Nonlinear Equilibrium Equations Research Articles

Geometrically nonlinear vibration of toroidal sandwich shells with auxetic honeycomb core under periodic/impulsive pressure

The main objective of this paper is to evaluate the dynamic behaviour of sandwich toroidal shells composed of a core with negative Poisson ratio (auxetic honeycomb material) and nanocomposite enriched coating layers under impulsive/periodic pressures considering the geometrical nonlinearity. The set of nonlinear equilibrium equations based on the first-order shear deformation theory are and the resultant nonlinear differential motion equations are approximately solved by the Galerkin method and 4th-order Runge–Kutta solution, respectively. The achieved responses show that the amplitude of outward deformations is intensively increased when the proposed sandwich shells are subjected to rectangular periodic and impulsive loadings which shows the destructive effects of this type dynamic loading. The performed parametric studies shows that these destructive effects can be controlled by changing the geometrical properties of the honeycomb cells and the volume fractions of the nanocomposites in the coating layers. It is notable that in the previous studies the nonlinear dynamic response of toroidal shells subjected to various periodic and impulsive pressures was not investigated. However, in the present work, the mentioned dynamic responses were achieved and investigated without spending time-consuming computational. In fact, the novelty of this work is to develop such a robust and efficient tool.

A method for finding distinct solutions of geometrically nonlinear structures

Geometrically nonlinear structures are extensively used as load-carrying systems or energy harvesting devices. The main characteristics of their behaviour include complex and/or multi-equilibrium paths. In this paper, a new method is proposed for analysing these structures wherein distinct deformation states of a structure under applied loads (forces or displacements) are directly obtained. Using the concepts of finite element approximations, the proposed method first establishes the nonlinear equilibrium equations for a structure undergoing large deformations. Then, contrary to the well-established incremental-iterative path-following methods that appeal to linearization concepts and use an arc-length constraint to solve the nonlinear equilibrium equations, the proposed method tries to directly find the solutions to these equations. To this end, a likelihood function is introduced based on the nonlinear equilibrium equations. This function is defined such that it reaches its maximum value at the solution(s) of the equilibrium equations. To maximize this likelihood function, the nested sampling algorithm is exploited. This algorithm explores the solution phase space thoroughly and finds the deformation state(s) that satisfy the nonlinear equilibrium equations. An application of the proposed method to analysing geometrically nonlinear truss structures is given and three truss structures are solved to demonstrate the capabilities of this method.

Systematic schemes for buckling analyses of a subsea bio-inspired non-circular FGM polyhedral liner with an arch invert

This study develops theoretical frameworks for the buckling failure of a non-circular thin-walled liner encased in a cracked pipeline with an arch invert. The buckling performance of the pipeline-liner system is improved by combining the polyhedral profiles and the Functionally Graded Materials (FGM). The liner is close-fitting to the inner surface of the pipeline. The energy function is obtained by assembling the thin-walled shell principle and the theory of minimum potential energy. Then, the nonlinear equilibrium equations and the buckling equations are derived to predict the buckling performances of the FGM polyhedral pipeline-liner system with an arch invert. Moreover, the present buckling behaviors of the system are compared with other available results, and good agreements are accomplished when the FGM polyhedral liner degenerates into an FGM circular liner or a homogeneous polyhedral liner. In addition, an enhancement coefficient is defined as the ratio of the buckling pressure of a polyhedral liner to a circular one. Finally, several key geometric and material parameters are examined to quantify their effects on the buckling behavior of the FGM polyhedral pipeline-liner system with an arch invert. These parameters are volume fraction exponents, polyhedral shapes, and thickness-to-radius ratios, respectively.

Systematic failure mechanism of an FGMs polyhedral arched liner under a fire disaster environment

This paper investigated the failure performance of an arched polyhedral liner yielding a fire disaster environment. The functionally graded materials (FGMs) are used in the liner to increase the ability of fire resistance. A concentrated load may occur at the invert of the arched liner due to the debris, which is considered in the study. The polyhedral arched liner deforms in a single-lobe shape, and this deformation is expressed by a trigonometric function. The nonlinear equilibrium equations are presented by applying the thin-walled shell theory and the principle of minimum potential energy, respectively. Based on the above investigations, two innovative contributions of the present study are addressed: (1) a novel FGMs polyhedral liner is introduced to rehabilitate the cracked non-circular tunnel in the combination of the mechanical loading and fire environment; and (2) the critical buckling load, as well as the load-displacement nonlinear equilibrium curves, is derived explicitly by solving the nonlinear equilibrium equations. The effect of the fire disaster on the buckling load is quantized to describe the destabilization behavior of the arched liner in a fire environment, respectively. Then, the present study is compared efficiently with other closed-form formulations when the polyhedral FGM arched liner simplifies to a circular homogeneous liner. It is shown that a smaller number of polyhedral edges and a higher content of ceramic are beneficial to resist external loadings and fire disasters in engineering practices. Finally, the effects of geometric parameters, volume fraction exponents, and temperature variations on the nonlinear buckling behavior of the polyhedral arched liner are evaluated.

An improved dog-leg method for form-finding of tensegrity structures

This paper presents a novel and highly efficient form-finding method for tensegrity structures. The method is based on converting the nonlinear equilibrium equations with the nodal coordinates vector as the variables into nonlinear least-squares problems. Subsequently, the Dog-Leg method is applied to solve these transformed problems. Furthermore, the Dog-Leg method is combined with the MBFGS (Modified Broyden-Fletcher-Goldfarb-Shanno) method to overcome the singularity problem of the stiffness matrix. The resulting improved Dog-Leg method, i.e., DL-MBFGS method, can effectively address form-finding problems not only in self-stressed and non-self-stressed tensegrity structures, but also for tensegrities subjected to external loads and other scenarios including mechanisms and large displacements problems, which shows its great versatility and robustness. Several numerical examples are presented to validate the accuracy and effectiveness of the proposed method.

Nonlinear inelastic analysis of space semi-rigid steel frames subjected to static load using plastic-zone method

A new hybrid element based on plastic-zone method is proposed for static nonlinear analysis of space semi-rigid steel frames. Four zero-length nonlinear rotational springs are added to the beam ends to model both in- and out-of-plane connection rotational behavior with appropriate semi-rigid connection models. The von Mises yield criteria are applied to govern material plasticity along the member length and throughout the cross-section using Newton–Cotes integration points. The modified generalized displacement control method is used to solve the system of nonlinear equilibrium equations. Some numerical examples are investigated to show the reliability and visual illustration of the proposed hybrid element.

Stability of higher-order lattice composite cylindrical shell reinforced with graphene platelets by means of a Chebyshev collocation-based semi-analytical approach

In this study, we present a novel analysis approach for lattice composite cylindrical shells reinforced with Graphene Platelets (GPL) nanoparticles. Our primary contribution lies in the investigation of these advanced structures, incorporating nanocomposite reinforcement, orthotropic inhomogeneity, and semi-analytical methods. The lattice composite comprises an anisogrid lattice laminated shell, reinforced with functionally graded GPL. We model this structure using a global continuous orthotropic deep shell approach, integrating the Halpin-Tsai and rule of mixtures homogenization strategies to estimate equivalent mechanical properties. We derive theoretical formulations utilizing Reddy's third-order shear deformation theory and nonlinear Sanders' kinematic assumptions, tailored for deep thick shells. Nonlinear equilibrium equations are obtained using Hamilton's principle and Hooke's constitutive law, leading to linearized bifurcation equations through adjacent-equilibrium and membrane pre-buckling analysis. Our stability analysis employs a semi-analytical method combining trigonometric expansion and Chebyshev collocation functions. Validation through parametric examples demonstrates the accuracy and efficiency of our approach, unveiling insights into the impact of lattice composite and geometric parameters on the stability response of these innovative structures.

Theoretical dynamic reduction for steady-state nonlinear vibration responses of complex jointed structures having hysteretic contact behavior

A novel nonlinear dynamic reduction method was developed to determine the steady-state vibration responses of complex jointed structures having hysteretic contact behavior. By using the harmonic balance method to reformulate nonlinear dynamic equilibrium equations into a set of nonlinear algebraic ones, a dynamic reduction technique based on nonlinearity transformation was theoretically developed to iterate nonlinear vibration solutions in the local coordinate that was related only to the degree-of-freedoms of nonlinear joints. Only odd-order harmonic components were truncated to approximate the hysteretic nonlinear contact forces of joint interfaces, which was conducive to further dimension reduction of the nonlinear algebraic equations and iteration matrix. Then, a nonlinear dynamic reduction solver was developed to associate the steady-state nonlinear vibration responses of overall structures with the dynamic characteristics of underlying linear substructures, nonlinear joint models and external excitations. Combined with finite element analysis, the steady-state nonlinear vibration responses of a complex assembled structure with four reduced-order nonlinear joint models were numerically simulated to validate the proposed nonlinear dynamic reduction method by comparing with the literature. The comparative results of nonlinear frequency response functions exhibited good agreement, and the proposed method indicated higher computational efficiency. The experimental investigations of a rubber isolator system were also performed to validate the proposed method, which demonstrated good performance.

Topology optimization of geometrically nonlinear structures using reduced-order modeling

High computational costs are encountered in topology optimization problems of geometrically nonlinear structures since intensive use has to be made of incremental-iterative finite element simulations. To alleviate this computational intensity, reduced-order models (ROMs) are explored in this paper. The proposed method targets ROM bases consisting of a relatively small set of base vectors while accuracy is still guaranteed. For this, several fully automated update and maintenance techniques for the ROM basis are investigated and combined. In order to remain effective for flexible structures, path derivatives are added to the ROM basis. The corresponding sensitivity analysis (SA) strategies are presented and the accuracy and efficiency are examined. Various geometrically nonlinear examples involving both solid as well as shell elements are studied to test the proposed ROM techniques. Test cases demonstrates that the set of degrees of freedom appearing in the nonlinear equilibrium equation typically reduces to several tenth. Test cases show a reduction of up to 6 times fewer full system updates.

Structural optimization of an encased bi-material FGP-GNF pipeline-liner system with novel polyhedral configurations

This article proposes a novel thin-walled polyhedral functionally graded porous (FGP) liner consolidated by graphene nanofillers (GNF) to rehabilitate the cracked pipeline. A buckle propagation may occur since the liner is subjected to hydrostatic pressure and temperature variational field simultaneously. A cosine function is developed to describe the deformed shape of the liner. Both pores and the GNF are distributed symmetrically on the cross-section of the liner, which also has a polyhedral profile to improve its bending stiffness. The nonlinear equilibrium equations are accurately obtained by introducing the thin-walled shell theory and the principle of minimum potential energy. The critical buckling pressure is obtained by solving the nonlinear equilibrium equations. Analytical solutions are compared successfully with other closed-form results to accomplish the verification when the FGP-GNF polyhedral liner reduces to an isotropic circular one. Furthermore, one improvement factor is proposed to analyze the effect of different polyhedral shapes on the buckling pressure of the encased polyhedral FGP-GNF liner. Finally, parameters are evaluated to further understand the buckling behavior of the FGP-GNF polyhedral liner, including the number of polygons, porosity coefficient, mass fraction, geometric shape of the GNF, radius-to-thickness ratio, temperature field, etc.

A Novel Geometric Nonlinear Analysis Method for Planar Beam Structures Based on Co- Rotation and Substructure Method

Nonlinear behavior analysis of structures using finite element methods requires multiple iterative solutions. Therefore, improving computational efficiency while ensuring the accuracy of nonlinear calculations is crucial. In this paper, a new geometric nonlinear analysis method for planar beam structures is proposed, based on the co-spinning method and the substructure method, using a substructure containing two planar beam elements as an example. The co-spinning method accurately calculates small deformations by subtracting rigid body displacement from the large displacement of the structure. The substructure method condenses the degrees of freedom of internal nodes into boundary nodes, greatly reducing the order of the nonlinear equilibrium equation group of the structure. The two methods are combined, and a calculation program is developed using the arc length method, which has good performance in solving nonlinear equation groups. Two classic examples are used to verify the correctness of the proposed method by comparing the results with existing literature.

Static and dynamic nonlinear behavior of a multistable structural system consisting of two coupled von Mises trusses

Recent years have witnessed a growing interest in multistable structures. In most cases it is attained by a sequence of bistable elements. However, little is known on their nonlinear static and dynamic responses. In this work a detailed static and dynamic nonlinear analysis of a multistable structural system consisting of two coupled bistable von Mises trusses is conducted to enlighten the complex behavior of these systems. For this, the exact nonlinear equilibrium equations and equations of motion in their dimensionless forms are obtained through the principle of stationary potential energy and Hamilton’s principle, respectively, considering a linear elastic material. Using continuation algorithms, the nonlinear equilibrium paths are obtained. The stability of each configuration is analyzed using the principle of minimum potential energy. Multiple equilibrium paths of the coupled system are identified, leading to several coexisting stable and unstable solutions, bifurcations and potential wells which are closely connected to the symmetries of the system. It is shown that the number of coexisting solutions increases exponentially with the number of bistable units. The effect of unavoidable imperfections is also clarified. The nonlinear dynamics and bifurcations of the system under harmonic forcing and static pre-load are then studied. Bifurcation diagrams, Poincaré maps and cross-sections of the basins of attraction are used to study the influence of coexisting attractors due to multiple potential wells, resonances and bifurcations. The coexisting responses can merge giving rise to several types of cross-well motions, the threshold value being closely connected with the limit point load of the coupled system. The increase in coexisting solutions leads to increasingly complex basins of attraction with broad fractal regions. On the one hand, the complexity of the bifurcation scenarios seems valuable in several applications to encode information, to respond to external forcing and to switch between different states. On the other hand, multiple attractors and their fractal basins can lead to the loss of global stability and dynamic integrity due to the small uncorrupted basins surrounding each attractor. Thus, the knowledge of the static and dynamic behavior of multistable systems is of great significance to either inducing or avoiding this phenomenon or making use of it.

Numerical Modeling of Lyapunov Exponents for Structural Damage Identification

The main purpose of this article is to discuss the use of the Lyapunov exponents to evaluate the integrity of structures. The use of such coefficients is examined in an analysis that considers the geometric and physical nonlinearities, aiming to ensure the applicability of the method in robust simulations. The material nonlinearity is modeled using the multilinear isotropic elastoplastic model together with a recently developed damage model. The nonlinear equilibrium equations solution is obtained using the positional finite element method. The Newmark time-marching procedure is implemented to evaluate the Lyapunov coefficients and a nonlinear predictor technique that needs a single data series is employed. A numerical example of a frame structure is presented to illustrate the methodology applicability. Its results show that the Lyapunov exponents can be used as indicative parameters of structural integrity, since its analysis was able to detect the occurrence of the destabilization of the structure with the dynamic jump and the presence of material failures. The non-linear predictor proved to be an efficient technique for obtaining the Lyapunov exponents, with a low computational cost. The methodology presented to monitor structural integrity was shown to be a promising alternative.

Flexural buckling and post-buckling analysis of tapered columns in transient fire

This paper presents time-varying flexural buckling and post-buckling responses of tapered columns under a transient fire of which the temperature is non-uniform along the length. First, the transient temperature field within the column is determined by solving a dimensionally reduced transient heat transfer problem considering the convection-radiation boundary conditions and temperature-dependent (TD) material properties. Subsequently, the von Kármán geometric nonlinear theory is applied to obtain the nonlinear equilibrium equation. The nonlinear heat transfer and equilibrium equations are solved by the differential quadrature (DQ) method. According to a yield criterion, the ultimate load at which the material failure occurs is predicted. The influences of a transient localised fire on the temperatures, buckling and post-buckling responses of tapered steel tubular columns are discussed.

Nonlinear thermal post-buckling analysis of rectangular FG plates using CUF

In this study, a numerical analysis examines the thermal post-buckling response of rectangular functionally graded (FG) plates. Since the post-buckling is analyzed in a thermal environment, temperature distribution across the thickness is computed by an accurate numerical method. Here, for the geometrically nonlinear analysis because of assuming large displacement, FEM is utilized according to Carrera Unified Formulation (CUF), a precise formulation and higher-order deformation theory. The principle of virtual work is employed to achieve the nonlinear equilibrium equations. The governing equations are solved by operating the arc-length method. The influence of parameters such as geometric aspect ratio, length-to-thickness ratio, boundary conditions, type of temperature distribution, and volume fraction index on the FG plates’ temperature-deflection path has been investigated. In addition, a comparison among the present study method with another reference about thermal post-buckling and also linear buckling analysis (LBA) is performed. Nonlinear post-buckling analysis shows that the bifurcation point occurs only at fully clamped boundary conditions and low aspect ratios. It proved the results of LBA in the mentioned state are verifiable. The outcomes confirm the usefulness of utilizing CUF on thermal post-buckling since the temperature curves in the figures are at a lower level compared to the similar reference.

Effect of graphene reinforcement distribution on energy release rate and vibration of thermally pre/post-buckled delaminated composite plates

The nonlinear thermal stability responses of functionally graded graphene-reinforced composite (FG-GRC) laminated plates with single or multiple through-the-width delaminations, are investigated in the article. The possibility of delamination growth is evaluated using the three-dimensional crack tip element (3D-CTE) method, which determines the energy release rate (ERR) at the delamination edge. Furthermore, the influence of delamination configurations and the types of graphene reinforcement distribution patterns, on the free vibration of FG-GRC delaminated plates in thermally pre/post-buckled regimes is evaluated. The von Karman geometrical nonlinearity is adopted in a solution based on the layerwise third-order shear deformation theory (TSDT). The nonlinear equilibrium equations derived by the minimum total potential energy principle are solved using the Ritz method in conjunction with the Newton–Raphson iterative procedure. A three-dimensional finite element model is also developed using ABAQUS to corroborate the accuracy of the theoretical results. Parametric studies reveal that while the FGX graphene distribution pattern improves the bending stiffness of delaminated composite plates more than FGA, the ERR at the near surface delamination edge in FGX pattern is twice as high as that in FGA. This implies that the possibility of delamination propagation in FGX plates can be higher than FGA. Additionally, while the fundamental frequency of the FGX plate with and without delamination exceeds the frequency of other graphene distribution patterns at the reference temperature, its natural frequency is the lowest among all patterns in the post-buckled regime.

Research on One-to-Two Internal Resonance of Sling and Beam of Suspension Sling–Beam System

An approach is presented to investigate the 1:2 internal resonance of the sling and beam of a suspension sling–beam system. The beam was taken as the geometrically linear Euler beam, and the sling was considered to be geometrically nonlinear. The dynamic equilibrium equation of the structures was derived using the modal superposition method, the D’Alembert principle and the Hamilton principle. The nonlinear dynamic equilibrium equations of free vibration and forced oscillation were solved by the multiple-scales method. We derived the first approximation solutions for the single-modal motion of the system. Numerical examples are provided to verify the correctness of formula derivation and obtain the amplitude–time response of free vibration, the primary resonance response, the amplitude–frequency response, and the amplitude–force response of forced oscillation. According to the analysis, it is evident that the combination system exhibits robust nonlinear coupling properties due to the presence of internal resonance, which are useful for engineering design.

Nonlinear static analysis of bi-directional functionally graded sandwich plates in thermal environments by a higher-order finite element model

This study addresses nonlinear static behaviours of bi-directional functionally graded (2D-FGM) sandwich plates lying on an elastic foundation in thermal environments for the first time. The plates comprise two 2D-FGM face sheets and a homogeneous core layer. The material properties of the face sheets change according to the power law in both the longitudinal and thickness directions. Furthermore, the material properties also possess temperature-dependent characteristics in the thermal environment. For the analysis purpose, a higher-order finite element model (FEM) based on Shi’s plate theory that takes into account the geometrically nonlinear von Kármán without regard to material nonlinearity is established. The minimum potential energy principle is used to establish the weak form of governing equations. The four-node rectangular element with the Lagrange and Hermitian interpolation functions is used to discretize the domain of the plate. The nonlinear equilibrium equations are solved by employing Newton–Raphson iterative procedure. The numerical examples comparing the obtained results with the published results in open literature have confirmed the accuracy and convergence of the proposed model. Finally, a few novel parametric studies are conducted to show the effects of the material properties, geometric parameters, temperature, elastic foundation stiffness, and boundary conditions on the nonlinear static behaviours of the 2D-FGM sandwich plates.

On the influence of edge damage on thermally induced buckling of stepped circular bi-laminates

The influence of edge damage on thermo-mechanical buckling of stepped circular bi-laminates is examined analytically. The nonlinear equations of equilibrium, the associated boundary conditions and the transversality conditions for the propagating interior boundaries of a region of sliding contact are derived via the Principal of Stationary Potential Energy as a propagating boundary problem in the calculus of variations. The problem is ultimately recast in a mixed formulation expressed in terms of the membrane forces and the transverse displacements of the composite structure. A loading parameter is identified and closed form analytical solutions to the resulting non-linear problem are determined. In addition, a stability criterion is established based on the second variation of the total potential energy of the structure, and qualitative behavior concerning contact is deduced analytically. Results of numerical simulations based on the analytical solutions established herein are generated and the phenomenon of “sling-shot buckling” is observed, whereby the structure first deflects in one direction upon heating (or cooling) and then, when a critical temperature is achieved, dynamically “sling-shots” from the current equilibrium configuration to an alternate equilibrium configuration possessing deflections in the opposite sense. In addition, the phenomenon of “buckle-trapping” is observed for stepped circular bi-laminates with hinged-fixed supports, as well as for clamped-fixed structures with relatively large damaged area. A robust mixture of contact behavior, including propagating sliding contact, edge contact and no contact, and the transition from one form of contact to another, is exhibited for both pre-buckling and post-buckling configurations. The influence of edge damage on critical load and characteristic behavior is assessed.

A Semi-Analytical Approach for the Linearized Vibration of Clamped Beams with the Effect of Static and Thermal Load

The geometric nonlinearity due to static and thermal load can significantly alter the vibration response of structures. This study presents a semi-analytical approach to illustrate the nonlinear vibration of clamped-clamped beams under static and thermal loads. The von Karman strain and Hamilton’s principle are employed to derive the nonlinear static equilibrium equation and nonlinear governing equation. The vibration equation’s coefficient is variable. The transfer-matrix method and local homogenization are used to solve the equation. The proposed method’s accuracy is validated by commercial software and literature. The numerical results indicate that uniform stress caused by thermal load only reduces the structural mode frequencies. The geometric nonlinearity of the structural static deformation affects both the mode frequencies and mode shapes. And the mode shapes cannot be approximated by harmonic functions. When the static deformation is significant, the structure’s local RMS response is substantially affected. The combined loads have a more significant impact on the acceleration response than the superposition of individual load effects.