Effects of Main Spar Geometry and Layup Parameters on Bend–Twist Response and Structural Performance Balance in Wind Turbine Blades

Abstract

Geometric and laminate modifications can increase bending-induced twists in wind turbine blade spars, but the associated changes in deformation, stress, mass, and vibration are not always assessed on a consistent basis. This study therefore compares the structural trade-offs of four established design parameters under a common finite-element framework. An equivalent thin-walled composite box-spar model based on the NREL 5 MW reference blade was developed using a midsurface representation with spanwise-varying cross-sections, zoned layups, and thickness tapering. Spar-cap thickness, shear-web inclination angle, spar-cap off-axis ply angle, and off-axis ply fraction were varied individually under consistent loading and boundary conditions. The results show that spar-cap thickness primarily influences flapwise deflection and structural mass. Shear-web inclination produces the largest twist-to-deflection response index, reaching approximately 1.17–1.21 °/m at ±20°, but also increases tip deflection and peak equivalent stress. Increasing the off-axis ply angle or fraction does not improve this response for the investigated layups. For the C2 configuration, the response index decreases to approximately 22% of the baseline value, accompanied by increased deflection and stress and reduced natural frequencies. These findings reveal parameter-dependent structural trade-offs under the prescribed model and loading conditions, providing a comparative basis for further blade-spar design and validation.

1. Introduction

Wind power has become an important part of the global renewable energy sector due to its distinct advantages. According to the Global Wind Report 2026 released by the Global Wind Energy Council (GWEC), 165 GW of new wind power capacity was installed worldwide in 2025, representing the a 40% increase over the previous record set in 2024. The global wind power industry continues to expand rapidly [1]. As wind turbine capacity increases, blades become larger, lighter, and more flexible [2,3]. Fiber-reinforced composites are widely used in large wind turbine blades because of their high specific strength and stiffness [3]. The main spar is the primary load-bearing structure of a wind turbine blade. It carries the main bending loads and governs the bending, torsional, and bend–twist coupling stiffness of the blade section. Changes in its geometry and layup therefore directly affect the blade’s bending, torsional, and coupled responses. Accordingly, clarifying the effects of main spar parameters on bending and torsional responses is important for the structural design and passive load alleviation of large flexible blades.

Bend–twist coupling refers to the simultaneous bending and torsional deformation of a blade under bending loads. It can be achieved through material coupling by exploiting composite anisotropy or through geometric coupling using structural features such as sweep and pre-bend [4,5]. In addition, adjusting the internal web position can also induce a bend–twist response, which is referred to as structural coupling [6]. In recent years, research has gradually shifted from bend–twist coupling mechanisms to passive load alleviation. Zhao et al. [7] investigated the load alleviation of large flexible blades by varying the fiber misalignment angle in the main spar and the onset location of coupling. Serafeim et al. [8] applied material coupling through rotation of the spar-cap laminates to the lightweight design of a 10 MW wind turbine blade. These studies showed that both geometric and material modifications can alter the bend–twist response, although their effects on structural performance differ.

For the structural and dynamic behavior of bend–twist coupled blades, Chen et al. [9] investigated the effects of cross-sectional stiffness parameters on aeroelastic responses using a 6 × 6 sectional stiffness matrix. Shakya et al. [10] examined the effects of laminate imbalance and fiber orientation on the flutter characteristics of composite blades. Ma et al. [11] further investigated the effects of blade pre-bend and sweep on aeroelastic loads. Regarding the internal configuration of the main spar, Wang et al. [12,13,14] studied the effects of double-web configurations, web offset, and composite layups on the bend–twist response, stress distribution, and buckling behavior of large blades. Their results revealed a clear interaction between web offset and layup configuration.

In composite layup and structural optimization, Dong et al. [15] developed a multi-objective optimization method for layup parameters. Fu et al. [16] experimentally investigated the effects of fiber orientation and hybrid ratio on the equivalent bend–twist coupling coefficient of carbon/glass hybrid composite blades. Li et al. [17] quantified the effects of main spar parameters on the sectional stiffness matrix and aeroelastic stability. Kou et al. [18] optimized the layup ratio and sequence of the spar cap for a 5 MW blade based on classical laminate theory. Tang et al. [19] further examined the relationship between layup parameters and flutter characteristics under multiple operating conditions. Hermansen and Lund [20] performed coupled material–thickness optimization, while Wu et al. [21] incorporated structural mass and reliability-related metrics into the multi-objective design of large flexible blades. In addition, Jang et al. [22] and Kim and Kang [23] investigated composite blade safety from the perspectives of progressive failure and fatigue crack propagation, respectively. He et al. [24] and Jeong and Park [25] further integrated aerodynamic, structural, and safety performance into multidisciplinary design. From the perspective of structural condition assessment, Civera and Surace [26] reviewed non-destructive inspection and structural health monitoring methods for wind turbines and summarized the general requirements for structural integrity evaluation. Although their work focuses on monitoring rather than bend–twist design, it provides relevant context for interpreting structural response indicators and considering their validation under practical operating conditions.

Previous studies have shown that spar geometry and composite layup can affect the bending, torsional, and dynamic responses of wind turbine blades. Several studies have also examined the combined effects of shear-web configuration and composite layup or performed structural and laminate optimization. However, differences in reference models, loading assumptions, and evaluation metrics hinder direct comparison of the relative effects of these design parameters and their associated performance trade-offs. A larger bending-induced twist does not necessarily indicate better overall structural performance, because it may be accompanied by increased deformation and stress or unfavorable changes in natural frequencies. A consistent comparative framework is therefore needed to distinguish parameter-dependent response characteristics and associated performance penalties under identical modeling and loading conditions.

To provide such a comparison, this study develops a parametric finite element model of the NREL 5 MW wind turbine blade main spar using a thin-walled composite midsurface representation with spanwise-varying cross-sections and zoned layups. Four design variables are investigated: spar-cap thickness, shear-web inclination angle, spar-cap ply angle, and off-axis ply content. Their effects on tip displacement, bending-induced twist, maximum equivalent stress, the first three natural frequencies, and structural mass are evaluated under consistent loading and boundary conditions. The comparison combines the direct response values, a range-normalized influence index, and a load-specific twist-to-deflection response index. The main contribution lies in identifying parameter-specific response characteristics and structural trade-offs within a common numerical framework, rather than proposing a new finite element or optimization methodology. Attention is given to the directional twist response induced by shear-web inclination, the mass penalty associated with increasing spar-cap thickness, and the stiffness and stress penalties caused by off-axis plies. The findings are limited to the parameter ranges investigated and prescribed loading conditions.

2. Materials and Methods

2.1. Reference Wind Turbine and Blade Spar Model

This study uses the blade of the NREL 5 MW reference wind turbine as the reference configuration [27,28]. The blade has a length of 61.5 m and the hub diameter is 3 m. The rotor diameter is 126 m, equal to twice the blade length plus the hub diameter. Table 1 summarizes the principal specifications of the reference wind turbine used in the present study.

Table 1. Main specifications of the NREL 5 MW reference wind turbine.

Parameter	Unit	Value
Rated power	MW	5
Rated rotor speed	rpm	12.1
Minimum rotor speed	rpm	6.9
Rotor diameter	m	126
Hub diameter	m	3
Blade length	m	61.5
Cut-in wind speed	m/s	3.0
Rated wind speed	m/s	11.4
Cut-out wind speed	m/s	25.0
Rated blade-tip speed	m/s	≈80
Design life	year	20
Blade mass	kg	17,740
Mass per unit length at blade root	kg/m	678.94
Mass per unit length at blade tip	kg/m	10.32

Note: The 17,740 kg value refers to the complete NREL 5 MW reference blade, whereas 737.37 kg is the mass of the baseline model used here. The ratio of approximately 4.16% only reflects the different structural scopes and does not represent the actual spar mass fraction or a model-validation error.

2.1.1. Geometric Reference Model of the Blade Spar

Detailed information on the spar-cap width, web position, local layup, and thickness distribution is not provided in the public NREL 5 MW blade data. Therefore, an equivalent box-spar model was developed using the published spanwise chord distribution and airfoil geometry. The model includes the upper and lower spar caps and the front and rear shear webs, but excludes the complete aerodynamic shell, leading- and trailing-edge structures, and other full-blade components. Nine control stations were selected within 0.05 L–0.95 L to describe the spanwise variation in the spar geometry.

The chordwise width of the spar box was defined as a fraction of the local blade chord:

b_s(z) = λ_bc(z)

(1)

where b_s(z) is the chordwise width of the spar box, (cz) is the local blade chord, and λ_b is the dimensionless relative spar-width coefficient. The baseline value of λ_b = 0.2 was adopted to define a consistent reference geometry for the subsequent parametric analysis.

The spar-box height was estimated from the local maximum airfoil thickness:

h_s(z) = λ_ht_rel(z)c(z)

(2)

where h_s(z) is the spar-box height (m), t_rel(z) is the dimensionless relative airfoil thickness,(cz) is the local blade chord (m), and λ_h is the dimensionless spar-height coefficient. The baseline value of λ_h = 0.7 was adopted to keep the equivalent spar box within the local airfoil envelope.

Table 2 lists the spanwise coordinates and calculated principal geometric dimensions of the nine control stations. The stations are located at x/L = 0.05, 0.10, 0.20, 0.30, 0.40, 0.50, 0.65, 0.80, and 0.95. Here, x denotes the spanwise distance measured from the blade root, and L = 61.5 m is the blade length. S1 and S9 are interior reference stations rather than the physical root and tip of the complete blade.

Table 2. Spanwise control-station coordinates and principal geometric dimensions of the baseline equivalent box spar.

Station	x/L	x (m)	Chord c (m)	Box Width bs (m)	Box Height hs (m)	Midsurface Width Bm (m)	Midsurface Height Hm (m)
S1	0.05	3.075	3.737	0.747	0.86	0.597	0.56
S2	0.1	6.15	4.089	0.818	0.981	0.698	0.741
S3	0.2	12.3	4.605	0.921	1.197	0.851	1.067
S4	0.3	18.45	4.458	0.892	1.025	0.837	0.915
S5	0.4	24.6	4.128	0.826	0.826	0.776	0.716
S6	0.5	30.75	3.748	0.75	0.637	0.705	0.537
S7	0.65	39.975	3.194	0.639	0.479	0.609	0.414
S8	0.8	49.2	2.641	0.528	0.343	0.508	0.298
S9	0.95	58.425	1.836	0.367	0.22	0.355	0.195

Note: x/L denotes the normalized station coordinate referenced to the blade root. b_s and h_s are geometric envelope dimensions, B_m and H_m are the corresponding box-spar midsurface dimensions. Values are rounded to three decimal places. The source workbook defines the rotor-centered radial coordinate as R = X + 1.5 m.

Figure 1 presents the parametric blade-spar model used in this study. Figure 1a shows the nine spanwise control stations (S1–S9) and the variable cross-sectional geometry.

Figure 1. Parametric modeling of the wind turbine blade spar: (a) geometric reference model and spanwise control stations; (b) thin-walled midsurface finite-element model.

2.1.2. Thin-Walled Midsurface Model of the Blade Spar

Based on the geometric reference model, the midsurfaces of the upper and lower spar caps and the front and rear shear webs were extracted while preserving the locations, widths, and heights of the control sections. These surfaces were used as the reference geometry for the composite shell elements, as shown in Figure 1b. The physical thickness of the spar caps and shear webs were subsequently defined in ANSYS (2022R1) ACP according to the material type, ply orientation, and number of plies.

This midsurface representation reduces the computational cost associated with layerwise solid modeling and avoids excessively high-aspect-ratio elements. At the same time, it preserves the key laminate and geometric features that govern the global bending and torsional responses of the blade spar. These features include region-wise laminate definitions, spanwise ply drop-offs, fiber orientations, and the dual-web configuration.

The equivalent box-spar midsurface model was discretized with four-node SHELL181 elements in ANSYS Mechanical. Laminate thicknesses, stacking sequences, and material orientations were defined in ANSYS ACP. For the selected M3 mesh, a global element size of 50 mm was used, with local refinement to 10 mm in the root region.

2.2. Composite Material System and Baseline Laminate Configuration

To provide sufficient spanwise bending stiffness while maintaining shear load-carrying capability, this study adopts a composite material system consistent with established large wind turbine blade spar designs. The laminate model was implemented in ANSYS Composite PrepPost (ACP), where the local material orientations, stacking sequences, and laminate regions were defined [25]. The upper and lower spar caps were modeled using M1 unidirectional carbon-fiber-reinforced polymer (CFRP), whose high longitudinal modulus provides the primary spanwise tensile and compressive load-carrying capability. The front and rear shear webs were modeled using M2 biaxial glass-fiber-reinforced polymer (GFRP) to provide in-plane shear resistance, torsional support, and overall stability of the box spar. Each material was defined in ACP using homogenized cured-ply properties. The NREL 5-MW reference report provides the reference turbine and blade structural context but does not specify the resin formulation of the equivalent composite materials adopted here. Accordingly, the present model does not identify a specific resin grade or model the reinforcing fibers and polymer matrix as separate constituents. The fiber volume fraction was not specified as an independent input parameter in the present model. Accordingly, the numerical results characterize the responses of the adopted equivalent composite material systems rather than experimentally measured behavior of manufactured blade-spar specimens. Table 3 summarizes the mechanical properties and baseline laminate definitions adopted in the model.

Table 3. Mechanical properties and baseline laminate definitions of the composite materials used in the blade spar.

Parameter	M1	M2
Material type	Unidirectional carbon-fiber-reinforced composite	Biaxial glass-fiber-reinforced composite
E1 (MPa)	114,500	13,600
E2 (MPa)	8390	13,300
G12 (MPa)	5990	11,800
ν12	0.27	0.49
ρ (kg⋅m−3)	1220	1780
Ply thickness (mm)	0.47	1
Application region	Upper and lower spar caps	Front and rear shear webs

Note: M1 and M2 are study-specific labels assigned to the equivalent composite material systems used in the finite-element model, rather than commercial material grades. The effective orthotropic properties are used to represent the composite plies, and the fiber and resin constituents are not modeled separately. The resin formulation and fiber volume fraction were not independently specified in the adopted model. Ref. [25] provides the reference turbine and blade structural context but does not specify the resin formulation or fiber volume fraction of M1 and M2.

For the laminate orientation, the local spanwise direction of the blade spar was defined as the 0° reference direction for the spar caps. The upper and lower spar caps in the baseline model used 0° plies as the primary load-carrying layers. Region-wise ply drop-offs were introduced along the span to reproduce the prescribed thickness variation in the baseline spar. The front and rear shear webs were modeled using the M2 biaxial GFRP fabric. In ACP, the Modeling Ply angle for the web laminates was set to 0°, which aligns the local material coordinate system of the fabric with the reference direction of the shear web. This setting does not imply that all constituent fiber directions in the web are oriented at 0°. The all 0° spar-cap layup was adopted as the baseline configuration because the spar caps primarily carry spanwise tensile and compressive loads induced by flapwise bending. This baseline configuration provides a consistent reference for evaluating the effects of off-axis reinforcement; it is not intended to represent an optimal layup for every loading condition. To investigate the influence of off-axis plies, spar-cap ply angles of ±30°, ±45°, and ±60°, together with different ±45° ply fractions, were examined in the subsequent parametric analyses.

At the laminate level, classical laminate theory distinguishes between stiffness and coupling terms. For balanced laminates, the extension–shear coupling terms A₁₆ and A₂₆ vanish. For symmetric laminates, the extension–bending coupling matrix B is zero. However, the bending–twisting coupling terms D₁₆ and D₂₆ may remain nonzero, depending on the through-thickness ply arrangement. Therefore, changes in the global bending-induced twist response of the equivalent box-spar model do not necessarily indicate changes in intrinsic laminate-level coupling stiffness.

Figure 2 illustrates the baseline laminate regions and local material orientations adopted for the blade spar.

Figure 2. Baseline laminate regions and local material orientations of the blade spar.

Table 4 summarizes the active ply counts and corresponding nominal laminate thicknesses of the upper and lower spar caps and the front and rear shear webs at each spanwise control station S1–S9.

Table 4. Ply counts and nominal laminate thicknesses at the nine spanwise control stations.

Station	Upper Spar Cap (Plies)	Lower Spar Cap (Plies)	Front Shear Web (Plies)	Rear Shear Web (Plies)	Total Active Plies	Spar-Cap Thickness (mm)	Shear-Web Thickness (mm)
S1–S2	16	16	4	4	40	7.52	4
S2–S3	14	14	4	4	36	6.58	4
S3–S4	12	12	3	3	30	5.64	3
S4–S5	10	10	3	3	26	4.7	3
S5–S6	8	8	3	3	22	3.76	3
S6–S7	6	6	2	2	16	2.82	2
S7–S8	4	4	2	2	12	1.88	2
S8–S9	4	4	1	1	10	1.88	1

Note: The ply counts represent the number of active plies in each structural region at the corresponding spanwise station. Total active plies are calculated as the sum of the ply counts in the upper and lower spar caps and the front and rear shear webs. The nominal single-ply thicknesses of M1 and M2 are 0.47 mm and 1.00 mm, respectively. Laminate thicknesses are calculated from the corresponding ply counts and nominal single-ply thicknesses.

2.3. Design Variables and Response Metrics

To compare the effects of different blade-spar design parameters under consistent modeling, loading, and boundary conditions, the thin-walled midsurface model was adopted as the baseline model. Four design variables were considered: spar-cap thickness, shear-web inclination angle, spar-cap off-axis ply angle, and off-axis ply fraction. These variables represent different structural tailoring mechanisms, namely sectional bending-stiffness control, geometric bend–twist coupling, and composite anisotropy tailoring. A single-factor-at-a-time approach was used for parametric analysis. When one design variable was varied, all other non-target parameters, including the remaining geometric and laminate parameters, material system, loads, and boundary conditions, were kept unchanged. This approach allows the effect of each design variable to be examined under consistent baseline conditions. However, it does not account for interaction effects among spar-cap thickness, shear-web inclination, and laminate configuration. Therefore, the reported response trends are specific to the selected baseline configuration and the investigated parameter ranges. Table 5 summarizes the investigated design variables and parameter levels.

Table 5. Design variables and investigated parameter levels of the blade spar.

Design Variable	Baseline	Investigated Levels
Spar-cap thickness scaling factor (kt)	1	0.8, 1.0, 1.2, 1.4
Shear-web inclination angle (θw)	0°	−20°, −10°, 0°, 10°, 20°
Spar-cap off-axis ply angle (θp)	0°	0°, ±30°, ±45°, ±60°
Off-axis ply fraction (C)	C0	C0, C1, C2

To quantify the effects of the design variables on the static, dynamic, and bend–twist responses of the blade spar, the following response quantities were evaluated: flapwise tip deflection (w_tip), tip twist angle (θ_tip), maximum equivalent stress in the root evaluation region (σ_root), the first three natural frequencies (f₁, f₂ and f₃), and structural mass (M). To quantify the global twist induced by flapwise bending, this study adopts a bend–twist response measure based on established evaluation approaches [29]. The bend–twist response index, K_BT, is defined as:

$$K_{\mathrm{BT}}={\displaystyle \frac{\left|{\theta }_{\mathrm{tip}}\right|}{w_{\mathrm{tip}}}}$$

(3)

where w_tip is the flapwise tip displacement magnitude at S9 in meters and θ_tip is the corresponding signed twist angle in degrees. In ANSYS Mechanical, θ_tip was obtained from the X-axis component of the Flexible Rotation result at RP-Tip-Center. Positive rotation follows the right-hand rule about the positive (X) axis. Accordingly, K_BT (°/m) represents the magnitude of load-induced twist per unit flapwise displacement. It is used as a global comparative response index rather than a measure of intrinsic sectional bend–twist coupling stiffness. Since the computed rotation was not decomposed into contributions from structural deformation, laminate anisotropy, and load eccentricity, its sign is considered separately only to distinguish the rotation direction. Without defined aerodynamic inflow and blade-pitch references, the rotation cannot be classified as twist-to-feather or twist-to-stall.

To enable a consistent cross-comparison of the four design variables, a relative influence metric was further introduced. Because the response quantities have different units and numerical ranges, their raw variations cannot be compared directly. The baseline response was therefore used as a common reference. For each design variable p and response quantity j, the maximum absolute deviation from the baseline is defined as:

$$A_{p,j}=\underset{k}{\max }\left|R_{p,k,j}-R_{0,j}\right|$$

(4)

where p denotes the investigated parameter level, R_p,k,j is the value of response j for design variable p at level k, and R_0,j is the corresponding baseline response.A_p,j therefore represents the maximum variation in response j induced by parameter p within the investigated range.

To compare the relative effects of different design variables on the same response quantity, A_p,j was normalized as:

$$S_{p,j}={\displaystyle \frac{A_{p,j}}{{\max }_p{A}_{p,j}}}$$

(5)

where 0 ≤ S_p,j ≤ 1 A value of S_p,j = 1 indicates that the corresponding design variable produces the largest variation in response j among the variables considered. The index reflects only the relative influence within the prescribed parameter ranges and should not be interpreted as a global or probabilistic sensitivity measure.

2.4. Baseline Analysis Cases and Verification Metrics

To establish a consistent numerical basis for the subsequent parametric analysis, the reference model was examined through static and free-vibration modal analyses. The material layup and root constraint were kept unchanged in all verification cases. The S1 section of the main spar was fully fixed. As defined in Section 2.1.1, the fixed support was applied at S1 through the named selection Root-edge. In the static analysis, the load was applied through a remote point located at the center of the S9 section to reduce local stress concentrations caused by direct point loading. In the user-defined Blade Coordinate System, the transverse load was applied along the positive (Y) (flapwise) direction, corresponding to force components of (0, 1000, 0) N. The deformable coupling allowed deformation of the scoped edge during load transfer. The blade-tip flapwise displacement w_tip, blade-tip torsion angle θ_tip, maximum stress, and root reaction force were extracted as the static response metrics. The stress metric was obtained as the maximum averaged equivalent stress on the top surface of the selected ply in the CAP_01 region. This quantity was used to compare the relative stress responses among the investigated configurations and was not treated as a ply-level failure index or a direct measure of composite strength.

The blade-tip torsion angle θ_tip represents the torsional response induced by transverse bending loads. This response reflects the combined effects of sectional geometric asymmetry and composite anisotropy and is therefore interpreted as load-induced bend–twist coupling, rather than the response to an independently applied pure torsional moment. The geometric center of the S9 section may not coincide with its shear center. Therefore, load eccentricity may contribute to the computed twist response. Since this effect was not quantified separately, the twist angle is treated as a comparative load-induced response rather than a direct measure of intrinsic bend–twist coupling.

Free-vibration modal analysis was performed using the same root constraint without external loading. The first three natural frequencies and their corresponding mode shapes were extracted. Therefore, the torsional response considered in this study represents load-induced bend–twist coupling rather than the response to pure torsional loading. Table 6 summarizes the baseline static and free-vibration analyses, including the prescribed conditions, extracted responses, and their use in subsequent parametric comparisons. The quantities listed in this section are used only to verify the reference model.

Table 6. Baseline analysis cases and response measures for the equivalent box-spar model.

Analysis Type	Boundary and Loading Condition	Extracted Quantities	Evaluation Purpose
Static analysis Transverse static analysis	S1 fully constrained; 1000 N transverse force remotely applied at the center of the S9 section	S9 displacement, S9 rotation, selected-ply equivalent stress, root reaction force	Compare load-induced bending, torsional, and stress responses
Free-vibration analysis	S1 edge fully constrained; no external load	f1, f2, f3 and corresponding mode shapes	Evaluate low-order dynamic characteristics

Note: The 1000 N transverse load was applied at S9 through RP-Tip-Center using a deformable remote-point connection scoped to Tip-edge (ACP Pre). The load was used only as a consistent reference condition for comparative analysis and does not represent the distributed loading of an operating wind turbine blade.

2.5. Mesh Convergence Analysis

To evaluate the effect of finite element discretization on the static and dynamic responses of the blade spar, a mesh strategy combining global mesh control with local root refinement was adopted. The geometry, material layup, boundary conditions, loading, and element type were kept unchanged. Three shell-element meshes were generated with global element sizes of 100, 70, and 50 mm and corresponding root-refinement sizes of 20, 14, and 10 mm. These meshes were used to assess the mesh convergence of the baseline model. Table 7 summarizes the three mesh configurations and the corresponding structural responses.

Table 7. Mesh configurations and corresponding key structural responses.

Parameter	M1	M2	M3
Global element size (mm)	100	70	50
Local element size at the root region (mm)	20	14	10
Number of elements	33,596	68,034	132,849
Number of nodes	33,619	68,051	132,876
wtip (mm)	890.38	895.95	897.04
θtip (°)	0.0035718	0.0035528	0.0035303
σroot (MPa)	35.991	37.813	39.418
f1 (HZ)	0.60341	0.60325	0.60315
f2 (HZ)	0.92433	0.92373	0.92339
f3 (HZ)	1.9843	1.9839	1.9837

Using the M3 solution as the reference, the relative difference between the M2 and M3 results was calculated as:

$${\delta }_R={\displaystyle \frac{\left|R_{M2}-R_{M3}\right|}{\left|R_{M3}\right|}}\times 100\%$$

(6)

where R_M2 and R_M3 denote the corresponding structural responses obtained with the M2 and M3 meshes, respectively.

The relative differences between M2 and M3 are 0.122% for the flapwise tip deflection and 0.637% for the tip twist angle. For the first three natural frequencies, the corresponding differences are 0.0166%, 0.0368%, and 0.0101%, respectively. All values are well below 5%. For the selected-ply stress metric defined in Section 2.4, the maximum averaged equivalent stress in the root evaluation region increased from 37.813 MPa for M2 to 39.418 MPa for M3. The relative difference was 4.072%, indicating greater mesh sensitivity in the local stress response than in global displacement and low-order modal responses. M3 was therefore retained for subsequent comparative analyses. However, mesh convergence alone does not validate composite failure predictions or demonstrate insensitivity to the prescribed boundary conditions.

For a direct comparison of convergence trends among responses with different magnitudes, each response was normalized by the corresponding M3 result:

$$R^{\ast }={\displaystyle \frac{R}{R_{\mathrm{M}3}}}$$

(7)

where R^* is the normalized response and R_M3 is the corresponding value obtained with the M3 mesh.

Figure 3 compares the normalized global responses obtained with the three mesh resolutions. All responses approach unity as the mesh is refined from M1 to M3, confirming satisfactory convergence of the global static and low-order dynamic responses. The flapwise tip deflection changes from 0.9926 to 1.0000, whereas the normalized tip twist approaches unity from 1.0118. The first three natural frequencies remain close to the M3 reference values for all three meshes. These results show that further mesh refinement has only a minor effect on the global stiffness and low-order dynamic characteristics of the spar. Accordingly, the M3 mesh was adopted for all subsequent static and modal analyses.

Figure 3. Mesh-convergence behavior of the normalized key structural responses.

2.6. Model Applicability, Numerical Verification and Limitations

Mesh convergence demonstrates the numerical stability of the finite element discretization, but it does not constitute an external validation of the local structure of the prototype blade. The publicly available NREL 5 MW data provide the blade dimensions, chord distribution, and overall structural characteristics. However, they do not include the complete local geometry and laminate details required to reproduce the box-type blade spar considered in this study. Therefore, the present model is treated as an equivalent parametric model for investigating structural response trends rather than as a high-fidelity reproduction of the prototype blade.

Furthermore, the NREL 5 MW reference data describe the structural properties of the complete blade, whereas the present model represents only an equivalent internal box spar. The omitted aerodynamic shell, leading- and trailing-edge structures, and other full-blade components contribute to the overall mass distribution, stiffness, and inertial properties. Their exclusion may therefore affect the predicted natural frequencies, mode shapes, and bending-induced twist response. Consequently, the published full-blade data cannot serve as direct validation targets for the present equivalent box-spar model. Independent quantitative validation against a geometrically and materially comparable spar model, with matched sectional stiffness or modal data, is not currently available.

All subsequent parametric analyses were conducted using the same baseline model, the M3 mesh, and identical loading and boundary conditions. This consistent numerical framework reduces the influence of modeling assumptions on the relative comparison of the design variables. The model retains the main features governing the global bend–twist response, including the variable cross-section, region-wise ply drop-offs, fiber orientations, and dual-web geometry. It is therefore suitable for the comparative analysis of deflection, twist, low-order natural frequencies, and stress trends addressed in this study.

However, no publicly accessible experimental dataset sufficiently matching the local spar geometry, laminate configuration, material properties, boundary conditions, and loading conditions of the present model was identified for direct quantitative validation. Accordingly, the present model has not been experimentally validated, and its results should be interpreted as comparative numerical trends under the prescribed modeling assumptions and loading conditions, rather than as experimentally confirmed predictions for a specific manufactured blade. The prescribed 1000 N transverse force was used only as a comparative reference load and should not be interpreted as evidence of aerodynamic load alleviation under actual operating conditions. In addition, the present modal analysis was conducted using temperature-independent material properties, without accounting for environmental variability and its potential influence on the predicted natural frequencies.

Future work will focus on establishing a more comprehensive validation framework for the present equivalent box-spar model. First, a higher-fidelity reference model will be developed. This model will include more complete structural geometry and laminate information. It will allow systematic comparisons of spanwise mass distribution, sectional bending and torsional stiffness, and modal characteristics. Furthermore, suitable experimental datasets will be sought where available. Dedicated static bending, torsional, and modal tests may also be conducted where feasible. The predicted displacements, twist angles, strains, natural frequencies, and mode shapes will then be compared with measured responses. These efforts will help further evaluate the accuracy and applicability of the equivalent box-spar model. Future investigations will also consider temperature-dependent composite material properties and modal responses under varying environmental conditions to distinguish design-induced frequency changes from environmental effects.

3. Results

3.1. Effect of Spar-Cap Thickness

The baseline spar cap uses segmented spanwise thickness tapering. Because the reference ply count and equivalent thickness vary along the span, a single absolute thickness is not suitable as the design variable. Therefore, a dimensionless thickness scaling factor, k_t, was introduced to uniformly scale the local reference thickness and was defined as:

$$k_t={\displaystyle \frac{t_{\mathrm{cap}}(r)}{t_{\mathrm{cap},0}(r)}}$$

(8)

where r denotes the spanwise position, t_cap,0(r) is the local spar-cap thickness of the baseline model, and t_cap(r) is the corresponding thickness for a given design case. Accordingly, k_t = 1 represents the baseline configuration.

In ANSYS ACP, the scaling defined by k_t was implemented by applying the same factor to the effective thickness of each spar-cap ply. The ply count, stacking sequence, material orientations, and material properties were kept unchanged. This treatment preserves the baseline laminate configuration and spanwise ply-drop pattern while allowing continuous thickness variation.

The effects of the spar-cap thickness scaling factor on the static, bend–twist, and modal responses are summarized in Figure 4.

Figure 4. Effects of the spar-cap thickness scaling factor on structural responses: (a) flapwise tip deflection; (b) maximum equivalent stress; (c) tip twist angle; (d) first three natural frequencies.

Figure 4a,b show that increasing k_t from 0.8 to 1.4 reduces the tip displacement from 1139.30 mm to 624.30 mm and the maximum stress from 49.381 MPa to 27.989 MPa. These changes correspond to reductions of 45.20% and 43.32%, respectively. Comparison between adjacent cases shows that the marginal reductions in both responses become smaller as spar-cap thickness increases, indicating diminishing improvement in static performance.

Figure 4c shows that its magnitude decreases from 0.004110° to 0.003166° as k_t increases, corresponding to a reduction of 22.97%. This reduction is considerably smaller than that in tip displacement, indicating that spar-cap thickness primarily affects bending stiffness and has a weaker influence on the torsional response. Figure 4d shows that the first three natural frequencies increased from 0.58308, 0.87532, and 1.9180 Hz to 0.63098, 0.99227, and 2.0744 Hz, respectively. The corresponding increases are 8.21%, 13.36%, and 8.15%. The second natural frequency is the most sensitive to spar-cap thickness. Within the range investigated, the stiffness increase therefore has a greater effect on the low-order natural frequencies than the associated increase in structural mass.

Overall, spar-cap thickness primarily governs global stiffness. Its effects on tip displacement and maximum stress are stronger than those on tip twist angle and low-order natural frequencies. However, increasing k_t from 1.0 to 1.4 raises the structural mass from 737.37 kg to 887.18 kg, an increase of 20.32%, while the improvement in static response gradually decreases. Spar-cap thickness should therefore be selected by balancing stiffness improvement against the associated mass penalty.

3.2. Effect of Shear-Web Inclination Angle

To evaluate the effect of main-spar cross-sectional geometry, the shear-web inclination angle, θ_w, was selected as the geometric variable. The midsurface position of the upper spar cap in the baseline box-shaped spar was kept fixed, while the lower spar cap was shifted chordwise at each control section. This produced equal inclination angles in the front and rear shear webs. The spar-cap width, spar-cap and shear-web thicknesses, material properties, layup configuration, and in-plane partitioning were kept unchanged. Five cases were considered, with −20°, −10°, 0°, 10°, and 20°. All cases used the M3 mesh. For each control station, the chordwise offset required to generate a prescribed shear-web inclination was calculated as:

δ_i = H_m,itanθ_w

(9)

where δ_i is the chordwise offset at the i control station, H_m,i is the corresponding midsurface height, and θ_w is the shear-web inclination angle. Table 8 presents the station-dependent offsets and local web lengths are listed in the Supplementary Materials.

Table 8. Chordwise lower-cap offsets at the nine control stations (mm).

Station	−20°	−10°	0°	+10°	+20°
S1	−203.8	−98.7	0.0	98.7	203.8
S2	−269.7	−130.7	0.0	130.7	269.7
S3	−388.4	−188.1	0.0	188.1	388.4
S4	−333.0	−161.3	0.0	161.3	333.0
S5	−260.6	−126.3	0.0	126.3	260.6
S6	−195.5	−94.7	0.0	94.7	195.5
S7	−150.7	−73.0	0.0	73.0	150.7
S8	−108.5	−52.5	0.0	52.5	108.5
S9	−71.0	−34.4	0.0	34.4	71.0

The external loading conditions were kept identical for all five inclination cases. RP-Tip-Center remained fixed, and the same 1000 N transverse force was applied in the positive (Y) direction. However, changing the cross-sectional geometry may shift the sectional shear center and alter the load eccentricity. Therefore, the computed twist may contain contributions from both geometry-dependent structural response and eccentric-load-induced torsion.

The corresponding structural responses are summarized in Figure 5.

Figure 5. Effects of the shear-web inclination angle on structural responses: (a) flapwise tip deflection; (b) maximum equivalent stress; (c) tip twist angle; (d) first three natural frequencies.

Figure 5a,b show that both tip displacement and maximum stress increase with the magnitude of θ_w, with an approximately symmetric response about 0°. The baseline tip displacement is 897.04 mm. At ±10°, it increases to 941.25 and 941.04 mm, approximately 4.9% above the baseline value. At ± 20°, it further increases to approximately 1086 mm, corresponding to an increase of about 21.1%. Maximum stress is more sensitive to θ_w. It increases from 39.418 MPa at 0° to approximately 51.46 MPa at ±10° and 66.78 MPa at ±20°, representing increases of about 30.5% and 69.4%, respectively. These results indicate that increasing the shear-web inclination magnitude reduces the effective resistance to global bending deformation and alters the internal load-transfer path. The stress response is therefore more sensitive to θ_w than the overall displacement.

Figure 5c shows that the shear-web inclination angle provides strong directional control of the bend–twist response. As θ_w increases from −20° to +20°, the tip twist angle changes from 1.2699° to −1.3096° and approaches zero near 0°, giving an approximately antisymmetric response. The mean absolute twist angle is about 0.449° at ±10° and increases to approximately 1.290° at ±20°. Thus, increasing |θ_w| markedly amplifies the torsional response. The baseline twist angle at θ_w = 0° is approximately −0.0035309°. Therefore, ratios normalized by the baseline twist magnitude are sensitive to the small denominator and should not be interpreted as proportional changes in intrinsic bend–twist coupling stiffness. As discussed in Section 2.4, the computed twist may also include contributions from structural asymmetry and load eccentricity. Thus, the twist sign is used only to distinguish the rotation direction. Without defined aerodynamic inflow and blade-pitch references, it cannot be directly interpreted as twist-to-feather, twist-to-stall, or aerodynamic load alleviation. Figure 5d further shows that the first three natural frequencies respond differently to θ_w. With no modal-order exchange observed, increasing θ_w from 0° to 20° reduces the first and third natural frequencies from 0.60315 and 1.9837 Hz to approximately 0.5447 and 1.7497 Hz, respectively, corresponding to decreases of 9.7% and 11.8%. In contrast, the second natural frequency increases from 0.92339 to approximately 0.9944 Hz, an increase of 7.7%. These opposite trends suggest that shear-web inclination redistributes directional stiffness and coupling rather than producing a uniform change in overall stiffness.

Overall, increasing the magnitude of the shear-web inclination produces pronounced changes in the signed tip rotation, but also affects deformation, comparative stress response, structural mass, and low-order dynamic characteristics. These changes should be interpreted as global load-induced responses to the modified geometry rather than as direct evidence of enhanced intrinsic bend–twist coupling. Possible effects of load eccentricity relative to the sectional shear center were not quantified separately. Therefore, the shear-web inclination should be selected by considering the overall structural performance trade-offs.

3.3. Effect of Spar-Cap Off-Axis Ply Angle

To investigate the effect of spar-cap layup angle on the structural response and dynamic characteristics of the blade. The main-spar geometry, material properties, total laminate thickness, boundary conditions, and external loads were kept unchanged. Four layup cases were considered:0°, ±30°, ±45°, ±60°. The structural mass remained constant at 737.37 kg for all cases, eliminating the influence of mass variation on the response. The resulting responses are summarized in Figure 6.

Figure 6. Effects of the spar-cap off-axis ply angle on structural responses: (a) flapwise tip deflection; (b) maximum equivalent stress; (c) tip twist angle; (d) first three natural frequencies.

Figure 6a,b summarize the static responses. The flapwise tip deflection increases from 897.04 mm for the baseline case to 967.55, 1019.4, and 1038.4 mm for the ±30°, ±45°, and ±60° cases, corresponding to increases of 7.86%, 13.64%, and 15.76%, respectively. This trend indicates a reduction in the effective spanwise bending stiffness as the fiber direction deviates from the primary load-carrying direction. The maximum equivalent stress shows a non-monotonic variation, increasing from 39.418 MPa at 0° to 49.060 MPa at ±30° and 52.365 MPa at ±45°, before decreasing to 49.281 MPa at ±60°. The ±45° case therefore produces the highest local stress level among the configurations investigated.

Figure 6c,d present the bend–twist and modal responses. The absolute tip twist angle increases by 17.61%, 18.32%, and 18.45% for the ±30°, ±45°, and ±60° cases, respectively, relative to the baseline. The variation becomes small beyond ±30°, indicating that a larger off-axis angle does not enhance the global bend–twist response under the present configuration. The first three natural frequencies also decreased with increasing off-axis ply angle. From 0° to ±60°, f₁, f₂ and f₃ decrease by 11.42%, 10.93%, and 7.20%, respectively.

Overall, increasing the magnitude of the spar-cap off-axis ply angle increases flapwise deformation and lowers the low-order natural frequencies, while the absolute tip twist does not increase under the prescribed loading conditions. The CLT results show that A₁₆ and A₂₆ are approximately zero and B is negligible for the investigated balanced and symmetric laminates, whereas D₁₆ and D₂₆ remain nonzero in the off-axis configurations. For the ±30°, ±45°, and ±60° cases, the corresponding (D₁₆, D₂₆) pairs are (55.46, 21.28), (44.30, 44.30), and (21.28, 55.46) × 10³ N·mm, respectively. These results indicate that changes in effective structural stiffness do not necessarily enhance the global bend–twist response. Nonzero laminate-level bending–twisting coupling terms do not, by themselves, guarantee increased global tip twist. These findings apply to the balanced and symmetric layups investigated under the prescribed loading conditions and should not be generalized to unbalanced laminates or other stacking sequences.

3.4. Effect of Off-Axis Ply Fraction

Building on the layup-angle analysis, this section further investigates the effect of off-axis ply content in the spar cap on the bend–twist coupling response. With the off-axis ply angle fixed at ±45°, three layup schemes, C0, C1, and C2, were defined by varying the proportions of 0° and ±45° plies in the root and mid-span primary load-bearing regions. C0 is the baseline case with an all-0° spar-cap layup. C1 corresponds to the ±45° case, in which four 0° plies in each of the M1_16, M1_14, M1_12, M1_10, and M1_8 regions were replaced by two +45° and two −45° plies. C2 was derived from C1 by further replacing four 0° plies with two additional +45° and two additional −45° plies in each of the M1_16, M1_14, and M1_12 regions, while the remaining regions were unchanged. The off-axis plies in both configurations were arranged in balanced and symmetric stacking sequences.

Table 9 presents the detailed ply counts, off-axis ply fractions, and stacking sequences for each laminate region in the C1 and C2 configurations.

Table 9. Regional ply counts, off-axis ply fractions, and stacking sequences for the C1 and C2 configurations.

Case	Laminate Group	Ply Counts (0°/+45°/−45°)	±45° Fraction (%)	Stacking Sequence
C1	M1–16	12/2/2	25	[0/0/0/+45/−45/0/0/0]S
	M1–14	10/2/2	28.57	[0/0/+45/−45/0/0/0]S
	M1–12	8/2/2	33.33	[0/0/+45/−45/0/0]S
	M1–10	6/2/2	40	[0/+45/−45/0/0]S
	M1–8	4/2/2	50	[0/+45/−45/0]S
	M1–6	6/0/0	0	[0/0/0]S
	M1–4	4/0/0	0	[0/0]S
	M1–4	4/0/0	0	[0/0]S
C2	M1–16	8/4/4	50	[0/+45/−45/0/+45/−45/0/0]S
	M1–14	6/4/4	57.14	[0/+45/−45/0/+45/−45/0]S
	M1–12	4/4/4	66.67	[0/+45/−45/+45/−45/0]S
	M1–10	6/2/2	40	[0/+45/−45/0/0]S
	M1–8	4/2/2	50	[0/+45/−45/0]S
	M1–6	6/0/0	0	[0/0/0]S
	M1–4	4/0/0	0	[0/0]S
	M1–4	4/0/0	0	[0/0]S

Note: The subscript S denotes symmetry about the laminate mid-plane. The off-axis ply fraction is calculated as ${\displaystyle \frac{\left(N_{+45°}+N_{-45°}\right)}{N_{\mathrm{total}}}}\times 100\%$.

To distinguish laminate-level stiffness from the global bending-induced twist response, the CLT stiffness matrices were examined for the five configurations. Table 10 lists the independent components of the extensional stiffness matrix A and bending stiffness matrix D, together with the maximum absolute value of the extension–bending coupling matrix B. The matrices were extracted from ANSYS ACP at comparable locations in the upper spar-cap M1–16 region using consistent local reference directions and the stacking sequences defined in Table 9.

Table 10. Local laminate stiffness matrix components of the investigated spar-cap configurations obtained from classical laminate theory.

Component	C0	±30°	C1 (±45°)	±60°	C2
A (103 N/mm)
A11	865.6642	782.0248	720.7186	681.7457	575.7731
A22	63.4316	80.0713	119.0442	180.3504	174.6568
A12	17.1265	50.6264	61.7930	50.6264	106.4595
A16	≈0	≈0	≈0	≈0	≈0
A26	≈0	≈0	≈0	≈0	≈0
A66	45.0448	78.5447	89.7113	78.5447	134.3777
B (N)
max |Bij|	9.02 × 10−10	7.28 × 10−10	6.11 × 10−10	7.28 × 10−10	3.78 × 10−10
D (106 N·mm)
D11	4.07947	3.77770	3.55650	3.41589	2.61729
D22	0.29892	0.35896	0.49958	0.72077	0.85993
D12	0.08071	0.20158	0.24187	0.20158	0.53129
D16	0	0.05546	0.04430	0.02128	0.09968
D26	0	0.02128	0.04430	0.05546	0.09968
D66	0.21228	0.33314	0.37343	0.33314	0.66286

Note: The A, B, and D matrices denote the extensional, extension–bending coupling, and bending stiffness matrices, respectively. The A and D matrices are symmetric; only their independent components are shown. All B-matrix entries are numerical residuals (max |Bij| < 10⁻⁹ N), so B ≈ 0. The ±30° and ±60° cases retain the C1 replacement-ply positions. D16 and D26 describe local laminate-level bending–twisting stiffness, not the sectional coupling stiffness of the entire box spar.

The three schemes use the same material system, total ply count, laminate thickness, and 0° layup configuration of the front and rear structural members. The structural mass remains constant at 737.37 kg, ensuring direct comparison among the three cases. The corresponding static and modal responses are summarized in Figure 7.

Figure 7. Effects of the off-axis ply fraction on structural responses: (a) flapwise tip deflection; (b) maximum equivalent stress; (c) tip twist angle; (d) first three natural frequencies.

Figure 7a,b show that increasing the off-axis ply content increases both tip displacement and maximum stress. The tip displacement increases from 897.04 mm for C0 to 1019.40 mm for C1 and 1095.40 mm for C2, corresponding to increases of 13.64% and 22.11% relative to C0. The increase from C1 to C₂ is 7.46%. Maximum stress shows a stronger response, increasing from 39.418 MPa for C0 to 52.365 MPa for C1 and 76.500 MPa for C2. These values correspond to increases of 32.85% and 94.07% relative to C0, respectively. The stress under C2 is also 46.09% higher than that under C1. These trends indicate that replacing part of the 0° plies with ±45° plies reduces the effective longitudinal load-bearing stiffness of the spar cap and redistributes the internal load, resulting in larger global deformation and higher local stress.

Figure 7c shows that increasing the off-axis ply content does not enhance the global twist response. The absolute tip twist angle decreases from 0.0035303° for C0 to 0.0028834° for C1 and 0.0009583° for C2. These reductions correspond to 18.32% and 72.85% relative to C0, respectively, while C2 is 66.76% lower than C1. Figure 7d shows a similar degradation in dynamic performance. From C0 to C2, the first three natural frequencies decreased by 18.96%, 17.77%, and 8.51%, respectively, with the first two modes showing greater sensitivity. Together with the increase in tip displacement, these reductions indicate an overall decrease in structural stiffness as the ±45° ply content increases.

In short, increasing the ±45° ply fraction redistributes the spar-cap stiffness and alters global structural performance. According to the CLT results in Table 10, from C0 to C2, the extensional stiffness A₁₁ decreases by 33.49% from 865.66 to 575.77 × 10³ N/mm, while the bending stiffness D₁₁ decreases by 35.84% from 4.079 to 2.617 × 10⁶ N·mm. Meanwhile, D₁₆ and D₂₆ increase from approximately to 99.68 × 10³ N·mm Despite these local coupling terms, the absolute tip twist decreases from 0.0035303° in C0 to 0.0009583° in C2 under the prescribed reference load. Thus, increasing the ±45° ply fraction does not enhance the global bend–twist response for the studied balanced and symmetric layups. This finding should not be generalized to unbalanced laminates or other stacking sequences not examined in this study.

4. Discussion

4.1. Relative Influence of Design Parameters on Structural Responses

This section evaluates the relative influence of different main-spar design parameters on structural responses. Based on Equations (4) and (5), heatmaps of the relative influence matrix were constructed for the four parameter groups and response variables. The relative influence index S ranges from 0 to 1. The value of S = 1 indicates that the corresponding parameter causes the largest variation in each response within the investigated range. It should be noted that S quantifies only the magnitude of influence and does not indicate the direction of change. The increase or decrease in each response must therefore be identified from the original simulation results.

Figure 8 shows clear response selectivity among the design parameters. Spar-cap thickness has the greatest effect on tip displacement, followed by off-axis ply content and shear-web inclination angle, indicating that global bending deformation is mainly governed by spar-cap geometry and structural stiffness. The shear-web inclination angle has the strongest effect on tip twist angle and therefore dominates the global bend–twist response within the investigated range. Off-axis ply content has the greatest effect on maximum stress and the first natural frequency, suggesting a strong influence of layup parameters on local load transfer and low-order dynamic stiffness. Structural mass is mainly controlled by spar-cap thickness, whereas the shear-web inclination angle has only a minor effect and the other two parameters have negligible influence.

Figure 8. Range-normalized relative influence matrix of the blade-spar design parameters on key structural responses.

The first three natural frequencies reported in this study were obtained under constant material properties and boundary conditions. Under actual operating conditions, temperature variations may alter the effective stiffness and modal behavior of composite wind turbine blades. Rezazadeh et al. [30] experimentally demonstrated that the modal properties of a scaled composite wind turbine blade vary with temperature, highlighting the importance of environmental effects in structural health monitoring. Therefore, the modal-frequency variations reported here should be interpreted as parameter-induced numerical responses under the prescribed modeling conditions. They should not be regarded as temperature-invariant indicators of in-service structural performance or damage.

In summary, no single design parameter dominates all structural responses; instead, the four variables exhibit clear response selectivity. The comprehensive evaluation should therefore consider the bend–twist response together with changes in stiffness, stress, natural frequency, and structural mass. However, the relative influence indices were derived from independent single-factor variations. Therefore, they do not account for interactions among the design variables. The resulting response rankings should not be directly extended to cases in which spar-cap thickness, shear-web inclination, and laminate parameters are varied simultaneously.

4.2. Bend–Twist Response and Structural Performance Trade-Offs

This section uses the bend–twist coupling response coefficient K_BT together with the main structural responses to compare the overall effects and performance trade-offs of different design parameters. Table 11 summarizes the bend–twist coupling response indices for representative cases.

Table 11. Bend–twist response indices for representative design cases.

Design Parameter	Representative Case	KBT (°/m)
Baseline model	——	0.00394
Spar-cap thickness	kt = 1.4	0.00507
Shear-web inclination angle	θw = −20°	1.16891
Shear-web inclination angle	θw = +20°	1.20567
Spar-cap off-axis ply angle	θp = ±60°	0.00277
Off-axis ply fraction	C2	0.000875

When the spar-cap thickness was increased to 1.4t₀, K_BT increased from 0.00394 °/m for the baseline case to 0.00507 °/m. Meanwhile, the model mass increased from 737.37 kg to 887.18 kg, corresponding to an increase of approximately 20.3%. Together with the reduced tip deflection and increased natural frequencies, these results show that a thicker spar cap enhances global stiffness and low-order dynamic performance, but at the expense of a substantial mass increase.

The shear-web inclination angle has the strongest influence on the bend–twist response. At −20° and +20°, K_BT reaches 1.16891 °/m and 1.20567 °/m, respectively, with opposite twist directions. However, the large ratios partly result from the very small baseline K_BT and should therefore be interpreted together with the absolute values. These ratios should not be interpreted as proportional enhancements of intrinsic bend–twist coupling stiffness. Increasing the inclination magnitude also increases tip displacement, maximum equivalent stress in the selected spar-cap ply region, and structural mass. Thus, the shear-web inclination angle should be selected by balancing bend–twist coupling capability against structural load-bearing performance.

The two types of layering parameters exhibit different trade-off characteristics. When the spar-cap off-axis ply angle increases to ±60°, K_BT decreases to 0.00277 °/m, compared with 0.00394 °/m for the baseline model. Increasing the off-axis ply fraction to C2 causes a more pronounced reduction, with K_BT decreasing to 0.000875 °/m, accompanied by increased local stress and lower low-order natural frequencies. The local ABD results and global finite element responses describe different aspects of the structural behavior. Nonzero D₁₆ and D₂₆ terms do not necessarily lead to greater global tip twist. The response also depends on the spanwise ply distribution, box-spar geometry, load position, and boundary conditions. Therefore, the present conclusions are limited to the balanced and symmetric layups and loading conditions examined. Other stacking sequences, including unbalanced layups, require separate evaluation.

To further illustrate the local stress response in the spar-cap region investigated for the ±45° ply modification, Figure 9 presents the in-plane shear-stress τ₁₂ distributions of selected analysis plies at the same U_03 spar-cap region in the C0 and C1 configurations under the prescribed flapwise bending load. As shown in Figure 9a, the selected 0° ply in the baseline configuration C0 exhibits a nonuniform shear-stress distribution, with τ₁₂ ranging from −0.38804 to 0.39851 MPa. In contrast, Figure 9b shows the corresponding local response of a selected off-axis ply in C1, where the shear stress ranges from −0.46449 to 1.3799 MPa. The different distributions illustrate the influence of the modified laminate orientation on the local stress response within the investigated spar-cap region. However, because these stresses are expressed in their respective ply material coordinate systems, the values should not be directly interpreted as a coordinate-independent measure of reinforcement effectiveness. As demonstrated by the global responses in Figure 7, increasing the ±45° spar-cap ply content in the investigated configurations is accompanied by increased tip displacement and maximum equivalent stress, together with a reduced absolute bending-induced twist angle. Therefore, the newly added contours provide a visualization of the local stress response in the region considered for ±45° reinforcement, rather than evidence of an independently optimized reinforcement layout.

Figure 9. In-plane shear-stress τ₁₂ distributions of representative analysis plies in the U_03 spar-cap region under the prescribed flapwise bending load: (a) the 0° spar-cap ply in the baseline configuration C0; (b) a selected ±45° spar-cap ply in the modified configuration C1.

The selected-ply stress was defined as the maximum averaged equivalent stress in the CAP_01 root evaluation region. Its relative difference between M2 and M3 was 4.072%. However, no independent boundary-condition sensitivity analysis was performed, and the local peak stress may be affected by the idealized fully fixed root constraint. The 94.07% increase in the C2 stress response represents a relative numerical change under the prescribed loading conditions, not a proportional reduction in material strength or structural safety margin.

The reported trade-offs were evaluated under the prescribed reference loading and idealized boundary conditions. Therefore, the numerical responses should not be interpreted as evidence of operational load alleviation or structural safety under realistic turbine loading. Since ply-level failure, buckling, and fatigue were not assessed, the equivalent stress is used only as a comparative response measure.

In summary, the design of each parameter should be based on overall performance trade-offs rather than on a single response metric. Spar-cap thickness is constrained by weight requirements, while the shear-web inclination angle is limited by load-bearing capacity. Off-axis ply angle and ±45° ply content are jointly constrained by stiffness, stress, and dynamic performance. Therefore, simply increasing the off-axis ply content is not an effective strategy for improving bend–twist performance. Nevertheless, these findings are based on single-factor comparisons. The interaction effects among spar-cap thickness, shear-web inclination, and laminate configuration have not yet been quantified. Further factorial or response-surface analyses are needed to assess whether the observed trends remain valid when multiple parameters vary simultaneously. Such analyses would also help identify feasible combined design configurations.

5. Conclusions

Using a consistent modeling framework with identical loading and boundary conditions, the effects of spar-cap thickness, shear-web inclination angle, spar-cap layup angle, and off-axis ply content on response selectivity and performance trade-offs were compared. The main conclusions are as follows:

(1)

The four design parameters exhibit distinct response selectivity. Spar-cap thickness mainly governs global bending stiffness and structural mass, while the shear-web inclination angle has the strongest influence on the bend–twist response and twist angle. Spar-cap layup angle and off-axis ply content primarily affect anisotropic stiffness distribution, local stress, and low-order dynamic characteristics. No single parameter simultaneously optimizes all response metrics.

(2)

Within the investigated range, the shear-web inclination angle is the most influential geometric parameter for the bend–twist response. At −20° and +20°, the bend–twist response indices reach approximately 1.17 °/m and 1.21 °/m, respectively, with opposite twist directions. However, larger inclination magnitudes increase tip displacement and maximum stress, slightly increase structural mass, and reduce some low-order natural frequencies. Therefore, shear-web inclination should be assessed by balancing bending-induced twist against deformation, comparative stress response, structural mass, and low-order dynamic performance.

(3)

Under the symmetric balanced layup, increasing the off-axis ply angle or content does not enhance the global bend–twist response and leads to stiffness and stress penalties. The twist-to-deflection response index decreases to 0.00277 °/m for the ±60° configuration and to 0.000875 °/m for C2, compared with the baseline value of 0.00394 °/m. These results are specific to the balanced and symmetric laminate configurations investigated under the prescribed loading conditions. The CLT results show that the off-axis configurations retain nonzero local bending–twisting stiffness terms D₁₆ and D₂₆, even when the extension–bending coupling matrix B is negligible. These local terms alone do not determine the global response of the equivalent box spar, which also depends on the stacking sequence, structural geometry, and loading conditions. Further comparisons with unbalanced or differently sequenced laminates are needed before broader conclusions about material coupling can be drawn.

(4)

The parameter–response relationships and trade-off patterns clarify the distinct roles of global stiffness, geometry-induced bend–twist coupling, and material anisotropy. Different parameters dominate different structural responses and involve distinct performance trade-offs. These relationships provide a deterministic basis for subsequent reliability analysis and multi-objective optimization of wind turbine blade main spars.