Path-following Analysis Research Articles

Koiter–Newton Reduced-Order Method Using Mixed Kinematics for Nonlinear Buckling Analysis

The Koiter–Newton method improves the computational efficiency of nonlinear buckling analysis; however, the construction of reduced-order models using fully nonlinear kinematics is still a tedious and time-consuming work. In this paper, the Koiter–Newton reduced-order method using mixed nonlinear kinematics is presented for the geometrically nonlinear buckling analysis of thin-walled structures. Strain energy variations up to the fourth order were achieved using mixed kinematics for the improved Koiter theory. Corotational kinematics, which is inconvenient for high-order variations, was applied to calculate the first- and second-order variations for the internal force and tangent stiffness, respectively, whereas the third- and fourth-order strain energy variations were facilitated by explicit algebraic formulations using updated von Kármán kinematics. A reduced-order model with 1+m degrees of freedom was established, of which m perturbation loads were considered to make the method applicable for buckling problems. The geometrically nonlinear response was traced using a predictor–corrector strategy by combining the nonlinear prediction solved by the reduced-order model and the correction using Newton iterations. Numerical examples of structures with various buckling behaviors demonstrate that the performance of the proposed method is not obviously affected by using simplified kinematics, and sometimes it even exhibits a superior capability for path-following analysis.

A novel reduced-order method using mixed nonlinear kinematics for geometrically nonlinear analysis of thin-walled structures

Reduced-order methods based on the Koiter perturbation theory are applicable only to structural buckling problems, and high-order strain energy derivatives influence the computational efficiency of the methods. In this paper, a novel reduced-order method is proposed for the geometrically nonlinear analysis of thin-walled structures with large deflections. Mixed nonlinear kinematics are developed for the reduced-order method using co-rotational and updated von Kármán formulations. The co-rotational kinematics are applied to determine the internal force and tangent stiffness, whereas the third- and fourth-order strain energy derivatives are calculated using the updated von Kármán kinematics. Reduced-order models with one degree of freedom is constructed based on the perturbation theory, the solutions of which are nonlinear predictors of the geometrically nonlinear response. The use of mixed nonlinear kinematics significantly reduces the computational cost of constructing a reduced-order model. Nonlinear predictors are corrected using internal force-based residuals to ensure the accuracy of the geometrically nonlinear analysis. A geometrically nonlinear response with favorable smoothness is efficiently achieved using large step sizes in the path-following analysis. Various numerical examples, including the stiffened plate and wing structure, are provided to validate the accuracy and efficiency of the proposed method.

Bifurcation and Path-Following Analysis of Periodic Orbits of a Fermi Oscillator Model

In this research, we offer a bifurcation analysis to describe impacting, stick, and grazing between a particle and the piston to better understand the nonlinear dynamics of a Fermi oscillator. The principles of hybrid dynamical systems will be utilized to explain the moving process in such a Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. We introduce the hybrid dynamical system and numerical continuation detection tool COCO. Physical and mathematical models are used to construct the obtained bounded mathematical model. The frequency, amplitude in oscillating base displacement, and the gap between the stationary boundary and the piston’s equilibrium position are the major parameters. We employ path-following analysis to illustrate the bifurcation that leads to solution destabilization. From the viewpoint of eigenvalue analysis, the essence of period-doubling and Fold bifurcation is revealed. Numerical continuation methods are used to perform a complete one-parameter bifurcation analysis of the Fermi oscillator. The presence of codimension-1 bifurcations of limit cycles, like period-doubling, fold, and grazing bifurcations, is shown in this work. Then, the one-parameter continuation analysis is tracked in two-parameter bifurcation diagrams that include a second important factor of interest. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.

A Box element to model semi-rigid connections in shell-based thin-walled structures analysis

In this study, we develop and propose a finite element (Box Element) designed specially to improve the shell-based simulation of elastic thin-walled structures. The Box Element (BE) allows a simple and automatic connection between beams and columns, as well as, between slabs and beams. The BE stiffness is assembled by bracing panels immersed in its numerical structure simulating the flexibility of connections. Its formulation is developed using an in house positional FEM code, whose 3D rotation degrees of freedom are replaced by generalized vectors, facilitating bending couplings and large rotation modeling. Instability and geometrical nonlinear path-following analysis are studied in examples, comparing results with literature benchmarks and proposing a practical application.

Arch bridges subject to pier settlements: continuous vs. piecewise rigid displacement methods

Settlements severely affect historic masonry arch bridges worldwide. There are countless examples of structural dislocations and ruins in recent years due to severe settlements at the base of pier foundations, often caused by shipworm infestation of wooden foundations or scouring and riverbed erosion phenomena. The present paper proposes an original way to approach the failure analysis of settled masonry arch bridges. The proposed method combines two different 2D numerical models for the prediction of masonry arch bridge capacity against settlements and for safety assessment. The first one is the Piecewise Rigid Displacement method, i.e. a block-based limit analysis approach using the well known Heyman's hypotheses; the second one is a continuous Finite Element approach. The case study of the four-span Deba Bridge (Spain, 2018) failure is presented with the aim to illustrate how the methods work. The failure analysis produced satisfactory results by applying both methods separately, in confirmation of their reliability. Their combination also allowed to obtain a significantly reduction in computational cost and an improvement of prediction accuracy. A sensitivity and a path-following analysis were also performed with the aim to demonstrate the robustness of the presented method. The obtained simulations highlighted that the results do not depend on the friction angle and that a proper prediction of the evolution of the structural behavior can be obtained only taking into account geometric nonlinearities. Such results demonstrate once again that in settled masonry arches geometry prevails over the mechanical parameters. The current study paves the way for the fruitful use of the proposed approaches for a wider range of applications, as, for example, the mechanism identification or the displacement capacity assessment of masonry structures under overloading as seismic loads.

A reduced model for nonlinear analysis and design of thin-walled structures prone to multi-modal buckling

A reduced model for nonlinear analysis and design of thin-walled structures prone to multi-modal buckling

Stick-slip suppression and speed tuning for a drill-string system via proportional-derivative control

This paper studies the problems of stick-slip mitigation and speed tuning for a lumped-parameter drill-string system by using a proportional-derivative feedback controller via path-following analysis. In this study, we consider two main control parameters, the weight-on-bit and the desired drill-bit speed. In particular, we determine the combinations of these two parameters for which the proposed control scheme is applicable, which is affected by the non-smooth nature of the system induced by bit-rock interaction. Our analysis using path-following techniques for non-smooth systems reveals the inherent coexistence of stick-slip vibration and constant rotation, and identifies a critical point where the drill-bit speed coincides with the desired angular speed. Furthermore, our analysis proposes a strategy that allows controlling the drill-bit speed to suppress stick-slip, by tuning the controller in a suitable manner.

Shear modelling of the beam-column joint in the nonlinear static analysis of r.c. framed structures retrofitted with damped braces

The adoption of a reliable shear model for predicting brittle failure modes of a reinforced concrete (r.c.) beam-column joint, beyond ductile flexural mechanisms at member level, is essential to retrofit r.c. framed buildings properly. The retrofitting of existing structures by means of the insertion of hysteretic damped braces (HYDBs) turns out to be a highly effective means of improving seismic response. In the present work, a Displacement-Based Design (DBD) procedure to proportion the HYDBs to attain, for a specific level of seismic intensity, a designated performance level has been revisited in order to take into account the effects of the nonlinear shear response of beam-column joints. To this end, two-, four- and eight-storey r.c. framed structures, representative of low-, mid- and high-rise r.c. framed buildings, are designed in line with a former Italian seismic code for a medium-risk seismic zone. These are then to be retrofitted by inserting HYDBs to attain performance levels imposed by the current Italian code in a high-risk seismic zone. A computer code for the nonlinear static analysis of r.c. framed structures has been developed, involving local shear response of beam-column joints. A path-following analysis based on the arc-length method has been adopted to obtain the pushover curves of primary and retrofitted test structures, with and without nonlinear shear modelling of the beam-column joints, and the HYDB response is idealized by a bilinear law on the assumption that buckling of the steel braces is prevented.

Bifurcation analysis of non-linear oscillators interacting via soft impacts

In this paper we present a bifurcation analysis of two periodically forced Duffing oscillators coupled via soft impact. The controlling parameters are the distance between the oscillators and the difference in the phase of the harmonic excitation. In our previous paper http://arXiv:1602.04214 (P. Brzeski et al. Controlling multistability in coupled systems with soft impacts [11]) we show that in the multistable system we are able to change the number of stable attractors and reduce the number of co-existing solutions via transient impacts. Now we perform a detailed path-following analysis to show the sequence of bifurcations which cause the destabilization of solutions when we decrease the distance between the oscillating systems. Our analysis shows that all solutions lose stability via grazing-induced bifurcations (period doubling, fold or torus bifurcations). The obtained results provide a deeper understanding of the mechanism of reduction of the multistability and confirmed that by adjusting the coupling parameters we are able to control the system dynamics.

Path-following analysis of the dynamical response of a piecewise-linear capsule system

The dynamical response of a piecewise-linear capsule system is studied by means of path-following techniques in this paper. As the capsule model belongs to the class of piecewise-smooth dynamical systems involving impact and friction, a special care is taken in order to divide the trajectory of the system into a smooth vector field in each disjoint subregion. Specifically we study a two-sided drifting system focusing on directional control and energy consumption. We aim to address two practical problems which are maximizing the rate of progression and directional control of the system by following a typical period-1 trajectory. The one-parameter analysis shows that two types of bifurcations, grazing bifurcation and boundary-intersection crossing bifurcation are found, and the maximal rate of progression is achieved when the capsule performs the oscillations without sticking phases. In our two-parameter study, the control parameters for which the rate of progression is maximal are identified using fixed value of power consumption, and the curves which divide the motion of the capsule between forward and backward progression are obtained.

Modelling and nonlinear static analysis of reinforced concrete framed buildings irregular in plan

Reinforced concrete (r.c.) existing structures with asymmetric plan may require the assessment of the seismic vulnerability directions in terms of displacement and strength. To this end, a computer code for the nonlinear static analysis of spatial framed structures is developed, adopting a path-following analysis based on the arc-length method to obtain the pushover curve for an assigned in-plan direction of the seismic loads. The seismic response of the r.c. frame members is simulated with a simplified lumped plasticity model, which includes a flat surface model of the bounding surface of the axial load–biaxial bending moment elastic domain, at the end sections of girders and columns where inelastic deformations generally occur. A case study of the existing town hall of Spilinga, a small town near Vibo Valentia (Italy), is investigated. The building, a two-storey r.c. structure with an L-shaped plan, was built at the beginning of 1960; in 2004, the Administration of Spilinga carried out a survey of the structure and its materials. To improve the knowledge level of the Spilinga building the present work uses, preliminarily, a simulated design with reference to the codes in force at the time of construction. Then, comparisons with a refined fibre model are made in order to: (a) verify the reliability of the flat surface model approximating the elastic domain of r.c. cross-sections; (b) calibrate the lumped plasticity model by means of nonlinear static analysis of the test structure. Finally, capacity domains in terms of top displacement and base shear are proposed, which allow an estimation of the in-plan directions of least seismic capacity at the serviceability (i.e. operational and immediate occupancy) and ultimate (i.e. life-safety and collapse prevention) limit states imposed by the Italian and European seismic codes.

複数の部材座屈と極限点が重複するアーチ状トラスの最悪不整

Imperfection sensitivity properties are investigated for an arch that has multiple member buckling at a limit point, which is classified as hilltop branching with multiple symmetric bifurcation points. The critical loads of imperfect structures are shown to be governed by a piecewise linear law of the imperfection parameter. Antioptimization problems are formulated to obtain the worst imperfection mode that most drastically reduces the critical load. A formula for the worst nodal imperfection as a linear combination of the critical modes is also presented. The validity of the formula is ensured by path-following analysis of arches with randomly generated imperfections.

Maximum loads of imperfect systems corresponding to stable bifurcation

Abstract A simple and computationally inexpensive approach is presented for obtaining the maximum load factor of an elastic structure considering reduction of load-carrying capacity due to inevitable initial imperfections. The structure has a stable bifurcation point if no initial imperfection exists. An antioptimization problem is formulated for minimizing the maximum loads reduced by the most sensitive imperfection within the convex bounds on the imperfections of nodal locations and nodal loads. The maximum loads may be defined by bifurcation points or deformation constraints. A problem of simultaneous analysis and design with energy method is formulated to avoid laborious nonlinear path-following analysis. The stable bifurcation point is located by minimizing the load factor under constraint on the lowest eigenvalue of the stability matrix. It is shown in the examples that a minor imperfection that is usually dismissed is very important in evaluating the maximum load of a flexible structure.

Mixed formulation and locking in path-following nonlinear analysis

The arc-length Riks strategy has rapidly become a standard tool for path-following analysis of nonlinear structures due to its theoretical ability to surpass limit points. The aim of this paper is to show that the failures in convergence that are occasionally experienced are not related to proper defects of the algorithm but come from a subtle ‘locking’ effect intrinsic to the nonlinear nature of the problem. As a consequence, its sanitization has to be pursued within a reformulation of the structural model. The use of a mixed (stress-displacement) variant of the algorithm, in particular, appears very promising in this respect. The topic is discussed with reference to the analysis of nonlinear frames using a mixed version of the nonlinear beam model discussed in [39]. It is shown that, with no extra computational cost and only a minor modification in coding with respect to a purely compatible formulation, it is possible to achieve a noticeable improvement in convergence and a real gain in both computational time and overall robustness of the algorithm.