Damage Identification of Gas Station Double Layer Grid Structure Based on Time Domain Response Sensitivity Analysis

Abstract

Gas station canopy grid structures develop local damage during service life, necessitating regular inspection and maintenance to prevent structural collapse. However, conventional field inspection remains inefficient and highly dependent on manual operation. This paper proposes a time domain response sensitivity methodology for damage assessment of structural members in gas station canopy grid structures. The proposed methodology advances time-domain sensitivity analysis to handle spatially complex grid structures with dense spectral characteristics, while proposing a calculation method for implementing intelligent sensing technology in field inspections that enables automated damage localization in practical canopy structures. Through analyzing time domain response sensitivity matrix, an optimal sensor placement method for spatial grid structures is presented. A double-layer spatial grid structure model is constructed to validate the time domain response sensitivity damage identification method and the optimal sensor placement method based on sensitivity analysis. The results show that the time domain response sensitivity damage identification method identifies the member damage in gas station canopy grid structural numerical model with satisfactory accuracy and efficiency, the optimal sensor placement methodology is suitable for damage identification of structural members.

1. Introduction

The canopy structure is a critical component of gas stations, typically constructed as a double-layer spatial grid structure. Although its construction scale is not as large as many conventional large-span spatial structures, it is numerous and widely distributed in cities, villages and highway service zones. Owner demands for rapid construction schedules and the unique operational requirements of the petroleum industry give rise to deficiencies in design and construction, resulting in initial defects within canopy grid structures [1,2]. During service, structural components continuously deteriorate due to environmental exposure, experiencing aging [3], corrosion [4], and degradation of mechanical properties [5,6]. Maintenance and operation are often not standardized [7]. The combined effect of these factors inevitably leads to progressive local damage within the canopy grid structure over time, as shown in Figure 1. The local damage subsequently compromises the overall structural performance, gradually reduces the ultimate load-bearing capacity, and creates potential safety hazards [8,9,10]. Structures with existing local damage exhibit reduced reliability under various loading conditions [11,12]. The risk of structural collapse increases significantly during extreme weather events, such as strong winds or heavy snowstorms [13]. The interaction between local damage and severe weather frequently contributes to safety incidents.

To effectively mitigate inherent structural hazards and reduce collapse risks during service life, regular inspection, assessment, and timely repair of local damage in gas station canopy grid structures are essential. Field inspection serves as an effective method for identifying local damage within these structures. Traditional field inspection involves manual visual examination of all structural members in the grid structure, followed by non-destructive testing (NDT) of sampled members [14,15]. Visual inspection is heavily reliant on subjective experience: members exhibiting obvious damage features—such as fracture, bending deformation, or severe corrosion—are readily identifiable, whereas concealed damage or subtle defects are frequently overlooked. Moreover, visual methods cannot precisely quantify damage severity. In recent years, computer vision technology has attracted significant research interest, demonstrating potential to replace traditional manual visual inspection in field applications. While computer vision technology can improve inspection efficiency compared to manual visual checks, it fails to adequately resolve the problem that damage degree cannot be accurately quantified [16,17]. To avoid the above problems in traditional field inspections, comprehensive damage identification requires full-coverage NDT of all structural members [18,19,20]. However, this approach remains labor-intensive and highly dependent on manual operation. Consequently, conducting thorough field inspections of gas station canopy grid structures remain challenging in practice, characterized by operational complexity, significant time requirements, high dependence on specialized personnel, low efficiency, and substantial costs.

Precise identification of local damage is critical for structural reliability assessment under various loading conditions during service life [21,22,23]. Intelligent sensing technology has been increasingly adopted for real-time online health monitoring of large-scale structures [24,25,26]. In the health monitoring of large-scale structures, using data from various sensors can effectively identify structural damage [27,28,29]. Moreover, comprehensive analysis of sensor data enables early warning of potential hazards in large-scale structures [30,31,32]. These findings from real-time structural health monitoring studies demonstrate that intelligent sensing technology holds promising application prospects for field inspection. Through rational deployment of sensors in structures, automated damage detection can be effectively achieved during field inspections. Compared with traditional field inspection, automated damage identification using intelligent sensing technology demonstrates significantly higher operational efficiency. Time domain damage identification approaches utilizing structural dynamic response information from sensors have been extensively researched, due to their effectiveness in dealing with limited and short duration measurements from a structure under operational conditions [33,34,35]. Law et al. [36] proposed a method for identifying coupling forces between substructures using acceleration responses under support excitation. Subsequently, structural local damage is identified through dynamic sensitivity analysis based on the identified coupling forces. The method presented in this study has been validated for damage identification in planar truss structures. Li et al. [37] presented a damage identification approach for bridge structures subjected to moving vehicular loads, independent of vehicle properties and moving force time histories. This approach is applicable to bridge structures. Liu et al. [38] enhanced sensitivity-based damage detection through singular spectrum analysis. Structural responses are decomposed, with sensitivity and computed response vectors projected into the corresponding subspace. Li et al. [39] adopted the covariance of strain response as a modal-parameter-dependent damage index to detect the local damage in the structure. The damage identification techniques employed in the two aforementioned studies are suitable for frame structures. Nabavi et al. [40] developed an efficient methodology for damage localization and severity quantification using time-domain structural responses through optimization techniques. The research has proven effective for damage detection in offshore platform structures. Guo et al. [41] proposed a substructure-based seismic damage detection method for isolated buildings using partial noise-contaminated measurements. This approach is applicable to shear wall structures. Baybordi et al. [42] developed an innovative explicit sensitivity equation for structural parameter estimation using time-domain measurements. The formulation establishes a direct linear relationship between response time histories and structural parameter variations. Baybordi et al. [43] proposed closed-form equations demonstrating high computational efficiency for time-domain response sensitivity computation. By directly populating the sensitivity matrix with measured responses, this approach enhances the accuracy and reliability of finite element model updating procedures. These two investigations have advanced the time-domain sensitivity methodology, yielding modified approaches suited for planar truss structures and offshore platform structures, correspondingly. In summary, while various damage identification methods based on time-domain response sensitivity have been successfully applied to different types of structures, this category of approaches has not yet been extended to more complex spatial grid structures with a larger number of members.

This paper proposes a damage assessment method for structural members in gas station canopy grid structures based on time domain response sensitivity. This method extends the applicability of time-domain sensitivity-based damage identification to spatial grid structures characterized by numerous structural elements and closely spaced modal frequencies. Furthermore, it proposes a calculation method for implementing intelligent sensing technology in field inspections, thereby enabling automated detection of damaged members in gas station canopy grid structures. Through analyzing time domain response sensitivity matrix, an optimal sensor placement method for spatial grid structures is presented. The time domain response sensitivity damage identification method and the optimal sensor placement method based on sensitivity analysis are numerically verified with simulation studies of a double-layer spatial grid structure.

2. Sensitivity-Based Damage Identification Algorithm

The equation of motion of a damped linear structure with multiple Dofs can be written as

$$\mathit{M}\ddot{\mathit{x}}+\mathit{C}\dot{\mathit{x}}+\mathit{Kx}=\mathit{LP}\left(t\right)$$

(1)

where $\mathit{M}$, $\mathit{C}$ and $\mathit{K}$ are the mass, damping and stiffness matrices of the structural system, respectively. $\mathit{P}\left(t\right)$ is the vector of external forces on the structure and $\mathit{L}$ is the mapping matrix for the external forces. $\ddot{\mathit{x}}$, $\dot{\mathit{x}}$ and $\mathit{x}$ are vectors of the acceleration, velocity and displacement responses of the structural system, respectively. The structure is assumed to exhibit Rayleigh damping for discussion as

$$\mathit{C}=a_1\mathit{M}+a_2\mathit{K}$$

(2)

where $a_1$ and $a_2$ are the damping coefficients. The proposed method can, however, adopt different damping models in the assessment.

To correlate the damage extent of structural members with physical stiffness degradation, assuming the fractional change in stiffness of the ith element in the structure as ${\alpha }_i$, a change in the global stiffness matrix can be described as

$$\mathsf{\Delta }\mathit{K}={\sum }_{i=1}^{\mathit{NE}}{\alpha }_i{\mathit{K}}_i$$

(3)

where ${\mathit{K}}_{\mathit{i}}$ is the stiffness matrix of the ith intact element with $0.0\le {\alpha }_i\le 1.0$, NE is the number of finite elements of the structure. Performing differentiation to both sides of Equation (1) with respect to the damage index ${\alpha }_i$, while considering Equation (2), we have

$$\mathit{M}{\displaystyle \frac{\partial \ddot{\mathit{x}}}{\partial {\alpha }_i}}+\mathit{C}{\displaystyle \frac{\partial \dot{\mathit{x}}}{\partial {\alpha }_i}}+\mathit{K}{\displaystyle \frac{\partial \mathit{x}}{\partial {\alpha }_i}}=-{\displaystyle \frac{\partial \mathit{K}}{\partial {\alpha }_i}}\mathit{x}-a_2{\displaystyle \frac{\partial \mathit{K}}{\partial {\alpha }_i}}\dot{\mathit{x}}$$

(4)

where ${\displaystyle \frac{\partial \ddot{\mathit{x}}}{\partial {\alpha }_i}}$, ${\displaystyle \frac{\partial \dot{\mathit{x}}}{\partial {\alpha }_i}}$ and ${\displaystyle \frac{\partial \mathit{x}}{\partial {\alpha }_i}}$ are the sensitivity vectors which can also be obtained by solving Equation (4) with a time-stepping integration method, such as the Newmark-β method. The length of the sensitivity vectors is the same as the number of measured data point. The sensitivity formulations for three distinct types of responses—displacement, velocity, and acceleration—are consolidated into a unified representation. The sensitivity matrix $\mathit{S}$ of the structure corresponding to the measured response can be determined from Equation (4) with

$$\mathit{S}=\left[\begin{array}{ccccc}{\mathit{S}}_{\alpha ,1}& \dots & {\mathit{S}}_{\alpha ,i}& \dots & {\mathit{S}}_{\alpha ,\mathit{NE}}\end{array}\right]$$

(5)

where ${\mathit{S}}_{\alpha ,i}$ is the response sensitivity with respect to the ith element.

The identification equation for the local damage of a structure with the response sensitivity matrix can be represented as

$$\mathit{S}\mathit{\alpha }+\mathrm{o}\left({\mathit{\alpha }}^2\right)={\ddot{\mathit{X}}}^{\mathit{cs}}-{\ddot{\mathit{X}}}^{\mathit{ms}}$$

(6)

where the measured acceleration response ${\ddot{\mathit{X}}}^{\mathit{ms}}$, the corresponding calculated acceleration response ${\ddot{\mathit{X}}}^{\mathit{cs}}$ and the acceleration sensitivity matrix $\mathit{S}$ in Equation (6) are known while the higher order term $\mathrm{o}\left({\mathit{\alpha }}^2\right)$ can be omitted. The proposed method can, however, adopt different measured responses in the assessment. The unknown damaged index vector $\mathit{\alpha }$ can be determined from Equation (6) with an iterative method described below, and Tikhonov regularization is used for optimizing the following objective function in the kth iteration as

$$\mathit{J}\left(\mathsf{\Delta }{\mathit{\alpha }}^k,{\lambda }_{\alpha }^k\right)={‖\mathit{S}\mathsf{\Delta }{\mathit{\alpha }}^k-({\ddot{\mathit{X}}}^{\mathit{cs}}-{\ddot{\mathit{X}}}^{\mathit{ms}})‖}^2+{\lambda }_{\alpha }^k‖\mathsf{\Delta }{\mathit{\alpha }}^k‖$$

(7)

where ${\lambda }_{\alpha }^k$ is the regularization parameter in the kth iteration obtained with the L-curve method.

The local damage of the structure can be detected with the following steps

Step 1: The measured responses ${\ddot{\mathit{X}}}^{\mathit{ms}}$ are collected from the damaged structure. Construct structural finite element model based on the original parameters. Set initial iteration number k = 0.

Step 2: The computed responses ${\ddot{\mathit{X}}}_k^{cs}$ and sensitivity matrix $\mathit{S}$ are obtained from the structural finite element model with Equation (1) and Equation (4), respectively. Then calculate the response difference ${\ddot{\mathit{X}}}_k^{cs}-{\ddot{\mathit{X}}}^{\mathit{ms}}$.

Step 3: Let k = k + 1 and then identify the kth local change increment of the stiffness $\mathsf{\Delta }{\mathit{\alpha }}^k$ with Equation (7).

Step 4: Update the finite element model and repeat Steps 2 to 4 until convergence in Equation (8) is met. The final damage index vector $\mathit{\alpha }$ which is used to represent the change in stiffness can be obtained as ${\mathit{\alpha }}^{k+1}={\sum }_{k+1}\mathsf{\Delta }{\mathit{\alpha }}^{k+1}$.

$$\begin{array}{cc}{\displaystyle \frac{‖{\mathit{\alpha }}^{k+1}-{\mathit{\alpha }}^k‖}{‖{\mathit{\alpha }}^{k+1}‖}}<Tol& \hspace{1em}Tol=10^{-7}\end{array}$$

(8)

The flowchart of the time domain response sensitivity damage identification method is shown in Figure 2.

3. Response Sensitivity Analysis

3.1. Feature Analysis of Response Sensitivity Matrix

To establish an optimal sensor placement strategy for members damage identification in gas station canopy grid structures, the feature of the response sensitivity matrix is investigated below.

The plane of the spatial grid structure is uniformly divided into n rectangular regions, using cells as fundamental units. The schematic diagram of the partitioned structural plane is shown in Figure 3, different colors represent different regions.

The motion equation of the ith region among n structural regions can be written as

$${\mathit{M}}_i{\ddot{\mathit{x}}}_i+{\mathit{C}}_i{\dot{\mathit{x}}}_i+{\mathit{K}}_i{\mathit{x}}_i={\mathit{L}}_i{\mathit{P}}_i$$

(9)

where the subscripts i denote the number of structural regions with $1\le i\le n$. The force vector ${\mathit{P}}_i$ includes external forces and structural internal forces between adjacent regions.

Assuming the local damage extent of the jth structural region as damage index vector ${\mathit{\alpha }}_j$, obviously $1\le j\le n$, the vector length is equal to the total number of members in the jth structural region, and each vector element quantifies the relative stiffness variation in a structural member within the jth partitioned region. The structural internal forces between adjacent regions are related to stiffness variation in all structural members. Due to the force vector ${\mathit{P}}_i$ includes the structural internal forces between adjacent regions, it is related to the damage index vector ${\mathit{\alpha }}_j$.

Performing differentiation to both sides of Equation (9), the motion equation of the ith region, with respect to the damage index vector ${\mathit{\alpha }}_j$, we have

$${\mathit{M}}_i{\displaystyle \frac{\partial {\ddot{\mathit{x}}}_i}{\partial {\mathit{\alpha }}_j}}+{\mathit{C}}_i{\displaystyle \frac{\partial {\dot{\mathit{x}}}_i}{\partial {\mathit{\alpha }}_j}}+{\mathit{K}}_i{\displaystyle \frac{\partial {\mathit{x}}_i}{\partial {\mathit{\alpha }}_j}}=-{\displaystyle \frac{\partial {\mathit{K}}_i}{\partial {\mathit{\alpha }}_j}}{\mathit{x}}_i-a_2{\displaystyle \frac{\partial {\mathit{K}}_i}{\partial {\mathit{\alpha }}_j}}{\dot{\mathit{x}}}_i+{\mathit{L}}_i{\displaystyle \frac{\partial {\mathit{P}}_i}{\partial {\mathit{\alpha }}_j}}$$

(10)

where ${\displaystyle \frac{\partial {\ddot{\mathit{x}}}_i}{\partial {\mathit{\alpha }}_j}}$, ${\displaystyle \frac{\partial {\dot{\mathit{x}}}_i}{\partial {\mathit{\alpha }}_j}}$ and ${\displaystyle \frac{\partial {\mathit{x}}_i}{\partial {\mathit{\alpha }}_j}}$ are the ith structural region response sensitivity matrices with respect to the damage index vector of the jth structural region.

In the global sensitivity matrix $\mathit{S}$ corresponding to the measured response, referenced earlier, each row vector contains all sensitivity values associated with a particular sensor location, while each column vector comprises all sensitivity values related to a specific structural member. Matrix reordering operations partition the acceleration sensitivity matrix according to the structural regions, where each partitioned submatrix corresponds to a structural region sensitivity matrix ${\displaystyle \frac{\partial {\ddot{\mathit{x}}}_i}{\partial {\mathit{\alpha }}_j}}$, as highlighted by the red rectangle in Figure 4.

The stiffness matrix of the ith structural region is only related to its own damage parameter, when $i\ne j$, the stiffness matrix ${\mathit{K}}_i$ is independent of the damage index vector ${\mathit{\alpha }}_j$. Thus, we obtain ${\displaystyle \frac{\partial {\mathit{K}}_i}{\partial {\mathit{\alpha }}_j}}=0$, and Equation (10) can also be written as

$${\mathit{M}}_i{\displaystyle \frac{\partial {\ddot{\mathit{x}}}_i}{\partial {\mathit{\alpha }}_j}}+{\mathit{C}}_i{\displaystyle \frac{\partial {\dot{\mathit{x}}}_i}{\partial {\mathit{\alpha }}_j}}+{\mathit{K}}_i{\displaystyle \frac{\partial {\mathit{x}}_i}{\partial {\mathit{\alpha }}_j}}=\left\{\begin{array}{ll}-{\displaystyle \frac{\partial {\mathit{K}}_i}{\partial {\mathit{\alpha }}_j}}{\mathit{x}}_i-a_2{\displaystyle \frac{\partial {\mathit{K}}_i}{\partial {\mathit{\alpha }}_j}}{\dot{\mathit{x}}}_i+{\mathit{L}}_i{\displaystyle \frac{\partial {\mathit{P}}_i}{\partial {\mathit{\alpha }}_j}}& i=j\\ {\mathit{L}}_i{\displaystyle \frac{\partial {\mathit{P}}_i}{\partial {\mathit{\alpha }}_j}}& i\ne j\end{array}\right.$$

(11)

In Equation (11), if $i\ne j$, only one term is considered on the right-hand side. If $i=j$, there are three terms are considered during the sensitivity solution. Obviously, the acceleration sensitivity matrices of uniformly divided region obtained from Equation (10) or Equation (11), ${\displaystyle \frac{\partial {\ddot{\mathit{x}}}_i}{\partial {\mathit{\alpha }}_j}}$ ($i\ne j$) have less contribution than ${\displaystyle \frac{\partial {\ddot{\mathit{x}}}_i}{\partial {\mathit{\alpha }}_i}}$ or ${\displaystyle \frac{\partial {\ddot{\mathit{x}}}_j}{\partial {\mathit{\alpha }}_j}}$ to the global sensitivity matrix $\mathit{S}$ corresponding to the measured acceleration response.

The response sensitivity in this work quantifies the rate of change in structural dynamic responses with respect to variations in structural stiffness, or equivalently, the influence of stiffness modifications on dynamic responses. Equation (11) demonstrates that the responses at measurement points of uniformly divided region are more significantly affected by stiffness variations within their own region, while the influence of stiffness changes in other regions, especially distant and non-adjacent ones, are generally less pronounced. This observation aligns with fundamental structural mechanics principles. Through row and column transformations, the global response sensitivity matrix is reconfigured into a partitioned block matrix. Based on the derived formulations and theoretical analysis, the distribution pattern of values within the reconstructed sensitivity matrix can be inferred: the largest-magnitude elements are expected to cluster predominantly within the submatrices along the main diagonal, as shown in Figure 5.

3.2. Numerical Verification for Sensitivity Matrix Features

A finite element model is developed with reference to typical practical engineering structures. The gas station canopy, featuring a double-layer spatial grid structure depicted in Figure 6, is adopted for this study. The structure is discretized using 576 spatial truss finite elements without internal nodes, interconnected by 162 hinge nodes, resulting in a total of 474 degrees of freedom. Hinge supports are applied at four intermediate nodes, indicated in Figure 6 by solid blue squares. The upper and lower chord members each have a length of 2.0 m, while the web members are 1.65 m long. All members have a diameter of 48 mm and a wall thickness of 3 mm, yielding a cross-sectional area of 0.000424 m². The structure spans a plane measuring 18 m × 16 m, with a height of 0.85 m. The first eight natural frequencies of the structure are 12.1 Hz, 12.7 Hz, 13.9 Hz, 15.0 Hz, 20.2 Hz, 21.9 Hz, 22.9 Hz and 28.5 Hz, respectively. Rayleigh damping is adopted for the system, and the two damping coefficients are $a_1=2.3332$ and $a_2=3.8555\times 10^{-4}$. The mass density and elastic modulus of material are, respectively, $7.8\times 10^3$ kg/m³ and 206 GPA.

A vertical external load is applied to the structural nodes, and the external load can be modeled as

$$F\left(t\right)=650\mathit{sin}(50\pi t)+600\mathit{sin}(110\pi t)+550\mathit{sin}(160\pi t)$$

to simulate excitations over a relatively wide range of frequencies. The sampling rate is 1000 Hz and the time duration of study is 0.5 s after the load application. The acceleration responses of the structure are calculated using the Newmark-β method as the “measured” responses.

The plane of the spatial grid structure is uniformly divided into 6 rectangular regions in this study, with region numbers marked on both sides of the structural plan view, as shown in Figure 7. A vertical dynamic load with an identical time history is applied sequentially to each region of the structure, and a total of six independent numerical simulations are conducted. The nodal application locations of this load are marked by red five-pointed stars in Figure 7. The expression for the time history of this vertical external load has been provided in the preceding section.

Under the original structural state, the acceleration sensitivity matrices are computed separately for loads applied to different regions. These matrices encompass information from all potential sensor locations and all structural elements. To eliminate the interference of the load on the sensitivity analysis, the structural sensitivity matrix is computed under the condition that the load acts simultaneously on all regions. To facilitate observation of the distribution patterns of the element values in the sensitivity matrix, every sensitivity vector norm with respect to single member stiffness change parameter at the different nodes is calculated. This vector incorporates information from all three directions movement and all sampling time points at the sensor location. Evidently, the norm of the sensitivity vector is positively correlated with the values of the sensitivity matrix elements at that measurement point. By replacing the vectors in the sensitivity matrix with their norms, a simplified form of the matrix is obtained. In this case study, this process yields seven simplified matrices of dimension 158 × 576. The number of rows corresponds to the total number of nodes minus the support nodes, while the number of columns equals the total number of structural member elements. Owing to structural symmetry, the features of sensitivity matrices are identical when the load is applied to regions 1, 2, 5, and 6, and likewise identical for loads applied to regions 3 and 4. Therefore, only three load cases need to be examined regarding the distribution of element values in the simplified matrices. To analyze the distribution pattern of element values in the simplified matrix, all elements are sorted in descending order, as shown in Figure 8.

As can be seen in Figure 8, the descending-order curves of the element values exhibit similar characteristics. To facilitate discussion, the value range of the elements in each matrix is divided into three consecutive intervals. The specific division is based on the proportion of the sum of element values within each interval of the total sum, set at 40%, 40%, and 20%, respectively. Accordingly, the elements are classified into three levels based on their respective intervals, where the values of Level 1 elements are greater than those of Level 2, which in turn are greater than those of Level 3. To facilitate discussion, Level 1 elements are defined as the significant sensitivity.

As shown in

Table 1. Proportion of Elements at Different Levels.

Load Applied Location	Proportion of Elements
	Level 1	Level 2	Level 3
Region 1	7.9%	29.6%	65.5%
Region 2	11.6%	34.1%	54.3%
All Region	12.5%	35.7%	51.8%

, for the case where the load is applied to Region 1, the value range of Level 3 elements in the simplified sensitivity matrix is from 0 to 0.6, accounting for 65.5% of the total number of elements. The value range of Level 2 elements is from 0.6 to 2.1, representing 29.6% of the total elements. Level 1 elements have a value range from 2.1 to 40.1 and constitute 7.9% of the total. When the load is applied to Region 3, the value range of Level 3 elements ranges from 0 to 0.8, making up 54.3% of all elements. Level 2 elements fall in the range of 0.8 to 2.4, accounting for 34.1%, while Level 1 elements span from 2.4 to 36.3 and represent 11.6% of the total. For the case where the load acts on all regions simultaneously, Level 3 elements cover a value range from 0.1 to 1.8, comprising 51.8% of all elements. Level 2 elements have values between 1.8 and 4.8, accounting for 35.7%, and Level 1 elements range from 4.8 to 63.4, representing 12.5% of the total number of elements.

Based on the above data analysis, common patterns can be summarized across all working conditions regarding the proportion and value ranges of elements at different levels in the simplified matrix. Level 2 elements and Level 3 elements significantly outnumber Level 1 elements, while the values of Level 1 elements are considerably larger than those of Level 2 and Level 3. Level 1 elements exhibit a wide value range but account for a small proportion of the total number, indicating that only a limited number of elements in the matrix possess highly significant sensitivity. In contrast, Level 3 elements have a narrow value range yet constitute over 50% of the total elements, suggesting that more than half of the elements exhibit insignificant sensitivity, with a considerable number of values approaching zero.

Row and column transformations are applied to the simplified sensitivity matrix, which rearranges the positions of the elements without altering their values, thereby partitioning the matrix according to the structural regions. Since the structural model is evenly divided into six regions, the matrix consists of 6 × 6 submatrices, totaling 36 blocks. For each submatrix, the average value of the elements, the proportion of Level 1 elements, and the proportion of Level 2 elements are computed. These three metrics are used to comprehensively evaluate the distribution of sensitivity within the partitioned matrix. If all three metrics of a submatrix exceed those of the global matrix, the submatrix is classified as having a high concentration of sensitivity. If one or two metrics exceed the global values, it is classified as medium; otherwise, it is considered low. The results for the three loading cases are shown in Figure 9. From the results, it can be observed that the load has an amplification effect on the sensitivity near its application area, thereby altering the distribution of significant sensitivity. When the load is applied to all regions simultaneously, this influence is eliminated, and the submatrix with high sensitivity concentration tends to distribute along and near the diagonal of the matrix.

Figure 10 illustrates the detailed distribution and proportion of elements at different levels within both high and low sensitivity concentration submatrices when the load is applied to all regions. It provides a clear visual comparison of the sensitivity concentration differences and the disparity in the content of Level 1 elements (i.e., significant sensitivity) between the two types of submatrices. The results confirm that the submatrices near the main diagonal of the matrix contribute more significantly to the overall sensitivity matrix, and that significant sensitivity is predominantly distributed along and near the reconstructed matrix diagonal.

4. Optimal Sensor Placement Based on Sensitivity

4.1. Significant Sensitivity Analysis

Significant sensitivity is critical for the successful damage identification of structural members. To facilitate discussion, the percentage of significant sensitivity in the sensitivity matrix is defined as the proportion of significant sensitivity. Among the regional sensitivity submatrices associated with structural members in the ith region, submatrix ${\displaystyle \frac{\partial {\ddot{\mathit{x}}}_i}{\partial {\mathit{\alpha }}_i}}$ along the main diagonal contains the highest concentration of significant sensitivity, as previously demonstrated through sensitivity analysis. Each column of the partitioned matrix corresponds to the structural members within the same region. In every column, the submatrix located along the main diagonal contains the highest concentration of significant sensitivity. Figure 11 illustrates the proportion of Level 1 elements in each submatrix when the load is applied to all regions. Although the proportion of Level 1 elements in some off-diagonal submatrices is higher than that in certain submatrices along the main diagonal, it should be noted that these off-diagonal submatrices do not exhibit the highest concentration of Level 1 elements within their respective columns. Therefore, they are not the submatrices containing the most significant sensitivity associated with the structural members in their corresponding regions.

The damage identification sensitivity matrix is ultimately determined by the selection of sensor locations. To achieve effective damage identification for members in ith region using a limited number of sensors, priority should be given to placing sensors within ith region. This ensures that the sensitivity matrix incorporates as many elements as possible from submatrix ${\displaystyle \frac{\partial {\ddot{\mathit{x}}}_i}{\partial {\mathit{\alpha }}_i}}$, thereby capturing a greater amount of significant sensitivity. Similarly, to balance the number of sensors and the damage identification effectiveness for members in each region, sensors should be distributed across all regions. This strategy maximizes the inclusion of significant sensitivity associated with structural members in every region into the damage identification sensitivity matrix with a limited number of sensors.

As fully demonstrated in the preceding sensitivity analysis, submatrix ${\displaystyle \frac{\partial {\ddot{\mathit{x}}}_i}{\partial {\mathit{\alpha }}_i}}$ contains the highest concentration of significant sensitivity among the regional sensitivity submatrices associated with the sensors in the ith region. This indicates that the responses at the sensor locations are most sensitive to stiffness changes in the members within their own region. This finding is further supported by the proportion of Level 1 elements in each submatrix, as shown in Figure 11. In the partitioned matrix, each row of submatrices corresponds to sensors within the same region, and the submatrix with the highest significant sensitivity in each row is located along the main diagonal.

Since the regions are artificially defined, if we progressively reduce the region size until focusing on a single sensor location and its adjacent structural members, the following conclusion can be drawn: the response at a sensor is most sensitive to stiffness changes in the surrounding members. As the significant sensitivity associated with each sensor is limited and primarily related to its neighboring members, the measured responses can only be effectively used to identify damage in structural members located near the sensor.

The area covered by each sensor for damage identification varies, primarily depending on the proportion of significant sensitivity. For the purpose of discussion, the proportion of significant sensitivity is adopted as a quantitative measure of the damage identification coverage. Figure 12 shows the distribution of significant sensitivity proportions across all potential sensor locations in the structure. Sensor locations at the structural edges exhibit a relatively high proportion of significant sensitivity, ranging from 15.6% to 37.9%. The proportion for sensors at other locations falls between 4.6% and 14.5%.

4.2. Sensor Layout Configuiation

The selection principles for measuring points can be summarized into two points:

• Sensors should be distributed across all regions.
• Sensor coverage for damage identification depends on the proportion of significant sensitivity.

Based on the two principles for sensor location selection, an optimal sensor placement methodology is proposed for damage identification in gas station canopy grid structures. This approach builds upon the characteristic analysis of response sensitivity, adopting the proportion of significant sensitivity as the optimization criterion. The development of a sensor placement scheme for damage identification of structural members in gas station canopy grid structures can be carried out in the following steps.

Step 1: Compute the proportion of significant sensitivity for all potential sensor locations in the structure.

Step 2: Sum all the proportional values of significant sensitivity and then compute the reciprocal of this total sum. This reciprocal provides a reference ratio between the number of actual sensors and the total number of potential sensor locations, thereby determining the required number of sensors.

Step 3: Select sensor locations with a high proportion of significant sensitivity, ensuring that the sensors are evenly and symmetrically distributed across all regions.

Step 4: Examine the final sensor placement scheme to verify complete coverage of the entire structure without any omitted areas.

Following the above procedure for sensor placement, multiple sensor layout schemes with identical sensor quantities will be generated in the numerical study. Two representative schemes are subsequently developed using this methodology.

The sum of the significant sensitivity proportions across all potential sensors reaches 1982.6%, far exceeding 100%. This indicates substantial redundancy in sensor coverage for damage identification. The reciprocal of this total sum can serve as a reference for selecting the number of sensors. In this numerical study, the reciprocal of 1982.6% is 5%, suggesting that using only 5% of the total 158 potential sensors is sufficient to cover the entire structure and ensure accurate damage identification.

By further considering the actual coverage capability of each sensor and ensuring that the damage identification areas of the selected sensors do not overlap while still achieving full structural coverage, a specific sensor placement scheme for the numerical case study is determined, as illustrated in Figure 13. The selection of sensor locations is not strictly fixed, provided they are evenly and symmetrically distributed throughout the structure. The locations and serial numbers of the sensors are indicated by red circles with numbers inside. Owing to structural symmetry, only three distinct types of sensor locations exist in each scheme. The specific proportions of significant sensitivity for these sensor types are listed in

Table 2. Significant Sensitivity Proportions in Sensor Arrangements.

Sensor Layout	Significant Sensitivity Proportion (Damage Identification Coverage)
	Sensor 1	Sensor 3	Sensor 4	Total
Configuration A	35.94%	8.16%	10.07%	180.22%
Configuration B	23.09%	6.08%	28.47%	172.22%

. The total proportions of significant sensitivity for all sensors in the two schemes are 180.22% and 172.22%, respectively, both exceeding 100%. With sensors distributed across all regions, the entire structure can be covered.

This research discusses in detail the proportion of significant sensitivity for each node and proposes a sensor optimization method using this proportion as the optimization criterion. The primary focus of this research is the establishment of this optimization criterion. The mind map of the optimal sensor placement method for damage identification in gas station canopy grid structures is shown in Figure 14.

5. Numerical Simulation of Damage Identification

5.1. Damage Identification of Spatial Grid Structure

The simulation example in Section 3.2 is adopted in this section, and two damage scenarios are studied in this section. In the gas station canopy grid structure, ten structural members are randomly selected as the to-be-identified damaged members, and their stiffness is reduced in the finite element model. By performing this procedure twice, two damage scenarios are formed. The acceleration responses of the structure are calculated using the Newmark-β method as the “measured” responses. Two damage scenarios as shown in Figure 15 are studied in this section. The damaged members in the two damage scenarios are marked in red in Figure 15, with their identification numbers labeled in red digits. The load is applied at the same location for both numerical simulation studies, which is indicated by a red five-pointed star in Figure 15. The sensor placement follows the two schemes presented in Figure 13.

For each damage scenario, the damaged members are divided into four groups based on the value of stiffness reduction. The detailed stiffness reduction values for all damaged members in both scenarios are provided in

Table 3. Damage Scenarios.

Damage Scenario	Damaged Member Number and Stiffness Reduction Value
	Group 1	Group 2	Group 3	Group 4
Scenario I	1, 4, 8 (−10%)	2, 9 (−12%)	3, 5, 7, 10 (−8%)	6 (−15%)
Scenario II	1, 3, 7, 9 (−10%)	2, 4 (−8%)	5, 10 (−5%)	6, 8 (−6%)

.

To evaluate the accuracy of the damage identification result, a calculated error is defined as

$$error\left(\alpha \right)={\displaystyle \frac{\left|\left.{\alpha }_{id}-{\alpha }_{real}\right|\right.}{\left|\left.1-{\alpha }_{real}\right|\right.}}\times 100\%$$

(12)

where ${\alpha }_{id}$ is the identified damage index and ${\alpha }_{real}$ is the real damage index.

Damage identification of all structural members is performed using incomplete response data. With the time domain response sensitivity damage identification method, the damage identification results of the two scenarios are shown in Figure 16 and Figure 17. Since the damaged members in the damage scenario are randomly selected, their numbering does not align with the element numbering in the finite element model. In the damage identification results for all elements, the serial numbers of the damaged members are marked with red digits in Figure 16 and Figure 17.

In the calculated damage indices for all structural members, undamaged members consistently show zero values, while the indices of damaged members precisely match the stiffness reduction values assigned in the simulated damage scenarios. The identified results are consistent with the real damaged scenarios, which indicate that the time domain response sensitivity damage identification algorithm is correct and accurate. The damage locations can be identified with satisfactory accuracy in both scenarios. The calculated error values of the identified results for damaged members are shown in

Table 4. Calculated error of the identified results.

Damaged Member Number	Sensor Configuration A		Sensor Configuration B
	Scenario I	Scenario II	Scenario I	Scenario II
1	0.0170%	0.0113%	0.0029%	0.0219%
2	0.0001%	0.0014%	0.0001%	0.0032%
3	0.0029%	0.0020%	0.0007%	0.0045%
4	0.0017%	0.0038%	0.0002%	0.0035%
5	0.0013%	0.0001%	0.0004%	0.0004%
6	0.0105%	0.0008%	0.0007%	0.0005%
7	0.0012%	0.0044%	0.0002%	0.0033%
8	0.0017%	0.0026%	0.0002%	0.0041%
9	0.0198%	0.0053%	0.0003%	0.0355%
10	0.0017%	0.0003%	0.0007%	0.0006%
Mean	0.0045%		0.0042%

. The identified error of damaged member 9 in Scenario Ⅱ reaches about 0.0355%. The mean damage identification errors of the two sensor placement schemes are similar, at 0.0045% and 0.0042%, respectively. This indicates that two representative schemes provide comparable efficacy in structural member damage identification when applied for sensor deployment. The number of sensors is sufficient to ensure accurate damage identification calculations. Significant sensitivity serves as an effective optimization criterion.

This study shows that the time domain response sensitivity damage identification method can be applied in assessing the gas station canopy grid structures with small damages distributed in structural multiple members by incomplete measured acceleration responses (5% of the response data is measured in this simulation study).

5.2. Uncertainty Analysis of Damage Identification

Measured structural responses from sensors are inevitably influenced by environmental conditions, leading to inaccuracies in the acquired data. Meanwhile, finite element models of structures often contain inherent element stiffness inaccuracies. These two aspects represent primary sources of uncertainty in structural damage identification. The following numerical studies will assess the performance of the time-domain sensitivity-based damage identification method for gas station canopy grid structures when considering measurement errors and element stiffness errors separately. The same two damage scenarios from Section 5.1 are adopted. To account for these uncertainties, the sensor configuration is expanded to measure 20% of the structural responses, with the specific sensor arrangement detailed in Figure 18.

Damage identification is performed under a 5% measurement noise condition, with the results presented in Figure 19. The results demonstrate that damaged members in both scenarios can be accurately located despite the presence of 5% measurement noise. Structural damage identification is conducted with a 2% variation in element stiffness within the finite element model, and the corresponding results are shown in Figure 20. The results indicate that the damaged members in both scenarios can be accurately located under the 2% stiffness error condition. Numerical simulations incorporating both measurement noise and stiffness errors demonstrate that the time-domain sensitivity-based damage identification method exhibits favorable robustness for gas station canopy grid structures.

6. Conclusions

This study presents a time-domain response sensitivity-based methodology for damage identification and optimal sensor placement in gas station canopy double-layer grid structures. The following key conclusions can be drawn from the numerical investigations:

The proposed time-domain response sensitivity damage identification method effectively detects and quantifies local damage in structural members, even with incomplete acceleration response measurements. The method exhibits high accuracy, with mean identification errors as low as 0.0042% and 0.0045% under two sensor configuration schemes. The method remains capable of accurately locating damaged members even when accounting for measurement noise and modeling stiffness errors, demonstrating strong robustness.

The sensitivity matrix possesses clear structural and physical interpretability. Numerical analysis of the sensitivity matrix reveals that the proportion of significant sensitivity in the diagonal submatrices reaches 28.9%, significantly higher than the global matrix average of 12.5%. Elements with significant sensitivity are predominantly concentrated in submatrices along the main diagonal of the regionally partitioned matrix. The sensitivity matrix feature analysis indicates that sensor responses are most sensitive to stiffness changes within their own structural region. Thus, the measured responses can only be effectively used to identify damage in structural members located near the sensor.

The significant sensitivity associated with each sensor is limited and primarily related to its neighboring members. Numerical analysis of the significant sensitivity demonstrates that sensor locations at the structural edges exhibit a relatively high proportion of significant sensitivity, ranging from 15.6% to 37.9%. The proportion of sensors at other locations falls between 4.6% and 14.5%.

An optimal sensor placement strategy is developed based on the proportion of significant sensitivity at potential sensor locations. Numerical simulations of two damage scenarios demonstrate that both sensor placement schemes derived from the proposed method achieve comparable performance in damage identification. The method enables effective full-structure coverage using only 5% of the total potential sensors (8 out of 158 nodes), maintaining high measurement efficiency without compromising identification accuracy.

The proposed framework offers an efficient and practical solution for structural field inspection of gas station canopy structures, with potential applicability to other similar spatial grid structures in petrochemical engineering and beyond. Furthermore, it proposes a calculation method for implementing intelligent sensing technology in field inspections, thereby enabling automated detection of damaged members in gas station canopy grid structures. Integrating the proposed member damage identification method with intelligent sensing-based damage detection techniques for supports and joints enables comprehensive automated inspection of spatial grid structures.