Seismic Response of Artificial Dams in Coal Mine Underground Reservoirs

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This study provides a simplified numerical framework for comparing the seismic response of different artificial dam structures in coal mine underground reservoirs, supporting preliminary but fast dam-type selection, dynamic safety assessment, and structural optimisation under underground reservoir operating conditions.

Abstract

Coal mine underground reservoirs are increasingly used for mine-water storage and reuse in ecologically fragile mining regions, but the dynamic response of artificial dam structures under coupled water-pressure and seismic loading remains insufficiently understood. This study develops a simplified two-dimensional frame-based dynamic model to compare flat slab, gravity, and arch-equivalent artificial dams. Two water pressure levels, 0.1 and 1.0 MPa, and two seismic intensities, PGA = 0.1 g and 0.5 g, were considered using four representative acceleration histories. The arch dam was represented by a vertical rectangular section with equivalent arch-action lateral restraint. Results show that water pressure primarily controls peak total displacement, whereas PGA mainly governs the dynamic displacement increment and absolute acceleration. Increasing water pressure from 0.1 to 1.0 MPa markedly amplified total displacement and tensile stress demand, while increasing PGA from 0.1 g to 0.5 g produced a clearer effect on dynamic increments than on total displacement. The arch-equivalent dam consistently showed the smallest displacement response, while the gravity-type dam developed higher tensile stress demand under high water pressure in the simplified model. Effective modal frequencies were relatively high, explaining the coexistence of small displacement demand and noticeable acceleration response. The results provide a mechanistic basis for artificial dam-type comparison and preliminary safety assessment in underground reservoirs.

1. Introduction

Coal mining in western and northwestern China is commonly constrained by the coexistence of abundant coal resources and scarce water resources. Large-scale and high-intensity extraction in these ecologically fragile regions can disturb overlying strata, induce groundwater loss, and aggravate the contradiction between coal production and water resource conservation [1,2]. Coal mine underground reservoirs have therefore been developed as an important technology for water-preserved coal mining, in which the goaf formed by longwall extraction is used as an underground storage space for mine water [3,4]. This technology provides a practical route for retaining, storing, and reusing mine water underground, thereby reducing surface discharge and supporting ecological protection in arid and semi-arid mining areas. In the Shendong mining area and other major coal-producing regions, underground reservoir technology has gradually evolved from a conceptual water-conservation approach into an engineering system involving goaf storage spaces, coal pillar dams, artificial dams, seepage-control measures, and operational water-level management [4,5]. Recent reviews have further emphasised that the long-term safety of underground reservoirs is governed not only by the storage capacity of goafs, but also by the stability and tightness of dam structures that retain the stored water [6].

The dam system of a coal mine underground reservoir generally consists of coal pillar dams and artificial dam structures. Coal pillar dams are reserved coal or rock masses that separate adjacent goaf reservoirs or working areas, while artificial dams are constructed concrete or cementitious retaining structures used to block water flow pathways between coal pillars, roadways, and storage spaces. A recent study by Xu et al. [7] systematically illustrated the typical underground reservoir dam system and classified artificial dams into representative flat slab, gravity, and arch-type structures; Figure 1 provides a useful schematic reference for the structural composition and dam-type idealisation adopted in the present work. Existing studies on coal pillar dams have mainly focused on tightness, fracture development, seepage stability, long-term permeability evolution, and optimum width design. For example, Zhang et al. [8] evaluated the long-term tightness of coal pillar dams under different reservoir connections and operating conditions, while Wang et al. [9] investigated the damage and failure evolution of coal pillar dams affected by water immersion. Other studies have examined fracture propagation and deformation evolution under combined mining and hydraulic conditions, highlighting that seepage channels may form when fractures penetrate the dam body or its surrounding strata [10,11]. These studies have established an important basis for understanding coal pillar dam stability, but the mechanical response of artificial dam structures, especially under coupled hydraulic and dynamic actions, remains less well-understood.

During reservoir operation, dam structures are subjected to complex service environments involving lateral water pressure, overburden pressure, mining-induced stress redistribution, wetting–drying cycles, seepage erosion, water immersion, and cyclic loading. These factors can affect dam stability through changes in strength, stiffness, permeability, damage accumulation, and local stress concentration. Laboratory and numerical studies have shown that water immersion can degrade coal or concrete-like dam materials and reduce tightness performance [8,9], while creep-seepage coupling may alter the permeability and deformation characteristics of coal pillar dams over time [12]. For artificial dams, hydro-mechanical damage modelling has been used to examine localised failure and stability under water pressure [13], and physical and numerical simulations have been conducted to determine ultimate water levels and deformation-failure patterns [14]. More recently, storage–drainage cycles and cyclic loading–unloading have been shown to affect the mechanical performance and stability evolution of artificial dams, indicating that reservoir operation should be treated as a time-dependent and cyclic loading process rather than a single static condition [15,16]. Despite these advances, most existing studies still emphasise static water-retaining performance, seepage safety, or material degradation, whereas the dynamic response of different artificial dam forms remains insufficiently explored.

This issue becomes increasingly important as coal mining extends to greater depths. With increasing burial depth, underground reservoir dams may experience higher in situ stresses, stronger mining-induced stress redistribution, larger water pressure, and more frequent dynamic disturbances. Adjacent working face extraction can impose additional stress concentration and deformation on reservoir dams, while roof breaking, fault slip, mine tremors, and regional earthquakes may generate dynamic loads that are superimposed on the existing static stress and hydraulic pressure fields. Yao et al. [17] experimentally investigated coal pillar dam damage under dynamic loading and water absorption conditions, demonstrating that dynamic disturbance can markedly affect damage and failure behaviour. Xu et al. [18] further studied the critical instability and dynamic loading laws of artificial dams, showing that displacement, velocity, and stress distributions are strongly affected by dynamic load intensity and source distance. These studies indicate that dynamic effects cannot be ignored in the safety assessment of deep underground reservoir dams. However, the majority of available dynamic studies have focused on either coal pillar dams or a single artificial dam configuration, and systematic comparisons between alternative artificial dam forms under coupled water-pressure and seismic loading are still limited.

The structural form of an artificial dam can strongly influence its deformation mode, stress distribution, and failure tendency. Flat slab dams are structurally simple but may be more susceptible to bending deformation and edge cracking under lateral hydraulic loading. Gravity-type dams can improve global stiffness through increased base thickness, whereas arch-type dams may redistribute lateral loads through arch-action and abutment restraint. Previous work has considered groove depth, dam size, hydro-mechanical damage, non-uniform biaxial loading, water-level capacity, and structural type effects in artificial dams [7,13,14,19,20,21]. Cao [21] investigated the aseismic performance of different artificial dam structures, and Xu et al. [7] further showed that dam type, hydraulic pressure, and top restraint jointly control displacement, stress concentration, and cracking patterns under static loading. Nevertheless, important gaps remain. First, the influence of dam type has rarely been assessed under a unified dynamic framework that simultaneously considers water pressure and seismic intensity. Second, the distinction between total displacement dominated by static hydraulic loading and dynamic displacement increment induced by seismic excitation has not been clearly quantified. Third, the mechanism by which structural stiffness and modal characteristics control the coexistence of small displacement demand and noticeable acceleration response remains insufficiently discussed.

To address these gaps, this study develops a simplified two-dimensional frame-based dynamic model to compare the behaviour of flat slab, gravity, and arch-equivalent artificial dams under coupled water-pressure and seismic loading. Two water pressure levels, 0.1 MPa and 1.0 MPa, and two seismic intensity levels, PGA = 0.1 g and PGA = 0.5 g, are considered. Four representative acceleration histories with different frequency characteristics are applied to evaluate the sensitivity of dam response to input motion. The arch dam is represented by a vertical rectangular side section with an equivalent arch-action lateral restraint, allowing the plan-view arch effect to be incorporated without incorrectly imposing arch curvature in the vertical side section. The analysis first evaluates effective modal frequencies under different water-pressure conditions and then compares peak total displacement, peak dynamic increment, peak absolute acceleration, tensile stress demand, and normalised response ratios among the three dam types. Through this approach, the study aims to clarify how water pressure, seismic intensity, and structural form jointly govern the dynamic response of underground reservoir artificial dams, and to provide a mechanistic basis for dam-type selection and safety evaluation in deep coal mine underground reservoirs.

2. Methodology

2.1. Model Idealisation and Structural Configurations

A simplified two-dimensional dynamic analysis framework was developed to compare the response of three representative dam structural forms under coupled water-pressure and seismic loading. The intention of the model is to provide a consistent comparative framework rather than a full three-dimensional simulation of a dam-rock-water system. The three investigated configurations are a flat slab dam, a gravity-type dam and an arch-type dam, as shown in Figure 2. The flat slab dam represents a vertical wall-like structure with uniform thickness. The gravity dam represents a section with increasing thickness towards the base. For the arch dam, the vertical side section is kept rectangular, while the plan-view arch effect is introduced through an equivalent lateral restraint and enhanced flexural stiffness. This avoids representing the arch as a curved vertical section, which would be physically inconsistent with the actual plan-view nature of arch action.

All models have a height of 5.0 m and an out-of-plane width of 6.0 m. The flat slab and arch dam use a constant thickness of 2.0 m, whereas the gravity dam thickness varies linearly from 2.5 m at the base to 1.5 m at the top. The sectional area and second moment of area are calculated as

$$A=bt, I={\displaystyle \frac{b{t}^3}{12}}$$

(1)

For the gravity dam, the thickness at height $z$ is defined as

$$t\left(z\right)=1.5+\left(2.5-1.5\right)\left(1-{\displaystyle \frac{z}{H}}\right)$$

(2)

For the equivalent arch dam, the initial flexural stiffness is increased using an arch-stiffness factor, and additional lateral springs are distributed along the height to represent abutment restraint:

$${\left(EI\right)}_{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}={\eta }_{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}EI, {\eta }_{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}=1.50$$

(3)

$$k_{x,i}=k_{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}{h}_i, k_{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}=8.0\times {10}^8\hspace{0.25em}{\displaystyle \frac{\mathrm{N}}{\mathrm{m}}}\hspace{0.25em}$$

(4)

where $h_i$ is the tributary height associated with node $i$. The bottom node is fully fixed, while the top node is restrained in horizontal translation but remains free in vertical translation and rotation. This top boundary condition approximates the lateral confinement from surrounding rock or structural cover without imposing a fully fixed upper boundary.

The equivalent arch-stiffness parameters should be interpreted as modelling parameters rather than calibrated material or structural constants. Because project-specific field measurements or three-dimensional arch dam calibration data are unavailable, the arch-stiffness factor and distributed lateral spring stiffness were selected to represent a moderate level of arch-action restraint within the simplified two-dimensional framework. The purpose of these parameters is to incorporate, in an approximate manner, the additional lateral load-transfer capacity and abutment restraint associated with arch action, while maintaining the same side-section geometry used for comparison with the flat slab dam. Therefore, the arch-equivalent model is used to evaluate relative response trends rather than to reproduce the exact behaviour of a specific arch dam. To examine the influence of these assumptions, a parameter-sensitivity analysis was conducted by varying the arch-stiffness factor and lateral spring stiffness.

The material is assumed to be reinforced concrete with an elastic modulus of 30 GPa, density of 2500 kg/m³, Poisson’s ratio of 0.20 and damping ratio of 5%. The tensile and compressive strengths are taken as 2.5 MPa and 30 MPa, respectively (

Table 1. Main model and material parameters.

Parameter	Symbol	Value
Dam height	$H$	5.0 m
Out-of-plane width	$b$	6.0 m
Flat slab thickness	$t$	2.0 m
Gravity dam thickness	$t\left(z\right)$	2.5 m at base to 1.5 m at top
Arch dam side-section thickness	$t$	2.0 m
Elastic modulus	$E$	30 GPa
Density	$\rho$	2500 kg/m3
Poisson’s ratio	$\nu$	0.20
Damping ratio	$\zeta$	5%
Tensile strength	$f_t$	2.5 MPa
Compressive strength	$f_c$	30 MPa
Cracked-stiffness ratio	$r_t$	0.35
Minimum softening stiffness ratio	$r_c$	0.25
Equivalent arch-stiffness factor	${\eta }_{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$	1.50
Equivalent arch spring stiffness	$k_{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$	$8.0\times {10}^8$ N/m per m height

). A simplified stiffness degradation model is used to represent tensile cracking and compressive softening. The tensile stress demand is estimated from the maximum element end moment, and the compressive stress demand is obtained by adding the overburden pressure to the bending-induced compressive stress:

$${\sigma }_t={\displaystyle \frac{M_{\mathrm{m}\mathrm{a}\mathrm{x}}c}{I}}, {\sigma }_c=p_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{b}\mathrm{u}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{n}}+{\displaystyle \frac{M_{\mathrm{m}\mathrm{a}\mathrm{x}}c}{I}}, c={\displaystyle \frac{t}{2}}$$

(5)

When ${\sigma }_t>f_t$, the effective flexural stiffness is reduced using an equivalent cracked-stiffness ratio, while compressive softening is activated when ${\sigma }_c>0.6f_c$, with the stiffness gradually reduced to a prescribed minimum value. It should be emphasised that this degradation rule is an equivalent engineering approximation rather than a full cyclic reinforced-concrete constitutive model. The method is intended to capture the first-order influence of tensile overstress and compressive stress demand on global flexural stiffness within the simplified frame-based framework. It does not explicitly model crack initiation and propagation, crack opening–closure, reinforcement yielding, bond-slip, stiffness recovery during unloading, or hysteretic energy dissipation under cyclic loading. Therefore, the calculated cracked elements and tensile stress exceedance should be interpreted as damage tendency indicators rather than as direct predictions of physical crack patterns. To assess the influence of this assumption, a parameter-sensitivity analysis was conducted by varying the cracked-stiffness ratio and the compressive softening threshold.

2.2. Dam Element Formulation and Global Assembly

Each dam is discretised using two-dimensional Euler–Bernoulli frame elements. Each node has three degrees of freedom, namely, horizontal displacement, vertical displacement and rotation, written as $\{u_i,w_i,{\theta }_i{\}}^T$. The element local stiffness matrix is

$${\mathbf{k}}_e=\left[\begin{array}{cccccc}{\displaystyle \frac{EA}{L}}& 0& 0& -{\displaystyle \frac{EA}{L}}& 0& 0\\ 0& {\displaystyle \frac{12EI}{L^3}}& {\displaystyle \frac{6EI}{L^2}}& 0& -{\displaystyle \frac{12EI}{L^3}}& {\displaystyle \frac{6EI}{L^2}}\\ 0& {\displaystyle \frac{6EI}{L^2}}& {\displaystyle \frac{4EI}{L}}& 0& -{\displaystyle \frac{6EI}{L^2}}& {\displaystyle \frac{2EI}{L}}\\ -{\displaystyle \frac{EA}{L}}& 0& 0& {\displaystyle \frac{EA}{L}}& 0& 0\\ 0& -{\displaystyle \frac{12EI}{L^3}}& -{\displaystyle \frac{6EI}{L^2}}& 0& {\displaystyle \frac{12EI}{L^3}}& -{\displaystyle \frac{6EI}{L^2}}\\ 0& {\displaystyle \frac{6EI}{L^2}}& {\displaystyle \frac{2EI}{L}}& 0& -{\displaystyle \frac{6EI}{L^2}}& {\displaystyle \frac{4EI}{L}}\end{array}\right]$$

(6)

The consistent element mass matrix is

$${\mathbf{m}}_e={\displaystyle \frac{\rho AL}{420}}\left[\begin{array}{cccccc}140& 0& 0& 70& 0& 0\\ 0& 156& 22L& 0& 54& -13L\\ 0& 22L& 4L^2& 0& 13L& -3L^2\\ 70& 0& 0& 140& 0& 0\\ 0& 54& 13L& 0& 156& -22L\\ 0& -13L& -3L^2& 0& -22L& 4L^2\end{array}\right]$$

(7)

The local matrices are transformed into the global coordinate system using the element transformation matrix ${\mathbf{T}}_e$, and the global stiffness and mass matrices are assembled as

$$\mathbf{K}=\sum _e{\mathbf{T}}_e^T{\mathbf{k}}_e{\mathbf{T}}_e,\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{M}=\sum _e{\mathbf{T}}_e^T{\mathbf{m}}_e{\mathbf{T}}_e$$

(8)

The lateral water pressure is converted into a uniform distributed line load using the out-of-plane width:

$$q_w=p_{w}b$$

(9)

The equivalent nodal load vector is obtained from the consistent load formulation. The vertical overburden pressure is not applied as an explicit nodal force in the dynamic equation; instead, it is included in the stress evaluation to account for the compressive stress state caused by overlying strata or structural cover. The adopted overburden pressure is 4.0 MPa.

2.3. Loading Programme, Static Analysis and Modal Analysis

Two lateral water-pressure levels are considered: 0.1 MPa and 1.0 MPa. The former represents a moderate pressure condition, whereas the latter represents a high confined-water-pressure condition. Four horizontal acceleration histories are used as input motions. They represent a low-frequency excitation, a high-frequency excitation, a mine-seismic-like decaying burst and a pulse-like excitation (Figure 3). Each acceleration history was scaled to two target PGA levels, 0.1 g and 0.5 g, using the following amplitude-scaling relationship:

$$a_g^{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}}\left(t\right)=a_g\left(t\right){\displaystyle \frac{\mathrm{P}\mathrm{G}\mathrm{A}\cdot g}{\mathrm{m}\mathrm{a}\mathrm{x}\left|a_g\left(t\right)\right|}}$$

(10)

Here, $a_g\left(t\right)$ is the original acceleration history, $a_g^{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}}\left(t\right)$ is the scaled acceleration history, PGA is the target acceleration level expressed in units of g, and g is gravitational acceleration.

To make the input definition more transparent, the spectral and duration characteristics of the four motions were quantified using dominant frequency, mean frequency, Arias intensity, and significant duration D_5–95. The four motions were designed to cover contrasting excitation characteristics, including low-frequency, high-frequency, short-duration mine-seismic-like, and pulse-like inputs. The dominant frequencies of EQ1, EQ2, EQ3, and EQ4 were approximately 1.17 Hz, 5.00 Hz, 12.00 Hz, and 2.50 Hz, respectively, confirming the intended differences in frequency content among the four excitation types. In addition, the significant duration D_5–95, calculated from cumulative Arias intensity, was used to distinguish relatively long-duration harmonic-type inputs from short-duration burst and pulse-type motions. This classification was adopted to ensure that the dynamic analysis considered different frequency contents, duration characteristics, and transient loading patterns within a controlled PGA-scaling framework, while maintaining a simplified and comparative seismic input definition.

For each water-pressure condition, nonlinear static analysis is first performed to obtain the initial displacement field and the corresponding degraded stiffness state. The reduced static equilibrium equation is

$${\mathbf{K}}_r{\mathbf{u}}_r={\mathbf{F}}_r$$

(11)

After each static solution, element end forces are recovered, stress demands are evaluated using Equation (5), and the effective flexural stiffness is updated. The iteration is terminated when the relative change in displacement satisfies

$${\displaystyle \frac{∥{\mathbf{u}}^{\left(k\right)}-{\mathbf{u}}^{\left(k-1\right)}∥}{∥{\mathbf{u}}^{\left(k\right)}∥}}<{10}^{-4}$$

(12)

Modal analysis is subsequently performed using the updated stiffness matrix so that the effective frequencies reflect the stiffness degradation induced by water pressure. The undamped eigenvalue problem is

$$\left({\mathbf{K}}_r-{\omega }^2{\mathbf{M}}_r\right)\mathsf{\varphi }=\mathbf{0}, f={\displaystyle \frac{\omega }{2\pi }}$$

(13)

Rayleigh damping is adopted in the dynamic analysis:

$${\mathbf{C}}_r=\alpha {\mathbf{M}}_r+\beta {\mathbf{K}}_r$$

(14)

The coefficients $\alpha$ and $\beta$ are determined from the first two effective modal frequencies and a target damping ratio of 5%:

$$\left[\begin{array}{cc}{\displaystyle \frac{1}{2{\omega }_1}}& {\displaystyle \frac{{\omega }_1}{2}}\\ {\displaystyle \frac{1}{2{\omega }_2}}& {\displaystyle \frac{{\omega }_2}{2}}\end{array}\right]\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]=\left[\begin{array}{c}\zeta \\ \zeta \end{array}\right]$$

(15)

The analysis programme consists of 48 dynamic cases, combining two water pressure levels, two PGA levels, three dam types and four input motions. The cases are summarised in

Table 2. Analysis cases used in the dynamic response study.

Category	Cases
Dam type	Flat slab dam; gravity dam; arch dam
Water pressure	0.1 MPa; 1.0 MPa
Seismic intensity	PGA = 0.1 g; PGA = 0.5 g
Input motion	EQ1_lowfreq; EQ2_highfreq; EQ3_mine_like; EQ4_pulse
Boundary condition	Bottom fixed; top horizontal translation restrained
Arch dam representation	Vertical rectangular side section with equivalent lateral arch-action restraint

.

2.4. Nonlinear Dynamic Analysis and Response Indices

The dynamic response is computed using the Newmark average acceleration method with $\beta =0.25$ and $\gamma =0.50$. The equation of motion for the relative response under horizontal base acceleration is

$${\mathbf{M}}_r{\ddot{\mathbf{u}}}_r\left(t\right)+{\mathbf{C}}_r{\dot{\mathbf{u}}}_r\left(t\right)+{\mathbf{K}}_r{\mathbf{u}}_r\left(t\right)=-{\mathbf{M}}_r\mathbf{r}{a}_g\left(t\right)$$

(16)

where $\mathbf{r}$ is the influence vector associated with horizontal base excitation. The influence vector assigns unit participation to horizontal translational degrees of freedom and zero participation to all other degrees of freedom. At each time step, the Newmark effective stiffness matrix is

$${\mathbf{K}}_{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathbf{K}}_r+{\displaystyle \frac{1}{\beta \Delta t^2}}{\mathbf{M}}_r+{\displaystyle \frac{\gamma }{\beta \Delta t}}{\mathbf{C}}_r$$

(17)

The dynamic analysis is carried out using a time step of 0.002 s and a total duration of 12.0 s. During the dynamic analysis, the effective element stiffness is updated based on the combined static and dynamic displacement field, allowing tensile overstress and stiffness degradation to influence the subsequent dynamic response.

The total lateral displacement is defined as the sum of the nonlinear static displacement and the dynamic displacement increment:

$${\mathbf{u}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\left(t\right)={\mathbf{u}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}}+{\mathbf{u}}_{\mathrm{d}\mathrm{y}\mathrm{n}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{c}}\left(t\right)$$

(18)

The peak total displacement and peak dynamic increment are evaluated as

$$u_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l},\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}=\underset{t}{\mathrm{m}\mathrm{a}\mathrm{x}}\left(\underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}}\left|u_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l},i}\left(t\right)\right|\right), u_{\mathrm{d}\mathrm{y}\mathrm{n},\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}=\underset{t}{\mathrm{m}\mathrm{a}\mathrm{x}}\left(\underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}}\left|u_{\mathrm{d}\mathrm{y}\mathrm{n}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{c},i}\left(t\right)\right|\right)$$

(19)

The absolute acceleration is obtained by adding the ground acceleration to the relative structural acceleration:

$$a_{\mathrm{a}\mathrm{b}\mathrm{s},i}\left(t\right)={\ddot{u}}_{\mathrm{d}\mathrm{y}\mathrm{n}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{c},i}\left(t\right)+a_g\left(t\right), a_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k},g}={\displaystyle \frac{1}{g}}\underset{t}{\mathrm{m}\mathrm{a}\mathrm{x}}\left(\underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}}\left|a_{\mathrm{a}\mathrm{b}\mathrm{s},i}\left(t\right)\right|\right)$$

(20)

Two displacement-based dynamic amplification factors are used to distinguish the total displacement response from the pure dynamic increment:

$${\mathrm{D}\mathrm{A}\mathrm{F}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={\displaystyle \frac{u_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l},\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}}{u_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c},\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}}}, {\mathrm{D}\mathrm{A}\mathrm{F}}_{\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}={\displaystyle \frac{u_{\mathrm{d}\mathrm{y}\mathrm{n},\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}}{u_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c},\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}}}$$

(21)

Peak tensile stress, peak compressive stress and maximum bending moment are extracted from the element end-force results over the full time history. In addition to absolute response quantities, the gravity and arch dam responses are normalised by the corresponding flat slab response under the same loading condition:

$$R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}={\displaystyle \frac{R_{\mathrm{d}\mathrm{a}\mathrm{m}}}{R_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}\hspace{0.25em}\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{b}}}}$$

(22)

This normalisation provides a direct measure of the relative performance of the gravity and arch-type configurations compared with the flat slab dam. A value smaller than unity indicates response reduction relative to the flat slab, whereas a value larger than unity indicates response amplification. The interpretation of the results is therefore based on both the absolute response values and the normalised response ratios.

2.5. Modal Verification and Interpretation

To further examine the dynamic characteristics of the simplified framework, an approximate modal-frequency verification was performed using equivalent Euler–Bernoulli beam theory. The first-order natural frequencies of equivalent cantilever and fixed-pinned beam systems were calculated and compared with the numerical modal results obtained from the frame-element model. The comparison showed generally consistent frequency levels under the adopted stiffness assumptions and boundary conditions.

It should be noted that the calculated modal frequencies are relatively high compared with those of conventional large-scale dam–foundation–reservoir systems. This is mainly because the present model represents a relatively small and stiff underground artificial dam system with a characteristic height of approximately 5 m, strong top and bottom constraints, and simplified equivalent beam idealisation. In addition, the current framework does not explicitly consider fluid–structure interaction, hydrodynamic added mass, surrounding rock flexibility, or interface deformation, all of which would reduce the effective system stiffness and modal frequencies in real engineering conditions. Therefore, the present modal analysis is primarily intended to support comparative dynamic assessment among different dam types within a consistent simplified framework, rather than to reproduce the exact modal characteristics of field-scale underground reservoir systems.

To evaluate the influence of the simplified stiffness degradation rule, a parameter-sensitivity analysis was performed for the cracked-stiffness ratio and the compressive softening threshold. The cracked-stiffness ratio was varied as r_t = 0.25, 0.35, and 0.50, while the compressive softening threshold was varied as 0.5 f_c, 0.6 f_c, and 0.7 f_c. The baseline values used in the main analysis were r_t = 0.35 and 0.6 f_c. The sensitivity analysis was not intended to calibrate a detailed cyclic damage model, but to examine whether the main comparative trends among dam types were robust against reasonable changes in the equivalent stiffness degradation parameters.

3. Results

3.1. Effective Modal Characteristics

The effective modal frequencies obtained after the nonlinear static stiffness update are shown in Figure 4 and summarised in

Table 3. Effective modal frequencies after nonlinear static stiffness updating.

Dam Type	Water Pressure (MPa)	Mode 1 (Hz)	Mode 2 (Hz)	Mode 3 (Hz)
Flat slab dam	0.1	173.3	196.3	522.1
Gravity dam	0.1	191.4	206.5	528.8
Arch dam	0.1	173.3	241.8	521.1
Flat slab dam	1.0	123.0	173.3	477.9
Gravity dam	1.0	140.4	191.4	495.9
Arch dam	1.0	155.8	173.3	521.1

. All three structural forms exhibit relatively high natural frequencies, with first-mode frequencies ranging from 173.3 to 191.4 Hz under 0.1 MPa water pressure and from 123.0 to 155.8 Hz under 1.0 MPa water pressure. This indicates that the dam models behave as stiff, short-height structural systems. The first-mode frequencies are considerably higher than the dominant frequency range of the imposed input motions, which provides an important basis for interpreting the subsequently observed combination of small displacement demand and noticeable absolute acceleration response.

The reduction in modal frequency under 1.0 MPa water pressure is most apparent for the flat slab dam and gravity dam. This reduction is associated with the combined effect of larger static deformation and stiffness degradation after tensile overstress. In contrast, the arch-equivalent model maintains a relatively high first-mode frequency under the high-water-pressure condition because the equivalent arch-action lateral restraint increases the effective lateral stiffness. The modal frequency was evaluated as:

$$f_i={\displaystyle \frac{{\omega }_i}{2\pi }}$$

(23)

where ${\omega }_i$ is the $i$th circular natural frequency and $f_i$ is the corresponding frequency in Hz.

In addition to the FEM modal analysis, an approximate first-order cantilever frequency was calculated using Euler–Bernoulli beam theory:

$$f_1={\displaystyle \frac{{1.875}^2}{2\pi }}\sqrt{{\displaystyle \frac{EI}{\rho A{H}^4}}}$$

(24)

where E is the elastic modulus, I is the equivalent section moment of inertia, $\rho$ is the material density, A is the equivalent section area, and H is the dam height. Taking the flat slab dam as an example, with E = 30 GPa, H = 5.0 m, A = 12.0 m², I = 4.0 m⁴, and $\rho$ = 2500 kg/m³, the approximate cantilever frequency is about 44.8 Hz. This value is lower than the FEM first-mode frequencies reported in this study because the numerical model adopts stronger boundary restraint and equivalent stiffness assumptions than a free cantilever. Therefore, the high FEM frequencies should be interpreted as the dynamic characteristics of the simplified stiff frame system rather than as direct field-scale modal frequencies of a fully coupled dam–rock–water system.

3.2. Peak Total Displacement Under Water Pressure–Seismic Coupling

The peak total displacement response is presented in Figure 5. The most pronounced trend is the dominant influence of water pressure. When the water pressure increases from 0.1 MPa to 1.0 MPa, the peak total displacement increases by more than one order of magnitude for all three dam types. For example, under PGA = 0.1 g, the mean peak total displacement of the flat slab dam increases from 0.0175 mm to 0.4122 mm, whereas that of the arch dam increases from 0.0116 mm to 0.2463 mm. By contrast, increasing PGA from 0.1 g to 0.5 g produces only a small change in total displacement, especially under 1.0 MPa water pressure.

This result demonstrates that the total lateral displacement is primarily governed by the static water pressure rather than by the seismic intensity. Across all loading cases, the displacement ranking is consistent:

$$u_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}\hspace{0.25em}\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{b}}>u_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}}>u_{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}$$

(25)

The flat slab dam exhibits the largest total displacement because of its relatively low lateral stiffness. The gravity dam shows an intermediate response due to its increased sectional thickness near the base. The arch dam exhibits the smallest total displacement because the equivalent arch-action restraint reduces lateral deformation.

3.3. Peak Dynamic Increment Induced by Seismic Excitation

The peak dynamic displacement increment is shown in Figure 6. Unlike the total displacement, the dynamic increment isolates the seismic contribution from the static water-pressure-induced deformation. The results show that PGA has a much clearer influence on the dynamic increment than on total displacement. Under 0.1 MPa water pressure, increasing PGA from 0.1 g to 0.5 g increases the mean dynamic increment of the flat slab dam from 0.0009 mm to 0.0043 mm. Under 1.0 MPa water pressure, the corresponding increase is from 0.0022 mm to 0.0110 mm.

The dynamic increment was calculated as:

$$u_{\mathrm{d}\mathrm{y}\mathrm{n},\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}={\mathrm{m}\mathrm{a}\mathrm{x}}_t\left({\mathrm{m}\mathrm{a}\mathrm{x}}_i\left|u_{\mathrm{d}\mathrm{y}\mathrm{n},i}\left(t\right)\right|\right)$$

(26)

where $u_{\mathrm{d}\mathrm{y}\mathrm{n},i}\left(t\right)$ is the horizontal dynamic displacement increment of node $i$. The dynamic increment also increases slightly when the water pressure changes from 0.1 MPa to 1.0 MPa, reflecting the influence of static-stress-induced stiffness degradation on the dynamic response. Nevertheless, the dam-type ranking remains consistent. The flat slab dam generally shows the largest dynamic increment, whereas the arch dam shows the smallest dynamic increment because of its equivalent lateral restraint.

3.4. Time–History Comparison Under Representative Severe Cases

Figure 7 compares the dynamic displacement increment histories under representative strong seismic loading with PGA = 0.5 g. Two water pressure levels are shown to clarify the influence of the initial static stress state on the subsequent seismic response. The time–history curves confirm the trends observed in the peak response matrices. For a given earthquake input, the flat slab dam generally develops the largest dynamic displacement increment, the gravity dam shows an intermediate response, and the arch dam remains the least deformable structural form.

The time–history comparison also shows that the frequency content and waveform characteristics of the input motion affect the timing and magnitude of the response peaks. The mine-like and pulse-like records generate short-duration response bursts, while the low-frequency and high-frequency harmonic records produce more continuous oscillatory responses. Despite these differences, the relative performance of the three dam types remains stable, which supports the robustness of the peak response trends.

3.5. Normalised Displacement Performance

To evaluate the relative performance of the gravity and arch dams, the response of each alternative dam type was normalised by the corresponding response of the flat slab dam under the same loading condition. The normalised response ratio was defined as:

$$R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}={\displaystyle \frac{R_{\mathrm{d}\mathrm{a}\mathrm{m}}}{R_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}\hspace{0.25em}\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{b}}}}$$

(27)

where $R_{\mathrm{d}\mathrm{a}\mathrm{m}}$ is the response quantity of the gravity or arch dam and $R_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}\hspace{0.25em}\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{b}}$ is the corresponding response of the flat slab dam. Figure 8 presents the normalised peak total displacement and normalised peak dynamic increment.

The gravity dam reduces total displacement only moderately relative to the flat slab dam, with normalised total displacement ratios of approximately 0.86 to 0.95. The arch dam provides a much larger reduction, with normalised total displacement ratios of approximately 0.60 to 0.66. A similar trend is observed for the dynamic increment. The gravity dam reduces the dynamic increment to approximately 0.78 to 0.97 of the flat slab response, whereas the arch dam reduces it to approximately 0.57 to 0.64. These results indicate that the equivalent arch-action restraint is more effective in controlling lateral displacement than the gravity-type sectional thickening used in the present simplified framework.

3.6. Acceleration Amplification and Tensile Stress Demand

The normalised peak absolute acceleration and normalised peak tensile stress are shown in Figure 9. These quantities provide additional insight beyond displacement response. Under 0.1 MPa water pressure, the acceleration ratios of the three dam types are close to unity, indicating similar acceleration response levels. Under 1.0 MPa water pressure, however, the gravity dam develops substantially larger peak absolute acceleration than the flat slab dam, particularly under PGA = 0.5 g, where the gravity-to-flat-slab acceleration ratio reaches approximately 1.65. The arch dam also shows a modest acceleration increase under the most severe case, but its acceleration amplification remains much smaller than that of the gravity dam.

The tensile stress response shows a different trend from displacement (

Table 4. Averaged peak responses over the four input motions.

Water Pressure (MPa)	PGA (g)	Dam Type	Total Displacement (mm)	Dynamic Increment (mm)	Peak Acceleration (g)	Tensile Stress (MPa)
0.1	0.1	Flat slab dam	0.0175	0.0009	0.1055	0.4857
0.1	0.1	Gravity dam	0.0167	0.0008	0.1050	0.3958
0.1	0.1	Arch dam	0.0116	0.0005	0.1034	0.4807
0.1	0.5	Flat slab dam	0.0200	0.0043	0.5276	0.5534
0.1	0.5	Gravity dam	0.0190	0.0040	0.5248	0.4513
0.1	0.5	Arch dam	0.0131	0.0028	0.5172	0.5480
1.0	0.1	Flat slab dam	0.4122	0.0022	0.3709	4.2199
1.0	0.1	Gravity dam	0.3546	0.0017	0.4609	7.0134
1.0	0.1	Arch dam	0.2463	0.0013	0.3502	4.1437
1.0	0.5	Flat slab dam	0.4184	0.0110	1.4986	4.2827
1.0	0.5	Gravity dam	0.3597	0.0088	2.4760	7.1069
1.0	0.5	Arch dam	0.2501	0.0063	1.5994	4.2057

). Under 0.1 MPa water pressure, the gravity dam has a lower tensile stress demand than the flat slab dam, while the arch dam is close to the flat slab dam. Under 1.0 MPa water pressure, the gravity dam develops the highest tensile stress demand, with tensile stress ratios of approximately 1.5 to 1.7 relative to the flat slab dam. The arch dam, despite having the smallest displacement, shows a tensile stress demand close to that of the flat slab dam. Therefore, a smaller displacement response should not be interpreted as universally superior performance across all response measures. In the present simplified model, the tensile stress result should be interpreted as a stress-demand or tensile-overstress indicator rather than as a detailed prediction of crack propagation.

4. Discussion

4.1. Interpretation of the Water-Pressure and Seismic Effects

The results demonstrate that water pressure and seismic intensity control different aspects of the dam response. The peak total displacement is mainly governed by the static water-pressure level, whereas the seismic intensity primarily controls the dynamic displacement increment and the absolute acceleration response. This distinction is important because the total response contains both the static deformation caused by water pressure and the incremental deformation induced by earthquake excitation. The relationship can be expressed as:

$$u_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\left(t\right)=u_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}}+u_{\mathrm{d}\mathrm{y}\mathrm{n}}\left(t\right)$$

(28)

When the water pressure is increased from 0.1 MPa to 1.0 MPa, the static component becomes dominant, and the resulting peak total displacement increases substantially. By comparison, increasing PGA from 0.1 g to 0.5 g has only a minor effect on total displacement under high water pressure because the dynamic increment remains small relative to the static displacement. However, the same PGA increase produces a clear increase in the dynamic increment. This explains why Figure 5 is mainly controlled by the water pressure level, whereas Figure 6 exhibits a much stronger dependence on PGA.

The high-water-pressure case also activates the stiffness degradation mechanism in the simplified model. Once the tensile stress exceeds the concrete tensile strength, the element flexural stiffness is reduced, which further increases the displacement response. Therefore, the large increase in total displacement under 1.0 MPa is not merely a linear consequence of the larger lateral load; it is also affected by stress-induced stiffness degradation. This mechanism is especially relevant for interpreting the strong contrast between the 0.1 MPa and 1.0 MPa displacement matrices.

4.2. Role of Equivalent Arch-Action Restraint

The arch dam was represented by a vertical rectangular side section with equivalent arch-action lateral restraint. This modelling strategy was adopted because the physical arch curvature of a real arch dam occurs in plan view rather than in the vertical side section. The equivalent lateral springs and increased flexural stiffness were therefore introduced to represent the additional restraint provided by abutments and arch action without introducing an artificial curved vertical centreline.

It should be emphasised that the equivalent arch-action parameters were not calibrated against a specific engineering case. They were introduced to provide a controlled representation of the additional lateral restraint and stiffness enhancement expected from arch action within the simplified frame-based model. The sensitivity analysis showed that increasing the arch-stiffness factor or lateral spring stiffness reduced the displacement response and increased the effective modal frequency of the arch-equivalent dam, as expected. However, within the tested parameter range, the relative displacement-reduction trend of the arch-equivalent dam compared with the flat slab dam remained unchanged. Therefore, the conclusions regarding the arch-type response should be interpreted as comparative trends under an equivalent arch-action representation, rather than as calibrated predictions for a particular arch dam.

The normalised displacement results in Figure 7 show that this equivalent arch-action restraint is effective in reducing both total displacement and dynamic displacement increment. The arch dam consistently has the smallest displacement response across all water-pressure and PGA combinations. Relative to the flat slab dam, the arch dam reduces peak total displacement to approximately 60–66% and reduces peak dynamic increment to approximately 57–64%. This reduction is larger than that achieved by the gravity dam, indicating that lateral restraint is more efficient than sectional thickening alone for controlling lateral displacement in the present simplified framework.

At the same time, the arch dam should not be interpreted as universally superior in all response indices. Although its displacement response is the smallest, its peak acceleration and tensile stress are not always the lowest. This highlights the importance of evaluating multiple response measures when comparing structural forms under coupled static and dynamic loading.

4.3. Coexistence of Small Displacement and Noticeable Acceleration

One of the most important observations is that the displacement demand remains small, while the absolute acceleration response can be noticeably amplified, particularly under high water-pressure and strong seismic excitation. This response pattern can be explained by the modal characteristics of the models. The first-mode frequencies are in the range of approximately 123–191 Hz after static stiffness updating, which is much higher than the dominant frequency content of the imposed input motions. The system therefore behaves as a stiff short-height structure, and significant displacement resonance is not expected.

A useful interpretation is obtained by considering the frequency ratio between the structural frequency and the dominant input frequency:

$${\eta }_f={\displaystyle \frac{f_1}{f_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}}}$$

(29)

where $f_1$ is the first natural frequency of the dam model and $f_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}$ is a characteristic frequency of the input motion. When ${\eta }_f$ is much larger than unity, displacement amplification is limited because the input frequency is far from the displacement-resonant range of the structure. However, the constrained structural system can still exhibit noticeable absolute acceleration because the inertia response is directly associated with the imposed base acceleration and the high stiffness of the system.

This interpretation is consistent with Figure 3 and Figure 8. The high modal frequencies explain why the dynamic displacement increments are small, while the acceleration ratios can exceed unity under the most severe high-pressure cases. The gravity dam exhibits the largest acceleration amplification under 1.0 MPa and PGA = 0.5 g, suggesting that the interaction between sectional stiffness distribution, stress-induced stiffness degradation and inertial response can lead to acceleration demand that is not directly reflected by the displacement response.

4.4. Tensile Overstress and Stiffness Degradation Under High Water Pressure

The tensile stress results indicate that high water pressure can push the structural response beyond the nominal tensile capacity of concrete in the simplified model. Under 1.0 MPa water pressure, peak tensile stresses reach approximately 4.1–4.3 MPa for the flat slab and arch dams and exceed 7.0 MPa for the gravity dam. Since the adopted tensile strength is 2.5 MPa, these values imply tensile overstress and trigger the stiffness degradation mechanism.

The stress-based stiffness reduction can be summarised as:

$$E{I}_{\mathrm{e}\mathrm{f}\mathrm{f}}=r_{t}E{I}_0, {\sigma }_t>f_t$$

(30)

where $E{I}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is the degraded flexural stiffness, $E{I}_0$ is the initial flexural stiffness, $r_t$ is the cracking-related stiffness ratio, ${\sigma }_t$ is the tensile stress demand and $f_t$ is the tensile strength. In the present model, the stiffness degradation is a simplified representation of tensile damage rather than a full crack propagation model.

The gravity dam develops the highest tensile stress under high water pressure in the present frame-based idealisation. To examine the influence of the beam idealisation on the tensile stress demand of the gravity dam, a simplified two-dimensional plane-stress benchmark was additionally conducted using the same gravity section, water pressure and boundary condition. The comparison showed that the beam-based tensile stress demand and the plane-stress maximum principal tensile stress were of the same order of magnitude, although local stress distributions differed between the two models. This confirms that the frame model can be used as a comparative stress-demand indicator, but it should not be interpreted as a detailed prediction of local tensile stress in the gravity section. A real gravity dam resists lateral loading through a more complex two-dimensional or three-dimensional stress redistribution mechanism involving self-weight, base compression, contact conditions and foundation interaction. These mechanisms are not fully represented by the vertical frame-element idealisation used here. Therefore, the result should be described as a high tensile stress demand within the present simplified framework, rather than as evidence that gravity dams are generally the most vulnerable structural form.

Finally, in a real three-dimensional arch dam, arch action would be expected to transfer part of the lateral water pressure into compressive thrust along the arch direction, thereby reducing bending demand compared with a flat slab dam. In the present simplified framework, however, the arch effect is represented by equivalent lateral restraint applied to a two-dimensional vertical section rather than by an explicit three-dimensional arch geometry. Consequently, the model captures the displacement-reducing effect of arch restraint, but the bending moment and tensile stress calculated from the vertical frame elements remain affected by local restraint reactions. The tensile stress of the arch-equivalent dam should therefore be interpreted as a simplified stress-demand indicator within the equivalent model, not as a direct representation of the true three-dimensional stress field of an arch dam.

4.5. Mechanistic Synthesis

The governing mechanisms are summarised in Figure 10. Water pressure mainly controls peak total displacement because it determines the initial static deformation and, under high pressure, can induce stiffness degradation. PGA mainly controls the dynamic displacement increment and peak absolute acceleration because these responses are directly associated with the imposed base acceleration. The arch-type dam exhibits the smallest displacement because of the equivalent lateral restraint, whereas the gravity-type dam develops higher tensile stress demand under high water pressure in the present simplified frame model. Finally, the high effective modal frequencies explain why the system can exhibit small displacement demand while still showing noticeable absolute acceleration response.

The comparative results further suggest that the structural advantage of the arch-equivalent dam becomes more pronounced under relatively high water-pressure conditions. Under the lower water-pressure case (0.5 MPa), the displacement and stress differences among the three dam types remain relatively limited. However, when the water pressure increases to 1.0–1.5 MPa, the arch-equivalent configuration shows a clearer reduction in global lateral displacement compared with the flat slab dam because of the additional restraint and load-redistribution effect associated with arch action. Therefore, within the simplified framework adopted in this study, the arch-type configuration appears more suitable for relatively high hydraulic loading conditions. Nevertheless, local edge stress concentration and boundary restraint effects should still be carefully considered in practical design.

The mechanistic interpretation also clarifies the role and scope of the simplified model. The model is most useful for comparing relative response trends and identifying dominant control mechanisms. It should not be regarded as a substitute for a full three-dimensional dam–foundation–water interaction model. Nevertheless, it provides a transparent framework for explaining how water pressure, seismic intensity and structural form jointly affect displacement, acceleration and tensile stress demand.

4.6. Limitations

This study provides a simplified comparative framework for evaluating the dynamic response of different dam structural forms under coupled water-pressure and seismic loading; however, several limitations should be acknowledged. First, the model is a two-dimensional frame-based idealisation and does not fully reproduce the three-dimensional behaviour of real dam–rock–water systems. In particular, the arch dam was represented using an equivalent lateral restraint rather than an explicit three-dimensional arch geometry, and the effects of realistic abutment deformation, surrounding rock interaction and dam–foundation contact were not explicitly simulated. For the gravity-type section, the Euler–Bernoulli beam idealisation cannot fully capture two-dimensional stress redistribution within the trapezoidal section. The calculated tensile stress should therefore be regarded as a conservative stress-demand indicator for comparative purposes, rather than as an exact local stress prediction. The implications of these simplifications should be explicitly recognised. By neglecting fluid–structure interaction and hydrodynamic pressure, the present model does not account for the added-mass effect of stored water or for the frequency-dependent dynamic water pressure acting on the dam surface during seismic excitation. This may lead to an underestimation of hydrodynamic demand and may also affect the effective modal properties of the coupled system. Similarly, the surrounding rock mass and dam–rock interface are represented only through idealised boundary restraints. As a result, rock-mass flexibility, interface deformation, local opening or sliding, and radiation damping are not explicitly considered. These factors would generally reduce the effective stiffness of the dam–rock system, alter stress transfer near the abutment and boundary regions, and potentially lower the modal frequencies compared with those obtained from the simplified frame model. Similar soil/rock–structure interaction effects have also been reported in previous seismic analyses of underground structures, where the interaction between the surrounding medium and the structure was shown to significantly influence the dynamic response and stress redistribution [22]. Therefore, the numerical values reported in this study should be interpreted as response indicators within a comparative simplified framework, rather than as direct predictions of absolute field-scale seismic response.

Second, the seismic inputs were idealised acceleration histories designed to represent different frequency characteristics, rather than site-specific earthquake or mine-seismic records. This assumption should be interpreted with caution because seismic signals in underground or shallow underground environments may differ from free-field surface motions due to wave scattering, local amplification, soil–structure interaction, and the presence of underground cavities or structural systems. As discussed by Zucca et al. [23], shallow underground structures can influence the recorded seismic signal, indicating that underground seismic input may not be equivalent to the corresponding free-field ground motion. Therefore, the input motions used in this study should be regarded as representative excitation signals for comparative analysis rather than as site-specific underground seismic records. Third, the stiffness degradation model uses an equivalent reduction in flexural stiffness after tensile overstress. This approach captures only the first-order influence of cracking-related stiffness loss on global response, but it does not adequately represent the full cyclic nonlinear behaviour of reinforced concrete dams. In particular, crack initiation and propagation, crack opening–closure, stiffness recovery during unloading, reinforcement contribution, bond-slip, hysteretic energy dissipation, contact opening–closure, and progressive damage accumulation are not explicitly simulated. Therefore, the cracked elements and tensile overstress results should be interpreted as simplified damage tendency indicators rather than as direct predictions of cyclic crack evolution. Future work should extend the present framework to three-dimensional dam–rock–water interaction models that incorporate hydrodynamic pressure, fluid–structure interaction, rock-mass deformation, dam–rock interface behaviour, recorded earthquake or mine-seismic motions, and more advanced concrete damage or fracture models to validate the response mechanisms identified in this study.

5. Conclusions

This study developed a simplified two-dimensional frame-based dynamic analysis framework to compare the response of flat slab, gravity and equivalent arch-restrained dam structures under coupled water-pressure and seismic loading. Two water pressure levels, two PGA levels and four representative input motions were considered. The results provide a mechanistic interpretation of how water pressure, seismic intensity and structural form jointly influence displacement, acceleration and tensile stress demand. The following conclusions can be obtained from the work.

• Water pressure is the dominant factor controlling peak total displacement. Increasing the water pressure from 0.1 MPa to 1.0 MPa produces a much larger increase in total displacement than increasing PGA from 0.1 g to 0.5 g. This occurs because the total response is strongly influenced by the static water-pressure-induced deformation. Under the high-pressure condition, tensile overstress and stiffness degradation further amplify the total displacement response.
• PGA mainly controls the peak dynamic displacement increment and peak absolute acceleration. Although the total displacement changes only slightly when PGA increases under 1.0 MPa water pressure, the dynamic increment increases clearly. This confirms that separating total displacement from dynamic increment is necessary for interpreting coupled water-pressure and seismic response.
• The arch-equivalent dam shows the smallest displacement response among the three dam types. The equivalent arch-action lateral restraint reduces both peak total displacement and peak dynamic displacement increment relative to the flat slab and gravity dam configurations. Within the present simplified framework, lateral restraint is therefore more effective for controlling lateral deformation than sectional thickening alone.
• High effective modal frequencies explain the coexistence of small displacement demand and noticeable absolute acceleration response. The first-mode frequencies of the dam models remain in the range of approximately 123–191 Hz after static stiffness updating, which is substantially higher than the dominant frequency range of the imposed input motions. This high-frequency, stiff-system behaviour limits displacement amplification but does not eliminate acceleration demand.
• Tensile overstress becomes important under 1.0 MPa water pressure. In the present simplified frame-based idealisation, the gravity-type section develops the highest tensile stress demand under high water pressure, while the arch dam maintains the smallest displacement response. This indicates that displacement performance and tensile stress demand should be evaluated together; a smaller displacement response does not necessarily imply the lowest stress demand.