Geometric Nonlinear Effects Research Articles

Wave Propagation in Prestressed Structures with Geometric Nonlinearities Through Carrera Unified Formulation

This paper deals with the analysis of wave propagation characteristics in various prestressed structures with geometric nonlinearities using the Carrera Unified Formulation (CUF). CUF provides a versatile platform to model a wide range of structures and nonlinearities that can take care of all wave propagation aspects. In this work, different geometric nonlinearities for which representative governing equations have been derived and numerical solutions have been obtained through a unified approach are considered. The study investigates in detail the effect of prestress and geometric nonlinearity on wave propagation behavior. The results indicate that prestress has a very influential effect on modal frequency and dispersion characteristics for wave propagation. Specifically, three CUF-modeled beams are considered herein, having a sandwich, metallic portal, and metallic box cross section, respectively. Initially, the principal cross-sectional modal shapes of the unstressed, linear, and full nonlinear (i.e., full three-dimensional Green–Lagrange strain matrix) beam with a prestress are investigated, among which torsional and flexural modes can be recognized. Afterward, the equilibrium curves of such structures for various geometrical nonlinear approximations are traced, highlighting that most types of nonlinearity induce a hardening behavior in the system, which increases with the preload, directly leading to a variation in modal frequencies. The dispersion relations of the full nonlinear structure examined as a function of the applied preload are further compared, enriching the investigation by exploiting wave finite element method capabilities. This knowledge paves the way toward the design and optimization of prestressed systems with enhanced acoustic performance, and that fosters the development of sound absorption, noise insulation, and structural isolation.

Third-order geometric stiffness formulation for improved mesh convergence of thin and wide spatial beams in the generalized strain beam formulation

AbstractThis paper presents the stiffness formulation of a beam element with the relevant third-order nonlinear geometric effects for relatively wide and thin rectangular beams, in particular when loaded in the plane and simultaneously deformed out of the plane. The element is initially straight in its undeformed configuration. The formulation is based on Timoshenko beam theory with nonuniform torsion and Wagner effects. The derivation is carried out by means of the Hellinger–Reissner variational principle with custom interpolation functions. The element is incorporated into the generalized strain beam formulation for multibody systems. Numerical simulations of precision flexure mechanisms show that the use of a single third-order element per flexible member can already yield adequate performance, at a significant reduction of the necessary degrees of freedom and the computation time, compared with using multiple second-order elements in the generalized strain beam formulation.

Biaxial bending strength of thin silicon dies in the ring-on-ring test by considering geometric nonlinearity and material anisotropy

The ring-on-ring test (RoR) is a standard biaxial bending test specified in ASTM C1499-19 and ISO 17167. This test has been utilized for characterizing the biaxial bending strength of silicon dies or wafers to eliminate the die edge chipping effect in the four-point bending test. However, judging from the literature, when testing thin silicon dies, the test is subject to geometric nonlinear effects. This study aims to investigate this nonlinear mechanics in the RoR test using experimental, theoretical, and numerical methods while considering silicon material anisotropy. A 2D-isotropy model of a nonlinear finite element method (NFEM) simulation with specimen elastic modulus of 130 GPa is utilized and verified by experiments and a 3D-anisotropy model in terms of deformation (or displacement) and stresses. Based on the 2D-isotropy NFEM solutions, the fitting equations of correction factors to the theoretical solution are proposed and implemented on determining the biaxial bending strength of 10 mm × 10 mm silicon dies ranging from 57 μm to 297 μm in thickness. It is found that those proposed fitting equations are independent on the test specimen thickness, radius, and materials but not on the radii of the loading and supporting rings. It has also been successfully demonstrated that the RoR test using the theory associated with the correction factor equations can be easy to use to determine the biaxial bending strength of the thin silicon dies that frequently failed in the nonlinear range.

Nonlinear analysis of three-dimensional hyperelastic problems using radial point interpolation method

Hyperelastic materials are primarily common in real life as well as in industry applications, and studying this kind of material is still an active research area. Naturally, the characteristic of hyperelastic material will be expressed when it undergoes large deformation, so the geometrical nonlinear effect should be considered. To analyze the behavior of hyperelastic material, the Neo-Hookean model is imposed in this study because of its simplicity. The model shows the nonlinear behavior when the deformation becomes large due to the nonlinear displacement-strain relation. This constitutive relation also gives a good correlation with experimental data. This study performs a meshless method, namely the radial point interpolation method (RPIM), to analyze the nonlinear behavior of Neo-Hookean hyperelastic material under a finite deformation state in three-dimensional space. The standard Newton-Raphson technique is applied to obtain the nonlinear solutions. Unlike mesh-based approaches, the meshless method shows its advantages in large deformation problems due to its mesh independence. In this paper, the numerical results of three-dimensional problems that undergo large deformation will be calculated and validated with solutions derived from previous studies. By investigating the obtained results, the superior ability of RPIM in hyperelastic problems can be proved.

A fast search method of buckling load for telescopic boom structure

The telescopic boom structures are extensively utilized in engineering applications involving large mobile cranes, aerial work platforms of significant height, and similar mechanical devices. These are predominantly fabricated using solid-webbed box types, latticed truss designs, or a combination thereof. Under load, they exhibit pronounced geometric nonlinear effects, with the relationship between load and displacement demonstrating marked nonlinear characteristics. This paper focuses on the geometric nonlinear modeling of slender multi flexible beam structures. The overall structural system is divided into several substructures, and the concept of multi flexible system modeling is borrowed to establish a follow-up connected body foundation on each substructure. By decomposing the node displacement and rotation in the substructure, the large displacement and large rotation of the node are decomposed into rigid body motion of the connected base and small displacement and small rotation relative to the connected base, effectively representing the large displacement and large rotation of the substructure as rigid body motion of the connected base. A modeling method for the geometric nonlinear analysis of slender and flexible multibody beam structures is proposed. By decomposing the nodal displacement and rotation within the substructures, the large displacement and rotation of the nodes are broken down into the rigid body motion of the body-fixed coordinate and small displacement and rotation relative to the body-fixed coordinate. This approach establishes the application conditions for calculating the structure’s virtual power of deformation using traditional beam elements with linear strain. A method for solving the instability load of slender and flexible multibody structures is proposed. This method integrates the characteristics of the system’s nonlinear equilibrium equations with respect to load, by deriving the system equations with respect to a load increment control parameter. This paper converts nonlinear equilibrium equations into first-order ordinary differential equation (ODE). The method proposed in this paper provides a more efficient solution for the buckling load of the telescopic boom. It can be used to systematically and quickly calculate the structure of the telescopic boom.

Dynamic Modelling and Performance Analysis of Deployable Telescopic Tubular Mast

AbstractThis work presents the longitudinal and transverse coupling vibrations of a deployable Telescopic Tubular Mast (TTM), a multi-stepped structure integrated into a spacecraft system, while considering the rigid-flexible coupling phenomenon. The model is derived using the principle of virtual work and discretized via the variable separation method. The von Kármán strain is employed to incorporate geometric nonlinear effects. Semi-analytical results for the shape functions and natural frequencies of the quasi-static multi-stepped boom are obtained using the extended transfer matrix method (ETMM). These natural frequencies are validated against results from Nastran, confirming the ETMM's accuracy. In addition, the model accounts for the continuously changing natural frequencies and shape functions during the deployment phase. Finally, the dynamic phenomena of the longitudinal and transverse displacements are analyzed at various deploying states, including locking and restart behaviors. The influence of the structural damping on the vibration evolution is also contained in the numerical analysis.

Systematic Experimental Evaluation of Aeroelastic Characteristics of a Highly Flexible Wing Demonstrator

This paper presents a comprehensive experimental analysis of the evolutionary modal characteristics of a highly flexible wing that exhibits bending–torsion coupling-driven instability. By implementing operational modal analysis on responses triggered by a combination of an external pulse-like stimulation and turbulence within the flow, this paper presents the airspeed-driven variations of the modal frequencies, damping ratios, and the underlying modal coupling behavior, leading to instability. This analysis is extended to varying the wing root pitch angles through which the effects of geometrical nonlinearity are exercised. Their effects are particularly noted on the hump feature of the airspeed-driven damping ratio locus of the mode responsible for instability. The decreasing critical damping ratio is shown to result in amplified turbulence-driven responses, which pose significant challenges to identification procedures by masking the visibility of other modes. Furthermore, through a novel technique used to analyze the modal coupling, the relative phase and magnitude properties of the coupled bending–torsion composition of the critical mode before and at flutter onset are evidenced experimentally. It is demonstrated that these relative participation measures provide a strong indication of the response content of the limit cycle oscillations that emerge after the flutter speed.

Finite Element Modeling and Calibration of a Three-Span Continuous Suspension Bridge Based on Loop Adjustment and Temperature Correction.

Precise finite element modeling is critically important for the construction and maintenance of long-span suspension bridges. During the process of modeling, shape-finding and model calibration directly impact the accuracy and reliability. Scholars have provided numerous alternative proposals for the shape-finding of main cables in suspension bridges from both theoretical and finite element analysis perspectives. However, it is difficult to apply these solutions to suspension bridges with special components. Seeking a viable solution for such suspension bridges holds practical significance. The Nanjing Qixiashan Yangtze River Bridge is the first three-span suspension bridge in China. To maintain the configuration of the main cable, the suspension bridge is equipped with specialized suspenders near the anchors, referred to as displacement-limiting suspenders. It is the first suspension bridge in China to use displacement-limiting suspenders and their anchorage system. Taking the suspension bridge as a research background, this paper introduces a refined finite element modeling approach considering the effect of geometric nonlinearity. Firstly, based on the loop adjustment and temperature correction, the shape-finding and force assessment of the main cables are carried out. On this basis, a nonlinear finite element model of the bridge was established and calibrated, taking into account factors such as pylon settlement and cable saddle precession. Finally, the static and dynamic characteristics of the suspension bridge were thoroughly investigated. This study aims to provide a reference for the design, construction and operation of the three-span continuous suspension bridge.

Nonlinear thermo-electro-mechanical responses and active control of functionally graded piezoelectric plates subjected to strong electric fields

Functionally graded piezoelectric plates (FGPPs) are often exposed to extreme thermal environments and subjected to large amplitude excitation and strong electric fields. Accurately predicting the nonlinear behaviors of FGPPs under large thermo-electro-mechanical loads remains a considerable challenge. This paper proposes a comprehensive nonlinear coupled analysis model that considers geometric nonlinearity, piezoelectric nonlinearity, and the temperature dependence of piezoelectric parameters. The total Lagrangian incremental finite element equations for FGPPs under thermo-electro-mechanical loads are derived using Hamilton's principle, by employing the first-order shear deformation theory and the von Kármán nonlinear strain-displacement relationship. The accuracy of the model is validated through comparative studies. Based on this multi-nonlinear model, we further investigate the effects of geometric nonlinearity, piezoelectric nonlinearity, and temperature dependence on the nonlinear static bending, dynamic responses, and active control of FGPPs. This study demonstrates that considering these factors enables accurate prediction of the static, dynamic, and active control behaviors of FGPPs.

Simultaneous effects of material and geometric nonlinearities on nonlinear vibration of nanobeam with surface energy effects

Simultaneous effects of material and geometric nonlinearities on nonlinear vibration of nanobeam with surface energy effects

Fracture modeling of curved composite shell structures using augmented finite element method

This paper presents an augmented finite element method for the fracture modeling of curved composite shell structures considering geometric nonlinear effects. We first derive the composite shell AFEM (CS-AFEM) formulation using a dynamic implicit algorithm, which enhances the numerical convergence when dealing with unstable crack propagations. Besides, the cohesive zone model featuring with shell kinematics is incorporated into the CS-AFEM to describe the quasi-brittle fracture behaviors of composite laminates. Furthermore, a coordinate transformation scheme is proposed to describe both the local material orientation and the crack evolving path in the curved shell structures. Finally, several benchmark examples are numerically investigated, and the capability of the proposed method in modeling the arbitrary evolving cracks in curved composite shell structures has been thoroughly demonstrated.

Feasible model-order reduction approach for analysis of composite rotor blades based on geometrically exact beam formulation

This paper presents a novel and practical reduced-order model (ROM) approach designed specifically for analyzing composite rotor blades, leveraging the geometrically exact beam formulation. The slender nature of rotor blades necessitates aero-structure coupled analysis to precisely predict their response, which involved numerous iterative computations. By capturing large displacements and rotations, this approach enhances the accuracy of blade behavior predictions under operational conditions. To address the computational demands of large-scale analyses, a ROM technique that effectively reduces system dimensionality while preserving critical structural features is introduced. The proposed method is then applied to the aero-structure interaction analysis of a hingeless hovering rotor trim, incorporating geometrical nonlinearity and centrifugal effects. The analysis results demonstrate that the ROM accurately captures the dynamics of these complex aeroelastic systems while significantly reducing computational costs. This approach exhibits significant potential for various aerospace applications, particularly in the design and analysis of structures undergoing geometrical nonlinear behavior and rotation-induced loads.

Stiff morphing composite beams inspired from fish fins.

Morphing materials are typically either very compliant to achieve large shape changes or very stiff but with small shape changes that require large actuation forces. Interestingly, fish fins overcome these limitations: fish fins do not contain muscles, yet they can change the shape of their fins with high precision and speed while producing large hydrodynamic forces without collapsing. Here, we present a 'stiff' morphing beam inspired from the individual rays in natural fish fins. These synthetic rays are made of acrylic (PMMA) outer beams ('hemitrichs') connected with rubber ligaments which are 3-4 orders of magnitude more compliant. Combinations of experiments and models of these synthetic rays show strong nonlinear geometrical effects: the ligaments are 'mechanically invisible' at small deformations, but they delay buckling and improve the stability of the ray at large deformations. We use the models and experiments to explore designs with variable ligament densities, and we generate design guidelines for optimum morphing shape (captured using the first moment of curvature), that capture the trade-offs between morphing compliance (ease of morphing the structure) and flexural stiffness. The design guidelines proposed here can help the development of stiff morphing bioinspired structures for a variety of applications in aerospace, biomedicine or robotics.

Optimization of Staging Height of RCC Overhead Water Tank in High Seismic Zones Using P-Delta Analysis

Abstract: This study investigates the P-delta effect, which is a secondary effect or a geometric non-linear effect in the analysis of a circular overhead water tank with a capacity of 100 KL using STAAD software in high seismic zones, i.e., Zone IV & Zone V with a focus on optimization of staging height and to quantify the effect of P-Delta. Initially, a linear static analysis was performed without considering the P-Delta effect to determine the shear forces, bending moments and lateral displacements due to all possible loads including seismic and wind loads acting on RCC overhead water tanks. Subsequently, the P-Delta effect has been considered to assess the structural behaviour of the RCC overhead water tank with varying staging heights. To optimize the staging height, the maximum lateral displacement has been considered as the governing criteria. Based on the analysis, the optimum staging heights can be observed at 30 m. and 33 m. in Seismic Zone V and Zone IV respectively

Nonlinear inelastic behaviors of concrete-filled steel tubular beam-column structures

A new advanced method combining stability function and distributed plasticity model has been developed using Fortran programming language to predict the nonlinear inelastic behavior of concrete-filled steel tubalar (CFST) under static loading. The advantage of this method is the ability to accurately study the nonlinear behavior using only one or two beam-column elements per member instead of using solid and shell elements as traditional methods, thereby improving the model analysis time. The Generalized Displacement Control (GDC) algorithm, capable of analyzing beyond the limit point, will be used to solve the nonlinear equilibrium equations instead of the traditional Newton-Raphson algorithm. The element stiffness matrix is integrated through the Gauss-Lobatto numerical integration framework, while the nonlinear geometric effects P-Δ and P-δ are considered using stability functions and a corresponding geometric matrix. The reliability and accuracy of the proposed method are verified by comparing the analysis results with experimental data. The obtained results have demonstrated that using beam-column elements for simulation, the proposed method still provides accurate results while significantly reducing computational resources. Therefore, this new method holds promise as a useful tool for practical design and analysis of statically loaded CFST structures.

Artificial neural network-based sequential approximate optimization of metal sheet architecture and forming process

Abstract This paper introduces a sequential approximate optimization method that combines the finite element method (FEM), dynamic differential evolution (DDE), and artificial neural network (ANN) surrogate models. The developed method is applied to address two optimization problems. The first involves metamaterial design optimization for metal sheet architecture with binary design variables. The second pertains to optimizing process parameters in multi-stage metal forming, where the discrete nature arises owing to changing tool geometries across stages. This process is highly nonlinear, accumulating contact, geometric, and material nonlinear effects discretely through forming stages. The efficacy of the proposed optimization method, utilizing ANN surrogate models, is compared with traditionally used polynomial response surface (PRS) surrogate models, primarily based on low-order polynomials. Efficient learning of ANN surrogate models is facilitated through the FEM and Python integration framework. Initial data for surrogate model training is collected via Latin hypercube sampling and FEM simulations. DDE is employed for sequential approximate optimization, optimizing ANN or PRS surrogate models to determine optimal design variables. PRS surrogate models encounter challenges in dealing with nonlinear changes in sequential approximate optimization concerning discrete characteristics such as binary design variables and discrete nonlinear behavior found in multi-stage metal forming processes. Owing to the discrete nature, PRS surrogate models require more data and iterations for optimal design variables. In contrast, ANN surrogate models adeptly predict nonlinear behavior through the activation function's characteristics. In the optimization problem of Metal Sheet Architecture for design target C, the ANN surrogate model required an average of 4.6 times fewer iterations to satisfy stopping criteria compared to the PRS surrogate model. Furthermore, in the optimization of multi-stage deep drawing processes, the ANN surrogate model required an average of 6.1 times fewer iterations to satisfy stopping criteria compared to the PRS surrogate model. As a result, the sequential global optimization method utilizing ANN surrogate models achieves optimal design variables with fewer iterations than PRS surrogate models. Further confirmation of the method's efficiency is provided by comparing Pearson correlation coefficients and locus plots.

Analysis of nonlinear bending behavior of nano-switches considering surface effects

Nano-switch structures are important control elements in nanoelectromechanical systems and have potential applications in future nanodevices. This paper analyzes the effects of surface effects, geometric nonlinearity, electrostatic forces, and intermolecular forces on the nonlinear bending behavior and adhesion stability of nano-switches. Based on the Von Karman geometric nonlinearity theory, four types of boundary conditions for the nano-switch structure were specifically calculated. The results show that surface effects have a significant impact on the nonlinear bending and adhesion stability of nano-switches. Surface effects increase the adhesion voltage of the nano-switch and decrease its adhesion displacement, and as the size of the nano-switch structure increases, the impact of surface effects decreases. A comparative analysis of the linear theory and the nonlinear theory results shows that the adhesion voltage predicted by the linear theory is smaller than that predicted by the nonlinear theory. The effect of geometric nonlinearity increases as the size of the nano-switch structure increases, as the distance between the electrodes increases, and as the aspect ratio of the nano-switch structure increases. These findings provide theoretical support and reference for the design and use of future nanodevices and nanoelectromechanical systems.

Updating Nonlinear Stochastic Dynamics of an Uncertain Nozzle Model Using Probabilistic Learning With Partial Observability and Incomplete Dataset

Abstract This article introduces a methodology for updating the nonlinear stochastic dynamics of a nozzle with uncertain computational model. The approach focuses on a high-dimensional nonlinear computational model constrained by a small target dataset. Challenges include the large number of degrees-of-freedom, geometric nonlinearities, material uncertainties, stochastic external loads, underobservability, and high computational costs. A detailed dynamic analysis of the nozzle is presented. An updated statistical surrogate model relating the observations of interest to the control parameters is constructed. Despite small training and target datasets and partial observability, the study successfully applies probabilistic learning on manifolds (PLoM) to address these challenges. PLoM captures geometric nonlinear effects and uncertainty propagation, improving conditional mean statistics compared to training data. The conditional confidence region demonstrates the ability of the methodology to accurately represent both observed and unobserved output variables, contributing to advancements in modeling complex systems.

Effects of structural geometric nonlinearities on the transonic aeroelastic characteristics of wing

The structural geometric nonlinearity has a great influence on the transonic aeroelastic characteristics of aircraft wings, which is an important problem for the aerodynamic and structural design of wing. In this paper, a CFD/CSD coupling method based on the high precision in-house solver is first established, and the reliability of the CFD and the CFD/CSD coupling method is then verified. Subsequently, the effects of geometric nonlinearity on the transonic aeroelastic characteristics of wing under different dynamic pressures are studied. The results show that: (1) At low dynamic pressure conditions, for geometric linear wing structure, limit-cycle oscillation occurs at various angles of attack, and the limit-cycle oscillation amplitude decreases with the increases of angle of attack. At low angle of attack, the wing deformation characteristics are dominated by first bending and first torsion, and the first torsion weaken as the angle of attack increases. For the geometric nonlinear wing structure, the limit-cycle oscillation occurs only at an attack angle of 0° As the angle of attack increases, the structural displacement eventually exhibits static deformation characteristics. (2) Under high dynamic pressure conditions: the displacement of geometric linear wing structure presents a divergent trend. For geometric nonlinear wing structure, the limit-cycle oscillation amplitude decreases first and then increases with the increase of the angle of attack. At low angle of attack, the deformation characteristics of the wing are dominated by first bending and first torsion. With the increase of the angle of attack, the first torsion weakens, while the second bending strengthen. With the further increase of the angle of attack, the second bending weaken and become co-dominated by the first bending and first torsion. It can be seen from the above results that the aeroelastic characteristics of both the geometric linear structure and the geometric nonlinear structure wings are significantly different. These findings provide valuable insights for the meticulous design of aircraft wings.

Damage detection of lightweight concrete dual systems reinforced with GFRP bars considering various building heights and earthquake intensities

Because of their distinct mechanical, rheological, and physical features, lightweight concrete and glass fiber reinforced polymers (GFRP) bars have a significant growth in the modern building sectors over the past decade. Minimizing subsequent quake damages, particularly in multistory residential or administrative RC structures, is a key concern for governments in earthquake zones. The dual frame-wall system is among the most appropriate structural systems for resisting seismic loads in multistory concrete structures. Therefore, this research aims to quantify the seismic damage of multistory dual frame-wall systems with varying heights of 12, 20, and 25 stories under mild, medium, and severe earthquake intensities. The inelastic damage analysis of RC buildings program is utilized to nonlinearly analyze 54 cases containing various materials such as normal concrete, lightweight concrete, steel reinforcement bars, and GFRP bars. Using lightweight concrete with half the density of normal concrete and pure linear-elastic GFRP bars in floor construction will significantly decrease the geometric nonlinearity effect created by the load-displacement (P-Δ) effect and dissipate a large amount of the kinetic energy generated by seismic excitation, resulting in a significant reduction in post-earthquake damage. When compared to the normal concrete, using lightweight concrete in floor construction reduced the mean story damage index by 14–20% as the building's height increased, while reinforcing these floors with GFRP bars resulted in substantial decreases in the overall damage index, reaching about 57–68% when both building's heights and seismic excitation intensities were considered.