resultant force of lateral earth pressure in …

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.1.6 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.2.1 Ka_h for Design Charts when =25° /=0 D=1/3 .............................. 66













vi









Table A. 4.6 Ka_h for Design Charts when =35° =2/3 D=1/2 ........................... 87


Table A.4.8 Ka_h for Design Charts when =35° =1 D=1/2 ............................. 89






Table A.5.6 Ka_h for Design Charts when =40° =2/3 D=1/2 .......................... 95





Table A.6.3 Ka_h for Design Charts when =45° =1/3 D=1/3 ......................... 100



vii


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


and Ka_h from classical Coulomb (=20° /=1/3 D=1/3) ............. 108




and Ka_h from classical Coulomb (=20° /=1 D=1/3) ............... 110


and Ka_h from classical Coulomb (=20° /=0 D=1/2) ................ 111

















viii







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























ix
























and Ka_hfrom classical Coulomb (=40° /=1 D=1/2) ................. 146







x


and Ka_h from classical Coulomb (=45° /=1 D=1/3) ............... 150




and Ka_h from classical Coulomb (=45° /=1/3 D=1/2)................ 152





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


(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1 ......................... 14


(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1 ......................... 16


(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1 ......................... 17


(a) / = 0; (b) / = 1/3; (c) / = 2/3; and (d) / = 1 ......................... 18


(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


D=1/2 (=20°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1 ................. 22


D=1/2 (=30°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1 ................. 23


D=1/2 (=30°): (a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1 ................. 24

xii


D=1/2 (=40°) :(a) /=0 (b) /=1/3 (c) /=2/3 (d) /=1 ................. 25


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


when /=1/3: (a) D=1/3 (b) D=1/2 ........................................................ 28


when /=2/3: (a) D=1/3 (b) D=1/2 ........................................................ 29


when /=1: (a) D=1/3 (b) D=1/2 ........................................................... 30


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


when /=2/3: (a) D=1/3 (b) D=1/2 ........................................................ 33


when /=1: (a) D=1/3 (b) D=1/2 ........................................................... 34


when /=0:(a) D=1/3 (b) D=1/2 ............................................................ 35


when /=1/3: (a) D=1/3 (b) D=1/2 ........................................................ 36


when /=2/3: (a) D=1/3 (b) D=1/2 ........................................................ 37


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


resultant inclination (Eq. 6 and Fig. 3): (a) =20; and (b) =30

(c) =40 ................................................................................................. 41


resultant inclination (Eq. 6 and Fig. 3): (d) =20; and (e) =30

(f) =40 .................................................................................................. 42


resultant inclination (Eq. 6 and Fig. 3): (d) =20; and (e) =30

(f) =40 .................................................................................................. 43


inclination when D=1/2: (a) =20° (b) =30° (c) =40° ........................ 44


inclination when D=1/2: (a) =20° (b) =30° (c) =40° ........................ 45


resultant inclination when D=1/2: (a) =20° (b) =30° (c) =40° ......... 46


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


(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


(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


(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


(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


(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


(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


(=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


(=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


(=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


(=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


(=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


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


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


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


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


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


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


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


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


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


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=1:2, Logspiral

Backslope V:H=1:2, Coulomb

=30°, =0°

41

22c


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





=40°, =0°

42

22d

22e


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



=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





=30°, =20°

43

22f


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





=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



=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





=30°, =0°, D=1/2

45

23c


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





=40°, =0°, D=1/2

46

24a

24b



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



=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





=30°,=20°,D=1/2

47

24c



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





=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





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





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





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





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





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





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





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





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





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


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


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


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


62


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


63


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


64


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


65


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


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


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


68


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


69


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


70


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


71


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


72


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


73


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


74


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


(=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


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)


109


(=20° /=2/3 D=1/3)

=20 /=2/3 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


110


(=20° /=1 D=1/3)

=20 /=1 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


111


(=20° /=0 D=1/2)

=20 /=0 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


112


(=20° /=1/3 D=1/2)

=20 /=1/3 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


113


(=20° /=2/3 D=1/2)

=20 /=2/3 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


114


(=20° /=1 D=1/2)

=20 /=1 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


115


(=25° /=0 D=1/3)

=25 /=0 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


116


(=25° /=1/3 D=1/3)

=25 /=1/3 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


117


(=25° /=2/3 D=1/3)

=25 /=2/3 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


118


(=25° /=1 D=1/3)

=25 /=1 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


119


(=25° /=0 D=1/2)

=25 /=0 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


120


(=25° /=1/3 D=1/2)

=25 /=1/3 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


121


(=25° /=2/3 D=1/2)

=25 /=2/3 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


122


(=25° /=1 D=1/2)

=25 /=1 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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)


123


(=30° /=0 D=1/3)

=30 /=0 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=30° /=1/3 D=1/3)

=30 /=1/3 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=30° /=2/3 D=1/3)

=30 /=2/3 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=30° /=1 D=1/3)

=30 /=1 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=30° /=0 D=1/2)

=30 /=0 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=30° /=1/3 D=1/2)

=30 /=1/3 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=30° /=2/3 D=1/2)

=30 /=2/3 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=30° /=1 D=1/2)

=30 /=1 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=35° /=0 D=1/3)

=35 /=0 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=35° /=1/3 D=1/3)

=35 /=1/3 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=35° /=2/3 D=1/3)

=35 /=2/3 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=35° /=1 D=1/3)

=35 /=1 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=35° /=0 D=1/2)

=35 /=0 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=35° /=1/3 D=1/2)

=35 /=1/3 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=35° /=2/3 D=1/2)

=35 /=2/3 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=35° /=1 D=1/2)

=35 /=1 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=40° /=0 D=1/3)

=40 /=0 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=40° /=1/3 D=1/3)

=40 /=1/3 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=40° /=2/3 D=1/3)

=40 /=2/3 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=40° /=1 D=1/3)

=40 /=1 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=40° /=0 D=1/2)

=40 /=0 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=40° /=1/3 D=1/2)

=40 /=1/3 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=40° /=2/3 D=1/2)

=40 /=2/3 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=40° /=1 D=1/2)

=40 /=1 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=45° /=0 D=1/3)

=45 /=0 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=45° /=1/3 D=1/3)

=45 /=1/3 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=45° /=2/3 D=1/3)

=45 /=2/3 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=45° /=1 D=1/3)

=45 /=1 D=1/3


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=45° /=0 D=1/2)

=45 /=0 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=45° /=1/3 D=1/2)

=45 /=1/3 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=45° /=2/3 D=1/2)

=45 /=2/3 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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


(=45° /=1 D=1/2)

=45 /=1 D=1/2


(v):(h) 0 2 4 6 8 10 12 14 16 18 20


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






)-.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



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^






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.');






soil');

disp(' ');

% ---------------------Input Data----------------------







disp(' ');


disp(' ');


max=0;

c1=0;

c2=0;

phi=m/180*pi;

sai=tan(phi);

delta=ratio*phi;

161


alfa=(al)/180*pi;

inc=0.1;

% -----------------------Calculate Ka-------------------


b1=i/180*pi;


b2=j/180*pi;



value=calPah(b1,b2,A,D,sai,omega,delta,Kh);



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


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;


% ezplot(X,Y,[angle1,angle2]),axis equal, grid on;

disp(X);

end

disp('Y coordinates of critical slip surface');


b=i;



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.');






soil');

disp(' ');

% ---------------------Input Data----------------------







disp(' ');


disp(' ');


max=0;

c1=0;

c2=0;

phi=m/180*pi;

sai=tan(phi);

delta=ratio*phi;


alfa=(al)/180*pi;

inc=1;

% -----------------------Calculate Ka-------------------


b1=i/180*pi;


b2=j/180*pi;

164



value=calPahn(b1,b2,A,D,sai,omega,delta,Kh);



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


angle1=c1/180*pi;

angle2=c2/180*pi;

disp('X coordinates of critical slip surface');

dif=(angle2-angle1)/10;


b=i;



disp(X);

end

disp('Y coordinates of critical slip surface');


165

b=i;



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






)-.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



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^



166




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.');






soil');

disp(' ');

% ---------------------Input Data----------------------







disp(' ');


disp(' ');


max=0;

c1=0;

c2=0;

phi=m/180*pi;

sai=tan(phi);

delta=ratio*phi;


alfa=(al)/180*pi;

inc=1;

167

% -----------------------Calculate Ka-------------------


b1=i/180*pi;


b2=j/180*pi;



value=calPahn(b1,b2,A,D,sai,omega,delta,Kh);



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...');


angle1=c1/180*pi;

angle2=c2/180*pi;



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






)-.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



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^






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.

resultant force of lateral earth pressure in …

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