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Undrained Stability of Unsupported Rectangular Excavations: Anisotropy and Non-Homogeneity in 3D

Undrained Stability of Unsupported Rectangular Excavations: Anisotropy and Non-Homogeneity in 3D Article Undrained Stability of Unsupported Rectangular Excavations: Anisotropy and Non-Homogeneity in 3D 1 2 3, 3 3 Van Qui Lai , Jim Shiau , Suraparb Keawsawasvong *, Sorawit Seehavong and Lowell Tan Cabangon Faculty of Civil Engineering, Ho Chi Minh City University of Technology (HCMUT), Vietnam National University Ho Chi Minh City (VNU-HCM), 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City 70000, Vietnam School of Engineering, University of Southern Queensland, Darling Heights, QLD 4350, Australia Department of Civil Engineering, Faculty of Engineering, Thammasat School of Engineering, Thammasat University, Pathumthani 12120, Thailand * Correspondence: ksurapar@engr.tu.ac.th Abstract: The stability of unsupported rectangular excavations in undrained clay is examined under the influence of anisotropy and heterogeneity using the three-dimensional finite element upper and lower bound limit analysis with the Anisotropic Undrained Shear (AUS) failure criterion. Three anisotropic undrained shear strengths are considered in the study, namely triaxial compression, triaxial extension, and direct simple shear. Special considerations are given to the study of the line- arly-increased anisotropic shear strengths with depth. The numerical solutions are presented by an undrained stability number that is a function of four dimensionless parameters, i.e., the excavated depth ratio, the aspect ratio of the excavated site, the shear strength gradient ratio, and the aniso- tropic strength ratio. To the authors’ best knowledge, this is the first of its kind to present the stabil- ity solutions of 3D excavation considering soil anisotropy and heterogeneity. As such, this paper introduces a novel approach for predicting the stability of unsupported rectangular excavation in Citation: Lai, V.Q.; Shiau, J.; undrained clays in 3D space, accounting for soil anisotropy and non-homogeneity. Notably, it de- Keawsawasvong, S.; Seehavong, S.; velops a basis to formulate a mathematical equation and design charts for estimating the stability Cabangon, L.T. Undrained Stability factor of such type of excavation, which should be of great interest to engineering practitioners. of Unsupported Rectangular |Excavations: Anisotropy and Keywords: stability; excavations; anisotropy; heterogeneity; finite element limit analysis Non-Homogeneity in 3D. Buildings 2022, 12, 1425. https://doi.org/ 10.3390/buildings12091425 Academic Editor: Giuseppina Uva 1. Introduction Received: 10 August 2022 Unsupported excavation does not require retaining wall systems, and it is considered Accepted: 8 September 2022 one of the affordable construction methods that are widely employed in many shallow un- Published: 10 September 2022 derground construction projects. Shallow underground structures such as pipelines, shallow tunnels, and underpasses can be constructed by utilizing this excavation technique. Other ex- Publisher’s Note: MDPI stays neu- amples may include the construction of piers, footings, retaining structures, raft foundations, tral with regard to jurisdictional mat foundations, and water tanks. An unsupported excavation during construction, if not claims in published maps and institu- tional affiliations. properly assessed, can lead to an eventual collapse of the excavation wall that could result in an injury or fatality. These unfortunate events can cost money and cause death. It is, therefore, imperative to assess the stability of such unsupported excavations to reduce the risk of soil failure, thereby improving site safety and preventing death. This study aims to contribute to Copyright: © 2022 by the authors. Li- reducing that risk by providing a novel approach that predicts the undrained stability of un- censee MDPI, Basel, Switzerland. supported rectangular excavations in anisotropic and non-homogeneous clays. This article is an open access article In general, the excavation can have either cylindrical, conical, or rectangular shapes. distributed under the terms and con- Griffiths and Koutsabeloulis [1] used a displacement-based elastoplastic finite element ditions of the Creative Commons At- analysis to study the stability of cylindrical excavations under axisymmetric conditions. tribution (CC BY) license (https://cre- The same problem was also examined by Britto and Kusakabe [2,3] using the plastic- ativecommons.org/licenses/by/4.0/). bound theorems. The recent development of finite element limit analysis (FELA) is a Buildings 2022, 12, 1425. https://doi.org/10.3390/buildings12091425 www.mdpi.com/journal/buildings Buildings 2022, 12, 1425 2 of 21 powerful numerical method based on the lower bound (LB) theorem, the upper bound (UB) theorem, and the finite element technique, as demonstrated in [4–10]. The axisymmetric FELA was employed by [11–16] to obtain stability solutions for vertical circular excavations. Re- cently, the stability of unsupported conical excavations was investigated by [17–21]. Among the various shapes of excavation, rectangular and cylindrical shapes are the most common in practice. Although cylindrical shapes may lead to smaller amounts of excavated material and they are more stable due to the arching effect [22], rectangular excavation is more widely used because it is less complex to build due to its shape and it follows a similar shape to most common subsoil structures being built within the excavation (e.g., footings, pile caps, piers, mat foundations). For the problem of unsupported rectangular excavations, stability so- lutions were reported by Ukritchon et al. [22] using 3D FELA. Their solutions are based on the Tresca failure criterion, which is limited to isotropic clays. However, it is common knowledge that soil, particularly clays, normally exhibits anisotropy and heterogeneity due to deposi- tional geologic processes. It is generally recognized that soil anisotropy can have a substantial influence on clay stability, e.g., [23–26]. Ladd [23,24] reported that partial strength anisotropy in natural clays is generated through the processes of deposition and sedimentation with fa- vored particle orientation. It was also demonstrated in the same paper that the anisotropic shear strengths of clays are very much dependent on the different shearing modes as well as the depositional axis. Thus, including anisotropy and non-homogeneity in the stability solu- tion of unsupported excavation will provide a more reliable and realistic solution to excava- tion problems. Some studies have explored the problem of excavation in anisotropic clays, e.g., [27,28], but most are braced or supported. Indeed, there are three undrained shear strengths that can be obtained in a labora- tory: (1) triaxial compression (TC), (2) triaxial extension (TE), and (3) direct simple shear (DSS). The three undrained shear strengths have contributed to the development of math- ematical forms of failure criteria for anisotropic soils, e.g., [9,10,25,29–33]. Recently, Krab- benhoft and Lyamin [34] developed a unique failure criterion for anisotropic clays, known as AUS (Anisotropic Undrained Shear), by adopting the Generalized Tresca (GT) criterion for undrained total stress analysis. Even though both Davis and Christian’s (DC) failure criterion [29] and the AUS failure criteria consider an empirical correlation of the un- drained strength (su) of clay in triaxial compression (TC), direct simple shear (DSS), and triaxial extension (TE), the explicit form of the DC failure criterion cannot be applied to 3D problems since it was developed under plane strain condition. Unlike the DC model, the AUS model was developed under 3D coordinates, which can be used to simulate 3D problems in the Cartesian coordinates. As a result, the AUS model is preferred in this paper to investigate the stability of 3D unsupported excavations. The AUS model has recently been included in the 3D FELA software, OptumG3 [35], and it has been successfully applied to the stability problems of plate anchors [36] and caissons [37]. Apart from the recent AUS studies, FELA has previously been adopted to report numerical results for various 3D geotechnical problems, such as determining the capacity of a rigid pile with a pile cap in Zhou et al. [38], the trapdoor stability problem in Shiau et al. [39], the bearing capacity of footings on slopes in Yang et al. [40], and the tunnel stability problem in Shiau and Al-Asadi [41–43]. A thorough search of the relevant literature shows that the undrained stability num- bers for unsupported rectangular excavations considering both anisotropy and heteroge- neity have never been reported in the literature. The most recent paper by Yodsomjai et al. [44], which has close similarities to this current study, tackled the undrained stability of unsupported conical slopes in anisotropic clays, which was similarly analyzed using the AUS failure criterion. However, due to its axisymmetric condition, it becomes a 2D plane strain problem rather than 3D. Other than the study undertaken by Ukritchon et al. [22] on the 3D undrained stability of unsupported rectangular excavations in non-homo- geneous clays, which is also similar to the current study but without considering soil ani- sotropy, most of the other studies in the literature that dealt with anisotropy and hetero- geneity in clays were related to other stability problems such as trapdoors [30], pile Buildings 2022, 12, 1425 3 of 21 bearing capacity [32], unlined square tunnels [33], anchors [36], and suction caissons [37]. Therefore, the aim of this paper is to study this underexplored subject on the 3D un- drained stability of unsupported rectangular excavations in clays with linearly increasing anisotropic shear strength. The stability solutions were formulated by a dimensionless stability number that is a function of four dimensionless parameters: the excavated depth ratio, the aspect ratio of the excavated site, the shear strength gradient ratio, and the ani- sotropic strength ratio. The selected failure mechanisms of this problem were examined to demonstrate the effects of all four dimensionless parameters. With the development of accurate design equations, the study would assist practicing engineers in determining the soil stability of unsupported rectangular excavations in clays with anisotropy and heter- ogeneity. 2. Statement of the Problem and Modelling Technique Figure 1 shows the problem of defining a 3D unsupported rectangular excavation. Due to the problem of symmetry, only a quarter of the model domain was used in the analysis. See Figure 1a for the model. The excavation depth is denoted by H, B is the ex- cavation width, and L is the length. (a) Problem geometry (b) linearly increasing anisotropic strength Figure 1. Statement of the problem. The AUS failure criterion with the associated flow rule was used to study the 3D soil stability of the unsupported rectangular excavation. The three anisotropic undrained shear strengths obtained from triaxial compression (suTC), triaxial extension (suTE), and Buildings 2022, 12, 1425 4 of 21 direct simple shear (suDSS) were the required strengths for this failure criteria. According to Krabbenhoft et al. [45], two anisotropic strength ratios can be defined using the three undrained shear strengths: (1) re = suTE/suTC and (2) rs = suDSS/suTC. The relationship between re and rs is the harmonic mean, which can be written as follows: 2r r = (1) 1 + r As shown in Equation (1), the parametric analysis only used one anisotropic strength ratio, which is re. Note that rs is a function of re, and the range of re should be between 0.5 and 1. A change in the re value may vary the AUS failure criterion’s failure surface, as shown in Figure 2 [34,45]. The form of the yield function of the AUS model with the har- monic mean of three undrained shear strengths can be expressed by Equation (2): Fr =− σσ + −12 σ −σ −s =0 (2) ()( ) ue 13 2 3 uTC where 1 ≥ 2 ≥ 3 are the principal stresses (positive in compression), and Fu is the yield function. It should be noted that the AUS failure criterion becomes the Tresca failure criterion when re = 1, meaning the isotropic state, i.e., suTC = suTE = suDSS. Note that, for the AUS failure criterion, three undrained shear strengths were considered to be an empirical correlation of the undrained strength (su) of clay in triaxial compression (TC) for suTC, direct simple shear (DSS) for suDSS, and triaxial extension (TE) for suTE. Figure 2. Generalized Tresca surface used in the Anisotropic Undrained Shear (AUS) failure criterion. The increasing shear strength with depth, i.e., heterogeneous soil behaviors, is an- other important factor when determining soil stability. This variation in shear strength has been considered by many researchers for the problems of the face stability of tunnels [10,31,33, 46], supported excavations [47], piles [48-49], floodwalls [50], and active trapdoors [30, 51]. This study considered three anisotropic undrained shear strengths that linearly increase with depth. Mathematically, they are expressed in Equations (3)–(5). (3) 𝑠 (𝑧) = 𝑠 +𝜌𝑧 (4) 𝑠 (𝑧) = 𝑠 +𝑟 𝜌𝑧 (5) 𝑠 (𝑧) = 𝑠 +𝑟 𝜌𝑧 Buildings 2022, 12, 1425 5 of 21 where suTC0, suTE0, and suDSS0 are the anisotropic undrained shear strengths at the ground level, z is the depth from the ground surface, and  is the linear strength gradient. See Figure 1b for the linear distributions of the three anisotropic undrained shear strengths. Using the dimensional analysis [52], a function combining four dimensionless param- eters that are variables of a stability number function can be expressed by Equation (6). γρ HBH B Nf == (, ,r,m= ) (6) sLB s uTC00 uTC where N is the stability number, B/L is the aspect ratio of the excavated site, H/B is the excavated depth ratio, re is the anisotropic strength ratio, and m is the strength gradient ratio. In the lower bound analysis, a four-node tetrahedron element is used, where six un- known nodal stresses are used for each node of tetrahedral elements. The statically ad- missible stress discontinuities are allowed to produce the continuity of normal and shear stresses along the interfaces of all the elements. The conditions of stress equilibrium, stress boundary condition, and the AUS failure criterion are all constraints in a typical LB anal- ysis, in which the objective function is to maximize the critical unit weight γ that yields an excavation collapse. In the upper bound theorem, a four-node tetrahedron element is also adopted for the upper bound analysis, where each node contains three unknown ve- locities that vary linearly within the tetrahedron element. The kinematically admissible velocity discontinuities are applied at the interfaces of all the elements. The material is set to obey the AUS failure criterion associated flow rule. The formulated objective function is to minimize the critical unit weight γ. The obtained critical γ from both LB and UB analyses were then used to compute the stability number in Equation (6). More details on the LB and UB FELA can be found in [5]. Figure 3 presents a typical 3D FELA mesh used for the analysis. The nodes around the sides of the model are fixed in the normal direction to the planes of the sides. The same boundary condition is applicable to the two symmetrical planes as well. At the bottom domain, the nodes are fixed in all directions. Both the ground surface and the excavation faces are free to move in all directions. The overall domain size is chosen to be sufficiently large such that the stability solutions are not affected by the boundary conditions, i.e., the effects of boundary size on the computed LB solutions are minimized by generating LB meshes with sufficient lateral and lower dimensions that produce a computed plastic yielding zone that does not intersect the boundary planes. Automatic adaptive mesh re- finement is one of the advanced features of the 3D program. This technique is based on Ciria et al. [53], where the numbers of elements in sensitive zones (i.e., with very high plastic shear strains) are increased through successive iterations with adaptive mesh re- finement. The required input for the adaptive scheme is the original and target number of elements, the number of adaptive iterations, and the control variable for error estimation (i.e., shear power in this paper). In this study, 5000 initial elements were employed, which was expanded to 10,000 elements after five iterations. Buildings 2022, 12, 1425 6 of 21 Figure 3. A typical FELA model and potential failure mechanism. Note that, the range of four dimensionless parameters in all studies of the paper are: (1) H/B = 0.5, 1, 2, 3, 4; (2) B/L = 1, 2/3, 1/2, 1/4, 1/8; (3) re = 0.5, 0.6, 0.7, 0.8, 0.9, 1; (4) m = 0, 4, 12, 25, 100. The ranges of H/B and B/L used in this study are based on the previous work by Ukritchon et al. [22]. For the range of re, Krabbenhoft et al. [45] suggested that the value of this parameter should be between 0.5 and 1, which corresponds to the natural ratios of compressive and tensile undrained shear strengths. The range of m or ρB/suTC0 constitutes the combined effect of the excavation size B, the compressive shear strength at the ground surface suTC0, and the linear strength gradient ρ. In practice, suTC0 and ρ depend on the geological nature of the sites where the excavated width B can range from 1 to 20 m in practice. Theoretically, the ρB/suTC0 parameter ranges from 0 (homogeneous case) to a large value (non-homogeneous case). The homogeneous cases correspond to a case with ρ = 0 and/or a very large value of suTC0. For the non-homogeneous cases, they represent the cases with a relatively low suTC0 and/or a relatively large value of ρ. 3. Comparison for Model Validation In the first step of the investigation, the stability numbers, N, determined by the rig- orous FELA solutions, were compared with the published results in Ukritchon et al. [22]. The comparison shown in Figure 4a is for the effect of H/B on the stability number N, as well as its effect on various B/L with isotropic (re = 1) and homogeneous (m = 0) clays. Note that re is the anisotropic strength ratio, and m is the strength gradient ratio. Moreover, note that the present solution is the average (Avg) results calculated from the UB and LB FELA solutions. In general, the stability number increases with the increasing depth ratio H/B. The increase can be either nonlinearly or linearly, depending on the value of B/L. When B/L is smaller (B/L = 1/4, 1/8), fewer 3D constraints are observed, and a linear rela- tionship between N and H/B is presented. Buildings 2022, 12, 1425 7 of 21 (a) m = 0 (b) m = 4 Figure 4. Comparison of stability numbers N (re = 1). Whilst in Figure 4b, the comparison is made for (re = 1 and m = 4). It is interesting to note that, for the large strength gradient ratio such as m ≥ 4, N increases linearly with an increase in H/B for all values of B/L. Overall, the numerical results have shown an excellent agreement between the two solutions. The neglectable numerical differences between the two results can be attributed to the use of the perfectly plastic Tresca failure criterion in Ukritchon et al. [22] as opposed to the AUS failure criterion, with re = 1 used in the present study. To the best knowledge of the authors, there are currently no other values of re to be compared since this is the first work to consider the stability of unsupported rectangular excavations in anisotropic and non-homogeneous soils. 4. Results and Discussions The effects of H/B on the stability number N are presented in Figure 5 for various values of re (the anisotropic strength ratio). Those shown in Figure 5a–f are for B/L = (0.25, 1.0) and m = (0, 12, 100). The numerical results have shown that the stability number N increases linearly with an increase in the excavation depth ratio H/B, except for the case of (B/L = 1.0 and m = 0). See Figure 5b for this special case of a square (B/L = 1.0) excavation in homogeneous (m = 0) clay, where N increases nonlinearly with the increasing H/B. One of the possible reasons could be attributed to the greater corner effects (geometrical Buildings 2022, 12, 1425 8 of 21 arching). Note that the rate of increase in N (i.e., the gradient) increases as the strength gradient ratio m increases. Furthermore, note that a decrease in the anisotropic ratio re results in a decrease in the stability number. The selected failure mechanisms (shear dis- sipation) are presented in Figure 6 for the different values of H/B = (0.5, 1, 2 3, 4). The comparison is based on the case of (re = 0.7, m = 4 and B/L = 1), and the results of the shear dissipation contour plots have shown a toe-failure mode for the shallow cases of H/B = (0.5 and 1). On the other note, for H/B > 1, a face-failure mode is obtained owing to the effect of the strength gradient ratio m. (a) B/L = 0.25 and m = 0 (b) B/L = 1 and m = 0 (c) B/L = 0.25 and m = 12 (d) B/L = 1 and m = 12 (e) B/L = 0.25 and m = 100 (f) B/L = 1 and m = 100 Figure 5. N vs. H/B for the various re (B/L = 0.25, 1.0 and m = 0, 12, 100). Buildings 2022, 12, 1425 9 of 21 (a) H/B = 0.5 (b) H/B = 1 (c) H/B = 2 (d) H/B = 3 (e) H/B = 4 Figure 6. Potential failure mechanisms—effect of H/B (re = 0.7, m = 4, and B/L = 1). Figure 7 shows the effects of B/L (the aspect ratio of the excavated site) on the stability number N for the various values of re (the anisotropic strength ratio). All of the values of m (m = 0, 4, 12, 25, 100) are considered for the chosen depth ratio H/B = 3, and they are presented in Figure 7a–e respectively. The numerical results have shown that N increases nonlinearly with the increasing B/L for all values of re. The gradient of the nonlinear curves becomes smaller as the strength gradient ratio m increases (see Figure 7a–d)—a linear re- lationship is observed for the case with m = 100. It is also noted that the stability number N decreases as the anisotropic strength ratio re decreases (transforming from isotropic to anisotropic soils). The comparison of five failure mechanisms for the various B/L = (1/8, 1/4, 1/2, 2/3, 1) is shown in Figure 8. The chosen plots are for H/B = 1 (re = 0.7, and m = 4). The shear dissipation contour plot of B/L = (1/8, 1/4) has shown a mechanism that resem- bles a 2D plane strain condition (see Figure 8a,b). As the value of B/L increases (so as the sta- bility number N), a stronger system is presented, owing to full 3D corner effects (see Figure 8e Buildings 2022, 12, 1425 10 of 21 for B/L = 1). Interestingly, a two-way failure mechanism is found in Figure 8e for B/L = 1. It should also be noted that the failure patterns are for the toe-failure mode in this shallow case of H/B = 1. (a) H/B = 3 and m = 0 (b) H/B = 3 and m = 4 (c) H/B = 3 and m = 12 (d) H/B = 3 and m = 25 (e) H/B = 3 and m = 100 Figure 7. N vs. B/L for the various re (H/B = 3.0 and m = 0, 4, 12, 25, 100). Buildings 2022, 12, 1425 11 of 21 (a) B/L = 1/8 (b) B/L = 1/4 (c) B/L = 1/2 (d) B/L = 2/3 (e) B/L = 1 Figure 8. Potential failure mechanisms—effect of B/L (H/B = 1, re = 0.7, and m = 4). Figure 9 shows the relationship between the stability number N and the strength gra- dient ratio m for the various values of re (the anisotropic strength ratio). The presentations are for B/L = (1/8, 1) and H/B = (0.5, 1.0, 4.0). In general, an increase in m results in an increase in N. A linear relationship between N and m is observed in all investigated cases. Same as the previous discussions, the smaller the re, the smaller the stability number N. The chosen case for the failure mechanism comparison is presented in Figure 10 for (re = 0.7, H/B = 1, B/L = 1) with different values of m = (0, 4, 12, 25, 100). It should be noted that the size of the failure zone decreases as m increases. As a result, the failure mechanism changes from a toe-failure mode to a face-failure mode when m is larger than 4. (a) H/B = 0.5 and B/L = 1/8 (b) H/B = 0.5 and B/L = 1 Buildings 2022, 12, 1425 12 of 21 (c) H/B = 1 and B/L = 1/8 (d) H/B = 1 and B/L = 1 (e) H/B = 4 and B/L = 1/8 (f) H/B = 4 and B/L = 1 Figure 9. N vs. m for the various re (H/B = 0.5, 1.0, 4.0 and B/L = 1/8, 1). (a) m = 0 (b) m = 4 (c) m = 12 (d) m = 25 (e) m = 100 Figure 10. Potential failure mechanisms—effect of m (re = 0.7, H/B = 1, and B/L = 1). Buildings 2022, 12, 1425 13 of 21 Figure 11 shows the relationships between the stability number N and the anisotropic strength ratio re for various values of m = (0, 4, 12, 25, 100). The plots are for the selected ratios of H/B = (0.5, 4) and B/L = (1/8, 1/2, 1). The numerical results have shown that the larger the m, the greater the stability number N. Overall, the stability number N varies linearly with the increase in the anisotropic ratio re. The rate of increase (gradient of the line) in N is dependent on the value of m. The larger the m, the greater the gradient of the line. Figure 12 shows a comparison of failure mechanisms among the various anisotropic ratios, re = (0.5–1). The comparison is for the excavation problem of (m = 4, H/B = 1, B/L = 1). The results have shown that the failure patterns are all in a toe-failure mode, and the variation of anisotropic ratio re does not seem to affect the failure size of the problem. The same conclusion can be drawn from Figure 13, where an additional study of m = 100 is presented. Indeed, as discussed previously, the face-failure mode is always the one ob- served for the large strength gradient ratio such as m = 100. It should be noted that all of the numerical results of this paper study are summarized in Tables 1–3. (a) B/L = 1/8 and H/B = 0.5 (b) B/L = 1/8 and H/B = 4 (c) B/L = 1/2 and H/B = 0.5 (d) B/L = 1/2 and H/B = 4 (e) B/L = 1 and H/B = 0.5 (f) B/L = 1 and H/B = 4 Figure 11. N vs. re for the various m (H/B = 0.5, 4.0 and B/L = 1/8, 1/2, 1). Buildings 2022, 12, 1425 14 of 21 (a) re = 0.5 (b) re = 0.6 (c) re = 0.7 (d) re = 0.8 (e) re = 0.9 (f) re = 1.0 Figure 12. Potential failure mechanisms—effect of re (m = 4, H/B = 1, and B/L = 1). (a) re = 0.5 (b) re = 0.6 (c) re = 0.7 (d) re = 0.8 (e) re = 0.9 (f) re = 1.0 Figure 13. Potential failure mechanisms—effect of re (m = 100, H/B = 1, and B/L = 1). Buildings 2022, 12, 1425 15 of 21 Table 1. Stability numbers, N (re = 1.0 and 0.9). re = 1 re = 0.9 m B/L B/L H/B H/B 1 2/3 1/2 1/4 1/8 1 2/3 1/2 1/4 1/8 0.5 4.559 4.372 4.234 3.959 3.860 0.5 4.299 4.128 4.027 3.764 3.460 1 5.291 4.953 4.677 4.153 3.955 1 4.958 4.661 4.420 3.931 3.759 0 2 6.420 5.969 5.553 4.637 4.125 2 5.968 5.560 5.201 4.384 3.917 3 7.170 6.707 6.279 5.135 4.362 3 6.657 6.243 5.849 4.833 4.137 4 7.770 7.280 6.862 5.606 4.632 4 7.216 6.774 6.388 5.262 4.378 0.5 9.259 9.008 8.758 8.362 8.255 0.5 8.749 8.495 8.288 7.870 7.898 1 15.758 14.648 13.891 12.832 12.437 1 14.778 13.806 13.189 12.151 11.775 4 2 30.562 28.105 26.095 22.663 21.168 2 28.283 26.251 24.639 21.455 19.996 3 45.966 42.212 39.161 33.462 30.330 3 42.365 39.288 36.776 31.691 28.730 4 61.106 56.160 52.214 44.704 39.942 4 56.528 52.402 48.986 42.238 37.914 0.5 18.213 17.662 17.175 16.735 16.561 0.5 17.156 16.730 16.444 15.785 15.516 1 34.964 32.841 31.468 29.458 28.713 1 32.745 31.043 29.716 27.916 27.309 12 2 69.931 65.641 62.571 57.045 53.981 2 65.470 61.913 59.186 53.975 51.219 3 104.996 98.597 93.842 85.662 80.372 3 98.151 92.897 88.782 81.116 76.146 4 140.088 131.336 125.132 114.290 107.172 4 130.918 124.024 118.326 108.284 101.830 0.5 32.254 31.504 30.613 30.072 29.207 0.5 30.536 29.713 29.120 28.351 28.198 1 64.231 61.253 59.346 56.412 55.183 1 60.475 57.966 56.147 53.315 51.845 25 2 128.460 122.685 118.597 111.345 107.223 2 120.981 115.931 112.269 104.690 101.445 3 192.627 183.920 178.077 167.298 160.784 3 181.659 174.002 168.429 158.561 152.492 4 256.720 245.260 237.102 223.242 214.300 4 241.932 231.924 224.580 211.538 203.318 0.5 111.392 109.320 107.744 106.143 105.742 0.5 105.461 103.419 101.948 100.684 100.869 1 223.302 218.275 215.000 209.382 206.320 1 211.114 206.404 203.672 198.231 196.114 100 2 446.620 436.122 429.972 415.484 411.352 2 423.437 413.457 407.257 396.090 389.648 3 671.124 654.846 645.300 627.836 617.555 3 632.489 620.067 611.184 594.366 585.180 4 894.322 873.216 859.562 837.478 823.428 4 845.674 827.874 813.880 793.196 780.282 Table 2. Stability numbers, N (re = 0.8 and 0.7). re = 0.8 re = 0.7 m B/L B/L H/B H/B 1 2/3 1/2 1/4 1/8 1 2/3 1/2 1/4 1/8 0.5 3.998 3.880 3.807 3.531 3.255 0.5 3.696 3.571 3.503 3.273 3.043 1 4.594 4.327 4.129 3.697 3.525 1 4.206 3.983 3.807 3.415 3.257 0 2 5.502 5.126 4.825 4.103 3.668 2 5.006 4.694 4.418 3.790 3.398 3 6.120 5.741 5.400 4.518 3.885 3 5.576 5.232 4.935 4.155 3.590 4 6.608 7.654 5.868 4.896 4.100 4 6.010 5.658 5.350 4.494 3.788 0.5 8.192 8.020 7.753 7.399 7.252 0.5 7.574 7.352 7.212 6.878 6.741 1 13.733 12.929 12.311 11.406 11.087 1 12.572 11.952 11.463 10.533 10.203 4 2 26.032 24.326 22.942 20.133 18.797 2 23.642 22.283 21.126 18.623 17.413 3 38.835 36.234 34.224 29.712 26.985 3 35.222 33.110 31.422 27.500 25.028 4 51.836 48.330 45.558 39.574 35.560 4 46.896 44.168 41.810 36.482 32.908 0.5 16.080 15.921 15.174 14.900 14.827 0.5 14.856 14.759 14.108 13.673 13.539 1 30.498 29.064 27.867 26.213 25.641 1 28.015 26.842 25.829 24.231 23.713 12 2 60.747 57.799 55.401 50.671 47.946 2 55.688 53.260 51.218 46.939 44.337 3 91.143 86.838 83.121 76.007 71.582 3 83.531 79.896 76.892 70.436 66.102 4 121.484 115.712 110.738 101.452 95.462 4 111.290 106.608 102.410 93.984 88.160 Buildings 2022, 12, 1425 16 of 21 0.5 28.631 27.955 27.377 26.578 25.636 0.5 26.442 25.683 25.257 24.657 24.445 1 56.445 54.317 52.686 50.016 49.108 1 51.882 50.107 48.741 46.161 45.508 25 2 112.831 108.652 105.268 99.169 95.067 2 103.890 100.505 97.455 91.487 87.891 3 169.331 162.911 157.857 148.704 142.529 3 155.756 150.548 146.213 137.873 132.387 4 225.288 231.924 210.434 198.498 190.700 4 207.658 200.882 194.796 183.884 176.652 0.5 98.804 97.060 95.754 94.274 92.989 0.5 91.379 89.479 88.734 87.369 82.820 1 197.710 193.955 190.910 183.274 183.608 1 182.879 179.463 176.887 171.749 169.835 100 2 395.618 387.909 382.134 372.104 364.822 2 365.556 359.051 353.148 340.475 337.916 3 594.897 581.319 573.506 558.114 548.117 3 549.108 538.401 530.817 517.062 502.274 4 794.012 775.738 763.674 744.168 731.800 4 733.122 719.498 706.896 689.530 677.690 Table 3. Stability numbers, N (re = 0.6 and 0.5). re = 0.6 re = 0.5 m B/L B/L H/B H/B 1 2/3 1/2 1/4 1/8 1 2/3 1/2 1/4 1/8 0.5 3.317 3.256 3.181 2.955 2.771 0.5 2.944 2.882 2.820 2.628 2.319 1 3.774 3.586 3.450 3.117 2.969 1 3.314 3.155 3.029 2.758 2.631 0 2 4.479 4.207 3.977 3.444 3.097 2 3.901 3.674 3.495 3.050 2.743 3 4.979 4.679 4.421 3.753 3.254 3 4.328 4.085 3.857 3.309 2.898 4 5.366 5.048 4.784 4.054 3.440 4 4.666 4.390 4.188 3.564 3.052 0.5 6.823 6.675 6.550 6.294 6.027 0.5 6.038 5.960 5.753 5.559 5.491 1 11.358 10.816 10.409 9.624 9.369 1 9.967 9.521 9.226 8.561 8.277 4 2 21.138 19.958 19.048 16.942 15.852 2 18.563 17.571 16.751 15.037 14.072 3 31.406 29.540 28.229 24.951 22.737 3 27.335 25.931 24.728 22.184 20.210 4 41.728 39.450 37.670 33.366 29.942 4 36.506 34.454 32.832 29.342 26.568 0.5 13.475 13.208 12.811 12.563 11.985 0.5 11.915 11.702 11.428 11.133 10.896 1 25.262 24.302 23.432 22.134 21.623 1 22.330 21.468 20.794 19.667 19.228 12 2 50.092 48.210 46.566 42.644 40.585 2 44.187 42.254 41.138 37.943 35.950 3 75.329 72.156 69.764 64.122 60.272 3 66.371 63.263 61.695 56.939 53.634 4 100.316 96.282 92.980 85.524 80.114 4 88.410 84.236 82.204 75.902 71.372 0.5 24.129 23.467 23.007 22.585 22.364 0.5 21.192 20.744 20.493 19.967 19.757 1 47.000 45.646 44.432 42.214 41.395 1 41.437 40.252 39.182 37.430 36.636 25 2 93.925 91.067 88.624 83.521 80.038 2 82.907 80.487 78.697 74.134 70.683 3 141.065 136.667 132.840 125.496 120.269 3 124.328 120.735 118.287 111.372 107.109 4 188.000 182.288 177.256 167.254 160.868 4 165.788 160.782 157.210 148.552 141.700 0.5 83.089 81.711 79.615 79.360 75.411 0.5 73.433 72.459 71.454 70.765 69.833 1 165.317 163.306 160.071 156.043 154.564 1 147.068 144.681 142.492 139.251 137.323 100 2 332.289 326.784 322.154 313.455 305.386 2 294.210 289.922 285.932 278.151 272.439 3 500.166 489.830 483.281 470.489 462.878 3 440.799 433.592 428.594 416.861 408.983 4 664.460 653.052 644.744 627.270 617.136 4 587.558 579.212 571.564 556.762 545.928 5. Design Equations A mathematical equation is developed and presented in this section by using a trial- and-error method of curve fitting. Nonlinear regression with multiple variables to the Avg bound solutions is employed to develop design equations for estimating the stability fac- tor of unsupported rectangular excavations in clays with anisotropy and heterogeneity, as shown in Equation (7). Buildings 2022, 12, 1425 17 of 21    ρρ BH B H B H ρB H      N=+aa + a +a + a −a + a +a (7)    12   348 56  7 sB L B s B s B       uTC0 uTC0  uTC0  where a1 to a8 are constant coefficients. To determine the optimal value of the constant coefficients (a1–a8), the nonlinear least square regression [54] is utilized. The sum of the squares of the deviation in N between the computed Avg solutions shown in Tables 1–3 and the approximate solutions from Equation (7) is then minimized to obtain the optimal values of constant coefficients. Note that, to achieve high accuracy, Equation (7) is a “step-wised” equation devel- oped for the different values of re. Using the complete data in Tables 1–3, the optimal val- ues of the coefficients a1 to a8 for the different values of re are computed and presented in Table 4. On the other hand, the comparisons of N between the computed Avg bound so- lutions and the approximate solutions from Equation (6) are shown in Figure 14a–f, respec- tively, for different values of re = (0.5 to 1.0). It is pleasing to see the highly accurate solutions of the equation development—the coefficient of determination (R ) = 99.99%. Table 4. Constant coefficients for the proposed design equation. Constant Coef- re ficients 0.5 0.6 0.7 0.8 0.9 1 a1 3.16697 3.3617 3.73044 4.1027 3.81764 4.28435 a2 1.35068 1.52276 1.66728 1.80244 1.92334 2.02793 a3 −0.69739 −0.55146 −0.64589 −1.47779 −0.09287 −0.55003 a4 0.46763 0.6014 0.72694 0.99956 1.01361 1.18849 a5 1.96752 2.24958 2.55588 3.08414 3.30362 3.39274 a6 0.92331 0.99065 1.14742 1.57205 1.5951 1.54619 a7 −0.07001 −0.07579 −0.081989 −0.12147 −0.13622 −0.13032 a8 −0.05667 −0.05068 −0.06182 −0.10530 −0.10229 −0.09884 R 99.99% 99.99% 99.99% 99.99% 99.99% 99.99% r = 0.6 r = 0.5 e 800 800 600 600 400 400 2 2 200 200 R = 99.99% R = 99.99% 0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 N, Proposed design equation N, Proposed design equation (a) re = 0.5 (b) re = 0.6 N, FELA solutions N, FELA solutions Buildings 2022, 12, 1425 18 of 21 1000 1000 r = 0.7 r = 0.8 800 800 600 600 400 400 200 2 200 2 R = 99.99% R = 99.99% 0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 N, Proposed design equation N, Proposed design equation (c) re = 0.7 (d) re = 0.8 1000 1000 r = 1.0 r = 0.9 800 800 600 600 400 400 2 200 2 R = 99.99% R = 99.99% 0 200 400 600 800 1000 0 200 400 600 800 1000 N, Proposed design equation N, Proposed design equation (e) re = 0.9 (f) re = 1.0 Figure 14. Predicted N vs. FELA N. 6. Conclusions Rigorous stability solutions of the unsupported rectangular excavation in anisotropic and heterogeneous clays have been successfully studied in the paper using 3D LB and UB FELA. The stability number (N) that is a function of the excavation aspect ratio, B/L, the excavated depth ratio, H/B, the strength gradient ratio, m = ρB/suTC0, and the anisotropic strength ratio, re, was presented throughout the paper. The following conclusions are drawn based on the study. 1. The stability number, N, increases with an increase in all of the investigated parame- ters of B/L, H/B, m, and re. The increases can be either in a linear or nonlinear relation- ship. The linear relationship was obtained for all investigated cases except for cases with smaller values of m, where a nonlinear relationship exists between N and B/L. 2. The failure patterns of unsupported rectangular excavation in anisotropic and heter- ogeneous clays are either in a toe-failure mode (for small values of H/B, i.e., H/B = 0.5, 1) or a face-failure mode (for large values of H/B > 1) due to the effect of the strength gradient ratio m. For large values of m > 4, the failure modes are predominately the face-failure mode. The variation in the anisotropic ratio, re, does not seem to affect the failure size of the unsupported rectangular excavation problem. 3. A new equation for predicting the stability number, N, of the unsupported rectangu- lar excavation in anisotropic and heterogeneous clays is proposed. With the coeffi- cient of determination (R ) being 99.99%, the proposed equation is highly accurate and useful for practical uses. The proposed study provides deeper contextualized insights into the understanding of 3D unsupported excavations in undrained clay under the influence of soil anisotropy N, FELA solutions N, FELA solutions N, FELA solutions N, FELA solutions Buildings 2022, 12, 1425 19 of 21 and heterogeneity. Future work directions may include the seismic stability performance as well as the soil random field probabilistic analysis. Author Contributions: V.Q.L.: Data curation, Software, Investigation, Methodology, Writin—orig- inal draft; S.K.: Methodology, Validation, Writing –original draft; S.S.: Formal analysis, Software, Methodology; J.S.: Methodology, Writing—review & editing, Project administration, Supervision, Funding acquisition; L.T.C.: Writing—review, revising & editing. All authors have read and agreed to the published version of the manuscript. Funding: This study was supported by Thammasat Postdoctoral Fellowship, Thammasat Univer- sity Research Division, Thammasat University. 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Geotechnol. 2021, 8, 557–568. https://doi.org/10.1007/s40515-021-00153-y. 45. Krabbenhøft, K.; Galindo-Torres, S.A.; Zhang, X.; Krabbenhoft, J. AUS: Anisotropic undrained shear strength model for clays. Int. J. Numer. Anal. Methods Geomech. 2019, 43, 2652–2666. 46. Keawsawasvong, S.; Ukritchon, B. Design equation for stability of shallow unlined circular tunnels in Hoek-Brown rock masses. Bull. Eng. Geol. Environ. 2020, 79, 4167–4190. https://doi.org/10.1007/s10064-020-01798-8. 47. Keawsawasvong, S., Ukritchon, B. Undrained basal stability of braced circular excavations in non-homogeneous clays with linear increase of strength with depth. Computers and Geotechnics, 2019, 115, 103180. Buildings 2022, 12, 1425 21 of 21 48. Keawsawasvong, S., Lai, V.Q. End bearing capacity factor for annular foundations embedded in clay considering the effect of the adhesion factor. International Journal of Geosynthetics and Ground Engineering, 2021, 7, 15. 49. Ukritchon, B., Keawsawasvong, S. Design equations of uplift capacity of circular piles in sands. Applied Ocean Research, 2019, 90, 101844. 50. Keawsawasvong, S., Ukritchon, B. Finite element analysis of undrained stability of cantilever flood walls. International Journal of Geotechnical Engineering, 2017, 11(4), 355-367. 51. Keawsawasvong, S.; Shiau, J. Stability of active trapdoors in axisymmetry. Undergr. Space 2022, 7, 50–57. https://doi.org/10.1016/j.undsp.2021.05.001. 52. Butterfield, R. Dimensional analysis for geotechnical engineers. 1999, 49, 357–366. https://doi.org/10.1680/geot.1999.49.3.357. 53. Ciria, H.; Peraire, J.; Bonet, J. Mesh adaptive computation of upper and lower bounds in limit analysis. Int. J. Numer. Methods Eng. 2008, 75, 899–944. https://doi.org/10.1002/nme.2275. 54. Devore, J.L. 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Undrained Stability of Unsupported Rectangular Excavations: Anisotropy and Non-Homogeneity in 3D

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Article Undrained Stability of Unsupported Rectangular Excavations: Anisotropy and Non-Homogeneity in 3D 1 2 3, 3 3 Van Qui Lai , Jim Shiau , Suraparb Keawsawasvong *, Sorawit Seehavong and Lowell Tan Cabangon Faculty of Civil Engineering, Ho Chi Minh City University of Technology (HCMUT), Vietnam National University Ho Chi Minh City (VNU-HCM), 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City 70000, Vietnam School of Engineering, University of Southern Queensland, Darling Heights, QLD 4350, Australia Department of Civil Engineering, Faculty of Engineering, Thammasat School of Engineering, Thammasat University, Pathumthani 12120, Thailand * Correspondence: ksurapar@engr.tu.ac.th Abstract: The stability of unsupported rectangular excavations in undrained clay is examined under the influence of anisotropy and heterogeneity using the three-dimensional finite element upper and lower bound limit analysis with the Anisotropic Undrained Shear (AUS) failure criterion. Three anisotropic undrained shear strengths are considered in the study, namely triaxial compression, triaxial extension, and direct simple shear. Special considerations are given to the study of the line- arly-increased anisotropic shear strengths with depth. The numerical solutions are presented by an undrained stability number that is a function of four dimensionless parameters, i.e., the excavated depth ratio, the aspect ratio of the excavated site, the shear strength gradient ratio, and the aniso- tropic strength ratio. To the authors’ best knowledge, this is the first of its kind to present the stabil- ity solutions of 3D excavation considering soil anisotropy and heterogeneity. As such, this paper introduces a novel approach for predicting the stability of unsupported rectangular excavation in Citation: Lai, V.Q.; Shiau, J.; undrained clays in 3D space, accounting for soil anisotropy and non-homogeneity. Notably, it de- Keawsawasvong, S.; Seehavong, S.; velops a basis to formulate a mathematical equation and design charts for estimating the stability Cabangon, L.T. Undrained Stability factor of such type of excavation, which should be of great interest to engineering practitioners. of Unsupported Rectangular |Excavations: Anisotropy and Keywords: stability; excavations; anisotropy; heterogeneity; finite element limit analysis Non-Homogeneity in 3D. Buildings 2022, 12, 1425. https://doi.org/ 10.3390/buildings12091425 Academic Editor: Giuseppina Uva 1. Introduction Received: 10 August 2022 Unsupported excavation does not require retaining wall systems, and it is considered Accepted: 8 September 2022 one of the affordable construction methods that are widely employed in many shallow un- Published: 10 September 2022 derground construction projects. Shallow underground structures such as pipelines, shallow tunnels, and underpasses can be constructed by utilizing this excavation technique. Other ex- Publisher’s Note: MDPI stays neu- amples may include the construction of piers, footings, retaining structures, raft foundations, tral with regard to jurisdictional mat foundations, and water tanks. An unsupported excavation during construction, if not claims in published maps and institu- tional affiliations. properly assessed, can lead to an eventual collapse of the excavation wall that could result in an injury or fatality. These unfortunate events can cost money and cause death. It is, therefore, imperative to assess the stability of such unsupported excavations to reduce the risk of soil failure, thereby improving site safety and preventing death. This study aims to contribute to Copyright: © 2022 by the authors. Li- reducing that risk by providing a novel approach that predicts the undrained stability of un- censee MDPI, Basel, Switzerland. supported rectangular excavations in anisotropic and non-homogeneous clays. This article is an open access article In general, the excavation can have either cylindrical, conical, or rectangular shapes. distributed under the terms and con- Griffiths and Koutsabeloulis [1] used a displacement-based elastoplastic finite element ditions of the Creative Commons At- analysis to study the stability of cylindrical excavations under axisymmetric conditions. tribution (CC BY) license (https://cre- The same problem was also examined by Britto and Kusakabe [2,3] using the plastic- ativecommons.org/licenses/by/4.0/). bound theorems. The recent development of finite element limit analysis (FELA) is a Buildings 2022, 12, 1425. https://doi.org/10.3390/buildings12091425 www.mdpi.com/journal/buildings Buildings 2022, 12, 1425 2 of 21 powerful numerical method based on the lower bound (LB) theorem, the upper bound (UB) theorem, and the finite element technique, as demonstrated in [4–10]. The axisymmetric FELA was employed by [11–16] to obtain stability solutions for vertical circular excavations. Re- cently, the stability of unsupported conical excavations was investigated by [17–21]. Among the various shapes of excavation, rectangular and cylindrical shapes are the most common in practice. Although cylindrical shapes may lead to smaller amounts of excavated material and they are more stable due to the arching effect [22], rectangular excavation is more widely used because it is less complex to build due to its shape and it follows a similar shape to most common subsoil structures being built within the excavation (e.g., footings, pile caps, piers, mat foundations). For the problem of unsupported rectangular excavations, stability so- lutions were reported by Ukritchon et al. [22] using 3D FELA. Their solutions are based on the Tresca failure criterion, which is limited to isotropic clays. However, it is common knowledge that soil, particularly clays, normally exhibits anisotropy and heterogeneity due to deposi- tional geologic processes. It is generally recognized that soil anisotropy can have a substantial influence on clay stability, e.g., [23–26]. Ladd [23,24] reported that partial strength anisotropy in natural clays is generated through the processes of deposition and sedimentation with fa- vored particle orientation. It was also demonstrated in the same paper that the anisotropic shear strengths of clays are very much dependent on the different shearing modes as well as the depositional axis. Thus, including anisotropy and non-homogeneity in the stability solu- tion of unsupported excavation will provide a more reliable and realistic solution to excava- tion problems. Some studies have explored the problem of excavation in anisotropic clays, e.g., [27,28], but most are braced or supported. Indeed, there are three undrained shear strengths that can be obtained in a labora- tory: (1) triaxial compression (TC), (2) triaxial extension (TE), and (3) direct simple shear (DSS). The three undrained shear strengths have contributed to the development of math- ematical forms of failure criteria for anisotropic soils, e.g., [9,10,25,29–33]. Recently, Krab- benhoft and Lyamin [34] developed a unique failure criterion for anisotropic clays, known as AUS (Anisotropic Undrained Shear), by adopting the Generalized Tresca (GT) criterion for undrained total stress analysis. Even though both Davis and Christian’s (DC) failure criterion [29] and the AUS failure criteria consider an empirical correlation of the un- drained strength (su) of clay in triaxial compression (TC), direct simple shear (DSS), and triaxial extension (TE), the explicit form of the DC failure criterion cannot be applied to 3D problems since it was developed under plane strain condition. Unlike the DC model, the AUS model was developed under 3D coordinates, which can be used to simulate 3D problems in the Cartesian coordinates. As a result, the AUS model is preferred in this paper to investigate the stability of 3D unsupported excavations. The AUS model has recently been included in the 3D FELA software, OptumG3 [35], and it has been successfully applied to the stability problems of plate anchors [36] and caissons [37]. Apart from the recent AUS studies, FELA has previously been adopted to report numerical results for various 3D geotechnical problems, such as determining the capacity of a rigid pile with a pile cap in Zhou et al. [38], the trapdoor stability problem in Shiau et al. [39], the bearing capacity of footings on slopes in Yang et al. [40], and the tunnel stability problem in Shiau and Al-Asadi [41–43]. A thorough search of the relevant literature shows that the undrained stability num- bers for unsupported rectangular excavations considering both anisotropy and heteroge- neity have never been reported in the literature. The most recent paper by Yodsomjai et al. [44], which has close similarities to this current study, tackled the undrained stability of unsupported conical slopes in anisotropic clays, which was similarly analyzed using the AUS failure criterion. However, due to its axisymmetric condition, it becomes a 2D plane strain problem rather than 3D. Other than the study undertaken by Ukritchon et al. [22] on the 3D undrained stability of unsupported rectangular excavations in non-homo- geneous clays, which is also similar to the current study but without considering soil ani- sotropy, most of the other studies in the literature that dealt with anisotropy and hetero- geneity in clays were related to other stability problems such as trapdoors [30], pile Buildings 2022, 12, 1425 3 of 21 bearing capacity [32], unlined square tunnels [33], anchors [36], and suction caissons [37]. Therefore, the aim of this paper is to study this underexplored subject on the 3D un- drained stability of unsupported rectangular excavations in clays with linearly increasing anisotropic shear strength. The stability solutions were formulated by a dimensionless stability number that is a function of four dimensionless parameters: the excavated depth ratio, the aspect ratio of the excavated site, the shear strength gradient ratio, and the ani- sotropic strength ratio. The selected failure mechanisms of this problem were examined to demonstrate the effects of all four dimensionless parameters. With the development of accurate design equations, the study would assist practicing engineers in determining the soil stability of unsupported rectangular excavations in clays with anisotropy and heter- ogeneity. 2. Statement of the Problem and Modelling Technique Figure 1 shows the problem of defining a 3D unsupported rectangular excavation. Due to the problem of symmetry, only a quarter of the model domain was used in the analysis. See Figure 1a for the model. The excavation depth is denoted by H, B is the ex- cavation width, and L is the length. (a) Problem geometry (b) linearly increasing anisotropic strength Figure 1. Statement of the problem. The AUS failure criterion with the associated flow rule was used to study the 3D soil stability of the unsupported rectangular excavation. The three anisotropic undrained shear strengths obtained from triaxial compression (suTC), triaxial extension (suTE), and Buildings 2022, 12, 1425 4 of 21 direct simple shear (suDSS) were the required strengths for this failure criteria. According to Krabbenhoft et al. [45], two anisotropic strength ratios can be defined using the three undrained shear strengths: (1) re = suTE/suTC and (2) rs = suDSS/suTC. The relationship between re and rs is the harmonic mean, which can be written as follows: 2r r = (1) 1 + r As shown in Equation (1), the parametric analysis only used one anisotropic strength ratio, which is re. Note that rs is a function of re, and the range of re should be between 0.5 and 1. A change in the re value may vary the AUS failure criterion’s failure surface, as shown in Figure 2 [34,45]. The form of the yield function of the AUS model with the har- monic mean of three undrained shear strengths can be expressed by Equation (2): Fr =− σσ + −12 σ −σ −s =0 (2) ()( ) ue 13 2 3 uTC where 1 ≥ 2 ≥ 3 are the principal stresses (positive in compression), and Fu is the yield function. It should be noted that the AUS failure criterion becomes the Tresca failure criterion when re = 1, meaning the isotropic state, i.e., suTC = suTE = suDSS. Note that, for the AUS failure criterion, three undrained shear strengths were considered to be an empirical correlation of the undrained strength (su) of clay in triaxial compression (TC) for suTC, direct simple shear (DSS) for suDSS, and triaxial extension (TE) for suTE. Figure 2. Generalized Tresca surface used in the Anisotropic Undrained Shear (AUS) failure criterion. The increasing shear strength with depth, i.e., heterogeneous soil behaviors, is an- other important factor when determining soil stability. This variation in shear strength has been considered by many researchers for the problems of the face stability of tunnels [10,31,33, 46], supported excavations [47], piles [48-49], floodwalls [50], and active trapdoors [30, 51]. This study considered three anisotropic undrained shear strengths that linearly increase with depth. Mathematically, they are expressed in Equations (3)–(5). (3) 𝑠 (𝑧) = 𝑠 +𝜌𝑧 (4) 𝑠 (𝑧) = 𝑠 +𝑟 𝜌𝑧 (5) 𝑠 (𝑧) = 𝑠 +𝑟 𝜌𝑧 Buildings 2022, 12, 1425 5 of 21 where suTC0, suTE0, and suDSS0 are the anisotropic undrained shear strengths at the ground level, z is the depth from the ground surface, and  is the linear strength gradient. See Figure 1b for the linear distributions of the three anisotropic undrained shear strengths. Using the dimensional analysis [52], a function combining four dimensionless param- eters that are variables of a stability number function can be expressed by Equation (6). γρ HBH B Nf == (, ,r,m= ) (6) sLB s uTC00 uTC where N is the stability number, B/L is the aspect ratio of the excavated site, H/B is the excavated depth ratio, re is the anisotropic strength ratio, and m is the strength gradient ratio. In the lower bound analysis, a four-node tetrahedron element is used, where six un- known nodal stresses are used for each node of tetrahedral elements. The statically ad- missible stress discontinuities are allowed to produce the continuity of normal and shear stresses along the interfaces of all the elements. The conditions of stress equilibrium, stress boundary condition, and the AUS failure criterion are all constraints in a typical LB anal- ysis, in which the objective function is to maximize the critical unit weight γ that yields an excavation collapse. In the upper bound theorem, a four-node tetrahedron element is also adopted for the upper bound analysis, where each node contains three unknown ve- locities that vary linearly within the tetrahedron element. The kinematically admissible velocity discontinuities are applied at the interfaces of all the elements. The material is set to obey the AUS failure criterion associated flow rule. The formulated objective function is to minimize the critical unit weight γ. The obtained critical γ from both LB and UB analyses were then used to compute the stability number in Equation (6). More details on the LB and UB FELA can be found in [5]. Figure 3 presents a typical 3D FELA mesh used for the analysis. The nodes around the sides of the model are fixed in the normal direction to the planes of the sides. The same boundary condition is applicable to the two symmetrical planes as well. At the bottom domain, the nodes are fixed in all directions. Both the ground surface and the excavation faces are free to move in all directions. The overall domain size is chosen to be sufficiently large such that the stability solutions are not affected by the boundary conditions, i.e., the effects of boundary size on the computed LB solutions are minimized by generating LB meshes with sufficient lateral and lower dimensions that produce a computed plastic yielding zone that does not intersect the boundary planes. Automatic adaptive mesh re- finement is one of the advanced features of the 3D program. This technique is based on Ciria et al. [53], where the numbers of elements in sensitive zones (i.e., with very high plastic shear strains) are increased through successive iterations with adaptive mesh re- finement. The required input for the adaptive scheme is the original and target number of elements, the number of adaptive iterations, and the control variable for error estimation (i.e., shear power in this paper). In this study, 5000 initial elements were employed, which was expanded to 10,000 elements after five iterations. Buildings 2022, 12, 1425 6 of 21 Figure 3. A typical FELA model and potential failure mechanism. Note that, the range of four dimensionless parameters in all studies of the paper are: (1) H/B = 0.5, 1, 2, 3, 4; (2) B/L = 1, 2/3, 1/2, 1/4, 1/8; (3) re = 0.5, 0.6, 0.7, 0.8, 0.9, 1; (4) m = 0, 4, 12, 25, 100. The ranges of H/B and B/L used in this study are based on the previous work by Ukritchon et al. [22]. For the range of re, Krabbenhoft et al. [45] suggested that the value of this parameter should be between 0.5 and 1, which corresponds to the natural ratios of compressive and tensile undrained shear strengths. The range of m or ρB/suTC0 constitutes the combined effect of the excavation size B, the compressive shear strength at the ground surface suTC0, and the linear strength gradient ρ. In practice, suTC0 and ρ depend on the geological nature of the sites where the excavated width B can range from 1 to 20 m in practice. Theoretically, the ρB/suTC0 parameter ranges from 0 (homogeneous case) to a large value (non-homogeneous case). The homogeneous cases correspond to a case with ρ = 0 and/or a very large value of suTC0. For the non-homogeneous cases, they represent the cases with a relatively low suTC0 and/or a relatively large value of ρ. 3. Comparison for Model Validation In the first step of the investigation, the stability numbers, N, determined by the rig- orous FELA solutions, were compared with the published results in Ukritchon et al. [22]. The comparison shown in Figure 4a is for the effect of H/B on the stability number N, as well as its effect on various B/L with isotropic (re = 1) and homogeneous (m = 0) clays. Note that re is the anisotropic strength ratio, and m is the strength gradient ratio. Moreover, note that the present solution is the average (Avg) results calculated from the UB and LB FELA solutions. In general, the stability number increases with the increasing depth ratio H/B. The increase can be either nonlinearly or linearly, depending on the value of B/L. When B/L is smaller (B/L = 1/4, 1/8), fewer 3D constraints are observed, and a linear rela- tionship between N and H/B is presented. Buildings 2022, 12, 1425 7 of 21 (a) m = 0 (b) m = 4 Figure 4. Comparison of stability numbers N (re = 1). Whilst in Figure 4b, the comparison is made for (re = 1 and m = 4). It is interesting to note that, for the large strength gradient ratio such as m ≥ 4, N increases linearly with an increase in H/B for all values of B/L. Overall, the numerical results have shown an excellent agreement between the two solutions. The neglectable numerical differences between the two results can be attributed to the use of the perfectly plastic Tresca failure criterion in Ukritchon et al. [22] as opposed to the AUS failure criterion, with re = 1 used in the present study. To the best knowledge of the authors, there are currently no other values of re to be compared since this is the first work to consider the stability of unsupported rectangular excavations in anisotropic and non-homogeneous soils. 4. Results and Discussions The effects of H/B on the stability number N are presented in Figure 5 for various values of re (the anisotropic strength ratio). Those shown in Figure 5a–f are for B/L = (0.25, 1.0) and m = (0, 12, 100). The numerical results have shown that the stability number N increases linearly with an increase in the excavation depth ratio H/B, except for the case of (B/L = 1.0 and m = 0). See Figure 5b for this special case of a square (B/L = 1.0) excavation in homogeneous (m = 0) clay, where N increases nonlinearly with the increasing H/B. One of the possible reasons could be attributed to the greater corner effects (geometrical Buildings 2022, 12, 1425 8 of 21 arching). Note that the rate of increase in N (i.e., the gradient) increases as the strength gradient ratio m increases. Furthermore, note that a decrease in the anisotropic ratio re results in a decrease in the stability number. The selected failure mechanisms (shear dis- sipation) are presented in Figure 6 for the different values of H/B = (0.5, 1, 2 3, 4). The comparison is based on the case of (re = 0.7, m = 4 and B/L = 1), and the results of the shear dissipation contour plots have shown a toe-failure mode for the shallow cases of H/B = (0.5 and 1). On the other note, for H/B > 1, a face-failure mode is obtained owing to the effect of the strength gradient ratio m. (a) B/L = 0.25 and m = 0 (b) B/L = 1 and m = 0 (c) B/L = 0.25 and m = 12 (d) B/L = 1 and m = 12 (e) B/L = 0.25 and m = 100 (f) B/L = 1 and m = 100 Figure 5. N vs. H/B for the various re (B/L = 0.25, 1.0 and m = 0, 12, 100). Buildings 2022, 12, 1425 9 of 21 (a) H/B = 0.5 (b) H/B = 1 (c) H/B = 2 (d) H/B = 3 (e) H/B = 4 Figure 6. Potential failure mechanisms—effect of H/B (re = 0.7, m = 4, and B/L = 1). Figure 7 shows the effects of B/L (the aspect ratio of the excavated site) on the stability number N for the various values of re (the anisotropic strength ratio). All of the values of m (m = 0, 4, 12, 25, 100) are considered for the chosen depth ratio H/B = 3, and they are presented in Figure 7a–e respectively. The numerical results have shown that N increases nonlinearly with the increasing B/L for all values of re. The gradient of the nonlinear curves becomes smaller as the strength gradient ratio m increases (see Figure 7a–d)—a linear re- lationship is observed for the case with m = 100. It is also noted that the stability number N decreases as the anisotropic strength ratio re decreases (transforming from isotropic to anisotropic soils). The comparison of five failure mechanisms for the various B/L = (1/8, 1/4, 1/2, 2/3, 1) is shown in Figure 8. The chosen plots are for H/B = 1 (re = 0.7, and m = 4). The shear dissipation contour plot of B/L = (1/8, 1/4) has shown a mechanism that resem- bles a 2D plane strain condition (see Figure 8a,b). As the value of B/L increases (so as the sta- bility number N), a stronger system is presented, owing to full 3D corner effects (see Figure 8e Buildings 2022, 12, 1425 10 of 21 for B/L = 1). Interestingly, a two-way failure mechanism is found in Figure 8e for B/L = 1. It should also be noted that the failure patterns are for the toe-failure mode in this shallow case of H/B = 1. (a) H/B = 3 and m = 0 (b) H/B = 3 and m = 4 (c) H/B = 3 and m = 12 (d) H/B = 3 and m = 25 (e) H/B = 3 and m = 100 Figure 7. N vs. B/L for the various re (H/B = 3.0 and m = 0, 4, 12, 25, 100). Buildings 2022, 12, 1425 11 of 21 (a) B/L = 1/8 (b) B/L = 1/4 (c) B/L = 1/2 (d) B/L = 2/3 (e) B/L = 1 Figure 8. Potential failure mechanisms—effect of B/L (H/B = 1, re = 0.7, and m = 4). Figure 9 shows the relationship between the stability number N and the strength gra- dient ratio m for the various values of re (the anisotropic strength ratio). The presentations are for B/L = (1/8, 1) and H/B = (0.5, 1.0, 4.0). In general, an increase in m results in an increase in N. A linear relationship between N and m is observed in all investigated cases. Same as the previous discussions, the smaller the re, the smaller the stability number N. The chosen case for the failure mechanism comparison is presented in Figure 10 for (re = 0.7, H/B = 1, B/L = 1) with different values of m = (0, 4, 12, 25, 100). It should be noted that the size of the failure zone decreases as m increases. As a result, the failure mechanism changes from a toe-failure mode to a face-failure mode when m is larger than 4. (a) H/B = 0.5 and B/L = 1/8 (b) H/B = 0.5 and B/L = 1 Buildings 2022, 12, 1425 12 of 21 (c) H/B = 1 and B/L = 1/8 (d) H/B = 1 and B/L = 1 (e) H/B = 4 and B/L = 1/8 (f) H/B = 4 and B/L = 1 Figure 9. N vs. m for the various re (H/B = 0.5, 1.0, 4.0 and B/L = 1/8, 1). (a) m = 0 (b) m = 4 (c) m = 12 (d) m = 25 (e) m = 100 Figure 10. Potential failure mechanisms—effect of m (re = 0.7, H/B = 1, and B/L = 1). Buildings 2022, 12, 1425 13 of 21 Figure 11 shows the relationships between the stability number N and the anisotropic strength ratio re for various values of m = (0, 4, 12, 25, 100). The plots are for the selected ratios of H/B = (0.5, 4) and B/L = (1/8, 1/2, 1). The numerical results have shown that the larger the m, the greater the stability number N. Overall, the stability number N varies linearly with the increase in the anisotropic ratio re. The rate of increase (gradient of the line) in N is dependent on the value of m. The larger the m, the greater the gradient of the line. Figure 12 shows a comparison of failure mechanisms among the various anisotropic ratios, re = (0.5–1). The comparison is for the excavation problem of (m = 4, H/B = 1, B/L = 1). The results have shown that the failure patterns are all in a toe-failure mode, and the variation of anisotropic ratio re does not seem to affect the failure size of the problem. The same conclusion can be drawn from Figure 13, where an additional study of m = 100 is presented. Indeed, as discussed previously, the face-failure mode is always the one ob- served for the large strength gradient ratio such as m = 100. It should be noted that all of the numerical results of this paper study are summarized in Tables 1–3. (a) B/L = 1/8 and H/B = 0.5 (b) B/L = 1/8 and H/B = 4 (c) B/L = 1/2 and H/B = 0.5 (d) B/L = 1/2 and H/B = 4 (e) B/L = 1 and H/B = 0.5 (f) B/L = 1 and H/B = 4 Figure 11. N vs. re for the various m (H/B = 0.5, 4.0 and B/L = 1/8, 1/2, 1). Buildings 2022, 12, 1425 14 of 21 (a) re = 0.5 (b) re = 0.6 (c) re = 0.7 (d) re = 0.8 (e) re = 0.9 (f) re = 1.0 Figure 12. Potential failure mechanisms—effect of re (m = 4, H/B = 1, and B/L = 1). (a) re = 0.5 (b) re = 0.6 (c) re = 0.7 (d) re = 0.8 (e) re = 0.9 (f) re = 1.0 Figure 13. Potential failure mechanisms—effect of re (m = 100, H/B = 1, and B/L = 1). Buildings 2022, 12, 1425 15 of 21 Table 1. Stability numbers, N (re = 1.0 and 0.9). re = 1 re = 0.9 m B/L B/L H/B H/B 1 2/3 1/2 1/4 1/8 1 2/3 1/2 1/4 1/8 0.5 4.559 4.372 4.234 3.959 3.860 0.5 4.299 4.128 4.027 3.764 3.460 1 5.291 4.953 4.677 4.153 3.955 1 4.958 4.661 4.420 3.931 3.759 0 2 6.420 5.969 5.553 4.637 4.125 2 5.968 5.560 5.201 4.384 3.917 3 7.170 6.707 6.279 5.135 4.362 3 6.657 6.243 5.849 4.833 4.137 4 7.770 7.280 6.862 5.606 4.632 4 7.216 6.774 6.388 5.262 4.378 0.5 9.259 9.008 8.758 8.362 8.255 0.5 8.749 8.495 8.288 7.870 7.898 1 15.758 14.648 13.891 12.832 12.437 1 14.778 13.806 13.189 12.151 11.775 4 2 30.562 28.105 26.095 22.663 21.168 2 28.283 26.251 24.639 21.455 19.996 3 45.966 42.212 39.161 33.462 30.330 3 42.365 39.288 36.776 31.691 28.730 4 61.106 56.160 52.214 44.704 39.942 4 56.528 52.402 48.986 42.238 37.914 0.5 18.213 17.662 17.175 16.735 16.561 0.5 17.156 16.730 16.444 15.785 15.516 1 34.964 32.841 31.468 29.458 28.713 1 32.745 31.043 29.716 27.916 27.309 12 2 69.931 65.641 62.571 57.045 53.981 2 65.470 61.913 59.186 53.975 51.219 3 104.996 98.597 93.842 85.662 80.372 3 98.151 92.897 88.782 81.116 76.146 4 140.088 131.336 125.132 114.290 107.172 4 130.918 124.024 118.326 108.284 101.830 0.5 32.254 31.504 30.613 30.072 29.207 0.5 30.536 29.713 29.120 28.351 28.198 1 64.231 61.253 59.346 56.412 55.183 1 60.475 57.966 56.147 53.315 51.845 25 2 128.460 122.685 118.597 111.345 107.223 2 120.981 115.931 112.269 104.690 101.445 3 192.627 183.920 178.077 167.298 160.784 3 181.659 174.002 168.429 158.561 152.492 4 256.720 245.260 237.102 223.242 214.300 4 241.932 231.924 224.580 211.538 203.318 0.5 111.392 109.320 107.744 106.143 105.742 0.5 105.461 103.419 101.948 100.684 100.869 1 223.302 218.275 215.000 209.382 206.320 1 211.114 206.404 203.672 198.231 196.114 100 2 446.620 436.122 429.972 415.484 411.352 2 423.437 413.457 407.257 396.090 389.648 3 671.124 654.846 645.300 627.836 617.555 3 632.489 620.067 611.184 594.366 585.180 4 894.322 873.216 859.562 837.478 823.428 4 845.674 827.874 813.880 793.196 780.282 Table 2. Stability numbers, N (re = 0.8 and 0.7). re = 0.8 re = 0.7 m B/L B/L H/B H/B 1 2/3 1/2 1/4 1/8 1 2/3 1/2 1/4 1/8 0.5 3.998 3.880 3.807 3.531 3.255 0.5 3.696 3.571 3.503 3.273 3.043 1 4.594 4.327 4.129 3.697 3.525 1 4.206 3.983 3.807 3.415 3.257 0 2 5.502 5.126 4.825 4.103 3.668 2 5.006 4.694 4.418 3.790 3.398 3 6.120 5.741 5.400 4.518 3.885 3 5.576 5.232 4.935 4.155 3.590 4 6.608 7.654 5.868 4.896 4.100 4 6.010 5.658 5.350 4.494 3.788 0.5 8.192 8.020 7.753 7.399 7.252 0.5 7.574 7.352 7.212 6.878 6.741 1 13.733 12.929 12.311 11.406 11.087 1 12.572 11.952 11.463 10.533 10.203 4 2 26.032 24.326 22.942 20.133 18.797 2 23.642 22.283 21.126 18.623 17.413 3 38.835 36.234 34.224 29.712 26.985 3 35.222 33.110 31.422 27.500 25.028 4 51.836 48.330 45.558 39.574 35.560 4 46.896 44.168 41.810 36.482 32.908 0.5 16.080 15.921 15.174 14.900 14.827 0.5 14.856 14.759 14.108 13.673 13.539 1 30.498 29.064 27.867 26.213 25.641 1 28.015 26.842 25.829 24.231 23.713 12 2 60.747 57.799 55.401 50.671 47.946 2 55.688 53.260 51.218 46.939 44.337 3 91.143 86.838 83.121 76.007 71.582 3 83.531 79.896 76.892 70.436 66.102 4 121.484 115.712 110.738 101.452 95.462 4 111.290 106.608 102.410 93.984 88.160 Buildings 2022, 12, 1425 16 of 21 0.5 28.631 27.955 27.377 26.578 25.636 0.5 26.442 25.683 25.257 24.657 24.445 1 56.445 54.317 52.686 50.016 49.108 1 51.882 50.107 48.741 46.161 45.508 25 2 112.831 108.652 105.268 99.169 95.067 2 103.890 100.505 97.455 91.487 87.891 3 169.331 162.911 157.857 148.704 142.529 3 155.756 150.548 146.213 137.873 132.387 4 225.288 231.924 210.434 198.498 190.700 4 207.658 200.882 194.796 183.884 176.652 0.5 98.804 97.060 95.754 94.274 92.989 0.5 91.379 89.479 88.734 87.369 82.820 1 197.710 193.955 190.910 183.274 183.608 1 182.879 179.463 176.887 171.749 169.835 100 2 395.618 387.909 382.134 372.104 364.822 2 365.556 359.051 353.148 340.475 337.916 3 594.897 581.319 573.506 558.114 548.117 3 549.108 538.401 530.817 517.062 502.274 4 794.012 775.738 763.674 744.168 731.800 4 733.122 719.498 706.896 689.530 677.690 Table 3. Stability numbers, N (re = 0.6 and 0.5). re = 0.6 re = 0.5 m B/L B/L H/B H/B 1 2/3 1/2 1/4 1/8 1 2/3 1/2 1/4 1/8 0.5 3.317 3.256 3.181 2.955 2.771 0.5 2.944 2.882 2.820 2.628 2.319 1 3.774 3.586 3.450 3.117 2.969 1 3.314 3.155 3.029 2.758 2.631 0 2 4.479 4.207 3.977 3.444 3.097 2 3.901 3.674 3.495 3.050 2.743 3 4.979 4.679 4.421 3.753 3.254 3 4.328 4.085 3.857 3.309 2.898 4 5.366 5.048 4.784 4.054 3.440 4 4.666 4.390 4.188 3.564 3.052 0.5 6.823 6.675 6.550 6.294 6.027 0.5 6.038 5.960 5.753 5.559 5.491 1 11.358 10.816 10.409 9.624 9.369 1 9.967 9.521 9.226 8.561 8.277 4 2 21.138 19.958 19.048 16.942 15.852 2 18.563 17.571 16.751 15.037 14.072 3 31.406 29.540 28.229 24.951 22.737 3 27.335 25.931 24.728 22.184 20.210 4 41.728 39.450 37.670 33.366 29.942 4 36.506 34.454 32.832 29.342 26.568 0.5 13.475 13.208 12.811 12.563 11.985 0.5 11.915 11.702 11.428 11.133 10.896 1 25.262 24.302 23.432 22.134 21.623 1 22.330 21.468 20.794 19.667 19.228 12 2 50.092 48.210 46.566 42.644 40.585 2 44.187 42.254 41.138 37.943 35.950 3 75.329 72.156 69.764 64.122 60.272 3 66.371 63.263 61.695 56.939 53.634 4 100.316 96.282 92.980 85.524 80.114 4 88.410 84.236 82.204 75.902 71.372 0.5 24.129 23.467 23.007 22.585 22.364 0.5 21.192 20.744 20.493 19.967 19.757 1 47.000 45.646 44.432 42.214 41.395 1 41.437 40.252 39.182 37.430 36.636 25 2 93.925 91.067 88.624 83.521 80.038 2 82.907 80.487 78.697 74.134 70.683 3 141.065 136.667 132.840 125.496 120.269 3 124.328 120.735 118.287 111.372 107.109 4 188.000 182.288 177.256 167.254 160.868 4 165.788 160.782 157.210 148.552 141.700 0.5 83.089 81.711 79.615 79.360 75.411 0.5 73.433 72.459 71.454 70.765 69.833 1 165.317 163.306 160.071 156.043 154.564 1 147.068 144.681 142.492 139.251 137.323 100 2 332.289 326.784 322.154 313.455 305.386 2 294.210 289.922 285.932 278.151 272.439 3 500.166 489.830 483.281 470.489 462.878 3 440.799 433.592 428.594 416.861 408.983 4 664.460 653.052 644.744 627.270 617.136 4 587.558 579.212 571.564 556.762 545.928 5. Design Equations A mathematical equation is developed and presented in this section by using a trial- and-error method of curve fitting. Nonlinear regression with multiple variables to the Avg bound solutions is employed to develop design equations for estimating the stability fac- tor of unsupported rectangular excavations in clays with anisotropy and heterogeneity, as shown in Equation (7). Buildings 2022, 12, 1425 17 of 21    ρρ BH B H B H ρB H      N=+aa + a +a + a −a + a +a (7)    12   348 56  7 sB L B s B s B       uTC0 uTC0  uTC0  where a1 to a8 are constant coefficients. To determine the optimal value of the constant coefficients (a1–a8), the nonlinear least square regression [54] is utilized. The sum of the squares of the deviation in N between the computed Avg solutions shown in Tables 1–3 and the approximate solutions from Equation (7) is then minimized to obtain the optimal values of constant coefficients. Note that, to achieve high accuracy, Equation (7) is a “step-wised” equation devel- oped for the different values of re. Using the complete data in Tables 1–3, the optimal val- ues of the coefficients a1 to a8 for the different values of re are computed and presented in Table 4. On the other hand, the comparisons of N between the computed Avg bound so- lutions and the approximate solutions from Equation (6) are shown in Figure 14a–f, respec- tively, for different values of re = (0.5 to 1.0). It is pleasing to see the highly accurate solutions of the equation development—the coefficient of determination (R ) = 99.99%. Table 4. Constant coefficients for the proposed design equation. Constant Coef- re ficients 0.5 0.6 0.7 0.8 0.9 1 a1 3.16697 3.3617 3.73044 4.1027 3.81764 4.28435 a2 1.35068 1.52276 1.66728 1.80244 1.92334 2.02793 a3 −0.69739 −0.55146 −0.64589 −1.47779 −0.09287 −0.55003 a4 0.46763 0.6014 0.72694 0.99956 1.01361 1.18849 a5 1.96752 2.24958 2.55588 3.08414 3.30362 3.39274 a6 0.92331 0.99065 1.14742 1.57205 1.5951 1.54619 a7 −0.07001 −0.07579 −0.081989 −0.12147 −0.13622 −0.13032 a8 −0.05667 −0.05068 −0.06182 −0.10530 −0.10229 −0.09884 R 99.99% 99.99% 99.99% 99.99% 99.99% 99.99% r = 0.6 r = 0.5 e 800 800 600 600 400 400 2 2 200 200 R = 99.99% R = 99.99% 0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 N, Proposed design equation N, Proposed design equation (a) re = 0.5 (b) re = 0.6 N, FELA solutions N, FELA solutions Buildings 2022, 12, 1425 18 of 21 1000 1000 r = 0.7 r = 0.8 800 800 600 600 400 400 200 2 200 2 R = 99.99% R = 99.99% 0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 N, Proposed design equation N, Proposed design equation (c) re = 0.7 (d) re = 0.8 1000 1000 r = 1.0 r = 0.9 800 800 600 600 400 400 2 200 2 R = 99.99% R = 99.99% 0 200 400 600 800 1000 0 200 400 600 800 1000 N, Proposed design equation N, Proposed design equation (e) re = 0.9 (f) re = 1.0 Figure 14. Predicted N vs. FELA N. 6. Conclusions Rigorous stability solutions of the unsupported rectangular excavation in anisotropic and heterogeneous clays have been successfully studied in the paper using 3D LB and UB FELA. The stability number (N) that is a function of the excavation aspect ratio, B/L, the excavated depth ratio, H/B, the strength gradient ratio, m = ρB/suTC0, and the anisotropic strength ratio, re, was presented throughout the paper. The following conclusions are drawn based on the study. 1. The stability number, N, increases with an increase in all of the investigated parame- ters of B/L, H/B, m, and re. The increases can be either in a linear or nonlinear relation- ship. The linear relationship was obtained for all investigated cases except for cases with smaller values of m, where a nonlinear relationship exists between N and B/L. 2. The failure patterns of unsupported rectangular excavation in anisotropic and heter- ogeneous clays are either in a toe-failure mode (for small values of H/B, i.e., H/B = 0.5, 1) or a face-failure mode (for large values of H/B > 1) due to the effect of the strength gradient ratio m. For large values of m > 4, the failure modes are predominately the face-failure mode. The variation in the anisotropic ratio, re, does not seem to affect the failure size of the unsupported rectangular excavation problem. 3. A new equation for predicting the stability number, N, of the unsupported rectangu- lar excavation in anisotropic and heterogeneous clays is proposed. With the coeffi- cient of determination (R ) being 99.99%, the proposed equation is highly accurate and useful for practical uses. The proposed study provides deeper contextualized insights into the understanding of 3D unsupported excavations in undrained clay under the influence of soil anisotropy N, FELA solutions N, FELA solutions N, FELA solutions N, FELA solutions Buildings 2022, 12, 1425 19 of 21 and heterogeneity. Future work directions may include the seismic stability performance as well as the soil random field probabilistic analysis. Author Contributions: V.Q.L.: Data curation, Software, Investigation, Methodology, Writin—orig- inal draft; S.K.: Methodology, Validation, Writing –original draft; S.S.: Formal analysis, Software, Methodology; J.S.: Methodology, Writing—review & editing, Project administration, Supervision, Funding acquisition; L.T.C.: Writing—review, revising & editing. All authors have read and agreed to the published version of the manuscript. Funding: This study was supported by Thammasat Postdoctoral Fellowship, Thammasat Univer- sity Research Division, Thammasat University. 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Journal

Buildings – Multidisciplinary Digital Publishing Institute

Published: Sep 10, 2022

Keywords: stability; excavations; anisotropy; heterogeneity; finite element limit analysis

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