Plastic Energy Density-Based Time-Dependent Seismic Performance Evaluation for Underground Structures

Abstract

Large underground structures such as subway stations are critical components of urban lifeline systems, but their seismic deformation and damage mechanisms remain not fully understood. This paper investigates the seismic response and damage mechanism of a typical two-story three-span underground frame structure in soft soil using three-dimensional nonlinear finite element analysis. The deformation transmission between the structure and surrounding soil is examined, the frequency response characteristics of the soil-structure system are revealed, and the energy dissipation and damage evolution are systematically analyzed. The results show that the predominant frequency of structural relative displacement is governed by both the free-field site and the structure itself. Soil deformation imposes forced displacement on the structure and suppresses its natural vibration. Seismic input energy is mainly dissipated by damping and plastic strain energy, with central columns and slabs as the primary plastic energy-dissipating components. The most severe damage occurs at the base of the lower columns. A plastic energy density-based damage index is proposed, which offers a supplementary perspective for time-dependent cumulative damage assessment of underground structures.

1. Introduction

Underground structures are an integral part of the modern metropolis, and their construction has expanded rapidly with urbanization to serve diverse functions. For a long time, large underground structures were considered seismically safe due to the confinement provided by surrounding soil. However, the Daikai subway station suffered catastrophic damage during the 1995 Great Hanshin earthquake [1,2]. Additionally, damage to underground structures was also observed in the 1976 Tangshan earthquake [3], the 1999 Parnitha earthquake [4], and the 1999 Chichi earthquake [5]. Moreover, a set of historical damage cases has been summarized by Lanzano [6]. These seismic damages exposed some important weak points of underground structures under strong earthquakes [7,8,9,10].

Damage to large underground structures can disrupt critical lifeline services (e.g., transportation and communication systems) and severely impact urban operations. Clarifying the seismic performance and damage mechanism of large underground structures is therefore critical in seismic-prone regions. Various methods have been proposed for seismic performance analysis of underground structures [11,12,13]. Among them, the integral three-dimensional (3D) simulation method, despite its high computational cost and complexity, is widely recognized as one of the most widely accepted approaches for large underground structures [14,15,16,17,18].

The seismic behavior of underground structures differs fundamentally from that of aboveground structures, as it is governed by seismic soil deformation rather than inertial forces [18,19]. The presence of underground structures restrains soil movement during earthquakes, while soil deformation in turn imposes forced displacement on the structure. Quantifying the respective influences of soil deformation and structural inertial forces is a key issue in underground seismic analysis. Dynamic soil-structure interaction (SSI) is a core factor governing the seismic response of underground structures [20,21]. Wang et al. [22] summarized the SSI effect between underground structures and soil. Parra-Montesinos et al. [23] evaluated SSI effects and drift demands on the Daikai subway station. Hashash et al. [24] characterized SSI between an underground box structure and surrounding soil. Xu et al. [25] analyzed response characteristics of shallow-buried underground structures under strong earthquakes considering SSI. Sayed et al. [26] simulated the SSI of the Daikai station using a multi-platform approach. Huang et al. [27] presented a framework for seismic fragility assessment of tunnels using an integrated SSI finite element model. However, research on the dynamic interaction between soil and large underground structures remains insufficient. In particular, the mechanism of deformation transmission between underground structures and soil, and how this mechanism evolves with increasing earthquake intensity, require further clarification.

Underground structures exhibit varying damage states under different earthquake intensities, and their damage mechanisms are complex [28,29]. Essentially, structural damage is driven by energy dissipation. Thus, analyzing structural energy responses offers a supplementary perspective for understanding underlying damage mechanisms [30,31,32]. Léger and Dussault [33] examined seismic energy dissipation in multi-degree-of-freedom (MDOF) structures. Wong [34] investigated energy transfer and dissipation in inelastic structures with tuned mass dampers. Quinde et al. [35] applied an energy-based approach to assess cumulative structural damage from low-cycle fatigue. Deniz et al. [36] proposed an energy-based seismic collapse criterion for ductile planar frames. Mahboubi et al. [37] proposed an input energy-based damage index for RC bridge piers. Aliakbari et al. [38] used an energy-based method for the seismic design of frames with metallic yielding dampers. Existing energy-based damage studies primarily focus on aboveground frame structures and shield tunnels, whereas research on the energy dissipation mechanisms and quantitative damage assessment of large-span underground frame structures (e.g., subway stations) remains limited. Traditional deformation-based indicators can only reflect instantaneous responses and cannot capture the cumulative damage effect throughout the earthquake process. Constrained by the surrounding soil, underground structures accumulate damage primarily through fatigue, and instantaneous deformation indicators tend to underestimate damage in the later stages of ground motion, necessitating further investigation. Energy-based indices offer a supplementary perspective for damage assessment of underground frame structures.

This study employs nonlinear 3D finite element (FE) analysis to investigate the deformation and damage mechanisms of underground structures. The deformation transmission between underground structures and the surrounding soil is investigated, and the influence of the resonance characteristics of underground structures is revealed. Furthermore, the damage mechanism of underground structures is carefully studied and evaluated through analysis of structural energy distribution and evolution. Compared with existing energy-based seismic damage indices, the novelty of this study is threefold. First, the element-level plastic energy density (ELPD) is adopted as the basic metric, which eliminates the interference of component volume differences and can accurately locate energy-concentrated vulnerable positions. Second, a global damage index weighted by element-level plastic energy dissipation is established; unlike volume-averaged methods, it highlights the controlling effect of energy-concentrated key components on overall structural safety, which is more consistent with the failure mode of underground frame structures. Third, the temporal evolution of plastic energy dissipation is revealed, enabling time-dependent cumulative damage assessment throughout the entire earthquake process, thereby compensating for the deficiency of traditional deformation-based indicators that only reflect instantaneous responses.

2. Theoretical Analysis of Underground Structures

2.1. Soil–Underground Structure Interaction System

The soil–underground structure interaction system analyzed in this study consists of a typical underground frame structure embedded in a soft-soil site. Seismic action on an underground structure arises from both its own dynamic characteristics and external forces exerted by surrounding soil. As established in prior studies, finite element analysis is one of the most effective methods for addressing dynamic soil–underground structure interaction under seismic excitation. Seismic excitation is applied at the bottom of the system, and the equations of motion and equilibrium are solved at discrete time increments. The governing equations of the entire interaction system can be simplified as:

$$\left[M\right]\left\{\ddot{x}\right\}+\left[C\right]\left\{\dot{x}\right\}+\left[K\right]\left\{x\right\}=-\left[M\right]\left\{I\right\}{\ddot{x}}_g$$

(1)

where $\left\{\ddot{x}\right\}$, $\left\{\dot{x}\right\}$, and $\left\{x\right\}$ are the nodal vectors of relative acceleration, velocity, and displacement, respectively. ${\ddot{x}}_g$ is the seismic input at the model bottom. $\left\{I\right\}$ is the influence vector which represents the relationship between input direction and model degrees of freedom; $\left[M\right]$, $\left[C\right]$, and $\left[K\right]$ are the matrices of mass, damping, and stiffness, respectively.

The AZIH seismic record, obtained from the 1985 Mexican earthquake, is adopted as the induced shaking. This record is selected because it caused severe damage to structures on soft soil sites during the event. It has a peak ground acceleration (PGA) of 0.162 g and a duration of approximately 70 s. Its acceleration time history and Fourier amplitude spectrum are shown in Figure 1a and Figure 1b, respectively. The predominant frequency of the AZIH record is 0.711 Hz, with dominant frequency components ranging from 0 to 15 Hz.

Figure 1. Acceleration time history and Fourier amplitude spectra of AZIH.

2.2. Numerical Simulation Method

The numerical soil–underground structure interaction model was built using the general finite element software ABAQUS2022, as shown in Figure 2. To ensure computational accuracy, a 3D simulation technique validated in prior studies is adopted [8,9,10], and five spans are truncated in the longitudinal direction (covering 5 column bays). The bilinear plastic model and the concrete damaged plasticity model were adopted to simulate the dynamic behavior of steel reinforcement and structural concrete, respectively. Eight-node linear solid elements with reduced integration (C3D8R) are adopted to simulate concrete components, whereas two-node three-dimensional truss elements (T3D2) are employed to represent steel reinforcements. Reinforcements are embedded in concrete using the embedded-region constraint. Similarly, the soil domain is meshed using C3D8R solid elements, and the total number of elements is 419,125. Specifically, the characteristic element size for structural components is approximately 0.2–0.5 m, and refined to 0.1 m at critical regions such as column ends and slab-column connections. Considering the mesh generation of the soil–underground structure interaction system, the vertical dimension of soil elements $l_e$ is determined by

$$l_e<{\displaystyle \frac{v_s}{8f_{NF}}}$$

(2)

where $f_{NF}$ is the cutoff frequency at 25 Hz and $v_s$ is the shear wave velocity. In the horizontal direction, the element size of soil is constrained such that the horizontal edge length does not exceed five times its vertical element dimension.

Figure 2. Finite element models of the integral SSI system, the structural concrete, the structural bars, and the partial SSI system.

The stress–strain relationship under dynamic loading is described by a nonlinear dynamic constitutive model based on the classic Davidenkov model. Details and validation of this model are available in prior work [8,39]; briefly, the tangent shear modulus over the full loading process is expressed as:

$$G^{t+∆t}={\displaystyle \frac{\partial {\tau }_{oct}^{t+∆t}}{\partial {\gamma }_{oct}^{t+∆t}}}=\left\{\begin{array}{l}{G}_{\max }{\left[1-\left[1+{\displaystyle \frac{2\cdot A\cdot B}{1+{\left|\frac{{\gamma }_{oct}}{{\gamma }_0}\right|}^{2B}}}\right]\cdot H\left(\left|{\gamma }_{oct}\right|\right)\right]}_{t+∆t}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{initial} \mathrm{loading}\hspace{1em}\\ G_{\max }{\left[1-\left[1+{\displaystyle \frac{2\cdot A\cdot B}{1+{\left|\frac{{\gamma }_{oct}-{\gamma }_{oct,c}}{2{\gamma }_0}\right|}^{2B}}}\right]\cdot H\left(\left|{\displaystyle \frac{{\gamma }_{oct}-{\gamma }_{oct,c}}{2}}\right|\right)\right]}_{t+∆t} \mathrm{unloading}/\mathrm{reloading}\end{array}\right.$$

(3)

where $G_{\max }$ is the maximum shear modulus of soil; ${\tau }_{oct}$ and ${\gamma }_{oct}$ are shear stress and shear strain of soil, respectively; $G^{t+∆t}$ and $G^t$ are the tangent shear modulus at the end of the current increment and the end of the previous increment, respectively; ${\gamma }_{oct,c}$ is the shear strain at the last load reversal; and $H\left(\gamma \right)$ is the shape function:

$$H\left(\gamma \right)={\left\{{\displaystyle \frac{{\left(\gamma /{\gamma }_0\right)}^{2B}}{1+{\left(\gamma /{\gamma }_0\right)}^{2B}}}\right\}}^A$$

(4)

where regression coefficients $A$, $B$, and ${\gamma }_0$ of the Davidenkov model for various Shanghai soft-soil strata are provided in Reference [8]. Such coefficients have been validated in previous seismic site response simulations [8,9,10].

The contact behavior between soil and the underground structure is modeled by the frictional surfaces, and a surface-to-surface contact technique is introduced in this simulation. As shown in Figure 3, in the normal direction, the contact pressure can be transferred mutually on different surfaces through the contact constraint; in the tangential direction, the contact shear stress is transferred between the boundary surfaces, and when the contact shear stress in the interface exceeds the critical shear stress ${\tau }_c$ (${\tau }_c={\mu }_{p_c}\cdot N_{p_c}$, where $N_{p_c}$ is the normal contact stress and ${\mu }_{p_c}$ is the friction coefficient, set to 0.40 for the soil-concrete interface [8,40]), slipping will occur.

Figure 3. Surface-to-surface contact behavior between soil and the structure.

To address the static-dynamic coupling boundary conditions, an artificial constrained boundary [8] was utilized, as shown in Figure 4. In the first step, geostatic analysis is conducted with horizontal constraints on site boundaries. Before the dynamic analysis step, the horizontal boundary constraints are replaced by the calculated horizontal reaction forces, while vertical constraints are retained. In the dynamic analysis, seismic waves are input at the bottom of the interaction system in the horizontal transverse direction (perpendicular to the longitudinal axis of the station), following the base-input excitation scheme. This study focuses on the relative laws under different seismic intensities without deconvolution, and this input method is widely used in the integral dynamic analysis of underground structures. In addition, to prevent the propagating wave from reflecting off the model, the side boundaries are placed approximately ten times the soil depth away from the structure, and the bottom boundary is located more than three times the structural height below the structure’s base slab [41]. Free-field calculations verify that this boundary condition yields satisfactory seismic wave transmission with negligible boundary reflection. In addition, symmetric constraints are applied on both longitudinal faces.

Figure 4. Transformation of the boundary condition during the analysis.

Referring to [13], a damping ratio of 3% relative to critical damping is assigned for the first vibration mode of the underground concrete structure. Stiffness-proportional Rayleigh damping is implemented herein. The coefficient η, which yields the target critical-damping fraction ξ₁ for the first mode, can be expressed as:

$$\eta =2{\xi }_1/{\omega }_1$$

(5)

where ${\xi }_1=3\%$ and ${\omega }_1$ is the first eigenfrequency.

Regarding the nonlinear seismic response of the soil site, soil hysteretic damping is intrinsically incorporated in the dynamic constitutive description, while extra viscous damping for soil is computed separately. The viscous damping matrix is established based on Rayleigh damping theory:

$$\left[C\right]={\alpha }_0\left[M\right]+{\alpha }_1\left[K\right]$$

(6)

In this study, only the stiffness matrix is involved in constructing the damping matrix $\left[K\right]$ according to Reference [13]. The corresponding damping coefficients are solved via:

$${\alpha }_0=0, {\alpha }_1=2{\xi }_1/{\omega }_1$$

(7)

where ${\xi }_1$ stands for the damping ratio of the fundamental mode, ranging from 2% to 5% (3% in this study), and ${\omega }_1$ is the natural frequency of the soil-structure interaction system.

Implicit dynamic analysis is performed using the Newmark-β integration algorithm, with an initial time step of 1 × 10⁻⁴ s, a maximum time step of 2 × 10⁻² s [8,12,15]; and a force residual convergence criterion of 5‰.

To conduct quantitative mesh convergence analysis, three mesh schemes coarse (0.4–1.0 m for structural components and 0.2 m refined at critical regions), medium (0.2–0.5 m for structural components and 0.1 m refined at critical regions) and fine (0.1–0.2 m for structural components and 0.05 m refined at critical regions) were designed for convergence verification. The key results under PGA = 0.40 g are compared as follows:

It can be observed from Table 1 that the relative errors in the maximum RD, total plastic energy, and ELPD response between the adopted medium mesh and fine mesh are all below 5%, satisfying the accuracy requirements for engineering calculations. Accordingly, these mesh convergence results verify the validity of the numerical simulation method employed in this study.

Table 1. Mesh convergence verification.

Quantity	Coarse Mesh	Medium Mesh	Fine Mesh	Relative Error
	279,415	419,125	628,690	(Coarse vs. Fine)	(Medium vs. Fine)
Maximum RD (m)	0.0494	0.0465	0.0448	10.27%	3.79%
Total plastic energy (J)	1.15 × 106	1.23 × 106	1.29 × 106	10.85%	4.55%
ELPD (J/m3)	904	981	1025	11.80%	4.29%

Furthermore, the enhanced element technique is adopted to suppress hourglass energy. Hourglass energy of the entire model is tracked throughout the analysis. For all loading cases, the ratio of hourglass energy to the total seismic input energy remains less than 5%. This value falls within the acceptable threshold for reduced-integration elements and exerts a negligible effect on the energy balance.

3. Interaction Behavior of Underground Structure and Soil

3.1. Prototype of Underground Structure and Soil

A typical two-layer, three-span reinforced concrete underground structure is adopted as a prototype to investigate soil-structure interaction behavior and energy-based damage mechanism. The cross-section dimensions and reinforcement layout of the station are shown in Figure 5. The width and height of the station are 20.40 m and 13.11 m, respectively. The size of the column is 0.7 m × 1.0 m, and the longitudinal distance between two columns is 8.0 m. The thicknesses of the top, middle, and bottom slabs are 0.90 m, 0.45 m, and 1.00 m, respectively. The top of the station is located at 2.75 m below the ground surface. In this simulation, the underground diaphragm wall, whose thickness is 0.8 m and depth is 30 m, is considered.

Figure 5. The cross-section dimension and distributed rebar of the station.

The structural concrete (subway station and underground diaphragm wall) is grade C35, with a Young’s modulus of 31.5 GPa, Poisson’s ratio of 0.18, and density of 2450 kg/m³. The reinforcement layout is shown in Figure 5b. Steel bars’ yield stress, Young’s modulus, Poisson’s ratio, and density are 0.30 GPa, 2.06 × 10² GPa, 0.30, and 7850 kg/m³, respectively. For CDP parameters, dilation angle ψ = 30°, eccentricity ε = 0.1, ratio of biaxial to uniaxial compressive strength σ_b0/σ_c0 = 1.16, ratio of the second stress invariant on the tensile meridian to that on the compressive meridian K_c = 0.667, viscosity parameter μ = 0.0005; uniaxial tensile strength f_t = 2.2 MPa, uniaxial compressive strength f_c = 35 MPa. Following the fracture-energy-based damage and stiffness degradation framework of continuum damage mechanics, isotropic damage elasticity coupled with isotropic tensile and compressive plasticity was adopted in the concrete model to represent the inelastic behavior. Two damage variables, $d_t$ and $d_c$, were introduced to account for tensile damage and compressive damage states, respectively. Referring to the Chinese code GB50011-2010 [42], the stress–strain relationship and damage variable–strain relationship of the C35 concrete are shown in Figure 6.

Figure 6. Characteristics of the concrete material model.

A bilinear model is used for steel reinforcement, with yield strength f_y = 300 MPa, hardening tangent modulus taken as 0.01E_s. A typical Shanghai soft soil site is selected, and the 70 m depth of soil deposit is simplified as nine horizontal soil layers [43]. The original profile properties of each soil layer are listed in Table 2. The location of the underground structure within the soil layers is shown in Figure 7. Therefore, the total model dimensions are 700 m (width) × 40 m (longitudinal) × 70 m (depth). The Davidenkov constitutive model parameters for different soil layers are given in Table 3.

Table 2. Original soil profile properties.

Layer	Type	Layer Thickness (m)	Density (kg/m3)	Poisson’s Ratio	Shear Wave Velocity (m/s)	Maximum Shear Modulus (MPa)
1	Fill	2	1960	0.44	110	24.20
2	Plain fill	4	1950	0.45	120	28.65
3	Silty clay	16	1940	0.4	150	44.54
4	Silty clay	14	1980	0.4	198	79.21
5	Stony clay	8	1950	0.38	230	105.26
6	Silty clay	10	1990	0.42	214	92.99
7	Silt	6	1980	0.4	343	237.70
8	Silty clay	4	1960	0.38	416	346.11
9	Gravel	6	1960	0.31	481	462.72

Figure 7. Location of the underground structure in the soil layers.

Table 3. Davidenkov constitutive model parameters for different soil types.

Soil Layer	Soil Type	A	B	γ0
1, 2, 5	Fill/Clay	1.01	0.81	2.00
3, 4, 6, 8	Silty clay	1.01	0.50	5.10
7	Silt	1.12	0.94	4.90

Structural deformation is a key indicator for evaluating the seismic performance of underground structures. Embedded in soil, underground structures are affected by both structural and soil properties. To investigate the deformation mechanism, both the modal analysis and time history analysis are performed. In the time history analysis, eight input PGA levels were considered: 0.035 g, 0.07 g, 0.10 g, 0.20 g, 0.40 g, 0.60 g, 0.80 g, and 1.0 g. It should be noted that the quantitative laws and specific thresholds obtained in this study are based on the selected AZIH record and Shanghai soft soil site, which are specific to this kind of soil-ground motion configuration and are not universally applicable.

As a preliminary study and for validation purposes, the seismic responses of the Daikai subway station that collapsed in the 1995 Great Hanshin earthquake were simulated as a reference using the three-dimensional numerical simulation method employed in this study. The specific seismic responses, including the maximum shear displacement, shear force, and axial force of the central column as well as their relative errors are summarized in the following Table 4 and compared with existing studies [23,44,45,46]. In addition, the relative error between the results of this study and the reference data is defined as:

Error(%) = [∣X_FE − X_ref∣/X_ref] × 100

(8)

where X_ref denotes the reference value from published literature, and X_FE denotes the finite element result obtained in this study.

Table 4. Numerical simulation comparison of seismic response of Daikai subway station.

	Shear Force		Deformation		Axial Force		Simulation Method
	Value (kN)	Error	Value (cm)	Error	Value (kN)	Error
This study	669	-	3.8	-	4898	-	3-dimensional
Huo et al. [46]	-	-	About 4	5%	4200	16.6%	2-dimensional
Parra-Montentesinos et al. [23]	730	8.36%	About 4	5%	4800	2.04%	2-dimensional
Chen et al. [44]	829	19.3%	3.2	18.7%	5103	4.02%	2-dimensional
He and Chen [45]	697	4.02%	3.2	18.7%	4561	7.38%	3-dimensional

In addition, the Mises stress contour plots and dynamic time-history response of the underground structure at the time of peak ground acceleration are shown in the following Figure 8, where the central column ends are the most vulnerable during an earthquake. These simulation results are consistent with the seismic investigations in previous studies [47]. These consistent simulation results verified the accuracy of the 3D nonlinear SSI numerical modeling framework, including the constitutive models, boundary treatments and solution algorithms, and confirmed the reliability of the numerical framework. However, this numerical verification should be distinguished from direct validation of the proposed damage index. The proposed ELPD-based damage index and its four-level damage thresholds are established based on the physical meaning of the CDP constitutive model and engineering experience. They are preliminary model-dependent classifications and have not been directly validated by the Daikai case or other experimental or field data. Further calibration and validation are required in future work.

Figure 8. Numerical simulation results of the Daikai subway station. (a) Mises stress contour plot of the Daikai subway station at the time of maximum peak ground acceleration. (b) Relative displacements between the top and the bottom of the central column. (c) Shear forces of the bottom of the central column. (d) Axial forces of the central column.

3.2. Modal Analysis and Comparison

Modal characteristics are critical dynamic properties of the soil–underground structure interaction system, and they help predict structural seismic behavior. Modal results, particularly the natural frequency of the interaction system, are also used to establish the damping model for subsequent time-history analysis. The fundamental frequency of the soil–underground structure interaction system is 0.884 Hz. The first mode shape (Figure 9) indicates that the structure undergoes primarily shear deformations. The free-field site has a fundamental frequency of 0.879 Hz, slightly lower than that of the interaction system.

Figure 9. First mode shape of the integral SSI system.

The underground structure has lower mass but higher stiffness than the excavated soil it replaces. According to the natural frequency empirical formula, reduced mass and increased stiffness raise the fundamental frequency of the integrated SSI system. However, since the underground structure occupies a small volume in the overall system, its influence on the system’s fundamental frequency is minimal. Notably, while this effect is negligible in the present study, it may become significant for extremely large underground structures.

To isolate the influence of surrounding soil on structural dynamic characteristics, modal analysis of the structure alone (with fixed base) was also performed (first mode shape shown in Figure 10), yielding a fundamental frequency of 2.48 Hz.

Figure 10. First mode shape of the underground structure.

Table 5 lists the predominant frequency of the relative displacement (RD) time-history response between the top and bottom slabs ($RD=u_{top-slab}-u_{bottom-slab}$, where u is the lateral displacement of the corresponding slab) and the fundamental frequency of the free-field site, at different input intensities. As PGA increases, soil rapidly enters a nonlinear state, reducing the free-field fundamental frequency. The predominant frequency of the RD time history is governed by both soil and structural properties, and lies between the two fundamental frequencies. At low input intensities (PGA ≤ 0.10 g), the predominant frequency of the RD time history is strongly influenced by structural properties. As intensity increases, it shifts rapidly toward the free-field fundamental frequency. At PGA = 0.20 g, the predominant frequency of the RD time history approximates the fundamental frequency of the free-field site, which is consistent with the expectation that structural natural vibration is largely suppressed. For PGA ≥ 0.20 g, combined with the characteristic that the interstory displacement frequency converges to the site fundamental frequency, it can be reasonably inferred from these frequency characteristics that lateral earth pressure from surrounding soil is likely to be the dominant seismic action of the structure, while structural inertial force has a relatively significant contribution at low PGA. It should be emphasized that this interpretation is a logical inference based on frequency convergence and deformation coefficient characteristics, rather than a directly proven conclusion via force decomposition. Direct quantitative force decomposition will be carried out in follow-up research to further verify this load transfer mechanism.

Table 5. Fundamental frequency of the free-field site and the predominant frequency of RD time history.

Type	PGA (g)
	0.035	0.07	0.10	0.20	0.40	0.60	0.80	1.0 g
Fundamental frequency of the free-field site (Hz)	0.836	0.799	0.771	0.707	0.616	0.571	0.537	0.518
Predominant frequency of RD time history (Hz)	2.17	1.852	1.785	0.801	0.625	0.595	0.588	0.581

The decrease in the predominant frequency of interstory displacement with increasing PGA is a combined result primarily attributed to nonlinear soil softening, with structural damage and SSI effects as secondary contributors. The increasing seismic intensity leads to the attenuation of soil shear modulus and the decrease in site fundamental frequency. Under high PGA, the structural response appears to be governed by site deformation, so the response frequency decreases synchronously with the site fundamental frequency. In addition, structural damage leads to a slight decrease in its own stiffness and natural frequency, which makes a minor contribution to the response frequency. The SSI effect is reflected in the suppression of structural natural vibration by soil; the suppression becomes stronger with higher intensity, driving the response frequency to converge to the site fundamental frequency.

3.3. Displacement Response Characteristics

The structural RD time histories of the upper layer, the lower layer, and the whole structure under different PGAs are plotted in Figure 11. Although RD amplitudes increase dramatically with PGA, the waveforms for the upper layer, lower layer, and whole station are nearly identical under the same input intensity. However, frequency components differ across input intensities.

Figure 11. Structural relative displacement time histories under different induced shaking intensities.

To illustrate this, the normalized Fourier spectra of the whole station’s RD time-history response at different intensities are shown in Figure 12. The predominant frequency of the whole station’s RD time history shifts toward lower values as PGA increases. Moreover, higher PGA leads to a narrower frequency bandwidth; high-frequency components are filtered out while low-frequency components are amplified.

Figure 12. Normalized Fourier spectra for structural relative displacement time history of the whole station.

3.4. Displacement-Based Interaction Behavior

The primary difference between underground and aboveground structures is that underground structures are dominated by surrounding soil pressure rather than inertial forces during strong earthquakes. Soil deformation imposes forced displacement on the structure. To clarify deformation transmission, the soil-structure interaction coefficient $\beta$ is defined as:

$$\beta ={∆}_{structure}/{∆}_{soil}$$

(9)

where ${∆}_{structure}$ and ${∆}_{soil}$ are the lateral deformation of the structure and the free-field site, respectively [see Figure 13]. When $\beta >1$, structural dynamic deformation is restrained by the surrounding soil (beneficial effect). When $\beta <1$, structural deformation is smaller than free-field deformation, meaning soil deformation imposes forced displacement on the structure (unfavorable effect).

Figure 13. Lateral deformation of the free-field and the underground structure.

Figure 13 compares the deformations of the underground structure and free-field site under different input intensities. The lateral deformation profiles exhibit similar shapes along the structural height, but free-field deformation is significantly larger than structural deformation, particularly at high input intensities. Quantitative values of ${∆}_{structure}$, ${∆}_{soil}$, and $\beta$ are listed in Table 6. For the relatively soft site in this study, β < 1 at all intensities, indicating that the soil imposes forced displacement on the structure. With PGA increasing from 0.035 g to 0.40 g, the soil-structure interaction coefficient β decreases rapidly from 0.875 to 0.576. When PGA exceeds 0.40 g, the nonlinear softening of soil becomes sufficiently developed, and β tends to stabilize at approximately 0.58, showing a two-stage variation characteristic of a rapid decrease first and then stabilization. This indicates that soil nonlinearity intensifies with increasing seismic intensity, and soil stiffness degrades more significantly and rapidly than structural stiffness. Consequently, the deformation gap widens, and the forced displacement effect of soil on the structure becomes more pronounced. Coupled with the aforementioned frequency convergence, the macroscopic response characteristics indicate that structural inertial forces contribute significantly at low PGA, while forced displacement induced by lateral soil pressure tends to govern structural response at high PGA. This deduction is based on observed macroscopic response characteristics, including frequency convergence and variation in the soil-structure interaction coefficient, rather than direct force decomposition. Direct quantitative force decomposition will be carried out in follow-up research to further verify this load transfer mechanism. In addition, for the present soft soil site and AZIH ground motion configuration, the transition value of PGA = 0.40 g is a case-specific observation rather than a universally applicable seismic threshold.

Table 6. Comparison of maximum lateral deformation of the underground structure and the free-field site.

PGA(g)	${∆}_{structure}$(m)	${∆}_{soil}$(m)	β	Relationship between β and PGA
0.035	0.0049	0.0056	0.875
0.07	0.0089	0.0117	0.761
0.10	0.0148	0.0197	0.751
0.20	0.0281	0.0393	0.724
0.40	0.0465	0.0807	0.576
0.60	0.0816	0.1385	0.589
0.80	0.114	0.1942	0.587
1.00	0.144	0.2462	0.585

4. Energy-Based Damage Mechanism of Underground Structure

4.1. Energy Balance Equations

This study adopts element plastic strain energy density (ELPD) instead of total plastic energy, which eliminates the influence of component volume differences and can locate vulnerable components with concentrated energy, providing a useful complementary approach for damage assessment of underground frame structures with significant differences in component sizes.

Furthermore, this study develops a global damage index weighted by plastic energy density, which overcomes the limitation of traditional volume-weighted methods that cannot capture the contribution of critical components. This formulation is more consistent with the failure mode of underground frame structures, where failure is governed by key components.

In addition, it reveals the temporal evolution law of plastic energy-dissipating components under different seismic intensities, realizing time-dependent damage assessment throughout the earthquake process, which can provide a basis for dynamic performance evaluation during earthquakes.

For large underground structures, their energy changes during earthquakes satisfy the first law of thermodynamics [48]:

$${\displaystyle \frac{d}{dt}}{\displaystyle \underset{V}{\int }\left(\frac{1}{2}\rho \overrightarrow{v}\cdot \overrightarrow{v}+\rho U\right)}dV={\displaystyle \underset{S}{\int }\overrightarrow{v}\cdot \overrightarrow{t}dS +}{\displaystyle \underset{V}{\int }\overrightarrow{f}\cdot \overrightarrow{v}dV}$$

(10)

where $\rho$ is the density of the structure; $\overrightarrow{v}$ is the velocity vector of the structure; $U$ is the internal energy per unit mass; $\overrightarrow{t}$ is the traction vector on the surface; $\overrightarrow{f}$ is the body force vector, and $\overrightarrow{n}$ is the normal vector on the boundary $S$. Integrating Equation (10) gives:

$${\displaystyle \underset{V}{\int }\frac{1}{2}\rho \overrightarrow{v}\cdot \overrightarrow{v}}dV+{\displaystyle \underset{V}{\int }\rho U}dV={\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}{\dot{E}}_{WF}}d\tau +constant$$

(11)

where ${\dot{E}}_{WF}={\displaystyle \underset{S}{\int }\overrightarrow{v}\cdot \overrightarrow{t}dS+}{\displaystyle \underset{V}{\int }\overrightarrow{f}\cdot \overrightarrow{v}dV}$ is the rate of work done to the structure by external forces and contact forces between the interfaces; $constant$ is a constant generated during the integration process and has no specific meaning. $E_K={\displaystyle \underset{V}{\int }\frac{1}{2}\rho \overrightarrow{v}\cdot \overrightarrow{v}}dV$ is the kinetic energy of the underground structure, and $E_U={\displaystyle \underset{V}{\int }\rho U}dV$ is the internal energy of the underground structure at time $t$, then, Equation (10) can be rewritten as:

$$E_K+E_U={\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}{\dot{E}}_{WF}}d\tau +constant$$

(12)

Practically, $E_U$ can be decomposed into:

$$E_U={\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\left({\displaystyle \underset{V}{\int }\dot{\epsilon }:\overrightarrow{\sigma }dV}\right)d\tau ={\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\left({\displaystyle \underset{V}{\int }\left({\sigma }^c+{\sigma }^v\right): \dot{\epsilon } dV}\right)}d\tau }={\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\left({\displaystyle \underset{V}{\int }{\sigma }^c: \dot{\epsilon } dV}\right)d\tau +} {\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\left({\displaystyle \underset{V}{\int }{\sigma }^v: \dot{\epsilon } dV}\right)d\tau } =E_I+E_V$$

(13)

where ${\sigma }^c$ is the stress calculated by the constitutive relationship of the material; ${\sigma }^v$ is the damping stress; $E_I$ is the part of energy determined by the material constitutive relation, and we continue to call it the internal energy in this study; $E_V$ is the damping energy. If we define $\dot{\epsilon } ={\dot{\epsilon }}^{el}+{\dot{\epsilon }}^{pl}$ (where ${\dot{\epsilon }}^{el}$ and ${\dot{\epsilon }}^{pl}$ represent the elastic and the plastic strain rates, respectively), $E_I$ is given by:

$$E_I={\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\left({\displaystyle \underset{V}{\int }{\sigma }^c:\left({\dot{\epsilon }}^{el}+{\dot{\epsilon }}^{pl}\right)dV}\right)}d\tau ={\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\left({\displaystyle \underset{V}{\int }{\sigma }^c:{\dot{\epsilon }}^{el}dV}\right)}d\tau +{\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\left({\displaystyle \underset{V}{\int }{\sigma }^c:{\dot{\epsilon }}^{pl}dV}\right)}d\tau =E_S+E_P$$

(14)

where $E_S$ is the elastic strain energy, $E_P$ is the plastic strain energy. Substituting $E_I$, $E_S$, and $E_P$ into Equation (10) gives the energy balance equation of the whole model:

$$E_S+E_P+E_V+E_K=E_{WF}+constant$$

(15)

Notice that $constant$ is a very small value and its influence is negligible. Equation (15) is the basis for the energy-based analysis of the integral SSI system using the finite element method.

It should be pointed out that a portion of $E_S$ cannot be recovered if damage occurs in the material. At that time, ${\sigma }^c$ can be expressed as:

$${\sigma }^c=\left(1-d\right){\sigma }^u$$

(16)

where $d$ is the damage parameter between 0 and 1, ${\sigma }^u$ is the “undamaged” stress. Therefore, we get:

$$E_S={\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\left({\displaystyle \underset{V}{\int }\left(1-d\right){\sigma }^u:{\dot{\epsilon }}^{el}dV}\right)}d\tau$$

(17)

In the unloading stage, the $d$ value remains at the value at time $t$, hence:

$$\begin{array}{ll}\hfill E_S& ={\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\left({\displaystyle \underset{V}{\int }\left(1-d_t\right){\sigma }^u:{\dot{\epsilon }}^{el}dV}\right)}d\tau +{\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\left({\displaystyle \underset{V}{\int }\left(d_t-d\right){\sigma }^u:{\dot{\epsilon }}^{el}dV}\right)}d\tau \hfill \\ & ={\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\left({\displaystyle \underset{V}{\int }\frac{\left(1-d_t\right)}{\left(1-d\right)}{\sigma }^c:{\dot{\epsilon }}^{el}dV}\right)}d\tau +{\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\left({\displaystyle \underset{V}{\int }\frac{\left(d_t-d\right)}{\left(1-d\right)}{\sigma }^c:{\dot{\epsilon }}^{el}dV}\right)}d\tau =E_E+E_D\hfill \end{array}$$

(18)

where $E_E$ is the recoverable strain energy; $E_D$ is the energy dissipated through damage.

4.2. Seismic Input and Dissipation Energy

To examine the relationship between input energy and seismic intensity, input energy at different PGAs is denoted as $E_{WF-PGA}$ ($\mathrm{PGA}=0.035 \mathrm{g}, 0.07 \mathrm{g}, 0.10 \mathrm{g}, 0.20 \mathrm{g}, 0.40 \mathrm{g}, 0.60 \mathrm{g}, \mathrm{and} 0.80 \mathrm{g}$, respectively), and the increased input energy $∆E_{WF-PGA}$ is defined by the ratio of $E_{WF-PGA}/E_{WF-0.035}$ (i.e., $∆E_{WF-0.07}=E_{WF-0.07}/E_{WF-0.035}$). In addition, $∆a_{PGA}$ represents the ratio of any given PGA to 0.035 g (i.e., $∆a_{0.07}=0.07 \mathrm{g}/0.035 \mathrm{g}=2$). Under different PGAs, the time histories of the input energy $E_{WF}$ are shown in Figure 14; the time history comparison of $∆E_{WF-PGA}$, $∆a_{PGA}$, and ${\left(∆a_{PGA}\right)}^2$ is shown in Figure 15. Specifically, the values of $∆E_{WF-PGA}$, $∆a_{PGA}$, and ${\left(∆a_{PGA}\right)}^2$ at the end of the induced shaking are given in Table 7.

Figure 14. Time histories of input energy under different PGAs.

Figure 15. Time histories of increased amplitudes for both input energy and peak acceleration.

Table 7. Comparison of increased amplitude between peak acceleration and input energy at the end of excitation.

PGA (g)	$\mathbf{∆}{\mathit{a}}_{\mathit{P}\mathit{G}\mathit{A}}$	($\mathbf{∆}{\mathit{a}}_{\mathit{P}\mathit{G}\mathit{A}}$)2	$\mathbf{∆}{\mathit{E}}_{\mathit{W}\mathit{F}-\mathit{P}\mathit{G}\mathit{A}}$	$\mathbf{∆}{\mathit{E}}_{\mathit{W}\mathit{F}-\mathit{P}\mathit{G}\mathit{A}}$/($\mathbf{∆}{\mathit{a}}_{\mathit{P}\mathit{G}\mathit{A}}$)2
0.07	2.00	4.00	2.62	0.66
0.10	2.86	8.16	4.17	0.51
0.20	5.71	32.65	13.28	0.41
0.40	11.43	130.61	44.07	0.34
0.60	17.14	293.88	94.18	0.32
0.80	22.86	522.45	154.05	0.29
1.0	28.57	816.32	225.44	0.28

Previous studies have shown that $∆E_{WF-PGA}$ is equal to ${\left(∆a_{PGA}\right)}^2$ for an elastic SDOF structure [49,50]. as the response of aboveground structures is dominated by inertial forces. However, Figure 15 and Table 7 show that $∆E_{WF-PGA}$ < ${\left(∆a_{PGA}\right)}^2$ for underground structures. $∆E_{WF-PGA}/{\left(∆a_{PGA}\right)}^2$ decreases with increasing PGA, but the rate of decrease slows down. This is because underground structures are subjected to two types of external forces: inertial force and earth pressure from surrounding soil. At low intensities (PGA < 0.10 g), SSI is weak and inertial forces dominate. As intensity increases, soil deformation grows rapidly, but structural stiffness restrains soil deformation, causing the soil to “push” the structure. Lateral earth pressure then becomes dominant, and SSI strengthens, reducing $∆E_{WF-PGA}/{\left(∆a_{PGA}\right)}^2$. When PGA exceeds 0.40 g, soil stiffness is severely degraded due to nonlinearity, slowing the decrease rate of $∆E_{WF-PGA}/{\left(∆a_{PGA}\right)}^2$. For the present soft soil site and AZIH ground motion configuration, this transition value is a case-specific observation rather than a universally applicable seismic threshold.

Figure 16 presents the time histories of seismic input energy ($E_{WF}$) and all energy components ($E_S$, $E_P$, $E_V$, and $E_K$) under different intensities. During earthquakes, input energy is converted to $E_K$ and $E_S$, or dissipated by $E_V$ and $E_P$. The energy response patterns are similar across all intensities. During the shaking, $E_K$ fluctuates significantly and is nearly zero at the end. $E_S$ stabilizes at a small positive value after fluctuations. Due to damage and permanent deformation, not all elastic strain energy is recoverable. $E_V$ and $E_P$ increases monotonically and reach their maxima at the end of excitation. $E_{WF}$ also fluctuates, and stabilizes at a value equal to the sum of all the components.

Figure 16. Time-history energy responses of underground structure under different PGAs.

The proportion of different energy components varies significantly with input intensity, as shown in Figure 17. At all intensities, $E_K$ and account for small proportions. Over 80% of the $E_{WF}$ is dissipated by $E_V$ and $E_P$. As PGA increases from 0.035 g to 1.0 g, the proportion of $E_V$ decreases from 99.96% to 44.28%, while $E_P$ rises from 0% to 37.61% and $E_S$ increases from 0.01% to 17.67%.

Figure 17. Proportion of different energy components at the end of induced shaking.

Specifically, at PGA < 0.10 g, the structure remains elastic, and most input energy is dissipated by damping. As PGA increases further, plastic zones appear and expand; then the plastic energy and unrecoverable part of elastic energy increase rapidly and share a considerable part of seismic input energy. At PGA > 0.60 g, severe structural damage slows the growth of $E_P$ and increases structural damping, reducing the rate of decrease in $E_V$.

4.3. Plastic Energy Dissipation and Distribution

Plastic energy magnitude reflects the degree of structural damage. For a complex structural system, total plastic energy alone cannot accurately reflect damage development in individual components. Thus, it is necessary to investigate the distribution of plastic energy dissipation across components. Figure 18 shows the distribution of plastic energy dissipation across structural components at the end of excitation for different PGAs. At low PGA, plastic energy dissipation is concentrated in central columns. As PGA increases, slabs and walls also begin to dissipate plastic energy. For PGA ≥ 0.20 g, wall-slab components become the primary plastic energy-dissipating parts, accounting for 80% of total plastic energy.

Figure 18. Distribution of $E_P$ at different parts of underground structures versus PGA.

As PGA increases from 0.035 g to 1.0 g, the proportion of plastic energy from central columns decreases from 97.87% to 21.96%, while sidewalls increase from 0.00% to 14.40%, and bottom slabs increase from 0.00% to 25.31%. The proportions for middle slab and top slab exhibit fluctuations. This trend occurs because central columns have lower stiffness and occupy a smaller volume than wall-slab components, with limited restoring force. At low intensities, plastic strain initiates in columns, which dissipate most plastic energy. At high PGA (>0.40 g), columns suffer severe damage and stiffness degrades dramatically, slowing their plastic energy growth. The primary dissipation then shifts from columns to wall-slab components.

The distribution of plastic energy dissipation at different parts of the underground structure at different times with PGA = 1.0 g is illustrated in Figure 19. Over the course of excitation, the component with the highest plastic energy dissipation shifts from central columns initially, to the top slab, and finally to the bottom slab. At high intensity, the primary energy-dissipating component evolves continuously throughout the seismic process.

Figure 19. Distribution of $E_P$ at different parts of the underground structure at different times (PGA = 1.0 g).

4.4. Plastic Energy Density-Based Evaluation

Cumulative plastic energy dissipation directly leads to structural failure. Thus, energy-based indices provide a useful complementary approach for damage assessment of underground frame structures, particularly for the period after peak input. To compare different evaluation indices for the seismic damage evaluation of underground structures, the ratios of structural response to the peak response at 25 s, 30 s, 40 s, 50 s, and 70 s for structural deformation and plastic energy under the 1.0 g shaking intensity are listed in Table 8. The ratio values for plastic energy are significantly higher than deformation ratios, indicating that energy-based evaluation indices can provide a supplementary perspective for describing the cumulative damage state throughout the seismic process.

Table 8. Ratio of structural response to the peak response for structural deformation and plastic energy.

	Times (s)
	25	30	40	50	70
Structural deformation	8.45%	1.26%	9.14%	6.05%	2.88%
Plastic energy	98.27%	99.97%	99.98%	99.99%	100%

To reflect the energy dissipation concentration of a single element, the plastic energy density is defined as:

$$ELPD=E_{P_{V0}}/V_0$$

(19)

where $ELPD$ is the plastic energy density of a specific element, $E_{P_{V0}}$ is the plastic energy dissipated by this element, and $V_0$ is the volume of this element.

The contour plots of the underground structure’s $ELPD$ at the end of the induced shaking under different PGAs are plotted in Figure 20. At PGA = 0.035 g, the maximum $ELPD$ is only 2.40 J/m³, with dissipation in very few locations. As PGA increases from 0.07 g to 0.20 g, plastic energy dissipation appears sequentially at the lower columns’ bottom and top, then the upper columns’ bottom, middle slab to sidewalls’ connection, top slab, and bottom slab. From 0.4 g to 0.6 g, dissipation extends to roof slab edges, columns’ bottom, and bottom slab edges. When PGA exceeds 0.6 g, structural damage becomes progressively more severe. For PGA > 0.20 g, the maximum $ELPD$ always occurs at the bottom of the lower column. The maximum $ELPD$ at 1.0 g reaches 5.71 × 10³ J/m³, approximately 2379 times the value at 0.035 g. In addition, when PGA ≤ 0.10 g, the structure is in a quasi-elastic working state, and the proportion of plastic strain energy to the total input energy is less than 1%. Only very slight local plastic strain occurs at stress-concentrated positions. This plastic deformation is caused by stress concentration with a negligible magnitude, which does not affect the overall elastic mechanical behavior of the structure and can be regarded as the elastic stage in engineering practice.

Figure 20. ELPD contour plots of the underground structure versus PGA.

Figure 21 summarizes the plastic damage evolution process based on plastic energy density. It can be seen that all structural connections experience plastic damage due to plastic energy dissipation. As PGA increases from 0.035 g to 1.0 g, plastic damage first appears at the top of upper columns (but does not develop rapidly), then at the base of lower columns, followed by the base of upper columns, the top of lower columns, and wall-slab connections. Damage at the lower columns’ bottom develops most significantly and eventually becomes the most severe region. Significant damage also occurs at slab-column and wall-slab connections.

Figure 21. Damage evolution process of the typical two-layer and three-span underground structure.

Additionally, the partial damage index ${Damage}_{part}^i$ is introduced to evaluate the structural damage and is defined by:

$${Damage}_{part}^i=\left({\displaystyle \sum _{j=1}^m{D}_j^i{V}_j^i}\right)/\left({\displaystyle \sum _{j=1}^m{V}_j^i}\right)$$

(20)

where ${Damage}_{part}^i$ is the partial damage index of part $i$ in the underground structure, $D_j^i$ is the damage variable of the $j^{th}$ (j = 1, 2, …, m) element in part $i$, which is the equivalent weighted value of concrete tensile damage factors, with D = 0 corresponding to no damage and D = 1 corresponding to complete material failure. $V_j^i$ is the volume of the $j^{th}$ (j = 1, 2, …, m) element in part $i$ (i = 1, 2, …, n). Then, the integral damage index can be expressed as:

$$Damag{e}_{whole}=\left({\displaystyle \sum _{i=1}^n{Damage}_{part}^i}\left({\displaystyle \sum _{j=1}^{m}ELP{D}_j^i{V}_j^i}\right)\right)/\left({\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^{m}ELP{D}_j^i{V}_j^i}}\right)$$

(21)

where $Damag{e}_{whole}$ is the integral damage index of the underground structure. The integral damage of the structure is defined to evaluate the damage status of the whole underground structure. It can be found that in Equation (21), the plastic energy of part $i$ (i = 1, 2, …, n) is used as the weight factor since the more seriously damaged part generally dissipates more plastic energy and has a larger plastic energy density. The weighting is plastic-energy-based at the element level, not volume-based. Unlike volume-averaged indices that dilute severe local damage, the proposed index weights each element by its plastic energy dissipation, such that energy-concentrated critical regions dominate the global damage value. This is consistent with the failure mode of underground frame structures, where failure initiates at key components such as column ends. The essence is using energy dissipation, not geometric volume, as the contribution measure. Therefore, $Damag{e}_{whole}$ can evaluate the integral damage of the underground structure reasonably. It should be clarified that the numerical boundaries for the following damage state classification are preliminary and model-dependent. They are determined based on the physical mechanism of the concrete damaged plasticity model and the specific numerical modeling scheme adopted in this study, and have not been calibrated against experimental data or field observations. Further calibration and validation are required in future work. Four damage states are defined as follows: (1) Slight damage ($Damag{e}_{part}$ < 0.1, $Damag{e}_{whole}<0.05$): concrete cracking, no reinforcement yielding, corresponding to the serviceability limit state; (2) Moderate damage (0.1 ≤ $Damag{e}_{part}$ < 0.4, $0.05\le Damag{e}_{whole}<0.20$): steel yielding, local concrete crushing, corresponding to the bearing capacity limit state; (3) Severe damage (0.4 ≤ $Damag{e}_{part}$ < 0.8, $0.20\le Damag{e}_{whole}<0.40$): core concrete crushing, steel hardening, corresponding to the near-collapse state; (4) Extremely severe damage ($Damag{e}_{part}$ ≥ 0.8, $Damag{e}_{whole}\ge 0.40$): complete component failure, corresponding to the collapse state. We fully acknowledge that this treatment has certain limitations. On the one hand, under high seismic intensities, significant compressive damage occurs at positions such as column bases and slab ends, and reinforcement yielding also affects the degradation of structural bearing capacity. A damage variable considering only tensile damage cannot fully reflect the multi-mode coupled failure state. On the other hand, an index based only on tensile damage will have deviations when evaluating the ultimate bearing capacity and near-collapse state of the structure. In addition, the damage classification thresholds are tied to the material parameters, element size, and modeling approach of the current study, and cannot be directly generalized to other engineering cases without independent calibration.

Table 9 lists the component damage indices and global damage index at the end of excitation for different PGAs. Both component and global damage indices increase with rising seismic intensity, and vary considerably across components. The bottom of the lower column has the largest partial damage index value, reaching 0.897 at 1.0 g (extremely severe damage). The top slab has the lowest damage index (0.181 at 1.0 g, making it the least damaged component. The global damage index reaches 0.338 at 1.0 g, indicating severe overall damage.

Table 9. Damage evaluation of the partial and the whole underground structure.

PGA (g)	${\mathit{D}\mathit{a}\mathit{m}\mathit{a}\mathit{g}\mathit{e}}_{\mathit{p}\mathit{a}\mathit{r}\mathit{t}}^{\mathit{i}}$						$\mathit{D}\mathit{a}\mathit{m}\mathit{a}\mathit{g}{\mathit{e}}_{\mathit{w}\mathit{h}\mathit{o}\mathit{l}\mathit{e}}$
	Lower Column	Upper Column	Sidewall	Middle Slab	Top Slab	Bottom Slab	Whole Station
0.035	0	0	0	0	0	0	0
0.07	0.00000	0.00003	0.00000	0.00003	0.00000	0.00000	0.00003
0.10	0.008	0.0016	0.0003	0.0002	0.0002	0.0005	0.0005
0.20	0.026	0.014	0.002	0.005	0.006	0.002	0.006
0.40	0.168	0.078	0.006	0.030	0.013	0.029	0.038
0.60	0.410	0.245	0.053	0.099	0.040	0.086	0.107
0.80	0.635	0.422	0.102	0.173	0.083	0.171	0.213
1.00	0.897	0.606	0.192	0.279	0.181	0.290	0.338

5. Conclusions and Limitations

5.1. Conclusions

This study investigates plastic energy density-based time-dependent seismic performance evaluation for underground structures via nonlinear 3D FE simulation. The deformation characteristics, soil-structure deformation transmission, energy distribution, and damage evolution are systematically analyzed. The main conclusions can be drawn as follows.

(1) The predominant frequency of the structural RD time history is governed by both the free-field site and the underground structure. At low PGA (<0.10 g), structural properties dominate the response frequency. As PGA increases, site influence rapidly becomes dominant. Since the structural natural frequency is higher than the free-field fundamental frequency, the response frequency decreases with increasing intensity and eventually converges to the free-field fundamental frequency. The lower story of the structure is more seismically vulnerable.

(2) Free-field soil deformation exceeds structural deformation (with similar profiles), and this gap widens with increasing PGA. Based on the observed macroscopic response characteristics, it can be inferred that at low PGA (<0.1 g), inertial force accounts for a substantial proportion of the total external force. As intensity rises, lateral earth pressure from surrounding soil becomes dominant. The soil-structure interaction coefficient β exhibits a two-stage variation characteristic of a rapid decrease first and then stabilization. Soil stiffness degrades more rapidly than structural stiffness, enlarging the deformation gap and strengthening the forced displacement effect of soil. It should be noted that this inference is based on frequency convergence and deformation coefficient characteristics rather than direct force decomposition, and that the transition values are specific to the present soft soil site and AZIH ground motion configuration.

(3) Over 80% of seismic input energy is dissipated by $E_V$ and $E_P$. As PGA increases, the proportion of $E_V$ decreases sharply while the proportion of $E_P$ rises rapidly. Under high seismic intensity (PGA = 1.0 g) throughout the earthquake process, the component with the largest plastic energy dissipation transfers from the central column to the top slab and finally to the bottom slab. From the perspective of total plastic energy proportion at the end of the earthquake, the sum of plastic energy dissipated by the columns and the slabs accounts for more than 80% of the total plastic energy of the structure.

(4) As PGA increases from 0.035 g to 1.0 g, plastic damage initiates sequentially at columns’ tops, columns’ bottoms, and wall-slab connections. Damage at the lower columns’ bottom section develops most rapidly and becomes the most severe region. The maximum $ELPD$ at 1.0 g is approximately 2379 times that of 0.035 g. The proposed damage index can serve as a complementary indicator for seismic damage assessment, but its thresholds are preliminary and model-dependent and require further calibration against experimental or field data.

5.2. Limitations and Applicability

(1) This study focuses on frame-shaped underground structures; the conclusions do not apply to other forms such as shield tunnels and column-free underground structures. Only a soft soil site is considered; the laws of deformation transfer and energy dissipation vary with different site classes.

(2) Only one far-field ground motion is adopted; the effects of near-fault pulse-like ground motions and vertical ground motions are not considered. In addition, the coupling effect of multi-dimensional ground motions should be considered in future research.

(3) The damage index and thresholds are established based on the physical meaning of the constitutive model using assumed material and contact parameters. The damage index has certain mesh dependency and constitutive model dependency. The proposed damage thresholds still need to be further calibrated and validated with rigorous experimental data.