Preamble

Vol. XX, No. X, XXX 20XX
NUCLEAR TECHNIQUES

Preliminary Study on Seismic Analysis of Graphite Cores for Gas-Cooled Microreactors

Lan Tianbao, Feng Tianyou, Huang Zhong, Sheng Feng
(Nuclear Engineering Academy, China Nuclear Power Engineering Co., Ltd., Beijing 100840, China)

Abstract

[Background] Graphite serves as a critical structural material in high-temperature gas-cooled reactors, and ensuring its integrity under seismic loading is essential for core safety. [Purpose] This study evaluates the seismic response of a gas-cooled microreactor core and verifies its compliance with ASME design codes. [Methods] Collision tests and simulations of small-sized graphite blocks were conducted to identify stiffness and damping parameters. A simplified core model with equivalent spring-damper elements was established and analyzed through time-history simulations. [Results] The Kelvin collision model effectively captured graphite block interactions, with stiffness ranging from 1.70×10⁹ to 2.20×10⁹ N/m for collision angles of 0.02°–0.05°. The maximum collision force between graphite components reached 1.21×10⁵ N. Stress evaluation based on the Weibull distribution indicated a failure probability below 0.5% under design-level seismic loads. [Conclusions] The proposed stiffness-equivalent method provides an efficient approach for simulating nonlinear core behavior, confirming the structural safety of the graphite core under seismic conditions.

Keywords: Graphite core; Seismic analysis; Stiffness; Damping; Failure probability

Introduction

Since its first application in the CP-1 (Chicago Pile-1) reactor in 1942, the structural integrity of nuclear graphite has remained a central concern in reactor safety research [1]. The development of fourth-generation Very High Temperature Reactors (VHTR) has further highlighted the importance of seismic performance in graphite cores [2]. In high-temperature gas-cooled reactors, graphite serves as the primary material for core structures such as reflectors and moderators [3]. Ensuring graphite integrity is crucial for overall core safety. Due to the nonlinear pin-key matching connections between graphite blocks, the entire structure behaves as a multi-body system that exhibits complex nonlinear dynamic characteristics under seismic loading, forming a coupled system of multi-body nonlinear forces and deformations [4]. Reliable experimental data are required to validate and support numerical simulations for structural seismic design. However, factors such as model size, mass, cost, and experimental facilities often necessitate the use of scaled or local models compared with numerical models. Ikushima et al. [5-6] and Iyoku et al. [7] conducted seismic evaluations of hexagonal graphite blocks used in high-temperature gas-cooled reactors, covering various models including 1:2 scale single-column graphite models, 1:2 partial planar models, and 1:2 vertical and horizontal section models. Theymann et al. [8] performed experimental studies to assess the seismic stability of pebble bed cores, including 1:2 scale side reflector models, 1:3 scale side reflector models, and 1:6 scale cylindrical core models. To verify structural safety under seismic conditions, China's High Temperature Test Reactor (HTTR) conducted a series of tests. Sun et al. [9] performed seismic studies on 1:2 scale double-layer structural models to examine deformation and analyze primary factors affecting the dynamic characteristics of graphite structures. Tian et al. [10] tested single-column graphite models to obtain fundamental frequencies and damping characteristics. Lai et al. [11] validated and optimized computational programs based on full-scale double-layer four-brick graphite assembly tests. Wang et al. [12] designed a 1:4 scale overall model of the graphite core support structure, analyzed dynamic characteristic data from seismic tests, and verified that the support structure met safety requirements. For large and heavy core structures, existing methods study local dynamic characteristics to extrapolate the response of the global model under seismic excitation.

The core components and reflector structure of gas-cooled microreactors use graphite materials. Nuclear graphite features high purity, thermal conductivity, irradiation stability, and thermal oxidation resistance, along with low thermal expansion coefficient and low elastic modulus, which help avoid large thermal stresses. With high neutron scattering cross-section, low neutron absorption cross-section, and excellent high-temperature mechanical properties and neutron irradiation resistance, graphite is a suitable moderator material. However, graphite material properties, influenced by manufacturing processes, exhibit brittle or quasi-brittle behavior [13-14]. During manufacturing, grain characteristics, high-temperature carbonization and graphitization, and cooling processes leave approximately 20% porosity in nuclear graphite [15], causing strength dispersion that makes probabilistic evaluation methods more appropriate than deterministic approaches. Under irradiation, graphite volume first contracts then expands, affecting component gaps that influence collision behavior. Consequently, seismic analysis must address two key characteristics: graphite volume changes under irradiation and material strength dispersion. This study incorporates irradiation-induced volume contraction into component gaps and adopts the Weibull distribution model from ASME codes to account for material dispersion in structural integrity evaluation.

The graphite core structure is a critical component in nuclear reactors, and its seismic performance is vital for safe operation. Under seismic loading, collisions between graphite components or with the reflector may cause cracking or loss of support function, necessitating safety evaluation of gas-cooled microreactor core structures. While most existing research focuses on vertical core structures, studies on horizontal core structures remain limited. Gas-cooled microreactors feature horizontal stacked structures with pin-key connections between graphite components and existing gaps. Seismic excitation induces numerous contact collisions, which are often simplified in numerical simulations due to excessive computational demands, typically as multi-rigid-body dynamics problems with springs and dampers. This paper presents seismic analysis of a horizontal core structure, establishing a stiffness-equivalent model through combined parameter identification experiments and simulations, performing seismic time-history analysis, and conducting preliminary investigation of core structural response.

1 Core Structure Description

The gas-cooled microreactor core adopts a horizontal arrangement, consisting primarily of three layers of hexagonal graphite blocks with 31 components per layer, as shown in [FIGURE:1]. Layers are positioned through pin-key connections, radially constrained by reflectors and axially constrained by front and rear reflectors. The reflectors are preloaded with springs, though this preload is conservatively ignored in seismic calculations. The core structure connects to a metal container via binding straps, though the metal container and reflector are not considered in this study.

[FIGURE:1] Horizontal layout of the core

2 Theoretical Methods

Seismic excitation causes collisions in graphite cores. For discrete masonry systems, directly establishing an elastoplastic collision numerical model for the entire core is often impractical due to high computational costs from contact identification between graphite blocks and the large number of degrees of freedom. Engineering applications require simplified equivalent models that capture overall structural response through contact collision relationships between graphite blocks and pin-keys. Common continuous contact force models include linear elastic, Kelvin, Hertz, and Hertz-damp models [16].

According to classical collision theory, the relationship between contact force $F$ and collision gap can be described as:

$$
F = k d^n + c d^m \dot{d} \quad \text{(2.1)}
$$

where $k$ and $c$ represent the equivalent stiffness and damping between colliding bodies, $e$ is the coefficient of restitution, $\dot{d}_0$ is the initial relative velocity, $d$ is the relative displacement during collision, and $\dot{d}$ is the relative velocity of colliding objects. Different values of $n$ and $m$ represent different collision models. When $n=1$ and $m=0$, the equation represents the Kelvin-Voight model. Adjusting $k$ can align the identified contact time with experimental results to obtain appropriate equivalent stiffness $k$ and damping $c$.

The primary experimental objective is to obtain acceleration and velocity variations during graphite block collisions, determine collision duration and pre-/post-collision relative velocities, and identify contact model parameters. Acceleration data are obtained from accelerometers mounted on graphite blocks, while velocity data come from acceleration integration. The test apparatus is shown in [FIGURE:2], with the excitation mechanism on the left and the collision block on the right. A pendulum connected by a parallelogram structure is released from a certain height to impact the active block specimen on the left, imparting initial velocity for collision with the passive block. Both blocks are placed on sliding rails to minimize friction. Buffer pads behind the blocks ensure safety after collision.

[FIGURE:2] Graphite block collision mechanism

To investigate potentially suitable oblique collision angles, finite element simulations were conducted for various cases with oblique angles from 0.01° to 0.1°, maintaining model dimensions consistent with experiments. Graphite material parameters used in the analysis are listed in [TABLE:1].

[TABLE:1] Mechanical parameter of the graphite

Parameters Value Young's Modulus (Pa) 9.6×10⁹ Poisson's ratio - Density (kg·m⁻³) 1.8×10³

Comparative results of collision acceleration and velocity curves are shown in [FIGURE:3] and [FIGURE:4] (P denotes passive block, A denotes active block). At small oblique angles (below 0.05°), no obvious dual acceleration peaks appear, while 0.08° and 0.1° cases exhibit two peaks. Smaller angles approach ideal head-on collision results, as seen in 0.01° and 0.02° cases. The 0.05° case shows consistent contact time between simulation and experiment, though peak acceleration differs significantly. Since acceleration peaks vary between measurement points and simulation centroid values, while contact time is more stable and better represents contact stiffness, contact time is adopted for stiffness identification. Experiments reveal that ideal head-on collisions rarely occur in practice, with line contact and local point contact being more common. Consequently, the 0.05° oblique collision case becomes the primary focus for subsequent simulation analysis and parameter identification based on simulation data.

[FIGURE:3] Comparison of acceleration between simulated and experimental graphite cube in head-on, 45° oblique, and oblique collisions (0.01~0.1°) case

[FIGURE:4] Comparison of velocity between simulated and experimental graphite cube in head-on, 45° oblique, and oblique collisions (0.01~0.1°) case

The results in [FIGURE:3] and [FIGURE:4] demonstrate that simulations can capture collision processes, though errors remain. The penalty function contact algorithm assumes a linear relationship between contact force and penetration. Excessive contact stiffness overestimates peak contact force, while insufficient stiffness introduces non-physical penetration. Friction is neglected in simulations and simplified into damping effects. Fixture stiffness and sensor installation in experiments are not precisely modeled, leading to idealized simulation boundary conditions. Real collisions involve certain angles, as perfectly head-on collisions are practically impossible, and these angles affect both experimental and simulation results. However, parameter optimization can make simulation results approach or remain conservative relative to experimental results to meet engineering design requirements.

2.2 Parameter Identification

Based on collision experimental and simulation results, finite element simulation is employed to identify collision stiffness and damping parameters for application to graphite component collisions, providing input for core calculations.

Drawing from graphite cube collision simulation and experimental comparison results, elastic model collision simulations of hexagonal graphite cores are conducted for small-angle oblique collisions. Ten cases from 0.01° to 0.1° at 0.01° intervals are analyzed, with collision schematics shown in [FIGURE:5].

[FIGURE:5] Collision mode of a pair of hexagonal graphite bricks: (a) Head-on collision 0°, (b) Oblique collision (0.01°~0.1°), (c) 45° oblique collision

Simulation trends align with graphite cube results: as collision angle increases, contact time extends, acceleration peak decreases, and post-collision relative velocity reduces, as shown in FIGURE:6. FIGURE:6 shows that except for 0.09° and 0.1° cases where post-collision relative velocity falls below head-on oblique collision values, oblique collision results lie between head-on and 45° oblique cases.

[FIGURE:6] The acceleration and velocity of the graphite brick during collision: (a) Acceleration Results, (b) Velocity Results

Detailed data are summarized in [TABLE:2]. Based on experimental data, graphite components rarely experience head-on collisions in motion, so the average values for 0.02°–0.05° collision angles are adopted as stiffness and damping parameters. Smaller collision angles yield larger equivalent stiffness, making these values more conservative.

[TABLE:2] Equivalent parameters for the collision of hexagonal graphite cores

Angle(°) Max Force(N) Max Accel(m/s²) Impact time(ms) Coeff. restitution Equivalent stiffness(N/m) Equivalent damp(Ns/m) 0 1.684e5 - - - 2.96e9 - 0.01 1.545e5 - - - 2.50e9 - 0.02 1.446e5 - - - 2.20e9 - 0.03 1.376e5 - - - 2.05e9 - 0.04 1.288e5 - - - 1.86e9 - 0.05 1.211e5 - - - 1.70e9 - 0.06 1.157e5 - - - 1.62e9 - 0.07 1.079e5 - - - 1.50e9 - 0.08 0.9881e5 - - - 1.32e9 - 0.09 0.9347e5 - - - 1.22e9 - 0.10 0.8981e5 - - - 1.15e9 - 45 0.4485e5 - - - 0.28e9 -

3 Stiffness-Equivalent Modeling

3.1 Stiffness Equivalence

To improve computational efficiency, a simplified model of the gas-cooled microreactor structure is established using a rigid-body model to capture responses under seismic excitation, including forces, displacements, and velocities. Graphite bricks are hexagonal prisms with holes and keys on their top and bottom surfaces, as shown in [FIGURE:7]. Horizontal displacement of graphite blocks is constrained by adjacent blocks and keys, though keys have limited load-bearing capacity and primarily serve installation and positioning functions. The simplification represents each graphite block as a central mass point connected to rigid beams, neglecting local stress and regional flexibility. This approach reduces model degrees of freedom while capturing overall displacement, velocity, and acceleration responses, enabling extraction of interaction forces between rigid graphite blocks for local stress analysis.

[FIGURE:7] Graphite structures and simplified rigid body models: (a) Graphite structures, (b) Simplified rigid body

The simplified rigid-body model uses springs, dampers, and gaps to characterize collisions between graphite components. Two primary contact modes exist: block-to-block collisions and pin-hole collisions between keys and holes. [FIGURE:8] illustrates the equivalent element representation for block-to-block collisions, where connections are implemented through contact elements that ensure compression-only behavior.

[FIGURE:8] Stiffness-equivalent simulation of collisions between graphite blocks

3.2 Irradiation Deformation Effects

Graphite components feature contact connections with预留 gaps to accommodate thermal expansion during high-temperature operation. However, with increasing neutron fluence over operational lifetime, graphite undergoes initial contraction followed by expansion. Structural contraction creates gaps detrimental to seismic performance, necessitating estimation of maximum gaps between components. Based on design neutron fluence, user-defined subroutines are developed to define graphite deformation behavior under neutron irradiation. The neutron fluence distribution is expressed as:

$$
N(x, y, z, t) = (0.136 + 1.605x - 2.36y - 2.36z) \times (t - 0.86) \quad \text{(3-1)}
$$

where $x$, $y$, and $z$ represent core spatial coordinates, $t$ is operational lifetime (<30 years), and $N$ denotes neutron fluence in dpa. For fast neutrons with $E>1$ MeV, approximately $1 \times 10^{25}$ n/m² corresponds to 1 dpa of irradiation damage. [FIGURE:9] shows the neutron fluence field in a column of hexagonal graphite prisms.

[FIGURE:9] Distribution of the overall irradiation field in a column of graphite hexagonal prism model

Assuming irradiated graphite behaves as a linear viscoelastic material, total strain comprises elastic strain, steady-state creep strain (secondary creep), transient creep strain (primary creep), and combined thermal/irradiation strain. For IG110 graphite mechanical properties, the constitutive model expression is derived based on Hooke's law and material property fitting formulas from literature [17], considering irradiation-induced dimensional change strain and creep strain. Calculations require only pre-irradiation elastic modulus $E$, Poisson's ratio $\nu$, and thermal expansion coefficient $\alpha$, with multiple state variables storing and updating elastic modulus and irradiation dose at each increment. The Jacobian matrix and stress components are continuously updated to obtain stress results throughout the calculation cycle, as shown in the flowchart in [FIGURE:10].

[FIGURE:10] The calculation flow of the developed user-defined material subroutine

Using the custom subroutine, maximum graphite component deformation is calculated. As shown in [FIGURE:11], the maximum displacement between node 1 and node 2 is 1.895 mm, leading to a conservative determination of the top space gap as 2 mm.

[FIGURE:11] Deformation at maximum shrinkage of graphite brick

[FIGURE:12] illustrates gap distribution in a horizontal graphite core structure model consisting of seven layers with 31 graphite components. The upper core edge gap is 2 mm, the lower core edge gap is 1 mm, and fuel assembly gaps are 1 mm, with upper and lower gaps.

[FIGURE:12] Core model gap distribution

The graphite core is modeled using ANSYS 18.0 commercial finite element software, comprising rigid beam elements (MPC184), contact elements (CONTA178), and mass point elements (MASS21) concentrating graphite component weight (30 kg per component including fuel). Boundaries connect to fixed constraints via contact elements. Equivalent parameters for relevant components are listed in [TABLE:3], simplifying the core discrete structure into a mass-point system linked by springs and dampers.

[TABLE:3] Input parameter

Part Equivalent stiffness (N/m) Equivalent damp (Ns/m) Brick-Brick 1.95e9 - Pin-Hole 0.11e9 -

3.3 Simplified 3D Model

Following simplification principles, the graphite core structure is reduced to an equivalent model consisting of three layers of rigid-body elements, as shown in [FIGURE:13], with 31 graphite components per layer. Connections in axial, lateral, and pin-hole directions are represented through simplified stiffness and damping.

[FIGURE:13] Simplified model of 3D core structure

4 Analysis Results

Time-history analysis is performed on the simplified 3D core model using the acceleration response shown in [FIGURE:14], extracted from a metal container model. Correlation coefficients between three-directional seismic time-history accelerations are below 0.3, with peak acceleration below 0.3g.

[FIGURE:14] Time history acceleration of three directions

Interaction forces between graphite blocks are extracted from contact elements. [FIGURE:15] shows collision forces at 10.22 s, when maximum collision forces occur. By extracting contact forces from elements, static strength analysis of graphite components can be performed.

[FIGURE:15] Collision force between graphite blocks at 10.22s

5 Stress Evaluation

Graphite component failure analysis employs a probabilistic failure model that predicts failure probability under given stress states by treating each group as a single chain, using a modified volume-normalized Weibull weakest-link failure criterion. This criterion combines the volume regularization method proposed by Schmidt et al. [18] and the failure probability calculation model by Hindley et al. [19], making it suitable for coarse-grained nuclear graphite failure probability prediction.

Parameters for the three-parameter Weibull distribution function are estimated based on experimental data from strength tests:

$$
P_f = 1 - \exp\left[-\left(\frac{\sigma - \sigma_0}{\sigma_c}\right)^m\right] \quad \text{(5.1)}
$$

where $P_f$ is failure probability, $\sigma$ is equivalent stress, $m$ is shape parameter, $\sigma_c$ is characteristic strength, and $\sigma_0$ is threshold strength. Parameters $m$, $\sigma_c$, and $\sigma_0$ are called Weibull three parameters.

The maximum equivalent stress generated by various loads during graphite lifetime must be less than the allowable stress corresponding to the reliability and load levels. Reliability and load levels for graphite are determined according to ASME ⅢD5 [20]. Equivalent stress is defined as:

$$
\sigma_{eq} = \sqrt{\sigma_1^2 + \sigma_2^2 + \sigma_3^2 - \sigma_1\sigma_2 - \sigma_1\sigma_3 - \sigma_2\sigma_3} \quad \text{(5.2)}
$$

where $\sigma_i$ ($i=1,2,3$) are principal stresses. When $\sigma_i$ is tensile stress, $f=1$; when compressive, $f$ is the ratio of average compressive strength to average tensile strength of graphite.

A UVARM subroutine is defined to output element equivalent stress variables. By applying uniformly distributed pressure from surface forces, equivalent stress is obtained, as shown in [FIGURE:16].

[FIGURE:16] Equivalent force distribution of hexagonal graphite under seismic loading

After calculating the stress field using finite element methods, principal stresses $\sigma_i$ and integration point volumes $v$ are output. Within the effective volume, integration point volumes and stresses are output. According to ASME requirements, volume grouping is performed first (e.g., total volume $V_I$ for group I), where $a$ represents the maximum particle size of the nuclear graphite. Stress gradients are limited to less than 7% within a group. The failure probability under 1 kN collision force is calculated as 0, with failure probabilities at different collision forces shown in [FIGURE:17]. The failure probability exceeds 0.5% when collision force surpasses 35 kN.

[FIGURE:17] Failure probability of hexagonal graphite under different collision forces

Conclusions

This study employed a stiffness-equivalent method to simplify the model, conducted graphite block collision tests to obtain stiffness and damping parameters, and simulated multi-body motion responses of graphite. The approach captures graphite displacement, velocity, and interaction forces while significantly reducing computational time. Final stress evaluation of graphite blocks confirms that strength meets ASME code requirements. The main conclusions are:

1) Ideal head-on graphite collisions rarely occur in practice; small-angle collisions are more realistic. Collision tests demonstrate that the Kelvin model can simulate graphite block collisions and provide stiffness-damping parameters.

2) Stiffness and damping parameters can be identified from collision acceleration and velocity curves. Data from elastic model collision simulations of hexagonal graphite bricks are extrapolated to obtain stiffness and damping parameters for actual hexagonal graphite bricks.

3) A global core model was established to extract interaction forces between graphite components under seismic loading. Local static analysis of graphite blocks confirms that graphite strength satisfies ASME code requirements.

Author Contributions

Lan Tianbao: Simulation, data analysis, experimental guidance, and manuscript writing. Feng Tianyou: Experiments and data processing. Huang Zhong: Research concept and theoretical guidance. Sheng Feng: Modeling guidance, project supervision, and review.

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