A Bayesian Method for Simultaneous Identification of Structural Mass and Stiffness Using Static–Dynamic Measurements

Abstract

This paper presents a Bayesian-based finite element model updating method that integrates static displacement measurements and dynamic modal data to simultaneously identify structural mass and stiffness parameters. By leveraging Bayesian inference, a posterior probability density function (PDF) is constructed by integrating static displacement and modal parameters, thereby effectively decoupling the identification of structural mass and stiffness. The Delayed Rejection Adaptive Metropolis (DRAM) Markov Chain Monte Carlo (MCMC) sampling algorithm is utilized to derive the posterior distributions of the updated parameters. To mitigate the computational burden associated with repetitive finite element (FE) analyses during large-scale MCMC sampling, a Kriging surrogate model is employed to efficiently approximate the time-consuming FE simulations. Numerical examples involving a cantilever beam and an actual concrete three-span single-box girder bridge illustrate that the proposed method accurately identifies simultaneous variations in mass and stiffness at multiple structural locations, effectively addressing parameter coupling and misidentification issues encountered when using either static or dynamic data alone. Moreover, the Kriging surrogate significantly improves computational efficiency. Experimental validation on an aluminum alloy cantilever beam further corroborates the effectiveness and practical applicability of the proposed method.

1. Introduction

In engineering practice, precise finite element (FE) models are crucial for structural design and analysis to be able to accurately capture both the static and dynamic characteristics of structures. However, significant discrepancies often exist between the static–dynamic properties of initial FE models and actual measurement data. To achieve more accurate FE models, it is imperative to update these models by incorporating experimentally measured structural response data [1]. Over the past two decades, FE model updating methodologies have seen substantial advancements, resulting in the development of numerous effective techniques and practical applications [2,3,4].

When it comes to static response, Ren et al. [5] harnessed the power of a sophisticated response surface methodology, leveraging vehicle-induced static load test data to refine the FE model of an intricate bridge structure. Wu et al. [6] meticulously updated the FE model of a simply supported concrete beam by incorporating measured static displacement data, thereby enhancing the model’s predictive prowess in capturing the beam’s static deformation behavior with remarkable precision. In a similar vein, Brownjohn et al. [7] drew upon static test data from a bridge structure to pioneer an innovative FE model updating method grounded in the least-squares approach, which markedly elevated the accuracy of predicting the bridge’s static response.

In contrast to static response data, dynamic response data are more frequently harnessed in the realm of model updating. Dynamic response indicators encompass a rich array of parameters, such as natural frequencies, mode shapes, frequency response functions (FRFs), the modal assurance criterion (MAC), modal flexibility, and modal strain energy [8,9,10]. Given that dynamic data acquisition can typically proceed without disrupting normal structural service conditions—such as real-time acquisition through advanced structural health monitoring systems—methods based on dynamic data have gained widespread adoption. For instance, Jaishi and Ren [11] pioneered a finite element model updating method grounded in modal flexibility matrices, which they successfully applied to the health monitoring of actual bridge structures. Meanwhile, Akbar et al. [12] leveraged frequency response function (FRF) data to meticulously quantify damage in severely compromised concrete beams, with their results unequivocally demonstrating the method’s success and robustness in damage quantification.

Moreover, integrated static–dynamic data model updating approaches have progressively garnered the attention of researchers. Yam et al. [13] conducted a comprehensive investigation into the sensitivity of static and dynamic parameters to damage in plate-like structures, proposing corresponding damage indicators to meticulously analyze their identification capabilities. Through rigorous numerical simulations and experimental tests, they verified the detection prowess of these indicators while offering invaluable guidance for parameter selection in damage detection. Hasançaebi et al. [14] devised an innovative artificial neural network (ANN)-based finite element model updating method tailored for reinforced concrete T-beam bridges. This method leveraged structural static displacements and dynamic frequencies to train ANNs, subsequently employing the refined models to accurately predict bridges’ intricate nonlinear behavior.

In the majority of existing finite element model updating studies, structural mass is conventionally assumed to be constant, with updates primarily targeting stiffness parameters or the elastic modulus [15,16,17]. A few pioneering studies [18] have ventured into the simultaneous updating of stiffness and mass-related parameters, such as elastic modulus and structural density. According to the fundamental principles of structural dynamics, however, both structural stiffness and mass play pivotal roles in determining natural frequencies and mode shapes [19]. Consequently, when changes occur in both mass and stiffness within specific structural regions or components, purely dynamic data-based updating introduces a coupling effect between mass and stiffness parameters—resulting in an infinite array of possible combinations for these parameters [20,21]. To tackle this coupling challenge, Cha et al. [22] proposed modifying structural stiffness matrices by incorporating known added masses. Dinh et al. [23] experimentally validated this mass addition method on a five-story frame structure, demonstrating its practical efficacy. Garbowski et al. [24] further enhanced finite element model updating and parameter identification—encompassing variations in material stiffness and density—by integrating dynamic modal analysis with targeted static measurements. They employed a trust region least-squares optimization algorithm within an inverse analysis framework, thereby providing more robust theoretical support for the structural assessment of concrete bridges. Zeng et al. [25] further extended the mass addition approach into stochastic domains, developing a sophisticated Bayesian framework utilizing DREAM sampling for the simultaneous updating of uncertain mass and stiffness parameters.

Despite the theoretical efficacy of the mass addition method in resolving mass–stiffness coupling, practical engineering applications encounter formidable challenges. For instance, the dynamic characteristic changes before and after mass addition may be imperceptibly subtle, measurement difficulties and associated costs can escalate significantly, and implementation schemes might pose substantial technical hurdles in real-world structures [26].

Guided by the principles of structural mechanics, static displacement responses are exclusively governed by structural stiffness and remain entirely independent of mass [27]. Consequently, integrating static and dynamic responses within a Bayesian stochastic finite element updating framework can effectively mitigate mass–stiffness parameter coupling. This paper introduces an innovative Bayesian finite element model updating method based on integrated static–dynamic response data. The approach first constructs posterior probability density function (PDF) expressions for structural parameters using static displacements, natural frequencies, and mode shapes. It then employs the Delayed Rejection Adaptive Metropolis (DRAM) algorithm for the Markov Chain Monte Carlo (MCMC) sampling of mass and stiffness correction factors. Samples are only accepted when both their static and dynamic responses meet predefined error thresholds, thereby rejecting coupled mass–stiffness solutions and ensuring physically meaningful parameter updates. To circumvent computationally intensive finite element iterations during MCMC sampling, this study incorporates Kriging surrogate models to significantly enhance computational efficiency.

This paper is organized with the following structure: Section 2 introduces Bayesian model updating fundamentals; Section 3 presents a novel error function resolving mass–stiffness coupling; Section 4 illustrates the effectiveness of the proposed method through the numerical analysis of a cantilever beam and the numerical simulation of an actual concrete three-span single-box girder bridge; Section 5 further verifies practical applicability via static loading tests and modal measurements on an aluminum alloy cantilever beam; and Section 6 gives the conclusion.

2. Bayesian Methodology for Decoupling Mass–Stiffness Coupling

In structural FE model updating, the stiffness parameters $\mathit{\alpha }=\left\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{n_{\alpha }}\right\}$ and mass parameters $\mathit{\beta }=\left\{{\beta }_1,{\beta }_2,\dots ,{\beta }_{n_{\beta }}\right\}$ represent uncertain structural properties that can collectively influence both static and dynamic behaviors. When relying exclusively on dynamic test data, such as natural frequencies and mode shapes, stiffness–mass coupling effects may emerge. This phenomenon allows for multiple combinations of $\mathit{\alpha }$ and $\mathit{\beta }$ to produce identical dynamic responses, thereby complicating the identification of physically unique solutions. Conversely, static test data, including static displacements and strains, alone are insufficient for calibrating mass parameters. To address these challenges, this study integrates static and dynamic data within a unified framework to simultaneously update stiffness and mass parameters, thereby mitigating the ambiguities induced by coupling effects.

Let the measured static–dynamic data be denoted by the vector $D$, which comprises a static subset $D_S$ (displacements) and a dynamic subset $D_D$ (natural frequencies and mode shapes). For a parameter vector $\mathit{\theta }=\left[\mathit{\alpha },\mathit{\beta }\right]$, the FE-predicted static and dynamic responses are represented by $R_S\left(\mathit{\theta }\right)$ and $R_D\left(\mathit{\theta }\right)$, respectively. By minimizing the discrepancies between $R_S\left(\mathit{\theta }\right)$ and $D_S$ and between $R_D\left(\mathit{\theta }\right)$ and $D_D$ through error minimization or likelihood maximization, the parameter vector $\mathit{\theta }$ can be effectively updated. This approach ensures the more robust and accurate identification of structural properties by leveraging the complementary strengths of both static and dynamic measurements.

2.1. Bayesian Updating Equations

A Bayesian framework is employed to address measurement uncertainties and incorporate prior knowledge effectively. According to the Bayesian theorem [4], the posterior probability density function is expressed as follows:

$$p\left(\mathit{\theta }\left|D\right.\right)\propto p\left(D\left|\mathit{\theta }\right.\right)p\left(\mathit{\theta }\right)$$

(1)

where $p\left(\mathit{\theta }\right)$ represents the prior distribution that encodes initial knowledge about the parameters and $p\left(D|\mathit{\theta }\right)$ denotes the combined likelihood of static and dynamic residuals. The likelihood function in Equation (1) is constructed from the residuals of both static and dynamic forces. Specifically, the static residuals are defined as follows:

$$e_S\left(\mathit{\theta }\right)=D_S-R_S\left(\mathit{\theta }\right)$$

(2)

where $D_S$ and $R_S\left(\mathit{\theta }\right)$ represent the measured and predicted displacements, respectively. Assuming that the static errors are Gaussian-distributed, uncorrelated, and have a known variance ${\sigma }_S^2$, the static likelihood function is formulated as follows [28]:

$${\mathcal{L}}_S\left(\mathit{\theta }\right)=\exp \left[-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{(D_S-R_S\left(\mathit{\theta }\right))}^2}{{\sigma }_S^2}}\right]$$

(3)

Similarly, the modal likelihood is expressed as follows [1]:

$${\mathcal{L}}_D\left(\mathit{\theta }\right)=\exp [-{\displaystyle \frac{1}{2}}\left({{\displaystyle \sum }}_{i=1}^{n_f}{\displaystyle \frac{{(f_i^{\left(\mathrm{test}\right)}-f_i\left(\mathit{\theta }\right))}^2}{{\sigma }_{f,i}^2}}+{{\displaystyle \sum }}_{j=1}^{n_{\varphi }}{\displaystyle \frac{\Vert {\varphi }_j^{\left(\mathrm{test}\right)}-{\varphi }_j\left(\mathit{\theta }\right){\Vert }^2}{{\sigma }_{\varphi ,j}^2}}\right)]$$

(4)

where $f_i^{\left(\mathrm{test}\right)}$ and $f_i\left(\mathit{\theta }\right)$ denote the i-th measured and computed natural frequencies, respectively; ${\varphi }_j^{\left(\mathrm{test}\right)}$ and ${\varphi }_j\left(\mathit{\theta }\right)$ represent the j-th measured and computed mode shapes; and ${\sigma }_{f,i}^2$ and ${\sigma }_{\varphi ,j}^2$ are the corresponding variances.

2.2. Integrated Static–Dynamic Likelihood

The composite likelihood function integrates both static and dynamic contributions as follows [29]:

$$p\left(D_S,D_D\left|\mathit{\theta }\right.\right)={\left[{\mathcal{L}}_S\left(\mathit{\theta }\right)\right]}^{w_S}\times {\left[{\mathcal{L}}_D\left(\mathit{\theta }\right)\right]}^{w_D}$$

(5)

where ${\mathcal{L}}_S\left(\mathit{\theta }\right)$ and ${\mathcal{L}}_D\left(\mathit{\theta }\right)$ correspond to the static and dynamic likelihoods, respectively, weighted by $w_S$ and $w_D$ based on data quality, reliability, or prior knowledge. The inclusion of ${\mathcal{L}}_S\left(\mathit{\theta }\right)$ eliminates non-unique solutions arising from stiffness–mass coupling, while ${\mathcal{L}}_D\left(\mathit{\theta }\right)$ constrains both the stiffness and mass parameters to enhance dynamic predictability. Together, these components narrow the parameter space to physically meaningful solutions.

Posterior sampling is conducted using the DRAM (DRAM) algorithm, which adaptively explores the parameter space to balance global search efficiency and convergence to high-posterior regions. Compared to gradient-based methods, MCMC techniques inherently handle multimodal distributions and measurement uncertainties, providing a robust framework for parameter estimation.

2.3. DRAM Sampling Method

MCMC sampling and Kriging models are employed to efficiently compute the predicted static response ${\mathit{R}}_{\mathit{S}}\left(\mathit{\theta }\right)$ and dynamic response ${\mathit{R}}_{\mathit{D}}\left(\mathit{\theta }\right)$ for each parameter vector $\mathit{\theta }$. This approach significantly accelerates the evaluation of likelihood functions. The overall workflow, illustrated in Figure 1, demonstrates the reliability and computational efficiency of the method through both numerical simulations and experimental validations.

The DRAM sampling algorithm [30] enhances the Bayesian updating process by integrating delayed rejection (DR) with an adaptive Metropolis (AM) strategy. This advanced MCMC technique incorporates both local and global adaptations, ensuring a thorough exploration of the parameter space. The key steps of the DRAM algorithm are as follows:

(1)

In the initial sampling phase of MCMC sampling, at iteration $t$, the current value of the Markov chain is assumed as ${\mathit{\theta }}_t$. A candidate sample ${\mathit{\theta }}_{t+1}^{*}$ is drawn randomly from a normal proposal distribution $\mathcal{N}({\mathit{\theta }}_t,{\sigma }_0^2)$, where ${\sigma }_0^2$ is the initial variance of the proposal distribution.

(2)

In the acceptance probability phase, the candidate sample ${\mathit{\theta }}_{t+1}^{*}$ is evaluated by computing its acceptance probability $p$, which determines whether it replaces the current state ${\mathit{\theta }}_t$ in the Markov chain. This acceptance probability is calculated using the Metropolis–Hastings criterion:

$$p=\min \left(1,{\displaystyle \frac{p({\mathit{\theta }}_{t+1}^{*}|D)}{p({\mathit{\theta }}_t|D)}}\right)$$

(6)

where $p(\mathit{\theta }|D)$ is the posterior probability proportional to the likelihood of the data and the prior distribution for $\mathit{\theta }$.

(3)

In the decision rule phase of the Metropolis–Hastings algorithm, a uniform random variable $u~u[0,1]$ is sampled from the interval [0, 1]. The candidate sample ${\mathit{\theta }}_{t+1}$ is accepted as the next state of the Markov chain ${\mathit{\theta }}_{t+1}={\mathit{\theta }}_{t+1}^{*}$ if $p>u$, where $p$ is the previously computed acceptance probability. Conversely, if $p<u$, the candidate is rejected and the chain retains its current state ${\mathit{\theta }}_{t+1}={\mathit{\theta }}_t$.

(4)

In the delayed rejection phase of adaptive MCMC sampling, if the initial candidate sample is rejected, a secondary candidate ${\mathit{\theta }}_{t+1}^{*}$ is generated using a refined proposal distribution $\mathcal{N}({\mathit{\theta }}_t,{\sigma }_1^2)$ which incorporates information from the rejected sample ${\sigma }_1^2$ to adjust its exploration strategy (e.g., reducing step size or reorienting search directions). The acceptance probability for is re-calculated as follows:

$$p_2=\min \left(1,{\displaystyle \frac{p({\mathit{\theta }}_{t+1}^{**}|D)(1-p({\mathit{\theta }}_{t+1}^{*}))}{p({\mathit{\theta }}_t|D)(1-p({\mathit{\theta }}_{t+1}^{**}))}}\right)$$

(7)

(5)

In the adaptive strategy phase, once the iteration count $\mathrm{t}+1$ exceeds a predefined non-adaptive period ${\mathrm{N}}_0$, namely, $\mathrm{t}+1>{\mathrm{N}}_0$, the covariance matrix of the proposal distribution is systematically updated to enhance sampling efficiency. The updated covariance matrix is computed as follows:

$$Co{v}_{t+1}=s_d\cdot \mathrm{Cov}({\mathit{\theta }}_1,{\mathit{\theta }}_2,\dots ,{\mathit{\theta }}_t)+s_d\cdot ϵI$$

(8)

where $s_d={\displaystyle \frac{2.42}{m}}$, $m$ is the parameter dimension, $ϵ$ ensures the numerical stability by preventing singularity, and $I$ is the identity matrix.

(6)

In the iteration and termination phase, the Markov chain is propagated through sequential steps that are repeated iteratively: proposal generation, acceptance probability calculation, decision rule application, and adaptive covariance updates. This iterative process continues until convergence of the Markov chain is achieved and sampling stability is confirmed.

The DRAM algorithm ensures efficient sampling by balancing global exploration through delayed rejection and focused adaptation via covariance updates. This adaptive strategy enables the Markov chain to converge effectively to high-probability regions of the posterior distribution, facilitating accurate and computationally efficient Bayesian updating. This approach is particularly advantageous in handling complex parameter spaces and multimodal distributions, ensuring robust parameter estimation in structural model updating.

3. Kriging Surrogate Modeling and Implementation Workflow

Direct FE evaluations for each MCMC-proposed parameter vector $\mathit{\theta }$ can be computationally prohibitive. To address this, Kriging surrogate models [28,29] are employed to approximate FE responses, thereby accelerating the Bayesian updating process while maintaining accuracy. The implementation of this approach follows the key steps below.

3.1. Initial Sampling and FE Computation

An initial set of N training samples ${\left\{{\mathit{\theta }}^{\left(k\right)}\right\}}_{k=1}^N,$ is generated within the parameter space using Latin Hypercube Sampling (LHS). The number of samples NN is determined based on the dimensionality of the parameters, prior ranges, and available computational resources. High-fidelity FE computations are then performed to obtain the corresponding static and dynamic responses, ${\mathit{R}}_S\left({\mathit{\theta }}^{\left(k\right)}\right)$ and ${\mathit{R}}_D\left({\mathit{\theta }}^{\left(k\right)}\right)$, respectively. Post-calculation, any outliers or numerical errors are filtered out to ensure data quality.

3.2. Kriging Model Training

Static and dynamic surrogate functions, ${\widehat{\mathit{R}}}_S\left(\mathit{\theta }\right)$ and ${\widehat{\mathit{R}}}_D\left(\mathit{\theta }\right)$, are defined to approximate the FE responses. For scalar outputs, such as displacements and natural frequencies, the Kriging model assumes the following form:

$$f\left(\mathit{\theta }\right)=\mu \left(\mathit{\theta }\right)+z\left(\mathit{\theta }\right)$$

(9)

where $\mu \left(\mathit{\theta }\right)$ represents the trend term, typically fitted using polynomial regression (constant, linear, or quadratic), and $z\left(\mathit{\theta }\right)$ is a zero-mean stationary Gaussian process characterized by the following correlation kernel:

$$R\left({\mathit{\theta }}_i-{\mathit{\theta }}_j;\varphi \right)=\exp (-{{\displaystyle \sum }}_{p=1}^d{\varphi }_p|{\mathit{\theta }}_{i,p}-{\mathit{\theta }}_{j,p}{|}^{r_p})$$

(10)

where $\varphi$ governs the correlation length scales and $d$ denotes the parameter dimension. The hyper-parameters, including the trend coefficients $\varphi$ and process variance ${\sigma }^2$, are identified through maximum likelihood estimation (MLE) or cross-validation using training data $\left\{{\mathit{\theta }}^{\left(k\right)},f^{\left(k\right)}\right\}$.

3.3. Surrogate Prediction and Validation

For a new parameter vector ${\mathit{\theta }}^{*}$, the Kriging model predicts the response as follows:

$$\widehat{f}\left({\mathit{\theta }}^{*}\right)=r^T\left({\mathit{\theta }}^{*}\right){\mathit{R}}^{-1}\left(\mathit{f}-1\widehat{\mathit{\beta }}\right)$$

(11)

where $r\left({\mathit{\theta }}^{*}\right)$ is the correlation vector between ${\mathit{\theta }}^{*}$ and the training points, $R$ is the training correlation matrix, and $\widehat{\beta }$ denotes the trend coefficients. The generalization capacity of the surrogate model is assessed using metrics such as the root mean squared error (RMSE), the $Q^2$ statistic, or cross-validation. Adaptive sampling techniques are employed to enrich the training data in regions where the surrogate model exhibits low accuracy, thereby refining the surrogate predictions.

4. Numerical Verification

4.1. A Numerical Cantilever Beam

To validate the proposed Bayesian model updating method that integrates static and dynamic measurements, a numerical cantilever beam model was meticulously analyzed. The beam, measuring 1.2 m in length with a cross-section of 0.1 m by 0.008 m, was crafted from aluminum alloy. The elastic modulus of the aluminum alloy was 66 GPa. Figure 2 vividly illustrates the loading configuration and sensor placement.

A baseline “healthy” model was established, with stiffness and mass reductions simulated by excising 30% and 40% of the material from regions 2 and 5, respectively. Static displacements at six strategically chosen measurement points were recorded under a vertical tip load of 7.9 N. Additionally, the first three natural frequencies and corresponding mode shapes, based on displacements at six critical nodes, were meticulously extracted. These results served as synthetic “true” data. To replicate real-world measurement conditions, 1% independent Gaussian white noise was superimposed on both the static and dynamic responses.

To evaluate the effectiveness of integrating static and dynamic data, three distinct updating strategies were rigorously examined:

(1)

This innovative approach entails simultaneous adjustments to stiffness and mass parameters by leveraging static displacements, natural frequencies, and mode shapes. By harmonizing these diverse data sources, this method aims to achieve a comprehensive and accurate parameter update. This strategy is defined as Case 1 in the following contents.

(2)

In this strategy, updates to stiffness and mass parameters are exclusively informed by dynamic data, specifically natural frequencies and mode shapes. This focused reliance on dynamic measurements allows for an in-depth exploration of structural behavior under dynamic conditions. This strategy is defined as Case 2.

(3)

This method initiates the calibration of stiffness parameters using static data, followed by subsequently updating mass parameters using dynamic data. By decoupling the updating process for stiffness and mass, this approach facilitates a detailed examination of identification performance when static and dynamic measurements are not employed concurrently. This strategy is defined as Case 3.

For each case study, the six stiffness parameters ${\mathit{\theta }}_1~{\mathit{\theta }}_6$ and the six mass parameters ${\mathit{\theta }}_7~{\mathit{\theta }}_{12}$ of the numerical beam were meticulously updated. The posterior distributions of these updating parameters were obtained through MCMC sampling. Figure 3 illustrates the Markov chain sampling process for all three cases, showcasing chain convergence across all scenarios. This convergence signifies that each case successfully identified parameter sets that effectively minimized discrepancies between simulated and measured responses, whether static or dynamic.

To quantitatively evaluate accuracy, mean values from stationary chain segments were meticulously compared against the ground-truth parameter reductions. Figure 4a juxtaposes the posterior means of the stiffness updating parameters, revealing that only Case 1—integrating static displacements and the first three dynamic modes—accurately quantified the 30% and 40% stiffness reductions in regions 2 and 5. In stark contrast, Case 2, which relied solely on dynamic data, and Case 3, which employed sequential static–dynamic updates, yielded either erroneous or imprecise stiffness estimates, particularly under conditions of strong mass–stiffness coupling.

Similarly, for the mass calibration depicted in Figure 4b, Case 1 alone accurately captured the true variations in mass parameters. In contrast, Case 2 yielded non-unique physical solutions, including spurious equivalents, due to the absence of static constraints. Meanwhile, Case 3 propagated initial stiffness errors, leading to unreliable mass updates. This sequential error amplification underscores the paramount importance of concurrent static–dynamic data fusion when addressing coupled parameter systems.

The comprehensive MCMC process encompassed tens of thousands of static–dynamic response analyses. Conducting direct high-fidelity FE computations for each analysis would have incurred prohibitively high computational costs. To address this challenge, Kriging surrogate models were employed to approximate static displacements, natural frequencies, and modal shapes, thereby significantly enhancing sampling efficiency. The stiffness and mass updating parameters using the FEM and Kriging models are displayed in Figure 5. From Figure 5, it can be seen that the means of the stiffness and mass updating parameters using the Kriging model can be consistent with those of the method using the FEM. However, in the presented case study, the total computation time was reduced from approximately two hours using direct FE analysis to just 15 min, while maintaining comparable identification accuracy and precision relative to full FE evaluations.

The innovative static–dynamic coupling updating methodology adeptly resolves stiffness–mass coupling ambiguities, magnificently bolstered by the Kriging surrogate models that slash computational costs by an order of magnitude. This dual prowess not only ensures the practical applicability of this approach to large-scale structural model updating tasks but also empowers simultaneous parameter identification even under conditions of measurement uncertainty.

4.2. Concrete Three-Span Single-Box Girder Bridge

In real-world bridges, it is not uncommon to encounter scenarios where parameters related to structural mass and stiffness evolve concurrently. The second numerical example in this paper demonstrates the efficacy of the proposed method in addressing such complex situations. A numerical model of a bridge, as illustrated in Figure 6a, served as the study object. The three-span continuous concrete beam bridge was modeled in ANSYS, using Solid65 elements, with the modulus of elasticity set to 35 GPa. Each span measured 16 m, resulting in a total bridge length of 48 m. The bridge featured a single-cell box girder cross-section, as detailed in Figure 6c. The bridge piers were positioned at span junctions according to the design drawings, with fixed constraints applied at the bridge ends and pier bases. The finite element model (FEM) comprised 3324 elements and 4726 nodes, with each node possessing 3 degrees of freedom (longitudinal, transverse, and vertical), leading to a total of 14,178 degrees of freedom. As illustrated in Figure 6b, the FEM analysis indicated that each mid-span was subjected to a concentrated load of 50 kN.

Nine measurement points were uniformly distributed along the top of the beam. Based on the length, the bridge was divided into 10 regions, as shown in Figure 6b, and the modulus of elasticity was assumed to be the same for each region. The initial design model served as the baseline model for the bridge. To simulate uncertainties during bridge construction, the actual elastic modulus for each region was assumed to deviate from its initial value, with specific values determined by the stiffness updating factors listed in

Table 1. Preset values of updating factors.

Element No.	${\mathit{\theta }}_1$	${\mathit{\theta }}_2$	${\mathit{\theta }}_3$	${\mathit{\theta }}_4$	${\mathit{\theta }}_5$	${\mathit{\theta }}_6$	${\mathit{\theta }}_7$	${\mathit{\theta }}_8$	${\mathit{\theta }}_9$	${\mathit{\theta }}_{10}$	${\mathit{\theta }}_{11}$
Preset Value	−0.2	−0.3	−0.05	−0.25	0.15	−0.3	−0.05	−0.25	−0.15	0.1	−0.15

(i.e., the ratio of the actual stiffness changes to the initial value). Additionally, it was considered that the actual density of the bridge material was 15% lower than the nominal value to account for inconsistencies in material density.

This scenario presented a significantly greater challenge compared to the first numerical example, as it involved the full coupling of all mass-related parameters concerning density and stiffness-related parameters corresponding to the modulus of elasticity. Based on these comprehensive parameters, the displacements and the first three mode shapes at nine measurement points, along with the first three natural frequencies of the bridge, were meticulously computed. To emulate measured responses, 3% Gaussian-distributed noise was introduced into these computations. The ultimate objective was to identify a total of eleven updating factors: ten for the modulus of elasticity across the regions and one for the density associated with the mass matrix.

Before performing model updating, it was necessary to establish a Kriging surrogate model for the bridge, given the large degree-of-freedom scale of the structure. Latin Hypercube Sampling (LHS) was utilized to randomly select 600 sets of random samples for stiffness correction factors within the ±40% range and 600 sets of density correction factor samples within the ±20% range. The corresponding responses were calculated using ANSYS software to construct and fit the Kriging surrogate model. Among these samples, 500 sets were used for model training, while the remaining 100 sets were used for validating the accuracy of the surrogate model.

Figure 7 compares the predictions of the Kriging surrogate model with the finite element analysis results on the validation set. To facilitate a direct comparison of the Kriging surrogate model’s predictive accuracy, all displacements and mode shapes in Figure 7 were normalized with respect to their corresponding values in the initial model. From Figure 7a, it is evident that the predicted displacements from the Kriging surrogate model exhibit remarkable consistency with those computed by ANSYS software, with data points clustering tightly along the 45-degree reference line. Figure 7b illustrates that the predicted frequencies from the Kriging surrogate model closely align with the validation set as well. Moreover, Figure 7c compares the modal assurance criterion (MAC) values between the predicted mode shapes from the Kriging model and the validation set. All the MAC values fall within the range of (0.99, 1), indicating an exceptional level of consistency between the mode shapes predicted by the trained Kriging surrogate model and the results from finite element analysis.

The Kriging surrogate model utilized in this study adeptly predicts the static displacements, frequencies, and mode shapes of the bridge, rendering it suitable for direct application in response prediction for the structure.

The simulated measurement responses were substituted into Equation (5) for MCMC sampling, with a total of 10,000 samples generated. After the sampling process, the initial 3000 samples were discarded as part of the burn-in period and the mean values of the remaining samples were computed, as illustrated in Figure 8. For comparative purposes, the updating results obtained using only the first three natural frequencies and mode shapes were also calculated.

Figure 8 clearly demonstrates that employing only modal response data to simultaneously update the bridge’s elastic modulus and mass parameters introduces significant errors due to the inherent coupling between mass and stiffness. In this approach, notable deviations are observed in the updating of parameters 6, 9, and 11, with parameter 9 exhibiting an error exceeding 30%. In contrast, when the proposed method is applied, the mean updated parameters show excellent agreement with their preset values. This superior accuracy is primarily attributed to the integrated use of both static displacement data and modal data, which effectively suppresses the emergence of pseudo-parameter combinations that, while potentially consistent with the measured response, are physically implausible. Consequently, this comprehensive data fusion approach constrains the solution space to physically realistic parameter sets, thereby ensuring enhanced precision and reliability in the parameter updating process.

5. Experimental Verification

To further validate the engineering feasibility and quantification accuracy of the proposed method, experimental tests were conducted on an aluminum alloy cantilever beam with material properties consistent with those used in the numerical study. Initially, the beam specimen was divided into six distinct regions representing an undamaged state, with the assumption that the pristine model parameters closely approximated the actual properties due to the tight fabrication tolerances characteristic of aluminum alloy structures. High-precision laser cutting was then employed on both sides of regions 2 and 5 to remove a controlled portion of material—corresponding to 30% and 40% reductions in stiffness and mass, respectively—to induce a genuine damage state, as illustrated in Figure 9. A calibrated vertical concentrated load of 7.9 N was applied at the free end, and static displacement measurements were sequentially acquired at six designated sensor locations using a trio of laser displacement transducers in two separate measurement sessions to capture both the displacement values and their associated uncertainties; the detailed measurement protocol is provided in Figure 10. Concurrently, under white-noise excitation, the vibration responses were recorded by six accelerometers, and the first three natural frequencies along with the corresponding mode shapes at the six sensor locations were obtained using an enhanced frequency domain decomposition (EFDD) method, as summarized in

Table 2. The measured frequencies of the aluminum alloy cantilever beam (Hz).

Mode	Initial Frequency (Hz)	Measured Frequency (No Added Mass)		Measured Frequency (With Added Mass)		Relative Error (Mean)
		Mean	CoV	Mean	CoV
1	4.71	4.42	0.0069	4.12	0.0061	−0.061
2	28.45	26.70	0.0078	26.7	0.0058	−0.061
3	85.95	74.76	0.0056	70.8	0.0046	−0.13
4	168.2	148.92	0.0072	141.6	0.0087	−0.09

and depicted in Figure 11. It should be noted that the number of static and dynamic measurement points was determined by the number of updated regions, with the underlying assumption that six measurements were adequate to accurately identify the parameters of these six regions.

Substantial discrepancies were observed between the experimental measurements and the predictions of the initial finite element model, necessitating model updating via the proposed framework. In this updating process, stiffness and mass parameters associated with the six regions were selected as design variables. Following the procedure outlined in the numerical case study in Section 4.2, 600 parameter combinations were generated using Latin Hypercube Sampling to compute the corresponding structural displacements, natural frequencies, and mode shapes. These responses were then utilized to train a Kriging surrogate model, effectively replacing the time-intensive finite element simulations. The static displacements, first three frequencies, and mode shapes were subsequently employed as measurement data to sample the posterior distribution via the MCMC method. The Kriging surrogate models were further leveraged to approximate both static and dynamic responses based on the finite element model, significantly accelerating the computational process. A total of 10,000 MCMC samples were generated, with the surrogates facilitating the rapid evaluation of the required responses.

For a comparative assessment, two existing approaches for simultaneous mass–stiffness updating were implemented on the experimental beam. The first method, the Improved Cross-Model Cross-Modal (ICMCM) technique [31], employed the identical dynamic measurement data as used in the proposed framework. These dynamic data include the first three natural frequencies and the six-node displacement mode shapes. The second approach, a recently introduced Bayesian-based added-mass method [25], implemented a calibrated mass augmentation of 40% in region 5, with modal data including the frequencies and mode shapes acquired prior to and following the mass attachment, as illustrated in Figure 11.

Figure 12 provides a comprehensive comparative assessment of the parameter updating results. As shown in Figure 12a, while the ICMCM method effectively detects regions of stiffness change, it exhibits absolute quantification errors up to 18%, primarily due to unresolved mass–stiffness coupling effects inherent in complex structural systems. Similarly, the added-mass approach successfully identifies stiffness reductions of 30% and 40% in regions 2 and 5, respectively, but its sensitivity to measurement uncertainties leads to false-positive indications of stiffness degradation, with errors ranging from approximately 3% to 7% in undamaged regions. In stark contrast, the proposed method synergistically integrates data from static displacements and dynamic modal responses. This dual information fusion not only decouples the interdependence between mass and stiffness but also exploits the predictive power of Kriging surrogate models to achieve remarkable computational efficiency. By confining the solution space to physically plausible parameter sets and eliminating illogical pseudo-parameter combinations that might still match measured responses, the approach attains superior identification accuracy with errors limited to within 2% in damaged zones. Moreover, it reduces computational costs by 85% compared to direct finite element simulations. This innovative framework, therefore, offers significant advantages in reliability and efficiency for structural health monitoring and damage assessment in complex engineered systems.

6. Conclusions

This study presents a novel Bayesian finite element model updating approach that leverages combined static and dynamic response data. This method initially formulates posterior PDFs for structural parameters by incorporating static displacements, natural frequencies, and mode shapes. Subsequently, it utilizes the DRAM algorithm to perform MCMC sampling of mass and stiffness updating parameters. The acceptance of samples is conditional on both the static and dynamic responses satisfying predetermined error limits, thereby eliminating coupled mass–stiffness solutions and ensuring physically plausible updating parameters. To mitigate the computational burden associated with FE iterations during MCMC sampling, this research employs Kriging surrogate models, which substantially improve computational efficiency.

Numerical case studies, including a cantilever beam and a real concrete three-span single-box girder bridge, demonstrate that the proposed approach effectively captures simultaneous changes in mass and stiffness across multiple structural locations. This addresses parameter coupling and misidentification challenges often encountered when relying solely on static or dynamic data. Furthermore, experimental validation using an aluminum alloy cantilever beam confirms the efficacy and practical utility of the proposed method.

The fusion strategy employed in this study addresses the inherent non-uniqueness of dynamic tests and the limited sensitivity of static assessments by actively rejecting non-physical stiffness–mass combinations during posterior sampling. This approach enhances the reliability of the model updating process and provides more plausible estimations of structural parameters. Furthermore, the integration of global dynamic measurements, which capture overall modal characteristics, with local static responses, which are more sensitive to localized damage, results in the high-precision identification of critical damage regions. This capability is clearly illustrated by the detection of 30% and 40% stiffness reductions in region 2 and region 5, respectively, demonstrating the method’s multi-scale adaptability. In addition to its identification accuracy, the framework incorporates Kriging surrogate models to approximate static–dynamic responses, thereby significantly reducing computational effort by approximately 87.5%, from 2 h to 15 min, without compromising the accuracy of parameter estimation.

Unlike other approaches that necessitate substantial mass modifications or invasive interventions, the proposed framework relies solely on routine static loading and vibration measurements. This non-intrusive methodology facilitates seamless integration with existing structural health monitoring systems for periodic and real-time assessments. Overall, the combined numerical and experimental results underscore the robustness, scalability, and engineering practicality of the unified static–dynamic data fusion approach in addressing large-scale problems characterized by significant stiffness–mass coupling. Future work will explore its application to complex bridge systems and other large-scale structures, incorporating adaptive sampling and sensitivity analysis to further enhance precision and computational performance.