Finite Element Model Updating Technique for Super High-Rise Building Based on Response Surface Method

Abstract

To establish a finite element model that accurately represents the dynamic characteristics of actual super high-rise building and improve the accuracy of the finite element simulation results, a finite element model updating method for super high-rise building is proposed based on the response surface method (RSM). Taking a 120 m super high-rise building as the research object, a refined initial finite element model is firstly established, and the elastic modulus and density of the main concrete and steel components in the model are set as the parameters to be updated. A significance analysis was conducted on 16 parameters to be updated including E1–E8, D1–D8, and the first 10 natural frequencies of the structure, and 6 updating parameters are ultimately selected. A sample set of updating parameters was generated using central composite design (CCD) and then applied to the finite element model for calculation. The response surface equations for the first ten natural frequencies were obtained through quadratic polynomial fitting, and the optimal solution of the objective function was determined using a genetic algorithm. The results of the engineering case study indicate that the errors in the first ten natural frequencies of the updated finite element model are all within 5%. The updated model accurately reflects the current situation of the super high-rise building and provides a basis for super high-rise building health monitoring, damage detection, and reliability assessment.

1. Introduction

With the continuous development of China’s economy and urban construction, super high-rise buildings have emerged rapidly. According to the Council on Tall Buildings and Urban Habitat (CTBUH), as of 2024, China has 25 of the top 50 super high-rise buildings, and the number of super high-rise buildings over 150 m has exceeded 3000. During the service of super high-rise buildings, due to the continuous influence of the environment and load, the structure suffers damages due to material aging, environmental corrosion, foundation settlement, and design defects. The accumulation of such damage further reduces the bearing capacity of the structure and threatens the overall safety. Therefore, health monitoring techniques that ensure safe operation and maintain the performance of the structure are of great practical significance and have broad application prospects.

In the health monitoring, damage detection, and optimization design of super high-rise structures, it is crucial to establish a finite element model that accurately represents the actual structural characteristics [1,2,3,4]. However, due to the simplified assumptions, material parameters, and boundary condition errors during the modeling process, there is often a discrepancy between the computed response of the finite element model and the measured response. Therefore, it is necessary to update the initial model to obtain a finite element model that truly reflects the actual structure.

Traditional updating methods, especially those that directly involve iterative calculations using complex finite element models, are often time consuming and inefficient. To address this problem, researchers introduced the concept of surrogate models, such as response surface models [5,6,7,8], Kriging models [9,10,11], and radial basis function models [12,13]. These methods establish explicit relationships between structural responses and design parameters, replace the iterative process of calling the finite element model, and thus significantly improve the efficiency of the updating process.

Currently, based on the target of updating, finite element model updating methods can be divided into matrix-based and parameter-based methods [14]. Among them, the response surface method, as a parameter-based updating technique, has been widely used in the finite element model updating of complex structures such as bridges due to its ease of implementation, low computational effort, and high accuracy [15,16]. Zhong et al. [17] used the response surface method to update the finite element model of Xiabaishi Bridge, and the calculated values of the updated model matched with the measured values, with a maximum relative error of no more than 6%. Ren et al. [18] proposed a structural dynamics finite element model (FEM) updating method based on response surfaces, and the practical application of the response surface method to a full-scale bridge demonstrated that the updated model was effective, converged quickly, and could replace the initial FEM. Fu and Kang et al. [19,20] conducted field experiments on a concrete continuous rigid-frame bridge and a composite box girder bridge, and the errors between the calculated values of the updated models and the measured values were all within a reasonable range. Cheng et al. [21] verified the validity of the FEM updating with a simplified model instead of a refined model in a shaped arch bridge, and the verified refined numerical model provided a theoretical basis for health monitoring, safety evaluation, and performance prediction of the bridge. Zhang et al. [22] proposed a method of model updating for a cable-stayed bridge based on frequency monitoring. By optimizing the response surface equations and objective function, the vibration frequency error of the updated model was significantly reduced, resulting in a more accurate simulation of the actual bridge structure. Fang et al. [23] updated a multi-scale model of a concrete-filled steel tube tied-arch bridge using the response surface method based on a radial basis function network. The results showed that the error of the updated model significantly decreased and the method was effective for multi-scale bridge models established using different programs. In recent years, finite element model updating techniques based on artificial intelligence and deep learning have been increasingly applied in practical engineering, laying the foundation for real-time structural state assessment and safety warnings [24,25].

In this study, based on the response surface method, a significance test was conducted on material parameters and selected highly significant parameters as updating parameters for response surface fitting, thereby establishing the functional relationship between the updating parameters and the response values. Subsequently, an objective function was constructed and optimized to determine the updated parameters. Using a 120 m super high-rise structure as the research subject, the updated finite element model more accurately reflects the dynamic characteristics of the actual structure, making it suitable for the health monitoring, damage detection, and reliability assessment of super high-rise structures.

2. Response Surface Method Model Updating Theory

Finite element model updating based on the response surface method [26] involves first selecting appropriate parameter values and response values, and then choosing representative samples within a reasonable range of structural parameter values using an experimental design method. Finite element software such as ABAQUS and ANSYS are used to calculate the response values of these samples, and an analysis of variance (ANOVA) is conducted to identify the parameters that have a significant effect on the response values. Next, an appropriate response surface function is selected for fitting and its accuracy is verified. An objective function is constructed to compare the calculated response surface values with the measured values, replacing the original numerical model with the response surface model for parameter optimization, so as to obtain the updated model parameters. These parameters are substituted back into the finite element model to obtain the updated model. The process of finite element model updating based on the response surface method is shown in Figure 1.

2.1. Experimental Design

Experimental design is used to determine the number of sample points and their spatial distribution through a reasonable experimental program, so as to improve the accuracy of the experimental results. Experimental design for the response surface method allows a high accuracy response surface model to be obtained using a small number of experimental data.

Commonly used experimental design methods include two-level factorial design, optimal design, central composite design (CCD), and Box–Behnken design (BBD). When the number of design parameters is large, the optimal design is the most widely used because the number of sample points can be adjusted according to the needs and adapted to various sample spaces. Among them, D-optimal design has the criterion of minimizing the estimation variance of the design samples, which makes it the most commonly used method currently. For constructing second-order response surfaces, the CCD is also widely used due to its good orthogonality and rotatability.

2.2. Parameter Significance Test

After determining the parameters to be updated in the structure, it is also necessary to perform further screening of the parameters, from which the parameters that have a significant effect on the response of the structure are selected to ensure the accuracy of the response surface fitting. ANOVA in statistical analysis is an important tool to study whether an independent variable significantly affects the dependent variable. A commonly used method is the F-test, with the calculation formula being as follows [27]:

(1)${\mathrm{F}}_{\mathrm{A}}={\displaystyle \frac{{\mathrm{S}}_{\mathrm{A}}^2}{{\mathrm{f}}_{\mathrm{A}}}}\times {\displaystyle \frac{{\mathrm{f}}_{\mathrm{E}}}{{\mathrm{S}}_{\mathrm{E}}^2}}~\mathrm{F}\left({\mathrm{f}}_{\mathrm{A}},{\mathrm{f}}_{\mathrm{E}}\right)$

where S_A is the sum of squared deviations due to factors, S_E is the sum of squared deviations due to the test error, f_A is the degrees of freedom for each factor, and f_E is the degrees of freedom for the test error.

The F-value obtained from the calculation of the design parameters is compared with the corresponding values in the F-test table, where α is the confidence interval, indicating the probability of error in judgement, and the commonly used judgement criteria are as follows: when ${\mathrm{F}}_{\mathrm{A}}\le 0.01$, the effect of the factor is extremely significant; when $0.01<{\mathrm{F}}_{\mathrm{A}}\le 0.05$, the effect of the factor is significant; when $0.05<{\mathrm{F}}_{\mathrm{A}}\le 0.1$, the effect of the factor has a certain effect; when $0.1<{\mathrm{F}}_{\mathrm{A}}$, the effect of the factor does not have a significant effect [27].

2.3. Response Surface Fitting and Accuracy Test

Commonly used response surface functions include polynomial functions, radial basis functions, and Kriging functions [14]. Due to the strong nonlinear capability and simplicity of polynomial functions, a complete quadratic polynomial is used to fit the complex nonlinear relationship between parameters and structural response. Its expression is as follows:

(2)$\mathrm{y}={\mathsf{\beta }}_0+{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{k}}{\mathsf{\beta }}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}+{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{k}}{\mathsf{\beta }}_{\mathrm{ii}}{\mathrm{x}}_{\mathrm{i}}^2+{\displaystyle \sum }_{\mathrm{i}<\mathrm{j}}{\mathsf{\beta }}_{\mathrm{ij}}{\mathrm{x}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{j}}$

where y is the response value, ${\mathsf{\beta }}_0$ is the constant term, ${\mathrm{x}}_{\mathrm{i}}$ and ${\mathrm{x}}_{\mathrm{j}}$ are the independent variable parameters, ${\mathsf{\beta }}_{\mathrm{i}},{\mathsf{\beta }}_{\mathrm{ii}}, {\mathrm{and}\mathsf{\beta }}_{\mathrm{ij}}$ are the linear, quadratic, and cross-term coefficients, respectively.

The commonly used response surface accuracy test metrics are the coefficient of determination R², the adjusted coefficient of determination (Adj-R²), and the root mean square error (RMSE). Their expressions are as follows:

(3)${\mathrm{R}}^2=1-{\displaystyle \frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\hat{\mathrm{y}}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}\right)}^2}{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{i}}-{\stackrel{-}{\mathrm{y}}}_{\mathrm{i}}\right)}^2}}$

(4)$\mathrm{Adj}-{\mathrm{R}}^2=1-{\displaystyle \frac{\left(1-{\mathrm{R}}^2\right)\left(\mathrm{n}-1\right)}{\mathrm{n}-\mathrm{p}-1}}$

(5)$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{i}}-{\stackrel{-}{\mathrm{y}}}_{\mathrm{i}}\right)}^2}$

where ŷ represents the calculated value of the response surface model, y represents the calculated value of the finite element model, ȳ represents the mean calculated value of the finite element model, n is the total number of samples, and p is the total number of variables. R² reflects the degree of response surface fitting; therefore, the closer it is to 1, the higher the fitting accuracy of the response surface. Adj-R² introduces the number of samples and variables. When added variables contribute to the response surface model, Adj-R² increases; otherwise, it decreases. The RMSE reflects the error after the response surface is fitted, and its value is better when closer to 0.

2.4. Objective Function Optimization

When constructing the objective function for the finite element model updating of super high-rise structures, natural frequency is commonly chosen as the dynamic response parameter. The objective function is typically set as the sum of the squared relative residuals between the response values and the measured values, as shown in the following equation:

(6)$\mathrm{F}={\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{n}}{\left({\displaystyle \frac{{\mathrm{f}}_{\mathrm{j}}}{{\mathrm{f}}_{\mathrm{jj}}}}-1\right)}^2$

where f_j represents the response value, and f_jj represents the measured value. After defining the objective function, genetic algorithms are developed to optimize the objective function, resulting in the updated parameter values.

3. Finite Element Model Updating

3.1. Environmental Vibration Testing

The MCC Technology Building is located in the Qianhai Guiwan area of Shenzhen, with a total construction area of 66,641 square meters. The building consists of two sections: one is a 119.95 m tall 29-story office building (the main tower) with a frame–core tube structure, and the other is a 24 m tall 5-story commercial building (the podium) with a steel-frame structure. The long side of the office building measures 48.3 m and the short side measures 30 m; the long side of the commercial building measures 62.8 m and the short side measures 22.05 m. The height of the first floor of the office building is 5.1 m, the height of the standard floor is 4.5 m, and the refuge floors are set on the 8th and 19th floors, with the height of the refuge floors being 3.9 m. The architectural rendering of the building and the office building standard floor plan are shown in Figure 2 and Figure 3.

During the project operation phase, the comfort of the building was monitored in real time to provide early warnings when environmental factors such as wind vibration caused building movement, thus reducing the discomfort of the occupants (Figure 4). Comfort monitoring employed accelerometers to measure data, which were used to identify modal parameters and obtain the dynamic characteristics of the structure. At the 8th, 19th, and 27th floors of the structure, two measurement points were arranged per floor, each equipped with an accelerometer in the X direction (east–west) and the Y direction (north–south). The sampling frequency was set to 100 Hz, with a measurement range of ±2 g, and the sampling duration was no less than 30 min. Based on the monitoring data, the stochastic subspace identification (SSI) method was used to identify the natural frequencies of the structure.

3.2. Finite Element Modeling

Based on the construction drawings of the structure, the ABAQUS model and the YJK model of the MCC Technology Building are established, respectively, as shown in Figure 5. In the ABAQUS model, B31 units are used for beams and columns, while S4R and S3R units are used for shear walls and floor slabs. After constructing the model, the total number of nodes is 60,363 and the total number of units is 50,588. In the YJK model, both beams and columns are modeled using fiber beam units, while shear walls and floor slabs are modeled using shell units. Meanwhile, during the establishment and analysis of the YJK model, the geometric dimensions and control indicators were separately checked against the drawings and compared with the relevant codes. The geometric dimensions include axis grid dimensions, floor elevations, and component cross-sectional sizes, while the control indicators include period ratio, stiffness ratio, stiffness-to-weight ratio, shear-to-weight ratio, and interlayer displacement angle. The results show that both the geometric dimensions and control indicators of the model meet the requirements of the drawings and codes. The material parameters of the main components in the ABAQUS model are shown in Table 1. To verify the accuracy of the model, the natural frequencies calculated from both models are compared, as shown in Table 2.

As shown in Table 2, the differences in the first 10 natural frequencies between the ABAQUS model and the YJK model of the structure are minimal, indicating that the ABAQUS model can be considered accurate.

3.3. Parameters Selection

For complex super high-rise structures, having too many parameters to update can lead to long computation times and low updating efficiency, while having too few parameters may lead to unrepresentative and meaningless updating results. Therefore, it is necessary to select representative parameters for model updating. The objective of this updating is to match the natural frequencies of the structure, and the parameters selected for updating include the elastic modulus and density of concrete and steel components. These parameters are selected according to the material parameters of the main structural elements during the establishment of the finite element model. The detailed settings of the updating parameters are shown in Table 3.

After determining the parameters to be updated, D-optimal design is used for experimental design of parameter samples. Under the two-level conditions of 16 parameters, the D-optimal design obtains a total of 142 samples, as shown in Table 4. The experimental data are subjected to analysis of variance (ANOVA) to test the significance of a total of 16 parameters in the model, E1-E8 and D1-D8, in the responses R1–R10 using the F-test. Among them, the responses R1-R10 are obtained through modal analysis of the finite element model, with the first ten natural frequencies selected from the modal analysis results as the responses. In the significance test, the relationship between the significance level p-value and the degree of significance is as follows:

(7)$\{\begin{array}{cc}p\le 0.01\hfill & \hfill \mathrm{E}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{l}\mathrm{y}\mathrm{S}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{t}\\ p\le 0.05\hfill & \hfill \mathrm{S}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{t}\\ 0.05<p\le 1\hfill & \hfill \mathrm{N}\mathrm{o}\mathrm{t}\mathrm{S}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{t}\end{array}$

The p-value of the significance level of each parameter in the response is calculated according to Equation (7) and is shown in Figure 6. From Figure 6, it can be observed that parameters E1, E3, E6, E8, D1, and D8 have significant effects on responses R1 to R10, with each parameter having p-values below 0.05 for at least seven modes of response. This indicates that these parameters have a considerable influence on the natural frequencies of the super high-rise structure and can be selected as parameters for model updating.

3.4. Response Surface Model and Accuracy Test

The response surface is fitted using a central composite design (CCD) with six parameters to generate a total of 86 samples, and the first ten orders of frequency are selected as the output response. Using a quadratic polynomial to fit the response surface model for the first ten orders of frequency of the structure and using the backward selection method to eliminate the terms that have less influence on the equations, the first four orders of the response surface equations of the structure are obtained as shown in Equations (8)–(11). The response surfaces of the elastic modulus E1 and density D1, E3 and D8, and E6 and E8 on the first four frequencies of the structure are shown in Figure 7, Figure 8, Figure 9 and Figure 10. As can be seen from the figures, the response surfaces are continuous and smooth, which preliminarily indicates a high fitting accuracy of the response surface.

(8)$\begin{array}{cc}\hfill \mathrm{R}1=0.259049& +0.43325\mathrm{E}1+0.027433\mathrm{E}3+0.139683\mathrm{E}6+0.000024\mathrm{E}8-0.391423\mathrm{D}1\hfill \\ & -0.001804\mathrm{D}8+0.089239\mathrm{E}1\mathrm{E}6-0.166094\mathrm{E}1\mathrm{D}1-0.388534\mathrm{E}{1}^2-0.149107\mathrm{E}{6}^2\\ & +0.341923\mathrm{D}{1}^2\end{array}$

(9) $\begin{array}{cc}\hfill \mathrm{R}2=0.362554& +0.57646\mathrm{E}1+0.042701\mathrm{E}3+0.200797\mathrm{E}6+0.00003\mathrm{E}8-0.544633\mathrm{D}1\hfill \\ & -0.002447\mathrm{D}8+0.115684\mathrm{E}1\mathrm{E}6-0.215937\mathrm{E}1\mathrm{D}1-0.524101\mathrm{E}{1}^2-0.208668\mathrm{E}{6}^2\\ & +0.469222\mathrm{D}{1}^2\end{array}$

(10) $\begin{array}{cc}\hfill \mathrm{R}3=0.403019& +0.921404\mathrm{E}1+0.017617\mathrm{E}3+0.02567\mathrm{E}6+0.000087\mathrm{E}8-0.349434\mathrm{D}1\hfill \\ & -0.004481\mathrm{D}8+0.128566\mathrm{E}1\mathrm{E}3+0.135893\mathrm{E}1\mathrm{E}6-0.374219\mathrm{E}1\mathrm{D}1-0.827367\mathrm{E}{1}^2\end{array}$

(11) $\begin{array}{cc}\hfill \mathrm{R}4=0.393445& +0.078596\mathrm{E}1-3.49907\times {10}^{-16}\mathrm{E}3-6.16151\times {10}^{-17}\mathrm{E}6+0.193491\mathrm{E}8\hfill \\ & +0.062138\mathrm{D}1-0.016033\mathrm{D}8-0.069787\mathrm{E}8\mathrm{D}1-0.006948\mathrm{E}8\mathrm{D}8-0.444905\mathrm{D}{1}^2\end{array}$

The accuracy of the response surface fitting is further evaluated using the coefficient of determination R², the adjusted coefficient of determination Adj-R², and the root mean square error (RMSE). The closer the values of R² and Adj-R² are to 1, and the closer the RMSE is to 0, the higher the fitting accuracy of the response surface. The results of the accuracy assessment for each response surface equation are shown in Table 5. As shown in the table, both coefficients of the fitted equations are greater than 0.96 and the RMSE is also close to 0, which indicates that the response surface equation is fitted with high accuracy.

3.5. Objective Function Optimization Solution

The first ten order frequencies are selected to construct the residual objective function, and the form of the objective function is set as shown in Equation (6). A genetic algorithm is used to find the model updating parameter values that minimize the objective function, and the updated parameter values are substituted back into the finite element model to obtain the dynamic response of the updated finite element model.

The genetic algorithm is written in Python, and the algorithm primarily completes biological genetic behaviors through the combination of three basic genetic operators: selection, crossover, and mutation. The selection operator uses tournament selection, the crossover operator uses single point crossover, and the mutation operator uses uniform mutation. The genetic algorithm aims to minimize the best fitness value during parameter setting, and gradually sets the gene length, population size, crossover rate, mutation rate, and number of iterations. The parameters of the genetic algorithm are set as follows: the population size is 100, the number of iterations is 200, the crossover rate is 0.8, the mutation rate is 0.01, and the gene length is 6. After running five times, the convergence curve of the objective function is as shown in Figure 11. The updated parameter values obtained after each run of the genetic algorithm are the same, and, based on the fact that the optimal objective function values obtained after the algorithm runs to convergence each time no longer change, it can be considered that the algorithm has found the optimal solution to the objective function.

The updated parameters are shown in Table 6. The variation in the updated parameters indicates that the parameters still have physical meaning. The updated parameters are substituted into the finite element model for calculation, and the calculated values are compared with the measured values, as shown in Figure 12 and Figure 13.

From the graph of frequency values before and after updating, it can be seen that the updated values are closer to the measured values compared to the initial values. From the error plots before and after updating, it can be seen that, although the relative error of the eighth order frequency is slightly increased and the percentage of error reduction is negative, the relative errors of the rest of the order frequencies are significantly decreased. The error comparison chart shows that although the relative error of the eighth frequency slightly increases (indicating a negative reduction rate), the relative errors of the other frequencies show a noticeable decrease. The average error reduction for the first ten frequencies of the structure is 4.22%, with a maximum error of 4.55% and a minimum error of 0.09%. The first three mode shapes in the X and Y directions calculated by the updated model are in good agreement with the measured mode shapes. The updated results show that the response surface method can accurately describe the relationship between the updated parameters and the structure’s natural frequencies. The method successfully achieves finite element model updating for the complex super high-rise structure, with error levels that meet practical engineering requirements.

The updated finite element model serves as a benchmark model for damage detection in super high-rise structures, and, combined with structural health monitoring data, enables the assessment of the safety and reliability of super high-rise structures. At present, many scholars use the finite element model to assess the state of super high-rise structures as well as to predict the remaining life in order to identify, locate, quantify, and diagnose structural damages so as to ensure the safe use of super high-rise buildings.

4. Conclusions

In this paper, the material parameters of the finite element model are optimized based on the response surface method, obtaining a model that can more accurately reflect the dynamic characteristics of super high-rise structures. The main research conclusions are as follows:

(1) The response surface method can accurately describe the relationship between the updated parameters and the structure’s natural frequencies. Compared with the traditional parametric finite element model updating method, the response-surface-method-based model updating only requires finite element calculations when obtaining sample points, while parameter optimization does not require repeated finite element calculations. This approach significantly reduces the computational cost of model iteration. The calculated frequency of the updated finite element model of the super high-rise structure based on the response surface method is close to the measured frequency, and the maximum relative error after updating decreases by 9.8% from the initial model, with the maximum error being 4.55% and the minimum error being 0.09%. The updated finite element model serves as a benchmark for subsequent structural damage detection and reliability analysis.

(2) The selection of parameters to be updated and the experimental design are crucial to the model updating results based on the response surface method. Through the significance analysis of the parameters to be updated, it is evident that the elastic modulus and density of concrete and steel components significantly affect the natural frequencies of super high-rise structures. However, due to the large number of main components in super high-rise structures, selecting appropriate parameters to be updated is often quite challenging. Therefore, it is especially necessary to set the parameters to be updated according to the material characteristics of the model. Meanwhile, more accurate significance analysis is a promising direction for further research, as it can contribute to constructing response surface models and improving the updating efficiency of complex models.

Author Contributions

Conceptualization, Y.W.; methodology, Y.W.; software, Y.W. and H.L.; validation, Y.W. and H.L.; formal analysis, D.Z.; investigation, D.Z. and H.L.; resources, H.L.; data curation, Y.W.; writing—original draft preparation, Y.W.; writing—review and editing, Y.W. and H.L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data of this study are available in the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. Flow of finite element model updating based on response surface method.

Figure 2. Architectural rendering.

Figure 3. Office building standard floor plan.

Figure 4. Environmental vibration test site diagram.

Figure 5. Structural finite element model diagram.

Figure 6. Significance analysis of parameters and response.

Figure 7. Schematic of first-order translational response surface in Y direction.

Figure 8. Schematic of first-order translational response surface in X direction.

Figure 9. Schematic of first-order torsional response surface in XY direction.

Figure 10. Schematic of second-order translational response surface in Y direction.

Figure 11. Convergence curve of the objective function.

Figure 12. Comparison of frequencies and relative errors before and after updating.

Figure 13. Comparison of the updated model with the test vibration modes.

Figure 13. Comparison of the updated model with the test vibration modes.

Table 1

Material parameters of main structural components.

Floor Number	Concrete Components				Steel Components
	Beam	Column	Wall	Brace	Beam	Column	Brace
29	C30	C30	/	/	Q345	Q345	/
28	C30	/	C30	/	Q345	Q345	Q345
25–27	C30	C30	C30	/	/	/	/
22–24	C30	C35	C35	/	/	/	/
18–21	C30	C40	C40	/	/	/	/
17	C30	C45	C45	/	/	/	/
14–16	C30	C45	C45	C45	/	/	/
13	C30	C50	C50	/	/	/	/
10–12	C30	C50	C50	C50	/	/	/
9	C30	C50	C50	/	/	/	/
6–8	C30	C55	C55	/	/	/	/
5	C30	C55	C55	/	Q345	Q345	/
1–4	C30	C60	C60	/	Q345	Q345	/

Table 2

Comparison of ABAQUS model and YJK model frequencies.

Order	Frequency
	f1/Hz	f2/Hz	Difference/%
1	0.31304	0.30792	1.64
2	0.43474	0.40909	5.90
3	0.52803	0.48352	8.43
4	0.74549	0.73752	1.07
5	0.77199	0.76126	1.39
6	0.92414	0.88244	4.51
7	0.98029	0.97363	0.68
8	1.31640	1.24388	5.51
9	1.44340	1.36488	5.44
10	1.54950	1.52681	1.46

Note: f₁ and f₂ represent the frequencies calculated from the ABAQUS and YJK models, respectively. The difference is calculated as (|ABAQUS frequency − YJK frequency|/ABAQUS frequency) × 100%.

Table 3

Parameter settings to be updated in the model.

Material Properties	Component Types	Parameters to Be Updated	Notation	Initial Value	Lower Value	Upper Value
Elastic modulus/GPa	Concrete	B1–29; C25–27, 29; W25–28	E1	30	24	36
		C22–24; W22–24	E2	31.5	25.2	37.8
		C18–21; W18–21	E3	32.5	26	39
		C14–17; Br14–16; W14–17	E4	33.5	26.8	40.2
		C9–13; Br10–12; W9–13	E5	34.5	27.6	41.4
		C5–8; W5–8	E6	35.5	28.4	42.6
		C1–4; W1–4	E7	36	28.8	43.2
	Steel	B1–5, 28–29; C1–5, 28–29; Br28	E8	206	165	247
Density /(kg·m−3)	Concrete	B1–29; C25–27, 29; W25–28	D1	2500	2000	3000
		C22–24; W22–24	D2	2500	2000	3000
		C18–21; W18–21	D3	2500	2000	3000
		C14–17; Br14–16; W14–17	D4	2500	2000	3000
		C9–13; Br10–12; W9–13	D5	2500	2000	3000
		C5–8; W5–8	D6	2500	2000	3000
		C1–4; W1–4	D7	2500	2000	3000
	Steel	B1–5, 28–29; C1–5, 28–29; Br28	D8	7800	6240	9360

Note: B1–29 represents beams on floors 1–29; C25–27, 29 represents columns on floors 25–27 and 29; W25–28 represents walls on floors 25–28; and Br14–16 represents braces on floors 14–16.

Table 4

The sample values of the experimental design.

Number	E1	E2	E3	E4	E5	E6	E7	E8	D1	D2	D3	D4	D5	D6	D7	D8	R1	R2	R3	R4
1	0.24	0.38	0.39	0.40	0.28	0.43	0.29	2.47	0.30	0.30	0.30	0.30	0.30	0.20	0.20	0.62	0.28	0.39	0.47	0.80
2	0.24	0.38	0.26	0.27	0.28	0.28	0.43	2.47	0.30	0.20	0.20	0.20	0.20	0.30	0.30	0.62	0.28	0.39	0.47	0.80
3	0.36	0.25	0.39	0.27	0.41	0.28	0.29	1.65	0.20	0.30	0.30	0.20	0.30	0.30	0.20	0.62	0.32	0.45	0.56	0.70
4	0.36	0.38	0.26	0.27	0.28	0.28	0.29	2.47	0.30	0.20	0.30	0.30	0.20	0.20	0.20	0.62	0.29	0.40	0.50	0.81
5	0.36	0.38	0.26	0.27	0.28	0.43	0.29	1.65	0.30	0.30	0.30	0.20	0.30	0.20	0.30	0.62	0.30	0.41	0.51	0.67
6	0.36	0.25	0.39	0.40	0.28	0.43	0.43	1.65	0.20	0.20	0.20	0.20	0.30	0.20	0.30	0.94	0.35	0.48	0.59	0.69
…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…
137	0.36	0.25	0.39	0.40	0.41	0.43	0.43	1.65	0.20	0.20	0.20	0.30	0.20	0.30	0.30	0.62	0.36	0.50	0.61	0.70
138	0.24	0.25	0.26	0.27	0.28	0.28	0.43	2.47	0.30	0.30	0.20	0.20	0.30	0.20	0.20	0.62	0.28	0.39	0.46	0.80
139	0.24	0.38	0.39	0.27	0.28	0.43	0.43	1.65	0.20	0.30	0.30	0.30	0.20	0.20	0.20	0.62	0.31	0.44	0.52	0.69
140	0.36	0.38	0.26	0.40	0.28	0.43	0.43	1.65	0.20	0.20	0.20	0.20	0.30	0.20	0.20	0.62	0.35	0.48	0.59	0.70
141	0.24	0.38	0.39	0.40	0.41	0.43	0.29	2.47	0.30	0.20	0.20	0.20	0.20	0.30	0.30	0.94	0.30	0.41	0.49	0.79
142	0.36	0.38	0.26	0.40	0.28	0.28	0.43	2.47	0.20	0.20	0.20	0.20	0.20	0.30	0.20	0.94	0.34	0.47	0.57	0.83

Note: The units for E1–E8 are 10⁻² GPa, the units for D1–D8 are 10⁻⁴ kg/m³, and the units for R1–R4 are Hz. Due to space limitations, only R1–R4 are shown in the table.

Table 5

Test of fitting accuracy of response surface model.

Order	R2	Adj-R2	RMSE
1	0.9998	0.9997	0.0003
2	0.9997	0.9997	0.0004
3	0.9994	0.9993	0.0008
4	0.9993	0.9993	0.0017
5	0.9999	0.9999	0.0006
6	0.9936	0.9926	0.0056
7	0.9791	0.9760	0.0065
8	0.9805	0.9776	0.0089
9	0.9671	0.9617	0.0134
10	0.9778	0.9745	0.0177

Table 6

Parameters change before and after updating.

Updated Parameters	E1	E3	E6	E8	D1	D8
Initial Value	30	32.5	35.5	206	2500	7800
Updated Value	24	38.1	34.6	247	2252	7021
Updated Range/%	20	17.2	2.5	19.9	9.9	10

Note: The units of E1, E3, E6, and E8 are Gpa, and the units of D1 and D8 are kg·m⁻³.

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