Post-yield Stiffness Ratio Research Articles

Residual Displacement Demand Evaluation from Spectral Displacement

In this study, residual displacement demands are investigated for SDOF systems with period range of 0.1-3.0 s for near-field and far-field ground motions. The effects of stiffness degradation and post yield stiffness ratio on residual displacements are investigated. The modified-Clough model is used to represent structures that exhibit significant stiffness degradation when subjected to reverse cyclic loading. An elastoplastic model is used to represent non-degrading structures. For inelastic time history analyses, Newmark’s step by step time integration method was adapted in an in-house computer program. Based on time history analyses, a new simple equation is proposed for residual displacement demand of a system as a function of structural period (T), ductility (µ), strain hardening ratio (α) and spectral displacement (Sd).

Analytical hysteretic model for reinforcing bars

This paper proposes an analytical hysteretic model for reinforcing bars considering the effects of inelastic buckling and low-cycle fatigue. A strain-dependent post-yield stiffness ratio formula is introduced into the traditional Bouc-Wen model to make it possess the asymmetrical characteristic, so that the compressive stress softening effect generally exhibited by buckling bars can be reasonably described. Meanwhile, the low-cycle fatigue degradation is also contained in the model. Based on the collected experimental hysteresis loops of reinforcing bars, the model control parameters are identified using the genetic algorithm. The corresponding prediction equations of control parameters in terms of physical parameters of the reinforcing bar, such as the slenderness ratio, yield strength and ultimate strength-to-yield strength ratio, are then established by regression analysis. Furthermore, the proposed model is implemented in OpenSees, an open-source computational platform. The effectiveness of the proposed model is verified by performing the comparison with the experimental results of both isolated reinforcing bars and reinforced concrete columns.

Constant Ductility Site‐Specific Yield Point Spectra for Seismic Design

Displacement‐based seismic design (DBSD) is an iterative process because the strength and stiffness of a structure are needed to be adjusted in order to achieve a specific performance level, which is extremely inconvenient for designers in practice. Yield point spectra‐based seismic design is treated as an alternative design method in which yield displacement as a basic parameter will not lead to an iterative process even though the lateral strength or stiffness of a structure changes during the whole design process. Along this line, this study focuses on investigating the yield point spectra (YPS) for structures located at different soil sites. YPS are computed for EPP systems under 601 earthquake ground motions. YPS for four soil sites are quantitatively analyzed by considering the influence of the vibration period, ductility factor, damping ratio, postyield stiffness ratio, and P‐delta effect. The results indicate that compared with the effects of the damping ratio, the effects of the postyield stiffness ratio and P‐delta effect on YPS are more profound. Finally, a prediction equation is proposed accounting for four soil sites and six ductility factors.

Constant‐ductility inelastic displacement ratios for displacement‐based seismic design of self‐centering structures

SummaryThis paper focuses on constant‐ductility inelastic displacement ratios of self‐centering single‐degree‐of‐freedom (SDF) systems with two different levels of energy dissipation capacity, in the presence of 5% viscous damping ratio. A statistical analysis is developed considering an earthquake database composed of 228 ground motions recorded in California with magnitudes greater than six and organized for NEHRP soil class, ground motion duration, and peak ground acceleration. The response of self‐centering SDF systems with large variability of initial periods, ductility levels, and postyield stiffness ratios is investigated and compared with the responses of SDF systems with bilinear plastic, Clough, and Takeda hysteresis. The inelastic demand variation with soil class, initial period, postyield stiffness ratio, unloading stiffness degradation, ductility level, and hysteretic behavior is highlighted. Simple and conservative analytical estimates of constant‐ductility inelastic displacement ratios for mean and 90th percentile values in terms of initial period, ductility level, and postyield stiffness ratio are proposed to allow the extension of the Displacement‐Based Design via Inelastic Displacement Ratio (CμDBD) to self‐centering structural systems.

Residual displacement ratios of SDOF systems subjected to ground motions recorded on soft soils

120 earthquake ground motions recorded on soft soil sites were employed to assess, through response history analysis of simplified systems, the residual displacement demand, Cr, defined as the ratio of the residual displacement to the maximum elastic displacement. Single degree of freedom systems were considered and the lateral strength ratio was parametrized to account for varying structural inelasticity. Four hysteretic laws were chosen to represent degrading and non-degrading performance while variation in the post-yield stiffness was considered. The analysis scheme enabled assessing the relationship of the residual displacement with the aforementioned structural characteristics. The residual displacement demand was found to be sensitive in the post-yield stiffness ratio. An equation was finally introduced to accommodate the reliable estimation of residual displacement ratio, the latter being beneficial for evaluating the seismic performance of existing structures built on soft soil sites.

A Simple Analysis Method for Estimation of Required Damping Ratio for Seismic Response Reduction

This paper proposes a simple analysis method to estimate the additional damping ratio required to achieve a target seismic performance level. The proposed method employs the Newmark and Hall′s design response spectrum and the simple version of the N2 method. With the help of combining the characteristic of the two methods, it is possible to approximately estimate additional damping ratios required to achieve target elastic or inelastic seismic responses of a building structure without iteration. To examine the validity and effective range of the proposed method, time history response analysis is conducted, and the results are compared with those of a numerical analysis in which the post-yield stiffness ratio is considered. The analytical results show that the post-yield stiffness ratio and the yield displacement are important factors that affect the validity of the proposed analysis method; it is also found that the proposed simple analysis procedure may be quite efficient in evaluating the performance point and the damping requirements for seismic response reduction of building structures.

Probabilistic inelastic seismic demand spectra for large-span planar steel structures subjected to vertical ground motions

Severe vertical ground motions (VGMs) may lead to detrimental seismic damages of large-span planar steel structures (LSPSSs), thus the inelastic seismic demand of LSPSSs under VGMs needs to be quantified so as to ensure the structural seismic reliabilities. In view that the uncertainties of VGMs greatly influence the seismic responses of LSPSSs, the VGMs-induced seismic demands of LSPSSs are investigated in a probabilistic way in this paper. Based on 680 strong VGM records, the vertical ductility demands (μ) of 2,100 equivalent single-degree-of-freedom (ESDF) models representing a series of LSPSSs are computed. It is revealed that the influences of VGM properties, including peak ground acceleration, epicentral distance, hypocenter depth, moment magnitude and local site condition, are tiny and irregular on the values of μ. Accordingly, the probabilistic seismic demand model for LSPSSs under VGMs is established regardless of the VGM properties in this paper. From the 1,428,000 computed values of μ, the VGMs-induced seismic demand of LSPSSs follows a positively skewed probabilistic distribution, and the distribution could be well fitted by the lognormal distribution model. The parameters of the lognormal distribution model for μ are simulated by two elementary functions subsequently, in which the effects of vertical strength reduction factor R, post-yield stiffness ratio α and elastic vibrating period T are accounted for. Using the lognormal model and the proposed functions, a group of probabilistic inelastic seismic demand spectra for LSPSSs under VGMs are generated. The established probabilistic spectra could provide the statistic properties of both the peak and the residual seismic demand of LSPSSs in association with pre-set values of T, R and α. Combining the proposed model with certain seismic hazard models that define the probabilistic characteristics of VGMs, the seismic reliability of LSPSSs could be quantified, and a proper structural seismic performance could be guaranteed.

A performance-based damage-control design procedure of hybrid steel MRFs with EDBs

A noteworthy feature of the hybrid steel moment resisting frames (MRFs) with energy dissipation bays (EDBs) is the damage-control behaviour characterised by concentration of plastic damages in the energy dissipation bay (EDB) under earthquakes. This paper presents a design methodology for conducting the damage-control design of hybrid steel MRFs with EDBs. First, the structural damage-control behaviour quantified by the classical bilinear kinematic hysteretic model with significant post-yielding stiffness ratio is clarified utilising the test results extracted from a large-scale quasi-static test programme. Then, based on the seismic energy balance of single-degree-of-freedom systems incorporating significant post-yielding stiffness ratios, the design philosophy and governing energy balance equations featuring the damage-control behaviour of low-to-medium rise hybrid steel MRFs with EDBs under earthquake ground motions are presented. Subsequently, a stepwise design procedure that can be used to search for a design strategy of a hybrid steel MRF with EDBs under expected ground motions is developed. Three low-to-medium rise prototype structures are designed by the proposed methodology, and the seismic responses of the systems are evaluated by pushover analyses and nonlinear response history analyses based on numerical models validated by the test results. The results indicate that all the prototype hybrid steel MRFs with EDBs can achieve the damage-control behaviour with the prescribed drift threshold, and hence the post-earthquake residual deformations are also mitigated. Since the proposed method is a direct-iterative design procedure, it also retains practical attractiveness and will facilitate the seismic design of hybrid steel MRFs with EDBs.

A simplified analytical model of dynamic behavior of hybrid coupled wall systems with steel coupling beams

This paper proposes a simplified analytical model to estimate the seismic response of hybrid coupled walls (HCW). The model tends to consider the inelastic behavior of steel coupling beams (SCB) and elastic behavior of wall piers. The two wall piers are simplified to one cantilever beam with equivalent stiffness. The steel coupling beams were simulated by rotational springs, and their actions on the walls were represented by concentrated moments distributed along the height. A bilinear hysteretic model is used to represent the inelastic behavior of SCBs. The Rayleigh-Ritz method is adopted to determine the seismic response of the simplified model. The performance of the simplified model is verified against finite element analysis of a 20-storey HCW building. The comparison shows that the simplification is reasonable to capture the main dynamic characteristics of HCW systems. Parametric studies are conducted and the results show that the yielding level and post-yielding stiffness of SCBs have significant influence on the dynamic behavior of HCW systems. To achieve the optimum performance, it is recommended to reduce the yielding level of SCBs by 40–60%, and to design the post-yielding stiffness ratio of SCBs in the range of 0.4–0.6.

Vertical ductility demand and residual displacement of roof-type steel structures subjected to vertical earthquake ground motions

The vertical component of seismic ground motions (VGM) may induce severe responses in roof-type structures such as mega canopies, large floors, etc. This paper proposes a ductility-based approach for estimating the inelastic responses of roof-type steel structures under VGMs. Due to the pre-loaded gravity, a roof-type steel structure exhibits obvious asymmetric hysteresis properties in the vertical direction. Accordingly, this kind of structure could be idealized as “one-side yielding” systems which only yield in the lower side of vibration, wherein the gravity effect and the VGM effect positively super-positions with each other. Based on the idealized model, the vertical ductility demand spectra and the vertical residual displacement spectra for roof-type steel structures are computed using 53 recorded strong near-fault VGMs. Taking into account the vertical strength reduction factor and the vertical post-yielding stiffness ratio, a vertical ductility demand model in the μ-T format is proposed for roof-type steel structures under VGMs. Meanwhile, the vertical residual displacement model is derived from the vertical ductility demand model, since the vertical residual displacement could be computed as the peak vertical displacement subtracted by the recoverable elastic displacement. An application of these proposed models is given in the paper. The vertical seismic responses of a large-span steel floor under VGMs are predicted based on the proposed models, and the obtained results (including both the peak and the residual responses in the vertical direction) show good accuracy as compared with the results given by nonlinear response history analysis.

Estimation of inelastic displacement ratio for base‐isolated structures

SummaryThis study develops a straightforward approximate method to estimate inelastic displacement ratio, C1 for base‐isolated structures subjected to near‐fault and far‐fault ground motions. Taking into account the inelastic behavior of isolator and superstructure, a 2 degrees of freedom model is employed. A total of 90 earthquake ground motions are selected and classified into different clusters according to the frequency content features of records represented by the peak ground acceleration to peak ground velocity ratio, Ap/Vp. A parametric study is conducted, and effective factors in C1 (i.e., fundamental vibration period of the superstructure, Ts; postyield stiffness ratio of the superstructure, αs; strength reduction ratio, R; vibration period of the isolator, Tb; strength of the isolator, Q; ratio of superstructure mass to total mass of the system, γm) are recognized. The results indicate that the practical range of C1 values could be expected for base‐isolated structures. Subsequently, effective parameters are included in simple predictive equations. Finally, the accuracy of the proposed approximate equations is evaluated and verified through error measurement, and comparisons are made in the analyses.

The Effect of LRB Parameters on Structural Responses for Blast and Seismic Loads

In the present paper, passive control technique such as base isolation system is studied under earthquake ground motions and underground blast-induced vibrations. The performance of the lead rubber bearing (LRB), idealized as Bouc–Wen model in mitigating the structural responses of a five-storey building model, is investigated. The earthquake ground motions are selected from ground motion database available on the portal COSMOS Virtual Data Centre, whereas the underground blast is modeled as an exponential decaying function as prescribed by Carvalho and Battista (Proc Inst Civ Eng Struct Build 156(3):243–253, 2003). The aim of the study is to analyze the effect of isolation parameters such as damping ratio, yield strength, post-yield stiffness ratio and yield displacement on the structural responses of the base-isolated building. Newmark’s step-by-step integration method is adopted to evaluate the structural responses of the building. It is observed that the LRB is very effective in reducing the structural accelerations and storey drifts induced in the building due to ground-induced vibration. The comparison of results show that high value of yield strength harvests low bearing displacement and low percentage reduction in the top floor absolute acceleration. In addition, the study also evaluates the energy dissipated by the isolated structure. The energy dissipated by the base-isolated (LRB) building subjected to blast-induced vibrations, show that an optimum value of yield strength is found to be in the range of 10–20% of the total weight of the structure.

Estimating the response of steel structures subjected to vertical seismic excitation: Idealized model and inelastic displacement ratio

Vertical seismic excitations (VSEs) could induce severe damage to building structures and infrastructural facilities, especially to those with large-spans or long cantilevers. Thus, there is a need in engineering practice for estimating the seismic response of structures under VSEs. In this paper, a tentative investigation on VSEs-induced inelastic response of steel structures is proposed. The structure is transformed into an equivalent single-degree-of-freedom (ESDF) system in an extended pushover process, in which the vertical component of response is included. Considering the influence of gravity, a novel ESDF model with asymmetric vibrating characteristic is established. In this ESDF model, the inelastic response only occurs at the “lower” side of vibration, whereas at the other side, the system keeps elastic. Based on the idealized ESDF model and 53 recorded strong VSEs, the vertical inelastic displacement ratio (IDR) is computed. The averaged IDRs are plotted against the elastic vibrating period T, and fitted by a piece-wise function to generate the IDR spectra, in which the effects of strength reduction factor R and of post-yield stiffness ratio α are accounted for by fitting coefficients. The overall process for estimating the VSE-induced inelastic responses of steel structures, using the extended pushover procedure and the proposed IDR spectra, is summarized. In the numerical example, seismic responses of a large-span space truss subjected to VSEs are computed by means of the nonlinear RHA method and the proposed IDR-based method, respectively, the results demonstrate that the IDR-based method could yield an anticipative estimation of structural responses under severe VSEs. For mid-span displacement, error of the IDR-based method is less than 13.6%; for quantity of yielded member, error is less than 14.2%.

Full-scale seismic testing of concrete building columns reinforced with both steel and CFRP bars

This paper presents an experimental study of an innovative concrete building column which is longitudinally reinforced with both steel bars and fiber-reinforced polymer (FRP) composite bars. The main objectives of this study include: (1) investigating the seismic failure mechanism of such FRP-steel reinforced concrete (FSRC) columns under relatively large gravity load and (2) analyzing the effects of the FRP bars on the seismic performance of the FSRC columns. To this end, a total of six full-scale cantilever columns with variable carbon FRP (CFRP) bar and steel reinforcement ratios were tested under combined constant axial load and lateral displacement reversals. The specimens had 400mm square sections and effective heights of 1800mm. Test results showed that adding additional CFRP bars into the conventional steel reinforced concrete (SRC) columns was efficient in improving the post-yield stiffness ratios and mitigating the residual displacements, while the hysteretic energy dissipation could be maintained. Failure modes of the tested FSRC specimens were characterized by crushing of FRP bars and buckling of steel bars at drift ratios larger than 2.4%. Furthermore, confining the FSRC column with external CFRP wraps was effective in delaying the crushing of internal CFRP bars and reducing the post-earthquake residual displacement.

Evaluating the Effect of Buckling-Restrained Brace Model on Seismic Structural Responses

Numerical simulation is an important measure to study the seismic performance of buckling-restrained braced frames (BRBFs). Practically, some simplified models, such as the elastic–plastic with kinematic hardening model and the elastic perfectly-plastic model, are used to simulate the behavior of buckling-restrained brace (BRBs). To provide structural engineers the reference of errors when simplified models are used, this paper comparatively evaluates the effect of the BRB model on seismic structural responses using the OpenSees software. A comparison is made on six-storey and 16-storey BRBFs with rigid beam-to-column connections; these are designed according to Chinese seismic design code. Moreover, the effects of the post-yielding stiffness ratio of frame [Formula: see text] and the stiffness ratio of BRB to frame [Formula: see text] on the errors are specifically investigated through a parametric study of both BRBFs. The results show that the seismic response average errors of the simplified models are mostly less than 5%, which satisfies the engineering requirements.

Residual displacement demands of conventional and dual oscillators subjected to earthquake ground motions characteristic of the soft soils of Mexico City

This paper presents a statistical evaluation of residual displacement (RDs) demands in conventional and dual single-degree-of-freedom (SDOF) oscillators subjected to a set of 220 earthquake ground motions recorded at soft soil sites of Mexico City. A dual SDOF oscillator represents a highly-dissipating energy system (e.g. buckling-restrained braces, BRBs) acting in parallel with a conventional system (e.g. flexible moment-resisting frames). To provide a context, RDs were normalised with respect to the corresponding: a) maximum transient displacements, and b) elastic spectral displacement. The effects of post-yield stiffness ratio (as affected by P-Δ effects), normalised period of vibration with respect to the predominant period of the ground motion, the type of hysteretic response, maximum displacement ductility, lateral strength ratio, type of transition (from elastic-to-plastic response), and damping ratio on normalised RDs were examined for the conventional oscillators. In addition, the effects of the stiffness ratio, lateral strength ratio, displacement ductility, post-yielding stiffness ratio, and type of hysteretic response of the primary and secondary parts of dual SDOF oscillators on normalised RDs were also evaluated. From the results of this investigation, it was observed that for dual SDOF systems the amplitude of normalised RDs is small when the primary part remains elastic. On the contrary, if the primary part exhibits inelastic response, normalised RDs might increase significantly and the post-yielding stiffness ratio of the secondary part plays a key role for constraining them (i.e. while a positive value reduces RDs significantly; a negative value is highly detrimental). Also, it was found that the type of hysteretic response of the primary part of a dual system has a significant effect on normalised RDs (e.g. it was found that a primary part with self-centring capacity, acting in parallel with a highly dissipation system (e.g. BRBs) as secondary part, is very effective for diminishing RDs). A discussion section is also offered to highlight the new findings of this study and differences on normalised RD demands between soft and stiff soils.

Influence of modeling assumptions and aftershock hazard level in the seismic response of post-mainshock steel framed buildings

Seismic response under strong aftershocks has become an important topic in performance-based earthquake engineering recently. This paper examines the influence of the modeling approach in the response of a case-study four-story steel framed building designed with modern seismic provisions when subjected to two sets of aftershocks representing different aftershock-hazard levels. It was assumed that the building models experienced specific post-mainshock residual drifts prior of being subjected to the aftershocks. Results shown that including additional sources of stiffness and strength (i.e. inclusion of the interior gravity framing and participation of the slab in the beams) in the building analytical model lead to a different response under aftershocks in terms of maximum and residual interstory drifts than that of the analytical model commonly used in previous studies. Particularly, additional overstrength and post-yield stiffness ratio help to constrain residual drift demands under strong aftershocks, and decrease the probability of imminent demolition due to excessive post-mainshock residual drifts.

Moment-Curvature Behaviors of Concrete Beams Singly Reinforced by Steel-FRP Composite Bars

A steel-fiber-reinforced polymer (FRP) composite bar (SFCB) is a kind of rebar with inner steel bar wrapped by FRP, which can achieve a better anticorrosion performance than that of ordinary steel bar. The high ultimate strength of FRP can also provide a significant increase in load bearing capacity. Based on the adequate simulation of the load-displacement behaviors of concrete beams reinforced by SFCBs, a parametric analysis of the moment-curvature behaviors of concrete beams that are singly reinforced by SFCB was conducted. The critical reinforcement ratio for differentiating the beam’s failure mode was presented, and the concept of the maximum possible peak curvature (MPPC) was proposed. After the ultimate curvature reached MPPC, it decreased with an increase in the postyield stiffness ratio (rsf), and the theoretical calculation method about the curvatures before and after the MPPC was derived. The influence of the reinforcement ratio, effective depth, and FRP ultimate strain on the ultimate point was studied by the dimensionless moment and curvature. By calculating the envelope area under the moment-curvature curve, the energy ductility index can obtain a balance between the bearing capacity and the deformation ability. This paper can provide a reference for the design of concrete beams that are reinforced by SFCB or hybrid steel bar/FRP bar.

Experimental study on concrete columns reinforced by hybrid steel-fiber reinforced polymer (FRP) bars under horizontal cyclic loading

Effective post-yield stiffness of reinforced concrete (RC) columns can significantly contribute to the seismic performance of RC structures. However, because of the elastoplastic properties of steel bars, the post-yield stiffness of an ordinary RC column can be very slight or even negative. Fiber reinforced polymer (FRP) can provide a high degree of ultimate strength, light weight, and protection from corrosion. By combining steel and FRP, a designable post-yield stiffness can be achieved for concrete structures reinforced with steel-FRP composite bars (SFCBs) or hybrid steel/FRP bars. This paper conducted cyclic loading tests on four concrete columns with different reinforcement types, including steel bars, hybrid steel/FRP bars, and SFCBs. The test results showed that (1) the columns reinforced with different bars had similar strain distributions from column base to cap prior to yielding. After yielding, the plastic deformation of the ordinary RC column concentrated at the column base and the loading capacity decreased with the increase of lateral drift because of the P-δ effect. (2) Unlike the negative post-yield stiffness of an ordinary RC column, the post-yield stiffness of a column with hybrid reinforcements was positive. As the post-yield stiffness ratio of the longitudinal reinforcement increased by 27 percent, the post-yield stiffness of the concrete column increased by 7.4 percent. Therefore, the corresponding displacement ductility could reach 11—much greater than that of the RC column (6.28). (3) As a result of the more robust hysteretic curve of the RC column, the equivalent viscous damping coefficients of the RC column were greater than those of the hybrid column, whereas the hybrid reinforced concrete columns could dissipate earthquake energy without a corresponding loss of strength.

Energy-factor-based damage-control evaluation of steel MRF systems with fuses

The primary objectives of this research are to investigate the energy factor response of steel moment resisting frame (MRF) systems equipped with fuses subject to ground motions and to develop an energy-based evaluation approach for evaluating the damage-control behavior of the system. First, the energy factor of steel MRF systems with fuses below the resilience threshold is derived utilizing the energy balance equation considering bilinear oscillators with significant post-yielding stiffness ratio, and the effect of structural nonlinearity on the energy factor is investigated by conducting a parametric study covering a wide range of parameters. A practical transformation approach is also proposed to associate the energy factor of steel MRF systems with fuses with classic design spectra based on elasto-plastic systems. Then, the energy balance is extended to structural systems, and an energy-based procedure for damage-control evaluation is proposed and a damage-control index is also derived. The approach is then applied to two types of steel MRF systems with fuses to explore the applicability for quantifying the damagecontrol behavior. The rationality of the proposed approach and the accuracy for identifying the damage-control behavior are demonstrated by nonlinear static analyses and incremental dynamic analyses utilizing prototype structures.