Experimental and Numerical Analysis of Wrinkling Behaviors of Inflated Membrane Airship Structures

Abstract

In this paper, the wrinkling behavior of an inflated cantilever beam is presented. An analytical solution for the load-bearing capacity of an inflated beam is proposed to predict the ultimate wrinkling force and critical wrinkling force of the inflated beam, and an iterative membrane properties method is used to simulate the wrinkling of membrane structures. The load-bearing capacities of an inflated beam, numerically simulated based on these two methods, are compared with experimental results. Good agreement between wrinkling using UMAT-modified M3D4 elements based on the IMP method and experiments was obtained. The effect of wrinkling on the stress distribution of the airship envelope under internal pressure is also explored based on a practice airship.

1. Introduction

Inflated membrane structures have less flexural and compressive strength than conventional structural systems for aerospace such as airship envelopes and civil engineering applications, due to their light weight and flexibility, so wrinkling can occur on membrane structures such as large-scale airships in some loading states. Much research has been conducted into the mechanical characteristics of inflated membrane structures using numerical and experimental approaches. Xue et al. [1] reported on a rectangular air-supported membrane structure fabricated using a PVC-coated polyester fabric membrane with dimensions of 38 m × 20 m × 7 m. They found that the initial pressure has a minimal effect on the collapse of air-supported membrane structures, while excessive cables accelerate the process of collapse. Liu et al. [2] studied the influence of air pressure on the structural behavior of an air-inflated beam considering the effect of beam length and loading conditions. It was found that different loading conditions created a similar deformed shape of the air inflated beam, as the loads transformed into the fixed end moment. Dai et al. [3] employed a dual-biprism-based stereo camera system to obtain image pairs of the membranes before and after deformation. They calculated the critical wrinkle force based on the tension field theory. It could be concluded that a stress change occurred between the membrane surface and its inner structure. Meitour et al. [4] presented the PS-DPS model to correct for the effects of wrinkles on membrane structures in the in-house finite element software Herezh++. They also performed a square cushion test and numerical modeling with this non-linear law. They estimated that the PS-DPS model was valid and could accurately take into account the wrinkles in flexible structures with all these linear and non-linear behaviors. Tian et al. [5] demonstrated a new numerical procedure based on a double Taylor series combining path-following techniques and discretization by a Trefftz method. Nakashino et al. [6] argued that an isogeometric membrane element based on a non-uniform rational basis spline (NURBS) model could account for membrane wrinkling based on tension field theory. They found that the convergence rate of the isogeometric membrane analysis with respect to refinement of the discretization was much better than that of a model based on tension field theory by using a numerical example. Venkata et al. [7] considered the possibility of harnessing various forms of instabilities such as wrinkling and necking using spatial material inhomogeneities in inflated hyperelastic auxetic membranes. They described the onset of wrinkling in an averaged way and also generated non-trivial instabilities at desired locations.

Wang et al. [8] conducted a theoretical evaluation of the natural frequencies of wrinkled inflated beams. The inflated pressure, the wrinkles, and the aspect ratio were three factors introduced into the vibration equation of the beam to predict the first-order natural frequency of the wrinkled inflated beams. Huang et al. [9] established a reduced FE model by using purpose-made codes linked to ABAQUS codes to implement simulations of wrinkling patterns of membrane structures under different loading states. Nguyen et al. [10] performed linear and nonlinear buckling analysis of an inflated beam made of orthogonal woven fabric using a three-dimensional Timoshenko beam model. Wong and Pellegrino [11] assessed the mechanism of wrinkling on membrane structures, devised a method that considered initial geometrical imperfections for pre-stressed membrane structures, and analyzed the linear and nonlinear buckling behavior of membrane structures by considering initial geometrical imperfections [12]. Kamaliya et al. [13] explored the effects of multiple creases and wrinkles on deployed thin Kapton surface accuracy by experimental and numerical methods. They found that double-creased membranes could achieve maximum surface correctness for any angular position, as the mirror effect folds shapes to most effectively reduce wrinkles. Tomoshi Miyamura [14] presented an experimental and numerical method to study the wrinkling of a stretched circular membrane under in-plane torsion, numerically simulated the tension behavior of an isotropic polyester film and an orthotropic PVC-coated textile based on bifurcation analysis, and measured the stresses of the wrinkled circular membranes. As membrane structures cannot resist compression, the other approach to the numerical simulation of wrinkling is based on tension field theory. The tension field theory can be used to explain how the minor principal stress is close to zero, whereas the major principal stress is greater than zero, for membrane structures under external loadings. The wrinkles are oriented in the direction of the local major principal stresses.

However, the second numerical simulation method cannot be performed directly using commercial software, as secondary development of commercial software or programming is necessary to simulate wrinkling of membrane structures. Previous researchers have only adopted one numerical simulation method to address the wrinkling of membrane structures, and their research has only involved problems related to planar membranes or inflated tube structures: they have not carried out studies on wrinkling and the load-bearing capacity of inflated beams based on the two aforementioned methods, i.e., the bifurcation buckling theory or tension field theory. Currently, the critical force when wrinkling starts to occur on the envelope, referred to as the critical wrinkling force, is commonly considered as a design parameter, whereas the ultimate force when wrinkling covers the circular cross-section of a beam, the ultimate wrinkling force, is usually ignored. The membrane structure will not be safe once the exterior load of the inflated beam reaches ultimate wrinkling loads, but it will still be available even if the exterior load of the inflated beam is beyond the critical wrinkling load and less than the ultimate wrinkling load. Thus, the mechanical behavior of an inflated beam after wrinkling occurs still needs to be further considered due to its complexity.

The engineering applications of inflatable membrane structures cover multiple fields, where their ultra-light weight, deployability, and adaptability are decisive advantages. In aerospace, they form the heart of modern airship envelopes/stratospheric platforms, cargo airships, inflatable space habitats, and planetary entry decelerators. The structural integrity of prototypes such as “Zhiyuan-1” depends directly on the wrinkle-resistant design of the envelope under aerodynamic and payload loads [15]. Emerging uses include inflatable antennas, solar sails, and medical expandable implants. Therefore, it is crucial that the wrinkling mechanisms of these film structures be explored to predict load-bearing capacity, prevent premature collapse, and optimize material utilization.

This paper presents two numerical approaches to simulate membrane wrinkling: the IMP method and its UMAT-enhanced variant. A mixed wrinkling criterion is adopted, and a UMAT subroutine is coded in Fortran and linked to ABAQUS through the user–material interface to establish a nonlinear numerical simulation framework for membrane wrinkling. The post-wrinkling mechanical characteristics of the flexible airship envelope are then investigated. Using an ETFE inflated tube as the benchmark, the validity of the numerical method is verified. Finally, practical design values for both the critical wrinkling load and ultimate load-bearing capacity of the flexible airship envelope are provided.

In Section 2, the load-bearing capacity of an inflated cantilever beam is numerically simulated using bifurcation buckling theory, and its analytical solution is obtained. Then, UMAT codes used to link to ABAQUS (2022) codes are used to simulate the wrinkling behavior and load-bearing capacity of the inflated beam based on the iterative membrane properties method as described in Section 3. In Section 4, an experiment to study the load-bearing capacities of an inflated cantilever beam under different pressures is described. Two numerical simulation methods and analytical solutions are compared with experimental results. Lastly, an airship envelope is considered as an example to study the effect of wrinkling on the stress distribution of the airship envelope under internal pressure.

2. Methodology

According to engineering beam theory, deformation of membrane structures can occur when the bending moment exceeds the critical bending moment, M_k, as determined by Equation (32). It can be numerically simulated using the iterative membrane properties (IMP) method based on tensile field theory. The IMP method is appropriate for isotropic material such as ETFE.

2.1. Variable Poisson’s Ratios Method

The variable Poisson’s ratio method is a type of wrinkling analysis method based on tensile field theory, with the assumption that the membranes have no pressure resistance. Stein and Hedgepeth [16] studied a partly wrinkled thin plate and found that there was a tensile area and a wrinkled area in the membrane structure. Plane stress can be expressed in a normal constitutive equation in the tensile area, whereas in the wrinkled area, the normal constitutive equation should be modified by importing a variable Poisson’s ratio λ to describe the stress states of the wrinkled area. In the wrinkled area, the variable Poisson’s ratio is greater than the actual Poisson’s ratio of the membrane, causing a uniaxial tensile state due to “excessive shrinking” of the membrane perpendicular to the wrinkling direction. Therefore, it should be assumed that the Poisson’s ratio in the boundary area between the tensile area and the wrinkled area of the membrane should be identical to the actual Poisson’s ratio of the membrane in order to obtain reliable results. The film must be assumed incapable of sustaining any compressive stress.

2.2. IMP Method Based on the Stein–Hedgepeth Theory

The Stein–Hedgepeth wrinkling theory can be suitable for elastic and isotropic membrane materials without bending stiffness or pressure resistance. If the major principal stress and minor principal stress are tensile stresses, membranes do not experience wrinkling. If the major principal stress and the minor principal stress are equal to zero, the membranes are slack. However, if the minor principal stress is equal to zero and the major principal stress is tensile stress, then wrinkling may occur in the membranes. For a planar membrane, the stress in the wrinkled area can be obtained using the strain compatibility equation. According to the stress-strain constitutive equation, the principal strain of membranes can be expressed as follows:

$${\epsilon }_1={\displaystyle \frac{{\sigma }_1}{E}}$$

(1)

$${\epsilon }_2=-\lambda {\displaystyle \frac{{\sigma }_1}{E}}$$

(2)

where λ is the Poisson’s ratio, and E is the elastic modulus. According to Equations (1) and (2):

$$\lambda =-{\displaystyle \frac{{\epsilon }_2}{{\epsilon }_1}}$$

(3)

With reference to Equation (3), the linear constitutive equation of the membranes can be rewritten as follows:

$${\displaystyle \frac{{\sigma }_x}{E}}={\displaystyle \frac{{\epsilon }_x+\lambda {\epsilon }_y}{1-{\lambda }^2}}={\displaystyle \frac{{\epsilon }_x-\frac{{\epsilon }_2}{{\epsilon }_1}{\epsilon }_y}{1-{\left(\frac{{\epsilon }_2}{{\epsilon }_1}\right)}^2}}={\displaystyle \frac{{\epsilon }_x{\epsilon }_1^2-{\epsilon }_1{\epsilon }_2{\epsilon }_y}{{\epsilon }_1^2-{\epsilon }_2^2}}$$

(4)

$${\displaystyle \frac{{\sigma }_y}{E}}={\displaystyle \frac{{\epsilon }_x+\lambda {\epsilon }_y}{1-{\lambda }^2}}={\displaystyle \frac{{\epsilon }_y-\frac{{\epsilon }_2}{{\epsilon }_1}{\epsilon }_x}{1-{\left(\frac{{\epsilon }_2}{{\epsilon }_1}\right)}^2}}={\displaystyle \frac{{\epsilon }_y{\epsilon }_1^2-{\epsilon }_1{\epsilon }_2{\epsilon }_x}{{\epsilon }_1^2-{\epsilon }_2^2}}$$

(5)

$${\displaystyle \frac{{\tau }_{xy}}{E}}={\displaystyle \frac{{\gamma }_{xy}}{2\left(1+\lambda \right)}}={\displaystyle \frac{{\gamma }_{xy}}{2\left(1-\frac{{\epsilon }_2}{{\epsilon }_1}\right)}}={\displaystyle \frac{{\epsilon }_1{\gamma }_{xy}}{2\left({\epsilon }_1-{\epsilon }_2\right)}}$$

(6)

where the strains can be expressed as follows:

$${\epsilon }_1^2={\displaystyle \frac{1}{4}}\left[\left({\epsilon }_x+{\epsilon }_y\right)\left({\epsilon }_x+{\epsilon }_y\right)+2\sqrt{{\left({\epsilon }_x-{\epsilon }_y\right)}^2+{\gamma }_{xy}^2}\left({\epsilon }_x+{\epsilon }_y\right)+\left({\left({\epsilon }_x-{\epsilon }_y\right)}^2+{\gamma }_{xy}^2\right)\right]$$

(7)

$${\epsilon }_1{\epsilon }_2={\displaystyle \frac{1}{4}}\left[{\left({\epsilon }_x+{\epsilon }_y\right)}^2-\left({\left({\epsilon }_x-{\epsilon }_y\right)}^2+{\gamma }_{xy}^2\right)\right]$$

(8)

$${\epsilon }_x{\epsilon }_1^2-{\epsilon }_1{\epsilon }_2{\epsilon }_y={\displaystyle \frac{\left({\epsilon }_x+{\epsilon }_y\right)}{4}}\left[2{\epsilon }_x^2+2{\epsilon }_x\sqrt{{\left({\epsilon }_x-{\epsilon }_y\right)}^2+{\gamma }_{xy}^2}-2{\epsilon }_x{\epsilon }_y+{\gamma }_{xy}^2\right]$$

(9)

$${\epsilon }_y{\epsilon }_1^2-{\epsilon }_1{\epsilon }_2{\epsilon }_x={\displaystyle \frac{\left({\epsilon }_x+{\epsilon }_y\right)}{4}}\left[2{\epsilon }_y^2+2{\epsilon }_x\sqrt{{\left({\epsilon }_x-{\epsilon }_y\right)}^2+{\gamma }_{xy}^2}-2{\epsilon }_x{\epsilon }_y+{\gamma }_{xy}^2\right]$$

(10)

$${\epsilon }_1+{\epsilon }_2={\epsilon }_x+{\epsilon }_y$$

(11)

$${\epsilon }_1^2-{\epsilon }_2^2=\left({\epsilon }_1-{\epsilon }_2\right)\left({\epsilon }_1+{\epsilon }_2\right)=\left({\epsilon }_1-{\epsilon }_2\right)\left({\epsilon }_x+{\epsilon }_y\right)$$

(12)

Therefore, Equations (4)–(6) can be simplified using Equations (7)–(12) as follows:

$${\displaystyle \frac{{\sigma }_x}{E}}={\displaystyle \frac{{\epsilon }_x}{2}}\left[ \mathit{cos}2\alpha +1\right]+{\displaystyle \frac{{\gamma }_{xy}}{4}}\mathit{sin}2\alpha$$

(13)

$${\displaystyle \frac{{\sigma }_y}{E}}={\displaystyle \frac{{\epsilon }_y}{4}}\left[1-\mathit{cos}2\alpha \right]+{\displaystyle \frac{{\gamma }_{xy}}{4}}\mathit{sin}2\alpha$$

(14)

$${\displaystyle \frac{{\tau }_{xy}}{E}}={\displaystyle \frac{{\epsilon }_x}{4}}\mathit{sin}2\alpha +{\displaystyle \frac{{\epsilon }_y}{4}}\mathit{sin}2\alpha +{\displaystyle \frac{{\gamma }_{xy}}{4}}$$

(15)

Thus, Equations (13)–(15) can be expressed as the following matrix:

$$\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_y\\ {\tau }_{xy}\end{array}\right\}={\displaystyle \frac{E}{4}}\left[\begin{array}{ccc}2\left[\mathit{cos}2\alpha +1\right]& 0& \mathit{sin}2\alpha \\ 0& 2\left[1-\mathit{cos}2\alpha \right]& \mathit{sin}2\alpha \\ \mathit{sin}2\alpha & \mathit{sin}2\alpha & 1\end{array}\right]\left\{\begin{array}{c}{\epsilon }_x\\ {\epsilon }_y\\ {\tau }_{xy}\end{array}\right\}$$

(16)

For membrane structures, Equation (16) can be used to modify the stiffness matrices that have been varied due to the occurrence of wrinkling.

The IMP method is implemented within the finite element iterative scheme and embeds the core assumption of the VPR method (namely, that wrinkling can be simulated by dynamically adjusting material properties) into an automated numerical framework. Thus, the VPR method provides the “material law” for each IMP iteration, while the IMP method handles the computational realization of membrane wrinkling.

2.3. Wrinkling Criterion

Membrane structures can exist in three possible states: taut state, slack state, and wrinkled state. Three wrinkling criteria are commonly used to describe the membrane state, which are the principal stresses, principal strains, and a combination of principal stresses and strains criteria. The wrinkling criterion of the combination of principal stress and strain can be used to assess the state of membrane structures, since it can provide the most accurate description of the membrane’s actual state [17,18]. Specifically, it refers to when a membrane is purely tensioned when the major principal stress σ₂ is greater than zero. However, when σ₂ is less than or equal to zero, the major principal strain ${\epsilon }_1$ can be considered as the parameter to identify purely tensioned membranes and wrinkled membranes. Thus, if ${\epsilon }_1$ is greater than zero, membranes can be called wrinkled membranes, whereas if ${\epsilon }_1$ is less than or equal to zero, membranes are identified as slack membranes. This can be described as follows:

• σ₂ > 0, membranes in taut state;
• σ₂ ≤ 0 and ${\epsilon }_1$> 0, membranes in wrinkled state;
• σ₂ ≤ 0 and ${\epsilon }_{1 }$ ≤ 0, membranes in slack state.

3. Wrinkling Analysis of Plane Membrane Structures

3.1. Experimental Study

3.1.1. Testing Setup and Specimen

In this experiment, a series of square plane membrane structures with a length of 250 mm is studied with the objective of understanding the wrinkling behaviors of plane membrane structures. Figure 1 shows the experimental setup and testing equipment for wrinkling analysis. The setup can support the membrane structures and clamp the edges of the membrane surface. After the membrane surface is imported prestress, it can support external loads. Therefore, two loading systems, including shear and tension loading systems, should be installed on the setup. Transparent support panels are used to support the membrane behaviors under shear and prestress force, and the scales are employed to measure shear in the membrane and tension in the cables, as shown in Figure 1. According to Figure 1a, the setup is manufactured into a steel cube frame with a length of 700 mm, and the support tracks on the frame can be formed into rectangular steel bars with a length of 30 mm and width of 2.5 mm. The membrane specimens are shown in Figure 2. The specimens in Figure 2 should be clamped at the edges of 1 and 2, as shown in Figure 1 and Figure 2. Figure 2 displays the membrane surface with a horizontal glue seam.

Table 1. Material property of membrane, cable, and glue.

Property	Membrane	Cable	Glue
Materials	Kapton	PBO	Epibond1590A/B
Density (kg/m3)	1390	1450	1090
Young’s modulus, E (N/mm2)	11,900	131,000	400,000
Poisson’s ratio	0.31	0.3	0.3

shows the material properties of the membrane, cable, and glue. The thickness of the membrane surface and glue are, respectively, 0.025 mm and 0.003 mm, and the diameter of the cable is 0.4 mm.

3.1.2. Loading Process

To vary prestress on the membrane surface, the tension of the cable and shear force on the clamping plate connected to the membrane edge could be adjusted. Figure 3 shows the testing process of the wrinkling analysis of membrane structures. According to Figure 3, the central section of tension cables can be connected to spring dynamometers, manufactured by Guangdong Kang Yu Control Systems Engineering Incorporated; these types of spring dynamometers are KYPS08, and their measuring range and precision are, respectively, 5 g and 10 kg. In this experiment, firstly, flatten the membrane surface on the glass support panel; then, align the membrane surface and clamp device and fix the membrane surface between the support panel and pressing plate. Finally, increase the tension of the cable connected to the support panel in order to prestress, as shown in Figure 3a. To apply shear force, the number of steel needles and horizontal displacement of the prestress plate connected to the membrane surface on the support panel can be varied as shown in Figure 1b. The applied shear force can be obtained by the spring dynamometers connected to the central section of tension cables. In the experiment, the specimen membrane was clamped along its upper and lower edges by distributing five clamping plates together with upper and lower pressure plates.

3.1.3. Measurement Methods

In this experiment, the Tianyuan 3D scan system (OKIO-V-400) is used as a non-contact measurement system developed by Beijing Ten Youn 3D Technology Limited, as shown in Figure 4. This scan system combines phase and stereo vision technology. The middle projector can project an optical grating on the object surface. Two cameras at two sides can shoot an aberrant raster image, then the phase at every point from two cameras can be obtained by using binary code light and the phase shift method. The coordinates in three-dimensional space can be calculated by matching points on the two images based on phase and epipolar, and a calibrated camera system, which can obtain the three-dimensional profile of the object surface.

Compared with other projection methods, the projector can obtain a more accurate projection grating fringe. The resolution of the optical grating is 500 nm, which means the width of 1 mm can contain 2000 grating fringes. For excellent projection conditions, the projector can obtain a contrast ratio of over 1:100. The number of grating fringes could not be directly counted due to deficient or discrete data. Gray coding [19,20,21] can be used to estimate the discrete number of gratings in this method. The space coordinates of each point can be obtained by using the phase shift method after the point phase on the images is obtained by using Gray coding. Each corresponding pixel can obtain one intensity vector, and then the relative phase $\Delta \phi$ of the pixel can be obtained by using the intensity vector. In one cycle of phase shift mode, the relative phase is the only one. The traditional phase formula under a phase shift of 90° according to traditional methods [22,23,24]:

$$\phi ={\mathit{tan}}^{-1}{\displaystyle \frac{I_1-I_3}{I_2-I_4}} -\pi <\phi <\pi$$

(17)

where $I_i$ ($i$ = 1, 2, 3, 4) is light intensity.

However, the absolute phase φ can be represented in Equation (18) by the number of optical gratings Z obtained from Gray coding and the relative phase $\Delta \phi$ obtained by the phase shift method [25]:

$$\phi =2\pi \cdot Z+\Delta \phi$$

(18)

Relevant parameters ($u,\nu , \mathrm{and} \phi$) and the corresponding object surface point (x, y, z) of recorded pixels (u, v) can be obtained from two calibrated cameras. However, as the error between the measured phase and actual phase is very large using the four-step phase-shift method, it cannot obtain the exact phase coordinate $\phi$. Therefore, to obtain the phase coordinate $\phi$ more accurately, the object phase transfer function (OPTF) should be decided first. The OPTF is a linear function in the ideal grating projection system, which represents the relation between the actual object phase $\mathsf{\Phi }$ and measured object phase $\phi$. It can be obtained in Equation (19):

$$\phi \left(\Phi \right)=\Phi$$

(19)

The fundamental principle of grating projection measurement is the triangulation measurement method, as shown in Figure 5a. In this method, when the optical grating from the projection reaches the object surface, each grating can be defined as one projection plane, as shown in Figure 5b. The distorted optical grating can be recorded by two cameras from two sides, and each pixel in the recorded images can be defined as a camera ray, which is a line formed by connecting the transillumination center and its corresponding light point on the object. According to calibrated results, phase values φ, and the other coordinates, the three-dimensional coordinate of the object point can be obtained by using the triangulation measurement method, to ignore the process of projector calibration and to prompt the measurement precision.

3.2. Numerical Modeling

For the numerical model, its first step is to apply initial stress by using the *INITIAL CONDITIONS, TYPE = STRESS* function in ABAQUS (2022), and then in the second step, the Lanczos method is used to carry out the eigenvalue analysis using the “EIGENSOLVER = LANCZOS” parameter in the “*BUCKLE” option. Some selected global deformation modes are combined and introduced into the structure as a geometrical imperfection. The imperfection is defined through the “IMPERFECTION” option provided by ABAQUS:

$$\Delta z={{\displaystyle \sum }}_i{\omega }_i{\varphi }_i$$

(20)

where φ_i is the ith eigenmode, and ω_i is a scaling factor whose magnitude is chosen as a proportion of the thickness of the membrane. Here, the value of ω_i is set to 25% of the thickness throughout the simulation. The third and final step is a geometrical nonlinear “*NLGEOM” incremental analysis. Prestress is also needed to provide a small, initial out-of-plane stiffness to the membrane, but the value should be set small enough so that the final results are not affected. The “STABILIZE” function is activated for this step.

A four-node, reduced-integration shell element (S4R5) was selected to mesh the membrane surface. To exploit the bilinear characteristics of the chosen S4R5 shell elements and to capture wrinkles for subsequent data processing, a uniform 1 mm × 1 mm mesh was adopted. Because the structure is subjected only to in-plane shear, and the edge frame is fabricated from homogeneous stainless steel, two-dimensional linear beam elements (B21) are employed for the frame, while the flexible edge cables are modeled with three-dimensional two-node truss elements (T3D2). In the actual specimens, the adhesive in the transverse seams is very thin (≈0.1 µm); therefore, the numerical model accounts only for membrane overlap and neglects the adhesive. During the testing, the upper and lower edges of the membrane specimen are gripped by five distributed fasteners acting through upper and lower clamping plates, resulting in intermittent edge restraint. This boundary condition is reproduced in the model by imposing piecewise boundary constraints. The physical parameters of the materials are listed in Table 1.

3.3. Validation

The wrinkling behaviors measured via a three-dimensional scanner of the type OKIO-V-400 are shown in Figure 6. The prestress level of plane membrane structures is 1.1 kg. According to Figure 6, the wrinkling behaviors of pane membrane structures with a horizontal glue seam can be observed, and the wave magnitudes of the membrane diagonal can be shown in Figure 6b. The wrinkle pattern along the diagonal under shear displacement of 1.6 mm from experimental and numerical results is given in Figure 7. According to Figure 7, the wrinkle pattern exhibits good agreement between the numerical and experimental results. The maximum sway amplitudes of the experimental and numerical results are both 1.8 mm.

Table 2. Wrinkling characteristics under different shear displacements.

Shear Displacement (mm)	Shear Force (kg)	Number of Wrinkles	Max-Z (mm)	Min-Z (mm)	E (Z) (mm)	D (Z) (mm)
0.6	1.060	3	0.761	−0.714	−0.0867	0.399
0.9	1.160	4	0.883	−0.598	0.0304	0.266
1.1	1.605	5	0.892	−0.849	−0.0293	0.320
1.6	2.665	5.5	1.165	−1.038	−0.0463	0.485
3.1	3.800	6	1.442	−1.168	−0.0213	0.569

shows the wrinkling characteristics under different shear displacements, including maximum displacement, minimum displacement, average displacement, and mean square error displacement. The shear displacements are 0.6 mm, 0.9 mm, 1.1 mm, 1.6 mm, and 3.1 mm, respectively. According to Table 2 and Figure 6, the magnitude of wrinkling with shear displacements increases from 0.6 mm to 3.1 mm, and wrinkling amplitudes of the membrane surface rise from 1.48 mm to 2.61 mm; the mean square error displacements of membrane structures in the z-direction go up from 0.27 mm to 0.57 mm. Therefore, it can be concluded that membrane surface wrinkles increase more obviously with shear displacements at the same prestress level in the membrane surface.

4. Load-Bearing Capacity of Inflated Cantilever Beam

4.1. An Analytical Solution

Inflated membrane structures such as airship envelopes are usually designed as cigar shapes, pencil shapes, or oval shapes, and their slenderness ratios are in the range of 4 to 6. These inflated membrane structures are made of flexible, lightweight, high-strength membranes, which can become stable structures after being inflated and are similar to the model of an inflated cantilever beam. Thus, the model of an inflated beam can be considered as the reference to validate the numerical simulation method of inflated membrane structures. The boundary conditions are illustrated in Figure 8. The left end is fully clamped. All translational and rotational degrees of freedom are fixed, while the right (free) end is subjected to either an applied load or a prescribed displacement without additional restraints.

In the following figure, θ₀ is the angle of the wrinkling area in relation to the location of σ₀ in the cross-section of the inflated tube, σ_m is the maximum stress of the tensile area on the cross-section of the inflated beam, σ₀ is the minimum stress of the compressive area on the cross-section of the inflated beam, F is the vertical load, p is the internal pressure of the inflated beam, and R is the radius of the cross-section of the inflated beam.

To obtain the analytical solution to the model, several assumptions are proposed as follows: (A) internal pressure in the inflated beam is constant; (B) the shape of the inflated beam can remain stable due to the internal pressure within it; (C) plane cross-section assumption, which means that the cross-section of the inflated beam can be kept in plane under internal and external loading; (D) stress distribution varies linearly in the direction of the cross-section; and (E) the membrane material of the inflated beam can be available in the elastic stage. The analytical model of the inflated cantilever beam is shown in Figure 8. According to the above assumptions and the three-dimensional Bernoulli beam theory, the circumferential and longitudinal stresses on the inflated beam are given as follows:

$$f_1=pR$$

(21)

$$f_2={\displaystyle \frac{pR}{2}}$$

(22)

where f₁ and f₂ are the circumferential and longitudinal tensile forces of the inflated beam, respectively, and p is the internal pressure of the inflated beam.

For the inflated cantilever beam, it is fixed on the left side, and a vertical load is applied, as seen in Figure 8. The longitudinal stress of the inflated beam σ_a can be expressed as Equation (23):

$${\sigma }_a={\displaystyle \frac{pR}{2t}}-{\displaystyle \frac{FxR}{I}}$$

(23)

$$I=\pi R^{3}t$$

(24)

where t is the thickness of the inflated beam membrane, R is the radius of the cross-section of the inflated beam, I represents geometric stiffness, and F is the vertical load.

The inflated beam is susceptible to wrinkling when σ_a is equal to zero. Thus,

$$x={\displaystyle \frac{\pi p{R}^3}{2F}}$$

(25)

Due to the assumed plane cross-section, the stress distribution varies linearly. σ_m is the maximum stress of the tensile area on the cross-section of the inflated beam, and σ₀ is the minimum stress of the compressive area on the cross-section of the inflated beam. The stress, σ, of the tensile area on the cross-section of the inflated beam is given in Equation (26):

$$\sigma ={\sigma }_0\left({\displaystyle \frac{1+\mathit{cos}\theta }{2}}\right)+{\sigma }_m\left({\displaystyle \frac{1-\mathit{cos}\theta }{2}}\right)$$

(26)

whereas for πpR³/2F < x < l, the stress, σ, of the tensile area on the cross-section of the inflated beam is described as follows:

$$\sigma =\left({\displaystyle \frac{\mathit{cos}{\theta }_0-\mathit{cos}\theta }{1+\mathit{cos}{\theta }_0}}\right){\sigma }_m \pi >\theta >{\theta }_0$$

(27)

$$\sigma =0 {\theta }_0>\theta >0$$

(28)

where θ₀ is the angle of wrinkling area in relation to the location of σ₀ in the cross-section of the inflated tube, θ₀ should not be equal to zero but should be close to zero, since the wrinkling on the cross-section of the inflated beam develops with load increase, but it could not fully cover the cross-section of the inflated beam.

Due to the equilibrium of the bending moment and stress in the longitudinal directions of the inflated beam, Equations (29) and (30) can thus be obtained:

$$Fx=-2{{\displaystyle \int }}_0^{\pi }\sigma h{R}^2\mathit{cos}\theta d\theta$$

(29)

$$p\pi R^2=2{{\displaystyle \int }}_0^{\pi }\sigma hRd\theta$$

(30)

Thus, the relationship between bending moments from the force, F, at the beam end, shown in Figure 8, and the angle of wrinkling can be deduced using Equations (26)–(30):

$${\displaystyle \frac{Fx}{p{R}^3}}={\displaystyle \frac{\pi \left(2\pi -2{\theta }_0+\mathit{sin}2{\theta }_0\right)}{4\left[\mathit{sin}{\theta }_0+\left(\pi -{\theta }_0\right)\mathit{cos}{\theta }_0\right]}} {\displaystyle \frac{\pi p{R}^3}{2F}}<x<l$$

(31)

Thus, the critical bending moment M_k and the ultimate bending moment M_u can be obtained as follows:

$$M_k=Fx={\displaystyle \frac{1}{2}}\pi p{R}^3 \theta =0$$

(32)

$$M_u=Fx=\pi p{R}^3 \hspace{1em}\hspace{1em}\theta =\pi$$

(33)

where R is the radius of the inflated beam, and p is the inner pressure of the inflated beam. The critical bending moment, M_k, represents the minimum bending moment when wrinkling starts to occur, and its corresponding beam edge force is the critical wrinkling force. The ultimate bending moment, M_u, refers to the maximum bending moment when wrinkling covers the circular cross-section of the inflated beam, and its corresponding beam edge force is the ultimate wrinkling force.

4.2. Numerical Simulation

In this section, numerical simulation is implemented by importing a FORTRAN user subroutine (named UMAT) linked to ABAQUS codes. Figure 9 displays the process of using UMAT codes. According to the flow chart, wrinkling analysis based on the IMP method can be numerically simulated by using the UMAT code to form a stiffness matrix and to solve the constitutive equations for integration points of all elements.

To verify the IMP method and the correctness of the UMAT program in numerically simulating wrinkling behaviors of the inflated beam, two FE models of an inflated cantilever beam made using M3D4 elements and UMAT-modified M3D4 elements are compared. An element length of 25 mm is selected. The geometric data for the numerical models are shown in Figure 10, and their material properties are listed in

Table 3. Material properties of ETFE.

Membrane	Density (kg/m3)	Elastic Modulus (MPa)	Poisson’s Ratio	Thickness (μm)
ETFE	1700	810	0.31	250

. The internal pressure of the inflated cantilever beam is 4 kPa. The parameters of the FE model are identical to those in Figure 10. The measurement locations of three points, A, B, and C, on the inflated beam are displayed in Figure 10. The models simulated using the M3D4 elements and UMAT-modified M3D4 elements are referred to as Models i and ii, respectively. The left side of the inflated cantilever beam is fully fixed. The load can be applied in the loading strip of the inflated beam, as displayed in Figure 10. For numerical analysis, the first step is to apply internal pressure at a magnitude of 4 kPa in the FE model. Then the loads are applied to the loading strip as the second step.

In the numerical simulation, the enclosed-air pressure was modeled in ABAQUS by applying a pressure function within a nonlinear analysis. The boundary conditions replicate those of the experiment: the left end is fully clamped.

The contour plot of the von Mises stress of Models i and ii at an internal pressure of 4 kPa is shown in Figure 11. As wrinkling of the models does not occur at this stage, the numerical results for Models i and ii at an internal pressure of 4 kPa are identical to each other. Thus, Model ii can be effectively used to estimate the mechanical characteristics of an inflated cantilever beam at the wrinkling stage. At the wrinkling stage, wrinkles gradually evolve from the left-side constrained area as the load is increasingly applied to the loading strip.

Contour plots of circumferential stress in most areas for Models i and ii are shown in Figure 12 and Figure 13. The positive magnitude stress in the upper inflated tube is tensile stress, and the negative magnitude stress in the lower inflated tube is compressive stress. The maximum circumferential compressive stress and maximum longitudinal compressive stress are present in both models and are 83.8 kPa and 2.9 kPa, respectively, in Model i, but they are 1.05 MPa and 2.8 MPa, respectively, in Model ii. The tensile stress of Model ii is 1.95 times that of Model i when the load reaches the ultimate load-bearing capacity of the models. The stress distributions of circumferential stress for Models i and ii are identical to each other, except for several elements having greater stress magnitudes for Model ii, as can be seen in Figure 12 and Figure 13. Figure 14 shows the load-deflection curves of Models i and ii. The ultimate load-bearing capacity of Model i is clearly 1.22 times that of Model ii, as can be seen in Figure 14, since the M3D4 membrane elements in Model i have a certain compressive stress-bearing capacity.

4.3. Experiments

To verify the numerical method of wrinkling simulation, a series of experiments has been performed. An inflated cantilever beam made of ethylene tetrafluoroethylene (ETFE) membrane was manufactured as a test model, and its material properties are displayed in Table 3. ETFE copolymer exhibits excellent performance of light transmission and a wide range of usage temperatures, which is appropriate for the building and aerospace industries. The dimensions of the inflated cantilever beam are shown in Figure 10. Figure 15 shows the test setups for recording the ultimate load-bearing capacity of the inflated cantilever beam. The fixed support of the inflated cantilever beam is welded onto a no. 20-type steel plate support, and the vertical plate and circular hollow section (CHS 426 × 12), at a length of 150 mm, onto a T-shaped fixed mount. The inflated tube is connected to the exterior polished circular hollow section by two clamps. The support is fixed via foundation bolts to the laboratory floor.

An air pressure controller is a key device that can affect the accuracy of the measurement of the load-bearing capacity of the beam. The pressure sensor shown in Figure 15a can convert the internal pressure of the inflated beam into an electronic signal picked up by a USB collection card, which is then converted by computer software into pressure magnitudes, and from these, it is evaluated whether the inflatable beam needs to be inflated continuously or not. After the inflated beam is fixed, it can be loaded continuously. The inflated beam should be checked for airtightness during the transition process from a flexible to a rigid beam as the first stage. Then, in the second stage, the magnitudes of the air pressure controller should be kept at a certain stable pressure when the internal pressure of the inflated beam reaches 3 kPa, 4 kPa, or 5 kPa, respectively. The load should be applied through the loading strip, as shown in Figure 15c. The locations of measurement points A, B, and C are displayed in Figure 10. In the final stage, loading increases very slowly, and if deformation of the inflated beam increases sharply, loading should be stopped. Wrinkling clearly evolves in the process of inflation, as shown in Figure 15d. This testing process is repeated three times, and the average values of experimental results are taken as actual values.

4.4. Results and Validation

The experimental load-displacement curves of the inflated beam at points A, B, and C under internal pressures of 3 kPa, 4 kPa, and 5 kPa are shown in Figure 16. According to Figure 16, the ultimate load-bearing capacity of the inflated beam under an internal enclosed-air pressure is, respectively, 52 N, 68 N, and 84 N. To verify the numerical method of wrinkling simulation, the load-displacement curves of the experimental model should be compared with those of Model ii at point C. Figure 16 displays the load-displacement curves for Model ii and the experimental model at points A, B, and C of the inflated beam with 3 kPa, 4 kPa, and 5 kPa enclosed-air pressure. The ultimate load-bearing capacity of Model ii using the UMAT-modified M3D4 elements is 68.6 N, as shown in Figure 16. According to Figure 16, a close agreement between the experimental and numerical results has been achieved for the inflated beam under three enclosed-air pressure states. The numerical results of the inflated beam at three internal pressures of 3 kPa, 4 kPa, and 5 kPa are, respectively, slightly less than the corresponding experimental results at their corresponding internal pressures.

From Equations (13) and (14), the ultimate wrinkling force can be obtained.

Table 4. Analytical, numerical, and experimental results of the load-bearing capacity of an inflated beam under 3 kPa, 4 kPa, and 5 kPa.

Internal Pressure (in kPa)	Analytical Results (in N)	Model ii (in N)	Experimental Results (in N)
3	54.2	49.5	52
4	72.2	66	68
5	90.3	82	84

shows the analytical, numerical, and experimental results of the load-bearing capacity of the inflated beam. According to Table 4, the ultimate wrinkling forces for the numerical and experimental results are very close to each other. For the ultimate wrinkling force, the discrepancy between the experimental model and numerical model under internal pressures of 3 kPa, 4 kPa, and 5 kPa is 4.8%, 2.9%, and 2.4%, respectively, which decreases with the internal pressure increase. However, the discrepancies between analytical and experimental results are, respectively, 4.2%, 6.2%, and 7.5%, which decline with the internal pressure increase. Thus, it is found that the numerical simulation method is effective in predicting the load-bearing capacity of an inflated membrane structure. For a higher internal pressure of the inflated beam, the accuracy of the numerical model is higher, whereas the analytical results decrease more.

5. Effect of Wrinkling on the Mechanical Performance of Flexible Airship Envelopes

5.1. Analysis of Load-Bearing Capacity

In the preceding numerical validation, the load-bearing test of the inflated cantilever beam demonstrated strong fidelity. The same methodology is now extended to a geometrically more intricate and engineering-relevant configuration—the airship. While the cantilever beam served as the benchmark for method verification, actual inflated membrane structures such as airships exhibit distinctive geometric characteristics—predominantly ellipsoidal envelopes with markedly higher slenderness ratios. A thorough understanding of wrinkling behavior within such configurations is therefore imperative to ensure structural integrity and operational performance. The effect of wrinkling on its mechanical performance of the flexible airship envelope as an inflated membrane structure can be studied based on the aforementioned simulation method.

Figure 17 shows the simplified ellipsoid airship envelope model of the long axis at a length of 30 m and of the short axis at a length of 8 m, according to the prototype model of a ZY-01 demonstration airship [15]. The internal pressure of this envelope model is 500 Pa. The boundary condition of this half airship envelope model is a fixed hinge constraint at the left side, as seen in Figure 17. A semi-circular load area with the width of 1000 mm on the main airship envelope is located away from the left end at a distance of 11,500 mm as shown in Figure 17a. The size of the FE model is meshed to a length of 200 mm. The skin thickness of the envelope model is 380 μm, and the elastic modulus of this membrane skin is 1380 MPa. The model created by the M3D4 element of ABAQUS(6.22) software is named Model A, whereas that created by the UMAT subroutine is named Model B. Then, a nonlinear analysis was performed. The internal pressure of 500 Pa was applied as an initial step, then the tensile force was applied at the load area as the second stage of the loading process to study the maximum bending moment of this envelope model.

Figure 18 and Figure 19, respectively, describe the vertical stress and deformation contour of Models A and B of the simplified airship envelope. According to Figure 18, maximum tensile von Mises stress occurs at the upper peak of the envelope model, ignoring wrinkling, and its magnitude is 8.1 MPa, whereas it is 10.8 MPa for the envelope model considering wrinkling. According to Figure 19, the maximum deformation of Model B is close to two times than that of Model A. Therefore, it is estimated that the neutral axis of the cross-section of the envelope model considering wrinkling moves upwards, and the envelope Model B at the upper peak is more vulnerable under external force than Model A. Figure 20 demonstrates the load-versus-deflection curves of envelope Models A and B at the upper peak. According to Figure 20, at the initial stage, it can be seen that the curves of Models A and B both vary linearly and are very close to each other. Then, the ultimate load-bearing capacity of Model A is greater than that of Model B, because Model B ignores the pressure bearing capacity, in accordance with Figure 20.

5.2. Effect of Wrinkling on the Stress Distribution

For an airship envelope, a hull and envelope with a suspended curtain are two common structural systems. The connection between the upper suspended curtain and envelope can be beneficial to reduce stress concentration in the envelope and cables composite structures. However, relevant research is still lacking in the field of composite structures of envelopes and suspended curtains. Thus, to facilitate the structural design of airships, the effect of wrinkling on the stress distribution of an airship with/without a suspended curtain should be studied.

5.2.1. Airship Without Suspended Curtain

An airship can fly and float due to self-weight and buoyancy, so the connection between the nacelle and envelope should be considered as constraint points to numerically simulate a static floating airship in an equilibrium system. Thus, the reaction against the resultant self-weight and buoyancy on the constraint points can be considered as an equilibrium force. To study the effect of wrinkling on the stress distribution of an airship, an FE model of a 75 m airship should be proposed. The dimensions and loading schematic diagram of a Zhiyuan-1 Airship are shown in Figure 20. The original line of a Zhiyuan-1 Airship is composed of three section polynomial functions [15].

The elastic modulus of the membrane material is 2000 MPa, and its thickness is 0.482 mm. A load at a magnitude of 15 kN should be applied to points C and D, respectively, on the envelope, as shown in Figure 21b. Due to the symmetry of an FE model of the airship, the envelope can be simplified into a half model; hence, two half loads at a magnitude of 7.5 kN are also applied to points C and D on the half-envelope model. The element size is decided at a length of 1.15 m. Point A should be simulated to be pinned at a translational degree of freedom in the x-, y-, and z-directions, whereas point B should be constrained at an out-of-plane translational and in-plane vertical degree of freedom, and its in-plane translational degree of freedom should be free. The FE Models i and ii of the airship envelope are, respectively, simulated by using M3D4 membrane elements and UMAT-modified M3D4 elements. Wrinkling can be imported to the envelope models by using UMAT-modified M3D4 elements and followed by a geometrically nonlinear analysis. The analysis should contain two loading steps; the first step is to apply internal pressure at a magnitude of 500 Pa, and a concentration load at a magnitude of 7.5 kN is applied as the second step.

Figure 22 shows the stress distributions of the circumferential and longitudinal stresses of Models i and ii at an internal pressure of 500 Pa. In the first loading step, stress distributions of Models i and ii are identical to each other due to no wrinkling. To propose the effect of wrinkling on the stress distribution of the airship, the load applied to points C and D is increased to 15 kN. Circumferential and longitudinal stresses for Models i and ii under the load of 15 kN can be obtained as seen in Figure 23. The contour plots of the circumferential and longitudinal stresses of Model i are shown in Figure 23. For Model i, its maximum pressure stresses in the circumferential and longitudinal directions are, respectively, 3.9 MPa and 12 MPa. Thus, airship envelopes have experienced wrinkling in the local area in accordance with the evaluation criterion of wrinkling seen in Section 3.1. Then, UMAT should be linked to ABAQUS codes to perform the effect of wrinkling on the stress distribution of the airship. Figure 24 displays the contour plots of the circumferential and longitudinal stresses of Model ii under a load of 15 kN. According to Figure 24, the maximum pressure stress of Model ii to i in the circumferential direction is reduced from 3.9 MPa to 0.2 MPa, and that in the longitudinal direction declines from 12 MPa to 0.8 MPa. Thus, the UMAT program is effective in redistributing stress in the local wrinkled area of the envelope.

5.2.2. Airship with Suspended Curtain

The details of an FE model of an airship with a suspended curtain are reported in the inventory data [15]. The geometrical data and simplified FE model of an airship with a suspended curtain are shown in Figure 25. Wrinkling may occur on the envelope under various loadings. In this part, two loading states are proposed for the airship models based on the IMP method, with two density differences of the envelope, internal and external, at 0.5 kg/m³ and 0.1 kg/m³, respectively, which are simulated by using an M3D4 membrane element and a UMAT-modified M3D4 membrane element, respectively named Models i and ii. The internal pressure of the airship envelope is 800 Pa.

Figure 26 shows the circumferential and longitudinal stresses of Model i at a density difference of 0.5 kg/m³. The circumferential and longitudinal stresses of Model ii at a density difference of 0.5 kg/m³ are displayed in Figure 27. According to Figure 26 and Figure 27, the maximum longitudinal pressure stresses of Model i and ii at a density difference of 0.5 kg/m³ are 1.39 MPa and 0.18 MPa, respectively, whereas their maximum circumferential stresses are identical to each other. Thus, longitudinal pressure stress of the envelope can be affected due to wrinkling. Figure 28 displays the circumferential and longitudinal stresses of Models i and ii at a density difference of 0.1 kg/m³. Contour plots of the circumferential and longitudinal stresses of Models i and ii are identical to each other. According to Figure 26, Figure 27 and Figure 28, wrinkling of the envelope can increase with a density difference increase, indicating that wrinkling can occur more easily at low altitudes.

6. Conclusions

In this paper, the iterative membrane properties method was introduced as a finite element method to numerically simulate the wrinkling of a membrane. A UMAT subroutine was also created and linked to ABAQUS codes to simulate the wrinkling of a membrane. The analytical solutions for the critical wrinkling force and ultimate wrinkling force of an inflated beam were obtained. Numerical simulation methods were verified by comparing with experimental load-bearing capacity and analytical results of an ETFE-membrane inflated beam with a length of 2.5 m. The following conclusions can be drawn:

• Wrinkling can also be numerically simulated by using wrinkling criteria for the iterative membrane property method. Numerical results for the load-bearing capacity of an inflated tube where M3D4 elements are employed are greater than those using UMAT-modified M3D4 elements, and the load-bearing capacity of a model based on the IMP method is closer to that of the experimental model. Thus, the load-bearing capacity of an inflated beam obtained using the IMP method should be adopted in structural design.
• The numerical results from Model ii using UMAT-modified M3D4 elements are closer to the experimental results than those from Model i. Thus, the outcomes of numerical Model ii are effective in predicting the load-bearing capacity. Moreover, intense wrinkling occurs more readily in the envelope at low altitudes.
• The effect of wrinkling on the stress distribution of the envelope model under external force could not be ignored. The effect of wrinkling on the circumferential stress of a flexible airship envelope could be ignored, whereas the effect of wrinkling on the longitudinal stress of a flexible airship envelope should be considered, depending on internal pressure and external force.
• For the analysis of the load-bearing capacity of an airship envelope, it can be seen that the curves of Models A and B both vary linearly and are very close to each other at the initial stage; however, the ultimate load-bearing capacity of Model A is greater than that of Model B, because Model B ignores the pressure bearing capacity.

For the proposed IMP-UMAT method, two limitations should be noted. Firstly, the model does not yet incorporate temperature-dependent material properties. Secondly, the study considered only quasi-static monotonic loading; the influence of repeated inflation–deflation cycles on wrinkle evolution and long-term durability has not been addressed.