Branch-switching Procedure Research Articles

Hill-top branching: Its asymptotically expanded and visually solved bifurcation equations

Hill-top branching (HTB) is an exceptional stability problem in which the bifurcation point (BP) coincides exactly with a limit point (LP). In such cases, snap-through and branching instability occur simultaneously. Then, HTB is often obscured in snap-through behavior and is often overlooked in stability design. The compound instability is in fact not well realized in the community of computational mechanics, when the rank deficiency and number of negative eigenvalues of the stiffness matrix are carelessly examined. No report of the relevant literature has established diagnosis of path branching or proposed a branch-switching procedure. Furthermore, few HTB models are available in the literature.This report proposes a set of asymptotic bifurcation equations to address this academically important and exciting issue. There might be a single or two critical eigenvectors of the singular stiffness matrix. Higher-derivatives of the stiffness matrix with respect to nodal displacements can be computed exactly using hyper-dual numbers (HDNs). The resulting simultaneous equations can be solved visually using graphical tools available in popular mathematical software packages such as MATLAB®. From the locations of existing real roots displayed on a graphic monitor, the crossover situation of path branching is visually clear above and below the hill-top. For the present study, three HTB examples are computed to verify the proposed algorithm strategy using HDNs and graphical tools. The obtained numerical results show excellent agreement with analytical and finite element predictions. The proposed asymptotically expanded and visually solved bifurcation equations reliably diagnose asymmetric and unstable symmetric HTB in structural stability problems and function well in computational engineering problems.

Branch-switching procedure for post-buckling analysis of thin-walled steel members at elevated temperature

The paper investigates the effects of the geometrical imperfections in buckling analyses of plated steel elements at elevated temperatures and provides an alternative branch-switching procedure to perform post-buckling analyses without introducing initial imperfections into the model. This procedure is appealing since the choice of appropriate imperfections in classical analyses is not straightforward, above all at elevated temperature. Several numerical analyses show that the choice of the imperfections is not trivial and that the buckling mode may vary with temperature. They also show that the proposed branch-switching procedure is an interesting preliminary tool to understand the instability behaviour of steel structural members.

A branch-switching procedure for analysing instability of steel structures subjected to fire

The paper describes the development of a two-dimensional (2D) co-rotational nonlinear beam finite element that includes advanced path-following capabilities for detecting bifurcation instability in ...

Post-buckling analysis of elastoplastic sandwich columns by means of an enriched 1D finite element model

An advantageous 1D finite element model is designed in the present paper so as to analyze in an efficient way the post-buckling behavior of sandwich beams, known to be commonly responsible for the final collapse of such structures. An enriched beam formulation is defined, where the relatively thin skin layers are modeled as Timoshenko–Reissner beams, as they may undergo large rotations at advanced post-buckled stages. As for the homogeneous core layer, its complex behavior is represented by specific kinematics involving hyperbolic functions. The numerical model is developed within a total Lagrangian formulation framework, considering purely elastic behavior for the skins and an elastoplastic core material. The 1D finite element program incorporates effective incremental control techniques, namely arc-length methods and branch-switching procedures, in order to cope with limit and bifurcation points due to material and geometric non-linearities. A series of incremental calculations is performed in the case of axially compressed columns, exhibiting both global and local buckling modes depending on the geometric and material features. Secondary bifurcations, giving rise to unstable post-buckled solutions, are encountered in most cases due to the operating modal interaction phenomena. The results are compared with reference numerical computations achieved using a 2D finite element customized program.

Elastoplastic buckling and post-buckling analysis of sandwich columns

Sandwich structures are widely used in many industrial applications thanks to their interesting compromise between lightweight and high mechanical properties. This compromise is realized thanks to the presence of different parts in the composite material, namely the skins which are particularly thin and stiff relative to the homogeneous core material and possibly core reinforcements. Owing to these geometric and material features, sandwich structures are subject to global but also local buckling phenomena which are mainly responsible for their collapse. The buckling analysis of sandwich materials is therefore an important issue for their mechanical design. In this respect, this paper is devoted to the theoretical study of the local/global buckling and post-buckling behavior of sandwich columns under axial compression. Only symmetric sandwich materials are considered with homogeneous and isotropic core/skin layers. First, the buckling problem is analytically addressed, by solving the so-called bifurcation equation in a 3D framework. The bifurcation analysis is performed using an hybrid model (the two faces are represented by Euler–Bernoulli beams, whereas the core material is considered as a 2D continuous solid), considering both an elastic and elastoplastic core material. Closed-form expressions are derived for the critical loadings and the associated bifurcation modes. Then, the post-buckling response is numerically investigated using a 2D finite element bespoke program, including finite plasticity, arc-length methods and branch-switching procedures. The numerical computations enable us to validate the previous analytical solutions and describe several kinds of post-critical responses up to advanced states, depending on geometric and material parameters. In most cases, secondary bifurcations occur during the post-critical stage. These secondary modes are mainly due to the modal interaction phenomenon and give rise to unstable post-buckled solutions which lead to final collapse.

Three-Dimensional Necking Bifurcation in Tensile Steel Specimens

Three-dimensional (3D) necking bifurcation in tensile steel specimens is simulated by a finite-element method. Although conventional necking analysis has tended to depend on material instability approaches with two-dimensional analysis assuming plane strain or axis symmetric state, 3D necking behavior is treated in the framework of only geometrical instability without material instability in this study. To explicate an essential bifurcation phenomenon, a branch-switching procedure is utilized after strictly detecting a bifurcation point in a perfect system. Ultimate localized modes of the tensile steel specimens with various rectangular cross sections are computed. In addition, actual tension experiments of the steel specimens, of which cross-sectional width-thickness ratios correspond to those in computations, are conducted, and excellent correlation with the computational results without the material instability is demonstrated. Thus, behavior from a uniformly deformed state to the ultimate localized mode, just before fracture, has been proved continuous geometrical instability instead of the material instability.

Postbuckling strength of an axially compressed elastic circular cylinder with all symmetry broken

Axially compressed circular cylinders repeat symmetry-breaking bifurcation in the postbuckling region. There exist stable equilibria with all symmetry broken in the buckled configuration, and the minimum postbuckling strength is attained at the deep bottom of closely spaced equilibrium branches. The load level corresponding to such postbuckling stable solutions is usually much lower than the initial buckling load and may serve as a strength limit in shell stability design. The primary concern in the present paper is to compute these possible postbuckling stable solutions at the deep bottom of the postbuckling region. Two computational approaches are used for this purpose. One is the application of individual procedures in computational bifurcation theory. Path-tracing, pinpointing bifurcation points and (local) branch-switching are all applied to follow carefully the postbuckling branches with the decreasing load in order to attain the target at the bottom of the postbuckling region. The buckled shell configuration loses its symmetry stepwise after each (local) branch-switching procedure. The other is to introduce the idea of path jumping (namely, generalized global branch-switching) with static imperfection. The static response of the cylinder under two-parameter loading is computed to enable a direct access to postbuckling equilibria from the prebuckling state. In the numerical example of an elastic perfect circular cylinder, stable postbuckling solutions are computed in these two approaches. It is demonstrated that a direct path jump from the undeformed state to postbuckling stable equilibria is possible for an appropriate choice of static perturbations.

Scaled corrector and branch-switching in necking problems

A new branch-switching procedure for necking problems by finite element method is proposed. The scaled corrector method adopted in this paper for computing the critical eigenvector of the tangent stiffness matrix, which is indispensable to bifurcation analysis, is quite simple and effective for large scale problems because eigenvalue analysis is not required. The numerical background of the approximate critical eigenvector calculated during equilibrium iterations and the branch-switching procedure in elasto-plastic bifurcation problems are described in detail. As numerical examples, necking bifurcation problems of plane strain/stress states in finite strains are demonstrated to validate the proposed branch-switching procedure.

A Toeplitz Jacobian Matrix/Fast Fourier Transformation Method for Steady-State Analysis of Discontinuous Oscillators

A semianalytical algorithm is proposed for the solutions and their stability of a piecewise nonlinear system. The conventional harmonic balance method is modified by the introduction of Toeplitz Jacobian matrices (TJM) and by the alternative applications of fast Fourier transformation (FFT) and its inverse. The TJM/FFT method substantially reduces the amount of computation and circumvents the necessary numerical differentiation for the Jacobian. An arc-length algorithm and a branch switching procedure are incorporated so that the secondary branches can be independently traced. Oscillators with piecewise nonlinear characteristics are taken as illustrative examples. Flip, fold, and Hopf bifurcations are of interest.

A Toeplitz Jacobian Matrix/Fast Fourier Transformation Method for Steady-State Analysis of Discontinuous Oscillators

A semianalytical algorithm is proposed for the solutions and their stability of a piecewise nonlinear system. The conventional harmonic balance method is modified by the introduction of Toeplitz Jacobian matrices (TJM) and by the alternative applications of fast Fourier transformation (FFT) and its inverse. The TJM/FFT method substantially reduces the amount of computation and circumvents the necessary numerical differentiation for the Jacobian. An arc-length algorithm and a branch switching procedure are incorporated so that the secondary branches can be independently traced. Oscillators with piecewise nonlinear characteristics are taken as illustrative examples. Flip, fold, and Hopf bifurcations are of interest.

Nonlinear stability-analysis of shell and contact-problems including branch-switching

In the analysis of nonlinear elastic shells often the stability and postbuckling behaviour governs the response. Here we discuss problems which also include contact constraints. A nonlinear cylindrical shell element is derived directly from the associated shell theory using one point integration and a stabilization technique. Within a general solution algorithm a simple but effective branch-switching procedure is presented. Additional considerations allow the treatment of bifurcation problems with contact constraints. Several examples of beam and shell problems show the performance of the developed algorithms and elements.

Stability problems in continuum mechanics and their numerical computation

In the present paper the theoretical and numerical treatment of static stability problems is discussed. For the structural stability behaviour path-following methods, strategies for detecting stability points and branch-switching procedures are investigated. Stability problems due to material damaging are treated with Gurson's model for the description of void growth in the context of plastic behaviour. For the stress update analysis a return mapping algorithm is used.

A quadratically convergent procedure for the calculation of stability points in finite element analysis

In this paper a new finite element formulation is given for the analysis of nonlinear stability problems. The introduction of extended systems opens the possibility to compute limit and bifurcation points directly. Here, the use of the directional derivative yields a quadratically convergent iteration scheme. The combination with arc-length and branch-switching procedures leads to a global algorithm for path-following.