Numerical Study of the Ultimate Bearing Capacity of Two Adjacent Rough Strip Footings on Granular Soil: Effects of Rotational and Horizontal Constraints of Footings

Abstract

In this paper, the numerical study of the ultimate bearing capacity (UBC) of two closely spaced strip footings on granular soil is investigated using the finite element method (FEM) and upper bound limit analysis (UBLA). Although the UBC of two adjacent footings has previously been studied in other experimental and numerical research, in all the previously reported studies, the footings were not allowed to rotate and move horizontally freely. Due to the deformation of the soil medium, two closely spaced footings are subjected to horizontal movements and tilting, even under central vertical loads. When the two adjacent footings are not permitted to rotate and move in the horizontal directions, the unwanted bending moment and horizontal force act on the footings. Indeed, the UBC of two closely spaced rough footings is evaluated under incorrect constraints in earlier research. In the present research, the UBC of two adjacent rough footings is evaluated with and without these incorrect constraints. The key finding of this study is that constraining the horizontal and rotational movement of the foundation artificially increases the UBC, which does not reflect field conditions. When foundations are permitted to rotate and move horizontally, there is no increase in UBC; however, there is an increased risk of differential settlement and structural instability.

1. Introduction

Determining the ultimate bearing capacity (UBC) of foundations is a complex challenge due to the unpredictable nature of soil behavior. Various researchers have proposed differing solutions using simplified assumptions to tackle this problem. The UBC of foundations is influenced by multiple factors, including the type of loading. For example, foundations subjected to eccentric and inclined loads require adjustments for effective foundation area and inclination factors in UBC calculations [1]. Such load eccentricities and inclinations notably decrease the UBC [2]. The implications of these load types on the UBC have been thoroughly explored in several studies [1,2,3,4,5,6].

The proximity of multiple footings, common in structures like offshore foundation systems for wind turbines and subsea systems, alters the UBC [7]. Furthermore, as urban populations continue to swell and cities expand through rapid development and urbanization, buildings are increasingly constructed in close proximity to one another. This spatial arrangement inevitably leads to interactivity among the foundations of these buildings. As a result, foundations situated in close proximity have the potential to interact, leading to alterations in failure mechanisms and bearing capacity. In situations where the distance between footings is minimal, the possibility of adverse adjacent interactions increases, potentially causing structure tilt and horizontal displacement and resulting in structural damage and reduction in serviceability [8]. The interaction among foundations is not confined solely to residential buildings but also encompasses silos and towers. Reports have documented the adverse effects of foundation interaction on these structures [9]. Several field observations, numerical analyses, and experimental studies have documented the significant impact of adjacent foundation interactions on the rotation, horizontal displacement of structure, and consequential structural damage [8,9,10]. Given the observable phenomenon of rotation and horizontal movement induced by the interaction between adjacent foundations, this study pioneers a fresh examination into how these factors collectively influence the ultimate bearing capacity. By delving into this nuanced dynamic, we aim to offer novel insights into the critical role played by rotation and horizontal movement in determining the overall bearing capacity of foundations. In modern urban environments, buildings are often constructed in close proximity to one another, resulting in potential interference among their foundations. Estimating the UBC of adjacent footings accurately is a critical step in the building design process. Field observations have indicated that two adjacent footings may experience tilting and horizontal movement over their service life [8]. In contrast to earlier investigations, this research introduces the concept of allowing for footings to rotate and horizontally move freely, under the premise that this adjustment could better simulate real-world conditions. Consequently, this study offers civil engineering designers a more precise and realistic means to estimate the UBC of two adjacent foundations.

Early theoretical and experimental investigations by Stuart [11] introduced the concept of the efficiency factor (ξγ), defining it as the ratio of the UBC of adjacent footings compared to an isolated footing. Stuart’s findings highlighted that footing interactions could significantly enhance the UBC, a notion further supported by subsequent studies [11,12,13,14]. However, prior experimental setups typically featured rigid connections between the footings and the loading frame, preventing any rotational or horizontal movements of the footings.

To conduct numerical analysis of foundations, various approaches such as limit analysis methods, the Finite Element Method (FEM), and finite difference methods (FDMs) can be employed [15,16,17,18,19,20,21,22]. Kumar and Kouzer [15] calculated the efficiency factors (ξγ) using the upper bound limit analysis (UBLA) in sandy soil. They simulated the rough footings as a rigid element, and footings were permitted to move only in the vertical direction. Ghazavi and Lavasan [16] employed the finite difference analysis to evaluate the performance of twin footings that are rested on reinforced sand. They considered the lateral resistance at the contact region between footings and soil to model the roughness of the footings. Indeed, footings were displaced only vertically downward without any horizontal movement. Mabrouki et al. [17] carried out a finite difference analysis to study the UBC of twin footings on granular soil under vertical central loads. Similar to the numerical studies mentioned previously, Mabrouki et al. [17] constrained the footings in the horizontal directions to simulate the roughness of footings. Schmüdderich et al. [18] carried out upper and lower band limit analysis to research the interfering influence of twin footings on sandy soil. Schmüdderich et al. [18] did not mention that two interfering footings were constrained in their numerical modeling, but simulation results show that both footings were constrained. Gupta and Sitharam [23] conducted a numerical study on the interference of closely spaced square footings on sand, with the bottom nodes of the footings constrained horizontally to simulate rough footing conditions. Anaswara and Shivashankar [24] investigated the behavior of two adjacent strip footings positioned on an unreinforced or reinforced granular bed overlying clay containing voids, employing a simulation of rigid rough footing by constraining the horizontal displacements of the footing nodes. Ghazavi et al. [25] conducted a comprehensive numerical analysis on the twin large circular footings situated atop a geocell-reinforced sand, with the interface between the soil and the footing being modeled as perfectly rough by restricting the displacement of nodal points representing the footing area in horizontal directions.

Table 1. Literature Overview on Bearing Capacity of Interfering Footings.

Study	Year	Method
Stuart [11]	1962	Numerical, limit equilibrium method
West and Stuart [12]	1965	Experimental, small-scale model
Das and Larbi-Cherif [13]	1983	Experimental, small-scale model
Kumar and Kouzer [15]	2008	Numerical, upper bound limit analysis
Ghazavi and Lavasan [16]	2008	Numerical, finite difference method
Kumar and Bhoi [14]	2009	Experimental, small-scale model
Mabrouki et al. [17]	2010	Numerical, finite difference method
Schmüdderich et al., [18]	2020	Numerical, upper and lower limit analysis
Gupta and Sitharam [23]	2020	Experimental and numerical, small-scale model and finite difference method
Anaswara and Shivashankar [24]	2021	Numerical, finite element method
Moghadasi et al. [9]	2023	Numerical, finite element method
Eslami et al. [8]	2024	Experimental and numerical, small-scale model and finite element method
Ghazavi et al. [25]	2024	Numerical, finite difference method
Moghadasi et al. [10]	2024	Experimental, small-scale model

provides a comprehensive overview of the existing literature concerning the bearing capacity of interfering footings.

In numerous focused numerical studies by Mabrouki et al. [17], Schmüdderich et al. [18], Kumar and Kouzer [15], Ghazavi and Alimardani Lavasan [16], Gupta and Sitharam [23], Anaswara and Shivashankar [24], and Ghazavi et al. [25], footings were modeled as rigid elements constrained horizontally to simulate their roughness, a common approach in analyzing the UBC under vertical loads. However, such constraints on footings might not mimic real-world conditions accurately, where footings might experience horizontal shifts and rotations due to subsoil deformations [8,11].

The review of the literature indicates that many studies on the UBC of twin rough strip footings under central vertical loads typically prevent footings from experiencing free horizontal movement and rotation. It has been observed that footings might undergo horizontal movements and rotations due to subsoil deformation, even under central vertical loads. Consequently, such constraints in the models can lead to additional bending moments and horizontal forces on the footings, potentially influencing the accuracy of the bearing capacity estimates.

Challenging the conventional modeling assumptions, this study proposes alternative hypotheses and employs innovative techniques. Unlike previous studies which constrain the rotation of footings under load and limit their horizontal movements to simulate roughness, this research allows for footings to rotate freely, based on the premise that removing these constraints might better replicate real-world conditions [26]. Furthermore, an accurate modeling approach is adopted where a surface-to-surface contact model, characterized by frictional and normal behaviors, is implemented at the contact region between the soil and the footings [27,28]. As previously mentioned, the interaction between foundations causes rotation and horizontal movement, as observed in the field. Therefore, this study examines the bearing capacity of adjacent foundations under conditions that allow for free rotation and horizontal movement. By considering these real-world conditions, the study aims to provide a more accurate evaluation of the UBC. Previous studies have not accounted for these real-world conditions, often reporting a significant increase in UBC due to foundation interaction. However, these results were derived under unrealistic assumptions, and relying on them for foundation design could jeopardize building stability and endanger the lives of residents. Therefore, it is crucial to evaluate the UBC under more realistic conditions to ensure safety and stability.

Utilizing the FEM and UBLA, this research aims to examine the influence of traditional constraints on the UBC of two adjacent rough strip footings on granular soil. By adopting these accurate methodologies and tools, the study seeks to provide a more accurate understanding of footing behavior under operational loads, thereby contributing to the refinement of modeling practices in geotechnical engineering.

2. Numerical Modelling

This section describes the computational methods and parameters used in the numerical analyses to assess the UBC of adjacent rough strip footings on granular soil under different constraint conditions.

2.1. Problem Definition

In the current study, two distinct strategies were employed for the numerical modeling of two adjacent rough strip footings, which are delineated as follows:

• Model I: The footings were not allowed to rotate or move horizontally freely, consistent with traditional modeling approaches noted in the literature (e.g., [17]).
• Model II: The footings were allowed to rotate and move horizontally without restriction.

Figure 1 illustrates these two modeling strategies. Each strip footing has a width, B, and there is a center-to-center distance, Δ, between them. Both footings are loaded vertically to failure simultaneously. In Model I, the connection of the loading frame to the footings is assumed to be rigid (Figure 1a). This setup prevents any rotational or horizontal movements under loading, thereby possibly introducing unwanted forces and moments to the footings. Conversely, in Model II, the load is transferred to the footings through a hinge and roller connection, allowing for free rotational and horizontal movements, eliminating the transfer of bending moments and horizontal loads from the loading shaft (Figure 1b). As stated previously, Numerical Model I is mostly used in the literature. The footings may be subjected to horizontal movements and tilting, even under central vertical loads. Thus, when Numerical Model I is used, unwanted force and bending moment are applied to the footings. The advantage of Model II, illustrated in Figure 1b, is that it allows for the footings to rotate and move horizontally freely, thereby preventing any bending moment and horizontal loads from being transferred from the loading shaft to the footings.

2.2. Numerical Modeling

To conduct the numerical study, PLAXIS 2D [29] and OptumG2 (V2, 12/02/2019) [30] software were employed for FEM and upper bound limit analysis, respectively.

2.2.1. Finite Element Modeling

The FEM program PLAXIS 2D was utilized to compute the efficiency factors (ξγ) for the evaluation of two adjacent rough footings. These footings, each having a width B, are situated on a surface of cohesionless soil and separated by a variable distance Δ, as depicted in Figure 1. In the numerical modeling, a comprehensive assessment of distances between two adjacent foundations was conducted, spanning from very close to far apart. This examination encompassed over ten different distances, meticulously ensuring the consideration of all possible scenarios of foundation placement in close proximity, thereby accurately reflecting real-world conditions.

Due to the symmetrical nature of the problem—both footings having the same width and being loaded simultaneously—only half of the problem domain was modeled. This approach not only simplifies the simulation but also ensures computational efficiency. The soil foundation modeled in the simulation has a depth and width of 20B and 25B, respectively, parameters chosen to effectively minimize the influence of finite boundary conditions (Figure 2).

In the simulation, the soil was represented using a linear elastic perfectly plastic model governed by the Mohr–Coulomb (M-C) failure criterion, complemented by an associated flow rule for depicting the yield surface. The mechanical behavior of the soil was further characterized through the generalized Hooke’s law. Notably, the modulus of elasticity was assumed to have a negligible impact on the calculated UBC [31], an assumption supported by empirical evidence. The mechanical properties, specifically Young’s modulus and Poisson’s ratio of the soil, were set to E = 80 MPa and υ = 0.35, respectively. Additionally, the friction angle of the soil was varied among three values: ϕ = 40°, 35°, and 30°, to assess their effect on the foundation’s behavior. In this study, we considered three friction angles: 30, 35, and 40 degrees, corresponding to loose, medium, and dense sand conditions. These values were selected based on their common use in validations and their support in numerous previous studies [12,13,15,17,18]. The modulus of elasticity has a minimal effect on the determination of UPC [31]. Therefore, the mechanical properties of the soil were set to reasonable and widely accepted values to ensure the reliability and relevance of the results. It is important to highlight that the soil type is granular, resulting in zero soil cohesion.

A rough interface condition was established between the footing and the underlying soil, matching the stiffness and strength parameters of the interface to those of the soil itself. This setup ensured a realistic interaction between the soil and the footing. To accurately capture stress concentrations, particularly in regions near the footings, a total of 4768 fifteen-node triangular elements were utilized in the finite element mesh, with increased density in critical areas. Figure 2 provides a detailed view of the finite element model and mesh specifics.

For the loading conditions, vertical displacement was applied to the rigid footing via a stiff loading shaft in Numerical Model I. In this model, the footing and the loading shaft were configured as very stiff plate elements rigidly connected to each other, with additional constraints preventing rotation and horizontal movement of the loading beam (Figure 3a). This setup ensured that the footing in Model I did not experience any rotational or horizontal shifts. Conversely, in Numerical Model II, the vertical displacement was directly applied to the center of the footings, allowing them complete freedom to rotate and move horizontally (Figure 3b), mimicking more realistic soil-structure interaction conditions.

2.2.2. Upper Bond Limit Analysis

The OptumG2 (V2, 12/02/2019) software was employed to calculate the UBC of the twin adjacent rough footings. A notable feature of OptumG2 is its capability to perform efficient calculations through automatic mesh adaptivity, which represents a significant advantage in handling complex geotechnical problems. In our simulations, the soil was modeled using a rigid–perfectly plastic Mohr–Coulomb (M-C) model, accompanied by an associated flow rule to accurately represent the yield behavior of the soil-footing system.

The footings themselves were simulated as rigid, weightless materials, ensuring that the structural properties of the footings did not affect the stress distribution in the soil. A perfectly rough interface was established at the contact region between the soil and the footings to realistically simulate the interaction forces and prevent any slip at the interface. The numerical model employed for the UBLA consisted of 100,000 elements, allowing for detailed and precise analysis. The model was subjected to ten adaptivity iterations, with shear dissipation serving as the adaptivity control mechanism. The configuration of the numerical model and its boundary conditions were kept consistent with those described in Figure 2, providing a direct comparison between the finite element and limit analysis methods.

3. Results and Discussion

In this section, the impacts of commonly utilized yet incorrect constraints on the UBC of two adjacent rough footings are numerically investigated. For this purpose, the UBC and efficiency factors (ξγ) of twin adjacent rough footings are computed using the numerical procedures described in Section 2. This involves subjecting the footings to vertical displacement, followed by calculating the resultant reaction force at the interface between the soil and footing. Subsequently, a graph illustrating the loading-displacement relationship is generated. The peak loading value is determined in accordance with the UBC value. Through the computation of the UBC for both isolated and twin adjacent footings, the efficiency factors (ξγ) are determined. These factors represent the ratio of the UBC of adjacent footings to that of an isolated footing, as previously defined. These results are subsequently compared with those reported in existing numerical studies and experimental results found in the literature.

3.1. Single Footing

The UBC of footing (q_u) was computed using FEM and UBLA, and then UBC factor N_γ was calculated as

$$N_{\gamma }=\frac{2q_u}{\gamma B}$$

(1)

where γ refers to the unit weight of soil and B refers to the width of the footing.

Table 2. Comparison of bearing capacity factor Nγ for isolated rough footings.

ϕ °	Terzaghi [32] a	Meyerhof [2] b	Vesic [3] c	Hansen [1] a	Martin [33] c	Mabrouki et al. [17] d	Kumar and Kouzer [15] e	Present Study
								FEM	UBLA
30	19.13	15.67	22.4	15.07	14.75	16.51	17.02	15.5	14.77
35	45.41	37.15	48.03	33.92	34.46	39.34	39.82	36.4	34.5
40	115.31	93.69	109.41	79.54	85.5	-	99.93	89	85.7

Note: ^a Limit equilibrium method; ^b Semiempirical solution; ^c Slip line method; ^d Finite difference method; ^e Upper bound limit analysis.

provides the value of UBC factors N_γ determined by the current study and other researchers. It can be seen that the value of calculated N_γ from the current numerical model is in good match with those reported by other researchers. This proves the validity of the present numerical modeling procedure.

3.2. Two Adjusted Footings: Numerical Model I

Figure 4 illustrates the calculated values of efficiency factors (ξγ) as a function of the spacing ratio (Δ/B) employing both the FEM and UBLA as outlined in Numerical Procedure I. The plots decisively demonstrate that the use of Numerical Procedure I in the simulations results in a substantial increase in bearing capacity due to the interaction between the footings. Additionally, the results presented in Figure 4 indicate that an increase in the friction angle enhances the efficiency factors (ξγ) at any given spacing. When the footings are positioned at very close spacings, an inverted arch forms between the footings, causing the two adjacent footings to behave as a single unit with a combined width of Δ + B. Consequently, the efficiency factors (ξγ) increase linearly with the spacing ratio from two to their maximum value, occurring at a specific spacing ratio.

Figure 5 presents a comparison of the calculated efficiency factors (ξγ) with those from published numerical and experimental studies in the literature [12,13,15,17,18]. It is observed that for three different values of the friction angle, ϕ = 40°, 35°, and 30°, the predicted efficiency factors (ξγ) align closely with the available numerical data. However, the present numerical study underestimates the efficiency factors (ξγ) in comparison to experimental results. A likely explanation for this discrepancy is the use of an associative flow rule in the numerical modeling, whereas the flow rule in actual soil behavior is generally non-associative (ψ ≠ ϕ). Moreover, difficulties in achieving convergence were encountered when attempting to incorporate a non-associative flow rule into the analysis. Another potential reason for the divergence between the experimental and numerical outcomes could be attributed to scale effects, influenced by factors such as the curvature of the Mohr–Coulomb Failure Envelope and particle size effects [34]. As demonstrated in Figure 4, when there is no separation between the two adjacent rough footings (effectively acting as a single unit), the experimentally derived efficiency factors (ξγ) are observed to be less than two. Consequently, the UBC factor Nγ decreases with an increase in the width of the footings, primarily due to scale effects.

As previously discussed, when footings are constrained from rotating and moving horizontally, bending moments and horizontal loads are transferred to the footings, resulting in a soil reaction that is both inclined and eccentric (refer to Figure 6). Conversely, when footings are permitted to rotate and move horizontally without restraint, they are subjected solely to central vertical loads.

The inclination angle (λ) and eccentricity (e) are computed at various distances between two adjacent rough footings using FEM. The soil reaction on the foundation base is resolved into components S (horizontal force) and P (vertical force), respectively, so that

$$\frac{S}{P}=tan\lambda$$

(2)

$$\frac{M}{P}=e$$

(3)

Figure 7 and Figure 8 illustrate the variation of tan (λ) and eccentricity (e) as a function of the distance between two adjacent footings. The variation of tan (λ) with the spacing ratio exhibits a trend similar to that observed for the efficiency factors (ξγ), an observation that aligns with the experimental findings reported by West and Stuart [12]. For all evaluated values of the friction angle (ϕ), the maximum values of tan (λ) and efficiency factors (ξγ) are observed at the same spacing ratio (Δ/B). As the distance between the two footings increases, both the inclination (λ) and eccentricity (e) approach zero.

3.3. Two Adjusted Footings: Numerical Model II

The efficiency factors (ξγ) for two adjacent rough footings are calculated using both FEM and UBLA according to Numerical Procedure II. As depicted in Figure 9, it is observed that the efficiency factors (ξγ) approximate a value close to one across all values of the spacing ratio (Δ/B).

For a more detailed comparison, the UBC of two adjacent rough footings is presented in

Table 3. The ultimate bearing capacity of two adjacent rough footings (kPa).

Analysis Method		Numerical Model I			Numerical Model II
	Δ/B	ϕ = 30°	ϕ = 35°	ϕ = 40°	ϕ = 30°	ϕ = 35°	ϕ = 40°
FEM	1	279	650	1600	145	340	842
	1.2	238	779	1950	142	332	826
	2	148	380	1135	141	329	825
	4	142	332	836	140	328	816
UBLA	1	266	622	1545	135	318	780
	1.2	195	753	1869	133	313	780
	2	138	347	996	133	312	779
	4	133	311	778	133	311	775

for Numerical Models I and II. In Numerical Model II, the maximum increase in the UBC due to the interfering effect is observed to be 3%, which is considered negligible. Conversely, in Numerical Model I, the UBC increases by more than 100%, reaching up to 170%. These results indicate that horizontal forces and bending moments significantly contribute to the increase in the UBC of twin adjacent rough footings in Numerical Model I. Stuart’s study [11] corroborates this finding, indicating that the horizontal component of the soil reaction in restrained footings causes interference between the foundations, thereby increasing their bearing capacity. An experimental study observed a tendency for footings to separate. If the footings are not restrained from separating or rotating, the interference effect diminishes, leading to a lower load at failure compared to when the footings are restrained [11]. The findings from Numerical Model II reveal that during the loading of footings, adjacent footings induce both tilting and horizontal displacement, consistent with field observations [8,9,10]. Such movements can result in differential settlement, cracking, diminished serviceability, and potentially structural failure. Consequently, positioning footings in close proximity does not enhance the UBC; instead, it exacerbates these detrimental effects on structural performance. Thus, it is imperative to exercise meticulous attention in the design of adjacent foundations to mitigate these adverse effects.

3.4. Failure Pattern

Figure 10 illustrates the incremental horizontal displacement arrows obtained from FEM analyses for Numerical Models I and II. As depicted in Figure 10a, a blocking effect is observed at very close distances between the footings. Consequently, a unit triangular elastic wedge is formed beneath the two adjacent rough footings, causing the twin footings to behave as a single footing. In Numerical Model I, due to the constraints on horizontal displacement and the roughness of the footings, no incremental horizontal displacement is observed in the triangular elastic wedge beneath the footings. Furthermore, the shape of the elastic wedge in Numerical Model I varies with the spacing ratio. At a spacing ratio of Δ/B = 0.2, a single triangular elastic wedge forms beneath the adjacent footings due to the blocking effect. For Δ/B = 1, nonsymmetrical elastic wedges are formed under the footings, as shown in Figure 10c. Additionally, at larger spacing ratios, the shape of the elastic wedges becomes symmetrical.

In Numerical Model II, the two adjacent rough footings were permitted to move horizontally without restraint. As a result, both the footings and the underlying elastic wedges shifted in the horizontal direction, leading to their separation. This movement and separation are clearly illustrated in Figure 10b,d,f. As shown in Figure 10 and Figure 11, the general shear failure patterns are formed under the footings. It is important to note that only the results corresponding to three distances between two adjacent foundations, selected from a range of over 10 distances, are presented in Figure 10 and Figure 11.

Figure 11 depicts the distribution of shear dissipation and the collapse mechanisms as derived from UBLA for Numerical Models I and II. The failure patterns identified by UBLA bear a schematic resemblance to those illustrated in Figure 10.

4. Conclusions

This study critically examined the UBC of two adjacent rough strip footings on granular soil by employing FEM and UBLA, addressing the conventional constraints typically applied in existing models. The literature reveals that many studies do not allow for free horizontal movement and rotation of footings under central vertical loads, which could lead to inaccuracies in UBC estimations due to unintended bending moments and horizontal forces arising from subsoil deformation. Challenging these traditional modeling assumptions, our research adopted accurate techniques and alternative hypotheses that permitted the footings to rotate and move horizontally freely, thereby aiming to more accurately replicate real-world conditions. This approach is underscored by the implementation of an accurate surface-to-surface contact model characterized by frictional and normal behaviors at the interface between the soil and the footings.

The findings demonstrate that when footings are restrained from tilting and separating horizontally, the UBC notably increases due to interference effects. However, such constraints also induce unwanted moments and horizontal loads, making the resultant soil reaction both inclined and eccentric. The key finding of this study is that the observed increase in UBC due to constraining the horizontal and rotational movement of the foundation does not accurately reflect field conditions. Incorporating this increased UBC into foundation design could jeopardize the stability of the structure. In contrast, when footings are allowed to tilt and move horizontally without restrictions, the efficiency factors (ξγ) remain at one for all values of the spacing ratio, indicating no increase in the safety factor against bearing capacity failure and suggesting that the footings act independently from a bearing capacity standpoint. When real conditions are considered—where the foundation is permitted to rotate and move horizontally—there is no increase in the UBC of adjacent foundations. Instead, this allowance leads to foundation rotation, which can have detrimental effects on buildings. By employing these methodologies and tools, the study provides a more refined understanding of the behavior of twin footings under operational loads, contributing significantly to the advancement of modeling practices in geotechnical engineering. The insights gained underscore the critical need for revising traditional modeling constraints to enhance the accuracy and reliability of UBC assessments in similar geotechnical applications.

This research assumes a granular soil type and focuses solely on investigating general shear failure of soil beneath the footings. Additionally, it considers the footings to be of strip type and operating under plane strain conditions. Future research could explore estimating the UBC of adjacent foundations without being constrained by the aforementioned limitations of the current study.