Simplified Frequency Estimation of Prefabricated Electric Poles Through Regression-Based Modal Analysis

Abstract

Prefabricated construction elements are widely used in both large- and small-scale projects, serving structural and infrastructural purposes. One notable application is in power transmission poles, which ensure the safe and efficient delivery of electricity. Despite their importance, limited research exists on the structural and modal behavior of reinforced concrete power poles. This study presents a comprehensive modal analysis of such poles, focusing on how factors like modulus of elasticity, height, and lower/upper inner and outer diameters influence dynamic performance. A total of 3240 finite element models were created, with reinforced concrete poles partially embedded in the ground. Modal analyses were performed to evaluate natural frequencies, mode shapes, and modal mass participation ratios. Results showed that increasing the modulus of elasticity raised frequency values, while greater pole height decreased them. Enlarging the lower inner and upper outer radii also led to higher frequencies. Regression analysis yielded high accuracy, with R² values exceeding 90% and an average error rate of about 6%. The study provides empirical formulas that allow for quick frequency estimations without the need for detailed finite element modeling, as long as the material and geometric properties remain consistent. The approach can be extended to other prefabricated structural elements.

1. Introduction

The construction industry has faced significant economic challenges, leading to a greater rise in construction costs compared to other sectors. In response, prefabrication techniques were developed to improve cost-efficiency through the mass production of standardized components. Elements such as columns, beams, slabs, and wall panels are manufactured under controlled factory conditions and widely used in large-scale projects. This approach supports off-site production, material standardization, and process consistency [1,2,3,4] (Amil and Aydin, 2004; Barlow et al., 2003; Pan et al., 2007; Thuesen and Hvam, 2011). Compared to traditional methods, prefabricated systems ensure higher construction quality by using components with uniform dimensions and minimal tolerance for error. Since most operations are carried out in factories, adverse seasonal effects are minimized, and production time is reduced through skilled labor and advanced equipment.

Concrete prefabrication is widely adopted in both large- and small-scale construction projects, serving structural, decorative, and infrastructural purposes [5,6] (Mtech Consult Limited, 2008; Seitablaiev and Umaroğulları, 2020). A prominent application is the production of energy transmission poles, which are vital for the safe and efficient delivery of electricity from production facilities to end-users. These poles help minimize energy losses, support high-voltage lines, and enhance safety, while contributing to a sustainable and balanced electricity distribution system. From both economic and environmental standpoints, they offer advantages over alternative solutions and are essential for meeting modern energy demands. The existing literature covers a range of materials used in pole design—including steel, concrete, and composites—focusing on their mechanical properties, durability, and cost-effectiveness [7,8] (Montgomery, 1999; Duggal, 2017). Studies also investigate performance under static and dynamic loads, maintenance needs, and environmental impacts, including effects on ecosystems and visual aesthetics [9] (Gonçalves et al., 2024).

The literature has addressed concerns related to potential impacts on natural habitats and the environmental footprint associated with the placement of these poles [10,11] (Vassiliou, 2009; Ezeonu and Anosike, 2017). Research has also investigated the use of composite materials in the design and construction of electricity poles to reduce their weight while enhancing durability [12,13,14] (Polyzois et al., 1999; Saboori and Khalili, 2011; Akoğlu, 2023). Despite the extensive studies available on the materials utilized in electricity pole design, their environmental effects, and their performance under load, research specifically focused on reinforced concrete electricity poles is notably scarce. While the existing literature predominantly emphasizes steel and composite materials, there remains a deficiency in the comprehensive analyses of the performance of reinforced concrete poles regarding static and dynamic loading, long-term durability, and maintenance requirements.

Due to the geometry of electric poles, creating a model is complex and difficult for engineers accustomed to classical designs. In addition, the placement of reinforcement and the spiral continuity of stirrups along the electric pole make it difficult to model with many existing commercial programs. Therefore, this study was carried out within the scope of university–industry collaboration to find the realistic frequency values and vibration patterns of electric poles that a manufacturing company currently faces, causing modeling problems. In addition, the study aims to eliminate this difficulty by examining the effects of geometric and material parameters on modal properties in detail. In addition, due to long-term modeling studies and the lack of technical staff performing modeling, the frequency values of electricity poles can be determined quickly with the estimation model according to different sizes, sections, and materials. Thus, this study aimed to create the closest model to the produced electricity poles, optimize the use of materials, and reduce the need for expensive experimental studies.

Theoretical and experimental modal analysis methodologies are employed to determine the mode shapes, damping ratios, and natural frequency values, which are recognized as the dynamic characteristics of engineering structures. The Theoretical Modal Analysis Method is currently executed using computational tools and structural analysis software.

In the analytical phase of this method, it is essential to develop a mathematical model that closely approximates the structural reality to facilitate analysis [15,16] (Güneş, 2017; Erkek et al., 2023). The finite element (FE) method, a numerical approach that considers material properties, support conditions, and geometric characteristics to ascertain the dynamic properties of structures, will be utilized to evaluate the mechanical properties of the system in question. The literature contains numerous studies dedicated to the modal analysis of reinforced concrete structures. Huang and Syu conducted a study on the free vibration and modal analysis of a typical tower crane frame utilizing SAP2000 V13 and ANSYS software. They developed a numerical model using three-dimensional beam elements for the vertical and horizontal frames, as well as rod elements for the connecting rods [17] (Huang and Syu, 2014). In a related study, Calayır et al. (2021) [18] explored the dynamic characteristics of masonry minarets through the FE model, focusing on a specific mosque minaret. Their analysis incorporated the interaction between the structure, foundation, and soil within the FE modeling framework. The FE solutions were achieved with the ANSYS 19 r1 FE package program. Erkek and Yetkin investigated a historical minaret that sustained damage from two significant earthquakes, with magnitudes Mw = 7.7 and Mw = 7.6, that struck the eastern region of Türkiye on 6 February 2023. They developed an FE model to evaluate the seismic performance of the mosque, examining its seismic responses under the earthquake’s influence [19] (Erkek and Yetkin, 2023). Currently, there is limited information available regarding the modal properties of the most commonly used reinforced concrete electric transmission poles. To address this gap in the literature, our study presents a detailed analysis of the modal characteristics of reinforced concrete electric transmission poles, considering various lengths, diameters, and material properties, along with an estimation model for the corresponding modal frequencies.

2. Material and Method

Centrifugal reinforced concrete electric poles are produced in our country following the “Electric High Current Installations Regulation” and the “Centrifugal Reinforced Concrete Pole Manufacturing Technical Specification” established by Turkish Electricity Distribution Inc. The manufacturing process is illustrated in Figure 1. The stages of production depicted in Figure 1 include the preparation of the electric pole reinforcement (Figure 1a), the placement of the reinforcement into specially designed molds (Figure 1b), the pouring of concrete into the molds (Figure 1c), the sealing of the molds (Figure 1d), the execution of the centrifugal process (Figure 1e), and the finalization of the produced electric poles (Figure 1f).

In the Electric High Current Installations Regulation, it is stipulated that “the safety coefficient based on the yield stress of the steel when calculating concrete poles must not be less than 1.5, and for the breaking test, the safety coefficient for failure must be no less than 2.” According to Turkish Electricity Distribution Inc., Ankara. Turkey, “the force (in kg) acting horizontally at the top of the pole represents the horizontal forces exerted on the pole during operation, measured 25 cm below the top and perpendicular to the pole’s axis”.

2.1. Design and Structural Features

Reinforced and prestressed poles shall be produced in the lengths and with minimum and maximum top forces outlined in

Table 1. Features of reinforced concrete electric poles manufactured according to TEDAŞ-MLZ/99-34.

Factor of Safety	Pole Length (m)	Minimum/Maximum Peak Force (kg)
Lower voltage (LV) and medium voltage (MV) joint grid poles within the city	9.3–10–11	300/3500
	12–13–14	300/3500
	15–16	300/3300
	17–18	500/3000
	19	500/2600
	20–21	500/2600
	22–23–24–25	600/2500
Large spaced overhead line poles	10–11–12–13–14	300/3500
	15–16	300/3300
	17–18	500/3000
	19	500/2600
	20–21	500/2600
	22–23–24–25	600/2500

below, based on the safety coefficients established for their intended application. Top forces will be increased incrementally in 100 kg intervals up to 600 kg and in 200 kg intervals thereafter.

This study focuses on the centrifugal reinforced concrete pole manufactured by KAMBETON A.Ş., which stands at a height of 9.3 m and has a load-carrying capacity of 300 kg. The total height of the electric pole is 9.3 m, with lower and upper outer diameters measuring 0.0875 m and 0.1575 m, respectively. The inner diameters at the lower and upper sections are 0.04 m and 0.0865 m, respectively. For the purpose of the designs, it was assumed that the first 120 cm of the pole would remain buried in the ground.

2.2. Finite Element Model

SE analysis is a computational technique used to predict how structures behave under various physical conditions. It is an analysis method based on dividing a model into smaller Fes [20] (Karthikeyan et al., 2020).

2.2.1. Workbench

ANSYS Workbench r1 2024 is a comprehensive software platform for engineering simulation. It provides a unified environment for modeling, analyzing, and visualizing engineering problems. Workbench software provides a range of tools for geometry creation, mesh generation, analysis, and post-processing. ANSYS Workbench can be used for both simple and complex linear or nonlinear analyses [21] (Lawrence, 2024).

Materials used for electric poles can be delineated as commands pertaining to the material properties of concrete and reinforcement, which are commonly utilized in the ANSYS Workbench program. The electric pole is a composite structure and given that both solid and link elements are employed in its modeling, establishing a connection between these components is essential [22] (Asif et al., 2023). In the ANSYS Workbench FE model, the SOLID65-3D element has been designated to represent the concrete [23] (ANSYS, 2025). The SOLID65 element is recognized for its efficacy in the 3D modeling of reinforced concrete solids. Its advanced capabilities in simulating cracking and crushing behavior in structurally reinforced concrete elements are notably effective. This element facilitates the accurate definition of nonlinear material properties, enabling the simulation of cracking in three vertical directions, crushing, and plastic deformation. The reinforcement element demonstrates both tensile and compressive capabilities. The LINK 180 element type, sourced from the ANSYS element library, is utilized to model vertical and horizontal steel bars and is applicable for both linear and nonlinear deformations within its plane. The described solid model possesses the ability to crack under tension and crush under compression. Each SOLD65 element, representing the concrete, consists of eight nodes, each possessing three degrees of freedom—accounting for translations in the x, y, and z directions [24,25] (Çelik et al., 2022; Venkatachalam et al., 2021). The geometry of the element, along with node locations and the coordinate system, is illustrated in the SOLID65 geometry in Figure 2.

The damping ratio is used in various studies in the range of 0.7–10% for uncracked reinforced concrete members [26,27,28] (Yan et al., 2007; Thyagarajan et al., 1998; Nguyen et al., 2020). In structural design applications, a viscous damping ratio of approximately 5% is generally accepted to represent the energy loss due to damping in reinforced concrete systems [29,30] (Riaz et al., 2023; Mostafaei et al., 2025). In this study, a viscous damping ratio of 5% was also accepted.

2.2.2. Modal Analysis

Modal analysis is conducted to explore the vibration characteristics, including natural frequencies and mode shapes, of a mechanical structure or component. It provides insight into how different parts of the structure move under dynamic loading conditions. This analysis presents various modes along with their corresponding natural frequencies, allowing for the determination of vibration frequencies. To evaluate these frequencies, modal analysis was carried out using the subspace iteration method, which is based on the Rayleigh–Ritz method and the power method. This approach has proven to be particularly effective for resolving large-scale structures with a limited number of vibrations and modes [31,32] (Ge et al., 2011; Muhammed et al., 2020).

Modal analysis is widely used to understand and predict the behavior of structures. However, since these methods are based on idealized assumptions, the results obtained analytically must be verified with experimental data. In this context, Ndambi et al. used experimental modal analysis in their study to demonstrate the accuracy of the modal analysis of the reinforced concrete beam. The beam length, cross-section, and reinforcement placement are shown in Figure 3 [33] (Ndambi et al., 2000).

The tests are prepared and conducted in the laboratory of the Civil Engineering Department of the K.U. Leuven. During the dynamic tests of the beam, they suspended it by means of elastic springs in order to simulate the free boundary conditions. They placed these springs at the theoretical nodes of the basic (first) bending mode of the beams. In the analytical model, the experimental model was taken into account; the concrete was modeled using the SOLID65 element and the reinforcement was modeled using the LINK180 element. The experimental and analytical model analysis results are given in

Table 2. Experimental and FE model frequencies of the validated beam and frequency differences.

Mode No.	Experimental Frequency (Hz)	FE Mode Frequency (Hz)	Frequency Difference (%)
1	20.48	20.79	1.52
2	57.07	56.51	−0.98
3	112.10	113.17	0.95
4	169.2	174.70	3.25
5	183.8	192.58	4.78

. It was determined that the mode shapes in the study conducted by [33] Ndambi et al. were similar to the analytical mode shapes. The concrete mode shape and the reinforcement mode shape are given in Figure 4. It was observed that the concrete and reinforcement worked together.

2.3. Modeling of Electricity Pole

The numerical model of the reinforced concrete electric pole was developed using the ANSYS Workbench platform. In the model, the concrete material was defined with a modulus of elasticity of 35 GPa and a density of 2400 kg/m³, while the reinforcement steel was assigned a modulus of elasticity of 200 GPa and a density of 7850 kg/m³. The FE representation of the pole, including the reinforcement detailing and overall geometry, is illustrated in Figure 5. The FE model was constructed using 2880–3802 elements and 20,280–28,392 nodes.

In this study, various parameters, including elasticity values, heights, and both lower and upper outer and inner diameters of the electric pole, were identified as significant variables. Figure 6 illustrates the flow chart developed for the purposes of this research. The elasticity values were established at 30, 32, 33, 34, 36, and 37 GPa. The height measurements were defined as 9.3, 10.1, 10.92, 11.73, and 12.54 m, reflecting the heights commonly utilized within the industry. The Lower Inner Diameter (Bid) was designated at 0.125, 0.25, 0.75, and 1 m, which are particularly favored in the sector. To ensure a consistent gradient between the top and bottom of the electric pole, the Lower Outer Diameter (Bod), Upper Inner Diameter (Tid), and Upper Outer Diameter (Tod) were calculated in relation to the Bid values.

This paper presents a workflow, as illustrated in the accompanying figure, designed to estimate model frequencies for electric poles without the need for detailed analysis. To this end, a sensitivity analysis was conducted utilizing the identified variables relevant to the electric pole, resulting in the acquisition of model frequencies. Subsequently, the data obtained from this analysis underwent a pre-processing stage to prepare it for the regression model. In the dataset analysis, all numerical data underwent standardization. This process resulted in normalized data with a mean of 0 and standard deviation of 1, thereby achieving an approximately normal distribution. This standardization process was crucial for optimizing model performance and ensuring consistent scaling across all variables. The dataset was then divided into training and testing subsets, following a 30–70% split, and estimation analyses were carried out. The estimations derived from the training dataset were rigorously compared with the testing dataset, yielding estimation accuracies for each frequency analyzed. All parameters in the study consisted of continuous numerical values representing the physical properties and dimensions of the electric poles. The standardization process was applied to both input features and target variables, ensuring consistent scaling throughout the dataset. This preprocessing step was fundamental for the performance of the ridge regression model and the reliability of the results. Ten-fold cross-validation is a resampling method employed to reduce bias in the training data and to determine if the model’s performance is attributable to randomness. The training data is divided into 10 segments, known as folds. In each iteration, one fold is designated as the validation set, while the other nine folds are utilized for training. After completing this process for all ten folds, the resulting model performances are assessed and averaged to confirm the model’s validity.

2.4. Ridge Regression Method

In this study, the ridge regression (RR) model was employed to estimate the frequencies obtained from the FE model more efficiently. The selection of ridge regression as the primary methodology in this study was based on its superior performance in preliminary analyses. When compared to alternative regression methods including linear regression, Lasso regression, and ElasticNet, ridge regression demonstrated the highest prediction accuracy while maintaining computational efficiency. This was particularly crucial given our large dataset of 3240 finite element models. Moreover, ridge regression’s ability to effectively handle multicollinearity among geometric parameters and generate practical prediction equations made it the most suitable choice for achieving the study’s objectives. RR is a form of linear regression specifically designed to address issues of multicollinearity. High correlations among independent variables due to multicollinearity can compromise the performance and accuracy of traditional linear regression models. Ridge regression mitigates this problem [34] (Walker and Birch, 1988). It is regarded as a more reliable and realistic regression method because it reduces the bias associated with the least squares method when estimating regression coefficients. While the least squares method treats variables equally, the relationships among them can vary significantly due to multicollinearity, leading to notable discrepancies between observed and actual values. RR demonstrates greater robustness than the least squares method when dealing with implausible data [35] (Harrington, 2012). Furthermore, when multicollinearity exists among independent variables, the ordinary least squares (OLS) regression method becomes less effective for calculating parameter estimates. In light of this, Hoerl A. E. introduced the ridge regression estimation in 1962 [36] (Envere et al., 2023).

Ridge regression (RR) has emerged as a valuable technique for analyzing multiple regression data that exhibit multicollinearity. By introducing a certain degree of bias into the regression estimates, RR effectively reduces standard errors, leading to more accurate estimates of the regression coefficients compared to the ordinary least squares method. The RR parameter is obtained by minimizing the sum of squared errors, which incorporates a constraint on the coefficients, encouraging them to approach zero. This method penalizes features with larger coefficients by adding a penalty that is proportional to the square of the coefficients’ magnitudes, thereby lessening the cost function [37] (Abhishek, 2021). The penalty term employed in RR is referred to as the L2 norm, mathematically represented by the following equation 1 [38] (Hoerl and Kennard, 1970):

$$\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t} \mathrm{F}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}=\sum _{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{y}}_{\mathrm{i}}-{\widehat{\mathrm{y}}}_{\mathrm{i}})}^2+\mathsf{\lambda }\sum _{\mathrm{j}=1}^{\mathrm{p}}{\mathsf{\beta }}_{\mathrm{j}}^2$$

(1)

where $\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2$, Residual Sum of Squared Errors. λ (lambda): regularization parameter. This parameter controls how much the model is penalized. β_j: They are the coefficients of the independent variables of the model. This formula reveals the main difference in ridge regression: as the regularization parameter λ increases, the coefficients become smaller, which makes the model more generalizable [39] (Hoerl, 2020).

Model Evaluation Metrics

In this study, several metrics were employed to assess the models, including Mean Absolute Percentage Error (MAPE), Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Correlation Coefficient (R²). These metrics have been used to estimate and compare the prediction accuracy and error rates of the models analyzed in the literature [40,41,42,43] (Karatas and Budak, 2024; Guvel et al., 2025; Dawid et al., 2025; Dawid et al., 2024). MAPE, an evaluation metric, expresses the error rate of the model as a percentage and is calculated as shown in Equation (2). The MAE is determined by taking the absolute value of the difference between each predicted value and the actual value, as illustrated in Equation (3). A lower MAE value, approaching zero, indicates enhanced prediction performance of the model. Another important metric is the RMSE, widely recognized for regression problems, which is computed as outlined in Equation (4). Similarly to the MAE, a lower RMSE signifies more accurate predictions. Since RMSE shares the same units as the dependent variable, it is frequently preferred over Mean Square Error (MSE) and MAE for evaluating the performance of regression models alongside other models. Additionally, the R² value, calculated using Equation (5), serves as a regression evaluation criterion that ranges from 0 to 1, with values closer to 1 indicating a superior model fit. The result derived from the R² value can also be interpreted as the proportion of variance explained by the model.

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\left|{\displaystyle \frac{{\mathrm{y}}_{\mathrm{i}}-{\widehat{\mathrm{y}}}_{\mathrm{i}}}{{\mathrm{y}}_{\mathrm{i}}}}\right|\ast 100$$

(2)

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\left|{\widehat{\mathrm{y}}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}\right|$$

(3)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\left({\widehat{\mathrm{y}}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}\right)}^2}$$

(4)

$${\mathrm{R}}^2=1-{\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{i}}-{\widehat{\mathrm{y}}}_{\mathrm{i}}\right)}^2}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{i}}-\overline{{\mathrm{y}}_{\mathrm{i}}}\right)}^2}}$$

(5)

where

$n=$ number of values in the dataset.

$y_i=$ actual values in the dataset.

${\widehat{y}}_i=$ predicted values in the dataset.

$\overline{y_i}=$ arithmetic mean of the actual values.

3. Results and Discussion

The electric pole was analyzed using a vertical finite element program with varying cross-sections and heights to determine its natural frequencies and mode shapes. The results indicate that the similarity of the mode shapes is more pronounced for the initial modes, gradually decreasing for subsequent ones. Consequently, the first ten modes were selected for this study. Additionally, modal analysis of the electric poles yielded the corresponding mode shapes, which were generally found to be similar. The mode shapes for the first ten modes of the 9.3 m tall electric pole are illustrated in Figure 7.

In Figure 8, the mode shapes for ten modes of electricity poles with varying heights (9.3 m, 10.1 m, 10.92 m, 11.73 m, and 12.54 m) are presented. The modal analysis revealed that the first eight modes exhibited similar translational behavior across the different heights. However, in Modes 9 and 10, the 9.3 m electricity pole displayed torsional behavior. Overall, it was observed that the mode shapes were largely similar, aligning well with the fundamental vibration theory of Euler–Bernoulli beams with lumped masses in bending vibration modes [44] (Ferroudji et al., 2021).

Upon examining the obtained frequencies, it was observed that Frequency (F) 1 and F2 are numerically very close to each other, as are F3 and F4; F5, F6 and F7; F8 and F9; and F10.

Consequently, similar numerical frequency values were consolidated as a result of the modal analysis. In this study, F1 and F2 are the mod x and y translations for the first bending mode, F3 and F4 are the mod x and y translations for the second bending mode, F5 is the mod first torsion mode, F6 and F7 are the mod x and y translations for the third bending mode, F8 is the mod x translations for the fourth bending mode, F9 is the mod y translations for the fifth bending mode, and F10 is the mod x translation for the sixth bending mode.

Additionally, the frequency values obtained from the modal analysis were categorized based on six different variables: Elastic, Height, Bid, Tid, Bod, and Tod. The statistical properties of these variables corresponding to each frequency value are presented in

Table 3. Statistical properties of the data.

	Mean	Std.	Min	25%	50%	75%	Max
Elastic (Gpa)	33.67	2.358	30	32	34	36	37
Height (m)	10.92	1.14	9.3	10.11	10.92	11.73	12.54
Bid (m)	0.164	0.044	0.094	0.131	0.156	0.2	0.25
Tid (m)	0.089	0.032	0.041	0.064	0.082	0.11	0.179
Bod (m)	0.082	0.014	0.062	0.075	0.087	0.1	0.1
Tod (m)	0.044	0.014	0.021	0.033	0.045	0.054	0.072
F1–2 (Hz)	2.70	1.16	0.73	1.83	2.49	3.35	7.73
F3–4 (Hz)	11.47	3.86	4.31	8.57	10.87	13.80	25.50
F5 (Hz)	28.56	9.46	10.55	21.46	27.05	34.20	66.29
F6–7 (Hz)	53.13	16.90	19.80	40.39	50.71	63.67	116.22
F8 (Hz)	78.44	21.20	32.07	62.48	76.15	92.42	152.90
F9 (Hz)	84.52	25.23	32.07	65.19	81.72	101.51	171.17
F10 (Hz)	102.54	25.80	47.32	83.31	99.50	119.41	198.99

. According to the data, the Elastic value ranges from 30 GPa to 37 Gpa, while the heights span from 9.3 m to 12.54 m. Furthermore, FE analysis indicated that each successive frequency value exceeds the preceding one when compared across the ten frequency values analyzed.

The correlations among the variables Elastic, Height, Bid, Tid, Bod, and Tod—representing distinct frequencies—along with their relationships to frequency values across various modes, are illustrated in Figure 9. Upon examining the high correlation values among these variables, several key observations can be made:

• A consistent and strong negative correlation exists between the ‘height’ variable and the ‘f’ variables across all modes. This indicates that as the ‘height’ increases, the ‘f’ values tend to decrease. Notably, the intensity of this negative correlation generally escalates with higher mode numbers.
• A robust positive correlation is observed between the ‘bid’ and ‘tod’ variables across all modes (r ≈ 0.85). Additionally, moderate-to-strong positive correlations are evident between ‘bid’ and ‘bod’ (r ≈ 0.63), as well as between ‘bid’ and ‘f’ (r ranging from approximately 0.55 to 0.75, with stronger correlations in the initial modes). These findings suggest that the variables ‘bid’, ‘bod’, ‘tid’, and ‘tod’ are interrelated.

Conversely, an examination of the lower correlations among the variables reveals the following:

• The variable ‘elastic’ displays correlations that are nearly zero with most other variables (specifically with ‘height’, ‘bid’, ‘bod’, ‘tid’, and ‘tod’) across all modes. A very weak positive correlation is noted only with the ‘f’ values (r ≈ 0.08 to 0.14), indicating that the property ‘elastic’ is largely linearly independent of the other parameters under consideration.
• The ‘height’ variable also lacks a significant linear relationship with the group comprising ‘bid’, ‘bod’, ‘tid’, and ‘tod’, as the correlations remain close to zero.

While the overall patterns of correlation are predominantly consistent across modes, minor variations in the strength of certain relationships do exist. The most pronounced alteration is seen in the strengthening of the negative correlation between ‘height’ and ‘f’ values in the higher modes. Furthermore, the positive correlation between ‘bid’ and ‘f’ values is most substantial in Modes 1–2 (r = 0.75) and experiences a slight decrease in the subsequent modes (r ≈ 0.55 to 0.67).

In this study, three regression models—ridge regression (RR), Lasso Regression, and ElasticNet Regression—were comparatively analyzed to efficiently estimate the frequencies obtained from the FE model, as presented in

Table 4. Regression analysis results.

	Ridge Regression (RR)				Lasso Regression (LR)				ElasticNet Regression (ENR)
	MAPE (%)	MAE	RMSE	R2	MAPE (%)	MAE	RMSE	R2	MAPE (%)	MAE	RMSE	R2
F1–2	8.42	0.1933	0.2655	0.9496	9.19	0.2327	0.3201	0.9266	17.03	0.4334	0.5803	0.7590
F3–4	5.54	0.5692	0.7468	0.9644	5.76	0.6062	0.7990	0.9593	9.07	1.0359	1.3827	0.8780
F5	5.37	1.3694	1.7801	0.9661	5.38	1.3833	1.7989	0.9653	8.86	2.5077	3.3262	0.8815
F6–7	5.13	2.5626	3.5479	0.9583	5.14	2.5740	3.5555	0.9581	9.14	4.7606	6.1872	0.8731
F8	7.74	5.9556	7.3659	0.8887	7.80	6.0140	7.4239	0.8869	10.38	7.8779	9.7891	0.8034
F9	5.24	4.3827	6.0895	0.9451	5.23	4.3767	6.0984	0.9449	9.30	7.5007	9.5521	0.8650
F10	6.44	6.4751	7.9178	0.9070	6.44	6.4704	7.9165	0.9070	7.92	7.9368	10.0789	0.8493

. The comparative analysis revealed that the ridge regression model demonstrated superior performance in terms of prediction accuracy and computational efficiency, thus being selected as the optimal approach for frequency estimation in this research. These estimates were considered successful, as the R² values exceeded approximately 0.90. The most accurate estimate was for the F5 frequency, which achieved an R² value of 0.9661, with an estimated error of around 5%. Other successful estimates include F3–4, F6–7, F1–2, F9, F10, and F8, in that order.

Alongside the analysis results, the performance of the models was assessed using both actual and estimated values. The graphs presented in Figure 10 illustrate that as prediction success increases, the data approaches the red line. Overall, prediction success is found to be high across all modes, with Mode 5 exhibiting the highest level of prediction success.

Figure 11 displays the prediction errors. For optimal prediction success, it is essential that prediction errors remain minimal. Therefore, values close to zero indicate that predictions are made with minimal error. The figure reveals that prediction errors are notably high in Modes 8–9 and Mode 10, while errors in the other modes are relatively lower.

The equations derived for estimating frequencies as a result of the RR analyses (Equations (6)–(11)) are presented below. It is important to consider the success and error rates of the estimations made for each equation in relation to the values mentioned previously. These equations represent regression models that predict the value of the dependent variable (f) across various modes, based on a set of independent variables (elastic, height, bid, tid, bod, and tod). Each equation delineates the quantitative relationship between these variables for a specific mode or group of modes.

$$\begin{gathered} f_{\mathrm{1,2}}=2.47+\left(0.16\ast elastic\right)+\left(-0.86\ast height\right)+\left(1.50\ast \mathrm{b}\mathrm{i}\mathrm{d}\right) \\ +\left(-0.70\ast \mathrm{t}\mathrm{i}\mathrm{d}\right)+\left(0.01\ast \mathrm{b}\mathrm{o}\mathrm{d}\right)+(0.23\ast \mathrm{t}\mathrm{o}\mathrm{d}) \end{gathered}$$

(6)

$$\begin{gathered} f_{\mathrm{3,4}}=10.66+\left(0.68\ast elastic\right)+\left(-3.72\ast height\ast \right)+\left(3.54\ast \mathrm{b}\mathrm{i}\mathrm{d}\right) \\ +\left(0.30\ast \mathrm{t}\mathrm{i}\mathrm{d}\right)+\left(0.20\ast \mathrm{b}\mathrm{o}\mathrm{d}\right)+\left(0.83\ast \mathrm{t}\mathrm{o}\mathrm{d}\right) \end{gathered}$$

(7)

$$\begin{gathered} f_5=26.69+\left(1.69\ast elastic\right)+\left(-9.26\ast height\right)+\left(6.29\ast \mathrm{b}\mathrm{i}\mathrm{d}\right)+\left(3.60\ast \mathrm{t}\mathrm{i}\mathrm{d}\right) \\ +\left(0.69\ast \mathrm{b}\mathrm{o}\mathrm{d}\right)+\left(1.78\ast \mathrm{t}\mathrm{o}\mathrm{d}\right) \end{gathered}$$

(8)

$$\begin{gathered} f_{\mathrm{6,7}}=49.79+\left(3.15\ast elastic\right)+\left(-16.57\ast height\right)+\left(10.88\ast \mathrm{b}\mathrm{i}\mathrm{d}\right) \\ +\left(6.04\ast \mathrm{t}\mathrm{i}\mathrm{d}\right)+\left(0.56\ast \mathrm{b}\mathrm{o}\mathrm{d}\right)+\left(4.10\ast \mathrm{t}\mathrm{o}\mathrm{d}\right) \end{gathered}$$

(9)

$$\begin{gathered} f_8=74.96+\left(4.53\ast elastic\right)+\left(-21.23\ast height\right)+\left(17.79\ast \mathrm{b}\mathrm{i}\mathrm{d}\right) \\ +\left(-4.85\ast \mathrm{t}\mathrm{i}\mathrm{d}\right)+\left(-0.39\ast \mathrm{b}\mathrm{o}\mathrm{d}\right)+\left(8.28\ast \mathrm{t}\mathrm{o}\mathrm{d}\right) \end{gathered}$$

(10)

$$\begin{gathered} f_9=142.75+\left(1.26\ast elastic\right)+\left(-15.58\ast height\right)+\left(229.84\ast \mathrm{b}\mathrm{i}\mathrm{d}\right) \\ +\left(161.20\ast \mathrm{t}\mathrm{i}\mathrm{d}\right)+\left(23.92\ast \mathrm{b}\mathrm{o}\mathrm{d}\right)+\left(349.26\ast \mathrm{t}\mathrm{o}\mathrm{d}\right) \end{gathered}$$

(11)

$$\begin{gathered} f_{10}=98.03+\left(6.15\ast elastic\right)+\left(-27.27\ast height\right)+\left(20.67\ast \mathrm{b}\mathrm{i}\mathrm{d}\right) \\ +\left(0.19\ast \mathrm{t}\mathrm{i}\mathrm{d}\right)+\left(3.73\ast \mathrm{b}\mathrm{o}\mathrm{d}\right)+(0.17\ast \mathrm{t}\mathrm{o}\mathrm{d}) \end{gathered}$$

(12)

When these equations are evaluated, the following is considered:

Variable Height: The coefficient for this variable is negative across all modes, with its absolute value increasing significantly as the mode number rises (from −0.86 to −27.27). This indicates that the function consistently decreases with an increase in height, and this diminishing effect becomes increasingly pronounced in higher modes.

Variable Elastic: This variable has a positive coefficient in every mode, with the coefficient rising notably as the mode number increases (from 0.16 to 6.15). This suggests that an increase in the elastic parameter leads to an increase in, with this effect becoming more substantial in higher modes.

Variable Bid: Likewise, the variable bid features positive coefficients in all modes, with these coefficients escalating as the modes progress (from 1.50 to 20.67). This illustrates that a larger bid size results in an increase in, and the effect intensifies in higher modes.

Variables Tid and Bod: The effects of these variables are more nuanced. The coefficient for tid is negative in certain modes (for example, −0.70 in Equation (6) and −4.85 in Equation (10)) but positive in others (such as 3.60 in Equation (8) and 6.04 in Equation (9)). This variability indicates that the influence of tid on can shift direction depending on the active mode. Similarly, the coefficient of bod can change sign (e.g., −0.39 in Equation (10) and +3.73 in Equation (11)), and its magnitude also varies.

Variable Tod: Generally, tod tends to have a positive coefficient, with its magnitude increasing as the modes progress (for instance, from 0.23 in Equation (6) to 8.28 in Equation (10)). However, it diminishes again in Equation (11) (to 0.17). This may suggest that while tod generally contributes to an increase, its impact weakens in the highest mode.

In this study, the electric poles used had heights ranging from 9.3 to 12.54 m and Bottom Inner Diameters (Bid) between 0.125 and 1 m. The Bottom Outer Diameter (Bod), Top Inner Diameter (Tid), and Top Outer Diameter (Tod) were calculated based on a consistent slope between the top and bottom of the electric pole. It should be noted that these regression equations are valid only within these height and diameter ranges, as the reinforcement plan changes with increasing height. Since electric poles manufactured in the industry typically fall within these ranges, these equations can be reliably used for frequency estimation in practical applications. However, when parameters fall outside these ranges, new analyses would be required. For future research, it is recommended to validate and test the applicability of the regression models presented here across a wider range of electric pole configurations. Additionally, validation of the models through actual field measurements would enhance reliability for practical applications.

In this study, the reinforcement and reinforcement ratio were assumed constant for various heights and diameters. The effects of changes in reinforcement diameter or additional reinforcement on frequencies and mode shapes are beyond the scope of this research. Furthermore, the boundary condition used in the analysis of the electric pole was defined as a fixed support, as specified in relevant regulations. However, different boundary conditions such as spring supports, solid foundation models, etc., could be employed in future studies. The impact of using different damping ratios (0–7%) on frequencies varies between 0.1% and 0.01%, indicating that damping has minimal effect on the system behavior or frequencies. Therefore, this study adopted the commonly used 5% damping ratio, which is frequently referenced in the literature.

4. Conclusions

This study investigates the influence of various factors—including the modulus of elasticity, height, lower and upper inner diameters, and lower and upper outer diameters—on the dynamic properties of electric poles. To conduct the numerical analysis, finite element models of 3240 distinct electric poles were developed. It was assumed that these poles were made of reinforced concrete and that a specific area was firmly anchored to the ground. Modal analyses were performed, and the poles were evaluated based on their natural frequencies and mode shapes. A prediction model was developed using ridge regression based on the analytically determined frequencies as dependent variables and Elastic modulus, Height, Bottom inner diameter (Bid), Top inner diameter (Tid), Bottom outer diameter (Bod), and Top outer diameter (Tod) as independent variables. To minimize bias and enhance prediction accuracy, 10-fold cross-validation was implemented during the ridge regression analysis. According to the results, the frequency predictions achieved acceptable accuracy with an approximate error rate of 5% and an R² value of approximately 0.90. The other results of this investigation are summarized below:

• The increase in the modulus of elasticity in all frequency types increased the frequency.
• It has been determined that as the height of the electricity poles increases, the natural frequency values decrease.
• It was determined that all frequency values increased as the Lower Inner Diameter and Upper Outer Diameter values increased.
• When the upper inner diameter value increases, the F3, F4, F5, F6, F7, F9, and F10 frequency values increase, but the F1, F2, and F8 frequencies decrease.
• When the lower outer diameter value increases, the F8 frequency decreases and the frequencies outside of these increase.
• When the regression analysis results are evaluated, it is seen that the R2 values of all frequencies are approximately above 90%. In addition, predictions were made with an error rate of approximately 6% at all frequencies.
• One of the primary limitations of this study is the absence of direct experimental or field validation of the developed regression model. The model was constructed solely on the basis of simulation data, a factor that must be acknowledged as a constraint. Nonetheless, the simulation dataset was meticulously generated through systematically controlled parametric analyses and was designed to encompass a comprehensive range of reinforcement configurations. The model demonstrated robust statistical performance, evidenced by an R² value exceeding 0.95, along with minimal values for RMSE, MAE, and MAPE, thereby indicating a high level of predictive accuracy within the specified input domain. Future work is planned to validate the model through experimental pullout tests, which will further substantiate its applicability in real-world contexts.
• Moreover, the natural frequencies of electric poles can be efficiently calculated using the aforementioned formulas, even in the absence of finite element (FE) analysis, albeit with varying degrees of error. It is crucial to recognize that these equations are contingent upon the specific material and geometric characteristics of the selected electric poles. In subsequent research endeavors, this methodology may be expanded beyond electric poles to encompass a diverse array of pre-production components, facilitating swift frequency estimations without the complexities typically associated with FE analysis.