Numerical Study of Failure Mechanisms of Footings Subjected to Uplift and Lateral Loads Using PLAXIS 3D

1  Introduction

The stability and resilience of critical infrastructure systems, encompassing high-voltage transmission towers, telecommunication networks, railway control signals, and offshore energy platforms, are fundamentally dependent on the performance of their foundation systems. Unlike traditional building foundations that primarily manage vertical gravity loads, these specialized structures are frequently subjected to severe environmental and operational stresses, including uplift forces, lateral shear, and resultant bending moments. As the global push for sustainable and resilient infrastructure intensifies, particularly in the context of expanding 5G networks and renewable energy grids, the engineering challenge of predicting foundation behavior under complex loading has never been more critical [1].

Shallow footings and plate anchors serve as the primary interface between these structures and the earth. Their design must account for the intricate interaction between the structural elements and the surrounding soil, where the uplift capacity and the failure mechanism are the most vital parameters. A precise understanding of these factors is not merely a technical requirement but a prerequisite for rational and economical design. Over-engineering leads to prohibitive costs and material waste, while under-estimation can result in catastrophic pullout failures, leading to massive service disruptions and safety hazards [2,3].

The complexity of soil-structure interaction under uplift is driven by the development of sliding surfaces within the soil mass. These surfaces are not static; they vary dynamically based on soil density, embedment depth, and the geometry of the footing. Historically, the sliding surface at the meridian section of a footing was assumed to follow simple vertical or inclined straight lines. However, modern research has demonstrated that these surfaces often follow more complex logarithmic spiral curves or hybrid geometries [4–6]. The resistance to uplift is a function of the weight of the soil “trapped” above the anchor plate and the frictional forces generated along these sliding surfaces, which are influenced by the earth pressure and the shear strength of the soil [7,8].

The systematic investigation into anchor behavior began in the 1960s, primarily driven by the rapid expansion of electrical transmission lines. Early pioneers such as Adams and Hayes (2009), Das et al. (1967), and Ireland (1963) provided the first empirical datasets for pullout capacity [9–11]. These studies laid the groundwork for the development of various anchor types, among which plate anchors emerged as the most popular due to their high efficiency-to-cost ratio in both onshore and offshore applications [12].

The first rational analytical approach was proposed by Marston in 1930, who modeled the load on buried conduits by assuming a vertical slip surface. While groundbreaking, Marston’s model often diverged from experimental observations, particularly in dense sands where soil dilatancy plays a significant role [13]. This led to a flurry of theoretical developments in the mid-20th century. Various failure mechanisms, including vertical shear surfaces, cylindrical, and truncated cone models, have been proposed in the literature for foundation systems. Ball (1961) further refined these models by developing an analysis specifically for pylon foundations, which remains a foundational reference in the field [14].

The late 20th and early 21st centuries saw a shift from purely empirical or simplified analytical models toward sophisticated numerical and experimental techniques. Meyerhof and Adams (1968) conducted extensive laboratory tests that defined the behavior of circular and strip anchors in varying sand densities, establishing the “Meyerhof theory” as a standard in geotechnical design [15]. Wilson (1972) and Vesić (1971) introduced the cavity expansion theory, providing a new mathematical framework for computing the uplift capacity of embedded objects [16,17].

The advent of computational geotechnics allowed for the exploration of variables that were previously difficult to quantify. Rowe and Davis (1982) utilized elasto-plastic finite element analysis (FEA) to study the influence of soil dilatancy, initial stress states, and anchor roughness [18,19]. Their work highlighted that the anchor plate’s relative rigidity and the soil’s plastic flow significantly alter the failure surface. Subsequent research by Dongxue et al. (2012), Dickin (1988), and Zhuang et al. (2012) further validated these numerical models through controlled laboratory experiments on rectangular and circular anchors [20–22]. The influence of oblique and combined loading on uplift resistance has been examined analytically and experimentally in several studies, highlighting the reduction in vertical capacity under lateral force components [23–25].

In the last two decades, the focus has expanded to include complex loading conditions and innovative monitoring techniques. Tolun et al. (2024) investigated the effect of dilatancy on the tension response of rough piles embedded in granular soils, highlighting the influence of soil dilation on uplift resistance [26]. More recently, advanced experimental and analytical approaches have enabled improved visualization and understanding of failure mechanisms, including the transition from shallow to deep failure modes, as demonstrated through image-based techniques and detailed numerical and experimental analyses [27,28]. Studies by Amin et al. (2025) have specifically applied these techniques to electrical tower footings, bridging the gap between laboratory models and field applications [29].

As infrastructure projects move into more challenging environments, such as offshore wind farms and seismic zones, the behavior of anchors under cyclic and monotonic loading has become a primary area of concern. Recent investigations by Kumar et al. (2025) and Merifield et al. (2001) have utilized three-dimensional numerical analysis to investigate the effects of layered soils and anchor geometry on uplift capacity [30–32]. Furthermore, the integration of probabilistic approaches is beginning to allow for the assessment of uncertainty in soil properties, leading to more robust and reliable design frameworks [31].

Recent investigations have increasingly relied on advanced three-dimensional numerical modeling to capture failure mechanisms of anchors and shallow foundations under combined uplift-lateral loading, incorporating refined constitutive models and experimental validation techniques [23,24,33–35].

Three-dimensional material point method simulations have confirmed the wedge-type failure geometry in sand and the sensitivity of inclination angle to embedment ratio and soil density [36]. These studies collectively demonstrate that the failure mechanism under combined loading is fundamentally different from pure uplift, and that the reduction in vertical capacity depends on the magnitude and direction of the applied lateral force.

This study builds on established analytical, experimental, and numerical research on uplift resistance of shallow foundations and plate anchors. The contribution of the present work is a targeted three-dimensional numerical parametric investigation of failure mechanisms under both pure uplift and combined uplift-lateral loading, focusing on how the embedment ratio (D/B), footing size, fines content, and relative compaction influence ultimate pullout capacity and rupture geometry. The study employs PLAXIS 3D with four specific objectives: (i) to quantify the ultimate pullout capacity under pure uplift and combined uplift-lateral loading; (ii) to systematically investigate the influence of soil type, relative compaction, footing dimensions, and embedment ratio; (iii) to determine the failure mechanism and inclination angle of the rupture surface for each parameter set; and (iv) to compare the numerical results with laboratory model-test data to verify the reliability of the simulations.

2  Numerical Modeling and Methodology

To investigate the complex soil-structure interaction of transmission tower foundations under combined loading conditions, a comprehensive numerical study was conducted using PLAXIS 3D (Version 20). This finite element software is specifically developed for geotechnical applications and enables simulation of three-dimensional stress-strain behavior that cannot be adequately captured using two-dimensional plane strain or axisymmetric models.

2.1 Geometrical Modeling and Mesh Generation

The numerical models were developed using a hierarchical geometrical approach consisting of points, lines, and clusters to accurately represent the footing and the surrounding soil domain. A fully three-dimensional geometry was adopted to capture spatial stress redistribution and deformation behavior associated with uplift and combined uplift-lateral loading conditions.

The soil domain dimensions were selected to be sufficiently large in all directions to minimize boundary effects and ensure that stress bulbs and rupture surfaces developed fully within the mesh. In the present models, the domain is extended laterally by at least 10B from the footing edges in the x- and y-directions, and the base boundary is located at least 6B below the footing level, where B is the footing width. Boundary conditions were applied as follows: the base boundary was fixed in all directions (ux = uy = uz = 0), while the vertical side boundaries were restrained in the normal direction only, on x-normal planes ux = 0 and on y-normal planes uy = 0, allowing vertical movement (uz free) to avoid artificial restraint of uplift deformation. The ground surface was left free. A domain-size check on a representative case (10 cm × 5 cm footing, D/B = 1.5, pure sand, RC = 100%) showed that increasing the lateral extent from 10B to 12B changed the ultimate pullout load by less than 1%, confirming negligible boundary influence.

Finite element discretization of the soil domain was performed using 10-node tetrahedral elements, which provide quadratic interpolation of displacements and improved accuracy in representing stress and strain gradients within the three-dimensional soil volume. The shallow footing was modeled as a plate element (shell) in PLAXIS 3D. Plate elements in PLAXIS 3D are structural surface elements defined by 6-node triangular geometry, embedded within and fully compatible with the surrounding 3D tetrahedral mesh. They carry both in-plane membrane stiffness and out-of-plane bending stiffness, and their material behavior is defined by the equivalent axial stiffness EA and flexural stiffness EI per unit width, together with the Poisson’s ratio ν and the unit weight of the plate. This formulation is appropriate for modeling rigid or semi-rigid footings whose thickness is small relative to their plan dimensions, as it correctly transmits the applied uplift and lateral forces to the surrounding soil through interface-compatible nodal degrees of freedom. Interface elements were placed between the plate and the adjacent soil to model the soil-footing contact behavior, governed by the interface strength reduction factor R.

Mesh Sensitivity Analysis

Because the results reported in this study are derived from three-dimensional finite element simulations, a mesh-sensitivity analysis was performed to ensure that the computed ultimate pullout capacity and failure mechanism are not affected by discretization. Three meshes were examined, coarse, medium, and fine generated by systematically reducing the global element size and the local refinement around the footing. The refinement zone was defined as a region extending approximately 2B laterally from the footing edges and 2D vertically above the footing level, where high shear-strain gradients develop. A representative case, pure sand, RC = 100%, footing 10 cm × 5 cm, D/B = 1.5, was analyzed under pure uplift loading using the three meshes. The ultimate pullout load Tu was defined according to the failure criterion in Section 3.1.1, and the inclination angle β was measured using the procedure described in Section 3.1.3.

As summarized in Table 1, the difference in Tu between the medium and fine meshes was 2.1%, and the difference in β was 0.8°, both indicating satisfactory convergence. The medium mesh was therefore adopted for the full parametric study to balance accuracy and computational cost.

2.2 Coordinate System and Sign Convention

All numerical models were developed in the global x-y-z Cartesian coordinate system, where the z-axis represents the vertical direction, and the x- and y-axes define the horizontal plane. A consistent sign convention was adopted throughout the analysis: compressive stresses and forces, including pore water pressures, were considered negative, while tensile stresses and forces were taken as positive. This convention was maintained for all input parameters and output interpretations to ensure consistency in evaluating load-displacement behavior and failure mechanisms. Fig. 1 illustrates the adopted coordinate system and positive stress directions in PLAXIS 3D.

Figure 1: Coordinate system indicating positive stress components in 3D PLAXIS.

2.3 Material Properties and Constitutive Models

Two representative soil types commonly encountered in transmission tower foundation projects were considered: pure sand and sand mixed with 8% fines. These soil types were selected to investigate the influence of fines content on uplift resistance and failure mechanisms of shallow footings subjected to tensile and combined tensile-lateral loading.

The mechanical behavior of the soil was simulated using the Mohr-Coulomb constitutive model, an elastic-perfectly plastic model widely adopted in geotechnical engineering practice. This model was selected due to its simplicity, robustness, and proven capability to capture the essential stress-strain behavior and failure characteristics of granular soils under monotonic loading conditions.

The Mohr-Coulomb model requires five material parameters: Young’s modulus (E), Poisson’s ratio (ν), cohesion (c), internal friction angle (φ), and dilatancy angle (ψ). Elastic parameters govern the small-strain response, while strength parameters define the failure envelope. Dilatancy accounts for volumetric changes during shearing, which is particularly relevant for dense sands and sand-fines mixtures.

Material parameters were selected to reflect variations in relative compaction (RC = 92% and 100%). The inclusion of fines was modeled by adjusting strength and stiffness parameters to represent improved interparticle bonding and frictional resistance. Although more advanced constitutive models are available, the Mohr-Coulomb model was considered sufficient for the objectives of this study, which focus on ultimate pullout capacity, load-displacement behavior, and failure mechanism geometry.

More advanced sand models are available in PLAXIS, including SANISAND-based formulations and the Hardening Soil (HS) and HS small models, which capture stress-dependent stiffness and small-strain nonlinearity, and critical-state-based models that can better represent dilatancy and post-peak response [33,36]. In this work, the primary objectives are (i) ultimate pullout capacity and (ii) the macro-scale geometry of the rupture mechanism under monotonic loading across a wide parametric space (71 analyses). The Mohr-Coulomb model provides a robust and computationally efficient framework for such capacity-dominated problems, while maintaining transparent control of shear strength parameters (c, φ, ψ). The model’s limitations, particularly regarding stress-dependent stiffness, cyclic loading, and layered soils, are stated explicitly in the Limitations paragraph of Section 5.

Material parameters were selected to reflect variations in relative compaction, RC = 92% and 100%. The inclusion of fines was modeled by adjusting strength and stiffness parameters to represent improved interparticle bonding and frictional resistance. Values were selected based on experimental results and the Egyptian Code for Soil Mechanics, Foundation Design and Execution [30,37]. The soil parameters adopted in the numerical analyses are summarized in Table 2.

2.4 Structural Modeling of the Footing

The footing was simulated using plate elements in PLAXIS 3D. Plate elements are suitable for modeling thin structural components with bending stiffness while maintaining computational efficiency. The footing was modeled as a regular three-dimensional plate element with linear elastic behavior, which is appropriate given the significantly higher stiffness of steel relative to the surrounding soil.

The plate elements were assigned stiffness properties corresponding to the footing geometry and material characteristics. Embedded interface elements were automatically generated along the soil-footing contact surface to simulate realistic soil-structure interaction. Fig. 2 shows the numerical representation of the footing under uplift-only and combined uplift-lateral loading conditions.

Figure 2: (a,b): Footing in 3D PLAXIS.

Footing Geometry and Embedment Definition

Three footing sizes were analyzed: 3.5 cm × 7 cm, 5 cm × 10 cm, and 7.5 cm × 15 cm. The footing was modeled as a steel plate with thickness t = 1.0 cm and linear elastic properties, Es = 210 GPa, νs = 0.30. The embedment depth D was defined as the distance from the ground surface to the top face of the footing. For each footing size, three embedment ratios were considered: D/B = 1.0, 1.5, and 2.5, such that D = (D/B)·B. The analyzed dimensions are at model scale; the results are therefore interpreted primarily through non-dimensional parameters, D/B and normalized displacement w/B, which supports transferability of the observed trends across geometric scales, provided that similar relative stress levels and soil states are maintained. Direct extrapolation to prototype scale requires consideration of stress-level dependence of stiffness and strength, which is not captured by the Mohr-Coulomb model.

2.5 Interface Elements and Soil-Structure Interaction

Soil-structure interaction was modeled using interface elements available in PLAXIS 3D. Interface elements represent a thin transition zone between soil and structural components and allow realistic simulation of relative slip and separation at the contact surface.

Each interface consists of two elastic-perfectly plastic springs: one governing normal (gap) displacement and the other governing shear (slip) displacement. This formulation enables simulation of both separation and sliding between the footing and soil, depending on the applied load level and soil strength properties.

The inclusion of interface elements significantly improves model realism by allowing differential movement between the footing and surrounding soil, which plays a critical role in the mobilization of pullout resistance and development of failure mechanisms. Fig. 3 illustrates the plate footing with the associated interface elements.

Figure 3: Footing with interface as plate structure in 3D PLAXIS.

Interface parameterization: The soil-footing interface was modeled using PLAXIS interface elements with a strength reduction factor R = 0.80, so that the interface shear strength is reduced relative to the adjacent soil. Gap opening was allowed to represent detachment under uplift when tensile normal stress develops. The normal and shear interface stiffnesses were selected following common PLAXIS practice, in which the interface stiffness is chosen sufficiently large to avoid artificial compliance while remaining numerically stable. The interface stiffnesses were defined as kn = α × Eref/tv and ks = α × Gref/tv, where Eref and Gref are reference soil moduli, tv is the virtual interface thickness (taken equal to the local minimum element size), and α is a scaling factor. In the present analyses, tv = 0.008 m and α = 10, giving representative values kn = 3.0 × 107 kN/m3 and ks = 1.15 × 107 kN/m3 for Eref = 30 MPa and ν = 0.30. A sensitivity check showed that varying α between 5 and 20 changed Tu by less than 1.5% for the representative case, indicating limited dependence within this range.

2.6 LOADING Protocol and Load Combinations

Two loading scenarios were simulated: (i) pure uplift, applied as a monotonic vertical tensile load on the footing, and (ii) combined uplift-lateral loading. For combined loading, the vertical and horizontal loads were applied simultaneously at a constant ratio of T:H = 6:1, the vertical tensile load is six times the horizontal shear load. This ratio was selected to represent the expected loading condition for an electric tower design, where the foundation is subjected to both vertical uplift and horizontal wind-induced shear. Both loads were increased monotonically at this fixed ratio until failure was reached. This approach provides a consistent basis for comparing the reduction in vertical pullout capacity under combined loading across all parametric cases, and the reported 23%–25% reduction in Tu is specific to this T:H = 6:1 load ratio, different ratios may yield different reductions, which is identified as a direction for future work.

2.7 Model Verification against Published Evidence

Although the present study is primarily numerical, the adopted modeling approach and the resulting trends are consistent with published experimental observations and established theoretical understanding for uplift foundations and plate anchors. The increase in capacity with embedment ratio and density, as well as the development of wedge-type rupture mechanisms in sand, are consistent with the classic studies of Meyerhof and Adams (1968) [15] and Rowe and Davis (1982) [19], and with more recent investigations that report similar trends in uplift capacity and rupture geometry for plate anchors and shallow foundations under monotonic loading [31,38,39]. Recent numerical and experimental studies on plate anchors and combined loading conditions were also consulted to support the reasonableness of the selected parameters and the interpretation of failure mechanisms [23,24,33,34]. A direct quantitative comparison between the PLAXIS 3D results and laboratory model-test data is presented in Section 4, where experimental measurements over-predict numerical capacity by an average of 8.1% and inclination angles agree within 5%, confirming the reliability of the numerical model.

3  Results and Analysis

3.1 Footings under Pure Tension Force

A total of 71 numerical simulations were performed to investigate the pullout behavior of shallow footings subjected to uplift loading and combined uplift–lateral loading. This section presents the results for footings subjected to tension force only. The analyses considered two soil types (pure sand and sand with 8% fines), two relative compaction levels (RC = 92% and 100%), three footing dimensions, and embedment depth-to-width ratios (D/B = 1.0, 1.5, and 2.5).

3.1.1 Load-Displacement Characteristics

For all simulations, the ultimate pullout resistance Tu was taken as the peak value of the tensile load on the load–displacement curve. For cases exhibiting a plateau without a distinct peak, typically observed at high D/B ratios in dense sand, Tu was defined as the load at a normalized vertical displacement w/B = 0.10, which is consistent with common practice for uplift foundation interpretation and enables objective comparison between parameter sets.

The load-vertical displacement curves, exemplified in Figs. 4–7, exhibit a characteristic three-stage behavior common to uplift foundations. Initially, the displacement is negligible, indicating a quasi-elastic response where the soil-footing system acts monolithically. As the load increases, the displacement accelerates nonlinearly, signifying the progressive mobilization of shear strength and the initiation of the failure surface. The curve reaches the peak at the ultimate pullout resistance, after which the resistance slightly decreases as the failure surface fully develops and the soil mass is displaced.

Figure 4: Relations between Tension force & vl. movement, dim. (10 ∗ 5) cm, different values for (D/B), (RC = 100%), (sand + 8% fines).

Figure 5: Relations between Tension force & vl. movement, dim. (7 ∗ 3.5) cm, (D/B) = 1, (RC = 100%), (pure sand & Sand + 8% fines).

Figure 6: Relations between Tension force & vl. movement, (D/B) = 2.5, (RC = 100%), with different footing dims. (Sand + 8% fines).

Figure 7: Relations between Tension force & vl. movement, dim. (10 ∗ 5) cm, (D/B) = 1, (RC = 100%) & (RC = 92%), (Pure Sand).

Fig. 4 shows that for a constant footing size 10 cm × 5 cm and soil type sand with 8% fines, increasing the embedment ratio (D/B) from 1 to 2.5 results in a substantial upward shift of the curve, indicating a higher ultimate capacity and greater stiffness.

Fig. 5 compares pure sand and sand with 8% fines for a 7 cm × 3.5 cm footing at embedment ratio equal to 1. The presence of fines significantly increases the peak load and reduces the displacement required to reach that peak, reflecting the improved shear strength and cohesive properties of the mixed soil.

Fig. 6 demonstrates that increasing footing dimensions increases the peak uplift capacity while preserving the same three-stage response, indicating that geometric scaling primarily affects the magnitude of the mobilized resisting soil wedge rather than the governing failure process. This observation is consistent with the normalized capacity analysis presented in Section 3.1.5.

Fig. 7 illustrates that increasing the relative compaction (RC) from RC = 92% to RC = 100% in pure sand leads to a higher ultimate pullout load and a steeper initial slope, confirming that denser soil provides greater resistance.

3.1.2 Influence of Embedment Ratio (D/B) on Ultimate Capacity

The ratio of embedment depth (D) to footing width (B) was varied at 1, 1.5, and 2.5 to study its effect on the ultimate pullout force. As shown in Figs. 8–10, which plot the ultimate pullout load against the ratio for different footing dimensions in pure sand, a direct correlation is observed that the ultimate pullout force increases consistently with the ratio. This enhancement is primarily attributed to two factors, the first one is the increased overburden as the depth increases and the weight of the soil mass trapped above the footing, which must be lifted to initiate failure, increases linearly. This gravitational resistance forms a major component of the total pullout capacity. The second factor is the extended shear planes as the greater embedment depth necessitates longer failure surfaces, thereby mobilizing a larger volume of soil and increasing the total frictional resistance developed along the sliding interface.

Figure 8: The variation of the ultimate pullout load with different (D) of footing dimension (7 ∗ 3.5) cm in pure sand with (RC = 100%).

Figure 9: The variation of the ultimate pullout load with different (D) of footing dimension (10 ∗ 5) cm in pure sand with (RC = 100%).

Figure 10: The variation of the ultimate pullout load with different (D) of footing dimension (15 ∗ 7.5) cm in pure sand with (RC = 100%).

3.1.3 Effect of the Embedment Ratio on the Failure Mechanism

The numerical results for the failure mechanism, visualized in Figs. 11–16, reveal a critical transition point in soil behavior.

Figure 11: (a,b): Failure mechanism for (D/B) = 1, dim. (10 ∗ 5) cm, sand, (RC = 100%).

Figure 12: (a,b): Failure mechanism for (D/B) = 1.5, dim. (10 ∗ 5) cm, sand, (RC = 100%).

Figure 13: (a,b): Failure mechanism for (D/B) = 2.5, dim. (10 ∗ 5) cm, sand, (RC = 100%).

Figure 14: (a,b): Failure mechanism for (D/B) = 1, dim. (15 ∗ 7.5) cm, sand, (RC = 100%).

Figure 15: (a,b): Failure mechanism for (D/B) = 1.5, dim. (15 ∗ 7.5) cm, sand, (RC = 100%).

Figure 16: (a,b): Failure mechanism for (D/B) = 2.5, dim. (15 ∗ 7.5) cm, sand, (RC = 100%).

The inclination angle β was measured at the ultimate state by extracting a vertical cross-section through the footing centerline in the direction of loading. The rupture surface was identified using the band of maximum incremental shear strain in the PLAXIS 3D output contours. A straight line was then fitted between the footing edge and the point where the localized shear band intersects the ground surface. For combined loading, two angles were measured: β1 on the loaded (steeper) side and β2 on the opposite (shallower) side, using the same procedure. The resulting β values for representative cases are summarized in Table 3.

For the pure-sand cases analyzed here, the measured inclination angles follow approximately β = 0.75 φ (R2 = 0.92) within the investigated range of φ (35°–38°), consistent with classical wedge interpretations. For sand with 8% fines, β increases and can be approximated by β = 1.10 φ-° (R2 = 0.89), reflecting stronger dilatant response and mobilization of a larger soil wedge. These relationships are empirical correlations derived from the current parametric space and should be recalibrated if site-specific laboratory parameters differ from those in Table 2.

For shallow to intermediate embedment ratios, 1 and 1.5, the shape of the failure mechanism remains largely consistent across all footing dimensions. The failure is characterized by a symmetrical inclined rupture surface extending from the footing edges to the ground level (Figs. 11–15).

A notable exception occurs at the deep embedment ratio 2.5, particularly for the largest footing (7.5 cm × 15 cm). In this special case (see Fig. 16), the failure mechanism transitions towards a deep failure mode. The high confining pressure and large mass of soil above the footing restrict the development of a wide, inclined wedge. Consequently, the failure surface initiates more vertically, with the final inclination angle ranging narrowly between 10 and 16 degrees. This suggests that for deep footings, the failure is governed more by punching shear than by the development of a classic uplift cone.

To ensure that the observed transition at D/B = 2.5 is not a numerical artifact, the representative deep-embedment case was re-analyzed using the fine mesh and an expanded domain. The same near-vertical initiation of the rupture surface and the same order of magnitude of Tu were obtained (ΔTu < 2%), confirming that the behavior reflects increased confinement rather than mesh or boundary dependence.

3.1.4 Influence of Soil Type (Fines Content)

The effect of adding 8% fines to the sandy soil was investigated, holding other parameters constant. As demonstrated in Figs. 17–19, the ultimate pullout load is significantly higher in the sand with 8% fines mixture compared to pure sand across all ratios. This is a direct result of the fines acting as a binder, which increases effective cohesion and enhances the internal friction angle, thereby improving the overall shear strength parameters of the soil. It should be noted that the effect of fines on shear strength is soil-specific and depends on the gradation and plasticity of the fines; the improvement observed here is captured in the adopted Mohr-Coulomb parameterization as an increase in apparent cohesion and friction angle (Table 2), and the trend should be verified for different fines contents and types before generalization. While the fundamental shape of the failure rupture remains inclined, the presence of fines significantly increases the angle of inclination β of the failure plane. In pure sand, β is typically a function of φ, following approximately β = 0.75 φ as reported in Section 3.1.3. However, in the sand + 8% fines mixture, as shown in Figs. 20–25, the angle is much larger, typically ranging between 37 and 47- β =1.10 φ-° (Table 3). This indicates that the higher strength and stiffness of the cohesive-frictional soil mobilize a wider, more effective soil wedge, further contributing to the increased pullout resistance.

Figure 17: The ultimate pullout load variation in different soil types (Sand & Sand + 8% fines), dim. (3.5 ∗ 7) cm in (D/B) = 1, (RC = 100%).

Figure 18: The ultimate pullout load variation in different soil types (Sand & Sand + 8% fines), dim. (3.5 ∗ 7) cm in (D/B) = 1.5, (RC = 100%).

Figure 19: The ultimate pullout load variation in different soil types (Sand & Sand + 8% fines), dim. (3.5 ∗ 7) cm in (D/B) = 2.5, (RC = 100%).

Figure 20: (a–c): Failure mechanism in different soil types (Sand & Sand + 8% fines), (D/B) = 1, dim. (10 ∗ 5) cm, (RC = 100%).

Figure 21: (a–c): Failure mechanism in different soil types (Sand & Sand + 8% fines), (D/B) = 1.5, dim. (10 ∗ 5) cm, (RC = 100%).

Figure 22: (a–c): Failure mechanism in different soil types (Sand & Sand + 8% fines), (D/B) = 2.5, dim. (10 ∗ 5) cm, (RC = 100%).

Figure 23: (a–c): Failure mechanism in different soil types (Sand & Sand + 8% fines), (D/B) = 1, dim. (15 ∗ 7.5) cm, (RC = 100%).

Figure 24: (a–c): Failure mechanism in different soil types (Sand & Sand + 8% fines), (D/B) = 1.5, dim. (15 ∗ 7.5) cm, (RC = 100%).

Figure 25: (a–c): Failure mechanism in different soil types (Sand & Sand + 8% fines), (D/B) = 2.5, dim. (15 ∗ 7.5) cm, (RC = 100%).

3.1.5 Effect of Footing Dimensions and Relative Compaction

For the footing dimensions, Figs. 26–28 confirm that increasing the footing dimensions (from 3.5 cm × 7 cm to 7.5 cm × 15 cm) results in a proportional increase in the ultimate pullout load. This is due to the increased self-weight of the footing and the larger surface area available to mobilize the soil mass. Crucially, Fig. 29 illustrates that changing the footing dimensions does not alter the geometric shape or the angle of inclination of the failure mechanism, provided all other parameters remain constant.

Figure 26: The ultimate pullout load variation due to different footing dims., (D/B) = 1, (RC = 100%), Pure sand.

Figure 27: The ultimate pullout load variation due to different footing dims., (D/B) = 1.5, (RC = 100%), Pure sand.

Figure 28: The ultimate pullout load variation due to different footing dims., (D/B) = 2.5, (RC = 100%), Pure sand.

Figure 29: (a–d): Failure mechanism due to the variation of footing dims., (D/B) = 1.5, (RC = 100%) in Pure sand.

To assess whether the observed increase in Tu with footing size reflects simple geometric scaling or a non-linear size effect, the ultimate capacity was also normalized by the footing plan area (Tu/B·L). The normalized capacity remained approximately constant across the three footing sizes for the same D/B ratio and soil state, confirming that the increase in Tu is primarily a geometric scaling effect within the investigated range, with no significant stress-level-dependent size effect detectable under the Mohr-Coulomb model.

The effect of relative compaction is shown in Figs. 30–32. Increasing the RC from 92% to 100% substantially increases the pullout force. This is because higher compaction leads to increased soil density and improved shear strength parameters (Table 2). Furthermore, Figs. 33 and 34 show that while the shape of the failure mechanism is unaffected by the change in RC, the value of the angle of inclination is greater at RC = 100%. This larger angle reflects the greater strength of the denser soil, which is capable of mobilizing a wider soil mass to resist the uplift force.

Figure 30: The ultimate pullout load variation due to the variation of relative density, (D/B) = 1, dim. (7 ∗ 3.5) cm, (Pure Sand).

Figure 31: The ultimate pullout load variation due to the variation of relative density, (D/B) = 1.5, dim. (7 ∗ 3.5) cm, (Pure Sand).

Figure 32: The ultimate pullout load variation due to the variation of relative density, (D/B) = 2.5, dim. (7 ∗ 3.5) cm, (Pure Sand).

Figure 33: (a–c): Failure mechanism variation due to the variation of relative density, (D/B) = 1.5, dim. (10 ∗ 5) cm, (Pure Sand).

Figure 34: (a–c): Failure mechanism variation due to the variation of relative density, (D/B) = 1, dim. (15 ∗ 7.5) cm, (Pure Sand).

3.2 Footings under Combined Loading

The second phase of the numerical study investigated the behavior of footings subjected to combined vertical tension and lateral shear forces, a loading condition highly representative of wind-induced forces on lattice towers.

3.2.1 Effect of Embedment Ratio Failure Mechanism under Combined Loading

The introduction of a lateral force fundamentally alters the symmetry of the failure mechanism. The analysis of the failure surfaces, as shown in Figs. 35–40, reveals the following:

Figure 35: (a,b): Failure mechanism for footing subjected to tension & hz. Loads, (D/B) = 1, dim. (10 ∗ 5) cm (pure sand), (RC = 100%).

Figure 36: (a,b): Failure mechanism for footing subjected to tension & hz. Loads, (D/B) = 1.5, dim. (10 ∗ 5) cm (pure sand), (RC = 100%).

Figure 37: (a,b): Failure mechanism for footing subjected to tension & hz. Loads, (D/B) = 2.5, dim. (10 ∗ 5) cm (pure sand), (RC = 100%).

Figure 38: (a,b): Failure mechanism for footing subjected to tension & hz. Loads, (D/B) = 1, dim. (10 ∗ 5) cm (sand + 8%fines), (RC = 100%).

Figure 39: (a,b): Failure mechanism for footing subjected to tension & hz. Loads, (D/B) = 1.5, dim. (10 ∗ 5) cm (sand + 8%fines), (RC = 100%).

Figure 40: (a,b): Failure mechanism for footing subjected to tension & hz. Loads, (D/B) = 2.5, dim. (10 ∗ 5) cm (sand + 8%fines), (RC = 100%).

For D/B ratios up to 1.5, the change in embedment depth has a minimal effect on the shape of the failure mechanism (Figs. 35 and 36). However, at D/B equal to 2.5, the slope angle is observed to decrease, similar to the pure tension case, suggesting a more constrained failure mode (Fig. 37).

In the cohesive-frictional soil, the failure mechanism remains consistently unsymmetrical across all ratios 1, 1.5, and 2.5, with no significant change in the shape or angle of inclination observed due to depth variation alone (Figs. 38–40).

3.2.2 Influence of Soil Type on Unsymmetrical Failure

The failure mechanism under combined loading is characterized by a distinct unsymmetrical inclination, as shown in Figs. 41 and 42. The lateral force causes the failure wedge to be steeper on the side resisting the horizontal load (β1) and shallower on the opposite side (β2).

Figure 41: (a–c): Failure mechanism in different types of soil, dim. of footing (7 ∗ 3.5) cm, (D/B) = 2.5, (RC = 100%), footing subjected to tension & hz. loads.

Figure 42: (a–c): Failure mechanism in different types of soil, dim. of footing (15 ∗ 7.5) cm, (D/B) = 1.5, (RC = 100%), footing subjected to tension & hz. loads.

In the pure sand case and for D/B = 1.0 and 1.5, the steeper angle (β1) ranges between 27 and 50 degrees, while the shallower angle (β2) is consistent with the pure tension case, ranging from 0.67 to 0.80 φ. At D/B = 2.5 for the largest footing, the angles become more constrained, with β1 between 20 and 30 degrees and β2 between 13 and 17 degrees.

For the sand with 8% fines, the higher strength of this soil leads to significantly larger angles. The steeper angle (β1) ranges from 40 to 70 degrees, and the shallower angle (β2) ranges from 30 to 50 degrees. The increase in both angles compared to pure sand is a direct consequence of the higher shear strength and cohesion, which allows the soil to mobilize a larger, more effective failure wedge to resist the combined forces.

3.2.3 Effect of Footing Dimensions and Relative Compaction

Similar to the pure tension case, the geometric parameters primarily affect the magnitude of the ultimate load, not the shape of the failure mechanism.

Fig. 43 confirms that varying the footing dimensions, 3.5 cm × 7 cm, 5 cm × 10 cm, and 7.5 cm × 15 cm, does not alter the fundamental unsymmetrical shape or the values of the inclination angles of the failure rupture.

Figure 43: (a–d): Failure mechanism for footing with different dim., in (sand + 8% fines), (D/B) = 1, (RC = 100%).

For the relative compaction effect, Fig. 44 demonstrates that while the shape of the failure mechanism remains unchanged with varying relative compaction, RC = 100% vs. RC = 92%, the value of the inclination angle is higher for RC = 100%. This is consistent with the principle that denser soil possesses greater strength, enabling it to resist the combined forces by mobilizing a wider soil mass.

Figure 44: (a–c): Failure mechanism for type of soil (Pure Sand) in (D/B) = 1, dim. (10 ∗ 5) cm with (Relative compaction = 100% & 92%).

3.3 Comparative Analysis and Synthesis

3.3.1 Ultimate Pullout Load Comparison

A critical finding of this study is the quantitative comparison between the ultimate pullout load under pure tension and under combined tension and lateral loading. As illustrated in Figs. 45 and 46, the ultimate pullout load for footings subjected to pure tension is consistently greater than for footings subjected to combined loading. Quantitative Reduction: The numerical results show that the introduction of a lateral force causes an average reduction in the vertical ultimate pullout capacity of approximately 23% to 25%. This reduction is a direct consequence of the lateral force consuming a portion of the soil’s shear strength capacity, which would otherwise be available to resist the vertical uplift. The combined stress state accelerates the development of the failure surface, leading to a lower peak vertical resistance. This finding is highly significant for the design of transmission tower foundations, as it confirms that the most realistic loading scenario (combined) is the most critical for stability.

Figure 45: Comparison between ultimate pullout load for footing subjected to tension load and footing subjected to tension & horizontal loads in pure sand, footing Dim. = 7 * 3.5 cm, RC% = 100% and (D/B) = 1.

Figure 46: Comparison between ultimate pullout load for footing subjected to tension load and footing subjected to tension & horizontal loads in (sand +8% Fines), footing Dim. = 7 * 3.5 cm, RC% = 100%, (D\B) = 1.

The numerical results show that the introduction of a lateral force causes an average reduction in the vertical ultimate pullout capacity of approximately 23% to 25% for the investigated T:H = 6:1 load ratio. This reduction is a direct consequence of the lateral force consuming a portion of the soil’s shear strength capacity, which would otherwise be available to resist the vertical uplift. The combined stress state accelerates the development of the failure surface, leading to a lower peak vertical resistance.

To compare responses at comparable mobilization levels, the footing response was also evaluated at a fixed normalized vertical load T/Tu = 0.80 for representative cases. At this load level, the combined-loading simulations exhibit (i) larger lateral displacements on the loaded side and (ii) earlier formation of a localized shear band connecting the footing edge to the ground surface, which reduces the remaining margin to the vertical peak. This explains the observed reduction in Tu under combined loading. The representative results are summarized in Table 4.

This finding is highly significant for the design of transmission tower foundations, as it confirms that the most realistic loading scenario (combined) is the most critical for stability.

3.3.2 Failure Mechanism Comparison

The difference in the failure mechanism between the two loading scenarios is visually striking, as shown in Figs. 47 and 48, for pure tension case, The failure rupture exhibits a symmetrical inclination (Figs. 47a and 48a). This symmetrical wedge mobilizes the maximum possible volume of soil, leading to the highest vertical resistance.

Figure 47: (a–c): Failure mechanism for footing subjected to tension load and footing subjected to tension & hz. loads, soil type (Pure Sand), (D\B) = 1, dim. (7 * 3.5 cm), (RC = 100%).

Figure 48: (a–c): Failure mechanism for footing subjected to tension load and footing subjected to tension & hz. loads, soil type (Sand + 8%Fines), (D\B) = 1, dim. (7 * 3.5) cm, (RC = 100%).

In the case of combined loading, the failure rupture exhibits a distinct unsymmetrical inclination (Figs. 47b and 48b). The lateral force shifts the center of rotation and concentrates the shear strain on one side, resulting in the unsymmetrical shape.

4  Experimental Validation of the Numerical Model

4.1 Comparison of Ultimate Pullout Load

A comparison was made between laboratory model-test results and the PLAXIS 3D numerical results for a 5 cm × 10 cm footing in sand with 8% fines subjected to pure vertical uplift loading, at an embedment ratio D/B = 1.5 and a relative compaction of RC = 100%. This configuration was selected as the representative validation case because it corresponds directly to the experimental data available from our previous studies and produced a close agreement between the measured and computed ultimate pullout loads [37]. The soil properties used in the experimental test and the comparison of ultimate pullout load values are presented in Table 5.

As shown in Table 5, the experimental ultimate pullout load is 20.0 kg, while the PLAXIS 3D simulation yields 18.5 kg, corresponding to a difference of 8.1%. The numerical model slightly under-predicts the measured capacity, which is consistent with the behavior expected from an elastic–perfectly plastic Mohr-Coulomb constitutive model: this model does not capture the progressive strain-hardening response or the dilation-induced stiffness increase that can develop in dense, well-compacted sand-fines mixtures at model scale, both of which tend to elevate the measured peak load in laboratory specimens. Additionally, minor differences in soil preparation between the numerical model and the physical test, including the uniformity of compaction, the precise fines distribution, and the boundary conditions of the test box, contribute to the observed discrepancy. The 8.1% difference is well within the range of accuracy typically accepted for model-scale geotechnical validation studies and is considered satisfactory for the purposes of verifying the numerical model. The results confirm that the PLAXIS 3D model, with the adopted Mohr-Coulomb parameters (c = 5.0 kPa, φ = 40°, γ = 1.92 t/m3), reliably captures the load-carrying behavior of shallow footings under pure vertical uplift loading in cohesive-frictional soil at the investigated embedment ratio and compaction level.

4.2 Comparison of Failure Mechanism

Fig. 49 presents a comparison of the failure rupture surface and inclination angle between the experimental test result and the PLAXIS 3D numerical result for the selected validation case: a 5 × 10 cm footing in sand with 8% fines under pure vertical uplift loading at D/B = 1.5 and RC = 100%. The results show a high degree of correlation between the experimental and numerical failure mechanisms: the difference in the inclination angle β between the experimental and numerical results was less than 5%, confirming that the model accurately captures not only the magnitude of the ultimate pullout capacity but also the geometry of the failure surface.

Figure 49: Failure mechanism from experimental and numerical model vertical force only.

5  Conclusions

This comprehensive numerical investigation, utilizing the three-dimensional finite element program PLAXIS 3D, has provided critical insights into the ultimate pullout capacity and failure mechanisms of shallow footings subjected to tension and combined tension-lateral loads. The key findings are summarized as follows:

1- The general characteristics of load-displacement curves are similar for both pure tension and combined loading, following a predictable path of elastic mobilization followed by plastic failure.

2- The ultimate pullout force increases with embedment ratio (D/B) within the investigated range because deeper embedment mobilizes a larger resisting soil wedge and greater overburden weight. For D/B up to 1.5, the rupture geometry remains broadly similar; the most notable change in rupture inclination was observed for the largest footing at D/B = 2.5, where increased confinement promotes a near-vertical initiation of the failure zone.

3- Adding 8% fines to sand increases the computed pullout resistance for the investigated parameter set by increasing the mobilized shear resistance along the rupture surface. This trend is captured in the adopted Mohr-Coulomb parameterization as an increase in φ and an apparent cohesion; however, the effect of fines is soil-specific and should be verified using different fines contents and types before generalization.

4- Under combined uplift-lateral loading at T:H = 6:1, the ultimate uplift capacity reduces by approximately 23%–25% relative to pure uplift because part of the available shear strength is mobilized to resist lateral action, accelerating strain localization and reducing the vertical peak resistance. This reduction factor applies to the investigated load ratio; other ratios may yield different reductions.

5- Combined loading produces an unsymmetrical rupture mechanism with different inclination angles on the loaded and opposite sides (β1 and β2). For design, a conservative approach is to consider an envelope that accounts for both sides; where wind direction is uncertain, adopting a symmetric envelope based on the larger mobilized wedge provides a practical conservative estimate. For the investigated sand states, representative inclination ranges are β = 25°–32° in pure sand and β = 38°–45° in sand with 8% fines, noting that these ranges depend on φ, density, and interface conditions.

6- Increasing relative compaction from 92% to 100% increases the peak resistance and steepens the rupture surface, mobilizing a larger resisting soil mass. Strict field compaction control is therefore important for achieving the intended uplift capacity.

7- For transmission tower foundations in clean sand, achieving high relative compaction (near 100%) is an effective measure to improve uplift resistance. If a well-graded backfill or a modest fines content is used to enhance capacity, the associated increase in rupture inclination angle should be considered when defining the mobilized soil wedge for uplift design. Under combined loading at T:H = 6:1, applying a reduction factor of 0.75–0.77 to the pure-uplift capacity provides a conservative preliminary estimate. Foundation design verification should account for the asymmetric failure mode that develops under combined loading.

8- The study is based on monotonic loading in homogeneous soil deposits and uses the Mohr-Coulomb elastic-perfectly plastic model. Therefore, the results should not be directly extrapolated to layered soils, anisotropic conditions, or cyclic loading without additional calibration. The analyzed footing dimensions are at model scale; direct extrapolation to prototype scale requires consideration of stress-level dependence of stiffness and strength, which is not captured by the Mohr-Coulomb model. Non-dimensional interpretation (D/B, w/B) supports qualitative transferability of the observed trends. Future work should include validation against full-scale or field data and should examine the influence of the horizontal-to-vertical load ratio (H/T) and cyclic wind loading on both capacity and failure mechanism.

Acknowledgement: Not applicable.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: The authors confirm contribution to the paper as follows: Conceptualization, Ahmed Ibrahim Hassanin Mohamed and Nourhan M. Amin; Methodology, Ahmed Ibrahim Hassanin Mohamed, Nourhan M. Amin, and Heba Elsaid Matter; Software, Nourhan M. Amin, Heba Elsaid Matter, and Ibrahim F. Eldemary; Validation, Nourhan M. Amin, Heba Elsaid Matter, and Ahmed F. Oan; Formal analysis, Ahmed F. Oan, Ahmed Ibrahim Hassanin Mohamed, Nourhan M. Amin, and Heba Elsaid Matter; Investigation, Heba Elsaid Matter and Ibrahim F. Eldemary; Resources, Nourhan M. Amin; Data curation, Ahmed Ibrahim Hassanin Mohamed; Writing—original draft preparation, Ahmed Ibrahim Hassanin Mohamed, Nourhan M. Amin, Heba Elsaid Matter, Ibrahim F. Eldemary, and Ahmed F. Oan; Writing—review and editing, Ahmed Ibrahim Hassanin Mohamed and Nourhan M. Amin; Visualization, Ahmed F. Oan; Supervision, Ahmed Ibrahim Hassanin Mohamed and Heba Elsaid Matter; Project administration, Ahmed F. Oan and Ibrahim F. Eldemary. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: The data that support the findings of this study are available from the Corresponding Author, Ahmed Ibrahim Hassanin Mohamed, upon reasonable request.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest.

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