Post-critical State Research Articles

Numerical and experimental analysis of stability of thin-walled composite structures subjected to eccentric load

The study investigates the effect of eccentric load on the stability and post-critical states of thin-walled composite top-hat cross-section columns made of CFRP under compression. The specimens were subjected to compressive load on the universal testing machine Zwick Z100 with regard to both axial and eccentric compressive loads. On the basis of the experimental studies, numerical models of thin-walled composite profiles were designed and verified by numerical analysis based on the FEM. The first stage of the conducted studies was performing a linear buckling analysis of stability of composite structures. The second stage of the analysis involved performing a nonlinear analysis of the thin-walled structures with geometric imperfections reflecting the lowest buckling modes. The obtained test results showed various effects of non-axial compressive load on the critical load values and stiffness of the construction in the post-critical state. In the case of eccentric loading in the direction parallel to the web of the top-hat profiles, the critical load values practically did not change – a drop in the force did not exceed 1.5%, as well as there was no change in the stiffness of the structure in the post-critical state. The realization of the eccentricity in the direction perpendicular to the profile web caused a drop in the critical force equal 43% and a significant decrease in the stiffness of the structure. On the other hand, implementation of the eccentricity load toward the web caused an increase in the critical force by more than 20%, despite very small eccentricity values of only 4mm were applied. This proved high sensitivity of the tested structures to the non-axial load in the direction perpendicular to the web of the profile. The FEM results and the experimental findings showed satisfactory agreement.

Analysis of the Problem of Stability of Thin Shells Compliant to Shear and Compression

The problem of stability of shells compliant to shear and compression is studied by the finite-element method. On the basis of relations of the geometrically nonlinear theory of thin shells compliant to shear and compression (six-mode version), we write the key equations for the determination of their initial postcritical state and formulate the corresponding variational problem. A numerical scheme of the finite-element method is constructed for the solution of the problems of stability of these shells. The order of the rate of convergence of the scheme proposed for the numerical solution of the problems of stability is investigated.

The influence of load eccentricity on the behavior of thin-walled compressed composite structures

The study investigates the effect of eccentric load on the stability and postcritical states of thin-walled CFRP channel section columns under compression. Test specimens are subjected to compression on a testing machine provided with a fixture for applying eccentric compressive loads. Loading forces, deflection and strains of the column walls and web are measured. The experiments also involve examination of the operating performance of a structure undergoing buckling and determination of its postcritical equilibrium paths describing the relationship between load and deflection. Based on experimental results, numerical models of composite structures are designed and verified by the FEM. The scope of the numerical analysis involves performing a linear analysis of stability, which allows for determining the buckling mode depending on the amplitude of compressive load eccentricity and corresponding critical loads. The next stage of the analysis involves performing a nonlinear analysis of the structures with implemented geometric imperfections reflecting the lowest buckling modes. Based on obtained results, postcritical equilibrium paths of the developed FEM models are determined. Obtained equilibrium paths are then compared with experimental characteristics of real structures. The numerical results and experimental findings show a satisfactory agreement.

Finite-element solution of the problems of the theory of shells amenable to shear and compression

For the convenience of applying numerical methods, in particular the finite element method, to solving tasks of the theory of thin shells amenable to shear and compression, all key equations for the research the stress-strain state within the considered shells under static and dynamic loading, detecting proper frequencies of free vibrations, and the initial post-critical state, have been noted in the matrix form. Numerical schemes for solving the problems of statics and dynamics, stability and vibrations of flexible shells amenable to shear and compression have been constructed. Cite as: I. Ye. Bernakevych, P. P. Vahin, I. Ya. Koziy, “Finite-element solution of the problems of the theory of shells amenable to shear and compression,” Prykl. Probl. Mekh. Mat. , Issue 16, 98–106 (2018) (in Ukrainian), http://doi.org/10.15407/apmm2018.16.98-106

On Post-Critical Behavior of a Beam on an Elastic Foundation

In this paper, we analyze the nonlinear equilibrium equation corresponding to the two-parameter bifurcation problem arising in the stability analysis of an elastic simply supported beam on the Winkler type elastic foundation for the case when bimodal buckling occurs. We perform the bifurcation analysis of the nonlinear problem, by using Lyapunov–Schmidt reduction, thus obtaining the number of the nontrivial solutions to the nonlinear problem and qualitatively characterizing the solution patterns. We also give the formulation of the problem and bifurcation analysis from the total energy viewpoint and determine the energy of each bifurcating solution. We assert that the solution with the smallest energy is the one that will be observed in the post-critical state. For specific choice of parameters, the bifurcating solution in the form of the second buckling mode has the smallest total energy. The numerical results illustrating the theory are also provided.

Effect of low-velocity impact damage location on the stability and post-critical state of composite columns under compression

This study investigates the effect of low-velocity impact damage location on the stability and post-critical states of thin-walled channel section columns under compression. The profiles were made of GFRP with a symmetric ply orientation relative to the mid-plane of the laminate. The damaged area (its shape and size) was defined based on the results of experiments performed on composite plates and reported in the literature. The composite damage was modelled using a simplified damage model (SDM), in which the thickness of individual plies of the laminate was reduced depending on the impact energy. The numerical analysis by the finite element method led to development of a series of numerical models of the thin-walled channel section column with the SDM by placing the damaged area in different places of the walls and web of the C column. The composite damage in the post-critical state was determined by the progressive damage criterion according to which composite damage initiation is described by the Hashin criterion whereas damage evolution is described by the energy criterion.

Energy dissipation in soil samples during drained triaxial shearing

The discrete-element method was used to simulate drained triaxial compression of large-scale, polydisperse numerical samples at a range of void ratios while tracing all relevant energy components. The frictional dissipation and boundary work are almost equal regardless of sample density. The volumetric work reaches a steady value at large strain. However, the distortional work increases continually as sample deformation continues post-critical state. There is a preferential orientation for frictional dissipation at around 45° to the major principal stress direction. This matches the orientation at which there is the largest number of sliding contacts. The work equations, which are fundamental in most commonly used constitutive models, are linear when plotted against deviatoric strain. The modified Cam Clay work equation substantially over-predicts the frictional dissipation for dense samples. An alternative, thermodynamically consistent work equation gives a much better description of frictional dissipation and is therefore recommended to ensure accuracy in modelling.

Large Anhysteretic Deformation of Shape Memory Alloys at Postcritical Temperatures and Stresses

Magnetic and nonmagnetic shape memory alloys (SMAs) exhibit thermoelastic martensitic transformations (MTs) which are hysteretic due to their first‐order nature. According to the thermodynamic Landau theory of phase transitions, which assumes ideal thermoelastic equilibrium at each point of the MT interval, the hysteresis is explained by the different limits of stability for austenite and martensite in the phase diagram. No interactions on the phase boundaries are taken into account. In the real alloys, the hysteresis of MT is related not only to the stability intervals of two phases but also to the processes of nucleation and growth of the resultant phase inside the parent phase. In turn, the features of these processes are related to the heights of energy barriers caused by the incompatibility of austenitic and martensitic lattices, crystal defects and some other physical factors. However, the defects, normally, play a minor role in the width of MT hysteresis if compared to the thermodynamic and crystallographic factors. A reduction of hysteresis of MT in SMAs, being crucial for technology, presents a challenging problem for science. A decrease of hysteresis width of MT was observed recently for the single crystals of ferromagnetic SMAs such Ni–Fe(Co)–Ga and Fe–Pd on approaching of their transformation paths to the critical point in stress–temperature phase diagram. Moreover, the superelastic and shape memory properties characterized by the nearly‐zero hysteresis width were observed in the postcritical transformational regime. Here we show that both the Landau‐type theory of ferroelastic phase transitions and neutron diffraction experiments carried out under axial compression describe the essential features of these properties. We also interpret the experimentally observed anhysteretic phenomena in Ni–Mn–Ga thin films and nanobeam actuators in terms of their postcritical state.

Fem and Experimental Analysis of Thin-Walled Composite Elements Under Compression

Thin-walled steel elements in the form of openwork columns with variable geometrical parameters of holes were studied. The samples of thin-walled composite columns were modelled numerically. They were subjected to axial compression to examine their behavior in the critical and post-critical state. The numerical models were articulately supported on the upper and lower edges of the cross-section of the profiles. The numerical analysis was conducted only with respect to the non-linear stability of the structure. The FEM analysis was performed until the material achieved its yield stress. This was done to force the loss of stability by the structures. The numerical analysis was performed using the ABAQUS® software. The numerical analysis was performed only for the elastic range to ensure the operating stability of the tested thin-walled structures.

The analysis of buckling and post buckling in the compressed composite columns

Purpose: The aim of the study was to analyse the work of a thin-walled C-shaped profile, made of a carbon-epoxy composite, which was subjected to unified axial compression. Design/methodology/approach: The scope of the study included the analysis of the critical and low post-critical state by the use of numerical and experimental methods. As a result of the experimental test, performed on the physical specimen, post-critical equilibrium path had been determined, on the basis of which, with use of the adequate approximation method critical load value was defined. The next stage of the research was devoted to numerical analysis based on the finite element method. The studies were carried out on a scope of the linear analysis of the eigenvalue problem, on the basis of witch the critical value of load for mathematical model was found. The next step of the numerical tests was covering the nonlinear analysis of the low post-critical state for the model with geometrical imperfection, corresponding to the lowest form of buckling. Findings: The result of the study was to determine the value of the critical load, on the basis of the experimentally obtained post-critical equilibrium paths of the structure, with use of two independent methods of Approximation: Koiter's method and the method of the vertical tangent. The results of the analysis were compared with the value of the critical load determined by using finite element method. Research limitations/implications: The obtained results of study provide the important information concerning the modelling techniques of the thin-walled structures made of composite materials, while confirming the adequacy of the numerical models developed both in the calculation of eigenvalue problem, as well as non-linear static analysis in the post-critical range. Originality/value: The research provided the necessary knowledge of the behaviour of the critical and low post-critical of the thin-walled structure made of modern orthotropic material (CFRP).

Evolutionary analysis of Fokker-Planck equation using multi-dimensional Finite Element Method

Response characteristics, stability and other problems of non-linear dynamic systems under randomly variable loading are mostly analyzed by means of Fokker-Planck equation. However, possibilities of its analytical investigation are very limited and rather restricted to steady state problems only. Thus a number of non-conventional problems are out of effect of this popular tool. So the Finite Element Method reveals to be a powerful or sometimes the only applicable approach for analysis of non-stationary problems, as for instance quasi-periodic response, post-critical states, unstable states in time limited period, special configuration of boundary conditions, etc. However, a number of specific problems must be overcome. They follow predominantly from the large multi-dimensionality of the Fokker-Planck equation, shape of the definition domain and special requirements on the nature of the solution form, which is out of a common practice of Finite Element employment. In particular: (i) selection of the element type, (ii) development of new original algorithms for multi-dimensional element mesh generation and (iii) working out the original procedures for governing differential and algebraic systems assembling and their subsequent solution. A couple of illustrative examples dealing with SDOF and TDOF systems under random excitation of additive and multiplicative types are presented.

NUMERICAL-RESEARCH ANALYSIS OF THE WORK OF COMPRESSED COMPOSITE THIN-WALLED ELEMENTS

Thin-walled, composite elements subjected to axial compression were the subject of conducted research. The samples were articulately propped against the edges of their cross-section. The scope of the research included a numerical analysis and experimental verification of both critical and weakly postcritical states. Numerical investigations were carried out in the elastic field, allowing for steady work of thin-walled structures until they reached the point of stability loss, with the simultaneous determination of critical forces values. For numerical calculations software ABAQUS® was used. The main issue of the research was to determine a convergence of the results obtained by numerical and experimental examination, in the range of operational allowable loads.

A combined spectral-amplitude-perturbation approach for systematic mode selection in thermal convection

Purpose– The post-critical convective state for Rayleigh-Benard (RB) convection is studied using a nonlinear spectral-amplitude-perturbation approach in a fluid layer heated from below. The paper aims to discuss these issues.Design/methodology/approach– In the spectral method the flow and temperature fields are expanded periodically along the layer and orthonormal shape functions are used in the transverse direction. A combined amplitude-perturbation approach is developed to solve the nonlinear spectral system in the post-critical range, even far from the linear stability threshold. Also, to leading order, the Lorenz model is recovered.Findings– It is found that very small Prandtl numbers (Pr < 0.1) can change the Nusselt number, when terms to O(ε5/2) and higher are considered. However, to lower orders the Prandtl number does not affect the results. Variation of the Nusselt number to different orders is found to be highly consistent. Comparison with experimental results is made and a very good qualitative agreement is observed, even far from the linear threshold.Originality/value– Unlike existing nonlinear formulations for RB thermal convection, the present combined spectral-perturbation approach provides a systematic method for mode selection. The number and type of modes to be included are directly related to the post-critical Rayleigh number. The method is not limited to the weakly nonlinear range.

Dynamic stability of a vertically excited non-linear continuous system

Easily deformable tall structures exposed to a strong vertical component of an excitation are endangered by auto-parametric resonance effect. This non-linear dynamic process in a post-critical regime might contribute to various damages caused by a kinematic excitation. Vertical and horizontal response components are independent on the linear level. However their interaction takes place due to non-linear terms in post-critical regime. Two generally different types of the post-critical regimes are presented: (i) post-critical state with possible recovery and (ii) exponentially rising horizontal response leading to a collapse. A special attention is paid to processes of transition from semi-trivial to post-critical state in case of time limited excitation period as it concerns the seismic processes. Solution method combining analytical and numerical approaches is developed and used. Its applicability and shortcomings are commented. A few hints for engineering applications are given.

The Distribution of Stress in a Sandwich Conical Shell Panel Under Combined Load

The paper presents an analysis of the stress distribution in the open, cone-shaped, sandwich shell panel. The shell under consideration consists of two load-carrying facings and a core layer of the soft type. The facings are made of an isotropic, compressible, work-hardening material and they are geometrically and physically symmetrical, so they are of equal thicknesses and the same material properties. The core layer resists transverse shear only. The shell under consideration is under combined external load in the form of lateral pressure and a longitudinal force. Different ratios of the longitudinal force to the lateral pressure are taken into account. In each case, the components of the effective stress as well as the effective stress values corresponding to the shell deformation were determined. To determine the stress state which occurs during the shell deformation, the stability equations have to be derived. The effective stress corresponding to the deformations in the shell under consideration can occur within the elastic, elastic-plastic or plastic range. In order to determine the stress distribution in the shell under consideration which corresponds to the on- and post-critical state, the constitutive relations of the Nadai-Hencky deformation theory, alongside the Huber-Mises-Hencky yield condition, are accepted. Analyses show that, in certain cases, there are very large differences between the values of the components of the stress state. In such cases, it seems to be possible to omit the components with small values.

A low-dissipative, scale-selective discretization scheme for the Navier–Stokes equations

A new Scale-Selective Discretization (SSD) procedure for the Navier–Stokes equations is proposed. The aim is to reduce the numerical dissipation of already existing numerical schemes to make the SSD scheme easily implementable to the existing CFD codes. In particular, the new procedure is designed to decrease the dissipation errors arising from the discretization of the convection term using upwind-biased convection schemes. Such dissipative errors reduce the quality of high-fidelity simulation approaches in fluid dynamics such as Large-Eddy Simulations (LES). The new discretization procedure is based on separating small and large scales of the flow using a high-pass filter. As a first pre-processing step the convecting velocity field ui is decomposed into a rapidly fluctuating part ui′ using the high-pass filter and a smooth part ui-ui′. After this the derivatives involving ui-ui′ may be discretized with a centered scheme whereas the derivatives involving ui′ can be discretized using an upwind method. The new procedure is tested in Navier–Stokes simulations by implementing the method into a second order accurate incompressible finite volume code based on the fractional step method. The numerical tests on the 2D lid-driven cavity at laminar conditions Re=2500 imply that the new method clearly improves the quality of the simulations. At Re=10,000 the SSD scheme captures the post-critical state of the cavity flow. The advantages of the new method are quantitatively assessed by studying a 2D temporally evolving shear layer. The results imply that the SSD scheme significantly reduces the numerical diffusion in contrast to the conventional upwind-biased schemes. Results from marginally resolved turbulent channel flow at Reτ=590 imply that the new scheme can be used for 3D simulations.

A study of the stable equilibrium of thin shells compliant to shear and compression

Using relations of the geometrically nonlinear theory of thin shells compliant to shear and compression (the six-modal variant), we have written the key equations for determining their initial post-critical state by the finite element method. A specific feature of this model lies in the semidiscretization of the vector of displacements of an elastic body with respect to its variable thickness, based on the Timoshenko–Mindlin kinematic hypotheses, with preservation of the total vector of rotations of a normal to the median surface. We have solved numerically the problem of the stability of a circular plate, clamped over its contour, under the action of radial compressive forces, distributed uniformly along the contour. We have also performed a comparative analysis of the numerical solutions obtained and data known from the literature.

The Basis of Rehabilitation and Treatment in the Acute Phase of Perinatal Cerebral Pathology

Full-term neonates with perinatal CNS damage and critical states at birth were investigated. The study was performed in three stages. At the first stage, 89 children underwent general assessment of status during the first three years of life; at the second, the structure and nature of CNS damage were studied in 69 children; at the third, the status of children during the first year of life was assessed using clinical and instrumented examinations, and some patients were treated with the peptide cortexin. Impairments to vital body functions due to severe posthypoxic states at birth were identified, these having significant effects on the patients’ quality of life at later ages, including the formation of neurological disability. These cases require appropriate correction of postcritical states during the neonatal period. Cortexin (5.0 mg i.m. for 10 days) was used as pathogenetic treatment. Use of Cortexin in children with moderate CNS damage led to virtually complete recovery by the end of the first year of life in 90% of patients.

Influence of residual stresses and geometric imperfections on the elastoplastic collapse of cylindrical tubes under external pressure

The buckling problem of a circular cylindrical shell has long been widely investigated due to its great importance in the design of aerospace and marine structures. Geometric imperfections and residual stresses are inevitable in practice and have been so far frequently considered in analytical and numerical predictions. But little attention has been paid until now on the combined influence of such initial defects on the critical and often unstable response of such elastoplastic structures. In this paper, a shell finite element is designed within the total Lagrangian formulation framework to deal with the elastoplastic buckling and post-buckling of thin cylindrical tubes under external pressure and axial compression. A specific experimental process will be introduced in order to measure residual stresses in the shell very accurately, so as to include them in the numerical calculations. The present formulation will enable us to describe the complete non-linear solutions, namely the critical pressures (bifurcation and limit (collapse) loads), the bifurcation modes and the bifurcated equilibrium branches up to advanced post-critical states. Comparisons will be made between numerical results and the experimental critical value and deformation patterns of a new generation profiler. Furthermore, the combined effects of geometric imperfections, residual stresses and plasticity will be analyzed.

The buckling of elongated rotating drill strings

There is a growing need for the development and implementation of new numerical methods for computer simulation of static and dynamic critical states of drill strings in connection with mastering deeper oil and gas resources. Current methods for analysis of extraordinary behavior of the drill strings, as a rule, concern their post-critical states related to essential change of their shape and contact interaction with borehole wall. In parallel with the post-critical regimes there also are critical modes of operation attended with incipiency of unstable equilibrium states and the DS buckling. Inasmuch as the critical states precede the post-critical ones, sometimes their prognosis can be more important. This paper considers the problem of quasi-static stability of elongated drill string tubes studied with allowance made for destabilizing effects of the longitudinal non-uniform preloading, action of torque, inertia forces of rotation and internal flows of mud. The constitutive equations are deduced, and the methods for their computer solving are proposed. As a result of the completed analysis the critical values of the characteristic parameters are found, and it is shown that the modes of incipient buckling possess complicated shapes associated with the tube spiral curving prevailing in the zone of bottom-hole-assembly.