Unstable Equilibrium Paths Research Articles

Buckling and post-buckling analysis of three-layer shallow arches with geometric imperfections and interlayer slip

In this paper, a semi-analytical solution is derived to capture the nonlinear stability and equilibria of shallow three-layered arches with flexible bond, which also accounts for the interlayer slip (i.e., the relative displacement between the layers). The member axis can have arbitrary geometric imperfections. With the equations presented, for the first time, the complete stable and unstable equilibrium paths can be predicted including multiple remote equilibria, which may occur in the presence of an imperfect member axis but cannot be determined with finite element simulations. Expressions for the direct prediction of critical loads for limit point buckling and bifurcation buckling are also provided, which may also be on the remote unconnected equilibrium paths. In several application examples, the effect of geometric imperfection and interlayer slip on the critical loads and nonlinear equilibria is revealed. Comparative finite element analyses with plane stress continuum elements demonstrate the accuracy and computational efficiency of the presented theory.

In-plane instability of shallow layered arches with interlayer slip

In this paper, a beam theory for predicting limit point buckling and bifurcation buckling of shallow arches composed of two layers flexibly bonded is presented. The flexibility of layer bond results in interlayer slip, which significantly affects the critical transverse loads. The presented theory is based on a layerwise assumption of the Euler–Bernoulli theory and a linear behavior of the interlayer. After establishing the equilibrium equations and boundary conditions, a numerical method for efficient solution of these equations is provided. In a first example, the presented theory is validated by comparative computations with a much more elaborate finite element analysis assuming a plane stress state. In several other examples, the effect of interlayer stiffness, load distribution and boundary conditions on the stable and unstable equilibrium paths of shallow arches with interlayer slip is investigated.

Interactive buckling of wide plates made of Functionally Graded Materials with rectangular stiffeners

A phenomenon of multimodal buckling of wide stiffened plates made of Functionally Graded Materials (FGM) subject to uniform compression in the elastic range is considered. This phenomenon occurs only when the eigenvalues corresponding to the global mode and the lowest local mode are comparable. Dimensions of wide plates with rectangular stiffeners were selected to make the eigenvalues close to one another. All simulations were conducted with the finite element method, where the behavior of a single segment of the stiffened plate was modeled.​ Symmetry conditions were assumed along the edges connecting the segments, whereas modified boundary conditions were assumed at both the ends to fulfill strictly the conditions of free support of the plate. A nonlinear problem was solved with the Riks algorithm to trace stable and unstable equilibrium paths. Values of the ultimate load-carrying capacity determined in ultimate points, plate modes in the ultimate state and under considerable overloading were analyzed. It was found that the cases of plates with stiffeners of the stiffness comparable to the stiffness of the skin were dangerous in particular, as the stiffener stability loss decreased significantly the ultimate load-carrying capacity of the whole structure. If the local mode of “stiffener buckling” is neglected by considering the corresponding imperfection, a considerable overestimation of the load-carrying capacity of the structure can take place, regardless of the fact if it is the lowest eigenmode. Strong over-rigidity of the stiffener causes that only the plate skin is subject to buckling and, thus, the load-carrying capacity of the whole structure improves. In the extreme situation, it leads to one-mode buckling. Thermal loads are neglected in this study.

Post-buckling path and free vibration of a symmetric laminated plate vertically coupled with fluid under in-plane load

In this paper, the effect of compressive in-plane load on the equilibrium paths and vibration characteristics of a vertical symmetric laminated plate coupled with fluid is studied theoretically both in the pre- and post-buckling state. The Third-order shear deformation theory and von Karman nonlinear strain displacement relationship are utilized to describe the nonlinear deformation of the plate consists of a number of fiber-reinforced layers. The Hamilton's principle is introduced to formulate the plate-fluid coupled governing equations. A two-step theoretical approach is developed. Firstly, based on the Cylindrical arc-length method, we solve the nonlinear static equations to trace the snap-back behavior and pass the limit points along the complex equilibrium paths obtaining the stable buckling deflection. Then the coupled vibration characteristics are calculated using the tangent stiffness at the stable positions. The present theoretical results are validated by the numerical results. The effects of added mass and hydrostatic load are compared and analyzed. It is noted that, in the post-buckling state, the number of limit points along the static unstable equilibrium path increases with the compressive in-plane load getting larger. The variation trends of the natural frequencies and modes under different in-plane loads in pre- and post-buckling state are investigated.

Experimental path-following of equilibria using Newton’s method. Part I: Theory, modelling, experiments

Modern numerical path-following techniques provide a comprehensive suite of computational tools to study the bifurcation behaviour of engineering structures. In contrast, experimental testing of load-bearing nonlinear structures is still performed using simple force control (dead loading) or displacement control (rigid loading). This means that established experimental methods cannot trace equilibrium manifolds in their entirety because structures snap to alternative equilibria at limit points in the forcing parameter and because branch switching to alternative equilibria cannot be controlled and performed reliably. To extend current testing methods, in Part I of this paper, we implement an experimental path-following method that uses tangent quantities (stiffness and residual forces) and Newton’s method to continue along stable and unstable equilibrium paths and traverse limit points. In addition to enforcing the displacement at primary load-introduction points, the overall shape of the structure is controlled via secondary actuators and sensors. Small perturbations of the structure using the secondary actuators allow an experimental tangent stiffness to be computed, which is then used in a control algorithm. As a pertinent test case, the experimental method is applied to a transversely loaded shallow circular arch. Due to the complexity of the test setup, the experiment is first designed using a virtual testing environment based on a surrogate finite element model. Experimental results demonstrate the robustness of the proposed experimental method and the usefulness of virtual testing as a surrogate, but also highlight that experimental efficiency and the effects of noise and sensor uncertainty is of particular concern. In Part II, we present perspectives on future research directions and novel testing capabilities that are enabled by extending the methodology to pinpointing of critical points, tracing of critical boundaries, and branch switching.

Nonlinear thermal dynamic buckling of FGM beams

Abstract Based on the nonlinear dynamic analysis, dynamic buckling and imperfection sensitivity of the FGM Timoshenko beam subjected to sudden uniform temperature rise are studied. Initial geometric imperfection of the beam is also taken into account. It is assumed that during deformation the beam is resting over a conventional three-parameter elastic foundation with softening/hardening cubic non-linearity. The analysis is performed with considering temperature dependency assumption of each thermo-mechanical property of the FGM beam. The governing nonlinear dynamic equations are derived based on the generalized Hamilton principle. In the spatial approximation of the problem, a set of ordinary differential equations in time is obtained by the conventional multi-term Ritz method. These equations are converted into a set of algebraic equations by utilizing the Newmark family of time approximation scheme. The obtained non-linear algebraic equations are solved via the well known Newton–Raphson iterative scheme. The Budiansky–Roth criterion is used to detect the unbounded motion type of dynamic buckling. Results reveal that for beams with stable post-buckling equilibrium path, no dynamic buckling occurs according to the Budiansky–Roth criterion. However, dynamic buckling may occur for the FGM beams resting on sufficiently stiff softening elastic foundation due to their unstable post-buckling equilibrium paths.

Nonlinear dynamic buckling of pinned–fixed shallow arches under a sudden central concentrated load

The structural behavior of a shallow arch is highly nonlinear, and so when the amplitude of the oscillation of the arch produced by a suddenly-applied load is sufficiently large, the oscillation of the arch may reach a position on its unstable equilibrium paths that leads the arch to buckle dynamically. This paper uses an energy method to investigate the nonlinear elastic dynamic in-plane buckling of a pinned–fixed shallow circular arch under a central concentrated load that is applied suddenly and with an infinite duration. The principle of conservation of energy is used to establish the criterion for dynamic buckling of the arch, and the analytical solution for the dynamic buckling load is derived. Two methods are proposed to determine the dynamic buckling load. It is shown that under a suddenly-applied central load, a shallow pinned–fixed arch with a high modified slenderness (which is defined in the paper) has a lower dynamic buckling load and an upper dynamic buckling load, while an arch with a low modified slenderness has a unique dynamic buckling load.

Non-linear buckling and postbuckling analysis of arches with unequal rotational end restraints under a central concentrated load

This paper presents an analytical study of the non-linear elastic in-plane buckling and postbuckling behaviour of pin-ended shallow circular arches having unequal elastic rotational end restraints under a central concentrated radial load. The principle of stationary potential energy is used to derive the differential equations of equilibrium, based on which the analytical solution for the non-linear equilibrium path is derived. It is found that the non-linear behaviour of an arch having unequal rotational end restraints is much more complicated than that of an arch with equal rotational end restraints. The arch may have a non-linear equilibrium path that consists of one or two unstable equilibrium paths and two or four limit points, and it may even have a non-linear looped equilibrium path in some cases. The number of limit points on the non-linear equilibrium path of an arch depends on its slenderness ratio and included angle, and on the stiffnesses of the unequal rotational end restraints. The switches in terms of an arch geometry parameter, which is introduced in the paper, are derived for distinguishing between arches with two limit points and those with four limit points, as well as for distinguishing between arches and beams curved in-elevation. The principle of conservation of energy at neutral equilibrium is used to derive the differential equations of buckling equilibrium, which are then used to investigate the buckling behaviour. It is found that an arch with unequal rotational end restraints cannot buckle in a bifurcation mode. Comparisons with finite element results show that the analytical solutions can accurately predict the non-linear buckling and postbuckling behaviour.

Nonlinear dynamic buckling of shallow circular arches under a sudden uniform radial load

The structural behavior of a shallow arch is highly nonlinear, and so when the amplitude of the oscillation of the arch produced by a suddenly applied load is sufficiently large, the oscillation of the arch may reach a position at its primary unstable equilibrium path or secondary bifurcation unstable equilibrium path, leading the arch to buckle dynamically. This paper presents an analytical study of the nonlinear dynamic in-plane buckling of a shallow circular arch under a uniform radial load that is applied suddenly and with an infinite duration. The principle of conservation of energy is used to establish the criterion for dynamic buckling of the arch, and the analytical solution for the dynamic buckling load is derived. It is shown that under a suddenly applied uniform radial load, a shallow pinned–fixed arch has a unique possible dynamic buckling load, while shallow pinned–pinned and fixed–fixed arches may have two possible dynamic buckling loads: a lower dynamic buckling load and an upper dynamic buckling load. The dynamic buckling loads of a shallow arch under a suddenly applied uniform radial load with infinite duration are found to be lower than their static counterparts, and to increase with an increase of the arch included angle and slenderness. The effect of static preloading on the dynamic buckling of an arch is also investigated. It is found that the pre-applied static load decreases the dynamic buckling load of the arch, but increases the sum of the pre-applied load and the dynamic buckling load.

Non-linear in-plane analysis and buckling of pinned–fixed shallow arches subjected to a central concentrated load

This paper is concerned with an analytical study of the non-linear elastic in-plane behaviour and buckling of pinned–fixed shallow circular arches that are subjected to a central concentrated radial load. Because the boundary conditions provided by the pinned support and fixed support of a pinned–fixed arch are quite different from those of a pinned–pinned or a fixed–fixed arch, the non-linear behaviour of a pinned–fixed arch is more complicated than that of its pinned–pinned or fixed–fixed counterpart. Analytical solutions for the non-linear equilibrium path for shallow pinned–fixed circular arches are derived. The non-linear equilibrium path for a pinned–fixed arch may have one or three unstable equilibrium paths and may include two or four limit points. This is different from pinned–pinned and fixed–fixed arches that have only two limit points. The number of limit points in the non-linear equilibrium path of a pinned–fixed arch depends on the slenderness and the included angle of the arch. The switches in terms of an arch geometry parameter, which is introduced in the paper, are derived for distinguishing between arches with two limit points and those with four limit points and for distinguishing between a pinned–fixed arch and a beam curved in-elevation. It is also shown that a pinned–fixed arch under a central concentrated load can buckle in a limit point mode, but cannot buckle in a bifurcation mode. This contrasts with the buckling behaviour of pinned–pinned or fixed–fixed arches under a central concentrated load, which may buckle both in a bifurcation mode and in a limit point mode. An analytical solution for the limit point buckling load of shallow pinned–fixed circular arches is also derived. Comparisons with finite element results show that the analytical solutions can accurately predict the non-linear buckling and postbuckling behaviour of shallow pinned–fixed arches. Although the solutions are derived for shallow pinned–fixed arches, comparisons with the finite element results demonstrate that they can also provide reasonable predictions for the buckling load of deep pinned–fixed arches under a central concentrated load.

ENERGY APPROACH FOR DYNAMIC BUCKLING OF AN UNDAMPED ARCH MODEL UNDER STEP LOADING WITH INFINITE DURATION

Performing a dynamic buckling analysis of structures is more difficult than carrying out its static buckling analysis counterpart. Some structures have a nonlinear primary equilibrium path including limit points and an unstable equilibrium path. They may also have bifurcation points at which equilibrium bifurcates from the primary equilibrium path to an unstable secondary equilibrium path. When such a structure is subjected to a load that is applied suddenly, the oscillation of the structure may reach the unstable primary or secondary equilibrium path and the structure experiences an escaping-motion type of buckling. For these structures, complete solutions of the equations of motion are usually not needed for a dynamic buckling analysis, and what is really sought are the critical states for buckling. Nonlinear dynamic buckling of an undamped two degree-of-freedom arch model is investigated herein using an energy approach. The conditions for the upper and lower dynamic buckling loads are presented. The merit of the energy approach for dynamic buckling is that it allows the dynamic buckling load to be determined without the need to solve the equations of motion. The solutions are compared with those obtained by an equation of motion approach.

Energy approach for dynamic buckling of shallow fixed arches under step loading with infinite duration

Shallow fixed arches have a nonlinear primary equilibrium path with limit points and an unstable postbuckling equilibrium path, and they may also have bifurcation points at which equilibrium bifurcates from the nonlinear primary path to an unstable secondary equilibrium path. When a shallow fixed arch is subjected to a central step load, the load imparts kinetic energy to the arch and causes the arch to oscillate. When the load is sufficiently large, the oscillation of the arch may reach its unstable equilibrium path and the arch experiences an escaping-motion type of dynamic buckling. Nonlinear dynamic buckling of a two degree-of-freedom arch model is used to establish energy criteria for dynamic buckling of the conservative systems that have unstable primary and/or secondary equilibrium paths and then the energy criteria are applied to the dynamic buckling analysis of shallow fixed arches. The energy approach allows the dynamic buckling load to be determined without needing to solve the equations of motion.

Compression Response of 2D Braided Textile Composites: Single Cell and Multiple Cell Micromechanics Based Strength Predictions

This article is concerned with the development of a finite element (FE) based micromechanics model for the prediction of compressive strength and post-peak compression response of 2D triaxial braided carbon fiber polymer matrix composites (2DTBC). This micromechanics based study was carried out on a series of single and multiple representative unit cell (RUC) 3-D FE models. The uniaxial compressive response, including unstable equilibrium paths, was studied using an arc-length method in conjunction with the ABAQUS commercial FE code. In the reported study, explicit account of the braid microstructure (geometry and packing) and the measured inelastic properties of the matrix (the in-situ properties) are accounted for via the use of the FE method. This enables accounting for the different length scales that are present in a 2DTBC. The computational model provides a means to assess the compressive response of 2DTBC and its dependence on various microstructural parameters. The model provides a means to compute the compression strength allowable for a 2DTBC structure. In particular, the dependence of compressive strength on the axial fiber tow properties and axial tow geometrical imperfections is discussed and shown to be significant in capturing the mechanism of damage development. Results are presented for 1, 4, 9, and 16 RUC representations of the 2DTBC, enabling to examine the dependence of compressive strength (or lack thereof) on the size of the region that is modeled. The predicted results are found to compare favorably against experiment.

Distributed Thermal Cracking of AC Pavement with Frictional Constraint

This paper develops a model to simulate the distributed thermal cracking of concrete structures with frictional constraint. This model is developed primarily for the thermal cracking asphalt-concrete (AC) pavement structures; however, with some modifications, it is also applicable to similar problems such as shrinkage cracking of concrete and cracking of reinforced concrete in uniaxial tension. This model reflects the multiscale nature of these problems: microcracking or damage on the mesoscale and localization or redistribution on the macroscale. Randomly distributed fictitious cracks are introduced to represent the inhomogeneities and damage in the material at the mesoscale. Friction is recognized as the mechanism leading to stress redistribution and, therefore, damage localization on the macroscale. When the problem is assumed to be one-dimensional and Coulomb friction is used, a semianalytical numerical scheme is developed. The formation of stress-free open cracks is due to the combination of continuous crack growth and unstable jumps, which involve a nonlinear stability analysis. Equilibrium solutions and stability conditions are given in the paper. Displacement controlled analysis is used to follow the unstable equilibrium path after the structure has lost stability. Numerical simulations clearly show that, with slight mesoscale inhomogeneities and in the presence of a constraining frictional force, microcracking or damage on the mesoscale localizes and finally leads to open cracks distributed at a spacing on the order of the macroscale.

Tubular Members. I: Stability Analysis and Preliminary Results

This paper presents a finite-element technique for the analysis of tubular member stability under combined external pressure and structural loads. A tube element is developed for the purposes of this research. While polynomial (quadratic) interpolation is used in the longitudinal direction, Fourier series expansions of the displacement components are adopted at the nodal cross sections. The formulation accounts for large inelastic deformation and recognizes initial imperfections and residual stresses. To trace unstable equilibrium paths, arc-length procedures are implemented. A simple, yet effective estimate of the contribution of external pressure to the tube element stiffness matrix, particularly significant in the analysis of slender tubes, is included. Preliminary results regarding the behavior of tubular members subjected to pressure and bending are reported and discussed. The effects of initial imperfections and residual stresses on the response to pressure along with bending are summarized. Finally, the influence of residual stresses on thrust-moment interaction in tubular beam-columns is examined briefly.

An incremental solution technique for unstable equilibrium paths of shell structures

Solution procedures based on arc-length control have been among the most successful to be applied in nonlinear structural analysis. However, in the course of applying this type of procedure to postbuckling analysis of pipe segments, the writers experienced difficulties with the occurrence of complex roots arising from the arc-length constraint equation and with determining the direction of the load increment. In this paper, modified formulations of the arc-length control criterion and the loading criterion are proposed to address these problems. Applications of the procedures are shown to be successful for postbuckling analysis of pipe segments. Their effectiveness in following and advancing on the equilibrium path for both snap-through and snap-back behavior is illustrated.

Large deformation buckle analysis of an inelastic ring under external pressure

A long circular cylindrical tube subjected to external pressure may collapse through the development of a propagating buckle. In this work, a finite element technique is applied to an inelastic ring for the prediction of the propagation pressure in deep-water pipelines. Under the framework of the updated Lagrangian formulation in a convected coordinate system, three-node isoparametric beam finite elements are employed. The modeling is based on large strain approximation, allowing large deformation, elastoplastic material behavior and contact between regions of the interior wall of the ring. For the purpose of tracing unstable equilibrium paths during large deformation of the ring, the beam element is equipped with a so-called pressure node. The constitutive description and finite element formulation are presented. Results of the analysis with an energy-balance argument are shown and compared with those of the experiment.

Finite element analysis of propagating buckles

AbstractA finite element technique is described for the analysis of propagating buckles in deep‐water pipelines, accounting for large deformation, elastoplastic material behaviour and contact between regions of the interior wall of the pipe. Within the framework of a Lagrangian formulation and a convected co‐ordinate system, nine‐node isoparametric shell finite elements are employed along with contact elements. They are equipped with a so‐called pressure node which facilitates the tracing of unstable equilibrium paths during the deformations of the pipe. For the analysis of a pipe surrounded by water, fluid‐structure interaction is treated in an approximate manner. Numerical results are presented and comparisons with experimental data are made, demonstrating the excellent performance of the technique.

A simple indicator and branch switching technique for hidden unstable equilibrium paths

The unfolding, via imperfections, on an asymmetric bifurcation may result in both stable and unstable equilibrium paths. In practive it it quite possible that an analysis will be performed on the stable path only without the analyst knowing of the adjacent unstable path. This process could lead to a disastrous design. In this paper we propose a simple indicator based on the local minimum of the smallest pivot of the tangent stuffness matrix, which can be used to warn the analyst of the danger. In addition if he wishes, he can employ a simple eigenmode injection technique to analyse the unstable path. Numerical examples are presented which prove that both the indicator and branching technique are efficient and reliable.

Finite element analysis of bifurcation in nonlocal strain softening solids

Progressive damage in brittle heterogeneous materials produces at the macroscopic level strain softening from which theoretical difficulties arise (e.g. ill-posedness and multiple bifurcation points). This characteristics behavior favours spurious strain localization in numerical analyses and calls for the implementation of localization limiters, for instance nonlocal damage constitutive relations. The issue of possible (stable or unstable) equilibrium paths, multiple localization zones, and of the detection of bifurcation points has, however, never been addressed in the context of nonlocal constitutive laws. We extend here the eigenmode analysis and perturbation method proposed by De Borst to the study of the bifurcation and post-bifurcation response of discrete nonlocal strain softening solids. Numerical applications on beams show that bifurcation and instability may occur in the post-peak regime. As opposed to the case of local constitutive relations, the number of possible solutions at a bifurcation point is restricted due to the constraint introduced by the localization limiter and these solutions are shown to be mesh independent.