Topology-optimized reinforcement of Terracotta Warriors validated by shaking-table testing and finite-element analysis

Abstract

The Terracotta Warriors, hailed as the ‘Eighth Wonder of the World’, face material degradation and structural instability. Conventional reinforcement brackets are inadequate for their diverse forms and uneven stress distributions. This study proposes a topology optimization-based reinforcement method. A customized bracket for a standing warrior replica was designed and validated via shaking table tests. Furthermore, aerospace-grade polycarbonate (PC) was introduced as an alternative material. Finite-element simulations under identical geometry and boundary conditions as steel brackets compared their dynamic responses. Results show the PC bracket maintains 90% light transmittance with appropriate hardness, achieving 39% average reinforcement efficiency, thus balancing exhibition esthetics and structural safety. This method is applicable to individualized support for large, brittle heritage objects including terracotta figures, ceramic sculptures, stone/wood statues, and thin-walled hollow artifacts in museums and in-situ displays.

Introduction

The Terracotta Warriors, hailed as the “Eighth Wonder of the World,” are a magnificent treasure of human civilization and mark the glorious beginning of realistic sculpture in ancient China^1,2. These artifacts were crafted around 240 BCE and have a history spanning over two millennia³. Due to prolonged burial, fire damage, tectonic activity, and subsurface weathering, the Terracotta Warriors are often unearthed in a fragmented state^4,5,6, and their restored structures exhibit complex and nonlinear mechanical behavior. With the continuous advancement of interdisciplinary integration, cultural heritage preservation is increasingly evolving into a multidisciplinary collaborative endeavor^7,8,9. In recent years, the issue of mechanical stability of cultural relics has drawn growing attention, as it directly affects the structural integrity and safety of the artifacts themselves^10,11,12,13. Improper handling during exhibition, reinforcement, or under external loading can easily lead to cracking or even structural failure¹⁴. In the field of cultural heritage conservation, the term “original damage” refers to all physical deterioration and structural damage that occurs to a cultural object during its production, use, and prolonged burial (or preservation), primarily prior to its excavation and before it undergoes systematic conservation and restoration; whereas “secondary damage” characterizes structural failures that emerge after restoration, induced by flawed restoration techniques, inadequate long-term storage conditions, or external loads (e.g., seismic events). The failure mechanisms involved in secondary damage are typically more complex than those of initial damage¹⁵. For cultural relics such as the Terracotta Warriors, secondary damage is often irreversible, posing severe threats to the structural safety of the heritage and resulting in irreparable losses to human civilization.

For terracotta warriors displayed in situ or exhibited in museums, their original structures—unlike smaller cultural relics such as porcelain or paintings—experience more substantial gravitational effects during long-term exhibition and remain susceptible to external loads such as seismic events¹⁶. Meanwhile, due to extended underground burial and environmental exposure during exhibition, their material properties gradually deteriorate under the combined effects of natural weathering and aging¹⁷. This degradation process leads to increasingly complex internal stress states and heightened sensitivity to self-weight and external disturbances, making them more prone to damage^18,19,20. To reduce the risk of structural instability, support and reinforcement systems are often employed to enhance overall stability²¹. However, current reinforcement practices primarily rely on expert experience and observations of artifact damage, which are then mechanically applied to similar types of cultural relics. This approach often overlooks the individual differences in geometric form and mechanical behavior among artifacts, resulting in insufficient conformity between the support systems and the artifacts themselves. This not only limits the effectiveness of reinforcement but may also introduce new structural risks.

Among the various types of Terracotta Warriors, standing figures—such as warrior and armored statues—constitute the overwhelming majority²². These figures are typically characterized by considerable height and a limited base support area, resulting in vertical load transmission and localized stress concentrations. Particularly during in situ display following restoration, the combined effects of self-weight and seismic loading significantly increase the risk of structural failure, highlighting their inherent vulnerability. Therefore, developing targeted and refined support designs for such artifacts is crucial to preventing secondary damage caused by internal stresses and external forces.

As a pivotal computational approach in structural mechanics, topology optimization aims to identify the optimal material distribution within a predefined design domain under given boundary conditions and performance constraints²³. This method leverages structural internal force characteristics and stress evolution patterns to achieve efficient material utilization through iterative optimization, thereby yielding structural configurations that combine lightweight properties with high load-bearing capacity. Although topology optimization has been widely adopted in advanced engineering sectors such as aerospace²⁴ and automotive industries²⁵, its potential remains underexplored in the field of cultural heritage conservation, particularly for the design of customized reinforcement structures. To address this gap, this study introduces a percentage-based topology optimization strategy into the reinforcement design of Terracotta Warriors, with the objective of developing support brackets that not only exhibit superior structural load-bearing capacity but also achieve mechanical compatibility with the complex and heterogeneous morphology of the artifacts. This design is expected to significantly mitigate stress concentration and secondary damage risks commonly associated with conventional experience-based support systems²⁶.

Although previous studies have extensively explored the structural analysis and reinforcement strategies for cultural heritage artifacts, most of them remain focused on mechanical performance evaluation and lack generalized and systematic approaches to reinforcement design. To address this gap, this study takes a replica of the Terracotta Warrior as the research object and establishes an integrated reinforcement design workflow that incorporates structural optimization²⁷, 3D laser scanning²⁸, reverse modeling²⁹, finite element analysis (FEA), and shaking table testing. Based on this workflow, an automated and adaptive support design framework has been developed, which enables targeted reinforcement of structurally vulnerable areas while effectively avoiding the mechanical mismatch and potential structural risks associated with experience-based reinforcement components. Furthermore, this study explores the feasibility of applying aerospace-grade materials in heritage reinforcement³⁰, selecting polycarbonate (PC) as a representative material. The mechanical performance of PC is evaluated using finite element simulations, and its engineering potential for cultural heritage protection is discussed based on its material advantages.

The main contributions of this study are listed below:

This study is the first to introduce structural optimization methods into the design of reinforcement supports for cultural heritage artifacts. A targeted and flexible support design approach is proposed, which can effectively accommodate the structural and mechanical differences of individual relics, while minimizing the risk of mechanical mismatch and secondary damage caused by conventional mass-produced supports.

Based on the mechanical characteristics of typical standing Terracotta Warriors, a percentage-based optimization strategy is developed for support design. Its reinforcement effectiveness is verified through finite element simulations, and the feasibility and performance advantages of the method are further validated through shaking table experiments.

The study explores the application of aerospace-grade materials in heritage support systems. Polycarbonate (PC) is selected as a representative material to construct a reinforcement framework, which is then compared with conventional steel supports in terms of economic efficiency index, demonstrating its mechanical adaptability and potential for engineering applications in cultural heritage preservation.

Methods

This study proposes an automated reinforcement bracket design method that integrates structural optimization with high-performance material selection, aiming to achieve customized support solutions adapted to the individual structural characteristics of cultural artifacts. The specific implementation workflow comprises the following key steps:

Digital Modeling: Employ 3D laser scanning and reverse modeling technologies to construct a high-precision physical model of the artifact;

Mechanical Analysis: Conduct mechanical performance evaluation based on the digital model to identify structurally vulnerable areas;

Initial Configuration: Establish an initial support structure targeting key vulnerable regions;

Iterative Optimization: Apply percentage-based optimization to iteratively refine the support configuration, using an economic efficiency indicator as the evaluation criterion for optimization outcomes;

Final Validation: Determine the optimal support design through FEA validation.

This method enables accurate identification of structural weaknesses and provides targeted reinforcement solutions. By enhancing mechanical compatibility between the bracket and the artifact, it ensures structural stability while significantly reducing the risk of secondary damage during reinforcement interventions. Consequently, it establishes a scalable and adaptable technical framework for the scientific conservation of large-scale fragile ceramic heritage objects. The flowchart of this method is shown in Fig. 1.

Fig. 1: Percentage-Based Optimization Design Process for Adaptive Supports.

Percentage‑based optimization workflow for adaptive support design. The flowchart illustrates the iterative process from 3D scanning, FEA and stress analysis, generation of an initial structure, percentage‑based optimization with economic weighting, to the final output of the optimized solution upon reaching the optimal criterion.

Percentage-based optimization for the structural reinforcement design of Terracotta Warrior brackets

Existing studies have shown that, during in-situ display, standing Terracotta Warriors commonly exhibit significant stress concentrations around the ankles and the lower hem of the robe³¹. This phenomenon primarily results from the considerable self-weight of the figure and the gradual reduction in cross-sectional area from top to bottom. In particular, the abrupt transition between the large cross-section of the robe and the narrower section at the leg junction forms a mechanically vulnerable zone. The combination of these factors leads to pronounced localized stress in critical structural regions, thereby increasing the risk of structural failure. To address the above-mentioned structural stability issues in a targeted and efficient manner, this study introduces a percentage-based optimization method for the design of reinforcement brackets, focusing on the lower portion of the Terracotta Warrior structure.

A key advantage of topology optimization lies in its high degree of adaptability, enabling the intelligent identification and preservation of critical load-bearing paths based on structural stress characteristics³². As a specific branch of topology optimization techniques, the percentage-based optimization method employs iterative FEA to progressively remove low-stress elements while retaining those under high stress³³. This process gradually achieves a more uniform stress distribution, enhancing structural efficiency while ensuring rational material usage.

In the structural reinforcement of the Terracotta Warriors, this method offers several notable advantages: (1) it significantly reduces the volume of the bracket system, minimizing visual obstruction for viewers; (2) it enables more precise and targeted mechanical reinforcement designs; and (3) it improves compatibility and mechanical coordination between the reinforcement system and the artifact itself. Furthermore, a distinctive feature of this method is its “smooth” iterative trajectory, wherein the number of elements removed in each optimization round decreases progressively. This effectively lowers the risk of sudden structural failure during the optimization process and enhances the overall stability and reliability of the design.

Moreover, by introducing an economic efficiency index factor, the method achieves a balance between material savings and load-bearing capacity, providing a practical and robust criterion for structural optimization. This parameter represents the balance point between material reduction and structural reinforcement reduction, indicating that the remaining structure has reached its maximum load-bearing capacity. In percentage-based optimization, the number of elements to be removed in each iteration is determined by the following equation:

$${D}_{n}=1-{D}_{n-1}\times (1-{R}^{n})$$

(1)

In this method, n represents the number of iterations, D_n denotes the percentage of elements removed from the optimization domain in the n-th iteration, and R_n indicates the element removal rate. The percentage-based structural optimization approach effectively avoids abrupt increases in the number of eliminated elements, thereby mitigating the risk of uncontrolled structural failure. In contrast to traditional methods, the number of low-efficiency elements gradually decreases with each successive iteration, ensuring that the number of elements removed in any given cycle is smaller than in the previous one. As the optimization progresses, the total number of structural elements is incrementally reduced, while the stress borne by each retained element correspondingly increases. This iterative process is characterized by stability, rapid convergence, and the ability to deliver structurally efficient and high-quality results.

$$\alpha =\frac{(E-{E}_{D})/E}{(F-{F}_{D})/F}$$

(2)

In this context, the economic efficiency index α serves as a quantitative measure to evaluate the trade-off between material reduction and structural performance. It is calculated based on the total number of elements in the design domain $E$, the number of elements removed during optimization ${E}_{D}$, the original load-bearing capacity of the structure $F$, and the load-bearing capacity after optimization ${F}_{D}$. This index reflects how effectively the structure retains its mechanical performance relative to the extent of material simplification.

This index captures the trade-off between the percentage of material removed and the corresponding loss in structural capacity. A declining value of α indicates that structural performance is deteriorating more rapidly than material savings are being achieved, suggesting that the current iteration may not offer an optimal material-utilization outcome. Consequently, the peak of the economic efficiency curve typically marks the optimal balance point between structural integrity and material efficiency. At this point, the optimization outcome achieves the greatest possible preservation of load-bearing capacity.

Data acquisition, reverse modeling, and initial support optimization design

This study employed two replica Terracotta Warriors manufactured from local clay obtained near the burial pits of the Mausoleum of the First Qin Emperor as experimental subjects. These replicas were produced using identical mold systems and fabrication techniques, demonstrating highly consistent parameters: average wall thickness of 30 mm, height of 187 cm, and individual mass of approximately 180 kg. The equivalence in critical physical parameters allows them to be regarded as homogeneous experimental constructs, effectively eliminating the impact of specimen variability on experimental results.

The Terracotta Warriors, characterized as hollow thin-walled structures, have an average wall thickness of approximately 20–30 mm, based on archeological statistics compiled over fifty years^22,31.

Although the unearthed fragments of the Terracotta Army can be used to obtain internal structural data, the complete replica adopted in this study is of a closed hollow structure, which poses a huge challenge for non-destructive internal detection and thus makes it difficult to establish a detailed internal morphological model. This technical bottleneck also exists when dealing with the early restored, structurally intact Terracotta Army specimens.

Currently, non-destructive methods for probing the internal structure of such large-scale hollow heritage objects remain limited, making it difficult to achieve detailed internal morphological modeling. Consequently, in both research and practical conservation work, a uniform wall thickness of 25–30 mm is generally adopted for FEA of the Terracotta Warriors. Experimental comparisons have shown that this range agrees well with actual conditions^22,31.

Point cloud data were acquired using an Artec3D Spider scanner. During the scanning process, Geomagic Wrap software was employed for real-time monitoring to ensure comprehensive and uniform data coverage. Raw point clouds contained 5,454,892 and 4,922,901 points, respectively; after outlier removal and decimation, the datasets were wrapped into watertight polygonal meshes. Post-acquisition, the point cloud data were processed in Geomagic Wrap, where large irrelevant external clusters were manually removed, and noise points and surface artifacts were extracted and eliminated. The refined, discrete point clouds were subsequently converted into continuous polygonal meshes through surface wrapping. Using the Mesh Doctor function, the mesh was repaired to correct self-intersections and high-refraction edges, and holes were subsequently filled. The triangular mesh was then converted into a NURBS surface using the Exact Surfacing tool³⁴. Through surface segmentation and grid generation, a solid model was finally exported in .igs format.

The resulting solid model was imported into HyperMesh for meshing. As the Terracotta Warrior can be considered a typical homogeneous thin-walled structure, shell elements were used for the meshing process. Given the model’s complex and irregular surface geometry, triangular shell elements were chosen to enhance adaptability and accuracy. The model was discretized with triangular shell elements (S3R) with a uniform thickness of 30 mm, consistent with the measured average wall thickness. The completed mesh model was exported in.inp format for subsequent FEA. The meshed components were then imported into the finite element solver. The mechanical properties of the replica material, derived from prior experimental tests, are summarized in Table 1. A static analysis step was first conducted to assess the structural behavior under gravitational loading. This was followed by an implicit dynamic analysis step to simulate the structural response to seismic loading using ground motion inputs. The data acquisition and reverse modeling workflow is illustrated in Fig. 2a and b.

Fig. 2: Initial bracket design process.

a Point cloud data acquisition of the terracotta warrior replica; b Reverse modeling workflow of the terracotta warrior; c Finite element analysis stress contour under self-weight loading for the replica; d Stress contour at maximum stress during seismic excitation; e Reinforcement target area (lower skirt region); f Initial bracket dimensions; g Finite element model of the initial bracket.

Table 1 Material properties of terracotta

In this study, both the finite element model of the Terracotta Army and the vibration table test adopted a unified coordinate system: the X-axis coincided with the main excitation direction of the vibration table (and the orientation of the Terracotta Army’s face), and was aligned with the front direction of the Terracotta Army; the Y-axis is oriented vertically upward (opposite to the direction of gravity); and the Z-axis serves as the transverse axis, forming a right-handed orthogonal coordinate system with the X- and Y-axes. The term “X-direction acceleration” used throughout this paper specifically refers to the acceleration component along the sensor X-axis/primary excitation direction. To replicate real-world boundary conditions, vertical upward (Y-direction) constraints were applied to the nodes at the bottom of the base plate to simulate contact between the Terracotta Warrior and the exhibition platform, while a gravitational load was introduced to represent the self-weight of the structure. During the dynamic analysis phase, acceleration time histories derived from the El Centro ground motion record were applied along the axis parallel to the facial orientation of the Terracotta Warrior to the same bottom nodes.

FEA was first conducted to investigate the mechanical behavior of the terracotta warrior replica under both gravitational and dynamic loads, with the dynamic load represented by the El Centro wave at a peak acceleration of 0.35 g (Fig. 2c and d). The results reveal that stress concentrations primarily occur at the junction between the skirt and legs, the peripheral regions of the skirt, the ankle areas, and the base plate. This stress distribution pattern aligns well with existing research findings and archeological observations. Based on these results, the reinforcement bracket was strategically positioned in these critical regions. An elliptical thin-walled frustum was constructed as the initial optimization structure based on the geometric profile of the lower robe edge. The frustum features a gradually increasing cross-sectional area from bottom to top, enhancing structural stability under external loads. The entire optimization process and structural simulations were implemented using the ABAQUS platform, with the detailed workflow illustrated in Fig. 2e, f and d.

During dynamic analysis, El Centro seismic waves with peak accelerations of 0.18 g, 0.35 g, and 0.70 g were applied as inputs, and the reinforcement effectiveness was comparatively evaluated through finite element simulations and shaking table tests.

Results

Optimization of the reinforcement bracket for the Terracotta Warrior

The initial elliptical bracket was topologically optimized using a percentage-based structural optimization method, implemented through Python scripting and ABAQUS secondary development. An element removal ratio of R_n = 5% was defined, meaning that in each iteration, the 5% of remaining elements with the lowest stress levels were eliminated. The convergence criterion for the optimization process was based on an economic efficiency index, which characterizes the loss of load-bearing capacity through the displacement at the top section of the structure under constant load.

Figure 3a illustrates the iterative progression and final topological configuration of the percentage-based optimization process. The optimized bracket features an eight-leg support structure, symmetrically distributed along the elliptical base in four directions. Figure 3b shows the variation curve of the economic efficiency index during optimization: the index rises continuously from the origin, indicating that while material usage is progressively reduced, the structural mechanical utilization rate keeps improving. At the 79th iteration, the economic efficiency index reaches its peak value of 8.78, marking the optimal balance between material removal and structural performance. Further elimination of elements beyond this point leads to a sharp decline in load-bearing capacity, demonstrating that excessive material reduction significantly diminishes material utilization efficiency.

Fig. 3: Optimization process and results.

a Bracket shape evolution during optimization; b Variation curve of economic efficiency index during optimization; c Bracket optimization procedure using BESO method; d Final bracket configuration.

To further validate this reinforcement configuration, a Bi-directional Evolutionary Structural Optimization (BESO) approach³⁵ was employed under the same conditions, targeting a final material retention rate of 20% (i.e., 80% material removal). The BESO procedure was also implemented using Python and ABAQUS, with 90 total iterations.

The final BESO-optimized structure closely resembled that produced by the percentage-based method—featuring eight symmetrically paired supports extending inward and downward from the elliptical base. The uniform load distribution in both cases confirms that this configuration provides the optimal mechanical support for the lower skirt region of a standing Terracotta Warrior. The BESO result is shown in Fig. 3c.

Based on the topological optimization results, the primary mechanical behavior of this bracket is to provide inward and outward support to the top through four directional support rods. To facilitate manufacturing and preserve the visual integrity of the exhibit, the final bracket features an elliptical ring at the top, supported by four solid iron rods positioned along the optimized load-bearing paths. The geometric configuration of the bracket is illustrated in Fig. 3d. This design not only maintains the fundamental structural logic revealed by the optimization—specifically, the radial load transfer from the elliptical base to the top ring—but also enhances visual esthetics while minimizing obstruction of the warrior’s body.

Finite element analysis of reinforcement performance for the Terracotta Warrior

To validate the reinforcement effectiveness of the bracket on the terracotta warrior structure, FEA was conducted on both unreinforced and reinforced specimens. Structural model data of the terracotta warrior replica were first acquired using a handheld high-precision 3D laser scanner (Artec Spider) for point cloud collection. The raw point cloud data underwent preprocessing in Geomagic Wrap software, including redundant point removal, external surface boundary elimination, and noise reduction to minimize interference from non-structural data and enhance modeling accuracy.

Based on this processed data, the point cloud was reconstructed into a triangular mesh, converting the discrete point set into a closed polygonal surface model. After mesh repair, a complete mesh model comprising approximately 100,000 triangular elements was obtained.

The mesh model was then segmented according to polygonal features to generate surface models, thereby completing the construction of a three-dimensional solid model of the terracotta warrior replica. The model was imported into ABAQUS. Two analysis steps were established: a static analysis step and an implicit dynamic analysis step. During the static analysis step, a gravitational load was applied to the structure. In the dynamic analysis step, El Centro seismic waves with peak accelerations of 0.18 g, 0.35 g, and 0.70 g were applied along the X-direction of the structure.

After the calculations, the stress distribution of the unreinforced Terracotta Warrior replica under self-weight is shown in Fig. 4a. The simulation results indicate that, under gravity load, the main stress concentration occurs in the lower skirt, ankle, and base plate regions, which is consistent with previous research findings. The maximum von Mises stress under self-weight was calculated as 0.045 MPa, far below the tensile strength of the terracotta material, suggesting that the structure remains in a safe state under gravity loading.

Fig. 4: FEA results (unreinforced vs. reinforced).

a shows the calculation results for the unreinforced terracotta warrior. b shows the shoulder acceleration curves of the unreinforced terracotta warrior under three types of seismic waves. c shows the finite element model of the reinforced terracotta warrior. d shows the shoulder acceleration curves of the reinforced terracotta warrior under three types of seismic waves.

Under dynamic loading, the X-direction acceleration responses at the left shoulder of the Terracotta Warrior were extracted for different PGA conditions. The results show that, at a PGA of 0.18 g, the peak acceleration was 0.4856 g; at a PGA of 0.35 g, the peak acceleration increased to 1.4620 g; and at a PGA of 0.70 g, it reached 2.5637 g. These results reveal a significant nonlinear amplification effect, with the structural acceleration response markedly increasing as the seismic intensity rises. The acceleration curves are presented in Fig. 4b.

For the terracotta warrior structure reinforced with supports, the bracket material employed conventional Q235 structural steel with the following mechanical parameters: elastic modulus of 210 GPa, Poisson’s ratio of 0.3, density of 7.85 g/cm³, and yield strength of 500 MPa. Since the strength of Q235 steel significantly exceeds that of the terracotta material used in the Terracotta Warriors, rigid body constraints were applied to the Q235 structural components during this simulation.

To accurately simulate the actual contact behavior between the bracket and the Terracotta Warrior structure, the bracket was installed in the lower skirt region of the warrior model during assembly. The interfacial interaction between the two was simulated using a general surface-to-surface contact, where the stiffer upper surface of the bracket was defined as the master surface, and the inner surface of the warrior’s skirt was defined as the slave surface. The normal behavior of the contact was set to “hard” contact, allowing separation after tension, while the tangential behavior was modeled using a penalty friction formulation. Given that the Terracotta Warrior is made of terracotta material, a friction coefficient of 0.3 was adopted based on existing typical material test results³⁶. This value accounts for the complexity of the terracotta material’s physical properties and the difficulty in precisely calibrating the friction parameters. To further improve computational efficiency and convergence, the “small sliding” formulation was employed under the expectation of relatively small relative displacements. Additionally, fixed boundary conditions were applied at the base of the bracket to simulate the ground anchorage effect in practical engineering, and all other parameter settings were consistent with the static analysis conditions.

The finite element model of the reinforced structure is shown in Fig. 4c. Subsequently, the reinforced model was subjected to El Centro seismic excitations with PGAs of 0.18 g, 0.35 g, and 0.70 g. The X-direction acceleration time histories at the left shoulder were extracted for analysis, and the acceleration responses of the reinforced structure are shown in Fig. 4d. The simulation results indicate that when the input PGAs were 0.18 g, 0.35 g, and 0.70 g, the corresponding peak accelerations at the shoulder were 0.23 g, 0.70 g, and 1.21 g, respectively. A summary of the reinforcement performance is provided in Table 4. The simulation results further confirm that the acceleration amplification effect of the reinforced structure has been significantly suppressed. The proposed bracket design effectively reduces the structural acceleration response under seismic excitation, achieving a reinforcement efficiency of up to 52.28%. The specific results are shown in Table 2.

Table 2 FEA—before/after reinforcement (steel bracket)

In heritage conservation using support brackets, the contact surface between the object and the bracket often experiences complex stress concentrations. Improper reinforcement can easily lead to local damage or even structural instability of the object. To investigate the mechanical influence of brackets designed via the percentage optimization method on the contact surface, this study compares and analyzes the mechanical behavior of the contact surface in both reinforced and unreinforced Terracotta Warriors under self-weight and three levels of seismic loading. Figure 5 shows the stress distribution within the skirt region. Table 3 shows the stress statistics for the lower part of the skirt.

Fig. 5: Stress contours at the lower skirt region (bracket support area).

a Stress contour of unreinforced structure; b Stress contour of reinforced structure.

Table 3 Stress distribution in the skirt region

Data from Fig. 5 and Table 3 indicate a substantial reinforcement effect of the bracket on the warrior’s skirt region, achieving an average reinforcement efficiency of 41.45%. Specifically, under self-weight conditions, the stress concentration at the leg-skirt junction is fundamentally improved. A critical transformation in the load transfer path is observed: it has shifted from being primarily focused on the junction to being shared jointly by the skirt and the junction. This shift signifies a more homogeneous stress distribution. When subjected to seismic excitations of three different intensities, the maximum stress values in the lower skirt area show a systematic reduction. Further analysis reveals that although the point of maximum stress remains at the leg-skirt junction, the reinforced skirt structure exhibits greater stability, with a substantial decrease in the stress gradient caused by vibrations. This proves that the bracket effectively alleviates the overall structural tilting tendency induced by seismic loads.

Shaking table test of reinforced Terracotta Warrior

To evaluate the effectiveness of the reinforcement bracket, shaking table tests were conducted using two replica Terracotta Warriors³⁷. To monitor the mechanical response during testing, an accelerometer was mounted on the lateral surface of each replica’s left shoulder and precisely aligned with the laboratory coordinate system. The location is the same as the acceleration extraction point on the shoulder in the FEA. The accelerometer arrangement is shown in Fig. 6a.

Fig. 6: Shaking table test of the reinforced Terracotta Warrior.

a Configuration of acceleration sensors on the Terracotta Warrior; b post-test results showing no significant damage; c shoulder acceleration curve of the unreinforced Terracotta Warrior replica; d shoulder acceleration curve of the reinforced Terracotta Warrior replica.

After completing the experimental setup, El Centro seismic waves with peak ground accelerations (PGAs) of 0.18 g, 0.35 g, and 0.70 g were applied to the shaking table to analyze the structural dynamic response under different seismic intensities. Each excitation lasted for 30 s, with a 10-min interval between tests to allow the structure to recover and release internal stresses. As the seismic waves were introduced, the deformation, stability, and vibration behavior of the specimens were closely observed, and high-precision sensors were used to record data on tilt angle, acceleration response, and vibration mode. The X-direction acceleration responses at the shoulders of both the unreinforced and reinforced Terracotta Warriors are shown in Fig. 6c and d.

The main experimental findings are summarized as follows: At a peak ground acceleration (PGA) of 0.18 g, neither the reinforced nor the unreinforced specimens exhibited any visible displacement, indicating stable structural behavior. When subjected to a PGA of 0.35 g, the unreinforced specimen displayed slight swaying, whereas the reinforced specimen remained stable with no observable movement. At the highest PGA of 0.70 g, the unreinforced specimen experienced significant oscillation, while the reinforced specimen showed only minor displacement without any signs of instability. The reinforcement results under different peak ground accelerations (PGAs) of seismic waves are summarized in Table 4.

Table 4 Shaking-table—Reinforcement efficiency (experiment)

Post-test inspections confirmed that no structural damage occurred in either specimen under any of the applied seismic intensities. These results demonstrate that the replica Terracotta Warriors can endure seismic excitations up to 0.70 g, and that the proposed reinforcement bracket markedly improves seismic stability. The post-test condition of the specimens is illustrated in Fig. 6b.

Comparative analysis with finite element results reveals that the average deviation between numerical simulations and experimental data across the three seismic intensity levels is 16.69%, demonstrating reasonable consistency between the two methodologies. Shaking table tests further verify that the bracket structure achieves a 50.40% reduction effect on seismic amplification. The discrepancies observed between simulated and experimental data primarily stem from inherent systematic errors in physical testing, which become particularly pronounced under low-amplitude conditions. These errors specifically include: fixture misalignment, interfacial contact imperfections, and geometric distortions introduced during reverse modeling processes.

Feasibility analysis of aerospace materials for the structural reinforcement of Terracotta Warriors

Although traditional steel support structures possess good mechanical properties, they exhibit significant limitations in cultural heritage reinforcement applications: excessive material hardness, over-strength, and lack of transparency, which can easily cause surface abrasion to fragile artifacts and adversely affect their visual presentation³⁸.

In contrast, aerospace materials, as cutting-edge achievements in modern materials science, offer innovative solutions for cultural heritage conservation. Their characteristics—lightweight, high strength, ease of forming, and transparent or translucent properties—align well with the esthetic and structural requirements of heritage preservation^39,40,41.

Among them, polycarbonate (PC), as a high-performance engineering plastic, demonstrates significant potential in structural reinforcement of heritage objects⁴². This material combines excellent optical and mechanical properties. According to existing research, a comparison of the mechanical properties between polycarbonate and Q235 steel is shown in Table 5.

Table 5 Comparison of material properties between Q235 steel and polycarbonate

As can be seen from Table 5, polycarbonate has a light transmittance of nearly 90% or higher, with a visual appearance close to ordinary glass, far superior to opaque materials like steel. It can be used to design invisible or nearly transparent support structures, greatly reducing visual interference from the support framework during artifact display. Its density is only 15% that of steel, making PC-based support structures much lighter and facilitating transportation and installation. Although the tensile strength and elastic modulus of polycarbonate are much lower than those of Q235 steel, they still exceed those of the Terracotta Warrior body material. This indicates that while meeting reinforcement requirements, PC can effectively mitigate issues associated with excessive strength. In terms of hardness, PC material is only 12% that of Q235 steel, while its elongation at break is significantly higher. This characteristic enables PC to reduce scratching and damage to contact surfaces during support and, when bearing loads, to better cushion external impacts due to its lower hardness.

Based on the above advantages, applying PC material to the structural support of Terracotta Warrior replicas offers a promising technical pathway for enhancing both the structural safety and esthetic presentation of cultural relics.

To systematically evaluate the application value of polycarbonate (PC) in reinforcement brackets for standing Terracotta Warriors and assess the feasibility of using aerospace materials in cultural heritage conservation, this study employed finite element simulation methods to compare the reinforcement performance of PC brackets and traditional steel brackets.

Using the same initial structural model, a percentage-based optimization method was applied with the material volume constraint set to Rn = 5%. The economic efficiency index was used as the optimization convergence criterion, and structural performance was characterized by the displacement at the top section under identical loading conditions.

The results (Fig. 7a–e) show that the reinforcement process reached its peak at the 79th iteration. The study found that the optimized reinforcement configuration obtained using polycarbonate is highly consistent with that using traditional Q235 steel. Considering both structural load-bearing capacity and exhibition esthetics, the four-directional support design is equally suitable for direct application with PC material. The economic efficiency curve is shown in Fig. 7f.

Fig. 7: The optimization process, results and economic benefit curve of PC brackets.

a–e Sequential stages of the percentage-based optimization process for the PC bracket design; f Economic efficiency index curve obtained during the optimization.

Based on the optimization results, the bracket structure was designed in the form of a hollow elliptical plate with four-corner supports. Similarly, this study analyzed the structural response of the Terracotta Warrior reinforced with a PC bracket under different seismic excitations. Figure 8 presents the shoulder acceleration response curves of the reinforced structure under El Centro ground motions with peak ground accelerations of 0.18 g, 0.35 g, and 0.70 g.

Fig. 8: Polycarbonate support for reinforcing the shoulder acceleration of the Terracotta Warriors.

The acceleration time-history curves recorded at the shoulder of the Terracotta Warrior structure after reinforcement with PC brackets, subjected to peak ground accelerations (PGA) of 0.18g, 0.35g, and 0.70g.

The average Reinforcement efficiency of the Terracotta Warrior bracket made from polycarbonate is approximately 39.00%, which is 11.3% lower than that of the bracket made from Q235 steel. However, considering the material’s properties and exhibition requirements, the polycarbonate bracket can still meet the seismic protection needs during in-situ display of the Terracotta Warriors. At the same time, it offers improved esthetics and enhances the overall visitor experience. It is minimally invasive to the original fabric of the artifacts and shows strong potential for practical application. (Table 6).

Table 6 FEA—before/after reinforcement (PC bracket)

Given the lower strength of polycarbonate material compared to Q235 steel, this study conducted a mechanical analysis of the complete heritage structure incorporating the PC bracket and extracted the stress results of the PC bracket itself. The analysis results are presented in Fig. 9. The results indicate that under self-weight and vibration loads, the polycarbonate bracket remained intact, with stress concentrations primarily occurring at the junctions between the four support legs and the top hollow elliptical ring, as well as on the top annular plate region. These results demonstrate that the polycarbonate bracket can effectively provide structural support for the Terracotta Warrior. Combined with its advantages such as transparency and lightweight properties, it demonstrates promising application prospects. Detailed stress data are provided in Table 7.

Fig. 9: Overall mechanical analysis of the PC-reinforced Terracotta Warrior replica.

The stress contour plots (nephograms) of the PC bracket reinforcement under self-weight condition and under seismic loading with PGA = 0.18g, 0.35g, and 0.70g.

Table 7 Stress Condition under PC Bracket Reinforcement (at Bracket Location)

Discussion

This study proposes a topology optimization-based approach—specifically utilizing a percentage-based optimization method—for designing adaptive reinforcement supports for cultural heritage artifacts. The methodology begins with a systematic assessment of the mechanical properties of the artifacts through 3D laser scanning and FEA. Initial support regions are then defined for structurally vulnerable areas, and an iterative percentage-based optimization process is applied to derive a support configuration highly compatible with the mechanical behavior of the artifacts.

Using a replica of a Terracotta Warrior as a case study, reinforcement bracket design, numerical simulations of mechanical performance, and shaking table tests were conducted. Dynamic response analysis demonstrated that the designed support structure effectively suppressed the acceleration response of the replica under seismic excitation, achieving a reinforcement efficiency of 52.28% and significantly enhancing its anti-overturning capacity during earthquakes. In terms of contact stress, the maximum stress in the region where the bracket interfaces with the lower part of the warrior’s skirt was reduced by an average of 41.45%, with a more uniform stress distribution. This indicates that the optimized bracket successfully redistributes loads, mitigating the risk of tensile or compressive failure and micro-cracking at the skirt-leg junction due to stress concentration.

Shaking table tests further validated the reinforcement effect. Under excitation from three types of seismic waves, neither the reinforced nor the unreinforced replica exhibited visible structural damage. However, at a peak acceleration of 0.70 g, the unreinforced specimen showed noticeable swaying. By extracting acceleration data from the left shoulder of the replica parallel to the direction of seismic waves, it was observed that the reinforced structure exhibited a 50.40% reduction in acceleration amplification, indicating that the bracket effectively suppresses dynamic response under seismic loading and reduces the risk of damage and overturning—a finding consistent with FEA results.

According to the finite element simulation results, under the conditions of self-weight and PGAs of 0.18, 0.35, and 0.70, the stress generated in the Terracotta Warrior replica remains below the material’s yield strength (<2.9155 MPa), indicating that the structure will not experience failure under these working conditions. Furthermore, with the addition of the support structure, the stress in the Terracotta Warrior, particularly at the contact areas, remains consistently below the material’s yield strength, and the stress values under each condition are reduced compared to those without the support. This demonstrates that the introduction of the support does not cause damage to the Terracotta Warrior structure but instead provides effective protection. These numerical simulation findings have been well validated through shake table tests.

Following qualitative and quantitative evaluation of the bracket’s reinforcement performance, this study further explored the feasibility of applying aerospace materials in heritage conservation. Using polycarbonate (PC) as a representative material, a transparent bracket was designed for the Terracotta Warrior replica through the same percentage-based optimization process. Results show that the PC bracket maintains an average reinforcement efficiency of approximately 39%, while offering high light transmittance and suitable material hardness. This not only avoids potential surface abrasion caused by metal materials but also minimizes visual intrusion into exhibition esthetics, aligning with the “minimal intervention” principle in cultural heritage conservation.

Since the proposed method is fundamentally based on structural mechanics analysis for targeted reinforcement, it holds broad applicability for a wide range of heritage objects beyond Terracotta Warriors.

Limitations of the Study: Despite the encouraging results, this study has several limitations that should be addressed in future research. Firstly, due to the hollow and thin-walled nature of the Terracotta Warriors, it remains challenging to accurately determine their internal structural details using non-destructive testing methods. Consequently, the finite element models were constructed using shell elements with a uniform thickness (25–30 mm) based on archeological statistics, which may not fully capture the actual heterogeneity or internal defects of the original artifacts. This simplification could affect the accuracy of stress distribution predictions, particularly in regions with complex internal geometries.

Furthermore, this study relied on replica specimens manufactured from local clay. Although these replicas are highly consistent in key physical parameters, they may not fully replicate the material degradation and micro-cracking present in the original Terracotta Warriors after millennia of burial. The mechanical properties used in the simulations were derived from laboratory tests on these replicas and, therefore, may not fully represent the behavior of the ancient terracotta material under long-term environmental exposure.

Moreover, although the shaking table tests validated the effectiveness of the reinforcement, they involved simplified boundary conditions and a limited range of seismic inputs (using only El Centro waves). The influence of multi-directional seismic excitations, soil-structure interaction, and long-term dynamic loads was not considered, which may limit the generalizability of the findings to real-world heritage conservation scenarios.

Finally, although the polycarbonate (PC) bracket demonstrated good reinforcement efficiency and esthetic advantages, its long-term durability and aging behavior under actual exhibition conditions—such as UV exposure, temperature fluctuations, and mechanical fatigue—were not evaluated. Further environmental aging tests are necessary to assess its suitability for permanent heritage reinforcement.