Effect of Temperature Changes on the EMA of Galvanized Steel Benchmark Structure

1. Introduction

In Civil Engineering structures, increasing the dimensions of the building and the loads to the elements will increase the dimensions of the elements in the system and thus an economic and suitable design is not possible. Therefore, structures are considered ductile under earthquake loads and structures are designed for reduced elastic earthquake loads. Damage to existing structures is inevitable since the earthquake is considered to be at the beginning of the design which will be damaged by showing the linear behavior. Damage to existing buildings is not caused by earthquakes. Excavations around the building, tunnel and subway constructions and blasting activities can also occur when conditions are encountered that adversely affect the stability of the building (Lenza et al. 2017). Therefore, the damage assessment in existing buildings should be monitored throughout the life of the building. The purpose of the study, load distribution or structural system will be changed, pre-earthquake or after the performance of the existing buildings to be evaluated and strengthened to perform and very detailed studies are obtained from these studies. For this reason, there is a need for a method to accelerate the damage detection studies applied to the risky buildings (De Canio et al. 2016). Operational Modal Analysis is the most preferred method that provides dynamic parameters to be measured by measuring the effects of ambient vibrations on structures.

The operational modal analysis method is the preferred method for determining the current performance levels of many types of structure due to its ease of application and its interactive solution with the ground. In order to determine the material performance, it is not necessary to take material samples from the building and to take ground samples in order to determine the building performance level (Nezhad and Poursha, 2015). The operational modal analysis method simplifies these detailed studies to provide faster and more accurate results. In order to determine the behavior of the structures under the effect of dynamic loads, it is necessary to determine the mode shapes, natural frequencies and damping ratios which are the dynamic parameters of the structures (Zhu et al. 2015; Basaran et al. 2015). In buildings, there are structural differences due to differences between project and application, damage or reinforcement. The dynamic characteristics of the building vary with these changes. By following the differences in the dynamic characteristics, it is possible to get an idea about the stiffness of a building and the change in the level of damage (Ni et al. 2015).

In general, operational modal analysis is used to determine the damage levels of the existing structures, to check the validity of the assumptions made while constructing the finite element model, to update the initial numerical model of the existing structures according to the experimental data, to determine the dynamic characteristics of the structures by the experimental modal analysis method when the numerical model of the existing structures cannot be formed and to follow the structural health is widely used in the process (Tseng et al. 1994; Aliev and Larin 1998; Ljung, 1999; Roeck, 2003). It was observed that three types of definitions were used in the engineering structures: modal parameter identification; structural-modal parameter identification; control-model identification methods are used. In the frequency domain the identification is based on the singular value decomposition of the spectral density matrix and it is denoted Frequency Domain Decomposition (FDD) and its further development Enhanced Frequency Domain Decomposition (EFDD). In the time domain there are three different implementations of the Stochastic Subspace Identification (SSI) technique: Unweighted Principal Component (UPC); Principal component (PC); Canonical Variety Analysis (CVA) is used for the modal updating of the structure (Friswell and Mottershead, 1995; Marwala, 2010). It is necessary to estimate sensitivity of reaction of examined system to change of random or fuzzy parameters of a structure. Investigated measurement noise perturbation influences to the identified system modal and physical parameters. Estimated measurement noise border, for which identified system parameters are acceptable for validation of finite element model of examine system. System identification is realized by observer Kalman filter (Kalman, 1960; Ibrahim, 1977; Juang et al.; 1993) and Subspace algorithms (Overschee and De Moor, 1996). In special case observer gain may be coincide with the Kalman gain. Stochastic state-space model of the structure are simulated by Monte- Carlo method. As a result of these theoretical and experimental studies, the importance of temperature change and humidity has emerged once again from the environmental factors affecting the modal parameters. The effects of temperature and humidity on modal parameters have been the subject of thorough examination in the last 15 years (Kasimzade and Tuhta, 2007a; 2007b; 2009). Many structures of steel, reinforced concrete and composite materials related to the effect of temperature and humidity change on modal parameters have been investigated. These examinations were conducted daily, weekly and monthly. Changes in temperature and humidity were evaluated in three different ways. The first is the ambient temperature and humidity, the other is the temperature and humidity of the building and the last is the surface temperature and humidity of the building. Analyzes were repeated for different situations. As a result of these analyzes, the relationship between the changes in the modal parameters for different temperature and humidity values was tried to be determined. It was clearly shown that modal frequencies decreased with the increasing of temperature (Kakar and Kakar, 2015).

The Quanser Shake Table is a bench-scale earthquake simulator ideal for teaching structural dynamics, control topics related to earthquake, aerospace and mechanical engineering and widely used in applications. In this study was investigated of possibility using the recorded micro tremor data on ground level as ambient vibration input excitation data for investigation and application Experimental Modal Analysis (EMA) on the bench-scale earthquake simulator (The Quanser Shake Table) for steel benchmark structures.

For this purpose, operational modal analysis of a steel benchmark structure for dynamic parameters was evaluated at two temperatures (+2 and +32 degrees). Ambient excitation was provided by shake table from the recorded micro tremor ambient vibration data on ground level. Enhanced Frequency Domain Decomposition is used for the output only modal identification.

2. Possible Effects of Temperature on SHM Systems through Dynamic Parameters

This section summarizes Structural Health Monitoring (SHM) systems. In this study, only the effect of temperature on dynamic parameters was examined in terms of application. SHM systems must have the ability to distinguish environmental effects. Otherwise, especially factors such as frequency changes due to temperature may interfere with measurements for damage detection. Therefore, the effect of temperature on frequency and damping ratios was tested experimentally in this study. Thus, it is aimed to contribute to the modal analysis methods used within the scope of SHM. SHM Systems include continuous observation of structural performances against damage effects occurring in structures over time and evaluation of structural health with the obtained measurements. However, it also records environmental effects. These systems are of vital importance in large and important critical structures. In these systems, vibration and dynamic behaviors are detected by accelerometers, deformations are measured by strain gauges (Sevieri and De Falco, 2020), and displacements are measured by LVDTs (Noel et al., 2017). Temperature effects from environmental factors are measured by thermocouple/heat sensors (Avci et al., 2021). Fiber optic sensors are used for high-accuracy long-distance measurements (Park et al., 2013). All these measured data are collected, filtered, and stored by wireless sensor networks (WSN) or wired systems. Data processing and analysis software are used to perform modal analysis of the structural system, determine frequency changes, and calculate the damage index. Among all these processes, determining temperature changes is especially important for the healthy interpretation of other measurements. At this point, compensation techniques have been developed to eliminate temperature effects with artificial intelligence and machine learning trained with statistical analysis. Thus, it can be determined with high accuracy whether the structure is really damaged or not. Errors that may occur in decisions such as maintenance, reinforcement or evacuation decisions are eliminated. The purpose of SHM systems is to detect structural damage early, reduce maintenance costs, perform real-time risk assessment, verify performance-oriented design and ensure safety assessment after dynamic effects. Therefore, possible deviations from this goal are reduced.

In SHM systems, dynamic-based methods based on vibration data such as modal frequency, shape and damping ratio analyses are used. Thus, if there is damage in the structure, it is determined that the frequency decreases and the mode shapes change. In SHM systems, static loading-based methods such as monitoring strain and displacement data are also evaluated. The finite element model is compared with the real structure. Statistical inferences are made from long-term measurement data. However, while performing all these operations, environmental effects such as temperature must be separated. Sohn et al. (2004), who proposed statistical methods to separate the effect of temperature on frequencies, classified SHM methods and discussed the effects of environmental factors such as temperature and humidity on signal variation in long-term field applications. Farrar and Worden (2007), who presented the basic concepts, development and classifications of SHM, especially focused on the effects of environmental changes on modal parameters and emphasized how these effects can be misleading in damage detection. Lin et al. (2011) used SHM systems for seismic monitoring and wind loads in the Taipei 101 building in Taiwan. The effects of wind gusts and the Wenchuan earthquake on the structure were monitored using 30 accelerometers in Taipei101. The obtained natural frequency and mode shapes showed the effects of temperature and environmental changes on the dynamic behavior. The system was able to monitor the performance of the structure under real-time seismic and wind loads via SHM. Tcherniak and Mølgaard (2015) performed a vibration stimulation with an actuator placed inside the blade in an experiment carried out on a 34 m long wind turbine blade. Despite the effect of ambient temperature, they were able to successfully detect pre-damage cracks with artificial neural networks through the measured vibrations. A system that can distinguish crack-free structures was established by including the temperature-vibration relationship in the SHM model. Thus, rotor blade vibrations and temperature effects were examined together and pre-crack detections were made accurately. Li et al. (2023) monitored the temperature and vibration behavior of the structure in real time using distributed fiber optic temperature and strain sensors on the Sutong Bridge in China. They showed that the measured mode frequencies without isolation showed a significant relationship with temperature and that temperature effects can cause significant deviations in SHM analyses. As a result, it can be said that although SHM systems are indispensable tools of modern engineering, environmental effects such as temperature compensation must be eliminated by methods such as statistical modeling and machine learning.

Especially in steel structures, temperature affects the structural stiffness by changing the Young modulus. This change can cause serious deviations in modal frequencies. For example, Serker and Wu (2009) investigated the changes in modal frequencies due to temperature by conducting experiments on a simply supported steel beam. The results showed that the frequency changes caused by moderate structural damage can be at the same or greater levels than the temperature changes. In other words, the frequency deviations due to temperature are large enough to cause confusion during damage detection. This is an indication that the reliability of modal-frequency based SHM methods applied without considering the temperature effect may be low. Xia et al. (2012) investigated the effects of temperature on vibration properties such as modal frequency, shape and damping in literature review and case studies. The analyses showed that temperature changes significantly changed the dynamic parameters in steel structures and thin members such as reinforced concrete slabs. It was determined that the changes in structural vibration properties due to temperature changes can be more significant than the changes caused by moderate structural damage. Luo et al. (2022) also emphasizes that modal variables should not be confused with temperature and states that based on literary data, temperature effects can be equivalent to structural damage. All of this shows that temperature effects should not be ignored in SHM analyses. In this case, devices used for SHM applications should model or normalize the temperature-data relationship instead of only monitoring frequency changes. Otherwise, temperature changes may lead to false damage detections.

In large-scale structures, the temperature distribution is irregular on the surface. It is a time-dependent process that includes internal-external differences. Therefore, reliability cannot be achieved with measurements in determining the temperature-frequency relationship. Because the surface/core temperature may be different in different regions of the steel structure; temperature transition occurs with a delay in time (Thermal gradients). In addition, if the supports in steel systems are displaced due to the effect of temperature, the modal behavior also changes (Change of boundary conditions). For these reasons, it is not possible to represent the real thermal situation on the entire structure by measuring only the air and surface temperatures. To overcome these problems, Bao et al. (2017) used fiber optic sensors as distributed temperature sensors to fully monitor both internal and external temperatures. Wang et al. (2020) monitored a laboratory-scale RC plate for 1 year and collected data on natural frequency, mode shapes and damping ratios depending on the average temperature values changing over time. Statistical/graded models (LR, ARX, ARIMA) such as the ARX+PSO approach were used to neutralize the temperature effect and detect damage. These models were trained with data obtained from the undamaged structure and defined with confidence intervals. The results showed that the expected damage levels were correctly determined with this method. This study is an example that shows that the temperature effect can be systematically and reliably separated in the SHM literature; moreover, the ARX+PSO combination can correctly define the changes on the mode frequencies as “damage” or “environmental change”. As a result, it was determined in the study that temperature changes create significant fluctuations in natural frequencies and their effect on the mode shape and damping is limited. In the study of Venglar & Lamperova (2021), it was observed that some natural frequencies increase with increasing temperature, while other modes decrease. For example, while the frequency of the first two modes increases, the frequency of the fourth mode decreases. This situation shows that there may be nonlinear behaviors in the temperature-frequency relationship. Wang et al. (2021) experimentally applied gradual temperature gradients ranging from 10–30°C on the bridge deck element, and the sample was tested under different support conditions. The same temperature loading conditions were modeled with the ABAQUS FE model through heat transfer simulation. The temperature field obtained as a result of FE was analyzed based on stress and possible modal properties and compared with experimental observations. Determination of the real temperature field with FE provides accuracy since it is based on the thermal distribution representing the entire thickness of the structure instead of just the air temperature. The application of heat transfer, thermal expansion and modal analysis stages together realistically captures the effect of temperature changes on modal frequencies. The fact that the obtained results are generally compatible with experimental data with deviations below 1% supports the usability of such models in practical SHM applications. Kamei et al. (2021) experimentally investigated the effect of the temperature increase rate on the modal behavior of a steel cantilever beam. In the study, the characterization of the dynamic changes in the period until isothermal (constant temperature) conditions were obtained was determined. Controlled temperature was applied to the structure at different heating rates (low, medium and high). Under each heating rate, the system was tested with modal analysis when the structure reached a certain temperature. After the relevant temperature was kept constant, vibration data were collected and natural frequencies were calculated. The heat increase was achieved with controlled heaters rather than external environment. As a result, a significant decrease was observed in natural frequencies due to temperature increase. However, even if the same final temperature was reached, different heating rates affected the modal responses of the system differently. In other words, the temperature change rate stood out as a parameter as important as the temperature itself. It was concluded that at high heating rates, transient thermal stresses and temperature gradients were greater, and therefore larger deviations occurred in frequencies. This study shows that in SHM applications, not only the instantaneous value of the temperature but also the transition process should be taken into account. Therefore, these results show that models such as ARX and ARIMA, which try to neutralize temperature effects, should also include heating curves and thermal lag effects.

Poudel et al (2024) modeled the temperature-frequency dependency by entering the real temperature distribution into the structural model with FE analyses performed together with experimental tests. First, the temperature distribution from 20 to 50°C on the real steel-concrete composite beam sample was calculated with heat transfer and thermal expansion analyses in the FE model. Then, the obtained temperature field was used in the modal analysis and the natural frequencies were evaluated. FE results coincided with the experimental frequency values with high accuracy (±0.06% deviation). Poudel et al (2024) based their study on a simply supported cantilevered steel beam. In the experiments, a decrease of approximately 0.44% was observed in the second bending mode frequency as the temperature increased by 1°C. This experimental study shows that both the frequency decreases as the temperature increases and the modal amplitude decreases. It is also understood from the study that the temperature increase rate may also be effective. In the light of all this information, it can be said that if the temperature effects are neglected, serious errors are inevitable in SHM systems. Distributed temperature measurement and modeling are required to establish the temperature-frequency relationship reliably. Experimental data show that the general trend is a decrease in frequency as the temperature increases, but an increase in the frequency of some modes can also be observed in some structure/geometry combinations.

3. Materials and Methods

3.1. Modal Parameter Extractions

The EFDD technique used in SHM system applications was preferred in this study and used as the basic data generator to understand the effect of temperature changes on frequency and damping ratios. Thus, the sensitivity of modal parameters, which are the basic data source of statistical or artificial intelligence-based compensation methods for eliminating temperature-related errors in SHM applications, was evaluated. Similar to studies such as Poudel et al. (2024) and Wang et al. (2020) that directly model the temperature-frequency relationship in SHM, this study also aimed to analyze the temperature effect directly through frequency shifts

The (FDD) ambient modal identification is an extension of the Basic Frequency Domain (BFD) technique or called the Peak-Picking technique. This method uses the fact that modes can be estimated from the spectral densities calculated, in the case of a white noise input, and a lightly damped structure. It is a non parametric technique that determines the modal parameters directly from signal processing (Lus et al. 2003). The FDD technique estimates the modes using a Singular Value Decomposition (SVD) of each of the measurement data sets. This decomposition corresponds to a Single Degree of Freedom (SDOF) identification of the measured system for each singular value (Lam and Yang, 2015; Brincker et al. 2000). The Enhanced Frequency Domain Decomposition technique is an extension to Frequency Domain Decomposition (FDD) technique. This t technique is a simple technique that is extremely basic to use. In this technique, modes are easily picked locating the peaks in Singular Value Decomposition (SVD) plots calculated from the spectral density spectra of the responses. FDD technique is based on using a single frequency line from the Fast Fourier Transform analysis (FFT), the accuracy of the estimated natural frequency based on the FFT resolution and no modal damping is calculated. On the other hand, EFDD technique gives an advanced estimation of both the natural frequencies, the mode shapes and includes the damping ratios (Jacobsen et al. 2006). In EFDD technique, the single degree of freedom (SDOF) Power Spectral Density (PSD) function, identified about a peak of resonance, is taken back to the time domain using the Inverse Discrete Fourier Transform (IDFT). The natural frequency is acquired by defining the number of zero crossing as a function of time, and the damping by the logarithmic decrement of the correspondent single degree of freedom (SDOF) normalized auto correlation function (Peeters, 2000; Bendat,1998).

In this study modal parameter identification was implemented by the Enhanced Frequency Domain Decomposition. The relationship between the input and responses in the EFDD technique can be written as, In this method, unknown input is represented with “x” (“t” ) and measured output is represented with “y” (“t” )

$$\left[G_{yy}\left(j\omega \right)\right]={\left[H\left(j\omega \right)\right]}^{*}\left[G_{xx}\left(j\omega \right)\right]{\left[H\left(j\omega \right)\right]}^T$$

(1)

Where “G” _”xx” “(jω)” is the r x r Power Spectral Density (PSD) matrix of the input. “G” _”yy” “(jω)” is the m x m Power Spectral Density (PSD) matrix of the output, “H(jω)” is the m x r Frequency Response Function (FRF) matrix, and * superscript T denote complex conjugate and transpose, respectively. The FRF can be reduced to a pole/residue form as follows:

$$\left[H\left(\omega \right)\right]={\displaystyle \frac{\left[Y\left(\omega \right)\right]}{\left[X\left(\omega \right)\right]}}=\sum _{k=1}^m{\displaystyle \frac{\left[R_k\right]}{{j\omega -\lambda }_k}}+{\displaystyle \frac{{\left[R_k\right]}^{*}}{{j\omega -\lambda }_k^{*}}}$$

(2)

Where n is the number of modes “λ” _”k” is the pole and, R_k is the residue. Then Eq. (1) becomes as:

$$\begin{gathered} G_{yy}(j\omega )=\sum _{k=1}^n\sum _{s=1}^n\left[{\displaystyle \frac{\left[R_k\right]}{{j\omega -\lambda }_k}}+{\displaystyle \frac{{\left[R_k\right]}^{*}}{{j\omega -\lambda }_k^{*}}}\right] \\ {G}_{xx}(j\omega ){\left[{\displaystyle \frac{\left[R_s\right]}{{j\omega -\lambda }_s}}+{\displaystyle \frac{{\left[R_s\right]}^{*}}{{j\omega -\lambda }_s^{*}}}\right]}^{\overline{H}} \end{gathered}$$

(3)

Where s the singular values, superscript is H denotes complex conjugate and transpose. Multiplying the two partial fraction factors and making use of the Heaviside partial fraction theorem, after some mathematical manipulations, the output PSD can be reduced to a pole/residue form as fallows;

$$\left[G_{\mathrm{yy}}\left(j\omega \right)\right]=\sum _{k=1}^n{\displaystyle \frac{\left[A_k\right]}{{j\omega -\lambda }_k}}+{\displaystyle \frac{{\left[A_k\right]}^{*}}{{j\omega -\lambda }_k^{*}}}+{\displaystyle \frac{\left[B_k\right]}{-{j\omega -\lambda }_k}}+{\displaystyle \frac{{\left[B_k\right]}^{*}}{{-j\omega -\lambda }_k^{*}}}$$

(4)

Where “A” _”k” is the k th residue matrix of the output PSD. In the EFDD identification, the first step is to estimate the PSD matrix. The estimation of the output PSD known at discrete frequencies is then decomposed by taking the SVD (singular value decomposition) of the matrix;

$$G_{\mathrm{yy}}(j{\omega }_i)={U_i{S}_{i}U}_i^{\overline{H}}$$

(5)

Where the matrix $U_i=\left[u_{i1},u_{i2},\dots ,u_{im}\right]$ is a unitary matrix holding the singular vectors $u_{ij}$ and $s_{ij}$ is a diagonal matrix holding the scalar singular values. The first singular vector $u_{ij}$ is an estimation of the mode shape. PSD function is identified around the peak by comparing the mode shape estimation $u_{ij}$ with the singular vectors for the frequency lines around the peak. From the piece of the SDOF density function obtained around the peak of the PSD, the natural frequency and the damping can be obtained.

3.2. Description of Steel Benchmark Structure

The Quanser shake table II is a uniaxial bench-scale shake table. This unit can be controlled by appropriate software was illustrated in Figure 1a,b. It is effective for a wide variety of experiments for civil engineering structures and models. Shake table specifications are as follows Table 1 (Quanser, 2008).

Steel benchmark structure is 1.00 m height. Thickness of elements is 0.0015 m. The structure dimensions are shown in Figure 2.

The steel system used is made of galvanized steel and its chemical composition is given in Table 2. Galvanized steel is a material obtained by protecting the steel surface with zinc coating. It provides advantages in long-term experimental studies or in outdoor conditions due to its resistance to corrosion. By choosing galvanized steel as the material in this study, it became possible to observe the effect of temperature effects directly on steel elasticity (Young’s Modulus). If a different quality steel that is prone to rust and age was used, this effect could be noisy. In addition, the surface reflectivity of galvanized steel reduces thermal transmittance; this can soften the effects of thermal gradient. Thus, problems such as temperature difference between the inner-core and outer surface occurred less in the tested system. Since the chemical composition of galvanized steel (Table 2) is low-carbon and low-alloyed, its mechanical properties are more sensitive to temperature. This makes it easier to capture dynamic frequency changes. Poudel et al. (2024) and Serker & Wu (2009) also stated that the temperature-frequency relationship can be observed more clearly with low-carbon steel or zinc-coated samples. Thanks to the galvanized coating, the outer surface of the structure is less affected by temperatures. This allows the internal structure to show a more stable frequency characteristic. In this way, galvanized steel offers the opportunity to clearly observe the expansion and elasticity changes caused by temperature. The thermal behavior of the material is of critical importance in order to distinguish the effects of temperature changes in SHM systems. Galvanized steel elements allow reliable vibration data to be collected without material deterioration (e.g. rust, surface roughness) in long-term SHM applications. The suppression of the moisture effect due to the coating also facilitates the separation of the mixed effects of temperature and moisture in SHM systems. In the study of Xia et al. (2012), the response of steel structures to temperature differed according to the material type. More stable mode shapes were observed in galvanized samples. Luo et al. (2022) points out that controlled surface systems such as galvanized are easier to model in SHM algorithms.

The coordinates of the point where the experiments were conducted in Samsun province of Turkey are 41.3623618; 36.1836511. Figure 3. shows the monthly average temperature changes in Samsun province of Turkey (Climate-Data.org., n.d.)., where the experiment was conducted, between 1991 and 2021. Since the largest temperature difference is between January and August, the dates of the experiments were determined in this way.

4. Analysis of Experimental Results

4.1. Experimental Modal Analysis of Steel Benchmark Structure at +2 °C Degrees

Experimental modal analysis of steel benchmark structure at different temperatures was performed. The temperature values ​​have been determined by considering the long annual data. According to these data, the first measurement was made on 13.01.2024 when the ambient temperature was +2 degrees. Ambient excitation was provided by the recorded micro tremor data on ground level. Three accelerometers (with both x and y directional measures) were used for the ambient vibration measurements, one of which were allocated as reference sensor always located in the first floor (they are shown by the red line in Figure 4a, b). Two accelerometers were used as roving sensors (they are shown by the black line in Figure 4a, b). The response was measured in two data sets (Figure 4a, b). For two data sets were used 3 and 5 degree of freedom records respectively (Figure 4a, b). Every data set (Figure 4a, b) was measured 100 min. The selected measurement points and directions are shown in Figure 4a, b.

The data acquisition computer was dedicated to acquiring the ambient vibration records. In between measurements, the data files from the previous setup were transferred to the data analysis computer using a software package. This arrangement allowed data to be collected on the computer while the second, and faster, computer could be used to process the data in site. This approach maintained a good quality control that allowed preliminary analyses of the collected data. If the data showed unexpected signal drifts or unwanted noise or for some unknown reasons, was corrupted, the data set was discarded and the measurements were repeated (Figure 5a, b).

Before the measurements could begin, the cable used to connect the sensors to the data acquisition, equipment had to be laid out. Following each measurement, the roving sensors were systematically located from floor to floor until the test was completed. The equipment used for the measurement includes three quanser accelerometers (with both x and y directional measures) and geosig uni-axial accelerometer, matlab data acquisition toolbox (wincon). For modal parameter estimation from the ambient vibration data, the experimental modal analysis (EMA) software ARTeMIS Extractor (1999) is used.

The simple peak-picking method (PPM) finds the eigenfrequencies as the peaks of nonparametric spectrum estimates. This frequency selection procedure becomes a subjective task in case of noisy test data, weakly excited modes and relatively close eigenfrequencies. Also for damping ratio estimation the related half-power bandwidth method is not reliable at all. Frequency domain algorithms have been the most popular, mainly due to their convenience and operating speed.

Singular values of spectral density matrices, attained from vibration data using PP (Peak Picking) technique are shown in Figure 6. Natural frequencies acquired from the all measurement setup are given in Table 3. The first five mode shapes extracted from experimental modal analyses are given in Figure 7. When all measurements are examined, it can be seen that there are best accordance is found between experimental mode shapes (+2 °Cand +32 °C degrees). When the experimentally identified modal parameters are checked with each other, it can be seen that there is a best agreement between the mode shapes in experimental modal analyses at +2 °Cand +32 °C degrees. (Table 5).

4.2. Experimental Modal Analysis of Steel Benchmark Structure at +32 °C Degrees

Experimental modal analysis of steel benchmark structure at different temperatures was performed. The temperature values ​​have been determined by considering the long annual data. According to these data, the second measurement was made on 20.08.2024 when the ambient temperature was +32 degrees.

Measurements at +32 degrees Celsius were repeated for this temperature. Singular values of spectral density matrices, attained from vibration data using PP (Peak Picking) technique are shown in Figure 8. Natural frequencies acquired from the all measurement setup are given in Table 4. The first five mode shapes extracted from experimental modal analyses are given in Figure 9. When all measurements are examined, it can be seen that there are best accordance is found between experimental mode shapes (+2 °C and +32 °C degrees). When the experimentally identified modal parameters are checked with each other, it can be seen that there is a best agreement between the mode shapes in experimental modal analyses at +2 °Cand +32 °C degrees. (Table 5).

Table 4. Experimental modal analysis result at the steel benchmark structure at +32 °C.

Mode number	1	2	3	4	5
Frequency (Hz)	2.017	5.725	6.828	7.770	8.987
Modal damping ratio (ξ)	0.672	1.822	1.035	0.551	0.670

Table 4. Experimental modal analysis result at the steel benchmark structure at +32 °C.

Mode number	1	2	3	4	5
Frequency (Hz)	2.017	5.725	6.828	7.770	8.987
Modal damping ratio (ξ)	0.672	1.822	1.035	0.551	0.670

Table 5. Comparison of experimental modal analysis results.

Mode number	1	2	3	4	5
Experimental frequency (Hz) +2°C	2.067	5.868	6.998	7.964	9.211
Experimental frequency (Hz) +32°C	2.017	5.725	6.828	7.770	8.987
Difference (%)	2.418	2.436	2.429	2.435	2.431

Table 5. Comparison of experimental modal analysis results.

Mode number	1	2	3	4	5
Experimental frequency (Hz) +2°C	2.067	5.868	6.998	7.964	9.211
Experimental frequency (Hz) +32°C	2.017	5.725	6.828	7.770	8.987
Difference (%)	2.418	2.436	2.429	2.435	2.431