Stability Assessment of an Underground Powerhouse Cavern Under Pseudo-Static and Dynamic Earthquake Loading

Abstract

This study examines the seismic stability of an underground powerhouse cavern located in the Lesser Himalayan region of Nepal. Both static and seismic loading conditions are analyzed using the finite element method (FEM) and the distinct element method (DEM). Rock mass properties are derived from field investigations and laboratory testing, while empirical correlations are applied to estimate rock mass strength and deformation modulus. Pseudo-static analyses are performed using the FEM-based software Rock and Soil-2-Dimensionsl (RS²) Version 11.027, and dynamic analyses are conducted using the DEM-based software Universal Distinct Element Code (UDEC) Version 5.0 to evaluate deformation and stress redistribution around the cavern. Seismic fragility curves are developed to quantify the probability of damage under varying seismic intensities. Results indicate that a peak ground acceleration (PGA) of 0.25 g increases cavern wall deformation by approximately 15–20 mm compared to static conditions. Fragility analysis shows a probability exceeding 68% for slight damage, while the probability of collapse remains low at approximately 1.7%. Seismic loading also significantly alters stress redistribution along the cavern boundary. Overall, the combined use of numerical modeling and fragility analysis provides a probabilistic framework for assessing seismic risk in underground caverns, offering valuable insights for the design and safety evaluation of hydropower projects in seismically active Himalayan regions.

1. Introduction

The Himalayan mountain range was formed by the subduction of the Indian plate beneath the Eurasian (Tibetan) plate approximately 55 million years ago [1,2]. As one of the youngest and most tectonically active mountain ranges in the world, the Himalayan rock mass is typically characterized by fractured and heterogeneous rock formations, often containing weak and highly deformable tectonic zones affected by significant weathering, fracturing, and shearing [3]. The active geotectonic conditions in the Himalayan mountains demand special attention during the construction of tunnels and caverns. Due to the ongoing subduction of the Indian plate beneath the Eurasian plate, earthquakes continue to occur frequently in the Himalayan region. Hence, the rock mass in this region is influenced by fracturing, faulting, folding, and schistosity, adding to the geological complexity [4]. Frequent earthquakes, due to the presence of faults along tectonic boundaries, have further emphasized the need for seismic analysis in the region. There are five major tectonic subdivisions in the Himalayan geological regions. These five tectonic subdivisions from south to north are the Gangetic plane (Terai), Siwalik zone, Lesser Himalayan zone, Higher Himalayan zone, and Tibetan-Tethys zone. These divisions span almost 2500 km following a NW-SE direction. The tectonic subdivisions are separated by the tectonic faults consisting of the Main Frontal Thrust (MFT) at the south, followed by the Main Boundary Thrust (MBT), Main Central Thrust (MCT), and South Tibetan Detachment Fault (STDF) at the north. The tectonic zones are characterized by lithology, tectonics, geological structures, and geological history and are made up of various rock types [5]. Figure 1 shows the major tectonic faults and geology of the Nepal Himalayas.

The typical operational (service) life of underground structures like tunnels and caverns may extend up to 150 years. During this period, these underground structures are subjected to a wide range of seismic events, from minor tremors to large-scale earthquakes. Traditionally, underground structures are considered safe under seismic loading. However, various studies have shown that the effect of earthquakes on tunnels and caverns cannot be completely ignored [6,7,8,9,10,11,12,13]. This is especially the case for the Himalayan region, which is tectonically active and experiences frequent seismic movement. In the Himalayan regions, major earthquakes of moment magnitude (M_w) around 8 occur almost every 80–100 years. The most recent major earthquake struck in Nepal in April 2015 with a magnitude of 7.8, followed by a major aftershock in May 2015 with a magnitude of 7.3 (Figure 1), which were classified as extreme intensity earthquakes. These major events shown in Figure 1 indicate the necessity of considering seismic loads during the design and construction of underground structures in the Himalayan region.

Figure 1. Tectonic thrust faults, geology, and occurrence of major earthquakes for the last 100-year period (geological map modified from Department of Mines [14]; major earthquake locations and magnitude overlayed from Chaulagain et al. [15]).

Figure 1. Tectonic thrust faults, geology, and occurrence of major earthquakes for the last 100-year period (geological map modified from Department of Mines [14]; major earthquake locations and magnitude overlayed from Chaulagain et al. [15]).

During the 2015 Gorkha earthquake, underground structures experienced relatively less damage compared to above-ground structures. An aftershock of magnitude 7.3 occurred in May 2015 with its epicenter near two major hydropower projects in Nepal. At the Khimti I Hydropower Plant, minor damage was reported, including the detachment of a 10 cm block of shotcrete from the roof of the powerhouse cavern. Similarly, at the powerhouse cavern of the Upper Tamakoshi Hydropower Project, a shotcrete block was found to have fallen. In the Melamchi Water Supply Tunnel, cracks appeared in the shotcrete linings at multiple locations due to earthquake shaking. At the originally proposed powerhouse location of the Super Dordi Hydropower Project, a small landslide occurred due to an earthquake. As a result, it was decided to construct an underground powerhouse instead of a surface (ground-level) powerhouse [16]. In addition, several hydropower surface structures were damaged, but deep tunnels reportedly remained functional. However, very few post-earthquake performance assessments of underground structures have been made in the Himalayan region, excluding visual observation, which indicates a need for rigorous and comprehensive impact assessment. Several other cases of underground structures in other regions indicate that underground structures are also susceptible to earthquake-induced damage. Notable examples include the 1995 Kobe earthquake in Japan, the 1999 Kocaeli earthquake in Turkey, the 1999 Chi-Chi earthquake in Taiwan, and the 2008 Wenchuan earthquake in China [17,18]. These events highlight the importance of incorporating seismic analysis in the design and construction of tunnels and caverns so that it is possible to evaluate their response to dynamic earthquake forces.

Hence, the main aim of this manuscript is to evaluate the stability condition of the underground powerhouse cavern subjected to seismic load for the Tanahu Hydropower Project (THP). The project is located near MBT in the Lesser Himalayan tectonic subdivision zone of Nepal. Field investigations, data collection from in-situ monitoring and insitu tests, laboratory test results, and past seismic records are used as input parameters for the assessment. Numerical, semi-analytical, and empirical relationships are exploited to assess the cavern’s stability under seismic load. The pseudo-static approach is employed to assess deformation in the cavern under a specific earthquake scenario. The fragility curves are generated to provide a broader probabilistic perspective on the vulnerability of underground structures across various seismic intensities. In addition, earthquake events are simulated in the Universal Distinct Element Code (UDEC) models to study stress redistribution around the excavation boundary of the underground powerhouse cavern. This study provides a framework to guide researchers and rock engineers in conducting comprehensive seismic stability assessments of underground caverns.

2. Description of the Case Project

Tanahu Hydroelectric Project (THP) is a daily peaking storage-type hydroelectric project with an installed capacity of 140 megawatts (MW). The project is located in the lower segment of the Seti River, which originates from the Annapurna Himalaya at an elevation of approximately 7555 m above sea level. The Seti River stretches about 120 km from its origin to the dam location and has a catchment area of 1502 km². The key civil structural components of the project are illustrated in Figure 2. The project facilitates (1) a reservoir (effective storage capacity of 295.1 million m³) by constructing a concrete gravity dam (140 m high), (2) an intake, (3) a headrace tunnel (1162 m), (4) a surge shaft (84 m high), (5) a penstock shaft (175 m), (6) an underground powerhouse cavern, and (7) a tailrace tunnel (86 m). In addition, the project consists of other civil structures such as a sediment flushing system, spillway, and access tunnels.

2.1. Geological Conditions

THP lies in Nepal’s Lesser Himalayas region, and the site consists of low- to medium-grade metasedimentary rocks consisting of slate, phyllite, dolomite, and marble. Figure 2a illustrates the regional geological features, and Figure 2b shows the hydropower components and geological conditions at the project. There are no regional faults or large-scale secondary faults that the underground structures cross, excluding rock boundaries and local small-scale shear faults. However, the slate rock is folded at the underground powerhouse cavern area due to past tectonic movements. The general lithology of the powerhouse area is characterized by light to dark grey, thinly to moderately foliated and folded phyllitic slates of medium strength. Soft and schistose phyllitic layers are within the moderately thick slate layers. The slate is slightly weathered to fresh rock mass. Localized dampness and localized dripping of water seen during excavation of the underground powerhouse cavern suggest that the rock mass has relatively low permeability.

There are three major joint sets in which foliation joints (J_f) are the dominant ones. The foliation joints strike SE-NW and dip at 40–50° towards the SW direction. The cross joint set (J₁) strikes in the NE-SW direction and dips 55–85° towards the SE direction. Similarly, joint set (J₂) strikes SE-NW and dips 55–65° towards the NE direction. The longitudinal axis of the cavern is oriented with an azimuth of N28°E (Figure 3). Following discontinuity conditions, the cavern has a favorable orientation with respect to the foliation joint set and cross joint set J₂ but is relatively unfavorable with respect to joint set J₁ (Figure 3).

2.2. Rock Mass Quality and Support System

The Q-system [21] of rock mass classification is used to evaluate rock mass quality in the powerhouse cavern.

Table 1. Rock mass classification by Q system.

Parameters	Symbol	Range	Typical Mean
Rock Quality Designation	RQD	60–40	50
Joint Set Number	Jn	6–9	8
Joint Roughness Number	Jr	1.5–2	2
Joint Alteration Number	Ja	1–4	2
Joint Water Reduction Factor	Jw	0.66–1	0.75
Stress Reduction Factor	SRF	1–2.5	2.5
$Q={\displaystyle \frac{RQD}{J_n}}\times {\displaystyle \frac{J_r}{J_a}}\times {\displaystyle \frac{J_w}{SRF}}$		0.44–20	1.875

summarizes Q-parameters obtained from the face mappings at the caverns. The rock mass quality at the cavern varies from very poor to good quality rock mass class, where Q-values range from 0.44 to 20. The typical mean value of Q is 1.87.

The underground powerhouse cavern has a dimension of 22 m in width, 45 m in height, and 89 m in length. The rock support system used in the cavern consists of a combination of rock anchors, rock bolts, and 20 cm thick M30 grade steel fiber shotcrete. The prestressed double corrosion protection (DCP) anchors, which are 10 and 25 m long and 47 mm in diameter, are spaced at 3 m center-to-center, and 6 m long, 32 mm diameter grouted rock bolts are spaced at 3 m in a staggered pattern (Figure 4). The design load of the DCP anchors is 942 kN. Load cells are installed on a total of 85 anchors to monitor the developed load during operation. Among these, four anchors have recorded peak loads marginally exceeding the design value. The maximum recorded load reached 963 kN at one location on the west wall, and the minimum recorded value is 252 kN on the north wall. However, no such damage in the anchors is seen, and hence they are considered acceptable for long-term functionality.

A total of 501 days was used to excavate the cavern. Multipoint borehole extensometers are installed in various locations of the cavern to monitor long-term deformation. The extensometers are installed in such a way that the blast-induced effect is at a minimum in the measurement readings. The maximum recorded deformation is found to be 51 mm in the southern wall of the cavern. Figure 5 presents the extensometer readings at four different locations in the powerhouse cavern. The relatively large change in deformation is observed in the southern wall after approximately 100 days of monitoring. It might be due to the excavation of the hydropower components in the vicinity of the wall, leading to loss of confinement and consequently an increase in the deformation magnitude.

2.3. Intact Rock and Rock Mass Properties

Field investigation and mineralogical analysis have confirmed phyllitic slate as the primary rock type in the cavern. X-ray diffraction (XRD) test revealed that the rock samples consist mainly of Quartz (≈36%), Feldspar (≈9%), and Phyllosilicates (≈53%). Intact rock samples are tested to determine specific weight (γ_r) and P-wave velocity (V_p), tensile strength (σ_t), uniaxial compressive strength (UCS), Young’s modulus (E_i), and Poisson’s ratio (υ) following the ISRM standard [22,23].

Unlike intact rocks used for laboratory testing, rock mass contains joints, and fissures and is weathered to some degree, which causes a considerable reduction in the rock mass strength and deformation modulus. Thus, while designing the underground space, it is important to estimate rock mass strength and rock mass deformation modules. The empirical relationships proposed by Barton [24] and Hoek & Diederichs [25] use the Q and GSI values to estimate uniaxial compressive strength (σ_cm) and deformation modulus (E_rm) of the rock mass. These empirical relationships depend upon Q and GSI values, which are both subjective and are sensitive to use in the assessment [26]. To overcome these limitations, Panthi [5,27] proposed relationships for the estimation of σ_rm and E_rm.

Table 2. Intact rock and rock mass properties at cavern location.

σci MPa	Eci GPa	σt MPa	Poisson’s Ratio (ν)	γ kN/m3	Vp m/s	a σcm MPa	b Erm GPa	c Grm GPa	ϕpeak Degrees	cpeak MPa	ϕresidual Degrees	cresidual MPa
38	14.25	6.15	0.25	27.18	4165	5.62	2.11	0.843	38	1.70	30.40	0.85

^a σ_cm = 1/60 × σ_ci^1.6 (Panthi) [27]. ^b E_rm = E_ci × (σ_cm/σ_ci) (Panthi) [5]. ^c G_rm = E_rm/(2 × (1 + ν)).

lists the estimated values of rock mass compressive strength and deformation modules for further analysis. In this study, the Mohr–Coulomb failure criterion is used, as dynamic loads can be approximately described by the Mohr–Coulomb criterion at a low confining pressure range [28]. The peak angles of internal friction and cohesion used are 38° and 1.7 MPa, respectively. Similarly, the residual angles of friction and cohesion are 30.4° and 0.85 MPa, respectively.

2.4. Joint Parameters

In discontinuum modeling, joint properties are important parameters that influence the stability of underground structures. Both normal and shear stiffness of the joints are calculated using Equations (1) and (2), respectively, following Barton [29] and Singh [30].

$$k_n={\displaystyle \frac{E_{ci}\times E_{rm}}{L\times \left(E_{ci}-E_{rm}\right)}}$$

(1)

$$k_s={\displaystyle \frac{G_{ci}\times G_{rm}}{L\times \left(G_{ci}-G_{rm}\right)}}$$

(2)

where k_n is the normal stiffness, k_s is the shear stiffness, L is the mean joint spacing, E_ci is the elastic modulus, G_ci is the shear modulus, and E_rm and G_rm are the deformation and shear modulus of the rock mass, respectively. For the fracture zone, normal stiffness and shear stiffness are estimated by using Equations (3) and (4) [31].

$$k_n={\displaystyle \frac{E_0}{t}}$$

(3)

$$k_s={\displaystyle \frac{G_0}{t}}$$

(4)

where E₀ and G₀ are the Young’s modulus and shear modulus of the infilling material, respectively, and t is the thickness of the fracture zone.

2.5. In-Situ Stress

In-situ stress conditions depend on the overlying strata, plate tectonics, and stress due to topographic effects. The stress condition at any location within the rock mass can be expressed by the magnitude and orientation of three principal stresses. To determine stress magnitudes, in-situ tests such as hydraulic fracturing and diametrical core deformation analysis are conducted at the cavern location.

The in-situ stress measurement established the maximum horizontal stress (S_Hmax) as 10.4 MPa (N10°E) and the minimum horizontal stress (S_Hmin) as 4.8 MPa (N80°W). The vertical stress is measured as 9.8 MPa. The relationship between the horizontal tectonic stress and vertical stress is expressed in Equations (5) and (6). Maximum horizontal tectonic stress (σ_tec,max) and minimum horizontal tectonic stress (σ_tec,min) are calculated as 10.55 MPa and 2.35 MPa, respectively. The longitudinal axis of the cavern aligns at N28°E, forming an angle of 18° with the maximum tectonic stress trend (Figure 6). The in-plane and out-of-plane horizontal stresses are computed using Equations (7) and (8), respectively, and the horizontal shear stress is calculated using Equation (9).

$$S_{Hmax}={\sigma }_{tec,max}+{\displaystyle \frac{\nu }{1-\nu }}\times S_{zz}$$

(5)

$$S_{Hmin}={\sigma }_{tec,min}+{\displaystyle \frac{\nu }{1-\nu }}\times S_{zz}$$

(6)

S_yy = S_Hmaxcos²θ + S_Hminsin²θ

(7)

S_xx = σ_Hmaxsin²θ + σ_Hmincos²θ

(8)

$$S_{xy}=S_{yx}=\left({\displaystyle \frac{S_{Hmax}-S_{Hmin}}{2}}\right)\mathit{sin}2\theta$$

(9)

In the above relationship, S_yy and S_xx are the in-plane and out-of-plane horizontal stresses, respectively. S_Hmax and S_Hmin are the maximum and minimum total horizontal stresses, and θ is the angle between the cavern axis and tectonic stress orientation. The calculated stresses are given in

Table 3. In-situ stress values in the powerhouse cavern location.

Parameters	Symbol	Values	Unit
Overburden	h	360.5	meters
Poisson’s ratio	ν	0.2	-
Density of rock	γ	27.18	kN/m3
Trend of tectonic stress	θtec	N10°E	-
Cavern trend	θc	N28°E	-
Angle between tectonic stress trend and cavern (in-plane) axes	θt	72	Degrees
Vertical stress due to gravity only	SZZ	9.80	MPa
Horizontal stress due to gravity only	SH,grv	2.45	MPa
Maximum total horizontal stress	SHmax	10.40	MPa
Minimum total horizontal stress	SHmin	4.80	MPa
In-plane horizontal stress	Syy	3.21	MPa
Out-of-plane horizontal stress	Sxx	9.64	MPa
Horizontal shear stress	Sxy = Syx	1.65	MPa

.

3. Seismic Assessment for Underground Powerhouse Cavern

3.1. Pseudo-Static Approach

In most underground openings, seismic design is primarily governed by ground deformations rather than inertial forces acting on the structure [32,33]. The effects of earthquake loading on underground structures are generally lower than those on surface structures due to the confinement provided by the surrounding rock mass. Consequently, peak ground acceleration (PGA) values are smaller underground, and such structures experience reduced dynamic amplification because of confining stress and seismic wave damping. As a result, the structural response is mainly controlled by the quasi-static deformation of the ground, making the pseudo-static approximation reasonable. Moreover, the key parameter governing the stability of an underground cavern is its permanent deformation rather than transient acceleration or stress waves. Therefore, pseudo-static methods are considered adequate due to their simplicity in estimating these parameters and their efficiency in reducing computational time.

Engineers often adopt the pseudo-static approach, in which inertia forces are neglected, and seismic loading is typically modeled as static far-field shear stresses or strains applied at the boundaries of the surrounding ground [17]. This approach enables the estimation of the ultimate seismic capacity of underground structures while significantly reducing computation time and simplifying the analysis process. Consequently, it has been widely applied in numerous studies, particularly in engineering design and evaluation, where it has demonstrated good agreement with field observations [34]. The main factors influencing earthquake-induced damage to underground structures are PGA and the magnitude of seismic wavelength. In this study, a 2D finite element model using RS² software is developed to implement the pseudo-static approach.

3.2. Fragility Curves

The pseudo-static approach provides a single deterministic response based on assumed seismic loads, but it does not account for the probability of damage under varying seismic intensities. To address this limitation, fragility curves are developed for assessing the seismic vulnerability of the structures. A fragility curve represents the probability of exceeding a certain damage level as a function of seismic intensity. Fragility curves are a useful tool for risk assessment on expected future events, as well as for the caverns to be built in the future.

Fragility curves establish the relationship between the intensity measure (IM) of damage and the probability of exceedance for various damage states (DS). Such curves help to quantify the seismic risk in underground structures, which are important for risk assessment. The selection of an appropriate intensity measure is crucial to represent the ground motion and structural response [35]. Equation (10) describes a common form of the fragility curve and gives the conditional probability of being in or exceeding a particular damage (DS_i) state for the given intensity measure (IM).

$$P_f\left[DS\ge DSi\right|IM]=\varphi \left({\displaystyle \frac{ln\left(IM\right)-ln\left({IM}_{DSI}\right)}{{\beta }_{tot}}}\right)$$

(10)

where DS is the level of damage, ϕ is the standard normal cumulative probability distribution function, IM_DSI is the median threshold value of the earthquake intensity parameter required to cause damage state, and β_tot is the lognormal dispersion parameter that depends upon three primary sources of uncertainty [36]. Equation (11) gives the expression for β_tot.

$${\beta }_{tot}=\sqrt{{\beta }_{DS}^2+{\beta }_C^2+{\beta }_D^2}$$

(11)

where β_DS is the standard deviation associated with the threshold of damage states and represents an uncertainty of damage states, β_c is the uncertainty associated with the response and resistance (capacity) of an underground opening, and β_D is the uncertainty associated with earthquake input motion (demand). β_DS is considered as 0.4 following Argyroudis & Pitilakis [36] and Hazus [37], whereas β_c is considered as 0.3 following Huang et al. [38]. The dispersion parameter β_D represents variability in structural response and damage-state exceedance due to uncertainties in seismic events and modeling assumptions. The β_D value needs to be determined based on the damage state conditions defined by cavern stability parameters. Hence, the damage parameters are defined using the deformation limit at the powerhouse cavern and the respective PGA. Hoek & Marinos [39] define the degree of damage to underground structures based on strain limits (

Table 4. Damage state conditions [39].

Damage State ID	Damage Condition	Strain Limits
DS0	No damage	<1%
DS1	Slight	1–2.5%
DS2	Moderate	2.5–5%
DS3	Extensive	5–10%
DS4	Collapse	>10%

).

Table 5. Damage state description for various PGA ranges [40].

Damage State ID	Condition	Condition	PGA Range
DS0	No Damage	Small cracks are developed with no rockfall.	<1%
DS1	Minor (Slight)	Linings start showing cracks with rockfall.	1–2.5%
DS2	Moderate	Lots of destructive cracks have developed.	2.5–5%
DS3	Extensive	Large cracks developed in the tunnel lining, falling of big rocks, and sinking of road surfaces. Tunnels are heavily damaged and remain useless without repair.	5–10%
DS4	Collapse	Serious cracks and clear deformation can be observed in the tunnel lining. Tunnels collapsed, and there is a need for reconstruction.	>10%

shows damage intensity classification, which is based on peak ground acceleration at the site location [40].

Recent studies [41,42] indicate that under cyclic and dynamic loading, foliated and brittle rock masses experience progressive damage through microcrack development, joint sliding, and stiffness degradation, even in the absence of squeezing. Accordingly, deformation-based performance thresholds provide a rational and conservative framework for assessing seismic-induced damage. Although originally formulated for squeezing conditions, the Hoek and Marinos [39] damage indicators are fundamentally strain-based and independent of rock lithology. In the present study, seismic-induced damage in the foliated slate is controlled by progressive deformation rather than abrupt brittle failure, justifying the use of strain-based criteria. The adopted indicators, therefore, provide a conservative and physically consistent assessment of deformation-driven damage under dynamic loading.

3.3. Dynamic Modeling

Earthquakes alter stress orientation and magnitude in underground structures [43,44]. To investigate stress redistribution, a numerical model is developed using UDEC. Seismic inputs are typically provided as acceleration or velocity time histories, commonly derived from recorded or synthetic ground motions. In UDEC, simulations are conducted using a dynamic implicit analysis with sinusoidal input motion (Equation (12)) at a constant frequency (f) and amplitude (A). The velocity amplitude (v_a) and displacement amplitude (u_a) are obtained using Equations (13) and (14), respectively.

u(t) = A sin(2πf t)

(12)

v_a = A/2πf

(13)

u_a = v_a/2πf

(14)

Rayleigh damping, also known as proportional or viscous damping, is used to define the damping values in numerical models. The damping matrix [C] is formed by linear combinations of the mass matrix [M] and the stiffness matrix [K] scaled by the mass-proportional coefficient (α) and stiffness-proportional coefficient (β) (Equation (15)). The damping ratio ξ at frequency ω (=2πf) is given by Equation (16).

[C] = α[M] + β[K]

(15)

ξ(ω) = α/2ω + βω/2

(16)

Following Equations (12)–(16), the seismic input parameters are determined for the UDEC model. The stress redistribution and variations in displacement magnitude along the excavation boundary are evaluated to assess the seismic effects on the powerhouse cavern.

Dynamic analysis is generally performed using either recorded time-history data or a simplified wave input. In this study, the cavern is located at a depth greater than 360 m within the rock mass, where the substantial overburden confinement and high stiffness of the surrounding material significantly suppress the propagation and amplification of high-frequency seismic waves. Consequently, the seismic response of such deep caverns is governed primarily by quasi-static inertial effects and overall ground deformation rather than transient accelerations that require a full time-history analysis [32]. The use of low-frequency sinusoidal input, therefore, represents a conservative and rational approximation for capturing the dominant deformation mechanisms, while the contribution of higher-frequency components to the overall response is expected to be marginal at this depth. Moreover, when time-history analysis is applied in UDEC, model convergence becomes challenging, and computational time increases considerably. Therefore, simplified sinusoidal wave input is adopted in the current study, as it provides a practical balance between accuracy and computational efficiency. The results obtained through this simplified input are considered adequate for assessing stress redistribution and deformation patterns under seismic loading. Such simplified approaches are widely accepted in the literature [45,46] for parametric or comparative studies involving deep rock conditions, where dynamic amplification effects are minimal. Hence, the use of simplified wave input is appropriate for achieving the objectives of this study.

Therefore, this research is divided into two main parts, where the first part involves the development of a fragility curve to assess the seismic vulnerability of the cavern, and the second part examines stress redistribution around the rock mass in the cavern during an earthquake event.

Table 6. Methodology used for the seismic analysis of the cavern.

	FEM Model	DEM Model
TOOL	RS2	UDEC
OBJECTIVE	Develop parameters required for the fragility curve.	Find the change in stress magnitude during an earthquake.
INPUT	Rock mass properties, in-situ stress, boundary conditions, cavern geometry, and support parameters.	Rock mass properties, in-situ stress, boundary conditions, cavern geometry, and joint properties.
SEISMIC LOADING	Pseudo-static conditions.	Dynamic analysis.
OUTPUT	Deformation with respect to PGA.	Variation of principal stress during an earthquake.
INTERPRETATION	Vulnerability of the cavern by developing fragility curves.	Discussion on the effect of stress redistribution in the cavern during the earthquake event.

summarizes the methodology adopted for seismic analysis.

4. Numerical Modeling

The combined use of RS² (FEM) and UDEC (DEM) is advantageous because of their complementary strengths. RS² efficiently captures the continuum-scale response of the cavern system, while UDEC is better suited for explicitly modeling jointed rock mass behavior and discontinuity-controlled deformation under dynamic loading. Conducting both analyses in a single platform would either compromise the probabilistic fragility analysis (if only DEM is used) or oversimplify joint-controlled dynamic behavior (if only FEM is used). Hence, the combined FEM–DEM approach adopted in this study enables a more realistic representation of both continuum and discontinuum responses.

4.1. Pseudo-Static Model

In a pseudo-static numerical model, seismic forces are calculated as the product of mass and acceleration, using seismic coefficients, which are dimensionless values representing the maximum earthquake acceleration as a fraction of gravitational acceleration. These coefficients are incorporated into numerical models, and the resulting deformations are computed as the response to seismic loading. At the cavern location, a horizontal seismic coefficient corresponding to a PGA of 0.35 g is considered following the Nepal National Building Code (NBC 105) [47]. Based on the recommendation by Power et al. [48], the ratio of ground motion from the surface to the cavern location is assumed to be 0.7, resulting in a horizontal PGA of approximately 0.25 g. Furthermore, the vertical seismic coefficient is assumed to be 50% of the horizontal coefficient, following Hynes-Griffin & Franklin [49] and Jeldes & Drumm [50].

The cavern is modeled in multiple stages, with sequential blasting simulated alongside the corresponding installation of support systems. Adopted support systems are explicitly modeled after excavation, as their interaction with the surrounding rock mass governs deformation, which is the primary output to evaluate the cavern fragility function under pseudo-static loading conditions. The rock mass properties provided in Table 2 and stress conditions given in Table 3 are used as inputs to the numerical models.

The bottom boundary of the numerical model is restrained in both horizontal and vertical directions, whereas the vertical boundaries are restrained only in the horizontal direction (Figure 7). This boundary condition ensures a stable equilibrium condition of the model while allowing the stiffness of the surrounding rock mass to be properly represented in the analysis. The pseudo-static seismic force acting on each element is calculated as the product of the seismic coefficient and the body force, where the body force is defined as the unit weight of the material multiplied by the element area [31]. RS² allows PGA values to be directly assigned as seismic coefficients, thereby enabling straightforward implementation of pseudo-static loading conditions.

To capture the influence of seismic loading orientation, the analyses were performed for four distinct cases, each representing a combination of horizontal and vertical components of PGA. These cases are defined as: (i) leftward horizontal with upward vertical, (ii) leftward horizontal with downward vertical, (iii) rightward horizontal with upward vertical, and (iv) rightward horizontal with downward vertical. The comparative assessment of the results indicated that the maximum tunnel deformation was observed when PGA was applied with a rightward horizontal component combined with an upward vertical component. This response can be attributed to the combined effect of the in-situ stress field orientation and tunnel geometry, which together amplify deformation when external dynamic forces act in directions that align with the weaker confinement zones of the surrounding rock mass. Since this loading orientation represents the most critical condition for tunnel performance, this case was selected for detailed evaluation in the subsequent analysis. Deformation is then computed under both static and pseudo-static conditions, with PGA values ranging from 0 to 1 g in 0.05 g increments. The displacements computed are used to determine β_D, which is an uncertainty associated with earthquake input motion (demand).

4.2. Dynamic Model

The topography, along with the cavern, joints, and shear fault, is modeled using UDEC. The excavation boundary is defined, and joints are introduced in the regions illustrated in Figure 8. The numerical model is used to investigate the intrinsic stress response of the jointed rock mass under seismic loading, independent of support influence. This modeling approach provides insight into rock mass stress and discontinuity-controlled mechanisms without interfering with the support effects. Joints are introduced in the surroundings of the cavern where stress magnitudes are affected under seismic loading. Horizontal displacement is restrained at the side boundaries, while both horizontal and vertical displacements are restrained at the model base. A free-field condition is assigned at the base of the model to simulate earthquake occurrence, which allows for calculating damping forces and represents the response of an infinite medium. In addition, viscous boundaries are applied in both x and y directions at the base to absorb outgoing waves and to prevent artificial reflections from the model boundaries. This approach is widely used in seismic simulations to ensure realistic ground motion behavior [51]. UDEC requires seismic inputs to be applied as stress or velocity boundary conditions at the base or sides of the model, rather than at the ground surface. In this study, a seismic wave in the form of a sinusoidal wave is selected and is applied to the base of the model.

For a PGA of 0.25 g, the wave amplitude is computed as 2.45 m/s². Correspondingly, the velocity and displacement amplitudes are calculated as 0.26 m/s and 0.028 m, respectively. A damping ratio (ξ) of 5% is adopted, which is a typical default value for dynamic analysis in the rock mass. The typical frequency of waves during the Gorkha earthquake ranged from 0.1 Hz to 2.5 Hz, with an average value of approximately 0.24 Hz [52,53]. Thus, the wave frequency range of 0.1 to 2.5 Hz is used to determine the Rayleigh damping coefficients α and β, which yield values of 60.42 × 10⁻³ and 6.12 × 10⁻³, respectively. The numerical model is run for both static and dynamic loading conditions. The change in stress magnitude across the excavation boundary of the cavern during the seismic loading is computed.

5. Results and Discussions

5.1. Vulnerability Assessment from Fragility Curves

The deformations computed under varying PGA values are shown in Figure 9. The maximum total deformation under static conditions is found to be 5.63 cm in the wall and 4.69 cm in the crown. When seismic loading with a PGA of 0.25 g is applied, the deformation increases to 7.22 cm in the wall and 6.66 cm in the crown.

To develop a fragility curve, β_D needs to be computed based on the deformations developed at varying intensities of PGA in the cavern. β_D is estimated using the maximum likelihood estimation (MLE) method using binary damage outcomes derived from RS² pseudo-static simulation. During simulation, the cavern is subjected to increasing PGA levels. The deformation values from the simulations are compared with predefined damage criteria (Table 4 and Table 5). For each damage state, the probability of exceedance is modeled using a cumulative lognormal distribution function. The log-likelihood function is constructed using observed exceedance displacement data at various PGA values, and an optimization process is employed to determine the best-fit values for β_D. The resulting β_D reflects the uncertainty in the damage threshold relative to seismic intensity in the underground opening. A lower β_D value indicates a sharper transition from safe to damaged state, whereas a higher value suggests a more gradual increase in damage probability. This parameter is critical in fragility-based risk assessments, as it directly influences the likelihood of damage under varying seismic loads. The value of total dispersion (β_tot) is calculated as 0.5001.

Figure 10 illustrates the probability of exceeding different damage states against deformation and PGA limits in the powerhouse cavern. The damage state conditions are defined following Table 4 and Table 5, respectively. At a PGA of 0.25 g, which is defined by NBC 105 [47] for the powerhouse location, the deformation in the cavern wall is found to be 12.7 cm. This deformation extent gives a 14 percent probability of exceeding the no damage state (DS0) and about a 0.8 percent probability of exceeding moderate damage in the cavern wall, while the other damage conditions are not reached (Figure 10a). Similarly, at a PGA of 0.25 g, the likelihood of exceedance of slight damage will be about 68 percent, moderate damage 17 percent, extensive damage 5 percent, and collapse 1.7 percent (Figure 10b).

Based on a study of 71 tunnels, Dowding & Rozan [6] reported minor to moderate damage in tunnels subjected to PGA values between 0.19 g and 0.50 g. Similarly, Power et al. [48] reported slight to heavy damage in underground structures experiencing PGA in the range of 0.2 g to 0.5 g. These findings suggest that the powerhouse cavern at THP could experience minor to moderate damage in the event of a strong earthquake.

These results have close agreement with the site conditions, where almost no deformation was reported by the site engineers. In addition, as seen in the fragility curves, the powerhouse cavern exhibits different vulnerability characteristics depending on the extent of deformation values and PGA values. Therefore, both the exceedance of deformation and PGA must be considered when assessing stability against seismic loads in an underground opening.

Considering the closure limits proposed by Hoek and Marinos [39], the factor of safety (FOS) is computed and is presented in Figure 11. The FOS decreases nonlinearly with increasing PGA for all closure limits. For conditions with no stability issues, i.e., closure of less than 1%, the FOS is approximately equal to 1 at PGA values between about 0.55 g and 0.6 g. For all closure limits, a noticeable curvature or change in slope occurs at PGA values of approximately 0.5–0.7 g. This behavior indicates significant rock mass yielding, mobilization of plastic deformation, and a possible transition from an elastic to an elasto-plastic cavern response across these PGA limits.

5.2. Stress Redistributions Due to Earthquake

Before excavation, the rock mass is in an equilibrium stress state. When a cavern is excavated, the equilibrium stress state is disturbed, which leads to stress redistribution around the periphery. This results in the buildup of tangential stress in certain areas, which may affect the stability of the cavern. The orientation of joints and the presence of fracture zones strongly influence how stress is redistributed due to reduced stiffness and strength of the rock mass. After the excavation, discontinuities can slip, rotate, or deform, altering the stress paths and shifting deep into the in-situ rock mass. During an earthquake event, dynamic stresses are superimposed on the existing static in-situ stress environment, and the time-dependent stresses fluctuate rapidly. This process causes cyclic increase and decrease in tangential and horizontal stresses in the periphery of the cavern. The fluctuation in stress magnitudes can lead to temporary overstressing, propagation of cracks in applied rock support, and increased deformation. Moreover, during seismic events, the loading direction and intensity change quickly, and the peak principal stresses can exceed those under static conditions, making the rock mass response more complex and challenging to predict.

The UDEC model is used to study the stress change behavior during the earthquake event. Figure 12a shows the UDEC model before computation, and Figure 12b shows the stress tensor obtained after running the numerical modeling. The in-plane horizontal stress (S_yy), out-of-plane horizontal stress (S_xx), and horizontal shear stress (S_xy) are retrieved from the model, and the major and minor principal stresses are calculated using Equations (17) and (18), respectively.

$${\sigma }_1={\displaystyle \frac{S_{xx}+S_{yy}}{2}}+\sqrt{{\left({\displaystyle \frac{S_{xx}-S_{yy}}{2}}\right)}^2+{\left(S_{xy}\right)}^2}$$

(17)

$${\sigma }_3={\displaystyle \frac{S_{xx}+S_{yy}}{2}}-\sqrt{{\left({\displaystyle \frac{S_{xx}-S_{yy}}{2}}\right)}^2+{\left(S_{xy}\right)}^2}$$

(18)

The FOS for the rock mass with the MC properties is given by Equation (19). Figure 13 shows the variation of FOS across different rock mass depths from the excavation face, approximately mid-height under both static and seismic loading conditions.

$$FOS={\displaystyle \frac{c+\left({\sigma }_1+{\sigma }_3\right)/2\times tan\varphi }{\left({\sigma }_1+{\sigma }_3\right)/2}}$$

(19)

Stresses are measured in the horizontal direction, extending both left and right from the excavation boundary. It is observed that under seismic loading, the major principal stress increases and the minor principal stress decreases near the excavation boundary relative to static conditions. At a distance of approximately one cavern width (≈20–25 m) from the excavation boundary, the major principal stress becomes constant, whereas the minor principal stress magnitudes gradually increase with an increase in distance from the excavation boundary.

Near the excavation face, higher stress concentration and loss of confinement lead to a progressive reduction in FOS. The highest FOS values occur at locations farther from the excavated face, where the rock mass remains largely undisturbed, and confinement remains relatively high. Under dynamic loading conditions, FOS values are consistently lower than those under static loading, with reductions ranging from approximately 8% to 24%. Despite this reduction, the overall shape of the FOS distribution remains similar for both loading conditions.

The similarity in the FOS profiles indicates that dynamic loading primarily reduces the available strength of the rock mass rather than significantly altering the stress redistribution pattern around the cavern. The reduction in FOS under dynamic conditions highlights the destabilizing effect of stress path changes and confinement loss on cavern stability. Furthermore, the results show that stability is not governed solely by the conditions at the excavation face; instead, a critical zone develops at a finite distance from the face where the FOS reaches its minimum. This behavior emphasizes the importance of considering spatial variations in stability and asymmetric confinement loss when designing support systems for underground structures.

An increased major principal stress (σ₁) and decreased minor principal stress mean that there is a further increase in stress anisotropy, which leads to the increase in maximum tangential stress (

Table 7. Tangential stress values in the powerhouse cavern location.

Loading Condition	a Max. Tangential Stress (σθ,max) MPa		b Min. Tangential Stress (σθ,min) MPa
	Left Wall	Right Wall	Left Wall	Right Wall
Static	15.97	11.76	−5.34	−3.83
Dynamic	22.27	17.63	−7.45	−5.78

^a σ_θ,max = 3σ₁ − σ₃. ^b σ_θ,min = 3σ₃ − σ₁.

). If maximum tangential stress exceeds rock mass strength, there is a high risk of stress-induced instability. Therefore, the applied support systems should be such that they are capable of withstanding additional deformation caused by seismic loading and absorbing stress-driven movements. On the other hand, a decreased minor principal stress and an increase in stress anisotropy may bring the minimum tangential stress to negative (to tensile failure).

5.3. Significance and Engineering Implications

The study of seismic influence on the stability of the underground caverns in the Himalayan region is limited. This manuscript addresses this gap by evaluating underground caverns in a region characterized by frequent seismic movements. A key research advancement is the application of fragility curve analysis, which introduces a probabilistic perspective on the likelihood of different damage states under varying seismic intensities, an approach rarely used in the context of Himalayan underground construction, including hydropower tunnels and caverns. By quantifying deformation response and stress redistribution under seismic loading, the study provides valuable insights into how underground caverns behave during earthquakes. The methodology also establishes a framework that can be adapted for assessing other hydropower and underground projects to be built in the Himalayan region.

This is among the first detailed seismic assessments of an underground cavern located in the Lesser Himalayan rock formation, utilizing both pseudo-static and dynamic simulations. It is emphasized here that stress redistribution during earthquakes continues into the post-seismic phase, which may result in additional loading on the applied rock support. The study demonstrates that caverns can withstand moderate seismic events if adequately supported. However, local overstressing may occur, which may require additional reinforcement. Overall, the study enhances understanding of underground seismic vulnerability in the Nepal Himalaya and contributes to safer and more resilient underground engineering practices.

5.4. Limitations and Recommendations for Future Work

The present study focuses on a single representative case; however, future research incorporating multiple studies enables the development of more comprehensive and region-specific guidelines for the seismic analysis of underground caverns in the Himalayan region. The methodology adopted in this work provides a good foundation for developing such guidelines. In this study, fragility curves are developed using a pseudo-static approach. To enhance the reliability of seismic performance assessment, a time-history dynamic analysis can be employed in future work to generate comparable fragility curves and to facilitate the systematic comparison with those derived from the pseudo-static approach. Furthermore, the current 2D numerical modeling can be extended to 3D analysis to accurately capture stress redistribution along the longitudinal direction and to account for the inherent geological and structural variability within the cavern system.

6. Conclusions

This study presents a comprehensive seismic assessment of an underground powerhouse cavern located in the Lesser Himalayas, integrating both FEM and DEM approaches. The main conclusions and contributions are summarized as follows:

• Seismic deformation assessment using pseudo-static FEM
  • Cavern wall deformations increased from 5.6 cm (static) to 7.2 cm (PGA = 0.25 g), and crown deformations increased from 4.7 cm to 6.7 cm, indicating that seismic loading moderately amplifies deformation.
  • The deformation data were successfully used to develop fragility curves, providing a probabilistic framework to quantify seismic vulnerability.
• Fragility-based vulnerability insights
  • Fragility analysis revealed a 68% probability of slight damage and only a 1.7% probability of collapse at the site-specific PGA of 0.25 g.
  • The approach demonstrates that strain-based damage indicators, originally developed for weak rock squeezing, can be effectively adapted to foliated and brittle rock masses under deformation-controlled seismic loading.
• Stress redistribution under dynamic seismic loading
  • UDEC modeling highlighted that major principal stresses increase while minor principal stresses decrease near the cavern boundary during earthquakes, leading to localized stress anisotropy.
  • Factor of Safety (FOS) profiles showed reductions of 8–24% under dynamic loading, emphasizing the need to account for stress redistribution and confinement loss in the design of underground supports.
• Engineering significance
  • This study provides a practical methodology for combining probabilistic fragility analysis with numerical modeling of stress redistribution in deep underground caverns.
  • The findings offer site-specific guidance for the design and monitoring of underground hydropower structures in seismically active Himalayan regions.
  • The integrated FEM–DEM approach demonstrates a computationally efficient yet physically representative framework, which can be adapted for similar underground projects in complex geological conditions.

In conclusion, the study provides both quantitative insights into cavern deformation and failure probabilities under seismic loading and guidance for engineering design and risk assessment in the construction of underground structures in the Himalayan regions.