1. Introduction
Inflated membrane structures are commonly fabricated using continuously woven or braided composite fabrics, which are inflated with a sufficient quantity of gas to maintain their shape and structural rigidity [
1]. The excellent physical characteristics including lightweightiness, compact transportation volume and rapid deployment widen their application in the civilian and aerospace fields [
2]. Additionally, inflated membrane structures also exhibit fail-safe tendencies (reversible wrinkling) which endow them to recover the original shape and loading capacity once the load is removed [
3].
In recent years, due to the requirements of maritime rescue and military logistics for long-distance delivery and rapid response, as shown in
Figure 1, the U.S. military took the lead in proposing floating inflated structures including the Lightweight Modular Causeway System (LMCS) [
4,
5] and the Rapidly Installed Breakwater (RIB) [
6]. The inflated structures provide unique solutions for temporary structures in marine engineering. These new-type structures possess significant load-carrying capacities per unit weight and the efficiency of storage, transportation and deployability is improved by the application of inflated bladders. However, nonlinear material characteristics and fluid-structure interactions meet significant challenges in designing these structures. In particular, the difficulty would be enlarged when analyzing the failure behaviour of inflated structures because of their pressure-dependent natural characteristic. Therefore, research on the bearing capacity of the inflated structures is one of the key factors to guide the design, and it also has great significance for promoting the application of the inflated composite fabric membrane structure in marine engineering.
Inflated membrane structures play the role of flexural members in engineering applications; thus, researchers always simplify them as “beams”. Up to now, some researchers have studied inflated beams’ bending behaviour. For instance, Wielgosz and Thomas [
7,
8] proposed a finite element model of the inflated beam based on the Timoshenko beam theory and simplified the inflated gas pressure as a following force. Apedo et al. [
9] proposed a 3D
FEM utilizing the virtual work principle to predict the bending deflection of orthotropic braided beams. The accuracy of the method was verified by comparing it with experiments. Subsequently, Nguyen et al. [
10] improved Apedo’s model [
9] by associating the equation with the state of inflated prestress. Furthermore, Elsabbagh [
11] improved the model of Apedo [
9] to analyse the effect of the internal pressure and the diameter on the wrinkling load of the inflated beam, and compared the result with the calculation example of Apedo, founding that the deviation in the linear state was satisfied. However, the above methods have not presented satisfying accuracy in the post-wrinkling stage.
Apart from the global bending behaviour, the structural failure induced by local wrinkling has also gained particular interest. Wrinkling refers to the formation of small, localized folds or creases in the fabric of the inflated membrane structure. It could occur as a result of changes in the tension or shape of the fabric [
12]. Stein and Hedgepeth [
13] proposed the membrane model to predict the wrinkling load, assuming that fabric compressive stresses were not admissible. The wrinkling and failure loads were only dependent on the internal pressure and the mean radius [
14]. Furthermore, Veldman [
15] developed a thin-shell model and accounted for the material properties to analyse wrinkling loads. Wang et al. [
16] improved the thin-shell model of Veldman and proposed the concept of the wrinkling factor to estimate the critical wrinkling load. Then, Ji et al. [
17] established a multi-scale wrinkling analysis model to investigate the bending stiffness and load–deflection behaviour of inflated beams subjected to bending loads and validated the accuracy of the prediction model by comparing it with experimental results. Wang et al. [
18] conducted a series of experiments to investigate the interaction between gas and membrane, and they found the wrinkling load of the inflated beam was sensitive to the gas-membrane coupling relationship. In the previous works, the coupling effect of internal gas and structures due to the external load of the inflated membranes is ignored. In fact, most of the above-mentioned studies simplified the inflated gas pressure by considering it to be a uniform surface pressure. However, the mechanical characteristics of inflated composite fabric membranes, unlike the traditional composite structure, exhibit high nonlinearity due to internal pressure dependence [
19]. Moreover, the internal pressure increases significantly with an increase in the load [
20]. In this study, thus, the fluid cavity method is adopted to simulate the coupling effect between internal gas and structure, allowing for an accurate prediction of the structural response under external loads.
The purpose of this work is to investigate the bending and failure behaviour of inflated composite fabric membranes and to characterize the effect of pressure, membrane material and the coupling effect of internal gas and structures. Thus, four-point bending experiments were conducted to reveal the structural response behaviour of the inflated membranes subjected to bending load. In parallel, the fluid cavity method was adopted to establish a FEM with the material properties determined from the tensile tests. Subsequently, the bending and failure behaviour was analysed by both experiments and FEM, and the effects of the internal pressure and wrinkles were also discussed. Once the FEM reliability has been verified by comparing it with the experimental result, a simulation plan would be designed by design-of-experiment (DoE) to fit an empirical expression of the ultimate bending loading. Finally, the empirical expression was adjusted and compared with numerical predictions to verify its reliability.
4. Conclusions
In this research, the bending and failure behaviour that the inflated composite membrane structure represents is investigated by experimental and numerical methods used in marine engineering. The reliability of the numerical model was verified by comparing it with the experimental result, then a simulation plan was designed by design-of-experiment (DoE) to develop an empirical expression for the ultimate bending load. The main conclusions are obtained as follows:
(1) Strong agreement was achieved between numerical results and experimental measurements for both global and local analyses. The FEM exhibited remarkable precision in characterizing the response of the inflated composite membrane structure under bending loads, with ultimate loads and equivalent structural stiffness of deviation ranging from 1.65% to 4.57% and 1.81% to 4.83%, respectively, relative to the experimental measurements.
(2) The bending mode was described from different stages observed during the loading process. Under the initial load, the deflected curve transited smoothly along the axis of the inflated sample, which shifted into the trilinear model with the load increasing to the wrinkling load. In parallel, the bearing capacity of inflated structures decreased significantly with the appearance of wrinkles, and the evolution of wrinkles would induce a special failure mode of inflated structures: that was, the instability of the structure caused by local buckling. This behaviour was also verified in the load–deflection curves which exhibited an initial linear response followed by softening behaviour.
(3) The conducted parametric analysis was based on the ANOVA of responses recovered from a CCF-DoE plan of simulations for ultimate load, which adjusted the expected model to a reduction cubic model by eliminating part of insignificant interaction terms. The deviation values between ultimate loads in the RMSE and those obtained from FEM were less than 5%. Therefore, the proposed design expression presented a viable alternative to the FEM for evaluating the ultimate loading of the inflated composite membrane structure for the interesting range.
In this study, we have solely focused on the ideal four-points bending conditions. However, due to the application of inflated membrane structures in complex marine engineering environments, diverse boundary and loading conditions require further investigation. Therefore, it is important to conduct further research on this matter to address the challenges faced by these structures when subjected to real engineering conditions.