Nonlinear Static Response Research Articles

Nonlinear Static and Dynamic Response of a Random Rubble Stone Masonry Building with Horizontal Seismic Bands

Nonlinear Static and Dynamic Response of a Random Rubble Stone Masonry Building with Horizontal Seismic Bands

Nonlinear static and dynamic response of a metastructure exhibiting quasi-zero-stiffness characteristics for vibration control: an experimental validation

This work introduces a novel metastructure designed for quasi-zero-stiffness (QZS) properties based on the High Static and Low Dynamic Stiffness mechanism. The metastructure consists of four-unit cells arranged in parallel, each incorporating inclined beams and semicircular arches. Under vertical compression, the inclined beams exhibit buckling and snap-through behavior, contributing negative stiffness, while the semicircular arches provide positive stiffness through bending-dominated behavior. The design procedure to achieve QZS is established and validated through finite element analysis and experimental investigations. The static analysis confirms a QZS region for specific displacement. Dynamic behavior is analyzed using a nonlinear dynamic equation solved using the Harmonic Balance Method, validated experimentally with transmissibility curves showing sudden jump down with effective vibration isolation. Parametric studies with varied payload masses and excitation amplitudes further verify the ability to of metastructure to attenuate vibrations effectively in low-frequency ranges.

Discrete mechanical model for nonlinear dynamical analysis of planar guyed towers considering the unilateral contact of cables

Towers are widely used for power line transmission, wind power plants, TV and radio broadcasting, and telecommunications. To enhance their stability, cables are often employed to anchor these towers to the ground. In this study, we investigate the nonlinear static and dynamic responses of a planar guyed tower in which the unilateral constraints on the cables are considered. A representative discrete mechanical model with two degrees of freedom is developed to simulate the central mast of the tower, and the cables are modeled as unilateral springs with linear stiffness. The nonlinear equilibrium equations are derived using an energy approach that incorporates the dissipative forces, total potential, and kinetic energies into the Euler-Lagrange equations. Unilateral cable contact is directly included in the nonlinear equilibrium equation for the guyed tower, allowing for numerical analysis without the need to evaluate the contact point at each time or load step. Several numerical strategies are employed to obtain nonlinear static equilibrium paths, bifurcation diagrams, phase portraits, and Poincaré sections. Our analyses provide novel results for the influence of unilateral cable contact in nonlinear static and dynamic analysis, evaluating the effects of unilateral contact and prestressing on the results. A parametric analysis reveals that cable contact affects nonlinear oscillations, bifurcation, and stability. Our numerical results indicate that unilateral cable contact introduces less structural stiffness compared to bilateral contact, thereby significantly affecting the static and dynamic stability of a planar guyed tower. This is evidenced by a decrease in the static limit load and alterations in the bifurcation diagrams, where unilateral contact destroys the trivial solutions, leading to periodic and quasi-periodic solutions at low levels of vertical load.

Multi-segmented fifth-order polynomial–shaped shells under hydrostatic pressure

The design and construction of submerged complex shells for new applications in offshore structures are increasingly popular. Therefore, the purpose of this study is to present the analytical model and Lagrange multipliers associated with the constraint equation for large displacement analysis of a fifth-order polynomial–shaped shell under hydrostatic pressure for the first time. The shell geometry can be computed using differential geometry with a fifth-order polynomial. The energy functional of the fifth-order polynomial–shaped shell is derived based on the principle of virtual work and written in the appropriate form. The nonlinear static responses of the fifth-order polynomial–shaped shell under hydrostatic pressure can be calculated using the nonlinear finite element method via the fifth-order polynomial shape function. This study develops the model using one-dimensional beam elements divided along the shell radius. To avoid the slope of a meridian curve at the equatorial plane approaching infinity, the shell is divided into two regions defined by different surface parameters. At the junction of two adjacent regions, the continuity requirements are established as the constraint conditions using Lagrange multipliers. The numerical results from the proposed methods are demonstrated and discussed, along with the effects of varied seawater depth, thickness, and elastic modulus on the deformed configuration and principal curvature at the deformed state. The results show that the nonlinear displacement is higher than the linear one in the case of the hydrostatic pressure, whereas the case of the internal pressure has an opposite result. For principal curvatures at the apex, the principal curvatures increase as the seawater depth increases, whereas the principal curvatures decrease when the thickness and elastic modulus increase.

Nonlinear Static and Dynamic Responses of a Floating Rod Pendulum

Abstract A novel nonlinear dynamics model is developed in this paper to describe the static and dynamic nonlinear behaviors of a rod pendulum partially immersed in still water. The pendulum is hinged above the water level and subject to nonlinear gravity, hydrostatic, and hydrodynamic loads, all of which are incorporated into the system dynamics. The nonlinear static behavior and stability of the pendulum have been characterized by analyzing the fixed points. It is found that Pitchfork bifurcation governs the relationship between the rod density (the control parameter) and the static equilibrium angle. The pendulum's nonlinear response to external harmonic torque is obtained using Harmonic Balance Method (HBM). The influence of system parameters, including hinge height, rod diameter, and rod density, on the nonlinear frequency response is examined. Upon altering the system parameters, particularly the rod density, it is found that the system exhibits either a softening or a hardening effect.

Static and dynamic response of pyramidal lattice sandwich plate with composite face sheets reinforced by graphene platelets

For the pursuit of lightweight structures with excellent mechanical performance, sandwich structures with lightweight cores and strengthened face sheets have become popular in the advanced engineering fields in recent years. In this paper, the nonlinear static and dynamic responses of the sandwich plate constructed with the aluminum pyramidal lattice core and functionally graded graphene platelets reinforced composite (FG-GPLRC) face layers are analyzed. The partial differential equations of the pyramidal lattice sandwich plate considering the geometric nonlinearity are achieved based on the Reddy’s third order shear deformation theory (TSDT), which are solved by the combination of finite element model, Raphson’s method and Newmark numerical integration method. Four different types of the impact loads including the step loading, triangular loading, half-sine loading and exponential loading with the same peak value and duration time are investigated. The influences of the thickness-to-truss radium ratio and the angle of the truss core, as well as the weight fraction and distribution patterns of graphene platelets (GPLs) over the face sheets under different impact loads and different boundary conditions are analyzed. According to the numerical results, it is concluded that the sandwich plate with higher weight fraction of GPLs, lower thickness-to-truss radium ratio, and higher angle of the truss core can exhibit better performance in both static and dynamic loading conditions. The FG-X distribution of GPLs presents the best dynamic performance as compared to the other GPL distributions. The combination of lightweight truss core with the advanced GPLRC face sheets is instructive for potential applications and further research.

Influence of the Uplifting Mechanism of Embedded Footings on the Nonlinear Static Response of Steel Concentrically Braced Frames

Influence of the Uplifting Mechanism of Embedded Footings on the Nonlinear Static Response of Steel Concentrically Braced Frames

Lateral load response and collapse probability of reinforced concrete shear walls retrofitted for repairability

AbstractThis paper demonstrates the efficacy of a new reinforced concrete shear wall seismic retrofit method through a series of nonlinear static and incremental dynamic analyses. Unlike traditional retrofit methods, the method investigated aims to convert conventional walls into self‐centering walls whose behavior is governed by rocking and flexure. The retrofit involves creating a cold joint at the foundation–wall interface, cutting some reinforcing bars to allow rocking, adding external post‐tensioning to enable self‐centering, and externally confining wall toes to prevent concrete crushing. The retrofit was applied to two building archetypes, each with two different shear wall designs. The four walls were retrofitted by varying retrofit parameters (portion of the vertical reinforcement bars cut, and external post‐tensioning amount). Nonlinear static and nonlinear response history analyses were performed using experimentally validated, computationally efficient models that simulate walls with fiber‐based beam–column elements. Incremental dynamic analysis was used to create collapse fragility functions for pre‐ and post‐retrofit walls. The results show that the retrofit is effective when some vertical reinforcement bars are left uncut across the foundation–wall interface. The retrofit is more effective for walls with vertical reinforcement distributed across cross‐section as compared to walls with reinforcement concentrated near boundary elements and for walls with structurally efficient amounts of reinforcement as compared to walls with higher amounts of reinforcement. This is attributed to the larger amount of reinforcement bars cut in walls with concentrated reinforcement layouts or heavy reinforcement amounts, leading to a larger loss of strength, recovery of which requires larger amounts of post‐tensioning.

Nonlinear dynamic analysis of opto-electro-thermo-elastic perovskite plates

AbstractPhotostrictive materials have attracted tremendous interest as the new generation of smart materials that can achieve a direct conversion from optical energy to mechanical energy. Understanding their nonlinear mechanical properties under light illumination is of paramount significance for their realistic optomechanical applications. This article proposes a novel opto-electro-thermo-elastic constitutive model that can consider the effects of photostriction, photothermal temperature, and electrostriction for metal halide perovskite crystals and investigates the nonlinear static and dynamic responses of the perovskite plates. The nonlinear governing equations are established based on the first-order shear deformation theory and von Kármán nonlinearity and are numerically solved by the differential quadrature method. A detailed parametric investigation is performed to analyze the effects of light and electricity on the nonlinear mechanical behaviors of perovskite plates. It is concluded that light illumination leads to the presence of optical stress and thermal stress in the perovskite plates, giving rise to increased static and dynamic deformations and stresses, as well as reduced postbuckling and free vibration characteristics. The research findings pave the way for the optomechanical applications of perovskite-based smart materials and structures.

Voltage-controlled non-axisymmetric vibrations of soft electro-active tubes with strain-stiffening effect

Material properties of soft electro-active (SEA) structures are significantly sensitive to external electro-mechanical biasing fields (such as pre-stretch and electric stimuli), which generate remarkable knock-on effects on their dynamic characteristics. In this work, we analyze the electrostatically tunable non-axisymmetric vibrations of an incompressible SEA cylindrical tube under the combination of a radially applied electric voltage and an axial pre-stretch. Following the theory of nonlinear electro-elasticity and the associated linearized theory for superimposed perturbations, we derive the nonlinear static response of the SEA tube to the inhomogeneous biasing fields for the Gent ideal dielectric model. Using the State Space Method, we efficiently obtain the frequency equations for voltage-controlled small-amplitude three-dimensional non-axisymmetric vibrations, covering a wide range of behaviors, from the purely radial breathing mode to torsional modes, axisymmetric longitudinal modes, and prismatic diffuse modes. We also perform an exhaustive numerical analysis to validate the proposed approach compared with the conventional displacement method, as well as to elucidate the influences of the applied voltage, axial pre-stretch, and strain-stiffening effect on the nonlinear static response and vibration behaviors of the SEA tube. The present study clearly indicates that manipulating electro-mechanical biasing fields is a feasible way to tune the small-amplitude vibration characteristics of an SEA tube. The results should benefit experimental work on, and design of, voltage-controlled resonant devices made of SEA tubes.

Three-dimensional dynamics of supported pipe conveying fluid with arbitrary initial spatial shape

In order to meet different requirements of engineering applications, pipes conveying fluid usually have the geometric characteristic of a spatial initial shape. In this paper, a three-dimensional theoretical model based on absolute node coordinate formulation (ANCF) is proposed to analyze the dynamical behavior of spatial pipes conveying fluid with arbitrary initial shape. By introducing the spatial Frenet frame as the local coordinate, the global coordinate of each point on the pipe centerline is intuitively represented first. Considering the tensile, torsional and bending deformations of the pipe, the governing equations of spatial pipes conveying fluid are derived by means of the generalized Lagrange equation. Then the process of solving nonlinear static deformation, natural frequencies and nonlinear dynamic responses of the pipe is given. Further, by comparing the results calculated by the ANCF method with those of previous studies on straight and curved pipes, the effectiveness of the proposed three-dimensional ANCF model is verified. Finally, the nonlinear responses and natural frequencies of spatial pipes with three different initial shapes are calculated and some of them are compared with the results obtained using finite element simulations. The results show that the three-dimensional ANCF model proposed in this study has good feasibility and wide application prospect in nonlinear dynamics analysis of fluid-conveying pipes with complex initial shapes.

A refined first-order shear deformation theory for large dynamic and static deflection investigations of abruptly/harmonically pressurized/blasted incompressible circular hyperelastic plates

In this paper, nonlinear large-deformation dynamic and static responses of the suddenly and harmonically pressurized or blasted compliant incompressible hyperelastic circular plates are investigated through the development of a novel enhanced circular plate theory that in contrast to the available plate theories, utilizes transverse normal strains whose order is much higher than those of the in-plane strains. For this reason, no shear correction factor is needed for the new plate theory. The deformation gradient and Green's strain tensors are first determined based on the first-order shear deformation plate theory as predictors and then, corrected by simultaneously including the transverse compliance and volume conservation concepts in the neo-Hookean framework of hyperelasticity. The equations of motion are obtained using Hamilton's principle. The resulting highly nonlinear equations that include both large deformation and constitutive nonlinearities are solved by a hybrid iterative/updating technique that utilizes a combination of the differential quadrature (DQ) spatial-discretization and Newmark's time-marching methods, after an exhaustive order analysis for the identification of the extremely small terms. The effects of the changes in geometric parameters, material properties, and loading rate and type are investigated on the dynamic lateral deflection of the plate and the approximate fundamental natural frequency. Results reveal that the contribution of the higher vibration modes in the responses is much more remarkable in the high-rate and high-frequency loadings. Furthermore, the deformed shape of the plate that indicates the superimposed effects of the higher vibration modes is plotted for successive time instants.

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.

Nonlinear static analysis of multi-segmented toroidal shells under external pressure

This paper investigates the nonlinear static responses of multi-segmented toroidal shells with egg-shaped cross sections. The discontinuity effect in terms of displacement and the slope at the toroidal shell edge must be constrained by the Lagrange multiplier technique. The structural shape of the multi-segmented toroidal shells can be described by the differential geometry of the shell with the six-center-combined toroidal shells. The energy functional of the multi-segmented toroidal shells can be written in the appropriate form based on the principle of virtual work. The numerical results of the displacement responses and the volume change can be obtained by the nonlinear finite element method via the fifth-order polynomial shape function and a direct iterative method. In this study, the Lagrange multiplier is the internal force at the toroidal shell junction sustaining the smooth curve of the multi-segmented toroidal shells under several loads. Finally, the results of the volume change and the displacement responses for various geometric parameters are presented and discussed.

Novel incremental procedure in solving nonlinear static response of 2D-FG porous plates

The manuscript presents a nonlinear mathematical model capable to investigate the nonlinear bending response of Bi-directional functionally graded (BDFG) plate resting on elastic foundations including large deformations in the sense of von Kármán. A unified higher order shear plate theory is used to implement the shear influence with parabolic distribution through thickness direction. The gradation of materials is portrayed by power law function through axial (x-direction) and thickness (z-direction). The governing equations consist of a set of four nonlinear coupled partial differential equations. The assumption that the material properties change in the x- direction complicates the governing equations since they have variable-coefficients. Winkler–Pasternak model is exploited to present the elastic foundations with normal and shear deformations. In this work, a novel incremental–iterative method is proposed, and some tricky computational issues are performed to ensure stable and fast convergence rates even with large incremental load steps. The differential/integral quadrature method (DIQM) is developed to numerically discretize the variable-coefficient governing equations on a rectangular plate with different boundary conditions. Parametric studies are presented to illustrate the impact of nonlinear strains, gradation indices, geometrical properties, and boundary conditions on the transverse displacement, normal and shear stresses.

A first-order equivalent static loads algorithm for optimization of nonlinear static response

The Equivalent Static Loads (ESL) algorithm for nonlinear static response structural optimization is modified to promote convergence to designs satisfying first-order optimality conditions. The modifications involve first-order estimates of the equivalent static loads considered in the sub-problems and algorithmic stabilization through a trust-region approach. The practical convergence properties of the original and modified algorithms are assessed through numerical experiments on a set of reproducible structural size optimization problems. The results demonstrate the capabilities of the proposed algorithm in finding optimized designs that satisfy first-order optimality conditions numerically with modest computational resources.

Nonlinear Thermomechanical Static and Dynamic Responses of Bidirectional Porous Functionally Graded Shell Panels and Experimental Verifications

AbstractThe present article examines the nonlinear static/dynamic behavior of the functionally graded porous shell panel with variable geometrical shapes exposed to thermomechanical load. The higher-order shear deformation theory (HSDT) is employed to develop a finite element (FE)-based mathematical model. The geometric nonlinearity is incorporated using Green–Lagrange nonlinear strains (GLNS). Voigt's micromechanical model, in association with power-law (GT-I), sigmoid (GT-II) and exponential (GT-III) kinds of material grading patterns, is adopted to calculate the graded panel's effective properties. Also, even (PRT-I) and uneven (PRT-II) distributions of porosity are considered in the present work. The temperature-dependent (TD) properties are adopted in association with variable temperature fields, i.e., uniform (TD-I), linear (TD-II), and nonlinear (TD-III) for the computation of flexural responses. To compute the desired nonlinear responses, the direct iterative technique is utilized. Convergence is used to validate the established model's stability and correctness is further verified by comparing the current numerical data to published and experimental results. The experiment was carried out by fabricating a few natural fiber-reinforced linearly varying layerwise panels for the test run. The study is further extended to investigate the influence of design parameters on nonlinear static and transient data (flexural/stress) of the functionally graded curved/flat panel considering thermal environmental conditions.

Koiter–Newton Based Model Reduction for Large Deflection Analysis of Wing Structures

Wing structures subjected to large deflections are prone to nonlinear load-deflection behavior. Geometric nonlinearities can arise due to the accompanying large rotations and in-plane deflections that manifest in the form of stiffening effects in the nonlinear static response. To account for these nonlinearities, reduced-order modeling techniques in combination with nonlinear finite element formulations have been previously proposed. However, these methods often have a limited range of validity due to linear eigenmode-based formulations with assumptions of small rotations. In this paper, a large deflection analysis framework based on the Koiter–Newton model reduction technique is presented. It is demonstrated that the reduced model in its basic form is ineffective for large deflection analysis. To resolve this, an incremental updating procedure is used for the reduced-order model that incorporates the necessary nonlinear effects. The model updating enables the computation of nonlinear response for a large range of deflections.

Nonlinear static and dynamic response prediction of bidirectional doubly-curved porous FG panel and experimental validation

Nonlinear static and dynamic response prediction of bidirectional doubly-curved porous FG panel and experimental validation

Nonlinear static and dynamic (deflection/stress) responses of porous functionally graded shell panel and experimental validation

The static and dynamic characteristics of functionally graded (FG) structures have been investigated in this article, considering full geometrical nonlinearity. The finite element (FE) solutions are obtained for the graded panel by modelling through higher-order kinematics (HSDT) and Green-Lagrange strain-displacement relations (GLNST). The desired graded panel properties are obtained through Voigt’s micromechanical approach, considering different grading patterns (exponential, EPL; sigmoid, SGM and power law, POL). Additionally, to maintain the generality of the graded structure, different porosity distribution types (even and uneven, EVP and UEP) are incorporated into the proposed mathematical model. Further, the numerical solutions are obtained using Newmark’s method’s direct iterative technique and constant integration steps. The solution sensitivities are verified for the developed algorithm via adequate convergence and comparison tests. In addition, the experimental validation is performed using the layerwise fabricated luffa-fibre reinforced FG plates to validate the proposed theoretical model’s accuracy. Lastly, the responses are computed using the newly derived model for the variable design-dependent parameters associated with the FG structural geometry and properties. The final deliverables, that is the nonlinear deflection responses (NDFR)/stress values, are discussed under mechanical loadings (static and dynamic).