Strain Analysis of Membrane Structures for Photovoltaic Integration in Built Environment

Abstract

The integration of photovoltaic (PV) systems into tensioned membrane structures presents a significant advancement for sustainable applications in the built environment. However, a critical technical challenge remains in the substantial strains induced by external loads, which can compromise both PV efficiency and the structural integrity of the membrane. Current design methodologies prioritize stress, deflection, and ponding analysis of tensioned membranes. Strain behavior of whole structures, a key factor for ensuring long-term performance and compatibility of PV-integrated membranes, has been largely overlooked. This study addresses this gap by examining the whole membrane structure designed for PV integration, with the aim of optimizing the membrane to provide suitable conditions for efficient energy transfer while minimizing membrane strains. For this purpose, it provides a comprehensive strain analysis for full-scale hyperbolic paraboloid (hypar) membrane structures under various design parameters and external loads. Employing the Finite Element Method (FEM) via Sofistik software, the research examines the relationship between load type, geometry, material properties, and patterning direction of membranes to understand their performance under operational conditions. The findings reveal that strain behavior in tensioned membrane structures is strictly influenced by these parameters. Wind loads generate significantly higher strain values compared to snow loads, with positive strains nearly doubling and negative strains tripling in some configurations. Larger structure sizes and increased curvature amplify strain magnitudes, particularly in parallel patterning, whereas diagonal patterning consistently reduces strain levels. High tensile-strength materials and optimized prestress further reduce strains, although edge type has minimal influence. By systematically analyzing these aspects, this study provides practical design guidelines for enhancing the structural and operational efficiency of PV-integrated tensioned membrane structures in the built environment.

1. Introduction

Creating a sustainable future for communities is a critical priority in addressing the global challenges of decarbonization and optimization of resources. One of the most promising strategies for achieving sustainability is the integration of renewable energy technologies (RET) into the built environment, an approach that aligns with the Sustainable Development Goals (SDGs) of the United Nations, particularly SDG 7 (Affordable and Clean Energy) and SDG 11 (Sustainable Cities and Communities) [1]. Among RETs, photovoltaic (PV) systems have gained particular attention in building integration due to their advantages in sustainability, energy generation, flexibility, aesthetic appeal, lightness, and adaptability across diverse applications [2]. The evolution of PV technology progressed beyond conventional systems to include lightweight, flexible, and customizable solutions, opening new opportunities for integration with non-traditional surfaces [3]. Initially dominated by silicon-based solar cells (first generation), PV systems evolved to incorporate thin-film technologies (second generation), such as amorphous silicon (a-Si), cadmium telluride (CdTe), and copper indium gallium selenide (CIGS) [2]. These systems offered reduced material usage and enhanced adaptability to non-traditional surfaces. The third generation is based on a range of advanced technologies, such as dye-sensitized solar cells (DSSCs), organic photovoltaic (OPV) cells, quantum dot solar cells, and perovskite solar cells. They introduced innovation in flexibility, transparency, colors, patterns, and lightweight design. Among them, perovskite and OPV cells are particularly suited for applications in tensioned membrane structures due to their tunable properties and compatibility with flexible substrates [3].

The idea of applying PV to structures appeared in the previous century. Its development is closely related to the improvements in the PV technology. Rigid and heavy PV panels could only be attached to rigid and heavy buildings. Thus, the application of PV systems in the built environment has predominantly focused on heavyweight systems, typically categorized as Building Applied Photovoltaics (BAPV) and Building Integrated Photovoltaics (BIPV) [4]. While these approaches address the integration of solar energy into architecture [5,6], they primarily rely on rigid, heavyweight technologies, limiting their suitability for geometrically complex or delicate structures [7]. The development of flexible PV technologies has significantly expanded the possibilities for the integration of PV systems in architecture due to their ability to conform to non-linear geometries and their lightweight construction, which reduces the load on supporting structures [4,8]. Flexible and lightweight PV technologies have expanded possibilities for architectural integration, particularly in preserving the architectural value of retrofit projects through non-invasive and adaptive designs [5]. The attached PV was a large step forward, but the real integration was achieved when membrane-printed PV were created. Despite advancements in parametric modeling, prototype development, and performance evaluation of flexible PV membranes, their structural integration into tensioned membranes remains underexplored [2].

The critical gap lies in the lack of strain-focused analyses for double-curved, tensioned membranes. Unlike conventional structures, which are often produced through mass customization, PV tensioned membranes exhibit unique mechanical behavior that requires a fully customized approach tailored to the specific project [9]. Their properties result in significant strain variations, which can negatively impact the energy efficiency of even flexible PV systems. Research in this area has primarily focused on PV surface-mounted systems or small-scale samples, neglecting the holistic performance of fully integrated systems under real-world conditions [4]. Furthermore, studies often focus on the technological optimization of PV materials rather than adapting structural designs to accommodate existing PV systems. These aspects hold significant potential for the built environment, as fully integrated PV membranes offer a transformative opportunity by reducing material redundancy [10], improving structural efficiency, and creating multifunctional surfaces that generate energy while maintaining the lightweight and tensile properties characteristics of the membranes [11]. Previous studies have demonstrated the feasibility of integrating flexible PV materials [12], such as thin-film technologies, directly into membrane substrates, offering advantages in adaptability to curved and dynamic geometries. Research in this field has focused on the compatibility of PV designs with advanced parametric models to enhance the energy efficiency of BIPV tensile membrane structures [10]. Tools such as Grasshopper, ANSYS, EASY, and ABAQUS have been employed to analyze the structural and optical performance of these systems [9]. Similarly, prototypes of lightweight membranes incorporating OPV technologies have been developed [13], exploring fabrication, printing, and pattern designs [14]. A critical aspect of this research refers to the optimization of the shape of PV tensile membrane structures through advanced form-finding processes. Dynamic relaxation, finite element analysis [15], and force density techniques [16] have been applied to refine geometries for maximizing energy efficiency [17]. Additionally, dynamic structural analyses are used for evaluating the stability and adaptability of curved shapes in inflatable membrane structures. However, these systems are typically surface-mounted rather than fully integrated into the membrane’s structure. Despite these advancements, technical challenges such as mechanical strain, thermal expansion, and material compatibility persist. This gap underscores the need for innovative approaches that move beyond conventional applications, enabling PV systems to become intrinsic to structural design [18]. A need for flexible and lightweight solutions for energy improvement has already been noticed [19]. Studies into the possible integration of PV and membranes have been published. Structural behavior of woven membrane with PV has been analyzed [20]. Membrane-embedded PV systems were examined in the context of structural and electrical performance [21]. Testing of PV-integrated membrane samples showed the relation between the strain and the electrical efficiency [22]. Physical models of ETFE membrane with PV [23] have been used for several studies regarding thermal [24], structural [25], electrical [26], and photothermal performance [27]. A study on the feasibility of flexible PV emergency shelter has been carried out [28]. Life cycle cost analysis [29] and carbon footprint assessment [30] of non-double-curved membrane-integrated PV have been conducted. There have still not been any studies in which the whole PV-integrated textile membrane structure is examined. From the published studies, it can be concluded that PV-membrane integration is possible on a smaller scale.

The integration of PV systems into double-curved tensioned membranes is currently constrained by differences in their mechanical properties. Unlike conventional structural designs, membrane structures are lightweight, span large areas, and function as both load-bearing and covering elements [31]. Their design relies on tension to resist external loads. These unique mechanical characteristics [32] result in geometric changes and large strains, which are incompatible with the design of most PV systems, even flexible ones, as such strains reduce energy conversion efficiency [33]. Flexible PV technologies can tolerate bending but not the large strains inherent to membrane structures [34]. These strains can severely impact their efficiency, creating a critical challenge for successful integration [31]. One of the possible ways to facilitate the integration that has not been scientifically explored so far is the strain analysis of the membrane structures, with the aim of limiting their strains.

This study addresses these gaps by introducing strain analysis into the design methodology for PV-integrated tensioned membranes. Unlike traditional approaches, which emphasize stress, deflection, and ponding, this research prioritizes the relationships between strain behavior and structural parameters such as geometry, material selection, prestress, and patterning. By analyzing how design decisions influence strain under external loads, this research develops actionable guidelines for optimizing the integration of PV systems into tensile membranes. The study limits its scope to textile membrane structures, excluding foil membranes, and prioritizes a methodology that focuses on the relationships and trends among structural parameters. By understanding how design decisions influence strain behavior under external loads, the study aims to provide scientifically grounded recommendations that enable sustainable PV-membrane integration. This integration offers mutual benefits: PV systems benefit from large, adaptable application areas, while membranes access renewable energy for on-site use or grid transfer. Beyond the systems themselves, this research supports broader environmental goals (e.g., SDG 7 “Affordable and Clean Energy”, SDG 11 “Sustainable Cities and Communities”, SDG 9 “Industry, Innovation, and Infrastructure”, and SDG 13 “Climate Action”), contributing to global decarbonization and climate neutrality initiatives. By fostering the use of renewable energy and advancing decarbonization efforts in innovative architectural contexts, this research supports the practical implementation of climate neutrality design solutions for the built environment.

2. Materials and Methods

The research investigates strain behavior in tensioned membrane structures intended for integration with PV through numerical simulations using Finite Element Method (FEM) software SOFiSTiK 2023 to model tensioned membrane structures and analyze their mechanical properties under various load conditions. In the software, a geometrically non-linear analysis has been carried out. The membrane material has been modeled as linear elastic and orthotropic, according to [35]. Prestress can be defined along the two principal directions of the material and is applied during the form-finding process. Supports are rigid and fixed, except for edge cables. The connection between the cable and the membrane does not allow for the membrane to slip. Quadrilateral membrane finite elements are used for the membrane, and their size is approximately 0.10 × 0.10 m. The elements are formulated to handle in-plane tensile forces, which dominate in tensioned membrane structures, and are compatible with geometrically nonlinear analysis, capturing large deformations accurately. Temperature effects, material creep and aging, and dynamic effects are not considered. Connections between the pieces of the membrane are not modeled; therefore, the model consists of one piece of membrane. This is common practice in membrane structures research and analysis, as cutting patterns are defined at a later stage of development. The study uses a hyperbolic paraboloid (hypar) shape membrane as the base model, which is commonly used in tensile membrane structures. By exploring six key design parameters (size, curvature, material properties, prestress intensity, patterning, and edge types) [36] and their influence on structural strains, the research aims to provide actionable recommendations for reducing strain and ensuring compatibility with strain-sensitive PV systems. A range of values for each parameter is tested to identify trends in the relationships between these parameters and the resulting strains under load. This approach ensures that the limitations of current PV technologies, particularly regarding strains, are respected in the design of PV-integrated membrane structures. Additionally, the findings remain relevant as PV technologies advance and their mechanical properties improve.

Based on this methodology, the study is structured in the following sections:

• Numerical simulation that explains the use of FEM software (in SOFiSTiK 2023) for modeling membrane structures and their loads (Section 2.1).
• Model setup that provides details about the initial model (hypar-shaped membrane, dimensions, material properties) and explains the choice of parameters and applied loads (Section 2.2).
• Variable parameters that describe the six parameters analyzed and their relevance in influencing strain under load (Section 2.3).
• Simulation of multiple scenarios that analyzes a series of 88 different models, each with varying values for the six parameters (Section 2.4).

2.1. Numerical Simulation

As a first step in the analysis, an initial numerical model of the membrane structure is created. It is used to test the mesh convergence and as a base point for defining the variable parameters. The initial model is created with a hypar geometry, based on [37], with a square base measuring 6.00 × 6.00 m and a height of 2.00 m. The material properties include a 0.001 m thick membrane with an elastic modulus of 600.00 kN/m, a shear modulus of 30.00 kN/m in both principal directions, and a Poisson’s ratio of 0.4. The warp direction runs between the two elevated corners. The membrane is discretized into finite elements of approximately 0.10 × 0.10 m, and the mesh has been verified for convergence under applied loads. The membrane’s edges are reinforced with cables of 0.012 m diameter, prestressed to 30.00 kN, and made from steel with an elastic modulus of 205.00 kN/mm². The analysis was performed using the Direct Parallel Sparse Solver (PARDISO) in SOFiSTiK 2023. The nonlinear solver in SOFiSTiK was configured specifically for the analysis of tensioned membrane structures, accounting for their inherent geometric nonlinearity through a third-order theory. The solver was set with a maximum iteration limit of 90 iterations, ensuring it could handle the iterative nature of solving nonlinear equilibrium equations. To achieve a high level of accuracy, a tolerance of 0.001 was specified, requiring the residuals to fall below 0.1% of their initial values. These settings provided a robust framework for capturing the equilibrium states of the system under applied loading and large deformations. In practice, the solver consistently achieved convergence well within the defined iteration limits, demonstrating its reliability and accuracy in resolving the nonlinear behavior of the structure. The initial model is visualized in Figure 1.

2.2. Model Setup

In addition to self-weight and constant prestress, tensioned membrane structures are subjected to external loads. Snow and wind loads are the two most common external loads acting on these structures. While recent research has highlighted the importance of concentrated loads on membranes [38], these are not considered here, as concentrated loads (typically representing a person standing on the membrane) are not relevant for PV-integrated systems.

The snow load is defined with an intensity of 0.60 kN/m², applied uniformly across the membrane in a vertical downward direction. The wind load is modeled with an intensity of 1.00 kN/m², acting perpendicular to the membrane surface with uplift action. Both loads are treated as static and are applied independently. The definitions of these loads align with parameters used in previous research [39], without a specific location set for the structure.

2.3. Variable Parameters

Starting from the defined initial model, a set of parameters was identified for analysis. The values of these parameters were varied to investigate their influence on membrane strains under load. Six parameters have been proven to affect the behavior of tensile membrane structures and are thus expected to also influence strains. These parameters are:

• Size of the structure (Section 2.3.1).
• Curvature of the structure (Section 2.3.2).
• Membrane material properties (Section 2.3.3).
• Prestress intensity (Section 2.3.4).
• Membrane patterning direction (Section 2.3.5).
• Membrane edges (Section 2.3.6).

The first step in designing any membrane structure is usually choosing the general shape of the membrane. In this study, hypar membranes are analyzed, although other shapes are commonly used, e.g., cone membrane shapes. Once the shape is selected, several other parameters need to be specified for the membrane to be defined. All the important parameters have been set as the six variables in this study, with only a few defined with constant values. All these parameters are determined by the designer of the tensile membrane structure. Designers have the flexibility to adjust these parameters to meet structural, aesthetic, and functional requirements. This flexibility also provides an opportunity to fine-tune them to reduce membrane strains under load, making integration with strain-sensitive PV systems feasible. The precise relationship between these parameters and resulting strains remains insufficiently understood, forming the focus of this research.

2.3.1. Size of the Structure

The original size of the membrane was 6.00 × 6.00 × 2.00 m. In order to analyze the influence of size on strains, different sizes had to be analyzed. While membranes can span large areas, it is not likely that individual membranes with integrated PV will be extremely large, due to the strain limitations of PV. Therefore, the span of 10 m has been selected as the maximal in this research. At the same time, membranes smaller than 2 m would not provide a large area for integration of PV. Therefore, a range of usable and realistic values has been selected. It is important to note that the curvature of the membrane must not change so that the size parameter can be independently examined. Five different sizes of the model were defined, two smaller and two larger than the initial model. These are:

• S1: 2.00 × 2.00 × 0.67 m.
• S2: 4.00 × 4.00 × 1.33 m.
• S3: 6.00 × 6.00 × 2.00 m.
• S4: 8.00 × 8.00 × 2.67 m.
• S5: 10.00 × 10.00 × 3.33 m.

In addition to selecting equal steps of increase, these sizes correspond to real-life applications of tensile membrane (Figure 2).

2.3.2. Curvature of the Structure

The geometry of the membrane is a big factor in defining how it will react to external loading. There are two simple ways to change the curvature: to increase the size of the base of the structure or to increase the height of the structure. In this research, the second option is selected, as this is what designers of the structure usually prefer. The height of the original structure is 2.00 m. For the upper limit, half of the length of the base side is selected. Flat membranes are not considered here, as they are prone to ponding, so an incremental step size of 0.50 m and a minimal height of 0.50 m was selected for the analysis. Analyzed heights of the structure are:

• H1: 0.50 m.
• H2: 1.00 m.
• H3: 1.50 m.
• H4: 2.00 m.
• H5: 2.50 m.
• H6: 3.00 m.

All the structures have a base of 6.00 × 6.00 m. Visual representation is given in Figure 3.

2.3.3. Membrane Material Properties

The selection of membrane material for a specific structure depends on various factors, including expected lifespan, aesthetic considerations, cost, expected loads, and resistance to adverse effects. This aspect is one of the key outcomes of structural analysis. In numerical simulations, membrane materials are usually modeled as linear elastic and characterized by the following properties: elastic modulus in warp and weft direction, shear modulus, and Poisson’s coefficient. Based on the guidelines provided in [37], six material configurations were taken to cover the whole range of membrane materials used in practice. According to the same reference, the rule states that the shear modulus can be defined in relation to the elastic modulus, as 1/20 of the value of the elastic modulus. The calculation value of Poisson’s coefficient has a negligible impact on the simulation results and is therefore treated as a secondary consideration in this analysis. The following values for the membrane material are chosen for the analysis to represent a comprehensive range:

• M1: Elastic modulus 100.00 kN/m, shear modulus 5.00 kN/m, Poisson’s coefficient 0.4.
• M2: Elastic modulus 300.00 kN/m, shear modulus 15.00 kN/m, Poisson’s coefficient 0.4.
• M3: Elastic modulus 600.00 kN/m, shear modulus 30.00 kN/m, Poisson’s coefficient 0.4.
• M4: Elastic modulus 1000.00 kN/m, shear modulus 50.00 kN/m, Poisson’s coefficient 0.4.
• M5: Elastic modulus 2000.00 kN/m, shear modulus 100.00 kN/m, Poisson’s coefficient 0.4.
• M6: Elastic modulus 5000.00 kN/m, shear modulus 250.00 kN/m, Poisson’s coefficient 0.4.

The elastic modulus is considered to be the same in the warp and weft directions of the membrane in all cases.

2.3.4. Prestress Intensity

Tensioned membrane structures rely on maintaining tension within the membrane. This is achieved by applying prestress. Prestress introduces a controlled level of tension to the membrane, ensuring it remains in tension even when subjected to external loads. Its intensity varies depending on design requirements and is determined through structural analysis. Existing recommendations define minimum values [40,41]. Prestress can differ between warp and weft directions of the membrane or remain equal, depending on the material’s anisotropic or isotropic properties. In this analysis, they are assumed equal in both warp and weft directions, as the material used has identical properties in both directions. The initial model adopts a prestress value of 3.00 kN/m. To assess the effects of varying prestress levels, four additional values are selected for analysis. The recommended minimum prestress for tensioned membranes is 1.00 kN/m. For this study, an incremental step of 1.00 kN/m is selected. Therefore, the analyzed prestress values are:

• P1: 1.00 kN/m.
• P2: 2.00 kN/m.
• P3: 3.00 kN/m.
• P4: 4.00 kN/m.
• P5: 5.00 kN/m.

It is worth mentioning that the unproportional change in prestress between the warp and weft directions can alter the geometry of the membrane structure. In this study, such changes were intentionally avoided to exclude the influence of curvature on this phase of the analysis. By maintaining equal prestress in both directions, the analysis isolates the effects of prestress intensity without introducing geometric variations that could complicate the interpretation of results.

2.3.5. Membrane Patterning Direction

The issue of patterning is very specific for woven membrane structures due to their orthotropic behavior. This behavior depends on the construction of the membrane, where the fibers are arranged in two perpendicular directions. Thus, the mechanical properties of the membrane vary depending on the orientation. During fabrication, the flat membrane material is cut into strips and joined to create the desired double-curved membrane surface. The material is cut along the warp fibers; therefore, the “patterning direction” effectively defines the orientation of the warp fibers in the membrane. In square-based hyperbolic paraboloid (hypar) membranes, the warp direction can be aligned with either one of the diagonals or one of the sides of the structure. When the membrane material has the same properties in both warp and weft directions, and the prestress intensities are equal, the number of patterning directions is reduced to two. This is because a 90° rotation of the membrane does not produce any structural difference. These two patterning directions are analyzed in this research and shown in Figure 4. They are effectively called:

• DP: diagonal patterning.
• PP: parallel patterning.

Figure 4. Analyzed patterning directions: (a) diagonal patterning DP; (b) parallel patterning PP (source: Authors’ elaboration).

Figure 4. Analyzed patterning directions: (a) diagonal patterning DP; (b) parallel patterning PP (source: Authors’ elaboration).

2.3.6. Membrane Edges

One of the main advantages of tensioned membrane structures is their lightness, both in the structural and visual sense. To achieve this, prestressed cables are used as edge supports. However, there are cases where it is necessary to have straight edges of the structure, and this cannot be achieved by using cables. One of the possible examples is when a membrane needs to be attached to a neighboring structure or another membrane. As both edge types are possible and frequently used, it is decided to include both of them in the analysis. They are marked with:

• FE: flexible edges.
• RE: rigid edges.

The difference in models with two types of edges is given in Figure 5. Straight edges are modeled as rigid and fixed. Flexible cable edges are more complex as they affect the overall geometry of the structure. The prestress of the cables should be set in accordance with the prestress of the membrane and the size of the structure. The cable sag from the initial model was kept constant throughout the analysis. This effectively meant that the prestress intensity of edge cables had to be changed alongside the prestress of the membrane and the change in structure size.

2.4. Simulation of Multiple Scenarios

In this research, six variable parameters are defined, each with specific values selected for analysis. The values of these parameters are varied independently, with each variation based on an initial reference model. The parameters and their respective variations are as follows:

• Structure size: 5 analyzed values (Section 2.3.1).
• Height of the structure: 6 analyzed values (Section 2.3.2).
• Membrane material properties: 6 analyzed values (Section 2.3.3).
• Prestress intensity: 5 analyzed values (Section 2.3.4).

These variations result in a total of 22 unique models.

Additionally, two analyzed parameters—type of edges and patterning direction—do not have numerical values but instead feature two distinct configurations each. The combinations of these configurations produce 4 variations (2 edge types × 2 patterning directions). Each of the 22 base models is combined with all 4 configurations of the categorical parameters, resulting in a total of 88 unique models. For each of these models, two different load cases are applied, resulting in a total of 176 analyzed load cases on 88 different models (88 models × 2 load cases). The results are presented in the next section.

3. Results

All numerical models of membrane structures have been loaded as planned, and resulting strains were recorded. Strains in two principal directions marked with S-I and S-II have been monitored for each load case. Both the layout of strain values and the extreme values have been analyzed. First, the results for the initial model are presented (Section 3.1). Then, results are given for the following parameters:

• Size of the structure (Section 3.2).
• Curvature of the structure (Section 3.3).
• Membrane material properties (Section 3.4).
• Prestress intensity (Section 3.5).
• Membrane patterning direction (Section 3.6).
• Membrane edges (Section 3.7).

Some graphical results are given in Appendix A.

3.1. Initial Model

The initial model served as a starting point for all other combinations. Therefore, its results are presented here in detail. Figure 6 shows the strains of the initial model, in both principal directions and under each applied load. The results are given for all points of the membrane. Extreme values can be read from the color scale on the left side of each graph. On the right and bottom side of each graph, a measurement scale is given. Membranes are presented in top view because this view provides the best visibility of results. The downside of this representation is that the height of the given membrane is not visible. In all graphs where the results are presented in this way, the elevated supports are located in the upper left and lower right corners.

3.2. Size of the Structure

The first analyzed parameter is the size of the membrane structure. The results for this parameter are given in Figure 7. Unlike the results for the initial model, here the values are not shown for each point of the membrane. Instead, extreme values of strain in S-I and S-II are given for both analyzed loads. In each graph, results for five different models are shown. The models differ in size, while other parameters are kept constant. In this way, the effect of changing size on resulting strains is isolated and can be analyzed independently. The horizontal axis shows the size of the model, while the vertical axis shows the value of strains. Four graphs are presented in Figure 7 because there are four combinations of values for patterning direction and type of edges.

3.3. Curvature of the Structure

The curvature of the membrane was the second analyzed parameter. The results are organized in a similar fashion as for the previous parameter. The main difference is that there are six analyzed values for the height of the structure, as opposed to five analyzed sizes. Here the heights are marked on the horizontal axis. The results for different curvatures of the model are given in Figure 8.

3.4. Membrane Material Properties

The membrane material was the next analyzed parameter. It also has six analyzed values, as the previous one. The results are presented using the same system. Extreme strain values under snow load are given in blue color, while the orange color is used for the results under wind load action. The obtained results are given as dots. Lines are used to connect the measured results in order to make the trends more visible. Lines in S-I are shown as continuous, and the lines in S-II as dashed. The results for different membrane materials are given in Figure 9.

3.5. Prestress Intensity

Prestress is the last analyzed parameter with numerical values. Therefore, the results of the analysis of this parameter are presented in the same way as the previous ones. There are five analyzed values, similar to the parameter of membrane size. The results are given in Figure 10. Once again, four graphs are given, as the results differ significantly in models with different patterning and edges. The analysis of the results is given in the discussion section of the paper.

3.6. Membrane Patterning Direction

The results of the parameter of patterning direction have already been presented with the results of all the previous parameters. Since it is obvious that this parameter influences the strains, it is decided that it is needed to show the results for different patterning in more detail. The results are shown for each point on the membrane. The diagonal and parallel patterning are shown one next to another. First, the results under snow load for both S-I and S-II are shown in Figure 11. Then, the results are shown for the wind load in Figure 12.

3.7. Membrane Edges

The results for the two analyzed edge types have also been already shown with the first five parameters. However, due to the different shapes the membrane takes, the results are presented here in more depth. Figure 13 and Figure 14 show models with flexible and rigid edges parallelly. Figure 13 presents the result under snow load and Figure 14 under wind load. In each graph, the most extreme value is shown in red color, regardless of whether its value is positive or negative.

4. Discussion

The results are discussed in this section in the same order as presented in the previous section. For certain aspects of the discussion, data provided in Appendix A are referenced.

First, the action of snow load is directly proportional to the vertical projection of the membrane. This relationship applies both in the numerical simulations and in real-world applications [42]. Specifically, changes in the vertical projection area, influenced by parameters such as size and edge type, result in proportional changes to the total snow load acting on the membrane. In contrast, wind load behavior differs as it is related to the surface area of the membrane. Consequently, wind load varies not only with parameters like size and edge type but also with the curvature of the structure.

Another important aspect is to note that all strains presented in this study represent the transition from the prestressed state to the loaded state of the membrane. The prestressing process itself induces strains that are not included in this analysis. If PV systems are installed after the membrane has been prestressed, these initial strains can be disregarded.

The mechanical behavior of woven membranes is extremely complex and cannot yet be fully replicated through numerical simulations. Therefore, the strain values should be interpreted as indicative guidelines rather than exact representations of real-world behavior. Although some of the resulting strains in this research are too high for practical application, they provide us with important information about the behavior of membranes. The primary contribution of this research is in identifying the relations between the analyzed parameters and the resulting strains under load. The findings are not intended to replicate physical testing results but rather to establish a framework for understanding these interactions. The ultimate output of this research is a set of design guidelines for future PV-integrated membrane structures, derived from the discussion that follows, which explores the influence of various parameters on strain behavior under load.

The initial model shows slightly higher positive than negative strains under snow load. For positive strains, the maximum values are concentrated in the central part of the membrane, and high strains stretch towards the high supports (Figure 6). Strains in the S-I principal direction are the lowest close to lower supports. In the S-II direction, the area of large negative strains is oriented from one low support towards the other. Maximum values occur next to the edges, close to low corners of the membrane. The lowest strains in S-II are close to high corners. Under wind load, negative strains have larger absolute values than positive strains. High positive strains are located in the central part of the membrane and spread towards the low supports. Minimal strain values in S-I are next to the high corners, thus making this arrangement resemble strains in S-I under snow load, rotated by 90°. Areas of high negative strains are very concentrated and located close to the middle of the membrane, towards the high corners of the structure. The lowest negative strains are around all four supports. The behavior of strains can be explained physically: under snow load, the fibers connecting the high supports get elongated due to the downward load. Fibers parallel to the diagonal connecting the low supports have an upward sag, and their length decreases under snow load. In contrast, wind loads act upward, causing significant stretching of fibers previously under reduced tension. The peculiar strains in S-II are explained by the wind load acting perpendicular to the membrane, thus creating the largest deflections in the middle and at the half distance between the middle and the high supports. Extreme negative strains occur at the midpoint between these points, as this is the location where the membrane will shorten the most. Extreme values of strains under wind load are significantly larger compared to the snow load. Positive strains are almost twice as high, while negative strains are almost three times bigger.

The size of the area to be covered is usually given to the designer, but this is not necessarily equal to the size of the structure. The designer has the freedom to divide this area into several smaller structures, in case it is justified by some reason. The question posed in this research is whether the size of the membrane structure has any impact on strains under load, and if so, what is the relation between these two? The size of the model proved to have a large influence on the value of strains, based on the obtained results provided in Figure 7. The first observation that can be made is that in all analyzed cases, the increase in the size of the structure leads to an increase in strains. This is valid for both applied loads. The reason for this is that not only does the span increase, but also the total applied load, as explained earlier in this section. The next trend that can be noted is that strains in both principal directions are always more extreme under wind load. This is explained by the higher intensity of wind load action. In models with parallel patterning, positive strains have higher absolute values compared to negative strains, regardless of the edge type. Extreme strains in models with diagonal patterning are visibly lower than the strains of models with parallel patterning. This shows the significant influence of patterning direction on strains and will be checked on the rest of the obtained results. Figure A1 in Appendix A shows the layout of strains under snow load of models with different sizes. The size does not affect the strain’s layout significantly. It can be observed that with the increase in size, the high strains, including both positive and negative strains, get more localized. This is more emphasized in S-II, where the position of negative strains does not change with the increase in size, but the extreme value gets higher than the rest of the strains, therefore the change of color on the biggest part of the largest models. Wind load results shown in Figure A2 show large positive strains along the diagonal connecting the two low supports. Maximal negative strains are located near the center of the membrane, symmetrically moved towards the corners.

Next analyzed was the curvature of the membrane, defined by different heights of the structure. In the case of this parameter, the behavior of the membrane differs considerably for models with different patterning directions. Models with parallel patterning react to increases in height with the increase in extreme positive and negative strains under both loads, as can be seen in Figure 8. This is not the case with models with diagonal patterning. At first, they show an increase in strains with the increase in height; however, further height increases lead to a reduction of strains. This behavior is not simultaneous in S-I and S-II strains. As it happens under both loads, it cannot be explained by differences in loads but rather by the combined effect of two factors. One is the increase in curvature, which should help the membrane resist external loads, and the other is the increase in the fiber length caused by the increase in height. The second factor is not as important in membranes with parallel patterning, since in this case the fibers are straight and not curved. Two observations made during the analysis of the first parameter are confirmed here: strains under wind load are more extreme than under snow load, and strains of models with diagonal patterning are lower than those with parallel patterning. The change of the edge type does not result in important changes in the extreme strain values. Figure A3 shows an interesting behavior of S-II strains under snow load. The position of the maximal negative strains moves with the increase in curvature, starting from the position next to the cables and close to the high corners, towards the center of the membrane, and then back next to the edge cables, but this time close to the low corners of the membrane. A similar movement of the position of maximal negative strains is observed under wind load, presented in Figure A4. In this case, the location of maximal negative strains moves from the edge close to the low corners towards the center of the membrane, and with an increase in curvature, it gets closer to the center.

While there are several factors influencing the choice of membrane material for any specific structure, it is the structural analysis that sets the minimum values for tensile strength. More resistant materials can be chosen for reasons other than structural. One of the possible reasons to select a material with higher mechanical properties may be the need to facilitate the integration of PV. This research aimed to answer whether this is justified, and the results showed clear trends. The change of membrane material results in a very significant change in maximal strains (Figure 9). With the increase in the membrane material mechanical values, the strains drop. This is especially noticeable in models with diagonal patterning. Among the graphs presented in Figure A5 and Figure A6, especially interesting are the ones marked as (f), showing the S-I strains of a model with the highest values of membrane material. Here, the position of the maximal strains is unlike any other presented model, once again showing the uniqueness and importance of the parameter of membrane material. The results under snow and wind load are similar in layout, only rotated by 90°. The position of high strains moves with the increase in membrane material values from the area near high corners to the center of the membrane and then to the area near the low corners in S-I. The converse trajectory is present in the S-II direction. Once again, the change of edge type does not result in noteworthy changes of strains. Strain in models with diagonal patterning is here as well, lower compared to the same models with parallel patterning.

The increase in prestress intensity is a parameter usually defined by the structural needs. When structural analysis shows that the membrane will lose tension under expected external load, the pretension value is increased. The results obtained by this research show that the increase in prestress intensity can also help in reducing strains under load (Figure 10). The largest positive strains under both loads show almost linear behavior with changes in the prestress. Their value is reduced slightly with the increase in prestress. Negative strains in models with diagonal patterning show more rapid decrease up until a point where further increase in prestress has negligible effect on them. In models with parallel patterning, negative strains show almost linear decrease with the increase in prestress. Figure A7 shows that under snow load the position of high strains does not change significantly with the increase in prestress value. However, the areas of high strains expand in both the S-I and S-II directions, making a larger part of the membrane more strained, although the value of maximal strains decreases gradually. Similar behavior occurs under wind load (Figure A8). The influence of edge type shows a somewhat larger impact in models with low prestress values, especially in the values of negative strains. In other cases, the effects of changing the edge type are very low.

Patterning is a necessity in woven tensile membrane structures. The direction of patterning can have a subtle esthetical impact on the appearance of the structure. It also affects the structural properties, such as deflections under load. Sometimes the patterning orientation is defined in a certain way to ease the installation of the membrane. Results shown in Figure 11 and Figure 12 provide results for the models that differ only in patterning direction. The presented results show significantly larger strains in models with parallel patterning. This is valid for both positive and negative strains. Under snow load strains, the parallel patterning is almost 4 times larger. Under wind load, positive strains are three times larger, and negative almost double with parallel patterning compared to the diagonal. Figure 11 and Figure 12 show that the behavior of the membrane is different under the two patterning directions, and this is depicted by differences in strain layouts. When using parallel patterning, extreme strains under both loads are close to the middle of the edges and, under wind load, additionally in the central part of the membrane. In models with diagonal patterning, principal strains are mostly aligned with the warp and weft direction. Textile membranes have fibers in two orthogonal directions. These fibers are load-carrying, and due to such structure of the membrane, it is numerically modeled as orthotropic. The fibers provide higher strength in their directions, compared to the off-fiber directions. The largest strains occur when the load is applied diagonally to the fibers. This should not be confused with diagonal patterning of the membrane, where the fibers are aligned with the diagonals of the structure. Diagonal patterning ensures that fibers are in the direction connecting the corners. When corner supports are connected by a stronger direction of the membrane, it is reasonable to expect lower strains. In addition, when fibers are connecting the corners of the hypar membrane, they have curvature, unlike in parallel patterning. This amplifies the lower straining of the membrane under load compared to parallel patterning, which is in line with the obtained results.

Different membrane edges affect the geometry and structural properties. Among the affected structural properties are the strains under loading. However, the results have shown that, compared to other analyzed parameters, this one has a low impact on the strain values. If we consider the strains given in Figure 13 and Figure 14, we can justify this conclusion. On the right side of the figures, the model with rigid edges is presented to compare it easily with the one with flexible edges. Only small differences in strain values can be noted when analyzing these results. In addition, the layout of the strains is very similar, except for the larger membrane area in models with rigid edges.

Guidelines for Strain Analysis of Photovoltaics-Integrated Tensioned Membrane Structures

Based on the findings of this study, the following design recommendations are classified according to different design stages for guiding the integration of PV systems into tensioned membrane structures:

• Strain management should be a critical focus during the conceptual and design stages to ensure compatibility between the PV system and the membrane structure. The following elements should be considered:
  • Strain limits determination:

    a.

    Establish strain limit values specific to the PV system to be integrated, based on its tolerances for mechanical deformation and electrical efficiency degradation.

    b.

    Define acceptable strain thresholds by quantifying the impact of strain on the electrical output and lifespan of the PV system.

    c.

    Obtain limit values for strains for the PV system that will be integrated with the membrane.

  • Strain-focused design analysis:

    a.

    Use strain analysis in the early design phase to evaluate the impact of anticipated external loads and optimize structural performance.

    b.

    If predicted strain values exceed the limits for the PV system, apply mitigation strategies listed below, such as geometry adjustments, material upgrades, or prestress optimization.

• Geometry of the membrane structure significantly affects strain behavior and should be carefully designed during the design stage, considering:
  • Size reduction as larger membrane structures generate higher strains.
  • Curvature control:

    a.

    Adjust the membrane curvature by changing the height of the structure.

    b.

    Increase in curvature does not result in unambiguous change in maximal strains. For membranes with parallel patterning, higher curvature tends to increase strain. For membranes with diagonal patterning, the relationship is less pronounced and may allow for optimized curvature without excessive strain.

• Patterning and edge design influence strain distribution but require several considerations in the design stage:
  • Patterning direction:

    a.

    Switch from parallel to diagonal patterning to achieve significant reductions in strain values under external loads.

    b.

    Diagonal patterning distributes forces more evenly, reducing stress concentrations and mitigating extreme strain values.

  • Edge type selection:

    a.

    The edge type (rigid or flexible) has minimal impact on strain levels and can be chosen based on other design or structural factors such as ease of assembly or aesthetic considerations.

• Material selection is fundamental in the design stage to ensure strain mitigation:
  • Tensile strength requirements:

    a.

    Set the minimal tensile strength according to the typical structural analysis of the membrane structure based on stresses and deflections.

  • Elastic and shear modulus:

    a.

    Use membrane materials with higher elastic and shear moduli to effectively reduce strains under load.

• Prestress optimization is essential in the design stage to ensure strain management:
  • Prestress intensities:

    a.

    Higher prestress intensities generally reduce strain magnitudes, particularly for positive strains, which show near-linear reduction with increased prestress.

    b.

    Negative strains may plateau beyond a certain prestress value, indicating diminishing returns at high prestress intensities.

  • Balancing prestress:

    a.

    Optimize prestress to balance strain reduction without overloading the membrane or creating unnecessary material stresses.

Also, the comparative influence of structural parameters shows the importance of different factors (

Table 1. Comparative influence of structural parameters (source: Authors’ elaboration).

Factors	Recommendations
Most significant	Material selection has the greatest influence on reducing strain levels, making it a primary focus in design optimization
	Changing patterning direction from parallel to diagonal has a substantial impact on strain mitigation
	Reducing membrane size is highly effective in lowering strain levels
Moderate impact	Increasing prestress levels provides strain reduction but shows diminishing returns beyond certain thresholds
	Decreasing curvature somewhat reduces strain in parallel-patterned membranes but has limited influence in diagonal-patterned membranes
Least significant	Edge type has a negligible effect on strain values, allowing flexibility in selection based on secondary criteria

).

5. Conclusions

This research analyzed the strains experienced by tensioned membrane structures under load, aiming to support the integration of photovoltaic systems onto these structures. Six key parameters—size; curvature; membrane material properties; prestress intensity; patterning direction; and edge type—were investigated for their influence on strain behavior under two calculation loads. Conducted as a parametric study using numerical models, the research analyzed a total of 88 configurations, with two load cases applied to each, resulting in 176 simulations. The objective was to establish the relationship between these parameters and strain behavior, culminating in actionable design recommendations for optimizing membrane structures for photovoltaic (PV) integration.

The study’s significance aligns with broader sustainability goals, particularly the integration of renewable energy technologies into the built environment as a pathway to achieving the United Nations’ Sustainable Development Goals. Lightweight and flexible PV technologies have revolutionized the potential for PV integration into complex architectural forms. However, the mechanical incompatibilities between tensioned membranes and PV systems—specifically the large strains experienced by the former—pose significant challenges for integration. This research addresses this gap by identifying design strategies to manage strains and enable the successful coupling of these technologies.

The results of this research provide critical insights into strain management for membrane structures. Larger membranes experience higher strain under load, so reducing their size can mitigate this issue. Although smaller structures may require more individual membranes to cover a given area, impacting the design, the designers are advised to use smaller membranes to reduce the strains. While curvature influences stresses and deflections, its role in strain management is complex. Only parallel-patterned membranes demonstrate a predictable reduction in strain with decreased curvature, limiting their reliability as a strain management tool. Higher elastic and shear moduli of the membrane significantly reduce strains. While this method has a substantial impact, it comes with increased material costs. However, as it does not alter the aesthetic design, it is often preferred for projects prioritizing visual integrity. Therefore, the use of membrane materials with higher mechanical properties is recommended for PV integration. Increased prestress reduces strains but must be carefully calibrated to avoid structural deterioration. While effective, the strain reduction achieved is lower compared to size reduction or increase in material properties. Diagonal patterning is highly effective in minimizing strain, with no additional cost or significant aesthetic compromise, making it a recommended option over parallel patterning. Thus, diagonal patterning should be the first choice for designers when designing PV-integrated membrane structures. The choice between rigid and flexible edges has minimal influence on strain and can therefore be selected based on other design or functional priorities without significant concern for strain behavior.

The integration of these findings into design processes offers new solutions for optimizing membrane structures to support PV systems. The research refrained from defining a specific strain threshold to support the ongoing advancements in flexible PV technology. Instead, it focused on identifying trends and relationships, ensuring the findings remain relevant as PV systems evolve.

This research highlights the potential of integrating PV systems into membrane structures, demonstrating how these designs can simultaneously serve as adaptable energy-generating surfaces and sustainable architectural solutions. This symbiotic relationship not only advances technical and design capabilities but also supports global efforts toward decarbonization and climate neutrality. By offering scientifically grounded recommendations, this research facilitates innovative design solutions for the built environment, bridging the gap between renewable energy integration and architectural innovation. It positions tensioned membrane structures as a sustainable solution for the energy demands of future cities, contributing to a more resilient and environmentally conscious built environment.