Post-critical Analysis Research Articles

On the role of different nonlinear damping forms in the dynamic behavior of the generalized Beck’s column

The influence of internal and external nonlinear damping forms on the dynamics of a generalized Beck’s column, namely a visco-elastic cantilever beam, subjected to conservative and non-conservative loads at its free end, is investigated. A variational principle provides the equations of motion of the system, which are properly recast into an integro-differential form. The linear stability analysis of the system is then carried out and bifurcation points are detected in the space of parameters associated with the conservative and non-conservative loads. Starting from Hopf’s bifurcation points, a post-critical analysis, based on the Method of Multiple Scales is directly performed on the continuous system, avoiding any a-priori discretization. This method provides the bifurcation equations whose analysis reveals the double nature of nonlinear damping, which can be beneficial or detrimental in terms of stable or unstable bifurcated equilibria. It is found that both the internal and external forms of nonlinear damping can turn a supercritical instability of the system into a subcritical one, thus revealing another destabilizing effect of damping, beyond the very well-known one occurring in the linear field. Numerical simulations, grounded on a Galerkin discretization of the original system, confirm the analytical findings.

Equilibrium of Λ-fractional liquid crystals

Stability of the Λ-fractional equilibrium states of nematic liquid crystals is discussed. Applying bifurcation theory in the Λ-fractional space, branching analysis of the liquid crystals is performed and their constant initial direction ceases to exist at a critical value of the external (electromagnetic) field. Post-critical analysis is performed, defining the variable orientation of the crystals. The analysis is transferred into the initial space, defining the true variable orientation of the liquid crystal field.

William Carlos Williams’ “The Young Housewife”: A Postcritical Reading vis‐à‐vis Shel Silverstein's The Giving Tree

AbstractUsing the framework of Rita Felski in her 2015 book The Limits of Critique, this essay offers a postcritical analysis of William Carlos Williams’ 1915 poem “The Young Housewife.” Its intention is to show how Williams’ poem or any poem can be approached through a variety of critical lenses, but that these may get in the way of more immediate, rewarding ways of reading. Shel Silverstein's well‐known 1964 short book The Giving Tree is similar at the level of “plot” to “The Young Housewife.” Taken in tandem, these two texts neatly exemplify the value of postcritical/non‐resistant reading.

Thermo-mechanical post-critical analysis of nonlocal orthotropic plates

A closed-form analytical solution for critical temperature and nonlinear post-critical temperature-deflection behaviour for nonlocal orthotropic plates subjected to thermal loading is presented. The long-range molecular interactions are represented by a nonlocal continuum framework, including orthotropy. The Von-Karman nonlinear strains are employed in deriving the governing equations. An approximate solution to the system of nonlinear partial differential equations is obtained using a perturbation type method. Series expansions up to second order of the associated field variables and the load parameter, dictating nonlinearity are employed. The behaviour in the post-critical regime is illustrated numerically by adopting an example of orthotropic Single Layer Graphene Sheet (SLGS), a widely acclaimed nano-structure, often modelled as plate. Post-critical temperature-deflection paths are presented with special emphasis on their post-critical reserve in strength and stiffness. Influence of aspect ratio and behaviour in higher modes are demonstrated. Implications of nonlocal interactions on the redistribution of in-plane forces are presented to show striking disparity with the classical plates. The obtained solution may serve as benchmark for verification of numerical solutions and may be useful in formulating simple design guidelines for plate type nanostructures liable to the thermal environment.

Numerical analysis of intralaminar damage evolution on various composite laminates

The paper presents the numerical modelling of progressive damage distribution on the layers of various configurations of composite laminates (cross-ply, angle-ply, balanced laminate and quasi-isotropic laminate) under tensile loading. The purpose of the paper is to study the effect of the stacking sequence and fibre orientation angles on the damage propagation on multi-layered composites. These damage initiations do not necessarily imply a complete loss of structural integrity of the composite laminate. Additional load can be supported due to the stress redistribution over the fibres. A post-critical analysis is carried out, in order to analyse the integrity or the degradation state of the layers of composite laminates. The progressive failure analysis is performed in ANSYS Composite Prep/Post, to enable the investigation of damage initiation and damage evolution on each ply with various fibre orientations of the composite laminates. The intralaminar failure is initially detected based on a failure criterion and the post-critical analysis is evaluated based on a damage evolution law. The results are presented in terms of representative tensile displacements, which lead to damage initiation and damage evolution on the layers of the analysed composite laminates. Moreover, the laminas with different fibre orientations are investigated for progressive loading steps, from the first local failure to the total collapse. The obtained results show that the damages distribution on the plies of the composite laminates are strongly influenced by the fibre orientations, but also by the interaction with the other laminas and stacking sequence.

Dynamic Relaxation Method for Load Capacity Analysis of Reinforced Concrete Elements

In this paper, an analysis method for the nonlinear behavior of reinforced concrete elements subjected to short-term static loads is proposed. The range of inelastic properties for the structural materials is considered, and the deformation processes of reinforced concrete bar elements are modeled. The structural material properties are modeled using the assumptions from plastic flow theory. The load capacity analysis method for the structural system is developed using the finite difference method. The dynamic relaxation method with critical damping allows for describing the static behavior of a structural element, which is used to solve the nonlinear equilibrium equations. To increase the effectiveness of the method for post-critical analysis, the arc-length parameter on the equilibrium path is included. Numerical experiments for a reinforced concrete beam and eccentrically loaded column are run. Comparative analysis with previously published experimental, numerical, and analytical results demonstrated that the proposed computational method is very effective.

Thermomechanics of Inelastic Thin-Walled Structural Members with Piezoelectric Sensors and Actuators Under Harmonic Loading (Review)

Models, combined numerical–analytical methods, and results related to study of the forced resonance vibrations and self-heating of thin-walled inelastic structural members with piezoelectric sensors and actuators under monoharmonic mechanical and electric loading are presented. The thermomechanical behavior of passive and piezoactive materials is described using the concept of complex characteristics that are assumed to depend on temperature and invariants of the strain tensor. The classical and refined thermomechanical theories are used to model the vibrations and self-heating of thin-walled structural members with sensors and actuators. Nonlinear coupled thermoelastic problems for thin-walled structural members are solved by iteration and numerical methods. The thermal failure of structural members is considered. Methods for determining the critical electrical and mechanical monoharmonic loads and methods of postcritical analysis are described. The effect of various factors on the effectiveness of active damping of the resonance vibrations of inelastic thin-walled structural members by piezoelectric sensors and actuators is studied

A qualified plate theory for rigid rotation in postcritical nonlinear analysis

ABSTRACTA new, simple method is presented for deriving the instability potential of thin plates, based on the equilibrium conditions and constitutive law for incremental forces considering the geometric nonlinear effect. First, the equations of equilibrium are established for the plate at the last calculated C1 and current deformed C2 states, from which the equilibrium equations for the incremental forces at C2 are derived. Then, the incremental forces are derived as the summation of two parts: (1) the part from the increase of the Cauchy stresses at C1 to the 2nd Piola-Kirchhoff stresses at C2, which are related to the strain increments and derivable from the linear constitutive; and (2) the part by the difference between the 2nd Piola-Kirchhoff and 1st Piola-Kirchhoff stresses, which are related to the cross-sectional forces at C1 and rigid displacements. Based on the previous concept, the rigid body rule and coordinate indifference of tensor expressions, the constitutive law for incremental forces including the geometric nonlinear effect is derived. Further, by the variational principle, the following instability potentials are derived: δUGB caused by the balance conditions for initial forces, δUGC caused by the constitutive law for incremental forces, and δUGR caused by the rotation of boundary moments. Consequently, the total instability potential is presented for the plate in the virtual work form. The present approach is self-explanatory, which requires only simple integration operations. Moreover, the two essential conditions are satisfied: equilibrium conditions for incremental forces and rigid body rule. Numerical examples are presented to demonstrate the rationality of the present plate theory compared with the conventional one.

Deformation modes for the post-critical analysis of thin-walled compressed members by a Koiter semi-analytic approach

In this paper a semi-analytic solution for the post-critical behavior of compressed thin walled members with generic cross sections is presented. It is based on the Koiter approach and the method of separation of variables. The buckling solution is exactly evaluated using a single sinusoidal function and the initial post-critical behavior is obtained as a decoupled sinusoidal series solution along the beam axis. A specialized integration scheme allows obtaining the solution with only a few terms and a very low cost, with respect to standard finite element analyses. This tool is then used to highlight the reason of the poor behavior of beam models enriched with cross section deformation modes in reconstructing the post-critical solution. Successful strategies are proposed in order to overcome these limitations. A series of numerical tests are reported.

Using Quality of Service Delivery to Evaluate Federal Government of Nigeria Policy on Public Service Outsourcing: A Case of Public Healthcare Institutions, Nigeria

It is a normal practice in all works of life to evaluate or assess the impacts or effects of policies made by individuals, groups and organizations, more especially, the public policies of governments. By assessing some of these policies, especially, nascent ones such as this one, gaps could be established between the policy framework and impacts such could create on the people or institutions which they are meant to serve. This could also serve as post critical analysis for the policy stakeholders. This study is set out to evaluate the public service outsourcing of the Federal Republic of Nigeria, using quality of service delivery questionnaires as the major assessment instrument. A descriptive survey research was adopted for the study, and correlation design as its method was used for data analysis. Pearson Moment Correlation Coefficient was used to determine its internal consistency. Three sets of questionnaire were designed to service the population sample size of 319 (Three Hundred and Nineteen) in the study areas. The SPSS assisted result from the study shows strong correlate of 0.828, at 0.01 level (2-tailed) between public service outsourcing and quality of service delivery.

T-Shaped Frame Critical and Post-Critical Analysis

Abstract The paper shows solution of a T-shaped frame, strength- ened with two linear springs, regarding critical and post-critical analysis. The solution is exact using the Euler elastic approach and the frame of reference, originated in the point of column axis inflexion. The derived Numerical results show the effect of the springs strengthening for the crit- ical and the post-critical system behaviour. The influence of the geometry change is analyzed, as well.

Post-critical analysis of ground resonance phenomenon: effect of stator asymmetry

In helicopter with hinged rotor, an unstable dynamic phenomenon known as ground resonance may occur, leading to the total destruction of the aircraft. The prediction of the stability boundary has been addressed for various helicopter configurations by using different methods. However, in order to develop control strategies for this phenomenon, the post-critical behavior of the system needs to be known. In this paper, the post-critical behavior of the ground resonance instability is investigated. For this purpose, the behavior of a helicopter with four identical hinged blades is investigated and fuselage asymmetry is taken into account. It is shown that, due to fuselage asymmetry, in addition to periodic solutions the system may exhibit quasiperiodic and chaotic response.

Post-critical analysis of highly deformable Joined Wings: The concept of snap-divergence as a characterization of the instability

Theoretical modeling of the aeroelastic static instability with the possibility of tracing post-critical branches (via arc length technique) in the framework of Joined Wings has never been presented before. A complete formulation of the numerical iterative method of solution of the aeroelastic equations is presented.The true critical condition is compared with the divergence speed evaluated by solving an eigenvalue problem about a steady state equilibrium, showing how this last approach may be unreliable and even nonconservative.This work also explores the theoretical implications of using mechanical loads to mimic the real loading conditions.A physical interpretation based on the aeroelastic effects, overconstrained nature of the system, and the bending/torsion coupling, is provided to interpret the conditions that lead to the snap-divergence.

Experimental buckling path of compressed bars under displacement control

Buckling and postcritical analysis of prismatic metal bars, including complex cycles of loading are considered in the paper. The postbuckling path analysis of bars loaded in the form of increasing displacements is discussed in detail. The results of experimental research are reported as a dependence of the force P versus lateral deflection f as well as total shortening of the rod u. It is stated that only the simultaneous analysis of both methods of presentation of results comprises a complete tool for developing the column response. Particularly, the response that appears in the first, linear elastic phase of loading can be observed on P-u diagrams. The increase of the compressing force causes visible shortening of the bar without its buckling. This important phenomenon is not usually included in stability considerations of prismatic rods.

Koiter asymptotic analysis of folded laminated composite plates

A mixed formulation, four node, flat shell element is proposed for the geometrically nonlinear analysis of laminated composite plates. The element is based on the linear analysis of isotropic folded plates, which is then generalized for nonlinear analysis by using a corotational formulation and Koiter’s asymptotic method. Numerical results are presented for buckling and post-critical analysis. The equilibrium paths compare well with those obtained by continuation methods but at a fraction of the computational cost.

Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads

The combined effects of conservative and non-conservative loads on the mechanical behavior of an unshearable and inextensional visco-elastic beam, close to bifurcation, are investigated. The equations of motion and boundary conditions are derived via a constrained variational principle, and the Lagrange multiplier successively condensed, to get integro-differential equations. These latter, with the mechanical boundary conditions appended, are put in an operator-form, amenable to perturbation analysis. A linear stability analysis is carried out in the space of the two loading parameter, displaying the existence of codimension-1 and codimension-2 bifurcations. The influence of both internal and external damping on this scenario is thoroughly investigated. A post-critical analysis is carried out around a double-zero bifurcation, by using an adapted version of the multiple scale method, based on fractional series expansions in the perturbation parameter. The integro-differential problem is directly attacked, so that any a priori discretization is avoided. Emphasis is given to the interaction between the two damping coefficients. This reveals the existence, also in the non-linear range, of a phenomenon of destabilization, so far known only in the linear range.

Modes of Stability Loss of Materials

Three problems of stability loss are investigated and corresponded buckling modes are discussed. The first one is the stability loss of a compressed transversely isotropic linearly elastic medium. The standard analysis based on the Hadamard condition is conducted to solve this problem. The critical compression could be uniquely defined from the bifurcation equations but not a wave length. So, the buckling mode remains generally indefinite. The second considered problem is the stability loss of a compressed half-space with a free surface. It could be shown that the waviness is localized near the free plane surface but as for an entire space the wave length and the buckling mode are indefinite. These problems are treated in linear and nonlinear statement. In linear approach the pre-buckling deformations are ignored. It is shown that for some values of parameters the linear approach leads not only to the numerical error but also to qualitatively incorrect results. The thisrd problem under investifation is the stability loss of an uniformly compressed plate lying on a soft elastic half-space. In this problem the wave length is uniquely defined. Using the nonlinear post-critical analysis it is shown that the buckling mode could be fully defined and is has a chessboard-like character.

BIFURCATION ANALYSIS OF DAMPED VISCO-ELASTIC PLANAR BEAMS UNDER SIMULTANEOUS GRAVITATIONAL AND FOLLOWER FORCES

The mechanical behavior of a non-conservative non-linear beam, internally and externally damped, undergoing codimension-1 (static or dynamic) and codimension-2 (double-zero) bifurcations, is analyzed. The system consists of a purely flexible, planar, visco-elastic beam, fixed at one end, loaded at the tip by a follower force and a dead load, acting simultaneously. An integro-differential equation of motion in the transversal displacement, with relevant boundary conditions, is derived. Then, the linear stability diagram of the trivial rectilinear configuration is built-up in the space of the two loading parameters. Attention is then focused on the double-zero bifurcation, for which a post-critical analysis is carried out without any a-priori discretization. An adapted version of the Multiple Scale Method, based on a fractional series expansion in the perturbation parameter, is employed to derive the bifurcation equations. Finally, bifurcation charts are evaluated, able to illustrate the system behavior around the codimension-2 bifurcation point.

Volume and surface stability of uniformly compressed transversely isotropic linearly elastic materials

The stability loss of a transversely isotropic linearly elastic medium is studied. The medium is uniformly compressed in both horizontal directions, and the initial stress in the vertical direction is equal to zero. The standard analysis based on the Hadamard condition is used. The bifurcation equation divides into two parts, and therefore, two kinds of buckling modes are possible. The critical initial compression is found, but the buckling modes remain indefinite (as the wave length so the relation between the wave numbers is arbitrary). The stability loss of a compressed half-space with a free surface is studied. Only one kind of buckling mode localized near the free surface is possible, and as for an entire space, the buckling mode and the wave length are indefinite. In these problems, linear as well as non-linear approaches are used. In the linear approach, the pre-buckling deformations are ignored. It is shown that for some values of parameters, the linear approach leads to qualitatively incorrect results. The stability loss of an uniformly compressed plate lying on a soft elastic half-space is studied. By using the non-linear post-critical analysis, it is shown that the buckling mode is a chessboard-like one.

Aerothermoelastic Post-Critical and Vibration Analysis of Temperature-Dependent Functionally Graded Panels

This paper deals with the aerothermoelastic post-critical and vibration characteristics of temperature-dependent functionally graded panels in a supersonic airflow. The structural formulation is based on the von Karman plate theory and material properties are assumed to be temperature-dependent and graded in the thickness direction according to power law distribution in terms of the volume fractions of the constituents. The two-dimensional panel under study is simply supported, for which the first order piston theory is used to account for the supersonic aerodynamic loading. The Galerkin method is applied to convert the partial differential governing equation into a set of ordinary differential equations. Panel vibration responses are investigated through time history responses, state-space trajectories, frequency spectra and the bifurcation diagrams of Poincaré maps. Moreover, post-critical behaviors are detected using numerical and analytical methodologies such as bifurcation diagrams of Poincaré maps, Lyapunov exponents and Lyapunov dimension. Finally, it is shown that the Lyapunov dimensions for stable and divergence conditions are zero value, while these values for limit-cycle and chaos vibration conditions are integer and non-integer quantity, respectively.