University of Central Florida STARS
Electronic Theses and Dissertations, 2004-2019
2011
Finite Depth Seepage Below Flat Apron With End Cutoffs And A Downstream Step
Arun K. Jain University of Central Florida
Part of the Engineering Commons Find similar works at: https://stars.library.ucf.edu/etd University of Central Florida Libraries http://library.ucf.edu
This Doctoral Dissertation (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations, 2004-2019 by an authorized administrator of STARS. For more information, please contact [email protected].
STARS Citation Jain, Arun K., "Finite Depth Seepage Below Flat Apron With End Cutoffs And A Downstream Step" (2011). Electronic Theses and Dissertations, 2004-2019. 1853. https://stars.library.ucf.edu/etd/1853 FINITE DEPTH SEEPAGE BELOW FLAT APRON WITH END CUTOFFS AND A DOWNSTREAM STEP
by
ARUN K. JAIN B.E. University of Jodhpur, 1990 M.E. J.N.V. University, 1995
A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Civil, Environmental, and Construction Engineering in the College of Engineering and Computer Science at the University of Central Florida Orlando, Florida
Summer Term 2011
Major Professor: Lakshmi N. Reddi
Arun K. Jain
ii
ABSTRACT
Hydraulic structures with water level differences between upstream and downstream are subjected to seepage in foundation soils. Two sources of weakness are to be guarded against: (1) percolation or seepage may cause under-mining, resulting in the collapse of the whole structure, and (2) the floor of the apron may be forced upwards, owing to the upward pressure of water seeping through pervious soil under the structure. Many earlier failures of hydraulic structures have been reported due to these two reasons.
The curves and charts prepared by Khosla, Bose, and Taylor still form the basis for the determination of uplift pressure and exit gradient for weir apron founded on pervious soil of infinite depth. However, in actual practice, the pervious medium may be of finite depth owing to the occurrence of a clay seam or hard strata at shallow depths in the river basin. Also, a general case of weir profile may consist of cutoffs, at the two ends of the weir apron. In addition to the cutoffs, pervious aprons are also provided at the downstream end in the form of (i) inverted filter, and (ii) launching apron. These pervious aprons may have a thickness of 2 to 5. In order to accommodate this thickness, the bed adjacent to the downstream side of downstream cutoff has to be excavated. This gives rise to the formation of step at the downstream end.
Closed form theoretical solutions for the case of finite depth seepage below weir aprons with end cutoffs, with a step at the downstream side are obtained in this research. The parameters studied are : (i) finite depth of pervious medium, (ii) two cut offs at the ends, and (iii) a step at the downstream end.
iii
The resulting implicit equations, containing elliptic integrals of first and third kind, have been used to obtain various seepage characteristics. The results have been compared with existing solutions for some known boundary conditions. Design curves for uplift pressure at key points, exit gradient factor and seepage discharge factor have been presented in terms of non- dimensional floor profile ratios.
Publications resulting from the dissertation are:
1. Jain, Arun K. and Reddi, L. N. “Finite depth seepage below flat aprons with equal end
cutoffs.” (Submitted to Journal of Hydraulic Engineering, ASCE, and reviewed).
2. Jain, Arun K. and Reddi, L. N. “Seepage below flat apron with end cutoffs founded on
pervious medium of finite depth.” (Submitted to Journal of Irrigation & Drainage
Engineering, ASCE).
3. Jain, Arun K. and Reddi, L. N. “Closed form theoretical solution for finite depth seepage
below flat apron with equal end cutoffs and a downstream step.” (Submitted to Journal of
Hydrologic Engineering, ASCE).
4. Jain, Arun K. and Reddi, L. N. “Closed form theoretical solution for finite depth seepage
below flat apron with end cutoffs and a downstream step.” (Submitted to Journal of
Engineering Mechanics, ASCE).
iv
I dedicate this dissertation to
God Almighty
Who has provided me with the strength and motivation to carry on this work
My father, Prof. Bal Chandra Punmia
Who has been my role model and inspiration to achieve my goals
My mother, Kewal Punmia
Who has dedicated her life bringing the best in me
My wife, Smita Jain
Who has encouraged me throughout this work and otherwise
and
My brother, Ashok K. Jain
Who has been taking care of me in all ups and downs of life.
v
ACKNOWLEDGMENTS
I would like to express my deepest appreciation to the Committee Chair Professor
Lakshmi Reddi for this guidance, support, and insight throughout the research work. I also thank him for his philosophical guidance and personal attention to re-build my strengths, through the difficult period I went through.
Thanks go out to the other committee members, Dr. Manoj Chopra, Dr. Scott Hagen, and
Dr. Eduardo Divo for their suggestions and comments.
I convey my thanks and appreciation to my family and friends for their understanding, encouragements, and patience. I would like to mention some who have helped me immensely – Rupesh Bhomia, Michel Machado, Mrs. Padma Reddi, Mrs. Kiran Jalota,
Justin Vogel, Laxmikant Sharma, Shantanu Shekhar, Mool Singh Gahlot, Juan Cruz and
Chandra Prakash Shukla. I would also like to thank Dr. Rose M. Emery, who has been one of the biggest supports during the most difficult times of this study. I thank my son,
Aseem Jain for his co-operation and patience. I extend my thanks to Indian Student
Association – UCF, and Jain Society of Central Florida, Altamonte Springs and Nurses and doctors at Florida Cancer Institute, Orlando for their everlasting care.
Last but not least, I am thankful to all colleagues, staff and faculty in the Civil,
Environmental, & Construction Engineering department who made my academic experience at the University of Central Florida a memorable, valuable, and enjoyable one.
vi
TABLE OF CONTENTS LIST OF FIGURES ...... x
LIST OF TABLES ...... xvi
CHAPTER 1 INTRODUCTION ...... 1
1.1 General ...... 1
1.2 Early Theories ...... 2
1.2.1 The Hydraulic Gradient Theory ...... 2
1.2.2 Bligh’s Creep Theory ...... 2
1.2.3 Lane’s Weighted Creep Theory ...... 3
1.2.4 Work of Weaver, Harza and Haigh ...... 4
1.3 Methods of Analysis ...... 5
1.3.1 Finite Difference Method (FDM) ...... 6
1.3.2 Finite Element Method (FEM) ...... 7
1.3.3 Boundary Element Method (BEM) ...... 9
1.3.4 Software Packages ...... 11
1.3.5 Method of Fragments ...... 12
1.3.6 Closed Form Analytical Solutions ...... 12
1.3.6.1 Khosla’s Analysis ...... 14
1.3.6.2 Work by Pavlovsky and other Russian Workers...... 14
1.3.6.3 Confined Seepage Research by others ...... 15
1.4 Scope of Present Investigations ...... 16
CHAPTER 2 FLAT APRON WITH EQUAL END CUTOFFS ...... 18
2.1 Introduction ...... 18
vii
2.2 Theoretical Analysis ...... 18
2.2.1 First Transformation ...... 21
2.2.2 Second Transformation ...... 29
2.3 Computations ...... 35
2.4 Design Charts ...... 45
2.5 Comparison with Infinite Depth Case ...... 50
2.6 Development of Interference Formulae ...... 50
2.7 Development of Simplified Equations for PE, PD and GEc/H...... 57
2.8 Conclusions ...... 59
CHAPTER 3 FLAT APRON WITH UNEQUAL END CUTOFFS AND A STEP AT
DOWNSTREAM END ...... 60
3.1. Introduction ...... 60
3.2. Theoretical Analysis ...... 60
3.2.1 First Transformation ...... 63
3.3 Computations and Results ...... 92
3.4 Design Charts ...... 105
3.5 Conclusions ...... 105
CHAPTER 4 FLAT APRON WITH UNEQUAL END CUTOFFS ...... 133
4.1 Introduction ...... 133
4.2 Theoretical Analysis ...... 133
4.2.1 First Transformation ...... 135
4.2.2 Second Transformation ...... 146
4.3 Computations and Results ...... 151
viii
4.4 Design Charts ...... 155
4.5 Conclusions ...... 168
CHAPTER 5 FLAT APRON WITH EQUAL END CUTOFFS AND A STEP AT
DOWNSTREAM END ...... 169
5.1 Introduction ...... 169
5.2 Theoretical Analysis ...... 169
5.2.1 First Transformation ...... 170
5.2.2 Second Transformation ...... 181
5.3 Computations and Results ...... 186
5.4 Design Charts ...... 189
5.5 Conclusions ...... 199
CHAPTER 6 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS ...... 200
6.1 Summary and Conclusions ...... 200
6.1.1 Flat Apron with Equal End Cutoffs ...... 200
6.1.2 Flat Apron with Unequal End Cutoffs and a Step at Downstream End ...... 202
6.1.3 Flat Apron with Unequal End Cutoffs ...... 204
6.1.4 Flat Apron with Equal End Cutoffs and a Step at the Downstream End ...... 206
6.2 Recommendations for Future Work ...... 208
APPENDIX A SCHWARZ – CHRISTOFFEL TRANSFORMATION ...... 211
APPENDIX B ELLIPTIC INTEGRALS ...... 215
REFFRENCES...... 220
ix
LIST OF FIGURES
Fig. 1.1 Generalized model development by finite difference and finite element method ...... 8
Fig. 2.1 Illustrations of the problem – Schwarz Christoffel transformations ...... 20
Fig. 2.2a Variation of B/D with σ and γ ...... 37
Fig. 2.2b Variation of B/D with σ and γ, for γ close to 1 ...... 37
Fig. 2.3a Variation of B/c with σ and γ ...... 38
Fig. 2.3b Variation of B/c with σ and γ, for γ close to 1 ...... 38
Fig. 2.4 Determination of σ and γ ...... 40
Fig 2.5 Design curves for PE ...... 46
Fig 2.6 Design curves for PD ...... 47
Fig 2.7 Design curves for q/kH ...... 48
Fig 2.8 Design curves for GEc/H ...... 49
Fig. 2.9 Interference Correction for PE...... 53
Fig. 2.10 Interference Correction for PD ...... 54
Fig. 2.11 Interference Correction for GE c/ H ...... 55
Fig. 3.1 Illustrations of the problem – Schwarz Christoffel transformations ...... 62
Fig. 3.2a Variation of PE with a/c2 (B/c2 = 10 & c1/c2 = 0.4) ...... 106
Fig. 3.2b Variation of PD with a/c2 (B/c2 = 10 & c1/c2 = 0.4) ...... 106
Fig. 3.3a Variation of PE1 with a/c2 (B/c2 = 10 & c1/c2 = 0.4) ...... 107
Fig. 3.3b Variation of PD1 with a/c2 (B/c2 = 10 & c1/c2 = 0.4) ...... 107
q Fig. 3.4a Variation of with a/c2 (B/c2 = 10 & c1/c2 = 0.4)...... 108 kH
c2 Fig. 3.4b Variation of G with a/c2 (B/c2 = 10 & c1/c2 = 0.4) ...... 108 E H
x
Fig. 3.5a Design curve for PE (c1/c2 = 0.2 & a/c2 = 0.1) ...... 109
Fig. 3.5b Design curve for PE for (c1/c2 = 0.4 & a/c2 = 0.1) ...... 109
Fig. 3.5c Design curve for PE (c1/c2 = 0.6 & a/c2 = 0.1) ...... 110
Fig. 3.5d Design curve for PE for (c1/c2 = 0.8 & a/c2 = 0.1) ...... 110
Fig. 3.6a Design curve for PE (c1/c2 = 0.2 & a/c2 = 0.2) ...... 111
Fig. 3.6b Design curve for PE for (c1/c2 = 0.4 & a/c2 = 0.2) ...... 111
Fig. 3.6c Design curve for PE (c1/c2 = 0.6 & a/c2 = 0.2) ...... 112
Fig. 3.6d Design curve for PE for (c1/c2 = 0.8 & a/c2 = 0.2) ...... 112
Fig. 3.7a Design curve for PD (c1/c2 = 0.2 & a/c2 = 0.1) ...... 113
Fig. 3.7b Design curve for PD for (c1/c2 = 0.4 & a/c2 = 0.1) ...... 113
Fig. 3.7c Design curve for PD (c1/c2 = 0.6 & a/c2 = 0.1) ...... 114
Fig. 3.7d Design curve for PD for (c1/c2 = 0.8 & a/c2 = 0.1) ...... 114
Fig. 3.8a Design curve for PD (c1/c2 = 0.2 & a/c2 = 0.2) ...... 115
Fig. 3.8b Design curve for PD for (c1/c2 = 0.4 & a/c2 = 0.2) ...... 115
Fig. 3.8c Design curve for PD (c1/c2 = 0.6 & a/c2 = 0.2) ...... 116
Fig. 3.8d Design curve for PD for (c1/c2 = 0.8 & a/c2 = 0.2) ...... 116
Fig. 3.9a Design curve for PE1 (c1/c2 = 0.2 & a/c2 = 0.1) ...... 117
Fig. 3.9b Design curve for PE1 for (c1/c2 = 0.4 & a/c2 = 0.1) ...... 117
Fig. 3.9c Design curve for PE1 (c1/c2 = 0.6 & a/c2 = 0.1) ...... 118
Fig. 3.9d Design curve for PE1 for (c1/c2 = 0.8 & a/c2 = 0.1) ...... 118
Fig. 3.10a Design curve for PE1 (c1/c2 = 0.2 & a/c2 = 0.2) ...... 119
Fig. 3.10b Design curve for PE1 for (c1/c2 = 0.4 & a/c2 = 0.2)...... 119
Fig. 3.10c Design curve for PE1 (c1/c2 = 0.6 & a/c2 = 0.2) ...... 120
xi
Fig. 3.10d Design curve for PE1 for (c1/c2 = 0.8 & a/c2 = 0.2)...... 120
Fig. 3.11a Design curve for PD1 (c1/c2 = 0.2 & a/c2 = 0.1) ...... 121
Fig. 3.11b Design curve for PD1 for (c1/c2 = 0.4 & a/c2 = 0.1) ...... 121
Fig. 3.11c Design curve for PD1 (c1/c2 = 0.6 & a/c2 = 0.1) ...... 122
Fig. 3.11d Design curve for PD1 for (c1/c2 = 0.8 & a/c2 = 0.1) ...... 122
Fig. 3.12a Design curve for PD1 (c1/c2 = 0.2 & a/c2 = 0.2) ...... 123
Fig. 3.12b Design curve for PD1 for (c1/c2 = 0.4 & a/c2 = 0.2) ...... 123
Fig. 3.12c Design curve for PD1 (c1/c2 = 0.6 & a/c2 = 0.2) ...... 124
Fig. 3.12d Design curve for PD1 for (c1/c2 = 0.8 & a/c2 = 0.2) ...... 124
q Fig. 3.13a Design curve for (c1/c2 = 0.2 & a/c2 = 0.1) ...... 125 kH
q Fig. 3.13b Design curve for for (c1/c2 = 0.4 & a/c2 = 0.1) ...... 125 kH
q Fig. 3.13c Design curve for (c1/c2 = 0.6 & a/c2 = 0.1) ...... 126 kH
q Fig. 3.13d Design curve for (c1/c2 = 0.8 & a/c2 = 0.1) ...... 126 kH
q Fig. 3.14a Design curve for (c1/c2 = 0.2 & a/c2 = 0.2) ...... 127 kH
q Fig. 3.14b Design curve for (c1/c2 = 0.4 & a/c2 = 0.2) ...... 127 kH
Fig. 3.14c Design curve for (c1/c2 = 0.6 & a/c2 = 0.2) ...... 128
Fig. 3.14d Design curve for (c1/c2 = 0.8 & a/c2 = 0.2) ...... 128
c2 Fig. 3.15a Design curve for G (c1/c2 = 0.2 & a/c2 = 0.1) ...... 129 E H
xii
c2 Fig.3.15b Design curve for G (c1/c2 = 0.4 & a/c2 = 0.1)...... 129 E H
c2 Fig. 3.15c Design curve for G (c1/c2 = 0.6 & a/c2 = 0.1) ...... 130 E H
c2 Fig.3.15d Design curve for G (c1/c2 = 0.8 & a/c2 = 0.1)...... 130 E H
c2 Fig. 3.16a Design curve for G (c1/c2 = 0.2 & a/c2 = 0.2) ...... 131 E H
c2 Fig.3.16b Design curve for G (c1/c2 = 0.4 & a/c2 = 0.2)...... 131 E H
c2 Fig. 3.16c Design curve for G (c1/c2 = 0.6 & a/c2 = 0.2) ...... 132 E H
c2 Fig.3.16d Design curve for G (c1/c2 = 0.8 & a/c2 = 0.2)...... 132 E H
Fig. 4.1 Illustrations of the problem – Schwarz-Christoffel transformations...... 134
Fig. 4.2a Design curves for PE for c1/c2 = 0.2 ...... 156
Fig. 4.2b Design curves for PE for c1/c2 = 0.4 ...... 156
Fig. 4.2c Design curves for PE for c1/c2 = 0.6 ...... 157
Fig. 4.2d Design curves for PE for c1/c2 = 0.8 ...... 157
Fig. 4.3a Design curves for PD for c1/c2 = 0.2 ...... 158
Fig. 4.3b Design curves for PD for c1/c2 = 0.4 ...... 158
Fig. 4.3c Design curves for PE for c1/c2 = 0.6 ...... 159
Fig. 4.3d Design curves for PE for c1/c2 = 0.8 ...... 159
Fig. 4.4a Design curves for PE1 for c1/c2 = 0.2 ...... 160
Fig. 4.4b Design curves for PE1 for c1/c2 = 0.4 ...... 160
Fig. 4.4c Design curves for PE1 for c1/c2 = 0.6 ...... 161
Fig. 4.4d Design curves for PE1 for c1/c2 = 0.8 ...... 161
xiii
Fig. 4.5a Design curves for PD1 for c1/c2 = 0.2 ...... 162
Fig. 4.5b Design curves for PD1 for c1/c2 = 0.4 ...... 162
Fig. 4.5c Design curves for PD1 for c1/c2 = 0.6 ...... 163
Fig. 4.5d Design curves for PD1 for c1/c2 = 0.8 ...... 163
Fig. 4.6a Design curves for GEc2/H for c1/c2 = 0.2 ...... 164
Fig. 4.6b Design curves for GEc2/H for c1/c2 = 0.4 ...... 164
Fig. 4.6c Design curves for GEc2/H for c1/c2 = 0.6 ...... 165
Fig. 4.6d Design curves for GEc2/H for c1/c2 = 0.8 ...... 165
Fig. 4.7a Design curves for q/kH for c1/c2 = 0.2 ...... 166
Fig. 4.7b Design curves for q/kH for c1/c2 = 0.4 ...... 166
Fig. 4.7c Design curves for q/kH for c1/c2 = 0.6 ...... 167
Fig. 4.7d Design curves for q/kH for c1/c2 = 0.8 ...... 167
Fig. 5.1 Illustrations of the problem: Schwarz – Christoffel transformations ...... 171
Fig. 5.2a Variational curve for PE for B/c = 10 ...... 190
Fig. 5.2b Variational curve for PD for B/c = 10 ...... 190
Fig. 5.3a Variational curve for PE1 for B/c = 10 ...... 191
Fig. 5.3b Variational curve for PD1 for B/c = 10...... 191
Fig. 5.4a Variational curve for discharge factor for B/c = 10 ...... 192
Fig. 5.4b Variational curve for exit gradient factor for B/c = 10 ...... 192
Fig. 5.5a Design curve for PE for a/c = 0.1 ...... 193
Fig. 5.5b Design curve for PE for a/c = 0.2 ...... 193
Fig. 5.7a Design curve for PE1 for a/c = 0.1 ...... 195
Fig. 5.7b Design curve for PE1 for a/c = 0.2 ...... 195
xiv
Fig. 5.8a Design curve for PD1 for a/c = 0.1 ...... 196
Fig. 5.8b Design curve for PD1 for a/c = 0.2 ...... 196
Fig. 5.9a Design curve for discharge factor for a/c = 0.1 ...... 197
Fig. 5.9b Design curve for discharge factor for a/c = 0.2 ...... 197
Fig. 5.10a Design curve for exit gradient factor for a/c = 0.1 ...... 198
Fig. 5.10b Design curve for exit gradient factor for a/c = 0.2 ...... 198
Fig. 6.1 Finite depth problems for future work ...... 210
Fig. A-1 Schwarz-Christoffel transformation ...... 213
xv
LIST OF TABLES
Table 2.1 Computed Values for 0.5 ...... 39
Table 2.2 Data Generated from Matching the Given Values of B/D and B/c Ratios ...... 41
Table 2.3 Comparison with Infinite Depth Case ...... 50
Table 2.4 Interference Values (I) ...... 52
Table 2.5 Values of Factors a and n...... 56
Table 2.6 Comparison of Values of PE from the Formula ...... 58
Table 3.2. Comparison with Malhotra's case of Infinite Depth...... 93
Table 3.1 Computed values for B/D = 0.2, a/c2 = 0.1 ...... 94
Table 4.1 Computed Values ...... 152
Table 4.2 Comparison with Malhotra's Case of Infinite Depth ...... 155
Table 5.1 Computed Values ...... 187
Table 5.2 Comparison with Malhotra's Case of Infinite Depth ...... 189
xvi
CHAPTER 1. INTRODUCTION
1.1 General
If a structure is built on a pervious soil and water level upstream of the structure is higher than it is downstream, enforced percolation or seepage will occur in the underlying permeable soil. The design of apron is intimately connected with the possibility of this percolation.
Two sources of weakness are to be guarded against: (1) percolation may cause undermining of the pervious granular foundation, which starts from the tail end of the work and may result in collapse of the whole structure, and (2) the floor or the apron may be forced upwards, owing to the upward pressure of water seeping through the pervious soil under the structure. The failure of the old Manufla regulator in Egypt was due to undermining, whereas the failure of Narora weir in India was due to excessive water pressure causing the floor of the weir to be blown upwards.
The two essentials to be considered in the design of impervious apron of a hydraulic structure are:
(i) Residual or uplift pressure at any point at the bottom of the floor, and
(ii) Exit gradient.
1
1.2 Early Theories
1.2.1 The Hydraulic Gradient Theory
The law of flow of water through permeable soil was enunciated for the first time in 1856 by H. Darcy who, as a result of experiments, found that for laminar flow conditions, the velocity of flow varied directly as the head and inversely as the length of path of flow. Later work on the flow was done in the United States by Allen Hazen (1892) and C.S. Slichter (1899), and in India by Col. J. Clibborn and J.S. Beresford. Clibborn and Beresford carried out experiments (1895-
97) with a tube 120 ft long and 2 ft internal diameter filled with Khanki sand, in connection, with the proposals for repairs to the damage to the Khanki weir on Chenab River. The hydraulic gradient theory for weir design, apparently originated between Sir John Ottely and Thoman
Higham, and was developed as a result of experiments by Col. Clibborn. With the publication of
Clibborn's experiments in 1902, the Hydraulic Gradient Theory was generally accepted in India.
According to the hydraulic gradient theory, the safety of a weir founded on a permeable soil medium depends upon its path of percolation that is the distance through which water would have to travel below the weir floor before it could rise up on the downstream, and cause scour.
Whether the masonry was laid horizontally or vertically was immaterial, so long as the current below the structure was exposed to friction for the same length of the soil.
1.2.2 Bligh’s Creep Theory
In 1907, Bligh, in his book `Practical Design of Irrigation Works' evolved a concept according to which the stability of a weir apron depended on its weight. But in 1910 edition of the book, he admitted the fallacy of this original concept and became converted to the `Hydraulic
Gradient Theory' of Ottley, Higham and Clibborn, and gave his `creep theory'. In Bligh's creep
2 theory; he stated that the length of the path of flow had the same effectiveness, length for length, in reducing the uplift pressures, whether it was along the horizontal or the vertical. He assumed the percolating water to creep along the contact of the base profile of the weir with the subsoil losing head enroute, proportional to the length of its travel. He called this loss of head per unit length as the percolation coefficient (C) and assigned a safe value to C for different types of soils.
Bligh's conclusions were based on the study of failure of only two dams, both on fine sand foundations and only experimental data available at the time were of Clibbron. From these meager data, Bligh evolved a simple formula which fitted neither Clibborn results with sheet piles nor those at Narora Weir. However, because of its simplicity, this theory found general acceptance. Some works designed on this theory failed while others stood, depending on the extent to which they ignored or took note of the importance of vertical cutoffs at the upstream and downstream ends.
1.2.3 Lane’s Weighted Creep Theory
Colman (1916) for the first time carried out tests with weir models resting on sand to determine the distribution of pressure under the weir base, and the relative effect of sheet piles at the upstream and down stream ends. These experiments established that vertical contacts are more effective than horizontal contacts, contradicting Bligh's assumptions.
Lane (1935), after analyzing over 200 dams on pervious foundations all over the World, advanced a theory known as `Weighted Creep Theory and accounted for the vertical and horizontal cutoffs in a modified way. Lane was of the view that water can occasionally travel along the line of creep because it is not easy to have close contact between the two surfaces, i.e.
3 the flat surface of the solid foundation of the weir and the pervious soil upon which it is founded.
In practice, an intimate contact between the vertical and steeply sloping surface is more possible than in horizontal or slightly sloping. Thus contact between earth and sheet pile is more intimate than for concrete foundation cast over flat bedding. This led to different weights to the vertical and horizontal creeps. According to Lane, the weighted creep distance is the sum of vertical creep distance (steeper than 45°), plus one third the horizontal creep distances (less than 45°).
Though a statistical examination of a large number of dams confirm the basic idea of relative creep effectiveness on the lines of Lane, but to try to suggest such simple ratios of their effectiveness was considered to be too arbitrary.
1.2.4 Work of Weaver, Harza and Haigh
The problem of uplift pressure on the base of a dam founded on pervious medium of infinite depth was mathematically analyzed by Weaver (1932). He showed that with no sheet piling, the path of flow are the lower halves of a family of confocal ellipses with foci at heel and toe, respectively and the equipotential lines are the conjugate family of confocal hyperbolas. This shows that the pressure gradient is no more a straight line as assumed by Bligh, but is sinusoidal.
Weaver, however, did not consider the exit gradient.
Harza (1934) and Haigh (1935) independently published two papers on similar lines. All these attempts were responsible for a shift from empirical approach towards the rational approach, for the design of base of weir aprons. They gave due weightage to exit gradient as controlling factor in stability and discussed the distribution of pressures which could be considered as safe.
4
1.3 Methods of Analysis
The process of seepage through porous media can be classified according to the dimensional character of the flow, the boundaries of flow region or domain, and the properties of the medium and of the fluid (Rushton and Redshaw, 1979). Any scenario in the seepage studies consists of a governing along with boundary conditions and initial conditions which control seepage in a particular problem domain. If the domain has a complex configuration, analytic solution seems to be difficult to contrive at and hence one may resort to approximating techniques for solution of the governing equation. The goal of these approximate methods is thus to find a function (or some discrete approximation to this function) that satisfies a given relationship between various derivatives on some given region of space and/or time, along with some boundary conditions along the edges of this domain.
Thus, these approximate methods provide a rationale for operating on differential equations that make up a model and for transforming them into a set of algebraic equations.
Using computer, one can solve large number of algebraic equations by iterative techniques or by direct matrix methods. The solution obtained can be compared with that determined from analytical solution, if one is available or with values observed in the field or from laboratory experiments. It should be noted that the numerical procedures (approximate methods) yield solutions for only a predetermined number of points, as compared to analytical solutions that can be used to determine values at any point in a problem domain. The next five sub-sections are devoted to different approximate methods which are chiefly used in seepage studies.
5
1.3.1 Finite Difference Method (FDM)
A finite difference method proceeds by replacing the derivatives in the governing differential equation with finite-difference approximations. This gives a large, but finite algebraic system of equations to be solved in place of differential equation, which can be done using computer. Thus, the method actually performs discretization of the flow domain by dividing into a mesh that is usually rectangular, where the potential or head is computed at the grid points by solving the differential equation in finite difference form throughout the mesh or the grid system. There are two common types of grids: mesh-centered and block-centered.
Associated with the grids are node points that represent the position at which the solution of the unknown values (head, for example) is obtained. The choice of grid to use depends largely on the boundary conditions.
Hence, this method, also known as relaxation method is a process of steadily improved approximation for the solution of simultaneous equations, and any problem that can be formulated in terms of simultaneous equations can, theoretically, be solved by this method. One of the earliest uses of this method was by Richardson (1911) who applied it for a masonry dam problem. Several other researchers (Shaw and Southwell, 1941; Van Deemter, 1951; Jeppson,
1968a, 1968b; Herbert, 1968; Cooley, 1971; Freeze, 1971; Bruch Jr. et al., 1972; Huntoon, 1974;
Ronzhin, 1975; Bruch Jr. et al., 1978; Gureghian, 1978; Caffrey and Bruch, 1979; Dennis and
Smith, 1980; Karadi et al., 1980; Walsum and Koopmans, 1984; Pollock, 1988; Das et al. 1994;
Naouss and Najjar, 1995; Korkmaz and Önder, 2006; Jeyisanker and Gunaratne, 2009; Igboekwe and Achi, 2011) have used FDM as method of analysis for different seepage and groundwater problems.
6
The summary of important components and steps of model development for FDM and finite element method (discussed in next sub-section) are shown in figure 1.1.
1.3.2 Finite Element Method (FEM)
There are two basic problems in calculus: one relating to integration and the other to differentiation. Whereas FDM approximates differential equations by a differential approach,
FEM approximates differential equations by an integral approach. Based on inverse property of differentiation and integration to one another, one would expect the two methods to be related and to converge to same solution, but perhaps from different directions (Faust and Mercer,
1980).
FEM is essentially a numerical method in which a region is subdivided into sub regions called elements, whose shapes are determined by a set of points called nodes. The flexibility of elements allows consideration of regions with complex geometry. The first step in applying the
FEM, as shown in figure 1.1, is to develop an integral representation of the partial differential equation. The next step is to approximate the dependent variables (head, for example) in terms of interpolation functions, which are called the basis functions, and are selected to satisfy certain mathematical requirements and for ease of computation. As the element is generally small, the interpolation function can be adequately approximated by a low-order polynomial, for example, linear, quadratic, or cubic. As an example, consider a linear basis function for a triangular element. This basis function describes a plane surface including the values of dependent variable
(head) at the node points in the element. Having basis functions specified and the grid designed, the integral relationships are expressed for each element as a function of the coordinates of all node points of the element. Next the values of the integrals are calculated for each element.
7
Fig. 1.1 Generalized model development by finite difference and finite element method
(Adapted from Faust and Mercer, 1980)
8
The values for all elements are combined, including boundary conditions, to yield a system of first-order linear differential equations in time (Faust and Mercer, 1980).
A number of researchers (Neuman and Witherspoon, 1970; Doctors, 1970; France et al., 1971;
McLean and Krizek, 1971; Desai, 1973, 1976; Ponter, 1972; France, 1974; Semenov and
Shevarina, 1976; Kikuchi, 1977; Choi, 1978; Christian, 1980; Florea and Popa, 1980; Nath,
1981; Aalto, 1984; Rulon et al., 1985; Lacy and Prevost, 1987; Tracy and Radhakrishnan,1989;
Rogers and Selim, 1989; Morland, and Gioda, 1990; Griffiths and Fenton, 1993, 1997, 1998;
Hnang, 1996; Karthikeyan et al., 2001; Simpson and Clement, 2003; Benmebarek et al, 2005;
Soleimanbeigi and Jafarzadeh, 2005; Im et al., 2006; Hlepas, 2008; Ahmed, 2009; Ahmed and
Bazaraa, 2009; Ahmed and Elleboudy, 2010) have successfully used FEM in a wide variety of groundwater flow and seepage problems.
FEM differs from FDM in two respects. In the first place the domain is discretized into a mesh of finite elements of any shape, such as triangles. In the second place, the differential equation is not solved directly but replaced by a variational formulation (Strack, 1989).
1.3.3 Boundary Element Method (BEM)
Boundary element method constitutes a recent development in computational mathematics for the solution of boundary value problems in various branches of science and technology. For flow through porous media, Liggett (1977) was the first to use BEM for finding location of free surface in porous media where the solution is desired at a limited number of points (for example, on a failure surface in determining slope stability) or in a limited area of the flow.
9
BEM is based on integral equation formulation of boundary value problems and requires discretization of only the boundary (surface or curve) and not the interior of the region under consideration. Unlike, the ‘domain type’ methods e.g. FDM and FEM, the order of dimensionality reduces by unity in boundary element formulation, thus simplifying the analysis and the computer code to a large extent by solving a small system of algebraic equations (Kythe,
1995). This method is suitable for problems with complicated boundaries and unbounded regions, offering greatly reduced nodes for the same degree of accuracy as in FEM.
Various researchers (Brebbia and Wrobel, 1979; Herrera, 1980; Hromadka, 1984;
Liggett, 1985; Gipson et al. 1986; Elsworth, 1987; Karageorghis, 1987; Chugh, 1988; Savant et al., 1988; Chang, 1988; Aral, 1989; Abdrabbo and Mahmoud, 1991; Chen et al., 1994;
Demetracopoulos and Hadjitheodorou, 1996; Tsay et al., 1997; Leontiev and Huacasi, 2001;
Abdel-Gawad and Shamaa, 2004; Shen and Zhang, 2008; Filho and Leontiev, 2009) have used
BEM for different groundwater and seepage problems.
Summarizing, the main features of the above mentioned three numerical methods are
(Bear et al., 1996):
1. The solution is sought for the numerical values of state variables which are at specified points in the space and time domains defined for the problem (rather than their continuous variations in these domains).
2. The partial differential equations that represent balances of the considered extensive quantities are replaced by a set of algebraic equations (written in terms of the sought, discrete values of the state variables at the discrete points in space and time).
10
3. The solution is obtained for a set of specified set of numerical values of the various model coefficients (rather than as a general relationship in terms of these coefficients).
4. Since a large number of equations must be solved simultaneously, a computer program is prepared.
1.3.4 Software Packages
In recent years, computer programming codes have been developed for almost all the classes of problems encountered in the management of ground water. Some codes are very comprehensive and can handle a variety of specific problems as special cases, while others are tailor-made for particular problems (Bear et al., 1996). To name a few, some commercial and other softwares are (along with the name of the authors of the software): MODFLOW
(McDonald and Harbaugh, 1988), WALTON (Walton, 1970), FEMWATER (Yeh and Ward,
1979), FLONET: FLOWTRANS (Guiger et al., 1994), FLOWPATH (Franz and Guiger, 1994),
TRACR3D (Travis, 1984) and SEEP2D (Fred Tracy of the U.S. Army Engineer Waterways
Experiment Station; Jones, 1999).
Many of these provide modeling for groundwater flow, while others deal with solute transport as well. Some deal with saturated cases only while a few deal with complexities of unsaturated flow cases, etc. A few models can be used for 3D flows, while others for 2D flow or both. It should be kept in mind that all these models rely on numerical (approximate) methods mentioned in previous three sub-sections, while some use a combination of these methods, thus inheriting the limitations of these methods.
11
1.3.5 Method of Fragments
In many practical cases of seepage analysis, the solution by exact methods either leads to extremely complicated relationships or is not possible at all. It was proposed by Pavlovsky
(1936, 1937) that it is possible to single out an extensive group of problems of this kind which are characterized by means of surfaces which are close to the isopiestic surfaces or to the surfaces generated by the streamlines. In this division, the seepage region is split into sections or fragments for each of it is possible to obtain, by some method, a theoretical solution. The solutions derived for individual fragments, in the seepage region are linked by given relationships, by means of which a solution is subsequently obtained for the seepage flow as a whole (Aravin and Numerov, 1965).
Hence, the accuracy of the solution obtained from this method will depend how closely the preassigned (hypothetical) isopiestic or stream-surfaces approximate the true ones.
Polubarinova-Kochina (1962), Harr (1962) and Reddi (2003) present an exclusive treatise on this method. Several researchers (Devison, 1937; Christoulas, 1971; Griffiths, 1984; Mishra and
Singh, 2005; Shehata, 2006; Sivakugan et al., 2006) have applied this method for obtaining solutions to complex configurations.
1.3.6 Closed Form Analytical Solutions
It generally refers to a solution that captures the entire physics and mathematics of a problem as opposed to one that is approximate. It is generally in terms of functions and mathematical operations from a given generally accepted set. The mathematical result will show the functional importance of the various parameters, in a way a numerical solution cannot, and is therefore more useful in guiding design decisions.
12
Ascertaining the importance of these solutions, Ilyinsky et al. (1998) state “….Numerical techniques have become of ever greater significance in solving practical problems of seepage theory since the introduction of powerful computers in the sixties. However, even so analytical methods have proved to be necessary not only to develop and test the numerical algorithms but also to gain a deeper understanding of the underlying physics, as well as for the parametric analysis of complex flow patterns and the optimization and estimation of the properties of seepage fields, including in situations characterized by a high degree of uncertainty with respect to the porous medium parameters, the mechanisms of interaction between the fluid and the matrix, the boundary conditions and even the flow domain boundary itself.” Another author,
Reddi (2003) mentions in his book – “…The early phases of development of the study of seepage in soils attracted the attention of several eminent mathematicians. As a result, we have a wealth of closed-form analytical solutions available to solve problems even with complicated flow domains. It is painfully obvious at times that these solutions are not being exploited in industry. Often, numerical models that require extensive computing times are used to solve problems for which simple analytic solutions exist in the literature. Numerical solutions obtained using a discretization of flow domain, although required in a number of complicated cases, are no match for analytical closed-form solutions (where available) in providing an insight into the nature of the problem.”
In the next few sub-sections analytical solutions for different seepage scenarios in confined seepage domain are discussed.
13
1.3.6.1 Khosla’s Analysis
Inspired by Weaver's theoretical analysis, Khosla, Bose and Taylor (1936) made outstanding contributions towards the design of weirs on permeable foundations. Khosla and his associates gave a generalized solution to the problem of a weir floor with an intermediate sheet pile and a step, founded on pervious medium of infinite depth. They also verified the results obtained from theoretical analysis, by conducting tests on electrical analogy models.
For more complicated weir profiles, Kholsa gave the `method of independent variables'.
In this method, a complex weir profile is splitted up into its elementary standard forms for which theoretical solutions were available. Each elementary form is then treated as independent of others and the pressures at key points are found. The key points are the junction points of the floor and the pile line of that particular elementary form, the bottom points of that pile line and the bottom corners in the case of depressed floor. The results at the junction points are then corrected for (i) mutual interference of piles, (ii) the floor thickness, and (iii) the slope of the floor. In all these cases, the depth of the pervious medium was assumed to be infinite.
Malhotra (1936) solved mathematically the problem of seepage below a flat apron, with two equal cutoffs, at either end and founded on pervious medium of infinite depth. His results compared favorably with those obtained from electrical analogy experiments and from Khosla's method of independent variables.
1.3.6.2 Work by Pavlovsky and other Russian Workers
A general theory, and large number of individual solutions of the conformal transformation problems, as applied to weir foundation design, were published by Prof. N.N.
Pavlovsky (1922, 33, 56) but as the text was in Russian, this work remained almost unknown to 14 the profession. Pavlovsky work was described in English by Leliavsky (1955), Harr (1962) and
Polubarinova Kochina (1962).
The fundamental principle adopted by Pavlovsky is Riemann's original theorem. He transformed both the profiles, i.e. true weir profile and the rectangular filed, on to the same semi- infinite plane. The plane thus serves as a link joining the two planes of the analysis into one consistent unit. Both the cases of apron founded on finite as well as infinite depth of pervious medium were analyzed by him. Polubarinova-Kochina (1962) outlined several cases of weir profiles analyzed by various Russian workers. Fil'chakov (1959, 1960) studied analytically finite depth seepage that includes several schemes of weirs with cutoffs.
1.3.6.3 Confined Seepage Research by others
The last few decades have seen tremendous growth in the approximate methods for seepage studies; however, studies with closed form solutions are very few. King (1967) numerically solved the analytical solution to the problem of seepage below depressed floor on pervious medium of finite depth, originally formulated by Pavlovsky. Chawla (1975) made analytical studies on stability of structures with intermediate filters. Seepage characteristics of foundations with a downstream crack were analyzed by Sakthivadivel and Thiruvengadachari
(1975). Kumar et al. (1982) analyzed the case of seepage flow under a weir, resting on isotropic porous medium of infinite depth, with a vertical sheet pile at the toe and a segmental circular scour. Chawla and Kumar (1983) found an exact solution for hydraulic structures with two end cutoffs, resting on infinite media. Elganainy (1986) solved analytically for the flow underneath a pair of structures with intermediate filters on a drained stratum. Muleshkov and Banerjee (1987) developed an analytical solution for seepage towards vertical cuts. Kacimov and Nicolaev (1992)
15 analytically solved the problem of steady seepage near an impermeable obstacle in terms of a model for 2-D seepage flow with a capillary fringe. Ijam (1994) obtained an exact solution for seepage flow below a hydraulic structure founded on permeable soil of infinite depth for a flat floor with an inclined cut-off at the downstream end. Banerjee and Muleshkov (1993) gave analytical solution for finite depth seepage into double walled cofferdams. Farouk and Smith
(2000) analytically solved the case of hydraulic structures with two intermediate filters. Salem and Ghazaw (2001) investigated the characteristics of seepage flow beneath two structures with an intermediate filter. Goel and Pillai (2010) studied the effect of downstream stone protection on exit gradient due to infinite depth seepage below a flat apron with an end cutoff. Bereslavskii
(2009) conducted analytical studies on the design of the iso-velocity contour for the flow past the base of a dam with a confining bed. Bereslavskii and Aleksandrova (2009) analytically modeled the base of a hydraulic structure with constant flow velocity sections and a curvilinear confining layer. Abdulrahman and Mardini (2010) used Pavlovsky's method of two stage transformation to analyze Khosla’s case of infinite depth seepage below flat apron with intermediate cutoff.
1.4 Scope of Present Investigations
Flat aprons of hydraulic structures are invariably provided with cutoffs at both upstream and downstream ends. The cutoff at downstream end of apron safeguards the structure both against exit gradient as well as downstream scour, though it increases the uplift pressure all along the upstream side. The cutoff at upstream end protects the apron against upstream scour and at the same time reduces uplift pressure all along the downstream side. In addition to the cutoffs at both the ends, pervious aprons are also provided at the downstream side of the end cutoff in the form of inverted filter and launching apron. These pervious aprons may have a thickness of 2 to
16
5. In order to accommodate this thickness, the bed to the downstream side of downstream cutoff has to be excavated. This gives rise to the formation of a step at the downstream end.
The investigations reported herein envisage a theoretical study of seepage characteristics below the weir aprons of various boundary conditions, including a step at downstream side. The depth of pervious medium has been taken to be finite. Closed form theoretical solutions have been found by following the procedure originally suggested by Pavlovsky.
A general case of weir profile results when two cutoffs of depths c1 and c2 are provided at the upstream and downstream ends, and when the depth of medium is finite. For a similar floor profile founded on pervious medium of infinite depth, Khosla did not provide any theoretical solution, but instead suggested the use of an empirical method of independent variables a method still followed in design offices. However, in the present analysis, a complete theoretical solution has been founded, considering three additional parameters: (i) two cutoffs, one at each end (ii) the finite depth of pervious medium, and (iii) a step at the downstream side end cutoff.
The resulting analytical solution in terms of implicit equations, containing elliptic integrals of first and third kind, have been used in obtaining various seepage characteristics such as uplift pressures at key points, seepage discharge factor and exit gradient factor. Design curves have been produced, in easy to use form, for these seepage characteristics, in terms of non- dimensional floor profile ratios.
17
CHAPTER 2. FLAT APRON WITH EQUAL END CUTOFFS
2.1 Introduction
Flat aprons are invariably used for majority of hydraulic structures. However, in order to control the exit gradient, cutoff at the downstream end is absolutely essential. Downstream cutoff is also essential for protection of the apron against scour at the tail end. Similarly, cutoff is provided at the upstream end to serve two purposes: (i) to protect the apron against scour at the upstream end, and (ii) to reduce the uplift pressure all along. Incidentally, cutoffs provided at both the ends also reduce the seepage discharge. Malhotra analyzed the case of flat apron with equal end cutoffs, founded on pervious medium of infinite depth. However the present case deals with the flat apron with equal end cutoffs founded on pervious medium of finite depth.
2.2 Theoretical Analysis
Fig. 2.1(a) shows the floor profile in z-plane. Fig. 2.1(c) shows the well known w-plane, relating and . The points in the z and w-planes are denoted by complex co-ordinates z x iy and wi respectively, wherei 1 . The problem is solved by determining the functional relationship w f( z ). This is achieved by transforming both the z-plane and w-plane onto an infinite half plane, t-plane, shown in Fig. 2.1 (b) thus obtaining the relationships z f1 (t ) and w f2 (t ).
18
Floor Parameters The floor of the hydraulic structure, shown in Fig. 2.1(a) has the following floor parameters:
(i) Length of the apron : B
(ii) Finite depth of pervious medium : D
(iii) Depth of upstream and downstream cutoffs : c
The resulting independent non-dimensional floor profile ratios are:
B (i) (Finiteness ratio) D
B and (ii) (Length - cutoff ratio) c
The dependent non-dimensional floor profile ratio is
c (iii) (Cutoff depth ratio) D
c c B where DBD
19
Fig. 2.1 Illustrations of the problem – Schwarz Christoffel transformations
20
2.2.1 First Transformation
The Schwarz-Christoffel transformation of the floor from z-plane to t-plane is
t dt z MN11 0 12 8 (t t1 ) ( t tt 28 ) t ...... ( )
Here 1 //22 1 12/
2 2 2 1
3 //22 3 12
4 //22 4 12
5 2 5 1
6 //22 6 12
7 0 7 1
8 0 8 1
Choosing t1 ,t2 ,t 3 ,t 4 ,t 5 ,t 6 ,t 7 1 and t8 1 we get
t dt z MN 110 1 1 1 1 (t )(t2 )(t1 )(t 2 )(t 2 )(t 1 )(t 2 11 )(t 1 ) 1
t (t22 )dt or MN11 (2.1) 0 (t21 ) (t 2 2 )(t 2 2 )
t [(t22 1) ( 1)] dt or MN11 0 (t2 1) ( t 2 2 )( t 2 2 )
21
2 ttdtdt ( 1) or z MN11 002 2 2 2 2 2 2 2 2 (t )( tt ) t (1)( t )( )
Putting T t/ so that dtdT , we get
dtdT (12 ) or MN11 222222 22 222222 (TTTTT )( )(1 )( )( )
2 dTdT (1 ) or MN11 22 2 2 2 2 2 2 (1)1 TTTTT (1 )(1)1
M or z 1 [(,)(1F mm N 22 ) (, ,)] (2.2 a) 1 where F ( ,m ) = Incomplete elliptic integral of the first kind with modulus m
( , 2 ,m ) = Incomplete elliptic integral of the third kind with parameter 2
2 = Parameter of third degree elliptic integral 2
m = Modulus /
= argument sin11Tt / sin , ( ) which varies with the position
of the point on the floor domain (i.e. varies with t).
In order to determine the values of various unknowns in Eq. 2.2 (a), let us apply boundary conditions at some salient points.
22
BOUNDARY CONDITIONS
(i) Point 0: At point 0, t = 0 and z = 0
sinsin11 Tt /0
Fm(0, )0 and 00,2 ,m
Hence from (2), we get
M 0 1 [0 0] N From which N 0 1 1
Hence Eq. (2 a) becomes
M z 1 [(,)(1F mm 22 ) (, ,)] (2.2)
B (ii) Point 4: (Point E): At point 4, z and z= 2
sin1Tt sin 1 ( / ) sin 1 ( / ) sin1 1 / 2
Fm(,) F , m K complete elliptic integral of the first kind. 2
2 2 and (,,) m ,, m 0 = complete elliptic integral of the third kind. 2
Hence from Eq. 2.2, we have
B M 1 [K (1 2 ) ] 2 0
2M or B 1 [K (1 2 ) ] (2.3) 0
2M or 1 []E (2.3 a)
23
2 where E [(1)K ] 0 (2.3 b)
(iii) Point 9: At point 9, z=iD and t
t
sin . Hence argument becomes i so that sini sinh
Making use of standard identity 161.02 (Bird and Friedman, 1971)
F(,) im iFm( ,)
where, tan11 (sinh ) tan 2
F(,) i m iF, m i K 2
i[ F ( , mm ) 22 ( , , )] i and (,,)im 2 []K2 12 12 0
2 where 0 ,, m complete elliptic integrate of third kind with modulus m' and 2 parameter 2 .
2 1 2 and m2 1 m2
Substituting these values in Eq. (2), we get
M (12 )i 1 2 iD iK 2 {} K 0 where 1
M (12 ) or D 1 KK {} 2 (2.4) 2 0 1
M or 1 []L (2.4 a)
24
(12 ) 2 where L KK 2 {}0 (2.4 b) 1
B (iv) Point 5: At point 5, z ic and t = 2
sin 111Tt sin ( / ) sin ( / ) .
Hence from Eq. 2.2
B M1 12 1 2 ic Fmm sin , (1 ) sin , , ) 2
B Substituting the value of from Eq. 2.3, we get 2
M1 2 M1 1 2 1 2 [Kic (1 ) 0 ] F sin , mm (1 ) sin , , )
M1 1 2 1 2 or ic Fsin , m Km (1 ) sin , , ) 0 (2.5)
In the above equation, the argument of the elliptic integral is given by
sin1 / or sin ( / ) .
But. Hence sin1. The argument is therefore complex.
1 Since sin, we have the following reduction formula: m
Fm(,) K iF(,) A m
2 2 2 and (,,) m 01 i F(,)(,,) A m 2 A m 1
sin2 1 ( / )2 1 2 2 where A= new amplitude sin1 sin11 sin msin mm(/)
25
mm22 2 1 11 22
Substituting in Eq. 2.5 we get
M 2 1 22 ic K iF(,) A m Ki F (1) A m01 (,) A 0 m 2 (,,) 1
M 12 1 222 or c F( A , mA )( , m , ) 2 1 (2.6) 1
M or 1 []G (2.6 a)
12 22 2 where G F( A , mA )( , m ,2 ) 1 (2.6 b) 1
B (v) Point 6: At point 6, z and t 2
sin11tm / sin( / ) sin (1/ ) .
Substituting in Eq. 2.2, we get
B M1 111 2 1 2 Fsin , mm (1 ) sin , , ) 2 mm (2.7)
From identity 111.09, Bird and Friedman,
1 1 Fmsin , K iK m
2 121 m and sin , ,m 00 i 22 m m
2 where 0 ,, m 2
26
2 0 ,,m 2
22m 2 m22
B M m2 1 2 K iK (1 ) 00 i 22 2 m
B Substituting the value of from Eq. 2.3, we get 2
M M m2 1 K (1) 2 1 K iKi (1 2 ) 0 0022 m
m2 2 or {K iK K } (1 )00 i 22 0 0 m
m2 2 or K (1 )0 220 m
Substituting the value of m / and , we get
K m2 2 K (/) 2 2 K (12 ) .(1 ) 2 m22 (/) 000
1/ 2 K 1 (1 2 ) (2.8) 0
Thus, we find that is not an independent variable. Instead, it depends on and . Hence, there are only two unknowns (i.e., and ) in the t-plane.
27
General Relationship: z = f1 (t)
From Eq. 2.3,
BB. M1 2 (2.9 a) 2[KE (1 ) 0 ] 2
Hence from Eq. 2.2
B[(,)(1 F mm 22 ) (, ,)] z 2 (2.9) 2[K (1 ) 0 ]
which is the required relationship z f1() t
Floor Profile Ratios
From Eqs. 2.3. and 2.4, we have
2M 1 [K (1 2 ) ] B 0 D M 12 1 KK 2 2 0 1
2[K (1 2 ) ] 2E or 0 12 L KK 2 2 0 1
(2.10 a)
From Eqs. 2.3. and 2.5., we get
2M 1 [K (1 2 ) ] B 0 2E (2.10 b) c M 22(1 ) G 1 22 F(,)(,,) A m 2 01 A m 1
Hence
c BBEEG22 (2.10 c) D D c L G L
28
2.2.2 Second Transformation
Refer Fig. 2.1 (c).
The Schwarz Christoffel equation for transformation from w-plane to t-plane is
t dt w MN 22 0 1 6 7 8 ()()()()t t1 t t 6 t t 7 t t 8
1 Here 1 6 7 8 2
t1 ; tt68 ;1 and t7 1
t dt Hence MN221/ 2 1/ 2 1/ 2 0 (t )1/ 2 ( t ) ( t 1) ( t 1)
t dt or MN 220 2 2 2 (tt )( 1)
Putting Tt/ so that dtdT , we get wN TT dTdT 2 002 2 2 2 2 2 M 2 (TTTT 1)( 1) (1 )(1 )
This is the elliptic integral of the first kind (F0) with modulus m0 =
wN 2 F0 (,) M 2 where sin11Tt sin ( / )
Boundary conditions
(i) Point 0: At point 0, w kH /2 and t = 0
T = t/ = 0
11 and sinT sin 0 0 and F0 ( ) = F0 (0, ) = 0
29
Hence from Eq. (2.11)
wN 2 0 M 2
From which NwkH2 /2 (2.12)
Thus, the value of the additive constant N2 is known.
(ii) Point 6: At point 6, w = 0 and t =
T = t / = / =1
11 sinsinT 1 and FK00, 2 2
Hence from Eq. 2.11
0 N2 N2 kH K0 or M 2 (2.13) M 2 KK002
Thus, the value of the multiplier constant M2 is also known.
(iii) Point 7: At point 7, w iq and t 1
1 Tt/ 1/ and sin11T sin
1 1 Fsin , K00 iK
Substituting in Eq. 2.11, we get
1 1 w MFN2 0sin , 2
kH kH kH K 0 or iq []K00 i K i 2KK00 2 2
30
kH K Hence q 0 (2.14 a) 2K0
q K Seepage discharge factor, 0 (2.14) kHK 2 0
General Relationship wft 2 ()
Substituting the values of M2 and N2 in Eq. 2.11, we get
kHkH w M 2 FNF 020( , )( , ) 22K0
kH or [ FK00 ( , )] (2.15) 2K0
which is the required relationship w f2 () t
Uplift pressure distribution
For the base of the apron, 0
w khx
(i) Point E (point 4) wkh E and t
Tt//
1 E sinT sin
kH 1 Hence from Eq. 2.15, khE FK00sin , 2K0
31
1 F0 sin , H or hE 1 2 K0
1 F0 sin, hE or PE 100 50 1 (2.16) H K0
(ii) Point C (point 3)
At point 3, wkh C and t
T = t / = – /
1 11 sin (T ) sin sin
11 FF00sin , sin ,
kH 1 Hence from Eq. 2.15, khC FK00 sin , 2K0
1 F0 sin , hC or PC 100 50 1 (2.17) H K0
From Eqs. 2.16 and 2.17, we note that PE + PC = 100% which is in conformity with the principle of reversibility of flow.
(iii) Point D (point 5)
w khD and t
32
1 Tt// and D sinsinT
kH 1 khD FK00sin , 2K0
1 F0 sin, hD From which PD 100 50 1 (2.18) H K0
(iv) Point D' (point 2)
Following the same procedure,
1 F0 sin , PD 50 1 (2.19) K0
so that PPDD100%
Exit Gradient
The Schwarz-Christoffel equation for transformation from z-plane to t-plane is
22 t ()t dt z MN (2.20) 11 0 2 2 2 2 2 (t 1) ( t )( t )
dz Mt()22 or 1 dt (t2 1) ( t 2 2 )( t 2 2 )
B where M1 2 2[K (1 ) 0 ]
33
dz B ()t 22 . (2.20 a) 2 22 2 2 2 dt 2[(1)K ] 0 (ttt 1) ()()
where K is the complete elliptic integral of the first kind with modulus / .
2 0 ,, complete elliptic integral of third kind with modulus / and 2 parameter 22 .
Similarly, the transformation equation between w and t plane is
dt w MN 22 2 2 2 ()(tt 1)
dw M kH or 2 where M 2 2 2 2 dt (tt )( 1) 2K0
dw kH 1 . (2.20 b) 2 2 2 dt 2K0 (tt )( 1)
where K0 is the complete elliptic integral of first kind with modulus
If u and v are the velocity components in x and y directions, we have
dwdw dt Complex velocity u iv . dzdt dz
For the exit surface, u 0 and vv E
dw dt iv . E dt dz
Now from Darcy law, vEE k G
v 1 dw dt G E .. E k ik dt dz
34
dw dt Substituting the values of and , we get dt dz
11kh 2[K (1 2 ) ] (t2 1) ( t 2 2 )( t 2 2 ) G ..0 E 2 2 2 22 ik2( Kt0 (tt ) )( 1)
22 2 2 GBE 1tt K (1 ) 0 or 2 22. H tKt 0 1
At the exit end, t
22 2 2 GBE 1 K (1 ) 0 2 22. (2.21) H K0 1
c GB c Now G . E E H HB
22(1 ) 22F(,)(,,) A m A m 22 2 2 2 01 c 1 K (1 ) 0 1 or GE 2 2. 2 2 H KK001 2[ (1 ) ]
(2.22 a)
2 2 2 c GG1 2 2 2 or GE 2 2 2 2 2 (1 )( ) (2.22) H 2KK00 1 2 ( )
2.3 Computations
Computations were carried out for the determination of the following:
(i) B/D
(ii) B/c
(iii) Uplift pressures at the key points E, D, C and D'.
(iv) Seepage discharge factor, q/kH and
35
(v) Exit gradient factor, GE c/H.
Since Eqs. 2.10 (a) and 2.10 (b) for B/D and B/c ratios, respectively, are not explicit in transformation parameters, direct solution of these equations for the parameters and , corresponding to given values of floor profile ratios (B/D, B/c) is not suitable and practical.
However, these equations can be solved for physical floor profile ratios B/D and B/c, for assumed values of and . Hence B/D and B/c were computed for values 01 . In the computer programme, was varied from 0.01 to 0.99 in the steps of 0.01. For each value of , parameter was varied from an initial value of to 0.99 in the steps of 0.01.
Parameter was computed from Eq. 2.8, for each set of values of and . Table 2.1 gives the specimen results for 0.5
In order to determine the values of and for a given pair of values of B/D and B/c ratios, both B/D and B/c were separately plotted with and . Fig. 2.2. (a), (b), show the variation of B/D with and , plotting on x-axis and on various nomographs. The starting point of each nomograph is not the same, as the minimum value of is .
Similarly, Figs. 2.3(a), (b) show variation of B/c with and . From both these Figs., values of and were determined for selected values B/D = 0.2, 0.4, 0.6, 0.8, 1, 1.25, 1.5, 1.75, 2,
2.25, 2.5 and 3 and selected values of B/c = 4, 6, 8, 10, 15, 20, 25 and 30.
36
Fig. 2.2a Variation of B/D with σ and γ
Fig. 2.2b Variation of B/D with σ and γ, for γ close to 1
37
Fig. 2.3a Variation of B/c with σ and γ
Fig. 2.3b Variation of B/c with σ and γ, for γ close to 1
38
Table 2.1 Computed Values for 0.5
B/D B/c c/D PE PD q/kH GE cH/ 0.5 0.51 0.505017 0.708 165.67 0.004 6.80 4.80 0.632 0.034 0.5 0.6 0.551977 0.787 17.19 0.046 20.33 14.18 0.570 0.099 0.5 0.7 0.60955 0.884 8.74 0.101 27.33 18.79 0.505 0.129 0.5 0.8 0.676933 0.997 5.74 0.174 32.23 21.78 0.439 0.147 0.5 0.9 0.76523 1.148 4.01 0.286 36.44 24.02 0.363 0.157 0.5 0.91 0.776333 1.167 3.86 0.302 36.88 24.22 0.354 0.158 0.5 0.92 0.788134 1.188 3.71 0.320 37.32 24.41 0.344 0.159 0.5 0.93 0.800776 1.210 3.56 0.339 37.78 24.61 0.334 0.159 0.5 0.94 0.814458 1.234 3.41 0.361 38.26 24.80 0.323 0.159 0.5 0.95 0.829468 1.260 3.26 0.387 38.77 24.99 0.311 0.159 0.5 0.96 0.846252 1.289 3.10 0.416 39.33 25.18 0.298 0.159 0.5 0.97 0.865561 1.323 2.92 0.453 39.95 25.36 0.282 0.158 0.5 0.98 0.888874 1.365 2.72 0.501 40.70 25.55 0.263 0.156 0.5 0.99 0.920035 1.420 2.47 0.575 41.73 25.73 0.235 0.153
The resulting values of and so obtained were plotted against each value of B/D and
B/c (Fig. 2.4) giving rise to double set of nomographs corresponding to the above mentioned values of B/D and B/c. Fig. 2.4 is helpful in determining the values of and corresponding to a given set of values of B/D and B/c. In order to obtain more precise values of and , further iterative programming was done, using the values of and so obtained from Fig. 2.4 as initial input parameters.
Table 2.2 shows the values of and for the above mentioned sets of B/D and B/c ratio. Computer programme was re-run using these values of and as input values to confirm the values of B/D and B/c ratios and also to compute the seepage characteristics, PE, PD, q/kH and GE c/H. The computed values of these characteristic are also shown in Table 2.2.
39
Fig. 2.4 Determination of σ and γ
40
Table 2.2 Data Generated from Matching the Given Values of B/D and B/c Ratios
B/D B/c c/D PE PD q/kH GE cH/ 2 0.100 0.048 0.324 0.185 45.38 31.13 0.791 0.212 4 0.050 0.090 0.239 0.165 37.84 25.97 0.893 0.179 6 0.033 0.109 0.210 0.160 32.73 22.61 0.934 0.156 8 0.025 0.120 0.196 0.158 29.18 20.25 0.957 0.141 0.2 10 0.020 0.127 0.188 0.157 26.55 18.49 0.971 0.129 15 0.013 0.136 0.177 0.156 22.18 15.52 0.990 0.109 20 0.010 0.141 0.171 0.156 19.43 13.63 1.00 0.096 25 0.008 0.144 0.168 0.156 17.49 12.29 1.006 0.086 30 0.007 0.146 0.166 0.156 16.04 11.28 1.010 0.079
B/D B/c c/D PE PD q/kH GE c/ H 2 0.200 0.098 0.597 0.365 45.25 30.73 0.572 0.206 4 0.100 0.180 0.454 0.323 37.68 25.74 0.675 0.176 6 0.067 0.217 0.403 0.313 32.58 22.44 0.716 0.155 8 0.050 0.237 0.378 0.309 29.03 20.11 0.739 0.139 0.4 10 0.040 0.250 0.363 0.307 26.41 18.36 0.753 0.128 15 0.027 0.267 0.343 0.306 22.06 15.41 0.772 0.108 20 0.020 0.276 0.333 0.305 19.32 13.54 0.782 0.095 25 0.016 0.282 0.327 0.305 17.40 12.21 0.788 0.086 30 0.013 0.285 0.323 0.305 15.95 11.21 0.792 0.079
B/D B/c c/D PE PD q/kH GE c/ H 2 0.300 0.153 0.790 0.534 45.05 30.09 0.446 0.197 4 0.150 0.269 0.630 0.467 37.43 25.39 0.550 0.172 6 0.100 0.320 0.568 0.452 32.33 22.17 0.592 0.152 8 0.075 0.348 0.536 0.447 28.80 19.88 0.614 0.137 0.6 10 0.060 0.366 0.516 0.444 26.19 18.16 0.628 0.126 15 0.040 0.389 0.490 0.441 21.87 15.26 0.647 0.106 20 0.030 0.402 0.478 0.440 19.15 13.40 0.656 0.094 25 0.024 0.409 0.470 0.440 17.24 12.09 0.662 0.085 30 0.020 0.414 0.465 0.440 15.81 11.10 0.666 0.078
41
B/D B/c c/D PE PD q/kH GE cH/ 2 0.400 0.214 0.905 0.685 44.78 29.24 0.358 0.186 4 0.200 0.356 0.762 0.594 37.12 24.94 0.465 0.167 6 0.133 0.418 0.698 0.574 32.02 29.83 0.506 0.149 8 0.100 0.451 0.664 0.567 28.50 19.59 0.528 0.135 0.8 10 0.080 0.471 0.643 0.563 25.92 17.91 0.542 0.124 15 0.053 0.500 0.615 0.560 21.62 15.05 0.560 0.105 20 0.040 0.514 0.600 0.558 18.93 13.22 0.570 0.092 25 0.032 0.522 0.592 0.558 17.04 11.93 0.575 0.084 30 0.027 0.528 0.586 0.558 15.62 10.95 0.579 0.077
B/D B/c c/D q/kH 2 0.500 0.283 0.964 0.808 44.48 28.28 0.292 0.173 4 0.250 0.440 0.853 0.700 36.75 24.43 0.401 0.162 6 0.167 0.508 0.795 0.676 31.66 21.44 0.442 0.145 8 0.125 0.544 0.763 0.668 28.16 19.26 0.464 0.132 1 10 0.100 0.566 0.743 0.663 25.59 17.61 0.477 0.121 15 0.067 0.596 0.714 0.659 21.33 14.81 0.495 0.103 20 0.050 0.611 0.700 0.658 18.67 13.02 0.505 0.091 25 0.040 0.620 0.691 0.657 16.80 11.75 0.510 0.082 30 0.033 0.626 0.685 0.657 15.40 10.79 0.514 0.076
B/D B/c c/D q/kH 2 0.625 0.383 0.992 0.915 44.11 27.03 0.228 0.157 4 0.313 0.539 0.924 0.804 36.27 23.75 0.340 0.155 6 0.208 0.608 0.879 0.777 31.18 20.91 0.381 0.140 8 0.156 0.645 0.851 0.767 27.70 18.81 0.402 0.128 1.25 10 0.125 0.667 0.833 0.762 27.15 17.21 0.415 0.118 15 0.083 0.696 0.808 0.758 20.94 14.48 0.433 0.100 20 0.063 0.711 0.795 0.756 18.31 12.73 0.442 0.089 25 0.050 0.719 0.787 0.755 16.47 11.49 0.448 0.080 30 0.042 0.725 0.782 0.755 15.09 10.55 0.451 0.074
42
B/D B/c c/D PE PD q/kH GE c/ H 2 0.750 0.492 0.999 0.972 43.92 25.90 0.177 0.138 4 0.375 0.630 0.963 0.877 35.80 23.08 0.293 0.147 6 0.250 0.694 0.930 0.851 30.69 20.38 0.334 0.135 8 0.188 0.728 0.909 0.840 27.22 18.35 0.355 0.123 1.5 10 0.150 0.748 0.895 0.836 24.69 16.80 0.368 0.114 15 0.100 0.775 0.874 0.831 20.53 14.14 0.385 0.097 20 0.075 0.788 0.863 0.829 17.94 12.44 0.394 0.086 25 0.060 0.796 0.856 0.828 16.13 11.22 0.399 0.078 30 0.050 0.801 0.851 0.828 14.77 10.31 0.402 0.072
B/D B/c c/D q/kH 4 0.438 0.710 0.983 0.926 35.38 22.45 0.256 0.140 6 0.292 0.766 0.961 0.902 30.22 19.86 0.296 0.130 8 0.219 0.795 0.945 0.893 26.75 17.90 0.317 0.119 10 0.175 0.813 0.934 0.888 24.24 16.39 0.329 0.111 1.75 15 0.117 0.836 0.918 0.883 20.12 13.80 0.346 0.095 20 0.088 0.847 0.909 0.882 17.56 12.14 0.355 0.084 25 0.070 0.854 0.903 0.881 15.79 10.96 0.360 0.076 30 0.058 0.858 0.900 0.881 14.45 10.06 0.363 0.070
B/D B/c c/D q/kH 4 0.500 0.777 0.993 0.958 35.03 21.89 0.225 0.134 6 0.333 0.823 0.979 0.937 29.79 19.38 0.266 0.125 8 0.250 0.848 0.968 0.929 26.32 17.47 0.286 0.116 10 0.200 0.862 0.960 0.925 23.80 16.00 0.298 0.107 2 15 0.133 0.881 0.947 0.921 19.72 13.47 0.315 0.092 20 0.100 0.891 0.940 0.919 17.19 11.85 0.323 0.082 25 0.080 0.896 0.936 0.918 15.44 10.69 0.328 0.074 30 0.067 0.900 0.933 0.918 14.13 9.82 0.331 0.068
43
B/D B/c c/D PE PD q/kH GE cH/ 4 0.562 0.832 0.997 0.977 34.79 21.40 0.199 0.127 6 0.375 0.868 0.989 0.960 29.41 18.94 0.240 0.121 8 0.281 0.888 0.981 0.953 25.91 17.07 0.260 0.112 10 0.225 0.900 0.975 0.950 23.40 15.63 0.272 0.104 2.25 15 0.150 0.915 0.966 0.946 19.34 13.16 0.288 0.089 20 0.133 0.922 0.961 0.945 16.84 11.57 0.296 0.079 25 0.090 0.927 0.958 0.944 15.11 10.44 0.301 0.072 30 0.075 0.930 0.956 0.944 13.82 9.59 0.304 0.066
B/D B/c c/D q/kH 4 0.625 0.876 0.999 0.988 34.66 20.98 0.177 0.121 6 0.417 0.903 0.994 0.976 29.09 18.54 0.218 0.117 8 0.312 0.918 0.989 0.970 25.54 16.69 0.238 0.109 10 0.250 0.927 0.985 0.967 23.02 15.28 0.250 0.101 2.5 15 0.167 0.939 0.979 0.964 18.98 12.86 0.266 0.087 20 0.125 0.945 0.975 0.963 16.51 11.30 0.273 0.077 25 0.100 0.959 0.972 0.962 14.80 10.20 0.278 0.070 30 0.083 0.951 0.971 0.962 13.53 9.36 0.281 0.065
B/D B/c c/D q/kH 8 0.375 0.957 0.996 0.988 24.94 16.06 0.204 0.102 10 0.300 0.963 0.995 0.986 22.35 14.65 0.215 0.096 15 0.200 0.970 0.991 0.984 18.34 12.32 0.230 0.083 3 20 0.150 0.973 0.989 0.983 15.90 10.82 0.237 0.074 25 0.120 0.975 0.988 0.983 14.23 9.76 0.241 0.067 30 0.100 0.976 0.987 0.983 12.99 8.95 0.244 0.062
44
2.4 Design Charts
Table 2.2 also gives the values of uplift pressures PE and PD at key points, as well as the values of seepage discharge factor (q/kH) and exit gradient factor GE c/H. These results are used to create design charts. Figs. 2.5, 2.6, 2.7 and 2.8 show design charts for PE, PD, GE c/H and q/kH respectively. From these charts, it is observed that when BD/0.2 , the values of PE, PD and GE c/H approach those for the infinite depth case. The corresponding values of infinite depth case, analyzed by Malhotra (1936), are shown by dotted curve on these charts.
45
Fig 2.5 Design curves for PE
46
Fig 2.6 Design curves for PD
47
Fig 2.7 Design curves for q/kH
48
Fig 2.8 Design curves for GEc/H
49
2.5 Comparison with Infinite Depth Case
Malhotra (1936) obtained theoretical solution for uplift pressure below flat apron with equal end cutoffs, founded on pervious medium of infinite depth. The results of the present theoretical solution for B/D=0.4 and B/D = 0.2 were compared with those of Malhotra's solution.
Table 2.3 gives the values of uplift pressure PE at point E and PD at point D for some selected values of B/c ratios. It is seen from Table 2.3 as well as from Fig. 2.5, 2.6 and 2.7 that when
BD/0.2, the results from both the analyses are practically the same.
Table 2.3 Comparison with Infinite Depth Case
%Pressure PE at point E % Pressure PD at point D Infinite Present Present Infinite Present Present B/c Depth case by Analysis Analysis Depth Analysis Analysis Malhotra for for case by for for B/D=0.2 B/D=0.4 Malhotra B/D=0.2 B/D=0.4 3 41.4 41.3 41.1 28.4 28.2 28.0 4 37.9 37.8 37.7 26.1 26.0 25.7 6 32.9 32.7 32.6 22.7 22.6 22.4 8 29.2 29.2 29.0 20.3 20.3 20.1 12 24.6 24.5 24.4 17.2 17.1 17.0 24 17.8 17.8 17.7 12.6 12.5 12.4
2.6 Development of Interference Formulae
Since 1936, the prevalent practice has been to use the design curve or equation given by
Khosla et al. for the case of apron with end cutoff and apply interference formula suggested by him to take into account the effect of presence of another cutoff at the upstream side when the depth of pervious medium is infinite. A similar effort has been made here to develop interference equations for the seepage characteristics to take into account the effect of presence of cutoff at
50 the upstream end. The equations for PE, PD and GE c/H for the case of apron with downstream cutoff, developed by Pavlovsky are quite explicit and are amenable to direct computations while the equations developed in the present analysis for the case of cutoff at both the ends are not explicit. However, the above seepage characteristics can first be computed by Pavlovsky's equations and the interference formulae given below can be applied to take into account the effect of upstream cutoff.
For any seepage characteristic (such as PE, PD, GE c/H), we have
()I [(PP )( ) ] PE E SPE DP
()I [(PP )( ) ] PD D SPD DP and ()I [(G c / H ) ( G c / H ) ] (2.23) GE c/ H E SP E DP where I stands for interference correction, SP stands for single pile case and DP stands for double pile case. Since computed values of these characteristics are available both for double pile case, as well as single pile case (Pavlovsky), the values of interference corrections (I) were found for B/D = 0.8, 1, 1.5 and 2 and for B/c = 4, 6, 8, 10, 12, 15, 20 and 30. Table 2.4 gives the computed interference values. Figs. 2.9, 2.10 and 2.11 shows curves between interference values
(I), on y-axis, B/c values on x-axis and B/D values on nomographs, for PPED, and GE c/. H
From these curves, equations for interference correction were developed in the following form:
n B ()I SC a (2.24) C
where Interference correction for any seepage characteristic
a = Multiplying factor
n = Power factor
51
Table 2.4 Interference Values (I)
B/D B/c ()I ()I (I)Q/KH ()I PE PD GE c/ H 4 5.547 3.465 0.057 0.022 6 3.136 2.047 0.042 0.014 8 2.070 1.381 0.033 0.009 10 1.495 1.009 0.027 0.007 0.8 15 0.822 0.564 0.019 0.004 20 0.537 0.371 0.014 0.003 25 0.385 0.267 0.012 0.002 30 0.293 0.204 0.010 0.001 4 5.794 3.644 0.056 0.023 6 3.259 2.135 0.041 0.014 8 2.146 1.434 0.032 0.010 10 1.547 1.046 0.026 0.007 1 15 0.849 0.583 0.018 0.004 20 0.554 0.383 0.014 0.003 25 0.397 0.276 0.011 0.002 30 0.303 0.211 0.009 0.001 4 6.556 4.127 0.052 0.026 6 3.610 2.364 0.038 0.016 8 2.352 1.571 0.030 0.010 10 1.685 1.138 0.024 0.008 1.5 15 0.917 0.629 0.017 0.004 20 0.595 0.412 0.013 0.003 25 0.426 0.296 0.010 0.002 30 0.324 0.226 0.009 0.002 4 7.938 5.202 0.046 0.027 6 4.044 2.647 0.035 0.017 8 2.541 1.674 0.028 0.011 10 1.784 1.182 0.023 0.008 2 15 0.942 0.629 0.016 0.005 20 0.598 0.400 0.012 0.003 25 0.419 0.280 0.010 0.002 30 0.313 0.209 0.008 0.002
52
Fig. 2.9 Interference Correction for PE
53
Fig. 2.10 Interference Correction for PD
54
Fig. 2.11 Interference Correction for GE c/ H
55
Thus, for a given seepage characteristic (such as PE), four such equations (one for each value of B/D ratio) were developed. Table 2.5 gives the factors a and n for various seepage characteristics, for each chosen value of B/D ratio.
Table 2.5 Values of Factors a and n
Factors for PE Factors for PD Factors for GE c/H B/D a n a n a n 0.8 42.955 1.463 25.516 1.413 0.157 1.374 1 45.135 1.469 27.056 1.422 0.167 1.383 1.5 52.406 1.494 31.424 1.447 0.191 1.406 2.0 71.284 1.597 45.860 1.585 0.202 1.405
To apply the interference correction, let us take a case of B/D= 1 and B/c= 10. For these values, c/D = 0.1. Hence from the direct solution of Pavlovsky's equations, we get the following values for the case of apron with cutoff at downstream end.
PE 27.14%,PD 18.66% and GE . c / H 0.128
From Table 2.5 the factors a and n for PE for B/D=1 are 45.135 and 1.469 respectively.
Hence
n 1.169 ()I PE a( B / c ) 45.135(10) 1.53%
Since the provision of pile at the upstream end reduces the pressure at E, the above interference correction is negative.
Hence ()PE DP (PIE ) SP ( ) SP 27.14 1.53 25.61%
Value of PE obtained by the present analysis of apron with cutoffs at both the ends is
25.59%. For this excellent agreement, the interference formula can be applied with sufficient
56 degree of accuracy. Similar computations can be done for pressure at point D (i.e. PD) and exit gradient factor GE. c/H without the need for solving the exact equations of implicit nature.
2.7 Development of Simplified Equations for PE, PD and GEc/H
In the design curves given in Figs. 5, 6 and 8, linear interpolation is required for intermediate values of B/D ratios. In order to avoid this, the following algebraic equations have been developed, based on computer generated values for various values of B/D and B/c.
68.74 PE (2.25) 0.037 0.422e0.035(BD / ) BB Dc
49.865 PD (2.26) 0.411/(BD / )0.024 0.114(BD / ) B ()e c
0.0378(BD / )0.4226 c 0.354 B and GE (2.27) H ()e 0.214(BD / ) c
In order to illustrate the use of the above equations, taking
B/D = 2 and B/c = 20, we get
68.74 PE 0.035(2) 17.27% (2)0.037 (20) 0.422e which agrees well with the value of 17.19% computed from rigorous analysis of this case.
49.865 PD 0.024 11.83% (2)0.114 2 (20) 0.411/(2) which agrees well with the value of 11.85% computed from rigorous analysis of this case.
0.354 0.4226 .(20)0.0378(2) 0.0816 ()e 0.214(2)
57 which agrees well with the value of 0.082 computed from rigorous analysis of this case.
Thus, the values computed by the use of above simplified equations match closely with the values computed from the rigorous analyses of this chapter. Table 2.6 gives the values of values of PE computed from the simplified formula as well as from rigorous analysis, along with the difference in two values for various pairs of values of B/D and B/c. From this table, we find that the difference between the computed and actual values is well within two percent.
Table 2.6 Comparison of Values of PE from the Formula
Calculated values of PE from the formula B/D → 0.6 0.8 1 1.25 1.5 1.75 2 6 32.37 31.85 31.42 30.94 30.52 30.14 29.77 8 28.59 28.11 27.7 27.26 26.86 26.49 26.14 B/c 10 25.97 25.52 25.13 24.7 24.32 23.96 23.63 15 21.81 21.4 21.05 20.66 20.31 19.98 19.67 20 19.26 18.89 18.56 18.2 17.87 17.56 17.27
Actual values of PE from rigorous analysis B/D → 0.6 0.8 1 1.25 1.5 1.75 2 6 32.33 32.02 31.66 31.18 30.69 30.22 29.79 8 28.8 28.5 28.16 27.7 27.21 26.75 26.32 B/c 10 26.19 25.92 25.59 25.15 24.69 24.24 23.8 15 21.87 21.62 21.33 20.94 20.53 20.12 19.72 20 19.15 18.93 18.67 18.31 17.94 17.56 17.19 % Difference = 100(Actual – Calculated)/Actual B/D → 0.6 0.8 1 1.25 1.5 1.75 2 6 -0.12 0.53 0.76 0.77 0.55 0.26 0.07 8 0.73 1.37 1.63 1.59 1.29 0.97 0.68 B/c 10 0.84 1.54 1.80 1.79 1.50 1.16 0.71 15 0.27 1.02 1.31 1.34 1.07 0.70 0.25 20 -0.57 0.21 0.59 0.60 0.39 0.00 -0.47
58
2.8 Conclusions
The general solution to the finite depth seepage under an impervious flat apron with equal end cutoffs is obtained using the method of conformal transformation. The results obtained from the solution of implicit equations have been used to present design charts for seepage characteristics, such as uplift pressures at key points, discharge factor and exit gradient factor, in terms of non-dimensional floor profile ratios. When the depth of permeable soil is large, the numerical solutions tend toward the values given by the established solution for infinite depth.
Since the equations derived for various seepage characteristics involve elliptic integrals, simplified algebraic equations have also been suggested, results from which are in excellent agreement with those obtained through rigorous solutions.
59
CHAPTER 3. FLAT APRON WITH UNEQUAL END CUTOFFS AND A STEP AT DOWNSTREAM END
3.1. Introduction
Flat aprons of hydraulic structures are invariably provided with cutoffs at both upstream and downstream ends. The cutoff at the downstream end of apron safeguards the structure both against exit gradient as well as scour though it increases uplift pressure all along the upstream side. The cutoff at the upstream end protects the apron against scour and at the same time reduces the uplift pressure all along the downstream side. In addition to these, pervious aprons are provided at the downstream side of end cutoff in the form of inverted filter and launching apron. In order to accommodate the thickness of these pervious aprons, the downstream bed is excavated resulting in the formation of a step at the downstream end.
3.2. Theoretical Analysis
Fig 3.1(a) shows the floor profile in z-plane, with relative values of stream function and potential function on the boundaries. Fig 3.1(c) shows the well known w-plane, relating and
. The points in z and w-planes are denoted by complex coordinates z = x + iy and w = + i respectively in which i 1 .The problem is solved by transforming both the z-plane and w- plane onto an infinite half plane, t-plane, shown in Fig. 3.1(b), thus obtaining z = f1 (t) and w = f2 (t).
60
Floor Parameters
The floor of the hydraulic structure, shown in Fig. 3.1(a), has the following floor parameters:
(i) Length of the apron : B
(ii) Finite depth of pervious medium : D
(iii) Depth of upstream cutoff : c1
(iv) Depth of downstream cutoff : c2
(v) Depth of downstream step : a
The resulting independent non-dimensional floor profile ratios are:
1. B/D 2. B/c2 3. c1/c2, and 4. a/c2
61
Fig. 3.1
Illustrations of the problem – Schwarz Christoffel transformations
62
3.2.1 First Transformation
The Schwarz-Christoffel equation for transformation of the floor from z-plane to t-plane is given by
dt z MN11 (3.1 a) 12 78 (t t1 ) ( t t 2 ) ...... ( t t 7 ) ( t t 8 ) where 1 //22 ; 1 12/ 5 2 ; 5 1
2 2 ; 2 1 6 //22 ; 6 12
3 //22 ; 3 12 7 0 ; 7 1
4 //22 ; 4 12 8 0 ; 8 1
Substituting the values of 's , we get
dt z MN 11 1/2 1 1/2 1/2 1 1/2 1 1 ()()()()()()()()tt1 tt 2 tt 3 tt 4 tt 5 tt 6 tttt 7 8
()()t t25 t t dt or MN11 (3.1) (tttt7 )( 8 )( tttttttt 1 )( 3 )( 4 )( 6 )
This is an elliptic integral.
Referring to Identity 254, Byrd and Friedman (1971),
()()t t t t Let sn2u 4 1 3 (i a) ()()t4 t 3 t t 1
()()t t t t Such that m2 4 3 6 1 (i b) ()()t4 t 1 t 6 t 3
Differentiating Eq. (i a)
()tt ()()t t t t 2 sn u cn u dn u du 41 13dt 2 ()tt43 ()tt 1
Substituting the values of sn u, cn u, dn u and noting that
63
cn u 1 sn2u and dn1snumu , 22 we get
()()()()t4 t 1 t t 3 t 4 t 1 t t 3 dt 2.du 1 ()()()()t4 t 3 t t 1 t 4 t 3 t t 1
()()()()()t t t t t t t t t t ()tt 2 1.4 3 6 1 4 1 3 4 3 1 ()()()()()()tttttttt416343 1 tt 41 tt 31
Simplifying and rearranging, we get
dtdu 2 (ii) ()()()()()()tttttttt1 3 4 6 tttt 4 1 6 3
Again from Eq. (i a)
tt tt 3 43sn2 u tt 1 tt41
t4 t 3 22 t 4 t 3 or tt 3 t snsn u tu1 t4 t 1 t 4 t 1
From which
tt43 2 t31 tusn tt t 41 tt43 2 1snu tt41
For any point having co-ordinate tm , we have
tt43 2 (t31 tmm ) ( t t ) sn u tt41 tt m tt43 2 1 sn u tt41
64
tt43 2 (t3 t 2 ) ( t 1 tu 2 )sn tt41 Hence ()tt 2 (a) tt43 2 1sn u tt41
tt43 2 (t3 t 5 ) ( t 1 tu 5 )sn tt41 ()tt 5 (b) tt43 2 1sn u tt41
tt43 2 (t3 t 7 ) ( t 1 tu 7 ) sn tt41 ()tt 7 (c) tt43 2 1 sn u tt41
tt43 2 (t3 t 8 ) ( t 1 t 8 ) sn u tt41 and ()tt 8 (d) tt43 2 1 sn u tt41
t t t t (t t ) ( t t )4 3 sn22 u ( t t ) ( t t ) 4 3 sn u ()()t t t t 3 2 1 2t t 3 5 1 5 t t Hence 254 1 4 1 (iii) ()()t t78 t t t4 t 322 t 4 t 3 (t3 t 7 ) ( t 1 t 7 ) sn u ( t 3 t 8 ) ( t 1 t 8 ) sn u t4 t 1 t 4 t 1
Let t3 t 2 a 1 ()t2 t 1 b 1
t3 t 5 a 2 ()t5 t 1 b 2
t3 t 7 a 3 ()t7 t 1 b 3
t3 t 8 a 4 ()t8 t 1 b 4
65
tt and 43sn2up tt41
()()a b p a b p Hence R.H.S. of Eq. (iii) 1 1 2 2 ()()a3 b 3 p a 4 b 4 p
b b p2 p() a b b a a a 1 21 2 1 2 1 2 2 b3 b 43 p 4 p 4() a 3 b 3 a 4 b a a
bb12 2 Numerator b3 b 43 p 4 p 4() a 3 b 3 a 4 b a a bb34
b b a a b b p() a b b a 3 4 1 2 3 4 p() a b a b a a 1 2 1 2 3 4 4 3 3 4 b1 b 2 b 1 b 2
a1 b 3 b 4 a 2 b 3 b 4 a 1 a 2 b 3 b 4 p a3 b 4 a 4 b 3 a 3 a 4 bb b b b b R.H.S. 121 1 2 1 2 bb b b p2 p() a b a b a a 343 4 3 4 4 3 3 4
b1 b 2 a 3b1 b 2 a 4 b 1 b 2 a 3 a 4 p a1 b 2 b 1 a 2 a 1 a 2 bb b b b b 12 3 4 3 4 b3 b 4()() a 3 b 3 p a 4 b 4 p
b b p 1 2 1 2 (iv) b3 b 4()() a 3 b 3 p a 4 b 4 p
b1 b 2 a 3 b1 b 2 a 4 where 1 = coefficient of p a1 b 2 b 1 a 2 bb34
b1 b 2 a 3 a 4 2 = constant term aa12 bb34
66
p AB Now let 12 (iv a) ()()a3 b 34 p a 4 b p a3 b 34 p 4a b p
p() Ab () Bb Aa Ba 4 34 3 ()()a3 b 3 p a 4 b 4 p
Hence we get 1 Ab43 Bb and 2 Aaba43
From the above the equations, we have
ab33 A 12 a3 b 4 b 3 aa 43 b 4 b 3 a 4
ab44 and B 12 b3 a 4 a 3 b 4 b 3 a 4 a 3 b 4
Substituting the values of 1 and 2, we get
a3 b 1 b 2 a 3b1 b 2 a 4 b 3 b 1 b 2 a 3 a 4 A a1 b 2 b 1 aa 21 a 2 abba3434 b 3 b 4 abba 3434 bb 34
1 22b1 b 2 a 4 a 3 b 3 b 3 b 1 b 2 a 3 a 4 abababbabba33123312123 baa 312 b3() a 3 b 4 b 3 ab 44 b 4
1 abab abba bba22 aab 3312 3312 123 123 b3() a 3 b 4 b 3 a 4
()()a b b a a b b a 2 3 2 3 3 1 3 1 b3() a 3 b 4 b 3 a 4
Similarly,
a4b1 b 2 a 3 b 1 b 2 a 4 b 4 b 1 b 2 a 3 a 4 B a1 b 2 b 1 a 2 a 1 a 2 baab3434 b 3 b 4 baab 3434 bb 34
1 a4 b 4 b 1 b 2 a 322 b 1 b 2 a 3 a 4 b 4 abababba44124412 bbabaa 124412 b4() b 3 a 4 a 3 b 4 b 3 b 3
67
1 abab abba bba22 aab 4412 4412 124 124 b4() b 3 a 4 a 3 b 4
()()a b b a a b b a 2 4 2 4 4 1 4 1 b4() b 3 a 4 a 3 b 4
Substituting the values of A and B in Eqs. (iv) and (iv a), we get
b ba b(a b b a b a a )(b a b b a b a ) ( )( ) R.H.S. of Eq. (4) 1 22 42 2 3 4 2 4 3 1 3 4 1 1 3 1 (v) bb34 bab 3343433( baa )( bp ) bba 4343444 ( aba )( bp )
= p + q + r
Substituting the values of a1, a2...... etc., we get
bb (t t )( t t ) p 12 2 1 5 1 (v a) b3 b 4()() t 7 t 1 t 8 t 1
()()a b b a a b b a q 2 3 2 3 3 1 3 1 b3()( a 3 b 4 b 3 a ) 4 a 3 b 3 p
(tttttttt )( )( )( ) [( tttttttt )( )( )( )] 35715137 37217132 (tttttttt71373781 )( )[( )( )( tttt 7138 )( )] t7 t 1 t 4 t 3 2 1 sn u t7 t 3 t 4 t 1
[][]tt tt tt tt tt tt tt tt tt tt tt tt tt tt tt tt 37 57 31 51 53 13 57 17 32 72 31 71 73 13 72 12 t7 t 1 t 4 t 3 2 (tttttttttttttttttttt7 1 )( 3 7 )[] 38 1 78 . 31 71 sn 73 13 78 18 u t7 t 3 t 4 t 1
(t t )( t t )( t t )( t t ) 3 1 7 5 2 7 3 1 t7 t 1 t 4 t 3 2 (t7 t 1 )( t 3 t 7 )( t 3 t 1 )( t 8 t 7 )1 sn u t7 t 3 t 4 t 1
(t t )( t t )( t t ) 7 5 3 1 2 7 t7 t 1 t 4 t 3 2 (t7 t 1 )( t 8 t 7 )( t 3 t 7 ) 1 sn u t7 t 3 t 4 t 1
(t t )( t t )( t t ) 7 5 7 2 3 1 (v b) 22 (t7 t 1 )( t 8 t 7 )( t 7 t 3 )[1 2 sn] u
68
2 tt7 tt 1 43 where 2 . tt73 tt 4 1
()()a b b a a b b a r 2 4 2 4 4 1 4 1 b4()() b 3 a 4 a 3 b 4 a 4 b 4 p
[(tttttttt )( ) ( )( )] [( tttttttt )( ) ( )( )] 3581 5138 38218132 (tttttttt81387138 )( )[( )( )( tttt 3781 )( )] t8 t 1 t 4 t 3 2 1sn u t8 t 3 t 4 t 1
[]tt tt tt tt tt tt tt tt 38 58 31 51 53 13 58 18 (t8 tt 1 )( 3 t 8 )[] tt 73 tt 13 tt 78 tt 18 tt 38 tt 78 tt 31 tt 71
tt tt tt tt tt tt tt tt 32 82 31 81 83 13 82 12 t8 t 1 t 4 t 3 2 1 . sn u t8 t 3 t 4 t 1
(t t )( t t )( t t )( t t ) 1 8 5 3 1 8 2 1 3 (t8 t 1 )( t 3 t 8 )( t 7 t 8 )( t 3 t 1 ) t8 t 1 t 4 t 3 2 1 sn u t8 t 3 t 4 t 1
(t t )( t t )( t t ) 8 5 8 2 3 1 (v c) 22 (t8 t 1 )( t 8 t 3 )( t 7 t 8 )[1 1 sn] u
2 t8 t 1 t 4 t 3 where 1 t8 t 3 t 4 t 1
Again, from Eqs. 3.1 and 3.3, we have
()()t t25 t t 2du z MN11 ()()t t78 t t ()()t4 t 1 t 6 t 3
2M1 ()()t t25 t t or du N1 ()()t4 t 1 t 6 t 3 ()()t t78 t t
2M1 or (R.H.S. of Eq.iii ) du N1 ()()t4 t 1 t 6 t 3
69
2M1 ()()()()()t2 t 1 t 5 t 1 t 7 t 5 t 7 t 2 t 3 t 1 2 or z uu m (,,)2 ()()t4 t 1 t 6 t 3 ()()()()()t7 t 1 t 8 t 1 t 7 t 1 t 8 t 7 t 7 t 3
()()()t8 t 5 t 8 t 2 t 3 t 1 2 (,,)u11 m N ()()()t7 t 8 t 8 t 3 t 8 t 1
2M1 ()()()()t2 t 1 t 5 t 1 t 3 t 1 t 8 t 5 t 8 t 2 2 or uu m (,,)1 ()()t4 t 1 t 6 t 3 ()()()()t7 t 1 t 8 t 1 t 7 t 8 t 8 t 3 t 8 t 1
()()t7 t 5 t 7 t 2 2 (,,)u21 m N ()()t7 t 1 t 7 t 3
()()tt t t Again, sn2 u 4 13 ()()tt4 t 31 t
()()t t t t u sn(1 , ) 4 1 3 Fm ()()t4 t 3 t t 1
()()t t t t where sin1 4 1 3 ()()t4 t 3 t t 1 and (,,)um2 (,,) 2 m
2M1 ()()()()()t2 t 1 t 5 t 1 t 3 t 1 t 8 t 5 t 8 t 2 2 z F(,)(,,) m 1 m ()()t4 t 1 t 6 t 3 ()()()()()t7 t 1 t 8 t 1 t 7 t 8 t 8 t 3 t 8 t 1
()()t7 t 5 t 7 t 2 2 (,,) 21mN (vi) ()()t7 t 1 t 7 t 3
Here, we observe from t-plane that
t11 t52
t21 t62
t3 t7 1
t4 t8 1
70
Substituting in Eq. (7), we get
2M1 ( 1 1 )( 2 1 ) ( 1 )(1)(1) 2 1 2 zF mm ( , )( , , ) 1 ( 12 )( ) (1)(1)1 11 2 (1)(1)
(121 )(1 ) 2 (,,)21mN (3.2 a) (1 1 )(1 )
22 or z M1[F(,) A 11 m 1 B (,,) 1 m 2 C (,,)] 1 m N (3.2)
2(1 1 )( 1 2 ) 1 where A1 (3.3a) (1 11 )(1 ) ()() 12
(1 )(1 1 )(1 2 ) 1 B1 (3.3b) (1 1 )(1 ) ()() 12
(1 )(1 1 )(1 2 ) 1 C1 (3.3c) (1 1 )(1 ) ()() 12
()()()() tt sn2u 11 (3.4a) ( )(tt 11 ) 2 ( )
()()()() tt sinsin1111 (3.4b) ( )(tt 11 ) 2 ( )
( )( ) 2 ( ) and m2 2 1 1 2 (3.4c) (1 )( 2 ) ( 1 )( 2 )
2 2 (,,) 1 m =Third degree elliptic integral with amplitude , parameter 1 and
modulus m
2 2 (,,) 2 m =Third degree elliptic integral with amplitude , parameter 2 and
modulus m
2 ()()t8 t 1 t 4 t 3 11 1122 1 . (3.5 a) (t8 t 3 ) ( t 4 t 1 ) 1 1 1 1
71
2 ()()t7 t 1 t 4 t 3 11 2 2 (3.5 b) (t7 t 3 ) ( t 4 t 11 ) 1
In Eq. 3.2 there are seven unknown parameters : , 1 ,, 2 , 1 , 2 M1 and N1 and hence seven equations are required to determine these, which can be obtained from boundary conditions at points 3, 4, 2, 5, 1, 6 and 9. The simultaneous solutions of these equations would yield the values of the unknown parameters in the t-plane in terms of floor parameters of z-plane.
Boundary conditions
(i) Point 3 (Point E1):
At point 3, z = – B/2 and t = –
()() sin111 sin (0) 0 2 ( 1 )
Fm(,) Fm(0, ) 0 and (,,) 2 m (0, 2 ,m ) 0
Hence from Eq. 3.2, B /2MN11[0]
B From which N (3.6) 1 2
(2) Point 4 (Point E):
At point 4, zB /2and t
()() sin111 sin 1 2 ( 1 ) 2
Fm(,) F, m K 2
2 22 and (,,) m ,,(,) mm 0 2
72 where 0 = complete elliptic integral of third kind.
Hence from Eq. 3.2
22 B /2M1[( A 1 K , ) B 1 0 1 ( m , )] C 1 0 / 2 2 m B
22 or B M1[( A 1 K , ) B 1 0 1 ( m , )] C 1 0 2 m M 1 R (3.7)
BB where M (3.8 a) 1 22 A1 K Bm 1 0( Cm 1 , )( 1 , ) 0 2 R
22 and R A1 K Bm 1 0( Cm 11 , )( 0 2 , ) (3.8 b)
(3) Point 7:
At point 7, z = and t = + 1. Hence the point that is mapped into t = + 1 is a simple pole of integrand (3.1).
iD i M1 Lt( t 1). f ( t ). dt t1
()()t t25 t t Here ft() , wheret7 1 (tttt7 )( 8 )( tttttttt 1 )( 3 )( 4 )( 6 )
()()t t t t (t 1) f ( t ) 25 (t t8 ) ( t t 1 )( t t 3 )( t t 4 )( t t 6 )
()()tt 12 (1)(t t 12 )( t )( t )( t )
(112 )(1 ) iM1 (1 1) (1 12 )(1 )(1 )(1 )
M1 (112 )(1 ) D MJ1 (3.9) 2 (1 12 )(1 )(1 )(1 )
D (1 )(1 ) where J 12 (3.9 a) M1 2 (1 12 )(1 )(1 )(1 )
73
(4) Point 6:
At point 6, z = B/2 + i a and t = 2
11()() 12 1 sin sin 2 ( 12 ) m
1 1 Fsin , m K iK m
2 11 222 im and sin , ,mmm 00 ( , )( , ) m m22
22m where 2 m22
Hence from Eq. 3.2, we get
2 B 22im ia M A()(,)(,) K iK B m m 2 1 1 1 0 122 0 1 m 1
2 22im B C (,)(,) m m 1 0 222 0 2 2 m 2
22m 22m where 2 1 and 2 2 1 22 2 22 m 1 m 2
22 B ia M1[ A 1 K B 1 0 ( 1 , m ) C 1 0 ( 2 , m )]
mm22 i M A K B (,)(,) 22 m C m 1 1 12 2 0 1 1 2 2 0 2 mm 12
Separating real and imaginary parts, we get
22 B M1[ A 1 K B 1 0 ( 1 , m ) C 1 0 ( 2 , m )] M 1 . R
(which is the same as Eq. 3.7)
74
mm22 and a M A K Bm Cm ( 22 , )( , ) (3.10 a) 1 1 12 22 0 2 1 1 0 2 mm 12 or a ML1 (3.10 b)
mm22 where L = A K B (,)(,) 22 m C m (3.10 c) 1 12 2 0 1 1 2 2 0 2 mm 12
22 mm22 A1 K Bm 1 Cm 0( 1 , )( 1 , ) 0 2 a mm2 22 2 S 12 (3.10) 22 B A1 K B 1 0(,)(,) 1 m C 1 0 2 m
(5) Point 1 (Point G):
At point 1, z = – B/2 and t = – 1
()() sinsin11 11 2 ( 12 )
Fm(sin1 , ) iK
12 i 22 and (sin , ,m ) [Km 0 (1 , )] 12
Hence from Eq. 3.2, we get
BB1 C M i A K B K 2(1 2 , m ) 1 K 2 (1 2 , m 1 1 122 1 0 1 2 0 2 2211 12
BC A K11 K 2(1 2 , m ) K 2 (1 2 , m 0 (3.11 a) 122 1 0 1 2 0 2 11 12
BC or A K11 K 2( 2 , m ) K 2 ( 2 , m 0 (3.11) 122 1 0 1 2 0 2 12
2 2 22 where 1 1 1 and 22 1 .
75
(5) Point 5 (Point D):
At point 5, z = B/2+ i c2 and t = + 2
()() sin1112 sin y 2 ( 12 )
1 where y 1 m
Hence from Identity 115.02, Byrd and Friedman (1971)
Fm( ,) K i F(,) m
22m and ( ,, 2 ) m (,)(,),, 2 m i F mm 0 22 11
y2 1 y2 1 where sinsin11 my my22
()() 121 2 ( ) sin1 12 2 ( ) ( )( ) 1 1 2 1 2 (1 )( 2 ) 2 ( 1 2 )
( )( ) 2 ( ) sin1 1 2 1 2 (1 )( 2 ) 2 ( 1 2 ) ( 12 )( ) (12 )( )
2 sin1 1 2 1 2 2 2 1 2 1 2 2
( )( )( ) ( )( )( ) sin111 2 2 sin 1 2 2 (1 )( 2 )( 2 ) ( 1 )( 2 )( 2 )
()() or sin1 22 ()()22
76
Hence from Eq. 3.2.
2 2 B 2 m ic M A K iF(,)(,)(,),, m B m i F mm 1 2 2 1 11 0 1 22 11 11
2 2 2 2 mB C1 0( 2 , m ) i F ( , mm ), , 11 22 2 22
Separating real and imaginary parts, we get
22 B M1[( A 1 K , ) B 1 0 1 ( m , )]C 1 0 2 m M 1 R
(as found earlier in Eq. 3.7)
B 2 m2 and c M[(,){},, F m A B Cm 11 2 1 1 1 1 22 11 11
2 m2 Cm2 ,, (3.12) 1 22 11 22 or MT1 (3.12 a)
2 2 where T F(,){}(,,) m A B Cm 11 2 Cm 2 (,)2 (3.12 b) 1 1 13 2 142 11 12
m2 m2 2 and 2 3 2 4 2 11 12
(7) Point 2 (Point D1):
At point 2, z = – B/2 + i c1 and t = – 1
()()()() sin1 1 sin1 i 1 1 2 ( 1 1 ) 2 ( 1 1 )
()() sin1 y , where y ix i 1 2 ( 11 )
77
Since y = i x, the upper limit of is imaginary.
Hence becomes i
sini ixor isinh i x
x sinh
F(,) im i Fm( ,)
i and (im ,, 2 ) [ F ( , mm )( ,1 22 , )] 12 where tan 11 (sinh ) tan x
()() or tan1 1 2 ( 11 )
Hence from Eq. 3.2.
B Bi ic M A iF(,) m 1 F (,) m 22 (,1 ,) m 1 1 11 1 2 2 11
Ci B 1 F( , mm ) 22 ( ,1 , 2 22 12 2
BCBC22 c M F(,)(,1 m Am ,)1 1 (,1 1 m 1 ,) 22 1 2 1 1 112 2 2 2 2 1 1 1 2 1 1 1 2
(3.13) or = M1 . P (3.13 a)
BCBC22 whereP F (,) m A 1 1 1 1 (,1,)22 m 1 2 (,1,) m 12 2 2 1 2 2 1 1 1 2 1 1 1 2
(3.13 b)
78
General Relationship: zf t 1()
From Eq. 3.8 (a)
B M 1 22 A1 K Bm 1 0( Cm 11 , )(0 2 , )
Hence from Eq. 3.2
B[F(,) A m Bm (,,) Cm 22 (,,)] B z 11 1 1 2 22 A1 K B 1 0(,)(,) 1 m C 1 0 2 m 2
which is the required relationship zf t 1()
Determination of Floor Profile Ratios
There are five parameters in z-plane:
(i) Length of the apron, B
(ii) Finite depth of pervious medium, D
(iii) Depth of upstream cutoff, c1
(iv) Depth of downstream cutoff, c2 , and
(v) Depth of downstream scour a
Consequently, we have the following non-dimensional floor profile ratios:
(1) B/D (2) Bc/ 2 (3) cc12/ , and (4) ac/ 2
From Eqs. 3.9 and 3.10a,
D D a M1 J M 1 L M 1() J L
Hence from above equation and Eq. 3.7, we get
BRMR 1 (3.14) DMJLJL1()
79
From Eqs. 3.7 and 3.12a,
BRMR 1 (3.15) cM21 TT
From Eqs. 3.13a and 3.12a,
c M P P 11 (3.16) c21 M T T
a a BR Also, S (3.17) cB22 cT where R, J, L and T are given by Eqs. 3.8b, 3.9a, 3.10c and 3.12b respectively.
Determination of 1 and 2
From Eq. 3.10, we have
22 mm22 A1 K B 1 0(,)(,) 1 m C 1 0 2 m mm2 2 2 2 a 12S 22 A1 K B 1 0(,)(,) 1 m C 1 0 2 m B
mm22 or A K B (,)(,) 22 m C m 1 12 2 0 1 1 2 2 0 2 mm 12
22 S[ A1 K B 1 0 ( 1 , m ) C 1 0 ( 2 , m )]
m2 or A{}(,)(,) K SK B 22 m S m 1 122 0 1 0 1 m 1
m2 C ( 22 , m ) S ( , m ) 0 122 0 2 0 2 m 2
Substituting the values of AB11, and C1 and cancelling common terms,
80
2 2(1 1 )( 1 2 ) ( 1 )(1 1 )(1 2 ) m 22 {K SKm }( S, ) m ( , ) 220 1 0 1 (1 1 )(1 11 )(1 )(1 ) m 1 ( )(1 )(1 ) m2 1 1 2 ( 22 ,m ) S ( , m ) 0 220 2 0 2 (1 1 )(1 ) m 2
ABC2111221( )( ) (1 )(1 221 ) (1 )(1 2 ) 0 (I)
2()K SK where A2 (3.18 a) (1)(1) 11
() m2 Bm S 1 m (,)(,)22 (3.18 b) 20 10 1 22 (1 1 )(1 ) m 1
() m2 Cm S 1 m (,)(,)22 (3.18 c) 20 2 0 2 22 (1 1 )(1 ) m 2
Again, From Eq. 3.11, we have
BC A K11 K 2( 2 , m ) K 2 ( 2 , m 0 122 1 0 1 2 0 2 12
2 2 22 (where 1 1 1 and 12 1 )
Substituting the values of AB11, and C1 and cancelling common terms,
2( )( ) ( )(1 )(1 ) 1 1 1 1 2K 1 1 2 .(,) K 22 m 2 1 0 1 (1 1 )(1 1 ) (1 1 )(1 ) 1
( )(1 )(1 ) 1 1 1 2 .Km 22 ( , ) 0 2 2 0 2 (1 1 )(1 ) 2 or ABC3111231( )( ) (1 )(1 231 ) (1 )(1 2 ) 0 (II)
2 K where A3 (3.19 a) (1 11 )(1 )
81
() BKm 1 22 (,) (3.19 b) 31 0 1 2 (1 11 )(1 )
() and C 1 K 22 (,) m (3.19 c) 32 2 0 2 (1 12 )(1 )
Thus, we get the following two simultaneous equations from which 1 and 2 can be computed for each pair of values of ( ,, 12 and S):
ABC2111221( )( ) (1 )(1 221 ) (1 )(1 2 ) 0 (I) and ABC3111231( )( ) (1 )(1 231 ) (1 )(1 2 ) 0 (II)
2 From (I), ()AAAABBBB21 () 211 212 212 2 21 22 212
()CCCC2 0 2 1 2 2 2 1 2
2 or 12122(ABCABCABCABC ) 22122 ( ) 12222 ( ) ( 21 22 ) 0
2 ABCABC2 2 2 2 1 2 2 or 1 2 1 2 0 ABCABC2 1 2 2 2 1 2 2 or 1 2 PP 1 1 2 2 0 (III)
ABC2 2 2 where P1 (3.20 a) ABC2 1 2 2
2 ABC2 1 2 2 P2 (3.20 b) ABC2 1 2 2
2 Also, from (II), ()()AAAABBBB31 311 312 312 3 31 32 312
(CCCC3 3 1 3 2 3 1 2 ) 0
2 or 13133(ABCABCABCABC ) 23133 ( ) 12333 ( ) ( 31 33 ) 0
82
ABCABC 2 or 3 3 3 3 1 3 3 0 1 2 1 2 2 ABC3 1 3 3 ABC3 1 3 3 or 1 2 QQ 1 1 2 2 0 (IV)
ABC333 where Q1 (3.21 a) AB3 133 C
2 AB3 133 C Q2 (3.21 b) AB3 133 C
Hence we have following two simultaneous equations:
1 2 PP 1 1 2 2 0 (III) and (IV)
where values of PPQ1, 2, and Q2 are now known.
Adding (III) and (IV), we get
1 2()PQPQ 1 1 2 2
PQ22 or 12 PQ11
1 PQ22 From which 2 . (V) 1PQ 1 1
Substituting the values of 2 and 12 in (IV), we get
1 PQPQ2 2 2 2 1 QQ12 0 1PQPQ 1 1 1 1
2 PQPQ2 2 2 2 or 1 1QQ 2 1 0 PQPQ1 1 1 1
83
2 QPQQPQPQ1()() 1 1 1 2 2 2 2 or 11 0 P1 QP 11 Q 1
2 or 13 13PQ 0
PQQPQQPQ2 2 1()() 2 2 2 1 1 where QP33and PQPQ1 1 1 1
1 PPQ 2 4 , taking positive root only. (3.22) 1 2 333
Substituting in Eq. (V)
1 2.Q3 2 .Q3 (3.23) 2 1 PPQ3 34 3
Thus, 1 and 2 are known from Eq. 3.22 and 3.23.
3.2.2 Second Transformation
Refer Figs. 3.1 (b) and 3.1 (c).
The Schwarz Christoffel equation for transformation from w-plane to t-plane is
dt w MN22 (3.24) 1 6 7 8 ()()()()t t1 t t 6 t t 7 t t 8
1 Here 1 6 7 8 2
Here t1 1;tt 6 2 ; 7 1; 8 1
dt MN22 (t 12 )( t )( t 1)( t 1)
dt or MN22 (3.24 a) (t 1)( t 21 )( t ) ( t 1)
84
Using Identity 254, Byrd and Friedman (1971) and noting that
abc1,, 21 and d = – 1 for the present purpose,
we have
()()b d t c ( 1)(t ) sn2u 21 (b c )( t dt ) ()(21 1)
2 ()()b c a d 2()12 m0 (a c )( b d ) (1 12 )(1 )
()()b d t c ( 1)(t ) and sin11 sin 21 (b c )( t d ) ( 21 )( t 1)
Differentiating,
11 2 sn u cn u dn u du 21dt 2 12 (1 t )
Substituting the values of sn u, cn u, dn u in the above and noting that cn
2 22 u 1 sn u and dnu 1 m0 sn u , we get
( )(1) t2 ( 1)( t ) ( 1)( t ) dt 21du 1 2 2 1 2 1 (2 1)(1 1 ) ( 1 2 )(1)tt ( 1 2 )(1)
2( ) (1 )(t ) 1 1 2 2 1 (1 1 )(1 2 )( 1 2 )(t 1)
( )(1 t )2 (1 )( t ) t t t t or 2du 12 2112122121 (12 )(1 1 ) ( 1 2 )(tt 1) ( 1 2 )( 1)
t t 1 2 t 2 1 1 1 (1 1 )(t 1)
85
( )(tttt 1)2 (1 )( ) (1 )( ) (1 )(1 ) or dt 2du 1 22 1 1 2 1 (12 )(1 1 ) ( 1 2 )(ttt 1) ( 1 21 )( 1) (1 )( 1)
(1t )( tt )( t )(1 ) or 2du 12 (1 21 )(1 )
dt 2du or (1ttt )( 12 t )()(1 ) (1)(1) 12
dtdu 2 or (t 1)( t 2 )( t 11 )( t 1) 2 (1 )(1 )
Substituting in Eq. 3.24 (a) we get
2M u w 2 1 du N 0 2 (1 12 )(1 )
wN 2 2 or u1 M 2 (1 12 )(1 )
2 1 or sn (sin ,m0 ) (1 12 )(1 )
2 or Fm00(,) (3.25) (1 12 )(1 )
(1 )(t ) where sin1 21 (3.25 a) (12 )(t 1)
Boundary Conditions
(i) Point 1: At point 1, w kH and t 1
sin1 (0) 0
Fm0 (,) Fm(0, ) 0
Hence from (ii) wN2 0
86
or N2 wkH (3.26)
(ii) Point 6: At point 6, w 0 and t 2
(1 )( ) sinsin11 (1) 2 2 1 (1 2 )( 2 1)2
Fm0 ( ,) FmK000, 2
2()12 where modulus m0 (1 12 )(1)
Hence from Eq. 3.25
0 kH 2 K0 M 2 (1 12 )(1 )
kH M 2 (1 12 )(1 ) (3.27) 2K0
Hence the transformation equation becomes
kH w [(,)]F 0 m 0 K 0 (3.28) K0
Seepage Discharge
At point 7, w iq and t 1
11(1 21 )(1 ) 1 7 sin sin (1 2 )(1 1) m 0
1 1 Fm0(,) 7 0 Fsin , m0 K 0 iK 0 m0
Hence from Eq. 3.28
87
kH iq []K000 iK K K0
K q kH 0 K0
q K or 0 (3.29) kH K0
Pressure Distribution
For the base of the apron, wkh x
kH w khx [(,)] F0 m 0 K 0 K0
h Let P x 100 % pressure at any point below the apron. x H
100 [(,)]F0 m 0 K 0 K0
Fm00(,) or 100 1 (3.30) K0
(i) At point 6 (Point R): t 2
11(1 2 )( 1 2 ) R sin sin 1 (1 2 )( 2 1) 2
Fm00(,)R F0, m 0 K 0 2
K0 Hence PR 100 1 0, as expected. K0
88
(ii) At point 4 (point E): t
1 (21 1)() E sin (3.31 a) ()(112 )
Fm00(,)E Hence PE 100 1 (3.31) K0
(iii) At point 3 (point E1): t
1 (1 21 )() E1 sin (3.32 a) ()(112 )
Fm0(,)E 1 0 and PE1 100 1 (3.32) K0
(iv) Point 1 (point G): t 1
G 0;Fm00 ( G , ) 0
PG 100% , as expected
(v) Point 5 (point D): t 2
1 (1 2 )( 2 1 ) D sin (3.33 a) (1 2 )( 2 1)
Fm00(,)D PD 100 1 (3.33) K0
(vi) Point 2 (point D1): t 1
1 (1 2 )( 1 1 ) D1 sin (3.34 a) (1 2 )(1 1 )
89
Fm0(,)D 1 0 PD1 100 1 (3.34) K0
Exit Gradient
The Schwarz-Christoffel equation for transformation from z-plane to t-plane is
dz ()()t t25 t t M1 dt ()()()()()()tttt7 8 tttttttt 1 3 4 6 where t1 1,,;ttt 2 1 34
t5 2;;1;1ttt 6 2 78
()()tt12 M1 (1)(1)(t t t 12 )( t )( t )( t )
()()tt12 or M1 2 2 2 (t 1) ( t )( t 12 )( t )
BB where M 1 22 A1 K B 1 0(,)(,) 1 m C 1 0 2 m R
22 and R A1 K B 1 0(,)(,) 1 m C 1 0 2 m
B ()()tt Hence . 12 R 2 2 2 (t 1) ( t )( t 12 )( t )
Similarly, the transformation equation between w and t plane is
dw M 2 dt 2 (t 1)( t 12 )( t )
kH where M 2 (1 12 )(1 ) 2K0
kH 1 (1 12 )(1 ) 2K 2 0 (t 1)( t 12 )( t )
90
If u and v are the velocity components in x and y directions, we have
dw dw dt Complex velocity u iv . dz dt dz
At the exit end, u = 0 and v = vE
dw dt iv . E dt dz
Now, from Darcy law, vE = k GE
v 1 dw dt G E .. (3.35) E kik dt dz
dw dt Substituting the values of and , we get dt dz
1 kH(1 )(1 ) R (t2 1) ( t 2 2 )( t )( t ) G 12 12 E 2 ik2( KB01( )(t 21)( t) t 12 )( t t )
GB Rt(1t2 ) (1 )(1 ) ()22 or E 12 2 H 2K0 ( t 1 )( t 2 ) 1t
Gc BTcc Now E 2 G . 22 where H E HBBR
Gc R Tt (1t2 ) (1 )(1 ) ()22 E 2 12 2 H 2K0 R ( t 1 )( t 2 ) 1t
c Tt(1t2 ) (1 )(1 ) ()22 or G 2 . 12 E 2 H 2K0 ( t 1 )( t 2 ) 1t
At the exit end, t 2
c T (1 2 ) (1 )(1 ) ()22 G 2 . 2 1 2 2 (3.36 a) E 2 H 2k0 ( 2 1 )( 2 2 ) 12
91
2 2 2 c2 T (1 1 )(1 2 )(1 2 )( 2 ) or GE . (3.36) H 2K0 ( 2 1 )( 2 2 )
3.3 Computations and Results
Eqs. 3.14, 3.15 and 3.16 for floor profile ratios B/D, B/c2 and c1/c2 are not explicit in transformation parameters , 1 and 2. Hence direct solution of these equations for the parameters , 1 and 2 corresponding to given set of values of floor profile ratios (B/D, B/c2 and c1/c2) are extremely difficult. However, these equations can be solved for physical floor profile ratios B/D, B/c2 and c1/c2 for some assumed values of , 1 and 2 and for some selected values of
step ratio (/)ac2 . Hence B/D, B/c2 and c1/c2 were first computed for values of 01 12 corresponding to pre-selected values of step ratios a/c2=0.1, 0.2, 0.3 and 0.4. The parameters 1 and 2 were computed from Eqs. 3.22 and 3.23, for each set of values of , 1 and 2 and step ratio a/c2. From these values of floor profile ratios, iterative procedures were developed to compute the values of , 1 and 2 (and then of 1 and 2) for chosen pair of values of B/D, B/c2 and c1/c2 ratios, corresponding to each pre-selected values of step ratio (a/c2). The corresponding values of various seepage characteristics, such as PE, PD, PE1, PD1, q/kH and GE c2/H were computed from Eqs. 3.31, 3.33, 3.32, 3.34, 3.29 and 3.36 respectively. Table 3.1 gives the resulting values.
Comparison with Infinite Depth case (when c1 = c2 = c and a/c2 = 0)
Malhotra (1936) obtained the theoretical solution for uplift pressure below flat apron with equal end cutoffs (c1 = c2 = c) for the case of infinite depth of pervious foundation. The results of the present theoretical solution for B/D = 0.4 and B/D =0.2 were compared with those of
Malhotra's solution. Table 3.2 gives the values of uplift pressure PE at point E (Fig. 3.1 a) and PD
92 at point D, for selected values of B/c ratios. It is seen from Table 3.2 that when BD/0.2 , the results of both the analyses are practically the same.
Table 3.2. Comparison with Malhotra's case of Infinite Depth.
% Pressure PE at E % Pressure PD at point D
Infinite Present Present Infinite Present Present B/c ratio Depth case analysis for analysis for Depth case analysis for analysis for by Malhotra B B by Malhotra B B 0.2 0.4 0.2 0.4 D D D D
3 41.4 41.3 41.1 28.4 28.2 28.0
4 37.9 37.8 37.7 26.1 26.0 25.7
6 32.9 32.7 32.6 22.7 22.6 22.4
8 29.2 29.2 29.0 20.3 20.3 20.1
12 24.6 24.5 24.4 17.2 17.1 17.0
24 17.8 17.8 17.7 12.6 12.5 12.4
Effect of step on downstream side
Fig. 3.2 (a) shows the effect of increase in step ratio (a/c2) on PE, the uplift pressure at point E when B/c2 = 10 and c1/c2=0.4. Fig. 3.2 (b) shows the effect of increase in step ratio on
PD. Similarly, Figs. 3.3 (a), 3.3 (b), 3.4 (a) and Fig. 3.4 (b) show the variation of PE1, PD1, q/kH and GE c2/H, respectively, with increase in step ratio (a/c2) at the downstream side, for B/c2=10 and c1/c2=0.4, for different values of B/D ratios.
93
Table 3.1 Computed values for B/D = 0.2, a/c2 = 0.1
B/D B/c2 c1/c2 a/c2 σ β1 β2 γ1 γ2 PE PD PE1 PD1 q/kH GEc2/H 0.2 5 0.2 0.1 0.1215 0.1335 0.1818 0.1456 0.2370 37.41 25.16 83.85 88.64 0.9641 0.1919 0.2 10 0.2 0.1 0.1376 0.1437 0.1677 0.1498 0.1956 27.11 18.53 87.92 91.48 0.9977 0.1431 0.2 15 0.2 0.1 0.1433 0.1474 0.1634 0.1515 0.1821 22.30 15.33 89.96 92.91 1.0090 0.1190 0.2 20 0.2 0.1 0.1463 0.1494 0.1614 0.1524 0.1754 19.39 13.36 91.23 93.81 1.0147 0.1039 0.2 5 0.4 0.1 0.1158 0.1400 0.1761 0.1640 0.2314 36.67 24.70 77.28 84.10 0.9537 0.1886 0.2 10 0.4 0.1 0.1346 0.1468 0.1647 0.1590 0.1927 26.83 18.35 82.96 88.03 0.9920 0.1418 0.2 15 0.4 0.1 0.1413 0.1495 0.1614 0.1577 0.1801 22.15 15.23 85.83 90.02 1.0051 0.1182 0.2 20 0.4 0.1 0.1448 0.1509 0.1599 0.1571 0.1739 19.29 13.30 87.61 91.27 1.0116 0.1034 0.2 5 0.6 0.1 0.1105 0.1467 0.1706 0.1827 0.2258 35.92 24.24 72.33 80.74 0.9434 0.1853 0.2 10 0.6 0.1 0.1317 0.1500 0.1618 0.1683 0.1898 26.56 18.17 79.20 85.42 0.9862 0.1404 0.2 15 0.6 0.1 0.1394 0.1516 0.1595 0.1638 0.1781 22.00 15.13 82.68 87.83 1.0011 0.1174 0.2 20 0.6 0.1 0.1433 0.1525 0.1584 0.1617 0.1724 19.19 13.23 84.85 89.34 1.0086 0.1029 0.2 5 0.8 0.1 0.1054 0.1536 0.1652 0.2014 0.2203 35.17 23.77 68.23 78.01 0.9331 0.1819 0.2 10 0.8 0.1 0.1289 0.1533 0.1590 0.1776 0.1869 26.28 17.99 76.05 83.27 0.9804 0.1391 0.2 15 0.8 0.1 0.1374 0.1538 0.1575 0.1700 0.1762 21.85 15.03 80.04 86.01 0.9971 0.1167 0.2 20 0.8 0.1 0.1418 0.1541 0.1569 0.1663 0.1709 19.09 13.16 82.54 87.73 1.0056 0.1024
94
Table 3.1 (contd.) Computed values for B/D = 0.5; a/c2 = 0.1
B/D B/c2 c1/c2 a/c2 σ β1 β2 γ1 γ2 PE PD PE1 PD1 q/kH GEc2/H 0.5 5 0.2 0.1 0.2976 0.3254 0.4319 0.3528 0.5424 37.24 24.92 84.07 88.81 0.6757 0.1890 0.5 10 0.2 0.1 0.3335 0.3473 0.4004 0.3610 0.4593 26.91 18.35 88.06 91.59 0.7096 0.1413 0.5 15 0.2 0.1 0.3463 0.3555 0.3908 0.3646 0.4308 22.12 15.18 90.07 93.00 0.7208 0.1176 0.5 20 0.2 0.1 0.3529 0.3597 0.3862 0.3665 0.4164 19.22 13.23 91.32 93.88 0.7265 0.1028 0.5 5 0.4 0.1 0.2846 0.3402 0.4200 0.3937 0.5319 36.49 24.45 77.55 84.34 0.6656 0.1855 0.5 10 0.4 0.1 0.3268 0.3544 0.3940 0.3815 0.4533 26.63 18.16 83.15 88.17 0.7039 0.1399 0.5 15 0.4 0.1 0.3418 0.3601 0.3865 0.3782 0.4266 21.96 15.07 85.98 90.14 0.7170 0.1168 0.5 20 0.4 0.1 0.3495 0.3632 0.3829 0.3768 0.4133 19.11 13.16 87.75 91.37 0.7235 0.1022 0.5 5 0.6 0.1 0.2722 0.3555 0.4083 0.4340 0.5215 35.72 23.96 72.62 81.02 0.6554 0.1821 0.5 10 0.6 0.1 0.3202 0.3617 0.3877 0.4017 0.4473 26.35 17.98 79.41 85.60 0.6982 0.1385 0.5 15 0.6 0.1 0.3374 0.3649 0.3822 0.3917 0.4225 21.81 14.97 82.85 87.97 0.7131 0.1160 0.5 20 0.6 0.1 0.3462 0.3667 0.3797 0.3869 0.4101 19.01 13.09 85.01 89.46 0.7205 0.1017 0.5 5 0.8 0.1 0.2604 0.3714 0.3970 0.4733 0.5113 34.93 23.48 68.51 78.31 0.6452 0.1785 0.5 10 0.8 0.1 0.3138 0.3690 0.3816 0.4218 0.4415 26.06 17.79 76.28 83.47 0.6926 0.1371 0.5 15 0.8 0.1 0.3330 0.3697 0.3779 0.4052 0.4184 21.65 14.86 80.23 86.16 0.7091 0.1152 0.5 20 0.8 0.1 0.3429 0.3703 0.3765 0.3970 0.4070 18.91 13.02 82.71 87.87 0.7175 0.1012
95
Table 3.1 (contd.) Computed values for B/D = 1; a/c2 = 0.1
B/D B/c2 c1/c2 a/c2 σ β1 β2 γ1 γ2 PE PD PE1 PD1 q/kH GEc2/H 1.0 5 0.2 0.1 0.5522 0.5940 0.7341 0.6329 0.8435 36.76 24.20 84.75 89.33 0.4674 0.1803 1.0 10 0.2 0.1 0.6020 0.6216 0.6920 0.6405 0.7604 26.30 17.79 88.49 91.91 0.5011 0.1359 1.0 15 0.2 0.1 0.6192 0.6319 0.6789 0.6443 0.7276 21.55 14.71 90.41 93.25 0.5122 0.1133 1.0 20 0.2 0.1 0.6279 0.6373 0.6725 0.6465 0.7102 18.70 12.82 91.62 94.09 0.5177 0.0992 1.0 5 0.4 0.1 0.5325 0.6159 0.7207 0.6876 0.8351 35.94 23.68 78.40 85.04 0.4579 0.1765 1.0 10 0.4 0.1 0.5925 0.6316 0.6841 0.6679 0.7540 26.00 17.59 83.72 88.62 0.4958 0.1344 1.0 15 0.4 0.1 0.6132 0.6386 0.6736 0.6627 0.7229 21.38 14.59 86.45 90.49 0.5084 0.1124 1.0 20 0.4 0.1 0.6235 0.6424 0.6685 0.6604 0.7066 18.58 12.74 88.15 91.67 0.5148 0.0986 1.0 5 0.6 0.1 0.5137 0.6383 0.7077 0.7368 0.8267 35.10 23.15 73.52 81.85 0.4483 0.1726 1.0 10 0.6 0.1 0.5831 0.6418 0.6764 0.6939 0.7477 25.70 17.38 80.06 86.13 0.4904 0.1328 1.0 15 0.6 0.1 0.6070 0.6451 0.6681 0.6803 0.7182 21.22 14.48 83.40 88.39 0.5048 0.1116 1.0 20 0.6 0.1 0.6190 0.6472 0.6644 0.6738 0.7029 18.47 12.66 85.49 89.82 0.5119 0.0980 1.0 5 0.8 0.1 0.4958 0.6611 0.6949 0.7807 0.8185 34.24 22.60 69.40 79.24 0.4385 0.1686 1.0 10 0.8 0.1 0.5739 0.6521 0.6687 0.7185 0.7415 25.39 17.18 76.98 84.07 0.4850 0.1313 1.0 15 0.8 0.1 0.6008 0.6517 0.6627 0.6973 0.7134 21.05 14.37 80.83 86.65 0.5011 0.1107 1.0 20 0.8 0.1 0.6144 0.6520 0.6603 0.6867 0.6991 18.36 12.59 83.24 88.28 0.5091 0.0974
96
Table 3.1 (contd.) Computed values for B/D = 1.5; a/c2 = 0.1
B/D B/c2 c1/c2 a/c2 σ β1 β2 γ1 γ2 PE PD PE1 PD1 q/kH GEc2/H 1.5 5 0.2 0.1 0.7381 0.7780 0.8921 0.8125 0.9552 36.24 23.35 85.63 90.00 0.3564 0.1699 1.5 10 0.2 0.1 0.7811 0.7988 0.8573 0.8152 0.9057 25.55 17.09 89.05 92.32 0.3897 0.1291 1.5 15 0.2 0.1 0.7963 0.8075 0.8466 0.8182 0.8830 20.82 14.10 90.86 93.57 0.4001 0.1079 1.5 20 0.2 0.1 0.8050 0.8131 0.8423 0.8210 0.8711 17.98 12.26 92.02 94.38 0.4046 0.0943 1.5 5 0.4 0.1 0.7191 0.7985 0.8834 0.8574 0.9514 35.34 22.77 79.50 85.94 0.3479 0.1658 1.5 10 0.4 0.1 0.7727 0.8080 0.8516 0.8383 0.9018 25.22 16.87 84.46 89.18 0.3848 0.1275 1.5 15 0.4 0.1 0.7905 0.8129 0.8421 0.8332 0.8795 20.65 13.99 87.04 90.93 0.3970 0.1070 1.5 20 0.4 0.1 0.7995 0.8159 0.8378 0.8311 0.8674 17.90 12.20 88.67 92.05 0.4029 0.0939 1.5 5 0.6 0.1 0.7009 0.8191 0.8750 0.8937 0.9477 34.39 22.17 74.67 82.89 0.3390 0.1615 1.5 10 0.6 0.1 0.7642 0.8170 0.8457 0.8590 0.8978 24.88 16.65 80.90 86.80 0.3798 0.1258 1.5 15 0.6 0.1 0.7852 0.8188 0.8379 0.8476 0.8763 20.46 13.86 84.10 88.93 0.3935 0.1060 1.5 20 0.6 0.1 0.7955 0.8200 0.8344 0.8419 0.8646 17.78 12.13 86.10 90.28 0.4003 0.0933 1.5 5 0.8 0.1 0.6836 0.8397 0.8668 0.9225 0.9442 33.41 21.54 70.50 80.36 0.3297 0.1570 1.5 10 0.8 0.1 0.7558 0.8261 0.8400 0.8776 0.8938 24.54 16.42 77.87 84.83 0.3748 0.1241 1.5 15 0.8 0.1 0.7798 0.8245 0.8338 0.8609 0.8730 20.28 13.74 81.60 87.26 0.3900 0.1051 1.5 20 0.8 0.1 0.7916 0.8243 0.8312 0.8523 0.8619 17.66 12.04 83.92 88.81 0.3977 0.0927
97
Table 3.1 (contd.) Computed values for B/D = 2.0; a/c2 = 0.1
B/D B/c2 c1/c2 a/c2 σ β1 β2 γ1 γ2 PE PD PE1 PD1 q/kH GEc2/H 2.0 5 0.2 0.1 0.8566 0.8867 0.9604 0.9108 0.9886 36.16 22.90 86.60 90.72 0.2846 0.1593 2.0 10 0.2 0.1 0.8870 0.8997 0.9383 0.9110 0.9654 24.73 16.33 89.63 92.74 0.3181 0.1222 2.0 15 0.2 0.1 0.8986 0.9063 0.9319 0.9135 0.9534 19.95 13.39 91.35 93.93 0.3272 0.1020 2.0 20 0.2 0.1 0.9039 0.9095 0.9287 0.9148 0.9462 17.19 11.64 92.43 94.68 0.3318 0.0893 2.0 5 0.4 0.1 0.8422 0.9019 0.9562 0.9398 0.9874 35.33 22.45 80.75 86.93 0.2764 0.1547 2.0 10 0.4 0.1 0.8800 0.9054 0.9344 0.9257 0.9631 24.43 16.13 85.20 89.75 0.3141 0.1206 2.0 15 0.4 0.1 0.8933 0.9090 0.9283 0.9225 0.9509 19.83 13.31 87.67 91.40 0.3252 0.1013 2.0 20 0.4 0.1 0.9005 0.9118 0.9261 0.9218 0.9443 17.09 11.57 89.24 92.47 0.3299 0.0888 2.0 5 0.6 0.1 0.8284 0.9168 0.9521 0.9607 0.9862 34.43 21.98 76.04 84.06 0.2676 0.1498 2.0 10 0.6 0.1 0.8737 0.9118 0.9308 0.9388 0.9611 24.09 15.92 81.73 87.48 0.3095 0.1189 2.0 15 0.6 0.1 0.8888 0.9125 0.9253 0.9313 0.9488 19.66 13.20 84.80 89.48 0.3224 0.1005 2.0 20 0.6 0.1 0.8968 0.9138 0.9233 0.9281 0.9421 17.00 11.51 86.75 90.77 0.3283 0.0884 2.0 5 0.8 0.1 0.8156 0.9313 0.9482 0.9754 0.9850 33.41 21.42 71.86 81.64 0.2584 0.1446 2.0 10 0.8 0.1 0.8677 0.9182 0.9274 0.9500 0.9592 23.75 15.70 78.76 85.58 0.3047 0.1170 2.0 15 0.8 0.1 0.8847 0.9163 0.9225 0.9395 0.9468 19.47 13.07 82.36 87.87 0.3194 0.0995 2.0 20 0.8 0.1 0.8934 0.9162 0.9208 0.9342 0.9402 16.90 11.44 84.62 89.35 0.3263 0.0878
98
Table 3.1 (contd.) Computed values for B/D = 0.2; a/c2 = 0.2
B/D B/c2 c1/c2 a/c2 σ β1 β2 γ1 γ2 PE PD PE1 PD1 q/kH GEc2/H 0.2 5 0.2 0.2 0.1207 0.1327 0.1799 0.1447 0.2303 36.71 24.22 83.70 88.54 0.9706 0.2077 0.2 10 0.2 0.2 0.1369 0.1430 0.1664 0.1491 0.1919 26.51 17.77 87.85 91.43 1.0019 0.1542 0.2 15 0.2 0.2 0.1428 0.1469 0.1625 0.1509 0.1794 21.78 14.68 89.91 92.88 1.0122 0.1279 0.2 20 0.2 0.2 0.1458 0.1489 0.1606 0.1520 0.1733 18.92 12.79 91.20 93.79 1.0172 0.1117 0.2 5 0.4 0.2 0.1151 0.1391 0.1742 0.1631 0.2247 35.98 23.77 77.08 83.96 0.9601 0.2040 0.2 10 0.4 0.2 0.1339 0.1461 0.1635 0.1583 0.1889 26.24 17.60 82.86 87.96 0.9961 0.1527 0.2 15 0.4 0.2 0.1408 0.1489 0.1605 0.1571 0.1775 21.63 14.58 85.77 89.98 1.0082 0.1271 0.2 20 0.4 0.2 0.1443 0.1505 0.1591 0.1566 0.1718 18.82 12.72 87.57 91.24 1.0141 0.1111 0.2 5 0.6 0.2 0.1097 0.1458 0.1687 0.1817 0.2191 35.23 23.32 72.08 80.58 0.9496 0.2004 0.2 10 0.6 0.2 0.1310 0.1493 0.1606 0.1676 0.1860 25.97 17.42 79.08 85.34 0.9903 0.1513 0.2 15 0.6 0.2 0.1388 0.1511 0.1585 0.1632 0.1755 21.48 14.49 82.60 87.78 1.0042 0.1263 0.2 20 0.6 0.2 0.1428 0.1520 0.1576 0.1612 0.1704 18.72 12.66 84.80 89.31 1.0111 0.1106 0.2 5 0.8 0.2 0.1047 0.1527 0.1634 0.2004 0.2137 34.48 22.86 67.95 77.82 0.9391 0.1967 0.2 10 0.8 0.2 0.1282 0.1526 0.1578 0.1769 0.1832 25.70 17.25 75.91 83.18 0.9844 0.1498 0.2 15 0.8 0.2 0.1369 0.1532 0.1566 0.1694 0.1736 21.34 14.39 79.95 85.95 1.0001 0.1254 0.2 20 0.8 0.2 0.1414 0.1536 0.1561 0.1658 0.1689 18.63 12.59 82.48 87.69 1.0080 0.1100
99
Table 3.1 (contd.) Computed values for B/D = 0.5; a/c2 = 0.2
B/D B/c2 c1/c2 a/c2 σ β1 β2 γ1 γ2 PE PD PE1 PD1 q/kH GEc2/H 0.5 5 0.2 0.2 0.2963 0.3241 0.4288 0.3515 0.5307 36.55 23.99 83.92 88.70 0.6814 0.2047 0.5 10 0.2 0.2 0.3323 0.3461 0.3981 0.3598 0.4520 26.32 17.60 87.99 91.54 0.7133 0.1523 0.5 15 0.2 0.2 0.3453 0.3545 0.3890 0.3636 0.4256 21.60 14.54 90.03 92.96 0.7237 0.1265 0.5 20 0.2 0.2 0.3520 0.3588 0.3847 0.3656 0.4123 18.75 12.66 91.29 93.85 0.7288 0.1104 0.5 5 0.4 0.2 0.2833 0.3389 0.4168 0.3924 0.5200 35.80 23.53 77.35 84.19 0.6711 0.2009 0.5 10 0.4 0.2 0.3256 0.3532 0.3917 0.3803 0.4460 26.05 17.42 83.05 88.10 0.7076 0.1508 0.5 15 0.4 0.2 0.3408 0.3591 0.3847 0.3772 0.4214 21.45 14.44 85.92 90.09 0.7197 0.1256 0.5 20 0.4 0.2 0.3486 0.3623 0.3814 0.3759 0.4091 18.65 12.59 87.70 91.34 0.7257 0.1099 0.5 5 0.6 0.2 0.2709 0.3542 0.4052 0.4326 0.5095 35.04 23.06 72.37 80.85 0.6607 0.1971 0.5 10 0.6 0.2 0.3191 0.3605 0.3854 0.4006 0.4400 25.77 17.24 79.29 85.52 0.7018 0.1493 0.5 15 0.6 0.2 0.3364 0.3638 0.3804 0.3907 0.4172 21.30 14.34 82.78 87.92 0.7158 0.1248 0.5 20 0.6 0.2 0.3453 0.3658 0.3782 0.3860 0.4060 18.55 12.53 84.96 89.42 0.7227 0.1093 0.5 5 0.8 0.2 0.2592 0.3700 0.3939 0.4719 0.4991 34.26 22.58 68.24 78.12 0.6503 0.1932 0.5 10 0.8 0.2 0.3127 0.3679 0.3793 0.4206 0.4341 25.49 17.06 76.14 83.38 0.6961 0.1478 0.5 15 0.8 0.2 0.3320 0.3686 0.3761 0.4042 0.4131 21.15 14.23 80.15 86.10 0.7118 0.1239 0.5 20 0.8 0.2 0.3420 0.3694 0.3749 0.3961 0.4028 18.45 12.46 82.65 87.83 0.7198 0.1087
100
Table 3.1 (contd.) Computed values for B/D = 1.0; a/c2 = 0.2
B/D B/c2 c1/c2 a/c2 σ β1 β2 γ1 γ2 PE PD PE1 PD1 q/kH GEc2/H 1.0 5 0.2 0.2 0.5513 0.5931 0.7317 0.6320 0.8348 36.09 23.31 84.60 89.23 0.4718 0.1957 1.0 10 0.2 0.2 0.6009 0.6206 0.6898 0.6395 0.7532 25.73 17.07 88.42 91.86 0.5041 0.1466 1.0 15 0.2 0.2 0.6182 0.6310 0.6770 0.6434 0.7219 21.05 14.09 90.37 93.22 0.5145 0.1220 1.0 20 0.2 0.2 0.6270 0.6364 0.6709 0.6456 0.7055 18.25 12.27 91.59 94.07 0.5196 0.1066 1.0 5 0.4 0.2 0.5315 0.6150 0.7182 0.6868 0.8259 35.28 22.81 78.20 84.90 0.4622 0.1915 1.0 10 0.4 0.2 0.5913 0.6306 0.6819 0.6669 0.7467 25.44 16.87 83.62 88.55 0.4987 0.1450 1.0 15 0.4 0.2 0.6121 0.6376 0.6717 0.6618 0.7172 20.89 13.98 86.38 90.44 0.5107 0.1210 1.0 20 0.4 0.2 0.6226 0.6415 0.6669 0.6596 0.7019 18.14 12.19 88.10 91.64 0.5167 0.1060 1.0 5 0.6 0.2 0.5127 0.6374 0.7051 0.7361 0.8171 34.44 22.29 73.28 81.68 0.4523 0.1873 1.0 10 0.6 0.2 0.5819 0.6408 0.6741 0.6930 0.7402 25.14 16.68 79.94 86.05 0.4933 0.1433 1.0 15 0.6 0.2 0.6059 0.6442 0.6662 0.6794 0.7124 20.72 13.87 83.32 88.34 0.5070 0.1201 1.0 20 0.6 0.2 0.6181 0.6463 0.6628 0.6729 0.6982 18.03 12.12 85.43 89.79 0.5138 0.1054 1.0 5 0.8 0.2 0.4947 0.6603 0.6922 0.7801 0.8085 33.59 21.76 69.13 79.05 0.4423 0.1829 1.0 10 0.8 0.2 0.5727 0.6511 0.6664 0.7177 0.7338 24.83 16.48 76.84 83.98 0.4879 0.1416 1.0 15 0.8 0.2 0.5998 0.6507 0.6608 0.6964 0.7076 20.56 13.76 80.75 86.59 0.5033 0.1192 1.0 20 0.8 0.2 0.6135 0.6512 0.6586 0.6859 0.6943 17.92 12.05 83.18 88.24 0.5110 0.1047
101
Table 3.1 (contd.) Computed values for B/D = 1.5; a/c2 = 0.2
B/D B/c2 c1/c2 a/c2 σ β1 β2 γ1 γ2 PE PD PE1 PD1 q/kH GEc2/H 1.5 5 0.2 0.2 0.7376 0.7776 0.8910 0.8121 0.9514 35.59 22.50 85.49 89.90 0.3598 0.1849 1.5 10 0.2 0.2 0.7805 0.7982 0.8560 0.8147 0.9014 25.00 16.40 88.98 92.27 0.3922 0.1395 1.5 15 0.2 0.2 0.7957 0.8069 0.8454 0.8176 0.8793 20.34 13.51 90.82 93.54 0.4021 0.1162 1.5 20 0.2 0.2 0.8044 0.8125 0.8412 0.8204 0.8679 17.55 11.73 91.98 94.35 0.4062 0.1015 1.5 5 0.4 0.2 0.7186 0.7981 0.8823 0.8571 0.9474 34.70 21.94 79.31 85.81 0.3512 0.1804 1.5 10 0.4 0.2 0.7720 0.8074 0.8502 0.8378 0.8973 24.67 16.19 84.36 89.11 0.3872 0.1377 1.5 15 0.4 0.2 0.7899 0.8123 0.8408 0.8327 0.8757 20.17 13.40 86.98 90.89 0.3988 0.1153 1.5 20 0.4 0.2 0.7989 0.8153 0.8367 0.8305 0.8641 17.47 11.68 88.62 92.02 0.4045 0.1011 1.5 5 0.6 0.2 0.7003 0.8188 0.8737 0.8935 0.9434 33.76 21.36 74.44 82.73 0.3421 0.1757 1.5 10 0.6 0.2 0.7635 0.8165 0.8443 0.8586 0.8931 24.34 15.98 80.78 86.72 0.3822 0.1359 1.5 15 0.6 0.2 0.7845 0.8182 0.8367 0.8471 0.8723 19.99 13.28 84.02 88.88 0.3953 0.1143 1.5 20 0.6 0.2 0.7949 0.8195 0.8333 0.8414 0.8612 17.36 11.61 86.04 90.25 0.4019 0.1004 1.5 5 0.8 0.2 0.6831 0.8394 0.8655 0.9223 0.9395 32.78 20.74 70.24 80.19 0.3326 0.1707 1.5 10 0.8 0.2 0.7551 0.8256 0.8385 0.8772 0.8890 24.01 15.76 77.73 84.74 0.3770 0.1341 1.5 15 0.8 0.2 0.7791 0.8240 0.8325 0.8604 0.8690 19.81 13.16 81.51 87.20 0.3918 0.1132 1.5 20 0.8 0.2 0.7909 0.8237 0.8300 0.8518 0.8585 17.24 11.53 83.86 88.77 0.3992 0.0997
102
Table 3.1 (contd.) Computed values for B/D = 2.0; a/c2 = 0.2
B/D B/c2 c1/c2 a/c2 σ β1 β2 γ1 γ2 PE PD PE1 PD1 q/kH GEc2/H 2.0 5 0.2 0.2 0.8564 0.8865 0.9600 0.9106 0.9874 35.49 22.04 86.46 90.62 0.2875 0.1740 2.0 15 0.2 0.2 0.8982 0.9060 0.9313 0.9132 0.9514 19.49 12.84 91.31 93.90 0.3288 0.1100 2.0 20 0.2 0.2 0.9036 0.9092 0.9281 0.9145 0.9444 16.78 11.14 92.40 94.65 0.3331 0.0962 2.0 10 0.2 0.2 0.8867 0.8994 0.9377 0.9108 0.9633 24.19 15.67 89.56 92.70 0.3202 0.1322 2.0 10 0.4 0.2 0.8796 0.9052 0.9337 0.9255 0.9609 23.90 15.48 85.10 89.68 0.3161 0.1305 2.0 5 0.4 0.2 0.8420 0.9018 0.9557 0.9397 0.9860 34.66 21.61 80.55 86.80 0.2791 0.1689 2.0 15 0.4 0.2 0.8929 0.9087 0.9277 0.9223 0.9488 19.37 12.75 87.61 91.36 0.3267 0.1093 2.0 20 0.4 0.2 0.9002 0.9114 0.9255 0.9215 0.9424 16.68 11.08 89.20 92.44 0.3312 0.0956 2.0 5 0.6 0.2 0.8282 0.9167 0.9516 0.9606 0.9847 33.78 21.16 75.81 83.90 0.2701 0.1634 2.0 10 0.6 0.2 0.8734 0.9115 0.9301 0.9386 0.9588 23.56 15.27 81.62 87.40 0.3115 0.1286 2.0 15 0.6 0.2 0.8884 0.9122 0.9246 0.9311 0.9466 19.20 12.64 84.73 89.43 0.3240 0.1083 2.0 20 0.6 0.2 0.8964 0.9135 0.9227 0.9278 0.9402 16.60 11.02 86.69 90.74 0.3296 0.0952 2.0 5 0.8 0.2 0.8154 0.9312 0.9477 0.9754 0.9834 32.80 20.65 71.61 81.47 0.2607 0.1577 2.0 10 0.8 0.2 0.8673 0.9180 0.9267 0.9498 0.9567 23.22 15.06 78.62 85.49 0.3066 0.1266 2.0 15 0.8 0.2 0.8844 0.9160 0.9218 0.9393 0.9446 19.02 12.52 82.28 87.81 0.3209 0.1073 2.0 20 0.8 0.2 0.8931 0.9159 0.9201 0.9340 0.9382 16.49 10.95 84.56 89.31 0.3275 0.0946
103
Table 3.1 (contd.) Computed values for B/c2 = 10; a/c2 = 0.3 and 0.4 and c1/c2 = 0.4
B/D B/c2 c1/c2 a/c2 σ Β1 β2 γ1 γ2 PE PD PE1 PD1 q/kH GEc2/H 0.2 10 0.4 0.3 0.1332 0.1454 0.1622 0.1576 0.1851 25.61 16.78 82.76 87.89 1.0004 0.1662 0.5 10 0.4 0.3 0.3244 0.3520 0.3893 0.3790 0.4384 25.42 16.61 82.94 88.03 0.7114 0.1642 1.0 10 0.4 0.3 0.5901 0.6295 0.6795 0.6658 0.7389 24.83 16.10 83.52 88.47 0.5019 0.1580 1.5 10 0.4 0.3 0.7712 0.8068 0.8487 0.8373 0.8924 24.08 15.45 84.26 89.04 0.3897 0.1503 2.0 10 0.4 0.3 0.8792 0.9049 0.9329 0.9252 0.9585 23.32 14.77 85.00 89.61 0.3183 0.1426 0.2 10 0.4 0.4 0.1325 0.1447 0.1608 0.1568 0.1810 24.91 15.88 82.65 87.81 1.0050 0.1831 0.5 10 0.4 0.4 0.3231 0.3507 0.3866 0.3777 0.4304 24.73 15.73 82.83 87.95 0.7155 0.1810 1.0 10 0.4 0.4 0.5889 0.6283 0.6768 0.6647 0.7304 24.17 15.25 83.40 88.39 0.5052 0.1744 1.5 10 0.4 0.4 0.7704 0.8060 0.8470 0.8366 0.8869 23.44 14.64 84.14 88.96 0.3925 0.1661 2.0 10 0.4 0.4 0.8788 0.9045 0.9320 0.9250 0.9557 22.70 13.99 84.90 89.54 0.3206 0.1578
104
3.4 Design Charts
The values of seepage characteristics PE, PD, PE1, PD1, q/kH and GE c2/H, obtained corresponding to some selected pairs of floor profile ratios (B/D, B/c2 and c1/c2) and selected values of step ratios a/c2 (= 0.1, 0.2, 0.3, 0.4) were used to prepare design charts for practical use in design office. Fig. 3.5 (a), (b), (c), (d) show design charts for PE for step ratio a/c2 = 0.1 and c1/c2 = 0.2, 0.4, 0.6 and 0.8 respectively, while Figs. 3.6 (a), (b), (c), (d) gives design charts for
PE for step ratio a/c2 = 0.2. Similarly, Figs. 3.7 and 3.8 shows design charts for PD, Figs. 3.9 and
3.10 show design charts for PE1, Fig 3.11 and 3.12 gives design charts for PD1, Fig. 3.13 and 3.14 gives the design charts for seepage discharge factor (q/kH), and Fig. 3.15, 3.16 show the design charts for exit gradient factor (GE c2/H).
3.5 Conclusions
The closed-form theoretical solution to the problem of finite depth seepage under an impervious apron with unequal end cutoffs followed with a step at downstream end is obtained using
Schwarz-Christoffel transformation in two stages. The results obtained from the solution of implicit equations involving elliptic integrals have been used to present design charts for various seepage characteristics such as uplift pressure at key points, seepage discharge factor and exit gradient factor in terms of non-dimensional floor profile ratios. It is seen that uplift pressures at all the four key points decrease with increase in step ratio. However, the seepage discharge factor increases by very little margin, while the exit gradient factor increases sharply with increase in the downstream step ratio.
Also, for a given cutoff ratio and step ratio, the uplift pressure at key points E and D as well as exit gradient increase with decrease in B/D ratio, and are maximum for infinite depth of pervious medium.
When the depth of permeable soil is large, the numerical solutions tend towards the values for infinite depth.
105
Fig. 3.2a Variation of PE with a/c2 (B/c2 = 10 & c1/c2 = 0.4)
Fig. 3.2b Variation of PD with a/c2 (B/c2 = 10 & c1/c2 = 0.4)
106
Fig. 3.3a Variation of PE1 with a/c2 (B/c2 = 10 & c1/c2 = 0.4)
Fig. 3.3b Variation of PD1 with a/c2 (B/c2 = 10 & c1/c2 = 0.4)
107
q Fig. 3.4a Variation of with a/c2 (B/c2 = 10 & c1/c2 = 0.4) kH
c2 Fig. 3.4b Variation of G with a/c2 (B/c2 = 10 & c1/c2 = 0.4) E H
108
Fig. 3.5a Design curve for PE (c1/c2 = 0.2 & a/c2 = 0.1)
Fig. 3.5b Design curve for PE for (c1/c2 = 0.4 & a/c2 = 0.1)
109
Fig. 3.5c Design curve for PE (c1/c2 = 0.6 & a/c2 = 0.1)
Fig. 3.5d Design curve for PE for (c1/c2 = 0.8 & a/c2 = 0.1)
110
Fig. 3.6a Design curve for PE (c1/c2 = 0.2 & a/c2 = 0.2)
Fig. 3.6b Design curve for PE for (c1/c2 = 0.4 & a/c2 = 0.2)
111
Fig. 3.6c Design curve for PE (c1/c2 = 0.6 & a/c2 = 0.2)
Fig. 3.6d Design curve for PE for (c1/c2 = 0.8 & a/c2 = 0.2)
112
Fig. 3.7a Design curve for PD (c1/c2 = 0.2 & a/c2 = 0.1)
Fig. 3.7b Design curve for PD for (c1/c2 = 0.4 & a/c2 = 0.1)
113
Fig. 3.7c Design curve for PD (c1/c2 = 0.6 & a/c2 = 0.1)
Fig. 3.7d Design curve for PD for (c1/c2 = 0.8 & a/c2 = 0.1)
114
Fig. 3.8a Design curve for PD (c1/c2 = 0.2 & a/c2 = 0.2)
Fig. 3.8b Design curve for PD for (c1/c2 = 0.4 & a/c2 = 0.2)
115
Fig. 3.8c Design curve for PD (c1/c2 = 0.6 & a/c2 = 0.2)
Fig. 3.8d Design curve for PD for (c1/c2 = 0.8 & a/c2 = 0.2)
116
Fig. 3.9a Design curve for PE1 (c1/c2 = 0.2 & a/c2 = 0.1)
Fig. 3.9b Design curve for PE1 for (c1/c2 = 0.4 & a/c2 = 0.1)
117
Fig. 3.9c Design curve for PE1 (c1/c2 = 0.6 & a/c2 = 0.1)
Fig. 3.9d Design curve for PE1 for (c1/c2 = 0.8 & a/c2 = 0.1)
118
Fig. 3.10a Design curve for PE1 (c1/c2 = 0.2 & a/c2 = 0.2)
Fig. 3.10b Design curve for PE1 for (c1/c2 = 0.4 & a/c2 = 0.2)
119
Fig. 3.10c Design curve for PE1 (c1/c2 = 0.6 & a/c2 = 0.2)
Fig. 3.10d Design curve for PE1 for (c1/c2 = 0.8 & a/c2 = 0.2)
120
Fig. 3.11a Design curve for PD1 (c1/c2 = 0.2 & a/c2 = 0.1)
Fig. 3.11b Design curve for PD1 for (c1/c2 = 0.4 & a/c2 = 0.1)
121
Fig. 3.11c Design curve for PD1 (c1/c2 = 0.6 & a/c2 = 0.1)
Fig. 3.11d Design curve for PD1 for (c1/c2 = 0.8 & a/c2 = 0.1)
122
Fig. 3.12a Design curve for PD1 (c1/c2 = 0.2 & a/c2 = 0.2)
Fig. 3.12b Design curve for PD1 for (c1/c2 = 0.4 & a/c2 = 0.2)
123
Fig. 3.12c Design curve for PD1 (c1/c2 = 0.6 & a/c2 = 0.2)
Fig. 3.12d Design curve for PD1 for (c1/c2 = 0.8 & a/c2 = 0.2)
124
q Fig. 3.13a Design curve for (c1/c2 = 0.2 & a/c2 = 0.1) kH
q Fig. 3.13b Design curve for for (c1/c2 = 0.4 & a/c2 = 0.1) kH
125
q Fig. 3.13c Design curve for (c1/c2 = 0.6 & a/c2 = 0.1) kH
q Fig. 3.13d Design curve for (c1/c2 = 0.8 & a/c2 = 0.1) kH
126
q Fig. 3.14a Design curve for (c1/c2 = 0.2 & a/c2 = 0.2) kH
q Fig. 3.14b Design curve for (c1/c2 = 0.4 & a/c2 = 0.2) kH
127
q Fig. 3.14c Design curve for (c1/c2 = 0.6 & a/c2 = 0.2) kH
q Fig. 3.14d Design curve for (c1/c2 = 0.8 & a/c2 = 0.2) kH
128
c2 Fig. 3.15a Design curve for G (c1/c2 = 0.2 & a/c2 = 0.1) E H
c2 Fig.3.15b Design curve for G (c1/c2 = 0.4 & a/c2 = 0.1) E H
129
c2 Fig. 3.15c Design curve for G (c1/c2 = 0.6 & a/c2 = 0.1) E H
c2 Fig.3.15d Design curve for G (c1/c2 = 0.8 & a/c2 = 0.1) E H
130
c2 Fig. 3.16a Design curve for G (c1/c2 = 0.2 & a/c2 = 0.2) E H
c2 Fig.3.16b Design curve for G (c1/c2 = 0.4 & a/c2 = 0.2) E H
131
c2 Fig. 3.16c Design curve for G (c1/c2 = 0.6 & a/c2 = 0.2) E H
c2 Fig.3.16d Design curve for G (c1/c2 = 0.8 & a/c2 = 0.2) E H
132 CHAPTER 4. FLAT APRON WITH UNEQUAL END CUTOFFS
4.1 Introduction
This chapter deals with the common case of flat apron with unequal cutoffs at the ends.
Downstream cutoff is provided for two purposes: (i) Protection of the apron against undermining due to excessive exit gradient, and (ii) Protection of the apron due to down stream scour.
Similarly, the upstream cutoff is provided for two purposes: (i) Reduction of uplift pressure all along downstream side, and (ii) Protection of the apron against upstream scour. In both these locations, the bottom of cutoff is provided below the corresponding deepest scour level.
However, since the depth of downstream deepest scour is always more than the depth of upstream deepest scour, the depth of the upstream cutoff is normally kept lesser than the depth of the downstream cutoff.
4.2 Theoretical Analysis
Fig. 4.1 (a) shows floor profile in z-plane, with relative values of stream function and potential function on the boundaries. Fig. 4.1 (c) shows the well known w-plane relating and
. The points in z and w-planes are denoted by complex coordinates z x iy and wi respectively, where i 1. The problem is solved by transforming both z-planes and w-planes on the infinite half-plane, t-plane, shown in Fig. 4.1 (b), thus obtaining z f1() t and w f2( t ).
133
Fig. 4.1 Illustrations of the problem – Schwarz-Christoffel transformations
134 Floor Parameters
The floor of the hydraulic structure, shown in Fig. 4.1(a) has the following floor parameters:
(i) Length of the apron : B
(ii) Finite depth of pervious medium : D
(iii) Depth of upstream cut off : c1
(iv) Depth of down stream cutoff : c2
The resulting independent non-dimensional floor-profile ratios are:
1. B/D, 2. B/c2, and 3. c1/c2.
4.2.1 First Transformation
The Schwarz - Christoffel equation of transformation of the floor from z-plane to t-plane is given by:
dt z MN11 (4.1 a) 12 78 (t t1 ) ( t t 2 ) ...... ( t t 7 ) ( t t 8 ) where 1 //22 ; 1 12/ 5 2 ; 5 1
2 2 ; 2 1 6 //22 ; 6 12
3 //22 ; 3 12 7 0 ; 7 1
4 //22 ; 4 12 8 0 ; 8 1
Substituting the values of 's , we get
dt z MN 11 1/2 1 1/2 1/2 1 1/2 1 1 ()()()()()()()()tt1 tt 2 tt 3 tt 4 tt 5 tt 6 tttt 7 8
135 ()()t t25 t t dt or z MN11 (4.1) (tttt7 )( 8 )( tttttttt 1 )( 3 )( 4 )( 6 )
where t1, t2 ...... t8 are the co-ordinates of the mapped points on t-plane, and M1,
N1 are constants to be determined.
Eq. 4.1 is an elliptic integral. Referring to Identity 254, Bird and Friedman (1971),
()()t t t t Let sn2 u 4 1 3 (4.2 a) ()()t4 t 3 t t 1
()()t t t t So that u sn(1 , ) 4 1 3 Fm (4.2 b) ()()t4 t 3 t t 1 where sn u = Jacobi's elliptic function, sinus amplitudinus
Fm(,) = Elliptic integral of first kind (Legendre's notation)
()()t t t t m = modulus 4 3 6 1 (4.2 c) ()()t4 t 1 t 6 t 3
()()t t t t = amplitude sin1 4 1 3 (4.2 d) ()()t4 t 3 t t 1
Differentiating Eq. 2 (a) and simplifying we get
dtdu 2 (4.3) (tttttttt1 )( 3 )( 4 )( 6 ) ( tttt 4 1 )( 6 3 )
From Eqs. 4.1 and 4.3, we obtain, after simplification.
()()t t t t 2du z MN25 11 ()()t t t t (t t )( t t ) 78 4 1 6 3
Carrying out the above integration, (as in chapter 3), we get
2M1 ()()()()t2 t 1 t 5 t 1 t 3 t 1 t 8 t 5 t 8 t 2 2 F(,)(,,) m 1 m ()()()()t t t t t t t t t t ()()t4 t 1 t 6 t 3 7 1 8 1 7 8 8 3 8 1
136 ()()t7 t 5 t 7 t 2 2 (,,)21mN (4.4) ()()t7 t 1 t 7 t 3
From Fig. 4.1(b), choosing ttttt1 1,,,, 2 1 345 , 2 ,
t6 2 t7 1 and t8 1 , we get from Eq. 4.4
2(M11 )( 1 2 ) 1 z Fm(,) ()() 12 (1 11 )( 1 )
(1 1 )(1 2 )(1 12 )(1 )(11 22 ) ( ,12 ,mm )( , , N 1 ) 2 (1 )(1 11 )(1 )(1 )
22 or z M1[ A 1 F (,) m B 1 (,,) 1 m C 1 (,,)] 2 m N 1 (4.5)
2 2 where (,,) 1 m and (,,) 2 m are elliptic integrals of third kind with
2 2 parameters, 1 and 2 respectively, given by
2 t8 t 1 t 4 t 3 11 2 2 tttt7 1 4 3 11 2 1 . and 2 . (4.6 a) t8 t 3 t 4 t 11 1 t7 t 3 t 4 t 11 1
2(1 1 )( 1 2 ) 1 A1 (4.7 a) (1 11 )(1 ) ()() 12
(1 )(1 1 )(1 2 ) 1 B1 (4.7 b) (1 1 )(1 ) ()() 12
(1 )(1 1 )(1 2 ) 1 C1 (4.7 c) (1 1 )(1 ) ()() 12
()() t = amplitude sin1 1 (4.8 a) 2 ( 1 t )
2 ( ) m = modulus 12 (4.8 b) ()()12
137 In Eq. 4.5, there are seven unknowns: ,,,,,, 1 2 1 2M 1 and N1 , the values of which are determined by applying boundary conditions at various points.
Boundary conditions
1. Point 3 (Point E1): At point 3, z = – B/2 and t = –
()() sinsin11 (0)1 0 2 ( 1 )
Hence Fm( , )0 and ( ,, 2 )m = 0
Hence from Eq. 4.5, we get NB1 /2 (4.9)
2. Point 4 (Point E): At point 4, z = B/2 and t =
()() sin111 sin (1) 2 ( 1 ) 2
Hence F(,), m F m K 2
2 2 2 and (,,),,(,) m m 0 m 2 where K = complete elliptic integral of first kind.
2 and 0 (,)m = complete elliptic integral of third kind.
Hence from Eq. 4.5, we obtain.
22 B M1[ A 1 K B 1 0 ( 1 , m ) C 1 0 ( 2 , m )] M 1 . R (4.10)
or M1 BR/ (4.10 a)
22 where R A1 K B 1 0( 1 , m ) C 1 0 ( 2 , m )] (4.10 b)
3. Point 7: At point 7, z and t = 1
Hence the point that is mapped into t = + 1 is a simple pole of integrand (1).
138 iD i M. Lt ( tf 1)t dt ( ) . 1 t1
()()t t25 t t Here ft() , wheret7 1 (tttt7 )( 8 )( tttttttt 1 )( 3 )( 4 )( 6 )
()()t t t t (1)(tf t ) 25 (t t81 ) ( t 3 t )( t t4 )( t t 6 )( t t )
()()tt 12 (1)(t t 12 )( t )( t )( t )
(112 )(1 ) iM1 (1 1) (1 12 )(1 )(1 )(1 )
(112 )(1 ) From which D MM11 J. (4.11) 2 (1 12 )(1 )(1 )(1 )
D (1 )(1 ) Where J . 12 (4.11 a) M1 2 (1 12 )(1 )(1 )(1 )
4. Point 6: At point 6, zB /2 and t 2
11()() 12 1 sinsin 2 ( 12 ) m
1 1 Fmsin , K i K m
1 im2 12 22 and sin , ,m 00 ,,mm 22 m m
Hence from Eq. 4.5, we get:
2 B 22im M A()(,)(,) K iK B m m 2 1 1 1 0 122 0 1 m 1
2 22im B C (,)(,) m m 1 0 222 0 2 2 m 2
139 22 B M1[( A 1 K , )(Bm 1 ,0 Cm )] 1 1 0 2
mm22 i M A K Bm Cm ( 22 , )( , ) 1 1 12 22 0 2 1 1 0 2 mm 12
Separating real and imaginary parts, we get
22 B M1[( A 1 K , ) B 1 0 1 ( m , )] C 1 0 . 2 m M 1 R
(which is the same as obtained in Eq. 4.10)
mm22 22 and A1 K Bm 1 2 Cm 22 0( 2 1 , )( 1 , ) 0 0 2 (4.12) mm 12
2 2 where 01(,)m and 02(,)m are complete elliptic integrals of third kind
2 2 with modulus 1 and 2 respectively, given by
22 22 1 ,m 2 2 ,m and 2 (4.13) 2 2 2 2 m 1 m 2
5. Point 1: At point 1, z = – B/2 and t = –
()() sinsin11 ( ) 11 2 ( 11 )
Fm(sin1 , ) iK
i and (sin12 , ,m ) [Km 22 (1 , )] 12 0
Hence from Eq. 4.5, we get, after simplification,
BC A K11 K 2(1 2 , m ) K 2 (1 2 , m ) 0 (4.14) 122 1 0 1 2 0 2 11 12
140 2 2 where 01(1, ) m and 02(1, ) m are the complete elliptic integrals of
2 2 third kind with parameters (1)1 and (1)2 respectively.
6. Point 5: At point 5, z = B/2 +i c2 and t = +
()() sinsin1112 y 2 () 12
1 where y 1 m
Hence from Identity 115.02, Bird and Friedman (1971),
Fm( ,) K i Fm(,)
22m 2 2 and (,,) m 0 (,)(,),, m i F mm 2 11 2
y2 1 y2 1 where sinsin11 my my22
()() 121 2 ( ) sin1 12 2 ( ) ( )( ) 1 1 2 1 2 (1 )( 2 ) 2 ( 1 2 )
( )( ) 2 ( ) sin1 1 2 1 2 (1 )( 2 ) 2 ( 1 2 ) ( 12 )( ) (12 )( )
2 sin1 1 2 1 2 2 2 1 2 1 2 2
11()( )( ) ()( )( ) sin1 2 2 sin 1 2 2 ()()(1 1 2 ) ()( 1 2 )( 2 )
141 ()() or sin1 22 ()()22
Hence from Eq. 4.5
B ic 2 2 2 m2 M A K iF(,)(,)(,),, m B2 m i F m 1 m 1 1 1 0 1 22 11 11
2 2 2 2 mB C1 0(,)(,),, 2 m i F m m 11 22 2 22
Separating real and imaginary parts, we get
22 B M1[ A 1 K B 1 0 ( 1 , m ) C 1 0 ( 2 , m )] M 1 R
(as found earlier in Eq. 4.10)
B 2 m2 and c M[(,){},, F m A B Cm 11 2 1 1 1 1 22 11 11
2 m2 Cm2 ,, (4.15) 1 22 11 22 or MT1 (4.15 a)
2 2 where T F(,){}(,,) m A B C 11 2 m Cm2 (,) 2 (4.15 b) 1 1 12 3 142 11 12
m2 m2 where 2 and 2 (4.15 c) 3 2 4 2 11 12
7. Point 2 (Point D1): At point 2, z = – B/2 +i c1 and t = –
()()()() sin11 1 sin 1i 1 1 sin 1 i x 2 ( 1 1 ) 2 ( 1 1 )
142 Thus, the upper limit of is imaginary, due to which becomes i.
Hence F(,) im i Fm( ,)
i 2 22 and (im ,, ) 2 F( , mm ),1 , 1
()() where = new amplitude tan1 11 (4.16 a) 2 ( 11 )
Hence from Eq (5), we obtain, after simplification
BCB 2 1 1 1 1 2 c1 M111 F( , m )( Am ,1 ,2 ) 2 2 1 1 1 2 1 1
2 2 2 Cm122 ( ,1 , (4.16) 12 or c1 = M1 . P (4.16 b)
BCBC22 whereP F (,) m Am 1 1 1 m 1 (,1,)22 1 2 (,1,) 112 2 2 2 2 1 1 1 2 1 1 1 2
(4.16 c)
Determination of 1 and 2
Parameters 1 and 2 are not independent parameters; instead, they depend on the values of , 1 and2. The values of 1 and 2 can be determined as under.
From Eq. 4.12
mm22 22 A1 K B 12 2 0( 1 , m ) C 1 2 2 0 ( 2 , m ) 0 mm 12
Substituting the values of A1, B1 and C1 and simplifying, we get
ABC2111221( )( ) (1 )(1 221 ) (1 )(1 2 )0 (4.17)
143 2K where A2 (4.17 a) (1)(1) 11
() m2 1 2 B2 .(22 , ) 01m (4.17 b) (1 11 )(1 ) m
() m2 1 2 and C2 .(22 , ) 02m (4.17 c) (1 12 )(1 ) m
Similarly, from Eq. 4.14, we get
ABC3111231( )( ) (1 )(1 231 ) (1 )(1 2 )0 (4.18)
2K where A3 (4.18 a) (1)(1) 11
() 1 1 22 B3 .(12 ,Km ) 1 0 1 (4.18 b) (1 11 )(1 ) 1
() 1 1 22 and C3 .2 Km 2 0 (1 2 , ) (4.18 c) (1 12 )(1 ) 1
Solving Eqs 4.17 and 4.18, we get
1 PPQ 2 4 (4.19 a) 1 2 3 3 3
2QQ 33 (4.19 b) 2 2 PPQ3 34 3 1
PQ22 where Q3 (4.19 c) PQ11
QPQQPQ1()() 2 2 2 1 1 and P3 (4.19 d) PQ11
ABC2 2 2 P1 ; (4.19 e) ABC2 1 2 2
144 2 AB2 122 C P2 (4.19 f) AB2 122 C
ABC333 Q1 ; (4.19 g) AB3 133 C
2 AB3 123 C Q2 (4.19 h) AB3 133 C
General Relationship: z = f1 (t)
From Eq. 4.10, we have
B M1 22 A1 K Bm 1 0( Cm 11 , )(0 2 , )
Hence from Eq. 4.5 we get
22 [A1 F (,) m B 1 (,,) 1 m C 1 (,,)] 2 m B z 22 (4.20) A1 K B 1 0( 1 , m ) C 1 0 ( 2 , m ) 2
which is the required relationship z = f1 (t)
Determination of floor profile ratios
There are four floor parameters in z-plane
(i) Length of apron, B
(ii) Finite depth of pervious medium, D
(iii) Depth of upstream cutoff, c1, and
(iv) Depth of down stream cutoff, c2
Consequently, we have the following non-dimensional floor profile ratios:
(1) B/D, (2) B/c2, and (3) c1/c2.
From Eqs. 4.10 and 4.11, we get
145 B [A K B ( 22 , m ) C ( , m )] 1 1 0 1 1 0 2 (4.21) D (1 )(1 ) 12 2 (1 12 )(1 )(1 )(1 )
R or = (4.21 a) J
From Eqs. 4.10 and 4.15, we have
B [A K B ( 22 , m ) C ( , m )] 1 1 0 1 1 0 2 (4.22) c 2222 2 BC1 1mm 1 2 F( , m )( A1 B 1 C 1 ) 2 , 2 , m 2 , 2 , m 1 1 1 1 1 2 1 2
B R or = (4.22 a) c2 T
Finally, from Eqs. 4.16 and 4.15
22 BCBC1 1 1 122 1 2 F(1 m )(,1 Am 11 ,)2 2 (,1 m 2 2 ,) 2 c 1 1 1 2 1 1 1 2 1 (4.23) c 2222 2 BC1 1mm 1 2 F( , m )( A1 B 1 C 1 ) 2 , 2 , m 2 , 2 , m 1 1 1 1 1 2 1 2
c P or 1 = (4.23 a) c2 T
4.2.2 Second Transformation
The transformation of w-plane to t-plane yields
dt w MN22 1 6 7 8 ()()()()t t1 t t 6 t t 7 t t 8
1 Here 1 6 7 8 2
t1 1;tt 6 2 ; 7 1; 8 1
146 dt w MN22 (4.24) (t 1)( t 21 )( t ) ( t 1)
This is an elliptic integral.
Using Identity 254, Byrd and Friedman (1971), and carrying out the above integration (as in chapter 3), we get.
2M 2 w F0(,) m 0 N 2 (4.25) (1 12 )(1 )
where Fm00( ,) = Elliptic integral of first kind with modulus m0
2()12 m0 (4.25 a) (1)(1) 12
(1 )(t ) = Amplitude sin1 21 (4.25 b) (21 )(t 1)
In order to find the values of constants M2 and N2, we apply boundary conditions at point 1 and 6.
Boundary Conditions
(i) Point 1: At point 1, w = – kH and t = –
1 sin (0) 0 and F0( , m 00 ) F (0, m ) 0
Hence from Eq. 4.25, NkH2 (4.26)
(ii) Point 6: At point 6, w = 0 and t =
(1 )( ) sin112 2 1 sin (1) (1 2 )( 2 1) 2
Fm00(,) F0, m 0 K 0 2
where K0 = Complete elliptic integral of first kind with modulus m0.
147 Hence from Eqs. 4.25 and 4.26, we get
kH M 2 (1 12 )(1 ) (4.27) 2K0
Substituting the values of M2 and N2 in Eq. 4.25, we get
kH w [(,)]F m00 K (4.28) K0
which is the desired relationship wft 2 ()
Seepage Discharge
At point 7, w = iq and t = + 1
11(1 21 )(1 ) 1 7 sin sin (1 2 )(1 1) m 0
1 1 Fm(,)70Fmsin K , iK0 0 0 m0
Hence from Eq. 4.28, we get
q K 0 (4.29) kH K0
22 where K0 = Complete elliptic integral of first kind, with modulus mm00 1
q/kH = Seepage discharge factor.
Pressure Distribution
For the base of the apron, w = – khx
kH w khx [(,)] F m00 K K0
h Let P x 100 % pressure at any point below the apron. x H
148 100 Px [F ( , m00 )] K K0
Fm(,) 0 or Px 100 1 (4.30) K0
(i) At point 6 (Point R): t 2
11(1 2 )( 1 2 ) R sinsin 1 (1 2 )( 2 1)2
Fm00(,)R FmK000, 2
K0 Hence PR 100 10 as expected K0
(ii) At point 4 (Point E): t =
1 (21 1)( ) E sin (4.31 a) (12 )(1 )
Fm00(,)E Hence PE 100 1 (4.31) K0
(iii) At point 3 (Point E1): t = –
1 (1 21 )( ) E1 sin (4.32 a) (12 )(1 )
Fm0(,)E 1 0 and PE1 100 1 (4.32) K0
149 (iv) At Point 1 (Point G): t = –
G 0 ;(Fm00 ,G ) 0
PG 100% as expected
(v) At point 5 (Point D): t =
1 (1 2 )() 2 1 D sin (4.33 a) ()(1 2 1) 2
Fm00(,)D PD 100 1 (4.33) K0
(vi) At point 2 (Point D1): t = –
1 (1 2 )( 1 1 ) D1 sin (4.34 a) (1 2 )(1 1 )
Fm0(,)D 1 0 PD1 100 1 (4.34) K0
Exit Gradient
From the first transformation, we have
dz B ()()tt . 12 (i) dt R 2 22 (t 1) ( t )( t 12 )( t )
Similarly, from the second transformation, we have
dw kH (1 )(1 ) 12 (ii) dt 2 K 2 0 (t 1)( t 12 )( t )
As shown chapter 3, the exit gradient GE is given by
1 dw dt G . E ik dt dz
150 dw dz Substituting the values of and simplifying, (as in chapter 3) we get dt dt
2 2 2 c2 T (1 1 )(1 2 )(1 2 )( 2 ) GE H 2(K02 1 )( 2 2 ) where = Exit gradient factor
4.3 Computations and Results
Eqs. 4.21, 4.22, 4.23 for floor profile ratios B/D, B/c2 and c1/c2 are not explicit in transformation parameters, , 1 and 2. Hence direct solution of these equations for the parameters, , 1 and 2 corresponding to given sets of values of floor profile ratios (B/D,
B/c2 and c1/c2) are not practical. However these equations can be solved for physical floor profile ratios B/D, B/c2 and c1/c2 for assumed values of , 1 and 2. Hence B/D, B/c2 and c1/c2 were first computed for values of 01.12 The parameters 1 and 2 were computed from Eqs. 4.19 (a) and 4.19 (b). From these values of floor profile ratios, iterative procedures were developed to compute the values of , 1 and 2 (and then of
1and 2) for chosen pairs of values of B/D, B/c2 and c1/c2 ratio. Table 4.1 gives the resulting values of , 1 and 2 for some chosen pairs of values of floor profile ratios along with seepage characteristics.
151 Table 4.1 Computed Values
B/D B/c2 c1/c2 1 2 PE PD PE1 PD1 q/kh GE c2/H 0.2 5 0.2 0.1222 0.1464 0.2435 38.06 26.03 83.99 88.74 0.9578 0.1787 0.2 5 0.4 0.1166 0.1649 0.2378 37.32 25.56 77.47 84.24 0.9476 0.1757 0.2 5 0.6 0.1112 0.1836 0.2323 36.46 25.09 72.56 80.90 0.9375 0.1727 0.2 5 0.8 0.1061 0.2024 0.2268 35.80 24.61 68.49 78.18 0.9273 0.1696 0.2 5 1 0.1013 0.2214 0.2214 35.03 24.13 64.97 75.87 0.9172 0.1665 0.2 10 0.2 0.1382 0.1504 0.1992 27.66 19.23 87.98 91.53 0.9937 0.1339 0.2 10 0.4 0.1352 0.1597 0.1963 27.39 19.05 83.05 88.09 0.9880 0.1326 0.2 10 0.6 0.1323 0.1690 0.1934 27.11 18.86 79.31 85.50 0.9823 0.1314 0.2 10 0.8 0.1295 0.1783 0.1905 26.83 18.67 76.18 83.36 0.9766 0.1301 0.2 10 1 0.1267 0.1877 0.1877 26.55 18.49 73.45 81.51 0.9709 0.1288 0.2 15 0.2 0.1439 0.1521 0.1846 22.79 15.93 90.00 92.94 1.0060 0.1115 0.2 15 0.4 0.1419 0.1582 0.1827 22.64 15.83 85.89 90.06 1.0021 0.1107 0.2 15 0.6 0.1399 0.1644 0.1807 22.49 15.73 82.75 87.88 0.9981 0.1100 0.2 15 0.8 0.1380 0.1706 0.1787 22.33 15.62 80.12 86.06 0.9942 0.1093 0.2 15 1 0.1360 0.1768 0.1768 22.18 15.52 77.82 84.48 0.9903 0.1086 0.2 20 0.2 0.1468 0.1529 0.1774 19.83 13.90 91.26 93.83 1.0123 0.0975 0.2 20 0.4 0.1453 0.1576 0.1759 19.73 13.83 87.66 91.13 1.0092 0.0970 0.2 20 0.6 0.1438 0.1622 0.1744 19.63 13.76 84.90 89.38 1.0062 0.0965 0.2 20 0.8 0.1423 0.1668 0.1729 19.53 13.69 82.60 87.77 1.0032 0.0961 0.2 20 1 0.1409 0.1715 0.1715 19.43 13.62 80.57 86.38 1.0002 0.0956 0.5 5 0.2 0.2988 0.3540 0.5535 37.89 25.78 84.21 88.91 0.6703 0.1759 0.5 5 0.4 0.2857 0.3950 0.5431 37.12 25.29 77.75 84.47 0.6604 0.1727 0.5 5 0.6 0.2733 0.4352 0.5329 36.35 24.80 72.85 81.17 0.6504 0.1695 0.5 5 0.8 0.2616 0.4745 0.5228 35.56 24.30 68.78 78.49 0.6404 0.1663 0.5 5 1 0.2504 0.5128 0.5128 34.76 23.80 65.24 76.20 0.6303 0.1630 0.5 10 0.2 0.3346 0.3621 0.4662 27.46 19.04 88.12 91.63 0.7060 0.1321 0.5 10 0.4 0.3279 0.3826 0.4603 27.18 18.85 83.24 88.24 0.7004 0.1308 0.5 10 0.6 0.3214 0.4029 0.4544 26.89 18.66 79.52 85.68 0.6948 0.1295 0.5 10 0.8 0.3149 0.4229 0.4485 26.60 18.46 76.41 83.56 0.6892 0.1282 0.5 10 1 0.3087 0.4428 0.4428 26.31 18.27 73.69 81.73 0.6836 0.1269 0.5 15 0.2 0.3473 0.3656 0.4359 22.60 15.77 90.11 93.03 0.7182 0.1101 0.5 15 0.4 0.3428 0.3792 0.4317 22.45 15.67 86.04 90.08 0.7143 0.1094 0.5 15 0.6 0.3384 0.3927 0.4276 22.29 15.56 82.93 80.02 0.7104 0.1086 0.5 15 0.8 0.3340 0.4061 0.4235 22.13 15.45 80.32 86.22 0.7065 0.1079 0.5 15 1 0.3297 0.4195 0.4195 21.97 15.34 78.03 84.66 0.7027 0.1072 0.5 20 0.2 0.3537 0.3674 0.4205 19.65 13.76 91.35 93.90 0.7243 0.0964 0.5 20 0.4 0.3504 0.3776 0.4173 19.55 13.69 87.79 91.40 0.7213 0.0959 0.5 20 0.6 0.3471 0.3878 0.4142 19.45 13.62 85.06 89.50 0.7184 0.0954
152 B/D B/c2 c1/c2 1 2 PE PD PE1 PD1 q/kh GE c2/H 0.5 20 0.8 0.3438 0.3979 0.4110 19.34 13.55 82.77 87.91 0.7154 0.0949 0.5 20 1 0.3405 0.4079 0.4079 19.24 13.47 80.76 86.53 0.7125 0.0944 1 5 0.2 0.5531 0.6336 0.8515 37.38 25.02 84.89 89.42 0.4633 0.1674 1 5 0.4 0.5334 0.6883 0.8434 36.56 24.49 78.59 85.17 0.4540 0.1639 1 5 0.6 0.5146 0.7375 0.8354 35.71 23.94 73.75 82.00 0.4445 0.1604 1 5 0.8 0.4967 0.7813 0.8276 34.84 23.38 69.66 79.41 0.4348 0.1567 1 5 1 0.4798 0.8198 0.8198 33.95 22.81 66.05 77.19 0.4250 0.1530 1 10 0.2 0.6030 0.6414 0.7671 26.84 18.45 88.56 91.96 0.4982 0.1269 1 10 0.4 0.5935 0.6688 0.7609 26.53 18.25 83.82 88.68 0.4930 0.1255 1 10 0.6 0.5841 0.6947 0.7547 26.22 18.04 80.18 86.21 0.4877 0.1241 1 10 0.8 0.5749 0.7193 0.7486 25.91 17.82 77.11 84.16 0.4824 0.1226 1 10 1 0.5659 0.7426 0.7426 25.59 17.61 74.41 82.39 0.4770 0.1212 1 15 0.2 0.6202 0.6453 0.7330 22.02 15.28 90.46 93.28 0.5100 0.1061 1 15 0.4 0.6141 0.6636 0.7284 21.85 15.16 86.51 90.53 0.5063 0.1052 1 15 0.6 0.6079 0.6812 0.7237 21.68 15.05 83.47 88.44 0.5026 0.1044 1 15 0.8 0.6018 0.6981 0.7191 21.51 14.93 80.92 86.70 0.4990 0.1036 1 15 1 0.5958 0.7144 0.7144 21.33 14.81 78.67 85.19 0.4953 0.1028 1 20 0.2 0.6287 0.6474 0.7146 19.12 13.33 91.65 94.11 0.5159 0.0929 1 20 0.4 0.6244 0.6613 0.7111 19.00 13.25 88.19 91.70 0.5130 0.0924 1 20 0.6 0.6199 0.6746 0.7074 18.89 13.17 85.54 89.86 0.5102 0.0918 1 20 0.8 0.6153 0.6875 0.7037 18.78 13.09 83.30 88.32 0.5074 0.0913 1 20 1 0.6108 0.7000 0.7000 18.67 13.02 81.33 86.98 0.5046 0.0908 1.5 5 0.2 0.7385 0.8128 0.9584 36.84 24.12 85.76 90.09 0.3532 0.1574 1.5 5 0.4 0.7195 0.8577 0.9550 35.93 23.53 79.68 86.06 0.3449 0.1536 1.5 5 0.6 0.7013 0.8939 0.9516 34.98 22.91 74.89 83.03 0.3361 0.1497 1.5 5 0.8 0.6841 0.9226 0.9482 33.98 22.27 70.75 80.53 0.3269 0.1456 1.5 5 1 0.6679 0.9450 0.9450 32.96 21.61 67.02 78.36 0.3173 0.1412 1.5 10 0.2 0.7817 0.8158 0.9096 26.06 17.22 89.11 92.36 0.3874 0.1204 1.5 10 0.4 0.7733 0.8388 0.9059 25.72 17.49 84.55 89.24 0.3825 0.1189 1.5 10 0.6 0.7648 0.8594 0.9021 25.39 17.27 81.01 86.88 0.3776 0.1173 1.5 10 0.8 0.7565 0.8780 0.8983 25.04 17.03 77.99 84.91 0.3726 0.1158 1.5 10 1 0.7483 0.8945 0.8945 24.69 16.79 75.31 83.20 0.3675 0.1142 1.5 15 0.2 0.7969 0.8187 0.8866 21.27 14.65 90.90 93.60 0.3983 0.1009 1.5 15 0.4 0.7911 0.8337 0.8831 21.09 14.53 87.10 90.97 0.3952 0.1001 1.5 15 0.6 0.7858 0.8480 0.8800 20.91 14.40 84.17 88.98 0.3917 0.0992 1.5 15 0.8 0.7805 0.8613 0.8768 20.72 14.27 81.68 87.31 0.3883 0.0983 1.5 15 1 0.7751 0.8737 0.8737 20.53 14.14 79.47 85.86 0.3848 0.0974 1.5 20 0.2 0.8056 0.8215 0.8742 18.38 12.74 92.05 94.40 0.4031 0.0883 1.5 20 0.4 0.8001 0.8316 0.8705 18.30 12.69 88.71 92.08 0.4014 0.0880 1.5 20 0.6 0.7961 0.8424 0.8678 18.18 12.61 86.15 90.32 0.3985 0.0874
153 B/D B/c2 c1/c2 1 2 PE PD PE1 PD1 q/kh GE c2/H 1.5 20 0.8 0.7922 0.8527 0.8651 18.06 12.52 83.98 88.85 0.3962 0.0868 1.5 20 1 0.7883 0.8625 0.8625 17.94 12.44 82.06 87.56 0.3936 0.0862 2 5 0.2 0.8567 0.9109 0.9896 36.78 23.68 86.73 90.81 0.2819 0.1471 2 5 0.4 0.8423 0.9398 0.9885 35.94 23.23 80.92 87.05 0.2738 0.1429 2 5 0.6 0.8286 0.9607 0.9874 35.02 22.73 76.25 84.20 0.2652 0.1384 2 5 0.8 0.8158 0.9754 0.9864 33.96 22.10 72.09 81.79 0.2563 0.1337 2 5 1 0.8039 0.9854 0.9854 32.60 21.21 68.11 79.59 0.2476 0.1292 2 10 0.2 0.8873 0.9113 0.9673 25.22 16.94 89.69 92.79 0.3162 0.1138 2 10 0.4 0.8803 0.9259 0.9651 24.92 16.74 85.29 89.81 0.3123 0.1124 2 10 0.6 0.8740 0.9389 0.9632 24.59 16.52 81.84 87.55 0.3077 0.1107 2 10 0.8 0.8680 0.9501 0.9614 24.24 16.30 78.88 85.67 0.3029 0.1090 2 10 1 0.8621 0.9596 0.9596 23.90 16.09 76.22 84.02 0.2979 0.1072 2 15 0.2 0.8989 0.9138 0.9552 20.38 13.91 91.39 93.95 0.3257 0.0953 2 15 0.4 0.8936 0.9228 0.9528 20.25 13.82 87.73 91.44 0.3237 0.0947 2 15 0.6 0.8891 0.9315 0.9508 20.09 13.71 84.87 89.52 0.3210 0.0939 2 15 0.8 0.8851 0.9397 0.9489 19.90 13.58 82.44 87.92 0.3179 0.0930 2 15 1 0.8812 0.9471 0.9471 19.71 13.45 80.27 86.52 0.3147 0.0920 2 20 0.2 0.9042 0.9151 0.9479 17.57 12.10 92.46 94.70 0.3305 0.0836 2 20 0.4 0.9008 0.9220 0.9460 17.48 12.03 89.28 92.50 0.3287 0.0831 2 20 0.6 0.8971 0.9283 0.9439 17.39 11.97 86.80 90.81 0.3271 0.0827 2 20 0.8 0.8938 0.9344 0.9420 17.28 11.89 84.68 89.39 0.3251 0.0822 2 20 1 0.8907 0.9403 0.9403 17.16 11.81 82.80 88.15 0.3228 0.0816
Comparison with Infinite Depth Case when c1 = c2 = c
Malhotra (1936) obtained theoretical solution for uplift pressure below flat apron with equal end cutoffs (i.e. when c1=c2=c) for the case of infinite depth of the pervious foundation. The results of the present theoretical solution for B/D = 0.4 and B/D = 0.2 were compared with those of Malhotra's solution. Table 4.2 gives the values of uplift pressure PE at point E and PD at point D, for some selected values of B/c ratios. It is seen from Table 4.2 that when BD/ 0.2 , the results of both the analyses are practically the same.
154 Table 4.2 Comparison with Malhotra's Case of Infinite Depth
PE PD Infinite Present Present Infinite Present Present B/c Depth case analysis for analysis for Depth case analysis for analysis for of Malhotra B/D = 0.2 B/D = 0.4 of Malhotra B/D = 0.2 B/D = 0.4 3 41.4 41.3 41.1 28.4 28.2 28.0 4 37.9 37.8 37.7 26.1 26.0 25.7 6 32.9 32.7 32.6 22.7 22.6 22.4 8 29.2 29.2 29.0 20.3 20.3 20.1 12 24.6 24.5 24.4 17.2 17.1 17.0 24 17.8 17.8 17.7 12.6 12.5 12.4
4.4 Design Charts
The values of seepage characteristics PE, PD, PE1, PD1, q/kH and GE c2/H, obtained corresponding to some selected pairs of values of floor profile ratios were used to prepare design charts for practical use in the design office. Figs 4.2(a), (b), (c) (d) shows the design charts for uplift pressure PE at key point E, for cut off ratios c1/c2 = 0.2, 0.4, 0.6 and 0.8 respectively. Similarly, Figs. 4.3, 4.4, 4.5, 4.6 and 4.7 show the design charts for pressure PD at point D, PE1 at E1, PD1 at D1, exit gradient factor GE c2/H and seepage discharge factor q/kH respectively.
155
Fig. 4.2a Design curves for PE for c1/c2 = 0.2
Fig. 4.2b Design curves for PE for c1/c2 = 0.4
156
Fig. 4.2c Design curves for PE for c1/c2 = 0.6
Fig. 4.2d Design curves for PE for c1/c2 = 0.8
157
Fig. 4.3a Design curves for PD for c1/c2 = 0.2
Fig. 4.3b Design curves for PD for c1/c2 = 0.4
158
Fig. 4.3c Design curves for PE for c1/c2 = 0.6
Fig. 4.3d Design curves for PE for c1/c2 = 0.8
159
Fig. 4.4a Design curves for PE1 for c1/c2 = 0.2
Fig. 4.4b Design curves for PE1 for c1/c2 = 0.4
160
Fig. 4.4c Design curves for PE1 for c1/c2 = 0.6
Fig. 4.4d Design curves for PE1 for c1/c2 = 0.8
161
Fig. 4.5a Design curves for PD1 for c1/c2 = 0.2
Fig. 4.5b Design curves for PD1 for c1/c2 = 0.4
162
Fig. 4.5c Design curves for PD1 for c1/c2 = 0.6
Fig. 4.5d Design curves for PD1 for c1/c2 = 0.8
163
Fig. 4.6a Design curves for GEc2/H for c1/c2 = 0.2
Fig. 4.6b Design curves for GEc2/H for c1/c2 = 0.4
164
Fig. 4.6c Design curves for GEc2/H for c1/c2 = 0.6
Fig. 4.6d Design curves for GEc2/H for c1/c2 = 0.8
165
Fig. 4.7a Design curves for q/kH for c1/c2 = 0.2
Fig. 4.7b Design curves for q/kH for c1/c2 = 0.4
166
Fig. 4.7c Design curves for q/kH for c1/c2 = 0.6
Fig. 4.7d Design curves for q/kH for c1/c2 = 0.8
167 4.5 Conclusions
The general solution to the finite depth seepage under an impervious flat apron with unequal end cutoffs is obtained using the conformal transformations. The results obtained from the solution of implicit equations have been used to present design charts for various seepage characteristics such as uplift pressures at key points, seepage discharge factor and exit gradient in terms of non-dimensional floor profile ratios. It is seen that uplift pressures at points E and D increase with the increase in the depth of downstream cutoff, while the reverse is true for pressures at points E1 and D1. However the exit gradient and seepage discharge decrease with increase in the downstream cutoff.
Also the uplift pressure at key points, as well as exit gradient, increases with decrease in
B/D ratio. When the depth of permeable soils is large, the numerical solutions tend towards the established solution for infinite depth.
168 CHAPTER 5. FLAT APRON WITH EQUAL END CUTOFFS AND A STEP AT DOWNSTREAM END
5.1 Introduction
Flat aprons with end cutoffs are invariably provided for various hydraulic structures such as weirs, barrages, head regulators, aqueducts, siphons etc. The purpose of downstream cutoff is to protect the apron against excessive exit gradient and downstream scour, though it increases the uplift pressure all along the upstream side. The purpose of upstream cutoff is to protect the apron against upstream scour, and to reduce the uplift pressure all along the down stream side. In many cases, the depths of both these cutoffs are kept equal. In addition to these cutoffs, pervious aprons are provided at the downstream side of end cutoff in the form of inverted filter and launching apron. The thickness of these pervious aprons may vary from 2 ft to 5 ft. In order to accommodate the thickness of these pervious aprons, the downstream bed is excavated resulting in the formation of a ‘step’ at the downstream end.
5.2 Theoretical Analysis
Fig. 5.1 (a) shows the floor profile in z-plane, with relative values of stream function and potential function on the boundaries. Fig. 5.1 (c) shows the well known w-plane, relating and The points in z and w-planes are denoted by complex co- ordinates z x iy and wi respectively where i 1 . The problem is solved by transforming both the z-plane and w-plane on to an infinite half plane, t-plane, shown in Fig. 5.1 (b), thus obtaining z = f1(z) and w = f2(t) and finally the desired relation
z = F(w).
169 Floor Parameters
The floor of the hydraulic structure shown in Fig. 5.1 (a) has the following floor parameters:
(i) Length of the apron : B
(ii) Depth of pervious medium : D
(iii) Depth of cutoffs : c
(iv) Depth of downstream step : a
The resulting independent non-dimensional floor profile ratios are:
1. Floor - depth ratio, B/D
2. Floor-cutoff ratio, B/c, and
3. Step ratio, a/c
5.2.1 First Transformation
The Schwarz-Christoffel transformation equation for the transformation of floor profile from z-plane (Fig. 1 a) to t-plane (Fig. 1 b) is
dt z M11 N (5.1 a) (t t )12 ( t t ) ...... ( t t )78 ( t t ) 1 2 7 8
where 1 //22 ; 1 12/ 5 2 ; 5 1
2 2 ; 2 1 6 //22 ; 6 12
3 //22 ; 3 12 7 0 ; 7 1
4 //22 ; 4 12 8 0 ; 8 1
170
Fig. 5.1 Illustrations of the problem: Schwarz – Christoffel transformations
171 Substituting the values of 's, we get
()()t t25 t t dt z MN11 (5.1) (tttt7 )( 8 )( tttttttt 1 )( 3 )( 4 )( 6 ) where t1, t2 ...... t8 are the co-ordinates of the mapped points on t-plane, and M1, N1 are the constants to be determined.
Eq. (5.1) is an elliptic integral. Referring to Identity 254, Byrd and Friedman (1971),
()()tt t t Let sn2 u 4 13 (5.2 a) ()()tt4 t 31 t
()()t t t t so that u sn(1 , ) 4 1 3 Fm (5.2 b) ()()t4 t 3 t t 1 where sn u = Jacobi's elliptic function, sinus amplitudinus
Fm(,) = Elliptic integral of first kind (Legendre's notation)
()()t t t t m = modulus 4 3 6 1 (5.2 c) ()()t4 t 1 t 6 t 3
()()t t t t =amplitude sin1 4 1 3 (5.2 d) ()()t4 t 3 t t 1
Differentiating Eq. 5.2 (a) and simplifying we get
dt2 du (5.3) (tttttttt1 )( 3 )( 4 )( 6 ) ( tttt 4 1 )( 6 3 )
From Eqs (5.1) and (5.3), we obtain, after simplification
()()t t25 t t 2du MN11 ()()t t78 t t (t4 t 1 )( t 6 t 3 )
Carrying out the above integration, (as in Chapter 3) we get
172 2M1 (t2 t 1 )( t 5 t 1 )( )( t 3 ) t 1 t 8 t 5 t 8 t 2 2 z F( , mm )( , , ) 1 ()()t4 t 1 t 6 t 3 (t7 t 1 )( t 8 t 1 )( )( t 7 ) t 8 t 8 t 3 t 8 t 1
()()t7 t 5 t 7 t 2 2 (,,)21mN (5.4) ()()t7 t 1 t 7 t 3
From Fig. 5.1(b), choosing t1 1,,,,, t 2 1 t 3 t 4 t 5 t 6 2 , t7 1 and t8 1 , we get from Eq. (5.4),
2(M111 )( ) Fm(,) ()() 12 (1 11 )( 1 )
()1 (1)(1) 22 (1)(1) (,,)(,,) 1m 2 m N 1 2 (1)(1 11 ) (1 )(1)
22 or z M1[ A 1 F (,) m B 1 (,,) 1 m C 1 (,,)] 2 m N 1 (5.5)
2 2 where (,,) 1 m and (,,) 2 m are elliptic integrals of third kind with parameters,
2 2 1 and 2 respectively, given by
2 t8 t 1 t 4 t 3 11 2 2 t7 t 1 t 4 t 3 11 2 1 . and 2 . ...(5.6) t8 t 3 t 4 t 11 1 t7 t 3 t 4 t 11 1
()() t = amplitude sin1 1 (5.7 a) 2 ( 1 t )
2 ( ) m = modulus 12 (5.7 b) ()()12
2(22 ) 1 A 1 (5.8 a) 1 2 (11 ) ()() 12
2 (1 )(1 ) 1 B1 (5.8 b) (1 1 )(1 ) ()() 12
173 2 (1 )(1 ) 1 C1 (5.8 c) (1 1 )(1 ) ()() 12
In Eq. (5.5), there are six unknowns: ,,,, 1 2M 1 and N1 , the values of which are determined by applying boundary conditions at various points.
Boundary Conditions
1. Point 3 (Point E1): At point 3, z = – B/2 and t = –
()() sinsin11 (0)1 0 2 ( 1 )
Fm( ,) Fm(0, ) 0 and (,,) 2 m (0, 2 ,m ) 0
B Hence from Eq. (5.5), we get N (5.9) 1 2
2. Point 4 (Point E): At point 4, z = +B/2 and t = +
()() sin111 sin (1) 2 ( 1 ) 2
F(,), m F m K 2
2 2 2 (,,),,(,) m m 0 m 2 where K = complete elliptic integral of first kind.
2 and 0 (,)m = complete elliptic integral of third kind.
Hence from Eq. (5.5),
22 B M1[ A 1 K B 1 0 ( 1 , m ) C 1 ( 2 , m )] M 1 . R (5.10)
174 22 where R A1 K Bm 1 0( Cm 1 , )( 1 , )] 0 2 (5.10 a) or M1 BR/ (5.10 b)
3. Point 7: At point 7, z and t = 1.
Hence the point that is mapped into t = + 1 is a simple pole of integrand (5.1).
iD i M. Lt ( tf 1)t dt ( ) . 1 t1
(t t25 )( t t ) Here ft() , where t7 1 (tttt7 )( 8 )( tttttttt 1 )( 3 )( 4 )( 6 )
(t t )( t t ) (1)tf t ( ) 25 , (t t81 ) ( t 3 t )( t 4 t )( t 6t )( t t )
(tt )( ) (t 1) ( t 12 )( t )( t )( t )
(1 )(1 ) or iD iM1 (1 1) (1 12 )(1 )(1 )(1 )
M (12 ) From which D 1 . MJ (5.11) 2 2 1 (1 )(1 12 )(1 )
1 2 where J . (5.11 a) 2 2 (1 )(1 12 )(1 )
B 4. Point 6: At point 6, z ia and t 2 2
11()()12 1 sin sin 2 ( ) m 12
1 1 Fmsin , K i K m
175 1 im2 12 22 and sin, , m 00 ,,mm 22 m m
Hence from Eq.( 5.5), we get, after simplification (as in Chapter 3):
mm22 22 a M1[ A 1 K B 12 2 0 ( 1 , m ) C 1 2 2 0 ( 2 , m )] M 1 . L . (5.12) mm 12
mm22 22 where L A1 K Bm 10 2 1 Cm 22 ( 210 , 2 )( , ) (5.12 a) mm 12
Adding Eqs. (5.11) and (5.12), we get
()Da DMJMLMJL 111 () (5.13)
Also, from Eq. 5.12, noting that MBR1 /
22 mm22 A1 K B 12 2 0(,)(,) 1 m C 1 2 2 0 2 m a mm 12 L S 22 (5.14) B A1 K B 1 0(,)(,) 1 m C 1 0 2 m R
22 22 2 1 ,m 2 2 ,m where 1 and 2 (5.14 a) 22 2 2 m 1 m 2
5. Point 1 (Point G): At point 1, z = – B/2 and t = –
()() sinsin11 ( ) 11 2 ( ) 11
1 Fm(sin , ) iK
i 12 22 and (sin , ,m ) 2 [Km 0 (1 , )] 1
Hence from Eq. (5.5), we get, after simplification,
B C A K1 K 2( 2 , m ) 1 K 2 ( 2 , m ) 0 (5.15) 122 1 0 1 2 0 2 11 12
176 22 22 where 11 1 and 12 1
2 2 and 01(,) m and 02(,) m are the complete elliptic integrals of third kind with
2 2 parameters 1 and 2 respectively.
B 6. Point 5 (Point D): At point 5, zic and t = + 2
()() sin111 sin y 2 ( 1 )
1 where y 1 m
Hence from Identity 115.02, Byrd and Friedman (1971),
Fm(,) K i F(,) m
22m 2 2 and 0 (,,) m 0 (,)(,),, m i F mm 2 12 1
()() where = new amplitude sin1 2 ()() 2
Hence from Eq. (5.5), we get, after simplification (as in Chapter 3).
2 B m2 c M F(,){},, m A B Cm 11 1 1 1 1 22 11 11
C 2 m2 12 22 ,,m (5.16) 11 22 or c MT 1
2 B 2 C where T F(,){}(,,)(,,) m A B C11 22 m 12 m ; (5.16 a) 1 1 122 3 4 11 12
177 m2 m2 2 and 2 (5.16 b) 3 2 4 2 11 12
B 7. Point 2 (Point D1): At point 2, z +i c and t = – 2
()()()() sinsinsin111 11ii x 2 ( 11 )2 ( )
Thus, the upper limit of is imaginary, due to which becomes i.
Hence F(,) im i Fm( ,)
i and (im ,, 2 ) F( , mm ),1 22 , 12
()() where = new amplitude tan1 1 ( 5.17) 2 ( 1 )
Hence from Eq (5.5), we obtain, after simplification (as in Chapter 3)
2 BB1C 1 1 M F( , m )( Am ,1 , ) 1 2 c 1 11 2 2 2 1 1 1 2 1 1
C 2 12 ( ,1 2 , m (5.18) 2 2 12 or c = M1 . P (5.18 a)
22 BBC1C 1 1 1 2 where P F(,) m A 1 (,1,)22 m (,1,) m 12 2 2 1 2 2 1 1 1 2 1 1 1 2
(5.18 b)
178 Determination of
Parameter is not an independent parameter; instead, it depends on the values of , 1 and 2. However, to start with, let the t-coordinates of points 2 and 5 be 1 and 2 respectively, though 1 = 2 =
From Eq. 5.14, we have
m2 A( K SK )( Bm , ) S ( m , 22 ) 110 10 1 22 m 1
m2 Cm S ( m 22 , ) ( , ) 0 1022 20 2 m 2
Substituting the values of AB11, and C2 and simplifying (as in Chapter 3), we get
ABC2111221( )( ) (1 )(1 221 ) (1 )(1 2 ) 0 or 1 2 PP 1 1 2 2 0 (I)
Similarly, substituting the values of A1, B1 and C1 in Eq. 5.15 and simplifying (as in
Chapter 3), we get or ABC3111231( )( ) (1 )(1 231 ) (1 )(1 2 ) 0
1 2 QQ 1 1 2 2 0 (II)
Adding Eqs. (I) and (II), we get
PQ22 12 PQ11
Making 1 = 2 = (as is really the case), we have
PQ 22 (5.19) PQ11
179 2 ABC222 AB2 122 C where P1 ; P2 AB2 122 C AB2 122 C
2 ABC333 AB3 133 C Q1 ; Q2 AB3 133 C AB3 133 C
2()K SK where A2 (1)(1) 11
() m2 B 1 (,)(,)22m S m 2 220 10 1 (1 1 )(1 ) m 1
2 ()1 m 22 C2 0(,)(,) 20m 2 S m (1 )(1 ) m22 1 2
2 K A3 (1 11 )(1 )
()1 22 B3 Km1 0(,) 1 2 (1 11 )(1 )
() C 1 Km22(,) 3 2 2 0 2 (1 12 )(1 )
General Relationship: z = f1 (t)
From Eq. 5.10 (a), we have
B M 1 22 A1 K B 1 0(,)(,) 1 m C 1 0 2 m
Hence from Eqs. (5.5) and (5.9), we get
22 B[ A F (,) m B (,,) m C (,,)] m B z 1 1 1 1 2 (5.20) 22 A1 K B 1 0(,)(,) 1 m C 1 0 2 m 2
which is the required relationship z = f1 (t)
180 Determination of floor profile ratios
From Eqs. (5.10) and (5.13), we get
B R = (5.21) D JL
From Eq. (5.10) and (5.16)
B R or = (5.22) c T
Also, from Eqs. (5.12) and (5.16)
a LSR = (5.23) c TT
where R, J, L and T are given by Eqs. 5.10 (a), 5.11 (a), 5.12 (a) and 5.16 (a) respectively.
5.2.2 Second Transformation
The transformation of w-plane (Fig. 5.1 c) to t-plane (Fig. 5.1 b) yields
dt w MN22 1 6 7 8 ()()()()t t1 t t 6 t t 7 t t 8
Here 1 6 7 8 1/ 2 and t1 1;t 6 2 ; t 7 1; t 8 1
dt w MN22 (5.24) (t 1)( t 21 )( t )( t 1)
This is an elliptic integral
Using Identity 254, Byrd and Friedman (1971) and carrying out the above integration, we get.
2M2 F0(,) m 0 N 2 (5.25) (1 12 )(1 )
181 where Fm00( ,) = Elliptic integral of first kind with modulus m0
2()12 m0 (5.25 a) (1)(1) 12
(1 )()t = amplitude sin1 21 (5.25 b) ()(21 1)t
In order to find the values of constants M2 and N2, we apply boundary conditions at points 1 and 6.
(i) Point 1: At point 1, w = – kH and t = –
1 sin (0) 0 and F000( , m ) F (0, m ) 0
Hence from Eq. 5.25, NkH2 (5.26)
(ii) Point 6: At point 6, w = 0 and t =
(1 )( ) sinsin11 (1) 2 2 1 (1 2 )( 2 1) 2
Fm00(,) F0, m 0 K 0 2
where K0 = Complete elliptic integral of first kind with modulus m0.
Hence from Eqs. 5.25 and 5.26, we get
kH M 2 (1 12 )(1 ) (5.27) 2K0
Substituting the values of M2 and N2 in Eq. 5.25, the transformation equation becomes:
kH w [(,)]F0 m 0 K 0 (5.28) K0
w f() t which is the desired relationship 2
182 Seepage Discharge
At point 7, w = iq and t = + 1
11(1 21 )(1 ) 1 7 sinsin (1 20 )(1 1) m
1 1 Fm070(,) Fm00sin, K 0 iK 0 m0
Hence from Eq. 5.28, we get
q K 0 (5.29) kH K0
2 where K0 = Complete elliptic integral of first kind, with modulus m0 and mm00 1
q/kH = Seepage discharge factor.
Pressure Distribution
For the base of the apron, w = – khx
kH w khx [(,)] F0 m 0 K 0 K0
h Let P x 100% pressure at any point below the apron. x H
100 Px [(,)]F0 m 0 K 0 K0
Fm00(,) or Px 100 1 (5.30) K0
183 (i) At point 6 (Point R): At point 6, t 2
11(1 2 )( 1 2 ) R sinsin 1 (1 2 )( 2 1)2
Fm00(,)R FmK000, 2
K0 Hence PR 100 10 , as expected K0
(ii) At point 4 (Point E): At point 4, t =
1 (21 1)( ) E sin (5.31 a) (12 )(1 )
Fm00(,)E Hence PE 100 1 (5.31) K0
(iii) At point 3 (Point E1): At point 3, t = –
1 (1 21 )( ) E1 sin (5.32 a) (12 )(1 )
Fm0(,)E 1 0 and PE1 100 1 (5.32) K0
(iv) At Point 1 (Point G): At point 1, t = –
11(1 2 )( 1 1 ) G sin sin (0) 0 (1 2 )( 1 1)
Fm00(G , ) 0
Hence PG 100[1 0] 100% as expected
(v) At point 5 (Point D): At point 5, t =
184 1 (1 21 )() D sin (5.33 a) ()(12 1)
Fm00(,)D PD 100 1 (5.33) K0
(vi) At point 2 (Point D1): At point 2, t = –
1 (1 21 )() D1 sin (5.34 a) ()(112 )
Fm0(,)D 1 0 PD1 100 1 (34) K0
Exit Gradient
From the first transformation, we have
dz B()() t t . dt R 2 22 (t 1) ( t )( t 12 )( t )
Similarly, from the second transformation, we have
dw kH (1 )(1 ) 12 dt 2 K 2 0 (t 1)( t 12 )( t )
The exit gradient GE is given by
1 dw dt G . E ik dt dz
dt Substituting the values of and and simplifying (as in Chapter 3), we get dz
B Rt(1t2 ) (1 )(1 ) 22 G 12 E 2 H 2K0 ( t )( t ) 1t
185 c Bc cT Now, G G (where ) E H E HB BR
c Tt(1t2 ) (1 )(1 ) 22 G 12 E 2 H 2(Kt0 )( t ) 1t
Substituting t 2 at the exit end, and simplifying, we get
c T (1 )(1 )(1 2 )( 2 2 ) G 1 2 2 2 (5.35) E 22 H 2K0 ()2 where GE cH/ = Exit gradient factor
5.3 Computations and Results
Eqs. 5.21 and 5.22 for floor profile ratios B/D and B/c are not explicit in transformation parameters , 1 and 2. Hence direct solutions of these equations for the parameters , 1 and 2 corresponding to given set of values of floor profile ratios (B/D and B/c) and given step ratio are extremely difficult. However, these equations can be solved for physical floor profile ratios B/D and B/c for assumed values of , 1 and 2 and desired value of step ratio (a/c). Hence B/D and
B/c ratios were first computed for values of 01,12 for some pre-selected values of step ratio (a/c). From these values of floor profile ratios, iterative procedures were developed to compute the values of , 1 and 2 for chosen pairs of values of B/D and B/c ratios, corresponding to each selected value of step ratio (a/c). Finally, computer program was executed using these values of , 1 and 2 as input values to confirm the values of these chosen floor profile ratios (B/D, B/c) and also to compute and the desired seepage characteristics PE, PD, PE1,
PD1 , q/kH and GE c/H. Table 5.1 gives the resulting values.
186 Table 5.1 Computed Values
B/D B/c a/c σ γ1 γ2 PE PD PE1 PD1 q/kH GEc/H 0.2 5 0.1 0.1006 0.2204 0.2150 34.40 23.30 64.68 75.68 0.9228 0.1785 0.2 10 0.1 0.1261 0.1870 0.1841 26.01 17.80 73.31 81.42 0.9747 0.1377 0.2 15 0.1 0.1355 0.1762 0.1743 21.70 14.92 77.73 84.42 0.9931 0.1159 0.2 20 0.1 0.1404 0.1710 0.1695 18.99 13.10 80.51 86.33 1.0026 0.1019 0.2 5 0.2 0.0999 0.2193 0.2083 33.72 22.40 64.38 75.48 0.9286 0.1929 0.2 10 0.2 0.1255 0.1862 0.1804 25.42 17.07 73.16 81.31 0.9786 0.1483 0.2 15 0.2 0.1350 0.1756 0.1716 21.19 14.29 77.64 84.36 0.9961 0.1246 0.2 20 0.2 0.1399 0.1705 0.1674 18.53 12.53 80.44 86.28 1.0050 0.1095 0.5 5 0.1 0.2493 0.5115 0.5012 34.14 22.98 64.95 76.01 0.6350 0.1750 0.5 10 0.1 0.3076 0.4417 0.4357 25.78 17.60 73.54 81.63 0.6869 0.1357 0.5 15 0.1 0.3287 0.4185 0.4143 21.50 14.76 77.94 84.60 0.7052 0.1144 0.5 20 0.1 0.3396 0.4071 0.4039 18.81 12.95 80.69 86.48 0.7146 0.1007 0.5 5 0.2 0.2480 0.5102 0.4889 33.47 22.10 64.65 75.81 0.6399 0.1893 0.5 10 0.2 0.3064 0.4405 0.4283 25.20 16.87 73.39 81.53 0.6903 0.1462 0.5 15 0.2 0.3277 0.4175 0.4090 20.99 14.13 77.84 84.53 0.7079 0.1230 0.5 20 0.2 0.3387 0.4062 0.3997 18.35 12.39 80.63 86.43 0.7168 0.1081 1.0 5 0.1 0.4788 0.8193 0.8103 33.35 22.04 65.77 77.00 0.4284 0.1646 1.0 10 0.1 0.5648 0.7418 0.7353 25.07 16.97 74.27 82.29 0.4796 0.1297 1.0 15 0.1 0.5948 0.7137 0.7088 20.88 14.25 78.57 85.13 0.4973 0.1098 1.0 20 0.1 0.6099 0.6993 0.6954 18.25 12.51 81.27 86.94 0.5063 0.0968 1.0 5 0.2 0.4777 0.8188 0.8000 32.71 21.21 65.47 76.80 0.4321 0.1784 1.0 10 0.2 0.5637 0.7410 0.7275 24.52 16.28 74.11 82.19 0.4824 0.1399 1.0 15 0.2 0.5937 0.7128 0.7028 20.39 13.65 78.48 85.06 0.4995 0.1182 1.0 20 0.2 0.6089 0.6985 0.6905 17.81 11.98 81.20 86.89 0.5081 0.1041 1.5 5 0.1 0.6674 0.9449 0.9407 32.38 20.89 66.75 78.18 0.3199 0.1523 1.5 10 0.1 0.7476 0.8942 0.8899 24.19 16.19 75.17 83.11 0.3696 0.1224 1.5 15 0.1 0.7745 0.8733 0.8698 20.09 13.61 79.38 85.80 0.3865 0.1041 1.5 20 0.1 0.7877 0.8621 0.8592 17.54 11.96 82.00 87.52 0.3950 0.0920 1.5 5 0.2 0.6668 0.9448 0.9358 31.77 20.11 66.46 77.99 0.3227 0.1656
187 Table 5.1 (contd.) Computed Values
B/D B/c a/c σ γ1 γ2 PE PD PE1 PD1 q/kH GEc/H 1.5 10 0.2 0.7468 0.8939 0.8850 23.66 15.53 75.02 83.01 0.3718 0.1322 1.5 15 0.2 0.7738 0.8728 0.8656 19.62 13.04 79.29 85.73 0.3883 0.1122 1.5 20 0.2 0.7870 0.8616 0.8557 17.12 11.45 81.93 87.47 0.3965 0.0990 2.0 5 0.1 0.8037 0.9854 0.9839 32.12 20.61 67.89 79.45 0.2493 0.1395 2.0 10 0.1 0.8618 0.9594 0.9572 23.41 15.50 76.08 83.93 0.2996 0.1150 2.0 15 0.1 0.8809 0.9470 0.9450 19.28 12.94 80.18 86.46 0.3161 0.0985 2.0 20 0.1 0.8904 0.9401 0.9384 16.78 11.36 82.74 88.11 0.3240 0.0872 2.0 5 0.2 0.8035 0.9854 0.9822 31.58 19.93 67.64 79.29 0.2512 0.1520 2.0 10 0.2 0.8614 0.9593 0.9547 22.88 14.86 75.93 83.83 0.3015 0.1244 2.0 15 0.2 0.8805 0.9468 0.9427 18.83 12.40 80.09 86.40 0.3176 0.1062 2.0 20 0.2 0.8900 0.9399 0.9364 16.38 10.87 82.67 88.06 0.3252 0.0939 0.2 10 0.3 0.1248 0.1854 0.1765 24.80 16.27 73.00 81.21 0.9827 0.1613 0.5 10 0.3 0.3052 0.4393 0.4206 24.59 16.09 73.23 81.42 0.6940 0.1591 1.0 10 0.3 0.5624 0.7402 0.7192 23.93 15.53 73.95 82.08 0.4853 0.1524 1.5 10 0.3 0.7460 0.8935 0.8795 23.10 14.82 74.86 82.90 0.3742 0.1442 2.0 10 0.3 0.8610 0.9592 0.9518 22.32 14.16 75.77 83.72 0.3035 0.1360 0.2 10 0.4 0.1241 0.1846 0.1725 24.12 15.40 72.83 81.09 0.9871 0.1777 0.5 10 0.4 0.3039 0.4380 0.4125 23.92 15.23 73.06 81.30 0.6978 0.1753 1.0 10 0.4 0.5611 0.7392 0.7102 23.28 14.70 73.78 81.96 0.4884 0.1682 1.5 10 0.4 0.7451 0.8931 0.8735 22.48 14.04 74.69 82.78 0.3767 0.1593 2.0 10 0.4 0.8605 0.9590 0.9486 21.70 13.40 75.60 83.61 0.3056 0.1505
Comparison with Infinite Depth Case when a/c = 0
Malhotra (1936) obtained theoretical solution for uplift pressure below flat apron with
equal end cutoffs, for the case of infinite depth of the pervious foundation. The results of the
present theoretical solution for B/D = 0.4 and B/D = 0.2 were compared with those of Malhotra's
solution. Table 5.2 gives the values of uplift pressure PE at point E and PD at point D, for some
188 selected values of B/c ratios. It is seen from Table 5.2 that when BD/0.2 , the results of both the analyses are practically the same.
Table 5.2 Comparison with Malhotra's Case of Infinite Depth
% Pressure PE at E % Pressure PD at point D B/c Infinite Present Present Infinite Present Present Depth case Analysis for analysis for Depth case Analysis for analysis for by Malhotra B/D = 0.2 B/D = 0.4 by Malhotra B/D = 0.2 B/D = 0.4 3 41.4 41.3 41.1 28.4 28.2 28.0 4 37.9 37.8 37.7 26.1 26.0 25.7 6 32.9 32.7 32.6 22.7 22.6 22.4 8 29.2 29.2 29.0 20.3 20.3 20.1 12 24.6 24.5 24.4 17.2 17.1 17.0 24 17.8 17.8 17.7 12.6 12.5 12.4
Effect of Step at Downstream end
Fig. 5.2 (a) shows the effect of increase in step ratio (a/c) on PE, the uplift pressure at point E, when the B/c = 10. Fig. 5.2 (b) shows the effect of increase in step ratio (a/c) on PD, the pressure at point D. Similarly, Fig. 5.3 (a), 5.3 (b), 5.4 (a) and 5.4 (b) show the variation of PE1,
PD1, q/kH and GE c/H respectively, with increase in step ratio a/c from 0.0 to 0.4, when B/c = 10, for different values of B/D ratios.
5.4 Design Charts
The values of seepage characteristics PE, PD, PE1, PD1, q/kH and GE c/H obtained corresponding to some selected pairs of floor profile ratios (B/D and B/c) and selected values of step ratio a/c (= 0.1, 0.2, 0.3, 0.4) were used to prepare design charts for use in design office.
Figs. 5.5 (a) and 5.5 (b) show the design charts for uplift pressure PE, for step ratio a/c = 0.1 and
0.2 respectively. Similarly, Figs. 5.6, 5.7, 5.8, 5.9, and 5.10 show design charts for PD, PE1, PD1, seepage discharge factor (q/kH) and exit gradient factor (GE c/H) respectively.
189
Fig. 5.2a Variational curve for PE for B/c = 10
Fig. 5.2b Variational curve for PD for B/c = 10
190
Fig. 5.3a Variational curve for PE1 for B/c = 10
Fig. 5.3b Variational curve for PD1 for B/c = 10
191
Fig. 5.4a Variational curve for discharge factor for B/c = 10
Fig. 5.4b Variational curve for exit gradient factor for B/c = 10
192
Fig. 5.5a Design curve for PE for a/c = 0.1
Fig. 5.5b Design curve for PE for a/c = 0.2
193
Fig. 5.6a Design curve for PD for a/c = 0.1
Fig. 5.6b Design curve for PD for a/c = 0.2
194
Fig. 5.7a Design curve for PE1 for a/c = 0.1
Fig. 5.7b Design curve for PE1 for a/c = 0.2
195
Fig. 5.8a Design curve for PD1 for a/c = 0.1
Fig. 5.8b Design curve for PD1 for a/c = 0.2
196
Fig. 5.9a Design curve for discharge factor for a/c = 0.1
Fig. 5.9b Design curve for discharge factor for a/c = 0.2
197
Fig. 5.10a Design curve for exit gradient factor for a/c = 0.1
Fig. 5.10b Design curve for exit gradient factor for a/c = 0.2
198 5.5 Conclusions
The closed-form solution to the problem of finite depth seepage under an impervious flat apron with equal end cutoffs, with a downstream step, is obtained using the conformal transformations. The results obtained from the solution of implicit equations have been used to present design charts for various seepage characteristics such as uplift pressures at key points, seepage discharge factor and exit gradient factor in terms of non-dimensional floor profile ratios.
It is seen that uplift pressures at points E, D, E1 and D1 decrease with the increase in step ratio.
However, the seepage discharge factor increases by very small margin, while the exit gradient factor increases sharply with increase in the downstream step. Also, for a given step ratio (a/c), the uplift pressures at points E and D, as well as exit gradient increase with decrease in B/D ratio, and are maximum for infinite depth of pervious medium. When the depth of permeable soil is large, the numerical solution tends towards the values for infinite depth.
199 CHAPTER 6 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
6.1 Summary and Conclusions
The present work envisages a theoretical study of seepage characteristics below the weir aprons of various boundary conditions, including a step at downstream side. The depth of pervious medium has been taken to be finite. Closed form theoretical solutions have been found by following the procedure originally suggested by Pavlovsky. The work has been sub-divided into four cases. Analysis and conclusions for each case are summarized below.
6.1.1 Flat Apron with Equal End Cutoffs
(a) Effect of increase in B/c ratio
1. For a given B/D ratio, the uplift pressures at point E and D decrease with increase in B/c
ratio, while the reverse is true for uplift pressures at points C and D'.
The decrease in B/c ratio corresponds to the increase in the depth of piles and increase in
pile penetration ratio. Hence greater depth of cut off results in an increase in uplift
pressure at key point E and D. However, greater depth of cutoff results in decrease in
pressure at key points C and D'.
2. For a given B/D ratio, the seepage discharge factor increases marginally with the increase
in B/c ratio. However, this increase is more for smaller values of B/D ratio and less for
higher value of B/D ratio.
3. For a given B/D ratio, the exit gradient factor decreases with increase in the B/c ratio.
However, the actual exit gradient increases. The increase in B/c ratio corresponds to the
decrease in depth of pile, which results in the increase in the exit gradient.
200 (b) Effect of depth of pervious medium
1. For given B/c ratio, uplift pressure at key points E and D increase with decrease in B/D
ratio. The reverse is true for uplift pressure at key point C and D'. For the downstream
half of the apron, the uplift pressures increases with decrease in B/D ratio while the
reverse is true for the upstream half of the apron.
2. For a given B/c ratio, the seepage discharge factor increases with the decrease in the B/D
ratio. As expected, the seepage discharge factor decreases with decrease in the depth of
pervious media.
3. For a given B/c ratio, the exit gradient factor and hence actual exit gradient, increases
with decrease in the B/D ratio. The infinite depth case gives the maximum exit gradient.
4. When BD/ 0.2, the values of uplift pressure ()PPED, and the exit gradient factor
(/GE c) H are quit close to the corresponding values for the infinite depth case. Hence
when the depth of pervious medium is equal to or more than five times the length of the
impervious apron, the depth of pervious medium may be assumed to be infinite.
5. The assumption of infinite depth of pervious medium results in greater values of uplift
pressure (,)PPED and exit gradient. Hence the current design practice of the use of design
curves for infinite depth case in safe though conservative.
201 (c) Use of Design Curves
The design curves for uplift pressures at key points E and D and for exit gradient factor
(/)GE c H will be useful for determining these without running the computer programme. The uplift pressures at key points C and D' can be found from the following relations based on principle of reversibility of flow:
PC 100 PE and PD 100 PD
6.1.2 Flat Apron with Unequal End Cutoffs and a Step at Downstream End
(a) Effect of increase in step ratio (a/c2)
1. Uplift pressure PE and PD at point E and D, respectively, decrease almost linearly with
increase in step ratio a/c2. The variation curves for various B/D ratios are almost parallel.
2. Uplift pressure PE1 and PD1 at point E1 and D1, respectively, also marginally decrease
linearly with increase in step ratio. Here also, the variation curves for various B/D ratios
are almost parallel.
3. There is very small increase in seepage discharge factor (q/kH) with the increase in step
ratio (a/c2). Here again, the curves corresponding to various values of B/D ratio are
almost linear and parallel.
4. There is very sharp increase in exit gradient factor GE c2/H (and hence the exit gradient
itself) with increase in step ratio (a/c2). The curves corresponding to various values of
B/D are almost parallel, though curvilinear.
202 b) Effect of increase in B/c2 ratio
1. For a given B/D ratio, the uplift pressure at point E and D decrease with increase in B/c2
ratio, while the reverse is true for uplift pressure at point E1 and D1. The decrease in B/c2
ratio corresponds to increase in the depth of downstream pile. Hence greater depth of
downstream cutoff results is an increase of uplift pressure at key points E and D.
However, greater depth of downstream cutoff results in decrease in pressure at point E1
and D1.
2. For a given B/D ratio, the seepage discharge factor increase only marginally with the
increase in B/c2 ratio.
3. For a given B/D ratio, the exit gradient factor decrease with increase in B/c2 ratio.
However, the actual exit gradient increases. The increase in B/c2 ratio corresponds to
decrease in depth of downstream pile, which results in the increase in the exit gradient.
(c) Effect of c1/c2 ratio
1. From Table 3.1, we find that for a given values of B/c2 ratio, uplift pressure PE, PD, PE1
and PD1 at all the four key point decrease with increase in c1/c2 ratio (i.e. with increase in
the depth of upstream pile).
2. The seepage discharge factor also decreases with the increase in c1/c2 ratio.
3. Similarly, for a given values of B/D ratio and B/c2 ratio, the exit gradient factor, and
hence the exit gradient decreases with increase in the c1/c2 ratio. Hence contrary to the
common belief, the depth of upstream cutoff also helps in controlling the exit gradient,
though at much flatter rate.
203 (d) Effect of depth of pervious medium
1. For a given B/c2 ratio, the uplift pressure at key point E and D increase with decrease in
B/D ratio. The reverse is true for uplift pressure at key point E1 and D1. For the
downstream half of the apron, the uplift pressures increase in decrease in B/D ratio while
the reverse is true for the upstream half of the apron.
2. For a given B/c2 ratio, the seepage discharge factor increases with decrease in B/D ratio.
As expected, the seepage discharge factor decreases with decrease in the depth of
pervious medium.
3. For a given B/c2 ratio, the exit gradient factor, and hence the exit gradient increases with
the decrease in B/D ratio. The infinite depth case gives maximum exit gradient.
4. When BD/ 0.2 , the values of uplift pressures (PE, PD) and the exit gradient factor (GE
c2/H) are quite close to the corresponding values for the infinite depth case. Hence when
the depth of pervious medium is equal or more than five times the length of impervious
apron, the depth of pervious medium may be assumed to be infinite.
5. The assumption of infinite depth of pervious medium results in higher values of uplift
pressure (PE, PD) and exit gradient. Hence the current design practice of use of design
curves for infinite depth case is safe though conservative.
6.1.3 Flat Apron with Unequal End Cutoffs
(a) Effect of increase in B/c2 ratio
1. For a given B/D ratio, the uplift pressures at points E and D decrease with increase in B/c2
ratio, while the reverse is true for uplift pressures at point E1 and D1. The decrease in B/c2
ratio corresponds to increase in the depth of downstream pile. Hence greater depth of
204 downstream cutoff results in an increase of uplift pressure at key points E and D. However,
greater depth of downstream cutoff results in decrease in pressures at point E1 and D1.
2. For a given B/D ratio, the seepage discharge factor increases marginally with the increase in
B/c2 ratio.
3. For a given B/D, ratio the exit gradient factor decrease with increase in B/c2 ratio. However,
the actual exit gradient increases. The increase in B/c2 ratio corresponds to decrease in depth
of downstream pile, which results in the increase in the exit gradient.
(b) Effect of c1/c2 ratio
1. From Table 4.1, we find that for a given value of B/c2 ratio, uplift pressure PE, PD, PE1 and
PD1 at all the four key points decrease with increase in c1/c2 ratio (i.e. with increase in the
depth of upstream pile).
2. The seepage discharge factor also decreases with the increase in c1/c2 ratio.
3. Similarly for a given values of B/D ratio and B/c2 ratio, the exit gradient factor, and hence the
exit gradient decreases with increase in the c1/c2 ratio. Hence contrary to the common belief,
the depth of upstream cutoff also helps in controlling the exit gradient, though at a much
flatter rate.
(c) Effect of depth of pervious medium
1. For a given B/c2 ratio, the uplift pressure at key point E and D increase with decrease in B/D
ratio. The reverse is true for uplift pressure at key point E1 and D1. For the downstream half
of the apron, the uplift pressures increase with decrease in B/D ratio while the reverse is true
for the upstream half of the apron.
205 2. For a given B/c2 ratio, the seepage discharge factor increases with decrease in B/D ratio. As
expected, the seepage discharge factor decreases, with decrease in the depth of pervious
medium.
3. For given B/c2 ratio, the exit gradient factor, and hence the exit gradient increases with the
decrease in B/D ratio. The infinite depth case gives maximum exit gradient.
4. When BD/0.2 , the values of uplift pressure (PE, PD) and the exit gradient factor (GE c2/H)
are quite close to the corresponding values for the infinite depth case. Hence when the depth
of pervious medium is equal or more than five times the length of impervious apron, the
depth of pervious medium may be assumed to be infinite.
5. The assumption of infinite depth of pervious medium results in higher values of uplift
pressures (PE, PD) and exit gradient. Hence the current design practice of use of design
curves for infinite depth case is safe though conservative.
6.1.4 Flat Apron with Equal End Cutoffs and a Step at the Downstream End
(a) Effect of increase in step ratio (a/c)
1. Uplift pressure PE and PD at point E and D, respectively, decrease almost linearly with
increase in step ratio a/c. The variation curves for various values of
B/D = 0.2, 0.5, 1, 1.5 and 2 are almost parallel.
2. Uplift pressures PE1 and PD1 at point E1 and D1, respectively, also marginally decrease
linearly with increase in step ratio. Here also, the variational curves for various B/D ratios
are almost parallel.
206 3. There is very minute increase in seepage discharge factor (q/kH) with the increase in step
ratio (a/c). Here again, the curves corresponding to various values of B/D ratio are almost
linear and parallel.
4. There is very sharp increase in exit gradient factor GE c/H (and hence the exit gradient
itself) with increase in step ratio (a/c). The curves corresponding to various values of B/D
ratio are almost parallel, though curvilinear.
(b) Effect of increase in B/c ratio
1. For a given B/D ratio, the uplift pressures at points E and D decrease with increase in B/c
ratio, while the reverse is true for uplift pressures at points E1 and D1. The decrease in B/c
ratio corresponds to increase in the depth of downstream pile. Hence greater depth of
downstream cutoff results is an increase of uplift pressure at key points E and D.
However, greater depth of downstream cutoff results in decrease in pressures at point E1
and D1.
2. For a given B/D ratio, the seepage discharge factor increases only marginally with the
increase in B/c ratio.
3. For a given B/D ratio, the exit gradient factor decreases with increase in B/c ratio.
However, the actual exit gradient increases. The increase in B/c ratio corresponds to
decrease in depth of downstream pile, which results in the increase in the exit gradient.
(c) Effect of depth of pervious medium
1. For a given B/c ratio, the uplift pressure at key points E and D increase with decrease in
B/D ratio. The reverse is true for uplift pressures at key points E1 and D1. For the
207 downstream half of the apron, the uplift pressures increase with decrease in B/D ratio
while the reverse is true for the upstream half of the apron.
2. For a given B/c ratio, the seepage discharge factor increases with decrease in B/D ratio.
As expected, the seepage discharge factor decreases with decrease in the depth of
pervious medium.
3. For a given B/c ratio, the exit gradient factor, and hence the exit gradient increases with
the decrease in B/D ratio. The infinite depth case gives maximum value of exit gradient.
4. When BD/0.2 , the values of uplift pressures (PE , PD) and the exit gradient factor (GE
c/H) are quite close to the corresponding values for the infinite depth case. Hence when
the depth of pervious medium is equal to or more than five times the length of impervious
apron, the depth of pervious medium may be assumed to be infinite.
5. The assumption of infinite depth of pervious medium results in higher values of uplift
pressures (PE, PD) and exit gradient. Hence the current design practice of use of design
curves for infinite depth case is safe though conservative.
6.2 Recommendations for Future Work
The present work formulates an exact solution for the problems considered in this dissertation. The solutions contain unknown parameters in implicit form in the equations derived. When the unknown parameters are three or more, the iteration for the computations becomes extremely difficult. Hence, an inverse method of solution of equations has to be employed for hydraulic structures having an intricate subsurface profile. The closed form solution method used here is more useful and practical when the pressure distribution along the boundaries is to be determined.
208 This method is applicable to all those situations where Laplace equation is applicable.
Hence, the method of solution is not applicable to heterogeneous soils, where the soil properties change from point to point. However, it is applicable for homogeneous anisotropic soils.
There are many seepage scenarios under hydraulic structures where analytical solution has not been attempted. The author proposes four different weir profiles with flat aprons on finite depth, where analytic solution could be found. These are shown in Fig. 6.1. The first weir profile shown in Fig. 6.1a is a flat apron with two unequal intermediate cutoffs. Another case (Fig. 6.1b) is an inclined apron with single end cutoff or with double end cutoffs. The inclined floors are provided for energy dissipation through formation of hydraulic jumps. The effect of width of concrete cutoff may have an appreciable effect on various seepage characteristics. This case, as shown in Fig. 6.1c is equally interesting. Finally, the effect of crack in a downstream end cutoff, leading to leakage offers a challenging problem for analytical-solution seekers (Fig. 6.1d).
The research for obtaining seepage characteristics under hydraulic structures was initiated for infinite depth cases. An impervious boundary may occur at a finite depth, which alters seepage characteristics in the seepage domain considerably. Only a few cases have been solved for finite depth of pervious medium; hence it is recommended to lay more emphasis on finite depth scenarios.
209
Fig. 6.1 Finite depth problems for future work
210 APPENDIX A. SCHWARZ – CHRISTOFFEL TRANSFORMATION
211
A seepage scenario is governed by Laplace equation. To determine the seepage characteristics, the solution of Laplace equation is required. Consider any region under seepage. This region has infinite sets of curvilinear orthogonal lines, namely the equipotential lines and the stream-lines.
If for any such curvilinear square, velocity potential (Φ) and stream function (Ψ) are known, then the Laplace equation is said to be solved at that point. Since, the Laplace equation is continuous in this region, values of Φ and Ψ for all curvilinear squares or for the whole flow-net must be known.
Let us assume that we have a known square that represents the seepage region. The values of Φ and Ψ on this square would then indicate a solution of Laplace equation for the region. A mathematical process can be found, which can relate this known rectangular field (Φ –
Ψ plane, also known as w – plane) to the real flow domain. That process could give us the change in the expressions of equipotential lines and stream-lines in the real domain. This leads us to the theorem by Schwarz and Christoffel. The theorem gives us a differential equation which embodies the nature of change in a function of real plane, say z-plane, bounded by any straight- sided figure in order that the boundary may be opened out into one straight line; the opening-out involving, of course, the distortion of the function.
Thus, if F1 is a function representing the flow region in z-plane, and if it can be transformed into a straight line in t-plane, this relationship can be written as
z = F1(t)
Similarly, if the known rectangular field in w-plane is represented by a function F2, and if it can be transformed into the same straight line (mentioned above) in the t-plane, following relationship is known:
w = F2(t)
212 Combining both the above equation, we have
z = f(w) where all the elements of flow in the real domain can be determined.
This transformation technique actually maps conformally a polygon consisting of straight lines onto a similar polygon in the lower half of t-plane, in a manner that the sides of the polygon in the z-plane extend through the real axis of t-plane. This conformal mapping technique is known as Schwarz-Christoffel transformation (here-after, called S-C transformation) method. It is a specific method of transformation from one plane onto the lower half of another plane.
Fig. A-1 Schwarz-Christoffel transformation
Consider a closed polygon ABCDE in z-plane. Its mapping on t-plane is accomplished by opening the polygon at some convenient point (say, P) between E and A (Fig. A-1a) and extending one side to t = +∞ to t = -∞ (Fig. A-1b).
The governing equation for S-C transformation is
dt z M N a b n t ta t t b ...... t t n
213 interior angle at point i in z-plane Where 1 , i
and ta, t b ,.... t n co-ordinates of the mapped points on t-plane.
Notes: 1) The interior angle at point P of the opening in the z-plane is π.
interior angle at P P 11 0 Since , point P has no part in the transformation.
However, the point of opening, P in the z-plane is represented in the lower-half of the t-
plane by a semi-circle of infinite radius.
2) The S-C transformation, in effect maps conformally the interior region of the polygon
ABCDE of the z-plane into the interior of the polygon bounded by ab, bc, cd,…. and a
semi-circle with a radius of infinity in the lower half of the t-plane.
3) The S-C transformation theorem is applicable only for mapping the sides of closed
polygon of z-plane into a straight line profile of t-plane, where the angles between the
sides of π.
4) The origin of the t-plane can be located at any convenient point.
5) Any three values of ta, t b , tt cn ,..... can be chosen arbitrarily, while the remaining n - 3
values of t-coordiantes are found from boundary conditions.
214 APPENDIX B. ELLIPTIC INTEGRALS
215 1. Types of Elliptic Integrals Integrals of the form R[ t ,( )] P t , dt where P(t) is a polynomial of the third or fourth degree and R is a rational function, are known as elliptic integrals because a special example of this type arose in the rectification of the arc of an ellipse.
Thus the integral
4 3 2 I Rt[ ,] at0 at1 at 2 atadt 3 4 is called the elliptic integral if the equation
4 3 2 a0 t a1 t a 2 t a 3 t a 4 0 (a0 and a1 not both zero) has no multiple roots and if R is a rational function of t and of the square root of the above equation.
Although some early work on these was done by Fragnano, Euler, Legendre and Landen, they were first treated systematically by Legendre, who showed that any elliptic integral may be made to depend on three fundamental integrals which he denoted by F( , m ), E ( , m ) and
( , 2 ,m ). These three integrals are called Legendres canonical elliptic integrals of the first, second and third kind respectively.
(i) Elliptic Integral of First Kind
The elliptic integral of first kind in canonical form is
y dt (1) 2 2 2 0 (1t )(1 m t ) where m is the modulus of the integral. Substituting t sin in the above equation, we have
216 d Fm(,) (1a) 22 0 1sinm where sin1 y amplitude of the integral. Fm( ,) is in Legendre notation and is the usual tabular notation. In many tables, the modulus is given as
m sin where is the modular angle.
When , we get complete elliptic integral of first kind: 2
/2 d K Fm, (1b) 2 22 0 1m sin
(ii) Elliptic Integral of Second Kind
The elliptic integral of the second kind in the canonical form of Legendre is defined as
y 1 mt22 dt (2) 2 0 1 t
Substituting t sin , we have
Em(,) 1 md22 sin (2a) 0
where is the amplitude and E (, m) is Legendre's notation for the elliptic integral of second kind. When we get complete elliptic integral of second kind:
/2 22 E E, m 1 m sin d (2b) 2 0
217 (iii) Elliptic Integral of Third Kind
The elliptic integral of the third kind in the canonical form of Legendre is:
y dt (ym ,,2 ) (3) 2 222 2 0 (1) (1ttm )(1) t where m is the modulus and 2 is the parameter.
Substituting t sin ,
d ( ,, 2 ) m (3a) 2 22 2 0 (1 sin ) (1 m sin )
where is the amplitude. When = /2, we get the complete elliptic integral of third kind.
/2 2 d 0 ,, m (3b) 2 2 2 2 2 0 (1 sin ) (1 m sin )
2. JACOBIAN ELLIPTIC FUNCTIONS
Let us consider the equivalence of the integrals
y dt d u (4) 2 2 2 22 0 (1t )(1 m t ) 0 1m sin
which are related by the substitution t = sin .
The Jacobian elliptic functions are defined by the relations:
sn u = sin , cn u = cos and dn u 1 m22 sin
or by the equivalent set
sn u = y cn u = 1 y2 and dn u 1 my22
218 The amplitude of the integral is written as
= am u
From Eqs. 4 we have
u = F ( , m) = sn–1 (y, m) (5)
where sn–1 (y, m) is the inverse of the Jacobian elliptic function `Sinus Amplitudinis’
(sn u).
There are in all twelve Jacobian Elliptic Functions. However, the above three functions are more commonly used and are related by the following equations:
sncn22uu 1
m2sn 22 u cn u
dncn2u m 2 2 u m2 (6) and m2sn 2 u cn 2 u dn2 u
The Jacobian elliptic functions are related to Legendre's Elliptic Integrals by the following equations:
Fm(,) du u sn1 ( y , m ) (7 a)
Em(,) dn2 u. du (7 b)
du (,,) 2 m (7 c) 122 sn u
219 REFFRENCES
Aalto, J. (1984). “Finite element seepage flow nets.” Int. J. Numer. Anal. Meth. Geomech. Vol. 8(3), pp. 297-303. Abdel-Gawad, Hossam A. and Shamaa, Mohamed T. (2004). “Seepage through earth dams with horizontal filters and founded on impervious foundation (numerical analysis with boundary element method).” Alexandria Engineering Journal, Vol. 43, No. 6, pp. 891-903. Abdrabbo, F. M. and Mahmoud, M. A. (1991). “Interruption of a seepage flow from a water source.” Applied Mathematical Modelling, Vol. 15, Issue 8, pp. 406-415. Abdulrahman, A. and Mardini, Jack (2010). “Seepage analysis under dams with vertical sheet pile using Schwarz-Christoffel Transformation.” International Journal of Geotechnical Engineering, Vol. 4, No.4, pp. 537-547. Ahmed, A. A. and Bazaraa, A. S. (2009). “Three-Dimensional Analysis of Seepage below and around Hydraulic Structures.” J. Hydrol. Eng., Vol. 14, no. 3, pp. 243-247. Ahmed, Ashraf A. (2009). “Stochastic analysis of free surface flow through earth dams.” Computers and Geotechnics, Vol. 36, Issue 7, pp. 1186-1190. Ahmed, Ashraf A. and Elleboudy, Azza M. (2010). “Effect of Sheetpile Configuration on Seepage beneath Hydraulic Structures.” ASCE Conf. Proc. on Scour and Erosion (ICSE-5), pp. 511-518. Aral, M. M. (1989). “Semianalytic boundary element solution of groundwater seepage problems.” Water Resources Research, Vol. 25, No. 7, pp. 1495-1503. Aravin, Vladimir Ivanovich and Numerov, S. N. (1965). Theory of fluid flow in undeformable porous media. Israel Program for Scientific Translations. Banerjee, S. and Muleshkov, A. (1992). "Analytical solution of steady seepage into Double-Walled coffer Dams." ASCE, Journal of Engineering Mechanics, Vol. 118, No. 3. Bear, J, Beljin, Milovan S. and Ross, Randall R. (1996). 'Fundamentals of Ground-Water Modeling' Chapter 9: EPA environmental assessment sourcebook, Edited by Boulding, Russell J., CRC Press. Benmebarek, N., Benmebarek, S., and Kastner, R. (2005). “Numerical studies of seepage failure of sand within a cofferdam.” Computers and Geotechnics, Vol. 32, Issue 4, pp. 264-273. Bereslavskii E. N. and Aleksandrova L. A. (2009). “Modeling of the base of a hydraulic structure with constant flow velocity sections and a curvilinear confining layer.” Russian Mathematics (Iz. VUZ), Vol. 53, No. 3, pp. 61–66.
220 Bereslavskii, E.N. (2009). “The design of the iso-velocity contour for the flow past the base of a dam with a confining bed.” Journal of Applied Mathematics and Mechanics, Vol. 73, Issue 4, pp. 426-433. Bligh, W.G. (1910). Practical Design of Irrigation Works, Second Edition. Van Nostrand. Brebbia, C. A. and Wrobel, L. C. (1979). “Boundary element method for fluid flow.” Advances in Water Resources, Vol. 2, pp. 83-89. Bruch Jr., John C., Frederick C. Sayle, Sloss, James M. (1978). “Seepage from a trapezoidal and a rectangular channel using variational inequalities.” Journal of Hydrology, Vol. 36, Issues 3-4, pp. 247-260. Bruch, John C., Jr., Fernandez Sainz and Leandro B. (1972). “Analytical Flow Nets in Channel Seepage Flows.” Water Resources Research, Vol. 8, Issue 2, pp. 519-524. Byrd, P. F. and Friedman, M.D. (1971). Hand Book of Elliptic Integrals for Engineers and Scientists, Second Edition, Springer Verlag. Desai, C. S. (1975). “Free surface seepage through foundation and berm of cofferdams.” Indian Geotechnical Journal, Vol. 5, No. 1, pp 1 – 10. Desai, C.S. (1976) “Finite element residual schemes for unconfined flow.” International Journal for Numerical Methods in Engineering, Vol. 10, pp. 1415-1418. Desai, C. S. and Beseghi, B. (1988). “Theory of residual flow procedure for 3-D free surface seepage.” Advances in Water Resources, Vol. 11, No. 4, pp. 192 – 203. Caffrey, J. and Bruch Jr., J. C. (1979). “Three-dimensional seepage through a homogeneous dam.” Advances in Water Resources, Vol. 2, pp. 167-176. Chang, Ching S. (1988). “Boundary-Element analysis for unconfined seepage problems.” Journal of Geotechnical Engineering, Vol. 114, No. 5, pp. 556-572. Chawla, A. S. (1975). “Stability of structures with intermediate filters.” Journal of Hydraulic Engineering. Vol. 101, No. 2, pp. 223-241. Chawla, A. S. and Kumar, A. (1983). "Stability of structure with two-end cutoffs." Journal of Institution of Engineers (India), Vol. 63, No. 5, pp. 272-279 Chen, J.T., Hong, H.-K. and Chyuan, S.W. (1994). “Boundary element analysis and design in seepage problems using dual integral formulation.” Finite Elements in Analysis and Design, Vol. 17, Issue 1, pp. 1-20. Choi, E. C. C. (1978). “A finite element study of steady state flow in an unconfined aquifer resting on a sloping bed.” Water Resources Research, Vol. 14, No. 3, pp. 391-394. Christian, J. T. (1980). “Flow nets from finite element data.” Int. J. Numer. Anal. Meth. Geomech., 4(2), pp. 191-196.
221 Christoulas, Demetrius G. (1971). “Approximate Solution to Flow Problem under Dams.” Journal of the Soil Mechanics and Foundations Division, Vol. 97, No. 11, pp. 1573-1592. Chugh, A. K. (1988). “Flow nets for zoned anisotropic media by the boundary element method.” Computers & Structures, Vol. 29, Issue 2, pp. 207-220. Clibborn, C. and Beresfod, J.S. (1902). "Experiment on the Passage of water through sand". Govt. of India, Central Printing office. Colman, J. B. T. (1916). "The action of water under dam", ASCE paper No. 1356. Cooley, R. L. (1971). “A finite difference method for unsteady flow in variably saturated porous media: application to a single pumping well.” Water Resource Research 7(6), pp. 1607–1625. Das, B., Steinberg, S., Weber, S., and Schaffer, S. (1994). “Finite difference methods for modeling porous media flows.” Transport in Porous Media. Vol. 17, No. 2, pp. 171-200. Demetracopoulos, A. C. and Hadjitheodorou, C. (1996). “Seepage from surface canals by boundary element method.” Journal of Irrigation and Drainage Engineering, Vol. 122, no. 1, pp. 40-48. Dennis, S. C. R. and Smith, F. T. (1980). “Steady flow through a channel with a symmetrical constriction in the form of a step.” Proceedings, Series A - Mathematical and Physical Sciences, Royal Society (London), Vol. 372, No. 1750, pp. 393-414. Desai, Chandrakant S. (1973). “Approximate solution for unconfined seepage.” Journal of the Irrigation and Drainage Division, Vol. 99, No. 1, pp. 71-87. Desai, Chandrakant S. (1976). “Finite element residual schemes for unconfined flow.” Int. J. Numer. Meth. Engng., 10(6), pp. 1415-1418. Devison, B. B. (1937). “A simplified scheme for the computation of confined ground water flow.” (In Russian). Trudy Gosudarstvennogo gidrologicheskogo institute, No. 5. Doctors, L. J. (1970). “An application of the finite element technique to boundary value problems of potential flow.” Int. J. Numer. Meth. Engng., 2(2), pp. 243-252. Elganainy, M. A. (1986). “Flow underneath a pair of structures with intermediate filters on a drained stratum.” Applied Mathematical Modelling, Vol. 10, Issue 6, pp. 394-400. Elsworth, D. (1987). “Boundary Element - Finite Element procedure for porous and fractured media flow.” Water Resources Research WRERAQ Vol. 23, No. 4, pp. 551-560. Farouk, M. I. and Smith, I. M. (2000). “Design of hydraulic structures with two intermediate filters.” Applied Math Modelling. Vol. 24, No. 11, pp. 779-794. Faust, C. R. and Mercer, J. W. (1980). “Ground-water modelling: numerical models.” Ground Water, Vol. 18, No. 4, pp. 395-409.
222 Filchakov, P. F. (1959, 60). "The Theory of Filtration beneath Hydro-technical Structures” (In Russian): Vols.1 and 2, Izd. AN UkrSSR, Kiev. Filho, O. Piermatei and Leontiev, A. (2009). “An optimization approach for unconfined seepage problem with semipermeable conditions.” Structural and multidisciplinary optimization, Vol. 39, Issue 6, pp. 581 -588. Florea, J., Seteanu, I. and Popa, R. (1980). “Finite Element Method in Solving Some Unusual Problems of Infiltration in Porous Media. Finite Elements in Water Resources. Volume 1.” Preprints of the Third International Conference (2 Vol.), Mississippi University, Oxford, May 19-23, 1980. pp. 2.239-2.248. France, P. W. (1974). “Finite element analysis of three-dimensional groundwater flow problems.” Journal of Hydrology, Vol. 21, Issue 4, pp. 381-398. France, P. W., Parekh, C., Peters, J. C., and Taylor, C. (1971). “Numerical analysis of free surface seepage problems.” Journal of the Irrigation and Drainage Division, Vol. 97, No. 1, pp. 165-179. Freeze, R. A. (1971). “Influence of the unsaturated flow domain on seepage through earth dams.” Water Resources Research, Vol. 7, No. 4, pp. 929-941. Gipson, G. S., Ortiz, J. C., Brewer, J. A. III, Walters, H. G., and Cook, R. M. (1986). “A graphics driven boundary element system.” Betech 86; Proceedings of the Second Boundary Element Technology Conference, Cambridge, MA, UK. Goel, A. and Pillai, N. N. (2010). "Variation of exit gradient downside of weirs on permeable foundations." The Pacific Journal of Science and Technology, Vol. 11, No.1, pp. 28-36. Griffiths, D. V. (1984). “Rationalized charts for the method of fragments applied to confined seepage.” Géotechnique, Vol. 35, Issue 3, pp. 375 –377. Griffiths, D. V. and Fenton, Gordon A. (1993). “Seepage beneath water retaining structures founded on spatially random soil.” Geotechnique. Vol. 43, no. 4, pp. 577-87. Griffiths, D. V. and Fenton, Gordon A. (1997). “Three-Dimensional Seepage Through Spatially Random Soil.” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 123, No. 2, pp. 153-160. Griffiths, D. V. and Fenton, Gordon A. (1998). “Probabilistic Analysis of Exit Gradients Due to Steady Seepage.” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 124, No. 9, pp. 789-797. Gureghian, A. B. (1978). “Solutions of Boussinesq's Equation for Seepage Flow.” Water Resources Research, Vol. 14, Issue 2, pp. 231-236. Haigh, F. F. (1935). "Design of weirs on sand foundations." Punjab Engineering Congress Paper No. 182. Harr, M.E. (1962). Ground Water and Seepage, McGraw Hill Book Co. Inc., New York. Harza, L.F. (1934). "Uplift and seepage under dams on sand." Proceedings, ASCE, Paper No. 1920.
223 Herbert, R. (1968). “Seepage under Sheet Piles.” Civ. Eng. Vol. 63, No. 746, pp. 977-980. Herrera, I. (1980). "Boundary Methods in flow problems." Proceedings Third International Conference on Finite Elements in Flow Problems, Banff, Canada, 10-13 June, Vol. 1, pp. 30-42. Hlepas, Georgette (2008). “Stabilized finite element methods for porous media flows.” Dissertation Abstracts International. Vol. 69, No. 6. Hnang, Tien-kuen (1996). “Stability analysis of an earth dam under steady state seepage.” Computers & Structures, Volume 58, Issue 6, pp. 1075-1082. Hromadka II, T. V. (1984). “A simple model of a phreatic surface through an earth dam”. Advances in Water Resources, Vol. 7, Issue 3, pp. 141-143. Huntoon, R. W., (1974). Finite difference methods as applied to the solution of groundwater flow problems, Wyo. Water Resour. Res. Inst., University of Wyoming, Laramie, WY, 98 pp. Igboekwe, Magnus. U. and Achi, N. J. (2011). “Finite Difference Method of Modelling Groundwater Flow.” Journal of Water Resource and Protection. 3(3), pp. 192-198. Ijam, Abbas Z. (1994). “Conformal analysis of seepage below a hydraulic structure with an inclined cutoff.” Int. J. Numer. Anal. Meth. Geomech. 18(5), pp. 345-353. Ilyinsky N. B., Kacimov, A. R. and Yakimov, N. D. (1998). “Analytical solutions of seepage theory problems. Inverse method, variational theorems, optimization and estimates (a review).” Fluid Dynamics, Vol. 33, No. 2, pp. 157-168. Im, D-K., Yeo, H-K., Kim, K-H. and Kang, J-G. (2006). “Suitability Analysis of Numerical Models Related to Seepage through a Levee.” Journal of Korea Water Resources Association. Vol. 39, Issue 3, pp. 241-252. Jeppson, R. W. (1968a). “Seepage from ditches—Solution by finite differences.” J. Hydr. Div., 94HY1, pp. 259–283. Jeppson, R. W. (1968b). “Seepage through dams in the complex potential plane.” ASCE Proc., J. Irrig. and Drainage Div. Vol. 94, IR1, PAP 5835, pp. 23-39. Jeyisanker, Kalyani and Gunaratne, Manjriker (2009). “Analysis of water seepage in a pavement system using the particulate approach.” Computers and Geotechnics, Vol. 36, Issue 4, pp. 641-654. Jones, N. L. (1999). SEEP2D Primer Manual. Utah: Environmental Modeling Research Laboratory, Brigham Young University. Kacimov, A. R. and Nicolaev, A. N. (1992). “Steady seepage near an impermeable obstacle.” Journal of Hydrology, Vol. 138, Issues 1-2, pp. 17-40. Karadi, Gabor M., De Miguel, Enrique and Krizek, Raymond J. (1980). “Analysis of leaky sheetpile by Summary Representation.” Int. J. Numer. Anal. Meth. Geomech. 4(3), pp. 199-213.
224 Karageorghis, A. (1987). “Numerical solution of a shallow dam problem by a boundary element method.” Computer Methods in Applied Mechanics and Engineering, Vol. 61, Issue 3, pp. 265-276. Karthikeyan M., Tan T-S. and Phoon K-K. (2001). “Numerical oscillation in seepage analysis of unsaturated soils.” Canadian Geotechnical Journal, 38(3), pp. 639-651. Khosla, A. N., Bose, N. K. and Taylor, E. M. (1936). "Design of weirs on permeable foundations" Publication No. 12, Central Board of Irrigation, India. Kikuchi, Noboru. (1977). “Seepage flow problems by variational inequalities: Theory and approximation.” Int. J. Numer. Anal. Meth. Geomech. 1(3), pp. 283-297. King, G. J. W. (1967). "Seepage under a rectangular dam." Journal of soil Mechanics and Foundation Division, ASCE, Vol. 93, No. SM2, pp. 45-64. Korkmaz, Serdar and Önder, Halil (2006). “Seepage from a Rectangular Ditch to the Groundwater Table.” Journal of Irrigation and Drainage Engineering. Vol. 132, No. 3, pp. 263-271. Kumar, D., Chandra, S. and Mishra, G. C. (1982). “Flow under weir on scoured bed.” Journal of Hydraulic Engineering. Vol. 108, No. 4, pp. 529-543. Kythe, Prem K. (1995). An introduction to boundary element methods, CRC Press. Lacy, Sara J. and Prevost, Jean H. (1987). “Flow through porous media: A procedure for locating the free surface.” Int. J. Numer. Anal. Meth. Geomech. 11(6), pp. 585-601. Lane, E. W. (1935). "Security from under seepage: Masonry dam on earth foundations." Trans. ASCE 60 (4), pp. 929-966. Leliavsky, S. (1955). Irrigation and Hydraulic Design Vol. 1, Chapman and Hall, London, England. Leontiev, A. and Huacasi, W. (2001). “Mathematical programming approach for unconfined seepage flow problem.” Engineering Analysis with Boundary Elements, Vol. 25, Issue 1, pp. 49-56. Liggett, J. A. (1977). “Location of free surface in porous media.” J. Hydraul. Div., Am. Soc. Civil Eng., 103, HY4, pp. 353-65. Liggett, J. A. (1985). “Multiple boundary conditions, free surface flow and the boundary element method.” Commun. Appl. Numer. Methods, 1(3), pp. 105-112. Malhotra, J. K. (1936). "Notes on Khosla’s Principle of Independent Variables I-Mutual interference of equal piles at ends of a floor." Memoir of the Punjab Irrigation Research Institute, Vol. II No. 22, Article reproduced in reference: Khosla et al. (1936). McDonald, M.G., and Harbaugh, A.W. (1988). A modular three-dimensional finite-difference ground- water flow model: U.S. Geological Survey Techniques of Water-Resources Investigations, Book 6, Chap. A1, 586 p.
225 McLean, F. G. and Krizek, R. J. (1971). “Seepage characteristics of imperfect cutoffs.” Journal of the Soil Mechanics and Foundations Division, Vol. 97, No. 1, pp. 305-312. Mishra, G. C. and Singh, A. K. (2005). “Seepage through a levee.” International Journal of Geomechanics. Vol. 5, No. 1, pp. 74-79. Morland, L. W. and Gioda, G. (1990). “A mapping technique for steady-state unconfined seepage analysis.” Int. J. Numer. Anal. Meth. Geomech. 14(5), pp. 303-323. Muleshkov, A. and Banerjee, S. (1987). “Seepage towards vertical cuts.” Journal of Geotechnical Engineering. Vol. 113, No. 12, pp. 1419-1431. Najjar, Y. and Naouss, W. (1999). "Finite Element based seepage design charts for sheet pile penetrating heterogeneous media." Transportation Research Record: Journal of the Transportation Research Board, Vol. 1963, pp. 64-70. Najjar, Y. and Naouss, W. (2004). "Seepage design charts for embedded dams resting on heterogeneous media." ASCE conf. Proc. 143(4), pp. 49-61. Naoss, W. and Najjar, Y. (1995). "Seepage design charts for flat bottom dams resting on heterogeneous media." Int. J. Numer. Anal. Meth. Geomech., Vol. 19, No. 9, pp. 637-51. Nath, B. (1981). “A novel finite element method for seepage analysis.” Int. J. Numer. Anal. Meth. Geomech. 5(2), pp. 139-163. Neuman, Shlomo P. and Witherspoon, Paul A. (1970). “Finite element method of analyzing steady seepage with a free surface.” Water Resources Research, Vol. 6, No. 3, pp. 889-897. Pavlovsky, N. N. (1922). The Theory of Ground Water Flow Beneath Hydrotechnical Structures, Petrograd, U.S.S.R. (In Russian). Pavlovsky, N. N. (1936). “Principles of Hydromechanical Computations for Senkov-Type Barrages.” (In Russian). Gidrotekhnicheskoe stroitel’stvo, No. 8-9. Pavlovsky, N. N. (1937). “Hydromechanical Computations for Senkov-Type Barrages.” (In Russian). ONTI. Pavlovsky, N. N. (1956). Collected works, Akad, Nauk, U.S.S.R. Leningrad. Pollock, D. W. (1988). “Semianalytical computation of path lines for finite-difference models.” Ground Water, 26(6), pp. 743-750. Polubarinova-Kochina, P.Ya. (1962). Theory of Groundwater Movement, translated by J.M. Roger de Wiest, Princeton Univ. Press, Princeton, N. J. Ponter, A. R. S. (1972). “The application of dual minimum theorems to the finite element solution of potential problems with special reference to seepage.” Int. J. Numer. Meth. Engng. 4(1), pp. 85-93.
226 Reddi, L. N. (2003). Seepage in soils: Principles and Applications, John Wiley and Sons. Richardson, L. F., (1911). “The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Application to the Stresses in a Masonry Dam.” Phil. Trans. Roy. Soc. Lond. A, 210, pp. 307-357. Rogers, J. S. and Selim, H. M. (1989). “Water flow through layered anisotropic bedded soil with subsurface drains.” Soil Science Society of America Journal. Vol. 53, no. 1, pp. 18-24. Ronzhin, I. (1975). “Analog analysis of seepage under dams.” Power Technology and Engineering. Vol. 9, Issue 7, pp. 618-620. Rulon, Jennifer J., Rodway, Ray and Freeze, R. Allan (1985). “The Development of Multiple Seepage Faces on Layered Slopes.” Water Resources Research, Vol. 21, No. 11, pp. 1625-1636. Rushton, K. R. and Redshaw, S. C. (1979). Seepage and groundwater flow: numerical analysis by analog and digital methods, Wiley. Sakthivadivel, R. and Thiruvengadachari, S. (1975). “Seepage characteristics of foundations with a downstream crack.” Journal of Hydraulic Research, Vol. 13, No. 1, pp. 57-77. Salem, A. A. S. and Ghazaw, Y. (2001). “Stability of two consecutive floors with intermediate filters.” Journal of Hydraulic Research, Vol. 39, No. 5, pp. 549-556. Savant, S. A., Reible, D. D., Ortiz, J. C. and Gipson, G. Steven. (1988). “A direct method for equipotential and streamline tracing in the boundary element method.” Engineering Analysis, Vol. 5, Issue 3, pp. 155-160. Semenov, V. V. and Shevarina, N. N. (1976). “Application of the finite-element method in calculating seepage in the foundations of hydraulic structures.” Power Technology and Engineering. Vol. 10, Issue 4, pp. 326 -331. Shaw, F. S. and Southwell, R. V. (1941). “Relaxation Methods Applied to Engineering Problems. VII. Problems Relating to the Percolation of Fluids through Porous Materials.” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 178, No. 972, pp. 1-17. Shehata, A. K. (2006). "Design of downstream blanket for overflow spillway founded on complex formation." Journal of Applied Sciences Research, 2 (12), pp. 1217-1227. Shen, Jun and Zhang, Jian. (2008). “Analysis and solution to boundary element of seepage in engineering.” Meitian Dizhi yu Kantan (Coal Geology & Exploration). Vol. 36, No. 3, pp. 59-62. Simpson, M. J. and Clement, T. P. (2003). “Comparison of finite difference and finite element solutions to the variably saturated flow equation.” Journal of Hydrology, Vol. 270, Issues 1-2, pp. 49-64.
227 Sivakugan, N., Rankine, K. J. and Rankine, K. S. (2006). “Study of drainage through hydraulic fill stopes using method of fragments.” Geotechnical and Geological Engineering. Vol. 24, No. 1, pp. 79-89. Soleimanbeigi, Ali and Jafarzadeh, Fardin (2005). “3D Steady State Seepage Analysis of Embankment Dams.” Electronic Journal of Geotechnical Engineering, Vol. 10, No. F. Strack, Otto D. L. (1989). Groundwater Mechanics. Prentice Hall, New Jersey. Tracy, Fred T. and Radhakrishnan, N. (1989). “Automatic Generation of Seepage Flow Nets by Finite Element Method.” Journal of Computing in Civil Engineering, Vol. 3, No. 3, pp. 268-284. Travis, B. J. (1984). “TRACR3D: A model of flow and transport in porous/fractured media.” Report LA- 9667-MS, Los Alamos Natl. Lab., Los Alamos, NM. Tsay, R., Chiou, Y-J., and Cheng, T-C. (1997). “Boundary element analysis for free surface of seepage in earth dams.” Journal of the Chinese Institute of Engineers, Vol. 20, No. 1, pp. 95-101. Van Deemter, J. (1951). “Results of mathematical approach to some flow problems connected with drainage and irrigation.” Applied Scientific Research, Springer Netherlands. 2(1), pp. 33-53. Walsum, P. E. V Van and Koopmans, R. W. R. (1984). “Steady two-dimensional groundwater seepage: numerical analysis in the ϕ-ψ-plane.” Journal of Hydrology, Vol. 72, Issues 3-4, pp. 331-354. Walton, W. C., (1970). Groundwater Resource Evaluation. New York: McGraw Hill, p. 664. Yeh, G. T. and Ward, D. S. (1979). “FEMWATER: A finite-element model of water flow through saturated-unsaturated porous media.” Report ORNL-5567. Oak Ridge Natl. Lab. Oak Ridge, TN.
228