Numerical Investigation of Seismic Soil–Structure–Excavation Interaction in Sand

Abstract

The dynamic loads affecting earth-retaining structures may increase in seismically active regions. Therefore, studying the soil–structure interaction among the soil, shoring systems, and adjacent structures is crucial. However, there is limited research on this important topic. This study investigates the seismic performance of a deep braced excavation and a nearby 10-story building in sandy soil formation. The main focus of this study is the consideration of the influence of varying foundation depths of adjacent structures on the seismic response of the shoring system and the performance of the shoring system and adjacent structure under different earthquake records. PLAXIS 2D software (Version 22.02) was used to carry out the numerical analysis. Sandy soil was modeled using the Hardening Soil with small-strain stiffness model (HS-small). Back analysis of observation data extracted from a real case study of a deep braced excavation in the central district of Kaohsiung City, adjacent to the O₇ Station on the Orange Line of the Kaohsiung MRT system in Taiwan, was used to validate the numerical analysis. Beyond model validation, a parametric study was conducted to address the effect of the foundation level of the building adjacent to the excavation on both the seismic behavior of the shoring system and the structure itself, using the Loma-Prieta (1989) earthquake record. The parametric study was further extended to assess the responses of the shoring system and the adjacent structure under the influence of the earthquake records of Loma-Prieta (1989), Northridge (1994), and El-Centro (1940). The results show that the maximum lateral displacement of the diaphragm wall occurred at the top of the wall in all studied cases. The maximum dynamic bending moment in the retaining structure was more than three times the static one on average. In contrast, the dynamic shear force was more than 2.85 times the static one on average. In addition, the dynamic axial force of the first and second struts was 1.38 and 3.17 times the static forces, respectively. The results also reveal large differences in the behavior of the shoring system and the adjacent structure between the different earthquake records.

1. Introduction

A growing population and urban activities within large cities call for underground space to establish urban services, transportation infrastructure, parking areas, or other engineering works that necessitate deep vertical excavations near existing buildings. Retaining structures are essential for inhibiting large and unsafe soil displacement in the areas around excavation. These retaining structures are constructed and supported at different levels by horizontal struts and/or ground anchors, preserving excavation sides and limiting damage to the surrounding structures and utilities.

Braced excavations generate significant changes in stress and strain of the surrounding soil, resulting in permanent displacements to adjacent structures and/or infrastructure and causing severe damage in some cases [1,2,3,4,5]. In order to prevent damage to adjacent structures, it is essential to control the lateral deflections of the retaining structures as the excavation deepens [6,7]. Most studies have focused on the analysis of the behavior of deep excavations under static conditions using distinct methodologies. Empirical and semi-empirical approaches are commonly used, depending on extensive databases of excavation projects from various locations worldwide [1,2,8,9,10,11]. Additionally, numerical methods have been extensively employed to investigate various parameters affecting deep excavations and surrounding structures, including soil elasticity, creep, and the soil–wall interface [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. Laboratory physical model tests were conducted by Chowdhury et al. [25] to investigate the static behavior of braced excavation.

Nevertheless, research has rarely been conducted on braced retaining structures subjected to seismic conditions. Callisto et al. [26] studied braced retaining walls numerically. Their results showed that propagation of the seismic waves through the soil causes quick mobilization of shear stress in the soil surrounding a propped cantilever wall. The wall experiences a significant increase in its lateral deflection and bending moment, along with an increase in the axial forces of the struts when compared to static conditions. Callisto and Soccodato [27] carried out numerical time history analysis, applying two earthquake time histories to cantilever retaining walls embedded in dry coarse-grained soil using FLAC 2D software (v. 5.0). It is suggested that critical horizontal acceleration can serve as a representation of the seismic resistance of a retaining wall’s permanent rigid body movement. This critical acceleration is calculated based on limit equilibrium analysis by an iterative method. A significant increase in bending moment during earthquakes is also observed owing to the instantaneous contact stress redistribution.

Konai et al. [28] investigated the seismic behavior of braced excavations with a single level of struts in dry sand using shaking table tests and numerical modeling using sinusoidal motions to simulate seismic loading. The results of the strut axial force, lateral displacement, and bending moment of the retaining structure from shaking table tests were compared to those of numerical models. The results demonstrated that in seismic conditions, wall stiffness has a larger influence on wall deflection and bending moments than strut forces.

Experimental studies have been conducted to interpret the behavior of different types of retaining structures under seismic loading [29,30,31,32,33,34]. Several centrifuge tests have been performed on cantilever and propped retaining walls in dry sand [35]. It has been observed that during earthquakes, no significant additional displacements would have occurred if the retaining wall had already been exposed to an earthquake that is more severe than the current one. Furthermore, it has been observed that the retaining wall undergoes nearly rigid permanent displacements even for maximum accelerations that are smaller than the critical limit equilibrium one.

Bahrami et al. [36] numerically studied the seismic performance of strutted diaphragm walls in dry sand using FLAC 2D software. The results indicated that under static loads, structures designed using the Peck method performed adequately. However, under seismic conditions, the wall bending moments and shear forces, as well as the struts’ axial forces, increased significantly, exceeding the permissible limits specified in ACI 318 [37] and AISCE [38]. Castaldo and De Iuliis [39] numerically studied the seismic vulnerability of existing buildings affected by adjacent deep excavations using PLAXIS 2D. By considering the non-linear dynamic properties of both soils and structures, it was found that the presence of adjacent excavation significantly increased the ductility demand and inter-story drift of nearby buildings. The use of Reduced Length Buckling-Restrained Braces (RLBRBs) as struts in deep excavations subjected to seismic loading was numerically investigated by Khademalrasoul et al. [40]. The results showed that incorporating RLBRBs in soft soils could enhance the overall stability of deep excavations under seismic loading. The effects of earthquakes on a 9.5 m deep anchored excavation with a diaphragm wall and an adjacent concrete building were analyzed using PLAXIS 2D software (Version 8) [41]. The results revealed that increasing the pre-stressing force of the anchor beyond a specific balanced value serves as a constant final anchor force after the earthquake. The SOIL-STRUCT tool was applied to assess a 10-story building adjacent to a deep excavation of 28 m depth and supported by a 0.7 m thick diaphragm wall [42]. The analysis results showed that roof displacements with smooth asymptotes are larger than those with concave downward asymptotes. Additionally, roof displacements are significantly influenced by an increase in the yield stress of steel, whereas this effect is less pronounced for concrete building frames. Chowdhury et al. [43] numerically investigated the behavior of a strutted retaining structure under seismic conditions using FLAC-2D (v.5.0). It was observed that during an earthquake, the wall’s lateral displacement is the most affected parameter, and the struts’ axial force is the least affected parameter. Yeganeh et al. [44] used the finite difference method in FLAC-2D (version 7.0) to investigate building–excavation interaction and structure models with a fixed base.

To better understand the excavation–structure interaction, this research investigates the performance of deep braced excavations in sand under seismic loads, considering the interaction with adjacent structures using a numerical approach. The hardening soil model with small-strain stiffness (HS-small) is employed to simulate the non-linear and hysteretic behavior of the soil in both static and dynamic conditions. Model validation is performed using a back analysis of a well-documented case study of a deep braced excavation in sandy soil, as cited in [45]. Following validation, a comprehensive parametric study is performed on deep braced excavation in sandy soil with an adjacent structure in static and dynamic cases. This study focused on the response of strutted diaphragm walls and surrounding structures under different earthquake records. The acceleration time histories of Loma-Prieta (1989), Northridge (1994), and El-Centro (1940) earthquakes are utilized in this study. Moreover, the foundation level is also investigated under the effect of the Loma-Prieta (1989) earthquake record. This study discusses the diaphragm wall lateral displacements, settlement trough behind the wall, straining actions of the diaphragm wall, axial forces of the struts, and maximum story displacement of the adjacent structure under static and seismic conditions to evaluate and assess the seismic performance of both the deep braced excavation and the adjacent structure.

2. Numerical Modeling

Plain strain analysis is used to model the soil behavior statically and dynamically. The HS-small strain model, which is implemented in PLAXIS 2D (Version 22.02) based on research of Benz [46], is used to model soil behavior to account for the stress dependency of stiffness and variance in the ratio of secant shear stiffness (G_s/G₀) with increasing shear strain (γ). The HS-small strain model uses the hyperbolic approximation of the stress–strain curve proposed by Hardin [47], which was modified then by Santos et al. [48], given as:

$$\tau =G_s\gamma = {\displaystyle \frac{G_{0 \gamma }}{1+0.385\frac{\gamma }{{\gamma }_{0.70}}}}$$

(1)

where τ is the soil shear stress, and G_s is the secant shear modulus related to the shear strain (γ). γ_0.7 is the shear strain when G_s is reduced to 70% of G₀. In the HS-small model, the stress dependency of the shear modulus G₀ is considered with the power law, which matches the one used for the other stiffness parameters, as indicated in Equation (2). The threshold shear strain γ_0.7 is taken independently of the mean stress, as follows:

$$G_0= G_0^{ref}{\left({\displaystyle \frac{c cos\varnothing - {{\sigma }^{\prime }}_3 sin\varnothing }{ cos\varnothing + p^{ref} sin\varnothing }}\right)}^m$$

(2)

Moreover, the model uses Mohr–Coulomb strength parameters (cohesion c and friction angle ϕ), along with soil stiffness, which is defined by three input values: E₅₀ (secant stiffness modulus at 50% deviatoric stress (q_f)), given by Equation (3), E_oed (1D oedometer loading modulus), and E_ur (unloading–reloading modulus).

$$E_{50}=E_{50}^{ref}{\left({\displaystyle \frac{c cos\varnothing -{{\sigma }^{\prime }}_3 sin\varnothing }{ cos\varnothing +p^{ref} sin\varnothing }}\right)}^m$$

(3)

where $E_{50}^{ref}$ is the secant modulus at the reference confining stress, p^ref (equal to 100 kPa), and ${{\sigma }^{\prime }}_3$ is the minor principal stress. Here, m is the exponent factor for stress dependency, and it is taken to be equal to 0.50 for sandy soil, as proposed by Benz [46]. The default values of E_oed = $E_{50}^{ref}$ and E_ur = 3 $E_{50}^{ref}$, as proposed by [49,50].

Despite using the HS-small constitutive model, which captures energy dissipation through hysteric behavior, a small Rayleigh damping of 2% is used in this study during the dynamic analysis to eliminate the high-frequency noise resulting from the numerical integration. In finite element modeling, the formulation of damping is often based on mass and stiffness and is defined as proposed by [51,52,53]:

$$C=\alpha \left[M\right]+\beta \left[K\right]$$

(4)

where [M] is the mass matrix, [K] is the stiffness matrix, and α and β are the Rayleigh damping coefficients, given by:

$$\alpha =2\varsigma {\displaystyle \frac{{\omega }_1{\omega }_2}{{\omega }_1+{\omega }_2}}$$

(5)

$$\beta ={\displaystyle \frac{2\varsigma }{{\omega }_1+{\omega }_2}}$$

(6)

where $\varsigma$ is the damping ratio, which is 2% in this study, and ω₁ and ω₂ are considered to be the target angular frequencies corresponding to the target circular frequencies f₁ and f₂, respectively. The first frequency, f₁, is equal to the fundamental frequency of the whole soil layer, which is defined as the frequency at which the most significant amplification can be expected, and it is calculated by Equation (7) based on the formula proposed by Kramer [54]. The second frequency is the closest odd number given by the ratio of the fundamental frequency of the input signal at the bedrock and the fundamental frequency of the whole soil layer, as proposed by Hudson [55].

$$fn={\displaystyle \frac{Vs}{4H}}$$

(7)

2.1. Meshing Considerations

The discretization of real models influences the responses of both linear and non-linear finite element models. Dynamic analysis requires that the maximum element dimension be limited to one-eighth to one-fifth of the shortest wavelength considered in the analysis, as suggested by other researchers [56,57]. In this research, 15-node triangular soil elements are used to model the soil continuum. The dimension of the elements is adjusted, and the local refinement of the meshes is implemented to obtain an appropriate value for the average length of the element dimension to increase the accuracy of the results and so that the seismic waves can propagate without numerical errors.

2.2. Boundary Conditions

During static analysis, the movement of side boundaries is restrained horizontally, while movements for the bottom boundary of the model are restrained horizontally and vertically. As for the dynamic analysis, a repeatable tied degree of freedom boundary condition, which was proposed by Zienkiewicz et al. [58], is utilized to simulate the side boundaries of the model to connect the nodes at the same elevation, causing the same displacements. On the other hand, a compliant base boundary condition, as proposed by Joyner and Chen [59], is applied at the bottom of the model to consider the absorption and application of the dynamic loads.

3. Verification of a Case Study on Deep Excavation in Sand [45]

A real case study of a deep braced excavation in Kaohsiung, Taiwan, presented by [45], as shown in Figure 1, is utilized to validate the numerical model before dynamic analysis to assess the model’s accuracy in capturing the behavior of real braced excavation. The excavation dimensions are 70 m in length and 20 m in width, with a 16.8 m depth. A 0.90 m diaphragm wall is used as a retaining structure, supported by four levels of steel struts spaced 5.50 m apart horizontally. The excavation is executed in medium-dense to dense silty–sandy soil with bands of low-plasticity clay (CL). Additionally, before excavation, the groundwater table is nearly 2.0 m below the ground surface and is lowered by 1.0 m before any excavation stage to provide a dry environment for the construction process. Details of the soil lithology, excavation levels, and strut levels have been provided in Hsiung and Dao [45].

PLAXIS 2D FE software (Version 22.02) developed by [60] is used to perform back analysis of the intended case study. A plane-strain FEM model is carried out by adopting a 15-node triangular element to simulate the deformations and stresses in the soil volume. Hardening soil with a small strain model (HS-small) is utilized to perform this study, as recommended by Hsiung and Dao [45], to represent the behavior of the sand layers, whereas the behavior of the clay bands is modeled using the elastic–perfectly plastic Mohr–Coulomb (MC) model. The soil properties analyzed in this study followed the recommendations of Hsiung and Dao [45]. The diaphragm wall and steel struts are modeled as linear elastic materials, adopting plate elements for the diaphragm wall and fixed-end anchors for the steel struts. A Poisson’s ratio of 0.2 is used for both the diaphragm wall and steel struts. The diaphragm wall elastic modulus is 24.8 × 10⁶, while the elastic modulus of steel struts is 2.1 × 10⁵ (MPa). The stiffness of the diaphragm wall and steel struts is decreased by 30% and 40%, respectively, from their nominal values to account for bending moment-induced cracks in the diaphragm wall, repeated use, and improper installation of steel struts, as suggested by Ou [61]. More detailed data about the soil properties, diaphragm wall, and steel struts have been provided by Hsiung and Dao [45].

The verification model’s dimension is extended to a depth of 60 m below the ground surface to comply with the site investigation program. The length of the side boundary of the model to the diaphragm wall is taken to be seven times the maximum excavation depth, as suggested by Khoiri and Ou [62] for deep excavations in sands, as shown in Figure 2a. The mesh discretization shown in Figure 2b is set to global medium, and local refinement is implemented to enhance the accuracy of the analysis.

Verification Results of the FEM Model

Figure 3 shows the comparison of the diaphragm wall lateral displacements resulting from the finite element simulation at the final stage of the excavation and the measured ones from the field data. The measured and predicted lateral displacements at the final stage of excavation showed good agreement. The measured lateral displacement for the first part of the wall, up to 4.0 m of the wall, is less than the predicted one by about 8.0 to 12.0%. However, the predicted and measured displacements seem to be identical up to the final excavation level at 16.80 m, with a difference not exceeding 1%. Moreover, the predicted displacement beyond the final excavation level is further increased by an increasing rate from 1.11% to about 11.40%. The results imply that the HS-small strain model achieved the best prediction of the wall lateral deflections and ground surface settlement in such simulations of these cases, and this is consistent with the findings of previous research [45,62,63]. The ground surface settlement adjacent to the diaphragm wall is also investigated and compared to that of field measurements, as shown in Figure 4. It is demonstrated that the HS-small strain constitutive soil model predicts the settlement in reasonable agreement with measured field data, especially beside the diaphragm wall. However, the predicted settlement seems to be larger than the measured ones as the excavation goes deeper to the final level, but the settlement trough still has the same pattern, as proposed by Hsieh and Ou [8] for braced excavations.

4. Dynamic Modeling of the Shoring System and Adjacent Structure

This research focuses on a braced excavation with 10.00 m depth that is modeled in a dense sandy soil with a 60.00 m deep overlying bedrock and a surrounding structure composed of a basement and nine replicated stories, each with two spans 5.00 m. Side boundaries of the model, as shown in Figure 5, which are chosen to be at 7B (where B is the width of excavation) from the left and right walls. According to ASCE 7-16 [64], the site classification is performed according to the SPT (N) values. For this study, the chosen N-value locates the soil in a seismic category (D), as indicated in ASCE 7-16 [64], and the shear wave velocity, Vs is chosen to be 200 m/s in this study. The soil properties used in this study are presented in

Table 1. Soil properties used in the parametric study of the dense sand layer using the HS-small model.

Depth of Layer (m)	60
Soil type	Dense sand
SPT (N-Value)	37
Soil model	Hardening soil model with small strain (HS-small)
Unsaturated unit weight, γunsat (kN/m3)	19
Saturated unit weight, γsat (kN/m3)	21
Rayleigh damping (α)	0.1634
Rayleigh damping (β)	1.662 × 10−3
Damping ratio	2%
Internal friction angle, ${\varnothing }^{\prime }(°)$	36
Cohesion, $\mathrm{C}\prime \left(\mathrm{K}\mathrm{P}\mathrm{a}\right)$	1
Dilatancy angle, Ψ $(°)$	6
Secant stiffness from drained triaxial test, ${\mathrm{E}}_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}\left(\mathrm{K}\mathrm{P}\mathrm{a}\right)$	50,000
Reference tangent stiffness for oedometer primary loading, ${\mathrm{E}}_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}\left(\mathrm{K}\mathrm{P}\mathrm{a}\right)$	50,000
Reference unloading/reloading stiffness, ${\mathrm{E}}_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}\left(\mathrm{K}\mathrm{P}\mathrm{a}\right)$	150,000
Unloading/reloading Poisson’s ratio, ν’ur	0.20
Exponent of stress-level dependency of stiffness, m	0.50
Failure ratio, Rf	0.90
At-rest earth pressure coefficient, K0	0.4122
Reference small-strain shear modulus, G0ref (kPa)	180,000
Shear strain magnitude at 0.722 G0, γ0.7	1 × 10−4
Analysis type	Drained
Rinter	0.67

. The shoring system and surrounding structure are modeled statically and dynamically using different earthquake records. Static and dynamic behavior is investigated.

In this analysis, linear elastic plate elements are used to model the diaphragm wall with an elasticity modulus of 26.00 GPa, which is calculated according to ACI-318 [37], and a Poisson’s ratio of 0.20. A Rayleigh damping of 5% is considered during seismic analysis.

Two levels of struts with horizontal spacing in the longitudinal direction equal to 5.00 m are used in this study. The first level is installed at 2.00 m from the ground surface, and the second one is installed at 6.00 m from the ground surface, as recommended by the design guide of Chowdhury et al. [15]. Linear elastic node-to-node anchors are used to model the struts with axial stiffness of 25 × 10⁶ and 50 × 10⁶ kN/m², respectively.

A reinforced concrete structure composed of two bays, each with 5.00 m width, with one basement with 3.25 m height and nine typical stories with 3.00 m height each, is set adjacent to the shoring system. The equivalent dead and live loads are considered to be 10 kN/m’/storey. The structural elements are simulated as linear elastic plate elements with axial and flexural stiffness to model columns, walls, beams, and slabs of an equivalent thickness of 0.30 m, while the raft foundation has a 1.00 m thickness. Modal analysis using CSI-SAP2000 (Computers and Structures Inc., Berkeley, CA, USA) identified the first and second modes of vibration as dominant, representing approximately 90% of the structure’s dynamic mass participation. These modes have time periods of 2.47 s and 1.07 s, respectively, following the regulations stated in ASCE [64]. Using these time periods and a 5% damping ratio, the Rayleigh damping constants α and β are calculated using Equation (8) as 0.1984 and 0.008556, respectively, as presented in the methodology proposed in references [52,65].

$${\varsigma }_i={\displaystyle \frac{1}{2}}({\displaystyle \frac{\alpha }{{\omega }_i}}+\beta {\omega }_i)$$

(8)

where α is the mass-proportional damping constant, β is the stiffness-proportional damping constant, ς_i is the damping ratio, and ω is the angular frequency, in rad/s. Interface elements are added to simulate soil–structure interaction between the shoring system, basement walls, the raft, and the surrounding soil. Similar results of recent research have been reported [66,67,68,69,70].

5. Parametric Study

The main objective of this study is to investigate the seismic performance of deep braced excavations in sand, considering the presence of a building adjacent to the excavation, to interpret the soil–structure interaction in such complex cases. All the studied parameters are summarized in

Table 2. Summary of the studied parameters.

Studied Parameter	Parameter Variables	Chosen Parameters
Foundation level of the structure adjacent to the shoring system	Df = 1.0 m	H = 10 m B = 20 m, Tw = 80 cm D = H = 10 m Loma-Prieta EQ
	Df = 3.0 m
	Df = 5.0 m
Earthquake records	Loma-Prieta (1989), Mw = 6.90	H = D = 10 m Tw = 80 cm B = 20 m
	Northridge (1994), Mw = 6.70
	El-Centro (1940), Mw = 6.90

H = Final excavation depth. B = Width of excavation. T_w = Diaphragm wall thickness. D = Diaphragm wall embedment depth. D_f = Foundation level of the adjacent structure.

.

First, static analysis is performed, considering that the diaphragm wall is wished-in-place and no deformation has accumulated from the adjacent wall construction. Then, the excavation starts from the pre-defined levels up to the final excavation level. After each excavation stage, the bracing system is also installed at the proposed locations. Eventually, dynamic analysis is carried out, imposing the seismic loads as a prescribed displacement in the horizontal direction (i.e., Global X-direction) using a dynamic multiplier that represents the time history of the seismic ground motions.

In the dynamic stage of construction, the dynamic boundary conditions and damping are executed before running the analysis, as explained in detail in Section 2.1 and Section 2.2. The implicit Newmark time integration scheme is employed to solve the dynamic equilibrium equations of the system. The dynamic time intervals are used as the total time of the earthquake record. The time steps are also adjusted to match the number of data points that represent the dynamic multiplier of the ground motion. To control the accuracy of the numerical integrations, the Newmark time integration coefficients α and β are chosen to achieve the following conditions.

$$\beta \ge {\displaystyle \frac{1}{2}},\hspace{1em}\alpha \ge {\displaystyle \frac{1}{4}}{\left({\displaystyle \frac{1}{2}}+\beta \right)}^2$$

(9)

The Newmark scheme with α = 0.25 and β = 0.5 (average acceleration method) is utilized, as recommended by PLAXIS [71].

Characteristics of the Input Ground Motions

Acceleration time histories of three well-known earthquakes of Loma-Prieta (1989), Northridge (1994), and El-Centro (1940), retrieved from the Pacific Earthquake Engineering Research Center (PEER) [72], are considered and applied on bedrock after the last stage of excavation, as shown in Figure 6, Figure 7 and Figure 8. The main characteristics of the earthquake records are given in

Table 3. Properties of input earthquake ground motions.

Earthquake Record	Moment Magnitude (Mw)	PGA (g)	Duration (s)	Predominant Period, Tp (s)	Arias Intensity, Ia (m/s)	Significant Duration, Ds 5–95 (s)
Loma-Prieta (1989)	6.90	0.37	39.90	0.22	1.35	11.37
Northridge (1994)	6.70	0.88	60.00	0.22	2.72	8.75
El-Centro (1940)	6.90	0.34	53.76	0.56	1.76	24.46

.

6. Analysis Results and Discussion

The parametric study aims to assess the shoring system design parameters and the behavior of adjacent structures in static and seismic analyses.

6.1. Effect of the Adjacent Structure’s Foundation Level on the Shoring System–Structure Interaction

6.1.1. Displacement–Time History (DTH)

Figure 9a–c present the displacement–time history for the top of the diaphragm wall and the top roof of the adjacent structure, respectively, for different foundation levels of the structure adjacent to the shoring system. The free-field ground surface response (point B in Figure 5) for different foundation levels, as shown in Figure 9a, indicates that the response for all foundation levels is the same as the original ground motion at the base, with a time difference of 0.33 s and maximum amplification ratios of 232%, 209%, and 203% for 1.00 m, 3.00 m, and 5.00 m depths, respectively. It was also noticed that permanent lateral displacements of the ground surface occur in the same direction as the original ground motions of 62.92 mm, 16.71 mm, and 0.56 mm for the depths of 1.00 m, 3.00 m, and 5.00 m, respectively. The results indicate that as the building foundation level increased, the permanent displacement of ground surface decreased significantly by 73.40% as the foundation depth increased from 1.00 to 3.00 m and by 96.60% as the foundation depth increased from 3.00 to 5.00 m.

Figure 9b demonstrates the response of the top of the diaphragm wall (point C in Figure 5) for different foundation levels. The results indicate a time difference of 0.23 s at the top point of the diaphragm wall and an amplification of the incident waves by 331%, 385%, and 388% for foundation depths of 1.00, 3.00, and 5.00 m, respectively. At the end of the earthquake, the crest of the diaphragm wall experienced a permanent lateral movement of −87.89, −128.03, and −138.55 mm for foundation levels of 1.00, 3.00, and 5.00 m, respectively. The response of the top point of the diaphragm wall indicates that the wall displacement at the top increased as the structure’s foundation level increased. As the foundation level increased from 1.00 to 3.00 m, the permanent displacement increased by 45.70%, and it increased by 8.20% as the depth increased from 3.00 to 5.00 m. Deeper foundations caused an increase in lateral displacement at the top of the diaphragm wall because of an amplification effect. The large amplification of seismic waves led to large lateral displacements of the foundation raft towards the excavation area, and this, in turn, exposed the diaphragm wall to extra lateral displacements. Moreover, ponding between the basement floor and the diaphragm wall may have caused this large displacement of the diaphragm wall.

Figure 9c shows the response at the top roof of the structure adjacent to the shoring system (point D in Figure 5). A significant increase in the displacement at the top roof was noticed as the foundation level of the structure increased. The structure underwent large permanent lateral displacements at the top roof at the end of ground shaking. The lateral displacements were amplified by 41.22, 36.37, and 25.80 times compared to the original ground motion for foundation levels of 1.00, 3.00, and 5.00 m, respectively. The top roof exhibited a permanent lateral displacement of −348.32, −406.54, and −446.15 mm at the end of shaking for foundation levels of 1.00, 3.00, and 5.00 m, respectively. As the foundation level increased from 1.00 to 3.00 m, the permanent displacement increased by 16.70%, and a 9.70% increase in permanent displacement was noticed as the depth increased from 3.00 to 5.00 m.

6.1.2. Settlement Trough Beside the Shoring System

Figure 10a,b represent the results of the settlement trough in static and dynamic cases for different foundation levels of the structure adjacent to the shoring system, respectively. The static settlement trough in Figure 10a shows that as the foundation level of the structure increased, the settlement trough decreased significantly. Moreover, the extent of the settlement trough decreased as the foundation level increased, and for a structure’s foundation level of 1.00 m, the settlement trough extended to five times the ultimate excavation depth (5 H) until no significant changes were noticed. On the other hand, for foundation levels of 3.00 m and 5.00 m, the settlement trough extended to two times the ultimate excavation depth (2 H) until no significant changes were noticed. The results indicate that the structure was exposed to differential settlement due to excavation and that the differential settlement decreased as the foundation level increased.

On the contrary, the dynamic settlement trough, as shown in Figure 10b, behaves differently from the static one. The results indicate that the settlement trough and the differential settlement of the structure increase as the foundation level of the structure increases from 1.00 m to 3.00 m, while as the foundation level increased from 3.00 m to 5.00 m, the settlement trough and differential settlement decreased once more. The dynamic trough extends to three times the final excavation depth (3 H), and beyond this distance, no significant changes in the settlement were noticed. The results of the dynamic trough indicate that the ground surface was exposed to uniform permanent vertical settlement of about 5.0 cm for all foundation levels.

Furthermore, for the adjacent structure, the deflection and cracked conditions of the structure components have a direct association with particular parameters, such as differential settlement, relative rotation, and the deflection ratio [61]. Rankine [73] provided guidelines for how the maximum settlement and maximum slope of a building affect its potential damage. The results of maximum static vertical settlement of the building adjacent to the excavation indicate that the building is in a moderate-to-high-risk category based on the Rankine guidelines [73]. However, in the case of dynamic vertical settlement, the building falls under the high-risk category, in which structural damage to the building is expected.

6.1.3. Lateral Displacement of the Shoring System

Figure 11a,b show the static and dynamic lateral displacements of the diaphragm wall for different foundation levels of the adjacent structure, respectively. The static lateral displacements of the diaphragm wall presented in Figure 11a show that the lateral displacements decreased significantly as the foundation level of the structure increased. The lateral displacement at the crest of the diaphragm wall decreased by 60.70% as the foundation level increased from 1.00 m to 3.00 m and decreased by 18.10% as the foundation level increased from 3.00 m to 5.00 m, respectively. However, the lateral displacement of the wall in the dynamic analysis behaves differently, as shown in Figure 11b. The results of the dynamic analysis show that the maximum lateral displacement occurred at the top of the diaphragm wall in all the studied cases. Moreover, as the structure’s foundation level increased, the lateral displacement of the wall increased significantly. As the structure’s foundation level increased from 1.00 m to 3.00 m, the dynamic lateral displacement at the top of the diaphragm wall increased by 47.10%, while as the foundation level increased from 3.00 m to 5.00 m, the lateral displacement increased by 9.50%. On the other hand, as the foundation level increased from 3.00 m to 5.00 m, the dynamic lateral displacements beyond the bottom of the excavation level decreased by 8.40%.

Figure 12 presents a comparison of the dynamic and static lateral displacements at the top of the diaphragm wall and at the bottom of the excavation level for different foundation levels of the structure adjacent to the shoring system. The dynamic lateral displacements at the top of the wall are 1.15, 4.32, and 5.77 times the static ones for the structure’s foundation levels of 1.00 m, 3.00 m, and 5.00 m, respectively. Moreover, the dynamic lateral displacement at the bottom of the excavation level is 1.43, 3.37, and 4.15 times the static ones for the structure’s foundation levels of 1.00 m, 3.00 m, and 5.00 m, respectively.

The significant increase in the dynamic values is attributed to an amplification of the seismic waves while being transmitted from the base of the model to the ground surface. This behavior implies the vital role of foundation levels of structures adjacent to deep excavation projects in the behavior of the shoring system in dynamic cases. Moreover, modeling the structure adjacent to the excavation as a real structure, not only as a surcharge load, is very important and shows the significance of seismic soil–structure interaction (SSSI) in complex problems, such as deep excavations.

The damage degree of the diaphragm wall can be identified based on the residual lateral displacement of the diaphragm wall, as proposed by Gazetas [74]. The results show that the maximum dynamic lateral displacement at the top of the diaphragm wall falls between 0.087 and 0.14 m, indicating noticeable damage to the wall, according to the findings of Gazetas [74].

6.1.4. Straining Actions of the Diaphragm Wall and Its Supporting Struts

The foundation level of the structure adjacent to the deep excavation affects the straining actions of the diaphragm wall and axial forces of the supporting struts as well. Static and dynamic bending moments along the wall are presented in Figure 13a,b, respectively. The result pattern of the static bending moments shown in Figure 13a for the foundation level of 1.00 m is different from other foundation levels of 3.00 m and 5.00 m, especially at the second strut location. The bending moment at the second strut location decreases as the foundation level increases. The maximum static bending moment occurs at 87.5% H for all foundation levels. Moreover, it was also noticed that the maximum moment beyond the bottom of the excavation level occurred at depths of 15 to 25% D. On the other hand, the dynamic bending moment shown in Figure 13b reveals that the maximum bending moment takes place at the location of the second strut.

Furthermore, the static bending moment increased by 37.51% as the foundation level increased from 1.00 m to 3.00 m, while it decreased by 11.48% as the foundation level increased from 3.00 m to 5.00 m. Moreover, the dynamic bending moments decreased by 12.17% as the foundation level increased from 1.00 m to 3.00 m, while they increased by 7.96% as the foundation level increased from 3.00 m to 5.00 m.

Figure 13c,d show the static and dynamic shearing forces along the diaphragm wall for different foundation levels of the structure, respectively. As shown in the figures for both static and dynamic cases, the maximum shear force takes place at the location of the second strut. The results show that in the static case, the maximum shear force decreased by 40.15% as the foundation level increased from 1.00 m to 3.00 m, while it increased by 9.23% as the foundation level increased from 3.00 m to 5.00 m. Moreover, the maximum dynamic shear force increased by 1.39% and 5.40% as the foundation level increased from 1.00 m to 3.00 m and from 3.00 m to 5.00 m, respectively.

Figure 14 illustrates the ratio of the maximum dynamic bending moments to the static ones. The maximum dynamic moment is 4.29, 2.74, and 3.34 times the static one for the foundation levels of 1.00 m, 3.00 m, and 5.00 m depths, respectively. In addition, the figure also presents the ratio of the maximum dynamic shear force to the maximum static shear force. The results show that the dynamic shear force is 2.02, 3.41, and 3.29 times the static one for foundation levels of 1.00 m, 3.00 m, and 5.00 m depths, respectively.

The effect of the structure’s foundation level on the struts’ axial forces in both static and dynamic cases is illustrated in Figure 15. The results show that the static axial force of the first and second struts decreased by 29.40% and 48.00% as the foundation level increased from 1.00 m to 3.00 m, respectively. Moreover, the axial force of the first and second struts decreased by 24.00% and 11.80% as the foundation level increased from 3.00 m to 5.00 m, respectively.

Figure 15 also demonstrates that the dynamic axial forces of the first and second struts decreased by 25.00% and 11.00% as the foundation level increased from 1.0 m to 3.0 m, respectively. Furthermore, the dynamic axial force of the first strut increased by 68.00% as the foundation level increased from 3.00 m to 5.00 m, while it decreased by 7.00% for the second strut as the foundation level increased from 3.00 m to 5.00 m.

Figure 16 shows the effect of dynamic forces on axial forces of the struts for different foundation levels of the adjacent structure, compared to the static ones. The results show that for the first strut, there is nearly no change in the axial force in dynamic cases for a foundation level of 1.00 m, while it increased by 1.06 and 2.33 times the static ones as the foundation levels of the structure increased to 3.00 m and 5.00 m, respectively. On the other hand, the dynamic axial force of the second strut is 2.20, 3.77, and 3.96 times the static ones for foundation levels of 1.00 m, 3.00 m, and 5.0 m, respectively.

6.1.5. Response of the Structure Adjacent to the Shoring System

Figure 17a depicts the lateral displacement of the structure adjacent to the shoring system because of excavation for different foundation levels of the structure. The results indicated in this figure show that as the depth of the foundation increased, the lateral displacement of the structure decreased significantly due to the lateral confinement of the soil to the structure’s embedded depth and rigid diaphragm wall, which prevents excessive lateral displacements. At the base of the structure, as the foundation level increased from 1.0 m to 3.0 m depth, the lateral movement decreased by 52.20%, and as the foundation level increased from 3.00 m to 5.00 m, the lateral displacement decreased by 24.30%. Furthermore, the lateral displacement at the top roof of the structure decreased by 55.50% and 9.20% as the foundation level increased from 1.00 m to 3.00 m and from 3.00 m to 5.00 m depth, respectively.

The results of the lateral displacement of the building under dynamic load are shown in Figure 17b. A large amplification of the seismic waves and the stress relief due to excavation and, consequently, large lateral displacement of the diaphragm wall caused a large displacement of the adjacent structure. At the structure’s base, the dynamic lateral displacement increased significantly by 127.80% as the foundation level increased from 1.0 m to 3.0 m depth. However, as the foundation level increased from 3.0 m to 5.0 m depth, the lateral displacement increased by 15.90%. Moreover, the lateral displacement at the structure’s top roof increased by 19.70% as the foundation level increased from 1.0 m to 3.0 m depth, while it increased by 22.90% as the foundation level increased from 3.0 m to 5.0 m depth.

Figure 18 presents the ratio of the lateral displacement in dynamic cases to that in static cases at the base and top of the structure for different foundation levels of the structure. At the base of the structure, the dynamic lateral displacement is 0.64 times the static one at the foundation level of 1.00 m, whereas it is 3.06 and 4.69 times the static one for foundation levels of 3.00 m and 5.00 m, respectively. On the other hand, the dynamic lateral displacement at the top roof is 2.30, 6.18, and 8.37 times the static ones for the foundation levels of 1.00 m, 3.00 m, and 5.00 m, respectively. The results show that the top roof of the structure experienced large lateral displacements in dynamic cases due to large amplifications of the seismic waves while being transmitted from the base to the roof.

6.2. Effect of Earthquake Records on Shoring System–Structure Interaction

6.2.1. Displacement–Time History (DTH)

Figure 19a–c show the displacement–time history of the original ground motion at the base of the model (point A), free-field ground surface displacement (point B), top point of the diaphragm wall (point C), and top roof of the structure adjacent to the shoring system (point D), all shown and presented in Figure 5, respectively.

The maximum displacement amplitude of the Loma-Prieta earthquake, as shown in Figure 19a, is amplified by 2.10, 2.94, and 2.97 times the base ground motion for ground surface in the free field, on top of the diaphragm wall, and on the roof of the structure, respectively. Furthermore, the ground surface, top of the diaphragm wall, and roof of the structure have a permanent displacement of 18.37, −60.73, and −148.39 mm, respectively, at the end of the earthquake.

The ground surface responses, top of the diaphragm wall, and top roof of the structure under the Northridge earthquake record are illustrated in Figure 19b. The free-field ground surface, top of the wall, and top roof of the structure amplified the displacements to 1.70, 2.28, and 5.58 times the original one that was imposed at the base of the model, respectively. The results indicate that the ground surface, top of the wall, and top of the structure undergo permanent displacements of 112.31, −47.95, and −215.48 mm at the end of ground shaking, respectively.

The results of the El-Centro earthquake record are presented in Figure 19c. The amplifications of the ground surface, top of the wall, and top of the structure are 1.97, 2.23, and 2.89 times the original displacement of the earthquake, respectively. Moreover, the ground surface, top of the wall, and top of the structure experience permanent displacements of 33.71, −93.80, and −230.45 mm at the end of ground shaking, respectively.

The amplification ratio and the permanent displacement at the end of the earthquakes’ ground motions are illustrated in

Table 4. Amplification ratio and permanent displacements at the end of earthquakes.

Earthquake Record		Loma-Prieta, Mw = 6.9	Northridge, Mw = 6.7	El-Centro, Mw = 6.9
Ground surface	A,max (%)	210	170	197
	Ux,per (mm)	18.37	112.31	33.71
Top of the diaphragm wall	A,max (%)	294	228	223
	Ux,per (mm)	−60.73	−47.95	−93.8
Top of the structure	A,max (%)	297	558	289
	Ux,per (mm)	−148.39	−215.48	−230.45

A_max: Maximum amplification ratio. U_x,per: Permanent displacement at the end of the earthquake.

. The results indicate that the Loma-Prieta earthquake had a minimal effect on the ground surface displacement, while the Northridge earthquake had the largest effect on the ground surface displacement. However, the El-Centro and Northridge earthquakes had a noticeable effect on the diaphragm wall and the adjacent structure.

6.2.2. Lateral Displacement of the Shoring System

Static and dynamic lateral displacements of the diaphragm wall using the previously mentioned earthquake records are shown in Figure 20. The results indicate that the lateral displacement in static analysis has a deep inward movement pattern, and the maximum displacement occurred near the final excavation depth. Otherwise, the maximum lateral displacement in dynamic analysis takes place at the top of the wall for all earthquake records.

The ratio of the maximum lateral displacement at the top of the diaphragm wall in the dynamic analysis to that of the static analysis for different earthquake records is given in Figure 21. The results indicate that the lateral displacement at the top of the diaphragm wall in the dynamic analysis is 3.92, 3.67, and 6.18 times the static ones for the Loma-Prieta, Northridge, and El-Centro earthquake records, respectively. The results show that the El-Centro earthquake affected the wall more than the other two earthquakes, which is attributed to the fact that the El-Centro earthquake has a long duration, high frequency, and long period of strong shaking. Moreover, the predominant period of the El-Centro record is larger than the other two records, which implies that the predominant frequency of El-Centro is small and may be near the natural frequency of the diaphragm wall; this causes large displacements of the diaphragm wall, which may be attributed to some sort of resonance phenomenon.

The results show that the maximum dynamic lateral displacement at the top of the diaphragm wall falls between 0.048 and 0.094 m, indicating negligible damage to the wall itself, with noticeable damage to related structures at struts, according to the findings of Gazetas [74].

6.2.3. Settlement Trough Beside the Shoring System

Settlements behind the diaphragm wall resulting from static and dynamic analyses during the aforementioned earthquakes are depicted in Figure 22. The maximum settlement occurs along the diaphragm wall and extends to approximately twice the ultimate excavation depth, beyond which no notable changes in settlements are seen.

Settlements under dynamic analysis showed 5.10, 7.0, and 7.50 times the maximum static settlement for the Loma-Prieta, Northridge, and El-Centro records, respectively. The results of the dynamic study demonstrate that the ground surface experiences a uniform settlement of approximately 50 mm, equivalent to 0.50% of the final excavation depth. Furthermore, due to the existing buildings in proximity to the wall and the lateral displacements of the shoring system, along with stress release from the excavation operation, the structure is subjected to differential settlement and tilting towards the excavation region.

The results of maximum dynamic vertical settlement of the building adjacent to the excavation indicate that the building falls in a high-risk category, in which structural damage to the building is expected based on the Rankine guidelines [73].

6.2.4. Straining Actions of the Diaphragm Wall and Its Supporting Struts

Figure 23a,b illustrate the resulting straining actions along the diaphragm wall in both static and dynamic analyses. Dynamic bending moments along the diaphragm wall, as shown in Figure 23a, have the same pattern for different earthquakes, but the amplitude changes due to earthquake intensity and duration. The bending moment under the Loma-Prieta earthquake has the smallest value compared to the other earthquake records. The bending moment distribution pattern in the dynamic cases looks different from that in the static case due to the nature of the dynamic loads, whose values and direction change with time. The maximum static bending moment takes place at 87.5% H from the top of the diaphragm wall, whereas the maxium dynamic bending moment above the bottom of the excavation level occurs at the position of the second strut level and below the bottom of the excavation level by nearly 30% D. It was noticed that below the excavation level, El-Centro has a larger effect than other earthquakes.

Figure 23b presents the shearing forces in static and dynamic cases along the diaphragm wall. It was noticed that the shearing forces of different earthquakes have the same pattern. The shearing forces under the Loma-Prieta earthquake have the smallest values compared to the other earthquake records. The maximum shearing forces occurred at the location of the second strut.

Figure 24 shows the ratio of the maximum dynamic straining actions to the maximum static ones. The maximum wall bending moments under the Loma-Prieta, Northridge, and El-Centro records are 1.90, 3.04, and 2.93 times the static ones, respectively. Moreover, the maximum wall shear forces under the Loma-Prieta, Northridge, and El-Centro records are 2.34, 3.24, and 3.56 times the static ones, respectively. This increase emphasizes the importance of the seismic design of such structures.

Figure 25 shows the resulting axial force of the struts after static and dynamic analyses. As expected, the seismic loads cause large axial forces in the struts. The second strut exhibits a larger axial force than the first one due to the large earth pressure and shearing forces of the wall at this location.

A comparison of the axial forces of the struts under static and dynamic analyses is shown in Figure 26. The axial force in the first level of struts is 1.52, 1.94, and 1.64 times the static one under the Loma-Prieta, Northridge, and El-Centro earthquakes, respectively. Moreover, the axial force in the second level of struts is 4.39, 7.04, and 6.55 times the static one under the Loma-Prieta, Northridge, and El-Centro earthquakes, respectively. The results indicate that the Northridge and El-Centro earthquakes have a larger effect on the struts’ axial force than the Loma-Prieta earthquake.

6.2.5. Response of the Structure Adjacent to the Shoring System

Figure 27 presents the structure’s lateral displacement in static and dynamic cases. The structure undergoes a lateral displacement in the static case due to the excavation process because of stress relief resulting from removing soil adjacent to the structure and lateral deformation of the diaphragm wall due to induced lateral earth pressure.

A comparison of the resulting maximum story displacement at the base and top of the structure in static and dynamic cases is shown in Figure 28. The lateral displacement of the base of the structure is 3.87, 1.75, and 4.12 times the static one under the Loma-Prieta, Northridge, and El-Centro earthquakes, respectively. However, the lateral displacement of the roof in the dynamic analysis is 4.78, 6.74, and 6.79 times the static one for the Loma-Prieta, Northridge, and El-Centro earthquakes, respectively.

The large lateral displacement of the structure in dynamic cases is due to a large lateral displacement at the top of the diaphragm wall in the dynamic analysis. The results emphasize the seismic soil–structure interaction between the soil, shoring system, and adjacent structures.

7. Conclusions

In this study, 2D numerical non-linear time history analysis using PLAXIS 2D is performed to investigate the seismic performance of deep braced excavation in sandy soil. This study focuses on the seismic soil–structure–excavation interaction to assess the seismic susceptibility of the excavation support system and the adjacent structure. The model is validated against a real case study of deep braced excavation in silty–sandy soil. The parametric study includes the following variables: foundation level of the structure adjacent to excavation and the response of the shoring system and adjacent structure under the effect of the acceleration time histories of Loma-Prieta, Northridge, and El-Centro. Based on the analysis and discussion of this research, the following conclusions are drawn:

(a) 

Seismic performance of the entire system:

• As the building’s foundation level adjacent to the excavation increases, the permanent ground surface displacement decreased significantly, while the displacement at the top of the diaphragm wall and top of the structure increased significantly.
• The Loma-Prieta earthquake had a minimal effect on the soil at the ground surface, while the Northridge earthquake had a greater effect on the ground surface. However, the El-Centro and Northridge earthquakes had a great effect on the diaphragm wall and the adjacent structure.
• Settlements under dynamic analysis were 5.10, 7.0, and 7.50 times the maximum static ones under the Loma-Prieta, Northridge, and El-Centro records, respectively.

(b) 

Seismic performance of the shoring system and its supporting elements:

• The dynamic moment increased by an average of 3.32 times the static one, while the average increase in shear force under dynamic conditions was 2.85 times that under static conditions.
• The average increases in the axial force of the two strut levels because of dynamic forces were 1.38 and 3.17 for the first and second struts, respectively, compared to static ones.
• The maximum bending moments under the Loma-Prieta, Northridge, and El-Centro records were 1.90, 3.04, and 2.93 times the static ones, while the maximum wall shear forces under the Loma-Prieta, Northridge, and El-Centro records were 2.34, 3.24, and 3.56 times the static ones, respectively.
• The axial forces in the first level of struts were 1.52, 1.94, and 1.64 times the static ones under the Loma-Prieta, Northridge, and El-Centro earthquakes, respectively. However, the axial force in the second level of struts was 4.39, 7.04, and 6.55 times the static one, respectively.

(c) 

Seismic performance of the structure adjacent to the shoring system:

• The dynamic lateral displacement at the top roof was 2.30, 6.18, and 8.37 times the static ones as the foundation level increased from 1.00 m to 5.00 m depth, respectively.
• The lateral displacements of the base of the structure were 3.87, 1.75, and 4.12 times the static ones under the Loma-Prieta, Northridge, and El-Centro earthquakes, respectively. On the other hand, the lateral displacements at the top roof in the dynamic analysis were 4.78, 6.74, and 6.79 times the static ones, respectively.