Nonlinear Equilibrium Paths Research Articles

Effect of a Bending Stiffness on the Load-Displacement Path of Von Mises Truss

Von Misses truss is one of the best examples to explain nature of non-linear solution and define the snap-through. Linear buckling analysis and nonlinear finite element approaches are compared in presented paper. At the present time theoretical models for the evaluation of the ultimate load assume a structure with imperfections. The peculiarities of the effect of the magnitude and mode of initial imperfections are investigated. Effect of member stiffness on the load level in critical point of nonlinear solution, as well as the relative position with respect to the critical load from buckling analysis are analyzed. To obtain the nonlinear equilibrium paths, Newton-Raphson iteration algorithm has been used. Obtained results are compared with those gained using ANSYS system.

Non-linear in-plane stability analysis of FG circular shallow arches under uniform radial pressure

In this paper, non-linear stability behavior of functionally graded (FG) circular shallow arches subjected to a uniform radial pressure is investigated by an analytical method. For this purpose, the classical single layer assumption is used to approximate the displacement field through the arch. Donnell׳s non-linear model for shallow shells is employed to derive the strain–displacement relations. The material properties vary smoothly through the thickness of the arch according to a power-law distribution. The governing equilibrium equations and the complete set of boundary conditions are extracted employing the principle of virtual displacements and variational calculus. Because of considerable pre-buckling deformations of shallow arches, the stability analysis is accomplished considering the pre-buckling deformations. The non-linear equilibrium paths are presented for two symmetric types of boundary conditions. Results show the influences of properties dispersion, geometrical characteristics, and boundary conditions on the stability behavior of the FG circular shallow arches. Also, the results of the paper are compared with the known data in literature.

A one-dimensional continuum with microstructure for single-wall carbon nanotubes bifurcation analysis

In this paper a one-dimensional continuum endowed with polynomial nonlinear hyperelastic constitutive functions is used in order to capture the necking and the kinking arising in carbon nanotubes. In view of some preceding works by Antman and Podio-Guidugli these phenomena are seen as cases of bifurcation from a trivial nonlinear equilibrium path. The bifurcation analysis is performed by means of the asymptotic method. For a sample case, the bifurcation point and the eigenmode are determined by solving a standard eigenvalue problem. The slopes of the trivial and the bifurcated equilibrium curves at the critical point are also determined and shown.

Nonlinear Equilibrium and Stability Analysis of Axially Loaded Piles Under Bilateral Contact Constraints

This paper presents a nonlinear stability analysis of piles under bilateral contact constraints imposed by a geological medium (soil or rock). To solve this contact problem, the paper proposes a general numerical methodology, based on the finite element method (FEM). In this context, a geometrically nonlinear beam-column element is used to model the pile while the geological medium can be idealized as discrete (spring) or continuum (Winkler and Pasternak) foundation elements. Foundation elements are supposed to react under tension and compression, so during the deformation process the structural elements are subjected to bilateral contact constraints. The errors along the equilibrium paths are minimized and the convoluted nonlinear equilibrium paths are made traceable through the use of an updated Lagrangian formulation and a Newton-Raphson scheme working with the generalized displacement technique. The study offers stability analyses of three problems involving piles under bilateral contact constraints. The analyses show that in the evaluation of critical loads a great influence is wielded by the instability modes. Also, the structural system stiffness can be highly influenced by the representative model of the soil.

Geometrical nonlinear formulation of a Molecular Mechanics model applied to the structural analysis of single-walled carbon nanotubes

In this paper, the post-critical behavior and buckling modes of single-walled carbon nanotubes are analyzed via a Molecular Mechanics model. The main target is to develop a general formulation for the model, which has been simplified under small strains assumption, and to implement a versatile tool for the structural analysis of carbon nanotubes in the framework of geometrical nonlinearity. For this purpose, a mechanical formulation able to reproduce any load configuration and supporting conditions has been derived by using an energy approach. Then, an incremental-iterative solution procedure has been implemented in order to trace several nonlinear equilibrium paths and to obtain the corresponding critical strains of clamped–clamped nanotubes under compressive, flexural and torsional loading distributions. The model shows a good numerical performance and results in agreement with previous atomistic works. Two interatomic potentials have been adopted in order to find out the influence of different constitutive relationships on the final nonlinear response. We have concluded that the choice of the potential function has no significant effect on the final buckling strains. Our results confirm that the final buckling response is strongly determined by geometrical imperfections in the nanotube, which can be well reproduced in the proposed model, but are much more difficult to handle in continuum models.

Closed form solutions for buckling and postbuckling analysis of imperfect laminated composite plates with piezoelectric actuators

Buckling and postbuckling behavior of symmetric laminated composite plates with surface mounted and embedded piezoelectric actuators subjected to mechanical, thermal, electrical, and combined loads is studied. Formulation is based on the classical laminated plate theory with von-Karman non-linear kinematic relations. Initial geometrical imperfections are also accounted, and finally applying Galerkin procedure, the resulting equations are solved to obtain closed form expressions for non-linear equilibrium paths. Temperature dependency of thermo-mechanical properties is considered. Three cases of simply supported boundary conditions are investigated. Effects of in-plane compressive loading, temperature dependency and independency of properties, electrical loading, lay-up configuration, and geometric imperfection are discussed. Results for various states are verified with the known data in the literature.

Interface behavior of grouted connection on monopile wind turbine offshore structure

The interface behavior of a concrete grouted connection is studied in a monopile wind turbine offshore structure. The grouted connection between transition piece and monopile is fixed by filling it with grout materials made of high-strength concrete. We observe a nonlinear equilibrium path from the grouted connection in experiments. For comparison with the experimental results, nonlinear finite element analysis of the grouted connection is performed, taking the effects of both material and geometrical nonlinearities into consideration. A finite element model of the grouted connection between the steel and concrete materials is realistically simulated using an interface element. The finite element results compares well with the experimental results. Thus, the finite element analysis showed that the numerical model based on the interface element is a good method for investigating the behavior of grouted connections of monopile offshore structures.

In-plane nonlinear buckling of circular arches including shear deformations

A numerical strategy is presented to trace the pre-buckling as well as the post-buckling nonlinear equilibrium paths for elastic circular arches in which the effects of shear deformations and the geometric nonlinearity due to large deformations are taken into account. Timoshenko beam hypothesis is adopted for incorporating shear. The constitutive relations including shear effects for stress and finite strain are based on a hyperelastic constitutive model. The finite strain equilibrium equations are developed for the circular arches. Based on the derived transformed equilibrium equations and the boundary conditions, the nonlinear buckling behaviour of circular arches is investigated using the trapezoid method with Richardson extrapolation enhancement. The results are validated using available experimental results in the literature, the finite element package ANSYS and other solutions in the literature. Parametric studies are performed on examples to identify the factors that influence the nonlinear buckling of circular arches. The shear deformation effects on the nonlinear buckling behaviour and the buckling mode are investigated for circular arches under many different loading conditions and various boundary conditions.

Iterative strategies associated with the normal flow technique on the nonlinear analysis of structural arches

A large part of the numerical procedures for obtaining the equilibrium path or load-displacement curve of structural problems with static nonlinear behavior is based on the Newton-Raphson iterative scheme to which are coupled the path-following methods. In this context, this study uses one technique, referred to as normal flow, in the process of obtaining the approximate nonlinear static response of structural systems. Basically, this technique is an adaptation made with in the Newton-Raphson iterative scheme in an attempt to speed up the nonlinear solution process and/or remove convergence problems. To overcome the critical points and to trace the whole nonlinear equilibrium path, three different strategies are used in association with the normal flow technique: the cylindrical arc-length, the minimum residual displacement norm and the generalized displacement. With this procedure, the performance of these strategies when associated with the normal flow technique is valued. Two arches with highly nonlinear load-displacement curves are used in the study. The results obtained demonstrated that the association of the generalized displacement strategy with the normal flow technique contributes to the improvement of the nonlinear solution methodology.

Nonlinear Analysis of Buckling & Postbuckling

Abstract The stability analysis of slender web loaded in compression was presented. To solve this problem, a specialized computer program based on FEM was created. The nonlinear finite element method equations were derived from the variational principle of minimum of potential energy. To obtain the nonlinear equilibrium paths, the Newton-Raphson iteration algorithm was used. Corresponding levels of the total potential energy were defined. The peculiarities of the effects of the initial imperfections were investigated. Special attention was focused on the influence of imperfections on the post-critical buckling mode. The stable and unstable paths of the nonlinear solution were separated. Obtained results were compared with those gained using ANSYS system.

Multiple unstable equilibrium branches and non-linear dynamic buckling of shallow arches

An arch under a suddenly-applied load will oscillate about its equilibrium position. If the suddenly-applied load is sufficiently large, the oscillation may reach a position on the unstable equilibrium branch of the arch, triggering its dynamic buckling. In many cases, arches are supported by other structural members or by elastic foundations which provide elastic types of rotational restraints to the ends of the arch. When the rotational end restraints of an arch are not equal to each other, the in-plane non-linear equilibrium path of the arch may have multiple unstable branches, which will influence the dynamic buckling of the arch significantly. This paper investigates effects of multiple unstable equilibrium branches on the non-linear in-plane dynamic buckling of a shallow circular arch under a suddenly-applied central concentrated load. Two methods based on the energy approach are used to derive the dynamic buckling loads. It is found that the number and magnitude of dynamic buckling loads are influenced significantly by the number of unstable equilibrium branches, by the stiffness of the unequal rotational end restraints, and by the included angle and slenderness ratio of the arch.

Postbuckling analysis of Timoshenko nanobeams including surface stress effect

A modified continuum model is developed to predict the postbuckling deflection of nanobeams incorporating the effect of surface stress. To have this problem in view, the classical Timoshenko beam theory in conjunction with the Gurtin–Murdoch elasticity theory is utilized to propose non-classical beam model taking surface stress effect into account. The geometrical nonlinearity is considered in the analysis using the von Karman assumption. By employing the principle of virtual work, the size-dependent governing differential equations and related boundary conditions are derived. On the basis of the shifted Chebyshev–Gauss–Lobatto grid points, the generalized differential quadrature (GDQ) method is adopted as an accurate, simple and computational efficient numerical solution to discretize the non-classical governing differential equations along with various end supports. Selected numerical results are worked out to demonstrate the nonlinear equilibrium paths of the postbuckling behavior of nanobeams corresponding to different values of beam thickness, buckling mode number, surface elastic constants, and various types of boundary conditions.

Non-linear thermo-elastic and buckling analysis of FGM shallow arches

The paper presents a theoretical investigation on the non-linear thermo-elastic behaviour of pin-ended functionally graded material (FGM) circular shallow arches. Temperature dependency of constituents is taken into account and the arch is subjected to a uniform temperature field. Classical arches theory along with the non-linear shallow shell theory of Donnell are postulated as the basic assumptions. The virtual displacement principle and calculus of variation techniques are implemented to obtain the equilibrium equations. An analytical solution is presented to trace the primary equilibrium path of a pin-ended arch under in-plane thermal loading. Afterwards, closed form expressions are yielded to obtain the radial and axial displacement, stress, strain and bending moment of the arch as a function of temperature parameter. Adjacent equilibrium criterion is used to extract the stability equation associated to the non-linear primary equilibrium path. A closed form solution is presented to estimate the fundamental thermal bifurcation points of the arch. Illustrative results examine the role of the various involved parameters such as power law index, opening angle and length to thickness ratio. Numerical results reveal that, in most cases, critical buckling temperature difference of the FGM shallow arches are too high, even for thin class of arches.

Distortional–global interaction in lipped channel columns

This work presents and discusses numerical results concerning the post-buckling behaviour (elastic and elastic–plastic) and ultimate strength of cold-formed steel lipped channel columns experiencing distortional–global interaction. The analyses cover columns with (a) geometries selected to ensure very close global and distortional buckling loads, and (b) the support conditions usually adopted in experimental investigations: fixed-ended and pin-ended columns. A fairly large number of columns are analysed, differing in the initial geometrical imperfection shape and/or yield stress – the various initial geometrical imperfection shapes are linear combinations of the competing buckling modes with amplitudes equal to (a) 10% of the wall thickness, t (D) and (b) L/1000 (G). For clarification purposes, the column pure global elastic post-buckling behaviours are also addressed. These results are (a) elastic (mostly) and elastic–plastic non-linear equilibrium paths, (b) figures and curves providing the evolution, along the elastic paths, of column deformed configurations and, for the elastic–plastic columns, (c) figures showing the plastic strain location and growth, as well as the natures of the failure mechanisms.

In-plane stability of preloaded shallow arches against dynamic snap-through accounting for rotational end restraints

Because the non-linear equilibrium path of a shallow arch has unstable branches, when the arch is subjected to a suddenly-applied load, it may lose its stability by dynamic snap-through owing to excessive oscillations reaching its unstable branch. In practice, arches are usually subjected to a certain level of static loading before the sudden load is applied. The static load may have a profound influence on the dynamic snap-though instability of the arch. In addition, the rotational end restraints of an arch provided by adjacent structural members or/and elastic foundations have significant influences on the geometric non-linearity of the arch, and consequently they influence its stability against dynamic snap-through under a suddenly-applied load. An energy approach is used to derive the dynamic snap-through buckling loads of statically preloaded shallow circular arches having rotational end restraints under a suddenly-applied uniform radial load. It is found that the statically preloaded arches can be susceptible to dynamic snap-through buckling. It is also found that the dynamic buckling loads of the arches are influenced significantly by the magnitude and difference of the relative flexibility of the rotational end restraints. Arches having equal rotational end restraints may have an upper and a lower dynamic buckling loads, whereas arches having unequal rotational end restraints have only a unique dynamic buckling load.

Special Issue on Stability of Composite Structures

The increased use of composites in aerospace and civil engineering has resulted in growing demand for engineers versed in the advanced design of composite structures. Composite structures are understood in the wide sense, including microstructured materials, composite structural shapes, composite members built up by subelements of various materials, laminate or sandwich members, small-scale composite structures, etc. Compositelike structural elements are increasingly used in traditional engineeringfields, such as civil engineering, mechanical engineering, or aeronautical engineering, and they have also found some new applications with some recent developments, such as biomedical engineering or biomechanics, nanotechnology, etc. This special issue is primarily based on the presentations given in the symposium on the stability of composite structures organized by the EngineeringMechanics Institute (EMI) Stability Committee during the ASCE EMI Annual Conference held in Boston on June 2–4, 2011. This symposium has provided a forum to discuss recent advances and address the future prospects in the area of the stability of composite structures. The symposium has covered a large number of topics related to the stability of composite structures including the following: • Buckling of composite members; • Plate buckling; • Buckling and vibration of thin-walled structures; • Shear and large displacement analysis; • Interactive buckling and nonlocal mechanics; • Shear effects for in-plane and out-of-plane analyses; • Inelastic buckling; • Anisotropic effects, microstructured materials, and stability problems; • Buckling of sandwich structures; • Stability of partially composite members and delamination effects; • Postbuckling; and • Dynamic buckling. After going through an external review process, the following papers have been accepted and are included in this special issue on the stability of composite structures. As indicated by the title of this special issue, most of the presented papers uniquely deal with the challenges in stability analysis brought by the complexity of composite materials and structures. Their specific advances by these contributions are briefly summarized as follows: 1. In “Local Buckling Analysis of Restrained Orthotropic Plates under Generic In-Plane Loading,” Qiao et al. investigated analytically and numerically the local buckling of composite orthotropic plates under generic in-plane loading and with rotationally restrained boundary conditions. They used both semianalytical and finite-element method (FEM) numerical approaches to characterize the stability behavior of laminated plates or panels of fiber-reinforced plastic structural shapes. The approximated semianalytical solution is obtained from the variationally based Rayleigh-Ritz method, and it is successively compared with available results published in the literature. 2. In “Modeling of Interactive Buckling in Sandwich Struts with Functionally Graded Cores,” Yiatros et al. studied the interactive buckling in sandwich struts with functionally graded cores. The postbuckling behavior is analyzed for this sandwich column composed of two Euler-Bernoulli extensible columns connected with a shear functionally graded core. Linearized and nonlinear analyses are presented for this nonlinear elastic structural problem, and the possibility of capturing secondary bifurcations for such nonlinear systems is numerically shown. This nonlinear model is compared successively to some FEM results based on two-dimensional analyses for the core behavior. 3. In “Buckling Loads of Two-Layer Composite Columns with Interlayer Slip and Stochastic Material Properties,” Schnabl et al. presented an efficient stochastic buckling model for studying the structural reliability of layered composite columns with interlayer slip between the layers and random material and loading parameters. The deterministic model is composed of two connected elastic columns with the shear effect neglected for each column. The nonderterministic approach is based on the exact buckling model, response surface method, and Monte Carlo simulations. The probability of failure for this stochastic buckling problem is computed, and it is shown to be very sensitive to loading and material distribution parameters. 4. In “Effect of Fiber Orientation on Buckling and First-Ply Failures of Cylindrical Shear-Deformable Laminates,” Cagdas and Adali investigated the effect of fiber orientation on buckling and first-ply failures of cylindrical shear-deformable laminates using an 8-node shell finite element (FE). The best ply angle is optimized for each stacking sequence to maximize the failure load. They identified that the rotational restraints at the curved edges have a pronounced effect on the failure load. 5. In “Thermal Postbuckling of Shear Deformable FGM Cylindrical Shells Surrounded by an Elastic Medium,” Shen incorporated the additional effect of an elastic medium on the thermal postbuckling of shear deformable functionally graded material (FGM) cylindrical shells. The surrounding elastic medium is modeled as a Pasternak foundation. The nonlinear governing equations are solved using a singular perturbation technique. The numerical results show that in some cases the FGM cylindrical shell with an intermediate volume fraction index does not have an intermediate buckling temperature and thermal postbuckling strength. In the case of heat conduction, the nonlinear equilibrium path for geometrically perfect FGM shells with simply supported boundary conditions is no longer of the bifurcation type. 6. In “Generalized Beam Theory to Analyze the Vibration of Open-Section Thin-Walled Composite Members,” Silvestre

Nonlinear elastic analysis and buckling of pinned–fixed arches

This paper presents a theoretical analysis of the linear and nonlinear elastic in-plane behaviour and buckling of pinned–fixed circular arches that are subjected to a radial load distributed uniformly around the arch axis. In contrast to most previously reported studies, the pinned support and fixed support of a pinned–fixed arch provide different boundary conditions at each of its ends. Because of this, the structural behaviour of a pinned–fixed arch is quite different from that of a pinned–pinned or a fixed–fixed arch. The linear and nonlinear analyses show that the uniform radial load produces both an unsymmetrical bending moment and an axial compressive force in pinned–fixed arches; particularly in shallow pinned–fixed arches, the unsymmetrical bending moments are significant prior to in-plane buckling. It is found that a pinned–fixed arch under a uniform radial load can buckle in a limit point mode, but cannot buckle in a bifurcation mode. This is quite different from pinned–pinned or fixed–fixed arches under a uniform radial load that may buckle in a bifurcation mode and in a limit point mode. Analytical solutions for the nonlinear equilibrium path and limit point buckling load of shallow pinned–fixed circular arches are derived. Comparisons with finite element results show that the analytical solutions can predict the nonlinear equilibrium path and the nonlinear buckling and postbuckling behaviour of shallow pinned–fixed arches accurately. Although these solutions are derived for shallow pinned–fixed arches, comparisons with finite element results have demonstrated that they can also provide reasonably good predictions for the nonlinear buckling load of deep pinned–fixed arches under a uniform radial load.

Aspect Ratio of Slender Web & Postbuckling Behaviour / Pomer Strán Štíhlej Steny A Pokritické Pôsobenie

Abstract Postbuckling analysis of slender web loaded in compression is presented. The non-linear FEM equations [14] are derived from the variational principle of minimum of total potential energy [13]. To obtain the non-linear equilibrium paths, Newton-Raphson iteration algorithm [11], [12] is used. Peculiarities of the effect of the initial imperfections [7], [8] on load-deflection paths are investigated with respect to aspect ratio of the web. Special attention is focused on the postbuckling mode of the web.

A numerical approach for equilibrium and stability analysis of slender arches and rings under contact constraints

Underground constructions, such as shafts, courtain walls, foundations, pipes and tunnels, use structural elements that are supported by a geological medium (soil or rock) or are used to support the geological medium loads. If the geological medium is unable to react under tension, the structural element is subjected to unilateral contact constraints and, during the deformation process, may loose contact with the surrounding medium at one or more regions. The present work proposes an alternative numerical methodology for the geometrically nonlinear analysis of structural systems under unilateral contact constraints. The nonlinear problem involves two different types of variables: the displacement field and the length and position of the contact regions. In order to solve the resulting algebraic nonlinear equations with contact constraints and obtain the structural equilibrium configuration, the present work proposes a two-level iteration solution strategy at each load step. The first solves the contact problem as a linear complementary problem using Lemke’s algorithm. The second updates the displacement field. A nonlinear beam-column element is used to model the structure, while a bed of springs is used to model the geological medium. The use of an updated Lagrangian formulation, together with continuation and optimization techniques minimize the errors along the equilibrium paths and enables one to trace convoluted non-linear equilibrium paths with a varying number of contact regions. Special attention is given to the behavior of curved unidimensional support systems such as arches and rings. The nonlinear behavior of four such support systems is studied. The results clarify the influence of the foundation position (above or below the structure) and its stiffness on the nonlinear behavior and stability of curved structures. Comparison of the present results with those found in literature demonstrates the accuracy and versatility of the proposed numerical strategy in the analysis of structural elements with unilateral contact constraints.

Buckling and lateral buckling interaction in thin-walled beam-column elements with mono-symmetric cross sections

Effects of axial forces on beam lateral buckling strength are investigated here in the case of elements with mono-symmetric cross sections. A unique compact closed-form is established for the interaction of lateral buckling moment with axial forces. This new equation is derived from a non-linear stability model. It includes first order bending distribution, load height level and effect of mono-symmetry terms (Wagner’s coefficient and shear point position). Compared to the so-called three-factors (C1–C3) formula commonly employed in beam lateral buckling stability, another factor C4 is added in presence of axial loads. Pre-buckling deflection effects are considered in the study and the case of doubly-symmetric cross sections is easily recovered. The proposed solutions are validated and compared to finite element simulations where 3D beam elements including warping are used. The agreement of the proposed solutions with bifurcations observed on the non-linear equilibrium paths is good. Dimensionless interaction curves are dressed for the beam lateral buckling strength and the applied axial load, where the flexural-torsional buckling axial force is a taken as reference.