Numerical Study of the Nonlinear Soil–Pile–Structure Interaction Effects on the Lateral Response of Marine Jetties

Abstract

This study presents three-dimensional finite element analyses of two marine structures subjected to lateral loading to approximate environmental forces (e.g., wind, waves, currents, earthquakes). The first structure is a marine jetty supported by twenty-four piles, representative of an existing structure in Cyprus, while the second is a simplified four-pile marine structure. Soil–pile interaction is modelled using nonlinear p-y, τ-z, and q-z springs that are distributed along the piles, while steel plasticity is also considered. This study examines the relationship between failure modes, deformation modes, and plastic hinge locations with soil behaviour and soil reaction forces. It also aims at investigating the behaviour of the above structures in lateral loading and quantifying the consequences of unrealistic assumptions such as soil and steel linearity or tension-resistant q-z springs. The results indicate that such assumptions can lead to the wrong prediction of failure modes, plastic hinges, and critical elements while emphasising the crucial role of soil nonlinearity and axial pile–soil behaviour on the structural response. It is demonstrated that the dominant nonlinear sources relevant to this study, whether soil nonlinearity, plastic hinge formation, or a combination of the two, are primarily influenced by the axial capacity of soil–pile foundation systems, particularly their tensile component.

1. Introduction

The response of bottom-fixed marine structures founded on pile foundations is highly nonlinear under the impact of dynamic loading from waves, wind, and earthquakes. Nonlinearities include structural material nonlinearity, soil nonlinearity, and P-Δ effects, all of which are interconnected through complex nonlinear soil–pile–structure interaction (SPSI) phenomena. Due to their complexity, these phenomena are usually accounted for through finite element (FE) analyses, which simulate SPSI problems in the time domain, rigorously considering the continuous interaction between the structure, piles, and soil in a single step [1,2]. In FE analyses, soil can be rigorously modelled using solid elements [3,4,5,6,7], pile–soil systems can be replaced by springs [8], or soil reactions on piles can be represented by a series of springs along the piles [9,10,11,12]. The latter springs are widely employed in offshore foundation systems due to their simplicity. Lateral p-y springs are commonly used to simulate the soil–pile interaction in the lateral deformation of the pile, with a widespread application in monopile foundations of wind turbines [13,14]. Springs acting in the vertical direction, namely τ-z and q-z springs, have been used to simulate the pile shaft and the pile tip behaviours, respectively. These springs are particularly important when the axial resisting mechanism is prominent [15], making them widely applicable in jacket foundation systems [16,17,18,19].

Previous studies [20,21] have demonstrated that the combination of p-y, τ-z, and q-z springs is essential for more complex marine structures, such as jetties. Koronides et al. [20] showed that in such structures, the response of the SPSI system can be significantly influenced by soil nonlinearity, arising from both lateral and axial pile responses. The linear behaviour of soil is confined to very small strain levels [22,23,24,25,26], which are likely exceeded under adverse dynamic loading in marine environments. Soil nonlinearity becomes even more pronounced in marine structures due to the influence of P-Δ effects, stemming from the inherent flexibility of these structures. When significant, P-Δ effects can negatively impact the structural response, increasing displacements, forces, and bending moments on the foundation piles [3,27,28].

Piles’ forces and bending moments are also influenced by the type and behaviour of surrounding soil. Zeng et al. [14] demonstrated that accurately simulating lateral soil resistance is crucial for correctly estimating both the magnitude and location of the maximum bending moments on monopiles. Chigullapally et al. [29] similarly found that increasing the ultimate lateral resistance in shallow soil layers can lead to an upward shift in the maximum bending moment experienced by a bridge pier. As demonstrated by Gerolymos et al. [30] through numerical studies on pile bridge foundations, low-stiffness soils, such as soft clay or loose sand, lead to larger pile displacements, activating the response of deeper pile segments. This results in substantial bending moments reaching deeper sections of the pile compared to stiff clay and dense sand soil conditions. Also, the maximum bending moment and the potential formation of a plastic hinge will occur at greater depths in low-stiffness soils. However, such soils can reduce the maximum bending moment and the amplitude of curvature. The smaller pile curvatures that are anticipated in soft soil conditions can explain the longer plastic hinges compared to those in stiffer soils, especially for cohesive soil, as observed by Goel [31] by examining the response of monopiles of marine structures.

Following the above and given that soil stiffness can vary with excitation amplitudes due to its nonlinear behaviour, the responses of SPSI systems differ under varying excitation levels. Various studies observed that the maximum bending moment on piles shifts to greater depths as the lateral load amplitude increases [3,6,31,32,33]. Wen et al. [3] claimed that this is due to the more pronounced nonlinear behaviour of shallower soil layers. Heidari and El Naggar [32] noted that this shift is more pronounced in loose sand than in dense sand, while it was similar for soft and stiff clay. The same study and a study by Chiou et al. [34] observed that this downward shift continues until the pile reaches its first yield, after which the maximum bending moment remains fixed at the same location despite further increases in lateral force. Memarpour et al. [33] concluded that the shift in the maximum bending moment with an increasing lateral load can be further amplified by the development of gaps in cohesive soils. According to the latter study, soil nonlinearity and gap formation reduce the lateral stiffness provided by soil resistance, leading to increased displacements and a downward shift in the critical section of the piles. Additionally, it is expected that the plastic hinge length increases with the magnitude of external force. For varied excitation levels, Goel [31] demonstrated that this effect is more pronounced in piles embedded in loose sand and soft clay, which exhibit greater nonlinear behaviour compared to stiffer soils.

In structures supported by multiple piles, the pile bending moments, axial forces, and shear forces differ from pile to pile, depending on their position with respect to the loading direction and the number of piles. Assuming lateral loading, part of the base moment is taken by the incremental compressive forces of leading piles and extension forces of the trailing piles [3,5,20,21]. The presence of substantial axial and shear forces can explicitly impact the moment capacity of piles due to the well-established behaviour of structural material under combined loading, V-M-H [5]. Also, incremental compressive strengths can increase the bending moments acting on trailing piles due to P-Δ effects. More importantly, the different axial forces carried by the piles in a foundation system affect the soil confining stress, which significantly affects the soil behaviour. An increase in soil confining stress around the leading (compressed) pile enhances the soil stiffness, which, as discussed earlier, can lead to higher bending moments acting on the pile. The maximum bending moment and the formation of plastic hinges are expected to occur at shallower depths for the leading piles compared to the trailing ones, as observed in the numerical studies of Wen et al. [3]. The same study also observed larger lateral reaction forces by the soil acting on the leading pile, which is a consequence of the increased soil stiffness.

Given that the axial pile loading impacts the piles and soil behaviour, it can be crucial for the failure mode of an SPSI system. As observed by Psychari and Anastasopoulos [5] and Koronides et al. [20], a failure mode can occur when the bearing axial capacity of both the compressive and tensile pile-soil systems is reached. Once this occurs, the axial forces of the piles closer to the centre of rotation adapt to sustain equilibrium and allow for further load, before they also fail. The role of pile–soil systems’ axial capacity on the deformation mode was demonstrated by Hamidia et al. [8], assuming a single-degree-of-freedom structure founded on springs, each one representing a pile–soil system. By comparing numerical results from numerical analyses with varied tensile strength for the soil–pile systems, the authors observed a rocking-dominated deformation mode when the tensile strength was low. In contrast, systems with higher tensile strength exhibited a swaying mode, promoting the damage to the superstructure rather than the foundation system. Similarly, other studies [3,5] have stated the relation of deformation modes with the axial capacity of piles and the magnitude of external force. Specifically, under weak loads, the swaying mode predominates, while the rocking mode becomes dominant when the load is large enough to mobilize the axial bearing capacity of the pile–soil systems in the foundation. Finally, larger external loads lead to mobilisation of the motion of deeper soil, allowing penetration of pile lateral and axial displacements [14], affecting the overall deformation of the SPSI system.

Most of the referenced studies examined the correlation between lateral and axial soil behaviour of piles on the SPSI response separately. However, studies exploring the simultaneous impact of both lateral and axial soil reactions on plastic hinge formation, critical structural elements, deformation modes, and the nonlinear behaviour of soil remain scarce. Furthermore, although it is well known that SPSI effects can significantly influence the response of structures, there are very few studies focused on bottom-fixed marine jetties. Such structures, which are often critical infrastructure for a country’s energy sector, are expected to exhibit strong SPSI effects due to their flexibility, the anticipated highly nonlinear behaviour of marine soils, large external loads, and significant anticipated displacements. The scarcity of research in this area is likely due to the limited availability of data, as these structures are typically owned by oil and gas companies.

The present study conducts three-dimensional (3D) finite element analyses of two marine structures subjected to monotonic lateral pushover loading, intending to approximate environmental forces, such as wind, waves, currents, and earthquake. The first structure is a marine jetty, while the second one is a simplified structure founded on four piles. The marine jetty is a representative of an existing structure located in Vasiliko, Cyprus, and founded on twenty-four piles driven in dense sand and Nicosia Marl. Geological and geotechnical data for the site were supplied by a marine infrastructure owner operating in the Vasiliko area. Soil–pile interaction is modelled using nonlinear p-y, τ-z, and q-z springs distributed along the piles, with steel plasticity and P-Δ effects also accounted for.

The objective of this research is to assess the behaviour of the two examined structures under lateral loading and quantify the impacts of unrealistic assumptions, such as linear soil and steel behaviour or tension-resistant q-z springs. The important relationship between soil nonlinearity, plastic hinge formation, failure modes, deformation modes, and soil reactions is emphasised and compared for the two examined structures. This study highlights the axial capacity of pile–soil systems as a key factor influencing the overall response and nonlinear behaviour of the examined marine structures. The formation of plastic hinges is shown to be strongly influenced by the axial capacity of the soil–pile foundation system, particularly its tensile component. When the tensile capacity is low, which in the present study is implicitly accounted for through considering tensionless q-z springs, the nonlinear behaviour of the system can be controlled by soil nonlinearity, with minimal impact on structural nonlinearity. However, for the case of high-foundation axial capacity, such as when tension-resistant q-z springs are assumed, plastic hinges are promoted to form and play a significant role in the nonlinear behaviour of the SPSI system. Also, this study shows that tensionless q-z springs promote foundation rotation and larger displacements, as opposed to the swaying deformation and smaller displacements observed in systems with tension-resistant or linear springs. These results underscore the importance of incorporating realistic soil behaviour and boundary conditions at pile tips in numerical analyses to achieve reliable predictions of deformation modes and potential damage locations in the examined marine structures.

2. Structural Characteristics

The studied structure is representative of an existing T-junction of a marine jetty, shown in Figure 1, located 1.2 km offshore of the Vasiliko area in Cyprus, as extensively described by Koronides et al. [20]. The safety of such marine structures is vital, particularly as they are often critical to a country’s operations, as is the case for Cyprus’s energy sector. The most important component of the marine jetty is a T-junction that connects two transverse components (trestles) along its short axis and one longitudinal trestle along its longer axis (Figure 1a). The T-junction, which is a bottom-fixed marine platform, is the main focus of the present study.

The examined T-junction structure features a reinforced-concrete (C35/45) deck that has a thickness of 1.30 m and dimensions of 32 m × 20 m. It is supported by a foundation system, consisting of 24 steel piles, which are inclined at a 1:3 rake. Their inclination directions and positions beneath the deck are shown in Figure 2. These piles feature a hollow circular cross-section, measuring 1067 mm in diameter and 27 mm in thickness. The assumed steel material, API L5 grade X65, is characterised by a yield stress (f_y) of 450 MPa and an ultimate (f_u) stress of 550 MPa.

The foundation piles have three different lengths, 42.7 m, 45.4 m, and 54.3 m, and were driven into the seabed using the impact method until their tips reached depths of 15.35 m, 17.85 m, and 26.35 m, respectively. As shown in Figure 2, the shorter piles are positioned near the perimeter of the deck, whereas the longest piles are positioned closer to its centre of gravity. The deck bottom is elevated 7.5 m above mean sea level, with the piles connected to the deck beneath this elevation using concrete plugs. These plugs have a length of 2.4 m and behave as a steel–concrete composite component that is considerably stiffer than the remaining bare steel pile segments.

Due to the expected interaction between the foundation soil, piles, and the T-junction, the SPSI system is investigated as a whole. The system is anticipated to encounter substantial lateral and vertical forces due to the T-junction’s structural role in connecting three trestle structures, operational demands, and the impact of wave, wind, and seismic loading. As a result of these loading conditions, a highly nonlinear response is anticipated from the system.

3. Site Conditions

The marine structure under examination is presumed to be located 1.2 km offshore in the Vasiliko area of Cyprus, a region critical for the country’s energy infrastructure. The seabed depth in this area was found to be 17.7 m. Geological and geotechnical data for the site were supplied by VTTV, the owner of the marine jetty [35].

As detailed in Koronides et al. [20], the in situ geotechnical investigation comprised borehole drilling and standard penetration tests (SPTs). Drilling was carried out through the rotary percussion method using a jack-up platform. During this process, SPT tests were conducted, and samples were collected using a split spoon barrel sampler. Additionally, using either a sampler driven by static pressure from drill rig or SPT hammer, undisturbed samples were retrieved from the rock formation. The retrieved samples underwent a series of lab tests, including unconfined compressive strength (UCS) tests, bulk and dry density tests, Atterberg limits, natural moisture, linear shrinkage tests, and particle size distribution. All tests adhered to British Standards [36,37].

Three soil types were identified by the geological investigation. These are fine-grained sandy/silty/clay soil, coarse-grained soil, and Nicosia Marl. Figure 3 presents the identified soil stratigraphy. The top 1.5 m of soil was found to be loose sand, with very low stiffness and was, thus, disregarded in the ensuing analyses. Below this, two subsequent layers comprised dense and very dense gravelly sand, respectively, each showing low plasticity, with a plasticity index (PI) between 0 and 15%. SPT tests were carried out in six offshore boreholes drilled along the jetty. For the upper gravelly sand layer, the tests revealed N-SPT values exceeding 48 and usually 60 per 30 cm of settlement. Notably, for the second gravelly sand layer and at a borehole located beneath the T-junction, 30 cm of settlement could not be reached after 100 blows, indicating very high stiffness for this layer. Using the above N-SPT values (N’) and Equation (1) given by Bowles [38] for gravelly sand, the elastic Young’s modulus (Esi) for the dense sand layer varies between 60 and 80 MPa, while for the very dense to poorly cemented sand, it ranges between 100 and 125 MPa. For the ensuing numerical analyses, Esi of 60 and 100 MPa are assumed for the upper and lower sand layers, respectively.

Esi (kPa) = 1200 · (N’ + 6)

(1)

Nicosia Marl was encountered beneath the dense sand layer. The top two meters of this Marl were weathered, followed by fresh Marl. Nicosia Marl is a soft rock formation, and it was found to behave similarly to stiff clay [39,40,41,42]. The geotechnical investigation [35] reported N-SPT values over 40, confirming the layer’s high stiffness. Soil classification studies indicated a liquid limit of 55% and a plasticity index of 30%. Based on the classification chart of British Standards [43], the combination of the above two properties classifies the Marl layer as high-plasticity clay (CH).

The UCS tests mentioned earlier were conducted on Marl samples retrieved from different depths across six boreholes in the vicinity of the jetty. As detailed in Koronides et al. [20], significant variation in the UCS values was observed, while the mean value was found to be 1180 kPa. In the absence of more precise data, the undrained shear strength of Nicosia Marl is assumed as 590 kPa, as inferred from the mean UCS value (Su = UCS/2). Unlike the undrained shear strength, the small strain Young’s modulus (Esi) cannot be inferred from UCS values. As Koronides et al. [20] stated, existing studies that correlate UCS with Esi for clayey rocks [44,45] were found unsuitable for a soft rock like Nicosia Marl and produced unreasonable results. This study adopts an empirical expression for Esi proposed by Reese et al. [46] for stiff clays, as follows:

Esi = k · H

(2)

where H represents the depth and k is a parameter dependent on Su. For an Su value of 590 kPa, the referred study suggests a k value of 2000 kN/m³.

For comparison reasons, this study adopts the approach suggested by Loukidis et al. [42], calculating Marl formation’s stiffness in relation to vertical effective stress (σ’_v) using the power law relationship:

$$\mathrm{Esi}={A·{\sigma \prime }_v}^n{{·p}_a}^{1-n}$$

(3)

where $p_a$ represents the atmospheric pressure equal, set at 100 kPa, while $A$ and $n$ are dimensionless parameters. The present study adopts $A$ = 395 and $n$ = 0.2, as calibrated by Loukidis et al. [42] against data obtained from lab tests on Nicosia Marl.

Figure 4 presents the Esi profiles for the Marl layers estimated from Equations (2) and (3). The Esi profile used in this study was derived from these estimates and is also depicted in the figure. It is assumed that the weathered layer of Marl exhibits lower Esi than the underlying fresh Marl, with uniform stiffness applied within each layer.

4. Numerical Model

The numerical simulations were performed using ABAQUS/Standard FE software [47]. These simulations involved modelling monotonic pushover loading on two marine structures by employing two FE models. The first model, SPSI^jetty, explicitly simulates the jetty’s T-junction described in Section 2. The second model, SPSI^8×8, simulates a simplified marine structure comprising an 8 m x 8 m deck and four foundation piles. A comparison between the two models aims to examine how assumptions about soil nonlinearity and steel plasticity affect the response of marine structures, depending on the complexity of the foundation system. As widely employed in previous studies [3,14,19], pushover simulations are conducted in this study to evaluate the overall nonlinearity of the response, identify nonlinear driving mechanisms, and determine the lateral loading capacity of the two SPSI systems. The nonlinear mechanisms incorporated in the analyses include soil nonlinearity, accounted for by nonlinear springs, as well as steel nonlinearity, accounted for by using a plasticity model for steel. Given the structure’s flexibility and the significant lateral loads, large structural displacements are anticipated. Therefore, all analyses incorporate geometric nonlinearity to account for P-Δ effects.

The SPSI^jetty model (Figure 5) comprises a deck of 1.3 m thickness, supported by 24 inclined piles that possess the same geometric characteristics as those depicted in Figure 2. The effects of the trestles attached to the three sides of the T-junction are neglected, allowing the deck to move laterally without constraints. The SPSI^8×8 model, which is depicted in Figure 6, comprises an 8 m × 8 m deck with the same thickness and elevation as the deck in the larger model. The lengths and cross-sectional area of the piles are identical to the SPSI^jetty’s piles that are closer to the geometrical centre (see Figure 2). Finally, the foundation soil conditions for both models are assumed to be those described in Section 3.

4.1. Structural Elements

The SPSI^8×8 and SPSI^jetty marine structures are discretised using identical element types and dimensions. The concrete deck is modelled with reduced-integration 20-node quadratic brick elements (C3D20R), having dimensions of 0.5 m in the x and y directions and 0.325 m in the z direction. Given the minimal bending anticipated, the concrete is assumed to behave elastically.

The piles are modelled using three-node quadratic beam elements (B32) with an approximate length of 0.50 m. Each pile consists of two segments with distinct properties. The upper 2.4 m of the piles corresponds to the steel–concrete composite section that connects the concrete slab with the steel piles (see Section 2). These pile segments are prescribed to behave elastically and are assigned to considerably higher stiffness than the bare steel segments. This increased stiffness is achieved by enhancing the cross-sectional area (A) and moment of inertia (I), ensuring that the axial rigidity (EA) and flexural rigidity (EI) are equivalent to those of the steel–concrete composite cross-section. Similarly, an equivalent density (ρ) is used. The bare steel segments behave either elastically (ELsteel) or elastoplastically (EPsteel). Beam elements representing piles’ elastoplastic steel are assigned to a plastic mechanical model featuring a strain-hardening behaviour with a 10% ultimate strain at failure (ε_u). Aiming to achieve a smooth transition from elastic to fully plastic states of steel, the stress–strain behaviour is determined following the approach proposed by Ramberg and Osgood [48]. Additionally, in accordance with ABAQUS guidelines, this behaviour is corrected to reflect true stress–strain relationships. The plastic properties used for the steel piles are detailed in

Table 1. Steel properties input in the present numerical analyses.

Acronym	Behaviour	fy (MPa)	fu (MPa)	εu (%)
ELsteel	Elastic	n/a	n/a	n/a
EPsteel	Elastoplastic	450	550	10

, and the corresponding stress–strain curves are shown in Figure 7.

During the design phase of the actual jetty, the composite pile segments were intended to achieve a moment connection between the piles and the concrete deck. In the numerical model, due to the lack of rotational degrees of freedom for the brick elements, the simulation of the pile–slab connections is achieved by extending the beam elements and embedding them into the brick elements. The embedded beam elements were positioned vertically to ensure common nodes and, thus, common displacement degrees of freedom with the surrounding (host) brick elements. The embedded elements have a length that matches the combined vertical (z-direction) dimension of two brick elements. As shown by previous studies [49], the moment capacity of the connections depends on the stiffness of the extended elements that are embedded within the brick elements. Assuming rigid connections, this study employs rigid beam elements for the embedded elements. All elastic properties of the structural elements input in the numerical analyses are listed in

Table 2. Elastic properties of structural elements input in the numerical analyses.

Material	E (GPa)	ρ (Mg/m3)	ν
Concrete	34	2.5	0.2
Steel	210	7.85	0.3
Rigid Extension	2E6	0.01	0.3

.

The SPSI^jetty numerical model comprises a total of 57,750 nodes, with 10,852 brick elements, 2296 beam elements (48 of which are rigid), and 5662 spring elements (refer to Section 4.2). The SPSI^8×8 model comprises 6645 nodes, with 1024 brick elements, 440 beam elements (8 of which are rigid), and 1280 spring elements (refer to Section 4.2). The adequacy of the chosen element sizes was verified by a previous study [20].

4.2. Soil–Pile Interaction

Soil–pile interaction is considered in both SPSI^jetty and SPSI^8×8 numerical models through the imposition of axial and lateral springs that simulate soil reactions. Specifically, one τ-z spring, acting in the z-direction of the model (see Figure 5 and Figure 6), is placed at every pile node beneath the elevation of the seabed to model shaft resistance. Two p-y springs, aligned with the x and y directions of the model, are assigned to the same set of nodes to simulate lateral soil resistance. Finally, a q-z spring acting in the z-direction is positioned at each pile tip node to represent base resistance. The shear effect on the soil–pile interface is accounted for through the τ-z springs, although these do not act along the interface. The piles are inclined at a 1:3 rake (18°), meaning that when a shaft reaction force T is applied along the z-axis, 0.95 T is directed along the pile–soil interface. This suggests a negligible difference in load transfer, regardless of whether the t-z springs are applied vertically or inclined. This claim is supported by Poulos and Davis [50], who demonstrated that for piles inclined at angles below 30°, the load distribution closely approximates that of vertical piles. The springs exhibit either linear or nonlinear behaviour, while their constitutive behaviour neglects sliding or detachment between the piles and the surrounding soil.

Existing nonlinear spring behaviours were derived for either monotonic or cyclic loading conditions. This study adopts springs suitable for monotonic loading, complying with the static incremental loading applied. Also, springs are typically distinguished based on their suitability for sand or clay conditions. Considering the stratigraphy shown in Figure 3, herein, springs appropriate for sand are assigned to nodes within the first 7 m below the seabed. Springs suitable for stiff clay are assigned to deeper nodes, where Marl formation exists, given that Marl behaves similarly to stiff clay [39,40,41,42].

4.2.1. Nonlinear Springs

The nonlinear relationships between shaft, T, and base, Q, forces, and vertical displacement are inferred from the subgrade reactions, τ-z and q-z, respectively, following the API [51] recommendations. Both subgrade reactions are determined assuming that the critical displacement, where the ultimate reaction occurs, is equal to 1% of the pile diameter. The ultimate reaction stresses, acting on the pile segments driven in sand, are computed following the DNV guidelines [52] and assuming medium dense to dense sand. For pile segments driven in cohesive soils, the ultimate stresses are computed based on API’s [51] guidelines. For the clay layers, the residual reaction stresses are decreased by 10% from the ultimate stress.

The nonlinear behaviour of the τ-z reaction stresses is assumed to be symmetrical for positive and negative vertical displacements, indicating identical behaviour for the upward and downward motions of the pile. Conversely, two variations in the q-z curves are assumed. One assumes zero resistance for positive (upward) vertical displacements, representing tensionless pile tip springs. The second variation assumes tension-resistant q-z springs with symmetrical behaviour between the negative (compressive) and the positive (tensile) vertical displacements.

The p-y springs that represent resistance from sand layers, i.e., the first 7 m below the seabed, were computed based on API [51] guidelines. The subgrade reaction for lateral loading, which is required for the computation of p-y, is estimated through the definition of Vesic [53]. The ultimate capacity in lateral pressure is computed based on modern provisions [51,54], assuming an angle of shearing resistance (φ’) of 30° The behaviour of springs below 7 m, which represent resistance from Nicosia Marl, is defined based on the guidelines proposed by Reese et al. [46] for subgrade reactions in stiff clay.

A detailed description of the computation of the nonlinear behaviour of springs can be found in Koronides et al. [20]. All soil and structural properties necessary for spring computations are provided in

Table 3. Soil properties used to determine spring behaviour.

Layer No.	Soil Type	Depth Below Seabed (m)	γ (kN/m3)	${\mathit{E}}_{\mathit{s}\mathit{i}}$ (MPa)	ν	φ’ (°)	Su (kPa)
1	Loose sand	0–1.5	Disregarded
2	Dense sand	1.5–3.0	19.4	60	0.3	30	-
3	Dense sand	3.0–7.0	19.4	100	0.35	30	-
4	Weathered Nicosia Marl	7.0–9.0	20.6	30	0.35	-	590
5	Fresh Nicosia Marl	9.0–26.35 (deepest pile tip)	20.6	45	0.35	-	590

and

Table 4. Structural properties used to determine spring behaviour.

$\mathit{D}$ (m)	${\mathit{E}}_{\mathit{p}}$ (MPa)	${\mathit{I}}_{\mathit{p}}$ (m4)
1.067	2.10 × 105	1.19 × 10−2

, respectively. As indicated in Table 3, the top 1.5 m of soil is excluded from consideration (i.e., no springs were imposed for this layer) because the geotechnical survey revealed loose sand with very low stiffness and high susceptibility to scouring. The properties for the remaining soil layers are derived from the results from in situ investigations and laboratory tests detailed in Section 2. All nonlinear force–displacement curves used in the numerical analyses can be found in Appendix A.

4.2.2. Linear Springs

The stiffnesses of the linear τ-z ($K_T$) and q-z ($K_Q$) springs are determined from the initial slope of their corresponding nonlinear counterparts. The p-y springs are assigned stiffness values ($K_P$) calculated by multiplying the subgrade reaction ($k_p$), as proposed by Vesic [53], by the pile diameter and node spacing. The calculated stiffness values for the linear springs are provided in

Table 5. Stiffness values of linear springs.

Layer No.	Soil Type	Depth Below the Seabed (m)	${\mathit{K}}_{\mathit{T}}$ (kN/m)	${\mathit{K}}_{\mathit{Q}}$ (kN/m)	${\mathit{K}}_{\mathit{P}}$ (kN/m)
1	Loose sand	0–1.5	Disregarded
2	Dense sand	1.5–3.0	345	-	962
3	Dense sand	3.0–7.0	795	-	14,882
4	Weathered Nicosia Marl	7.0–9.0	9373	-	4487
5	Fresh Nicosia Marl	9.0–11.0	10,000	-	6962
		11.0–15.35 (pile tip) *	9676	556,236	6235
		11.0–17.85 (pile tip) *	10,217	556,236	6182
		11.0–26.35 (pile tip)	10,858	556,236	5937

* Applicable solely to the SPSI^jetty model, which is supported by piles of three different lengths.

.

4.3. Loading

Dead and live loads are represented by a vertical load (V), while dynamic forces from wind, waves, currents, and earthquakes, along with lateral loads from the adjacent trestles, are approximated by a single horizontal load (Fx). V and Fx loads are statically applied as concentrated forces at the centre of the deck of both SPSI^8×8 and SPSI^jetty marine structures, as illustrated in Figure 5 and Figure 6. This configuration exploits the deck’s rigidity to simplify the complex dynamic conditions impacting the structural response. The vertical load is incrementally applied before the horizontal load, taking values of 3.9 MN for SPSI^8×8 and 39 MN for SPSI^jetty. After the vertical load is applied, the horizontal load is also applied incrementally until either it reaches 80 MN or the examined marine structures attain their lateral load capacity.

4.4. Numerical Model Verification and Result Interpretation

The numerical models of both SPSI^8×8 and SPSI^jetty marine structures were verified by Koronides et al. [20] through equilibrium checks and comparisons with existing studies. Additionally, the reliability of these models is verified in the present study, as they produce results consistent with findings from previous studies employing similar boundary conditions, as detailed in Section 6. Although experimental data would be valuable for further validating the numerical analyses, such data are currently unavailable.

The numerical results of the study conducted by Koronides et al. [20] highlighted the importance of soil nonlinearity and assumptions regarding the tension capacity of pile tips on the response of the examined structures. To further explore the effects of the above factors on the formation of plastic hinges on piles and structural ductility in marine structures, parametric analyses are carried out herein. In these analyses, springs, which represent the unbounded soil domain, are either linear or nonlinear, and q-z springs are assumed either tension-resistant or tensionless. Additionally, steel can behave either elastically or elastoplastically, following the stress–strain behaviours shown in Figure 7. The purpose of the parametric analyses is to quantify the impact of certain unrealistic assumptions, namely linear soil behaviour, tension-resistant q-z springs, or elastic steel, on the predicted behaviour of the examined SPSI system.

The response is evaluated through applied lateral force (Fx) versus output horizontal displacement of the deck centre (d) curves. These force–displacement (Fx-d) curves can provide insight into the system’s load capacity, stiffness, and nonlinear behaviour. The presence of a plateau in these curves indicates a ductile behaviour, whereas the absence of a plateau before failure indicates brittle behaviour. The system’s ductility can be influenced by the number of plastic regions formed on the steel columns. The sequence of the formation of these regions is also illustrated to assess the impact of assumptions regarding soil behaviour on steel plasticity. Finally, the effect of steel plasticity on the reaction forces of springs is discussed.

5. Results

5.1. SPSI^8×8 Marine Structure

The impact of soil nonlinearity and steel plasticity on the response of the smaller SPSI^8×8 marine jetty is demonstrated in Figure 8, which shows the output Fx-d curves. The figure verifies the more flexible response of the systems with nonlinear springs. This was the case until the system with nonlinear springs failed. As evidenced by Koronides et al. [20], who studied the same SPSI configuration, the failure took place due to the inability of the system to maintain vertical and moment equilibrium. This is due to the −x piles (see Figure 6c) reaching their axial capacity, as the sum of shaft (T) and base (Q) reaction spring forces reached their limits, as illustrated in Figure 9. The limits of T, Q, and lateral (P) reaction forces are calculated as the residual reaction force multiplied by the number of springs considered.

In the case of nonlinear soil, incorporating steel plasticity in the numerical model is found to have a marginal impact on both the response (Figure 8) and reaction forces (Figure 9). This is attributed to the fact that plasticity was not triggered during the analysis incorporating elastoplastic steel, with the exception of a load increment prior to failure. However, the formation of these plastic regions did not contribute to the failure of the system. The failure mode is identical to that predicted by the analysis with elastic steel and is associated with the −x piles reaching their axial capacity, as mentioned earlier.

Unlike the analyses with nonlinear springs, the analyses with linear springs are found to be strongly impacted by the assumption regarding the behaviour of steel. As indicated by Figure 8, the EPsteel analysis predicted plastic hinges to be gradually formed on the piles, which softened the response of the SPSI^8×8 marine jetty compared to the response produced by the ELsteel analysis. Additionally, EPsteel and ELsteel analyses with linear springs produced significantly different T, Q, and P forces, as shown in Figure 10. Due to springs’ linear behaviour, spring reaction force magnitudes are not constrained; thus, T, Q, and P increase linearly when steel behaves elastically. However, in the analyses incorporating elastoplastic steel, these forces are constrained by the formation of plastic regions. As illustrated in Figure 10, the accumulation rate of T, Q, and P forces decreased dramatically after the formation of the first plastic hinges. Since spring reaction forces are the only resisting mechanisms, this behaviour results in the system reaching its capacity for lateral loading, as indicated by the plateau observed in Figure 8.

The areas on the piles where plastic regions are formed during the EPsteel analyses are illustrated in Figure 11, which depicts the distribution of the accumulation of plastic strains, defined as equivalent plastic strains (PEEQ) in ABAQUS. The presented distributions refer to the last converged increment of the analyses with linear and nonlinear springs, while the numbers indicate the order of the formation of the hinges. In both analyses, the first plastic hinges formed were at the top of the piles, specifically at the interface between plain steel and the stiffer steel–concrete composite section below the concrete slab. In the case of nonlinear soil, these plastic hinges were formed very locally, and the accumulated plastic strains were very small (Figure 11b). This occurred because, as previously mentioned, the analysis failed to converge shortly after the formation of the plastic hinges, which hindered their propagation and prevented the development of significant plastic strains.

Conversely, when springs are linear, the plastic hinges at the top of the piles propagate, resulting in a larger area of plastified steel and much larger plastic strains. Additionally, plastic hinges formed near the seabed level due to constraints imposed by soil resistance through the springs. Notably, plastic hinges were created at the +x piles before forming at the −x piles, as illustrated in Figure 11a. The creation of two plastic hinges per pile creating a plastic hinge mechanism gradually led the system to its capacity, as indicated by the plateau in Figure 8. The small incremental lateral force placed after the creation of these hinges is attributed to the hardening behaviour of steel, shown in Figure 7.

The above discrepancy in the failure modes between EPsteel analyses with linear and nonlinear springs demonstrates that assuming linear soil behaviour can lead to unrealistic failure predictions. Such assumptions may also result in the incorrect identification of critical elements, which is crucial for ensuring a rapid response in critical energy infrastructures, like the one examined, especially during extreme events.

5.2. SPSI^jetty Marine Structure

The previous section demonstrated the impact of soil nonlinearity and steel plasticity on the simplified SPSI^8×8 marine structure, showing that its capacity was influenced by the axial capacity of the piles in tension. As discussed by Koronides et al. [20], this is linked to the behaviour of the q-z springs at the pile tips. This section explores the interaction of these factors and their impact on the response of the more complex jetty structure described in Section 2, with an emphasis on plastic yielding and failure modes.

Figure 12 presents the force–displacement response of the SPSI^jetty marine structure, as predicted by analyses with linear and nonlinear springs. For the nonlinear case, q-z springs were assigned to either a tensionless or tension-resistant behaviour. As expected, soil nonlinearity makes the response more flexible, significantly decreasing the system’s capacity compared to when linear springs are used. The capacity increases when q-z springs can sustain tensile forces. Additionally, soil nonlinearity significantly affects the magnitude of the externally applied lateral force at which plastic hinges form on the piles. In fact, when soil is nonlinear, plastic hinges formed at much smaller applied forces and larger lateral displacements compared to when the soil behaves linearly.

As expected, plastic hinges make the response more flexible and decrease the load capacity of the system compared to when the steel behaves linearly. The discrepancies in system stiffness and capacity observed in analyses with elastic and elastoplastic steel are attributed to the different forces that can be carried by the springs, which are the only resisting mechanisms. Figure 13 presents the sum of T, Q and P forces of the centre (C) and rear (R) in-plane (+x^C, +x^R, −x^C, −x^R) piles, whose positions are shown in Figure 5b, computed by EPsteel analyses with linear and nonlinear springs (nonlinear q-z springs are assumed tensionless). As the figure indicates, much larger reaction forces accumulate on linear springs compared to nonlinear springs. When springs behave linearly, the reaction forces increase linearly with increasing lateral force, until steel plasticity is mobilised. When plastic hinges form at the +x^C piles, the accumulation rate of all T, Q and P reaction forces for these piles decreases, while it increases for the +x^R and −x^C piles.

Unlike the analysis with linear springs, the analysis with nonlinear springs predicts reaction forces that accumulate nonlinearly with increasing excitation force, as indicated by their deviation from the results of the linear analysis. Due to the tensionless nature of q-z springs, the Q forces of the piles in tension, −x^C and −x^R, a drop to zero well before failure takes place. At this stage, the shaft resistance of both centre and rear in-plane piles increases, compared to the linear analyses, to maintain axial force and bending moment equilibrium. At larger applied forces, the accumulated T and Q axial reaction spring forces acting on all the in-plane piles remain constant with increasing applied lateral force, indicating that these piles reached their axial capacity. When this occurs, the incremental bending moments and axial forces are sustained by the springs of the out-of-plane piles, the reaction forces of which are not presented for brevity. Additionally, Figure 13 illustrates that reaching the axial capacity of the in-plane piles leads to a decrease in the accumulation rate of P forces from the springs attached to the centre piles. At the same time, there is a significant increase in the P forces acting on the rear +x piles.

For nonlinear springs, plasticity is activated after the in-plane piles reach their axial capacity. Thus, plasticity is initiated as the system approaches its overall bearing capacity. This is also indicated in Figure 12, which illustrates that the initial plastic hinges form as the force–displacement curve approaches its plateau. Similar to the observations for linear springs, plastic hinges redistribute the incremental P forces. They reduce the accumulation rate for the +x piles and increase it for the −x piles. Overall, steel plasticity appears to marginally affect the reaction forces of the springs acting on the in-plane piles (Figure 13), while it slightly impacts the system’s bearing capacity under lateral load (Figure 12).

Therefore, steel plasticity is not a significant factor for the response of the examined SPSI system with tensionless q-z springs, as the soil’s axial behaviour primarily governs the system’s nonlinearity. However, this may change if the tensile axial soil resistance increases, such as in the case where q-z springs are assumed to be tension-resistant. As illustrated in Figure 14, tension-resistant springs allow for the development of tensile forces on pile tips, as opposed to the zero tensile forces predicted by the analysis-utilised tensionless springs. Unlike the compressive axial reaction forces, T and Q, which are not affected, when tension-resistant springs are used, the tensile shaft reaction forces, T, are mitigated under the same lateral force. This enables the system to sustain larger lateral forces until its axial capacity is reached. Since the system’s failure with tensionless springs is closely linked to its axial capacity, this explains the higher bearing capacity observed in the analysis using tension-resistant q-z springs compared to the tensionless springs, as shown in Figure 12. Due to the larger strength imposed by the system-utilised tension-resistant q-z springs, the plastic hinges were created at larger lateral excitation force. However, unlike in the analysis with tensionless springs, the first hinges formed before the system’s axial capacity was reached. This is evidenced by the creation of hinges well before the plateau in the force–displacement curve in Figure 12, as well as by Figure 14, which shows hinges forming before the sum of T and Q reached their limit. As a result, steel plasticity significantly influenced the nonlinear behaviour of the system with tension-resistant q-z springs. This is in contrast with the results obtained from the analysis with tensionless springs, where plasticity only occurred after the system had nearly reached its axial and bearing capacity.

The above observations demonstrate that failure modes predicted by numerical analyses are strongly associated with the assumptions made on the behaviour of springs. In the realistic scenario with nonlinear springs, failure is controlled by spring (i.e., soil) nonlinearity. Conversely, when unrealistic linear springs are assumed, failure is predicted to occur through a plastic hinge mechanism. To demonstrate the impact of the assumed springs’ behaviour on the failure mode, the following figures present the sequence of plastic hinge formation in both plan and side views. The side views show the examined marine structure displaced from its initial position, with displacements exaggerated by a factor of five.

Figure 15 shows the results from the analysis with linear springs. The figure demonstrates very small displacements of piles below the seabed, indicating the impact of non-degradable stiffness of the springs due to their assigned linearity. As a consequence of the constraints imposed by the springs, the first two plastic hinges form at the +x^C piles at an elevation below the seabed. These hinges are followed by another two plastic hinges at the same elevation on the −x^C piles and subsequently by four plastic hinges near the head of the +x piles. Precisely, the latter hinges form at the interface between the plain steel and the much stiffer composite sections (see Section 2). The +x^C piles are the most critical elements, under the assumption of linear springs. They accumulate the largest plastic strains and are subjected to significant bending, resulting in large local plastic deformations. The latter can explain the failure mechanism of the SPSI^jetty.

A significantly different deformation mode and plastic hinge formation are predicted by the analysis with nonlinear springs and tensionless q-z springs, as illustrated by Figure 16. The figure demonstrates significant vertical displacements of the buried pile segments and horizontal displacements due to bending in the segments above seabed. Specifically, the in-plane −x piles exhibit an upward motion due to tension, whereas +x piles exhibit a downward motion due to compression. The out-of-plane piles, inclined perpendicular to the excitation load, exhibited limited deformation and bending, demonstrating their minimal role in the overall response. Thus, no plastic hinge formed on the out-of-plane piles, whereas two plastic hinges were created on each in-plane pile at failure.

As shown in Figure 16b, the plastic regions developed at depths much deeper than those where the shallowest springs are attached, in contrast to the case of linear soil, where plastic regions were developed to the level of these springs (Figure 15b). Notably, the hinges on the tensile, −x, piles formed at a greater depth below the shallowest spring compared to those formed on the compressive, +x, piles. This difference is not clearly illustrated in the deformed structure due to the vertical motion of the piles, which also affects the elevation of the shallowest spring representing the seabed level. Plastic regions are also created near the head of the piles, where piles become much stiffer and elastic.

The first steel yielding is observed at the heads, precisely at the interface between the plain steel and composite section, of two +x^R piles. These were followed by plastic hinges forming at the +x piles at an elevation below seabed and at the heads of −x^R piles. In subsequent increments of the analysis, two plastic hinges are created at each in-plane pile. This state does not lead to a mechanism due to the presence of the out-of-plane piles and the hardening behaviour of steel. As mentioned earlier, the model fails due to spring plastification rather than a plastic hinge mechanism.

A similar deformation shape and failure mode are predicted by the nonlinear analysis with tension-resistant q-z springs, as shown in Figure 17. As expected, the tensile −x piles exhibited smaller upward displacements (Figure 17b) compared to the case with tensionless q-z springs (Figure 16b). This difference becomes appreciable by comparing the displaced shapes shown in Figure 16b and Figure 17b, particularly considering the latter one corresponding to a larger applied lateral load. Another notable difference is that plastic hinges form over a greater number of increments, allowing for the redistribution of bending moments and forces on piles and showing a more ductile behaviour. The higher ductility is further evidenced by the smoother transition from the elastic to the fully plastic stage in the system’s force–displacement response, as shown in Figure 12. The assumption of tension-resistant pile tips, compared to the more brittle tensionless behaviour, contributed to this enhanced ductility. Additionally, the formation of plastic hinges before reaching the axial capacity of the springs (Figure 14) underscores their crucial role in the system’s nonlinear behaviour, as already mentioned.

The above contrasts with the analysis using tensionless q-z springs, where steel yielding had a minimal impact on the system’s nonlinearity or bearing capacity. Plastic hinges were formed on steel piles when the system’s stiffness had already been degraded due to spring yielding. These hinges were primarily a result of bending, as both axial and shear forces were very small in comparison with their respective ultimate capacities. The axial forces (N) and bending moments (M) acting on the piles are impacted by the behaviour of the springs. Figure 18 presents N and M versus the horizontal displacements computed at the top of the platform centre. The presented N and M are calculated for eight cross-sections shown in Figure 15, −x^CT, −x^CL, −x^RT, −x^RL, +x^CT, +x^CL, +x^RT, +x^RL. Here, −x and +x indicate cross-sections on compressive and tensile piles, respectively; C and R refer to cross-sections at centre and rear piles; and T and L specify cross-sections at the pile top and at a lower position, respectively. Each of these cross-sections developed a plastic hinge during the analysis; therefore, the presented bending moments are expected to be the largest acting on the pile where each cross-section belongs.

Figure 18a verifies that all cross-sections on the −x piles take tensile forces, as opposed to the compression forces acting on the cross-sections of +x piles. Although the axial forces are much smaller than the ultimate axial force, they reach a maximum value. This is associated with the axial capacity of the springs attached on the pile of each cross-section. The maximum axial forces in the cross-sections of −x piles are lower in absolute value than those in +x piles, due to the additional axial capacity provided by the compressive q-z spring attached to the tip of the latter piles.

Conversely, as illustrated in Figure 18b, the bending moments acting on the top of the −x and +x piles are the same for most of the cross-sections examined. They differ when referring to cross-sections below the seabed level due to the different impacts of soil springs. In fact, the absolute values of bending moments acting on the −x^CL and −x^RL are smaller than those of +x^CL and +x^RL. The former corresponds to cross-sections that developed plastic hinges on the compressive piles; thus, they are deeper in the soil than the latter cross-sections, which developed hinges at shallower depth. The fact that the −x^CL and −x^RL cross-sections are deep justifies the negligible moment observed at small displacements. They develop larger bending moments when the axial springs approach their capacity, promoting vertical as well as lateral displacements of the pile. As indicated by the plateaus in Figure 18a, the axial springs attached to the -x^R piles reach capacity at a smaller displacement than those attached to the −x^C piles. This justifies the difference in bending moments observed between the deep cross-sections on these piles, −x^CL and −x^RL. For the remaining cross-sections, the central and rear piles develop equivalent bending moments at corresponding elevations. For each pile, the bending moments at the two examined elevations, T and L, exhibit opposite signs due to the change in pile curvature influenced by the springs at the L position. Nevertheless, the bending moments at all examined cross-sections tend to converge to a common absolute value, which corresponds to the plastic moment capacity of the steel cross-sections.

6. Discussion

This study demonstrated that when soil is assumed to be linear, numerical analyses simulating marine structures subjected to environmental loads can predict unrealistic failure modes, aligning with a similar conclusion made by Abhinav and Saha [19]. The failure mode of both examined SPSI^8×8 and SPSI^jetty marine structures was controlled by a plastic hinge mechanism, associated with plastic hinges formed at areas with stiffness contrast. These are near the top of the piles, where piles are connected with the concrete platform, and below the seabed, where springs impose restrictions in the axial and the two lateral directions.

Likewise, analyses using nonlinear springs indicated that plastic hinges are likely to form near and below the seabed, aligning with similar findings from previous studies [3,30,55]. However, the impact of steel yielding on the SPSI response differs. When more reasonably soil modelled with nonlinear springs and tensionless q-z, steel yielding had a negligible impact on both the failure mode and the nonlinear response of the structure. The failure is strongly associated with the axial capacity of the pile–soil systems, particularly the soil’s axial resistance provided by the vertical τ-z and q-z springs. This is because plastic hinges formed only after all axial springs attached to the in-plane piles had yielded and after the stiffness of the system had significantly degraded. A similar failure mode was observed when using tension-resistant q-z springs, although the contribution of the plastic hinges to the system’s behaviour differed. The increased tensile axial capacity of the individual pile–soil systems allowed the foundation to sustain larger base moments and shear forces, enabling steel yielding before the foundation reached its full axial capacity. Consequently, the nonlinear behaviour of this system was attributed to both soil and steel nonlinearity. These underscore the importance of the axial capacity of piles in determining the bearing capacity and the nonlinear behaviour of marine structures. Additionally, they demonstrate that the unrealistic assumption of tension-resistant q-z springs can lead to an unconservative overestimation of the system’s capacity in lateral loading.

When unrealistic linear elastic springs were used, the axial displacements were minor, and the deformation mode resembled a fixed-base structure (refer to Figure 15b). On the contrary, when soil was nonlinear and q-z tensionless, the compressive piles exhibited a downward motion and the tensile ones an upward motion (refer to Figure 16b). With tension-resistant springs, though, the upward displacements of the tensile piles were significantly reduced under the same lateral load. The susceptibility of the foundation to axial displacements significantly affects the system’s deformation mode. When axial displacements are restricted due to high soil strength, such as the case with linear springs, a sway deformation mode was predicted for the jetty. Conversely, a deformation mode characterised by a prominent rotation around the seabed level was observed when the axial soil resistance was minimised, which was the case when nonlinear τ-z, q-z, and p-y springs and tensionless q-z springs were used. This observation aligns with the findings of Hamidia et al. [8], who showed that the rocking deformation mode of a single-degree-of-freedom structure founded on a pile foundation system is promoted by decreasing the tensile length of the individual pile–soil systems. It also aligns with Wen et al. [3], who showed that a laterally loaded jacket structure can exhibit a rocking-dominated deformation when the lateral loads are sufficiently large to induce upward motion in the tensioned piles.

As shown by the results of the analysis with the more realistic assumptions (i.e., nonlinear springs and tensionless q-z), the formation of plastic hinges did not significantly impact the SPSI behaviour. This indicates that the linear elastic assumption for structural elements, commonly adopted in geotechnical problems, is valid for the examined structures. However, this may not be the case for SPSI systems with higher tensile strength for soil–pile systems. As indicated by analyses with tension-resistant q-z springs, which increased the tensile strength of these systems, steel yielding significantly contributed to the nonlinear response of the system and the failure mode. These observations suggest that the impact of structural yielding on the behaviour of SPSI systems may vary depending on the axial strength of the soil–pile systems.

This study demonstrated that higher soil strength results in the formation of plastic hinges and peak plastic strains, indicating critical element locations, at shallower depths below the soil surface. This conclusion drawn by examining the response of the SPSI^8×8 and SPSI^jetty marine structures aligns with observations made in previous studies by examining the response of onshore laterally loaded structures founded on monopiles [14,29,30]. This conclusion explains the discrepancy observed in linear soil analyses, which inaccurately predict plastic hinge formation at shallower depths relative to predictions derived from nonlinear soil analyses (refer to Figure 15b and Figure 16b). It also explains the different depths of the plastic hinges on the −x and +x piles, as observed in the nonlinear analysis with tensionless q-z springs. Specifically, the hinges on the tensile, −x, piles formed at a greater depth below the seabed compared to those on the compressive, +x, piles. This occurs because the pile–soil systems under tension exhibit reduced strength, as the constraint from the q-z springs vanishes under tensile forces. Therefore, it can be concluded that the overall strength of the pile–soil system, rather than soil strength alone, is the key factor influencing the depth of plastic hinges.

A consequent outcome from the above observations is that accurately identifying critical structural elements necessitates incorporating appropriate soil constitutive behaviour into the numerical model, aligning with similar conclusions from Wen et al. [3]. As demonstrated by the comparison between the SPSI^8×8 and SPSI^jetty systems, the locations of the yielded springs that caused failure are influenced by the number of piles in the foundation system. For the case of the SPSI^8×8 structure founded on four piles, which can resemble a foundation system of a jacket structure, failure occurred abruptly when the axial springs attached on the two piles that are in tension reached their tensile capacity. This is because the other two compressive piles cannot sustain vertical equilibrium. On the contrary, for the SPSI^jetty, which is founded on a foundation system with twenty-four piles, the yielding of springs resulted in the redistribution of bending moments and forces across the piles. Once the axial capacity of springs attached on individual piles was reached, the pile’s axial forces remained constant (Figure 18a), while the bending moments of deep cross-sections increased significantly (see −x^CL and −x^RL cross-sections in Figure 18b). The described failure mode and ability for redistribution of shear forces and bending moments across the piles allowed for a more ductile behaviour for the SPSI^jetty compared to the brittle failure of the SPSI^8×8. Moreover, the system’s ductility is enhanced with an increase in the tensile axial strength of the pile–soil systems, as indicated by the analyses with tension-resistant q-z springs.

The axial force and bending moment redistribution on the piles led to a redistribution of the T, Q, and P reaction forces of springs, ensuring that equilibrium was satisfied. As shown in Figure 13, the τ-z springs attached on the compressive centre piles, +x^C, yielded first. After this point, vertical equilibrium was maintained through redistribution of the additional axial forces to the q-z springs of the same piles and the τ-z and q-z springs of the tensile centre, −x ^C, piles. Eventually, failure occurred shortly after the axial τ-z and q-z springs attached on all in-plane piles yielded. Finally, T reaction forces were found to depend on the ability of the pile tip to sustain tensile forces. When tension-resistant q-z springs were incorporated in the numerical model, the T reaction forces of the tensile pile were mitigated in comparison with the ones predicted by analyses with tensionless springs. However, the assumption of a tension-resistant pile tip, often made in FE analyses, where the pile tip is assumed to be compatibly connected to solid soil elements, is unrealistic. Consequently, the underestimation of T forces can be unconservative and lead to an underestimation of the system’s vertical displacements.

7. Conclusions

This study presented 3D finite element analyses of two marine structures subjected to lateral pushover loading to approximate environmental forces (e.g., wind, waves, currents, earthquake). The first structure was a marine jetty founded on twenty-four piles, while the second was a simplified marine structure supported by four piles. The simulated jetty is a representative of an existing structure located in Vasiliko, Cyprus. The behaviour of these structures to lateral loading was investigated, accounting for SPSI effects through the imposition of τ-z, q-z, and p-y springs, while steel columns were assumed to exhibit an elastoplastic behaviour. The axial capacity of pile–soil systems was shown to play a pivotal role in determining the bearing capacity and nonlinear behaviour of marine structures, suggesting the need for appropriate modelling of pile–soil interaction behaviour in numerical analyses. It was also shown that unrealistic assumptions such as soil linearity or tension-resistant q-z springs can result in incorrect numerical predictions. These include an unconservative overestimation of a system’s capacity and inaccurate predictions of failure modes, plastic hinge formation, critical elements, deformed shape, and soil reaction forces.