin(b1)+2/3*tan(omega)+1/3*A*exp(-b2*sai)*sin(b2)
)-.41666666666666666666666666666667e-1*(9.*exp(-3.*b1*sai)*sin(b1)*sa
i^2+9.*exp(-3.*b1*sai)*sin(3.*b1)*sai^2-54.*exp(-1.*b2*sai)^2*exp(-1.
*b1*sai)*sin(b1)*sai^2+27.*exp(-1.*b2*sai)^2*exp(-1.*b1*sai)*sin(-1.*
b1+2.*b2)*sai^2-27.*exp(-1.*b2*sai)^2*exp(-1.*b1*sai)*sin(b1+2.*b2)*s
ai^2+27.*exp(-1.*b2*sai)^3*sin(3.*b2)*sai^2-9.*exp(-3.*b2*sai)*sin(b2
)*sai^2-9.*exp(-3.*b2*sai)*sin(3.*b2)*sai^2+27.*exp(-1.*b2*sai)^3*sin
(b2)*sai^2+24.*sai*exp(-3.*b2*sai)*cos(b2)-24.*sai*exp(-3.*b1*sai)*co
s(b1)-3.*exp(-1.*b2*sai)^2*exp(-1.*b1*sai)*sin(b1+2.*b2)-6.*exp(-1.*b
2*sai)^2*exp(-1.*b1*sai)*sin(b1)+9.*exp(-3.*b1*sai)*sin(b1)-9.*sin(b2
)*exp(-3.*b2*sai)+exp(-3.*b1*sai)*sin(3.*b1)+3.*exp(-1.*b2*sai)^2*exp
(-1.*b1*sai)*sin(-1.*b1+2.*b2)+3.*exp(-1.*b2*sai)^3*sin(b2)+3.*exp(-1
.*b2*sai)^3*sin(3.*b2)-1.*sin(3.*b2)*exp(-3.*b2*sai))*Kh*A^3/(9.*sai^
2+1.)-1/2*Kh*tan(omega)*(A*exp(-b1*sai)*cos(b1)-2/3)-Kh*tan(omega)*(A
*exp(-b1*sai)*cos(b1)-A*exp(-b2*sai)*cos(b2)-1)*(1/2*A*exp(-b2*sai)*c
os(b2)+1/2*A*exp(-b1*sai)*cos(b1)-1/2)-1/2*Kh*(A*exp(-b1*sai)*cos(b1)
-A*exp(-b2*sai)*cos(b2)-1)*(A*exp(-b2*sai)*sin(b2)-A*exp(-b1*sai)*sin
(b1)-tan(omega))*(2/3*A*exp(-b2*sai)*cos(b2)+1/3*A*exp(-b1*sai)*cos(b
1)-1/3))/(A*exp(-b1*sai)*cos(b1)-D+tan(delta)*(A*exp(-b1*sai)*sin(b1)
+D*tan(omega)));
158
C1.3 Ka.m
clear;
disp('This program is designed to calculate active earth pressure
coefficient Ka.');
disp('All input parameters should be normalized.');
disp('Note: phi=friction angle of reinforced soil');
disp(' omega=batter of wall (angle between inclined wall and
vertical direction)');
disp(' delta=friction angle between facing blocks and reinforced
soil');
disp(' ');
% ---------------------Input Data----------------------
batter=input('Please input the batter of wall, omega (degree): ');
al=input('Please input the back slope angle, alfa (degree): ');
D=input('Please input the height of resultant force Pa, D: ');
m=input('Please input soil friction angle phi (degree): ');
ratio=input('Please input the ratio of delta and phi (e.g.: 0~1): ');
Kh=input('Please input seismic coefficient, Kh: ');
disp(' ');
disp('Calculating Ka, please wait...');
disp(' ');
% -----------------Process Input Data-------------
max=0;
c1=0;
c2=0;
phi=m/180*pi;
sai=tan(phi);
delta=ratio*phi;
omega=(batter)/180*pi;
alfa=(al)/180*pi;
inc=0.1;
% -----------------------Calculate Ka-------------------
for i=-m:inc:90-m-inc
b1=i/180*pi;
for j=i+inc:inc:90-m
b2=j/180*pi;
159
A=((1-tan(omega)*tan(alfa))/(exp(-b1*sai)*(cos(b1)+sin(b1)*tan(alfa))
-exp(-b2*sai)*(cos(b2)+sin(b2)*tan(alfa))));
value=calPah(b1,b2,A,D,sai,omega,delta,Kh);
X2=A*exp(-b2*sai)*sin(b2)-A*exp(-b1*sai)*sin(b1)-tan(omega);
if value>max & A>0 & X2>0 & value<=1
max=value;
c1=i;
c2=j;
VA=A;
VX2=X2;
end
end
end
max=max*2;
if max>=1.01
disp('Error occured, check program or parameters.');
flag=1;
else
disp('Ka= ');
disp(max);
flag=0;
end
% -----------------Display Critical Slip Surface-------------------
if flag==0
disp('Generating Critical Slip Surface, please wait...');
syms b angle1 angle2;
angle1=c1/180*pi;
angle2=c2/180*pi;
X=VA*exp(-b*sai)*sin(b)-VA*exp(-angle1*sai)*sin(angle1);
Y=-VA*exp(-b*sai)*cos(b)+VA*exp(-angle1*sai)*cos(angle1);
ezplot(X,Y,[angle1,angle2]),axis equal, grid on;
disp('Completed.');
end
160
C1.4 ressa1
function toplam()
for
p = 357/(tana+cos35);
p*sin35 = 1000*tana +p*tanacos35;
end
end
tana=?
C1.5 sur
clear;
disp('This program is designed to generate data for critical slip
surfaces.');
disp('All input parameters should be normalized.');
disp('Note: phi=friction angle of reinforced soil');
disp(' omega=batter of wall (angle between inclined wall and
vertical direction)');
disp(' delta=friction angle between facing blocks and reinforced
soil');
disp(' ');
% ---------------------Input Data----------------------
batter=input('Please input the batter of wall, omega (degree): ');
al=input('Please input the back slope angle, alfa (degree): ');
D=input('Please input the height of resultant force Pa, D: ');
m=input('Please input soil friction angle phi (degree): ');
ratio=input('Please input the ratio of delta and phi (e.g.: 0~1): ');
Kh=input('Please input seismic coefficient, Kh: ');
disp(' ');
disp('Calculating Ka, please wait...');
disp(' ');
% -----------------Process Input Data-------------
max=0;
c1=0;
c2=0;
phi=m/180*pi;
sai=tan(phi);
delta=ratio*phi;
161
omega=(batter)/180*pi;
alfa=(al)/180*pi;
inc=0.1;
% -----------------------Calculate Ka-------------------
for i=-m:inc:90-m-inc
b1=i/180*pi;
for j=i+inc:inc:90-m
b2=j/180*pi;
A=((1-tan(omega)*tan(alfa))/(exp(-b1*sai)*(cos(b1)+sin(b1)*tan(alfa))
-exp(-b2*sai)*(cos(b2)+sin(b2)*tan(alfa))));
value=calPah(b1,b2,A,D,sai,omega,delta,Kh);
X2=A*exp(-b2*sai)*sin(b2)-A*exp(-b1*sai)*sin(b1)-tan(omega);
if value>max & A>0 & X2>0 & value<=1
max=value;
c1=i;
c2=j;
VA=A;
VX2=X2;
end
end
end
max=max*2;
if max>=1.01
disp('Error occured, check program.');
flag=1;
else
disp('Ka= ');
disp(max);
flag=0;
end
% ----------------Generate Data for Slip Surface--------------------
if flag==0
syms b angle1 angle2;
angle1=c1/180*pi;
angle2=c2/180*pi;
disp('X coordinates of critical slip surface');
dif=(angle2-angle1)/10;
162
for i=angle1:dif:angle2
b=i;
X=VA*exp(-b*sai)*sin(b)-VA*exp(-angle1*sai)*sin(angle1);
% ezplot(X,Y,[angle1,angle2]),axis equal, grid on;
disp(X);
end
disp('Y coordinates of critical slip surface');
for i=angle1:dif:angle2
b=i;
Y=-VA*exp(-b*sai)*cos(b)+VA*exp(-angle1*sai)*cos(angle1);
% ezplot(X,Y,[angle1,angle2]),axis equal, grid on;
disp(Y);
end
end
163
C.2 3rd version of Matlab used to generate equivalent log spiral Ka.
C2.1 surn.m
clear;
disp('This program is designed to generate data for critical slip
surfaces.');
disp('All input parameters should be normalized.');
disp('Note: phi=friction angle of reinforced soil');
disp(' omega=batter of wall (angle between inclined wall and
vertical direction)');
disp(' delta=friction angle between facing blocks and reinforced
soil');
disp(' ');
% ---------------------Input Data----------------------
batter=input('Please input the batter of wall, omega (degree): ');
al=input('Please input the back slope angle, alfa (degree): ');
D=input('Please input the height of resultant force Pa, D: ');
m=input('Please input soil friction angle phi (degree): ');
ratio=input('Please input the ratio of delta and phi (e.g.: 0~1): ');
Kh=input('Please input seismic coefficient, Kh: ');
disp(' ');
disp('Calculating Ka, please wait...');
disp(' ');
% -----------------Process Input Data-------------
max=0;
c1=0;
c2=0;
phi=m/180*pi;
sai=tan(phi);
delta=ratio*phi;
omega=(batter)/180*pi;
alfa=(al)/180*pi;
inc=1;
% -----------------------Calculate Ka-------------------
for i=-m:inc:90-m-inc
b1=i/180*pi;
for j=i+inc:inc:90-m
b2=j/180*pi;
164
A=((1-tan(omega)*tan(alfa))/(exp(-b1*sai)*(cos(b1)+sin(b1)*tan(alfa))
-exp(-b2*sai)*(cos(b2)+sin(b2)*tan(alfa))));
value=calPahn(b1,b2,A,D,sai,omega,delta,Kh);
X2=A*exp(-b2*sai)*sin(b2)-A*exp(-b1*sai)*sin(b1)-tan(omega);
if value>max & A>0 & X2>0 & value<=1
max=value;
c1=i;
c2=j;
VA=A;
VX2=X2;
end
end
end
max=max*2;
if max>=1.01
disp('Error occured, check program.');
flag=1;
else
disp('Ka= ');
disp(max);
flag=0;
end
% ----------------Generate Data for Slip Surface--------------------
if flag==0
syms b angle1 angle2;
angle1=c1/180*pi;
angle2=c2/180*pi;
disp('X coordinates of critical slip surface');
dif=(angle2-angle1)/10;
for i=angle1:dif:angle2
b=i;
X=VA*exp(-b*sai)*sin(b)-VA*exp(-angle1*sai)*sin(angle1);
% ezplot(X,Y,[angle1,angle2]),axis equal, grid on;
disp(X);
end
disp('Y coordinates of critical slip surface');
for i=angle1:dif:angle2
165
b=i;
Y=-VA*exp(-b*sai)*cos(b)+VA*exp(-angle1*sai)*cos(angle1);
% ezplot(X,Y,[angle1,angle2]),axis equal, grid on;
disp(Y);
end
end
C2.2 calPahn.m
function calPahn=Pah(b1,b2,A,D,sai,omega,delta,Kh)
calPahn=(-1/24*(18*exp(-3*b1*sai)*cos(b1)*sai^2+54*exp(-b2*sai)^3*cos
(b2)*sai^2-54*exp(-b2*sai)*cos(b2)*exp(-b1*sai)^2*sai^2-27*exp(-b2*sa
i)*exp(-2*b2*sai)*cos(3*b2)*sai^2+18*exp(-3*b2*sai)*cos(3*b2)*sai^2-1
8*exp(-3*b1*sai)*cos(3*b1)*sai^2+27*exp(-b2*sai)*exp(-2*b1*sai)*cos(-
2*b1+b2)*sai^2+27*exp(-b2*sai)*exp(-2*b1*sai)*cos(2*b1+b2)*sai^2-18*e
xp(-3*b2*sai)*cos(b2)*sai^2-27*exp(-b2*sai)*exp(-2*b2*sai)*cos(b2)*sa
i^2-24*sai*exp(-3*b1*sai)*sin(b1)+24*sai*exp(-3*b2*sai)*sin(b2)-3*exp
(-b2*sai)*exp(-2*b2*sai)*cos(3*b2)-3*exp(-b2*sai)*exp(-2*b2*sai)*cos(
b2)+6*cos(b2)*exp(-3*b2*sai)+6*exp(-b2*sai)^3*cos(b2)+2*cos(3*b2)*exp
(-3*b2*sai)-2*exp(-3*b1*sai)*cos(3*b1)+3*exp(-b2*sai)*exp(-2*b1*sai)*
cos(2*b1+b2)+3*exp(-b2*sai)*exp(-2*b1*sai)*cos(-2*b1+b2)-6*exp(-b2*sa
i)*cos(b2)*exp(-b1*sai)^2-6*exp(-3*b1*sai)*cos(b1))*A^3/(9*sai^2+1)-1
/2*tan(omega)*(1/3*tan(omega)+A*exp(-b1*sai)*sin(b1))-(A*exp(-b1*sai)
*cos(b1)-A*exp(-b2*sai)*cos(b2)-1)*tan(omega)*(A*exp(-b1*sai)*sin(b1)
+1/2*tan(omega))-(1/2*A*exp(-b2*sai)*sin(b2)-1/2*A*exp(-b1*sai)*sin(b
1)-1/2*tan(omega))*(A*exp(-b1*sai)*cos(b1)-A*exp(-b2*sai)*cos(b2)-1)*
(2/3*A*exp(-b1*sai)*sin(b1)+2/3*tan(omega)+1/3*A*exp(-b2*sai)*sin(b2)
)-.41666666666666666666666666666667e-1*(27.*exp(-1.*b2*sai)^2*exp(-1.
*b1*sai)*sin(-1.*b1+2.*b2)*sai^2-27.*exp(-1.*b2*sai)^2*exp(-1.*b1*sai
)*sin(b1+2.*b2)*sai^2-9.*exp(-3.*b2*sai)*sin(3.*b2)*sai^2-9.*exp(-3.*
b2*sai)*sin(b2)*sai^2+27.*exp(-1.*b2*sai)^3*sin(3.*b2)*sai^2+27.*exp(
-1.*b2*sai)^3*sin(b2)*sai^2+9.*exp(-3.*b1*sai)*sin(3.*b1)*sai^2+9.*ex
p(-3.*b1*sai)*sin(b1)*sai^2-54.*exp(-1.*b2*sai)^2*exp(-1.*b1*sai)*sin
(b1)*sai^2+24.*sai*exp(-3.*b2*sai)*cos(b2)-24.*sai*exp(-3.*b1*sai)*co
s(b1)-3.*exp(-1.*b2*sai)^2*exp(-1.*b1*sai)*sin(b1+2.*b2)-6.*exp(-1.*b
2*sai)^2*exp(-1.*b1*sai)*sin(b1)+3.*exp(-1.*b2*sai)^3*sin(3.*b2)+9.*e
xp(-3.*b1*sai)*sin(b1)+exp(-3.*b1*sai)*sin(3.*b1)+3.*exp(-1.*b2*sai)^
3*sin(b2)-9.*sin(b2)*exp(-3.*b2*sai)+3.*exp(-1.*b2*sai)^2*exp(-1.*b1*
sai)*sin(-1.*b1+2.*b2)-1.*sin(3.*b2)*exp(-3.*b2*sai))*Kh*A^3/(9.*sai^
2+1.)-1/2*Kh*tan(omega)*(A*exp(-b1*sai)*cos(b1)-2/3)-Kh*tan(omega)*(A
*exp(-b1*sai)*cos(b1)-A*exp(-b2*sai)*cos(b2)-1)*(1/2*A*exp(-b2*sai)*c
166
os(b2)+1/2*A*exp(-b1*sai)*cos(b1)-1/2)-1/2*Kh*(A*exp(-b1*sai)*cos(b1)
-A*exp(-b2*sai)*cos(b2)-1)*(A*exp(-b2*sai)*sin(b2)-A*exp(-b1*sai)*sin
(b1)-tan(omega))*(2/3*A*exp(-b2*sai)*cos(b2)+1/3*A*exp(-b1*sai)*cos(b
1)-1/3))/((A*exp(-b1*sai)*cos(b1)-D)*cos(omega)-(A*exp(-b1*sai)*sin(b
1)+D*tan(omega))*sin(omega)+tan(delta)*(A*exp(-b1*sai)*cos(b1)-D)*sin
(omega)+tan(delta)*(A*exp(-b1*sai)*sin(b1)+D*tan(omega))*cos(omega));
C2.3 kan.m
clear;
disp('This program is designed to calculate active earth pressure
coefficient Ka.');
disp('All input parameters should be normalized.');
disp('Note: phi=friction angle of reinforced soil');
disp(' omega=batter of wall (angle between inclined wall and
vertical direction)');
disp(' delta=friction angle between facing blocks and reinforced
soil');
disp(' ');
% ---------------------Input Data----------------------
batter=input('Please input the batter of wall, omega (degree): ');
al=input('Please input the back slope angle, alfa (degree): ');
D=input('Please input the height of resultant force Pa, D: ');
m=input('Please input soil friction angle phi (degree): ');
ratio=input('Please input the ratio of delta and phi (e.g.: 0~1): ');
Kh=input('Please input seismic coefficient, Kh: ');
disp(' ');
disp('Calculating Ka, please wait...');
disp(' ');
% -----------------Process Input Data-------------
max=0;
c1=0;
c2=0;
phi=m/180*pi;
sai=tan(phi);
delta=ratio*phi;
omega=(batter)/180*pi;
alfa=(al)/180*pi;
inc=1;
167
% -----------------------Calculate Ka-------------------
for i=-m:inc:90-m-inc
b1=i/180*pi;
for j=i+inc:inc:90-m
b2=j/180*pi;
A=((1-tan(omega)*tan(alfa))/(exp(-b1*sai)*(cos(b1)+sin(b1)*tan(alfa))
-exp(-b2*sai)*(cos(b2)+sin(b2)*tan(alfa))));
value=calPahn(b1,b2,A,D,sai,omega,delta,Kh);
X2=A*exp(-b2*sai)*sin(b2)-A*exp(-b1*sai)*sin(b1)-tan(omega);
if value>max & A>0 & X2>0 & value<=1
max=value;
c1=i;
c2=j;
VA=A;
VX2=X2;
end
end
end
max=max*2;
if max>=1.01
disp('Error occured, check program or parameters.');
flag=1;
else
disp('Ka= ');
disp(max);
flag=0;
end
% -----------------Display Critical Slip Surface-------------------
if flag==0
disp('Generating Critical Slip Surface, please wait...');
syms b angle1 angle2;
angle1=c1/180*pi;
angle2=c2/180*pi;
X=VA*exp(-b*sai)*sin(b)-VA*exp(-angle1*sai)*sin(angle1);
Y=-VA*exp(-b*sai)*cos(b)+VA*exp(-angle1*sai)*cos(angle1);
ezplot(X,Y,[angle1,angle2]),axis equal, grid on;
disp('Completed.');
168
end
C2.4 calSurn
function calSurn=Sur(b1,b2,A,D,sai,omega,delta,Kh)
calSurn=(-1/24*(18*exp(-3*b1*sai)*cos(b1)*sai^2+54*exp(-b2*sai)^3*cos
(b2)*sai^2-54*exp(-b2*sai)*cos(b2)*exp(-b1*sai)^2*sai^2-27*exp(-b2*sa
i)*exp(-2*b2*sai)*cos(3*b2)*sai^2+18*exp(-3*b2*sai)*cos(3*b2)*sai^2-1
8*exp(-3*b1*sai)*cos(3*b1)*sai^2+27*exp(-b2*sai)*exp(-2*b1*sai)*cos(-
2*b1+b2)*sai^2+27*exp(-b2*sai)*exp(-2*b1*sai)*cos(2*b1+b2)*sai^2-18*e
xp(-3*b2*sai)*cos(b2)*sai^2-27*exp(-b2*sai)*exp(-2*b2*sai)*cos(b2)*sa
i^2-24*sai*exp(-3*b1*sai)*sin(b1)+24*sai*exp(-3*b2*sai)*sin(b2)-3*exp
(-b2*sai)*exp(-2*b2*sai)*cos(3*b2)-3*exp(-b2*sai)*exp(-2*b2*sai)*cos(
b2)+6*cos(b2)*exp(-3*b2*sai)+6*exp(-b2*sai)^3*cos(b2)+2*cos(3*b2)*exp
(-3*b2*sai)-2*exp(-3*b1*sai)*cos(3*b1)+3*exp(-b2*sai)*exp(-2*b1*sai)*
cos(2*b1+b2)+3*exp(-b2*sai)*exp(-2*b1*sai)*cos(-2*b1+b2)-6*exp(-b2*sa
i)*cos(b2)*exp(-b1*sai)^2-6*exp(-3*b1*sai)*cos(b1))*A^3/(9*sai^2+1)-1
/2*tan(omega)*(1/3*tan(omega)+A*exp(-b1*sai)*sin(b1))-(A*exp(-b1*sai)
*cos(b1)-A*exp(-b2*sai)*cos(b2)-1)*tan(omega)*(A*exp(-b1*sai)*sin(b1)
+1/2*tan(omega))-(1/2*A*exp(-b2*sai)*sin(b2)-1/2*A*exp(-b1*sai)*sin(b
1)-1/2*tan(omega))*(A*exp(-b1*sai)*cos(b1)-A*exp(-b2*sai)*cos(b2)-1)*
(2/3*A*exp(-b1*sai)*sin(b1)+2/3*tan(omega)+1/3*A*exp(-b2*sai)*sin(b2)
)-.41666666666666666666666666666667e-1*(27.*exp(-1.*b2*sai)^2*exp(-1.
*b1*sai)*sin(-1.*b1+2.*b2)*sai^2-27.*exp(-1.*b2*sai)^2*exp(-1.*b1*sai
)*sin(b1+2.*b2)*sai^2-9.*exp(-3.*b2*sai)*sin(3.*b2)*sai^2-9.*exp(-3.*
b2*sai)*sin(b2)*sai^2+27.*exp(-1.*b2*sai)^3*sin(3.*b2)*sai^2+27.*exp(
-1.*b2*sai)^3*sin(b2)*sai^2+9.*exp(-3.*b1*sai)*sin(3.*b1)*sai^2+9.*ex
p(-3.*b1*sai)*sin(b1)*sai^2-54.*exp(-1.*b2*sai)^2*exp(-1.*b1*sai)*sin
(b1)*sai^2+24.*sai*exp(-3.*b2*sai)*cos(b2)-24.*sai*exp(-3.*b1*sai)*co
s(b1)-3.*exp(-1.*b2*sai)^2*exp(-1.*b1*sai)*sin(b1+2.*b2)-6.*exp(-1.*b
2*sai)^2*exp(-1.*b1*sai)*sin(b1)+3.*exp(-1.*b2*sai)^3*sin(3.*b2)+9.*e
xp(-3.*b1*sai)*sin(b1)+exp(-3.*b1*sai)*sin(3.*b1)+3.*exp(-1.*b2*sai)^
3*sin(b2)-9.*sin(b2)*exp(-3.*b2*sai)+3.*exp(-1.*b2*sai)^2*exp(-1.*b1*
sai)*sin(-1.*b1+2.*b2)-1.*sin(3.*b2)*exp(-3.*b2*sai))*Kh*A^3/(9.*sai^
2+1.)-1/2*Kh*tan(omega)*(A*exp(-b1*sai)*cos(b1)-2/3)-Kh*tan(omega)*(A
*exp(-b1*sai)*cos(b1)-A*exp(-b2*sai)*cos(b2)-1)*(1/2*A*exp(-b2*sai)*c
os(b2)+1/2*A*exp(-b1*sai)*cos(b1)-1/2)-1/2*Kh*(A*exp(-b1*sai)*cos(b1)
-A*exp(-b2*sai)*cos(b2)-1)*(A*exp(-b2*sai)*sin(b2)-A*exp(-b1*sai)*sin
(b1)-tan(omega))*(2/3*A*exp(-b2*sai)*cos(b2)+1/3*A*exp(-b1*sai)*cos(b
1)-1/3))/((A*exp(-b1*sai)*cos(b1)-D)*cos(omega)-(A*exp(-b1*sai)*sin(b
1)+D*tan(omega))*sin(omega)+tan(delta)*(A*exp(-b1*sai)*cos(b1)-D)*sin
(omega)+tan(delta)*(A*exp(-b1*sai)*sin(b1)+D*tan(omega))*cos(omega))
169
REFERENCES
[1] Baker, R. (1981). “Tensile strength, tension cracks, and stability of slopes.” Soils
and Foundations, 21(2), 1-17.
[2] Huang, C-C, Chen, Y-H,(2004) “Seismic stability of soil retaining walls situated
on slope.” Journal of geotechnical & geoenvironmental engineering, 130(1),
January 2004/45
[3] Leshchinsky, D., and San, K.-C. (1994). “Pseudostatic seismic stability of slopes:
Design charts.” Journal of Geotechnical Engineering, 120(9), 1514-1532.
[4] Leshchinsky, D., Zhu, F., and Meehan, C.L. (2010). “Required unfactored strength
of geosynthetic in reinforced earth structures” Journal of Geotechnical and
Geoenvironmental Engineering, 136(2), 281-289.
[5] NCMA (National Concrete Masonry Association). (1997) . Design manual for
segmental retaining walls, 2nd
Edition, J.G. Collin editor, Herndon, VA.
[6] Seed, H.B., and Whitman, R.V (1970). “Design of earth retaining structures for
dynamic loads.” Lateral stresses in the ground and design of earth retaining
structures, ASCE. New York, 103-147.
[7] Zhu, F. (2008). “Geosynthetic reinforced earth structures: effects of facing units and
force distribution functions”. A thesis submitted to the Faculty of the
University of Delaware in partial fulfillment of the requirements for the degree
of Master of Civil Engineering.