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Large deflection analysis is crucial for understanding and predicting the mechanical performance of flexible beam structures, which can be used to analyze metamaterial unit cells simplified into flexible beam structures. This paper investigates the large deflection behavior of an inclined cantilever beam with its freedom end subjected to a dead load. Firstly, considering the geometric nonlinearity of the beam and the influence of boundary conditions, a mathematical model of the beam is established and solved. Secondly, equations for the deflection curve and strain energy are derived, expressed in the semi-analytical form of elliptic functions. Then, a program is developed using Riemann integration combined with the bisection method to iteratively obtain the final calculation results. Finally, the calculation results of this paper are compared with those obtained by the nonlinear finite element method, thereby validating the accuracy of the proposed algorithm.

Thin rectangular plates are considered basic structures in various sectors like aerospace, civil, and mechanical engineering. Moreover, isotropic and laminated composite plates subjected to transverse normal loading and undergoing small and large deflections have been extensively studied and published in the literature. Yet, it seems that the particular case of long thin plates having a high aspect ratio appears to be almost ignored by various scholars despite its engineering importance. The present study tries to fill this gap, yielding novel findings regarding the structural behavior of long thin plates in the small- and large-deflection regimes. In contrast to what is normally assumed in the literature, namely that a long plate with a high aspect ratio can be considered an infinitely long plate, the present results clearly show that the structural effects of the ends continue to exist near the remote ends of the long plate. An innovative finding is that long plates would (only on movable boundary conditions for the large-deflection regime) exhibit a larger mid-width displacement in comparison with deflections of infinitely long plates. This innovative higher deflection appears for both small and large-deflection regimes for both all-around simply supported and all-around clamped boundary conditions. This new finding was shown to be valid for both isotropic and orthotropic materials and presents a novel engineering approach for the old assumption well quoted in the literature that a relatively long plate on any boundary condition can be considered an infinite plate. Based on the present research, it is recommended that this assumption should be used carefully as the largest plate mid-deflection might occur at finite aspect ratios.

Setal arrays provide insects with the good capability to walk fast and stably on rough surfaces, making them important research objects in the field of low-friction bionic design. The low-friction mechanism of the setal array is commonly attributed to the reduction of the actual contact area. However, the effect of the significant structural large deflection deformation of the seta on its friction behavior on rough surfaces has not been comprehensively studied. In this study, an analytical model that can describe the true deformation process and friction behavior of the seta on rough surfaces was developed. Accordingly, the friction behavior of the setal array sliding on rough surfaces was systematically analyzed. It’s been found that the large deflection deformation of a single seta reduces friction by decreasing the apparent tangential and normal stiffness. The superposition of the setae asynchronous deformation at different positions in the array reduces the vibration of the array’s apparent friction force and improves motion stability. These friction properties can be further optimized by modulating the seta density. The applicability of high-density setal arrays in bionic transportation is also experimentally demonstrated. This paper provides an important guide to revealing the biological friction mechanism and design of the bionic structures.

Abstract Functionally graded materials (FGMs) have various mechanical advantages and are naturally occurring, such as bamboo. Although bamboo is hollow and tapered, it remains resilient under a wind load. Based on this mechanical rationale, this study focuses on a large deflection of a tapering structure and the hollowing effect of a functionally graded beam. A theoretical analysis is conducted on a nonlinear bending of a slender and tapered hollow beam made of an axially FGM subjected to a uniformly distributed load undergoing a large deflection. To this end, governing equations are derived and a parametric study is conducted to investigate the effect of the inhomogeneous material, load magnitude and tapering and hollowing ratios on the large deflection of the beam. A linear analysis is conducted to examine the bending stress of the tapered beam and sets of deflection curves and angles along the beam are obtained and compared with values obtained from previous studies. As a result, the hollow and inhomogeneous nature of the axially FGM improves its rigidity against a wind load. This study provides insights into the potential use of axially FGMs to obtain more efficient and sturdier structural designs.

A circular non-touch mode capacitive pressure sensor can operate in both transverse and normal uniform loading modes, but the elastic behavior of its movable electrode plate is different under the two different loading modes, making its input–output analytical relationships between pressure and capacitance different. This suggests that when such a sensor operates, respectively, in transverse and normal uniform loading modes, the theory of its numerical design and calibration is different, in other words, the theory for the transverse uniform loading mode (available in the literature) cannot be used as the theory for the normal uniform loading mode (not yet available in the literature). In this paper, a circular non-touch mode capacitive pressure sensor operating in normal uniform loading mode is considered. The elastic behavior of the movable electrode plate of the sensor under normal uniform loading is analytically solved with the improved governing equations, and the improved analytical solution obtained can be used to mathematically describe the movable electrode plate with larger elastic deflections, in comparison with the existing two analytical solutions in the literature. This provides a larger technical space for developing the circular non-touch mode capacitive pressure sensors used for measuring the static gas pressure (belonging to normal uniform loading).

A circular capacitive rainfall sensor can operate from non-touch mode to touch mode; that is, under the action of enough rainwater, its movable electrode plate can form a circular contact area with its fixed electrode plate. Therefore, the weight of rainwater is borne by only its movable electrode plate in non-touch mode operation but by both its movable and fixed electrode plates in touch mode operation, and the total capacitance of its touch mode operation is much larger than that of its non-touch mode operation. Essential to its numerical design and calibration is the ability to predict the deflection shape of its moveable electrode plate to determine its total capacitance. This requires the analytical solution to the fluid–structure interaction problem of its movable electrode plate under rainwater. In our previous work, only the analytical solution for the fluid–structure interaction problem before its movable electrode plate touches its fixed electrode plate was obtained, and how to numerically design and calibrate a circular non-touch mode capacitive rainfall sensor was illustrated. In this paper, the analytical solution for the fluid–structure interaction problem after its movable electrode plate touches its fixed electrode plate is obtained, and how to numerically design and calibrate a circular touch mode capacitive rainfall sensor is illustrated for the first time. The numerical results show that the total capacitance and rainwater volume when the circular capacitive rainfall sensor operates in touch mode is indeed much larger than that when the same circular capacitive rainfall sensor operates in non-touch mode, and that the average increase in the maximum membrane stress per unit rainwater volume when the circular capacitive rainfall sensor operates in touch mode can be about 20 times smaller than that when the same circular capacitive rainfall sensor operates in non-touch mode. This is where the circular touch mode capacitive rainfall sensor excels.

Thin plates subjected to transverse load and undergoing large deflections have been widely studied and published in the literature. However, there is still a lack of information and understanding about the membrane stresses created under large deflections and their associated Airy stress function, as displayed in the well-known von Kármán equations set. The present study aims at providing explicit expressions for the membrane stresses, the deflections, and the Airy stress function for a general square plate area vertically uniformly loaded to reach large deflection state. This was obtained by using the results of a high-fidelity finite element analysis applied on a lateral loaded simply supported thin square plate, which are then casted to yield approximate Fourier series expressions for the membrane stresses, deflections, and the Airy stress function. The stress map figures provide a good understanding of the critical points on the plate, while the explicit mathematical expressions enabled the calculation of deflections and stresses for the entire plate area. Among other interesting findings, the presence of relatively high tensile and compressive membrane stresses existing near the plate edges was revealed, which might lead to potential failure hazards. The derivatives of the deflections and the Airy stress function enabled the validation of the large deflections von Kármán equations set for the investigated case, and it turned out that the generated expressions for the stresses and the lateral deflection based on a high-fidelity finite element model satisfy the second equation with a good accuracy, while the first one remains to further be investigated. Moreover, using the generated explicit equations, the load influence on the deflections and stresses was also analyzed to yield general novel expressions for the medium and very large deflections states. To generalize the investigated case, the stresses and the deflections were non-dimensionalized so they can be used for any material and plate dimensions.

In this paper, the previously proposed shaft-loaded blister test technique for the synchronous characterization of the surface and interface mechanical properties of a thin-film/substrate system is further studied theoretically. The large deflection problem of the steady shaft-loaded blistering thin film is reformulated by surrendering the small-rotation-angle assumption of the membrane, which was previously adopted in the out-of-plane and in-plane equilibrium and radial geometric equations. A new and more accurate analytical solution to this large deflection problem is presented and is used to improve the previously presented synchronous characterization theory. The new analytical solution is numerically compared with the previous analytical solution to confirm the superiority of the new analytical solution over the previous analytical solution. An experiment is conducted to verify the beneficial effect of the improved synchronous characterization theory on improving the characterization accuracy.

Transonic/supersonic wind tunnels are indispensable equipment for advanced aircraft to operate across subsonic, transonic, and supersonic regimes. The deformation of the flexible nozzle is the key to accurately controlling the Mach number of transonic wind tunnels. However, solving the deformation of flexible wall plates remains challenging due to the highly nonlinear relationship between wall loading and deformation, as well as the lack of simple yet effective mathematical models under complex boundary conditions. To accurately describe the deformation of flexible wall plates and improve computational efficiency, this study systematically investigates the deformation characteristics of flexible walls in two orthogonal directions and proposes an orthogonal beam function (OBF) model for characterizing small-deflection deformations. For large-deflection deformations in a flexible wall, an elliptic integral (EI) solution is introduced, and the OBF model is correspondingly modified. Experimental validation confirms that the OBF model effectively describes large-deflection deformations in a flexible wall. This research contributes to solving large-deflection deformation in flexible wall plates, enhancing both computational efficiency and accuracy.

Large deflection of rectangular plates under transverse pressure is described by Föppl–von Kármán equations, which have only approximated solutions. One of these methods is the separation into a small deflection plate and a thin membrane described by a simple third order polynomial expression. The present study presents an analysis to obtain analytical expressions for its coefficients by using the plate’s elastic properties and dimensions. To validate the non-linear relationship between the pressure and the lateral displacement of the multiwall plate, a vacuum chamber loading test is used to measure the plate’s response, with a large number of plates and length–width combinations. In addition, to further validate the analytical expressions, several finite element analyses (FEA) were performed. It has been found that the polynomial expression fairly describes the measured and calculated deflections. This method allows the prediction of plate deflections under pressure as soon as the elastic properties and the dimensions are known.