Geometric Nonlinearity Research Articles

Chained Spatial Beam Adomian Decomposition Model: A Novel Model of Flexible Slender Beams for Large Spatial Deflections

Abstract The main element of compliant mechanisms and continuum robots is flexible slender beams. However, the modeling of beams can be complicated due to the geometric nonlinearity becoming significant at large elastic deflections. This paper presents an explicit nonlinear model called the spatial beam Adomian decomposition model (SBADM) for intermediate spatial deflections of a slender beam with uniform, bisymmetric sections subjected to general end-loading. Specifically, the elongation, bending, torsion, and shear deformations of the beams are modeled based on Timoshenko's assumptions and Cosserat rod theory. Then, the quaternion transformation and Adomian decomposition are used to solve the nonlinear governing differential equations for the beam by truncating the higher-order terms, yielding an explicit expression for spatially deflected beams within intermediate deflection ranges. Simulations demonstrate the accuracy and time-wise efficiency of the SBADM, as well as its advantages over the state-of-the-art. In addition, this paper also introduces a discretization-based scheme called the chained SBADM (CSBADM) for large spatial deflections of flexible beams. Real-world experiments with two different configurations have also been performed to validate the effectiveness of the CSBADM. The results indicate that the CSBADM can accurately calculate the load-displacement relations for large deformed beams.

Stability Analysis of Construction Factors for Partially Cable-Stayed Bridges with Multiple Towers and High Piers

Partially cable-stayed bridges have the characteristics of continuous rigid-frame bridges and cable-stayed bridges, making them a novel composite bridge system. This study focuses on the construction project of a multi-tower high-pier curved partially cable-stayed bridge to investigate the bridge’s stability during construction. The Midas/Civil software was used to establish a model for key construction stages of the bridge, considering structural linear elasticity and geometric nonlinearity. The study examines the impact of static wind loads, asymmetric construction of the main girder, closure sequence, and the load and detachment of the hanging basket on the bridge’s stability during construction. The results indicate that static wind loads have a significant impact on structural geometric nonlinearity, with a maximum reduction of 4.99%. Asymmetric construction at both ends of the main girder can cause structural instability and should be avoided. The geometric nonlinearity stability coefficient for the hanging basket load decreased by 10.83% during the maximum no-cable stage and by 7.84% during the cable stage, significantly affecting the stability during construction. A bridge closure sequence of side-span, secondary midspan, and midspan provides the most stable condition during the construction phase. The results of this study can inform the construction of similar partially cable-stayed bridges.

Computational analysis of wildland fire resistance for transmission line tower

Finite element (FE) analysis of the transmission towers under a fire event and the accompanying wind is presented. Several key aspects (including temperature-dependent non-linear material properties, geometric non-linearity, member eccentricity due to the single-leg bolted connection, and the temperature-dependent connection behaviour in both the axial and the rotational directions) are considered. A quasi-static analysis is also employed in the FE model using the explicit solver available in Abaqus. The member temperature variation during a realistic fire event is derived analytically based on the fire intensity and the resultant vertical gas temperature distribution so that the effect of the realistic wildland fire can be input as a member temperature profile. First, fire design guidance in terms of the most critical heating and wind pattern, as well as the effect of fire-affected heights are provided based on the case study results. Then, further fire analysis is carried out to investigate the most critical fire scenario and to evaluate the vulnerability of the tower during a realistic fire event. Lastly, an evaluation of a commonly used spray-on thermal insulation and its effectiveness in reducing the member temperature and preserving ultimate strength during a catastrophic wildland fire event are presented. The FE simulations revealed that a 45-degree wind associated with partial side heating leads to a more critical degradation of the overall strength, whereas the effect of fire height is less obvious. In terms of the fire scenario, a longer fire duration associated with a slower wind is more critical for the fire resistance/rating.

Nonlinear vibrations and static bifurcation of a hyperelastic pipe conveying fluid

This study investigates the free and forced nonlinear vibrations, along with the static bifurcation behavior, of a fluid transmission hyperelastic pipe supported by clamped ends. Both geometrical and material nonlinearities are taken into account. The mathematical model is grounded in Euler–Bernoulli beam theory, incorporating von Kármán nonlinearity to capture the complexities of the system. For the hyperelastic materials, the strain energy function proposed by Yeoh is employed. Considering that the lowest critical velocity of the fluid flow is associated with the first mode, after deriving the governing equation using Hamilton’s principle, Galerkin discretization is applied to obtain the reduced order model in the first mode. Analysis of static bifurcation and stability is performed and critical velocity values of fluid flow are determined. The analysis of free and forced vibrations of the hyperelastic pipe is conducted using the method of multiple time scales, specifically focusing on the sub-critical velocity range of fluid flow, which corresponds to the unbuckled state of the pipe. This approach allows for a comprehensive examination of how fluid flow velocity influences the frequency characteristics of free vibrations, as well as the frequency response in the vicinity of the main resonance. The results obtained from this study can significantly enhance our understanding of the design and application of hyperelastic pipes in various fluid transfer scenarios. By elucidating the dynamic behavior and critical parameters associated with these pipes, the findings can inform engineering practices and contribute to the optimization of systems that utilize hyperelastic materials for fluid conveyance.

Harmonic response analysis of a circular plate subjected to moving point loads using Green’s function approach

Dynamics of circular plates under moving loads are an important class of problems in manufacturing industry, for example, cutting thin annular plates using point cutting tools and wood/bar/pipe cutting saws. These tools are subjected to excessive vibration during operation. Its dynamic behavior needs to be analyzed rigorously. In the present work, dynamics of a circular plate subjected to moving harmonic point load is analyzed using Kirchhoff’s plate hypothesis. In order to capture the geometric nonlinearity, the von Kármán strain displacement relation is used. Constitutive relations are developed using a complex modulus. Further, extended Hamilton’s principle is used to obtain the nonlinear coupled governing equations in radial, tangential, and transverse directions. Governing equations are discretized using Galerkin’s method. The novelty in the present work is to develop Green’s function incorporating viscous damping to obtain the plate response subjected to arbitrary moving load for general boundary conditions. The proposed method is capable to obtain the plate response due to single/multi point moving loads. In order to verify the accuracy of the proposed method, Green’s function solution is compared with the numerical solution obtained from the RK4 method and shows good agreement. Further, resonance instabilities are studied, which occur when a single point load rotates with critical angular velocity. Subsequently, the response of the circular plate under three-point loads, each having a magnitude of 1/3rd of the single moving point load moving circularly in a constant phase has been discussed. In the end, a parametric study of hysteresis damping on the dimensionless deflection of the plate is carried out.

Enhancing energy capacity of the car’s hood door under excitation frequency via functionally graded triply periodic minimal surface material: Application of machine learning and mathematical simulation

This study investigates the nonlinear dynamic response and vibration characteristics of a car’s hood door, focusing on its energy capacity under various excitation frequencies. The hood door is modeled using a functionally graded triply periodic minimal surface (FG-TPMS) material, which offers superior mechanical properties and lightweight design. The analysis is conducted using a higher-order shear deformation theory (HSDT), which accounts for shear deformations more accurately than classical theories. The incorporation of von Karman nonlinear terms captures the geometric nonlinearity due to large deformations, providing a realistic simulation of the hood door’s behavior under dynamic loads. To solve the complex equations of motion derived from the HSDT and von Karman terms, a fourth-order Runge–Kutta method is employed. This numerical method ensures accurate time integration and stability of the solution. The dynamic response is further analyzed to determine the energy absorption capacity of the hood door material, crucial for safety and durability in automotive applications. Validation of the numerical model is performed using previous published articles and a hybrid machine learning algorithm, which combines data-driven approaches with traditional physics-based models. This hybrid validation enhances the accuracy and reliability of the predicted responses, ensuring that the proposed model can be effectively used in the design and optimization of car components. The results demonstrate the potential of FG-TPMS materials in automotive applications, offering improved energy absorption and vibration control, which are essential for enhancing vehicle safety and performance.

Static and dynamic characteristics of electrostatically coupled MEMS beams under asymmetric actuation conditions

Abstract Electrostatically actuated microbeams have gained prominence due to their applications as micro-actuators and ultrasensitive sensors. In this study, we investigate the dynamic behavior of electrostatically coupled microbeams, specifically focusing on the effects of asymmetric actuation conditions. Our investigation encompasses both static and dynamic characteristics of the coupled microbeam system, considering various gap ratios between the beams and the fixed electrodes. Within the coupled system, two doubly clamped microbeams are positioned between two fixed electrodes. We develop a reduced-order model (ROM) to investigate static and dynamic responses of the coupled system. To validate the ROM, we rigorously compare its results with finite element (FE) simulations. Our findings reveal that the gap ratios play a pivotal role in shaping the system’s response. Notably, first two natural frequencies are important for design of micro-resonators and they exhibit complex variation with respect to the applied DC voltage. Moreover, we explore resonance frequency tunability and investigated interplay between geometric nonlinearity and nonlinear electrostatic actuation forces. These insights contribute significantly to the efficient design of MEMS resonators.

Nonlinear forced vibration of the FGM piezoelectric microbeam with flexoelectric effect

In this paper, based on the extended dielectric theory, Euler beams theory and von Karman’s geometric nonlinearity, a nonlinear FGM piezoelectric microbeam model is established with flexoelectric effect. The governing equations, initial conditions and boundary conditions are obtained by applying Hamilton’s principle and then solved by combining the differential quadrature method (DQM) and iteration method. The innovation of this paper is to construct a nonlinear forced vibration model of piezoelectric microbeams. The coupling response between the inverse flexoelectric effect and the inverse piezoelectric effect is investigated. Various effects are examined, including the functional gradient index m and transverse distributed load q affecting the distribution of electric potential. Results indicated that the functional gradient index m, beam thickness h, and span-length ratio L/h have a significant impact on the dimensionless deflection of the FGM microbeam. The influence of the flexoelectric effect on dimensionless deflection increases with the decrease of scale. In addition, transverse load q and the functional gradient index m also have a significant impact on the distribution of electric potential. This paper will provide useful theoretical guidance for the design of micro-sensors and micro-actuators.

A study on solid-shell finite element formulations applied to nonlinear thermoelastic analysis of thin-walled structures

The solid-shell elements have shown great advantages in three-dimensional (3D) finite element (FE) simulation of thin-walled structures. To overcome the numerical lockings for 3D element with a large span-thickness ratio, the assumed natural strain (ANS) method combined by either the hybrid stress (HS) or enhanced assumed strain (EAS) formulations are commonly used. In this work, two types of finite element formulations of the solid-shell elements are developed for nonlinear thermoelastic analysis of thin-walled structures, which are termed as the “ANS+HS” and “ANS+EAS” formulations. Accordingly, two eight-node solid-shell elements (CSSH8 and CSSE8) are developed based on the “ANS+HS” and “ANS+EAS” formulations, respectively. The Green–Lagrange displacement–strain relation is applied to involve the geometrical nonlinearities, and both the thermal expansion and temperature-dependent material properties are considered. Nonlinear thermoelastic equilibrium equations are constructed in the framework of the two types of finite element formulations using the Hellinger–Reissner and conventional variational principles, respectively. The proposed method can easily realize three different coupling analyses for thermal–mechanical loads by modifying the parameters of nonlinear thermoelastic equilibrium equations. Numerical examples demonstrate that the proposed method using CSSH8 and CSSE8 elements traces the nonlinear thermoelastic response of the isogrid stiffened panel as accurately as the shell, solid-shell, and solid elements of ABAQUS. Furthermore, the path-following capability of the proposed method using the CSSH8 with “ANS+HS” formulation is slightly superior to that using the CSSE8 with “ANS+EAS” formulation, but both better than ABAQUS.

Evaluating the Seismic Resilience of Above-Ground Liquid Storage Tanks

Historical seismic events have repeatedly highlighted the susceptibility of above-ground liquid storage steel tanks, underscoring the critical need for their proper design to minimize potential damage due to seismic forces. A significant failure mechanism in these structures, which play essential roles in the extraction and distribution of various raw or refined materials—many of which are flammable or environmentally hazardous—is the dynamic buckling of the tank walls. This study introduces a numerical framework designed to assess the earthquake-induced hydrodynamic pressures exerted on the walls of cylindrical steel tanks. These pressures result from the inertial forces generated during seismic activity. The computational framework incorporates material and geometric nonlinearities and models the tanks using four-node shell elements with two-point integration, specifically Belytschko shell elements. The Arbitrary Lagrangian–Eulerian (ALE) method is employed to accommodate substantial structural and fluid deformations, enabling a full simulation of fluid–structure interaction through highly nonlinear algorithms. Experimental test data are utilized to validate the proposed modeling approach, particularly in replicating sloshing phenomena and identifying stress concentrations that may lead to wall buckling. The study further presents results from a parametric analysis that varies the height-to-radius and radius-to-thickness ratios of a typical anchored flat-bottomed tank, examining the seismic performance of this common storage system. These results provide insights into the relationship between tank properties and mechanical behavior under dynamic loading conditions.

Structural Analysis Technique and Its Validation for Large Range Mirror Refocusing System for Spaceborne Deployable Telescopes

Large aperture deployable telescopes with numerous onboard instruments are popular in the space industry for astronomy. Scientific data in various wavelength bands, from 0.6µ to 28µ, is provided by these instruments and is crucial for deep space exploration. The deployment mechanisms handle the alignment and functionality at the telescope level only. At instrument level, there is always a need for a refocusing system which can cater individual need of instrument for focusing independent to other instruments nearby. In this paper, we present a structural analysis technique for such a type of large range mirror refocusing mechanism, similar to the Near Infrared Spectrograph (NIRSpec) payload of the James Webb Space Telescope (JWST). The method relies on geometric non-linearity for large deflection, in which the structure's stiffness matrix changes as a function of loading. We have designed, analyzed structurally, and tested a 4mm range slider crank based compliant mechanism. The outcomes of linear and non-linear static analysis, computed numerically, have been compared. There are noticeable significant variations in one direction between the two analyses' results. The results are validated using appropriately designed and realized test setups, like measurements of displacement, changes in focus and image position of the optical system.

Investigation of the nonlinear dynamics of thin sandwich shells composed of functionally graded materials with double curvature in thermal environments

Double curvature structures play a crucial role as load-bearing components across various engineering fields. Functionally graded materials, blending metals and ceramics, boast superior material characteristics, fueling their expanding applications. This study pioneers the non-linear dynamic analysis of sandwich shells that feature functional gradient compositions in thermal settings. We assume that both the upper and lower shells of these double curvature structures are crafted from functionally graded materials, with a ceramic core. This variation in material exhibits a ceramic layer on the inner surface and a metallic layer on the outer surface, showing properties that vary along the shell thickness in a power-law gradient. Non-linear dynamic equations are derived using third-order shear theory, encompassing geometric non-linearity and shear deformation. Employing the Galerkin method, we discretize the equations of motion into a non-linear dynamic system with five degrees of freedom, and subsequently give an analytical expression for the nonlinear naturalfrequency by means of multiscale analysis. Our discussion examines the impacts of structural parameters, porosity volume fraction, volume fraction index, and temperature differences on the non-linear/linear frequency ratio of doubly curved sandwich shells. Calculations reveal a sharp rise followed by a rapid decline in the non-linear/linear frequency ratio with increasing structural aspect ratio b/a. Conversely, it decreases with increasing thickness-to-length and radius-to-length ratios, with temperature differences initially reducing and later increasing it. These findings offer practical insights for designing functional gradient double curvature sandwich shells.

Geometrically nonlinear vibrations of plate embedding a large circular acoustic black hole

The concept of acoustic black hole (ABH) designates a decrease in bending stiffness and a localized added damping within a thin structure aiming to localize and efficiently dissipate vibrations. The small thickness results in large amplitude vibrations, which suggests to take into account geometrical nonlinearities. The subject has been previously studied on a beam with a one dimensional ABH. Previous research studied a geometrically nonlinear plate embedding a one dimensional ABH. This study aims at solving the Von Karman equations for a variable thickness plate, via a modal projection using the eigenmodes obtained with a finite element model. The use of a finite element model in this context allows to take into consideration more complex geometries. The methodology is then applied to a rectangular plate embedding a large circular ABH with added damping. The convergence of the nonlinear coefficients is studied and time domain computations show the relative importance of the geometrical nonlinearity.

Influence of the vertical seismic component on the response of continuous RC bridges

This paper investigates the influence of vertical seismic accelerations on the seismic response of RC bridges through numerical simulations using an enhanced non-linear numerical model. Results confirm that the incorporation of vertical accelerations, either through actual records or scaled horizontal records, can considerably modify the seismic response and the collapse mechanism. In the case of actual vertical records, the vertical component significantly contributes to premature structural deterioration, intensifying demand and accelerating failure mechanisms. On the other hand, the study underscores the inadequacy of using scaled horizontal records to represent vertical accelerations, as suggested by some seismic codes, as it not only distorts seismic response evaluation but also alters failure modes. The analysis of vertical vibration reveals higher displacements, increasing flexural demand on the deck, and leading to a progressive loss of vertical support at the central column. The research establishes the need to accurately account for vertical seismic accelerations in bridge design evaluations, as their impact on structural response and failure mechanisms cannot be underestimated. The work highlights the importance of a highly detailed 3D numerical model in assessing traditional parameters and capturing complex collapse mechanisms arising from material and geometric nonlinearities.

Nonlinear Dynamics Research of Ground Undulatory Fin Robot with Flexible Deformation and Frictional Contact.

The unique rigid-flex connection between the fin-rays and fin-surface in a bionic undulatory fin robot endows the fin-surface with both active flexibility and load-bearing capacity, enabling this robot to perform amphibious motions in underwater, terrestrial, and even marshy environments. However, investigations into dynamic modeling problems for the undulatory fin robot, considering the impact of nonlinear deformation and frictional contact on ground locomotion performance, are scarce. Given this, based on the absolute nodal coordinate formulation (ANCF), this paper presents an efficient and accurate nonlinear dynamic model for this robot to elucidate the fin's flexible deformation and motion law. This model considers material, geometric, and boundary nonlinearities, utilizing ANCF thin plate elements and reference nodes to individually describe the fin-surface and fin-rays of the undulatory fin. Then, by using the master-slave technique, a frictional contact formulation for the fin and the ground is proposed. Furthermore, we conduct in-depth research and analysis on the formation and undulatory motion of the undulatory fin, encompassing its static deformation, static contact deformation, and frictional contact motion, and successfully obtain its responses under various conditions. Research indicates that during fin-surface motion, longitudinal sliding or a tendency for sliding at the contact points results in the undulatory fin moving in a crawling gait. The proposed theoretical model correctly captures the fin's complex nonlinear deformations and frictional characteristics and reveals its ground locomotion mechanism, whose effectiveness and superiority are validated through numerical examples and experiments.

Geometrical Nonlinearities on the Bearing Capacity in Clay: A Validation Data Set for Numerical Tools

Geometrical Nonlinearities on the Bearing Capacity in Clay: A Validation Data Set for Numerical Tools

Discrete macro - models of nonlinear interlocking mechanisms in the out-of-plane failure of masonry walls

AbstractHistorical unreinforced masonry (URM) constructions are generally vulnerable to out-of-plane (OOP) failures due to the absence of rigid floors and poor connections between orthogonal walls. That leads to the activation of rocking mechanisms of external walls, whose ultimate force and displacement are affected by complex nonlinear interactions with sidewalls. These interactions are often neglected in the engineering practice, potentially leading to significant approximations, as demonstrated by experimental and numerical studies available in the literature. As a novel contribution to the field, this paper presents an upgraded discrete macro-element model (DMEM) to predict the rocking capacity of OOP loaded URM walls interacting with sidewalls. Considering both the onset and the evolution of the rocking mechanism of the front wall, interlocking effects with the sidewalls are first simulated through frictional resistances using the macro-block model (MBM) and the nonlinear kinematic approach of limit analysis. Then, the upgraded DMEM is implemented on the basis of the equivalence between the continuous distribution of these forces, introduced as a further novelty of the paper, and the discrete distribution of lateral elastic-plastic links, accounting for mechanical and geometrical nonlinearities. The results of the two models are discussed in terms of both frictional resistance-displacement and pushover curves, referring to a case study of a front wall belonging to a two-storey URM building. The wall response is also compared with the results derived from the original source of the case study and analysed by changing the number of nonlinear links to define different levels of accuracy.

Nonlinear Dynamic Characteristics of Smart FG-GPLRC Sandwich Varying Thickness Truncated Conical Shell with Internal Resonance for First Three Order Modes

This paper examines the 1:1:1 internal resonant nonlinear dynamic characteristic of the simply supported varying thickness functionally graded graphene platelets reinforced composite (FG-GPLRC) smart truncated sandwich conical shell subject to the combined effects of transverse load and in-plane force. The truncated smart sandwich conical shell is composed of an FG-GPLRC varying thickness core and two magneto-electro-elastic face layers, whose material properties and constitutive relations are individually identified by the rule of mixture, improved Halpin-Tsai approach and generalized Hooke's law. Utilizing the first-order shear deformation theory (FSDT), von Karman's geometrical nonlinearity, Hamilton's principle and Galerkin technique, the 3DOF dimensionless nonlinear dynamic formulations for the truncated smart FG-GPLRC conical shell are established. The multiple-scale technique is applied to developing the averaged equations for the truncated smart FG-GPLRC conical shell under combined resonance. The frequency-response and force-response curves, Poincare maps, phase portraits, time history diagrams, bifurcation and maximum Lyapunov exponent diagrams can be portrayed by the nonlinear equation solver and Runge-Kutta approach. The effects of the damping and tuning parameters, transverse and in-plane forces on the 1:1:1 internal resonant nonlinear dynamic characteristic of truncated smart varying thickness FG-GPLRC sandwich conical shell are examined.

Snap-through eversion of axisymmetric shells under contact indentation

In this paper, we systematically investigate the stability of an axisymmetric shell and the snap-through eversion induced by indentation through a discrete numerical approach. To capture the intricate interplay between the geometric and boundary nonlinearities during contact actuation, we employ the discrete axisymmetric shell model accompanied by the incremental potential formulation for our analysis. Our results reveal that the indentation response of a spherical shell can be classified into three groups, i.e. monotonous, monostable and bistable behaviours, whose boundaries can be characterized by a simple scaling law. We further discover that, for bistable shells, the snap-through eversion happens at a critical state where the configurations are universal, which is independent of the indenter size and can be captured by a simple geometric model. One interesting prediction of our model is that, with increasing indenter size, the contact state between the shell and indenter changes from conformal contact to partial separation, which is validated by a finite element method simulation. Our findings may provide explanations for some biophysical phenomena (e.g. cell fusion) and can also guide optimal designs of intelligent structures (e.g. soft actuators and soft robots).

Exploring FEA Techniques for Bending Rectangular Glass Plates into Parabolic Shapes

Abstract: A structural Finite Element Analysis (FEA) was conducted, incorporating both linear and nonlinear analysis. The investigation considered geometric nonlinearity, material nonlinearity, and boundary nonlinearity, also known as contact analysis. All boundary conditions, loadings, and material properties used in the analysis were based on real-world conditions. Experimental validations were performed to confirm the accuracy of the FEA results, providing strong confidence in the investigation's findings.