Damage Detection on Steel Truss Bridges Using Modal Based Methods

1. Mode Shape (MS) Damage Index

This method is a displacement-based approach that evaluates damage by comparing the mode shapes of intact and damaged structures (Maia et al., 2003; Rucevskis & Wesolowski, 2010). The damage index is defined as:

• Δv_i – mode shape (MS) damaged index for the considered mode,

• $v_i^d$ – modal displacement of the damaged structure,

• v_i – modal displacement of the intact structure,

• i – measurement point.

The fundamental idea of the mode shape (MS) damage index is based on the direct comparison of modal displacements between intact and damaged structures. Damage is indicated at locations where significant differences in modal displacement occur. To summarize the results for all modes, the following index is used:

• N – total number of mode shape measurement,

• Δv_i – mode shape (MS) damaged index for each mode.

2. Mode Shape Slope (MSS) Damage Index

This algorithm uses changes in the slope of the mode shape (Rucevskis & Wesolowski, 2010). Local disturbances in experimental measurements may introduce irregularities in the mode shape, which appear as peaks in the slope profile and can be associated with damage locations. To reduce the influence of measurement noise, the damage index is averaged over multiple modes.

• $\Delta v_i^{\prime }$ – mode shape slope (MSS) damaged index for the considered mode,

• $v_i^{\prime \text{d}}$ – modal slope of the damaged structure,

• $v_i^{\prime }$ – modal slope of the intact structure,

• i – represents the measurement point.

• v_i+1 – modal displacement at the subsequent point (i+1),

• v_i−1 – modal displacement at the preceding point (i−1),

• h – distance between two consecutive measurement points.

If there are multiple mode measurements, the index is used.

3. Mode Shape Curvature (MSC) Damage Index

In this method, damage is identified based on changes in mode shape curvature (second spatial derivative) between intact and damaged structure (Rucevskis & Wesolowski, 2010). The underlying concept is that the curvature of a healthy structure is generally smooth and can be approximated using a polynomial function. By comparing the curvature of the damaged structure with that of the intact structure, localized damage can be detected (Janeliukstis et al., 2019).

• $\Delta v_i^{″}$ – mode shape curvature (MSC) damaged index for the considered mode,

• $\Delta v_i^{\prime \prime \text{d}}$ – modal curvature of the damaged structure,

• $v_i^{″}$ – modal curvature of the intact structure,

• v_i+1 – modal displacement at the subsequent point (i+1),

• v_i−1 – modal displacement at the preceding point (i−1),

• h – distance between two consecutive measurement points.

• If there are multiple mode measurements, the index is used.

4. Mode Shape Curvature Square (MSCS) Damage Index

The damage index is defined as follows:

• $\Delta {\left(v_i^{″}\right)}^2$ – mode shape curvature square (MSCS) damaged index for the considered mode,

• ${\left(v_i^{\prime \prime \text{d}}\right)}^2$ – modal curvature square of the damaged structure,

• ${\left(v_i^{″}\right)}^2$ – modal curvature square of the intact structure.

If there are multiple mode measurements, the index is used.

Among all the algorithms mentioned above, Rucevskis and Wesolowski (Rucevskis & Wesolowski, 2010) proposed the mode shape curvature square (MSCS) method for detecting damage locations in their research, as this method is deemed capable of identifying damage locations using only the mode shape of the tested damaged structure. All methods will be applied in this study to detect damage locations in steel truss bridges with several proposed damage scenarios.

Vibration-based damage detection is applied in experimental tests on steel truss bridges. This technique is part of non-destructive testing. It is proposed for global damage identification testing with various complexities (Fan & Qiao, 2011). The fundamental idea behind vibration-based damage detection is that damage can lead to a decrease in structural stiffness, thereby altering the dynamic properties of the structure, including mode shapes, modal damping, and natural frequencies (Fan & Qiao, 2011; Nick & Aziminejad, 2021). the design of the study, including whether it is experimental, observational, etc.

3.1. Model and Material Properties

To verify the validity and effectiveness of the modelled damage algorithm, a numerical modal analysis based on the Finite Element (FE) method was conducted using ABAQUS software. In this study, a symmetric damage model was applied to steel angle members, including various forms of cutting, buckling, and the removal of diagonal members in the main bridge truss at specific locations. The geometric configuration of the steel angle members is shown in Figure 2.

Figure 2:

The dimensions of the steel angle profile used are 30 × 30 × 3 mm. The damage scenarios applied to the steel angle members were introduced in the diagonal members. Figure 2 represents types of damage to structural elements of steel truss bridges that commonly occur in practice. Specifically, the geometry of the damage shown in Figure 2(a) consists of a cut with a length of 10 cm and a depth of 3 cm, which was applied in the experimental tests to the diagonal members of the main truss at positions 0.5L and 0.2L, respectively (see Figure 5 and Table 1). In addition, buckling damage with a bending angle of 165° was applied to the diagonal members of the main truss and tested alternately at the same positions (0.5L and 0.2L) (see Figure 5 and Table 1). Likewise, the damage scenario involving the removal of a diagonal member in the main truss was also tested alternately at these two positions (see Figure 5 and Table 1).

The cut damage is represented as a consequence of corrosion of the cross-sectional profile, which leads to a reduction in the stiffness of the affected member (Figure 3). The buckling damage represents conditions caused by impact loads that result in member bending (Figure 4). This type of damage is emphasized because many cases of bridge damage and even collapse occur not due to design or construction errors, but as a result of accidents, such as vehicles (cars or trucks) colliding with the structure (e.g., cars, trucks colliding with the structure) (Antonia, Menga Terje, Kanstad Daniel et al., 2022; Hardono et al., 2022). Meanwhile, the scenario involving the removal of a diagonal member is presented as an extreme case representing the loss of structural stiffness due to deterioration over the service life.

Figure 3:

The dimensions of both profiles are 650 mm. The material properties determined experimentally are: Young's modulus E = 200,000 MPa and density ρ = 7800 kg/m³. The geometry of the steel frame bridge specimen to be tested is as follows: span length, L = 5000 mm, bridge width, l = 840 mm, and bridge height, h = 700 mm. A summary of the model in this study is presented in Table 2 and Figure 5.

Figure 4:

Figure 5:

The finite element model for the steel frame bridge structure consists of three-dimensional structural elements (ABAQUS). Each node has three degrees of freedom: translation along the X and Y axis, and rotation about the Z axis. For the healthy elements, a constant stiffness EI is assumed for all elements, while the damaged elements are modeled by reducing the stiffness of the selected elements. The reduction in stiffness is achieved by decreasing the cross-sectional area in the damaged regions of the profile, which consequently reduces the moment of inertia.

3.2. Experimental Set-up

Experimental activities were conducted to obtain the dynamic behaviour of the structure in response to applied dynamic loads. The dynamic behaviour of the structure is assessed based on modal properties recorded by accelerometer sensors, including natural frequency and acceleration. Frequency and acceleration are used to evaluate the health condition and determine the location of damage in the bridge structure. This assessment is expressed in a modal-based curvature index. Referring to the working principle of curvature assessment, the peak values indicate the location where damage occurs.

In the experiment, two conditions of the structure will be tested: the intact bridge structure and the damaged one. The test specimen of the steel frame bridge is designed at a lab scale of 1:12 (Figure 6). A steel angle profile was chosen for use in the design of this lab-scale test specimen (Table 1).

Figure 6:

The general experimental setup consisted of a bridge specimen, accelerometer sensors, a USB gateway for data acquisition, the SensorConnect application for reading the recorded signals, and a computer system (Figure 7). A limited number of six accelerometer sensors were installed at specific locations along the structure. This limited number and spatial distribution of sensors directly affect the spatial resolution and the accuracy of mode shape determination, particularly in curvature estimation. To investigate and mitigate the impact of this limitation, a sensor relocation strategy was implemented at all measurement points along the main truss over the bridge span. This approach enhances spatial resolution, strengthens curvature-based analysis, and improves the reliability of mode-shape-based methods by producing sharper curvature peaks for damage localization.

Figure 7:

Several key points related to the experimental procedure are as follows: The dynamic load was applied using a concrete cylinder (Figure 8) dropped from a height of 7.5 cm at mid-span (0.5L), positioned between the longitudinal girders. Accelerometer sensors were installed on the main truss on both sides of the bridge and were placed sequentially at fixed positions of 0.4L, 0.5L, and 0.6L along the span (measured from the support), allowing all monitoring points to be recorded through a sensor relocation method.

The accelerometer sensors were arranged using a relocation method, in which sensors on the right and left sides of the bridge were tested at consecutively shifted locations to improve spatial resolution and the reliability of curvature-based analysis. During testing using the dropping-weight method, vibrations of the bridge were recorded by the accelerometer sensors, after which the data were transmitted via the USB gateway and displayed on a computer screen in the form of time-domain signals. Vibration data were recorded with a duration of 3 seconds for each test, using accelerometers with a sampling frequency of 256 Hz, and only the z-axis channel was analysed. The time-domain data were then converted into the frequency domain using the SensorConnect application. The time-domain measurement data were imported and used in the analysis to obtain displacement through numerical integration, by converting velocity into displacement at each measurement point derived from the accelerometer results. This method was applied to seven test conditions: intact, cutting element on center (Dmg-CC), buckling element on center (Dmg-BC), empty element on center (Dmg-EC), cutting element on edge (Dmg-CE), buckling element on edge (Dmg-BE), and empty element on edge (Dmg-EE) (Table 1). Each model was tested three times to evaluate the accuracy of the obtained data.

Hinge-roller boundary conditions were simulated by placing the bridge specimen on two pipes representing the hinge and roller supports. After impact, the vibration responses were recorded by the accelerometer sensors.

The hinge–roller boundary conditions were simulated by placing the bridge specimen on two pipes representing the hinge and roller supports. During the experiment, the bridge specimen was excited using a dropping weight. The excitation was applied by a concrete cylinder with a diameter of 15 cm and a height of 30 cm, dropped from a height of 7.5 cm (Figure 8). Following the impact, vibration responses were recorded by the sensors.

Figure 8:

After measurements were taken at each location, the USB gateway automatically transferred the recorded data to the SensorConnect application, where the time-domain signals were displayed and subsequently transformed into the frequency domain using the Fast Fourier Transform (FFT) algorithm. This procedure was repeated until all scanning points had been measured. The natural frequencies and acceleration responses, representing the dynamic properties of the structure, were then extracted from the Fast Fourier Transform (FFT) results.

4.1. Dmg-CC Analysis

The FEM model was adjusted to match the experimentally tested bridge specimen in terms of geometry and section properties. The adjustment was performed through manual tuning of material and boundary parameters to minimize the difference between the numerical and experimental responses, particularly the global displacement and modal frequencies. This procedure ensures that the FEM reliably represents the bridge behaviour. The comparison between experimental and numerical results is presented in Figure 11 (Curvature Index of Each Method on the DMG-CC Model), demonstrating good agreement and validating the adopted FEM.

Figure 9 presents the damage model of a diagonal member with a cut section located at mid-span. The displacement at each measurement point was observed, and the obtained displacement values were adjusted to match the measurement locations in the experimental test. The results of the finite element analysis are shown in Figure 9, while Figure 10 displays the DMG-CC model used in the experimental testing.

Figure 9:

Figure 10:

Figure 11:

The modal-based method was examined on the Dmg-CC damage model, with the introduced damage location highlighted in grey. Figure 11 shows that all damage index methods (MS, MSS, MSC, and MSCS) successfully identified the damage position according to the intended scenario. These three methods exhibited similar curvature index profiles, as observed in both numerical analyses and experimental mode shape measurements. Based on the curvature indices derived from numerically computed mode shapes, the highest peak appeared at the damage location, confirming that the proposed damage index methods are effective in detecting and localizing structural damage.

To evaluate the accuracy of the finite element method (FEM) results compared with the experimental data, error metric values were calculated (Table 1). Sum of Squared Errors (SEE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE) are statistical approaches used to measure the total deviation between the observed (actual) and predicted values (Hrehova et al., 2025). Smaller error metric values indicate a better agreement with the experimental results.

Table 3 presents the deviation between the responses of the intact and damaged structures. The squared displacement difference is defined as the square of the difference between the displacement index obtained from Abaqus and the experimentally measured modal displacement index, expressed as (i_FE − i_Exp)². Smaller values of the error metrics (SSE, MSE, and RMSE) indicate better agreement between the finite element analysis results and the experimental data. Based on these metrics, the mode shape slope (MSS) and mode shape curvature (MSC) exhibit relatively small error values, indicating that the MSS and MSC approaches provide high accuracy in detecting damage at the mid-span.

4.2. Dmg-BC Analysis

Figure 12 represents the structural modelling of buckling damage in the middle-span diagonal member. The modal displacement at each measurement point is observed. The measured displacement values are adjusted to match the experimental measurement points. The results of the finite element analysis are presented in Figure 12. Figure 13 displays the Dmg-BC model used in the experimental testing.

Figure 12:

Figure 13:

Figure 14:

Figure 14 illustrates the modal displacement of the Dmg-BC structure, which exhibits a buckling damage in the diagonal member at mid-span. The introduced damage location is highlighted in grey. The results in Figure 14 show that the mode shape slope (MSS) and mode shape curvature (MSC) damage indices, obtained from both numerical calculations and experimental mode shape measurements, successfully identify the damage location with good accuracy. Meanwhile, the mode shape (MS) and mode shape curvature square (MSCS) methods show a reduction in curvature at mid-span, indicating a possible discrepancy between the experimental data and the finite element analysis results. The displacement deviation between the experimental measurements and the finite element analysis results needs to be evaluated for accuracy using error metric methods.

Table 4 illustrates the deviation between the responses of the intact and damaged structures. The mode shape slope index (MSSi) and mode shape curvature index (MSCi) methods exhibit low error values, as indicated by the error metrics obtained from the three evaluated methods. These results demonstrate that the mode shape slope (MSS) and mode shape curvature (MSC) approaches are highly accurate in identifying buckling damage in the mid-span diagonal member.

4.3. Dmg-EC Analysis

Figure 15 presents the specimen model with a damage scenario in which the diagonal member at mid-span is removed. This scenario aims to assess the capability of the modal-based method in accurately identifying the damage location. The modal displacements at each measurement point are recorded and adjusted to align with the corresponding points from the experimental measurements. The results of the finite element analysis are displayed in Figure 15.

Figure 15:

Figure 16:

Figure 17:

The modal displacement of the Dmg-EC structure is presented in Figure 17, with the introduced damage location marked in grey. The results shown in Figure 20 indicate that neither the mode shape (MS) nor the mode shape slope (MSS) damage index, obtained from both numerical calculations and experimental mode shape measurements, can accurately detect the damage location. The mode shape curvature (MSC) and mode shape curvature square (MSCS) damage index methods successfully indicate the damage location according to the established scenario. From the curvature indices obtained using mode shape information calculated numerically, it is observed that the largest peak value appears at the damage location, indicating that the proposed damage index methods can effectively detect and locate the damage.

Table 3 confirms the effectiveness of the modal-based approach in detecting the removal damage of the mid-span diagonal member. Based on the error metric values, the mode shape slope (MSS) and mode shape curvature (MSC) methods show lower error values, indicating that these methods provide high accuracy compared to the experimental results. This finding is consistent with the previous case, which demonstrated that the mode shape slope (MSS) and mode shape curvature (MSC) methods are reliable in identifying damage locations in mid-span members of steel truss bridges. However, further evaluation of all four methods is required to verify their performance in detecting damage in elements located at one-fifth of the span.

4.4. Dmg-CE Analysis

Figure 18 shows the structural modelling and finite element analysis results with a cut section located at 1/5 of the span. The modal displacement at each measurement point was observed, and the measured displacement values were adjusted to match the measurement points used in the experimental test. Meanwhile, Figure 19 presents the experimental testing of the Dmg-CE model.

Figure 18:

Figure 19:

Figure 20 shows the modal displacement of the Dmg-CE structure, with the introduced damage location marked in grey. The results presented in Figure 20 indicate that the mode shape slope (MSS) and mode shape curvature (MSC) damage indices can effectively detect the damage location at 1/5 of the span (Node 0.8). Based on the curvature index obtained from the mode shape information calculated numerically, the highest peak value appears at the damage location in both the experimental and numerical results. The mode shape slope (MSS) and mode shape curvature (MSC) approaches demonstrate that the proposed damage index method can effectively detect and identify the damage location. To evaluate the accuracy level of the four methods used in the damage location detection analysis, error metric calculations were performed.

Figure 20:

Table 6 verifies the capability of the mode shape slope (MSS) and mode shape curvature (MSC) methods in identifying the cut section located at one-fifth of the span. The error metric results indicate that all four methods produce low error values, demonstrating a high level of accuracy compared to the experimental results. These findings further confirm that the mode shape slope (MSS) and mode shape curvature (MSC) methods are reliable in detecting damage locations at the one-fifth span, consistent with the modal displacement analysis, which also shows that both methods perform well in identifying damage in this region. However, the mode shape (MS) and mode shape curvature square (MSCS) methods are found to be less reliable in detecting the damage location in this case.

4.5. Dmg-BE Analysis

Figure 21 shows the structural modelling and finite element analysis results with buckling damage on the diagonal member located at 1/5 of the span. The modal displacement at each measurement point was observed, and the measured displacement values were adjusted to match the measurement points used in the experimental test. Figure 22 presents the experimental testing of the Dmg-BE model.

Figure 21:

Figure 22:

Figure 23 illustrates the modal displacement distribution of the structure with buckling damage occurring in the diagonal member located at 1/5 of the span, where the introduced damage area is highlighted in grey. The results presented in Figure 23 demonstrate that the mode shape slope (MSS) and mode shape curvature (MSC) damage indices effectively identify the damage location at one-fifth of the span (Node 0.8). Based on the curvature index derived from the numerically calculated mode shape, the highest peak value appears precisely at the damage location in both the experimental and numerical results. These findings indicate that the mode shape slope (MSS) and mode shape curvature (MSC) methods provide high accuracy and efficiency in detecting and identifying structural damage. To further evaluate the accuracy of all four modal-based damage detection methods, error metric analyses were performed.

Figure 23:

Table 7 shows that the mode shape slope (MSS) and mode shape curvature (MSC) methods demonstrate excellent capability in identifying buckling damage on the diagonal member located at 1/5 of the span. Based on the error metric results, all four methods produce relatively small error values, indicating a high level of accuracy in detecting the damage location. These findings also reinforce the consistency between the numerical analysis and experimental data, showing that the mode shape slope (MSS) and mode shape curvature (MSC) methods are reliable in detecting buckling damage on the diagonal member at 1/5 of the span, whereas the mode shape (MS) and mode shape curvature square (MSCS) methods are less effective in accurately identifying the damage location. This confirms the reliability of the mode shape slope (MSS) and mode shape curvature (MSC) methods in identifying buckling damage in the structure.

4.6. Dmg-EE Analysis

Figure 24 presents the structural modelling and finite element analysis results with the removal of the diagonal member located at 1/5 of the span. The modal displacement at each measurement point was observed, and the displacement values obtained from the analysis were adjusted to match the measurement points used in the experimental test. Meanwhile, Figure 25 shows the experimental testing of the Dmg-EE model.

Figure 24:

Figure 25:

Figure 26 shows the modal displacement distribution of the structure with damage caused by the removal of the diagonal member located at one-fifth of the span, where the introduced damage area is highlighted in grey. The results presented in Figure 27 indicate that the mode shape slope (MSS) and mode shape curvature (MSC) damage indices effectively identify the damage location at one-fifth of the span (Node 0.8). Based on the curvature index obtained from the numerically calculated mode shape, the highest peak value appears precisely at the damage location in both the experimental and numerical results. These findings confirm that the mode shape slope (MSS) and mode shape curvature (MSC) methods provide high accuracy and efficiency in detecting and identifying structural damage. Furthermore, to evaluate the accuracy level of all four modal-based damage detection methods, an error metric analysis was conducted.

Figure 26:

Table 8 confirms the effectiveness of the modal-based approach in detecting damage caused by the removal of the diagonal member located at one-fifth of the span. Based on the obtained error metric values, all four methods (MS, MSS, MSC, and MSCS) show low error values, indicating that these methods have a high level of accuracy compared to the experimental results. However, the mode shape slope (MSS) and mode shape curvature (MSC) methods, as demonstrated in the modal displacement analysis, show greater reliability in identifying the damage location of the diagonal member at one-fifth of the span of the steel truss bridge. From these results, it can be concluded that the mode shape slope (MSS) and mode shape curvature (MSC) methods are sufficiently reliable in detecting damage locations at one-fifth of the span.