Scaling of strength reduction factors for degrading elasto‐plastic oscillators

The inelastic (design) spectra characterizing a seismic hazard are generally obtained by the scaling‐down of the elastic (design) spectra via a set of response modification factors. The component of these factors, which accounts for the ductility demand ratio, is known as the strength reduction factor (SRF), and the variation of this factor with initial period of the oscillator is called an SRF spectrum. This study considers scaling of the SRF spectrum in the case of an elasto‐plastic oscillator with strength and stiffness degradation characteristics. Two models are considered: one depending directly on the characterization of source and site parameters and the other depending on the normalized design spectrum characterization of the seismic hazard. The first model is the same as that proposed earlier by the second author, and is given in terms of earthquake magnitude, strong‐motion duration, predominant period, geological site conditions, ductility demand ratio, and ductility supply‐related parameter. The second model is a new model proposed here in terms of the normalized pseudo‐spectral acceleration values (to unit peak ground acceleration), ductility demand ratio and ductility supply‐related parameter. For each of these models, least‐square estimates of the coefficients are obtained through regression analyses of the data for 956 recorded accelerograms in western U.S.A. Parametric studies carried out with the help of these models confirm the dependence of SRFs on strong‐motion duration and earthquake magnitude besides predominant period and site conditions. It is also seen that degradation characteristics make a slight difference for high ductility demands and may lead to lower values of SRFs, unless the oscillators are very flexible. Copyright © 2004 John Wiley & Sons, Ltd.     

Scaling of ductility and damage‐based strength reduction factors for horizontal motions

The conventional approach of obtaining the inelastic response spectra for the aseismic design of structures involves the reduction of elastic spectra via response modification factors. A response modification factor is usually taken as a product of (i) strength factor, R_S, (ii) ductility factor, R_μ, and (iii) redundancy factor, R_R. Ductility factor, also known as strength reduction factor (SRF), is considered to primarily depend on the initial time period of the single‐degree‐of‐freedom (SDOF) oscillator and the displacement ductility demand ratio for the ground motion. This study proposes a preliminary scaling model for estimating the SRFs of horizontal ground motions in terms of earthquake magnitude, strong motion duration and predominant period of the ground motion, geological site conditions, and ductility demand ratio, with a given level of confidence. The earlier models have not considered the simultaneous dependence of the SRFs on various governing parameters. Since the ductility demand ratio is not a complete measure of the cumulative damage in the structure during the earthquake‐induced vibrations, the existing definition of the SRF is sought to be modified with the introduction of damage‐based SRF (in place of ductility‐based SRF). A parallel scaling model has been proposed for estimating the damage‐based SRFs. This model considers damage and ductility supply ratio as parameters instead of ductility demand ratio. Through a parametric study on ductility‐based SRFs, it has been shown that the hitherto assumed insensitivity of earthquake magnitude and strong motion duration may not be always justified and that the initial time period of the oscillator plays an important role in the dependence of SRF on these parameters. Further, the damage‐based SRFs are found to show similar parametric dependence as observed in the case of the ductility‐based SRFs. Copyright © 2000 John Wiley & Sons, Ltd.     

Collapse period of degrading SDOF systems

Seismic demand estimation for a structure is a critical issue for seismic performance assessment so that the potential damage can be estimated realistically. Many researchers proposed simplified methods to estimate the demand of a structure under strong ground motions. However, most of them did not consider degradation and collapse potential of the structures. Even some of theme considered the degradation effect, stiffness and strength degradation effects were considered separately without collapse potential caused by dynamic instability. In this study, collapse potential of SDOF systems caused by dynamic instability with stiffness and strength degradation has been investigated. Nonlinear time history analyses were performed, using an energy-based, strength and stiffness degraded hysteretic model that considers the collapse potential, with 160 earthquake acceleration time histories. An equation was proposed for the estimation of collapse period of SDOF systems as a function of certain strength reduction factor, ductility level and post-capping stiffness ratio. Finally, effects of parameters of the considered hysteretic model and local site conditions on the collapse period were investigated.     

Inelastic displacement ratios for evaluation of structures built on soft soil sites

This paper summarizes the results of a comprehensive statistical study aimed at evaluating peak lateral inelastic displacement demands of structures with known lateral strength and stiffness built on soft soil site conditions. For that purpose, empirical information on inelastic displacement ratios which are defined as the ratio of peak lateral inelastic displacement demands to peak elastic displacement demands are investigated. Inelastic displacement ratios were computed from the response of single‐degree‐of‐freedom systems having 6 levels of relative lateral strength when subjected to 118 earthquake ground motions recorded on bay‐mud sites of the San Francisco Bay Area and on soft soil sites located in the former lake‐bed zone of Mexico City. Mean inelastic displacement ratios and their corresponding scatter are presented for both ground motion ensembles. The influence of period of vibration normalized by the predominant period of the ground motion, the level of lateral strength, earthquake magnitude, and distance to the source are evaluated and discussed. In addition, the effects of post‐yield stiffness and of stiffness and strength degradation on inelastic displacement ratios are also investigated. It is concluded that magnitude and distance to the source have negligible effects on constant‐strength inelastic displacement ratios. Results also indicate that weak and stiffness‐degrading structures in the short spectral region could experience inelastic displacement demands larger than those corresponding to non‐degrading structures. Finally, a simplified equation obtained using regression analyses aimed at estimating mean inelastic displacement ratios is proposed for assisting structural engineers in performance‐based assessment of structures built on soft soil sites. Copyright © 2006 John Wiley & Sons, Ltd.     

Yielding oscillator subjected to simple pulse waveforms: numerical analysis & closed‐form solutions

Numerical and analytical solutions are presented for the elastic and inelastic response of single‐degree‐of‐freedom yielding oscillators to idealized ground acceleration pulses. These motions are typical of near‐fault earthquake recordings generated by forward rupture directivity and may inflict damage in the absence of substantial structural strength and ductility capacity. Four basic pulse waveforms are examined: (1) triangular; (2) sinusoidal; (3) exponential; and (4) rectangular. In the first part of the article, a numerical study is presented of the effect of oscillator period, strength, damping, post‐yielding stiffness and number of excitation cycles, on inelastic response. Results are presented in the form of dimensionless graphs and regression formulas that elucidate the salient features of the problem. It is shown that conventional R–µ relations may significantly underestimate ductility demand imposed by near‐fault motions. The second part of the article concentrates on elastic‐perfectly plastic oscillators. Closed‐form solutions are derived for post‐yielding response and associated ductility demand. It is shown that all three ground motion histories (i.e. acceleration, velocity, and displacement) control oscillator response—contrary to the widespread view that ground velocity alone is of leading importance. The derived solutions provide insight on the physics of inelastic response, which is often obscured by the complexity of numerical algorithms and actual earthquake motions. The model is evaluated against numerical results from near‐field recordings. A case study is presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Inclusion of P–Δ effect in displacement‐based seismic design of steel moment resisting frames

A procedure for treating the P– Δ effect in the direct displacement‐based seismic design of regular steel moment resisting frames with ideal elastoplastic material behaviour is proposed. A simple formula for the yield displacement amplification factor as a function of ductility and the stability coefficient is derived on the basis of the seismic response of an inelastic single degree‐of‐freedom system taking into account the P– Δ effect. Extensive parametric seismic inelastic analyses of plane moment resisting steel frames result in a simple formula for the dynamic stability coefficient as a function of the number of stories of a frame and the column to beam stiffness ratio. Thus, the P– Δ effect can be easily taken into account in a direct displacement‐based seismic design through the stability coefficient and the yield displacement amplification factor. A simple design example serves to illustrate the application of the proposed method and demonstrate its merits. Copyright © 2007 John Wiley & Sons, Ltd.     

平面不对称结构非弹性地震反应分析的平扭耦联Bouc-Wen模型

综合考虑强度退化、刚度退化、捏拢效应等典型滞回特性的影响,建立了双向地震作用下平面不对称结构非弹性地震动力响应分析的平扭耦联Bouc-Wen模型.该模型采用圆形屈服面来描述双向抗侧恢复力之间的耦合效应,而采用锥体或球体屈服面来描述双向抗侧和抗扭恢复力之间的耦合效应,并结合69条强震记录,定量地分析了平扭耦联效应对平面不...     

Hysteretic model development and seismic response of unbonded post‐tensioned precast CFT segmental bridge columns

The study investigated the cyclic behavior of unbonded, post‐tensioned, precast concrete‐filled tube segmental bridge columns by loading each specimen twice. Moreover, a stiffness‐degrading flag‐shaped (SDFS) hysteretic model was developed based on self‐centering and stiffness‐degrading behaviors. The proposed model overcomes the deficiency of cyclic behavior prediction using a FS model, which self‐centers with fixed elastic and inelastic stiffnesses. Experimental and analytical results showed that (1) deformation capabilities of the column under the first and second cyclic tests were similar; however, energy dissipation capacities significantly differed from each other, and (2) the SDFS model predicted the cyclic response of the column better than the FS model. Inelastic time‐history analyses were performed to demonstrate the dynamic response variability of a single‐degree‐of‐freedom (SDOF) system using both models. A parametric study, performed on SDOF systems subjected to eight historical earthquakes, showed that increased displacement ductility demand was significant for structures with a low period and low‐to‐medium yield strength ratio and reduced displacement ductility demand in these systems was effectively attained by increasing energy dissipation capacity. Copyright © 2008 John Wiley & Sons, Ltd.     

Seismic design of yielding structures on flexible foundations

This paper introduces a simple method to consider the effects of inertial soil–structure interaction (SSI) on the seismic demands of a yielding single‐degree‐of‐freedom structure. This involves idealizing the yielding soil–structure system as an effective substitute oscillator having a modified period, damping ratio, and ductility. A parametric study is conducted to obtain the ratio between the displacement ductility demand of a flexible‐base system and that of the corresponding fixed‐base system. It is shown that while additional foundation damping can reduce the overall response, the effects of SSI may also increase the ductility demand of some structures, mostly being ductile and having large structural aspect ratio, up to 15%. Finally, a design procedure is provided for incorporation of the SSI effects on structural response. Copyright © 2015 John Wiley & Sons, Ltd.     

Damage index based on stiffness degradation of low‐rise RC walls

Widely used damage indices, such as ductility and drift ratios, do not account for the influences of the duration of strong shaking, the cumulative inelastic deformation or energy dissipation in structures. In addition, the formulation and application of most damage indices have until now been based primarily on flexural modes of failure. However, evidence from earthquakes suggests that shear failure or combined shear‐flexure behavior is responsible for a large proportion of failures. Empirical considerations have been made in this paper for evaluating structural damage of low‐rise RC walls under earthquake ground motions by means of a new energy‐based low‐cycle fatigue damage index. The proposed empirical damage index is based on the results of an experimental program that comprised six shake table tests of RC solid walls and walls with openings; results of six companion walls tested under QS‐cyclic loading were used for comparison purposes. Variables studied were the wall geometry, type of concrete, web shear steel ratio, type of web shear reinforcement, and testing method. The index correlates the stiffness degradation and the destructiveness of the earthquake in terms of the duration and intensity of the ground motions. The stiffness degradation model considers simultaneously the increment of damage associated to the low‐cycle fatigue, energy dissipation, and the cumulative cyclic parameters, such as displacement demand and hysteretic energy dissipated. Copyright © 2014 John Wiley & Sons, Ltd.     

Performance‐based seismic design via yield frequency spectra‡

Yield frequency spectra (YFS) are introduced to enable the direct design of a structure subject to a set of seismic performance objectives. YFS offer a unique view of the entire solution space for structural performance. This is portrayed in terms of the mean annual frequency (MAF) of exceeding arbitrary ductility (or displacement) thresholds, versus the base shear strength of a structural system having specified yield displacement and capacity curve shape. YFS can be computed nearly instantaneously using publicly available software or closed‐form solutions, for any system whose response can be satisfactorily approximated by an equivalent nonlinear single‐degree‐of‐freedom oscillator. Because the yield displacement typically is a more stable parameter for performance‐based seismic design compared with the period, the YFS format is especially useful for design. Performance objectives stated in terms of the MAF of exceeding specified ductility (or displacement) thresholds are used to determine the lateral strength that governs the design of the structure. Both aleatory and epistemic uncertainties are considered, the latter at user‐selected confidence levels that can inject the desired conservatism in protecting against different failure modes. Near‐optimal values of design parameters can be determined in many cases in a single step. Copyright © 2016 John Wiley & Sons, Ltd.     

Seismic performance of steel reinforced ultra high-strength concrete composite frame joints

To investigate the seismic performance of a composite frame comprised of steel reinforced ultra high-strength concrete (SRUHSC) columns and steel reinforced concrete (SRC) beams, six interior frame joint specimens were designed and tested under low cyclically lateral load. The effects of the axial load ratio and volumetric stirrup ratio were studied on the characteristics of the frame joint performance including crack pattern, failure mode, ductility, energy dissipation capacity, strength degradation and rigidity degradation. It was found that all joint specimens behaved in a ductile manner with flexural-shear failure in the joint core region while plastic hinges appeared at the beam ends. The ductility and energy absorption capacity of joints increased as the axial load ratio decreased and the volumetric stirrup ratio increased. The displacement ductility coefficient and equivalent damping coefficient of the joints fell between the corresponding coefficients of the steel reinforced concrete (SRC) frame joint and RC frame joint. The axial load ratio and volumetric stirrup ratio have less influence on the strength degradation and more influence on the stiffness degradation. The stiffness of the joint degrades more significantly for a low volumetric stirrup ratio and high axial load ratio. The characteristics obtained from the SRUHSC composite frame joint specimens with better seismic performance may be a useful reference in future engineering applications.     

Inelastic displacement demand of bridge columns considering shear–flexure interaction

Reinforced concrete bridge columns exhibit complex hysteretic behavior owing to combined action of shear, bending moment, and axial force under multi‐directional seismic shakings. The inelastic displacement of columns can be increased by shear–flexure interaction (SFI). This paper develops a simple yet reliable demand model for estimating the inelastic displacement and ductility based on the nonlinear time history analyses of 24 full‐size columns subject to a suite of near‐fault ground motions. A coupled hysteretic model is used to simulate the shear‐flexure interactive (SFI) behavior of columns and the accumulated material damage during loading reversals, including pinching, strength deterioration, and stiffness softening. Guided by rigorous dimensional analysis, the inelastic displacement responses of bridge columns are presented in dimensionless form showing remarkable order. A dimensionless nonlinearity index is derived taking into account of the column strength, ground motion amplitude, and softening or hardening post‐yield behavior. Strong correlation is revealed between the normalized inelastic displacement and the dimensionless structure‐to‐pulse frequency, the dimensionless nonlinearity index as well as the aspect ratio. Two regressive equations for displacement and ductility demands are proposed and validated against the simulation results. The SFI effects are discussed and included explicitly through the aspect ratio in the proposed model. This study offers a new way to realistically predict the inelastic displacement of columns directly from structural and ground motion characteristics. Copyright © 2010 John Wiley & Sons, Ltd.     

Collapse capacity of soil‐structure systems under pulse‐like earthquakes

In this paper, a comprehensive study is carried out to examine the possibility of dynamic instability produced in soil‐structure systems using an ensemble of 50 pulse‐like records. A number of structural models with various vibration periods varying from 0.1 to 2 s are used in this study. The superstructure is simulated as a non‐linear SDOF oscillator with a two‐segment backbone curve having negative post‐yield stiffness. The soil is idealized based on the cone model concept widely used for practical purposes. The results of this investigation demonstrate that as the pulse period increases, the collapse relative lateral strength ratio decreases and probability of dynamic instability enhances. Moreover, soil flexibility makes the system dynamically more unstable, and as the non‐dimensional frequency increases, the collapse relative lateral strength ratio highly reduces. Additionally, the aspect ratio has insignificant effects on the collapse relative lateral strength ratio. Furthermore, comparison of the collapse relative lateral strength ratios resulting from pulse‐like motions with those obtained from studies under non‐pulse‐like motions (Miranda and Akkar; FEMA 440) for fixed‐base conditions shows that high‐velocity pulses exacerbate the dynamic instability problem and decrease the collapse relative lateral strength ratio. Copyright © 2014 John Wiley & Sons, Ltd.     

Midbroken reinforced concrete shear frames due to earthquakes. A hysteretic model to quantify damage at the storey level

A nonlinear hysteretic model for the response and local damage analyses of reinforced concrete shear frames subject to earthquake excitation is proposed, and, the model is applied to analyse midbroken reinforced concrete (RC) structures due to earthquake loads. Each storey of the shear frame is represented by a Clough and Johnston hysteretic oscillator with degrading elastic fraction of the restoring force. The local damage is numerically quantified in the domain [0,1] using the maximum softening damage indicators which are defined in closed form based on the variation of the eigenfrequency of the local oscillators due to the local stiffness and strength deterioration. The proposed method of response and damage analyses is illustrated using a sample 5 storey shear frame with a weak third storey in stiffness and/or strength subject to sinusoidal and simulated earthquake excitations for which the horizontal component of the ground motion is modeled as a stationary Gaussian stochastic process with Kanai-Tajimi spectrum, multiplied by an envelope function.     

Nonlinear seismic analysis of reinforced concrete frames using the force analogy method

A method of nonlinear seismic analysis for RC framed structures considering full‐range factors, including stiffness and strength degradation, geometric nonlinearity, and structural member failure, is established based on the fundamental concept of the force analogy method. The strong material nonlinearity, large geometric deformation, and internal forces redistribution due to the member failure can be depicted by the proposed local plastic mechanisms, the rotation hinges at the member ends and the slide hinges assigned to the columns, of which the measurement relationships are moment versus plastic rotation and shear force versus shear plastic deformation, respectively. They are capable of evaluating the exact response of RC structures. Because only unchanging initial stiffness matrices are used through the whole computation process, the state‐space formulation was used for solving the equations of motion. The advantages of the force analogy method, such as high efficiency and stability, are still retained. The exactness of the proposed local plastic mechanisms is verified against a group of tests data, and the application of the proposed procedure is performed to an RC framed structure to simulate the full‐range nonlinear response by increasing the excitation step by step until failure of partial structural members appear. Copyright © 2014 John Wiley & Sons, Ltd.     

Influence of deterioration modelling on the seismic response of steel moment frames designed to Eurocode 8

This paper assesses the influence of cyclic and in‐cycle degradation on seismic drift demands in moment‐resisting steel frames (MRF) designed to Eurocode 8. The structural characteristics, ground motion frequency content, and level of inelasticity are the primary parameters considered. A set of single‐degree‐of‐freedom (SDOF) systems, subjected to varying levels of inelastic demands, is initially investigated followed by an extensive study on multi‐storey frames. The latter comprises a large number of incremental dynamic analyses (IDA) on 12 frames modelled with or without consideration of degradation effects. A suite of 56 far‐field ground motion records, appropriately scaled to simulate 4 levels of inelastic demand, is employed for the IDA. Characteristic results from a detailed parametric investigation show that maximum response in terms of global and inter‐storey drifts is notably affected by degradation phenomena, in addition to the earthquake frequency content and the scaled inelastic demands. Consistently, both SDOF and frame systems with fundamental periods shorter than the mean period of ground motion can experience higher lateral strength demands and seismic drifts than those of non‐degrading counterparts in the same period range. Also, degrading multi‐storey frames can exhibit distinctly different plastic mechanisms with concentration of drifts at lower levels. Importantly, degrading systems might reach a “near‐collapse” limit state at ductility demand levels comparable to or lower than the assumed design behaviour factor, a result with direct consequences on optimised design situations where over‐strength would be minimal. Finally, the implications of the findings with respect to design‐level limit states are discussed.     

Influence of connection typology on seismic response of MR‐Frames with and without ‘set‐backs’

The work presented is aimed at the investigation of the influence of beam‐to‐column connections on the seismic response of MR‐Frames, with and without ‘set‐backs’, designed according to the Theory of Plastic Mechanism Control. The investigated connection typologies are four partial strength connections whose structural details have been designed to obtain the same flexural resistance. The first three joints are designed by means of hierarchy criteria based on the component approach and are characterized by different location of the weakest joint component, leading to different values of joint rotational stiffness and plastic rotation supply and affecting the shape of the hysteresis loops governing the dissipative capacity. The last typology is a beam‐to‐column connection equipped with friction pads devoted to the dissipation of the earthquake input energy, thus preventing the connection damage. An appropriate modelling is needed to accurately represent both strength and deformation characteristics, especially with reference to partial‐strength connections where the dissipation of the earthquake input energy occurs. To this aim, beam‐to‐column joints are modelled by means of rotational inelastic springs located at the ends of the beams whose moment‐rotation curve is characterized by a cyclic behaviour which accounts for stiffness and strength degradation and pinching phenomena. The parameters characterizing the cyclic hysteretic behaviour have been calibrated on the base of experimental results aiming to the best fitting. Successively, the prediction of the structural response of MR‐Frames, both regular frames and frames with set‐backs, equipped with such connections has been carried out by means of both push‐over and Incremental Dynamic Analyses. Copyright © 2016 John Wiley & Sons, Ltd.     

Seismic behaviour of shallow foundations: Shaking table experiments vs numerical modelling

The capability of a simplified approach to model the behaviour of shallow foundations during earthquakes is explored by numerical simulation of a series of shaking table tests performed at the Public Works Research Institute, Tsukuba, Japan. After a summary of the experimental work, the numerical model is introduced, where the whole soil–foundation system is represented by a multi‐degrees‐of‐freedom elasto‐plastic macro‐element, supporting a single degree‐of‐freedom superstructure. In spite of its simplicity and of the large intensity of the excitation involving a high degree of nonlinearity in the foundation response, the proposed approach is found to provide very satisfactory results in predicting the rocking behaviour of the system and the seismic actions transmitted to the superstructure. The agreement is further improved by introducing a simple degradation rule of the foundation stiffness parameters, suitable to capture even some minor details of the observed rocking response. On the other hand, the performance of the model is not fully satisfactory in predicting vertical settlements. Copyright © 2007 John Wiley & Sons, Ltd.