Wind Turbine Tower Structural and Cost Optimisation within Acceptable Structural Integrity

Emeka H. Amalu ^* , Justice O. Nwaneto

1. Department of Engineering, School of Computing, Engineering and Digital Technologies (SCEDT), Teesside University, Middlesbrough, Cleveland, TS1 3BA, UK

* Correspondence: Emeka H. Amalu 

Academic Editor: Andrés Elías Feijóo Lorenzo

Received: June 26, 2025 | Accepted: February 08, 2026 | Published: February 13, 2026

Journal of Energy and Power Technology 2026, Volume 8, Issue 1, doi:10.21926/jept.2601003

Recommended citation: Amalu EH, Nwaneto JO. Wind Turbine Tower Structural and Cost Optimisation within Acceptable Structural Integrity. Journal of Energy and Power Technology 2026; 8(1): 003; doi:10.21926/jept.2601003.

© 2026 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.

1. Introduction

The rapid global expansion of wind energy necessitates reliability engineering to ensure system safety, as wind power which is clean and increasingly cost-competitive is emerging as a strong alternative to traditional coal and nuclear energy sources [1]. Wind turbine is a device that is used to harness wind energy to generate mechanical power which in turn is transformed to electrical energy. The height of tower of wind turbine and the angular rotation of the blade determine the magnitude of power output of the system. Most contemporary wind turbines have three blades mounted on a horizontal axis shaft that is coupled to a vertical tower. A key challenge to tower reliability is local collapse. It is caused by instability occasioned by buckling failure. To address this challenge, researchers and designers are utilising multidisciplinary design-based optimization methodologies to boost the dependability and robustness of wind turbine structures [2]. As turbine tower ought to be very tall to harness substantial wind energy, a critical challenge becomes how to design a lightweight tall wind turbine structure [3]. It is reported that structural stability of wind turbine tower is a crucial design consideration [4,5].

Figure 1 below shows a wind turbine comprising blades, nacelle and tower structure. The nacelle contains the drivetrain-gearbox, generator and transformer. Wind-induced blade rotation drives the shaft, activating the generator to produce electricity, while the tower structurally supports all components.

Figure 1 Components of a typical wind turbine structure.

Wind turbine towers are typically made from tubular steel, assembled from 20-40 m tapered sections joined by flanged bolts. Tapering reduces weight while enhancing structural stability.

While numerous studies focus on rotor and blade design, fatigue strength, and optimisation, few address the structural reliability of support towers despite rising failures and cost competitiveness demands [6,7,8,9,10,11,12,13]. Ref. [7] assesses failure rates and reliability of floating offshore wind turbines, they developed a model and Bayesian network to assess failure rates and reliability of floating offshore wind turbines, identifying key influencing factors. Their findings show tower failures comprise 36% of support structure failures, the highest among components, with tower collapse accounting for 83% of these, and other failures making up the remaining 17%. Ref. [8] assesses the fatigue reliability of offshore wind turbine with stochastic availability. They focused on foundation fatigue in the structural reliability whilst implementing a failure probability model to assess the availability of offshore wind turbine. In Ref. [10], a proven methodology for reliability-based design optimization (RBDO) of floating wind turbine (FWT) systems is developed, presented, and applied to the spar-buoy FWT. The study enhances deterministic design optimization (DDO), which targets a less over-dimensioned floating structure by aiming for a global system performance within the defined limits for safe operation, by incorporating reliability criteria and accounting for environmental uncertainties. Ref. [11] studies the effect of environmental modelling and inspection strategy on the optimal design of floating wind turbines. The researchers assessed trade-offs between design costs and inspection costs with different design fatigue factors (DFFs). The support structure of interest includes tower and platform. Ref. [12] analysed fatigue reliability of jacket support structure for offshore wind turbines, considering effects of corrosion. They modelled welded multi-planar tubular joints designed for 70 m deep North Sea site, accurately predicting dynamic responses to wind and wave loads using a decoupled analysis approach.

As demand drives the trend toward taller towers sharing key design techniques becomes essential. This study presents methods for optimising the cost and structural integrity of a 90 m wind turbine tower using Taguchi Design of Experiment method. It focuses on key parameters: wall thickness, top-to-bottom diameter ratio, and steel grade. The research is timely, given that WT towers account for 14-20% of total turbine cost [14], tower failures can cause significant downtime, and wind turbine capacity building is vital to achieving the Net Zero agenda by 2050.

2. States of the Investigation

Techniques of application of wind thrust on tower and methods of parameter optimisation led to variation in the results of tower design. This research reviews the techniques, selects and implements appropriate techniques and methods. Discussion on these is presented thus:

2.1 Application of Wind Thrust on Turbine Tower

The simplest model for application of wind thrust on a tubular tower considers the tower as prismatic; the trust is applied as a uniformly distributed load over the height of the tower [15,16]. This method is presented in Figure 2a. Another model is a step function of uniformly distributed load. It is denser at the top of the tower and lightest at the bottom of the tower. This application is shown in Figure 2b [17,18]. A third technique implemented in Ref. [16] has 3 uniformly distributed wind loads applied in three shell segments which is summed up as total pressure acting on the tower. As reported in Refs. [14,18,19,20,21], some researchers considered the curvature of the vertical profile of the wind and developed mathematical model for the real structural response. The method is presented in Figure 2c. Other researchers [22] obtained drag coefficient for the tower for each height differential, an approach which appears more complex and rigorous but more practical. The effects of implementation of these techniques are yet to be fully evaluated, and the possible error associated with their use remain unknown [23]. Improving on the challenge, Edwin. H [23] adapted the three methods presented and developed an expression for calculating wind thrust on tower. Proposed model has tower drag coefficient (C_e) that depends on Reynolds number. On review of the models, this study models the wind thrust on tower as a triangular distributed profile. The profile of the load model is presented in Figure 3. The model assumes the wind speed increases uniformly with height of the tower.

Figure 2 Different wind load profiles for various proposed models from Ref. [23].

Figure 3 Triangular distributed wind load showing resultant force F_WT and its location on the tower.

2.2 Methods of Analysis Employed

Numerous researchers including [19,24,25,26,27,28,29] implemented various optimisation techniques in their investigations on optimisation of structural integrity of wind turbine tower in operation. Present study employs the Taguchi design of experiment (DOE) technique to generate the designs which are simulated in ANSYS Finite element analysis software. Taguchi has been demonstrated as a reliable technique in parameter design optimisation.

2.3 Ethical Statement

All procedures in this study were conducted in accordance with ethical standards. No human or animal subjects were involved, and no personal or sensitive data was collected. The research complies with institutional national and international ethical guidelines for scientific integrity and responsible research conduct.

3. Methodology

Three key methods are employed in this investigation. These are Taguchi design of experiment in conjunction with signal to noise (S/N) ratio analysis, finite element analysis (FEA) and analytical methods. This section discusses the research paradigm in sub-section 3.1, Taguchi design in sub-section 3.2, FEA in sub-section 3.3 and analytical method in sub-section 3.4.

3.1 Research Paradigm

This study follows the research framework in Figure 4, beginning with a literature review to identify strategies supporting NET Zero. Wind turbine technology, backed by stakeholders, is key to this goal. Instability-induced tower failures are noted. Three optimisation methods are applied, using Taguchi designs in ANSYS, validated analytically and based on deflection, stress, and buckling responses to evaluate tower structural reliability. The research demonstrates how capacity can be built in delivering cost effective tower at acceptable reliability.

Figure 4 Research paradigm showing the schematic of the research design.

3.2 Taguchi Design of Experiment

This technique is used in this investigation because it has been demonstrated and reported as a successful optimisation method employed to empirically understand the impacts of design variables on product performance [30,31,32,33]. It utilises standard orthogonal arrays in its designs. Three structural reliability influencing factors of wind turbine tower are considered at three levels. The factors are tower wall thickness, top-to-bottom diameter ratio, grade of steel material. They are designated as A, B and C, respectively. Utilising Taguchi design, L₉ orthogonal array was selected. Table 1 presents the tower reliability factors considered, while Table 2 shows the nine (9) design points and the standard model generated using the design. The volumes of the shells of the nine designs are equal. The standard model is a 5 MW wind turbine.

Table 1 Turbine tower reliability influencing factors and their levels.

Table 2 Taguchi L₉ experimental designs showing the reliability influencing factors.

The structural reliability of wind turbine towers (WTTs) is significantly affected by geometrical and material design parameters. Among the most critical as explored in this work are wall thickness, top-to-bottom diameter ratio, and the material grade used in tower fabrication.

The wall thickness influences a tower’s resistance to buckling and fatigue. Study by Ref. [34] shows that increased thickness improves loadbearing capacity (strength) but adds cost and mass, affecting transport and manufacturing process. The diameter ratio between the tower’s top and base affects structural stiffness and dynamic response. A balanced ratio optimises load transfer and deflection control, as noted by Refs. [35,36].

Material choice, particularly the grade of structural steel, directly impacts yield strength, fatigue resistance [37]. Incorporating these parameters into reliability-based design frameworks enhances safety while maintaining cost-effectiveness [14,35,37,38].

3.3 Numerical Method Employing Finite Element Analysis (FEA)

Efficient finite element models are needed to predict accurately stress histories of slender structures such as wind turbine tower [39]. The technique utilises individual nodes to study system properties such as stress, deformation and strain energy. Finite element analysis (FEA) has been employed my many researchers including [24,25,40] in comparable investigations. ANSYS software was employed by [28,41,42] in the analysis of structural components with acceptable reliable results. Thus, ANSYS software is used in this study. NX12 software is used to create models of the tower and static structural package in ANSYS mechanical workbench is used to carry out the simulation. The simulation was run in a Windows 10 64bit operating system computer which has RAM size of 32 GB, storage capacity of 320 GB, processor intel^® core i7-8700 and CPU processing speed of 3.20 GHz. One-hour 5 mins was used to generate 10 mm mesh on one model and circa 30 minutes was utilised to run the simulation on the computer.

3.3.1 Model Creation Meshing

A tapered cylindrical tubular CAD model of the tower is created using NX12 software. The model has constant shell thickness and is shown in Figure 5. The dimensions of the tower are obtained, in parts, from the works of C. Desmond [42]. The standard tower is made of steel of density 7850 kg/m³, elastic modulus and Poisson’s ratio of 210 GPA and 0.3, respectively. Its height is 90 m and it has maximum 5 m diameter at the bottom. The diameter linearly decreases to 2.5 m at the top. The weight of the nacelle and rotor blades is circa 314,840 kg. The details and features of the turbine are presented in Table 3. It has 3 sections but modelled as a single unit owing to scope of investigation.

Figure 5 Model of the standard tower in NX12.0 environment.

Table 3 Details of a standard 5MW wind turbine specification [42].

As meshing is an important process in performing and obtaining accurate simulation in finite element modelling, effort was made to obtain adequate mesh. Implementing element sizing with optimisation, adequate mesh shown in Figure 6 is obtained. It is a tetrahedral element mesh in 3D mesh type. It consists of a total of 723,606 elements and 1,405,891 nodes and yielded average element quality of 0.82, orthogonal quality of 0.71 and skewness of 0.30. These values a consistent with standard good quality mesh.

Figure 6 Adequate mesh of tower showing tetrahedral element mesh.

3.3.2 Materials and Their Properties

Three steel grades- S275, S355 and AISI 4130 are evaluated based on economy, environment impact, and mechanical properties. Key criteria include yield strength, stiffness, fatigue strength, constructability, and corrosion resistance. S275, a low-carbon general-purpose steel, offers good machinability and weldability but has lower strength and limited corrosion resistance due to its low nickel and chromium content. The steel grades as mentioned above were evaluated for wind turbine tower design based on strength and cost. S275 has a minimum yield of 275 MPa and costs 0.55 pence/kg; it’s a low-carbon, general-purpose steel with good weldability but lower corrosion resistance. S355, a high-strength low-alloy structural steel under EN10025-2004, offers 355 MPa yield strength, improved performance, and costs 0.60 pence/kg. AISI 4130 (Chromalloy steel), a chromium-molybdenum alloy, provides superior toughness and strength, ideal for high-stress applications, at 0.88 pence/kg. Table 4 presents their detailed mechanical properties, sourced from EUROCODE EN1993-1-1 and Thomas net (Thomasnet.com). Wind turbine towers are built in multiple sections for ease of transport and assembly. These are joined using fasteners-commonly carbon steel bolts grade 8.8, with a density of 7.85 kg/m³, Young’s modulus of 200 GPa, 660 MPa yield strength, and 830 MPa tensile strength. Their corrosion resistance is critical for structural integrity and long-term reliability.

Table 4 Relevant mechanical properties of the graded of steel used in this study.

3.3.3 Loads and Boundary Conditions

The loads on the tower include the weights of nacelle, rotor, blades, tower self-weight, and aerodynamic loads. The latter include rotor thrust during normal operation, extreme wind loads, and moments induced by rotor blades. Some of these loads are obtained from a structural load document provided by the turbine manufacturer. Wind speed, pressure, self-weight, and rotor thrust loads are obtained using appropriate formulas. The loads are mainly static while others are dynamic in nature. Dynamic load response of tower is outside the scope of this investigation.

Some key assumption made in the modelling of the tower and application of these loads include: (i) tower structural model is represented by an equivalent long, tapered, hollow cantilever cylinder having uniform cross-sectional properties with 100 mm thickness from top to base; (ii) tower is cantilevered to the ground and is carrying a concentrated mass at its free end approximating the inertia properties of the nacelle/rotor unit. The mass is assumed to be rigidly attached to the top of the tower; (iii) tower material is linearly elastic, isotropic, and homogeneous; (iv) distributed aerodynamic loads are restricted to a simplified triangular distributed load which has axially directed forces representing the wind load in form of pressure; (v) structural analysis is confined to deflection, stress and buckling of tower caused by wind load on tower, weight of nacelle and rotor hub as well as rotor thrust; (vi) wind loads are applied lateral on the surface of the tower and perpendicular to its axial in the direction of the wind; (vii) the wind turbine is assumed to respond linearly and aero-elastically to strong winds. This assumption allows the wind load to be calculated using the well-established aero-elastic analysis method [29].

In wind turbine tower analysis, the distributed aerodynamic load is often simplified to a triangular distribution because wind speed increases with height, making loads greater at the top. This approximation captures the vertical wind gradient effectively and simplifies analytical and numerical modelling it reduces computational complexity while maintaining acceptable accuracy in estimating structural response, particularly for preliminary design and parametric studies [23,43,44]. Research study by Raikwar et al. [45] Confirm that triangular loading provides reliable results for tower deformation, bending stress under static and dynamic wind conditions.

Nonlinear effects such as flutter and the associated bending-twist coupling are thus implicitly ignored in the study. It is implemented by applying a single equivalent force at two-third of total height from the ground. Figure 7 shows the FE model with loads implementation in ANSYS workbench environment. The turbine tower is fixed at the bottom, a compressive load of 3085.4 kN is applied on the top, wind load is applied laterally to the tower.

Figure 7 Wind turbine tower model showing forces application in ANSYS workbench.

3.4 Analytical Methods

Analytical method is employed to generate the values of parameters implemented in the simulation. It is also employed to validate the FEA results. Specifically, theoretical equations are employed to determine the wind speed which is used to compute the wind load on tower and rotor thrust. These are implemented in the simulation. The studies on moment, deflection and stress induced in the towers are supported by development and adoption of relevant theoretical equations. The detailed discussion on these is presented thus:

3.4.1 Determination of Maximum Wind Speed and Load on Tower and Rotor Thrust

This study adopts wind conditions characteristic of Northeast England, specifically Newcastle upon Tyne and Durham. Wind conditions vary due to local topography, with instantaneous wind speed and direction fluctuating more than hourly averages. In Newcastle, the windier period spans 5.5 months (11 October-28 March), peaking on 24 January at 26.8 km/h. the calmer season lasts 6.5 months (28 March-11 October), with the calmest day on 26 July at 16.5 km/h. Annual average wind speeds in Durham and Newcastle range from 7-10 m/s, reaching peaks of 15 m/s [xcweather.co.uk]. A representative mean wind speed of 9.5 m/s at 10 m height is used for wind load analysis in this investigation.

The power law presented in Equation (1) and reported in Ref. [18] is used to determine the wind speed for structures that do not exceed 100 m in height. The expression is based on wind speed that increases with height because of ground friction.

\[ V(z)=V_r\left(\frac{z}{z_r}\right)^\alpha \tag{1} \]

Where: V(z) is velocity to be determined at a particular height; V_r is basic mean speed measured at a known height (10 m) from wind measuring sensor. Z is the height at which the wind speeds are to be obtained. z_r is the height at which the measurements are obtained, α is the wind shear exponent. The values of α for different regions from Ref. [18] is presented in Table 5. Average wind shear power exponent (α) of 0.2 is chosen for this analysis as this represents the terrain found in Northeast region of England. the mean wind speed value of 9.5 m/s is used which represents the region as shown in Figure 8 below. According to IEC 61400-1 [46], this falls in the first category of wind class and the associated extreme wind speed in 50-year return period is 70.0 m/s. Implementing this mean wind speed yields a maximum wind velocity of 15 m/s at hub height, which is used in further calculations in this study.

Table 5 Comparison of the parameters for roughness and power.

Figure 8 Power coefficient for the ideal bare wind turbine as a function of the thrust coefficient, from Mahmoud and Gedalya [47].

3.4.2 Wind Load on Tower

Relations used to model wind load on tower are based on the knowledge that stoppage of wind by surface transformed its dynamic energy to pressure. The pressure in turn produces force. The mathematical models are therefore given in Equations (2) and (3):

\[ F=P\times A \tag{2} \]

\[ P\times A=\frac{1}{2} \rho v^{2} C_{D} A \tag{3} \]

Where: F_W,T is wind force (N) on tower, A is surface area (m²) impacted directly by the wind = $\frac{2\pi rH}{2} $, P is pressure (Pa), ρ is density of air (kg/m³), V is maximum wind speed (m/s) in the selected area, C_D is drag coefficient of the tower. Density of air is 1.22 kg/m³. Combining Equations (2) and (3) and the surface area, the formula for distributed wind load is:

\[ F_{W,T}=1.2HD_o\left(\frac{\rho V^2}{2}\right) \tag{4} \]

Where: H is height of tower at any point where the force is determined and D_o is the outside diameter. The 90 m tower in this study is divided into three 30 m sections. Similar technique is used by Ref. [13]. A 60 m height represents the point (measured from base area) on tower where the triangular distributed force in Equation (4) acts. This is at two-thirds of the whole tower height.

3.4.3 Wind Thrust on Rotor and Blade Assembly

The thrust F exerted by the wind on the rotor can be determined by implementing Equation (5) reported in Ref. [20].

\[ F_{W,R}=\frac{1}{2}\times\rho\times A\times C_T\times v^2 \tag{5} \]

Where: Density (ρ) is the density of air, A is the blade swept area, V is the wind speed across the wind turbine, C_T is the thrust coefficient. Designing at power coefficient of 0.5, the thrust coefficient C_T is 0.6 as presented in Figure 8 and demonstrated by Betz limit on wind turbine efficiency.

3.4.4 Gravitational Loads

In addition to loads due to rotor thrust and wind, the weight of the entire wind turbine system is considered in this investigation. The key contributors are the weights of the rotor, blade, nacelle, and self-weight of the tower. The total weight can be computed using Equation (6):

\[ F_G=M_Rg+M_Ng+M_Tg \tag{6} \]

Where: g is gravity, M_R is mass of rotor, M_N is mass of nacelle, M_T is mass of tower. A point mass M, which is a combination of M_R and M_N in Equation (6) is located at the top of the tower to represent the total mass of rotor, blades, hub, and nacelle housing.

3.4.5 Bending Moment Induced on Tower

The tower is subjected to bending moments caused by main two forces. These are the wind load attacking semi-circular surface area of the tower and the wind thrust on rotor. Dynamic tower loads are caused by blade moments. Blades bring about three key moments. These are the flap-wise, lead-lag and torsional moments. These loads are transferred to the tower during operation. The flap-wise moment is more significant than the other two sources. In this study, it is assumed that the turbine has teetered rotor which does not support transfer of flapping moment to the hub or the tower as reported in parts by [48]. Consequently, only aerodynamic rotor thrust at the top and triangular distributed load acting at the two-third of the tower length is considered and modelled using Equation (7):

\[ M_B=\left(F_{w,R}+\frac{2F_w}{3}\right)L \tag{7} \]

Where: L is the vertical height of tower measure from the ground. Values of 90 m and 60 m from the tower base are used for rotor thrust moments and wind triangular distributed load, respectively.

3.4.6 Deflection of Tower

Magnitude of deflection experienced by tower in operation is a critical design consideration. Deflection of tower is caused by lateral loads induced by wind force and weight of the structure. These loads cause displacement of free end of tower. Schematic representation of the forces acting on the tower is shown in Figure 9. The mechanics of the deflection response of the tower can be modelled using Equation (8) developed in the cause of this investigation:

\[ \delta_{total}=\delta_F+\delta_W=\frac{F_{w,R}h_t^3}{3EI}+\frac{W_xh_t^4}{8EI} \tag{8} \]

Where: δ_F is deflection due to rotor thrust (force), δ_W is the deflection due to wind load, F_w,R is the wind force on rotor, E is the elastic modulus of tower steel material, I is the area moment of inertia of a uniform hollow cylinder, h_t represents the height of the tower.

Figure 9 Schematic representation of all loads in a turbine tower. Adapted from [23].

I is expressed thus:

\[ I=\frac{\pi}{64}(D^4-d^4) \tag{9} \]

Where: D and d are the effective outside and inside diameters of cylindrical tower, respectively. Moment of inertia of the structure is computed by implementing I_B as base diameter and I_t as top diameter. The base of tapered cylindrical tower with bottom diameter of 5 m and top diameter of 2.5 m is I = (I_B + I_t)/(2) = 2.5815 m⁴.

The magnitude of δ_w is obtained using the Equation (10):

\[ \delta_w=\frac{Pa^2(3L-a)}{6EI} \tag{10} \]

3.4.7 Stress on the Wind Turbine (WT) Tower

Like deflection, the magnitude of stress induced in a wind turbine tower is a critical design consideration. Loads acting on the tower induce stress in it. The loads include bending moments from aerodynamic wind load and rotor. Hegseth and Erin [49] and Hernandez-Estrada et al. [23] showed that the most extreme thrust conditions in wind turbine towers are produced when maximum rotor speed is reached with nominal wind speed. Modelling stress in tower has been carried out by researchers which include Uys et al. [50] and Bang, Kim, and Lee [28]. They estimated wind load distribution up to the height of the tower based on a prescribed velocity profile. Uys et al. [50], applied three different line-loads [Nm^-1] along the tower while Bang, Kim, and Lee [28] applied a series of point loads (N), i.e., discrete loading, corresponding to the incremental positions where the wind velocity was approximated by Ref. [28]. Liu [29] in his analysis of a 1.5 MW wind turbine discussed the stress response of contemporary 40 mm thickness steel design.

In this study, two different forces representing wind load on rotor and distributed wind load are applied at 90 m and 60 m height, respectively. Equations (7), (11), (12) and (13) are implemented to model the maximum stress in tower.

\[ \sigma_m=\frac{F_g}{A_b}+\frac{M_T}{Z} \tag{11} \]

\[ Z=\frac{\pi(R^4-r^4)}{4R} \tag{12} \]

\[ \sigma_{bmax}=\frac{M_T}{Z} \tag{13} \]

Where: F_g is the weight of the wind turbine system (nacelle and rotor blades), A_b is the cross-sectional area of hollow cylindrical shape of the tower base, M_T is the total bending moment due to the distributed wind load and rotor thrust force, Z is 14 m³ and the section modulus of hollow cylinder in bending. σ_bmax is the maximum bending stress at the base of the tower.

3.4.8 Buckling of Wind Turbine Tower

The critical load F_cr on the tower is modelled both analytically and by simulation. Employing Equation (14) stated in Ref. [34], the critical load for tower - fixed at the base and free at the other end is modelled thus:

\[ F_{cr}=\frac{\pi^2EI}{4L^2} \tag{14} \]

Where: I represents the moment of inertia and E is young’s modulus of elasticity.

\[ I=\pi\left(\frac{D^4}{64}-\frac{d^4}{64}\right) \tag{15} \]

Where: I_B is base diameter and I_t is top diameter. For tapered cylindrical tower of 5 m bottom diameter and 2.5 m top diameter, obtain:

\[ I=(I_B+I_t)/(2)=2.5815m^4 \tag{16} \]

The set-up of the simulation and its implementation in ANSYS is shown in Figure 10. The figure shows the location of fixed support at the base C, applied force and moment at the top A and B.

Figure 10 Schematic representation of loads and support on a wind turbine tower.

4. Results and Discussion

This section presents and discusses the results of the investigation in four sub-sections. These are study on deflection, stress, instability and optimisation of wind turbine tower.

4.1 Study on Deflection of Wind Turbine Tower

Table 6 below presents the simulation deflection results for all models, with FEA schematics for the standard model, models 1 and 9 shown in Figure 11, Figure 12, Figure 13, while Figure 14 illustrates these deflection results in the form of a bar chart. The standard tower model’s free end shows a maximum deflection of 430.64 mm. Model 1 exhibits the least deflection at 376.82 mm (-12.6% from the deflection of the standard model), while Model 9 records the highest at 491.57 mm (+14.15%), indicating a trend of increasing deflection with model numbers.

Table 6 Showing the deflections for all models including standard model.

Figure 11 Schematic of FEA deflection of the standard WT Tower model.

Figure 12 Schematic of FEA deflection of model 1 WT tower.

Figure 13 Schematic of FEA deflection of model 9 WT tower.

Figure 14 Representation of FEA results of all the towers.

Anlytical method was employed to validate the results of the simulation. Equation (17) was implemented in the standard model.

\[ \delta_{total}=\delta_F+\delta_W=\frac{F_{w,R}h_t^3}{3EI}+\frac{Pa^2(3L-a)}{6EI} \tag{17} \]

Where: Rotor blade swept area (A) is 12470.6 m², Wind velocity at the hub is 15 m/s, thrust coefficient (C_T) is 0.6, Rotor thrust (force) is 1026.951 kN. The deflection of the tower due to wind on rotor (rotor thrust) is found to be δ = 460.34 mm. When varying distributed wind force (F_WT) on tower, is considered which is equal to 27.793 kN, deflection due to F_WT of 6.459 mm was obtained. Implementation of the Equation (17) produces total deformation magnitude of 467 mm. The percentage difference between the values generated from FEA and analytical is 7.78%. The difference is considered trivial considering the complexity of the problem and the challenge in application of the analytical method. Implementation of this method in the standard model produces 29% variation from maximum deformation recommended by IEC and ACI and 40.9% from maximum deflection recommended by Eurocode 1993. The magnitude of deflection is within 1% of the length of tower. Thus, the tower is found reliable.

4.2 Study on Stress Induced in Wind Turbine Tower

The FEA stress responses of the wind turbine (WT) tower models to the induced load are shown in Figure 15, Figure 16 and Figure 17. The Figure 15 represents the standard model, Figure 16 depicts model 9, while Figure 17 presents model 1. Each figure shows at (a) the stress on the whole tower model and at (b) the critical section of each model. It is observed that the stress is maximum at the base of the tower model as well as along the tower side in constant hit by the wind and its adjacent side. Figure 18 presents the stress magnitudes in all the models, while Figure 19 displays the percentage difference of each stress magnitude from the standard model. The former shows that model 1 acquired the least stress of magnitude 66.33 MPa at circa -10.3% difference from the stress accumulated by the standard model. Model 9 acquires the highest stress of 82.36 MPa at 11.403% difference from the standard model.

Figure 15 Equivalent (von-Mises) stress distribution in standard model, showing (a) whole WT tower (b) critical section of the WT tower.

Figure 16 Equivalent (von-Mises) stress distribution in Model 9, showing (a) Whole WT torwer (b) Critical section of the WT tower.

Figure 17 Equivalent (von-Mises) stress distribution in model 1, showing (a) whole WT torwer (b) critical section of the WT tower.

Figure 18 Plot of models and their maximum stress values.

Figure 19 Plot of models’ maximum stress percentage difference from standard model.

Anlytical method was employed to validate the results of the simulation. The Equations (18) and (19) were implemented in the standard model.

\[ \sigma_m=\frac{F_g}{A_b}+\frac{M_T}{Z} \tag{18} \]

\[ Z=\frac{\pi(R^4-r^4)}{4R}\, \,or\,\,Z=\frac{I}{y} \tag{19} \]

Where: I is moment of inertia, y is distance from centroid to top or bottom edge of the cylinder.

Using Equation (19) and calculating for base and top diameters yields average Z to be 1.14 m³. Recall that F_g is the weight of the wind turbine system (nacelle and rotor blades), A_b is the cross-sectional area of tower base, M_T is the total bending moment due to the distributed wind load and rotor thrust force, Z is the section modulus of hollow cylinder. R and r are the external and internal radii of the Tower. Implementation of Equations (7), (11), (12) and (13) at 90 m height of tower, 0.5 m top-to-bottom diameter ratio, and nacelle and rotor blades weigh of 3.1 MN, produces stress of magnitude 83.03 MPa. The percentage difference between the values generated from FEA (73.93 MPa) and analytical is 11%. The percentage difference is within acceptable range.

4.3 Study on Instability of Wind Turbine Tower

Result of simulation of buckling response of the three vital towers is presented in Figure 20. These are model 1, model 9 and the standard model. Models 1 and 9 are used because they represent the best and worst models, respectively. Inference is based on the results obtained from the deformation and stress responses of the models. The standard model serves as the control. Figure 20a shows the schematic of the distribution of deformation on the tower. The top of the tower accumulated maximum deformation. Figures 20b, 20c, 20d depicts that model 1 has deformation magnitude of 1.137 at load multiplier value of 37.593, Model 9 has deformation magnitude of 1.0035 at load multiplier value of 28.581 and standard model has deformation magnitude of 1.0612 at load multiplier value of 32.756.

Figure 20 Tower eigenvalue buckling deformation showing (a) Distribution of deformation, (b) Model 1 bucking deformation, (c) Model 9 bucking deformation, (d) Standard model bucking deformation.

Utilising the load valued in Figure 7 and implementing the compressive load of 3.0854 × 10⁶ and the first eigenvalue load multiplier factors, the critical buckling loads, P_CR, of the three models is obtained. Computed P_CR magnitudes of Model 1, Model 9 and standard model are approximately 116 MN, 88 MN and 101 MN, respectively. The results demonstrate that model 1 is most robust to failure caused by Instability.

From analytical consideration, Equation (14) is employed. The critical load, F_cr, is computed thus:

\[ F_{cr}=\frac{3.143^2\,*\,210\,*\,10^9\left(\frac{N}{m^2}\right)\,*\,2.5815m^4}{4\,*\,90^2m^2} \tag{20} \]

\[ F_{cr}=165MN \]

The simulation result is at 42% variance from analytical result, but it is acceptable considering the complex nature of the problem and the simplification done. More importantly, the objective of this work package is to compare the buckling responses of the models and rank them. Model 1 is found to sustain more buckling load than the standard model.

Since the F_cr depends on the material’s stiffness or flexural rigidity and not on its yield stress it means that a column made of high-strength steel offers no advantage over one made of lower-strength steel. This implies that the critical buckling load for standard tower model will be the same for the other nine models. Equation (20) implies that F_cr will tend to increase as the moment of inertia of the cross section increases. The tower structure is loaded eccentrically because the load due to blades acts at some distance (e) away from the centroid of the tower. Therefore, the maximum deflection during buckling can be determined using Equation (21) [51].

\[ \delta=e\left[\sec\left(\frac{P}{EI}*\frac{L}{2}\right)-1\right] \tag{21} \]

And the maximum stress computed using secant formula, Equation (22).

\[ \sigma=\frac{P}{A}\left[1+\frac{ec}{r^2}*\sec\left(\frac{2L}{2r}*\sqrt{\frac{P}{EA}}\right)\right] \tag{22} \]

Where: r is radius of gyration, $r=\sqrt{\frac{I}{A}}$; E is modulus of elasticity of material, P is eccentric load applied on tower, e is eccentricity, L is length of tower, A is cross-sectional area, c is distance from centroid axis to the outer layer of tower where the maximum compressive stress occurs. As tower slenderness ratio (L/R) increases, the tower will tend to fail at or near the Euler buckling load and this knowledge informs the incorporation of this study in this research.

4.4 Study on Optimisation Modelling of Wind Turbine Tower

The turbine tower is optimised against the standard model using three key parameters. These are cost, magnitude of deflection and amount of stress accumulated in the tower. The optimisation hypothesis is the lower the magnitudes of the response of the model, the better is the model. The optimal model will have the least magnitude of deflection, stress and cost. Table 7 presents the amount of these parameters and their ranking. Figure 21 presents composite bars which details the share of cost, deflection, and stress in each of them. The distribution is used to rank the models. The bars in Figure 22 are used to represent the ranking. It can be seen in the figure that Model 1 has lowest height of bars expected for deflection. It has lowest effective rank when all three parameters are considered. Effective rank is the rank which considered the three parameters, cost, deflection, and stress to produce the final rank. Consequently, Model 1 is the optimal wind turbine tower.

Table 7 Optimisation parameters of wind turbine tower.

Figure 21 Values of Cost, Deflection and Stress in models.

Figure 22 Ranking of models for optimisation.

4.5 S/N Ratio Analysis

To assess the significance of the influencing parameters, the Taguchi method employs the signal-to-noise (S/N) ratio, a statistical performance measure that accounts for both the mean and variability and is defined by the quality characteristic of the quality being optimised [30,52,53]. The deflection values obtained from the simulations, as presented in Table 8 below, were used to compute the signal-to-noise (S/N) ratios to determine the optimal combination of influencing parameters (factors). In S/N ratio analysis, the objective is to maximise the S/N ratio, which is achieved by the highest point of each parameter, and which corresponds to improved performance robustness. In this study, deflection is the objective function; therefore, the “smaller-the-better” S/N criterion is adopted, as the simulations aim to minimise the deflection of the wind turbine tower structure under operational wind loading. The equation used to calculate the S/N ratio is presented below. The S/N ratio for the smaller the better is:

\[ S-N=-10log\left(\frac{\Sigma y^2}{n}\right) \tag{23} \]

Where: y = value of deflection at each experiment (Simulation no), n = number of trial run at each setting which is 1 for each experiment.

The amount of deflection for each simulation and the S/N ratio values is presented in Table 8 below.

Table 8 Summary of results of Deflection and corresponding S/N values.

The S/N ratio for each factor for each level is calculated as follows. For tower thickness (factor A), Level 1, 2, 3 calculations are:

Level 1 = (8.496 + 8.328 + 7.897)/3 = 8.240

Level 2 = (7.596 + 7.309 + 7.005)/3 = 7.303

Level 3 = (6.744 + 6.466 + 6.168)/3 = 6.459

Similarly, the results of S/N ratio calculations for other factors(parameters) for each level is presented in Table 9 below.

Table 9 Response table of S/N ratio for each level of each factor.

With the use of Minitab software, the main effects plots for means and S/N ratios were obtained which is presented in Figure 23 and Figure 24 below.

Figure 23 Main effect plots for S/N ratio.

Figure 24 Main effect plots for mean response.

To determine the influence of the parameters on wind turbine tower deflection, S/N ratio analysis is performed, and parameters are ranked using delta statistics, where the highest delta indicates the most significant effect and the lowest indicates the least, as shown in the S/N response Table 9 above.

Wall thickness was identified as the most significant factor, followed by the top-to-bottom diameter ratio, while steel grade material had the least influence. The main effect plots for mean response and S/N ratio (Figure 23 and Figure 24) indicate that the optimal deflection occurs with the A1, B1, and C2 parameter combination, corresponding to a wall thickness of 95 mm, a top-to-bottom diameter ratio of 0.45, and S355 steel grade.

5. Conclusions

This research has shown that the reliability of a wind turbine tower can be improved while achieving a cost-effective design. It has demonstrated implementation of reliability engineering coupled with Taguchi design of experiment, finite element analysis and analytical methods in delivering an optimal cost-effective and safe tower design. Considering tower deflection, stress, and buckling responses magnitude to the applied load, it is concluded that deflection is the most critical safety design factor in tower integrity optimisation.

Model 1 is found optimal because it deflected least at a magnitude of 376.82 mm; accumulated least magnitudes of maximum stress at 66.329 MPa and supported highest buckling load. In comparison with the load responses of the standard model, it demonstrated 9.3% decrease in deflection magnitude, 7.62% decrease in maximum stress magnitude and 14.9% increase in critical buckling load magnitude.

It ranks lowest in cost among the other models because it is made of S275 grade of steel that cost circa £454,357.064. In addition, the model has equal shell volume of 1.03201 mm³ with the other models and yet achieves least magnitude of maximum deflection and stress.

Main effects plots of the S/N ratios reveal clear trends for all three WT tower design variables. Wall thickness (Parameter A) exhibits the most influence on deflection, followed by the top-to-bottom diameter ratio (Parameter B). The influence of Material selection (Parameter C) was quite negligible in relation to deflection. Hence, based on the S/N ratio results, the optimal design configuration corresponds to a specific combination of wall thickness, top/bottom diameter ratio and steel material type, that minimises deflection while maintaining structural integrity. The recommended setting of A1, B1, C2 is in close approximation to optimal setting of A1, B1, C1, as material selection has least influence on tower deflection.

The results demonstrate that robust optimisation using the Taguchi method provides valuable insight into the relative importance of wind turbine tower design parameters. The dominance of wall thickness and diameter ratio suggests that early-stage geometric optimisation offers greater structural integrity of the tower. However, in terms of cost-saving potential, material substitution has significant influence (Seen from cost breakdown in Table 7) when making design decisions.

From a practical perspective, the proposed framework enables designers to rapidly screen feasible tower configurations before engaging in more computationally intensive optimisation or detailed reliability analyses. This methodology implemented is particularly suited for preliminary design stages, where design flexibility is high and cost decisions have the greatest impact. The knowledge supports capacity building in tower technology needed to support the technological growth aimed at delivering Net Zero by 2050.

Recognising that the derived and developed analytical models used for tower deflection and tower buckling load estimation in this work utilised the average of section properties at smallest and largest sections of the tower, the models are being further refined. In the refinement process, the authors are considering the variation of the cross-sectional properties of the tapered tower along the length - especially the moment of inertia. This work utilised the FEA results to draw conclusions from the findings of the investigation. The credibility of this work and its approach is rooted in its key objective, which is to compare the FEA results on deflection and buckling load responses of the models and rank them. Thus, the analytical model is only used to validate the results of the simulation. The refined analytical models will be published in a separate article where its credibility will be validated using the FEA simulation method.

5.1 Limitations and Future Extensions

While the present study focuses on static structural performance and simplified cost modelling, future work may extend the framework to include fatigue damage, dynamic response, and lifecycle cost assessment. Integration with probabilistic reliability analysis and site-specific wind conditions would further enhance the applicability of the approach to real-world wind turbine projects.

Author Contributions

Emeka H. Amalu: writing, conceptualization, Methodology, formal analysis, – review and editing. Justice O. Nwaneto: writing – original draft, formal analysis, Software, editing. All authors have read and approved the published version of the manuscript.

Competing Interests

The authors have declared that no competing interests exist.

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