Seismic Performance and Parameter Optimization of Traditional Chinese Timber Structure Reinforced with Friction Dampers

Abstract

To effectively enhance the seismic performance of traditional Chinese timber structures, this study proposes a reinforcement method utilizing friction dampers. Based on the working mechanism of friction dampers and the extended discrete element theory, an analytical model for timber structures equipped with these dampers was developed and validated through shake table tests. Subsequently, dynamic analyses were conducted to systematically evaluate the enhanced seismic energy dissipation capacity of the ancient timber structures by the reinforcement of friction dampers. The friction coefficient (μ), bolt pre-tension strain (ε), and action distance (l) were selected as key parameters. A multi-objective optimization function was constructed using the weighted sum method, enabling a multi-objective parameter optimization analysis for the friction dampers to identify the optimal parameter combination under specific conditions. The results indicate that the established extended discrete element model effectively simulates the dynamic characteristics of the structure. The installation of friction dampers significantly enhanced the structure’s energy dissipation capacity and substantially reduced the peak displacement. However, due to the initial stiffness introduced by the dampers, the lateral stiffness of the column frame increased markedly, leading to a significant amplification of the acceleration response, with a maximum increase in peak acceleration reaching 77%. The multi-objective optimization analysis revealed that with weighting coefficients λ_a = λ_b = 0.5, the optimal damper parameter combination is μ = 0.36, ε = 102 με, and l = 268 mm. Under these conditions, the structural displacement response decreased by 38.5%, while the acceleration response increased by 93.7%. It is noted that the derived optimal design solutions are pertinent to the specific structural typology and ground motions considered.

1. Introduction

Chinese ancient architecture boasts an exceptionally long history and profound cultural heritage. As a vital component of Chinese civilization, it embodies significant historical, cultural, and artistic value [1,2]. These structures extensively utilize wood as the primary building material, with timber accounting for over 50% of the material used in various building types. Statistics indicate that timber structures account for more than 70% of China’s ancient architectural heritage [1,2]. Renowned for their unique construction methods and structural systems, these buildings employ mortise-tenon joints between components [3,4,5], columns that are directly placed on stone bases [6,7], and incorporate design features such as bracket sets (dougong) [8,9,10] and large roofs, demonstrating exceptional seismic performance.

However, the distinctive material properties of wood make it susceptible to contraction, cracking, insect infestation, and decay under natural and human-induced factors [11]. These physical changes can significantly degrade the mechanical properties of the timber, consequently compromising the overall seismic performance of ancient timber structures. Numerous earthquake damage studies reveal that ancient timber structures are vulnerable to irreversible damage during seismic events, often suffering severe deterioration or even complete collapse under strong ground motions [12,13,14]. Zhang et al. [15] conducted shake table tests on a model of a single-story hall-style timber structure with a central bay. The results indicated insufficient lateral resistance of the timber frame, characterized by significant residual deformation and severe column frame inclination. Under strong seismic action, excessive rotational deformation of the mortise-tenon joints led to their failure, ultimately causing the structure to collapse.

The failure of mortise-tenon joints is the direct cause of collapse in ancient timber structures. Consequently, clarifying the mechanical behavior and failure mechanism of these joints is a prerequisite for effective reinforcement. In recent years, scholars have conducted extensive research on the mechanical mechanisms and seismic performance of mortise-tenon joints. Yang et al. [16] developed a nonlinear analytical model for mortise-tenon joints with gaps, based on embedding theory and the assumption of wood’s orthotropic nature. Chen et al. [17] investigated the seismic performance of both spatial and planar hoop-head mortise-tenon joints through cyclic loading tests. The experimental results demonstrated that the joints exhibit favorable seismic performance. All specimens failed within the joint region, exhibiting distinct failure modes depending on the joint configuration and tenon size. Lu et al. [18] derived a theoretical method for estimating the rotational behavior of mortise-tenon joints. Their work quantitatively analyzed the changing position of the rotation center and the influence of bending deformation. Based on this analysis, simplified formulas for calculating the equivalent elastic stiffness and peak bending moment of the joints were proposed. Ogawa et al. [19] derived and experimentally validated a theoretical estimation method for the mechanical performance of mortise-tenon joints using gap as a parameter. Its predicted moment-deformation angle relationship agrees well with experimental results. Numerical analysis further quantified the significant influence of joint gap size on mechanical properties, confirming that the gap is a key influencing factor. Tanahashi et al. [20,21,22,23] proposed the Elasto-plastic Pasternak Model (EPM) to quantify the embedment behavior of wood, and applied this framework to analyze the restoring force characteristics of crosspiece and T-type joints, establishing a unified method for evaluating the elasto-plastic performance of traditional timber joints.

Based on investigations into the performance of mortise-tenon joints, scholars have conducted extensive research on techniques to enhance the seismic performance of these joints, aiming to improve the overall seismic behavior of ancient timber structures. Gan et al. [24] utilized steel plates to reinforce mortise-tenon joints. Through pseudo-static tests and numerical simulations, the moment-rotation hysteresis curves, skeleton curves, energy dissipation capacity, and stiffness degradation curves were compared and analyzed to investigate the influence of the steel plate reinforcement on the joint’s seismic performance. Dai et al. [25] proposed using fan-shaped shear dampers to strengthen dovetail mortise-tenon joints. By employing low-cycle repeated load tests, the hysteretic response, skeleton curve, stiffness degradation curve, and equivalent viscous damping coefficient curve of the test models were examined. It was determined that the joint damper effectively controlled the joint pull-out length and enhanced energy dissipation, strength, and rotational stiffness. Yi et al. [26,27] applied viscoelastic dampers to reinforce damaged joints. The results indicated that after reinforcement, crack propagation was controlled, and the joint’s bearing capacity and energy dissipation performance were significantly improved. Additionally, the reinforced joints were found to exhibit smaller tenon pull-out lengths, reduced strength degradation, excellent load-bearing capacity, and sufficient energy dissipation capacity. Shimoyama et al. [28] developed a knee-brace friction damper for mid-rise timber frame buildings. Time history seismic response analysis was conducted to investigate the performance of the damper to improve the seismic performance of a four-story timber building. Then, a static cyclic loading test on a beam-column joint with/without a prototype friction damper was carried out to verify the effectiveness of the damper. It is found that the damper mitigates slip behavior and increases the equivalent viscous damping ratio and equivalent stiffness. To significantly enhance the energy dissipation capacity of mortise-tenon joints within their limited rotation range, Zhang et al. [29,30] developed a displacement-amplified rotational friction damper (DARFD), whose positive control effect on the structural dynamic response was validated through experiments and numerical analysis. In addition, other advanced passive control strategies, such as the negative stiffness-based KDamper and its extended versions [31,32], have shown promise in seismic retrofit of multi-story structures [33].

While the efficacy of the DARFD as a reinforcing device for the mortise-tenon joint has been preliminarily validated through component-level experiments in our previous work [29], its impact on the global seismic performance of a complete traditional timber structural system remains inadequately explored. Furthermore, a systematic methodology for optimizing the damper’s key parameters to balance conflicting structural response targets (e.g., displacement reduction versus acceleration control) is still lacking.

To address this gap, this study investigates the seismic performance and parameter optimization of traditional Chinese timber structures reinforced with friction dampers, building upon the experimental findings from Reference [29] and employing a combined approach of numerical simulation and theoretical analysis. An extended discrete element model for the ancient timber structure equipped with friction dampers is developed. The relative displacement and absolute acceleration responses of the structure, both with and without the dampers, are compared and analyzed to evaluate the enhancement in seismic performance attributable to the friction dampers. Subsequently, treating relative displacement and absolute acceleration as coupled optimization objectives, a multi-objective parameter optimization is conducted using the weighted sum method. This analysis aims to propose the optimal mechanical parameters for the dampers under different weighting combinations.

2. Performance and Mechanism of the Friction Damper

This section details the fundamental mechanics and seismic performance of the Displacement-Amplified Rotational Friction Damper (DARFD), and the working principle and force-displacement relationship of the damper were derived.

2.1. Construction of the Friction Damper

To significantly enhance the energy dissipation capacity of mortise-tenon joints within their safe rotation range, Reference [29] proposed a displacement-amplified rotational friction damper (DARFD) tailored to the characteristics of these joints, as illustrated in Figure 1a. This damper is primarily assembled from steel components, comprising connecting rods, intermediate connectors, gourd-shaped friction plates, and end connectors. All main connecting members are connected by Grade 8.8 M20 bolts, which also provide the clamping force necessary to generate normal pressure on the friction plates. The friction plates, pre-formed from ceramic fiber composite material, feature a unique “gourd” contour that allows them to be embedded into the holes of the connecting pieces. This design ensures that the shims rotate in phase with the connectors, preventing relative slip. Furthermore, the stacked plates significantly increase the effective friction area, enabling the damper to achieve high energy dissipation efficiency even under minor rotational displacements.

2.2. Performance and Working Mechanism of the Damper

The friction damper is installed at the mortise-tenon joint of the timber structure. When the joint undergoes rotational deformation due to external factors, the linkage mechanism drives the stacked friction plates to produce relative rotation, thereby achieving energy dissipation and seismic mitigation, as illustrated in Figure 1b. The uniquely designed gourd-shaped friction plates not only ensure in-phase rotation with the connecting members but also maintain a standard annular contact surface between the shims during the rotational process. This design facilitates subsequent theoretical analysis and engineering design. The calculation diagram for the resisting moment of the friction shims is shown in Figure 1c. Reference [29] derived the calculation formula for the damping moment (M) of the friction damper:

$$M_{\max }={\displaystyle {\int }_0^{2\mathsf{\pi }}{\displaystyle {\int }_{R_1}^{R_2}(n-1)\mu r}}\cdot \sigma r\left(dr\right)\left(d\alpha \right)={\displaystyle \frac{2\mu P(n-1)\left({R_2}^3-{R_1}^3\right)}{3\left({R_2}^2-{R_1}^2\right)}}$$

(1)

$$P=\sigma \cdot \pi \cdot (R_2^2-R_1^2)=\epsilon E{A}_{Bolt}$$

(2)

where σ represents the stress on the friction plate, μ is the friction coefficient of the friction plate, R₂ represents the radius of the outer ring of the annular friction contact surface, R₁ is the radius of the circular hole of the annular friction contact surface, and P denotes the pre-compression force provided by the bolt to the friction plate, n is the number of friction plates, ε is the bolt pretension strain, A_Bolt is the cross-sectional area of bolt.

When the intermediate connector drives the friction plates through a unit rotation, the resulting resisting moment is M_max, the axial force in the intermediate connector is F, and the moment relative to the mortise-tenon joint is M_D. From the static equilibrium condition, the following relationship is obtained:

$$F={\displaystyle \frac{M_{\max }}{l}}={\displaystyle \frac{2\mu P(n-1)(R_2^3-R_1^3)}{3l(R_2^2-R_1^2)}}$$

(3)

$$M_D=FL={\displaystyle \frac{2\mu P(n-1)(R_2^3-R_1^3)}{3(R_2^2-R_1^2)}}{\displaystyle \frac{L}{l}}$$

(4)

where L represents the vertical distance from the axis of the upper connector of the damper to the edge of the beam, and l denotes the distance from the axis of the bolt hole in the intermediate connector to the rotational center (effective action distance of the damper).

As expressed in Equation (4), the key parameters governing the performance of the friction damper are the friction coefficient μ, the bolt pre-tension strain ε, and the action distance l. Consequently, these parameters are identified as the critical design variables for the subsequent parameter optimization analysis.

To investigate the energy dissipation performance of the DARFD, Reference [29] designed and fabricated four sets of specimens by varying the friction plate materials and the bolt pre-tension strain, and pseudo-static tests were conducted. The test utilized displacement control, with the joint rotation serving as the primary control parameter. The details of each specimen, including their designations and corresponding control parameters, are summarized in

Table 1. Specimen number and control parameters.

Specimen	Material of the Friction Plate	Friction Coefficient μ	Pre-Tension Strain με
MC-1	Steel fiber composite material	0.40	150
MC-2			200
MC-3	Ceramic fiber composite material	0.50	150
MC-4			200

. The schematic of the experimental loading setup is illustrated in Figure 2.

Pseudo-static tests were conducted on the DARFDs, and the hysteresis curves under various parameters are shown in Figure 3. Overall, the hysteresis loops exhibit a parallelogram shape with full and plump characteristics, demonstrating notable symmetry. Clear slip segments are observed when unloading to zero during both positive and negative unloading phases, attributable to gaps present at the connections. Once the loading displacement reaches the yield displacement, the hysteresis curves stabilize and remain relatively steady. Furthermore, with a continued increase in the loading displacement, no significant degradation in the peak load is observed. A summary of the experimental results for the test specimens is provided in

Table 2. Test results of DARFDs.

Specimen	Positive Loading (M/kN m)		Error/%	Reverse Loading (M/kN m)		Error/%
	Calculation	Test		Calculation	Test
MC-1	0.911	0.953	4.41%	−0.911	−0.974	6.47%
MC-2	1.454	1.487	2.22%	−1.454	−1.539	5.52%
MC-3	1.361	1.355	0.44%	−1.361	−1.383	1.59%
MC-4	2.223	2.401	7.41%	−2.223	−2.361	5.84%

.

The hysteresis loop of the friction damper exhibits an overall parallelogram shape, which is full and plump with pronounced symmetry. To facilitate extended discrete element modeling and subsequent dynamic analysis, the restoring force model of the DARFD is simplified based on the results of quasi-static tests, as illustrated in Figure 4.

3. Establishment and Verification of the Dynamic Analysis Model

To evaluate the impact of friction damper reinforcement on the seismic performance of ancient timber structures, this study adopts the shake table test model from the literature [15] as a prototype. An extended discrete element model was developed using the timber structure analysis software Wallstat5112 [34]. The model accuracy was validated by comparing it with the shake table test results, following which dynamic analyses were conducted.

3.1. Shaking Table Model Parameters

Based on the dimensions and construction techniques for the palace-style timber structure using second-grade timber, as recorded in the Yingzao Fashi [35], the research group designed and fabricated a single-bay, single-story test model with a geometric scale ratio of 1:3.52. Unidirectional shake table tests were conducted on this model to quantitatively study its structural characteristics and dynamic response [15]. The primary structural material selected for the model was Russian red pine. The structure employed traditional dovetail joints for connections. A column-top lintel, with cross-sectional dimensions of 1600 mm × 180 mm × 75 mm, was adopted at the top of the columns. Its lower part was connected to the column heads using tenon joints, while its upper part interfaced with the bracket sets. The column bases utilized bluestone slabs and were anchored to the shake table platform via four anchor bolts. To accurately simulate the dead load effect of the roof, a concrete mass block measuring 2400 mm × 2400 mm × 250 mm was configured atop the structure. The detailed configuration of the model structure is illustrated in Figure 5.

3.2. Establishment of Analysis Model

The analytical model for the ancient palatial-style timber structure equipped with friction dampers is illustrated in Figure 6. In traditional timber structures, beams and columns are typically connected by semi-rigid mortise-tenon joints. These joints are capable of transmitting axial forces, shear forces, bending moments, and torques, while also undergoing corresponding deformations. Consequently, within the extended discrete element analysis framework, these mortise-tenon joints are commonly simplified using nonlinear tension-compression springs combined with nonlinear rotational springs [36]. Under seismic action, the bracket sets primarily exhibit shear deformation in the horizontal direction and compressive deformation in the vertical direction. Thus, the mechanical behavior can be characterized by a biaxial spring mechanism: a bilinear shear spring in the horizontal direction models their shear resistance, while tension and compression springs in the vertical direction simulate their load-bearing capacity [37]. Beam and column members are modeled as an “elasto-plastic rotational spring + rigid rod”. The deformational characteristics of the member are captured by elasto-plastic rotational springs in the ends. The initial stiffness and ultimate moment capacity of these rotational springs are calculated based on the specimen’s geometric parameters, as well as the wood’s modulus of elasticity and flexural strength. The components of the friction damper are connected by bolts, with energy dissipation occurring through the relative sliding between the friction plates. Based on the damper’s construction features, boundary conditions, and working mechanism, it can be idealized as a rotational spring. A schematic representation of these model simplifications is shown in Figure 6a.

To facilitate extended discrete element modeling and analysis, the damper was simplified as an equivalent tension-compression spring under the condition of maintaining the moment consistency of the damper relative to the mortise-tenon joint. An extended discrete element analysis model of the palace-style ancient timber structure equipped with friction dampers was established, as shown in Figure 6b. The restoring force model of the nonlinear spring used in the modeling process is shown in Figure 7. The research group conducted quasi-static tests on dovetail mortise-tenon joints at the column head to investigate the connection characteristics and stiffness of the joints. The joint stiffness was determined through theoretical and experimental methods [38]. The shear stiffness and vertical compression stiffness of the bracket set were obtained via axial compression tests and quasi-static tests [39]. The spring parameters for the mortise-tenon joints and the bracket set joints can be set according to the respective experimental results.

3.3. Analysis Model Comparison and Verification

• Comparison of Natural Frequency

During the experiment, impact loads were applied to the test model by manual hammering to induce free vibration. The free vibration signal was acquired from a column head in the northeast direction, and spectral analysis of this signal yielded a natural frequency of 2.05 Hz for the experimental model [40]. In this study, the natural frequency of the numerical model was determined using the white-noise excitation method. By inputting a randomly distributed white-noise excitation and observing the same measurement point, the natural frequency during the free vibration phase of the numerical simulation was calculated via Fourier transform to be 1.85 Hz. The relative error compared to the experimentally measured natural frequency is merely 9%, which falls within the acceptable engineering tolerance, effectively validating the reliability of the proposed numerical model.

2.

Validation of seismic response

To further verify the accuracy of the established discrete element analysis model, the same ground motion input method as used in the shake table test was adopted in the numerical simulation. The El Centro wave, Lanzhou wave, and Taft wave were selected as the excitation sources for dynamic analysis. A comparison of the structural responses between the numerical simulation and the experiment was conducted, as shown in Figure 8 and Figure 9. The results demonstrate that the responses obtained from the discrete element analysis correlate well with the measured results from the shake table tests, showing strong consistency in overall trends, patterns, and frequencies. This indicates that the proposed discrete element model for the ancient timber structure effectively simulates the dynamic response of the prototype structure under seismic action.

3.

Validation of power spectral density (PSD)

To complement the time-history analysis and provide a more comprehensive assessment of the model’s dynamic characteristics, a frequency-domain validation was performed. The Power Spectral Density (PSD) of both the displacement and acceleration responses was computed for the experimental and numerical models under the three ground motions (El Centro-0.05g, Lanzhou-0.05g, and Taft-0.05g), as presented in Figure 10 and Figure 11, respectively.

The PSD comparisons reveal a consistent agreement between the simulated and experimental results across all seismic inputs. Firstly, a dominant spectral peak is distinctly observed in the low-frequency range (approximately 0–5 Hz) in all plots, which corresponds to the fundamental vibration mode of the structure. The frequency of this primary peak aligns closely between the simulation and experiment, corroborating the natural frequency match established earlier and confirming the model’s accuracy in capturing the global stiffness and mass distribution. Beyond the fundamental peak, the PSD curves demonstrate excellent agreement in their overall shape and the distribution of vibrational energy across the frequency spectrum. It reinforces that the model faithfully reproduces the structure’s frequency content and dynamic response, thereby significantly increasing confidence in its predictive capability for the subsequent parametric and optimization analyses conducted in this study.

4. Dynamic Response Analysis

A comprehensive analysis of the seismic performance of the timber structure retrofitted with friction dampers. The dynamic responses—including inter-story displacement, absolute acceleration, and energy dissipation—of the retrofitted structure are systematically compared against those of the uncontrolled structure under a suite of ground motions.

4.1. Layout Scheme of DARFDs

Based on shaking table tests, the literature [6,40] revealed that the deformation of ancient timber structures under horizontal seismic action is primarily concentrated on the column frame layer. Therefore, reinforcement should be carried out at the mortise-tenon joints, as illustrated in Figure 6a. When retrofitting the structure with friction dampers, two configuration schemes can be adopted: dampers installed on a single side (unilateral configuration) and dampers installed symmetrically on both sides (symmetric configuration), as shown in Figure 12a and Figure 12b, respectively. When the dampers are arranged asymmetrically on a single side of the structure, the input of unidirectional ground motion will induce a torsional effect in the structure. The specific response is depicted in Figure 12c. The corresponding torsional rotation angle θ of the structure is given by Equation (5):

$$\theta ={\displaystyle \frac{{\Delta }_2-{\Delta }_1}{L}}$$

(5)

where θ represents the torsional angle of the structure (unit: rad); ∆₂ is the length of the large torsional side (unit: mm); ∆₁ is the length of the small torsional side (unit: mm); L represents the length of the structural model (unit: mm).

Identical ground motion inputs were applied to the numerical models with both unilateral and symmetrical damper configurations, enabling a direct comparison of the structural displacement responses under the different arrangement schemes, as detailed in Figure 13. The results indicate that when the dampers are installed unilaterally, a significant discrepancy in relative displacement is observed between measuring Point 1 and measuring Point 2. Based on the data from Figure 13a and calculations using Equation (5), the torsional rotation angle θ of the structure is determined to be 0.01 rad for the unilateral damper configuration, whereas it is 0 rad for the symmetrical configuration. Consequently, a symmetrical damper arrangement method is proposed. This method effectively mitigates the eccentricity between the center of mass and the center of rigidity by balancing the distribution of structural stiffness, thereby suppressing the torsional effect.

4.2. Reinforcement Efficiency Analysis

To systematically evaluate the reinforcement efficiency of the DARFDs in the ancient timber structure, this study comprehensively considered the spectral characteristics and intensity levels of ground motions. The El Centro, Lanzhou, and Taft ground motion records were selected as input excitations to construct multi-dimensional seismic analysis conditions and conduct dynamic analysis. By inputting these ground motion records into the numerical model, the structural displacement and acceleration responses were extracted for comparative analysis of the damper’s control effect.

4.2.1. Comparison of Displacement Response

The displacement responses of the model, both with and without the DARFDs, under different ground motions are compared in Figure 14. The result indicates that after the installation of the DARFDs at the mortise-tenon joints, the energy dissipation capacity of the structure is significantly enhanced, leading to a noticeable reduction in the peak displacement. To further quantify the controlling effect of the damper reinforcement on the structural displacement response, a comparison of the peak displacement responses before and after reinforcement is presented in

Table 3. Comparison of structural responses of unreinforced and reinforced.

Condition	Displacement			Absolute Acceleration
	UR/mm	R/mm	η/%	UR/Gal	R/Gal	η/%
Taft-0.05g	4.71	3.37	30%	35.71	55.46	−55%
El Centro-0.15g	12.73	7.57	41%	61.92	125.51	−77%
Lanzhou-0.3g	39.14	11.61	70%	103.82	185.3	−30%

Where η represents the reinforcement efficiency of the damper, η = −((R − UR)/UR) × 100%. A positive value indicates a reduction, while a negative value indicates an increase.

. Under seismic excitations of varying intensities, the maximum reduction in the peak displacement after strengthening reaches 70%. This demonstrates that the friction damper effectively restrains the relative displacement at the mortise-tenon joints through frictional energy dissipation, thereby reducing the overall structural deformation.

4.2.2. Comparison of Acceleration Response

The comparison of the absolute acceleration responses of the timber structure model before and after reinforcement with the DARFD is shown in Figure 15. It can be clearly observed that the acceleration response of the structure increased significantly after the installation of the DARFD at the mortise-tenon joints. This phenomenon is attributed to the increased lateral stiffness of the column frame layer, resulting from the initial stiffness of the damper, which alters the dynamic characteristics of the structure. To quantitatively analyze the impact of the damper reinforcement on the structural acceleration response, a comparison of the peak acceleration responses before and after reinforcement is presented in Table 3. The comparative data indicate that under seismic excitations of varying intensities, the peak absolute acceleration of the structure increased notably after the installation of the friction dampers, with a maximum increase of 77%. These results reveal that the DARFD enhances energy dissipation and reduces displacement (i.e., increased lateral stiffness and friction), concurrently amplifying the acceleration response. This conflict between deformation control and acceleration suppression forms the central design challenge addressed in the following parameter optimization chapter.

5. Dual-Objective Parameter Optimization Analysis

While effectively reducing the inter-story displacement, the friction damper also significantly increases the absolute acceleration (see Table 3). This indicates that displacement (D) and acceleration (A) are two conflicting design objectives. Therefore, this study establishes a multi-objective optimization framework aimed at finding the best balance between these competing targets. As identified in Section 2, the key factors influencing the damping force of the friction damper are the friction coefficient μ, the bolt pre-tension strain ε, and the damper action distance l. Therefore, these parameters are selected as the critical parameters for the subsequent optimization analysis, aiming to determine the optimal parameter combination that yields the damper’s best performance.

5.1. Selection and Definition of Optimization Variables

As established in Section 2, the key parameters influencing the damping force of the friction damper are the friction coefficient μ, the bolt pre-tension strain ε, and the damper action distance l. The friction plates are manufactured from a metal fiber composite material. An excessively low friction coefficient μ can compromise the energy dissipation capacity and displacement control effectiveness, whereas an excessively high μ may lead to interface overheating, accelerating wear, and degradation of the contact surfaces. Therefore, a balance must be struck between the seismic mitigation performance and the long-term durability of the damper. The normal contact force is generated by the bolt pre-tension, quantified by the bolt pre-tension strain ε. Research on the fatigue optimization of high-strength bolts indicates that increasing the pre-tension may reduce the fatigue resistance of bolts [41,42]. Furthermore, the damper’s action distance l determines the effect of displacement amplification. Based on this analysis, a correlation matrix concerning μ, ε, and l is constructed as follows:

$$x={\left[\mu ,\epsilon ,l\right]}^T\in R^3$$

(6)

Considering both the damper’s performance and design constraints, the feasible domain for the parameters μ, ε, and l is defined as follows:

$$\left\{\begin{array}{c}0.1\le \mu \le 0.4\\ 100\le \epsilon \le 200 \left(\mu \epsilon \right)\\ 100\le l\le 500 \left(mm\right)\end{array}\right.$$

(7)

5.2. Optimization Problem Definition and Mathematical Model Establishment

From the above analysis, the DARFD studied in this paper can effectively reduce the inter-story displacement of the ancient timber structure during reinforcement, but it also amplifies its absolute acceleration response. Therefore, this study establishes a multi-objective optimization framework. The peak relative displacement (D) and peak absolute acceleration (A) of the structure are taken as the optimization objectives to conduct multi-objective parameter optimization. This aims to effectively reduce displacement while limiting the increase in acceleration, thereby accurately identifying the optimal parameter combination for the friction damper. For solving multi-objective parameter optimization problems, methods include the constraint method, the analytic hierarchy process, and the linear weighting sum method. The linear weighting sum method is currently more widely applied [43,44]. Consequently, this paper will perform an optimization analysis for the damper’s optimal parameter combination using the weighting sum method.

The multi-objective optimization problem is defined as minimizing both the displacement response D and the acceleration response A, which are functions of the damper parameters:

$$\left\{\begin{array}{c}\min D\left(x\right)=f_1\left(\mu ,\epsilon ,l\right)\\ \min A\left(x\right)=f_2\left(\mu ,\epsilon ,l\right)\end{array}\right.$$

(8)

where D represents the peak of relative displacement of the structure (mm); A is the peak of absolute acceleration (m/s²); μ is the friction coefficient of the damper; ε is the pre-tension strain of the bolt (με); and l represents the action distance of the damper (mm).

To eliminate dimensional differences, the range method (min-max normalization) was employed to normalize the two objective parameters, mapping the objectives to the interval [0, 1]:

$$\widehat{A}={\displaystyle \frac{A}{A_0}} \widehat{D}={\displaystyle \frac{D}{D_0}}$$

(9)

To simplify the optimization problem, this study adopts a multi-objective reduction strategy based on the linear weighting method. By introducing weighting coefficients, a comprehensive objective function J(x) is constructed, which is then solved using single-objective optimization. The comprehensive objective function J(x) is defined as follows:

$$J\left(x\right)={\lambda }_a\widehat{D}+{\lambda }_b\widehat{A}$$

(10)

Among them, the comprehensive objective function must satisfy:

$${\lambda }_a+{\lambda }_b=1$$

(11)

$${\lambda }_a,{\lambda }_b\in \left[0,1\right]$$

(12)

Therefore, the optimization problem and the selection of control parameters can be expressed as:

$$F\mathrm{ind}({\mu }^{*},{\epsilon }^{*},l^{*})=\arg \min J(x), \mathrm{subject} \mathrm{to}:\mu \in (0.1,0.4),\epsilon \in (100,200),l\in (100,500)$$

(13)

5.3. Numerical Sampling and Optimal Selection

For multi-dimensional variables, Latin Hypercube Sampling (LHS) serves as an efficient stochastic sampling method. Accordingly, this study employs LHS to generate 300 combinations of the key parameters: the friction coefficient μ, the bolt pre-tension strain ε, and the damper action distance l. The generated parameter sets are subsequently incorporated into an extended discrete element model for dynamic analysis. The peak relative displacement (D) and peak absolute acceleration (A) under various working conditions are extracted to serve as the basis for parameter optimization. Owing to the scope limitations of this paper, the optimization analysis is conducted specifically for the parameters under the excitation of the Lanzhou ground motion record at an intensity level of 0.15 g.

By combining Equation (9) with Equation (10), the comprehensive target value formula can be obtained as:

$$J\left(x\right)={\lambda }_a{\displaystyle \frac{D}{D_0}}+{\lambda }_b{\displaystyle \frac{A}{A_0}}$$

(14)

where D₀ represents the relative displacement baseline value of the unreinforced model, taken here as 12.73 mm, and A₀ represents the absolute acceleration baseline value of the unreinforced model, taken here as 0.062 g. Furthermore, the weighting coefficients λ_a and λ_b are preliminarily assumed to be 0.5 each.

A systematic analysis of the initially generated 300 parameter combinations revealed the presence of partially overlapping or excessively proximate parameters within the sample space. To ensure the validity and reliability of the experimental data, 100 sets of typical parameter combinations with significant distinctions were selected as the research sample. The analysis using the weighted sum method yielded the comprehensive objective function values for the friction coefficient μ, bolt pre-tension strain ε, and damper action distance l under different combinations, as illustrated in Figure 16a. The color gradient distribution in the figure reveals the correlation patterns of the multi-objective optimization. Cool-toned colors indicate that the comprehensive objective function values converge towards the ideal threshold, corresponding to the optimal matching state between the displacement control effect and the acceleration penalty effect of the friction damper. In contrast, the increasing warm-toned trend reflects system responses deviating from the design objectives, suggesting limitations in parameter combinations that favor a single objective. It is noteworthy that the global Pareto front solutions, marked by red pentagrams, achieve a dynamic balance between relative displacement and absolute acceleration under the specific constraint condition of λ_a = 0.5 and λ_b = 0.5.

Based on the optimal parameter combination derived from the weighted sum method described above, its status as a Pareto front solution was verified by extracting the Pareto frontier, as specifically illustrated in Figure 16b. In the figure, the red dashed line represents the Pareto frontier. The green triangle denotes the parameter solution corresponding to the minimum displacement, the pink triangle indicates the parameter solution for the minimum acceleration, and the red five-pointed star the location of the weighted optimal solution. The verification against the Pareto frontier confirms that the selected optimal parameter combination lies on the Pareto frontier, thereby validating the accuracy of the optimal parameter solution.

As shown in

Table 4. Optimal parameter combination of DARFDs.

μ	ε/με	l/mm	D/mm	A/g	Displacement Reduction	Acceleration Increase
0.36	102	268	7.83	0.12	38.5%	93.7%

, when equal importance is assigned to displacement and acceleration responses (λ_a = λ_b = 0.5), the resulting balanced solution corresponds to parameters μ = 0.36, ε = 102 με, and l = 268 mm. Under this configuration, the structural displacement response is reduced by 38.5%, but the acceleration response is increased by 93.7%. This quantifies the performance trade-off outcome for the specified design preference.

This optimization method offers significant advantages for practical engineering applications, as it allows for the adjustment of weighting coefficients for different indicators based on actual requirements.

Table 5. The optimal solutions under different weighting coefficients.

λa	λb	μ	ε/με	l/mm	D/mm	A/g
0.2	0.8	0.11	121	442	10.01	0.08
0.3	0.7	0.11	121	442	10.01	0.08
0.4	0.6	0.15	133	486	9.57	0.09
0.5	0.5	0.36	102	268	7.83	0.12
0.6	0.4	0.39	162	102	5.32	0.18
0.7	0.3	0.39	162	102	5.32	0.18

presents a comparative analysis of the optimal solutions derived under different weighting coefficients. The results demonstrate that the design preference (expressed via the weighting coefficients) directly determines the final ‘optimal’ parameters. As greater emphasis is placed on displacement control (increasing λ_a), the optimization favors a higher friction coefficient and a smaller action distance, which leads to a greater reduction in displacement but also a more severe amplification of acceleration. This intuitively shows that engineers can select the most suitable design solution from this Pareto front based on specific project priorities (e.g., greater concern for structural deformation versus the acceleration safety of interior artifacts/decorations).

6. Conclusions

This study established an extended discrete element analysis model for an ancient timber structure equipped with friction dampers. The accuracy of the model was validated by comparing its results with shake table tests, followed by dynamic analyses to evaluate the effectiveness of the friction dampers. Furthermore, a multi-objective parameter optimization was conducted using the weighted-sum method. The main conclusions are as follows:

• An extended discrete element model integrating the friction dampers within a full palace-style timber structural system was developed. The results indicated a reasonable agreement with experimental data, with a computed natural frequency of 1.85 Hz (9% error compared to tests) and simulated responses showing consistent trends with measurements under various ground motions, supporting the use of the model for the subsequent parametric analysis within the defined scope.
• The installation of friction dampers significantly enhanced the structural energy dissipation capacity, reducing the peak displacement by up to 70%. However, the system-level analysis revealed a critical side effect: the added initial stiffness of the dampers markedly increased the lateral stiffness of the column frame, leading to a substantial amplification of the absolute acceleration response, with a maximum increase of 77%. This clarifies the inherent trade-off between displacement control and acceleration amplification associated with this retrofit strategy.
• A multi-objective parameter optimization framework was implemented. The peak relative displacement (D) and peak absolute acceleration (A) of the structure were selected as the optimization objectives. Under the condition of weighting coefficients λ_a = λ_b = 0.5, the optimal parameter combination for the damper, determined based on the comprehensive objective value, is μ = 0.36, ε = 102 με, and l = 268 mm. Under this specific set of conditions and objectives, the structural displacement response was reduced by 38.5%, while the acceleration response increased by 93.7%. This framework illustrates a principled approach to tailor damper design based on specific performance priorities.

This study focuses on a specific palace-style timber model and limited ground motions. Future research could extend the analysis to a wider range of structural typologies and seismic inputs, investigate the long-term durability of the dampers, and integrate local damage indices into a multi-level optimization framework.