Post-critical State Research Articles

Stability analysis of shells of revolution under pressure conditions

The subject of the present paper is shells of revolution subjected to the external pressure. A family of shells of revolution, with positive and negative Gaussian curvature, were taken into consideration. The volume V s and length L of the shell were constant. Three different r c / t ratios were investigated: 300, 504 and 700. The goal is to investigate elastic stability of shells of revolution. For that reason two kinds of analyses were performed using ABAQUS system. These are linear eigenvalue buckling prediction and non-linear post-buckling analysis. As a result of the linear analysis the influence of a meridional radius of curvature on the critical load was determined and presented on the graph. Non-linear analysis allowed to create equilibrium paths showing the behaviour of shells in a post-critical state. As can be seen from a number of plots convex barrelled shells are not stable after exceeding the critical load. Some concave barrelled shells on the contrary have stable equilibrium paths in the post-critical range.

Elastoplastic bifurcation and collapse of axially loaded cylindrical shells

In this paper, a shell finite element is designed within the total Lagrangian formulation framework to deal with the plastic buckling and post-buckling of thin structures, such as cylindrical shells. First, the numerical formulation is validated using available analytical results. Then it is shown to be able to provide the bifurcation modes—possibly the secondary ones—and describe the complex advanced post-critical state of a cylinder under axial compression, where the theory is no longer operative.

Analytical and experimental modelling of non-linear aeroelastic effects on prismatic bodies

This paper deals with the aeroelastic instability and non-linear post-critical behaviour of a two degree-of-freedom oscillatory system. The main goal of the study is an introduction of a realistic non-linear theoretical model making possible a detailed non-linear analysis of post-critical states of an aeroelastic system. Two component self-induced vibration of a body is described by a system of two simultaneous non-linear differential equations. The coupling of the resultant motion is due to linear and non-linear aeroelastic effects only, while the elastic forces are taken independent. Differential system is auto-parametric and represents a combination of Rayleigh or Van der Pol with Duffing types of differential equations. The qualitative analysis is based on the uplift coefficients and torsional moments as the functions of the flow velocity and structural response components. The effects of non-linearities are tested on the model where the motion of one degree of freedom is prevented. The analysis is verified experimentally by the new generation technique allowing excessive amplitudes of the oscillation.

Stress analysis of flexible noncircular cylindrical shells with hinged edges for different critical loads

An analytic nonlinear boundary-value solution is found and used in analysis of the precritical and postcritical stress states of a flexible long cylindrical shell with variable curvature and hinged longitudinal edges under nonuniform loading

On materially and geometrically non-linear analysis of reinforced concrete planar frames

A family of new beam finite elements for geometrically and materially non-linear static analysis of reinforced concrete planar frames is derived, in which strain measures are the only interpolated unknowns, and where the constitutive and equilibrium internal forces are equal at integration points. The strain-localization caused by the strain-softening at cross-sections is resolved by the introduction of a `short constant-strain element'. Comparisons between numerical and experimental results on planar frames in pre- and post-critical states show both good accuracy and computational efficiency of the present formulation.

Theory of Elastic Stability: Analysis and Sensitivity

1.Introduction 2.Equilibrium States in Non-Linear Elasticity 3.Perturbation Techniques 4.Stability of an Equilibrium State 5.Critical States 6.The Limit Point 7.Bifurcation States 8.Equilibrium of Systems With Imperfections 9.Influence of Non-Linear Material Behavior on Buckling States 10.Imperfection Sensitivity of Critical Points 11.Design Sensitivity of Bifurcation States 12.Design Sensitivity of Post-Critical States 13.Bifurcation States:The W-Formulation 14.Mode Interaction 15.Sensitivity of Systems with Mode Interaction A.Displacements, Strains, and Stresses B.Energy of Bars and Trusses

Elastica of an euler rod with clamped ends

The stability of the postcritical states of equilibrium of a flexible rod with clamped ends loaded by an axial force is analyzed. It is shown that the existing Lagrange elliptic-integral solution has bifurcation points and branches of solution that have not been investigated thus far.

Plastic Model for Concrete in Plane Stress State. I: Theory

A rate-independent constitutive model for plain concrete on the basis of the bounding surface plasticity theory is formulated. Plastic strain rate is described by three independent plastic mechanisms without using a plastic potential surface. The actual plastic surface degenerates to the current stress point. The bounding surface shrinks in the postcritical state. Separate criteria of loading/unloading are used for the deviatoric and hydrostatic portions of stress rate. Plastic modulus functions are formulated separately for virgin loading/unloading/reloading cases on the basis of experimental data. Three different descriptions of shrinkage of the bounding surface are proposed. The important features of concrete behavior, namely, stiffness degradation, contraction/dilatation phenomenon, and different behavior of concrete in compression and in tension are satisfactorily rendered by this model. The presented model can describe concrete behavior in the plane stress state in the full range of loading including cyclic loadings.

On the rotating rod with variable cross section

Stability of a heavy rotating rod with a variable cross section is studied by energy method. Bifurcation points for the system of equilibrium equations are analyzed. It is shown that for the case when the rotation speed exceeds the critical one, the trivial solution ceases to be the minimizer of the potential energy, so that rod loses stability, according to the energy criteria. Also, a new estimate of the maximal rod deflection in the post-critical state is obtained.

Sensitivity of post-critical states to changes in design parameters

Sensitivity of post-buckling paths is studied in the context of the general theory of elastic stability of discrete structural systems. It is assumed that the sensitivity of the critical state itself has been computed and a formulation is developed to account for sensitivity of the curvature of the post-critical states when there are changes in design parameters. A linear fundamental path is considered. Explicit expressions are obtained for the sensitivities and they take the form of perturbation expansions. Only first order sensitivity of post-critical paths has been developed. A simple example of an angle section column with deformable cross-section illustrates that although a critical state may be insensitive to changes in certain design parameters, the post-critical response may be highly sensitive. In the first example presented (an axially loaded angle section column), the postbuckling response even changes from stable to unstable depending on the values of the design parameter considered.

Postbuckling of One-Dimensional Elements with Thin-Walled Angle Section Using Simplified Models*

ABSTRACT The present work is a detailed study of the initial post-critical behavior of one-dimensional elements with an angle section, using simplified models. The emphasis of the work is on the influence that the deformation of the cross section has on the critical and post-critical states. The model assumes that the cross section deforms entirely by rotation of the legs about the hinge at the apex. Three structural types are considered: the column under an axial load, the circular arch under a central load, and the circular ring under uniform radial pressure Critical states are assumed to be distinct, in the sense that mode interaction effects may be neglected. The critical states for all cases analyzed are unaffected by angle deformation, while the post-critical states are affected to different degrees, depending on the structural configurations. A parameter that defines post-critical strength has been consistently defined for the cases of the column and the arch. The unstable post-critical behavior found for short angle columns makes this type of structure sensitive to imperfections.

On the control of deformation process by plastic strain

The incremental relations are derived between stress rate and the second order plastic strain rate (or plastic strain path curvature). Such relations could be used in setting control of the deformation process by plastic strain either in an experimental test or in numerical procedure, especially when critical and post-critical states occur for stress or strain control.

Generalized Stokes Eigenfunctions: A New Trial Basis for the Solution of Incompressible Navier-Stokes Equations

The present study focuses on the solution of the incompressible Navier-Stokes equations in general, non-separable domains, and employs a Galerkin projection of divergence-free vector functions as a trial basis. This basis is obtained from the solution of a generalized constrained Stokes eigen-problem in the domain of interest. Faster convergence can be achieved by constructing a singular Stokes eigen-problem in which the Stokes operator is modified to include a variable coefficient which vanishes at the domain boundaries. The convergence properties of such functions are advantageous in a least squares sense and are shown to produce significantly better approximations to the solution of the Navier-Stokes equations in post-critical states where unsteadiness characterizes the flowfield. Solutions for the eigen-systems are efficiently accomplished using a combined Laoczos-Uzawa algorithm and spectral element discretizations. Results are presented for different simulations using these global spectral trial basis on non-separable and multiply-connected domains. It is confirmed that faster convergence is obtained using the singular eigen-expansions in approximating stationary Navier-Stokes solutions in general domains. It is also shown that 100-mode expansions of time-dependent solutions based on the singular Stokes eigenfunctions are sufficient to accurately predict the dynamics of flows in such domains, including Hopf bifurcations, intermittency, and details of flow structures.

On the failure of static stability analyses of nonconservative systems in regions of divergence instability

The stability of perfect bifurcational discrete dissipative systems under follower loads in regions of existence/non-existence of adjacent equilibria is thoroughly re-examined in the light of recent progress in nonlinear dynamics. A general theory for such nongradient systems described by autonomous ordinary differential equations is developed. Conditions for the existence of adjacent equilibria, the stability of precritical, critical and postcritical states, as well as for different types of local bifurcations are established. Focusing attention on the interaction of geometric nonlinearities and vanishing damping, new findings contradicting widely accepted results of the classical (linear) analysis are discovered. In a small region of adjacent equilibria near a compound branching point, which is explicitly determined, an interaction of two consecutive postbuckling modes occurs related to the following phenomena : in case of vanishing damping, loss of stability may occur via a Hopf (dynamic) bifurcation prior to static (divergence) buckling. Moreover, the critical states of divergence instability may be associated with a double zero Jacobian eigenvalue satisfying also the conditions of a Hopf (local) bifurcation. Besides local (dynamic) bifurcations, global bifurcations are also found. An example is used to illustrate the qualitative findings.

Stability of laminated orthotropic plates upon slippage of layers

The article deals with a laminated orthotropic plate. Between the layers there is rigid contact as well as contact with frictionless slippage. On the end faces at the level of the neutral surface of the slipping layers the plate is loaded by tangential loads. The model of loss of stability is based on the hypotheses on displacements of the plate that take transverse shear strain into account. Normal displacements of the plate do not depend on the transverse coordinate. The hypotheses derived by iteration make it possible to satisfy the conditions of interlaminar contact, both rigid and with frictionless slippage, with respect to displacements and stresses, and also the conditions on the outer surfaces of the plate. With the aid of Lagrange's variational principle we obtained differential equations under conditions of the precritical and the postcritical state, and also the corresponding boundary conditions. The equations of the postcritical state are written in terms of displacements. On the basis of the presented equations the problem of the stability of a homogeneous orthotropic rectangular plate with pin support along the periphery is solved. It is shown that the results obtained with the suggested model coincide with the known results. The article also examines a square three-layered orthotropic plate under conditions of frictionless slippage contact between the layers and pin support along the periphery. The central layer is loaded by tangential loads on the end faces at the level of its neutral surface. It is shown that when slippage is taken into account, the critical forces are substantially reduced.

Static and dynamic, local and global, bifurcations in nonlinear autonomous structural systems

Discrete damped or undamped gradient systems described by nonlinear autonomous ordinary differential equations (ODE) of motion are examined in detail. Emphasis is given to the relationship between static with possibly existing dynamic bifurcations. Criteria for dynamic bifurcations and stability of precritical, critical, and postcritical states associated with the nature of Jacobian eigenvalues are also presented. Cases of discrepancies between local and global dynamic analysis regarding the stability of equilibriums are reported for the first time. Finally, using a simple energy criterion, exact dynamic buckling loads for vanishing but nonzero damping are readily obtained.

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.

The relationships between post-failure state and compression strength of sudetic fractured rocks

The behaviour of the Sudetic crystalline rocks in a pre- and a post-critical state of stress is the topic of this work. The pre-and post-critical failure energy ratio has been discussed, and the regression lines of residual strength of fractured rocks has been detrmined, based on which the residual strength (Rrez) and the standard compression strength (Re) relationship was empirically determined as Rrez=0.18Rc+5.6;

Fracture of composites under compression along periodically placed parallel circular stratifications

Questions of the fracture of composite materials by compression along existing stratifications are quite urgent in the mechanics of composites. Investigations of the questions mentioned are reflected in numerous publications of domestic and foreign authors [2-4, 12, 13, 22, 23, 25-27, 29, 33-35, etc.]. Two stages of the fracture process are separated in the investigations, the initial stage characterized by buckling of a local nature in the neighborhood of defects, and defect development in thepost-critical (post-buckling) state. The first stage is examined in the overwhelming majority of researches within the framework of approximate schemes when local buckling of the material with defect-stratifications is treated as the Euler buckling of a compressed strip (plate) separated by defects with arbitrarily assigned clamping conditions (more often stiff framing and more rarely simple support). The second stage, the investigation of stratification development in the post-critical state, is ordinarily carried out on the basis of the theory of cracks in a beam approximation (see [ii, 14, 15, 18-20, 28, 30, 32] and subsequent papers). Let us also note that all the investigations executed are concerned mainly with near-surface single defects.

Complementarity problem and the existence of the post-critical equilibrium state of a thin elastic plate

Using the concept of a conically bounded set, we prove a variational principle for functionals defined on a locally compact pointed convex cone. Applying this principle to the nonlinear complementarity problem, we study the existence of the post-critical equilibrium state of a thin elastic plate, subjected to unilateral conditions.