Geometric Nonlinear Behavior Research Articles

Large-Amplitude Vibrations of Spider Web Structures

Spider silk, as a natural material, shows exceptional performance in its properties. The combination of the superior properties of spider silk and the geometry of spider structures make the spider web very resilient. A spider web structure can be considered as a cable-like structure with inappreciable torsional, bending and shear rigidities. An investigation emphasising on natural frequencies and corresponding mode shapes with and without the consideration of geometric nonlinearity is presented in this paper. This study is the world’s first discovery of large amplitude free vibration behaviours of spider web structures. Large deformable finite element 3D models of spider web structures have been developed and validated. By using the energy method, the variational model of the spider web structures have been established to further extend the finite element model, consisting of the strain energy due to axial deformation, kinetic energy due to the spider web movement and the virtual work caused by the self-weight per unit unstretched length. The emphasis of this study is placed on the linear and geometric nonlinear behaviour of the spider web structures considering different structural patterns and material properties. To determine the large-amplitude free vibrational behaviours, a series of pretension load is applied to the first step in Abaqus to initiate the nonlinear strain-displacement relationships enabling a precursor to free vibrations. The parametric studies stemming from structural patterns (the number of radial and capture threads), elastic modulus, density, and inertia moment have been highlighted. The insight will help engineers and scientists to adapt the concept of spider webs, their geometric properties, and damage patterns for the design of any structural membranes, preventing any failure from dynamic resonances and nonlinear phenomena.

Geometric Non-Linear Analysis of Auxetic Hybrid Laminated Beams Containing CNT Reinforced Composite Materials.

In the current work, a novel hybrid laminate with negative Poisson’s ratio (NPR) is developed by considering auxetic laminate which is composed of carbon nanotube-reinforced composite (CNTRC) and fiber-reinforced composite (FRC) materials. The maximum magnitude of out-of-plane NPR is identified in the case of (20 F/20 C/−20 C/20 C) S laminate as well. Meanwhile, a method for the geometric non-linear analysis of hybrid laminated beam with NPR including the non-linear bending, free, and forced vibrations is proposed. The beam deformation is modeled by combining higher-order shear-deformation theory (HSDT) and large deflection theory. Based on a two-step perturbation approach, the asymptotic solutions of the governing equations are obtained to capture the linear and non-linear frequencies and load-deflection curves. Moreover, a two-step perturbation methodology in conjunction with fourth-order Runge–Kutta method is employed to solve the forced-vibration problem. Several key factors, such as CNT distribution, variations in the elastic foundation, and thermal stress, are considered in the exhaustive analysis. Theoretical results for some particular cases are given to examine the geometric non-linearity behavior of hybrid beam with NPR as well as positive Poisson’s ratio (PPR).

Preliminary study on flutter scaling of high-aspect ratio wings

As a result of the increasing demand for aircraft performance, the high-aspect-ratio wing (HARW) becomes the primary structural type of the emerging aircraft as it improves the lift-to-drag ratio. However, HARWs result in higher deflections, leading to a geometrical nonlinear behavior and nonlinear aeroelastic problems. In this work, the preliminary study on aeroelastic behavior of two classic types of HARWs, namely, Goland wing and High Altitude Long Endurance (HALE) wing, are investigated. Aeroelastically scaled models using two scaling methods, called classic scaling method and modified scaling method, are applied and compared with each other. Froude number is considered in the modified scaling method. Both static aeroelastic and flutter analyses are conducted on each scaled model by using different scaling methods. The results show that the Froude number is important to the HALE wing, where both static deflects and flutter velocity obtained by the scaled model agree well with the full model when the Froude number similarity is met. Thus, the Froude number may be the key similar parameter for the HALE wing scaling model design. For the Goland wing scaling model, the Froude number could be ignored.

Geometrical Nonlinear Effects in the Dynamic Response of Sandwich Beams

AbstractThe geometrical nonlinear dynamic response of sandwich beams is studied using a dynamic high-order nonlinear model. The model is derived using the variational principle of virtual work and uses the Extended High-Order Sandwich Panel Theory approach with consideration of two interfaces between the three layers. A first-order shear deformation theory is adopted for the face sheets, while the kinematic assumption of high-order small deformations that account for out-of-plane compressibility are considered for the core layer. The nonlinearity of the dynamic model is introduced by considering geometrically nonlinear kinematic relations in the face sheets. The nonlinear kinematic relations and the dynamic modeling aim to evaluate the effects of the two features and their coupling on the response. The nonlinear dynamic response of sandwich beams is studied through two numerical cases and comparison of the nonlinear results with their linear counterparts. The first case looks into the coupling of the global geometrical nonlinear behavior with the dynamic behavior. The second case focuses on the local instability of the face sheets and its interaction with the compressibility of the core in the dynamic response of soft core sandwich beams. The comparison of linear and nonlinear dynamic response in the two cases sheds light on the coupling of the geometrical nonlinear and dynamic behaviors. Among other features, the latter is expressed by nonlinear attractors, higher modes response, nonlinear frequency response, and significant wrinkling response.

Critical buckling load of pile in liquefied soil

Buckling instability has been recently identified as a possible mechanism of pile failure in liquefiable deposits and this failure mechanism is not explicitly mentioned in most design codes. To carry out routine design and checking, it is necessary to reliably estimate the critical buckling load of pile for a given liquefiable site. However, the existing calculation methods for the critical buckling load of pile in liquefied soils do not consider the influence of geometric imperfections and nonlinear behavior of the pile. In this paper, an efficient approach using the Beam on Nonlinear Winkler Foundation (BNWF) model is proposed to calculate the critical buckling load of pile in liquefied soils considering geometric imperfections and nonlinear behavior of the pile. The method is verified and validated using the finite element method and results from centrifuge tests. Furthermore, parametric analysis has been carried out to understand the influence of buckling load for different soil relative density, initial geometric imperfections of pile, flexural rigidity of pile, and pier height. It is found that the critical buckling load of pile in liquefied soils increases with the increase of soil relative density and flexural rigidity of pile, and decreases with the increase of initial geometric imperfections of pile and pier height. Finally, a simplified estimation method based on Euler buckling theory is provided for predicting critical buckling load of pile in liquefied soils and an example is taken to show the application.

Geometric nonlinear dynamic analysis of tapered steel members

In this study, a theoretical analysis is presented for estimating the in-plane geometric nonlinear elastic stability behavior of steel members with tapered elements under dynamic loads. Beam-column approach is adopted for modeling the structural members as beam-column elements. The formulation is based on the Eulerian description taking into consideration the influence of axial force on bending stiffness. The changes in member chord length due to axial deformation and flexural bowing are also considered. In the dynamic analysis, the system mass properties have been represented using both lumped and consistent mass matrices. The consistent mass matrix is derived in three components: translational, rotational, and axial inertia. The formulation of the mass matrices in local and global coordinate systems of tapered members, which incorporates geometric nonlinearity, has been presented. A parametric study is conducted to examine the effects of number of tapered elements, tapering the prismatic members, time step size, and tapering ratio. The dynamic responses of tapered members have been shown to be significantly affected by these parameters.

Numerical and analytical investigation of cyclic behavior of steel ring dampers (SRDs)

In this study, the cyclic behavior of steel ring damper (SRD) was investigated through numerical and analytical methods. The present study was carried out to evaluate ductility and energy dissipation and providing an analytical equation to estimate the yielding capacity of SRDs. For this purpose, extensive parametric studies were performed using the finite element method. Parametric studies included investigating the effect of thickness, length, and diameter of steel ring on the behavior of SRD. Nonlinear static analysis was used to analyze finite element models under cyclic loading. Geometrical nonlinear behavior of materials and large deformations were considered in the modeling. Also, validation experiments were performed on experimental specimens for finite element models. Using the lower bound limit analysis and considering the effects of strain hardening, an equation was obtained to estimate the capacity of the steel ring. A good agreement was observed between results obtained from the measurement of yielding capacity using analytical equations and parametric studies. Also, yield force was found to be directly related to the length and diameter of the steel ring and yield stress. Results of parametric models and analytical relations showed that by increasing steel ring diameter, ductility and energy dissipation decreases and increases, respectively. Also, by increasing the diameter to thickness ratio (D/t), yield force, ductility, and energy dissipation decreased, respectively. • The cyclic behavior of steel ring damper (SRD) was investigated. • The numerical model of SRD was developed to predict the hysteresis behavior. • The effects of key parameters on the performance of the SDRs were studied. • Using the analytical method, was obtained the capacity of the SRDs. • A good agreement was observed between the results of numerical and analytical methods.

Topology optimization of compliant mechanisms considering stress constraints, manufacturing uncertainty and geometric nonlinearity

This paper proposes and investigates two formulations to topology optimization of compliant mechanisms considering stress constraints, manufacturing uncertainty and geometric nonlinearity. The first formulation extends the maximum output displacement robust approach with stress constraints to incorporate the effects of geometric nonlinear behavior during the optimization process. The second formulation relies on the concept of path-generating mechanisms, where not only the final configuration is important, but also the load–displacement equilibrium path. A novel path-generating formulation is thus proposed, not only to achieve the prescribed equilibrium path, but also to take stress constraints and manufacturing uncertainty into account during the optimization process. Although both formulations have different goals, the same main techniques are employed: density approach to topology optimization, augmented Lagrangian method to handle the large number of stress constraints, three-field robust approach to handle the manufacturing uncertainty, and the energy interpolation scheme to handle convergence issues due to large deformation in void regions. Several numerical examples are addressed to demonstrate applicability of the proposed approaches. The optimized results are post-processed with body-fitted finite element meshes. Obtained results demonstrate that: (1) the proposed nonlinear analysis based maximum output displacement approach is able to provide solutions with good performance in situations of large displacements, with stress and manufacturing requirements satisfied; (2) the linear analysis based maximum output displacement approach provides optimized topologies that show large stress constraint violations and rapidly varying stress behavior under uniform boundary variation, when these are post-processed with full nonlinear analysis; (3) the proposed path-generating formulation is able to provide solutions that follow the prescribed control points, including stress robustness.

Popular benchmarks of nonlinear shell analysis solved by 1D and 2D CUF-based finite elements

This research work deals with the analysis of elastic shell structures in the large displacement and rotation field adopting one-dimensional (1D) and two-dimensional (2D) unified models. Namely, higher order beam and shell theories accounting for geometrical nonlinearities are formulated by employing a unified framework based on the Carrera unified formulation (CUF) and a total Lagrangian approach. Thus, a finite element (FE) approximation is used along with a Newton-Raphson method and an arc-length path-following approach to perform nonlinear analyses. Low- to higher order beam and shell theories are used to evaluate the nonlinear equilibrium path, and results are compared between the two models, with reference solutions coming from literature or 2D and three-dimensional (3D) models from NASTRAN. Convergence analyses show how CUF 1D models are able to describe the geometrical nonlinear behavior of analyzed structure with a lower number of degrees of freedom (DoFs) than 2D and 3D models.

Theoretical and experimental modal analysis of circular cross-section shaft

Experimental Modal Analysis (EMA) is very helpful in engineering design and manufacturing of machine components. In this paper, modal parameters which are natural frequencies, mode shapes and damping ratios are extracted for free-free boundary conditions circular shaft, using EMA, then two disks are added to this shaft as second case study. EMA has been verified as effective and accurate tool, despite of increasing geometrical complexity and nonlinear behaviour of the structure. The two well-known excitation techniques (i.e. impact hammer and shaker) are used for this purpose. For validation, natural frequencies and mode shapes are determined analytically and numerically by Finite Element Modelling (FEM) ANSYS 15 workbench software, and then compared with results obtained experimentally. Noticeable remarks about these approaches are listed. Coherence plots are compared between impact and shaker tests. Digital Signal Processing (DSP) settings like windows type and sample (record) time are compared. The recommended DSP settings mentioned in this work can be used as a general guidance for researchers and people who are working in modal analysis.

Simulation of screw connected built-up cold-formed steel back-to-back lipped channels under axial compression

The objective of this paper is to validate finite element (FE) modeling protocol for screw connected, back-to-back built-up cold-formed steel (CFS) columns. The protocol is developed and validated using results from previously conducted experiments. The effort is motivated by two applications: (1) to augment experimental findings on built-up CFS columns, particularly for fastener demands and (2) to provide a simulation path for modeling the built-up CFS columns that are used as shear wall chord studs. Shell FE-based models were created in ABAQUS and include monotonic loading, nonlinear geometric and material behavior, geometric imperfections based on laser scanned measurements of tested specimens and a contact model that includes friction. Additionally, the screw fasteners were integrated into the modeling protocol using user-defined element subroutines capable of reproducing strength and stiffness deterioration under monotonic load as well as the pinching that occurs when screw fasteners are subjected to cyclic loads. Monotonic, concentric compression tests on 17 back-to-back CFS columns using two cross-section sizes and varying fastener layouts with sheathing conditions were simulated. Deformations, strength, and collapse mechanisms obtained by the models were in close agreement with the experimental results. An assessment of the loading demand on screw fasteners reveals the conservatism in built-up column fastener layout and design as required by the North American Specification for the Design of Cold-Formed Steel Structural Members (AISI S100-16 Section I1.2). Also, under the tested semi-rigid column end conditions, there is a little boost in axial capacity with the addition of member end fastener groups (EFGs) at the top and bottom of the columns. Numerical models are also used to assess the cyclic performance of axially-loaded columns so that chord stud buckling limit states could be captured in seismic simulations of CFS-framed shear walls in future work.

Nonlinear mixed finite element formulations for the analysis of planar curved beams

In the present paper, two mixed type functionals are presented to analyze the geometrical nonlinear bending behavior of slender planar curved beam by Finite Element Method (FEM). Bernoulli-Euler beam theory is taken into consideration within the context of the small strains but large displacements. Non-linear bending of curved beams in-plane has been investigated in both local and Cartesian coordinate systems. The Euler-Lagrange equations of the Functional I and Functional II which give the equilibrium and constitutive equations are in relation with the Green-Lagrange strain tensor and the first-order approximation (consistent linearization) of the Green-Lagrange strain tensor, respectively. The details of this process are given in Appendix. In the formulation of the problem in Cartesian coordinate, in addition to the boundary terms, that expression has been used, also to obtain the strain quantities which should be in the constitutive expressions in-prior to the equilibrium equations for the first time in this work. For the solution of the non-linear equations of the problems “incremental formulation” has been used. The validity, reliability and robustness of the proposed functionals have been presented by solving sample problems found in the literature.

Creating better dynamic relaxation methods

Purpose The purpose of this study is to demonstrate that using appropriate values for fictitious parameters is very important in dynamic relation methods. It will be shown that a better scheme can be made by modifying these terms. Design/methodology/approach Former research studies have proposed diverse values for fictitious parameters. These factors are very essential and highly affect structural analyses’ abilities. In this paper, the fictitious masses in ten previous well-known schemes are replaced with each other. These formulations lead to the extra 41 different new procedures. Findings To compare the skills of the created processes with those of the ten previous ones, 14 benchmark problems with geometrical nonlinear behaviour are analysed. The performances’ evaluations are based on the number of iterations and analysis time. Considering these two criteria, the score of each technique is found for the ranking assessments. Research limitations/implications To solve a static problem by using a dynamic relaxation (DR) scheme, it should be first converted to a dynamic space. Using the appropriate values for fictitious terms is very important in this approach. The fictitious mass matrix and damping factor play the most effective role in the process stability. Besides, the fictitious time step is necessary for improving the method convergence rate. Practical implications Different famous DR procedures were compared with each other previously. These solvers used their original assumptions for the imaginary mass and damping. So far, no attempt has been made to change the fictitious parameters of the well-known DR methods. As these fictitious factors highly affect structural analyses’ efficiencies, these solvers are formulated again by using new parameters. In this study, the fictitious masses of ten previous famous methods are replaced with each other. These substitutions give 51 different procedures. Originality/value It is concluded that the present formulations lead to more effective and favourable methods than the solvers with previous assumptions.

Inelastic compressive buckling behavior of a cylindrical shell at elevated temperature: Case study

A cylindrical shell is a structural form that is widely used in various industries such as for ocean structures, bridges, piers, and buildings. The purpose of this study includes investigating inelastic buckling behavior of a cylindrical shell under uniaxial compression at an elevated temperature and considering geometrically and materially non-linear analysis with imperfections. Selected parameters of the analytical model include fire exposure condition, slenderness ratio, and D/t (diameter/thickness ratio); compressive buckling behavior characteristics of a cylindrical shell were obtained using ABAQUS (a commercial FE Program). The critical buckling strength and mode shape of the cylindrical shell were analyzed through 3D modeling to present basic data for reasonably efficient design. Results indicate that it is necessary to consider nonlinear material and geometric behavior when compared to the case of the slender column.

Nonlinear Analysis of Spatial Cable of Long-Span Cable-Stayed Bridge considering Rigid Connection

To determine the cable sag effect and solve the rigid connection problem at cable ends of a long-span cable-stayed bridge, a two-node spatial catenary cable element with arbitrary rigid arms is developed. Using the finite rotation formula of the space vector and a differential method, the incremental relation between displacement and force at both ends of the rigid arm is given. Then, explicit expression of the tangent stiffness matrix of the element with arbitrary rigid arms is derived based on the catenary equations. Two numerical examples are provided to verify the validity of the new element. A long-span cable-stayed bridge application model is established, and the cables are simulated using three methods. The results show that the rigid end effect has influence on displacements, bending moments and rigidity and should not be ignored. The catenary cable element with arbitrary rigid arms can be used to simulate the geometric nonlinear mechanical behavior of the cables and can well solve the rigid connection problem.

Two-way coupled CFD fire and thermomechanical FE analyses of a self-supporting sandwich panel façade system

A self-supporting sandwich panel façade system under fire is studied. First, a thermomechanical FE model, which comprises the complete façade system, and incorporates material degradation and geometrical nonlinear behaviour except for the insulation material and connections, is loaded by a fire temperature curve. Eurocode design rules then predict the screw connections of a sandwich panel will fail in shear. Secondly, an existing programme, FDS-2-Abaqus, is extended to allow its two-way coupled analyses, in which CFD fire simulations are updated for changes in the thermomechanical FE model, to be applicable to the façade system. Again, these simulations show the shear failure of the screws. Parameter studies show differences in system behaviour for improved screw properties; a fuel-controlled vs. ventilation-controlled fire; and different panel thicknesses. Interestingly, as thermal bowing of the panel retards screw failure, and thicker sandwich panels bow less than thinner panels, thicker façade panels will decrease failure time. This and other insights obtained, and the predicted failure of two tiny but critical screws within 80 s, as compared to 150 min lasting sandwich panels in standard fire tests, stresses the need to study complete systems under realistic fires, rather than to study individual components in standard fire tests. Future research will focus on detailed FE models of the connections; full-scale fire experiments; CFD temperature measurement points at the façade outside; detailed modelling of the insulation; and the effects of high temperature creep.

COMPARATIVE STUDY OF NON-LINEAR SIMULATIONS OF A REINFORCED CONCRETE SLENDER COLUMN USING FINITE ELEMENT METHOD AND P-DELTA

This paper analyzes the non-linear geometric behavior of reinforced concrete slender columns. This approach is due to the fact that there is a tendency to reinforced concrete slender constructions, which may have significant second order effects. This research aimed at comparing different formulations for the analysis of non-linear behavior of reinforced concrete slender columns by comparing results from simulated problem (slender column with ten load scenarios) between the Finite Element Method (FEM) and the Iterative Process P-DELTA(P-Δ). Numeric results revealed that the Iterative Process P-Δ presented different results from FEM and that the second order effects are significant for reinforced concrete slender column problems.

Theoretical and Applied Insights on Pistons Buckling According to DNV Regulation

Pistons are fundamental structural elements in any engineering practices such as mechanical, civil, aerospace, and offshore engineering. Their strength strongly depends on buckling load, and such information is a major requirement in the design process. Euler's linear buckling equation is the most common and most used model in design. It is well suited for linear elastic members without geometrical imperfections and nonlinear behavior. Several analytical and experimental investigations of typical hydraulic cylinders have been carried out through the years but most of the available standards still use a linear approach with many simplifications. Pistons are slender beams with not-uniform cross section, which need a stronger effort than the classical Euler's approach. The present paper aims to discuss limitations of current DNV standards for piston design in offshore technologies when compared to classical numerical approaches and reference results provided by the existing literature.

Higher-order finite element models for the static linear and nonlinear behaviour of functionally graded material plate-shell structures

In this work, finite element formulations based on higher order shear deformation theories are used for the nonlinear static analysis of Functionally Graded Material plate-shell type structures. Linear and geometric nonlinear behaviour of the plate-shell type structures are considered. For the nonlinear analysis, the incremental equilibrium path is obtained using the updated Lagrangian procedure and Newton-Raphson incremental-iterative method, incorporating the automatic arc-length method for the cases of snap-through occurrence. The finite element models are based on a non-conforming triangular flat plate/shell element with 3 nodes and 8 or 11 degrees of freedom per node. The solutions of some illustrative plate-shell examples are performed, and the results are presented and discussed with numerical alternative models.

Combined Effects of Geometric and Material Non—linearities on One Dimensional Structural Members

Combined effects of geometric and material non-linearities on a uniform column subjected to an axial compressive load are presented in the present note. A simple, direct iterative numerical method has been proposed to study the geometric and material non-linear behavior of columns subjected to varying boundary conditions. Introduction of material non-linearity in the large deflection analysis of columns subjected to an axial compressive load reveals a reduction in Euler stress obtained when compared to the effect of geometric non-linear analysis and increase in the same when compared to the eflect of material non-linear analysis. A convergence study has been carried outfor the results obtained from the proposed iterative method to prove the efficacy.