Over the years, Structural Health Monitoring (SHM) has gained increasing importance as a means to prevent structural failures that can lead not only to economic losses but, more critically, to the loss of human life. Unfortunately, the literature reports numerous catastrophic events, both in civil and mechanical engineering contexts, that have resulted in significant damage and casualties.

Although certain cases—such as the failure of an electrical transmission tower—might appear less critical at first glance, such events can have extensive consequences. Indeed, the collapse of a single electric tower, which is often subjected to harsh environmental conditions and various types of degradation, can cause the failure of an entire power line and so major disruptions to economic activity and everyday life [1].

To mitigate such risks, it is essential to apply preventive damage detection analyses and to develop methods that are both simple and rapid, ensuring timely intervention when continuous monitoring is not feasible.

Over the past decades, a vast number of approaches have been proposed across different research fields. Among them, vibration-based methods have proven to be particularly effective for SHM and damage detection due to the high sensitivity of modal parameters to structural deterioration. Specifically, several techniques have been developed based on changes in natural frequencies, mode shapes, modal strain energy, modal flexibility, matrix updating (stiffness and mass), and various nonlinear approaches [2,3].

In recent years, several authors have shown increasing interest in combining modal parameters for damage detection with methods from other research fields. For instance, one rapidly growing area is the integration of Artificial Intelligence (AI) techniques into SHM applications [4], or the use of Convolutional Neural Networks (CNN) [5,6]. The Statik and Dynamik research group at Leibniz University Hannover has also explored, for example, on a real-world structure, the use of artificial neural networks and autoencoders in particular [7], as well as transfer learning techniques to transfer damage-related information, based on modal features, from a source structure with known damage to a target structure without such information, enabling a more complete dataset for both [8]. They have also investigated a multi-objective optimization approach combined with stochastic model updating for damage localization [9] and the application of the autocovariance function in this context [10].

These studies are cited here merely as representative examples, illustrating how this line of research has become increasingly active, with many other groups worldwide exploring similar hybrid approaches.

Nevertheless, although these methods have shown promising results, they often require significant computational effort, as well as reference data or supporting simulations. In addition, such approaches typically demand considerable time to train complex algorithms or to perform model-updating procedures. Unfortunately, this time is not always available in practical applications, where prompt detection and intervention are crucial to ensure structural safety and prevent further deterioration.

Within this framework, one of the vibration-based methods that has attracted considerable attention due to its simplicity, low computational cost, and minimal data requirements is the curvature method, first introduced by Pandey et al. [11]. This technique relies solely on the measurement of a displacement field associated with the structural response. Based on the relationship between structural stiffness and curvature, the method involves extracting the structure’s mode shapes and computing their second derivative, which can be estimated using finite difference schemes [11,12] or enhanced through other tools, such as the Continuous Wavelet Transform (CWT). The latter approach, as demonstrated by [13,14], significantly improves the accuracy and robustness of curvature estimation.

Regardless of the specific computational technique employed, the curvature method enables fast and effective damage localization without requiring a reference dataset, numerical model, or destructive testing. For these reasons, this approach has been extensively investigated, resulting in a large body of literature.

Among all the published works, some curvature-based techniques employ a reference model or data of the undamaged structure, using damage indices that compare the healthy and damaged states. They have been widely tested in either numerical [15–17] or experimental studies [18–22]. However, such information is often unavailable, especially in the case of historical structures or aging mechanical components, where no prior data exists. Therefore, it is essential to expand and refine reference-free approaches. Other authors have tested the curvature method under a variety of conditions [23–25], and even extended it by using higher-order derivatives (e.g., third and fourth derivative-based methods) [26,27].

Nevertheless, the curvature method has been widely employed and adapted in numerous studies, leading to the publication of a substantial number of papers in the field.

Hoping that a concise overview of the most relevant contributions has been provided, the aim of the present work is to demonstrate, through a real-world case study, that an extension of the curvature method, toward the use of the first derivative alongside the traditional second derivative (as previously introduced and discussed by the authors in another study), can also be of significant interest.

This is especially relevant since the first derivative is generally easier to estimate than higher-order derivatives, which are more sensitive to experimental noise. In some previous works, the first derivative has been incorporated into damage detection analyses, typically through specific damage indices that also depend on the second derivative, thereby increasing computational time. Alternatively, if used alone, the first derivative was mostly tested with information related to a reference healthy state, in numerical simulations, or on beam or plate-like structures [28–31].

Therefore, the objective of this study is to demonstrate the applicability and the necessity of using this extended approach, showing—through a real-world experimental dataset rather than a simulated one—that the use of the first derivative of the torsional mode shape can lead to simpler and more effective results, depending on the structure’s characteristics and geometry. Furthermore, the method retains the key advantages of being reference-free and non-destructive, making it particularly suitable for practical SHM applications.

The manuscript is organized as follows: first, a theoretical description of the proposed and applied method is provided. Section 3 presents the real-world case study and the corresponding experimental campaign, while Section 4 discusses the obtained damage localization results. Finally, the main conclusions are drawn.

It is well known that the second derivative of the mode shape is sensitive to damage localization, as it reflects the relationship between variations in structural stiffness and curvature.

Specifically, consider a beam with a circular cross-section undergoing pure bending deformation and subjected to a continuous bending moment. At a given instant in time, the following equation Eq. (1) can be expressed: (1)χ(x)=υ′′(x)=Mb(x)EJwhere Mb is the bending moment, E represents Young’s modulus, and J is the geometric moment of inertia of the beam’s cross-section. Damage at a specific cross-section of the beam results in a local reduction of bending stiffness, leading to a change in the structural curvature χ(x) at that location. As the curvature can be described through the use of the second derivative, this effect becomes evident through the appearance of a sudden peak in the second derivative of the mode shape, υ^{\prime}^{\prime}(x), at the damage position [11]. In fact, Eq. (1) shows a clear inverse proportionality between the structure stiffness and the second derivative.

Similarly, an equation analogous to Eq. (1) can be derived for a beam with a uniform circular cross-section subjected to a continuous external torque Mt(x). In particular, the theory of pure torsion states that in this case, the torsion angle θ(x) per length unit x, is directly related to the applied continuous torque Mt(x) and the torsional rigidity GJ0 as shown in Eq. (2) [32]: (2)θ(x)x=Mt(x)GJ0

Therefore, when damage occurs at a generic cross-section of the beam, the resulting sudden reduction in torsional stiffness produces a corresponding discontinuity in the rotation angle θ(x), as described in Eq. (2). By evaluating how θ(x) varies along the beam through its first derivative, Eq. (3), it becomes possible to observe a distinct peak in the derivative curve. This peak reliably indicates the location of the stiffness change and thus the position of the damage. (3)dθ(x)dx=Mt(x)GJ0

This implies that, in the case of pure torsional deformation of the beam, through Eq. (3) a relation equivalent to the one described by the curvature method can be derived, leading to the same result in damage localization but analyzing the pure torsional mode that is directly related to the rotation (θ) of the structure. Therefore, similarly to the bending case, which was the most commonly used and the only one originally introduced by Pandey et al. [11], considering Eq. (3) extends the concept to situations in which torsion is the dominant deformation mechanism, making it applicable to torsional mode shapes as well.

This extension is particularly important because pure bending modes are not always guaranteed in real-world structures. Geometric and material characteristics may lead to predominantly bending or torsional modes, or even to coupled flexural–torsional modes in which one component prevails over the other. Therefore, it becomes essential to also consider the relationship described in Eq. (3) and to carefully evaluate each case. When bending deformation dominates the mode shapes, the curvature method is appropriate, whereas when torsion is predominant, the first-derivative method should be employed [33].

Moreover, the lower sensitivity of the first derivative, compared to the second, in exhibiting spurious peaks due to noise in the measured data must be taken into account. Therefore, it is essential to employ both approaches when dealing with structures that exhibit bending, torsional, or even coupled behaviors, and to carefully analyze and select the most reliable results.

Therefore, the main objective of this work is to demonstrate the validity of this approach using data acquired from a real-world structure and, in particular, to highlight the advantages of employing the first derivative instead of the second one for damage localization in the tested structure.

The structure under investigation, described in detail in Section 3, is a slender truss structure. Consequently, it can be reasonably approximated by a beam with a circular cross-section. In this context, it is worth noting that the rotation θ(x) corresponds to the tangential displacement w(x), up to a constant equal to the radius R of the beam’s cross-section, according to w=Rθ. Hence, the rotational displacement fields were approximated by the axial displacements w(x).

Moreover, in this study, both Eqs. (1) and (3) were calculated aided by the use of CWTs (Continuous Wavelet Transforms). This choice is because derivatives and CWT have similar performance, and they cannot be considered two distinct and completely different mathematical tools. Moreover, they showed better performance in the case of data corrupted by external noise [13,14]. Eq. (4) shows the relations utilized. (4){dydx≈kh1∗yd2ydx2≈kh1∗(h1∗y)where the symbol “∗” denotes the numerical convolution operator between two finite sample streams (h₁ and y), h₁ is the differentiator filter, and y is the data stream.

The method was tested on a real-world structure, the Leibniz University Test Structure for MOnitoring (LUMO). This structure was made available by the Institute for Statics and Dynamics (ISD) research group of Leibniz University Hannover, and is used for experimental validation of Structural Health Monitoring (SHM) methods [34].

The method outlined in Section 2 was applied to the identified mode shapes. Specifically, the proposed concept of using the first derivative of the first torsional mode shape, supported by the application of the Continuous Wavelet Transform (CWT), to localize the damage, was implemented and compared with the traditional curvature-based method, which applies the second derivative to the second bending mode shape.

Due to its geometry, the LUMO structure exhibited mode shapes that were almost completely decoupled. Fig. 7a,b, respectively, show the damage localization results obtained from the torsional and bending mode shapes for the D1 damage case. Similarly, the results for the D2 and D3 cases are presented in Figs. 8 and 9, respectively. In all graphs, the regions affected by damage are highlighted.

As shown in Figs. 7–9, the method based on the use of the first derivative successfully and clearly localized the damage in at least two of the three analyzed cases, D1 and D2. For the third case, D3, a pronounced peak, even if not so much higher than the others, is observed near the damaged region; however, it is not perfectly contained within it. In contrast, when applying the curvature-based method, a peak is visible only for the D1 case.

To better understand the superior performance of the first derivative for the analyzed truss structure, an index was introduced to quantify the accuracy of the two damage-localization methods. Specifically, the accuracy was evaluated by measuring the deviation between the peak in the derivative and the actual damaged area. This deviation, expressed in meters, is reported as the index δ in Table 6.

As can be seen from Table 6, with the use of the 1st derivative method, the peak related to the localization of damage is almost always contained in the highlighted damaged area, while the use of the 2nd derivative leads to a huge error, especially considering that for D2 and D3 cases there is not a clear peak that was more visible than the others.

Building on the well-known curvature method used for damage detection and localization, typically applied to bending mode shapes, the goal of this study was to extend this approach by demonstrating not only how torsional mode shapes and their first derivatives can be used to detect and locate damage, but also that, in some cases, this extended method provides better results than the classical curvature technique. Consequently, it becomes important to incorporate this approach into damage detection analyses based on mode-shape derivatives, and in particular to employ both methods when dealing with real-world structures. This is because, in practical applications, pure bending modes are not always guaranteed; geometric and material characteristics may lead to mode shapes that are predominantly bending or predominantly torsional. In such cases, considering the first derivative (slope) of the torsional mode shape becomes essential. Moreover, the lower sensitivity of the first derivative, compared to the second derivative, to spurious peaks induced by noise in experimental data must also be considered as another advantage.

To achieve this goal, an appropriate experimental campaign was designed and conducted on a real-world truss structure. The collected data allowed for the identification of the first and second bending modes and the first torsional mode shape of the structure. Visual results of the obtained derivatives, as well as an introduced accuracy index, demonstrate how applying the first- and second-derivative methods revealed that using the first derivative on the torsional mode shape produced better results, enabling damage detection and clear damage localization in at least two of the three cases analyzed.

Moreover, a clear advantage of the introduced method was that the described results were obtained using ambient vibration data acquired over a duration of only 10 min per test, with 32 non-equidistant measurement points.

Finally, a multi-setup approach was tested and compared with the single-setup configuration, providing an alternative strategy for conducting measurements when a large number of sensors is not available.

The method was tested on a truss structure in operating conditions, so subjected to environmental conditions, showing the robustness of the method against ambient and measurement noise. Future research could extend its applicability to different types of structures.