Geomechanics and Geoengineering: An International Journal

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Stability charts for closely spaced strip footings on
cohesive-frictional soils
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Journal: Geomechanics and Geoengineering: An International Journal

Manuscript ID TGEO-2022-0083
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Manuscript Type: Technical note

Date Submitted by the

15-Jun-2022
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Author:

Complete List of Authors: Noo-Iad, Dulpinit; Thammasat University

Shiau, Jim; University of Southern Queensland
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Chim-Oye, Weeraya; Thammasat University

Banyong, Rungkhun; Thammasat University
Keawsawasvong, Suraparb; Thammasat University, Department of Civil
Engineering
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Bearing capacity, Interference, Cohesive-frictional, Footing, Limit

Keywords:
analysis
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Page 1 of 40 Geomechanics and Geoengineering: An International Journal

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1 Technical Note
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8 3 Stability charts for closely spaced strip footings
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10 4 on cohesive-frictional soils
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12 5 Dulpinit Noo-Iad1, Jim Shiau2, Weeraya Chim-Oye3, Rungkhun Banyong4, Suraparb Keawsawasvong5,*
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16 7 1Graduate Student, Department of Civil Engineering, Thammasat School of Engineering, Thammasat
17 8 University, Pathumthani, Thailand 12120 (Email: dulpinit91@gmail.com)
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19 9 2Associate Professor, School of Civil Engineering and Surveying, University of Southern Queensland, QLD,
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10 Australia 4350 (Email: jim.shiau@usq.edu.au)
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11 3Associate Professor, Department of Civil Engineering, Thammasat School of Engineering, Thammasat
24 12 University, Pathumthani, Thailand 12120 (Email: sweeraya@engr.tu.ac.th)
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26 13 4Graduate Student, Department of Civil Engineering, Thammasat School of Engineering, Thammasat
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14 University, Pathumthani, Thailand 12120 (Email: rungkhun.ban@gmail.com)
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15 5Lectuer, Department of Civil Engineering, Thammasat School of Engineering, Thammasat University,
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31 16 Pathumthani, Thailand 12120 (Email: ksurapar@engr.tu.ac.th)
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33 17 * The corresponding author
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36 18 Abstract
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38 19 The bearing capacity of closely spaced footings, just like in the problem of pile groups, is
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40 one of the important topics in geotechnical engineering research. In this paper, three
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43 21 efficiency factors that describe the bearing capacity effects of closely spaced footings are
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45 22 developed and incorporated in the traditional Terzaghi’s bearing capacity equation. Using
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47 23 the advanced finite element limit analysis of upper and lower bound theorems, both the two
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24 closely spaced strip footings and the multiple closely spaced strip footings on cohesive-
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52 25 frictional soil with surcharge effect are proposed in the study. The finding is the efficiency
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54 26 factors are significantly influenced by the internal frictional angle and the spacing ratio.
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56 27 Various comparisons with published solutions are carried out. Failure mechanisms of closely
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3 28 spaced footings are investigated whilst design charts produced with a wide range of practical
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29 parameters. The study should be of great interest to foundation engineering practitioners.
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9 30 Keywords: Bearing capacity; Interference; Cohesive-frictional; Footing; Limit analysis
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Page 3 of 40 Geomechanics and Geoengineering: An International Journal

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31 1 Introduction
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6 32 Pile foundations are placed adjacent to one another to efficiently transfer the loads to
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8 33 underneath. As a result, the overlapped impacts in soils caused by neighboring footings
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10 34 which cannot be overlooked. Stuart (1962) was the first research to propose an effectiveness
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35 factor that compensates for the interfering impact of two neighboring strip footings in sand.
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15 36 The effectiveness factor was established by him as the ratio of the ultimate bearing capacity
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17 37 of two neighboring footings to a single specific footing. Das and Larbi-Cherif (1983) also
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38 carried out the laboratory model test to investigate the efficiency factor of the same problem
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22 39 as Stuart (1962). Upper bound techniques and stress characteristics were employed by
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24 40 Kumar and Ghosh (2007a; 2007b) to analytically derive the efficiency factors of two
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26 41 adjacent strip footings on cohesionless soil, where all possible values of the internal friction
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29 42 angle of sand, as well as the full range of the distance between two footings, were considered.
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32 43 The numerical technique called the lower bound finite element limit analysis was
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34 44 used by Kumar and Kouzer (2008) to compute the solutions of the efficiency factors of twin
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45 neighboring strip footings on unconsolidated soil. By also considering the influence of
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39 46 surcharge loading, the efficiency factors of two footings on cohesive-frictional soils were
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41 47 presented by Mabrouki et al. (2010) using the finite difference method software FLAC. Pal
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48 et al. (2016) and Lavasan et al. (2018) also investigated the bearing capability of two nearby
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46 49 bordered strip footings in sands. As the center-to-center spacing is tight enough, each footing
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48 50 has an interfering impact on other footing in the many evenly spaced strip footings system.
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50 51 The interfering impact of many footings has only been explored in a few earlier studies.
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53 52 Graham et al. (1984) assessed the interfering impact of three tightly spaced strip footings in
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55 53 sand utilizing the characteristics techniques and small-scale model testing. Moreover, by
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57 54 applying the upper bound (UB) and lower bound (LB) finite element limit analysis (FELA),
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55 Kouzer and Kumar (2008), Kumar and Bhattacharya (2010), and Yang et al. (2017)

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3 56 suggested the effectiveness factors of numerous evenly spaced strip footings in cohesionless
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57 soils. Ghazavi and Dehkordi (2021) provide a comprehensive assessment of the interference
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8 58 impact on the performance of shallow strip footings.
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11 59 According to Terzaghi’s bearing capacity approach, the bearing capacity factors
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60 consist of the cohesion factor Nc, the surcharge factor Nq, and the unit weight factor N. They
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16 61 represent the effects of soil cohesion, unit weight, and surcharge loading, respectively. In all
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62 published literature, only Mabrouki et al. (2010) proposed the numerical FLAC solutions of
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63 the efficiency factors of two footings by considering all impacts of soil cohesion, unit weight,
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23 64 and surcharge loading, although their solutions were limited to  = 20˚ to 40˚. The goal of
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25 65 this study was to determine the efficiency factor of two unique interference situations on
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66 cohesive-frictional soil including in the cases of two tightly spaced strip footings and
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30 67 numerous tightly spaced strip footings by also considering the surcharge loading and the full
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32 68 range of  = 5˚ to 45˚. The efficiency factors are quantitatively studied using sophisticated
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69 upper bound (UB) and lower bound (LB) finite element limit analysis (FELA). The stringent
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37 70 UB and LB solutions may be utilized in order to enable and create a set of strip footings that
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39 71 are based on cohesive-frictional soil and take surcharge loading into account.
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44 73 2 Problem Statement and FELA
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47 74 The first problem considered here is for the two interfering footings. Each footing
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49 75 has the same width B and is subjected to a limit vertical pressure of pu (i.e., the bearing
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52 76 capacity of the footing). As shown in Fig. 1(a), the edge-to-edge distance of the two
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54 77 interfering footings is defined as s. The surcharge pressure is denoted by q. The line of
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56 78 symmetry in the middle of the domain is indicated by the dashed line AE. Owing to the
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79 problem symmetry, the simulation utilizes just half of the domain and the symmetrical
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Page 5 of 40 Geomechanics and Geoengineering: An International Journal

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3 80 boundary condition (the left-hand side boundary) requires the nodes to move vertically only
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81 (Figs. 1b). At the right-hand side boundary (or the far side), the same condition as the left-
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8 82 hand side is applied. The bottom border is restricted with no movement allowed in both
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10 83 directions, while the upper boundary is unrestricted.
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13 84 The second problem is for the multiple interfering footings. Each footing has the
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15 85 same width B and the limit vertical pressure of pu applied at each footing as shown in Fig.
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86 2(a). The edge-to-edge footing distance is defined by s and the surcharge pressure is denoted
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20 87 by q. It is interesting to note that the symmetrical planes are represented by the dashed lines
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22 88 CD and AE, which can be represented the problems domain as seen in Figs. 2(b). Same as
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89 in the two interfering footings, the boundary condition for the two symmetric planes requires
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27 90 the nodes to be fixed in the horizontal direction only. The rest of the boundary conditions
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29 91 are identical comparing with the case of two overlapping footings.

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32 92 Since the upper bound (UB) and lower bound (LB) approximations may be utilized
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34 93 to reflect the exact collapse load, limit analysis is most beneficial (Sloan, 2013). Sloan
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94 established the first studies of linear programming, known as FELA (1988; 1989). Lyamin
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39 95 and Sloan (2002a; 2002b) and Krabbenhoft et al. (2007) further expanded the approach to
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46 98 stability solutions in the geotechnical fields (Izadi and Chenari, 2021; Shiau and Al-Asadi,
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48 99 2020a-e; 2022; Shiau et al., 2021a-d; 2022; Keawsawasvong and Lai, 2021;
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50 100 Keawsawasvong and Shiau, 2021; Yodsomjai et al., 2021; Keawsawasvong et al., 2022a-c;
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101 Das and Chakraborty, 2022; Ukritchon and Keawsawasvong, 2019; 2020).
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55 102 The evaluation of the LB approach employs three-node triangular components. The
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60 104 For creating the continuity of normal and shear stresses, as well as the interfaces of all the

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3 105 elements, the statically allowable stress discontinuities are permitted. In a typical LB
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106 analysis, the stress equilibrium requirements, the stress boundary condition, and the Mohr-
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8 107 Coulomb failure criterion are all restrictive, with the propose of maximizing the collapse
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10 108 load of problems. The upper bound theorem necessitates a kinematically acceptable velocity
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109 field with external work larger than or equal to plastic shear dissipation. Six-node triangular
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15 110 components are employed in the UB method's formulation. The horizontal (u) and vertical
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19 112 These two theorems are excellently suited to second-order cone programming nonlinear
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22 113 programming optimization problems (SOCP). The constraints employed in this approach are
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26 115 OptumG2 (OptumCE, 2020) provide more insights on the formulation.
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29 116 In the FELA of OptumG2, the underlying soil is modelled by volume elements and
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31 117 complies a Mohr-Coulomb material that is stiff and completely plastic. The concrete footings
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34 118 are modelled by rigid volume elements. The contact surface among footings and soils is
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36 119 given as a completely rough texture. The adaptive mesh refinement approach proposed by
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38 120 Ciria et al. (2008) is employed, resulting in the adoption of a significant number of
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121 components in particular sensitive locations with high incremental shears throughout the
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43 122 steps of mesh adaptive refinement. For all models in this investigation, five adaptive
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124 phase containing about 10,000 elements (or fifth step). Therefore, Figs. 1(c) and
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130 3 The Efficiency Factor
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131 The superposition equation proposed by Terzaghi (1943) may be used to determine
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11 133 the surcharge loading as expressed in Eq. (1).
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14 134 pu  cN c  qN q  0.5 BN (1)
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17 135 Where the soil cohesion represented as c, soil unit weight represented as , the surcharge
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136 loading represented as q, while B denotes as the width of the footing, and Nc, Nq, and N
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22 137 denote as the bearing capacity factors for a single strip footing as seen in Fig. 3. It is indeed
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27 139 internal friction angle , and obtained from FELA of a single strip footing. The surcharge
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29 140 loading q can be calculated by q = Df, where Df is the depth of embedment. In Eq. (2), the
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38 pu  cN c  qN q  0.5 BN
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55 145 For the bearing capacity of two interfering footings and multiple interfering footings
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 Geomechanics and Geoengineering: An International Journal Page 8 of 40

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3 148 where the three efficiency factors are denoted by c, q, and . These factors are literally
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6 149 ratios of the ultimate bearing capacity of two interfering footings or multiple interfering
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8 150 footings (pu,m) to the ultimate bearing capacity of a single isolated footing (pu). The
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10 151 procedure for determining each efficiency factor individually is expressed in Eqs. (4a-4c)
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22 154   (4c)
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The efficiency factors are now functions of the soil internal friction angle  and the
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25 155
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156 spacing ratio s/B, as expressed in Eq. (5).
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c and q and   f   , 
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34 158 The impacts of the spacing ratio (s/B) and the soil internal friction angle  on the
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36 159 efficiency factors of interfering strip footings are explored and reported in the pattern of
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160 design diagrams within the current investigation. The suggested efficiency factor may be
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41 161 applied to easily calculate the bearing capacity for a collection of strip footings in cohesive-
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43 162 frictional soil assuming surcharge loading is taken into account.
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46 163 4 Two Interfering Footings
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49 164 Figs. 4-6 show the link among the three efficiency factors c, q, and , as
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165 well as the distance ratio s/B respectively for the different values of the internal friction angle
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54 166  = 5˚ to 45˚. In Fig. 4, the cohesion efficient factor begins from c = 1 at s/B = 0 (i.e. a single
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56 167 footing) and it increases linearly with a small rise in s/B. This occurs in any values of , and
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59 168 the relationship between c and s/B may be written as a linear equation c = (s/2B) + 1.
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3 169 Depending on the internal soil friction angle , as the distance ratio s/B rises, the cohesion
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6 170 efficiency factor c declines rapidly at certain stages after reaching the apex, finally returning
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8 171 to unity (c = 1) at various s/B. It is to be noted that (c = 1) could represent a single footing
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172 or two footings that are placed far away from each other.
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13 173 Interestingly, numerical results of the surcharge efficient factor q in Fig.5 have
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16 174 shown the same results as in those of c. The linear equation is also found to be q = (s/2B)
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175 + 1. Lavasan et al. (2018) have made a comparable investigation as well. In typically, the
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176 higher the internal friction angle , the higher the efficiency factors' value.
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177 Fig. 6 shows that the unit weight efficiency factor  starts from  = 2 at s/B = 0 (i.e.
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26 178 a single footing) and it increases nonlinearly with a small increase in s/B. This occurs for all
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31 180 + s/B)2/2. After reaching a peak at certain respective s/B, also depending on the value of ,
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33 181 the unit weight efficient factor  decreases dramatically and eventually returns to unity at
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36 182 various s/B. The overall trend is very similar to c, as discussed above.
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183 Fig.7(a-c) show the comparisons of c for the different values of  = 30˚, 35˚, and
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41 184 40˚. For the case of  = 30˚ in Fig. 7(a), finite difference results (FDM) presented by
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43 185 Mabroukei et al (2010) are also equal to the present FELA results. Note that there were only
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46 186 the existing results of c by Mabroukei et al (2010) for the cases of  = 30˚, 35˚, and 40˚ in
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48 187 the past. As can be seen in Fig. 7(b) and Fig. 7(c) for  = 35˚, and 40˚, respectively, the
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188 differences between the present study and those by Mabroukei et al (2010) become larger
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53 189 when  is large.
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56 190 Comparisons of q for the different values of  = 35˚, and 40˚ are presented in Fig.
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58 191 8(a-c). In Fig. 8(a), for  = 30˚, finite difference results (FDM) of Mabroukei et al (2010)
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3 192 and Lavasan et al. (2018) are in good agreement with the present FELA ones. Nevertheless,
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193 those reported by the limit equilibrium and the upper bound (KEM, Mechanism II) methods
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8 194 (Lavasan et al. 2018) are well under the agreed curve and the results cannot be used in
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10 195 practice. As  increases, the differences become greater - see Fig.8(b) and Fig.8(c) for  =
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13 196 35˚, and 40˚ respectively.
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197 Comparisons of the efficiency factor  for  = 35˚, and 40˚ are made with those
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18 198 published by Schmudderich et al (2020), Mabrouki et al (2010), Kumar and Kouzer (2008),
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20 199 and Kumar and Ghost (2007). It can be observed from Fig.9 that, the present solutions agree
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200 well with all other published solutions at larger s/B ratios. The results of Kumar and Ghost
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27 202 higher than ours.
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30 203 The failure mechanisms for c (c ≠ 0,  = q = 0), q (q ≠ 0, c =  = 0), and  ( ≠
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32 204 0, c = q = 0) are presented in Figs. 10-12. The plots are for  30˚. For brevity, only the
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205 plots of shear dissipation contours for the various distance ratios s/B are shown. When S/B
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37 206 = 0, the problem of two footings turns to be a single footing with 2B, where the failure is in
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39 207 the pattern of Prandtl types. Noting the symmetrical domain in the figures, the overlapping
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42 208 impacts are particularly noticeable at minimal s/B values. This overlapping action may
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44 209 improve the capacity of the footing, but the downside would be the possible uneven footing
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46 210 settlement. On the other note, as expected, the larger the ratio s/B, the less the footing
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211 interference it is. Prandtl types of failure mechanisms are obtained for large values of s/B, as
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54 213 5 Multiple Interfering Footings
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214 The variation of efficiency factors c, q, and  with s/B for multiple interfering strip
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3 216 values of the internal friction angle  = 5˚ to 45˚. The initial value of the efficiency factors
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6 217 at s/B = 0 is an intriguing result that differs from the previous two interfering strip footings.
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8 218 All three efficiency factors c, q, and  have infinite values as s/B approaches zero since
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219 the problem turns to be a single footing with B being infinity compressing everywhere on
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13 220 soil surface. Once the ratio of s/B is increased leads to a decreasing of the efficiency factors
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15 221 since there is a gap for soil masses to be moved. Depending on the value of , the three
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18 222 efficiency factors c, q, and  decrease significantly to unity at different s/B values ~ a
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20 223 hyperbola type of smooth curve is presented. Numerical results in this study have also shown
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22 224 that the larger the value of  leads to a higher value of the efficiency factors.
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25 225 A comparison of the efficiency factor  of multiple interfering strip footings is

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28 226 presented in Fig. 16. The comparison is for  = 30˚. In general, the current analysis and
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30 227 earlier reported solutions are found to be in extremely excellent agreement. The numerical
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228 results of Yang et al. (2017) are remarkably similar to the findings of the current
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229 investigation, whilst Kouzer and Kumar (2008) predict larger values, and Kumar and
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37 230 Bhattacharya (2010) have lower values than ours.
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40 231 The failure mechanisms of multiple interfering strip footings are demonstrated in
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42 232 Figs. 17-19 for c (c ≠ 0,  = q = 0), q (q ≠ 0, c =  = 0), and  ( ≠ 0, c = q = 0), respectively.
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45 233 The chosen comparison is for   45˚. It's worth noting that the multiple footings'
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47 234 domains have symmetrical planes on both the left and right sides, the resulting efficiency
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49 235 factors are significantly larger than the ones in two interfering footings. Consequently, it
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52 236 may indicate that due to the lateral resistance produced by surrounding footings,
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54 237 the overlapping has a favorable influence on the overall bearing capacity of many footings.
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56 238 The impact of the two essential factors  and s/B for the problem of numerous interfering
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59 239 strip footings on cohesive-frictional soil are confirmed in this investigation.
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 Geomechanics and Geoengineering: An International Journal Page 12 of 40

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240 6 Examples
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241 This section includes some examples that explain how to utilize the generated data
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9 242 to assess the uniform bearing capacity of closely spaced footings employing the expression
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11 243 provided in Equation (3). The following examples show the results of applying the
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13 244 superposition principle.
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16 245
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246 Example 1: Frictional soil with surcharge loading (two footings)
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19 247
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21 248 Two strip footings have the same width B = 1.00 m and the edge-to-edge distance of
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23 249 the two interfering footings s = 0.30 m. The design parameters are given as: the unit weight
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250 γ = 18 kPa and the soil internal friction angle ϕ = 40°. The soil cohesion c is zero in this
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28 251 example. The surcharge loading q = 18 kPa. Given ϕ = 40°, the value of Nc, Nq and Nγ from
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30 252 Figure 3 are equal to 74.77, 63.84 and 84.09, respectively. Efficiency factor c =1.15, q =
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33 253 1.15 and  = 2.65 can be obtained from Figs. 4, 5 and 6, respectively, for ϕ = 40° and s/B
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iew

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36 254 = 0.3. Substituting these into Equation (3), the bearing capacity of two closely spaced
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38 255 footings can be then calculated as: pu,m = (18×1.15×63.84) + (0.5×18×1×2.65×84.09) =
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41 256 3,327.13 kPa.

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43 257
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45 258 Example 2: Cohesive-frictional soil with surcharge loading (two footings)

46 259
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48 260 In this example, two strip footings has the same width B = 1.50 m and s = 0.60 m.
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50 261 The design parameters are given as: the unit weight γ = 18 kPa and the soil internal friction
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53 262 angle ϕ = 40°. The soil cohesion c = 15 kPa in this example. The surcharge loading q = 18
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55 263 kPa. Given ϕ = 40°, the value of Nc, Nq and Nγ from Figure 3 are equal to 74.77, 63.84 and
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264 84.09, respectively. Efficiency factor c =1.20, q = 1.20 and  = 2.9 are respectively
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60 265 obtained from Fig. 4, 5 and 6 for the case of ϕ = 40° and s/B = 0.4. Using Equation (3), the
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Page 13 of 40 Geomechanics and Geoengineering: An International Journal

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3 266 bearing capacity of two closely spaced footings can be then calculated as: pu,m =
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267 (15×1.2×74.77) + (18×1.2×63.84) + (0.5×18×1.5×2.9×84.09) = 6,016.93 kPa.
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8 268
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10 269 Example 3: Cohesive-frictional soil with surcharge loading (multiple footings)
11 270
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13 271 In this example, the multiple strip footings have the same width B = 1.50 m and
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15 272 equally spaced strip footings s = 3.00 m. The design parameters are given as: the unit weight
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18 273 γ = 18 kPa and the soil internal friction angle ϕ = 35°. The soil cohesion c = 15 kPa in this
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20 274 example. The surcharge loading q = 20 kPa. Given ϕ = 35°, the value of Nc, Nq and Nγ from
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275 Figure 3 are 45.85, 33.17 and 34.21, respectively. Efficiency factor c =1.47, q = 1.46 and
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276  = 1.00 are respectively acquired from Fig. 13, 14 and 15 for ϕ = 35° and s/B = 2.00.
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28
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277 Substituting these into Equation (3), the bearing capacity of multiple closely spaced footings
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278 can be then calculated as: pu,m = (15×1.47×45.85) + (20×1.46×33.17) +
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33 279 (0.5×18×1.5×1.00×34.21) = 2,441.39 kPa.
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35 280
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281 7 Conclusions
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42 282 The purpose of this investigation was to investigate the impact of closely spaced
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44 283 footings on cohesive-frictional soil. The objective of this study was to calculate the
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46 284 efficiency factors c, q, and  and that may be employed to determine the bearing capacity
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49 285 of closely spaced footings which similar to how pile group efficiency is calculated. These
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51 286 have been verified that the efficiency factors c, q, and  are a function of the internal
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287 frictional angle  and the spacing ratio s/B using sophisticated finite element limit analysis.
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56 288 The two critical factors  and s/B were shown to have a significant impact on the failure
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58 289 mechanisms. The present findings are comparable to prior solutions, boosting user
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 Geomechanics and Geoengineering: An International Journal Page 14 of 40

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3 290 confidence and allowing for the creation of design charts encompassing a variety of the two
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291 considered parameters for practical applications. The study adds to a growing body of
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8 292 literature on the stability of closely spaced soil structures. Future research work is needed
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10 293 for settlement design, considering the possibility of uneven settlement due to the overlapping
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294 effect.
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15 295
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296 Acknowledgements
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20 297 This research was supported by Thammasat University Research Unit in Structural
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22 298 and Foundation Engineering, Thammasat University.

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24
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25 299
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27 300
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29 301 Authors’ Contributions

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31
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33 302 Dulpinit Noo-Iad acquired methodology, software and contributed to investigation,
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35 303 conceptualization, writing—original draft and data curation.

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38 304 Suraparb Keawsawasvong acquired methodology, and contributed to investigation,
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305 conceptualization, writing—original draft.

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44 306 Weeraya Chim-Oye acquired methodology, software and contributed to investigation,
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46 307 conceptualization, writing—original draft and data curation.
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49 308 Jim Shiau provided resources, acquired supervision, contributed to writing—review and
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52 309 editing.
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54 310
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59 312
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3 313 Availability of Data and Material
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314 The data and materials in this paper are available.
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10 315
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13 316 Declarations
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317 Conflict of interest The authors declare that they have no conflicts of interest to this work.
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319 References
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321 Ciria, H., Peraire, J., Bonet, J. 2008. Mesh adaptive computation of upper and lower bounds
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8 322 in limit analysis. International Journal for Numerical Methods in Engineering 75, 899–
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10 323 944.
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12 324 Das, B.M., Larbi-Cherif, S., 1983. Bearing capacity of two closely spaced shallow
13 325 foundations on sand. Soils Found 23 (1), 1–7.
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15 326 Das, S., Chakraborty, D. Effect of Soil and Rock Interface Friction on the Bearing Capacity
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17 327 of Strip Footing Placed on Soil Overlying Hoek-Brown Rock Mass International
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20 329 Ghazavi, M., Dehkordi, P.F. 2021. Interference influence on behavior of shallow footings
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48 345 Keawsawasvong, S., Shiau, J., Ngamkhanong, C., Lai, V.Q., Thongchom, C. Undrained
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53 348 Keawsawasvong, S., Shiau, J., Yoonirundorn, K. Bearing capacity of cylindrical caissons in
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17 359 Kumar, J., Ghosh, P., 2007b. Upper bound limit analysis for finding interference effect of
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19 360 two nearby strip footings on sand. Geotech. Geol. Eng. 25 (5), 499–507.
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45 375 capacity for two interfering strip footings on sands. Comput. Geotech. 37, 431–439.
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48 377 https://optumce.com/ (accessed on 10 April 2020).
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3 384 Shiau, J., Al-Asadi, F. 2020a. Two-dimensional tunnel heading stability factors Fc, Fs and
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387 stability factors. Computers and Geotechnics, 119, 103345.
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7 419 Terzaghi K. Theoretical soil mechanics. New York: Wiley; 1943.
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420 Ukritchon, B., Keawsawasvong, S. Stability of retained soils behind underground walls with
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12 422 Geotechnical and Geological Engineering, 2019, 37(3), 1609-1625.
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14 423 Ukritchon, B., Keawsawasvong, S. Undrained stability of unlined square tunnels in clays
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17 425 Engineering, 2020, 38(1), 897-915.
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19 426 Yang, F., Zheng, X.C., Sun, X.L., Zhao, L.H. 2017. Upper-bound analysis of N and failure
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21 427 mechanisms of multiple equally spaced strip footings. Int J Geomech 17(9), 06017016.
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22 428 Yodsomjai, W., Keawsawasvong, S., Lai, V.Q. Limit analysis solutions for bearing capacity
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24 429 of ring foundations on rocks using Hoek-Brown failure criterion. International Journal
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26 430 of Geosynthetics and Ground Engineering 2021; 7, 29.
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433 Figures
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6 435 𝑝𝑢 𝑝𝑢
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436 A 𝑞
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11 438 B s/2 s/2 B
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Plane of symmetry c, , 
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450 Fig. 1. Caption: Two interfering footings: (a) problem statement; (b) model domain; (c)
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56 451 typical adaptive mesh.
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58 452 Fig. 1. Alt Text: Figure of the problem definition and mesh example of the problem of two
59 453 interfering footings.
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3 454
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5 455
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A C
8 457
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10 458 𝑝𝑢 𝑝𝑢 𝑝𝑢
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14 B s B/2 B/2 s/2 s/2 B
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20 464 Plane of symmetry c, ,  Plane of symmetry
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465 D
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48 471 Fig. 2. Caption: Multiple interfering footings: (a) problem statement; (b) model domain;
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50 472 (c) typical adaptive mesh.
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473 Fig. 2. Alt Text: Figure of the problem definition and mesh example of the problem of
54 474 multiple interfering footings.
55 475
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41 478
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43 479 Fig. 3. Alt Text: Figure showing the classic solutions of bearing capacity factors Nc, Nq, N
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480 for a single footing on cohesive-frictional soils.
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35 485 ( = 5° - 45°).
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37 486 Fig. 4. Alt Text: Figure showing the efficiency factor c for two interfering footings on
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40 487 cohesive-frictional soils.
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3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Fo

19
20
21
rP

22
23
24 503
ee

25
26 504 (a) ϕ = 30°
27
28
rR

29
30
31
ev

32
33
34
iew

35
36
37
38
39
40
On

41
42
43
44
ly

45
46
47
48 505
49
50 506 (b) ϕ = 35°
51
52
53
54
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Page 27 of 40 Geomechanics and Geoengineering: An International Journal

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15
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17
18
Fo

19
20
21
rP

22 507
23
24 508 (c) ϕ = 40°
ee

25
26 509 Fig. 7. Caption: Comparison of efficiency factor c .
27
28
rR

29 510 Fig. 7. Alt Text: Figure showing the efficiency factor c for two interfering footings on
30
31 511 cohesive-frictional soils.
ev

32
33
34
512
iew

35
36
37
38
39
40
On

41
42
43
44
ly

45
46
47
48
49
50
51
52
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54
55
56
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 Geomechanics and Geoengineering: An International Journal Page 28 of 40

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2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Fo

19
20
21
rP

22
23
24
513
ee

25
26
514 (a) ϕ = 30°
27
28
rR

29
30
31
ev

32
33
34
iew

35
36
37
38
39
40
On

41
42
43
44
ly

45
46
47
48
49 515
50
51 516 (b) ϕ = 35°
52
53
54
55
56
57
58
59
60

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Page 29 of 40 Geomechanics and Geoengineering: An International Journal

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3
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5
6
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8
9
10
11
12
13
14
15
16
17
18
Fo

19
20
21
rP

22 517
23
24 518 (c) ϕ = 40°
ee

25
26 519 Fig. 8. Caption: Comparison of efficiency factor q .
27
28
rR

29 520 Fig. 8. Alt Text: Figure showing the comparison of efficiency factor q for two interfering
30
31
ev

32 521 footings on cohesive-frictional soils

33
34
iew

35 522
36
37
38
39
40
On

41
42
43
44
ly

45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60

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 Geomechanics and Geoengineering: An International Journal Page 30 of 40

1
2
3
4 3.0
5 Present study - (FELA)
6
Schmüdderich et al., 2020 - (FELA)
7
8 2.5 Mabrouki et al., 2010 - (FLAC)
9
10 Kumar & Kouzer, 2008 - (UB)
11
2.0 Kumar & Ghosh, 2007 - (MOC)
12
13
14
15
16 1.5
17
18
Fo

19
20 1.0
21 0 1 2 3 4
rP

22 s/B
23 523
24 524
525 (a) ϕ = 35°
ee

25
26
27 526
28
rR

29 4.0
30 Present study - (FELA)
31
Schmüdderich et al., 2020 - (FELA)
ev

3.5
32
33 Mabrouki et al., 2010 - (FLAC)
34 3.0
iew

35 Kumar & Kouzer, 2008 - (UB)

36
Kumar & Ghosh, 2007 - (MOC)
37 2.5
38
39
2.0
40
On

41
42 1.5
43
44
ly

45 1.0
46 0 1 2 3 4 5
47 s/B
48
527
49
50
528 (b) ϕ = 40°
51
52 529
53
54 530 Fig. 9. Caption: Comparison of efficiency factor  .
55
Fig. 9. Alt Text: Figure showing the comparison of efficiency factor  for two interfering
56
57 531
58
59 532 footings on cohesive-frictional soils.
60

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Page 31 of 40 Geomechanics and Geoengineering: An International Journal

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2
3
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5
6
7
8
9
10
11
12
s/B = 0
13
14
15
533
16
17
18
Fo

19
20
21
rP

22
23
24
ee

25
26 s/B = 1
27
28
rR

29 534
30
31
ev

32
33
34
iew

35
36
37
38
s/B = 3
39
40
On

41 535
42
43 536
44
ly

45
46
47
48
49
50
51 s/B = 15
52
53
54 537
55
56 538 Fig. 10. Caption: Shear dissipation contours of two interfering footings (c,   30o ).
57
58
59
539 Fig. 10. Alt Text: Figure showing the failure mechanisms of shear dissipation contours for
60 540 two interfering footings (c,   30o ).

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 Geomechanics and Geoengineering: An International Journal Page 32 of 40

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2
3
4
5
6
7
8
9
10
11
s/B = 0
12
13
14 541
15
16
17
18
Fo

19
20
21
rP

22
23
24 s/B = 1
ee

25
26
27 542
28
rR

29
30
31
ev

32
33
34
iew

35
36
37
s/B = 3
38
39 543
40
On

41
42
43
44
ly

45
46
47
48 s/B = 10
49
50 544
51
52 545 Fig. 11. Caption: Shear dissipation contours of two interfering footings (q,   30o ).
53
54 546 Fig. 11. Alt Text: Figure showing the failure mechanisms of shear dissipation contours for
55
56
547 two interfering footings (q,   30o ).
57
58 548
59
60

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Page 33 of 40 Geomechanics and Geoengineering: An International Journal

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2
3
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5
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8
9
10
11
12
13 s/B = 0
14
15
16 549
17
18
Fo

19
20
21
rP

22
23
24
ee

25
26
27 s/B = 1
28
rR

29 550
30
31
ev

32
33
34
iew

35
36
37
38
39
40
s/B = 3
On

41
42 551
43
44
ly

45
46
47
48
49
50
51
52 s/B = 12
53
54
552
55
56
553 Fig. 12. Caption: Shear dissipation contours of two interfering footings ( ,   30o ).
57
58
59 554 Fig. 12. Alt Text: Figure showing the failure mechanisms of shear dissipation contours for
60 555 two interfering footings ( ,   30o ).

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 Geomechanics and Geoengineering: An International Journal Page 34 of 40

1
2
3 556
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Fo

19
20
21
rP

22
23
24
ee

25
26
27
28
rR

557
29
30 558
31
ev

32 559 Fig. 13. Caption: Variation of c with s/B for multiple interfering footings.
33
34
560 Fig. 13. Alt Text: Figure showing the efficiency factor c for multiple interfering footings
iew

35
36
37 561 on cohesive-frictional soils
38
39
40
On

41 562
42
43
44
ly

45
46
47
48
49
50
51
52
53
54
55
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Page 35 of 40 Geomechanics and Geoengineering: An International Journal

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Fo

19
20
21
rP

22
23
24
ee

25
26
27
28
rR

29 563
30
31 564
ev

32
33 565 Fig. 14. Caption: Variation of q with s/B for multiple interfering footings
34
566 ( = 5°- 45°).
iew

35
36
37 567 Fig. 14. Alt Text: Figure showing the efficiency factor q for multiple interfering footings
38
39
40 568 on cohesive-frictional soils.
On

41
42
43 569
44
ly

45
46
47
48
49
50
51
52
53
54
55
56
57
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 Geomechanics and Geoengineering: An International Journal Page 36 of 40

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4
5
6
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8
9
10
11
12
13
14
15
16
17
18
Fo

19
20
21
rP

22
23
24
ee

25
26
27
28 570
rR

29
30 571
31
Fig. 15. Caption: Variation of  with s/B for multiple interfering footings
ev

32 572
33
34 573 ( = 5°- 45°).
iew

35
36 574 Fig. 15. Alt Text: Figure showing the efficiency factor  for multiple interfering footings
37
38
39 575 on cohesive-frictional soils.
40
On

41
42 576
43
44
ly

45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60

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Page 37 of 40 Geomechanics and Geoengineering: An International Journal

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8
9
10
11
12
13
14
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18
Fo

19
20
21
rP

22
23
24
ee

25
26 577
27
28 578 Fig. 16. Caption: Comparison of efficiency factor  ( = 35°).
rR

29
30
31 579 Fig. 16. Alt Text: Figure showing the comparison of efficiency factor  for multiple
ev

32
33
34 580 interfering footings on cohesive-frictional soils.
iew

35
36
37 581
38
39
40
On

41
42
43
44
ly

45
46
47
48
49
50
51
52
53
54
55
56
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 Geomechanics and Geoengineering: An International Journal Page 38 of 40

1
2
3 582
4
583
5
584
6
7 585
8 586
9 587
10
11 588
12 589 s/B = 0.5 s/B = 1
13 590
14
591
15
16 592
17 593
18
Fo

19 594
20
21 595
rP

22
23
596
24
597
ee

25
26
27 598
28
rR

29 599
30
31 600
s/B = 3 s/B = 5
ev

32
33 601
34
iew

35 602
36
37
603
38
39
40 604
On

41
42 605
43
44 606
ly

45
46 607
47
48 608
49
50 609
s/B = 25
51
52
610
53
54
55 611 Fig. 17. Caption: Shear dissipation contours of multiple interfering footings (c,  = 45).
56
57 612 Fig. 17. Alt Text: Figure showing the failure mechanisms of shear dissipation contours for
58 613 multiple interfering footings (c,  = 45).
59
60

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Page 39 of 40 Geomechanics and Geoengineering: An International Journal

1
2
3 614
4
5
615
6
7
8
616
9
10 617
11
12 618
13
14 619 s/B = 0.5 s/B = 1
15
16 620
17
18
Fo

621
19
20
622
21
rP

22
23 623
24
624
ee

25
26
27 625
28
rR

29 626
30
31 627
ev

32 s/B = 3
33
s/B = 5
628
34
iew

35
629
36
37
38 630
39
40 631
On

41
42 632
43
44 633
ly

45
46 634
47
48
635
49 s/B = 25
50
51
636
52
53 637
54
55 638 Fig. 18. Caption: Shear dissipation contours of multiple interfering footings (q,  = 45).
56
57 639 Fig. 18. Alt Text: Figure showing the failure mechanisms of shear dissipation contours for
58
59 640 multiple interfering footings (q,  = 45).
60

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 Geomechanics and Geoengineering: An International Journal Page 40 of 40

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2
3 641
4
5
6
7
8
9
10
11
12
13
14
15 s/B = 0.5 s/B = 1
16
17
18
Fo

19
20
21 642
rP

22
23 643
24
ee

25 644
26
27 645
28
rR

29 646
30
31 647
ev

32 s/B = 5 s/B = 7
33
34
648
iew

35
36 649
37
38 650
39
40 651
On

41
42 652
43
44 653
ly

45
46
654
47
48
49 655
50 s/B = 10
51 656
52
53 657
54
55 658 Fig. 19. Caption: Shear dissipation contours of multiple interfering footings ( ,  = 45).
56
57
58
659 Fig. 19. Alt Text: Figure showing the failure mechanisms of shear dissipation contours for
59 660 multiple interfering footings ( ,  = 45).
60

40
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