Geometric Nonlinear Behavior Research Articles

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.

Precise analysis of geometrical nonlinear mechanical behavior of shear deformable laminated curved composite beams in local coordinates via mixed type finite elements

In the present article, geometrically nonlinear analyses of curved laminated composite beams are carried out using Mixed Finite Element Method (MFEM). Timoshenko beam theory is used to consider shear influence which significantly effects the structural behavior of composite layers. Energy-based functional is generally needed to perform geometric nonlinear analyzes of layered composites. However, obtaining energy-based functional in geometric nonlinear theory is quite challenging due to the difficulty of solving mathematically nonlinear differential equations. In this study, this situation is overcome and a functional suitable for performing geometric nonlinear analyzes of layered composite materials is obtained. Incremental formulation procedure is utilized as a solution method for nonlinear finite element analyzes. This is an innovative application of the non-linear mixed finite element method and has the ability to perform geometric nonlinear analysis of curved laminated composite beams with perfect precision. By obtaining the functional in local coordinates, it is possible to solve special problems in various fields such as construction, machinery, aerospace and biomechanics.

Insight into single-helix intelligent shape memory polymer cables: modeling and optimization under finite sliding contact

This pioneering study focuses on the finite element analysis (FEA) of thermomechanical properties of shape memory polymer (SMP) wire ropes and their components under both small- and finite-sliding contact deformation. To validate the FEA, we need to validate both geometric modeling and non-linear material behavior. Owing to intricate geometry, as well as excessive wire interactions in the structure, this part is studied by simulating a 1 × 37 steel wire rope and then comparing it with existing experimental data. To evaluate the response of non-linear material behavior, we employ the available numerical results to model the thermomechanical property of an SMP rectangular bar under a uniaxial test and then verify both constrained and unconstrained recovery behavior. After rigorous validation, two configurations of 1 × 7 and 1 × 27 SMP cables are modeled based on the thermo-visco-hyperelastic constitutive framework for acrylate polymer systems. Upon exerting an axially tensile load on these 1 × 7 and 1 × 27 SMP wire ropes, the response of force and shape recovery, as well as the normal and shear stress distributions, are measured under constrained and unconstrained conditions. For a deeper physical understanding, the influences of different temperature rates (5 and 1 °C min−1), inter-wire sliding frictional coefficient (0.1–0.6), and multiple-shape programming on the stress-strain-temperature relations of these SMP cables are also investigated. Furthermore, based on optimizing two cable factors of diameter and helix angle, and using the design of experiments method, the specific energy of a 1 × 6 SMP cable is maximized. Under different thermomechanical loadings, this study tries to cast light on the remarkable features and possible potential applications of these newly developed SMP actuators which may foster unparalleled advancements in various industries.

Non-Linear Behavior of Double-Layered Grids

Abstract This study presents a numerical approach to an analysis of the mechanical behavior of double-layered tensegrity grids. We present a comparative study on the behavior of tensegrity grids through geometric nonlinear analysis (GNA) and combined nonlinear analysis (CNLA) (geometric and material), considering the possible effect of evolution in the elasto-plastic domain of the cable elements. The effect of the relaxation of cable on the amplification of the displacement of these grids was taken into account. The updated Lagrangian formulation, which modifies the Newton-Raphson iterative scheme with incremental loading, was adopted. We have developed a numerical computational model specific to tensegrity structures that simulates the geometric and material nonlinear behavior. The reliability of the calculation tool developed has been validated. Additionally, the results of the application of the numerical model on a grid, which was generated based on demi-cuboctahedral tensegrity cells, are presented.

Robust Two-Layer Control of DC Microgrids With Fluctuating Constant Power Load Demand

This paper proposes a cascaded control structure for the regulation of DC Microgrids (MG). It is well-known that the negative impedance characteristics of a constant power load (CPL) adversely affect the stability of the network, and can cause problems such as voltage collapse or damage the electronic components. To mitigate this, we propose a two-layer control structure, where at the inner layer, the proposed controller achieves fast tracking of the supplied reference points and ultimate boundedness of the trajectories in a desired set. The outer layer generates the inner layer reference points, accounts for system constraints, and introduces robustness of the voltage dynamics to unknown perturbations of the CPL demand. For the first time, an investigation of the nonlinear geometric behaviour of the CPL is carried out to derive necessary conditions that ensure boundedness of the network dynamics and feasible regulation to a desired equilibrium set. Finally, Control Lyapunov functions are formulated to prove the stability and estimate the region of attraction of the closed loop dynamics. A simulated scenario of a meshed MG network is presented to confirm the validity of the results.

Non-Linear Behaviour and Analysis of Innovative Suspension Steel Roof Structures

Suspension structures are one of the most effective roof load-bearing structures for medium to long spans. Their shape under symmetric loads is usually a square parabola or a curve close to it. The biggest drawback of such structures is their increased deformability under asymmetric loads. So-called rigid cables are used to solve this problem. However, the production of such rigid cables with a curvilinear shape is complicated, and their maintenance also has drawbacks due to the above-mentioned shape. To avoid these shortcomings, straight-line suspension structures have been used. This paper proposes a new form of combined suspension roof structures consisting of main load-bearing straight suspension elements supported by cable struts. For the main suspension elements, the bending stiffness is accepted, taking into account the operational requirements of the structure. This article analyses the behaviour of such a combined suspension structural system in symmetric conditions with an innovative approach. The arrangements of this system are discussed. The calculation of the forces and displacements of this structure and its elements is presented, taking into account the geometrical nonlinear behaviour. The distribution of the forces in the rigid elements and node displacements of the structure are discussed. The proposed new form of a combined cable-supported roof structure was shown to be more effective in terms of weight than the standard parabolic-shaped suspension structure.

Assessment on flexural performance of monolithic glass considering spatial and depth characteristics of scratches

Glass products are widely used in building envelopes. Fracture can be triggered accidentally from the glass surface damage such as scratch. This study proposed an approach to assess the flexural performance of the aged monolithic glass panel, which has scratch with spatial and depth information detected via non-contact sensing. The coaxial double ring (CDR) test was conducted for experimentally evaluating the flexural strength on a total number of 160 scratched annealed (AN) and fully tempered (FT) glass specimens. Finite element (FE) analysis was performed to find the exact stress state by considering the geometric non-linear behaviour of glass panel under large deflection. Test and numerical results demonstrate that the scratch depth characterized by a confocal microscope and spatial location can both influence the failure load of FT glass, whereas AN glass is only sensitive to the latter. With combining linear elastic fracture mechanics (LEFM) theory, the relationship between the scratch depth and the residual strength of FT glass can be accurately determined. The spatial and depth characteristics of scratch can then be integrated for the failure load assessment of monolithic FT glass panel. The assessment framework can contribute to the accurate non-destructive assessment on aged architectural glass.

Memeplex-based memetic algorithm for the multi-objective optimal design of composite structures

Hybrid construction is proposed to decrease costs in lightweight structures using fiber-reinforced plastics (FRP) composite materials. Minimum weight, minimum strain energy (stiffness), and minimum energy variability of the structural system response are the objectives of the robust design optimization approach applied to composite shell structures with stiffeners. The trade-off depends on given stress, displacement, and buckling constraints imposed on composite structures considering non-linear geometric behavior. The design variables are ply angles and ply thicknesses of shell laminates, the cross-section dimensions of stiffeners, and the variables related to the selection of materials and their distribution at the laminate level and subsequently of the laminates along the structure. A Multi-Objective Memetic Algorithm (MOMA) applies multiple learning procedures exploring the synergy of different cultural transmission rules. An enlarged virtual population with a dual age-dominance nature captures and updates the Pareto curve. The concept of memeplex controls the meme selection and their propagation along the hybrid genetic and cultural evolution path. The success of memetic learning procedures is analyzed by measuring each meme's relative and absolute success events according to different approaches, depending on the reference time used in evolutionary history. Results show that MOMA is promising in multi-objective optimization of composite hybrid structures.

On the influence of free space in topology optimization of electro-active polymers

This study investigates the impact of the surrounding free space on the topology optimization (TO) of electro-active polymers (EAPs). It is well understood that, under the application of an electric field, the deformation of an EAP is not solely determined by the field distribution within the body, but also by the distribution in the free space surrounding it. This is particularly true for electronic EAP, which are emerging as leading candidates for developing artificial muscles. Our study specifically focuses on understanding the influence of the free space in the context of density-based TO. We model the free space as an extended void region around the design domain. Our numerical experiments focus on EAP actuators and take into account their geometrical nonlinear behavior. The results show that incorporating the surrounding free space has a significant impact on the performance of the optimized EAPs with low electric permittivity. This makes it essential to consider in real-world applications.

Nonlinear Stress-Free-State Forward Analysis Method of Long-Span Cable-Stayed Bridges Constructed in Stages

Structural analysis and construction control of staged-construction processes are major subjects in the context of modern long-span bridges. Although the forward and backward analysis methods are able to simulate situations, their main disadvantage is that they usually apply the stage superposition principle. In the actual construction process, due to changes made to the plan, the construction process needs to be adjusted at any time, and it is difficult to implement the construction process in complete accordance with the established plan. As a result, the existing simulation method based on the incremental structural analysis of each construction stage has poor adaptability to such adjustments. In this study, considering the strong geometric nonlinear behavior of the long-span cable-stayed bridge construction process, the geometrically nonlinear mechanical equations of the staged-construction bar system structure were derived. The minimum potential energy theorem was used by introducing the concept of the stress-free-state variable of the structural elements. The equation reflects the influence of the change in the stress-free-state variables of structural elements on the completion state of the structure. From the analysis of the geometrical condition that the equilibrium equation holds, the stress-free installation condition of the closing section of the planar beam element structure was obtained. A new simulation method for long-span cable-stayed bridge construction has been proposed, which is called the stress-free-state forward analysis. This method can directly obtain the intermediate process state of cable-stayed bridge construction without performing stage-by-stage demolition calculations, and causing the internal force and deformation of the completion state to reach the design target state. This method can realize the simulation of multi-process parallel operation in construction, and solves the problem of automatic filtering of temporary loads. To illustrate the application of the method, a long-span cable-stayed bridge was analyzed.

Computational methodology for the optimal design of steel truss frames integrating MATLAB and FEA platforms

Seeking to define more robust and executable structures, this article presents a computational framework for the optimal design of steel truss frames, which integrates two software of great prominence in structural engineering, MATLAB and ANSYS. Computational interfaces were created to enable the automation of the iterative optimization process. The optimization algorithm and computational interfaces were developed on the MATLAB platform, while structural analyses were performed with the ANSYS Mechanical APDL platform. Some details of the computational implementation are presented herein. The objective is to minimize the cost of structures by determining the optimal positioning of nodal coordinates and the optimal choice of commercially available structural profiles - shape and size optimization, respectively. For shape optimization, a computational model was defined to automatically generate the geometry of the structure, maintaining some intrinsic relations of symmetry and collinearity between elements. The design constraints are critical nodal displacements, stresses, and slenderness of elements, following the requirements of ABNT NBR 8800:2008. To exemplify the application of the proposed methodology, this article presents the optimal design of trusses for roofs of industrial sheds, considering its non-linear geometric behavior, typical of this type of structure.

Finite element analysis of unreinforced masonry walls with different bond patterns

Masonry is the oldest building material, yet it is also the least understood due to the non-linear and composite nature of masonry, which consists of brick units, mortar, and unit-mortar contact. In this paper, the response of a two-dimensional masonry wall with a window opening subjected to an in-plane lateral pushover loading is simulated by varying the interface properties of brick such as crushing, elastic, cracking, and shear properties. The simplified micro-modeling technique with the Engineering Masonry model for bricks and linear stiffness properties for the interfaces in the bed and head joints is employed to investigate the geometric nonlinear behavior of the masonry wall. The pushover curves obtained from the numerical simulations indicate that there is a significant influence on the lateral load response of the wall due to elastic, crushing, and shear parameters while the cracking parameters have less impact on the ductile capacity of the structure. Moreover, the study is also extended to examine the effect of bond patterns such as English, Stretcher, Flemish, and Header bond with varied aspect ratios of 1,1.5 and 0.75. In all four bond patterns, it was observed that the walls with lower aspect ratios exhibited higher strength. Further, in comparison to the other bond patterns, walls with the Flemish bond pattern demonstrated higher strengths at both lower and higher aspect ratios.

On the nonlinear analysis of pre-strained shape memory alloy beams

We present a theoretical and numerical investigation of the transverse dynamic response of monolithic Shape Memory Alloy (SMA) beams under the effect of axial pre-strains. The SMA beam is taken as benchmark structure to evaluate the effect of (axial) pre-strain on either the free or the forced dynamics. The combination of the static stress, produced by the pre-loading conditions, with the dynamic stress, associated with the vibratory response, affects the occurrence of the stress-induced SMA phase transformation hence providing a mechanism to passively tune the dynamic response. Particular attention is given to the effect that different pre-strain levels have on the overall ability to dissipate mechanical energy, that is on the effective damping. The numerical model of the SMA beam accounts for both material and geometric nonlinear behaviors. The nonlinear material behavior is due to the SMA phase-transformation and it is modeled via the one-dimensional improved Brinson’s model (including tension–compression asymmetry), while the geometric nonlinear behavior is due to large displacements occurring during the phase transformation and it is accounted for via von Kármán assumptions. The resulting model is solved numerically via the finite element method and used to evaluate both the free and the forced response of the beam. The free response analysis suggests the existence of optimal pre-strain levels to achieve maximum damping capacity, while the forced response highlights the occurrence of nonlinear dynamic features, such as bifurcation. In both cases, it is found that the level of pre-strain can significantly affect the dynamic response as well as the effective damping. The results presented in this work provide useful guidelines to understand the dynamics of continuous SMA structures under pre-strain and the ability to tune either their resulting dynamics or dissipation properties.

Improved identification of stiffness coefficients of non intrusive nonlinear geometric reduced order models of structures

Significant, successful efforts have been devoted in recent years to the non intrusive construction from finite element software of reduced order models of the nonlinear geometric response of structures. A key assumption in these models is that the elastic restoring forces can be represented as cubic polynomials of their generalized coordinates. However, there has been only very limited discussion of whether this representation is appropriate, how deviations from this polynomial form affect the identification of its coefficients and ultimately the prediction of the structural response, and how current identification strategies can be improved.These questions are the focus of the present investigation which considers first a simple structural model that exhibits the potential for snap-through and thus represents an interesting validation case with strongly nonlinear geometric behavior. It is shown that the force–displacement relationship which can be derived in closed form is not exactly a cubic polynomial but can be very closely modeled by one. However, the match between exact curve and polynomial fit is very sensitive to small changes in the linear, quadratic, and cubic stiffness coefficients and thus the prediction of the response may range from excellent to significantly in error depending on how the identification of the coefficients is performed. In particular, it is found that a cubic approximation obtained by Taylor series around the undeformed configuration is not very accurate post snap-through.Based on these findings, two strategies are proposed to improve the identification of the stiffness coefficients of these reduced order models. The first one is a tuning procedure in which a limited number of key coefficients of the model are adjusted to fit additional static response data. The second approach performs the identification of the stiffness coefficients at a series of different levels, as opposed to one as currently performed, and seeks the most appropriate value of each coefficient based on a “small change” criterion.The applicability of these two novel strategies is demonstrated on several challenging structural models: a clamped–clamped curved beam undergoing snap-through as well as a curved panel and a hat stiffened one. It is found that the reduced order models identified via the tuning or multi-level strategies not only lead to more accurate predictions of the structural response as compared to the original finite element model but also are more computationally stable than their counterparts identified with the current single-level strategy.

Geometrically nonlinear analysis of sandwich panels with auxetic honeycomb core and nanocomposite enriched face-sheets under periodic and impulsive loads

In this paper the shallow sandwich panels are considered to be made of the nanocomposite enriched face-sheets and the auxetic honeycomb core so that its Poisson's ratio is negative. In order to evaluate the geometrical nonlinear dynamic behavior of such sandwich panels, various periodic and impulsive loadings are considered. For the sake of solving the governing nonlinear differential equations system, an analytical approach on the basis of the Galerkin method is conducted which yields a closed form differential equation of motion which is numerically solved by the fourth-order Runge–Kutta method concept. The precision of the proposed analytical method is approved using comparison of the results related to some special examples and the relevant ones achieved by the other analytical and numerical methods in the references. The results of verification examples showed that the proposed method yields to the results with errors of less than 2% comparing to the results of previously published papers. The influences of different geometrical and mechanical parameters on the dynamic nonlinear responses of the proposed sandwich panels undergoing the various impulsive and periodic pressures are also evaluated in the framework of wide range of numerical study. The numerical studies revealed that by using the proposed method, performing the nonlinear dynamic analysis of the proposed panels without time-consuming computations is possible.

Data-driven computing for nonlinear problems of composite structures based on sub-domain search technique

The distance-minimizing data-driven algorithm brings new insight in computing nonlinear problems of composite structures, where the calculation is directly carried out by finding the material data that closest to equilibrium constraints. However, it is challenging to efficiently find the desired material data in a tremendous database. Towards this end, a sub-domain search technique is proposed to construct a sub-database with adaptive size for each computational step, which enables to significantly speed up the data search process in the data-driven algorithm. The databases of the composite materials are collected by the offline homogenization over the representative volume element. Several examples, including the nonlinear geometric behavior of sandwich beam, the internal instability of fiber reinforced composite plate and the large deformation of composite beam, are performed to validate the reliability and efficiency of the proposed technique. The results show that compared with the classical multiscale finite element method, the data-driven method with the sub-domain search technique is able to predict the nonlinear behaviors of composite structures with good accuracy and efficiency.

Effect of adhesive de-bond and crack in adherent plate on single lap joint with bi-adhesive

PurposeThis paper aims to study the benefits of use of bi-adhesive (combination of two different adhesives) over conventional single adhesive in bonded lap joints. Characterise damage severity due to cohesive and adherent failure as feedback for operating load levels that assist in developing damage tolerance design of the adhesively bonded joints.Design/methodology/approachSingle lap joint where the adherent plate is made up of aluminium alloy joined together with bi-adhesives is analysed. The nature of adhesives ranges from brittle, elastic-plastic, moderately ductile to largely ductile. Numerical analysis is performed considering the material and geometric non-linear behaviour of the joint. The optimum bond ratio of bi-adhesives and the effect of the location of adhesive on the stress distribution are studied. The cohesive zone modelling (CZM) is adopted to account for the cohesive failure of the joint. The adherent plate failure is also addressed by modelling and studying the behaviour of the crack at different locations in the plate using modified virtual crack closure integral (MVCCI).FindingsThe results obtained from the stress analysis show some important characteristic behaviour of the bi-adhesive joint. Although bi-adhesive is expected to result in improved joint strength, the purpose gets defeated if a brittle adhesive is used at the corners and ductile adhesive at the middle. The joint strength based on CZM, evaluated for a single adhesive, is in good comparison with the experimental results from the literature. Also, the location of the crack in the adherent plate plays a significant role in the failure of the joint.Originality/valueEstimating joint strength for the bi-adhesive model using CZM and evaluating damage severity in the presence of de-bond and crack in the bi-adhesive lap joint model assists in developing robust damage tolerance design models of such joints.

Peculiarities of calculation and design of smooth orthotropic panels with considering geometric nonlinear behavior

When calculating the stability and bearing capacity of the panels of the caisson of the wing of an aerospace aircraft, it is necessary to take into account the presence of a heat-shielding coating, which also acts as an elastic base (EB) for the skin. The subject of this work is thin smooth orthotropic rectangular panels with an elastic base that does not perceive compressive and shear flows acting on the panel. The aim of the work is to develop methods for evaluating and designing composite panels associated with an elastic foundation, taking into account the possible geometrically nonlinear behavior under loads close to the design level. Note that the indicated level of loading can be realized only in full-scale static tests in the experimental justification of the strength of the wing structure. The paper presents analytical solutions of a geometrically nonlinear problem by the Bubnov-Galerkin method for rectangular orthotropic panels, taking into account the ER under the action of compressive and tangential forces. Based on the solutions obtained, methods for determining the thicknesses of orthotropic panels are proposed, taking into account two possible criteria: either reaching the ultimate strength stresses in a possible supercritical state, or reaching the limiting values of the deflection amplitude with geometrically nonlinear behavior. The last specified condition is a consequence of the requirements for the strength of the adhesive bond between the orthotropic skin and the heat-shielding layer. The paper considers smooth rectangular orthotropic panels with rigid support along the long sides and hinged support along the short sides.

Surrogate model for a mechanical metamaterial undergoing microstructure instabilities and phase transformations

This paper addresses the modeling of periodic metamaterials with microstructure instabilities. Such instabilities are produced by bistable elastic mechanisms, i.e., snap-through deformation mechanisms, inducing phase transformations. Metamaterials with these features can be used for energy dissipation in elastic regimes between other applications.A new surrogate model of the bistable mechanism to replace High-Fidelity Finite Element simulations is developed in two sequential stages. In the first stage, a semi-analytical model of the bistable structural element is presented. Following an approach taken from the literature, the analysis of the bistable element is generalized to include load cases that had not been considered in the original study. These loading conditions naturally happen in functionally 2D and 3D metamaterials with complex microarchitectures. In a second stage, a frame element with linear kinematics and small deformations is described. The constitutive relations of the frame element, i.e., axial stress and moment in terms of axial strain and curvature, follow the same equations of the semi-analytical model derived in the first stage. A Variational Principle of Energetic Equivalence supplies the link between the semi-analytical model of the curved beam and the frame element. In this way, the response of this frame element, of only six degrees of freedom, reproduces the non-linear geometric behavior of the bistable element.The surrogate model is particularly efficient for simulating a large number of unit cells of the metamaterial and with the capacity to reproduce its behavior under general loading conditions. Thus, it is appropriate to assessing the metamaterial limit behavior, typically concerning phenomena such as energy dissipation and hysteresis under closed load cycles.

Two-step incremental procedure associated with the normal flow technique applied to trusses

To achieve the nonlinear structural behavior, there is a need to trace of their equilibrium path in the space of load-displacement. Truss systems are commonly implemented in several structural systems, including high-span bridges and bracing of the supporting structure of tall buildings. Our study adapts a two-step method with cubic convergence into an incremental-iterative procedure to analyze the geometric nonlinear behavior of trusses. The solution method is combined with the Linear Arc-Length path-following technique. To find the approximate root of nonlinear equation system in the two-step method, two formulas are used. Structures are discretized using the Positional Finite Element Method and all truss bars are assumed to remain linear elastic. The correction of the nodal coordinates subincrement vector is performed using the Normal Flow technique. A computational algorithm was implemented using the free program Scilab. Our numerical results show that, when compared to the standard and modified Newton-Raphson algorithms, the new algorithm decreases the number of iterations and the computing time in the nonlinear analysis of trusses. Equilibrium paths with force and/or displacement limit points are obtained with good precision.