In Situ Testing and Finite Element Analysis of a Discontinuous Mortise and Tenon Stone Bridge Under Natural Excitation

Abstract

To study the dynamic response of multi-span mortise and tenon stone bridges under natural excitation, a bluestone multi-span stone bridge with a main span of 2.56 m in southern China was taken as the research object. Based on the collected pulsating signals of bridge piers and slabs, the natural frequencies and damping ratios of the main span bridge slab and pier were analyzed using the half-power broadband method (HPBM) and random decrement technique (RDT). Modal analysis was conducted using ANSYS, and the results were compared with those obtained from on-site experiments for further performance analysis. The research results of this article indicate that the natural frequency range of the 2.56-m bridge slab identified by measured signals is 48–49 Hz, and the damping ratio range is 33.33–36.61%. The natural frequency of the central pier is 75–76 Hz, and the damping ratio range is 26.39–27.83%. Through finite element modal analysis, the natural frequency of the bridge slab is 54.401 Hz, with an error of 10.5%. The natural frequency of the overall stone bridge is about 82.2 Hz, with an error of about 8.2%. The validated finite element model was subjected to normal water flow impact and erosion simulation. The results indicate that under erosion with fewer particles and lower flow rates, the upstream pier bottom at the center receives the highest relative erosion mass and displacement per unit area. The bridge deck near the main span also experienced relative displacement. Therefore, in the subsequent protection work, special attention should be paid to these components.

1. Introduction

Traditional Chinese stone bridges are predominantly arch bridges, but several mortise and tenon stone beam bridges have been discovered in Jiangyong County, Yongzhou City, Hunan Province. The construction method for mortise and tenon structures is simpler compared to that of stone arch bridges, yet it offers greater load-bearing capacity and superior durability compared to wooden bridges. Currently, Jiang Xiangyuan and Yang Hongxin [1,2,3] have provided significant references to the historical and cultural background, structural forms, and construction periods of mortise and tenon stone bridges in southern Hunan. They believe that further studies on these bridges could address existing gaps in bridge history. This article focuses on a specific mortise and tenon stone bridge, validates the accuracy of the finite element model using measured data, and employs the model for erosion simulations, thereby contributing to the study of the dynamic characteristics of such bridges.

A mortise and tenon stone bridge is a system composed of multiple components, including piers, bridge slabs, and bridges, which exhibit complex responses due to the influence of natural excitation. Because most mortise and tenon stone bridges are ancient bridges, creating effective and accurate finite element models is crucial for proposing methods for protecting and restoring ancient stone bridges. In previous studies, most scholars have conducted extensive experimental and numerical research on brick and stone arch bridges, and some scholars have also proposed simplified modeling methods for multi-span arch bridges and used local (single-span or local few-span) methods to study the overall collapse or erosion behavior of arch bridges. For example, Tubaldi E et al. [4] modeled the masonry components (i.e., arch barrels and piers) of a multi-span masonry elevated bridge and verified that the modeling strategy adopted could accurately simulate experimental results. Finite element models were applied to analyze the influence of geometric features and mechanical parameters on the mechanical properties of the bridge body. Bi KM et al. [5] modeled and analyzed two spans of a collapsed multi-span simply supported bridge, and then explored the causes of the overall continuous collapse, proposing mitigation methods for multi-span bridge collapse. Rezaiguia A et al. [6] considered a multi-span, continuous, orthotropic, rectangular thin plate model of a three-span bridge as a linear rigid intermediate support, which is used to quickly and accurately determine the natural frequencies and modal shapes of orthotropic multi-span plates, and applied this method to study the dynamic interaction between bridges and vehicles. For example, Li HL et al. [7] conducted dynamic tests on continuous box girder bridges to determine the natural frequency and vibration response under subway train loads. By comparing numerical analyses with on-site measurements, they validated the dynamic model of the subway train–continuous girder bridge coupling system and utilized it for further dynamic analyses. Shi YL et al. [8] focused on the flat push bridge, validated the accuracy of the finite element model through comparison with measured data, and developed a flexible multibody dynamics model for this bridge type. Mao JX et al. [9] investigated the steel bridge tower of the Dashengguan Yangtze River Highway Bridge in Nanjing, China, using the random reduction method to identify modal parameters such as natural frequency and damping ratio, and analyzed how various environmental conditions affect these modal parameters based on empirical data. Some scholars have proposed evaluation methods for the overall mechanical performance of a bridge by conducting erosion analysis on the erosion models of bridge piers under different working conditions. For example, Majid P et al. [10] used the random field finite element method (SFEM) to evaluate the probabilistic safety of a historical bridge, and after determining the material parameters and loads, conducted static analysis on the bridge. Through mechanical performance analysis, the main stress and deflection distribution of the bridge were determined. Li YH et al. [11] have determined the degradation laws of materials under various erosion environments, conducted fluid erosion analysis and seismic vulnerability analysis of simply supported beam bridges, and proposed a damage assessment method based on the simply supported beam bridge model and bridge body. Xu JL et al. [12] compared and demonstrated the damage analysis of chloride ions on traditional integral cast-in-place bridge piers, and proposed that compared to integral bridge piers, the vulnerability of segmental bridge piers varies unevenly over time, especially the seismic resistance of the entire bridge pier deteriorates with material degradation.

In summary, the current analysis of bridge structures mainly focuses on modern bridges and brick and stone arch bridges, with limited performance analysis of mortise and tenon stone bridges. Therefore, based on existing laboratory conditions and literature research methods, this paper uses vibration signals of bridge slabs and piers under vertical excitation to compare the first-order natural frequencies of local and global finite element models and determine an accurate finite element model of ancient mortise and tenon stone bridges for the next step of erosion simulation. The research results provide valuable insights into the modeling methods and protection and restoration plans for such mortise and tenon stone bridges.

2. Structural Overview

The ancient stone bridge under study is situated in Jiangyong County, Yongzhou City, Hunan Province, China, as shown in Figure 1a. This bridge was constructed using techniques similar to those employed for the Shoulong Bridge and Hanlu Bridge, both also located in Jiangyong County, and built during the Song Dynasty (960–1279). It features multiple layers of bluestone joined with mortise and tenon joints, without any adhesive used for securing the bridge plate and piers. On-site measurements reveal that the bridge is approximately 8.5 m long, 1.3 m wide, and has piers approximately 1.8 m high, as depicted in Figure 1b,c. Due to prolonged environmental exposure, the bridge has experienced damage. Overall, the bridge’s age, structure, and materials exhibit notable representative characteristics. Thus, researching the dynamic characteristics and erosion of such stone bridges is of considerable importance.

3. Dynamic Characteristic Testing

3.1. Test Plan

To study the dynamic characteristics of the ancient stone bridge while minimizing human influence and potential damage to the structure, the natural excitation method was employed to collect pulsation signals from various sections of the bridge. Measurement points were established at the center of the bridge piers and plates on both sides, with a single sensor placed at each point to record the pulsation signals. A total of 12 sensors were used, and their placement is shown in Figure 2. In the experiment, the signal acquisition equipment includes the YZKJ-DIAK dynamic signal acquisition system and 991B accelerometer, as shown in Figure 3a,b, as well as a computer for processing the measured signals, as shown in Figure 3c. Data were collected during the afternoon periods when wind speeds were low and pedestrian traffic was minimal, with a sampling frequency of 2048 Hz and a testing duration of 5.5 min.

3.2. Time-Domain Analysis

Due to the use of mortise and tenon joints in the bridge slabs, there is no direct connection between the bridge slabs. Therefore, sensor 9 at the mid-span is selected to identify the local modal parameters of the stone bridge, and sensor 4, which approximates the overall natural frequency of the stone bridge is selected to identify the overall modal parameters of the stone bridge, and the positions of the red and blue dotted frames shown in Figure 2. Based on the frequency spectrum recognition results of vibration signals, after determining the true natural frequency, the original signal is subjected to signal preprocessing such as eliminating trend terms, bandpass filtering, and signal smoothing, among which the Butterworth filter is used for filtering. The passband setting of the filter is:

$$\left(f_1-0.08f_1,f_1+0.08f_1\right)$$

(1)

The 30-s pulsating signal under normal wind conditions was selected to create a filtered time history diagram, as shown in Figure 4.

3.3. Random Decrement Technique

3.3.1. RDT Algorithm Principle

The pulsating response $y\left(t\right)$ of a measuring point under any known excitation is shown in the following formula [13,14]:

$$y\left(t\right)=y_{0}D\left(t\right)+{\dot{y}}_{0}V\left(t\right)+{{\displaystyle \int }}_0^{t}h\left(t-\tau \right)f\left(\tau \right)d\tau$$

(2)

In the above formula:

$$D\left(t\right)={\displaystyle \frac{1}{\sqrt{1-{\xi }^2}}}{e}^{-\xi {\omega }_{0}t}\cos \left({\omega }_{d}t-\theta \right)$$

(3)

$$V\left(t\right)={\displaystyle \frac{1}{{\omega }_d}}{e}^{-\xi {\omega }_{0}t}sin\left({\omega }_{d}t\right)$$

(4)

$${\omega }_d=\sqrt{1-{\xi }^2}{\omega }_0$$

(5)

$$\theta =ta{n}^{-1}\left({\displaystyle \frac{\xi }{\sqrt{1-{\xi }^2}}}\right)$$

(6)

In the above formula, $y_0$ is the initial displacement of the system vibration, ${\dot{y}}_0$ is the initial velocity, $\xi$ is the damping ratio, and $\omega$ and ${\omega }_d$ are the circular frequencies of undamped and damped vibrations, respectively. $D\left(t\right)$ represents the free vibration response of the system with $y_0=1$ and $\dot{y_0}=0$. $h\left(t\right)$ is the unit impulse response function. $V\left(t\right)$ is the free vibration response of the system with $y_0=0$ and ${\dot{y}}_0=0$. $f\left(t\right)$ is an external incentive.

Setting a constant $x$ to intercept the signal $y\left(t\right)$ yields a series of different intersection times $t_i\left(i=1,2,\dots ,N\right)$. The response $y\left(t-t_i\right)$ starting at time $t_i$ can be expressed as:

$$y\left(t-t_i\right)=y\left(t_i\right)D\left(t-t_i\right)+\dot{y}\left(t_i\right)V\left(t-t_i\right)+{{\displaystyle \int }}_{t_i}^{t}h\left(t-\tau \right)f\left(\tau \right)d\tau$$

(7)

Moving a set of time starting points $t_i$ of $y\left(t-t_i\right)$ to the coordinate origin does not affect the randomness of the signal $y\left(t\right)$, resulting in a set of random subsample functions $x_i\left(t\right)\left(i=1,2,\dots ,N\right)$ with initial values A, which can be expressed as:

$$x_i\left(t\right)=AD\left(t\right)+\dot{y}\left(t_i\right)V\left(t\right)+{{\displaystyle \int }}_{t_i}^{t}h\left(t-\tau \right)f\left(\tau \right)d\tau$$

(8)

Due to the stationary random vibration with 0 mean values of $f\left(t\right)$, system vibration displacement $y\left(t\right)$, and velocity $\dot{y}\left(t\right)$ under natural excitation, the statistical average of $x_i\left(t\right)$ can be expressed as:

$$x\left(t\right)={\displaystyle \frac{1}{N}}{\displaystyle \sum }_{i=1}^N{x}_i\left(t\right)\approx E\left[AD\left(t\right)+\dot{y}\left(t_i\right)V\left(t\right)+{{\displaystyle \int }}_{t_i}^{t}h\left(t-\tau \right)f\left(\tau \right)d\tau \right]\approx AD\left(t\right)$$

(9)

The $x\left(t\right)$ obtained in the formula is the free vibration response with an initial displacement equal to x and an initial velocity equal to 0.

This article uses the random decrement method to identify the first-order natural frequency and damping ratio. The pulsating signals of sensors 9 and 4 are processed as follows: the signal amplitude is truncated to 1.5 times the standard deviation (S) of the input signal, and the amplitude interval is taken as $\tau$. The range of truncated amplitude $k_j$ is 1.0~2.0 times the standard deviation. Finally, the sub-signal segments are superimposed and averaged to obtain the random decrement signal [15]. By fitting the vibration attenuation curve, the damping ratio is obtained. The fitting formula is as follows:

$$y\left(t\right)=a·e^{-\zeta \omega t}sin\left(\omega t+\alpha \right)$$

(10)

In the formula, $a$ is the acceleration amplitude at $t=0$, that is, $a=y\left(0\right)sin\left(\alpha \right)$. $\zeta$ is the damping ratio, and $\omega$ is the circular frequency.

3.3.2. RDT Recognition Result

Based on the random decrement method, a program [16] was developed using MATLAB to obtain the attenuation vibration curve and determine the natural frequency $f$. Then, the circular frequency was calculated using the formula $\omega =2\pi f$. Finally, Formula (10) was used to determine the damping ratio. The results show that the first-order natural frequencies of the same bridge slab collected at different periods are 49.172 and 49.209 Hz, respectively, and the damping ratios are 35.32% and 36.11%, respectively. The first-order natural frequencies of the same bridge pier at different periods are 75.294 and 75.358 Hz, respectively, with damping ratios of 26.39% and 27.83%, respectively. The envelope curves and attenuation vibration curves of the same bridge slab and pier in different collected signals are shown in Figure 5 and Figure 6.

3.4. Half-Power Broadband Method

3.4.1. HPBM Algorithm Principle

The displacement frequency response function of a system with one degree of freedom is expressed as [17,18]:

$$H_d\left(\omega \right)={\displaystyle \frac{1}{m\left({\omega }_0^2-{\omega }^2+2j{\omega }_0\omega \right)}}$$

(11)

As the frequency response function is complex, taking this formula as an example, it can be expressed in the form of amplitude and phase as follows:

$$H_d\left(\omega \right)=\left|H_d\left(\omega \right)\right|e^{j\phi \left(\omega \right)}$$

(12)

Among them, the expressions for amplitude and phase are respectively represented as:

$$\left|{\omega }_d\left(\omega \right)\right|={\displaystyle \frac{1}{m\sqrt{{\left({\omega }_0^2-{\omega }^2\right)}^2+{\left(2\zeta {\omega }_0\omega \right)}^2}}}$$

(13)

$$\phi \left(\omega \right)=arctan\left({\displaystyle \frac{-2\zeta {\omega }_0\omega }{{\omega }_0^2-{\omega }^2}}\right)$$

(14)

To obtain the circular frequency ${\omega }_0^{max}$ corresponding to the peak value of the amplitude–frequency curve, the method of finding the extremum is used. The specific implementation method is as follows:

$${\displaystyle \frac{d\left|H\left(\omega \right)\right|}{d\omega }}=0$$

(15)

$$-\left({\omega }_0^2-{\omega }^2\right)+2{\zeta }^2{\omega }_0^2=0$$

(16)

From the above equation, it can be concluded that:

$${\omega }_0^{max}=\omega ={\omega }_0\sqrt{1-2{\zeta }^2}$$

(17)

Due to the relatively small damping calculated by Random Reduction Method (RDT), it can be considered that:

$${\omega }_0^{max}={\omega }_0$$

(18)

The frequency corresponding to the peak of the amplitude–frequency characteristic curve can be determined as the system’s natural frequency. The damping ratio of the system can be calculated by the power corresponding to the half power point. At the half power point, the frequency amplitude can be expressed as:

$${\displaystyle \frac{1}{m\sqrt{{\left({\omega }_0^2-{\omega }^2\right)}^2+{\left(2\zeta {\omega }_0\omega \right)}^2}}}={\displaystyle \frac{1}{\sqrt{2}}}{\left|H_d\left(\omega \right)\right|}_{max}={\displaystyle \frac{1}{\sqrt{2}}}{\displaystyle \frac{1}{2m\zeta {\omega }_0^2\sqrt{1-{\zeta }^2}}}$$

(19)

By solving Formula (19), two approximate solutions can be obtained:

$${\omega }_a\approx {\omega }_0\sqrt{1-2\zeta }\approx {\omega }_0\left(1-\zeta \right)$$

(20)

$${\omega }_b\approx {\omega }_0\sqrt{1+2\zeta }\approx {\omega }_0\left(1+\zeta \right)$$

(21)

By combining Formulas (20) and (21), we can obtain:

$$\zeta ={\displaystyle \frac{{\omega }_b-{\omega }_a}{2{\omega }_0}}$$

(22)

In a system with multiple degrees of freedom, any element $H_d^{pq}\left(\omega \right)$ of the displacement frequency response function matrix is represented as the excitation at position $q$, the unknown symbol ${\Phi }_{ir}$ is the $i^{th}$ component of the $r^{th}$ order mode vector, and the frequency response function of the displacement response at position $p$ is:

$$H_d^{pq}\left(\omega \right)={\displaystyle \sum }_{i=1}^N{\displaystyle \frac{{\varphi }_{pi}{\varphi }_{qi}}{M_i\left({\omega }_i^2-{\omega }^2+2j{\zeta }_i{\omega }_i\omega \right)}}$$

(23)

According to Formula (23), the natural frequency response function of a multi-degree-of-freedom system can be represented by the superposition of the frequency responses of multiple single-degree-of-freedom systems with natural frequencies equal to the original system’s natural frequency. Therefore, the single-degree-of-freedom system half-power broadband method can be used to identify the modal frequency and damping ratio of the multi-degree-of-freedom system.

3.4.2. HPBM Recognition Result

A program was developed using MATLAB R2022a based on the half-power broadband method to identify the pulsating signals of ancient stone bridge slabs and piers [16]. The results show that the first-order natural frequencies of the same bridge slab with different collected signals are 48.188 and 48.190 Hz, respectively, and the damping ratios are 33.33% and 33.33%, respectively. The first-order natural frequencies of the same bridge pier are 75.953 and 75.594 Hz, respectively, and the damping ratios are 27.78% and 27.78%, respectively. These findings are similar to the results obtained using the random decrement method described in Section 3.3, although the determined damping ratio is relatively large [19]. The comparison between these two methods is shown in Figure 7 and Figure 8.

4. Modal Analysis

4.1. Parameter Set-Up

To further study the mechanical properties of stone bridges, this paper considers the performance degradation of bluestone materials and establishes an ANSYS 2021R1 finite element model. Because this stone bridge is only connected by mortise and tenon structures, considering that the friction coefficient of limestone (bluestone belongs to the limestone category) measured in the literature [20] is in the range of 0.6–0.8 under different wet conditions, the indirect contact of the components of the overall stone bridge model is selected as “frictional” contact, and the finite element modal analysis under friction coefficients of 0.6 and 0.8 is analyzed. Through on-site research, the boundary conditions of the ancient stone bridge have been determined. Therefore, the red box in Figure 9 is fixed for selecting the bridge deck and pier bottom for modal analysis. The grid size is set to 50.00 mm, with a total of 34,616 elements. Considering the impact of long-term natural excitation and material aging on the mechanical properties of stone bridge structures, the physical performance parameters of bluestone [20] were adjusted using the coefficients specified in the current Chinese standard GB/T 39056-2020 Technical Code for Maintenance and Strengthening of Masonry Structures on Ancient Buildings [21]. Therefore, the elastic modulus of the stone is multiplied by a correction factor of 0.75. The physical performance parameters of the bluestone used are shown in

Table 1. Material mechanical performance parameters.

	Density (kg/m3)	Elastic Modulus (GPa)	Poisson’s Ratio
Bluestone	2610	39.6	0.22

.

4.2. Modal Analysis and Validation

The constraint condition of the bridge slab model is to constrain the displacement in the x, y, and z directions at one end, while only constraining the displacement in the z direction at the other end. Comparing the modal analysis results with the measured values, the results show that the first-order natural frequency is 54.401 Hz with an error of 10.5%. Considering that the material properties are only obtained after referencing the literature, the error is relatively large. In addition to fixing the bottom of the bridge piers, the overall constraint of the ancient stone bridge also fixes the bridge slab shown in Figure 9. The modal analysis results of the two friction coefficients [22] were compared with the measured values of the bridge pier. The results showed that: (1) When the friction coefficient was 0.6, the first-order natural frequency of the stone bridge model was 82.261 Hz, with an error of 8.16%. (2) When the friction coefficient is 0.8, the first-order natural frequency of the stone bridge model is 82.304 Hz, with an error of 8.20%. It can be seen that the modal analysis results of the stone bridge are less affected by the friction coefficient. Therefore, in the subsequent water flow impact, only the case where the friction coefficient is 0.6 was considered. Considering the material properties, the modeling of the stone bridge did not take into account local damage, resulting in significant errors. The vibration modes of the bridge slab are shown in Figure 10a, and the overall vibration modes of the stone bridge with different friction coefficients are shown in Figure 10b,c. Among them, the ratio of the mode participation coefficient to the first-order mode participation coefficient in the X direction is 1. The comparison between simulation results and actual measurement results is shown in

Table 2. Comparison of frequency recognition results.

		RDT (Hz)	HPBM (Hz)	Mean (Hz)	Simulation (Hz)	Relative Error (%)
Slab	Signal 1	49.172	48.188	48.690	54.401	10.5
	Signal 2	49.209	48.190
Pier	Signal 1	75.294	75.953	75.550	82.261	8.16
	Signal 2	75.358	75.594		82.304	8.20

. It should be noted that Figure 11 includes a fine line diagram of the original position, and the displacement present in the legend of the modal analysis results is not the actual vibration displacement of the bridge slab/stone bridge. The vibration mode diagram (i.e., relative displacement) of the bridge slab/stone bridge in the results is the focus of this article.

5. Fluid Simulation

5.1. Calculation of Relevant Parameters

Using FLUENT 2021R1 software to simulate and analyze the erosion of ancient stone bridge piers, this study explores the distribution of erosion under different working conditions and provides protection suggestions, providing a reference for future research on ancient stone bridges. This article uses the standard $k-\epsilon$ model to analyze the impact of water erosion. By setting the turbulence energy intensity and hydraulic diameter, the specific calculation formula is as follows:

$$I\cong 016{\left(R_e\right)}^{-1/8}$$

(24)

where $R_e$ is the Reynolds number, calculated as follows:

$$R_e={\displaystyle \frac{\rho vd}{\eta }}$$

(25)

In the formula, $\rho$ is the fluid density, $v$ is the fluid velocity, $d$ is the hydraulic diameter, and $\eta$ is the viscosity coefficient of the fluid.

5.2. Computation Model

ANSYS software was used to establish a numerical model and perform mesh partitioning. The three-dimensional numerical model is shown in Figure 11. The specific dimensions of the bridge piers are described in the previous text, and this article sets the distance between the bridge piers and the left and right sides, as well as the distance between the sand inflow inlet and the right side, to be 3 m. The boundary conditions for the bottom, left and right sides, and pier walls of the computational domain are WALL, and the DPM model attribute is “reflect” wall surface. The upper boundary condition of the computational domain is WALL, and the DPM model attribute is the “escape” wall. According to relevant literature [23,24,25], the injection method of sand particles is surface incidence, and the particles are inert particles (i.e., do not exchange heat with the flow field) with a density of 2650 kg/m³. The turbulence model adopts the Standard k-s model and calculates the Reynolds number and hydraulic diameter setting inlet surface through Section 5.1.

5.3. Erosion Simulation Results

To study the impact and erosion of normal water flow, based on the current situation and literature [25], a water depth of 1 m is designed, the sediment concentration of the river is 0.5 kg/m³, and the particle flow velocity and water flow velocity are both 1.2 m/s, with a particle size of 0.5 mm. Taking into account the gravity acceleration, the erosion rate calculation was carried out using the FINNIE model, and it was found that the highest erosion rate was 1.00 × 10⁻⁷ kg/m² at the center of the bottom of the bridge pier on the upstream side. The erosion rate was not significant at other wall positions. Considering the small number of particles in the calculation design and the particularity of the stone bridge position (there is a newly built bridge less than 20 m upstream), the erosion model of the stone bridge was less damaged. The cloud map of the erosion model of the stone bridge is shown in Figure 12.

Due to the positive impact of the water flow direction on the bridge body, the piers on the upstream side are subjected to high-speed water flow, resulting in greater static pressure. On the other hand, the piers on the downstream side are only subjected to relatively high static pressure on both sides due to the upstream side piers. The static pressure cloud diagram of the bridge body is shown in Figure 13a. By applying the obtained static pressure to the pier surface of the bridge body (as shown in Figure 13b for the static pressure application diagram), the overall displacement of the bridge body under the impact of normal flowing water is calculated as shown in Figure 13c. Although the displacement of the piers in the figure is relatively obvious, the maximum displacement is only 1.2843 × 10⁻⁶ m. The central upstream pier with displacement should be the key component for subsequent protection. Due to the fixed constraints between the two-end bridge plates and both banks, the displacement of the two-end bridge plates is the lowest, and the bridge plate near the main span should also be the focus of attention.

6. Conclusions

This study is based on the dynamic in-situ test results of a mortise and tenon stone bridge. The modal analysis results of local and global finite element models were compared using the random reduction method and the half-power broadband method. A finite element model of the stone bridge was proposed, and the influence of erosion on the mechanical properties of the stone bridge was further analyzed. The specific contributions are as follows:

(1)

The modal parameters of the main span slab and central pier of the mortise and tenon stone bridge were identified using the random reduction method and half-power broadband method. The results show that the natural frequency of the main span bridge slab is between 48–49 Hz, and the damping ratio is between 33.33% and 36.61%. The natural frequency of the central bridge pier is between 75–76 Hz, and the damping ratio is between 26.39% and 27.83%.

(2)

The main span bridge slab of the mortise and tenon stone bridge is set as a simply supported beam model. Through ANSYS finite element modal analysis, the first-order natural frequency of the main span bridge slab is obtained to be 54.401 Hz, with an error of 10.5% compared to the measured results. The natural frequency of the overall stone bridge is about 82.2 Hz, with an error of about 8.2%. Due to the overly ideal shape of the finite element model of the overall stone bridge and the fact that the identified results are for bridge piers, significant errors have occurred.

(3)

The validated finite element model was subjected to water flow impact and erosion simulation. The analysis results showed that the upstream pier located in the center was subjected to the maximum unit area erosion mass and static pressure, and the bridge slab near the main span also experienced displacement. Therefore, in the subsequent protection work, special attention should be paid to these components to provide a reference for the modeling method and protection work of such mortise and tenon stone bridges.