Microsoft Word - numero_65_art_09_4315.docx A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 112 Crack simulation for the cover of the landfill – A seismic design Abdoullah Namdar, Mehran Karimpour-Fard* School of Civil Engineering, Iran University of Science and Technology (IUST), Narmak, Tehran, Iran ab_namdar@mail.iust.ac.ir, Karimpour_mehran@iust.ac.ir Omer Mughieda Department of Civil Engineering Abu Dhabi University, Abu Dhabi, UAE omer.mughieda@adu.ac.ae Filippo Berto Department of Chemical Engineering Materials Environment, Sapienza - Università di Roma, Via Eudossiana, 18 00184 Roma, Italy filippo.berto@uniroma1.it Nurmunira Muhammad Faculty of Civil Engineering Technology, Universiti Malaysia Pahang, Malaysia aira.munira@gmail.com ABSTRACT. The stability of the landfill is an environmental issue. The collapse of the landfill causes environmental pollution and influences human life. In the present study, the crack on the cover of the landfill was simulated. Rankine’s theory and the Phantom Node Method were used for the simulation length of the crack and the mechanism of the crack propagation in the nonlinear extended finite element method (NXFEM). Artificial Neural Networks (ANNs) based on Levenberg-Marquardt Algorithm and Abalone Rings Data Set mode were used to predict displacement in critical points of the model. The vibration mechanism of the landfill was changed in each model. During applying seismic load on the model, the optimized thickness of the clay cover on the landfill was discussed. The thickness of the landfill cover controls the seismic response of the landfill. The numerical simulation shows differential displacement of the landfill impacts on the crack propagation and the need for the appropriate design of the cover thickness of the landfill. KEYWORDS. Landfill; Crack, Rankine’s theory, Phantom Node Method Displacement, Cover Thickness. Citation: Namdar, A., Krimpour-Fard, M., Mughieda, O., Berto, F., Muhammad, N., Crack simulation for the cover of the landfill – A seismic design, Frattura ed Integrità Strutturale, 65 (2023) 112-134. Received: 22.04.2023 Accepted: 23.05.2023 Online first: 26.05.2023 Published: 01.07.2023 Copyright: © 2023 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. https://youtu.be/By6P42sgWNs A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 113 INTRODUCTION growing amount of municipal solid waste (MSW) is being produced due to the industrialization of many urban areas [1], resulting in a massive accumulation of MSW and new landfill construction. Due to changes in the local climate, MSW's physical, chemical, and biological processes can also be affected. As a result of MSW accumulation, high temperatures are generated, which are associated with climate change [2]. During the cover process for landfills with clay, topsoil tends to be more influenced by temperature, and this happens more often in tropical regions, where the soil is exposed to higher temperatures [3]. As temperatures increase in the landfill, cracks form in the clayey cover [2–5], which extend to the body of the landfill [4-5], causing a change in the moisture content of the landfill. Also, during the rainy season, the amount of leachate increases. In the event of seismic loading, landfill cover cracks extend more rapidly, resulting in a reduction in landfill seismic stability. The landfill's seismic stability is related to local site conditions [6]. Due to the seismic load on the landfill, the damage is divided into major, significant, moderate, minor, slight, or no damage. The crack initiates and occurs in the landfill cover if the damage is moderate. Once the type of damage is recognized, geosynthetics can be utilized to improve landfill seismic stability [7]. Seismic loading causes landfill displacement, and it has been suggested that this phenomenon be quantified through Newmark's one-dimensional analytical sliding-block method [8]. Many researchers have applied the nonlinear finite element method to calculate the nonlinear displacement of the embankment model [9]. Using the appropriate method to simulate a landfill's seismic response accurately is essential. Further, it must be noted that the material's mechanical properties and the model's geometry properties have an impact on the displacement mechanism, and these properties need to be understood correctly by evaluating the input data accurately [10–12]. It is well known that the calculation of seismic deformation caused by an earthquake is prone to some error levels [13–15]. The impact of seismic loading on the displacement of the landfill has been systematically investigated to identify the seismic stability of the landfill [16–18]. Landfill displacement could be minimized using geomembrane [19–20], lattice drainage geocomposite, and flexible polyester [20]. Aside from these methods, clay soil is also used to cover landfills [21, 22]. The compacted soil liner (CSL) is required to control a hydraulic conductivity of (≤ 1 × 10−7 cm/s); for this purpose, clay with a thickness in the range of 0.6 to 1.5 m is required [24, 25]. Displacement prediction is needed before applying the seismic mitigation method to improve landfill seismic stability. Nonlinear numerical simulations can identify displacement at any point in the modeled landfill. Due to the nonlinearity of the seismic load, numerical simulation results need to be validated using statistical analysis. The current study used artificial neural networks to predict embankment displacement [11–12, 26]. The prediction is made from the validation and optimization results of the numerical simulation [27–29]. The main objective of the present study is to consider the impact of cracked landfill covers with different thicknesses on the displacement of two critical points in the model. In order to predict displacement at selected points in the model, the Levenberg-Marquardt algorithm was used, which considers Rankine's theory and the phantom node method in crack simulation and propagation. Figure 1: The entire numerical simulation steps. A A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 114 MODELING oil-like materials (SLM) such as MSW are highly deformable under seismic loading. The crack initiation on the landfill cover was simulated and shown in Fig. 1, which illustrates the entire numerical simulation procedure. Numerical modeling was conducted to design the clay landfill cover under seismic loading. Fig. 2 shows the model components, MSW and clay. 170 (m) 25 ( m ) 15 ( m ) 25 (m) 25 (m)45 (m) 45 (m)30 (m) Clayey subsoil Landfill Clayey landfill cover X = U1 = UR2 = UR3 = 0 Z Y Z Y X Y Z = U3 = UR1 = UR2 = 0 Y = U 2 = U R 1 = U R 3 = 0 Y = U 2 = U R 1 = U R 3 = 0 Subsoil cross section Initial crack path Figure 2: The landfill model with a clay cover, clay subsoil, and boundary conditions used in the numerical simulation. CRACK SIMULATION he initial crack in the landfill clay cover is designed based on Rankine’s theory (1857) [30]. The crack and solid zones are two significant parts of the model. The ratio of the crack height to the total thickness of the landfill cover for each part changes with respect to the model’s mechanical properties. In fact, when the two models have the same geometry but are constructed from two different MSW materials, the crack, and solid zones are changed according to the types of materials used. It is assumed that landfill leachate is confined under undrained conditions. Figure 3: Theoretical crack initiation for the clayey cover of the landfill. S T A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 115 According to Rankine’s theory, kp and ka are the passive and active earth pressure coefficients and are presented in Eqns. 1 and 2. The pressure and force in the active and passive states have been explained in Eqns. 1-12 [31].       1  sin 1  sinpk (1)       1  sin 1  sinak (2) From Eqns. 1 and 2, Eqn. 3 can be written.  1 p a k k (3) where the Rankine active state in the slip plane is,   θ 45   2a (4) where the Rankine passive state in the slip plane is,    θ 45    2p (5) The lateral earth pressure for the Rankine active state is,      ' '    x a z aa K K z (6) The lateral earth pressure for Rankine passive state is,      ' '    x p z pp K K z (7)       sat w (8) The lateral earth force in Rankine’s active state is,     0 2 00 1   2 H a a aP K z K H (9) The lateral earth force in Rankine’s passive state is,     0 2 00 1   2 H p p pP K z K H (10) For the undrained condition in a fully saturated clay, the active and passive pressures are calculated using the parameter cu, (ϕu is equal to zero) and the total unit weight γsat. When enough deformation occurs the state of plastic equilibrium occurs. Eqns. 11 and 12 have been used to identify the length of the crack, the solid zone, and the crack initiation zone [32]. A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 116         2  /   uZ Cracked zone c (11)   0            zS Solid zone H Z (12) Based on the theoretical concepts, the crack initiates in association with the mechanical properties of the clay. The crack propagation caused by applying an external load on the model was simulated. Fig. 3 illustrates the crack and solid zone in the landfill cover. SEISMIC DATA ccording to the literature report [33], to achieve acceptable results multidirectional seismic acceleration needs to be applied to the model in the numerical simulation. By applying one direction of seismic acceleration to the model, the underestimated strain, stress, and displacement would be obtained in the numerical simulation. Fig. 4 illustrates the multidirectional acceleration history (m/s2) used in NFEM. The seismic acceleration at 0°, 90°, and 360° of the Abra Pampa Earthquake, which occurred on 22 Feb 2022, with 6.0 MWW, in coordinates -22.6625 and -66.2673 in a depth of 242.3 km [34], has been used as the input data in the nonlinear numerical simulation. The seismic accelerations applied to models 1 and 2 are equal. The seismic acceleration in each direction has minimum and maximum domains. The combination of these three accelerations needs to be applied to the model. The critical duration of seismic acceleration at 0°, 90°, and 360° is shown in Fig. 4. The shaking over 0.05 (g) in positive and negative directions in all directions is the most unpleasant and needs to be considered in the nonlinear numerical simulation. Tab. 1 shows peak ground acceleration, displacement, and velocity at 0°, 90°, and 360°. Figure 4: The multidirectional acceleration history (m/s2) used in NFEM [34]. Direction Peak ground acceleration [g] Peak ground velocity [cm/sec] Peak ground displacement [cm] 0° - 0.0222 0.7732 0.0334 90° -0.0203 - 0.8904 0.0419 360° 0.0165 0.492 0.0257 Table 1: Earthquake data [34]. A A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 117 MECHANICAL PROPERTIES SW mechanical properties are reported in the literature. Tab. 2 illustrates the mechanical properties of MSW used to simulate landfills [35–37]. MSW is a highly deformable material with a high compressibility level. The MSW elasticity modulus has been calculated using Eqn. 13.   2   1E G v (13) Material Friction angle, ϕ (deg) Cohesion, C (kPa) Poisson’s ratio, ν Unit weight, γ (kN/m3) Modulus elasticity, E (MPa) Shear modulus, G (MPa) MSW 34 22 0.33 14.93 5.61 2.11 Table 2: Mechanical properties of the MSW [35-37]. The angle of internal friction for clay under undrained conditions is assumed to be equal to zero (ϕu = 0) [38]. The unit weight of undrained clay (γc) is 18 (kN/m3). The undrained modulus of elasticity (Eu) is 12.5 MPa. Poisson’s ratio νc of 0.3 and undrained shear strength equal to 25 kPa have been reported in the literature [39]. The model has three parts, the subsoil, the landfill, and the cover of the landfill. The landfill subsoil and cover were designed using clay. The Landfill has been simulated using MSW. NONLINEAR EXTENDED FINITE ELEMENT METHOD AND CRACK PROPAGATION he XFEM provides a situation in that discontinuities act independently, and different types of discontinuities are generating [40]. The crack simulation using XFEM was reported in the literature [41-42]. Eqn. 14 presents the standard form of XFEM [43-44].               * *       .  . h i i i i i I i I u x N x u N x N x a (14) The phantom node method for modeling fracture is introduced. This technique involves intersecting elements by a crack, erasing the crack, and replacing it with two new elements without modifying the original location of the element. This process does not alter the physical characteristics of the material [45], and later it was upgraded by using a more advanced technique [46]. In the present work, using ABAQUS, the phantom node method has been adopted to simulate crack propagation in the landfill cover. Figure 5: The phantom node method in mesh generation [47]. Fig. 5 explains nodes, elements, and meshes generation by the phantom node method [47], in this process the elements have independent displacement without sharing nodes [48]. Considering Fig. 5, according to the phantom node method, to explain the discontinuous displacement for overlapping elements can be written as [47], M T A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 118           ,   Ω .    Ω A A B B N X U x U X N X U x (15)         ,    Γ  A B sU X N X U U X (16) where Γs presents the crack path, ,ΩA and ΩB refer to the shaded area the portion of a new element A and B respectively. The N is the standard for finite element method shape functions and UA and UB are the displacements in nodes A and B. Fig. 2 illustrates the boundary condition adopted and applied in the numerical simulation. Two models of the landfill with the cracked cover are simulated. The smallest mesh size is selected for landfill cover. The mesh sizes of 2000 (mm), 4000 (mm), and 6000 (mm) have been selected for landfill cover, landfill, and subsoil of the model respectively. The geometry of the mesh is shown in Fig. 6. The MSW has been covered with clay. In model 1 the cover of the landfill is 3 (m), and in the second model, the cover of the landfill is increased to 6 (m). The selected nodes in numerical simulation for models 1 and 2 are shown in Fig. 6. In these two critical points the displacement, stress, and strain will be analyzed in the next part of the present study. Figure 6: Mesh design in the numerical simulation. ARTIFICIAL NEURAL NETWORKS rtificial neural networks (ANNs) are frequently employed for geotechnical engineering problem prediction, assessment, and solution [12, 26]. ANNs are employed in data prediction, categorization, association, and filtering [49]. The Levenberg-Marquardt algorithm was used in this study to perform ANNs for classification and prediction. The four key criteria for predicting displacement are seismic acceleration, stress, strain, and fracture length. The thickness of the landfill cover was varied in the numerical simulation, but the seismic acceleration and model geometries were fixed for models 1 and 2. In MATLAB, Artificial Neural Networks (ANNs) were utilized to forecast displacement in the Y direction of the chosen object. The ANNs in different layers were done, and the numerical simulation results were checked, validated, and predicted based on the training to reach the best outcomes for displacement occurrence. Eqns. 17-18 introduce the basic concept of the Levenberg-Marquardt Algorithm [50]. If   ,  , are unknown parameters at points  ,   ,   , i i ix y z                ,  , ,  ,   ,   ,  ;  ,  , ,    ,   ,   , i i i i i i if H x y z h x y z (17) A A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 119 Eqn. 17, can be minimized to       2 1   ,  , ,    n iS f (18) The Hessian matrix is presented in Eqns. 19-20. If f: Rn →R presents a function for input X  Rn and output f (X)  R. If partial derivation f is available, subsequently the Hessian matrix H is a square matrix [51].                                         2 2 2 2 1  2 1 1 2 2 2 2 2 1 22 22 2 2 1 2                        n f n n n n f f f x x x xx f f f H x x x xx ff f x x x x x (19)       2     f ij i j f H x x (20) where J presents the Jacobian matrix the Hessian matrix H is   TH J J (21) If “e” is selected as a vector of network errors, the gradient will be calculated from Eqn. 22,  T kg J e (22) The Updated weights in the Levenberg-Marquardt algorithm can be as         1 1    T T k kW W J J I J e (23) To approach second-order training without employing the Hessian matrix, the Levenberg-Marquardt technique was presented [52]. In light of the statistical idea offered in the literature [53], Eqns. 24-25 are proposed. d is the acquired nonlinear displacement in the numerical simulation, dp is the projected displacement using ANNs, and oD is the mean value of obtained nonlinear displacement in these two equations. The accuracy of displacement prediction was assessed using Eqns. 24- 25.      2 1 1         n p i MSE d d n (24)               2 12 2 1         1         n pi n oi d d R d D (25) A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 120 RESULTS AND DISCUSSIONS nonlinear numerical simulation was performed to predict the nonlinear displacement at two critical points in the landfill. The results of the nonlinear numerical simulation showed that the speed and shape of the deformation in each model have distinct behaviors. MSW is a highly deformable material. Because of this characteristic, when MSW is subjected to seismic acceleration, it has a different seismic response compared to clay. Fig. 7 illustrates the nonlinear shear stress-strain for a particular node. These points have been selected for monitoring displacement in the subsoil, the clayey cover of the landfill, and the main body of the landfill. Fig. 8 illustrates the multidirectional nonlinear acceleration- displacement in the Y direction for a selected node in models 1 and 2. Figs. 7 and 8 reveal the MSW-clay interface's impact on the model's seismic response. In addition, landfill covers of different thicknesses exhibit higher seismic resistance. In addition, models 1 and 2 have different deformability. The unique deformation pattern for each model was observed. According to the nonlinear shear stress-strain curve, model 2 collapses with lower deformation. Increasing the clay landfill cover by more than 3 meters reduces seismic stability. The displacement at the crest and toe of the landfill is dissimilar. The clay landfill cover acts as a surcharge on the landfill and increases its displacement. The seismic frequency causes fill excitation to impact landfill stability considerably [16]. The Abra-Pampa earthquake was simulated and applied to the landfill in the present work. This earthquake has a high frequency. The high frequency of the earthquake accelerates the model collapse before a high displacement level occurs. Due to the seismic load applied to the landfill, a crack develops on the landfill cover during the moderate damage stage [7]. Figs. 7 and 8 show that the landfill is subject to significant damage within a short period of time. In this mode of landfill damage, crack propagation does not occur in the landfill. The numerical simulations presented in Figs. 7–9 agree with the literature report. The landfill damage occurred based on the model's internal load interaction, the MSW's texture, the surcharge's magnitude, and the landfill design. In addition, the materials' surface interaction controls the seismic load transfer and dissipation patterns. Figure 7: The nonlinear shear stress (MPa) -strain for the selected nodes in models 1 and 2. A A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 121 By comparing the stress-strain curve, it was possible to determine the reduction of stiffness in model 2. The shear and compressive strain in model 1 at node 4 are larger than in model 2 at the same point. In the loading and reloading stages of models 1 and 2, at embankment and subsoil, stress-strain characteristics follow two different patterns. In both models, the stress-strain rate is higher in the subsoil than in the embankment. The nature of MSW-clay landfill cover interaction causes the development of distinct seismic responses and seismic differential displacement for each model. In addition, the numerical simulation results revealed that the model's vibration mechanism is associated with the mechanical properties and material design of landfill construction. Due to the characteristics of MSW, two surfaces of cracks interact with each other, which affects displacement and deformation in the landfill. Figure 8: The multidirectional nonlinear acceleration (g) – displacement (mm) in Y direction for the selected node. Figure 9: The displacement (mm) after 1 second of applied acceleration (g). A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 122 Fig. 9 illustrates the displacement in the whole model after 1 second of applying seismic acceleration (g) to the model. The crack shows different morphology at each stage of the numerical simulation. The nature of the seismic acceleration (g) determines the failure mechanism in both models. In addition, the landfill clay cover thickness controls the crack propagation mechanism. With attention to the difference between lower and higher displacement levels in models 1 and 2, it seems that model 1 has a higher collapsing speed. Fig. 9 shows that the model's failure pattern occurs based on the model's lateral sliding. These similarities in the type of failure in models 1 and 2 occur due to the nature of the seismic acceleration. The compacted soil liner (CSL) needs to maintain a hydraulic conductivity of (≤ 1 × 10−7 cm/s). For this purpose, clay with a thickness in the range of 0.6–1.5 m is required [24–25], but from the seismic design perspective, the appropriate clay thickness needs to be recommended. Based on this analysis, it can be concluded that the landfill cover thickness needs to be designed to restrict landfill collapse and control leachate as well. No Important Vertical Peak Acceleration (g) Vertical Peak Stress (MPa) Vertical Peak Strain Initial crack length (mm) Vertical Displacement (mm) 1 -0.01317 0.0013 0 2778 0.26664 2 0.01669 -0.0017 -6.57439E-6 2778 0.56451 3 0.01484 -0.00195 -3.985E-6 2778 -0.33746 4 -0.01374 -0.00157 1.58766E-6 2778 -0.32985 5 -0.02729 0.00205 -4.85409E-6 2778 0.30903 6 0.02255 4.3911E-4 -1.27384E-6 2778 -0.35656 7 0.01522 -0.00239 -8.50539E-6 2778 -0.40129 8 -0.01426 -0.00158 3.63012E-6 2778 0.26985 9 0.01475 -0.00168 -4.72572E-6 2778 -0.34188 10 0.016 0.00166 -3.00413E-6 2778 -0.40806 11 -0.01697 0.00147 -7.22713E-6 2778 0.33844 12 -0.01728 -0.00145 -4.9782E-6 2778 -0.52061 13 0.02336 6.57155E-4 2.68205E-6 2778 -0.37203 14 -0.01757 0.00143 2.82432E-6 2778 -0.39019 15 0.01848 -0.00205 -3.12113E-6 2778 0.30296 16 -0.02286 0.0018 -1.99101E-5 2778 -0.34633 17 0.01837 0.00115 -1.35034E-5 2778 0.28762 18 0.01679 0.00142 2.23796E-6 2778 0.32446 19 0.01536 -8.78402E-4 -4.34029E-6 2778 -0.39414 20 -0.01761 8.09171E-4 -9.24661E-6 2778 -0.38885 21 0.01656 -0.00324 4.23915E-6 2778 0.30545 22 0.01925 0.00297 -3.49117E-6 2778 0.32431 23 -0.01507 0.00208 -2.76953E-5 2778 0.56847 24 -0.02218 -4.41188E-4 2.85844E-5 2778 0.25819 25 0.0212 -6.62765E-4 -1.92921E-6 2778 0.26605 26 -0.01928 -0.00184 -2.7582E-5 2778 -0.34673 27 0.01607 -0.00183 -1.40253E-5 2778 -0.40151 28 -0.02097 0.00202 2.64986E-5 2778 -0.39267 29 0.02514 0.00187 -1.5879E-5 2778 0.34885 30 -0.02796 -8.60109E-4 3.01826E-5 2778 -0.42581 31 0.0249 -2.60599E-4 -2.90391E-5 2778 -0.32551 32 0.01613 -0.00144 1.00611E-5 2778 -0.55136 Table 3: Data used in ANNs for model 1 at node 4. A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 123 Tabs. 3, 5, 7, and 9 present the peak values of acceleration (g), stress (MPa), strain, and initial length of the crack (mm) obtained from numerical simulation and theoretical concepts. ANNs based on Levenberg-Marquardt algorithms have been used to predict and validate displacement in the Y direction at the critical point. Due to the huge number of outputs produced in the nonlinear numerical simulation, the peak value of the data has been selected and presented. The Levenberg-Marquardt algorithm in the Abalone Rings Data Set mode was used to validate and predict displacement accuracy in the ANNs process. The test data and results are validated based on the number of training observations in the numerical simulation. In the ANNs, three layers have been created, and mean squared error (MSE) and R have been obtained. Displacement prediction accuracy has been assessed using Eqns. 24 and 25. Figs. 10, 12, 14, and 16 show the regression analysis for ANNs outcome. Tabs. 4, 6, 8, and 10 present R2 and MSE values Model 1 and 2 prediction accuracy are associated with R2 and MSE. According to statistical analysis, the prediction results are acceptable. Figs. 11, 13, 15, and 17 show the accuracy of the predicted displacement quality. - Training Validation Test Number of layers in ANNs R2 076992 0.76983 0.7849 3 MSE 4.2268 4.1089 4.4893 3 Table 4: R2 and MSE results of ANNs for model 1 at node 4. Figure 10: Regression analysis in Y direction of displacement prediction for model 1 at node 4. A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 124 Figure 11: Prediction of displacement in Y direction for model 1 at node 4. No Important Vertical Peak Acceleration (g) Vertical Peak Stress (MPa) Vertical Peak Strain Initial crack length (mm) Vertical Displacement (mm) 1 6.50455E-4 7.92998E-5 -2.69808E-4 2778 0.73177 2 -0.00114 -8.40098E-5 -4.01878E-4 2778 0.70107 3 0.00895 -1.85242E-4 3.51639E-4 2778 0.69455 4 -0.0102 2.81983E-4 -5.36135E-4 2778 0.81327 5 0.02278 -5.08319E-4 4.98431E-4 2778 1.3219 6 -0.02165 3.04166E-4 -4.38268E-4 2778 1.34771 7 -0.02481 -8.2488E-4 5.30683E-4 2778 1.35579 8 0.02566 0.00103 -5.29435E-4 2778 0.99117 9 -0.03778 -0.00121 3.71622E-4 2778 0.94504 10 0.03811 0.00146 -5.83039E-4 2778 1.06602 11 0.0669 -0.0022 6.05269E-4 2778 0.76792 12 -0.0665 0.00204 -4.51155E-4 2778 0.66681 13 -0.08605 0.00298 4.98234E-4 2778 0.73003 14 0.08561 -0.00308 -3.44556E-4 2778 0.66487 15 -0.08479 -0.00367 4.23566E-4 2778 0.68248 16 0.08437 0.0042 -5.32241E-4 2778 1.0571 17 0.09081 -0.00429 6.147E-4 2778 1.1631 18 -0.08763 0.00419 -2.75271E-4 2778 1.07451 19 0.08611 -0.00464 5.82747E-4 2778 0.85982 20 -0.08454 0.00457 -6.60034E-4 2778 0.71614 21 0.09715 -0.00405 5.67388E-4 2778 0.80439 22 -0.09355 0.00453 -5.97126E-4 2778 -0.76087 23 0.09343 -0.00451 5.29893E-4 2778 -0.85938 24 -0.09165 0.00494 -5.78005E-4 2778 -1.14395 25 0.08607 0.00446 -5.11607E-4 2778 -1.23868 26 -0.08637 -0.0041 4.74269E-4 2778 -0.99735 27 -0.08832 0.00403 -5.55682E-4 2778 -0.66179 28 0.08978 -0.00461 -6.29915E-4 2778 -0.88863 29 -0.09711 0.00433 5.37781E-4 2778 0.81259 30 0.09345 -0.00411 -5.07329E-4 2778 1.16592 31 6.50455E-4 -0.00415 4.85554E-4 2778 1.55336 32 -0.00114 0.00398 -5.35886E-4 2778 1.18583 Table 5: Data used in ANNs for model 1 at node 12. A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 125 - Training Validation Test Number of layers in ANNs R2 0.74499 0.72808 0.73441 3 MSE 4.6616 5.384 4.1858 3 Table 6: R2 and MSE results ANNs for model 1 at node 12. Figure 12: Regression analysis in Y direction of displacement prediction for model 1 at node 12. Figure 13: Prediction of displacement in Y direction for model 1 at node 12. 0 5 10 15 15 Epochs 100 101 102 Best Validation Performance is 5.384 at epoch 9 Train Validation Test Best A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 126 No Important Vertical Peak Acceleration (g) Vertical Peak Stress (MPa) Vertical Peak Strain Initial crack length (mm) Vertical Displacement (mm) 1 0.00971 2.47041E-4 -7.65975E-5 2778 -0.29082 2 -0.00975 0.00103 -8.99527E-5 2778 -0.31459 3 0.01633 -0.00125 -1.34125E-4 2778 0.32941 4 0.0084 9.33859E-4 -8.16568E-5 2778 -0.28061 5 0.00994 -8.29053E-4 7.25802E-5 2778 0.45407 6 -0.0091 -0.00101 -8.99114E-5 2778 -0.30253 7 0.00852 -1.1407E-4 1.1234E-4 2778 0.47963 8 -0.00948 -0.00109 -1.02844E-4 2778 0.2593 9 -0.00944 0.00117 1.19955E-4 2778 0.27489 10 0.01398 9.86499E-4 9.62973E-5 2778 -0.38786 11 -0.0123 -8.92837E-4 8.3464E-5 2778 0.25863 12 0.00883 0.0011 9.44113E-5 2778 -0.38438 13 -0.01279 -0.00112 -1.16197E-4 2778 -0.2903 14 0.00864 0.00173 7.87148E-5 2778 -0.29552 15 -0.01205 -0.00152 -1.28119E-4 2778 0.41783 16 0.02132 0.00121 8.95447E-5 2778 0.37678 17 -0.01493 -0.00109 -1.32094E-4 2778 0.38465 18 0.01589 4.41696E-4 8.43624E-5 2778 -0.37205 19 -0.00939 0.00144 -1.21311E-4 2778 0.34543 20 0.01944 0.00144 1.8665E-5 2778 -0.39702 21 -0.02093 -2.79313E-4 9.00824E-5 2778 -0.4609 22 -0.00972 -6.36387E-4 1.0187E-4 2778 0.29921 23 0.0115 0.00107 -2.72714E-5 2778 -0.34462 24 -0.01642 2.5362E-4 8.46645E-5 2778 -0.44756 25 0.00904 -6.92872E-4 1.38161E-4 2778 -0.42403 26 -0.0101 -8.69604E-4 -2.31076E-5 2778 0.51274 27 0.00861 -0.00132 7.77794E-5 2778 0.3122 28 0.01371 0.0018 -1.00841E-4 2778 0.34447 29 -0.01745 8.44321E-4 7.84988E-5 2778 0.45651 30 0.00829 8.44321E-4 -1.17335E-4 2778 0.31465 31 -0.01455 9.8809E-4 -8.36407E-5 2778 -0.33765 32 0.01097 0.00104 8.99673E-5 2778 -0.37807 Table 7: Data used in ANNs for model 2 at node 8. A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 127 - Training Validation Test Number of layers in ANNs R2 0.77459 0.79535 0.72027 3 MSE 4.1458 7.0726 4.7931 3 Table 8: R2 and MSE results ANNs for model 2 at node 8. Figure 14: Regression analysis in Y direction of displacement prediction for model 2 at node 8. Figure 15: Prediction of displacement in Y direction for model 2 at node 8. 5 10 15 20 25 Target 5 10 15 20 25 Training: R=0.77459 Data Fit Y = T 5 10 15 20 25 Target 5 10 15 20 25 Validation: R=0.79535 Data Fit Y = T 5 10 15 20 Target 5 10 15 20 Test: R=0.72027 Data Fit Y = T 5 10 15 20 25 Target 5 10 15 20 25 All: R=0.76996 Data Fit Y = T 0 5 10 15 20 25 30 35 40 40 Epochs 100 101 102 Best Validation Performance is 4.0726 at epoch 34 Train Validation Test Best A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 128 No Important Vertical Peak Acceleration (g) Vertical Peak Stress (MPa) Vertical Peak Strain Initial crack length (mm) Vertical Displacement (mm) 1 -0.06228 0.00364 3.87191E-4 2778 0.49187 2 0.06384 -0.00374 -4.15158E-4 2778 0.81999 3 -0.08703 -0.00484 -5.59286E-4 2778 -0.20928 4 0.08813 0.00498 5.89115E-4 2778 0.96371 5 -0.0911 -0.00513 -5.89344E-4 2778 -0.67519 6 0.09609 0.00539 6.65428E-4 2778 0.60006 7 0.08894 -0.00395 5.04661E-4 2778 -1.59723 8 -0.08411 0.00451 -4.57753E-4 2778 -1.32669 9 0.07111 0.00402 5.3166E-4 2778 1.05664 10 -0.07159 -0.00393 -4.78772E-4 2778 -0.68074 11 -0.07212 -0.00371 -4.42859E-4 2778 1.52171 12 0.06941 0.00386 5.06271E-4 2778 1.18099 13 0.06717 -0.00381 -5.26548E-4 2778 -0.56321 14 -0.06425 0.00397 5.02833E-4 2778 0.69203 15 0.06058 -0.00362 4.38587E-4 2778 -1.17325 16 -0.06356 0.0034 -4.2151E-4 2778 -1.63809 17 0.06251 -0.00364 -4.58134E-4 2778 -1.32084 18 -0.06359 0.00334 4.37875E-4 2778 0.62052 19 -0.06205 -0.00407 -5.43534E-4 2778 -0.45421 20 0.06243 0.00304 -4.31483E-4 2778 1.38419 21 -0.0695 0.00335 4.65704E-4 2778 1.3113 22 0.066 -0.00347 4.06485E-4 2778 -0.39103 23 -0.06804 0.00304 -4.43368E-4 2778 0.78891 24 0.06476 -0.00411 -4.31212E-4 2778 -0.95975 25 0.07177 -0.00363 4.97951E-4 2778 0.32627 26 -0.0713 0.00394 -4.34561E-4 2778 -1.36753 27 -0.07217 -0.00427 4.53222E-4 2778 -1.07084 28 0.0667 0.00341 -4.10302E-4 2778 0.90474 29 -0.06252 0.00442 -4.36056E-4 2778 2.11807 30 0.06705 -0.00368 4.91303E-4 2778 1.01959 31 0.06841 0.00309 -4.46954E-4 2778 -0.72574 32 -0.07263 -0.00333 4.76944E-4 2778 -1.69437 Table 9: Data used in ANNs for model 2 at node 12. A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 129 - Training Validation Test Number of layers in ANNs R2 0.78406 0.78312 0.71877 3 MSE 3.9838 4.4124 4.7717 3 Table 10: R2 and MSE results ANNs for model 2 at node 12. Figure 16: Regression analysis in the Y direction of displacement prediction for model 2 at node 12. Figure 17: Prediction of displacement in Y direction for model 2 at node 12. 5 10 15 20 25 Target 5 10 15 20 25 Training: R=0.78406 Data Fit Y = T 5 10 15 20 Target 5 10 15 20 Validation: R=0.78312 Data Fit Y = T 5 10 15 20 Target 5 10 15 20 Test: R=0.71877 Data Fit Y = T 5 10 15 20 25 Target 5 10 15 20 25 All: R=0.7741 Data Fit Y = T 0 5 10 15 20 25 27 Epochs 100 101 102 103 Best Validation Performance is 4.4124 at epoch 21 Train Validation Test Best A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 130 THE CRACK MORPHOLOGY OF LANDFILL he MSW in each region possesses different mechanical properties. The present simulation work has been done numerically for designing the landfill clay cover. Figs. 18 and 19 illustrate the tension crack in the landfill clayey cover. In Fig. 18, a single initial tension crack in the clayey cover of the landfill is created. This crack we have simulated in the present work, while in Fig. 19, the initial multidirectional tension crack in the clayey cover of the landfill has been developed. Considering the clayey landfill cover for controlling leachate, as presented in the literature [24-25], the seismic stability design of the landfill plays an important role by considering the thickness of the clayey landfill cover when constructing the landfill. Figure 18: The single initial tension crack in the landfill [4]. In each region, MSW has different mechanical properties depending on how it is produced, how it is compacted during landfill construction, how temperature affects the mechanical properties of MSW, and how much moisture it contains due to climate and leachate drainage, etc. Predictions of landfill seismic stability are supported by simulations of landfill design. Landfill sliding and shear failures can be predicted based on landfill design. Fig. 19 shows how the initial tension crack of the landfill can interact during the seismic loading on the landfill. The crack interaction causes landfill failure acceleration. Figure 19: The initial multidirectional tension crack in the cracked zone of the landfill [5]. T A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 131 CONCLUSION A simulation was conducted to simulate the displacement of the landfill covered by cracked clay under seismic loading. Rankine’s theory, the Phantom Node Method, and the Levenberg-Marquardt Algorithm were used to simulate, predict, and validate the numerical simulation results. Stress, strain, and length of the crack were selected to predict displacement at critical points. ANNs were used to predict displacement in the Y direction for the selected nodes. The following findings were achieved in the present study:  In each model, the crack initiation, crack shape modification, deformation speed, and shape are distinct. These phenomena are associated with clay landfill cover. The crack propagation in the landfill model is related to the damage magnitude.  In response to nonlinear seismic acceleration, crack morphology changes and impacts seismic acceleration transmission and landfill seismic stability. The nature of seismic acceleration (g) in both studied models determines the failure mechanism. 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NOMENCLATURE E Modulus elasticity Eu Undrained modulus elasticity ϕ Friction angle ϕu Undrained friction angle ψ Dilatancy angle C Cohesion CU Undrained shear strength γ Unit weight γc Undrained unit weight ν Poisson’s ratio νc Undrained poisson’s ratio G Shear modulus Z0 Cracked zone Sz Solid zone Kp Passive earth pressure coefficients Ka Active earth pressure coefficients θa Rankine active state of slip planes θ p Rankine passive state of slip planes pP Lateral earth force in passive state aP Lateral earth force in active state   ' x a Lateral earth pressure in passive state   ' x p Lateral earth pressure in active state  ' z Effective stresses   Effective density  sat Saturated density  w Water density  iN x Standard finite element shape functions iu Standard finite element unknowns  *  iN x Partition of unity A. Namdar et alii, Frattura ed Integrità Strutturale, 65 (2023) 112-134; DOI: 10.3221/IGF-ESIS.65.09 134   N x Global enrichment function ia Unknowns associated with the enrichments Γs Crack path ΩA Shaded area is portion of a new element A ΩB Shaded area is portion of a new element B N Standard finite element method shape functions UA Displacement in node A UB Displacement in node B d Obtained nonlinear displacement dp Predicted displacement by ANNs oD Mean value of obtained nonlinear displacement. 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