Nonlinear response estimates of RC frames using linear analysis of SDOF systems

An approximate‐simple method for nonlinear response estimates of reinforced concrete frames subjected to near‐field and far‐field records is presented in this paper. The approximate method is based on equivalent single‐degree‐of‐freedom and linear multi‐degree‐of‐freedom models. In this procedure, the nonlinear maximum roof displacement is estimated using an effective period factor and elastic response spectrum with an equivalent damping. The effective period factor was proposed for far‐field and near‐field ground motion records. For regions of high seismicity, the maximum roof displacement can be estimated by applying an effective period factor of 2.3 and 2.1 for near‐field and far‐field records, respectively, and 9% damped displacement response spectrum. For regions of moderate seismicity, a lower effective period factor of 1.9 and 1.8, for near‐field and far‐field records, respectively, can be applied to estimate the maximum roof displacement. A relationship between linear and nonlinear response of multi‐degree‐of‐freedom systems was also proposed to obtain estimates of the maximum inter‐story drift of nonlinear responding reinforced concrete frames. In addition, the effects of number of ground motion records used in the analyses on the scatters of results were investigated. The required number of ground motions to produce a reliable response was proposed. Copyright © 2013 John Wiley & Sons, Ltd.     

Modelling damage potential of high‐frequency ground motions

Damage to building structures due to underground blast‐induced ground motions is a primary concern in the corresponding determination of the safe inhabited building distance (IBD). Because of the high‐frequency nature of this category of ground motions and especially the presence of significant vertical component, the characteristics of structural response and damage differ from those under seismic type low‐frequency ground motions. This paper presents a numerical investigation aimed at evaluating reinforced concrete (RC) structure damage generated by underground blast‐induced ground excitation. In the numerical model, two damage indices are proposed to model reinforced concrete failure. A fracture indicator is defined to track the cracking status of concrete from micro‐ to macrolevel; the development of a plastic hinge due to reinforcement yielding is monitored by a plastic indicator; while the global damage of the entire structure is correlated to structural stiffness degradation represented by its natural frequency reduction. The proposed damage indices are calibrated by a shaking table test on a 1: 5‐scale frame model. They are then applied to analyse the structural damage to typical low‐ to high‐rise RC frames under blast‐induced ground motions. Results demonstrate a distinctive pattern of structural damage and it is shown that the conventional damage assessment methods adopted in seismic analysis are not applicable here. It is also found that the existing code regulation on allowable peak particle velocity of blast‐induced ground motions concerning major structural damage is very conservative for modern RC structures. Copyright © 2003 John Wiley & Sons, Ltd.     

A new smooth hysteretic model for ductile flexural‐dominated reinforced concrete bridge columns

A new smooth hysteretic model is proposed for ductile, flexural‐dominated reinforced concrete bridge columns. Four columns designed per modern seismic codes were tested using monotonically increasing and variable‐amplitude cyclic loading protocols and ground motion loading to develop the model. Based on the test results, hysteretic rules for damage accumulation and path dependence of reloading were constructed. For damage accumulation, unloading stiffness degradation is correlated with the maximum displacement and hysteretic energy dissipation, while reloading stiffness degradation is set equal to the unloading stiffness degradation. Pinching severity is related to the residual displacement in the direction opposite to the loading direction. Strength deterioration is correlated with the damage index and does not occur until the damage index reaches a threshold, after which the deterioration is proportional to the increase of the damage index. For path dependence of reloading, reloading paths are classified into primary paths and associate paths. The primary paths are those that start from a residual displacement that is equal to or larger than the previous maximum one. The associate paths are those that do not belong to primary paths and tend to be directed towards certain points. Reloading without load reversal is assumed to be linear. Comparison with the results of pseudo‐dynamic tests using three consecutive ground motions showed that the proposed model closely matched the test results. Copyright © 2017 John Wiley & Sons, Ltd.     

Seismic retrofit of reinforced concrete frames using buckling‐restrained braces with bearing block load transfer mechanism

A new method of retrofitting reinforced concrete (RC) frames with buckling‐restrained braces (BRBs) to improve frame strength, stiffness and energy dissipation is proposed. Instead of typical post‐installed anchors, load is transferred between the BRB and RC frame through compression bearing between an installed steel frame connected to the BRB, and high‐strength mortar blocks constructed at the four corners of the RC frame. This avoids complex on‐site anchor installation, and does not limit the allowable brace force by the anchor strength. Cyclic displacements of increasing amplitudes were imposed on two RC frame specimens retrofitted with different BRB strength capacities. In one of the frames, the bearing blocks were reinforced with wire mesh to mitigate cracking. A third RC frame was also tested as a benchmark to evaluate the retrofit strength and stiffness enhancements. Test results indicate that the proposed method efficiently transferred loads between the BRBs and RC frames, increasing the frame lateral strength while achieving good ductility and energy‐dissipating capacity. When the bearing block was reinforced with wire mesh, the maximum frame lateral strength and stiffness were more than 2.2 and 3.5 times the RC frame without the BRB respectively. The BRB imposes additional shear demands through the bearing blocks to both ends of the RC beam and column member discontinuity regions (D‐regions). The softened strut‐and‐tie model satisfactorily estimated the shear capacities of the D‐regions. A simplified calculation and a detailed PISA3D analysis were shown to effectively predict member demands to within 13.8% difference of the measured test results. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonlinear behaviour of RC frames under repeated strong ground motions

This paper presents an extensive parametric study on the inelastic response of eight reinforced concrete (RC) planar frames which are subjected to forty five sequential ground motions. Two families of regular and vertically irregular (with setbacks) frames are examined. The first family has been designed for seismic and vertical loads according to European codes while the second one only for vertical loads, to study structures which have been constructed before the introduction of adequate seismic design code provisions. The whole range of frames is subjected to five real seismic sequences which are recorded by the same station, in the same direction and in a short period of time, up to three days. In such cases, there is a significant damage accumulation as a result of multiplicity of earthquakes, and due to lack of time, any rehabilitation action is impractical. Furthermore, the examined frames are also subjected to forty artificial seismic sequences. Comprehensive analysis of the created response databank is employed in order to derive important conclusions. It is found that the sequences of ground motions have a significant effect on the response and, hence, on the design of reinforced concrete frames. Furthermore, it is concluded that the ductility demands of the sequential ground motions can be accurately estimated using appropriate combinations of the corresponding demands of single ground motions.     

Nonlinear dynamic analysis of reinforced concrete shear wall structures

During strong ground motions, structural members made of reinforced concrete undergo cyclic deformations and experience permanent damage. Members may lose their initial stiffness as well as strength. Recently, Los Alamos National Laboratory has performed experiments on scale models of shear wall structures subjected to recorded earthquake signals. In general, the results indicated that the measured structural stiffnesses decreased with increased levels of excitation in the linear response region. Furthermore, a significant reduction in strength as well as in stiffness is also observed in the inelastic range. Since the in-structure floor response spectra which are used to design and qualify safety equipment have been based on calculated structural stiffness and frequencies, it is possible that certain safety equipment could experience greater seismic loads than were specified for qualification due to stiffness reduction.In this research, a hysteresis model based on the concept of accumulated damage has been developed to account for this stiffness degradation both in the linear and inelastic ranges. Single and three-degrees-of-freedom seismic Category I structures were analysed and compared with equivalent linear stiffness degradation models in terms of maximum displacement responses, permanent displacement, and floor response spectra. The results indicate significant differences in response between the hysteresis model and equivalent linear stiffness degradation model at PGA levels of greater than 0.8 g. The hysteresis model is used in the analysis of reinforced concrete shear-wall structures to obtain the in-structure response spectra. Results of both cumulative and one shot tests are compared.     

Inelastic spectra for infilled reinforced concrete frames

In two companion papers a simplified non‐linear analysis procedure for infilled reinforced concrete frames is introduced. In this paper a simple relation between strength reduction factor, ductility and period (R–µ–T relation) is presented. It is intended to be used for the determination of inelastic displacement ratios and of inelastic spectra in conjunction with idealized elastic spectra. The R–µ–T relation was developed from results of an extensive parametric study employing a SDOF mathematical model composed of structural elements representing the frame and infill. The structural parameters, used in the proposed R–µ–T relation, in addition to the parameters used in a usual (e.g. elasto‐plastic) system, are ductility at the beginning of strength degradation, and the reduction of strength after the failure of the infills. Formulae depend also on the corner periods of the elastic spectrum. The proposed equations were validated by comparing results in terms of the reduction factors, inelastic displacement ratios, and inelastic spectra in the acceleration–displacement format, with those obtained by non‐linear dynamic analyses for three sets of recorded and semi‐artificial ground motions. A new approach was used for generating semi‐artificial ground motions compatible with the target spectrum. This approach preserves the basic characteristics of individual ground motions, whereas the mean spectrum of the whole ground motion set fits the target spectrum excellently. In the parametric study, the R–µ–T relation was determined by assuming a constant reduction factor, while the corresponding ductility was calculated for different ground motions. The mean values proved to be noticeably different from the mean values determined based on a constant ductility approach, while the median values determined by the different procedures were between the two means. The approach employed in the study yields a R–µ–T relation which is conservative both for design and performance assessment (compared with a relation based on median values). Copyright © 2004 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.     

Estimating the seismic displacement of friction pendulum isolators based on non‐linear response history analysis

A procedure for developing equations that estimate the isolator displacement due to strong ground motion is applied to buildings isolated with the friction pendulum system. The resulting design equations, based on rigorous non‐linear analysis, offer an alternative to the iterative equivalent‐linear methods used by current U.S. building codes. The governing equations of the system are reduced to a form such that the median normalized displacement of the system due to an ensemble of ground motions is found to depend on only the isolation period—a function of the curvature of the isolator—and the friction force at incipient slip normalized by peak ground velocity. The normalization is effective in minimizing the dispersion of the normalized displacement for an ensemble of ground motions, implying that the median normalized displacement is a reliable estimate of response. The design equations reflect the significant (20 to 38%) increase in displacement when the excitation includes two lateral components of ground motion instead of just one component. Equivalent‐linear methods are shown to underestimate by up to 30% the exact median displacement determined by non‐linear response history analysis for one component of ground motion, and building codes include at most a 4.4% increase for a second component. Copyright © 2003 John Wiley & Sons, Ltd.     

Investigation of story ductility demands of inelastic concrete frames subjected to repeated earthquakes

The present study focuses on the influence of repeated earthquakes on the maximum story ductility demands of three-dimensional inelastic concrete frames. A comprehensive assessment is conducted using generic frames with 3-, 6-, 12-, and 18-story structures. Each is assumed to have behaviour factors of 1.5, 2, 4, and 6 referring to Eurocode 8. Stiffness and strength degrading hysteresis rule to represent reinforced concrete structure is considered in the plastic hinge of members. Twenty ground motions are selected, and single, double, and triple events of synthetic repeated earthquakes are considered. Some interesting findings are provided showing that repeated earthquakes significantly increase the story ductility demand of inelastic concrete frames. On average, relative increment of maximum story ductility demand is experienced 1.4 and 1.3 times when double and triple events of repeated earthquakes are induced, respectively. Empirical relationships are also provided to predict these increments where their efficiency is presented examining characteristic 3- and 8-story reinforced concrete buildings.     

Conditional spectrum‐based ground motion selection. Part I: Hazard consistency for risk‐based assessments

The conditional spectrum (CS, with mean and variability) is a target response spectrum that links nonlinear dynamic analysis back to probabilistic seismic hazard analysis for ground motion selection. The CS is computed on the basis of a specified conditioning period, whereas structures under consideration may be sensitive to response spectral amplitudes at multiple periods of excitation. Questions remain regarding the appropriate choice of conditioning period when utilizing the CS as the target spectrum. This paper focuses on risk‐based assessments, which estimate the annual rate of exceeding a specified structural response amplitude. Seismic hazard analysis, ground motion selection, and nonlinear dynamic analysis are performed, using the conditional spectra with varying conditioning periods, to assess the performance of a 20‐story reinforced concrete frame structure. It is shown here that risk‐based assessments are relatively insensitive to the choice of conditioning period when the ground motions are carefully selected to ensure hazard consistency. This observed insensitivity to the conditioning period comes from the fact that, when CS‐based ground motion selection is used, the distributions of response spectra of the selected ground motions are consistent with the site ground motion hazard curves at all relevant periods; this consistency with the site hazard curves is independent of the conditioning period. The importance of an exact CS (which incorporates multiple causal earthquakes and ground motion prediction models) to achieve the appropriate spectral variability at periods away from the conditioning period is also highlighted. The findings of this paper are expected theoretically but have not been empirically demonstrated previously. Copyright © 2013 John Wiley & Sons, Ltd.     

Seismic reliability functions for multistorey frame and wall‐frame systems

Seismic reliability functions of multistorey frame systems are expressed as values of Cornell's βindex in terms of two alternative measures of the earthquake intensity, normalized with respect to the yield displacement or to the deformation capacity of a simplified model of the global behaviour of the system obtained by pushover analysis. The safety margin is defined as the difference of the natural logarithms of the intensity that leads to collapse and that assumed to act on the system. The problem of defining a deformation capacity for a multistorey system is circumvented in this manner. The method proposed is illustrated through its application to several reinforced concrete rigid frames, including both column‐and‐beam and wall‐frame systems. Ground motion excitations are representative of those recorded at soft soil sites in the Valley of Mexico. A comparison is made of the reliability functions obtained on the basis of the gross section or the cracked section of reinforced concrete members. The results show that the reliability functions do not only depend on the expected values of the normalized intensity, but also on its dispersion, which is sensitive to the ratio of the fundamental period of the system to the dominant period of the ground motion. Some comments are presented about the establishment of reliability‐based seismic design criteria for generic systems. Copyright © 2006 John Wiley & Sons, Ltd.     

Shaking table test and numerical simulation of a 1/2‐scale self‐centering reinforced concrete frame

Self‐centering reinforced concrete frames are developed as an alternative of traditional seismic force‐resisting systems with better seismic performance and re‐centering capability. This paper presents an experimental and computational study on the seismic performance of self‐centering reinforced concrete frames. A 1/2‐scale model of a two‐story self‐centering reinforced concrete frame model was designed and tested on the shaking table in State Key Laboratory of Disaster Reduction in Civil Engineering at Tongji University to evaluate the seismic behavior of the structure. A structural analysis model, including detailed modeling of beam–column joints, column–base joints, and prestressed tendons, was constructed in the nonlinear dynamic modeling software OpenSEES. Agreements between test results and numerical solutions indicate that the designed reinforced concrete frame has satisfactory seismic performance and self‐centering capacity subjected to earthquakes; the self‐centering structures can undergo large rocking with minor residual displacement after the earthquake excitations; the proposed analysis procedure can be applied in simulating the seismic performance of self‐centering reinforced concrete frames. To achieve a more comprehensive evaluation on the performance of self‐centering structures, research on energy dissipation devices in the system is expected. Copyright © 2015 John Wiley & Sons, Ltd.     

Structural pounding between adjacent buildings subjected to strong ground motions. Part II: The effect of multiple earthquakes

The effect of collision between adjacent reinforced concrete building frames under multiple earthquakes is investigated in this paper. The four planar frames and the nine different pairs of adjacent reinforced concrete structures of the first companion paper are also examined here, under five real seismic sequences. Such a sequence of earthquakes results in a significant damage accumulation in a structure because any rehabilitation action between any two successive seismic motions cannot be practically materialised because of lack of time. Various parameters are investigated, such as the maximum horizontal displacement of top floor, ductility of columns, permanent displacements and so on. Furthermore, four different separation gaps between the building frames are considered to determine their influence on the behaviour of these frames. It is concluded that in most of the cases, the seismic sequences appear to be detrimental in comparison with the single seismic events. Copyright © 2013 John Wiley & Sons, Ltd.     

Accuracy of nonlinear static procedures for the seismic assessment of shear critical structures

The nonlinear behavior of reinforced concrete (RC) members represents a key issue in the seismic performance assessment of structures. Many structures constructed in the 1980s or earlier were designed based on force limits; thus they often exhibit brittle failure modes, strength and stiffness degradation, and severe pinching effects. Field surveys and experimental evidence have demonstrated that such inelastic responses affect the global behavior of RC structural systems. Efforts have been made to consider the degrading stiffness and strength in the simplified nonlinear static procedures commonly adopted by practitioners. This paper investigates the accuracy of such procedures for the seismic performance assessment of RC structural systems. Refined finite element models of a shear critical bridge bent and a flexure‐critical bridge pier are used as reference models. The numerical models are validated against experimental results and used to evaluate the inelastic dynamic response of the structures subjected to earthquake ground motions with increasing amplitude. The maximum response from the refined numerical models is compared against the results from the simplified static procedures, namely modified capacity spectrum method and coefficient method in FEMA‐440. The accuracy of the static procedures in estimating the displacement demand of a flexure‐critical system and shear‐critical system is discussed in detail. Copyright © 2015 John Wiley & Sons, Ltd.     

Effects of reinforcement slippage on the non‐linear response under cyclic loadings of RC frame structures

This paper discusses the importance of including the bond‐slip effects in assessing the response under cyclic loads of reinforced concrete frames. The discussion is based on analyses performed using numerical models which are simple, computationally efficient and capable of representing the salient features of reinforced concrete frames under both static and dynamic loads. The numerical models comprise a displacement‐based, reinforced concrete frame element with bond‐slip and a rigid beam column joint element with bond‐slip. Two applications illustrate the model accuracy and show the importance of including bond‐slip. The first application considers a reinforced concrete beam‐column subassemblage experimentally tested under cyclic loads. The second application considers the shaking table test of a two‐story one‐bay reinforced concrete frame In both cases the analytical results correlate well with the experimental results in terms of strength, displacement demands and hysteretic energy dissipation. Furthermore, the paper shows how the analyses that include bond‐slip yield a better correlation with the experimental results with respect to the analyses that assume a perfect bond. Copyright © 2003 John Wiley & Sons, Ltd.     

剪切型结构的抗震强度折减系数研究

为了研究剪切型结构抗震强度需求的变化规律,本文基于单自由度体系的非线性时程分析,研究了不同场地条件下延性折减系数与位移延性系数和结构自振周期的关系;采用修正等效单自由度体系位移延性折减系数的方法,研究了剪切型多自由度体系的延性折减系数;以基于中国建筑抗震规范设计的代表不同抗震能力要求的RC框架结构为分析对象,通过静力弹塑性分析,研究了RC框架结构的体系超强能力。分析结果表明场地类别、位移延性水准和结构振动周期对单自由度体系的延性折减系数有显著的影响;多自由度体系的抗震延性折减系数明显比其相应的等效单自由度体系的抗震延性折减系数小;RC框架结构的超强系数一般随结构楼层数的增加而减小,随抗震设防烈度的增大而减小,内框架的超强系数比边框架的超强系数大。     

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.     

Pseudodynamic response of torsionally unbalanced two‐storey test structure

Two one‐way eccentric, two‐storey, one‐by‐one‐bay reinforced concrete (RC) structures are pseudodynamically tested under unidirectional ground motions. Theoretical considerations about the effect of torsional coupling on modal periods and shapes agree with modal results of the test structure, considering member stiffness is equal to the secant stiffness to yielding in skew‐symmetric bending. Modal periods of such an elastic structure are in fair agreement with effective periods inferred from the measured response at the beginning of a test of a thoroughly cracked structure and at the end of the test. A time‐varying stiffness matrix and a non‐proportional damping matrix fitted to the test results may be used to reproduce the measured response approximately by modal superposition and identify the role of the four time‐varying modes. Flexible side columns sustained very large drift demands simultaneously in the two transverse directions and suffered significant but not heavy, damage at lap‐splices. RC‐jacketing of the flexible side columns practically eliminated the static eccentricity between the floor centres of twist and mass as well as the torsional response. Inelastic time‐history analysis with point‐hinge member models, using as elastic stiffness the secant stiffness to yielding and neglecting post‐ultimate‐strength cyclic degradation of resistance in members with plain bars and poor detailing, predicted fairly well the response until the peak displacements and member deformations occurred. After that, it underestimated displacement peaks and the lengthening of the apparent period and missed the gradual drifting of the response towards a permanent offset. Copyright © 2007 John Wiley & Sons, Ltd.     

Influence of dynamic soil–structure interaction on the nonlinear response and seismic reliability of multistorey systems

A set of reinforced concrete structures with gravitational loads and mechanical properties (strength and stiffness) representative of systems designed for earthquake resistance in accordance with current criteria and methods is selected to study the influence of dynamic soil–structure interaction on seismic response, ductility demands and reliability levels. The buildings are considered located at soft soil sites in the Valley of Mexico and subjected to ground motion time histories simulated in accordance with characteristic parameters of the maximum probable earthquake likely to occur during the system's expected life. For the near‐resonance condition the effects of soil–structure interaction on the ductility demands depend mainly on radiation damping. According to the geometry of the structures studied this damping is strongly correlated with the aspect ratio, obtained by dividing the building height by its width. In this way, for structures with aspect ratio greater than 1.4 the storey and global ductility demands increase with respect to those obtained with the same structures but on rigid base, while for structures with aspect ratio less than 1.4 the ductility demands decrease with respect to those for the structures on rigid base. For the cases when the fundamental period of the structure has values very different from the dominant ground period, soil–structure interaction leads in all cases to a reduction of the ductility demands, independently of the aspect ratio. The reliability index β is obtained as a function of the base shear ratio and of the seismic intensity acting on the nonlinear systems subjected to the simulated motions. The resulting reliability functions are very similar for systems on rigid or on flexible foundation, provided that in the latter case the base rotation and the lateral displacement are removed from the total response of the system. Copyright © 2006 John Wiley & Sons, Ltd.