Post-critical Behavior Research Articles

Experimental Investigation of Compressed Thin-Walled Steel Members

The paper presents fundamental information about realized experimental-theoretical research to determinate the load-carrying capacities for thin-walled compressed steel members with quasi-homogenous and hybrid cross-sections. The webs of such members are stressed in the elastic-plastic region. This continuous research joins on previous research of the first author of the paper. The aim of this research is to investigate and analyse the elastic-plastic post-critical behaviour of thin web and its interaction with flanges. The experimental program, test members and their geometrical parameters and material properties are evident from table 1 and table 2 as well as from figure 1 and figure 2. The test arrangement and failures of the test members are illustrated on Figures 3, 4 and 5. Some partial results are presented in Table 3 of the paper, too.

Aeroelastic flutter enhancement by exploiting the combined use of shape memory alloys and nonlinear piezoelectric circuits

In this paper, the combined effects of semi-passive control using shunted piezoelectric material and passive pseudoelastic hysteresis of shape memory springs on the aerolastic behavior of a typical section is investigated. An aeroelastic model that accounts for the presence of both smart materials employed as mechanical energy dissipation devices is presented. The Brinson model is used to simulate the shape memory material. New expressions for the modeling of the synchronized switch damping on inductor technique (developed for enhanced piezoelectric damping) are presented, resulting in better agreement with experimental data. The individual effects of each nonlinear mechanism on the aeroelastic behavior of the typical section are first verified. Later, the combined effects of semi-passive piezoelectric control and passive shape memory alloy springs on the post-critical behavior of the system are discussed in details. The range of post-flutter airflow speeds with stable limit cycle oscillations is significantly increased due to the combined effects of both sources of energy dissipation, providing an effective and autonomous way to modify the behavior of aeroelastic systems using smart materials.

Critical and post-critical behaviour of two-degree-of-freedom flutter-based generators

Energy harvesting from flow-induced vibrations is a recent research field, which considers a diverse range of systems, among which two-degree-of-freedom flutter-based solutions were individuated as good candidates to obtain high energy performance. In the present work, numerical linear analyses and wind-tunnel tests were conducted on a flat-plate sectional model. The aim is to identify some design guidelines for generators exploiting the classical-flutter instability, through the investigation of the critical condition and the response during the post-critical regime. Many sets of governing parameters of interest from the energy-harvesting point of view were considered, including high levels of heaving damping to simulate the operation of a conversion apparatus. In particular, eccentricity of the elastic centre and small downstream mass unbalance can be introduced as solutions aiming at optimal operative ranges. The collected results suggest the high potentiality of flutter-based generators, and a significant enhancement of performance can be envisaged. Moreover, they contribute to improve the knowledge of the flutter excitation mechanism and to widen the dataset of measurements in the post-critical regime.

A finite element approach to model galloping vibrations of iced suspended cables

The modeling of iced suspended cables under wind excitation is developed basing on aerodynamic elements superimposed on corotational beam finite elements. Eccentricity of the ice coating is accounted for and aerodynamic loads include the aerodynamic moment. The proposed formulation, applied to a suspended cables of the literature, highlights the importance of the eccentricity of the ice coating on the critical conditions for galloping and on the post-critical behavior.

Stability and post-critical behavior of a two-degrees of freedom aero-elastic system in a cross flow

The linear deterministic approach of aero-elastic stability of a slender prismatic beam in a cross flow has been widely investigated in the past. Due to character of the system and regulations in practice (civil engineering, aircraft design, etc.) it can be used for determination of the lowest flow velocity of various types of instability onset. Several two degrees of freedom (TDOF) theoretical models have been worked out and experimentally validated. Nevertheless, it comes to light that a random component appearing in an external excitation can influence significantly the effective aeroelastic stability of this system. Combination of deterministic and random types of external excitation is taken into account regarding simultaneous vortex shedding and turbulence in the shear layer around the profile. Additive excitation includes deterministic (more or less harmonic) and random parts. Multiplicative excitation has a random character only and acts as a perturbation of aeroelastic and elastic parameters of the system. Some attempts for investigation of such a system have been done in the past considering noises as Gaussian white noises. Experimental measurements, however, show that these noises have character of narrow band random processes with distinctly expressed maximum in relevant spectral density. To overcome the limitation of white noises in the Fokker-Planck equation, an extended definition of correlation coefficients is used when constructing the Ito system. Then the original narrow band processes emerge in the relevant Ito system only as a couple of input process spectral density values in significant points of the system resonance curves.

Wind-tunnel tests on the post-critical response of classical-flutter-based generators

Harvesting energy from two-degree-of-freedom flutter requires large-amplitude steady-state oscillations. Nevertheless, the post-critical behaviour of this aeroelasic instability is still an open issue. The present study attempts to improve its scientific understanding and to supply information for the design of more unstable solutions for energy-harvesting applications. In this framework, the energy extraction was simulated through additional damping in the transverse motion component. Then, the influence of some governing parameters was explored through wind-tunnel tests and parametric linear analyses. It was found that a small mass unbalance downstream of the elastic centre is fundamental to anticipate the instability and to obtain unstable systems also in the presence of high damping levels. Moreover, the still-air frequency ratio is the main responsible for the magnitude of the pitching-to-heaving amplitude ratio. An increment of the heaving damping significantly affects the motion amplitude and conditions the optimal position of the elastic axis. Finally, the post-critical response seems to be little affected by the flow turbulence, suggesting the high potentiality of flutter-based generators for applications in real-flow environments.

Stability and Postcritical Behavior of Cylindrical Composite Shells with Local Imperfections Under External Pressure

An approach to solving the problem of the nonlinear deformation of orthotropic cylindrical shells is proposed. On the surface of the shell, there is a local deflection bounded by segments of the coordinate lines. The Timoshenko–Mindlin shell theory, the Byskov–Hatchinson asymptotic method, and the continuous continuation method for solving nonlinear algebraic equations are used. To determine the critical loads and deformation paths, it is necessary to estimate the number of interacting modes sufficient to achieve satisfactory accuracy. Examples of analyzing composite shells with an initial local deflection of positive or negative amplitude are given

Post-buckling analysis of variable-angle tow composite plates using Koiter's approach and the finite element method

The design of lightweight structures is often driven by buckling phenomena. Increasing demands for fuel-efficient aircraft structures makes post-buckled designs attractive from a structural weight perspective. However, the need for reliable and efficient design tools that accurately model the emerging nonlinear post-buckled landscape, potentially one containing multiple branches, remains. With this aim, a previously derived flat shell element, MISS-4, is extended to the geometrically nonlinear analysis of variable-angle tow (VAT) composite plates using Koiter's asymptotic approach. The curvilinear fiber paths in VAT lamina open the design space for tailoring the buckling and post-buckling capability of plates and shells. A finite element implementation of Koiter's asymptotic approach allows the pre-critical and post-critical behavior of slender elastic structures to be evaluated in a computationally efficient manner. Its implementation uses a fourth-order expansion of the strain energy, and requires both the structural modeling and finite element discretization procedures to be, at least, of fourth order. The corotational approach adopted in the MISS-4 element readily fulfills this requirement by starting from a linear finite element discretization. VAT plates with prismatic fiber variations and different loading conditions are analyzed using the MISS-4 element and numerical results of the post-buckling paths are presented. The computed equilibrium paths are compared to benchmark results using the commercial finite element package ABAQUS, and strong asymptotic solutions of the differential equations. The results document the good accuracy and reliability of the proposed modeling approach, and also highlight the importance of multi-modal analysis when multiple buckling modes coincide as is the case in long plates, shells and other optimized thin-walled structures.

Postcritical behavior of a discrete Nicolai column

The Nicolai paradox, concerning the loss of stability of a column subjected to an evanescent follower torque, so far analyzed in the literature in the linear field, is here studied in the nonlinear regime. A paradigmatic, discrete, minimal mechanical system is introduced, roughly describing the column. The bifurcation equations ruling the dynamics around a semi-simple Hopf bifurcation are derived via the Multiple Scale Method, and the postcritical behavior of the system is investigated. A slight asymmetry of the model, which splits the critical eigenvalues, is also taken into account. It is shown that, while the trivial equilibrium configuration of the system is rendered unstable by vanishingly small circulatory forces, nonlinearities can, under certain circumstances, limit the amplitude of the oscillations. The existence of unsafe initial conditions is discussed, and the dangerous effects of the follower torque, also in nonlinear regime, are highlighted.

A study of stability and post-critical behaviour of thin-walled composite profiles under compression

A study of stability and post-critical behaviour of thin-walled composite profiles under compression

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.

Dynamic behavior of three-dimensional composite beam under flapwise excitation

In this paper, dynamic response of a symmetrically laminated composite beam is studied under harmonic base excitation. The base is subjected to flapwise excitation tuned to the primary resonance in the presence of 2:1 internal resonance between the out-of-plane bending motion and the in-plane bending and torsional motions. In literature, modified modulation equations of composite beam have been derived and the stability of fixed points has been investigated in frequency and forced responses. However, post-critical behavior of the modulation equations is studied in this study. In bifurcation diagrams sketched near primary and internal resonances, it appears that detuning the flapwise excitation amplitude causes phenomena like jumps, period doubling, multi and quasi-periodic solutions to occur.

Hard loss of stability of Ziegler’s column with nonlinear damping

The paper is devoted to discuss the effects of nonlinear damping on the post-critical behavior of the Ziegler’s column. The classical Ziegler’s double-pendulum is considered in regime of finite displacements, in which, moreover, nonlinear damping of Van der Pol-type is introduced at the hinges. A second-order Multiple Scale analysis is carried out on the equations of motion expanded up to the fifth-order terms. The nature of the Hopf bifurcation, namely, supercritical or subcritical, as well as the occurrence of the ‘hard loss of stability’ phenomenon, are investigated. The effects of the nonlinear damping on the amplitude of the limit-cycle are finally studied for different linearly damped columns.

Post-critical Behavior of Three-dimensional Composite Beams Near the Autoparametric Resonance under Flapwise Excitation

The bifurcation analysis of a composite beam subjected to harmonic flapwise base excitation is studied when the ratio of flapwise and chordwise internal resonances is 1:2. Results are obtained by the numerical solution of modulation equations. Chracteristics of the response are investigated in terms of time history, phase portraits diagrams and bifurcation diagram of poincare maps. It is observed that periodic and multi-periodic motions, jump and stable motion are appeared near the internal and external tunnig frequency parameters

Nonlinear hysteretic damping effects on the post-critical behaviour of the visco-elastic Beck’s beam

The effects of nonlinear hysteretic damping on the post-critical behaviour of the visco-elastic Beck’s beam are discussed in this paper. The model consists of an inextensible and shear-undeformable cantilever beam, internally and externally damped, loaded at the free end by a follower force. Equations are derived in finite kinematics by taking as the configuration variable the rotation of the cross-section, instead of the usual deflection v of the beam axis, thus making the description of the contact internal actions simpler. The linear stability analysis is carried out and results of the literature are re-obtained. Then, a multiple-scale approach is developed in order to evaluate the nature, super- or subcritical, and the amplitude of the limit-cycle, occurring close to the Hopf critical load. The effects of the nonlinear damping on the amplitude of the limit-cycle are finally discussed for different linear damping coefficients.

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.

Helical buckling and postbuckling of pre-stressed cylindrical tubes under finite torsion

In this paper we analyze bifurcation and postbifurcation of a finite deformation boundary-value problem for a residually-stressed elastic body using the finite element method. In particular, the torsion of a circular cylindrical tube subject to radial and circumferential residual stresses is studied. The material model is given by a residual-stress dependent nonlinear elastic constitutive law in terms of invariants. Firstly, the numerical methodology able to capture the elastic stability of tubes under torsion is developed without including residual stress. The bifurcation point is initially estimated reducing the original problem to an algebraic eigenvalue problem. A nonlinear analysis is performed to capture the postcritical behavior. Secondly, prestress is introduced. The residual-stress dependent nonlinear elastic constitutive law is implemented in a finite element code. The dependence of bifurcation and postbifurcation behavior of tubes under torsion on residual stresses is illustrated and compared with results when there is no residual stress. Analytical solutions involve a complicated mathematical machinery that deals with specific material models and loads as well as, usually, a perfect geometry. Here, we provide a highly flexible approach that is implemented in a commercial finite element code, which allows this methodology to be easily used on other problems under different loading conditions and geometries. The numerical results obtained for the (thin) tubes are compared with the (analytical) solution for the torsion of an infinitely long circular cylindrical membrane to validate the methodology.

Stress constraints in simple bodies undergoing large strains: a variational approach

We consider a simple body that is hyperelastic in the large-strain regime until the 3-covector defining the first Piola–Kirchhoff stress, once it has been projected on the appropriate second-rank tensor space, reaches a threshold indicating critical states. No information is given on the post-critical behaviour. We determine the existence of equilibrium configurations according to the constraint. Such configurations can have a concentration of strain in regions with vanishing volume. The related stress appears naturally as a measure over the deformation graph. Once it is restricted to the regular part of the deformation, such a measure determines the first Piola–Kirchhoff stress tensor and may also be concentrated over sets with vanishing volume projections on the reference and current placements. These configurations in space can be interpreted as dislocations or dislocation walls. We analyse explicitly specific cases.

Stability, bifurcation and post-critical behavior of a homogeneously deformed incompressible isotropic elastic parallelepiped subject to dead-load surface tractions

We study the equilibrium homogeneous deformations of a homogeneous parallelepiped made of an arbitrary incompressible, isotropic elastic material and subject to a distribution of dead-load surface tractions corresponding to an equibiaxial tensile stress state accompanied by an orthogonal uniaxial compression of the same amount. We show that only two classes of homogeneous equilibrium solutions are possible, namely symmetric deformations, characterized by two equal principal stretches, and asymmetric deformations, with all different principal stretches. Following the classical energy-stability criterion, we then find necessary and sufficient conditions for both symmetric and asymmetric equilibrium deformations to be weak relative minimizers of the total potential energy. Finally, we analyze the mechanical response of a parallelepiped made of an incompressible Mooney–Rivlin material in a monotonic dead loading process starting from the unloaded state. As a major result, we model the actual occurrence of a bifurcation from a primary branch of locally stable symmetric deformations to a secondary, post-critical branch of locally stable asymmetric solutions.

Dynamic Buckling Response of Long Plates for the Prediction of Local Plate Buckling of Corrugated Core Sandwich Columns

An analytical model predicting the dynamic local buckling failure of plates with a large dimension in the longitudinal direction compressed at a constant rate was proposed. The model began with the hypothesis that the proposed analytical approach could be an alternative methodology to approximate the dynamic local plate buckling response of constituent plates of corrugated core sandwich columns. Prior to the model development, four preliminary finite-element (FE) simulations were conducted to observe the typical dynamic response of the sandwich columns having thin core web plates or thin face sheets. From the simulations, several wrinkles with a regular pattern were generated, and then one of the wrinkles grew excessively to a failure. Accordingly, the proposed model considered an imaginary patch plate on a long plate simulating a face sheet or a core web plate. The size of the patch plate was predefined so as to encompass the major growing wrinkle, and the out-of-plane displacement was calculated till load drop. The verification of the proposed model was followed by comparison with the FE calculations. The model was satisfactory in predicting maximum forces and times-to-failure, but some discrepancies were found when postcritical behavior and plasticity were involved. The sources of the discrepancies were discussed.