## Abstract

A model is proposed based on a Fourier series method to analyse the interactive bending wrinkling behaviour of inflated beams. The whole wrinkling evolution is tracked and divided into three stages by identifying the bifurcations of the equilibrium path. The critical wrinkling and failure moments of the inflated beam can then be predicted. The global–local interactive buckling pattern is elucidated by the proposed theoretical model and also verified by non-contact experimental tests. The effects of geometric parameters, internal pressure and boundary conditions on the buckling of inflated beams are investigated finally. The results reveal that the interactive buckling characteristics of an inflated beam under bending are more sensitive to the dimensions of the structure and boundary conditions. We find that for beams which are simply supported at both ends or clamped and simply supported, boundary conditions may prevent the wrinkling formation. The results provide significant support for our understanding of the bending wrinkling behaviour of inflated beams.

## 1. Introduction

As one of the most typical membrane-like structures, inflated beams have been the subject of interest in recent years owing to their distinctive features. They are of much lower weight and cost, can be folded easily and deployed quickly. Hence, they are widely applied to terrestrial and aerospace structures [1]. However, an inflated beam is a typical thin-wall structure and instability phenomena, including global buckling, local wrinkling or both, may easily occur [2,3]. Investigations on the instability phenomena of these structures can be separated into two categories: global buckling analysis and local wrinkling analysis.

The global buckling, as its name suggests, analysis mainly focuses on a global buckling mode and critical buckling load. Euler–Bernoulli's kinematics has been widely adopted because of its simplicity and ability correctly to determine the structural response in many practical situations. Pioneering work was carried by Comer & Levy [4] who provided analytical expressions for the deflection in an inflated cantilever beam by means of Euler–Bernoulli's kinematics. Then, Main *et al*. [5] conducted experiments on cantilever-inflated beams and compared the results with Comer & Levy's theory [4]. Suhey *et al*. [6] considered a pressurized tube under uniformly distributed load in ocean aquaculture and theoretical results were also obtained assuming Euler–Bernoulli's kinematics. However, this theory did not consider shear and inflation pressure effects on inflated beams. In order to improve the previous formulations, many other authors preferred to use the Timoshenko beam theory, which is more appropriate for thin-walled beams. Fichter [7] published a paper in which the internal pressure appears in the deflection expression based on the minimization of total potential energy. By taking into account the deformed state of the beam, Wielgosz & Thomas [8,9] gave analytical expressions for inflated beams and panels using Timoshenko beam theory. In their approach, the force generated by internal pressure was treated as a follower force, which accounted for pressure stiffening effects. Le Van & Wielgosz [10] improved Fichter's [7] theory by using the principle of virtual work in the context of the total Lagrangian formulation and proposed solutions for the bending and buckling of an inflated tube. Davids & Zhang [11] developed Fichter's [7] results by considering the pressure work during the volume change in isotropic Timoshenko beams and accounted for fabric wrinkling via a moment-curvature nonlinearity. Apedo *et al*. [12] developed efficient numerical techniques for predicting the nonlinear load–displacement response of inflated beams by proposing a three-dimensional Timoshenko beam with a homogeneous orthotropic woven fabric. Barsotti & Ligarò [13] assessed the nonlinear elastic response of an inflated cylindrical beam through a simple mechanical model. The geometrical nonlinearities in the model due to both the cross-sectional ovalization and wrinkling were carefully considered.

Distinct from the global buckling analysis, local wrinkling is assumed to occur when the maximum compressive stress reaches the uniform compression critical value for a circular tube. Three models are used to describe the wrinkling of the inflated beam: the membrane model, the thin-shell model and the shell-membrane model. Haugthon & McKay [14] adopted the membrane model to predict the wrinkling characteristics of the inflated beam where the wrinkles occurred when the axial compression reached zero. In the membrane model, both the wrinkling and the collapse moments are independent of the material properties, and are only related to the structural cross-section size and inflation pressure. Then, Veldman [15] regarded the inflated beam as a thin shell and predicted the wrinkling and collapse moments utilizing the buckling critical load of a shell. In fact, the membrane and thin-shell models correspond to two perfect states, respectively, which are different from experimental results [16–18]. Therefore, Wang *et al*. [18] proposed a shell-membrane model using a modified factor based on the membrane and thin-shell models. Compared with previous models, the shell-membrane model showed a better correlation with the experimental results in predicting the wrinkling and collapse loads of inflated beams.

Inflatable structures do have additional stiffness, which comes from initial pressure, but do not contain the stiffness that comes from material and shape. Both buckling and wrinkling mean a deformation process in which a structure subjected to high compressive stress undergoes a sudden change in morphology at a critical load. When the beam is compressed axially, beyond a certain critical load it spontaneously bows outwards, which is called global or Euler buckling. Compared with global buckling, the characteristics of local wrinkling are wavelike and periodic. The above review of literature indicates that the global buckling analysis focuses on the prediction of global buckling characteristics, such as the critical bifurcation and collapse loads, the wrinkling region and the buckling mode. By contrast, the local wrinkling analysis focuses on the local wrinkling characteristics, such as the wrinkling amplitude, the wrinkling wavelength, the wrinkling number and the wrinkling configuration. Furthermore, most of the experiments on the instability phenomena concentrate on the detection of the global buckling characteristics [19,20] or of the local wrinkling characteristics [21]. Few papers focus on both of these features in theoretical or experimental studies. Consequently, there is a need to develop theoretical and experimental methods for describing the instability phenomena that couples global and local wrinkling characteristics. A Fourier series method, which corresponds to the periodic and sinusoidal characteristics of wrinkles correctly, is able to account for instability phenomena, such as bending instabilities of circular tubes [22,23], buckling of sandwich structures [24] or of a Winkler foundation beam [25].

In this paper, the Fourier series method is introduced to the Fichter's [7] theory to derive a Fourier model, which can characterize the interactive bending wrinkling behaviour of inflated beams. The nonlinear governing equations of the Fourier model are then solved by a Newton trust region method [26]. The whole wrinkling evolution is tracked using the proposed model and the predicted buckling patterns are verified by the non-contact experimental tests. Finally, the effects of geometric parameters, internal pressure and boundary conditions on the buckling of an inflated beam are investigated.

## 2. Methodology

### (a) Classical model of inflated beam

Inflated beams are considered to be thin-walled cylinders which depend on internal pressure for much of their load-carrying ability as shown in figure 1. The axial displacement (in the *x* direction), transverse displacement (in the *y* direction) and rotation angle about the *z*-axis are represented by *u*, *v* and *ω*, respectively. The parameters *L*, *r* and *t* are respectively the length, cross-sectional radius, and wall thickness of the inflated beam. In-plane means that a steady moment *x*-axis. According to Fichter's [7] theory, a system of equilibrium equations can be obtained from the principle of virtual work following the Timoshenko model:
_{str} and Π_{ext} represent the strain energy and external energy, respectively, while Π_{pres} represents the potential energy of the internal pressure due to deformation. We make the simplification of neglecting the circumferential strain and twist around the longitudinal axis and thus the strain energy Π_{str} of the inflated beam can be expressed as:
*ϵ*, γ and *k* are respectively the axial, shearing and bending strain of the inflated beam. As the inflated beam is made from isotropic polyurethane thin film, the strains and the corresponding loads (*N*, *S*, *M*) are defined in equations (2.3)–(2.4)
*E* and *G* are the elasticity and shear modulus, respectively. In addition, if the inflated beam is usually made from fabric, equation (2.4) can be replaced by an orthotropic constitutive equation to describe the orthotropic wrinkling behaviour in principle. However, in this manuscript, we make the simplification of neglecting the circumferential strain in this Fourier model. Therefore, the expression of orthotropic constitutive equation is almost similar to isotropic equation (2.4). The potential energy Π_{pres} is then given by
*V* is the change in enclosed volume due to deformation, *p* is the internal pressure. In addition, the external work by the applied load Π_{ext} is then given by
*f*_{1}, *f*_{2}, *f*_{3},

### (b) Fourier model of inflated beam

The Fourier series method in this part is adopted to investigate the interactive bending wrinkling behaviour of the inflated beam. The deformation response of the system is considered as the sum of a slowly varying mean field and a nearly periodic fluctuation. All the unknowns in §2a are written in the form of Fourier series
*U*(*x*) is the unknown field which contains functions *u*, *v* and *ω*. The Fourier coefficients *U _{j}*(

*x*) represent the envelope for the

*j*th harmonic, and the coefficients of the zeroth harmonic are real while the others are complex. The frequency

*q*is defined as

*q*= (π/

*L*)

*n*, where

*n*is the wavenumber.

The Fourier coefficients *U _{j}*(

*x*) vary slowly and are assumed to be constant over a period

*a*(

*x*) and

*b*(

*x*) are Fourier series with slowly varying Fourier coefficients as in equation (2.7), the following identities hold

By applying the above identities to equations (2.3)–(2.4), the force and strain fields of the new Fourier model are obtained as follows:

In the same way, the strain energy and potential energy can be expressed as

Of course, a more refined picture could be constructed by the inclusion of more Fourier terms, which would also results in a correspondingly more complex and difficult task to ensure numerical convergence. We consider just three envelopes (*j* = −1, 0, 1) as these are sufficient to reveal interactive buckling phenomena. *U*_{0} is identified as the mean value and represents the global deformation, while *U*_{1}, which is conjugated with *U*_{−1}, represents the amplitude of fluctuation for the local wrinkling. Then the energy can be expressed as:

As expected, equations (2.20)–(2.21) include the mean field *U*_{0} and the amplitude of the fluctuation *U*_{1}, which depicts the interactive instability of the inflated beam. On the other hand, the term

In this paper, the wrinkling behaviour of an inflated beam is studied under pure bending: moments are applied at both ends of the beam. As there is no applied external axial force and the applied moments are constant, the assumptions *v*_{1}(*x*) is real, which only considers amplitude modulation and disregards the evolution of phase. Then, the transverse displacement can be expressed as

By substituting equations (2.20)–(2.21) into equation (2.1), the final buckling governing equation of the inflated beam can be deduced as

### (c) Numerical implementation

The nonlinear governing equations (2.22), (2.29)–(2.32) are solved through an incremental Newton–Raphson numerical procedure and enhanced to enable the tracing of post-buckling equilibrium paths through a Newton trust region method [26]. In order to apply this method, the quadratic model *m* should be considered at first

Here, *g _{k}* is the gradient of the objective function

*f*(

*x*) at the current iteration point

_{k}*x*.

_{k}*s*is the unknown functions and the step size is required to be bounded by the trust region radius

*δ*. Both the Cauchy point (the minimizer of

_{k}*m*in the steepest descent direction) and the Newton point are utilized. The incremental step is then given by a combination of the Cauchy step with the Newton step, and the unknowns are determined until the norm of the scaled residual is less than a given tolerance. The computation is started by a damped Newton step. Here, the damping factor, which determines the first Newton step, is set to 0.5 to ensure good convergence. Meanwhile, an initial displacement imperfection is introduced as a geometric imperfection by applying a very small out-of-plane displacement at the centre to trigger global buckling. The value of the displacement imperfection is 10

^{−5}, and the numerical results show insensitively to the initial imperfection as long as it is small enough (approx. 10

^{−5}–10

^{−6}), which is also demonstrated by others [17,29].

## 3. Results and discussion

In this section, we will discuss the interactive bending wrinkling behaviour of the simply supported inflated beam under bending without restriction on axial displacement (*v*_{0}) and local wrinkling (*v*_{1}) using the numerical method in §2c. According to the following experiments, the material and structural parameters of the inflated beam are listed in table 1.

### (a) Theoretical prediction and experimental verification

The values of moment and deflection values are normalized by *t*, respectively, in order to be given in dimensionless terms. Here, the critical wrinkling moment of shell model

Then, the dimensionless moment and deflection can be obtained as

Figure 2*a* illustrates the deformation patterns of the inflated beam under two different applied moments (*m* = 0.37 and 0.93). It shows that the structure is unstable when the applied moment is 0.37 and global bending deformation dominates the pattern without any local wrinkles. As the applied moment increases to 0.93, local wrinkling is observed and the buckling pattern is the typical interactive buckling between the global bending deformation and local wrinkling. Figure 2*b* shows a magnified view of the local wrinkles regardless of the global deformation. As seen from the diagram, local wrinkles distribute along the whole inflated beam and the amplitude of these wrinkles is almost the same except for those near the boundary. Wrinkles at the edges are smaller than those in the middle owing to the boundary conditions which restrict the evolution of local wrinkles.

In order to verify the theoretical results, a non-contact testing scheme is proposed to measure the buckling pattern of the inflated beam. In the experiment, an inflated beam, which is made from polyurethane thin film, is fixed onto the bending test apparatus (as shown in figure 3). The properties of polyurethane thin film are Young's modulus of 200 MPa, Poisson's ratio of 0.3 with 50 µm thickness. The length and cross-sectional radius are 0.942 and 0.025 m, respectively. The bending moment is carried out by rotating a support disc gradually. The buckling pattern is captured using VIC-3D and T-Scan laser tracker instruments (as shown in figure 3). The global bending deformation belongs to a macroscale and is captured using VIC-3D instrument based on the Digital Image Correlation (DIC) technique, which is an effective and reliable method to test the displacement field with an approving precision [31]. There are four main steps in the VIC-3D wrinkling measurement: (i) random speckles made on the inflated beam surface, (ii) CCD camera calibration, (iii) capture photos, and (iv) images post-processing. The local wrinkling configuration belongs to a microscale and is captured using T-Scan laser tracker instrument based on the laser interferometer technique. The instrument can obtain the wrinkling characteristics accurately with a measurement precision of 1 µm and a speed of 7000 points per second.

The interactive bending wrinkling pattern is observed and tested in the experiment (as shown in figure 4). The buckling pattern of the inflated beam under bending behaves as an interactive feature of the global bending deformation and local wrinkling configuration. The experimental results of the global deformation and local wrinkling configuration under 0.25 N m moment are respectively shown in figure 4*a*,*b* The tested global deformation (0.045 m) agrees with the theoretical result (0.041 m) within a difference of 9.7% (as listed in table 2). In addition, the ratio of the mean wrinkling amplitude *A* to the mean wrinkling wavelength λ is defined as *γ* = *A*/λ to characterize the local wrinkling level. Large *γ* ratio changes mean the high wrinkling level. Obtained from figure 4*b* and table 2, the tested ratio *γ* is 0.23, which is close to the theoretical result (0.20).

In addition, the global deformations and the ratio *γ* = *A*/λ (denoting the wrinkling level) under different applied moments are tested and compared with theoretical predictions in table 2. Accompanied by the increasing moment, the ratio *γ* is increased, indicating the rise in the wrinkling level. The tested global deformations are close to the theoretical results within a mean difference of 9.9%. The difference between the experimental and theoretical results may be because the Fourier model is established on a natural state rather than the deformed configuration and the effects of the follower loads are neglected. Moreover, some inevitable limitations in test techniques may also cause the differences between the experiment and theory.

### (b) Wrinkling evolution analysis

Figure 5 shows the moment–deflection response of the inflated beam under bending. Observed from figure 5, there are two critical points, which are named as the critical wrinkling point (marked as A) and the failure point (marked as C). The critical wrinkling load corresponds to the point where a nonlinear equilibrium path departs from the initial linear one. The failure point is determined as the following intersection between the bifurcation path and the primary path. Based on the critical points in the equilibrium path, the wrinkling evolution can be divided into three stages, which are named as I, II, III. Stage I (OA) is named as a pre-bucking stage and a linear response characterizes the primary path. In this stage, without any local wrinkles *M*_{w}. When the applied moment reaches *M*_{w}, an obvious bifurcation occurs at the point A and the primary path turns to the bifurcation path. Then the wrinkling evolution goes into Stage II, which is identified as the post-buckling stage. The wrinkling strain (the last term in equation (2.22)), which has an abrupt increase due to the local wrinkling formation, is adopted to reflect the first bifurcation. The local wrinkling phenomenon appears in this stage and the pattern characterizes as a typical interactive buckling between the global deformation and local wrinkling. Note that the applied moment has slight effects on the structural deflection in the initial post-wrinkling stage (AB). From the view of the energy equilibrium (equation (2.1)), the wrinkling strain energy offsets the increase of external energy. Therefore, in this stage, the wrinkles grow in a relatively stable state and the global bending deformation remains almost stable. In response to the increasing applied moment, the global deformation and local wrinkling increases simultaneously, which results in a continued monotonic increase in the moment–deflection curve (BC).

When the moment exceeds the failure point C (the failure moment *M*_{f} = 1.19), the wrinkles eventually localize and cause catastrophic failure in the form of sharp local kinks (as shown in figure 5) which promotes the wrinkling evolution into the failure stage (CD). Recall that the results of the Fourier model are obtained under the assumption that the wrinkling configuration has a slow and nearly periodic fluctuation. Thus, the assumption is invalid in the last stage due to the existence of sharp local kinks.

### (c) Critical wrinkling moment and failure moment

In this part, the predicted critical wrinkling and failure moments are compared with the results from previous models, including the membrane model [14], the thin-shell model [15] and the shell-membrane model [18]. The comparisons (in figure 6) show that the critical wrinkling (0.78) and failure moments (1.19) lie among the results predicted by the shell and membrane models.

In the Fourier model, the structural failure corresponds to the occurrence of sharp local kinks before the wrinkles distribute around the whole cross-section of the inflated beam. However, in the membrane and thin-shell models, the structure fails when the wrinkling angle reaches 2*π* (the whole hoop wrinkles) which does not comply with the practical case. Different from the membrane and thin-shell models, the failure of the shell-membrane inflated beam occurs when the wrinkling angle approaches 1.5*π*, which shows a better correlation with the experimental results and the Fourier model.

### (d) Internal pressure and structural size

The effects of the internal pressure and structural size on the wrinkling behaviour are carried out and shown in figure 7. The moment–deflection responses of the inflated beam with 0.025 m radius for different internal pressures are shown in figure 7*a*. The increasing pressure results in a slight reduction of the deflection and a slight increase of the critical wrinkling and failure moment. Specifically, for a 40% variation on the assumed pressure of 5 kPa, the critical wrinkling moment changes only by 9.52% and the failure moment by 6.45%. It also shows that the critical wrinkling and failure moments are proportional to the internal pressure due to the improvement of the transverse shear stiffness, which is described by equation (2.32). This is in line with the conclusions in the literature [8].

Figure 7*b* depicts the moment–deflection responses of the inflated beam with internal pressure of 5 kPa for different radii (0.015, 0.025 and 0.035 m). An increasing radius results in a large reduction of the deflection and an obvious increase of the critical wrinkling and failure moment due to the improvement of structural stiffness. Specifically, the critical wrinkling and failure moments have a more than three times improvement for a 40% increase on the assumed 0.025 m radius. The effects of radius on the critical wrinkling and failure moments are more significant compared with the cases of internal pressure. In addition, for a slender inflated beam, the sharp local kinks appear quickly after bifurcation due to a slower bending stiffness and a rapid wrinkling evolution. In such cases, the inflated beam rapidly fails soon after it is wrinkled.

### (e) Boundary conditions

Figure 8 shows the buckling patterns with the same applied moment (0.25 N m) under four cases with different boundary conditions: simply supported and sliding simply supported (S-SL), simply supported and simply supported (S-S), clamped and sliding simply supported (C-SL), clamped and simply supported (C-S) (note that we refer to sliding which is sometimes elsewhere referred to as rolling). The results in figure 8 reveal that the buckling pattern of the inflated beam under bending is closely relevant to the boundary condition. The symmetrical boundary conditions lead to a symmetrical buckling pattern and the same situation is met for non-symmetrical boundaries.

Global deformation dominates the configuration under the S-S or C-S boundary conditions and there is no wrinkle in these two cases. The configuration under the S-SL or C-SL boundary condition behaves with a typical interactive pattern between the global bending and local wrinkling. Compared with the other two boundary conditions (S-S and C-S), the S-SL and C-SL boundary conditions have no restrictions on the axial displacement (

## 4. Conclusion

In this study, the Fourier series method is introduced to the Fichter's theory to derive a Fourier model, which can characterize the interactive bending wrinkling behaviour of inflated beams. The whole wrinkling evolution can be tracked using the proposed model and the predicted buckling patterns are verified by the experimental tests. The bending wrinkling behaviour of the inflated beam is closely related to its internal pressure, relative dimensions and boundary conditions. Several important conclusions are drawn as follows.

The wrinkling evolution can be divided into three stages based on the critical wrinkling point and the failure point in the equilibrium path. In the pre-buckling stage, global bending dominates the deformation pattern of the inflated beam without any local wrinkles and a linear response characterizes the primary path. In the post-buckling stage, the local wrinkling phenomenon appears and the pattern characterizes as a typical interactive buckling between the global deformation and local wrinkling. Finally, the formation of sharp local kinks transforms the wrinkling pattern into the failure stage and accelerates the collapse of the structure.

In the Fourier model, the predicted critical and failure moments are respectively 0.78 and 1.19 and the failure wrinkling angle is amount to 1.52*π*. Moreover, the effects of radius on the critical wrinkling and failure moments are more significant than that of internal pressure. The buckling pattern is quite sensitive to the boundary condition. A larger radius and internal pressure of the inflated beam with an S-S or C-S boundary condition will be a desired consideration to control wrinkling characteristics and evolution. In addition, the inflated beam is made from isotropic polyurethane thin film in the experiment. Therefore, the isotropic constitute equation is adopted here and the orthotropic mechanical behaviour of inflated beam made from fabric will be focused on investigation in the future.

## Authors' contributions

Y.P.L. conceived the mathematical models, performed the experimental measurements and drafted the paper. C.G.W. proposed the idea of the study, funded and supervised the project. H.F.T. and M.K.W. revised the draft. All authors regularly discussed the progress during the entire work.

## Competing interests

We declare we have no competing interests.

## Funding

This work is supported by National Natural Science Foundation of China, 11172079 and 11572099; Program for New Century Excellent Talents in University, NCET-11-0807; Natural Science Foundation of Heilongjiang Province of China, A2015002; the Fundamental Research Funds for the Central Universities, HIT.BRETIII.201209 and HIT.MKSTISP.2016 29.

## Acknowledgement

The support from NSF is gratefully acknowledged. The authors also would like to thank four anonymous reviewers for their encouragement and helpful comments on an earlier version of the paper.

- Received June 21, 2016.
- Accepted August 4, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.