Seismic Assessment of Concrete Gravity Dam via Finite Element Modelling

Abstract

The failure of large gravity dams during an earthquake could lead to calamitous flooding, severe infrastructural damage, and massive environmental destruction. This paper aims to demonstrate reliable methods for evaluating dam performance after a seismic event. The work included a seismic hazard analysis and nonlinear finite element modelling of concrete cracking for two large dams (D1 and D2, of 35 and 90 m in height, respectively) in Eastern Canada. Dam D1 is located in Montreal, and Dam D2 is located in La Malbaie, Quebec. The modelling approach was validated using the Koyna Dam, which was subjected to the 1967 Mw 6.5 earthquake. This paper reports tensile cracks of D1 and D2 under combined hydrostatic and seismic loading. The latter was generated from ground motion records from 11 sites during the 1988 Mw 5.9 Saguenay earthquake. These records were each scaled to two times the design level. It is shown that D1 remained stable, with minor localised cracking, whereas D2 experienced widespread tensile damage, particularly at the crest and base under high-energy and transverse inputs. These findings highlight the influence of dam geometry and frequency characteristics on seismic performance. The analysis and modelling procedures reported can be adopted for seismic risk classification and safety compliance verification of other dams and for recommendations such as monitoring and upgrading.

1. Introduction

Concrete gravity dams are crucial infrastructures, providing essential services and supporting economic development such as water supply, flood control, and hydropower. In Canada, more than 900 large dams are currently in operation, over 60% of which were constructed before 1970 [1,2]. These aging structures were designed using older engineering standards that did not fully account for seismic forces, especially in Eastern Canada. Greater Montreal is classified as a region with moderate seismic hazards [3]. However, the 1988 Saguenay earthquake (Mw 5.9) emphasised the need to reassess this classification and raised concerns about the structural safety of existing dams [4]. As most of these dams remain operational without significant retrofitting, there is an urgent need to develop reliable methods for assessing their seismic performance for public safety [5,6].

The purpose of this paper is to propose a nonlinear finite element modelling framework for evaluating the seismic behaviour of concrete gravity dams, like those in Eastern Canada and similar ones elsewhere. Dams are massive structures and were initially designed and analyzed using the pseudo static method [7,8]. With advances in our understanding of dynamic analysis, researchers recognised the importance of considering the interaction between the dam body, reservoir, and foundation to more accurately represent the real behavior of the dam–reservoir system. Finite element method (FME) is a power tool, and many commercial software programs, such as SAP 2000, ANSYS (25.2), LS-DYNA (R15.0.2), ABAQUS (2025), etc., have been developed for this purpose. In addition to FEM, Hariri-Ardebili et al. [9] proposed a Random Finite Element Method (RFEM) for the seismic analysis of gravity dams, in which both the tensile strength and the modulus of elasticity of concrete were treated as random variables over the body of the dam. This methodology offers a more realistic representation of real-world conditions. However, the model has not yet been calibrated or validated. Researchers have used different FEMs to examine the performance of concrete gravity dams under earthquake excitations. Although they have reached a good accuracy, they might be not suitable for dam failure analysis, as noted by Haghani et al. [10]. To address this limitation, they developed a new approach, i.e., the extended finite element method (XFEM), on MATLAB (25.1). The XFEM is able to take into account the opening and closing of cracks and their effect of the dam’s higher mode frequencies. Since no commercial software packages are currently available to implement either RFEM or XFEM, FEM remains a standard tool for dam analysis.

Among commercial software programs (SAP 2000, ANSYS (25.2), LS-DYNA (R15.0.2), ABAQUS (2025), OpenSees (3.7.1), etc.), ABAQUS (2025) is widely used for dam analysis [11,12,13] in academia research given its ability to capture the nonlinear behavior of concrete under strong earthquake loading through different concrete damage/crack models. Cracks in concrete dams are typically evaluated using two main modelling approaches: the discrete crack model and the continuum crack model. When using the discrete crack model, the finite element mesh has to be monitored and adjusted according to the crack propagation as the loading progresses. Given the complexity of modelling gravity dams, the use of the discrete crack mode is very limited due to the high computational costs and time requirements. The continuum crack model is generally classified into two categories, i.e., the damage mechanics approach and the smeared crack model. The latter is considered convenient since it does not require re-meshing or the addition of new degrees of freedom during the analysis, as explained in Alijani-Ardeshir et al. [14]. The smeared crack model considered in ABAQUS (2025) is represented by the concrete damage plasticity model (CDP). The CDP model is the most widely and extensively used material model in the nonlinear analysis of concrete dams reported by Gorai and Maity [7]. One of the most recent studies on the ABAQUS CDP model is presented in Ribeiro and Léger [15]. In particular, the authors examined the so-called Carpinteri Gravity Dam considered in Carpinteri et al. [16] under different discrete and continuum concrete crack models. A thorough review of experimental studies and FE analyses of the Carpinteri Gravity Dam was also presented. Furthermore, their study provides comprehensive details on the modelling and simulation of the dam using the Abaqus CDP model.

This study uses ABAQUS (2025) and the concrete damaged plasticity (CDP) model [17,18] to simulate structural damage and dynamic response under real-world earthquake excitation. Using two large concrete gravity dams in Quebec, D1 in Montreal and D2 in La Malbaie (Figure 1), this study aims to explore how geometry, mass, and modal properties affect damage localisation and displacement response. Specifically, this research addresses the following questions:

• How does seismic load intensity affect the location and severity of tensile cracking?
• How do differences in dam geometry, stiffness, and natural frequency influence seismic vulnerability?
• Can the use of site-specific, scaled ground motions improve damage prediction accuracy compared to synthetic or unscaled inputs?

The seismic assessment of gravity dams presents several technical challenges. It is complicated and computationally demanding to quantify the nonlinear behaviour of concrete, including cracking under tension and crushing under compression [17,18]. Additional factors, such as dam–reservoir–foundation interaction, transient stress redistribution, and damping effects, further complicate the problem [19]. Traditional linear static analyses fail to capture the initiation and propagation of damage under seismic loads [20,21]. These limitations may be overcome by using the CDP model, which simulates both material degradation and structural response [22,23]. However, accurate simulations require a careful selection of ground motions, robust mesh design, and validation against experimental or benchmark data [24].

Historically, dam safety analyses have relied on simplified analytical methods or empirical seismic design provisions. Past earthquake-induced dam failures have shown the inadequacy of such traditional approaches. The 1967 Koyna earthquake in India caused severe cracking in a gravity dam, highlighting the need for seismic design in dam engineering [25,26]. Similar damages occurred during the 1990 Manjil earthquake in Iran and the 2011 Tōhoku earthquake in Japan [27]. These events prompted the shift toward numerical modelling techniques like the finite element method (FEM), which enable dynamic response simulations and crack tracking [28,29]. In Quebec, studies have started to develop fragility curves and conduct nonlinear time–history analysis, but few have applied full-scale, validated models with real ground motions [20,30].

The novelty of this paper lies in using real-world seismic records, both longitudinal and transverse components, from multiple sites, scaling each of the ground motions, and realistically capturing its impact on the dam. This approach produces more accurate results than using synthetic or unscaled records [20,21]. Additionally, this paper offers a comparison of seismic vulnerability between two dams with significantly different geometries, masses, heights, stiffnesses, and shapes. Such analyses are beyond conventional performance metrics [31,32]. The comparison reveals time-dependent displacement histories and spatial crack propagation patterns under both original and scaled acceleration. This study incorporates design ground motion’s value with respect to the dam location and soil classes in alignment with the National Building Code of Canada (NBCC) [33].

Specifically, the comparison aims at quantifying how increasing seismic load intensity affects the location and severity of cracking in the upstream heel, crest, and downstream face of the dam in question. These zones are critical [34,35]. Thus, this study helps create valuable knowledge to inform dam structural upgrades, regulatory improvements, and readiness planning [36].

A proper validation is essential for any modelling approach before its use in practical engineering applications. This is true particularly in the context that performance-based seismic assessment remains limited for older dams in Eastern Canada [32], in spite of good progress made in the past. In the following, the finite element modelling methods are used, along with validation using real-world seismic records and scaling based on modal frequencies. Then, the modelling results are presented, with details about crack propagation and performance and the most vulnerable zones of D1 and D2. Next, the implications of the results as well as new contributions from this study are discussed before conclusions are drawn.

2. Methodologies

2.1. Dam Geometry and Finite Element Discretisation

The dimensions of D1 and D2 [22] are as follows: D1 has a top width T = 5 m, a base width B = 27.5 m, an upstream water depth h = 32 m, a total structural height of H = 35 m, a downstream face slope 1:s = 0.75:1, a crest elevation (or freeboard) c = 3 m above the reservoir water level, and a downstream crest height d = 5 m (Figure 1b). For D2, T = 5 m; B = 70 m; h = 86 m; H = 90 m; 1:s = 0.79:1; c = 4 m; d = 8 m. D1 and D2 represent low and high dams, respectively, in the areas of moderate and high seismicity in Eastern Canada.

The seismic response of the dam was computed using the dynamic equation of motion in finite element form, considering mass, damping, stiffness, and earthquake record. A detailed formulation can be found in Chopra [37]. The two-dimensional (2D) finite element meshing for the two dams employed CPE4R elements, 4-node bilinear plane strain elements with reduced integration [21,24,38,39]. D1 was modelled with 403 nodes and 360 elements, while D2 had 819 nodes and 760 elements, ensuring mesh convergence and reliable simulation of stress concentrations. The selected mesh configurations were based on a balance between computational efficiency and solution accuracy and were informed by similar studies involving dam fracture under seismic loading [35]. The final mesh ensured reliable simulation of nonlinear behavior while maintaining practical run times for the large number of earthquake input cases considered.

It is important to note that 2D analyses are the most commonly used approach for the design and analysis of dams. The presence of construction joints, which are essential in the design of conventional concrete dams, further justifies the use of 2D modelling. Žvanut [40] performed both 2D and 3D analyses on an arch dam and concluded that the results were in good agreement. A similar conclusion was also drawn in Azevedo [41]. Therefore, 2D analysis was adopted in this study.

2.2. Concrete Damaged Plasticity Model

Since concrete has extremely low tensile strength (about 10% of its compressive strength), it is susceptible to structural cracking. This study used the CDP model for the following reason: it effectively simulates both the cracking and post-failure behavior of concrete [18,23,42]. More importantly, it incorporates the principles of isotropic damage elasticity, combined with both tensile and compressive plasticity. Although the CDP model has been extensively used in dam analysis, it has several drawbacks, such as mesh sensitivity, convergence issues, and the assumption of isotropic damage in simulations, as reported by Li and Wu [43], Fakeh et al. [44], and Ramadam et al. [45]. To evaluate this, a mesh sensitivity analysis was performed using coarse (403 nodes), medium (576 nodes), and fine (1491 nodes) discretisation. The difference in crest displacement between the coarse and medium meshes was only ~1.2%, and tensile damage distributions were nearly identical. Although the finest mesh showed slightly higher damage, this is consistent with the expected behavior of softening models. These results confirm that the coarse mesh is sufficiently refined for the simulations, while significantly reducing computing costs. To mitigate convergence issues, a viscosity regularisation parameter of 0.0001 was specified, which introduces a small rate dependence to stabilise post-peak softening behavior [17,18]. In addition, a stiffness-proportional damping coefficient (β = 0.003367) was applied, while the mass-proportional term (α) was set to zero. This damping strategy reduces excessive oscillations in the low-frequency range without altering the inertial response, improving both the stability and realism of the simulations.

Fracture energy dissipation is another issue that requires attention for the analysis. It is acknowledged that ABAQUS’s tension stiffening approach can introduce unreasonable mesh sensitivity into the results if no reinforcement is considered in modelling. The fracture energy approach proposed by Hillerborg [46] in the development of the CDP model is considered to be adequate for practical applications since the brittle behavior of concrete is better characterised by a stress–displacement response instead of a stress–strain response. Furthermore, the study by Bhattacharjee and Léger [47] reveals that the fracture energy dissipation does not have a significant influence on the total strain energy in the dam under investigation. It is also recognised that CDP models fail to accurately capture changes in material behavior under dynamic load conditions, such as variations in material properties, strain rate effects, and cyclic loading response. To address these issues, Patra et al. [48] developed a modified CDP model (CDPM) based on a Lagrangian formulation and conducted six simulations. The difference in the results of dam displacement between the CDP and CDPM models was about 5–6%. Thus, the original ABAQUS CDP model was adopted for their study as well as the present study.

The behaviour of concrete is characterised by linear elasticity followed by strain softening in tension (Figure 2a) and stress hardening followed by strain softening in compression (Figure 2b) [49]. For concrete under tension, the stress–strain response is elastic until the initial (i.e., maximum) tensile stress, ${\sigma }_{\mathrm{t}0}$, is reached.

Under uniaxial tensile loading, concrete exhibits a linear elastic response until it reaches its initial tensile strength, ${\sigma }_{\mathrm{t}0}$, which marks the onset of micro-cracking. Beyond this point, stiffness begins to degrade, and the stress–strain curve enters a strain-softening regime, reflecting progressive crack development [50,51]. This degradation in stiffness is mathematically described by the scalar damage variable, $d_t$, resulting in the following relationship:

$${\sigma }_{\mathrm{t}}=(1-{\mathrm{d}}_{\mathrm{t}}){\mathrm{E}}_{\mathrm{o}}({\epsilon }_{\mathrm{t}}-{\epsilon }_{\mathrm{t}}^{\mathrm{p}\mathrm{l}})$$

(1)

where ${\sigma }_{\mathrm{t}}$ is the tensile stress (MPa); ${\mathrm{E}}_{\mathrm{o}}$ is the initial elastic modulus (MPa); ${\epsilon }_{\mathrm{t}}$ is the total tensile strain; and ${\epsilon }_{\mathrm{t}}^{\mathrm{p}\mathrm{l}}$ is the plastic strain in tension, representing irreversible crack-induced deformation. The damage variable ${\mathrm{d}}_{\mathrm{t}}$ (ranging from zero for undamaged to one for fully damaged material) quantifies the extent of tensile degradation.

In compression, the concrete initially behaves elastically up to the yield stress, ${\sigma }_{\mathrm{c}0}$, followed by a phase of stress hardening until the ultimate compressive strength, ${\sigma }_{\mathrm{c}\mathrm{u}}$, is reached. Beyond this point, the material undergoes strain softening, signifying crushing and structural degradation. This compressive behaviour is described using a similar formulation:

$${\sigma }_{\mathrm{c}}=(1-{\mathrm{d}}_{\mathrm{c}}){\mathrm{E}}_{\mathrm{o}}\left({\epsilon }_{\mathrm{c}}{-\epsilon }_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}\right)$$

(2)

where ${\sigma }_{\mathrm{c}}$ is the compressive stress; ${\epsilon }_{\mathrm{c}}$ is the total compressive strain; and ${\epsilon }_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}$ is the plastic strain in compression. The scalar variable, ${\mathrm{d}}_{\mathrm{c}}$, analogous to ${\mathrm{d}}_{\mathrm{t}}$, captures the extent of compressive damage. To compute the plastic strains used in these damage formulations, the following expressions are applied:

$${\epsilon }_{\mathrm{t}}^{\mathrm{p}\mathrm{l}}={\epsilon }_{\mathrm{t}}^{\mathrm{c}\mathrm{r}}-{\displaystyle \frac{{\mathrm{d}}_{\mathrm{t}}}{1-{\mathrm{d}}_{\mathrm{t}}}}{\displaystyle \frac{{\sigma }_{\mathrm{t}}}{{\mathrm{E}}_{\mathrm{o}}}}$$

(3)

$${\epsilon }_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}={\epsilon }_{\mathrm{c}}^{\mathrm{i}\mathrm{n}}-{\displaystyle \frac{{\mathrm{d}}_{\mathrm{c}}}{1-{\mathrm{d}}_{\mathrm{c}}}}{\displaystyle \frac{{\sigma }_{\mathrm{c}}}{{\mathrm{E}}_{\mathrm{o}}}}$$

(4)

where ${\epsilon }_{\mathrm{t}}^{\mathrm{c}\mathrm{r}}$ and ${\epsilon }_{\mathrm{c}}^{\mathrm{i}\mathrm{n}}$ represent the cracking strain in tension and inelastic strain in compression, respectively. These expressions ensure consistency between the evolving damage variables and the actual mechanical state of the material.

Together, these formulations allow the CDP model to capture the nonlinear, inelastic behaviour of concrete under both tensile and compressive regimes, which is critical for assessing structural damage in gravity dams subjected to seismic loading. The material properties used in this study are given in

Table 1. Key parameters of concrete used in the computations.

Parameter	Symbol	Value	Units
Concrete compressive ultimate stress	${\sigma }_{\mathrm{c}\mathrm{u}}$	25	MPa
Concrete initial yield compressive stress	${\sigma }_{\mathrm{c}0}$	11	MPa
Concrete initial tensile stress	${\sigma }_{\mathrm{t}0}$	3	MPa
Concrete mass density	$\rho$	240	kg/m3
Young’s modulus	E	25,000	MPa
Poisson’s ratio	ν	0.15

.

2.3. Tensile and Compressive Damage

This study implemented nonlinear concrete modelling and quantified the degradation of stiffness due to micro-cracking in tension and crushing in compression, using two scalar damage parameters, (1) tensile damage, ${\mathrm{d}}_{\mathrm{t}}$, and (2) compressive damage, ${\mathrm{d}}_{\mathrm{c}}$, in order to track the material deterioration under loading. Tensile softening was implemented using the strain-based tension stiffening law, where stress is defined as a function of cracking strain. This formulation implicitly governs the fracture energy as the area under the stress-cracking strain curve multiplied by the characteristic element length. While this approach is mesh-dependent, it remains widely used in dam analyses and has been reported in several studies [12,43,47]. The adopted strain-based softening law is known to be mesh-sensitive. The value of compressive damage is assumed to be zero. The values of ${\mathrm{d}}_{\mathrm{t}}$ ranged from zero (undamaged) to one (complete loss of load-carrying capacity). These two damage parameters were essential for simulating the progressive failure of concrete under seismic and cyclic loads. They depend on plastic strain, temperature, $\theta$, and another potential field-dependent variable, ${\mathrm{f}}_{\mathrm{i}}$, i.e., ${\mathrm{d}}_{\mathrm{t}}={\mathrm{d}}_{\mathrm{t}}({\epsilon }_{\mathrm{t}}^{\mathrm{p}\mathrm{l}},\theta ,{\mathrm{f}}_{\mathrm{i}})$ and ${\mathrm{d}}_{\mathrm{c}}={\mathrm{d}}_{\mathrm{c}}({\epsilon }_{\mathrm{c}}^{\mathrm{p}\mathrm{l}},\theta ,{\mathrm{f}}_{\mathrm{i}})$. In this study, they are expressed as

$${\mathrm{d}}_{\mathrm{t}}=1-{\displaystyle \frac{{\sigma }_{\mathrm{t}}}{{\sigma }_{\mathrm{t}0}}}$$

(5)

$${\mathrm{d}}_{\mathrm{c}}=1-{\displaystyle \frac{{\sigma }_{\mathrm{c}}}{{\sigma }_{\mathrm{c}\mathrm{u}}}}$$

(6)

Values of ${\sigma }_{\mathrm{t}0}$ and ${\sigma }_{\mathrm{c}\mathrm{u}}$ are listed in Table 1. Note that ${\sigma }_{\mathrm{t}0}$ and ${\sigma }_{\mathrm{c}\mathrm{u}}$ are given in Equations (1) and (2), respectively. The tensile damage parameter, ${\mathrm{d}}_{\mathrm{t}}$, vs. the cracking strain, ${\epsilon }_{\mathrm{t}}^{\mathrm{c}\mathrm{r}}$, is illustrated in Figure 3. It should be noted that Equations (5) and (6) are simplified scalar formulations within the CDP model [17,18]. They assume isotropic stiffness degradation and therefore do not capture anisotropic cracking, cyclic degradation, or strain rate effects. While these simplifications are widely adopted in seismic dam studies [7,12,13,15], they represent an idealisation of real concrete behavior.

2.4. Crest Displacement

To understand the dynamic behaviour of the dams, a frequency analysis was first performed to calculate their natural periods. This modal analysis revealed that D1 and D2 had fundamental periods of 0.091 s and 0.212 s, respectively. These values were essential for scaling the input ground motions and assessing resonance effects. Following seismic loading, the primary response parameter studied was crest displacement, which reflects the horizontal movement of the dam’s top relative to the ground. It serves as a key indicator of seismic performance, as excessive crest displacement can signal potential structural instability or failure. The horizontal displacement at a given note in the dam can be calculated as

$${\mathrm{u}}_{\mathrm{c}}\left(\mathrm{t}\right){=\mathrm{u}}_{\mathrm{d}}\left(\mathrm{t}\right)-{\mathrm{u}}_{\mathrm{g}}\left(\mathrm{t}\right)$$

(7)

where ${\mathrm{u}}_{\mathrm{d}}$ is absolute displacement of the dam, i.e., the total motion with respect to a fixed global reference frame, and ${\mathrm{u}}_{\mathrm{g}}$ is the ground displacement, i.e., the input motion at the foundation. Their difference, ${\mathrm{u}}_{\mathrm{c}}$, represents the relative displacement of the dam crest with respect to the moving base, which is the relevant measure of the dam’s structural response.

2.5. Selection of Earthquake Records

To evaluate the seismic performance of the two dams, this study utilised 22 acceleration time–history records from the 1988 Saguenay earthquake, consisting of 11 longitudinal and 11 transverse components recorded at 11 different ground motion stations across Quebec (

Table 2. Ground motion characteristics, including spectral acceleration, S_a, and spectral velocity, S_v, from the 1988 Saguenay earthquake records.

Station	Site Name	Lat.	Long.	Record Length	Sa	Sv	Sa/Sv	Sa	Sv	Sa/Sv
ID		(° N)	(° W)	(s)	(m/s2)	(m/s)	(s−1)	(m/s2)	(m/s)	(s−1)
						D1			D2
1	St-Ferreol	47.117	70.850	48.64	1.840	0.021	9.129	0.330	0.102	3.245
2	Quebec	46.814	71.208	39.04	0.546	0.005	10.995	0.157	0.046	3.428
5	Tadoussac	48.146	69.713	38.96	0.808	0.010	8.224	0.029	0.0004	79.240
7	Baie-St-Paul	47.441	70.499	17.74	1.314	1.314	0.102	0.247	0.075	3.310
8	La-Malbaie	47.654	70.153	29.65	1.904	1.904	0.102	0.299	0.076	3.956
9	St-Pascal	47.528	69.803	39.06	0.694	0.694	0.102	0.113	0.029	3.880
10	Riviere-Ouelle	47.433	70.017	33.26	0.680	0.004	18.014	0.100	0.030	3.372
14	Ste-Lucie	46.133	74.183	17.75	0.190	0.002	8.830	0.029	0.008	3.646
16	ChocoutimiNord	48.441	71.059	33.99	3.247	0.035	9.426	0.191	0.066	2.876
17	St-Andre	47.101	67.759	28.35	4.241	0.059	7.374	0.159	0.053	3.022
20	Les Eboulements	47.478	70.323	20.65	1.918	0.023	8.418	0.270	0.082	3.280

). For clarity in the Results and Discussion sections, records are referenced using simplified site codes: for example, Site 1L (longitudinal) and Site 1T (transverse), where the number corresponds to the recording site, and the suffix L or T refers to the longitudinal or transverse component, respectively. Seismic analysis was carried out using both the original (unscaled) ground motion records from the earthquake event and the same records scaled to match the design spectrum at the fundamental period T1 of the dam. The design spectrum for the dam site was developed in accordance with National Building Code of Canada (NBCC) and Canadian Highway Bridge Design Code (CHBDC) for 5% damping, site class C (hard rock).

These records were originally provided in their unscaled form; later, each earthquake record was scaled to two times the design level. By using both unscaled and scaled inputs, this study enables a comprehensive understanding of dam behaviour across a range of seismic intensities. Similar methods have been validated in previous research for improved accuracy in nonlinear response simulations [31,51].

3. Results

3.1. Case for Validation

To verify the modelling approach adopted in this study, a benchmark analysis was performed using the well-documented case of the Koyna Dam, which experienced significant damage during the 1967 Maharashtra earthquake (Mw 6.5). Peak ground accelerations recorded at the site were 0.49 g (stream direction), 0.63 g (cross-stream), and 0.34 g (vertical), where g is the gravitational acceleration (equal to 9.81 m/s²). Observed damage included horizontal cracking on both the upstream and downstream faces. The model results were compared with published numerical studies [52,53].

3.2. Natural Frequencies of Dams

To assess the dynamic behaviour of the dams under seismic loading, a frequency analysis was performed using modal extraction in Abaqus. The first four natural frequencies for D1 and D2 are presented in

Table 3. Natural frequencies (in radian/s) of D1 and D2.

Mode	D1	D2
1	68.453	29.699
2	161.504	63.521
3	192.517	80.629
4	234.720	103.481

. Results indicate that D1, having a narrower base and greater stiffness, exhibited significantly higher frequencies than D2. Specifically, D1′s fundamental mode is approximately 2.3 times higher than that of D2. Both dams primarily oscillate in the horizontal direction at the crest in their first mode, which is typical for gravity dams. Higher modes showed nodal points at increasing elevations along the dam height. These dynamic properties were subsequently used to identify potential resonance and scaling of the Saguenay earthquake records.

3.3. Tensile Damage

The tensile damage contours presented in this study represent the spatial distribution and severity of cracking within the dam structure under seismic loading. In the ABAQUS CDP model, the ${\mathrm{d}}_{\mathrm{t}}$ parameter is a scalar value ranging from zero to one, where a value of zero indicates undamaged, fully intact concrete, and a value of one denotes complete loss of tensile strength and stiffness in that region. Intermediate values of ${\mathrm{d}}_{\mathrm{t}}$ reflect progressive damage, with values around 0.01–0.1 corresponding to initial microcracking, values between 0.2 and 0.5 indicating moderate crack development and stiffness reduction, and values above 0.6 representing extensive cracking and severe material degradation. These contours are essential for visualising not only the location of cracks but also the intensity and evolution of damage across different zones of the dam.

3.4. Seismic Response of D1: Original Acceleration

Among the unscaled ground motion simulations, the most critical response for D1 was observed under the longitudinal component recorded at Site 8 (La Malbaie). The maximum horizontal displacement at the crest reached 11.74 mm at 10.3 s (Figure 4). Interestingly, tensile damage was triggered earlier in the response, peaking at a value of $5.00\times {10}^{-3}$ (Figure 5). The damage was confined to the upstream heel, consistent with typical stress concentration zones for gravity dams subjected to lateral loading. This case highlights the role of cumulative stress accumulation over time in inducing damage, even when peak acceleration occurs later in the motion sequence.

Other notable simulations included Sites 7T and 20L. Site 7T produced a crest displacement of −10.42 mm and a peak damage of $1.05\times {10}^{-3}$, while Site 20L showed −8.89 mm displacement with damage peaking at $3.09\times {10}^{-3}$. In all cases, tensile cracking remained localised near the heel, and no structural instability was observed. The consistent localisation of damage indicates that D1 performs well under moderate seismic excitation, with damage primarily controlled by localised stress amplification near the base.

3.5. Seismic Response of D1: Scaled Acceleration

Under scaled input motions, D1 exhibited a significantly intensified response, with widespread tensile damage observed in the majority of simulations. The most severe case corresponded to Site 2L (Quebec City), where crest displacement peaked at 43.74 mm at 18.26 s (Figure 6). The tensile damage contour (Figure 7) revealed a wide and deep cracking zone at the upstream heel, with peak values reaching $4.84\times {10}^{-1}$ and minor damage on the downstream wall with a value of $1.01\times {10}^{-1}$. The distribution of the damage along the base implies a dominant shear failure mechanism driven by amplified inertial loads. Early crack initiation preceding peak displacement suggests fatigue-like degradation under cyclic loading conditions.

Other records also induced notable responses. For instance, Site 1L showed damage at both the heel and crest with a displacement of −36.88 mm, and Site 7T induced transverse cracking at the downstream face along with $8.25\times {10}^{-1}$ peak heel damage. Site 9L generated moderate displacement (−25.59 mm) but notable downstream wall cracking, indicating stress reversal effects. Simulations from Sites 10L and 14L showed mixed-mode failures, involving both crest and base cracking. Collectively, these observations indicate that D1 is susceptible to flexural and shear cracking when subjected to higher energy input, with damage localisation strongly influenced by excitation direction.

3.6. Seismic Response of D2: Original Acceleration

Compared to D1, D2 exhibited more frequent and distributed tensile damage under the original Saguenay earthquake records. The most critical case occurred under transverse motion at Site 1T (St-Ferreol), which caused a maximum crest displacement of 85.04 mm at 13.92 s (Figure 8). Tensile damage initiated at 13.78 s and reached its peak, $1.60\times {10}^{-1}$, shortly after (Figure 9). The cracking was distributed across both the heel and crest regions, suggesting that base shear and top flexure jointly influenced the failure mechanism.

Additional simulations, such as those for Sites 9T and 20L, recorded crest displacements of 38.77 mm and −47.26 mm, respectively. Site 9T displayed clear but limited cracking at the heel with ${\mathrm{d}}_{\mathrm{t}}=6.24\times {10}^{-3}$, whereas Site 20L exhibited damage of up to $7.30\times {10}^{-2}$, again localised at the base. These results affirm that D2′s larger geometry and lower natural frequencies increase its vulnerability to long-period excitations, leading to stress accumulation at both ends of the structure.

3.7. Seismic Response of D2: Scaled Acceleration

Under scaled seismic loading, D2 underwent extensive structural damage, with several simulations exhibiting near-complete material degradation in critical zones. The most severe response was observed in the simulation for Site 9L, where the crest displacement reached −115.91 mm at 16.65 s (Figure 10). Tensile cracks developed across the entire crest region with peak damage of $9.0\times {10}^{-1}$, followed by damage propagation down the downstream wall (Figure 11). The damage pattern suggests a flexural failure mechanism driven by amplified inertial rotation and crest uplift.

Simulations for Sites 14L, 16L, and 20L also showed high displacements (>85 mm) and dual-zone cracking at both the crest and base. For example, Site 14L produced an early displacement of 108 mm within 0.4 s and caused simultaneous crest and mid-height cracking, reflecting a short-period energy spike. Site 16L triggered peak crest damage with limited base cracking, while Site 20L produced balanced cracking at both ends of the dam body. These results collectively confirm that D2 is more vulnerable than D1 under design-level seismic excitations, primarily due to its lower stiffness and increased mass, leading to higher inertial demand and structural deformation.

3.8. Comparison of Dam Behaviour

Overall, the results demonstrate contrasting seismic performance between D1 and D2. D1 had a higher stiffness and compact geometry and thus experienced relatively low displacements and highly localised damage in most cases. Tensile failure was generally confined to the upstream heel and occurred under amplified input motions with fatigue-like behaviour. In contrast, D2, which was characterised by a broader base and lower natural frequencies, showed more widespread damage. Both unscaled and scaled records led to cracking at the heel and crest, driven by base shear and flexural effects. The damage was particularly severe in scaled scenarios, where multiple simulations exceeded 100 mm crest displacement, and tensile damage approached material failure limits.

This comparative analysis underlines the importance of modal characteristics in determining seismic response. Lower natural frequencies, greater mass, and higher slenderness ratios contribute to greater inertial forces and larger structural deformation. These findings support the use of frequency-matched input motions and nonlinear damage modelling (e.g., CDP) in the seismic assessment of gravity dams, as also recommended in recent studies [54,55].

It should be noted that the present study was carried out under the condition that the reservoir is empty. However, the stresses and strains induced by the normal water level were considered prior to the application of the seismic load, which placed the dam in a prestressed condition. Studies by Patra et al. [48], Xu et al. [56], and Alijani-Ardeshir et al. [14] on dam–reservoir and dam–reservoir–foundation interaction have shown that the dominant periods of the dam increase slightly by about 10% when the reservoir is full compared to when it is empty. This suggests that the empty-reservoir condition represents a more critical scenario compared to the full-reservoir case.

4. Discussion

Structural geometry and dynamic properties directly influence a dam’s seismic vulnerability. D1 had higher natural frequencies, resulting in lower design amplification and relatively modest crest displacements. Tensile cracking remained limited to the upstream heel and only occurred under scaled loading, supporting previous findings that short-period, stiff structures primarily fail in localised tension zones [37]. Conversely, D2 had lower stiffness and broader geometry, which led to resonance with long-period ground motions, yielding higher crest displacements and extensive cracking patterns. The results from scaled records (e.g., Sites 9T, 14L) demonstrate classic flexural response behaviour, where both the heel and crest zones undergo simultaneous tension, a phenomenon supported by experimental studies on large dam models [57].

These findings affirm the suitability of the CDP model for capturing nonlinear degradation, especially under repeated or scaled seismic loads. The observed lag between crest displacement and peak damage further validates fatigue-based mechanisms as critical factors in crack propagation. Moreover, damage patterns in D2 under original records indicate that moderate earthquakes can still yield progressive failure if modal properties align with ground motion characteristics. Overall, the results suggest that a modal analysis should precede seismic safety assessments and that damage localisation can inform practical reinforcement strategies, such as heel anchoring or crest stiffening. The simulations consistently revealed the upstream heel to be the most vulnerable zone for tensile cracking. However, under higher intensity or scaled records, additional cracking emerged at the crest and downstream face, highlighting the need to address multi-zone failure potential in seismic safety planning.

The proposed seismic assessment framework offers a practical and adaptable procedure for evaluating the structural safety of concrete gravity dams under earthquake loading. The nonlinear finite element approach using the CDP model was successfully applied to two dams, D1 and D2, each featuring distinct geometries and frequency characteristics. The same modelling approach was also validated using the well-documented Koyna Dam, reinforcing its generalisability. This workflow, comprising dam geometry modelling, mesh generation, material property assignment, loading application, and damage analysis, can be readily applied to other dams globally. Only the dam geometry, concrete parameters, and soil/foundation conditions need to be updated, making this a flexible method for engineers and researchers aiming to assess seismic vulnerability in diverse regions.

Compared to previous studies, this work incorporates important methodological differences. Alembagheri [31], for instance, employed linear time–history simulations with empirical damage criteria based on crest displacements and demand/capacity ratios. While useful for fast screening, such approaches require post-processing assumptions about failure states and cannot simulate the full range of nonlinear behaviour, including crack initiation, propagation, and residual capacity. The methods in this study have eliminated the above-mentioned gap by using a constitutive model capable of directly capturing inelastic concrete behaviour under seismic loading. Similarly, fragility-based studies [30,38] often rely on simplified lumped-mass models or reduced-order finite element formulations, which may neglect spatial damage distribution or realistic material degradation. In contrast, the use of full-continuum models in this paper provides the spatial resolution of cracking patterns, enabling the identification of critical zones like the upstream heel and dam crest.

A key advantage of the proposed method lies in its generalisability. Once dam-specific parameters are defined, the same modelling procedure can be applied elsewhere. The use of the computational tools makes the process accessible for engineering teams, as the tool offers prebuilt CDP modules and efficient meshing capabilities. However, this study does come with limitations. Most notably, dam–foundation and soil–structure interaction (SSI) effects were not explicitly modelled; both dams were assumed to be rigidly fixed at the base. While this simplification is common in many nonlinear dam studies, it may lead to overestimation of internal stress and underrepresentation of energy dissipation through the foundation. Future studies should incorporate SSI and hydrodynamic pressure effects to refine results, especially for dams founded on softer soils or deep reservoir bases.

Regionally, this study helps fill an important gap in the seismic safety literature. Eastern Canada has received relatively little attention in dam-focused seismic assessments, despite the existence of hundreds of aging structures, many of which were built before modern seismic codes. While seismicity in this region is considered moderate, events such as the 1988 Saguenay earthquake have revealed the potential for damaging ground motions. By applying scaled acceleration records from that event, this study provides a site-specific evaluation using realistic ground motion inputs, aligned with Canadian seismic hazard maps. It also offers a foundation upon which future Eastern Canadian dam studies, particularly those exploring fragility curves or probabilistic risk, can build. The comparative analysis between D1 and D2 further emphasises how modal properties, geometry, and input directionality influence seismic response, offering insight for regional prioritisation of dam retrofitting needs.

This study also aligns closely with CDA’s performance-based requirements. The CDA emphasises realistic modelling approaches, use of site-specific seismic inputs, and classification of structural response into meaningful damage states. The methods in this study satisfy all three. Design acceleration values were selected according to National Building Code of Canada (NBCC) seismic hazard criteria based on site class, and each earthquake record was scaled to match the fundamental period of the respective dam. This period-based scaling ensures resonance conditions are properly captured, a factor critical for evaluating crest displacements and damage progression. Moreover, the reported damage results, expressed using CDP scalar parameters, allow classification into negligible, repairable, or severe damage categories, directly supporting CDA-based seismic safety evaluations [1].

This study has contributed both technically and regionally. It presents a detailed and validated nonlinear modelling approach that improves on traditional and semi-empirical methods by capturing true material behaviour and spatial damage patterns. It addresses the lack of dam-specific seismic evaluations in Eastern Canada and provides a repeatable framework that others can adopt or extend. Although this study focused on two representative dams in Eastern Canada, the modelling framework is general and can be readily applied to dams of different geometries and material properties by updating the finite element model. For seismic hazard environments, the framework can incorporate ground motions from past earthquakes recorded in the region, which may be spectrally scaled to the target soil class and return period using the National Building Code of Canada (NBCC) design spectra. In this study, rigid foundation conditions were assumed to isolate the superstructure response; however, incorporating dam–foundation–reservoir interaction represents an important extension for future work. While limitations such as neglecting soil–structure interaction remain, the presented results form a robust basis for future fragility analysis, performance-based design, and seismic retrofitting decisions across similar moderate seismic regions. Lessault et al. [58] reported a range of earthquake scenarios for the Montreal region. In future studies, it would be interesting to scale up the scenarios specifically for seismic evaluations of dams in the region.

5. Conclusions

This paper presents a detailed seismic performance evaluation of two distinct Canadian gravity dams, D1 in Montreal and D2 in La Malbaie (Figure 1), via nonlinear finite element modelling. The investigation focuses on tensile damage localisation and crest displacement under both original and scaled Saguenay earthquake records (Table 2), with a special emphasis on crack initiation and progression as governed by dam geometry. The following conclusions have been reached:

• With a narrower base and greater stiffness, dam D1 exhibited significantly higher natural frequencies than dam D2 (Table 3). The fundamental mode of D1 was approximately 2.3 times higher than that of D2, resulting in a stiffer and more compact structural profile. D1 demonstrates strong seismic resilience; even under the most critical acceleration from station 8L (Figure 1, Table 2), it exhibited a very small tensile damage parameter value, ${\mathrm{d}}_{\mathrm{t}}$ = $5.03\times {10}^{-3}$, at the upstream heel. The corresponding crest displacement for this case was only 11.74 mm. In contrast, D2 showed a crest displacement of 85.04 mm under the same excitation, and the maximum tensile damage reached ${\mathrm{d}}_{\mathrm{t}}=1.60\times {10}^{-1}$, revealing two cracking zones at the upstream heel and around the crest region.
• For the scaled acceleration, the maximum crest displacement of D1 was 43.74 mm, being significantly lower than that of D2 (−115.91 mm). The tensile damage distribution in D1 showed widespread cracking concentrated at the upstream heel, with a peak value of ${\mathrm{d}}_{\mathrm{t}}=4.84\times {10}^{-1}$. Additionally, minor damage was observed along the downstream wall, reaching a value of ${\mathrm{d}}_{\mathrm{t}}=1.01\times {10}^{-1}$. In contrast, under the same scaled records from station 9L, D2 exhibited the most severe response; the crest displacement reached −115.91 mm. Extensive tensile cracking initiated along the entire crest region, reaching a peak tensile damage value of ${\mathrm{d}}_{\mathrm{t}}=9.0\times {10}^{-1}$, and subsequently propagated downward along the downstream face (Figure 11).
• Structural geometry and modal response characteristics play a decisive role in seismic vulnerability. D1′s behaviour reaffirms the benefits of compact and stiff configurations for seismic resistance, while D2 serves as a case where the kind of distributed failure can arise in massive structures lacking sufficient frequency separation from incoming ground motion.

While this study focused on two case study dams, the framework is applicable to other dams in terms of updating geometry, material parameters, and regional seismic inputs. Future extensions will incorporate dam–foundation–reservoir interaction to further enhance realism. The insights gained from this comparative analysis offer practical implications for the seismic design and retrofitting of gravity dams. Measures such as base anchoring, crest stiffening, and tailored reinforcement detailing should be prioritised for dams exhibiting modal resonance with expected regional ground motions. This study supports the incorporation of nonlinear damage modelling and record scaling in fragility assessment frameworks, particularly for critical infrastructure in seismically active zones.