Post-critical Path Research Articles

Postcritical Imperfection Sensitivity of Functionally Graded Piezoelectric Cylindrical Nanoshells Using Boundary Layer Solution

The nonlocal (NL) and the strain gradient (SG) based equivalent continuum theories were proposed for nanomechanics by accommodating the long-range molecular interactions and to bypass the computationally expensive atomistic simulations. Nanostructures are often made of smart materials (e.g., piezoelectric, piezoceramic, and flexoelectric) for multifunctional properties. Applications of nanostructures in critical systems deserve accurate analysis. The buckling and postcritical behavior of an axially loaded, piezoelectric, nonlocal strain gradient (NLSG) thin cylindrical shells with functionally graded elastic properties were presented in earlier studies following Donnell’s approach, leading to a set of stiff, nonlinear, partial differential equations, accommodating the prebuckling nonlinearity and large postcritical deflection. This study extends the previous work by accommodating geometric imperfections in the formulation. Furthermore, a consistent thickness-wise distribution for the electric potential is also adopted following the Maxwell equation. The boundary layer (BL) concept is employed to solve the resulting nonlinear equations via asymptotic expansions of the regular and the BL fields. The solution is numerically illustrated on moderately short shells, illustrating the influence of the geometric imperfections. The critical load remains imperfection sensitive, yet the imperfections change an unstable postcritical path into a stable one. The influence of the functional gradation and the external electric fields on the imperfection sensitivity behavior are illustrated.

A boundary layer solution for the post-critical thermo-electro-mechanical stability of nonlocal-strain gradient Functionally Graded Piezoelectric cylindrical shells

Continuum mechanics is extended to the Nonlocal (NL) and the Strain Gradient (SG) theories by incorporating the long-range molecular interactions. Nanostructures are often made of smart materials (e.g. Piezoelectric) for multifunctional attributes. The NL-SG theory based analysis/design of nanostructures offers convenience comparing the computationally expensive atomistic simulation. This study investigates the thermal post-buckling of a piezoelectric, NLSG thin cylindrical shell with thickness-wise gradation of elastic properties. The governing equations are derived following the Donnell's approach including the pre-buckling nonlinearity, large post-critical deflection and geometric imperfections. A thickness-wise cosine distribution of the electric potential is employed following the Maxwell equation. The Boundary Layer (BL) concept is employed to solve the nonlinear equations using asymptotic expansions of the regular and the BL field variables. The solution is illustrated on moderately short shells to show the prominent SG but nominal NL effects. The interactions result in conservative critical temperature but non-conservative post-critical path in contrast with the classical shell. The influence of functional gradation and the electric fields are illustrated. Increasingly sharper displacement gradient around the BL of the SG shell suggests the increasing relevance of the BL theory. The proposed solution may be useful for analysis, design and active instability control by adaptive feedback voltage.

Robustness assessment of semi-rigid steel multi-storey frames

In recent years, the interest in knowledge and evaluation of the post-critical path of structures has increased mainly due to the recent occurrence of terrorist attacks and other exceptional events. In steel structures, joints play an essential role in the structural behaviour. Therefore, an adequate resistance is vital to prevent progressive collapse. This study presents a numerical evaluation of steel structures with bolted flush endplate semi-rigid joints using models based on finite element method focused on their post-critical nonlinear response. The current investigation addresses the structural behaviour of steel joints centred on the collapse of an internal column scenario. Furthermore, the structure robustness and the steel joints structural response was evaluated, as well as the development of internal stresses. The results were also estimated and compared with the main robustness design codes. The results indicated that the presence of elements able to withstand substantial tension loads is conditioned to an increase of the structural robustness to prevent progressive collapse. Within this perspective, the results also indicated that an increase of the bolt diameter led to an enhanced rotation capacity and ability to sustain gravity loads. Besides, it was noticed that most of the evaluated joints were able to meet the codes criteria to prevent progressive structural collapses.

Optimization of cylindrical shells stiffened by rings under external pressure including their post-buckling behaviour

Optimization of a simply supported cylindrical shell stiffened by inner rings is considered in this paper. The shell is loaded by external pressure. The first critical buckling load is maximized. A material volume and a slope of a post-critical path are assumed as optimization constraints. The inner rings are made from the material obtained from the shell by decreasing its thickness. This way the volume of the material remain constant. The structure is modelled and solved by the finite element method (FEM). The linear and nonlinear stability analyses are done in the ANSYS software. An effect of geometric imperfections on a shape of equilibrium path is discussed. The optimization was performed numerically using the modified particle swarm optimization method (MPSO). The results are compared with a reference cylindrical shell with no rings. The single ring placed in the middle of the shell is good enough for stabilization of a post-buckling path for a shorter shell, regardless of its thickness. It is necessary to use three rings optimally distributed along the shell length, for a longer shell. The additional optimization profit was obtained by varying internal diameters of the rings. The proposed concept of shell stiffening by inner rings eliminates a major disadvantage of smooth cylindrical shells, namely an unstable post-buckling path.

Design sensitivity of post-buckling states including material constraints

This paper reports theoretical studies on the design sensitivity of problems with geometrically non-linear behavior leading to buckling and post-buckling of thin-walled structural members. For the class of problems considered, buckling occurs in the form of a stable bifurcation, and it is assumed that changes in the design parameters do not break the bifurcation behavior. The specific focus of the research is the first yield or first failure of the material as part of the sensitivity study of equilibrium states along the post-critical path. The investigation employs a discrete model of a structure in terms of generalized coordinates (suitable for finite element analysis) and a single load parameter; and perturbation techniques to classify the critical state and to approximate the post-critical path. The problem of material behavior is modeled by means of constraints on the post-critical path, based on a yield criterion. For simplicity, the presentation uses the von Mises yield criterion, but other more complex criteria, such as those employed in composite materials (first-ply failure) can also be represented. Two forms of the constraints are formulated, and the problem of sensitivity with respect to changes in a design parameter is discussed. A simple example of a circular plate is presented to illustrate the use of the formulation for the sensitivity with respect to a single design parameter.

Elastic limit state flexural–torsional postbuckling analysis of bars with open thin-walled cross-sections under axial thrust

This work deals with the elastic limit state flexural–torsional postbuckling analysis of simply supported bars with open thin-walled asymmetric cross-sections under axial thrust. As it is well known stocky bars with the above type of cross-sections always fail by flexural–torsional buckling in the case of asymmetric cross-sections (whose centroid does not coincide with the shear centre), while in the case of monosymmetric cross-sections the failure may occur either through flexural (Euler) buckling or flexural–torsional buckling depending on the geometric characteristics of the bars. In all the above three cases the critical state is associated with postbuckling strength. In this paper attention focuses on the first yielding occurring at the initial part of the post-critical path of flexural–torsional buckling. This is, in case of bars made from ideal elastic–ideal plastic material, associated with the maximum combined normal stress, due to axial compression, bending and warping, which, along with the nonlinear equilibrium equation, yield the maximum (ultimate) elastic load-carrying capacity. The elastic limit state postbuckling analysis given here is demonstrated with the aid of equal-leg angles commonly used in trusses.

Evaluation of simple bifurcation points and post-critical path in large finite rotation problems

The main aspects of an algorithm for the exact evaluation of single bifurcation points along the nonlinear equilibrium trajectory of flexible mechanisms with large 3-D rotations are discussed. A first version of this algorithm was presented elsewhere [1]. In this work we extend its range of application to problems involving large finite rotations. Several flexible mechanisms and structures presenting bifurcation points are analyzed. Solutions of perfect bifurcation cases are compared to solutions of the literature, which were computed by geometric perturbation.

Optimal design of elastic-plastic structures with post-critical behaviour

This work deals with the optimal design of nonlinear structures where both geometric and path-dependent material nonlinearities are considered. Postcritical behaviour is allowed. Critical and postcritical constraints are considered. Constraints on local and global stability have been introduced. The classical critical load constraint against global instability is given with numerical advantage by a new method using a displacement constraint. The total Lagrangian description and a continuum variational formulation are used for the response and design sensitivity analysis. A continuation algorithm is used to implement the postcritical path. The path-dependent sensitivity problem is addressed by an incremental strategy. A direct differentiation approach is used to derive the response sensitivities with respect to both cross-section and configuration design. A finite element technique models the structure. A mathematical programming approach is used for the optimization process. Numerical examples are performed on three-dimensional truss structures.

Prediction and measurement of the post-critical behavior of fiber-reinforced composite columns

Experimental measurements on axially loaded columns made of fiber-reinforced composite materials are presented. Experimental load/deflection plots at twelve flange points and three web points allow for an accurate description of the deformations, including the post-critical behavior. Data processing by two methods is described and a comparison is made between the two methods. In one method, the incipient buckling data are used, while in the second method, the post buckling data are used. The effect of damage accumulation is illustrated. The behavior, including the curvature of the post-critical path, is predicted by using stability theory. Comparisons between predicted and measured values of critical load and post-critical path curvature are presented.

Element behavior in post-critical plane frame analysis

Depending on the type of uni-modal (cuspoid) catastrophe which governs the local behavior of a structure at a simple critical point, correct prediction of the post-critical path requires an accurate estimation of a number of terms in the Taylor series expansion of the total potential function at the critical point. This imposes certain conditions on the accuracy of the adopted non-linear elements. These conditions are mainly related to the quality of the kinematic assumptions which underlie the strain definition used in the model. If these assumptions are not accurate enough, the fourth and higher order terms in the Taylor expansion of the total potential function will not be correctly represented. As a consequence, the model will fail to predict correctly even the initial post-buckling behavior whenever the criticality at hand is more complex than the fold (limit point, asymmetric bifurcation). This proves to be the case for the so-called ‘technical’ beam models. This inability is inherited by any constructed finite element model, regardless of the interpolation function used in its definition and of the number of elements used in the discretization. An improved non-linear model based on the treatment by Antman (Bifurcation problems in non-linearly elastic structures, in: P.H. Rabinowitz, ed., Application of Bifurcation Theory, Academic Press, NY, 1977), is adopted, and several finite elements are developed on the basis of this model. These elements are tested for a number of problems for which the critical behavior is governed by fold, cusp and butterfly singularities. The numerical results outline the importance of the ‘small’ kinematic terms, especially in conjunction with the occurrence of higher-order uni-modal singularities. Some risks for numerical locking are pointed out and remedied. The recommended element leads to a computational cost, which is comparable to an implemented ‘shallow arch’ element.

Instability of Discrete Pseudo-Conservative Structural Systems

Critical and postcritical states of pseudo-conservative discrete structural systems are studied by means of a new formulation leading to a classification of critical states and to an approximate form of the postcritical equilibrium path. The nonlinear equilibrium equations are derived from the total potential energy function of a classical system, but with the addition of at least one control parameter. The follower force effect is thus included by nonlinear constraints to the equilibrium equation. The nonlinear equations are solved by perturbation techniques. Finally the theory is applied to investigate the instability of some simple mechanical models.

Post-buckling of elastic cone-cylinder and sphere-cylinder complex shells

A study is presented of the elastic buckling of pressurized complex shells of revolution which are formed by the intersection of cylindrical with spherical or conical shells. The studies are based on the general theory of elastic stability using an asymptotic approximation to the secondary path. An axisymmetric finite element is used for the discretization of the shell. First, the behaviour of the individual components is presented for simply supported and clamped boundary conditions. Next, the complex shells are investigated in terms of the buckling load, the meridional and circumferential shape of the buckling mode, and the initial characteristics of the post-buckling path. The results show that the bifurcation loads of complex shells are lower than those of the individual components; but the post-critical path does not decrease as severely as in the isolated shells.

Initial Postcritical Behavior of Thin‐Walled Angle Section Columns

This work deals with the overall flexural instability of an angle section column under axial load, taking into account deformations of the cross section as buckling occurs. A nonlinear differential formulation with two variables is considered. The perturbation method is used to follow the initial postbuckling behavior from the bifurcation point. Deformations of the cross section do not modify the Euler critical load, but they may significantly affect the nature of the postcritical path. Unlike the well-known behavior of columns with solid cross sections, the angle section may exhibit unstable postbuckling deformations typical of the light-gauge sections, as reported in previous studies on thin circular arc sections.

A note on incrementation schemes in non-linear structural analysis

The prediction of the non-linear behavior of engineering structures by the finite element method, wherein instabilities (buckling, bifurcation, etc.) do not occur, has received considerable attention in the past. In contrast to that, it was not until recently that the development of so-called continuation algorithms has been actively pursued. Precisely, these algorithms aim to circumvent inherent numerical difficulties encountered in the vicinity of critical-points of the load-deflection diagram and to continue the analysis beyond these points (‘post-critical path’). The purpose of this paper is twofold. First, numerical methods currently in use for pre-and post-buckling analysis are briefly reviewed. Second, alternative algorithms are proposed and their computational efficiency and reliability are evaluated in comparison with each other and with existing algorithms as well. Numerical solutions for several typical test studies are presented in the last part of the paper.

Pre- and postbuckling finite element analysis of thin-walled one-dimensional structures (trusses, cables, etc.)

In recent years, the development of so-called continuation algorithms has become the subject of intense research efforts. Basically, these algorithms aim to circumvent inherent numerical difficulties exhibited by the standard algorithms for the solution of nonlinear problems in the proximity of instability points, and to continue the step-by-step solution process beyond these points (“postcritical path”). The purpose of this paper is twofold. First, alternative continuation algorithms currently in use are briefly reviewed and discussed. Second, their computational efficiency and reliability is evaluated by studying the pre- and postcritical response of several thin-walled engineering structures (like trusses, cables, etc.).

On the Stability of Equilibrium Paths Associated With Autonomous Systems

The postcritical behavior and stability distribution on the equilibrium paths emanating from a divergence point associated with an autonomous system are studied within a state-space formulation. The analysis concerning the stability of equilibrium paths is based on the eigenvalues of the Jacobian evaluated at arbitrary equilibrium points in the vicinity of a critical point. Explicit conditions of stability and instability concerning the initial and postcritical paths are obtained through a perturbation approach. It is shown that at an asymmetric point of bifurcation an exchange of stabilities between two paths occurs in complete analogy with conservative systems. Similarly, a symmetric point of bifurcation involves a postcritical path which is totally stable (unstable) if the initial path is unstable (stable).

Stability and post-critical growth of a system of cooling or shrinkage cracks

When a system of parallel equidistant cooling cracks propagates into an elastic halfspace, it reaches at a certain depth of the cracks a critical point of instability, and the equilibrium path of the system bifurcates. Further extension of equally long cracks is unstable and impossible. The stable post-critical path consists of extension of every other crack upon further cooling, initially with a crack jump at constant temperature, while the intermediate cracks stop growing and gradually diminish their stress intensity factor until it becomes zero. This represents a second critical state at which these intermediate cracks suddenly close over a finite length at no change in temperature and at constant length of the leading cracks. Subsequently, as the cooling front further advances, the leading cracks grow at equal length until they again reach a critical state, at which every other crack stops growing, and the process in which the crack spacing doubles is repeated. In this manner, the spacing of the opened cooling cracks fluctuates around roughly the one-half value of the cooling penetration depth.