Numerical Analysis of Deformation Behavior in the Double-Layer Flexible Photovoltaic Support Structure

Abstract

Flexible photovoltaic (PV) support systems, referring to cable-supported structural systems that carry conventional rigid PV modules rather than flexible thin-film modules, have attracted increasing attention as a promising solution for photovoltaic construction in complex terrains due to their advantages of broad-span design and simplified installation. However, the deformation behavior of flexible PV supports remains insufficiently understood, which restricts its application and engineering optimization. To address this issue, a three-dimensional finite element model of a flexible PV support system was developed using an in-house Python code to investigate its deformation characteristics. The model discretizes the structure into beam and cable elements according to their mechanical properties, and the coupling relationship between their degrees of freedom is established by means of a multi-point constraint. The validation of the proposed model is confirmed by comparison with theoretical solutions. Simulation results reveal that the deformation of flexible PV supports is more sensitive to horizontal loads, indicating that their overall deformation performance is primarily governed by lateral rather than vertical loading. Furthermore, dynamic analyses show that higher loading frequencies induce noticeable torsional de-formation of the structure, which may compromise the stability of the PV panels. These findings provide valuable theoretical guidance for the design and optimization of flexible PV support systems deployed in complex terrains.

1. Introduction

Flexible photovoltaic (PV) support systems, referring specifically to cable-supported structural systems for conventional rigid PV modules rather than flexible photovoltaic cells or module frames, do not require extensive foundation work, enabling them to span gullies and adapt to uneven terrain easily [1]. These structural characteristics help these systems significantly enhance land use in steep-slope regions through efficient installation processes, resulting in reduced costs, mainly due to fewer foundations, simplified installation procedures, and reduced material consumption [2]. Previous engineering and techno-economic studies have reported that cable-supported flexible PV support systems can achieve noticeable reductions in balance-of-system costs compared with conventional rigid supports; however, a detailed cost analysis per installed wat is beyond the scope of the present study. Substantial differences exist between flexible and rigid PV supports. The conventional rigid support consists of foundations, posts, beams, and rails. The beam-column system of conventional rigid supports provides high stiffness and excellent resistance. However, the conventional rigid support (Figure 1a) requires extensive site leveling [3], resulting in poor site adaptability and low construction efficiency. However, flexible supports (Figure 1b) employ suspension cables as primary load-bearing members. In these structures, the cable tension distributes loads from PV modules and other components [4,5]. These systems demonstrate superior deformation adaptability and long-span deployment capabilities, potentially reducing the number of foundations and steel required, making them suitable for complex mountainous and hilly terrains. The original photovoltaic module support systems were mainly rigid bracket systems [6,7,8,9]. Baumgartner first proposed photovoltaic module systems supported by single-layer cables [10,11]. Subsequently, the research was focused on the static and nonlinear response of the flexible structure. Huang et al. [12] and Gasparini [13] investigated the nonlinear deformation behavior of flexible suspension cables under external loads and evaluated the effects of large deflections. Di et al. [14] proposed analytical expressions for a single cable and modified them for a double-layer prestressed flexible truss supporting PV modules, assuming geometric nonlinearity and small-strain. Zhang et al. [15] proposed equilibrium equations for the deformation curve, the maximum displacement, and the maximum tension of wire ropes under varying loads. These studies show that an increased prestress enhances the stiffness of suspension cables in flexible PV supports. Unlike rigid members, suspension cables possess low bending stiffness and rely primarily on axial tension to resist external loads. This flexibility enables cables to dissipate wind-induced energy through large deformation and geometric nonlinearity, thereby reducing stress concentration and mitigating resonant amplification under fluctuating wind loads [8,16,17]. Several experimental and numerical investigations have examined the dynamic response of flexible PV supports. For instance, Lou et al. [18] performed finite-element simulations, using synthetic turbulent wind loads to analyze the static and dynamic responses of cable-suspended PV arrays. They found that wind fluctuations significantly affect cable deformation. Abiola-Ogedengbe et al. [19] and Jubayer et al. [20] studied the wind pressure distribution on individual PV modules through wind tunnel testing and numerical simulations. Their findings indicate that wind direction and panel tilt angles significantly affect surface wind loads. They identified the critical wind angles associated with maximum lift and overturning moments. Zou et al. [21] conducted wind-tunnel tests on multi-span double-layer cable PV systems and found that wind-induced responses dominate in the panels’ windward rows. Cai et al. [22] investigated wind-resistance design for flexible PV supports. They analytically estimated the standard deviation of dynamic wind responses based on the standard deviation of static responses.

In practical engineering applications, the durability of structural cables is a critical concern, particularly with respect to corrosion in outdoor environments. To mitigate corrosion-related degradation, cable-supported flexible PV support systems commonly adopt corrosion-resistant cable technologies, such as galvanized steel strands, epoxy-coated or polyethylene-sheathed cables, and stainless-steel cables in aggressive environments. In addition, protective measures including surface coatings, sealed anchorage systems, and regular inspection and maintenance are typically employed to ensure long-term structural performance. In the present study, the cables are assumed to be adequately protected against corrosion, and corrosion effects are therefore not explicitly considered in the numerical model. Li et al. [23] combined wind tunnel and numerical modeling data to determine the spatial distribution of wind pressures on flexible PV arrays, proposed differential load coefficients for interior versus edge panels, and evaluated gust response factors. Wu et al. [3] used wind tunnel tests and finite element analysis for a novel cable-truss flexible PV support and found an improved stiffness of 10.3% while withstanding wind speeds up to 36 m/s. They discovered that cable tension, span, and spacing are the key factors for enhancing structural stability. He et al. [24] performed wind-tunnel testing and found that flexible PV supports undergo large-amplitude vertical and torsional oscillations when wind speeds exceed critical thresholds, particularly under head-on (180°) wind conditions. Abdollahi [25] analyzed that under extreme wind conditions, some components must exceed the safe load-bearing capacity of the PV module. Several studies employed finite element modeling to investigate the dynamic characteristics of flexible PV supports. Zhu et al. [26] employed two-way FSI simulations and found that increasing cable pre-tension in flexible PV supports reduced natural periods and reaction force vibration coefficients but increased displacement vibration coefficients. Xie [27] and Wang [28] investigated the natural vibration modes and wind-induced responses of single-layer flexible PV supports. They found that such structures are prone to vertical and torsional vibrations, whereas the influence of fluctuating wind loads on lateral deflections is relatively limited.

This study focuses on the numerical investigation of the deformation behavior of flexible PV supports. A three-dimensional finite element model is developed with precise beam–cable coupling using the multi-point constraint (MPC) method. The suspension cable theory is used to derive analytical expressions for cable deflection under prestress and geometric nonlinearity. The study investigated the spatial mismatch between displacement and stress peaks under static loading, excessive deflections in upper versus lower cables, and the heightened vulnerability of end spans. Furthermore, the time-history evolution from double-to single-peak deflection under constant headwind loading is examined. The vibration-suppression mechanism of middle brackets is investigated, which provides rotational restraint and sensitivity to amplitude and duration. These insights contribute to the development of design methodologies and optimization strategies for the flexible PV support in mountainous applications.

2. Numerical Modeling and Theoretical Derivation

2.1. Parametric Modeling of the Flexible PV Support System

To establish a unified finite element dynamic formulation for the multi-span flexible PV support system, it is necessary to derive the governing equations based on the principle of virtual work prior to introducing the multi-point constraint (MPC) coupling. The MPC method is employed to enforce kinematic compatibility between beam and cable elements by constraining selected nodal degrees of freedom, thereby ensuring displacement continuity and internal force equilibrium at their shared nodes and avoiding artificial stiffness discontinuities. The virtual work principle provides a consistent foundation for discretization and element assembly of beams and cables within the structure.

The dynamic equilibrium of the continuum [29] can be expressed in terms of virtual work as Equation (1), where ε and σ denote the strain and stress tensors, and ρ, c_d, b, and $\overline{t}$ represent the density, damping coefficient, body force, and surface traction, respectively.

$${\displaystyle {\int }_V\delta }\epsilon :\sigma dV={\displaystyle {\int }_V\delta }u\cdot (\rho \ddot{u}+c_d\dot{u}+b)dV+{\displaystyle {\int }_S\delta }u\cdot \overline{t}dS$$

(1)

Equation (1) can be written as a finite element integral point by numerical discretization:

$${\displaystyle {\int }_V\delta }u\cdot (\rho \ddot{u}+c_d\dot{u}+b)dV={\displaystyle \sum _e{\displaystyle \sum _{\alpha }{w}_{e\alpha }}}\delta u(x_{e\alpha })\cdot [\rho \ddot{u}(x_{e\alpha })+c_d\dot{u}(x_{e\alpha })+b(x_{e\alpha })]$$

(2)

$${\displaystyle {\int }_V\delta }\epsilon :\sigma dV={\displaystyle \sum _e{\displaystyle \sum _{\alpha }{w}_{e\alpha }}}\delta \epsilon (x_{e\alpha }):\sigma (x_{e\alpha })$$

(3)

where δu denotes the virtual displacement vector, ρ is the material density, c_d represents the viscous damping coefficient, and b is the body force vector.

The subscript e denotes the element index, and α indicates the Gauss integration point within the element. w_α is the numerical integration weight associated with the Gauss point x_eα. ε and σ represent the strain tensor and stress tensor, respectively.

Accordingly, δu(x_eα) and δε(x_eα) indicate the virtual displacement and virtual strain evaluated at the Gauss integration point x_eα.

In matrix form, the virtual work balance of the entire structure is expressed as Equation (4), where M, C, and K denote the global mass, damping, and stiffness matrices, respectively, and F(t) represents the external load vector.

$$\delta u^T\left(M\ddot{u}+C\dot{u}+Ku-F\left(t\right)\right)=0$$

(4)

Equations (5) and (6) represent the discrete virtual work formulation, where N and B are the shape-function and strain-displacement matrices, and D is the constitutive matrix. This equation forms the foundation for assembling global mass, damping, and stiffness matrices.

$$\delta q^T\left(M\ddot{q}+C\dot{q}+Kq-f\left(t\right)\right)=0$$

(5)

$$\begin{array}{l}M={\displaystyle \sum _e{\displaystyle {\int }_{{\mathrm{\Omega }}_e}\rho N^{T}Nd\mathrm{\Omega }}}\\ K={\displaystyle \sum _e{\displaystyle {\int }_{{\mathrm{\Omega }}_e}{B}^{T}DBd\mathrm{\Omega }}}\\ f\left(t\right)={\displaystyle \sum _e\left({\displaystyle {\int }_{{\mathrm{\Omega }}_e}{N}^{T}bd\mathrm{\Omega }+{\displaystyle {\int }_{{\mathrm{\Gamma }}_{t,e}}{N}^T\overline{t}d\mathrm{\Gamma }}}\right)}\end{array}$$

(6)

The beam element [30] and the cable element satisfy the virtual work balance equation, and the general form can be expressed as follows:

$$\delta q_e^T\left(M_e{\ddot{q}}_e+C_e{\dot{q}}_e+K_e{q}_e-{\mathrm{f}}_e\right)=0$$

(7)

The virtual work balance of a finite element, either a beam or a cable element, can be expressed in the unified form of Equation (7), where M_e, C_e, and K_e denote the element mass, damping, and stiffness matrices, respectively.

For both beam and cable elements, the element-level mass and stiffness matrices are constructed following standard finite element procedures based on isoparametric interpolation and numerical integration. Specifically, the element mass matrix M_e and stiffness matrix K_e are expressed as

$$M_e={\displaystyle \int \rho N^{T}NdV,K_e={\displaystyle \int B^{T}DBdV}}$$

(8)

where ρ denotes the material density, N is the shape function matrix, B is the strain–displacement matrix, and D is the constitutive matrix corresponding to Euler–Bernoulli beam theory for beam elements and truss theory for cable elements, respectively.

For cable elements, the stiffness matrix additionally includes a geometric stiffness component associated with the updated axial force, which accounts for the effect of large deformation. Numerical integration of the element matrices is performed using standard Gauss quadrature. These formulations ensure consistency with the virtual work principles and enable straightforward reproduction of the proposed element models.

This formulation is applicable to both Euler–Bernoulli beam elements, in which bending deformation is considered while shear deformation is neglected, and truss elements, which carry axial force only. In the present formulation, the beam components of the flexible PV support system are modeled using standard Euler–Bernoulli beam elements, whereas the cable components are modeled as truss elements. Accordingly, the element mass matrix M_e, damping matrix C_e, and stiffness matrix K_e adopt their conventional finite element forms derived from Euler–Bernoulli beam theory and truss theory, respectively, ensuring consistency with the virtual work formulation and enabling straightforward reproduction of the proposed elements.

Geometric nonlinearity in the present model is primarily considered through the large-displacement behavior of the suspension cables. As the structural configuration evolves under external loading, the axial force of the cable elements is updated incrementally and contributes to the geometric stiffness associated with the current deformation state. In this study, an updated Lagrangian formulation is adopted to account for the continuously changing geometry of the flexible PV support system during deformation.

The global governing equations are solved using an incremental time-integration strategy, in which equilibrium is enforced at each time step based on the updated configuration. This numerical framework enables the coupled static and dynamic responses of the beam–cable system to be captured while maintaining computational efficiency and numerical stability.

To ensure displacement compatibility at the beam–cable–support interfaces, the Multi-Point Constraint (MPC) [31] method is introduced. Differences in kinematic definitions between beam and cable elements would otherwise cause degree-of-freedom inconsistency or stiffness discontinuity at shared nodes.

To address the mismatch of nodal degrees of freedom between beam elements and truss elements, a linear constraint relationship is introduced:

$$Gu=0$$

(9)

where G denotes the MPC matrix and u is the global displacement vector containing all nodal degrees of freedom. By introducing the Lagrange multiplier λ, the global stiffness equation can be extended as follows:

$$\left[\begin{array}{cc}K& G^T\\ G& 0\end{array}\right]\left[\begin{array}{c}u\\ \lambda \end{array}\right]=\left[\begin{array}{c}F\\ 0\end{array}\right]$$

(10)

This formulation ensures displacement compatibility and internal force equilibrium between beam and cable elements at their connection interfaces, thereby preventing abrupt stiffness variations and unreasonable local stress concentrations. When combined with dynamic analysis, the constraint conditions are likewise incorporated into the second-order motion equation [32]:

$$M\ddot{u}+C\dot{u}+Ku=F\left(t\right)+G^T\lambda$$

(11)

where M, C and K represent the mass, damping, and stiffness matrices, respectively, and F(t) denotes the external load vector. Within this framework, the structural response under both static and dynamic loading can be simultaneously considered.

In this study, the numerical model was developed to investigate the structural response of a single-layer, four-row, eight-span flexible PV support system under both static and dynamic loading conditions. The model was constructed using an in-housing Python script (Python 3.11) to automate component assembly, with cross-sectional properties assigned in accordance with the technical specifications provided in

Table 1. Cross-sectional properties of components in flexible PV support.

Component Type	Subcomponent	Element Type	Cross-Section Property [mm]
Edge Support	posts	Beam	ø102 × 4.0
	bracing rod	Beam	ø38 × 3.0
	tie rod	Truss	ø16
Intermediate Support	post	Beam	ø133 × 4.5
	inclined beam	Beam	□70 × 50 × 4.0
	diagonal brace	Beam	ø38 × 4.0
	strut	Beam	ø114 × 4.0
Middle Bracket	stabilizing rope	Truss	ø10
	L-section steel	Beam	∟50 × 50 × 5.0

Cross-section notations: “ø” indicates diameter × thickness (e.g., ø102 × 4.0 means 102 mm diameter, 4.0 mm thickness); “□” indicates width × height × thickness for rectangular sections; “∟” indicates leg length × leg length × thickness for angle sections. Beam Element” and “Truss Element” refer to finite element modeling types used in structural simulation (beam for bending, truss for axial loads).

. The script also facilitates the generation of geometric topology, material property assignment, and the application of boundary conditions. Figure 2 presents a schematic of the numerical model of the flexible PV support, also illustrates its individual components. Transversely, the system comprises edge support brackets, intermediate support brackets, and stabilizing brackets; longitudinally, it adopts a dual-cable configuration consisting of upper and lower suspension cables with a horizontal spacing of 1.4 m and a vertical spacing of 0.8 m.

2.2. Theoretical Solution for Single Cable Deflection

2.2.1. Fundamental Assumptions

The analytical model for the suspension cable used in this study is based on classical catenary theory. To derive the deflection profile of a prestressed flexible suspension cable, the following assumptions are adopted:

1. The suspension cable is assumed to be ideally flexible, carrying neither bending moments nor shear forces.

2. The small-slope approximation (|y`| ≪ 1) is adopted, while retaining second-order terms of arc length to capture geometric nonlinearity.

3. The cable material is assumed to exhibit linear elasticity, with the axial force–elongation relationship given by N = EAε.

4. An initial prestress is introduced to enhance the stiffness of the suspension cable structure.

2.2.2. Force Analysis and Deflection Equation Derivation

To analyze the mechanical behavior, we consider a typical single-span flexible suspension cable under prestress and a uniformly distributed load. A cartesian coordinate system is established with the origin at the left support of the cable. The cable has an initial horizontal prestress (σ₀), a cross-sectional area (A), and is subjected to a distributed load q along its arc length (see Figure 3).

Let the two neighboring points on the cable be M (x, y) and M′ (x + Δx, y + Δy). An infinitesimal segment is analyzed to derive equilibrium equations (see Figure 4).

From the vertical force equilibrium, it follows that

$${\displaystyle \frac{d^{2}y}{d{x}^2}}={\displaystyle \frac{q}{H}}$$

(12)

where q denotes the uniformly distributed vertical load per unit length, and H is the horizontal component of the cable force.

For the cable segment from the left support to position x, the vertical equilibrium yields the shear force:

$$V\left(x\right)={\displaystyle {\int }_0^{x}qdx}=qx,\tan \theta ={\displaystyle \frac{V}{H}}\approx y^{\prime }$$

(13)

Thus, the following governing equation can be obtained:

$$y^{\prime }\left(x\right)={\displaystyle \frac{V}{H}}={\displaystyle \frac{q}{H}}x+C_1$$

(14)

The deflection equation is then obtained as

$$y\left(x\right)={\displaystyle \frac{q}{2H}}{x}^2+C_{1}x+C_2$$

(15)

By applying the boundary conditions y (0) = 0 and y (L) = 0, the coefficients C₁ and C₂ can be determined.

$$\left\{\begin{array}{c}y(0)=0\Rightarrow C_2=0\hfill \\ y(L)=0\Rightarrow {\displaystyle \frac{q}{2H}}{L}^2+C_{1}L=0\Rightarrow C_1=-{\displaystyle \frac{qL}{2H}}\hfill \end{array}\right.$$

(16)

The final deflection equation of the suspension cable is obtained as

$$y\left(x\right)={\displaystyle \frac{q}{2H}}x\left(L-x\right)$$

(17)

The maximum mid-span deflection w_max is obtained as

$$w_{\max }=y\left({\displaystyle \frac{L}{2}}\right)={\displaystyle \frac{q}{2H}}L\left(L-{\displaystyle \frac{L}{2}}\right)={\displaystyle \frac{q{L}^2}{8H}}$$

(18)

Under external loading, the deformation of the suspension cable exhibits a parabolic shape; Thus, the parabolic expression for the cable deflection can be derived as

$$y\left(x\right)=4w_{\max }{\displaystyle \frac{x}{L}}\left(1-{\displaystyle \frac{x}{L}}\right)$$

(19)

The deformation of the suspension cable consists of both geometric effects and material elongation. The geometric effect arises from changes in curvature that alter the arc length of the cable, while the material elongation results from elastic stretching induced by variations in axial force.

Under the condition that both ends of the cable are anchored and the horizontal projection length remains unchanged, the deformation compatibility equation is given as

$${\displaystyle {\int }_0^L\frac{1}{2}{\left(\frac{dy}{dx}\right)}^{2}dx=\frac{H-H_0}{EA}}L$$

(20)

where E is the elastic modulus of the suspension cable, A is its cross-sectional area, H₀ is the initial horizontal tension, and H is the horizontal component of the cable force after deformation.

By substituting Equation (15) into Equation (16), the incremental increase in geometric arc length can be obtained as

$${\displaystyle {\int }_0^L\frac{1}{2}{\left(\frac{dy}{dx}\right)}^{2}dx=}{\displaystyle {\int }_0^L\frac{1}{2}{\left(\frac{dy}{dx}\right)}^{2}dx=}{\displaystyle \frac{8{w_{\max }}^2}{3L}}$$

(21)

The deformation compatibility equation of the suspension cable can be rewritten as:

$$H=H_0+{\displaystyle \frac{8EA}{3L^2}}{w_{\max }}^2$$

(22)

By combining Equation (18) with Equation (14), the balance equation is obtained as

$$k{w_{\max }}^3+H_0{w}_{\max }-{\displaystyle \frac{q{L}^2}{8}}=0$$

(23)

where k = 8EA/3L², H₀ = σ₀A.

Define W = qL²/8; normalizing Equation (19) then yields

$${w_{\max }}^3+m{w}_{\max }+n=0$$

(24)

where m = H₀/k and n = W/k.

Let

$$w_{\max }=2\sqrt{{\displaystyle \frac{m}{3}}}\sinh u$$

(25)

Using the identity sinh(3u) = 3sinh(u) + 4sinh³(u), substituting Equation (21) into Equation (20) simplifies the expression to

$$\sinh \left(3u\right)={\displaystyle \frac{3\sqrt{3}n}{2\sqrt{m^3}}}$$

(26)

By expressing the inverse function explicitly and restoring the original parameters, the maximum mid-span deflection w_max is expressed as

$$w_{\max }=2\sqrt{{\displaystyle \frac{H_0}{3k}}}\sinh \left[{\displaystyle \frac{1}{3}}\ln \left({\displaystyle \frac{3\sqrt{3}W\sqrt{k}}{2\sqrt{{H_0}^3}}}+\sqrt{{\left(1+{\displaystyle \frac{3\sqrt{3}W\sqrt{k}}{2\sqrt{{H_0}^3}}}\right)}^2}\right)\right]$$

(27)

2.3. Model Validation and Theoretical Consistency

To validate the accuracy of the numerical simulation of the flexible PV support system, an independent single-cable numerical model was established. The theoretical solution is derived from finite-displacement theory for suspension cables under uniformly distributed loading, explicitly accounting for initial prestress, span length, cable length, and elastic modulus. When the cumulative length of PV panels mounted on a single-span cable is approximately equal to the span length, the panel loads can be reasonably idealized as a uniformly distributed load [33]. Under identical external loading, Figure 5 compares the numerical and theoretical results of single-cable deformation at three initial prestress levels (200 MPa, 400 MPa, and 800 MPa). The numerical and theoretical results exhibit good agreement, with the maximum deviation below 5%. This discrepancy primarily arises from finite element discretization and the omission of higher-order geometric nonlinear terms in the analytical solution. As the initial prestress increases, the maximum cable deflection decreases and the curve flattens, consistent with the theoretical prediction that the increase in horizontal tension suppresses the vertical displacement of the cable. These results demonstrate that the developed numerical model can accurately capture the deformation behavior of suspension cables subjected to uniformly distributed loads.

3. Static Response of the Flexible PV Support System

This study investigated the static response of the flexible PV support system. Compared with rigid support, PV modules mounted on flexible structures endure larger displacements and impose higher structural demands [34,35]. Specifically, flexible PV supports sustain long-term environmental loads such as mountainous wind and snow; hence, an in-depth understanding of their static response characteristics is essential.

In conventional rigid PV support structures, wind resistance is typically governed by the strength and stiffness of columns, beams, and foundations, and excessive wind loads are primarily limited by member strength or foundation capacity. In contrast, for cable-supported flexible PV support systems, wind-induced responses are more closely associated with large deformation, displacement compatibility, and cable tension variation, rather than strength failure alone. Therefore, the admissible wind performance of flexible PV supports is controlled by deformation and serviceability considerations, which differs fundamentally from that of rigid support systems.

3.1. Stress and Deformation Under Static Loads

3.1.1. Stress Analysis of Flexible PV Support System

Under wind, snow, and panel’s self-weight loads, the overall stress in the high-position group (ES-5–8) is greater than that in the low-position group (ES-1–4), as shown in Figure 6. The high posts provide a base maximum-rapid attenuation along height and experience compression and bending effects, producing high stress concentrations at the upper ends where they connect to stabilizing rods and diagonal braces. In contrast, the low posts experience axial tension and uplift, with peak stress occurring at the column bases, indicating that the required tensile strength and peak stress are higher in the high-position group. The transverse connecting rods display a concentrated stress distribution at the weak ends at mid-span, with maximum stress at the end nodes where they intersect the high and low posts. This reflects their principal role in transferring horizontal shear and local bending moments between the cables and posts, while the mid-span exhibits relatively low stress to maintain the structure’s overall lateral stability.

In mountainous terrain, the mechanical performance of edge supports in flexible PV systems significantly depends on the elevation differences across the array. To quantify this effect, a diagonal-rod activation index (β_d) is introduced, defined as the ratio of the absolute mean stress in the forward diagonal rods to the sum of the absolute mean stresses in forward and backward diagonal rods (Equation (26)). A β_d approaching unity indicates that the forward diagonal rods dominate the load-bearing contribution. For ES-1–8, a diagonal-rod activation index approaches 1, indicating that the system operates in a single-diagonal load-bearing mode dominated by the forward diagonal rods. This irregularity occurs because the wind direction increases the vertical load component along the forward diagonal rods, thereby enhancing their tensile response while simultaneously decreasing the contribution of the backward diagonal rods. Consequently, the supports at high-position edge experience greater stress concentrations at critical nodes and sustain a larger structural resistance.

$${\beta }_d={\displaystyle \frac{\left|{\sigma }_{t+}\right|}{\left|{\sigma }_{t+}\right|+\left|{\sigma }_{t-}\right|}}\in \left(0,1\right)$$

(28)

Under static loading, the peak stress of intermediate support varies with elevation, but the overall stress patterns remain consistent. Figure 7 shows the stress contour plots of the intermediate supports, indicating a node-dominated stress distribution. It can be observed that stress concentrated at the connection nodes is significantly lower along the member mid-spans. This pattern develops because undulating terrain increases the inclination of the suspension cables, thereby enhancing the vertical force components transferred to the joints and inducing larger bending moments and shear forces at the nodes. In contrast, mid-span segments mainly carry axial force and experience lower stress. Terrain undulation increases the inclination of the suspension cables, thereby increasing the vertical force components and bending moments at the nodes, demonstrating higher peak stress in the elevated group. Overall, the stress distribution dominates at the knee nodes, defined as the intersection of the post, diagonal brace, and strut. For example, in IS-1, the post exhibits a peak stress of approximately 134 MPa at the knee node, which decreases with the column height. This reflects compression-bending behavior governed by eccentric reaction and horizontal force components. The inclined beam develops shear-bending–torsion coupling peaks at both ends, with attenuation toward the mid-span. Both ends of the diagonal braces also exhibit stress concentrations, with the longer brace reaching around 142 MPa compared to approximately 123 MPa for the shorter brace, indicating axial compression-bending behavior and stress sensitivity to end restraints and initial imperfections existing within the structure. The strut primarily experiences axial force, with higher stress at its ends (a maximum of about 73.2 MPa) than at the mid-span. Stress peaks concentrate at the knee nodes, whereas displacement peaks occur at the ends of inclined beams and mid-sections of slender members. This reveals a spatial mismatch: nodes exhibit high stiffness, small displacements, and high stress, while spans show low stiffness, large displacements, and lower stresses. Such findings indicate that the performance of the nodes primarily governs structural safety.

3.1.2. Deformation of the Flexible PV Support System

This study analyzed the displacement of suspension cables of the flexible PV support system. Cables are highly flexible axially loaded components; therefore, they determine the serviceability of PV panels. In contrast, the supporting members exhibit localized stiffness concentrations under static loading, resulting in small displacements but high stress; therefore, these are considered more appropriate for stress analysis. Under combined static loading, Figure 8 shows the displacement field of the support system, where the cables form smooth deflection curves along the terrain slope. Terrain undulations cause an uneven distribution of deflection across spans, with slightly larger displacements in the end spans than in the interior spans. Notably, the knee nodes do not indicate the peak displacement; however, they sustain the highest stress, indicating a spatial mismatch between displacement peaks and stress peaks. From an engineering perspective, serviceability depends on the mid-span deflection of the main spans. At the same time, the end nodes of the transverse connecting rods at the edge supports and the knee nodes of the intermediate supports provide additional strength and prevent failure.

Figure 9 shows the vertical mid-span deflections of the paired suspension cables (upper and lower) across all spans, indicating that the deflection of the upper cable is consistently greater than that of the lower one. This mechanism can be described by the simplified parabolic cable theory and load distribution, where the mid-span deflection approximately satisfies the following expression:

$${u_y}^{\max }\approx {\displaystyle \frac{q{L}^2}{8H}}$$

(29)

where q is the equivalent uniformly distributed load acting on the suspension cable, and H is the prestressing force. When the tilted panels experience wind pressure in the forward direction, the combined effects of panel self-weight, snow load, and normal wind load generate a resultant moment about the lower edge. It should be noted that, in the present study, PV modules are treated as non-load-bearing components from a structural perspective. The PV panels do not provide compression resistance or contribute to the global stiffness of the support system; instead, they act solely as sources of external loads, including self-weight, snow load, and wind pressure, which are transferred to the cable-supported structure through the mounting connections. This results in a higher equivalent vertical load on the upper cable than on the lower one. Suppose the initial horizontal forces in the two cables are comparable. Once the relationship f_upper/f_lower ≈ q_upper/q_lower > 1 is satisfied, it follows the stable “upper greater than lower” pattern, as shown in Figure 8. In addition, elevation differences among supports alter the vertical force components at the nodes and the cable incidence angle, thereby increasing the vertical load on the upper cable in higher-elevation spans and further amplifying its deflection. Considering span-to-span variation, the end spans exhibit relatively larger deflections due to fewer boundary constraints and lower effective horizontal stiffness. Figure 8 shows a pronounced end-span deformation.

The observed mismatch between displacement and stress peaks helps control displacement primarily from the cable side. This may include increasing the prestress of the upper cable, optimizing the panel tilt angle and load-sharing width, introducing counter-tensioning cables within the span, or adding mid-span cross ties. In contrast, stress control must be addressed on node detailing and end restraints, such as enhancing nodal rotational stiffness, incorporating stiffeners and rounded transitions, applying prestress to long diagonal braces, or introducing intermediate lateral restraints.

3.2. Load Sensitivity Analysis of the Flexible PV Support System

Sensitivity analyses were performed for wind, snow, PV module weight, and temperature. From Figure 10, it can be observed that the mid-span deflection increases with each of these factors and exhibits mild nonlinearity. However, incremental slopes yield a different ranking from the original values; for example, temperature and PV module weight show the steepest slopes (highest sensitivity), wind exhibits moderate sensitivity, and snow shows the least sensitive factor. Notably, temperature modifies cable tension through thermal expansion, rapidly changing effective stiffness and deflection. PV module weight acts as a sustained vertical load, thereby increasing sag in the flexible system. In contrast, wind includes horizontal components that increase axial force and partially stiffen the structure, yielding a moderate slope. At the same time, additional snow acts as a vertically distributed load, thereby inducing comparatively minor incremental deflection due to the associated increase in horizontal cable force enhances geometric stiffness with the sag.

Stress Sensitivity Analysis

Using the same approach, the study analyzed the effects of wind, snow, PV-module weight, and temperature on structural stress. Each factor’s intensity increased from 0% to 100%, and stress responses were recorded at critical locations. Figure 11 shows that wind, snow, and PV-module weight cause the maximum von Mises stress with increased load intensity, whereas the temperature curve decreases with increasing temperature. The relative changes indicate that temperature and PV-module weight exhibit the strongest sensitivity, wind is moderately sensitive, and snow is the least sensitive over the full range.

The negative temperature trend occurs due to thermal expansion of prestressed cables. As the temperature rises, the cable length increases, and the structure accommodates this increase by reducing horizontal cable force (prestress) and increasing sag rather than developing compressive thermal stress. Relaxing cable tension reduces the axial forces transmitted to joints and adjacent members, thereby decreasing maximum von Mises stress at critical nodes. Conversely, at lower temperatures, cable contraction increases effective prestress, raising stress at key regions. PV module’s weight is a relatively significant factor, increasing linear stress by directly increasing the vertical component carried by cables and nodes. Wind produces a smaller incremental effect because lateral components redistribute axial forces and partially stiffen the system. Snow, modeled as an additional vertically distributed load, yields the least incremental stress growth in this configuration.

4. Dynamic Response of the Flexible PV Support System

This study investigated the PV support system’s dynamic response under external dynamic loading. Analyzing the dynamic behavior of such flexible structures can predict potential structural damage and provide a theoretical foundation for optimizing flexible PV support systems.

4.1. Application of Dynamic Wind Loads

The dynamic wind loads acting on PV panels are nonuniform because the load distribution coefficients vary across discrete panel locations. To more accurately capture this effect, this study used standard local shape coefficients [36,37] to calculate wind load on individual PV panels. The calculated loads were distributed equally to the upper and lower suspension cables, ensuring the rationality of wind load transfer and the accuracy of the numerical simulation. The gust factor, wind pressure coefficients, and the resultant wind loads on the cables are listed in

Table 2. Calculation results of loads on the cables.

Gust factor at height z	1.6
Parameter	Near End		Middle				Far End
Simplified Wind Load Coefficient	−1	−1.7		−1.5		−2.3		−2.2
Height Adjustment Factor for Wind Pressure	1.3
Reference Wind Pressure (kN/m2)	w
System Component	Lower cable				Upper cable
Resultant Wind Load (N)	FLower				FUpper

.

4.2. Influence of Wind Load on Dynamic Cable Response

4.2.1. Effect of Wind-Load Amplitude

An explicit dynamic analysis was performed to compare the response of flexible PV support under wind loads with amplitudes of 1 × A and 2 × A. The wind action was applied using a trapezoidal time-history: 0–1 s linear ramp-up, 1–3 s constant amplitude, and 3–4 s linear ramp-down (Figure 12). Figure 13 shows vertical deflection contours at selected times for the two amplitudes. At 1 s, once the load reached the plateau, a pronounced sagging shape was formed; end-span deflections were clearly larger than the interior spans, indicating a boundary-dominated flexibility. At 3 s, under sustained constant loading, peak deflections increased further, and the initial asymmetric profile evolved into a nearly single-peak, quasi-symmetric profile. Comparing amplitudes shows that the 2 × A case yields sub-linear growth: the deflection amplitude relative to 1 × A is consistently less than 2, with sublinearity more pronounced at the end spans. After unloading at 4 s, the system exhibits significant rebound with negligible residual deformation.

Figure 13 shows that the flexible PV support comprises eight spans. Span 1 is the left-end span. Spans 2 and 3 are interior spans adjacent to the end. Spans 4 to 7 are the remaining interior spans, which function like spans 2 and 3. Span 8 is the right-end span, which functions like span 1. To quantify amplitude effects, this study defined the following indices (evaluated at the same span, time, and point):

$$k={\displaystyle \frac{\left|w_{2A}\right|}{\left|w_{1A}\right|}}$$

(30)

where k is the amplitude-to-response amplification factor, which quantifies the increase in vertical deflection resulting from a change in wind-load amplitude. w is the vertical deflection, defined as the peak deflection of the span at a given time (

Table 3. Mid-span peak deflections and amplification factors.

Time	Span Number	Lower Cable-1 × A	Lower Cable-2 × A	k	Upper Cable-1 × A	Upper Cable-2 × A	k
1 s	Span 1	0.975	1.159	1.19	1.481	1.973	1.33
	Span 2	0.555	0.772	1.39	0.716	1.004	1.40
	Span 3	0.506	0.734	1.45	0.683	0.987	1.45
3 s	Span 1	0.851	1.124	1.32	1.409	1.885	1.34
	Span 2	0.586	0.784	1.34	0.809	1.100	1.36
	Span 3	0.712	0.685	0.96	0.506	0.895	1.77

).

$$\rho ={\displaystyle \frac{\left|w_{upper}\right|}{\left|w_{lower}\right|}}$$

(31)

ρ is the upper-to-lower deflection ratio, which characterizes the relative magnitude of deflection between the upper and lower suspension cables at the same span, time, and location. Here, w_upper and w_lowr are the deflections of the upper and lower suspension cables, respectively, under identical span, time, and other conditions (

Table 4. Upper-to-lower deflection ratio.

Parameters	1 × A (1 s)	2 × A (1 s)	1 × A (3 s)	2 × A (3 s)
ρ1	1.52	1.70	1.66	1.68
ρ2	1.29	1.30	1.38	1.40
ρ3	1.35	1.34	0.71	1.31

).

$${\chi }_{edge}={\displaystyle \frac{\left|w_1\right|}{\left(\left|w_2\right|+\left|w_3\right|\right)/2}}$$

(32)

χ_edge is the edge-span amplification ratio, which characterizes the relative disadvantage of the end span compared to the average of interior spans 2 and 3. Here, w₁, w₂, and w₃ are the deflections of the end span (left-edge) and the two representative interior spans, respectively (

Table 5. Edge-span amplification ratio.

Parameters	1 × A (1 s)	2 × A (1 s)	1 × A (3 s)	2 × A (3 s)
${\chi }_{\mathrm{edge}}$ lower	1.84	1.54	1.31	1.53
${\chi }_{\mathrm{edge}}$ upper	2.12	1.98	2.14	1.89

).

Figure 14 shows the transverse deflection curves of spans 1 to 3 at 1 s and 3 s. At 1 s, the peak deflections of the end span (span 1) are 0.975 m (1 × A) and 1.159 m (2 × A) for the lower cable (k = 1.19), and 1.481 m (1 × A) and 1.973 m (2 × A) for the upper cable (k = 1.33). For the interior spans2 and 3, k₂ ranges from 1.39 to 1.45, indicating values slightly higher than the end span but still below 2. The upper-to-lower deflection ratio $\rho$ at the end span reaches 1.52 (1 × A) and 1.70 (2 × A), whereas the interior spans remain around 1.29 to 1.35. This verifies that under identical prestress, the upper cable consistently exhibits greater deflection than the lower one, and the disparity becomes more pronounced at the end span as load amplitude increases. Considering the peak values, the edge-span amplification ratio χ_edge is 1.84/1.54 for the lower cable and 2.12/1.98 for the upper cable, highlighting the poor performance of the end span. At 3 s, the deflection curves further stabilize, with the peak positions approximately approaching mid-span. At this stage, k for span 1 is 1.32 (lower cable) and 1.34 (upper cable), while interior spans mostly fall between 1.34 and 1.36. The upper-to-lower ratio $\rho$ decreases slightly compared with 1 s, exhibiting 1.66 (1 × A) and 1.68 (2 × A) at the end span, and 1.31–1.40 for the interior spans. The edge-span amplification ratio remains greater than unity. Still, it shows a modest reduction relative to 1 s (lower and upper cables ratios are 1.31/1.53 and 2.14/1.89, respectively), reflecting a proportionally stronger response of the interior spans at higher amplitudes. Notably, for span 3’s lower cable at 3 s, a slight peak-position drift combined with geometric stiffening produces an apparent k as 0.96; however, when compared at the exact in-span location, the expected trend of k > 1 is still satisfied.

Figure 15 shows that when the wind-load amplitude increases from 1 × A to 2 × A, the mean stress during the plateau stage (1–3 s) rises from 230.94 MPa to 436.39 MPa, indicating an 89.0% increase and a sublinear amplification. The corresponding peak stress within the plateau increases from 259.21 MPa (at 2 s) to 476.93 MPa (at 1.5 s), with an 84.0% increment. In comparison, the fluctuation amplitude (maximum and minimum values during the plateau) increases from 57.06 MPa to 70.58 MPa, indicating a 23.7% rise. Under the same loading condition, the plateau mean stress at IS-2 is about 38–40% higher than that at IS-1, indicating that the knee node at IS-2 is consistently less favorable. Overall, load amplitude emerges as the dominant factor governing structural stability. When the amplitude increases from 1 × A to 2 × A, stress at critical nodes rises prominently, accompanied by geometric stiffening and partial energy dissipation within the structure. It should be noted that the stress levels reported herein correspond to the dynamic stress response under amplified wind loading conditions and remain below the admissible stress limits of the structural steel cables considered in this study. Although increased load amplitude leads to a pronounced rise in stress at critical nodes, no immediate material yielding or structural failure is indicated within the investigated loading range. Nevertheless, the observed stress amplification highlights the potential risk of serviceability degradation and long-term fatigue damage under sustained or extreme wind excitation.

An increase in load amplitude results in larger displacements; however, the overall structural response remains sub-linear. This occurs because greater deflection increases the cables’ axial force, leading to geometric stiffening, while inherent energy dissipation constrains displacement growth. Under initial prestress conditions, the upper cable exhibits greater deflection than the lower cable. This difference depends on the load path and effective vertical stiffness, which tends to stabilize as loading increases. The end spans are more susceptible due to weaker one-sided boundary restraints and reduced effective horizontal stiffness. With increasing amplitude and sustained loading, the relative response of interior spans becomes more pronounced, slightly reducing the difference between end and interior spans. From a design perspective, it is essential to prioritize control of displacement and stress limits in the end spans and the upper cable. Effective mitigation strategies include increasing end-support stiffness, optimizing the upper cable’s incidence angle, and moderately increasing prestress, thereby reducing peak deflections and moderating differential responses.

4.2.2. Effect of Wind-Load Frequency

This study investigated the influence of dynamic wind-load frequency on the structural response of the flexible PV support system. Two sets of equal-amplitude with different-frequency load time histories were constructed, corresponding to the fundamental frequency f (denoted as 1 × f) and its first harmonic (2 × f). Figure 16 shows their corresponding time-history curves. In both cases, the load amplitude and plateau duration ratio were identical, and the loading frequency varied. Subsequently, a transient dynamic analysis was performed to evaluate the system’s global deformation and the transverse deflection distribution of individual spans at representative time instants. Here, the end span (span 1) and an interior span (span 2) were selected as representative cases.

Figure 17 shows that when the amplitude remains constant, increasing frequency moderately reduces the stress at critical nodes, indicating the mean plateau stress changes from 230.94 MPa to 217.56 MPa (−5.8%), the peak stress reduces from 259.21 MPa (at 2 s) to 242.66 MPa (at 1.5 s) (−6.4%), and the fluctuation amplitude decreases from 57.06 MPa to 39.99 MPa (−30.0%). In both frequency cases, IS-2 remains higher than IS-1, indicating that frequency is a secondary modulation factor. While higher frequency lowers the mean and peak stress at the knee node and further decreases them during the unloading stage, the potential adverse impact of increased cycle frequency on fatigue life under long-term wind action remains. Therefore, structural design and verification must primarily focus on amplitude control, with frequency effects treated as supplementary. Under the 1 × f condition (Figure 18a), the structure exhibits overall sagging during the plateau stage, with end-span displacements significantly larger than the interior spans, while displacements gradually recover during unloading. Under the 2 × f condition (Figure 18b), the overall deflection amplitude decreases, and the recovery of structural displacement becomes faster; however, localized deformation in the interior spans becomes more evident during mid-loading. In both cases, a slight torsional response appears, indicating an asynchronous displacement between the upper and lower suspension cables. This is more noticeable at mid-loading under the 2 × f condition, where the upper cable rebounds earlier than the lower cable, producing downwind torsional deformation of the span’s cross-section. These findings suggest that higher-frequency loads contain stronger higher-order harmonic components, which couple the system’s torsional modes and amplify the torsional response. These findings suggest that higher-frequency loads contain stronger higher-order harmonic components, which couple the system’s torsional modes and amplify the torsional response.

Table 6. Comparison of span deflection extrema at representative stages under two loading frequencies.

Stage	Span 1 (1 × f)	Span 1 (2 × f)	Rate of Change	Span 2 (1 × f)	Span2 (2 × f)	Rate of Change
early loading	1.506	1.482	−1.6%	0.716	0.641	−10.5%
mid-loading	1.439	1.349	−6.3%	0.815	0.942	+15.6%
unloading	0.468	0.292	−37.6%	0.315	0.255	−19.0%

compares the overall span deflections under the two loading frequencies, i.e., the peak vertical displacements of the end span 1 and the interior span 2. For consistency, the time instants of 1 s, 3 s, and 4 s in the 1 × f case were aligned with 0.5 s, 1.5 s, and 2 s in the 2 × f case, corresponding to the early loading stage, mid-loading stage, and early unloading stage, respectively. Under the 2 × f condition, the displacement extrema of span 1 were smaller than those under the 1 × f condition, with a remarkable reduction occurring in the early unloading stage, decreasing by 37.6%. This demonstrates that higher-frequency loading can effectively suppress end-span deflections. In contrast, span 2 exhibited a 15.6% increase in displacement during the mid-loading, indicating that higher-frequency excitation triggers localized deformation in interior spans. The comparison reveals that during unloading, the decrease in displacement in both spans was significantly higher under the 2 × f condition. This indicates that increased loading frequency decreases structural response, suggesting that high-frequency wind loads more rapidly alter the system’s dynamic equilibrium and promote faster energy dissipation.

Variations in the dynamic frequency of wind loads significantly affect the coupling between wind load energy and the structural modes. Under the 1 × f condition, the response primarily depends on low-order bending modes, resulting in larger displacements at the end spans. When the frequency increases to 2 × f, harmonic components of the load become more prominent, exciting higher-order modes, particularly those involving torsional effects. Therefore, the low-order bending response of the end spans gradually reduces, while higher-order responses in the interior spans and local connection regions are amplified. In this case, the overall structural deflection decreases, whereas localized torsional effects intensify, indicating that high-frequency loading exerts a greater influence on the system’s local dynamic characteristics.

5. Discussion

This study systematically investigated the deformation behavior of flexible photovoltaic (PV) support systems by integrating theoretical derivation with high-fidelity numerical simulations. The key findings of this study are as follows:

First, a pronounced spatial mismatch between displacement peaks and stress concentrations was identified. While serviceability is primarily affected by mid-span deflections of suspension cables, structural safety relies on the stress concentrations at nodal regions. This behavior contrasts with conventional rigid PV support, where displacement and stress distributions typically coincide. These findings align with published studies on cable-truss structures, highlighting the importance of nodal detailing and prestress optimization in meeting global stability.

Second, under dynamic wind loading, flexible PV supports exhibit distinct nonlinear responses. The transition from a transient double-peak to a stable single-peak deflection, and the consistently larger deflections of upper cables relative to lower ones, reflect the inherent asymmetry of load transfer. A similar asymmetry has been reported in published wind-tunnel studies of flexible PV systems, where panel tilt and wind incidence angles greatly influenced aerodynamic loading. The results obtained in this study clarify the mechanistic role of upper-lower cable interaction in dual-cable configurations, a topic rarely addressed in existing research.

Third, the comparative analysis of wind and snow loads highlights the importance of horizontal loads in inducing the structural deformation of flexible PV systems. Unlike rigid systems, where vertical loads typically control design, flexible systems redistribute forces so that snow loads primarily amplify deflection through geometric nonlinearity. In contrast, wind loads induce a self-stabilizing effect through axial force redistribution. This distinction emphasizes the need for wind-resistant design of flexible PV systems deployed in mountainous or coastal terrains.

These findings significantly impact the deployment of solar power infrastructure in complex environments. Flexible PV support offers a cost-effective alternative where rigid support is impractical. By lowering foundation requirements and enabling long-span deployment, they expand solar photovoltaics onto underused lands, boosting the scalability of renewable energy. However, this study’s results also highlight the need for design methodologies that incorporate nonlinear dynamic effects, nodal fatigue resistance, and asymmetric load sharing.

Future research must focus on the full-scale experimental validation to capture environmental complexities such as turbulence, temperature variation, and multidirectional wind. Moreover, optimization frameworks must integrate structural mechanics with aerodynamic parameters to determine optimal prestress levels, panel tilt angles, and cable spacing. The integration of flexible supports with emerging bifacial and tracking PV modules offers a promising solution, as these technologies may adjust load distributions and dynamic responses. Advancing these research areas will enable flexible PV support systems to mature into a technology capable of aiding the global shift to large-scale renewable energy deployment. Future work will focus on integrating fatigue damage models with the present numerical framework to enable fatigue life prediction of cable-supported flexible PV support systems under long-term cyclic wind loading.

6. Conclusions

In this work, to address the deformation analysis of the flexible photovoltaic support system, a finite element model is developed using an in-housing Python code to couple the degree of freedom of the beam and the cable element. The proposed numerical model is well validated by the comparison results with the theoretical solution. Based on this numerical model, the mechanical response and deformation performance of the flexible photovoltaic support structure are simulated under different loading conditions. The main findings and optimization suggestions are summarized as follows:

(1) The numerical results indicate that the maximum displacement and strain occur in the prestressed cables, implying that the global deformation of the flexible support structure is dominated by the tensile behavior of the cables.

(2) The results demonstrate that the deformation behavior of the flexible photovoltaic support system is predominantly dominated by horizontal loads, such as wind, rather than vertical loads. This finding indicates that the flexible support structure behaves as a horizontally load-controlled system.

(3) The dynamic analysis reveals that while increasing the amplitude of the dynamic load merely enlarges the displacement response, an increase in loading frequency induces torsional behavior of the prestressed cables, which may elevate the risk of hidden cracks in photovoltaic panels. Thus, enhancing the stiffness of the mid-span stabilization system is therefore recommended to suppress such torsional effects.

(4) The stress analysis under quasi-static and dynamic loading conditions reveals that stress concentration consistently occurs at the connection points of the lower support system. This finding suggests that enhancing the strength and stiffness of the joints is essential to mitigate fatigue and creep failures of the flexible photovoltaic structure over its long-term service life.

In addition, extending the proposed modeling framework to incorporate fatigue damage accumulation and fatigue life prediction under long-term environmental loading is identified as an important direction for future research.