Simulative study of the interactions between two neighboring shallow foundations

 Corresponding author: Mbuh Moses Kuma and Kouamou Nguessi Arnaud. Copyright © 2025 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0. Simulative study of the interactions between two neighboring shallow foundations Mbuh Moses Kuma 1, *, Nsahlai Leonard 2, Penka Jules Bertrand 2, Kouamou Nguessi Arnaud 1, *, Tchemou Gilbert 3, Agandeh Elvis 1 and Phonchu Claret Abong 2 1 Department of Civil Engineering and Forestry Techniques, Higher Technical Teacher Training College, University of Bamenda, Cameroon. 2 Department of Civil Engineering and Architecture, National Higher Polytechnic Institute (NAHPI), University of Bamenda, Cameroon. 3 Department of Civil Engineering, Higher Technical Teacher Training College, University of Douala, Cameroon. World Journal of Advanced Research and Reviews, 2025, 26(03), 1794-1805 Publication history: Received on 19 April 2025; revised on 25 May 2025; accepted on 27 May 2025 Article DOI: https://doi.org/10.30574/wjarr.2025.26.3.0675 Abstract Foundations are designed to receive both imposed loads and superimposed loads and evenly distribute these loads to the bearing soil. Construction of load bearing elements can necessitate that two shallow foundations should have a close proximity. This close proximity gives birth to interactions studied numerically in this article. This study investigates the influence of horizontal and vertical space separation on the settlement of the foundations in soft clay soil. In this study, the Numerical method of research Cast3m was used. A geometrical model with mechanical characteristics to translate exactly a real case was built. At the end of the simulation, the space separation versus settlement behavior followed an exponential increase whereby, beyond a space separation to base ratio greater than or equal to 0.7 there was no interaction. The closer the two foundations, the more the horizontal displacements at the extremities of both foundations-both foundations tend to function like a single footing when they are closely spaced. This study is a breakthrough in foundation interactions for the case of clay soil as many researchers have concentrated their studies on sandy soil. Civil Engineers, Architects should consider the effect of space separation influence carried out in this study when doing conception and analysis of structures. Keywords: Simulation; Space Separation; Settlement; Interactions; Shallow Foundations; Stresses 1. Introduction A foundation is that part of a structure that receives loads and evenly distributes it to the bearing soil. Foundations placed at very close proximity experience interactions which can engender various behaviors. Studying interaction between two neighboring shallow foundation is a great scientific advancement in foundation engineering in order to overcome problems associated with the foundation and the environment (soil). The safety of a foundation and the soil beneath it, totally depends on the factors taken into consideration before and after the realization. The advantage of taking into consideration the interaction between two neighboring shallow foundation includes; maximizing space separation between foundations, excess stresses and settlement. The concept of interaction between two neighboring foundation was first visited by Stuart on the effect of closely continuously spaced foundations [1]. West and Stuart applied the method of stress characteristics to establish a solution for the interference of a footing on sand soil [2]. Their outcomes showed that the efficiency factor (ξ); which is a function of spacing to width of the foundations and soil friction angle values were smaller compared to those obtained by Stuart [2]. The downside of their research is that they only configure a solution for a soil having friction angle of 35°. Selvadurai and Rabbaa [3] investigated the interference of three closely spaced strip foundations on Ottawa and silica sand and World Journal of Advanced Research and Reviews, 2025, 26(03), 1794-1805 1795 came out with the conclusion that; interference initiated when spacing is of ratio 𝑆/𝐵 < 3, where B is the base of the foundation and S, the space separation between the two foundations [2],[3]. Furthermore, Graham et al. [4] investigated the interference of three closely spaced strip foundations on Ottawa and silica sand using the same method suggested by West and Stuart [2]. The results show that the method of stress characteristics is applied to designate the interference of the outer foundations on the bearing capacity of the central footing and it is not a suitable theory for two closely spaced footings. This may justify why West and Stuart obtained lower efficiency factors (ξ). The result given by Graham et al [4] was that, the interaction depends on soil friction angle and efficiency factors versus spacing as given by [4], [5]. Kumar and Ghosh provided the failure mechanisms beneath two rigid continuous foundations which coincided well with the assumption of Stuart [6]. Moreover, several types of research are reported on the bases of analytical approach, probabilistic approach, and upper bound limit analysis that the bearing capacity of neighbored foundation increases as the spacing between them is reduced [7]. Lee and Eun [8] carried out field circular plate test on sand and concluded that failure stress of the soil beneath neighboured footing is higher than isolated footing; however, larger settlements occur beneath neighboured footing. Srinivasan and Ghosh carried out several laboratories scaled model tests of circular footings in dry dense homogeneous sand and concluded that efficiency factors (ξ) are found to be maximum at S/B = 0.5 [9], with B as the base of the foundation and S the space separation between the two foundations. Reddy et al [14] carried their test on square and circular footing model on medium dense sand and they reached the conclusion that, the closeness of footings improve the response of foundations both in terms of settlement and ultimate bearing capacity; nevertheless, increase in settlements are being observed at between B ≤ S ≤ 6B. Srinivasan and Ghosh carried out their investigation on two layers of sand (weak layer underline by strong layer) and they reached the conclusion that, the bearing capacity and the developed settlement at failure declined with an increase in the depth of the upper weak layer and the efficiency factors (ξ) are found to be maximum at S/B = 0.5. [9] [10]. Against the above background, the main objective of this research was to study the mechanical behavior of soft clay soil between and beneath two neighboring foundations influenced by the load it receives and space separation (vertical and horizontal) between them and how to limit or reduce excessive soil stress and settlement. 2. Materials and method 2.1. Method In the framework of this research, a numerical approximation to the behavior model of two neighboring shallow foundations in a soft clay soil was proposed. CAST3M software as numerical tool was adopted. CAST3M is a computer code for the analysis of structures by finite element method [15]. CAST3M presents a complete system, integrating not only the functions of calculation themselves, but also the functions of constructing models (preprocessor) and processing the results (post-processor) [15]. CAST3M made it possible for us to deal with problems of linear elasticity in the statics and dynamics fields (extraction of eigenvalues), nonlinear problems (elasto-viscoplasticity), step by step dynamic problems, etc [15]. In order to convert the names of the objects into data-processing entities usable by the program, it was necessary to have an interface. It was the GIBIANE language which made it possible for us to communicate directly with the program [15]. 2.2. Some features of the CAST3M software 2.2.1. Mesh Size Mesh size is an essential aspect of CAST3M. The mesh size refers to the spatial discretization of the computational domain into smaller elements for performing numerical simulations. To affix a Mesh size in CAST3M you go by the word DENS and after that, you input the value you want. In the case of this paper, we used 0.25 as the mesh size. The Figure 1 below show the discretization of triangular shape of three nodes. World Journal of Advanced Research and Reviews, 2025, 26(03), 1794-1805 1796 Figure 1 Discretization (meshing) of triangular shape of three nodes 2.2.2. Convergence Convergence is the process of ensuring that the numerical solution obtained from iterative calculations approaches the correct solution as the computational process progresses. 2.2.3. Finite Element. Triangular finite elements of three nodes were used. 2.3. Presentation of model 2.3.1. Physical geometry Three models characterized this study: The two foundations were placed at the same level, then the right foundation was varied vertically while fixing the left foundation and vice versa. The Figure 2, 3 and 4 show the three models of study. ( deta-x = horizontal space separation (m) , deta-y = vertical space separation (m) ). Figure 2 Foundations at the same level Figure 3 Vertically varying right foundation World Journal of Advanced Research and Reviews, 2025, 26(03), 1794-1805 1797 Figure 4 Vertically varying left foundation 2.4. Boundary conditions The Figure 5 below shows the design of boundary conditions. The model has three sub-behavioral laws (Fig 6): concrete, soil and interface. Figure 5 Boundary conditions Figure 6 Three sub-behavioral laws of model L20 = the line showing the left edge of the soil L2 = the line showing the right edge of the soil, UX = horizontal displacement. UY = vertical displacement. • CL1 = BLOQ DEPLA L1; the base of the foundation is rigid. • CL2 = BLOQ UX L2; the soil on the adjacent sides do not move in the horizontal direction. • CL3 = BLOQ UX L20; the soil on the adjacent sides do not move in the horizontal direction 2.5. Behavior models 2.5.1. MAZAR’S BEHAVIOR MODEL This behavior model was formulated by [Jacky MAZARS, 1984] that permits us to describe the elastic damage behavior of concrete. It is a 3D isotropic model, formulated according to a damage criterion in deformation and describing a dissymmetry tension-compression. The model has three sub-behavioral laws defined below 2.6. Foundation Model 2.6.1. Mazar’s deformation (Elasto-plastic) Mazar elasto-plastic model, is a constitutive model used here to describe the stress-strain behavior of soil under both elastic and plastic deformations. This model is employed to capture the complex behavior of soils, it incorporates both elastic and plastic components to represent the behavior of soil under loading. The elastic portion of the model accounts World Journal of Advanced Research and Reviews, 2025, 26(03), 1794-1805 1798 for reversible, recoverable deformations, while the plastic portion captures irreversible deformations associated with soil yielding and failure. It was developed by the geotechnical engineer Shlomo Mazar 𝐷𝑒= 1−𝜀𝑑𝑜(1−𝐴𝑐) 𝜀 −𝐴𝑐(𝐵𝑐(𝜀𝑑𝑜−𝜀 )) Where: (𝜀𝑑𝑜)= Threshold deformation in tension (𝐴𝑐) = Compression Parameter (𝐵𝑐) = Compression Parameter 𝐷𝑒 = elastic damage 𝜀 = strain tensor 2.6.2. Clay Soil Model CAM-CLAY (Elasto-plastic) Cam-Clay model is a classic elasto-plastic constitutive model used to describe the behavior of cohesive soils. The model is a fundamental tool in geotechnical engineering for predicting the stress-strain behavior of clayey soils. 𝑃′𝑐= 𝑃′𝑐𝑜[1+𝑒𝑖 𝜆−𝑘𝜀𝑣 𝑝] 2.7. Interface (Mhor Coulomb) The interface, also known as Mohr-Coulomb failure criterion, is a fundamental concept in the field of geotechnical engineering and soil mechanics. 𝜏 = 𝑐 +𝜎𝑡𝑎𝑛𝜑 Where: 𝜏 = 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑠 𝜎 = 𝑛𝑜𝑟𝑚𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠 𝜑 = 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 2.8. Material The Table 1, 2 and 3 shows the characteristics of Mazar’s deformation model, the characteristics of soft clay soil of Bonaberie in Douala and the concrete-soil interface characteristics. Table 1 Foundation (concrete) Characteristics of Mazar’s deformation model [12] Parameters Symbol Numerical value Young’sModulus E (Mpa) 27000 Volumic mass 𝝆 ( 𝐾𝑔/𝑚3) 2500 Poisson’s Ratio 𝝑 0.2 Threshold deformation in tension (𝜺𝒅𝒐) 𝑲𝑻𝑹𝑶 1×10−4 Compression Parameter (𝑨𝒄) 𝑨𝑪𝑶𝑴 1.4 Compression Parameter (𝑩𝒄) 𝑩𝑪𝑶𝑴 1900 Tension Parameter 𝑨𝑻𝑹𝑨 0.8 Tension Parameter 𝑩𝑻𝑹𝑨 17000 Correction parameter for shearing 𝑩𝑬𝑻𝑨 1.06 Note: the symbols; KTRO, ACOM, BCOM, ATRA, BTRA, BETA These values are inputed in the simulation process in CAST3M, found in the algorithm. World Journal of Advanced Research and Reviews, 2025, 26(03), 1794-1805 1799 Table 2 Characteristics of soft clay soil of Bonaberi-Douala Parameters Symbol Numerical value Young’s Modulus E (Mpa) 24 Volumic mass 𝜌 ( 𝐾𝑔/𝑚3) 1575 Poisson’s Ratio 𝜗 0.43 Indice of voids 𝐸𝑂 0,37 Coefficient of Friction 𝑀 0.6 Internal angle of friction 𝜑 30° Cohesion 𝐶𝑂𝐻𝐸 (𝑘𝑃𝑎) 40kPa Pre-consolidation pressure 𝑃𝑂 (𝑘𝑃𝑎) 20kPa Elastic slope 𝐾𝐴𝑃𝐴 0.02 Plastic slope 𝐿𝐴𝑀𝐷 0.1 Shear Modulus 𝐺1(MPa) 15.4 Table 3 Concrete-Soil Interface Characteristics [13] Parameters Symbol Numerical value Second Normal stiffness constant EF 2×1015𝑁/𝑚2 Threshold deformation ECN 1000% Cohesion COHE 10kPa Angle of friction FRIC 20 Maximum resistance in tension FTRC 0 2.9. Time interval of load application A load intensity of 50KN was applied. This loading was done following the Cast3m operator: PASAPAS, which takes into consideration time as a parameter. The Figure 7 below shows the load factor-time graph. Figure 7 Load factor-time graph World Journal of Advanced Research and Reviews, 2025, 26(03), 1794-1805 1800 The above graph represents time loading graph, with loads in KN and time in seconds, indicating that a load of 1KN is spread over a period of 1second and the results beneath were registered. 3. Results and discussion 3.1. Results After a rigorous inputting of all the necessary data requested by the cast3m software, we had the following results. The different cases; stress-displacement graphs and stress-space separation graphs and settlement-space separation graphs. The pictures of horizontal Displacement, Vertical Displacement (Settlement) and Vertical Stress were observed (Fig. 8 and 9). Horizontal Displacement of clay soil under load Vertical Displacement of clay soil under load Vertical stresses of clay soil under load Figure 8 Behavioural out coming from the simulation Figure 9 Settlement horizontal position graph World Journal of Advanced Research and Reviews, 2025, 26(03), 1794-1805 1801 From the settlement-space separation graph (CASE 1), a transition zone of interactions and a permanent zone of least interactions was observed. This zone is delimited by a space separation of 1 meter. Mathematically the graph follows an exponential decay with an empirical equation as shown below. 𝑦 = 𝐴𝑒−𝛽𝑥 +𝐵 −−−−(𝐸𝑞.1) Where 𝐴 = ∆ℎ𝑚𝑎𝑥 −∆ℎ𝑚𝑖𝑛 ,𝐵 = ∆ℎ𝑚𝑖𝑛 𝛽 = 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡; ∆hmin;minimum settlement for which there is no interaction ∆hmax ;maximum settlement for closes space separation The equation can be written: 𝑆 𝐵≥2 3~0.7 − − − − − − − − − −(Eq.2) Where S = Space separation between foundation B = Width of foundation The expression above, gives the threshold space separation above which there is least interaction between both foundations. 3.2. Shear stress diagram of the soil results from the foundation Plan of consideration is the horizontal plane; the Figure 10 show the shear stress-horizontal position. Figure 10 Shear stress-horizontal position graph From the Maximum Shear Stress-Space Separation Graph, the Shear stress in the soil between both foundations increases and attains a limit value. This graph mathematically represents an exponential increase which can be mathematically written in the form: 𝑦 = 𝜏(1−𝑒𝛽𝑥) −− − − − −(Eq.3) Where 𝑠ℎ𝑒𝑎𝑟𝑠𝑡𝑟𝑒𝑠𝑠 𝑣𝑎𝑙𝑢𝑒 = 𝜏𝑚𝑎𝑥 , 𝛽 = 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡; The expression above gives the threshold space separation above which there is least or no Shear stress interaction between both foundations (Fig 11). World Journal of Advanced Research and Reviews, 2025, 26(03), 1794-1805 1802 Figure 11 Settlement-vertical position graph The combined Settlement-Vertical position graphs (CASE 2) and Stress-Vertical Position graphs clearly interprets to us the various Mid Stress-Displacement graphs: From these combined graphs that, as the vertical position of the right footing increases, there is a decrease in the settlement and stress under the left footing. This ratio of decrease, becomes significantly very small as the horizontal gap between both foundations increase. This phenomenon becomes constant for a horizontal space separation of 1m (Fig 12). Figure 12 Shear stress-horizontal position graph It is observed that for a very small separation, shear stress increases as the depth of the right foundation increases and becomes constants at a vertical difference of 0,5m; The rate at which this transition occurs decreases as the space separation increases and becomes constant for a space separation (det-x) of 1,00m. (det-x = 1.00 is the thresh space separation above minimal interaction) The Figure below (Fig 13) is a graph resulting from a combined settlement-vertical position graph for CASE 3.