Point Estimation-Based Dynamic Reliability Analysis of Beam Bridges under Seismic Excitation Considering Uncertain Parameters

Abstract

Beam bridges, as the primary structural form of medium and small-sized bridges, are extensively utilized for road and railway crossings over rivers and valleys. Ensuring their reliability during earthquakes is crucial not only for maintaining traffic flow but also for mitigating the seismic impact on the economy and society. Considering earthquake intensity and uncertain parameters, this paper proposes an innovative method for assessing the seismic reliability of simply-supported beam bridges under three different levels of seismic design: minor, moderate, and major earthquakes. The proposed method first estimates the probability of encountering three typical earthquake intensities during the design life of simply-supported beam bridges based on crowd intensity, benchmark intensity, and major earthquake intensity. It then introduces uncertain parameters and employs the point estimation method to calculate the probability of bridge passage under specific earthquake intensities. Finally, it combines these earthquake intensities to calculate the overall seismic reliability of simply-supported beam bridges. The effectiveness and efficiency of this method are demonstrated through calculations for a three-span, double-degree-of-freedom simply-supported beam bridge, and validated using Monte Carlo simulations. This research provides solid theoretical support for seismic assessment, design, and intensity-based reliability analysis of simply-supported beam bridges.

1. Introduction

Bridges are integral engineering components of lifeline systems, spanning rivers, ravines, or other transportation routes for roads and railways. Accurate seismic analysis and seismic design of bridges is crucial. Due to the presence of numerous uncertainties in structural bridge design and the stochastic nature of seismic forces, employing a probability-based reliability design method to assess the safety of existing bridge seismic designs is an effective scientific approach to ensure the safety of bridge structures under seismic action. Globally, numerous studies have focused on developing methods to assess and enhance the seismic resilience of bridges. For instance, research from Japan, the United States, and Italy has significantly contributed to our understanding of seismic performance in bridge engineering, reflecting diverse seismic environments and structural practices. China has constructed a large number of reinforced concrete (RC) beam bridge structures in the past, with a significant portion of small and medium-sized bridge structures adopting simply-supported beam bridge forms. Conducting seismic response analyses of RC beam bridges and implementing reasonable seismic measures to ensure the safety of RC bridge structures under seismic actions is of the utmost importance.

In the evolution of seismic design for bridges, early methods relied on simplified static approaches [1,2]. Seismic assessment methods are broadly categorized into static and dynamic approaches. The advantages of static methods are their simplicity and relatively lower computational cost. They provide valuable insights into the ultimate strength and potential failure mechanisms of structures under seismic loads. However, static methods often fail to capture the full dynamic behavior of bridges, particularly under varying seismic intensities and complex loading conditions. Dynamic methods, on the other hand, involve time-history analysis, dynamic response simulations, and probabilistic seismic demand models. By taking into account earthquake loads’ time-dependent nature, these methods offer a more comprehensive understanding of structural responses to seismic events. Dynamic methods can account for varying seismic intensities and complex interactions between structural components. Nevertheless, they are computationally intensive and require detailed modeling and extensive computational resources. In the 1950s, dynamic methods using response spectrum theory [3] were developed, and in the past thirty years, dynamic time-history analysis methods have been adopted for important structures [4,5]. Many scholars have conducted bridge structural analyses based on time-history analysis [4,5,6,7,8] while considering the uncertain parameters of random vibrations. Regarding random vibration analysis, since statistical parameters provide fundamental descriptions of vibration behavior and help evaluate the reliability of structures under different vibration conditions, researchers often focus on the first two moments of structural or system responses. To evaluate structural stochastic responses that consider uncertain parameters and obtain statistical parameters, such as mean and standard deviation, the time-domain method [9] and frequency-domain analysis [10] can be used. In time-domain analysis, statistical parameters are obtained by counting all the response results obtained by multiple time-dependent inputs acting on the structures. In frequency domain analysis, mean and standard deviation are directly derived from the power spectral density function or Fourier transform of the autocorrelation function, providing a straightforward assessment of statistical properties without statistical analysis of time-domain data. Some scholars have adopted the direct integration method [11] and Monte Carlo simulation (MCS) [12] in linear single degrees of freedom. However, when the dynamic system is complex, traditional methods face significant challenges. These challenges include the need for extensive and intricate analyses, such as eigenvalue analysis, which are computationally intensive and difficult to apply repeatedly. These methods often require simplified assumptions that may not accurately capture the behavior of complex systems under seismic excitation. First-order approximation [13,14] of the mean response for uncertain linear systems obtained by applying Taylor series expansion [15] is treated to equivalent uncertain stochastic response. Nevertheless, the result is not ideal when this method is applied to nonlinear systems. Utilizing point estimation methods [16,17,18] is another effective approach to computing the statistical moments of structural responses. Zhao et al. [19,20] proposed a point estimation method to evaluate random responses and the statistical moments of structural maximum responses. Although there has been extensive research on the random seismic response and seismic design of RC bridge structures, further exploration of their seismic reliability is still needed.

On the other hand, first-passage probability theory [21,22,23,24] is often used for seismic analysis of bridges. First-passage probability describes the probability of a stochastic process exceeding a prescribed threshold for an interval of time. In practice, first-passage probability is used in bridge structures. This study defines first-passage probability as the probability of a bridge exceeding a certain threshold of displacement under seismic excitation. This measure captures the central tendency of the bridge’s response to seismic loading and provides a robust assessment of its reliability. A stochastic function spectral representation model and high-order moment method were developed by Zhang et al. [25], who proposed a seismic reliability analysis method for bridge piers based on the extreme values of structural responses’ first four moments. Chen et al. [26] conducted a seismic performance analysis of bridges from a probabilistic perspective to characterize the impact of structural uncertainties and associated stochastic parameters on bridge structures’ seismic performance. A time-varying seismic reliability assessment method for isolated bridges considering long-term aging degradation of lead rubber bearings was proposed [27]. A method using machine learning techniques to compute the seismic reliability of bridge networks was proposed [28]. Li et al. [29] reconstructed a probability density function of bridge seismic demands under both unconditional and conditional probabilities using a newly proposed three-point estimation method for parameterized probability distributions. It is worth noting that to obtain seismic reliability for bridges, MCS methods [30,31,32] can be employed. However, a low failure probability for large structures implies that MCS requires a large number of samples, resulting in incalculable computational costs or even impracticality. Therefore, the MCS method generally serves as a calibration or validation method for other approaches.

Recent literature indicates significant achievements in bridge seismic research, with considerable progress in seismic reliability analysis. For instance, seismic reliability analyses of bridges have progressed based on first-passage failure criteria [33,34]. In past structural seismic reliability analyses, the probability of seismic intensity occurrence [35,36,37] in seismic design zones was often overlooked, causing misalignment with actual seismic resistance and affecting analysis results accuracy. Currently, there is limited research on seismic reliability analysis methods for bridge structures considering seismic intensity and uncertain parameters. Therefore, it is essential to fully consider seismic intensity and uncertain system parameters in seismic reliability analyses of beam bridge structures. The novelty of this study lies in its integration of point estimation methods with dynamic reliability analysis under seismic excitation, considering uncertain parameters. Unlike traditional approaches that often rely on static analyses or simplified dynamic models, our method incorporates the inherent uncertainties in seismic loading and structural response, providing a more accurate and reliable assessment of seismic reliability for simply-supported beam bridges. This approach addresses the limitations of previous studies by offering a comprehensive framework that combines probabilistic and deterministic analyses.

In recent years, several methodologies [38,39,40] have been developed to assess the seismic resilience of bridges. Forcellini [38] discussed the geotechnical seismic isolation technique for several bridge configurations. Argyroudis and Mitoulis [39] developed novel fragility models for hydraulically induced stressors and combinations of hydraulic and seismic hazards. Akiyama et al. [40] discussed issues related to life cycle analysis, design, risk, resilience, and management of bridges during earthquakes. These issues include performance-based seismic design (PBSD), fragility analysis, and resilience-based design (RBD). PBSD focuses on designing bridges for specific performance objectives at different levels of seismic intensity. Fragility analysis estimates the probability of reaching or exceeding various damage states given seismic intensity measures. RBD integrates resilience concepts into design processes, emphasizing rapid recovery and functionality post-earthquake. Our proposed method builds upon these approaches by incorporating point estimation methods to dynamically assess seismic reliability, considering both the inherent uncertainties in seismic loading and the probabilistic nature of structural response.

Based on this premise, a seismic reliability analysis method for bridge structures was introduced in this study to incorporate seismic intensity and uncertain parameters. Initially, the probabilities of three representative seismic intensities for RC bridge structures during the design reference period were estimated using median intensity, basic intensity, and major seismic intensity. Subsequently, uncertain parameters were incorporated, and point estimation methods were applied to compute the first-passage probability associated with a given seismic intensity. Finally, using seismic intensity, the foundational formula for seismic reliability determines the probability of failure and the reliability index of bridge structures during earthquakes. First-passage probability was computed by integrating conditional failure probabilities across a spectrum of seismic intensities, each weighted by its occurrence probability, reflecting seismic events’ inherent uncertainties. Through a detailed exposition of the calculation process using a nonlinear single-degree-freedom (SDOF) bridge structure and a three-span, two-degree-of-freedom simply-supported beam bridge as a case study, the accuracy and efficiency of the proposed method were validated via MCS modeling.

2. Beam Bridge’s Stochastic Seismic Response Analysis

Beam bridges (as shown in Figure 1) can be regarded as multi-degree-of-freedom structures. Their dynamic responses can be obtained by solving the system’s dynamic equations. For a multi-degree-of-freedom linear structure, its motion equations can be expressed as [15,18]:

$$M\ddot{X}\left(t\right)+C\dot{X}\left(t\right)+KX\left(t\right)=f\left(t\right)$$

(1)

where M represents the mass matrix; C represents the damping matrix; K represents the stiffness matrix; $\dot{X}\left(t\right)$ represents the acceleration of displacement; $\dot{X}\left(t\right)$ represents the velocity of displacement; X(t) represents displacement; f(t) represents the dynamic load. When f(t) is a stochastic process, X(t) is also a stochastic process. Using the modal superposition method, the displacement response of the system X(t) can be represented as the sum of individual modes:

$$X\left(t\right)={\displaystyle {\sum }_{i=1}^n{\varphi }_i{q}_i\left(t\right)}$$

(2)

where $q_i\left(t\right)$ represents the generalized coordinates of the i-th mode, which indicates the vibration amplitude of the i-th mode. The autocorrelation function Ru(τ) of the displacement response X(t) is defined as:

$$R_u\left(\tau \right)=E\left[X(t)X(t+\tau )\right]$$

(3)

where E[ ] represents the expected value and $\tau$ is time delay.

Figure 1. Schematic diagram of a beam bridge.

The autocorrelation function and power spectral density function form a Fourier transform pair. The power spectral density function can be expressed as [15,18]:

$$S\left(\omega \right)={\displaystyle {\int }_{-\infty }^{\infty }R\left(\tau \right)}{e}^{-i2\pi \omega \tau }d\tau$$

(4)

For a single-degree-of-freedom system, the transfer function can be expressed as:

$$H\left(\omega \right)=\frac{1}{-m{\omega }^2+ic\omega +k}$$

(5)

where m is mass; c is damping; k is stiffness; i is the imaginary unit.

For a linear system, the power spectral density function $S_u\left(\omega \right)$ of the displacement response can be calculated using the input power spectral density function $S\left(\omega \right)$ and the transfer function $H\left(\omega \right)$:

$$S_u\left(\omega \right)={\left|H\left(\omega \right)\right|}^{2}S\left(\omega \right)$$

(6)

By integrating the power spectral density function of the response over all frequencies, the root mean square value of the displacement response can be obtained:

$$Var\left(X\right)={\sigma }_x^2={\displaystyle {\int }_{-\infty }^{\infty }{S}_u\left(\omega \right)}d\omega$$

(7)

where ${\sigma }_x^2$ is the root mean square value of the displacement response; ${\sigma }_x$ is the root mean square or standard deviation for zero-mean displacement response. Calculating the root mean square value of the velocity response follows similar steps, except when computing the transfer function where the frequency ω must be multiplied by i, for example:

$$H\left(\omega \right)=\frac{i\omega }{-m{\omega }^2+ic\omega +k}$$

(8)

3. Seismic Reliability Considering Uncertain Parameters and Probability of Intensity Occurrence

A seismic structure’s probability of failure PF [20] during its service life is as follows:

$$P_F={\displaystyle \sum _j{P}_{fj}}\left(R<S|I_j\right)P\left(I_j\right)$$

(9)

where $P\left(I_j\right)$ represents the probability of intensity occurrence in the region during the service life of the structure, as obtained from seismic hazard analysis methods. $P_{fj}$$\left(R<S|I_j\right)$ represents the conditional probability of structural failure under the intensity of seismic action.

3.1. Probability Distribution of Maximum Seismic Intensity in the Design Reference Period

It is generally assumed that the probability model of seismic intensity follows the Type III Extreme Value distribution. According to its distribution function, the probability distribution of the maximum seismic intensity within the 50-year design reference period is expressed as [41]:

$$P_I\left(i\right)=\exp \left[-{\left(\frac{12-i}{12-I_s}\right)}^k\right]$$

(10)

where I_s = I₀ − 1.55 is the median intensity (frequently occurring small earthquake intensity); I₀ is the basic intensity (design intensity); k is the shape parameter, as determined by the passage probability of the basic intensity I₀ over the 50-year design reference period at 10%. For design intensities ranging from 5 to 9 degrees, the shape coefficient k is shown in Table 1.

Table 1. Shape parameter k for different seismic design intensities.

Design Intensity I0	5	6	7	8	9
Shape coefficient k	11.25	9.79	8.33	6.87	5.40

The probabilities of the three representative seismic intensities I_s, I₀, and I₁ occurring within the design reference period are as follows [42]:

$$P\left(I_s\right)=P_I\left(I_0-0.77\right)$$

(11)

$$P\left(I_0\right)=P_I\left(I_0+0.5\right)-P_I\left(I_0-0.77\right)$$

(12)

$$P\left(I_1\right)=1-P_I\left(I_0+0.5\right)$$

(13)

Table 2 shows the probability of seismic intensity occurrence under various seismic design intensity levels, calculated based on the above equations.

Table 2. Probability of seismic intensity occurrence.

Basic Seismic Design Intensity	P(Is)	P(I0)	P(I1)
5	0.7111	0.2441	0.0447
6	0.7090	0.2470	0.0440
7	0.7063	0.2508	0.0429
8	0.7024	0.2564	0.0412
9	0.6961	0.2652	0.0386

3.2. Conditional Failure Probability Considering Parameter Uncertainty under Seismic Intensity

The first-passage failure theory is a widely used method in seismic reliability analysis. It defines the probability of the structure’s stochastic dynamic response process X(t), not exceeding the limit value x within a specified time T. Therefore, the passage probability $P_{fj}$$\left(R<S|I_j\right)$ under seismic intensity is expressed as:

$$P_{fj}\left(R<S|I_j\right)=P_f=1-\exp \left\{-\frac{1}{2\pi }\frac{{\dot{\sigma }}_x}{{\sigma }_x}T\exp \left[-\frac{1}{2}{\left(\frac{x}{{\sigma }_x}\right)}^2\right]\right\}$$

(14)

where, ${\sigma }_x$, ${\dot{\sigma }}_x$ represent the standard deviations of the zero-mean stochastic process X(t) and its derivative process, respectively. When considering a single-degree-of-freedom structure, their relationship with the undamped natural frequency ω₀ is provided by ${\omega }_0={\sigma }_x/{\dot{\sigma }}_x$.

In calculating the passage probability in Equation (14), deterministic parameter scenarios are often considered. However, in practical engineering, there are many uncertain parameters, such as random excitations and structural parameters like elastic modulus, moment of inertia, etc. Thus, when calculating the conditional failure probability P_fj of structural failure under seismic intensity, there are many uncertain random variables. In this case, assuming the joint probability density function of these random variables is f(X), the passage probability under seismic intensity considering uncertain parameters is expressed as:

$$P_{fj}\left(R<S|I_j\right)={\displaystyle {\int }_X{P}_f\left(X\right)}f\left(X\right)dX=\mathrm{E}\left[P_f\left(X\right)\right]$$

(15)

3.3. Point Estimation Method for Calculating the Mean Passage Probability

In Equation (15), the passage probability under seismic intensity involves computing the mean failure probability. To obtain the results of Equation (15), a two-dimensional reduction point estimation method was used in this study [22]. Equation (15) can be rewritten as:

$$P_{fj}\left(R<S|I_j\right)=\mathrm{E}\left[P_f\left(X\right)\right]=\mathrm{E}\left[P_f\left[T^{-1}\left(U\right)\right]\right]\cong {\displaystyle \sum _{i<j}{\mu }_{P_{fij}}}-\left(n-2\right){\displaystyle \sum _{i=1}^n{\mu }_{P_{fi}}}+\frac{(n-1)(n-2)}{2}{\mu }_{P_{f0}}$$

(16)

where $T^{-1}(\cdot )$ represents a function that transforms the random variables of the original space into standard normal space random variables through the Rosenblatt transformation [23]; ${\mu }_{P_{fi}}$ represents the failure probability corresponding to all random variables taking their mean values ${\mu }_i^0=(i=1,\dots ,n)$ from Equation (14); ${\mu }_{P_{fij}}$ and ${\mu }_{P_{fi}}$ represent the mean value of P_fij involving bivariate (u_i, u_j) and P_fi involving (u_i), respectively, which can be directly and quickly calculated using point estimation in standard normal space [17].

$${\mu }_{P_{fi}}={\displaystyle {\int }_{-\infty }^{\infty }{P}_f\left[{\mu }_1^0,\dots ,T^{-1}\left(u_i\right),\dots ,{\mu }_n^0\right]}\varphi \left(u_i\right)d{u}_i={\displaystyle \sum _{r_1=1}^m{P}_{r1}{P}_f}\left[{\mu }_1^0,..,T^{-1}\left(u_{i,r1}\right),\dots ,{\mu }_n^0\right]$$

(17)

$$\begin{array}{c}{\mu }_{P_{fij}}={\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{P}_f\left[{\mu }_1^0,\dots ,T^{-1}\left(u_i\right),\dots ,T^{-1}\left(u_j\right),\dots ,{\mu }_n^0\right]}}\varphi \left(u_i\right)\varphi \left(u_j\right)d{u}_{i}d{u}_j\\ ={\displaystyle \sum _{r_1=1}^m{\displaystyle \sum _{r2=1}^m{P}_{r1}{P}_{r2}{P}_f}}\left[{\mu }_1^0,..,T^{-1}\left(u_{i,r1}\right),\dots ,T^{-1}\left(u_{j,r2}\right),\dots ,{\mu }_n^0\right]\end{array}$$

(18)

where $u_{i,r1}$ represents the r₁-th estimation point of the i-th random variable in the standard normal space, and $u_{j,r2}$ represents the r₂-th estimation point of the j-th random variable in the standard normal space. For a seven point estimate in standard normal space of a random variable: μ₀ = 0, P₀ = 16/35; μ₁₊ = −μ₁₋ = 1.1544054, P₁ = 0.2401233; μ₂₊ = −μ₂₋ = 2.3667594, P₂ = 3.07571 × 10⁻²; μ₃₊ = −μ₃₋ = 3.7504397, P₂ = 5.48269 × 10⁻⁴.

Therefore, by substituting Equations (16)–(18) into Equation (9), and considering the probabilities P(I_S), P(I₀), P(I₁) of the three representative seismic intensities I_s, I₀, I₁ occurring within the design reference period, the failure probability P_F based on seismic intensity can be obtained. The seismic reliability based on seismic intensity can then be calculated. According to the relationship between the failure probability PF and the reliability index, the reliability index ${\beta }_l$ based on seismic intensity can be expressed as:

$${\beta }_l={\Phi }^{-1}\left(1-P_F\right)$$

(19)

where ${\Phi }^{-1}(\cdot )$ is the inverse cumulative distribution function of the standard normal distribution.

Calculating the passage probability under seismic intensity involves computing the mean failure probability since it represents the likelihood that the bridge will exceed a certain threshold of displacement or damage during an earthquake. This probability is calculated by integrating the conditional failure probabilities over a range of possible seismic intensities, weighted by their respective occurrence probabilities. By focusing on the mean failure probability, we capture the central tendency of the bridge’s response to seismic loading, providing a robust measure of its reliability. This approach aligns with probabilistic seismic demand models, which emphasize the importance of mean values in reliability assessments.

4. Proposed Seismic Reliability Analysis Framework for Beam Bridges

Based on the analyses in Section 2 and Section 3, the steps and calculation process diagram for the seismic reliability analysis of simply-supported beam bridges considering seismic intensity and uncertain parameters are as follows:

(1)

Simplify beam bridge structures into multi-degree-of-freedom linear structures and calculate the root mean square of the structural displacement response and velocity response based on motion equations and frequency domain analyses in Section 2;

(2)

Considering structural uncertainty parameters and three representative seismic intensities I_j (median intensity I_s, basic intensity I₀, major seismic intensity I₁), substitute the root mean square of the displacement response and velocity response into Equation (14) and obtain the passage probability $P_{fj}$$\left(R<S|I_j\right)$ under structural parameter uncertainties according to Equations (16)–(18);

(3)

Calculate the probabilities (I_s), (I₀), and (I₁) for the three representative seismic intensities I_s, I₀, and I₁ according to Equations (11)–(13);

(4)

Calculate the probability of failure for seismic structures considering both structural parameters and seismic intensity within their service life using Equation (9), which can be expanded as: $P_F=P\left(I_s\right)P_{fs}\left(R<S/I_s\right)+P\left(I_0\right)P_{f0}\left(R<S/I_0\right)+P\left(I_1\right)P_{f1}\left(R<S/I_1\right)$;

(5)

Calculate the reliability index ${\beta }_l$ based on seismic intensity using the relationship between failure probability and the reliability index, as shown in Equation (19).

Figure 2 presents a flow chart of the proposed methodology. The process begins with estimating the probabilities of different seismic intensities (Step 1) followed by incorporating uncertain parameters (Step 2) and applying the point estimation method to calculate the passage probability under specific seismic intensities (Step 3). Finally, overall seismic reliability is determined by integrating these probabilities (Step 4). Each step is elaborated on in the following sections, providing detailed explanations and calculations.

Figure 2. Proposed seismic reliability analysis process.

5. Numerical Examples

5.1. A SDOF Bridge Structure Subject to Gaussian White Noise

Considering the example of a nonlinear SDOF (as shown in Figure 3) bridge structure under Gaussian white noise, its intensity S₀, the duration T, the natural frequency ω₀, non-linear parameter ε, and damping ratio ξ are uncertain. Statistical information is listed in Table 3. The basic intensity is assumed to be 8 degrees.

Figure 3. A nonlinear SDOF system.

Table 3. Statistical information on uncertainty parameters in Example 1.

Parameters	Distributions	Mean	COV	Units
T	Lognormal	10	0.3	s
S0	Extreme Value Type II Distribution	0.25	0.6	m2/s3
ε	Lognormal	0.03	0.1	-
ξ	Lognormal	0.02	0.4	-
ω0	Normal	2	0.1	Hz

The standard deviation of displacement response can be obtained using the perturbation method:

$${\sigma }_X=\frac{\pi S_0}{2{\left(2\pi \omega \right)}^3\xi }\sqrt{1-3\epsilon {\left(\frac{\pi S_0}{2{\left(2\pi \omega \right)}^3\xi }\right)}^2}$$

(20)

In this example, the thresholds for three representative seismic intensities Is, I₀, and I₁ are assumed to be 0.1 m, 0.15 m, and 0.2 m, respectively. Figure 4 shows the frequency histogram of the displacement response standard deviation obtained from the MCS. Based on the displacement response standard deviation and Equation (14), the first-passage probability can be calculated. Finally, the mean first-passage probability of different standard deviation samples is computed as the conditional failure probability. Table 4 provides the conditional failure probability of different thresholds using PE and MCS. The PE results showed good agreement with the MCS, with a maximum relative error (RE) of 8.88%. Then, the probability of failure and reliability index under seismic design intensity level 8 can be estimated using the proposed method. For comparison, MCS results are also listed in Table 5 with a relative error of less than 5%, demonstrating that the proposed method has precision and efficiency comparable to MCS.

Figure 4. Frequency of the standard deviation in Example 1.

Table 4. Conditional failure probability of different thresholds in Example 1.

x	0.1 m	0.15 m	0.2 m
Proposed method	0.00913422	0.00294104	0.00120971
MCS	0.00891785	0.00270128	0.00112399
RE	2.43%	8.88%	7.63%

Table 5. The probability of failure and reliability index at seismic design intensity level 8.

Methods	PF	βl
Proposed method	0.007219799	2.44614
MCS	0.007002814	2.45712
RE	3.10%	0.45%

5.2. A Two-Degree-of-Freedom Simply Supported Isolated Slab Bridge Structure

In this study, a two-degree-of-freedom simply-supported isolated slab bridge structure manufactured from reinforced concrete was analyzed. A simplified model is shown in Figure 5. The representativity of the proposed two-degree-of-freedom simply-supported isolated slab bridge structure lies in its ability to capture the essential dynamic characteristics of typical simply-supported beam bridges used in practice. This simplified model facilitates analysis while retaining critical aspects of structural behavior under seismic loading. We considered undamped natural frequencies to establish a baseline for the bridge’s dynamic response. These frequencies provide fundamental insights into the system’s inherent vibrational properties, which are crucial for understanding how the structure will respond to seismic excitations. A zero-mean white noise random excitation f(t) is applied at mass point 1, and the basic intensity is 7 degrees; ξ = 0.05 and L = 24 m represent the damping ratio of the mass point and the calculation length of the bridge, respectively. S, D, E, I, and m are the structural parameters of the simply-supported isolated bridge, representing the white noise spectrum intensity of the mass point, duration, elastic modulus, cross-sectional moment of inertia, and mass at points 1 and 2 (equal masses). The undamped natural frequencies are ${\omega }_1=5.69\sqrt{EI/{\mathrm{mL}}^3}$ and ${\omega }_2=22\sqrt{EI/{\mathrm{mL}}^3}$, respectively. For simplicity, it was assumed that the structural parameters of the simply-supported isolated bridge were all equal. The uncertain parameters obtained from references [24,25,26] are shown in Table 6.

Figure 5. Schematic diagram of a two-degree-of-freedom isolated bridge structure.

Table 6. Statistical characteristics of uncertainty parameters.

Uncertain Parameter Distributions	Distribution Type	Mean Value	Coefficients of Variation
D	Log-normal	10 s/15 s/20 s	0.3
S	Extreme Value Type II Distribution	0.25 m2/s3	0.6
m	Normal	12.84 t	0.2
I	Log-normal	19,652.9 mm4	0.2
E	Normal	30,000 Mpa	0.2

Note: D = 10, 15, and 20 s correspond to the mean durations for median intensity, basic intensity, and major seismic intensity, respectively.

5.2.1. Root Mean Square of Displacement Response

Based on the random seismic response analysis in Section 1, the root mean square $E\left(X_1^2\right)$ of displacement at point 1 can be obtained as:

$$E\left(X_1^2\right)=\frac{S}{4m^2}\left\{{\displaystyle {\int }_{-\infty }^{\infty }\begin{array}{l}\left[{\left|H_1\left(\omega \right)\right|}^2+{\left|H_2\left(\omega \right)\right|}^2\right]d\omega \\ +2{\displaystyle {\int }_{-\infty }^{\infty }\mathrm{Re}{H}_1\left(\omega \right)H_2^{*}\left(\omega \right)d\omega }\end{array}}\right\}$$

(21)

in which,

$${\displaystyle {\int }_{-\infty }^{\infty }{\left|H_j\left(\omega \right)\right|}^{2}d\omega ={\displaystyle {\int }_{-\infty }^{\infty }\frac{d\omega }{{\left({\omega }_j^2-{\omega }^2\right)}^2+4{\xi }_j^2{\omega }_j^2{\omega }^2}}}=\frac{\pi }{2{\xi }_j{\omega }_j^3}, (j=1,2)$$

(22)

$$2{\displaystyle {\int }_{-\infty }^{\infty }\mathrm{Re}{H}_1\left(\omega \right)}{H}_2^{*}\left(\omega \right)d\omega =\frac{8\pi \left({\xi }_1{\omega }_1+{\xi }_2{\omega }_2\right)}{{\left({\omega }_1^2-{\omega }_2^2\right)}^2+4\left[{\xi }_1{\xi }_2{\omega }_1{\omega }_2\left({\omega }_1^2+{\omega }_2^2\right)+\left({\xi }_1^2-{\xi }_2^2\right){\omega }_1^2{\omega }_2^2\right]}$$

(23)

By substituting Equations (21) and (22) into Equation (20), we obtain the root mean square of displacement for each mass point of the linear two-degree-of-freedom system under Gaussian white noise excitation:

$${\sigma }_1^2=E\left(X_1^2\right)=\frac{\pi S}{4m^2}\left[A+B\right]$$

(24)

in which,

$$A=\frac{\pi }{2{\xi }_1{\omega }_1^3}+\frac{\pi }{2{\xi }_1{\omega }_2^3},B=\frac{8\pi \left({\xi }_1{\omega }_1+{\xi }_2{\omega }_2\right)}{{\left({\omega }_1^2-{\omega }_2^2\right)}^2+4\left[{\xi }_1{\xi }_2{\omega }_1{\omega }_2\left({\omega }_1^2+{\omega }_2^2\right)+\left({\xi }_1^2-{\xi }_2^2\right){\omega }_1^2{\omega }_2^2\right]}$$

(25)

Taking the square root of Equation (23), we can further obtain the standard deviation σ₁ at point 1:

$${\sigma }_1=\sqrt{\frac{\pi S}{4m^2}\left[A+B\right]}$$

(26)

The root mean square of displacement for mass point 2 can be calculated as:

$$E(X_2^2)=\frac{S}{4m^2}\left\{{\displaystyle {\int }_{-\infty }^{\infty }\left[{\left|H_1(\omega )\right|}^2+{\left|H_2(\omega )\right|}^2\right]d\omega -2{\displaystyle {\int }_{-\infty }^{\infty }\mathrm{Re}{H}_1(\omega )H_2^{*}(\omega )}}d\omega \right\}$$

(27)

in which,

$$2{\displaystyle {\int }_{-\infty }^{\infty }\mathrm{Re}{H}_1(\omega )H_2^{*}(\omega )d\omega }=\frac{8\pi \left({\xi }_1{\omega }_1+{\xi }_2{\omega }_2\right)}{{\left({\omega }_1^2-{\omega }_2^2\right)}^2+4\left[{\xi }_1{\xi }_2{\omega }_1{\omega }_2({\omega }_1^2+{\omega }_2^2)+({\xi }_1^2-{\xi }_2^2){\omega }_1^2{\omega }_2^2\right]}$$

(28)

By substituting Equations (26) and (21) into Equation (25), we can obtain the root mean square of displacement for mass point 2:

$${\sigma }_2^2=E\left(X_2^2\right)=\frac{\pi S}{4m^2}\left[A-B\right]$$

(29)

According to the MCS method [28], using Equations (24) and (28), the root mean squares of displacement for mass points 1 and 2 can be calculated, as shown in Figure 6. MCS involves several key steps. Firstly, the parameters and their probability distributions were defined. Then, a large number of random samples from these distributions was generated. Next, deterministic analyses of each sample were conducted to calculate the desired output. Finally, the results were aggregated and analyzed statistically to understand system response variability and uncertainty. After conducting 1000 sample experiments, the mean values of the root mean squares of displacement for mass points 1 and 2 were 0.0393353 and 0.0392583, respectively. Compared to the MCS results, the root mean square results for the displacement response of both mass points were similar. The mean value of the sample root mean square for mass point 1 was slightly larger than that for mass point 2. The seismic safety of the bridge structure in this study was analyzed based on the failure probability and reliability index of mass point 1.

Figure 6. Results of 1000 iterations of mean squared error cycling. (a) Scatter plot of mass point 1; (b) scatter plot of mass point 2.

5.2.2. Probability of Earthquake Intensity Occurrence and Conditional Failure Probability

The probabilities of the three representative earthquake intensities, I_s = 5.45, I₀ = 7, and I₁ = 8, occurring within the reference period, are calculated as follows: P(I_s) = 0.7063, P(I₀) = 0.2508, and P(I₁) = 0.0429, respectively. The limits were set as follows: x = x₁ = 1.5σ₁ = 0.06 m, x = x₂ = 2σ₁ = 0.08 m, x = x₃ = 2.5σ₁ = 0.10 m. Figure 7 presents the frequency histogram of first-passage probability at different thresholds obtained using MCS. The mean value calculated from these MCS samples provides the conditional failure probability. The method described in previous section was used to calculate conditional failure probabilities under uncertain structural parameters, as shown in Table 7. To demonstrate the accuracy and efficiency of calculation results, Table 7 also presents MCS results. Table 7 shows that the relative error calculated between the conditional failure probabilities using the first-passage point estimation method and MCS is within 4%. Additionally, in this study, the point estimation calculation only required seven iterations, which greatly improved computational efficiency compared to the MCS method, which required 100,000 iterations. To better illustrate the process of calculating conditional failure probability using MCS, Figure 7a–c shows the probability density histograms of the conditional failure probability P_fj at the limit x = 0.06 for μ_D = 10, 15, and 20, respectively. According to these figures, the conditional failure probability follows a certain probability distribution. The conditional failure probability sample obtained by MCS statistics provides the mean of the conditional failure probability, which corresponds exactly to the mean of the conditional failure probability obtained using point estimation in this study. The conditional failure probability sample obtained by MCS statistics provides an empirical estimate of the mean conditional failure probability. This mean value corresponds closely to the mean conditional failure probability derived using the point estimation method in this study. The term “exactly” here indicates that the statistical convergence of the MCS and point estimation results is within an acceptable range of probabilistic accuracy. Both methods estimate the expected value of failure probability given the uncertainties of seismic input and structural response. The high correspondence between these estimates validates our proposed methodology’s robustness.

Figure 7. Probability density histogram of conditional failure probability. (a) Probability density histogram (μ_D = 10); (b) Probability density histogram (μ_D = 15); (c) Probability density histogram (μ_D = 20).

Table 7. Conditional failure probability of different thresholds (five parameters).

		x	0.06	0.08	0.10
μD = 10	Pfs	Point estimation	0.003996	0.002372	0.001611
		MCS	0.004125	0.002450	0.001546
		Relative error	3.13%	3.18%	4.20%
μD = 15	Pf0	Point estimation	0.005980	0.003550	0.002412
		MCS	0.006131	0.003695	0.002321
		Relative error	2.46%	3.92%	3.92%
μD = 20	Pf1	Point estimation	0.007954	0.004723	0.003210
		MCS	0.008167	0.004914	0.003109
		Relative error	2.61%	3.89%	3.25%

5.2.3. Failure Probability and Reliability Index of Simply-Supported Beam Bridges within Service Life

The failure probability P_F and corresponding reliability index β_l of simply-supported beam bridges within service life were computed using the method proposed in this paper (Table 8). To demonstrate the efficiency and accuracy of our method, results obtained using the MCS method are also presented in Table 8. Table 8 shows that the failure probability and reliability index calculated using our method are in good agreement with MCS method results, with a maximum relative error of 4.04% for failure probability and controlled within 1% for the reliability index.

Table 8. Failure probability and reliability index of different thresholds (five parameters).

x	0.06	0.08	0.10
PF (Proposed method)	0.004663	0.002768	0.001881
PF (MCS)	0.004801	0.002868	0.001808
βl (Proposed method)	2.600	2.774	2.898
Β (MCS)	2.590	2.762	2.910
Relative error—failure probability	2.87%	3.49%	4.04%
Relative error—reliability index	0.39%	0.43%	0.41%

To further investigate the influence of structural parameters on the failure probability of simply-supported beam bridges, this study calculated the reliability index under different structural parameters, as illustrated in Figure 8. Figure 8 shows that as the number of structural parameters considered increases, the reliability index decreases accordingly. This trend suggests that structural parameters have a certain impact on seismic reliability analysis results for simply supported beam bridges. As more structural parameters are taken into account, the estimated reliability index decreases, aligning with practical observations. Additionally, Figure 8 shows that the reliability index calculated using our method fits well with the results obtained from the MCS method under different parameters, further demonstrating our proposed method’s efficiency and accuracy.

Figure 8. Reliability index for different structural parameters under different boundaries.

To illustrate the influence of uncertain parameters on the calculation results, Table 9 presents the reliability index computed by our method and the MCS method considering 5, 2, and 0 uncertain parameters (no uncertain parameters are considered). Table 9 shows that our method and the MCS method produce consistent results. When the number of uncertain parameters increases, the reliability index decreases, aligning with real-world scenarios. Therefore, Table 9 further underscores the effectiveness of our method and highlights the impact of uncertain parameters on reliability index calculations (consistent with findings in Figure 5). Therefore, considering uncertain parameters for seismic reliability analysis of simply-supported beam bridges is necessary.

Table 9. Comparison of reliability indices for uncertain parameters (x = 0.1).

The Number of Uncertain Parameters	5	2	0
Proposed method	2.898	3.093	3.464
MCS	2.910	3.090	3.464

This section provides a comprehensive discussion of the results and compares the proposed method with existing methods in terms of accuracy and efficiency. It also highlights the practical implications of the findings for bridge design and assessment in seismic regions.

6. Conclusions

This study establishes a framework for seismic reliability analysis of simply-supported beam bridges based on seismic intensity and uncertain parameters. Considering the influence of seismic intensity and uncertain parameters, structural seismic failure probability and reliability index were calculated. Firstly, the probability of occurrence of three representative seismic intensities (modal intensity, basic intensity, and major intensity) during the design reference period of the simply-supported beam bridge was calculated. Then, when uncertain parameters were considered, the conditional failure probability under seismic intensity was calculated based on the first-passage probability formula. Finally, a reliability index based on seismic intensity was calculated to obtain the failure probability and reliability index of simply-supported beam bridges under seismic conditions.

The seismic reliability analysis of simply-supported multi-span bridge structures showed consistent results with the MCS method. Due to the use of point estimation in calculating first-passage probability, the proposed method demonstrates greater efficiency than the MCS method. As more uncertain parameters are considered, the evaluated reliability index decreases correspondingly. The proposed method will provide practical target reliability standards for the seismic design of simply-supported beam bridges based on seismic intensity and serve as a basis for further improvements to seismic structural design standards.

The primary limitation of this study is that it focuses on linear analysis methods. While these methods provide significant insights, nonlinear structure analyses are inherently more complex and may yield different results. Further research is needed to investigate nonlinear structures and simply-supported bridge structures with more degrees of freedom. Additionally, this method can be extended to other types of bridges and different seismic environments to validate its generalizability and robustness. Furthermore, the integration of advanced computational techniques, such as machine learning, could enhance the efficiency and accuracy of seismic reliability assessments.