Bilinear Hysteretic Model Research Articles

A unified nonlinear dynamic model for bolted flange joint disk-drum structures under different interface states: Theory and experiment

Bolted flange joint disk-drum structures (BFJDDSs) are key components in aero-engines, however, bolt looseness inevitably occurs due to extreme service conditions. This can cause the bolted joint interface to exhibit complex contact behaviors, which significantly complicate the vibration characteristics of BFJDDSs. Existing dynamic models for BFJDDSs have not considered the effect of bolt looseness (slip and separation), so their vibration behaviors are not well understood. In this study, a unified nonlinear dynamic model for BFJDDSs considering different interface states (stick, slip, and separation) is proposed, which is effective under both bolt looseness and non-looseness (stick) conditions. The bi-linear hysteretic model combined with the piecewise linear model is used to simultaneously consider different interface states. The Kirchhoff plate theory, the Sanders’ shell theory, and the Euler-Bernoulli beam theory are used to derive the energy functions of the disk, drum, and flange, respectively. Then, the Lagrange equations are employed to derive the governing equations of BFJDDSs. Modal and nonlinear forced vibration experiments are carried out on a BFJDDS to prove the correctness of the proposed mathematical model. Research results show that the proposed mathematical model is able to achieve good predictions of nonlinear dynamic properties of BFJDDSs under both bolt looseness and non-looseness conditions. This model is also able to well predict the jumping phenomenon observed in the experiment.

Spectral incremental dynamic methodology for nonlinear structural systems endowed with fractional derivative elements subjected to fully non-stationary stochastic excitation

A novel spectral incremental dynamic analysis methodology for analysing structural response in nonlinear systems with fractional derivative elements is presented, aligning with modern seismic design codes, like Eurocode 8. Drawing inspiration from the concept of fully non-stationary stochastic processes, the vector of the imposed seismic excitations is characterised by time and frequency evolving power spectra stochastically compatible with elastic response spectra of specified damping ratio and ground acceleration. The proposed method efficiently determines the nonlinear system time-dependent probability density functions for the non-stationary system response amplitude by employing potent nonlinear stochastic dynamics concepts, such as stochastic averaging and statistical linearisation. Unlike traditional incremental dynamic analysis curves found in the literature, the herein proposed method introduces a three-dimensional alternative counterpart, that of stochastic engineering demand parameter surfaces, providing with higher-order statistics of the system response. An additional noteworthy aspect involves the derivation of response evolutionary power spectra as function of spectral acceleration, offering a deeper insight into the underlying system dynamics. Besides its capabilities, the method maintains the coveted element of a particularly low associated computational cost, increasing its attractiveness and practicality among diverse applications of engineering interest. Numerical examples comprising the bilinear hysteretic model endowed with fractional derivative elements subject to an Eurocode 8 elastic design spectrum demonstrate the capabilities and reliability of the proposed methodology. Its accuracy is assessed by juxtaposing the derived results with germane Monte Carlo Simulation data.

Reliability-based ductility reduction factors surfaces using the generalized bojorquez ground motion intensity measure IBg

In this work, reliability-based ductility reduction factors surfaces are obtained through uniform annual failure rate (UAFR) spectra. For this aim, several nonlinear single degree of freedom (SDOF) systems that include various hysteretic behavior models, constant ductility performance and structural reliability levels are subjected to two sets of ground motions recorded on Mexico City. The first set includes 50 ground motion records located on stiff soils, while the second set correspond to 50 soft soil ground motions. In order to compute the UAFR spectra, the ground motion records were scaled at different values of a particular case of the generalized intensity measure IBg named INp for spectral acceleration. The study discusses the influence of different post-yielding stiffness on UAFR spectra obtained for the selected Bojorquez intensity measure, through the bilinear hysteretic model. Additionally, the Takeda model is used to represent structures that exhibit the effect of stiffness degradation. This study is aimed to compute the effect to obtain ductility reduction factors for various uniform annual failure rates, different degrees of post-yielding stiffness, and different ductility capacity for various nonlinear structures. The analyses of the results indicate that strength reduction (through ductility reduction factors) is mainly influenced by parameters such as ductility, failure rate, period of the system and soil conditions. Finally, reliability-based ductility reduction factor surfaces for stiff and soft soil are provided for different UAFR. It is important to say that this is the first time that ductility reduction factors are proposed based on IBg or INp as intensity measure.

Development and experimental evaluation of cam-grip type compression-free energy dissipative braces

Recent efforts have focused on developing novel energy dissipation braces that allow only tensile yielding to overcome certain limitations of conventional braces and buckling-restrained braces (BRBs). Such limitations include insufficient information regarding the damaged state of braces in a post-earthquake scenario and the challenges associated with replacing heavy restraints to prevent steel cores from buckling. Generally, the available techniques adopt a saw-tooth mechanism that grips the core and only mobilizes the tensile forces in the brace. However, the discrete nature of saw-tooth systems may result in a slippage at the onset of the tensile loading cycle, thereby reducing the efficiency of the energy dissipation mechanism. This study introduces a novel Direction-based Force Control Mechanism (DFCM) utilizing a cam-type grip that ensures continuous gripping under reversed-cyclic loading. Using this mechanism, a new type of so-called compression-free energy dissipative braces (CEDBs) are developed and tested under reversed-cyclic loading. For this purpose, six CEDB specimens were fabricated and installed in a steel frame sub-assemblage. The frame-brace system was then subjected to quasi-static lateral drifts with increasing amplitudes to test the efficiency of the proposed device. The experimental results demonstrate that the proposed bracing system only mobilizes the tension forces with significantly low slippage with about 1 % drift (Δ ≈ 2 mm). The system also exhibits a stable energy dissipation mechanism and reliable hysteretic properties under multiple cycles of large inelastic deformations. Furthermore, the damaged steel cores can be conveniently observed and replaced after a severe shaking event, while the same DFCM can be repeatedly used. Therefore, the proposed bracing system emerges as a cost-effective option for enhancing the lateral performance of structures subjected to dynamic loadings and strong ground motions. Besides, an idealized hysteretic model for this system was developed and validated; two examples of specimens were conducted as uniaxial elements with the proposed bilinear hysteretic; based on experimental results, it is shown that the proposed bilinear hysteretic model can capture important mechanical properties and energy dissipation of the specimens with reasonable accuracy of both CEDB specimens which the average difference between experiment results and Idealized CEDBs in yield force is about 10.87 %, the ultimate force is about 1.90 %, and the corresponding cumulative energy dissipation is about 6.98 %.

A framework for nonlinear peak response evaluation from linear response spectra of base-isolated structures using the bilinear hysteretic model

This paper proposes a simple computational framework that can account for both properties of the base-isolated structure using the bilinear hysteretic model and reliably evaluate the nonlinear peak response without nonlinear response history analysis. The proposed computational framework requires the construction of new non-proportional damping models that match the base-isolated structure to accurately decompose the nonlinear multi-degree-of-freedom structure into multiple single-degree oscillators with intrinsic frequencies and damping ratios by equivalent linearization and complex-mode superposition methods. Each oscillator is excited by a response spectrum matching its own damping ratio, and this procedure is iteratively computed until the displacement error of the isolation layer achieves a preset value. Simultaneously, the peak nonlinear response of interest is calculated using the complex-mode-superposition complete square combination rule. The effect of peak response on nonlinear parameters is analyzed using a typical numerical example, and the relative error values for the applicable range are presented. Parametric research demonstrates that the new non-proportional damping matrix accurately reflects the modal characteristics of the base-isolated structure and prevents Rayleigh damping from suppressing the first-order modal characteristics. The particular example evaluation shows that the computational framework can effectively evaluate the nonlinear peak response of the BIS, with the isolation layer displacement error controlled within +5 %, the bottom shear force error within +20 %, and the top acceleration error between −10 % and +25 %.

Global method for nonlinear seismic analysis of embedded base-isolated nuclear island buildings in layered sites

Owing to the increasing scarcity of high-quality sites, the seismic analysis of base-isolated nuclear power plants (NPPs) under complex non-lithology sites has gained important engineering significance. In this study, a global method for nonlinear numerical analysis was developed for the seismic evaluation of base-isolated NPPs in complex layered sites. The seismic analysis of layered sites is realised using the equivalent linear method and one-dimensional wave propagation theory, the radiation damping of the infinite domain is modelled with a visco-elastic artificial boundary, and the nonlinear behaviour of the isolators is fully considered using a bilinear hysteretic model. Using this method, the influence of soil–structure interaction (SSI) and soil nonlinearity on the behaviour of base-isolated NPPs under the design basis earthquake and beyond design basis earthquake was analysed and compared, in terms of the floor response spectra (FRS) and inter-story displacement angle. In addition, the influence of embedment is also studied. The results show that in layered sites, the SSI significantly amplifies the response values for base-isolated NPPs, and as the ground motion input increases, the soil nonlinearity more effectively inhibits the response of base-isolated NPPs. In addition, embedding can significantly reduce the response, but it increases the FRS at some frequency domains. The isolation effect of base-isolated NPPs decreases significantly when SSI effects are considered, and the soil nonlinearity and embedment have little impact on the base isolation capacity. The method proposed in this study can be easily applied to study base-isolated NPPs in layered sites.

Uniaxial hysteretic spring models for static and dynamic analyses of self-centering rocking shallow foundations

Self-centering rocking shallow foundations have been proved to be a potential alternative to conventional foundations that provide superstructures with “fixed-base” constraints. Through appropriate design of foundations to undergo controlled rocking motions, they serve as “seismic fuses” (associated with a rocking isolation mechanism) which enable their supporting structures to remain even undamaged during intense shaking events without compromising the overall resilience of the soil–structure interaction (SSI) system. Recent progress in rocking shallow foundations has heightened the need for simple and efficient models in seismic design and evaluation of structures with rocking isolation. To this end, the paper proposes uniaxial hysteretic rocking spring models to enrich modeling of rocking-dominated foundation response in engineering practice. Two types of models are developed, including a bilinear hysteretic model (BHM) and a smooth hysteretic model (SHM). These models are defined by dimensionless backbone curves and recentering ratio based hysteretic rules obtained by performing monotonic and cyclic push-over analyses using a nonlinear hysteretic Winkler foundation (NHWF). The current implementation is based on square or strip footings on saturated clay foundations. Using the developed models, seismic performance of practical SSI systems consisting of rigid shear walls or deformable bridge piers on rocking shallow foundations are examined.

Seismic performance evaluation of base-isolated nuclear power plant reactor building considering aging effect of isolation device

ABSTRACT The base isolation (BI) device is widely used for reducing seismic energy transmitted to the upper part of the structures. The rubber property of lead rubber bearing (LRB) BI devices degrades with time and environmental conditions, affecting the controlling performance of BI devices. This study investigates the degradation of capacity of base-isolated nuclear power plant (BI-NPP) considering the aging of rubber material used in the LRB BI device. An NPP reactor building with BI devices is numerically modeled. To capture the nonlinear behavior of the BI device, a bilinear hysteretic model is considered. Assuming different ages, the nonlinear incremental dynamic analysis is conducted using a set of 25 earthquakes and the seismic outcomes are evaluated through fragility analysis. The seismic capacity is extracted from the fragility parameters in terms of High Confidence of Low Probability of Failure (HCLPF) capacity. The outcome shows that the capacity degraded remarkably up to 21% for 100 years of service life. Finally, the seismic capacity degradation model is proposed through the investigation, which will be helpful to evaluate the capacity of the BI-NPP structure for any instant of time.

A study of the dynamic seismic performance index of building in Indonesia

ABSTRACT Recent earthquakes have exposed the vulnerability of buildings in Indonesia. An evaluation is needed to confirm the seismic performance of the structures. A seismic index is a method to determine building performance. This research aims to evaluate the performance using a dynamic seismic performance index dIs by considering Indonesia’s seismic hazard. The nonlinear incremental dynamic analysis with a bilinear hysteretic model was used in the analysis. The estimation method using the equivalent linearization and using the characteristic displacement response is proposed to estimate a dynamic ductility index dF. The dynamic seismic performance index tends to increase with an increase in the critical ductility factor. The dynamic ductility index dF represents a modification factor (R) that also increases with an increase in the critical ductility. The accuracy of the equivalent linearization is more accurate in the lower period, but the characteristic displacement method gives the closest estimate in the higher period.

Application of seismic isolation for multi-story buildings in moderate seismicity areas like Vietnam

Multi-story buildings are important components of urban infrastructures due to their undeniable advantages. However, their structures are characterized by the mass distributed in height and a great slenderness leading to its earthquake vulnerability. Further, their typical structures designed by the conventional method provide low earthquake resistance. The seismic isolation bearings are considered as an effective solution for the structure located in regions of high and moderate seismic activity. The paper presents the procedure to calculate the isolated building structure located in the moderate seismicity areas such as Vietnam. The bilinear hysteretic model is employed to determine the properties of the isolator through the simplified procedure. Three-dimensional model of a typical 11-story building structure is analyzed to evaluate the effect of the isolator on seismic responses of building. The modal analysis and time history analysis are performed using Etabs V17, with the input record taken according to the earthquake ground motion in Hanoi-Vietnam, calculated by the Vietnamese code. The result shows the high effectiveness of isolator on reducing the seismic effect.

Critical Response of Multi-Story Damped Bilinear Hysteretic Shear Building Under Multi Impulse as Representative of Long-Period, Long-Duration Earthquake Ground Motions

The critical response is investigated for a multi-story damped bilinear hysteretic shear building model under a multi impulse as a representative of long-period long-duration earthquake ground motions. The critical response for an elastic-plastic single-degree-of-freedom (SDOF) model under a multi impulse was derived in the previous papers. However, it is difficult to derive the critical response for a multi-degree-of-freedom (MDOF) model under a multi impulse because the phase lag among masses induces a complicated situation. In place of deriving an explicit critical response, a criterion on the criticality of the multi impulse for the elastic-plastic MDOF model is derived and the properties of the critical responses are clarified. In the analysis, a new concept of ‘Multi Impulse Pushover (MIP)’ is introduced and the progressive performance evaluation for the increasing input level is conducted. The analysis of energy response (input energy, dissipated energy, etc.) and the comparison with the response to the corresponding sinusoidal wave are made to enhance the reliability and practicality of the proposed method using the multi impulse as a substitute of sinusoidal waves which are representatives of long-period long-duration earthquake ground motions.

First-excursion stochastic incremental dynamics methodology for hysteretic structural systems subject to seismic excitation

A novel efficient stochastic incremental dynamics methodology considering first-excursion probability for nonlinear structural systems subject to stochastic seismic excitations in alignment with contemporary aseismic codes provisions is developed. To this aim, an approximate nonlinear stochastic dynamics technique for conducting first-passage probability density function (PDF) based stochastic incremental dynamic analysis is developed. Firstly, an efficient stochastic iterative linearization methodology is devised achieving convergence of the equivalent system damping ratios with the damping premises of the excitation response spectrum leading to a coherent determination of a robust scalable intensity measure (IM) which bears direct relation to its damaging potential. Subsequently, utilizing the stochastically derived time-varying forced vibrational system properties in conjunction with a combination of deterministic and stochastic averaging treatment the first-excursion PDF is efficiently determined for each and every of the considered limit-state rules (LSs). Lastly, an incremental mechanization analogous to the one used in normal incremental dynamic analysis (IDA) is proposed to ensure the necessary compatibility for applications in the fields of structural and earthquake engineering. The back-and-forth twisting pattern of IDA curves which is related with the multiple points satisfaction of the very same limit-state rule encourages the study of the problem from a first-passage perspective considering timing in addition to the intensity of the excitation variable. The selected engineering demand parameter (EDP) of the first-excursion time constitutes an excellent response related variable with twofold meaning; it performs structural behavior monitoring considering intensity and timing information whereas it is inherently coupled with limit-state requirements. The developed stochastic dynamics technique provides with reliable higher order statistics (i.e., PDF) of the chosen EDP. A structural system comprising the bilinear hysteretic model serves as a numerical example for demonstrating the reliability of the proposed first-excursion PDF-based stochastic incremental dynamics methodology. Nonlinear response time-history analysis involving a large ensemble of Eurocode 8 spectrum compatible accelerograms is conducted to assess the accuracy of the proposed methodology in a Monte Carlo-based context.

Dynamic analysis of a multi-span simply supported prestressed concrete bridge with restrainers and seismic isolation devices

The Nonlinear F.E. code Structural Analysis Program (SAP) in which the primary nonlinear characteristics of bearings, impact elements and steel restrainer cables considered herein in order to investigate the dynamic analysis of a multi-span simply supported MSSS prestressed concrete bridge, equipped with steel restrainer cables and Lead Rubber Bearings LRB devices including Soil Structure Interaction (SSI) effects. A MSSS bridge with restrainer cables and lead rubber bearings at the two abutments and intermediate bents located in North Algeria is selected according to RPOA for seismic design category 1 is considered in this study. A detailed 3D nonlinear analytical model of study bridge subjected to three components identical seismic excitation including pounding elements, restrainer cables and bearing devices at expansion joints is developed. The nonlinear characteristics of theses boundary elements are represented by bilinear hysteretic models. Under strong seismic excitations, the large longitudinal displacements result the collision between bridge decks or even unseating of these decks at expansion joints of MSSS bridge. Finally, the study reveals among others that in order to prevent deck unseating resulting from restrainer failures and subsequent bridge collapse, particular attention should also be given to proper design of nonlinear characteristics of restrainers and bearing devices.

Design and development of low-cost HDRBs seismic isolation of structures

This paper presents a simplified procedure for the preliminary seismic design of high damping rubber bearings (HDRBs) for seismic isolation of structures. The procedure is consistent with the ASCE 7-16 specified seismic design for isolated structure, and starts defining the seismic base shear coefficient (CS) demand for superstructure, reduced by an appropriate response modification factor (RI) for isolated structure. The target time period for isolated structure is identified from the (CS/RI) versus time period plot, corresponding the available base shear capacity of structure. The required lateral shear stiffness of HDRBs is calculated, assuming a reasonable height-to-length ratio of bearing. The preliminary design suggests the required dimensions (height, width and length) of bearing, required thickness of elastomers and number of steel shim plates. The proposed procedure was validated for the seismic isolation of three-story deficient RC frame structure, located in a high seismic zone with a design basis PGA of 0.44 g. Representative reduced scale bearings were locally produced and tested on shake table, using an acceleration time history record of 1994 Northridge earthquake, through multi-levels excitations ranging from 0.10 to 1.0 g. The hysteretic and shear force–deformation response curves were analysed to develop constitutive relationship and hysteretic model for HDRBs. The derived hysteretic model parameters were incorporated in SeismoStruct, using plastic and bi-linear hysteretic models, for testing and validation against the shake table tests. The numerically predicted to experimentally observed peak displacements/forces exhibited a little over-prediction using bi-linear hysteretic model while a little under-prediction using plastic hysteretic model. Combining results from both the hysteretic models, and averaging, minimized the average error to less than 0.10%. Comparative seismic analysis of both as-built and isolated frames was carried out for both design basis earthquakes DBE and maximum considered earthquake MCE (considered as 3/2 of DBE), showing promising behaviour of the simplified design procedure and locally produced HDRBs.

Fragility analysis of nonproportionally damped inelastic MDOF structural systems exposed to stochastic seismic excitation

A novel methodology for conducting efficiently fragility analysis considering nonproportionally damped inelastic multi degree-of-freedom (MDOF) structural systems subject to stochastic seismic excitations defined by an advanced stochastic model consistent with magnitude-epicentral distance earthquake properties is developed. To this aim, an approximate stochastic dynamics technique for determining the system response amplitude probability density functions (PDFs) is developed. Firstly, relying on statistical linearization and state-variable formulation the complex eigenvalue problem is addressed through the time-domain. Secondly, utilizing the forced vibrational modal properties of the linearized MDOF system in conjunction with a combination of deterministic and stochastic averaging treatment, the MDOF system modal response amplitude process PDFs are determined. The modal participation factors are then defined for the complex-valued mode shapes and the total response amplitude process PDFs are provided in physical coordinates. Subsequently, appropriate limit states are related with the higher order statistics of the engineering demand parameters (i.e. that of the PDF) for quantifying structural system related fragilities. Nevertheless, due to the vector-valued nature of the adopted intensity measure, depicting system fragilities takes the form of three-dimensional fragility surfaces. The associated low computational cost renders the proposed methodology particularly useful for efficient structural system fragility analysis and related performance-based engineering design applications. A multi-storey building structure comprising the bilinear hysteretic model serves as a numerical example for demonstrating the reliability of the proposed fragility analysis methodology. Nonlinear response time-history analysis involving a large ensemble of compatible accelerograms is conducted to assess the accuracy of the proposed methodology in a Monte Carlo-based context.

Mechanical behavior of lead rubber bearings under and after nonproportional plane loading

SummaryLead rubber bearings, which have been extensively applied in many seismic isolation designs for buildings, infrastructures, and facilities worldwide, were tested under unilateral reversal loading as well as nonproportional plane loading including circular, figure‐eight, and square orbits in this study. The test results indicate that unlike the unilateral hysteretic behavior, the bilateral one of lead rubber bearings is too complicated to be characterized adequately by a simplified bilinear hysteretic model. It is mainly attributed to the bilateral coupling effect, which can be clearly observed from the abnormal deformation of the mesh pattern drawn on the rubber cover during the tests. In addition, after being subjected to nonproportional plane loading, the tested bearings reveal visible permanent twisting deformation. The profiles of the cut bearings present the fracture of the inside lead plugs. Even so, the further unilateral reversal loading test results prove that the fracture might not affect the whole hysteretic behavior and mechanical properties very much. The applicability, robustness, and generalization of adopting three previously developed analytical models for describing the coupled bilateral hysteretic behavior of lead rubber bearings are further demonstrated by comparing their predictions with the nonproportional plane loading test results. Although the coefficients are identified from unilateral reversal loading tests, the three analytical models can still have an acceptable prediction capability.

Uncertainties in dynamic response of buildings with non-linear base-isolators

Dynamic response of base-isolated buildings under uni-directional sinusoidal base excitation is numerically investigated considering uncertainties in the isolation and excitation parameters. The buildings are idealized as single degree of freedom (SDOF) system and multi-degrees of freedom (MDOF) system with one lateral degree of freedom at each floor level. The isolation system is modeled using two different mathematical models such as: (i) code-recommended equivalent linear elastic-viscous damping model and (ii) bi-linear hysteretic model. The uncertain parameters of the isolator considered are time period, damping ratio, and yield displacement. Moreover, the amplitude and frequency of the sinusoidal base excitation function are considered uncertain. The uncertainty propagation is investigated using generalized polynomial chaos (gPC) expansion technique. The unknown gPC expansion coefficients are obtained by non-intrusive sparse grid collocation scheme. Efficiency of the technique is compared with the sampling method of Monte Carlo (MC) simulation. The stochastic response quantities of interest considered are bearing displacement and top floor acceleration of the building. Effects of individual uncertain parameters on the building response are quantified using sensitivity analyses. Effect of various uncertainty levels of the input parameters on the dynamic response of the building is also investigated. The peak bearing displacement and top floor acceleration are more influenced by the amplitude and frequency of the sinusoidal base excitation function. The effective time period of the isolation system also produces a considerable influence. However, in the presence of similar uncertainty level in the time period, amplitude and frequency of the sinusoidal forcing function, the effect of uncertainties in the other parameters of the isolator (e.g., damping ratio and yield displacement) is comparatively less. Interestingly, the mean values of the response quantities are found to be higher than the deterministic values in several instances, indicating the need of conducting stochastic analysis. The gPC expansion technique presented here is found to be a computationally efficient yet accurate alternative to the MC simulation for numerically modeling the uncertainty propagation in the dynamic response analyses of the base-isolated buildings.

Responses of Building Base Isolated With High Damping Rubber Bearings

Seismic isolation is one of the most efficient techniques to protect structures against earthquakes. Rubber bearings are suitable for low-rise and medium-rise buildings due to its durability and easy fabrication. This paper presents the hori-zontal response of a six-storey base-isolated building using high damping rubber bearings (HDRB) under two ground motions of earthquakes as types I and II in JRA (2002) by finite element analysis. In this analysis, these bearings are mod-elled by the bilinear hysteretic model which is indicated in JRA and AASHTO. Comparison of horizontal response including base shear force and roof level acceleration between the two cases: base-isolated building and fixed-base building is carried out to evaluate the effectiveness of the use of HDRB on the protection of buildings from earthquakes. The numerical results show that the peak value of roof floor acceleration of the fixed-base building is two times higher than that of the base-isolated building, and the floor accelerations depend on the peak values of ground acceleration. In addition, the step-by-step design procedure for deter-mining the size of HDRBs used for buildings is also presented in this paper.

Nonlinear oscillations and dynamic stability of an elastoplastic pyramidal truss

Pyramidal space trusses are currently used in many present-day engineering applications, either as main parts or as a constitutive element. The present work studies, using a corotational finite element formulation considering large displacements and rotations and an elastoplastic material behavior with isotropic strain hardening, the nonlinear dynamic behavior of a pyramidal space truss. A bilinear hysteretic model with kinematic hardening is used to represent the material nonlinearity. An appropriate strain measure in such processes is the logarithmic strain. However, the engineering strain and the quadratic strain measures have been usually used to perform the elastic stability analysis of such structures. Thus, in the present work results using both the quadratic and the logarithmic strain measures are compared. First, the static nonlinear behavior is investigated in order to understand the influence of nonlinearities and static preload on the potential energy landscape, which controls the dynamics of the truss. These structures exhibit a two-well potential function that leads to coexisting in-well and cross-well solution branches, resulting in several coexisting periodic, quasi-periodic and chaotic attractors. The influence of the constitutive law, strain measure, truss geometry and support flexibility on the static stability is then investigated. Based on these results, a detailed parametric analysis of the nonlinear truss response under vertical load is conducted using bifurcations diagrams of the Poincare map to investigate the effect of the constitutive law, strain measure, truss geometry and load control parameters on the bifurcation scenario, coexisting attractors and dynamic buckling loads. The effect of the strain hardening parameter is particularly emphasized, allowing a more comprehensive picture of the dynamical behavior exhibited by the bistable system. Properties of the response are further illustrated by samples of time and phase plane responses and the related Poincare section plots. In addition to known behaviors, a rich class of solutions and bifurcations, including jump phenomena, symmetry-breaking, period-doubling cascades, fold and chaos is detected.

Input energy spectra for self-centering SDOF systems

This study proposes a procedure for determining design input energy spectra for self-centering single-degree-of-freedom (SDOF) systems using a characteristic flag-shaped hysteretic model. The procedure for determining design input energy spectra considers the influence of ground motion factors, hysteretic models, damping ratios, and ductility factors. First, nonlinear dynamic time history analyses are performed using 40 representative ground motion records to establish the relationship between input energy spectra and critical parameters. Based on this, normalized design input energy spectra and relevant parameters are proposed. The results of this study indicate that the bilinear hysteretic model is not suitable to accurately predict the input energy spectra for self-centering systems. Meanwhile, both the damping ratio and ductility factor have a significant influence on the resulting input energy spectra throughout the entire period range. The proposed procedure can be used to predict earthquake input energy spectra for self-centering SDOF systems.