Investigation of the Effect of Coating Light Steel Container Houses with Nano-TiO₂ on Dynamic Parameters Using OMA

Abstract

In recent years, the integration of nano titanium dioxide (TiO₂) into building materials has become a popular research topic due to its superior mechanical, photocatalytic and self-cleaning properties. In this study, the dynamic behavior of a light steel container house model coated with nano-TiO₂ is investigated using Operational Modal Analysis (OMA). The effects of TiO₂ on the natural frequencies, damping ratios and mode shapes of the light steel container house model are investigated. The Stochastic Subspace Identification-Unweighted Principal Component (SSI-UPC) method is used to extract the modal parameters from the ambient vibration data. The results show that the TiO₂ coating significantly increases the stiffness and improves the damping properties by increasing the natural frequencies of the light steel container house model. The findings indicate that nano-TiO₂ coatings can increase the structural integrity and durability of light steel container houses. This study provides a foundation for future research on nano-reinforced coatings in light steel structural systems.

1. Introduction

In recent years, developing nanotechnology has significantly increased the mechanical, chemical and environmental performances of materials used in civil engineering [1]. Especially titanium dioxide (TiO₂) is widely used in innovative building materials due to its properties, such as photocatalytic activity, self-cleaning ability, air purification and increased durability [2]. While traditional concrete and cement-based building materials are negatively affected by external factors such as air pollution, UV rays and abrasion, nano-TiO₂-added materials both increase the mechanical strength and enable the construction of sustainable structures [3]. Studies show that TiO₂ increases the compressive strength of concrete, reduces crack formation and provides long-term durability [4]. In addition, TiO₂ has been found to promote environmental sustainability by decomposing air pollutants (NOx, SOx, VOC) [5]. Furthermore, thanks to self-cleaning concrete and exterior coatings, maintenance costs are reduced and building life is extended [6].

Light steel is a widely used material in construction, automotive, aerospace and industrial applications, offering great advantages due to its high strength and workability [7]. However, when exposed to environmental factors, it encounters problems such as corrosion, wear and surface oxidation [8]. Therefore, advanced coating solutions are of great importance for making light steel long-lasting and durable. Titanium dioxide (TiO₂) coatings offer innovative solutions for light steel surfaces due to their high hardness, photocatalytic properties and chemical resistance [9]. One of the most important advantages of TiO₂ coatings is that they provide high corrosion resistance. When applied to a steel surface, it reduces the contact of the steel with air and moisture by creating a barrier against oxidation [10]. In addition, the photocatalytic activity of TiO₂ decomposes organic pollutants and rust layers, ensuring that the surface remains clean and durable [11]. In addition, TiO₂ coatings increase mechanical strength by increasing wear resistance on mild steel surfaces. Studies have shown that nano-TiO₂-reinforced coatings significantly increase surface hardness and wear resistance [12]. In addition, the hydrophobic properties of TiO₂-coated mild steel surfaces prevent water and other liquids from adhering to the surface, increasing the environmental resistance of the material [13]. Lightweight steel structures have a wide range of applications in modern civil engineering, and are widely preferred in residential, commercial and industrial buildings due to their high strength to weight ratio, fast installation time and sustainability advantages [14]. However, the dynamic behavior of light steel structures is significantly different from traditional reinforced concrete and steel structural systems, and these differences should be examined under earthquake, wind and vibration loads [15]. Dynamic parameters include the characteristics of a structure, such as natural frequencies, mode shapes, damping ratios and vibration amplitudes [16,17,18,19]. The low mass density and thin-walled elements of light steel systems cause them to have low natural frequencies and high mode shapes [20]. This situation requires a detailed engineering analysis in terms of wind-induced vibrations, seismic loads and human-induced dynamic effects [21]. In recent years, studies have investigated the behavior of lightweight steel structures under earthquake loads and examined the nonlinear effects of mode shapes and damping ratios [22]. In particular, the buckling and fatigue strengths of structural elements produced from cold-formed steel profiles are of critical importance in terms of dynamic stability [23]. The finite element method (FEM) and vibration tests are widely used in the dynamic analysis of lightweight steel structures [24]. These methods are applied to more accurately estimate structural damping capacity, vibration amplitudes and behavior under different loading scenarios [25].

Operational modal analysis (OMA) is an important technique used in the process of experimental investigation and modeling of structural dynamics [26,27]. It has great importance in determining the vibration modes and frequencies of structures, especially in engineering fields. OMA is an approach to the description of dynamic systems, usually supported by measurement techniques such as structural monitoring, vibration testing and frequency response function. In recent years, this analysis technique has been developed to overcome the limitations of traditional modal analysis methods and has been applied to a much wider application area [28,29,30]. While operational modal analysis involves experimental studies by observing the natural vibrations of the structure, using only the vibrations due to environmental conditions without applying an external stimulus (e.g., a shock or vibration source) makes this technique particularly advantageous [31,32,33]. This feature of OMA offers a much less invasive approach compared to traditional modal analyses, thus allowing it to be easily used in various application areas, especially in industrial structures. Non-destructive structure property determination methods have been developed in recent years, and many experimental studies have been conducted in many different countries of the world in this field [34,35].

The Stochastic Subspace Identification-Unweighted Principal Component (SSI-UPC) method aims to increase the accuracy rate by reducing the computational load compared to the traditional covariance-driven (COV-SSI) and data-driven (DATA-SSI) methods [36]. One of the main reasons for this is the reduction of unnecessary noise and errors by applying unweighted principal component analysis (PCA) [37]. SSI-UPC allows the automatic and reliable determination of modal parameters [38]. Traditional SSI methods are sensitive to external noise. SSI-UPC provides more reliable results thanks to the noise filtering mechanisms [39]. The compression of Hankel matrices and the optimization of singular value decomposition (SVD) make it easier to work with large data sets by reducing processing time [40]. SSI-UPC is widely used to determine the natural frequencies, damping ratios and mode shapes of large-scale engineering structures [41]. The SSI-UPC method is increasingly adopted for structural health monitoring and performance evaluation of large-scale structures in the field of engineering. It is considered an ideal method, especially for remote sensing and big data-based analysis [42]. Also, [43,44,45] new studies on operational modal analysis (OMA) were used.

The aim of this study is to reveal the effects of nano-TiO₂ coating application on the dynamic parameters of the container house model manufactured from light steel by using an innovative and non-destructive method, operational modal analysis. Thus, it will be seen how well nano-TiO₂ coatings can meet the rigidity and reinforcement demands of light steel container houses.

2. Materials and Methods

2.1. The Stochastic Subspace Identification Technique (SSI)

The stochastic subspace identification (SSI) method is a time-domain approach that directly utilizes raw time data, eliminating the need to convert it into correlations or spectra. The SSI algorithm determines the state-space matrices from measurements through robust numerical techniques. Once the mathematical representation of the structure (the state-space model) is obtained, it becomes easy to compute the modal parameters. The theoretical foundations are provided by Overschee and De Moor [36] and Peeters [46]. The model for vibrating structures can be expressed through a set of linear, constant coefficient and second-order differential equations [46]:

$$m\ddot{u}\left(t\right)+c\dot{u}\left(t\right)+ku\left(t\right)=F\left(t\right)=df\left(t\right)$$

(1)

In this context, $m, c, k$ represent the mass, damping, and stiffness matrices, respectively. $F\left(t\right)$ denotes the excitation force, while $u\left(t\right)$ is the displacement vector at continuous time t. d is an input influence matrix that characterizes the locations and types of known inputs $f\left(t\right)$. The state-space model, which originates from control theory, is also widely used in mechanical and civil engineering to calculate the modal parameters of a dynamic structure with a general viscous damping model [8]. The equation of motion (1) is transformed into the state-space form of first-order equations—i.e., a continuous-time state-space model of the system is formulated as

$$\dot{z}\left(t\right)=A_{c}z\left(t\right)+B_{c}f\left(t\right)$$

(2)

$$A_c=\left[\begin{array}{cc}0& I\\ -m^{-1}k& -m^{-1}c\end{array}\right]$$

(3)

And Equation (4):

$$B_c=\left[\begin{array}{c}0\\ m^{-1}d\end{array}\right]$$

(4)

$$z(t)=\left[\begin{array}{c}u\left(t\right)\\ \dot{u}\left(t\right)\end{array}\right]$$

(5)

where $A_c$ is the state matrix, $B_c$ is the input matrix and $z\left(t\right)$ is the state vector. The number of elements in the state-space vector corresponds to the number of independent variables required to describe the state of the system. If it is assumed that the measurements are taken at only one sensor location, and that these sensors can be accelerometers, velocity transducers, or displacement transducers (accelerometers), then the observation equation is as follows:

$$y\left(t\right)=C_a\ddot{u}\left(t\right)+C_v\dot{u}\left(t\right)+C_{d}u\left(t\right)$$

(6)

where $y\left(t\right)$ are the outputs and $C_a, C_v, C_d$ are the output matrices for acceleration, velocity and displacement, respectively. With these definitions,

$$C=[C_d-C_a{m}^{-1}k{C}_v-C_a{m}^{-1}c]$$

(7)

$$D=C_a{m}^{-1}d$$

(8)

Equation (6) can be transformed into

$$y\left(t\right)=Cz\left(t\right)+Du\left(t\right)$$

(9)

where $C$ is the output matrix and $D$ is the direct transmission matrix. Equations (2) and (6) together form a continuous-time deterministic state-space model. “Continuous time” means that the expressions can be evaluated at each time instant $t\in \mathbb{R}$, and deterministic means that the input–output quantities $u\left(t\right),y\left(t\right)$ can be measured exactly. However, this is not realistic in practice: measurements are typically taken at discrete time instants $k\Delta t$, where $k\in \mathbb{N}$ with $\mathsf{\Delta }t$ is the sample time, and noise always affects the data. After sampling, the state-space model becomes

$$z_{k+1}=A{z}_k+B{u}_k$$

(10)

$$y_k=C{z}_k+D{u}_k$$

(11)

where $z_k=z\left(k\mathsf{\Delta }t\right)$ is the discrete-time state vector, $w_k$ represents the process noise due to disturbances and modeling imperfections and $v_k$ is the measurement noise resulting from sensor inaccuracies. The stochastic noise is included, and as a result, we obtain the following discrete-time combined deterministic-stochastic state-space model:

$$z_{k+1}=A{z}_k+B{u}_k+w_k$$

(12)

$$y_k=C{z}_k+D{u}_k+v_k$$

(13)

The $w_k,v_k$ vectors are non-measurable, but it is assumed that they represent white noise with a zero mean. If this white noise assumption is violated—meaning that the input contains dominant frequency components in addition to white noise—these frequency components cannot be distinguished from the eigenfrequencies of the system. As a result, they will manifest as eigenvalues in the system matrix $A$.

$$E\left[\begin{array}{cc}\left(\begin{array}{c}{w}_p\\ v_p\end{array}\right)& \left(\begin{array}{cc}{w}_q^T& v_q^T\end{array}\right)\end{array}\right]=\left(\begin{array}{cc}Q& S\\ S^T& R\end{array}\right){\delta }_{pq}$$

(14)

where $E$ is the expected value operator and ${\delta }_{pq}$ is the Kronecker delta. In structural health monitoring (SHM), the available vibration information typically consists of the responses of a structure excited by operational inputs, which are often immeasurable. Due to the absence of input information, it is not possible to differentiate the deterministic input $u_k$ from the noise terms $w_k,v_k$, as discussed in [47]. If the deterministic input term $u_k$ is modeled by the noise terms $w_k,v_k$, the discrete-time purely stochastic state-space model of a vibrating structure is derived as

$$z_{k+1}=A{z}_k+w_k$$

(15)

$$y_k=C{z}_k+v_k$$

(16)

2.2. Materials

2.2.1. The Container House Model

The container house model consists of a lower and upper chassis. It is manufactured from four corner posts connecting the lower and upper chassis. The wall and roof cladding consists of 1.5 mm light steel DKP sheet metal on the outside and 2.5 mm EPS filler panels on the inside. The Young modulus of the light steel DKP sheet is 210 Gpa, the Poisson ratio is 0.3 and the unit volume weight is 7800 kg/m³. The Young modulus of the EPS filler panel is 150 Mpa, the Poisson ratio is 0.28 and the unit volume weight is 15 kg/m³. The container house model is anchored to the ground. The profile thickness on the lower and upper chassis is 3.5 mm. Chassis profiles with a thickness of 4 mm were used on the corner posts. A total of 120 chassis profiles were used on the corner posts, another 120 chassis profiles were used on the upper chassis and 105 chassis profiles were used on the lower chassis. Chassis profiles are given in Figure 1 as corner, lower and upper.

The dimensions of the light steel container house model are 2.4 m on the X axis, 8.1 m on the Y axis and 2.6 m in height from outside to outside. The dimensions of the light steel container house model on the plan are given in Figure 2.

The light steel container house model with accelerometers mounted on it is given in Figure 3.

2.2.2. Accelerometers and Dataloggers

The next step after preparing the steel garage model for operational modal analysis is the calibration of accelerometers and dataloggers. In this study, four 3-axis accelerometers and two 4-channel dataloggers were used. The locations of the accelerometers were determined using FEMtools 3.3 software [48]. Since there were sufficient accelerometers, the reference accelerometer was not used and the measurement was performed in one go. The accelerometers were fixed to the relevant nodes with hot glue and strong, double-sided tape. Testlab Network and MATLAB 2023a software [49] were used for accelerometer calibrations and data conversion with Fast Fourier Transform (FFT). The technical specifications of the accelerometers used for operational modal analysis are given in

Table 1. The technical specifications of the accelerometers.

Name	SENSEBOX 7021/22/23+
Number of Axes	3
Type	FBA-Force Feedback
Maximum Acceleration Measurement Range	±3 g
Output Noise Performance	130 ng/√Hz
Frequency Range	0.1–120 Hz
Sensibility	2400 mV/g
Shock Resistance	2000 g
Supply Voltage	+6 ± 15VDC
Working Temperature	−40 °C~+65 °C

.

The technical specifications of the dataloggers used for operational modal analysis are given in

Table 2. The technical specifications of the data loggers.

ADC Resolution	24 Bit
Synchronization	Simultaneous sampling—independent ADC per channel
Sensor Input Channel Count	4
Dynamic Measurement Range	138 dB
Sampling Rate (Max.)	4 Channel version up to 16 kHz/channel (standard−2 kHz)
Input Signal Type	Differential (best performance)/single ended
Analog Filter	Anti-aliasing/low-pass
DSP	Oversampling/downsampling/decimation/digital filter
Sensor Alert Options	+5 V, +12 V, −12 V (standard)
Sensor Connector Structure	IP67, push-pull, round, 9 pin
Digital I/O	4 Inputs/4 Outputs
CPU	ARM-cortex
Working Temperature	−20 °C~+55 °C
Power Input	12 VDC/40 watts max

.

The accelerometers and dataloggers used in the study are given in Figure 4.

Artemis Modal Pro 4.0. software [50] was used for parameter estimation. The layout plans of the accelerometers are given in Figure 5.

3. Results

3.1. Operational Modal Analysis Model of Light Steel Container House

Measurements were made at approximately 14 degrees Celsius and 57% humidity. Similar measurements were repeated to increase the data set security and for optimal purposes. Data sets that had unwanted noise (corrupt data) were discarded. During the measurement, the measurement time trigger command was used, and the light steel container house was stopped 250 m away at the time of the measurement. Among the repeated measurements, consistent data sets were taken as the basis many times. The measurement step was selected as 200 Hz based on previous studies. In calculating the measurement period, the “Introduction to Operational Modal Analysis” book [51] was taken into account. The measurement time relation taken from the Introduction to Operational Modal Analysis book is given in Equation (17).

$$t_{min}=10/\xi$$

(17)

Here, $t_{min}$ represents the minimum measurement period, and $\xi$ represents the theoretical damping ratio of the structure type. In this study, the damping ratio for steel structures was taken as 0.02. According to Equation (17), the minimum measurement period was found to be 500 s. In order to make the measurements more reliable, the measurement period was taken as 600 s. The placement of channel numbers is given in Figure 6. The accelerations obtained from each channel for data processing as a result of the measurement are given in Figure 7. Odd-numbered channels represent the X-direction, and even-numbered channels represent the Y-direction.

The data were processed in Artemis Modal Pro software. Detrending and filtering processes were performed. Lowpass was selected as the filtering method in the collection and processing of data. The SSI-UPC (Stochastic Subspace Identification-Unweighted Principal Component) method, which is a time domain method, was used to obtain the parameters. The spectral density matrices are given in Figure 8.

The mode shapes of the first five modes of the light steel container house model, which were obtained using operational modal analysis, are given in Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13, respectively.

The modal analysis results for the first five modes of the light steel container house model obtained using operational modal analysis are given in

Table 3. The operational modal analysis results for the light steel container house.

Mode	1	2	3	4	5
Frequency (Hz)	2.470	3.951	4.326	4.781	6.775
Mode Shape	Translation	Torsion	Translation	Torsion	Torsion
Damping (%)	1.152	0.584	1.937	3.912	0.754

.

3.2. Operational Modal Analysis Model of Light Steel Container House with Coating TiO₂

The entire outer surface of the light steel container house model is coated with titanium dioxide (TiO₂) using the cold spray method. The surface is expected to dry during each application. Approximately 1 h of curing is required in order to prepare a surface for application of titanium dioxide. After these setups, operational modal analysis tests are followed by curing to obtain the experimental dynamic characteristics, similar to previously used properties in order to obtain comparative measurements. A titanium dioxide layer was applied in a single layer. After curing, measurements were made every 50 cm² of the model with a paint measuring device. The measurement results ranged between 100 and 120 microns [52]. While 1 h is sufficient for curing, measurements were taken after 3 h to ensure the reliability of the operational modal analysis results. The locations of the accelerometers were determined before the coating was applied, and data were taken from the same point. Measurements were made at approximately 13 degrees Celsius and 56% humidity. The measurement time is unchanged, and is 600 s. The accelerations obtained from each channel for data processing as a result of the measurement are given in Figure 14.

The data were processed in Artemis Modal Pro software. Detrending and filtering processes were performed. Lowpass was selected as the filtering method in the collection and processing of the data. The SSI-UPC (Stochastic Subspace Identification-Unweighted Principal Component) method, which is a time domain method, was used to obtain the parameters. The spectral density matrices are given in Figure 15.

The mode shapes of the first five modes of the house TiO₂-coated model of the light steel container, which were obtained using operational modal analysis, are given in Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20, respectively.

The modal analysis results for the first five modes of the TiO₂-coated model of the light steel container house, obtained using operational modal analysis, are given in

Table 4. The operational modal analysis of the light steel container house coated with TiO₂.

Mode	1	2	3	4	5
Frequency (Hz)	3.322	5.607	5.930	6.266	7.962
Mode Shape	Translation	Torsion	Translation	Torsion	Torsion
Damping (%)	1.804	1.109	1.623	1.973	2.235

.

3.3. Comparison Results

The comparison of the frequency values of the first five modes obtained using operational modal analysis for both models (non-coated and coated) is given in

Table 5. Comparison of frequency values (non-coated and coated).

Mode	1	2	3	4	5
Non-coated (Hz)	2.470	3.951	4.326	4.781	6.775
Coating (Hz)	3.322	5.607	5.930	6.266	7.962
Difference (Hz)	0.852	1.656	1.604	1.485	1.187
Difference (%)	34.49	41.91	37.08	31.06	17.52

.

The comparison of the mode shapes of the first five modes obtained using operational modal analysis for both models (non-coated and coating) is given in

Table 6. Comparison of mode shapes directions (non-coated and coating).

Mode	1	2	3	4	5
Non-coated	Translation	Torsion	Translation	Torsion	Torsion
Coating	Translation	Torsion	Translation	Torsion	Torsion

.

The comparison of the damping ratio of the first 5 modes obtained by operational modal analysis for both models (non-coated and coated) is given in

Table 7. Comparison of damping ratios (non-coated and coated).

Mode	1	2	3	4	5
Non-coated (%)	1.152	0.584	1.937	3.912	0.754
Coated (%)	1.804	1.109	1.623	1.973	2.235
Difference (%)	0.652	0.525	−0.314	−1.939	1.481
Difference ratio (%)	56.60	89.90	−16.21	−49.57	196.42

.

All experimental measurements were made under identical conditions as specified for both models. Thus, the aim was to minimize the error margins in the comparison data obtained and to ensure the reliability of the results. In addition, when the results obtained are examined, it is seen that they are in line with the literature and theoretical expectations.

When the acceleration values from all channels were compared, it was observed that the accelerations in the X-direction decreased by approximately 60%, and the accelerations in the Y-direction decreased by approximately 86% with the TiO₂ coating.

When the natural frequencies for Mode 1 were examined, an increase of 34.49% was observed in the model in which the light steel container house was coated with TiO₂. This observed increase also shows the increase in rigidity. When the Mode 1 shapes were compared, no change was observed. When the damping ratios in Mode 1 were compared, an increase of 56.60% was observed. These changes in Mode 1 show that the TiO₂ coating increases the safety of the structure.

When the natural frequencies for Mode 2 were examined, an increase of 41.91% was observed in the model in which the light steel container house was coated with TiO₂. This observed increase also shows an increase in rigidity. When the Mode 2 shapes were compared, no change was observed. When the damping ratios in Mode 2 were compared, an increase of 89.90% was observed. These changes in Mode 2 show that the TiO₂ coating increases the safety of the structure.

When the natural frequencies for Mode 3 were examined, an increase of 37.08% was observed in the model in which the light steel container house was coated with TiO₂. This observed increase also shows an increase in rigidity. When the Mode 3 shapes were compared, no change was observed. When the damping ratios in Mode 3 were compared, a decrease of 16.21% was observed. Although there was a decrease in the damping ratio, there was also a higher percentage increase in the frequency ratio. Therefore, these changes in the third mode show that the TiO₂ coating positively affects structural safety.

When the natural frequencies for Mode 4 were examined, an increase of 31.06% was observed in the model in which the light steel container house was coated with TiO₂. This observed increase also shows an increase in rigidity. When the Mode 4 shapes were compared, no change was observed. When the damping ratios in Mode 4 were compared, a decrease of 49.57% was observed. Although there was a significant increase in stiffness due to an increase in frequency, a significant decrease in the damping ratio was observed proportionally. However, a ratio close to the 2% damping ratio predicted for steel was obtained based on the damping value. Therefore, it can be said that the TiO₂ coating gave positive results in terms of structural safety in the fourth mode.

When the natural frequencies for Mode 5 were examined, an increase of 17.52% was observed in the model in which the light steel container house was coated with TiO₂. This observed increase also shows an increase in rigidity. When the Mode 5 shapes were compared, no change was observed. When the damping ratios in Mode 5 were compared, an increase of 196.42% was observed. These changes in Mode 5 show that the TiO₂ coating increases the safety of the structure.

4. Conclusions

As a result of the study, it was concluded that coating the light steel container house model with TiO₂ significantly increased rigidity. Although there were reductions in damping ratios in some modes, the overall effect was positive as it did not cause a change in the mode shapes. In addition, there was a positive effect in the decrease in the accelerations coming to the nodes, which also provided evidence of the increase in rigidity. Considering the mass of TiO₂ applied to the structural system for reinforcement, it was observed that it was quite low compared to the structural system. In the literature and in theory, it is seen that increasing rigidity increases the dynamic load-bearing capacity, and that with an increase in the damping ratio, the structural system can absorb the acting dynamic forces more effectively. Given its experimental findings, this study demonstrates the usability of TiO₂ coatings as a method for reinforcing light steel container houses in order to increase their structural load-bearing capacity and dynamic safety. In addition, it can be said that the TiO₂-coating application is a practical method because it is applied by cold spraying. As a result of this study, it was concluded that coating light steel container houses with TiO₂ is recommended as a reinforcing method.