Modification of Ductility Reduction Factor for Strength Eccentric Structures Subjected to Pulse-Like Ground Motions

This article investigates the ductility reduction factors for RC eccentric frame structures subjected to pulse-like ground motions. The structural models are with the strength eccentricities which are much disadvantageous than the stiffness eccentricities during the inelastic response range. A method to determine the ductility reduction factors of the strength eccentric structures is suggested by modifying those of reference symmetric structures through an eccentricity modification factor. The four factors of strength eccentricity ratio, ductility ratio, story number and velocity pulse of ground motions, are investigated to gain insight into this modification factor. It shows that the ductility reduction factors of the eccentric structures are clearly smaller than those of the symmetric structures. The eccentricity modification factor is mainly affected by the strength eccentricity and the ductility ratio, decreasing with the increment of the eccentricity or the decrement of the ductility ratio in a medium eccentricity range. The earthquake pulse-like effect and the eccentricity have coupling influence on the modification factor, while the effect of story number is not apparent. Based on the results of a comprehensive statistical study a simplified expression is suggested, which can estimate the eccentricity modification factors for both pulse-like and nonpulse-like ground motion cases.     

Definition of Seismic Input for Out-of-Plane Response of Masonry Walls: I. Parametric Study

Response of masonry walls to out-of-plane excitation is a complex, yet inadequately addressed theme in seismic analysis. The seismic input expected on an out-of-plane wall (or a generic “secondary system”) in a masonry building is the ground excitation filtered by the in-plane response of the walls and the floor diaphragm response. More generally, the dynamic response of the primary structure, which can be nonlinear, contributes to the filtering phenomenon. The current article delves into the details and results of several nonlinear dynamic time-history analyses executed within a parametric framework. The study addresses masonry structures with rigid diaphragm response to lateral loads. The scope of the parametric study is to demonstrate the influence of inelastic structural response on the seismic response of secondary systems and eventually develop an expression to estimate the seismic input on secondary systems that explicitly accounts for the level of inelasticity in the primary structure in terms of the displacement ductility demand. The proposed formulation is discussed in the companion article.     

YIELDING OSCILLATOR UNDER TRIANGULAR GROUND ACCELERATION PULSE

An analytical solution is presented for the response of a bilinear inelastic simple oscillator to a symmetric triangular ground acceleration pulse. This type of motion is typical of near-fault recordings generated by source-directivity effects that may generate severe damage. Explicit closed-form expressions are derived for: (i) the inelastic response of the oscillator during the rising and decaying phases of the excitation as well as the ensuing free oscillations; (ii) the time of structural yielding; (iii) the time of peak response; (iv) the associated ductility demand. It is shown that when the duration of the pulse is long relative to the elastic period of the structure and its amplitude is of the same order as the yielding seismic coefficient, serious damage may occur if significant ductility cannot be supplied. The effect of post-yielding structural stiffness on ductility demand is also examined. Contrary to presently-used numerical algorithms, the proposed analytical solution allows many key response parameters to be evaluated in closed-form expressions and insight to be gained on the'response of inelastic structures to such motions. The model is evaluated against numerical results from actual near-field recorded motions. Illustrative examples are also presented.     

Non-Iterative Equivalent Linearization Based on Secant Period for Estimating Maximum Deformations of Existing Structures

The capacity spectrum method of ATC-40 uses the secant period as the equivalent period of equivalent linear systems. Therefore, it results in a direct graphical comparison. The maximum inelastic displacement and acceleration demands of structures can be simultaneously obtained from the intersection of the demand and capacity diagrams. However, for evaluation of existing structures, the demands need to be determined through iterations since the equivalent period and damping of the equivalent linear systems currently available are both a function of the (displacement) ductility ratio, which is unknown and is the target of evaluation. In addition, the equivalent damping used in the capacity spectrum method is independent of periods of vibration. It may lead to poor estimations of maximum responses especially for short-period systems. This article proposes two equivalent linear systems based on the secant period to estimate the maximum displacement and acceleration responses of existing structures. Both the recommended equivalent period and damping are defined by the strength ratio (elastic lateral strength/yield lateral strength), rather than the ductility ratio. Because the strength ratio of existing structures is a known parameter, the maximum displacement and acceleration responses of these structures can be determined without iterations. Besides, effects of periods of vibration on the equivalent linear systems are also included in this study. The equivalent damping is derived from statistical analyses for bilinear single-degree-of-freedom (SDOF) systems with different periods of vibration, strength ratios and post-yield stiffness based on 72 earthquake ground motions recorded on firm sites. Procedures and examples for applications of the proposed equivalent linear systems on nonlinear static analysis procedures are also provided.     

Direct Displacement-Based Seismic Design of Eccentrically Braced Steel Frames

A series of eccentrically braced frames (EBF) are designed and subjected to nonlinear analyses to highlight ambiguities and differences in current seismic design provisions for EBF structures. This provides motivation to implement better guidance for the checking of local displacement demand considerations and move towards a displacement-based design approach. A recently proposed direct displacement-based design (DDBD) procedure for EBFs is then described and further developed in this article through the calibration of a spectral displacement reduction factors that relate the displacement of an inelastically responding structure to that of the equivalent linear representation used in the DDBD of EBFs. Such an expression is calibrated as part of this study using an experimentally validated numerical model also proposed here for the EBF links such that the actual hysteretic behavior of the links is well represented. The DDBD guidelines are applied to EBF systems from 1–15 stories in height and their performance is verified via nonlinear dynamic analyses using two different sets of design spectrum compatible ground motions. The results of the study indicate the robustness of the proposed DDBD method in limiting the interstory drifts to design limits for a variety of EBF systems with short links, thus demonstrating that the proposed DDBD method is an effective tool for seismic design of EBFs.     

Reduction Factors for Seismic Design Spectra for Structures with Viscous Energy Dampers

A simple mathematical expression is proposed to estimate spectra reduction damping factors for seismic design of systems with viscous dampers. The expression is obtained from the ratios between ordinates of uniform hazard spectra associated with two different return intervals (50 and 125 years), corresponding to sites with different types of soil within the Valley of Mexico. The expression proposed depends on the dominant period of the soil, and on both the vibration period and damping ratio of the structural system. Values of the damping factors proposed here are comparable to those recommended by different authors and seismic design building codes.     

A PROCEDURE FOR ESTIMATING INPUT ENERGY SPECTRA FOR SEISMIC DESIGN

In this paper, the damage potential of an earthquake ground motion is evaluated in terms of the total power of the acceleration of the ground motion. By assuming an appropriate spectral shape for the input energy spectrum, and using the well-known Parseval theorem for evaluating the total power of a random signal, the peak amplification factor for the equivalent input energy velocity spectrum can be determined. It is shown that the peak amplification factor for the input energy spectrum depends on the peak-ground-acceleration to peak-ground-velocity ratio and duration of the strong motion phase of the ground motion. Values for the equivalent input energy velocity amplification factor vary from about 2 to 10 for most of the recorded ground motions used in this study. Although a considerable scatter of data is observed in this study, the peak amplification factor predicted by the Fourier amplitude spectrum of the ground acceleration provides a fairly good estimate of the mean value of the peak input energy compared to that determined from inelastic dynamic time history analyses, particularly for systems with high damping and low lateral strength. The peak amplification factor derived in this paper provides a more consistent approach for estimation of seismic demand when compared to an earlier empirical expression used for the formulation of duration-dependent inelastic seismic design spectra, even though only a slight difference in the required lateral strength results from the use of the new formula.     

Ductility-Equivalent Viscous Damping Relationships for Beam-to-Column Partial-Strength Steel Joints

This paper seeks to contribute to the development and improvement of displacement-based design procedures, proposing improved ductility-equivalent viscous damping relationships for steel moment-resisting framed structures with dissipative beam-to-column partial-strength joints. These relationships can be used directly in procedures like the Direct Displacement-Based Seismic Design (DDBD) that uses effective stiffness, ductility-equivalent viscous damping relationships, and period-displacement relationships in a performance-based design approach. To this end, a finite element model of a steel beam-to-column sub-assemblage, characterized by an extended end-plate, is developed in ABAQUS. The model, which is validated against monotonic and cyclic experimental data obtained in previous research, is employed to carry out non-linear time-history (NLTH) analyses, using real records scaled to target several levels of ductility demand. A procedure is then proposed and applied to determine the ductility-equivalent viscous damping relationships of the sub-assemblages. The equivalent linearization technique is applied to the non-linear responses obtained in the NLTH analyses, using an elastic single degree of freedom structure and the elastic displacement spectra. The influence of joints mechanisms and of the dynamic characteristics of the structure in the equivalent viscous damping is investigated, and an expression for ductility-equivalent viscous damping is proposed. The proposed expression represents a clear improvement in relation to the existing expressions available in literature.     

Probabilistic Characteristics of Seismic Ductility Demand of SDOF Systems with Bouc-Wen Hysteretic Behavior

This study investigates probabilistic characteristics of the peak ductility demand of inelastic single-degree-of-freedom systems. The hysteretic behavior of structural systems is represented by the Bouc-Wen model, which takes various hysteretic curves with degradation and pinching behavior into account, and a prediction equation of the peak ductility demand is developed. The application of the developed equation in reliability analysis of structures subject to earthquake loading is illustrated. The results indicate that the effects due to degradation and pinching behavior on the peak ductility demand as well as the reliability of structures can be significant, especially for stiff structures.     

Seismic Design of MRF-EBF Dual Systems with Vertical Links: EC8 vs Plastic Design

Despite the fact that Eccentrically Braced Frames with Vertical Links (VL-EBFs), also referred to as inverted Y-scheme, are codified in Eurocode 8, the issues related to their seismic response and design have not been widely investigated, so that design criteria commonly applied for Eccentrically Braced Frames with Horizontal Links (HL-EBFs) are commonly applied. However, the Theory of Plastic Mechanism Control (TPMC) has been recently extended to the case of VL-EBFs. The aims of this article, on one hand, are to provide a further validation of the recently proposed design procedure, based on TPMC, and, on the other hand, are to compare the seismic performance of dual systems composed by a moment-resisting part and VL-EBF part designed by means of TPMC with those occurring when Eurocode 8 design criteria are applied. The validation of the proposed design procedure is carried out by means of Incremental Dynamic Analyses (IDA). The main purpose of such analyses is the check of the fulfilment of the design goal of TPMC, i.e., the development of a pattern of yielding consistent with the collapse mechanism of global type. Such mechanism is universally recognized as the one leading to the highest energy dissipation capacity. In case of MRF-EBF dual systems, it is characterized by the yielding of all the links and all the beams at their ends. Conversely, all the columns and the diagonal braces remain in elastic range. Obviously, exception is made for the base sections of first story columns. In particular, two case studies are analyzed which are characterized by a different number of stories. Each building structure is designed according to both TPMC and Eurocode 8 provisions. The seismic response obtained is investigated by both push-over and IDA analyses. The attention is focused on the pattern of yielding obtained, the maximum interstory drift demand, the link plastic rotation demand and sharing of the seismic base shear between the moment-resisting part and the bracing part of the structural system. The results obtained point out improvement of the seismic response, compared to Eurocode 8 provisions, achieved by means of TPMC.     

Hysteretic Model for Flexure-shear Critical Reinforced Concrete Columns

A model for predicting the cyclic lateral load-deformation response of flexure-shear critical reinforced concrete (RC) columns subjected to combined axial load and cyclic shear is proposed. Strength deterioration in the primary curve due to the effect of shear after yielding is considered by a modification coefficient. Rules for unloading and reloading branches of the hysteretic curve are obtained from regression analysis of test results. Unloading stiffness is fitted as a function of displacement ductility and secant stiffness of the point with maximum displacement in the primary curve. Pinching is simulated by changing the slope of reloading branch. Path-based cyclic strength deterioration is incorporated in the proposed model. In the expression of cyclic strength deterioration, the effects of aspect ratio and axial-load ratio are considered. Comparison between the predicted cyclic response and experimental results indicates that the proposed model can predict the observed hysteretic response of flexure-shear critical RC columns well.     

Behavior Factor for Performance-Based Seismic Design of Plane Steel Moment Resisting Frames

Simplified expressions to estimate the behavior factor of plane steel moment resisting frames are proposed, based on statistical analysis of the results of thousands of nonlinear dynamic analyses. The influence on this factor of specific structural parameters, such as the number of stories, the number of bays, and the capacity design factor of a steel frame, is studied in detail. The proposed factor describes the seismic strength requirements in order to restrict maximum storey ductility to a predefined value. Interrelation studies between maximum storey ductility and the Park-Ang damage index are also provided for the damage-based interpretation of the performance levels under consideration. Realistic design examples serve to demonstrate the ability of the proposed factor to convert conventional force-based design to a direct performance-based seismic design procedure.     

CALIBRATION OF FORCE REDUCTION FACTORS OF RC BUILDINGS

A comprehensive study is undertaken to assess and calibrate the force reduction factors (R) adopted in modern seismic codes. Refined expressions are employed to calculate the R factors “supply” for 12 buildings of various characteristics represent a wide range of medium-rise RC buildings. The “supply” values are then compared with the “design” and “demand” recommended in the literature. A comprehensive range of response criteria at the member and storey levels, including shear as a failure criterion, alongside a detailed modelling approach and an extensively verified analytical tool are utilised. A rigorous technique is employed to evaluate R factors, including inelastic pushover and incremental dynamic collapse analyses employing eight natural and artificial records. In the light of the information obtained from more than 1500 inelastic analyses, it is concluded that including shear and vertical motion in assessment and calculations of R factors is necessary. Force reduction factors adopted by the design code (Eurocode 8) are over-conservative and can be safely increased, particularly for regular frame structures designed to lower PGA and higher ductility levels.     

A Multi-Mode Method for Estimation of Floor Response Spectra

A new simplified procedure for estimation of floor response spectra (FRS) is proposed. This methodology enriches the most common procedures using nonlinear response-history analysis to predict FRS by including a direct multi-mode technique to estimate FRS. A novel feature of the procedure is that the coupling effect is considered to establish equivalent modal systems and the FRS are developed by incorporating capacity spectrum method in conjunction with ductility-based FRS for each modal system. Both the proposed method and the traditional method are applied to three steel moment frame structures, and a reasonable accuracy is demonstrated.     

Numerical Evaluation of Topographic Effects at the Nicastro Ridge in Southern Italy

Numerical site response analyses were carried out on the Nicastro ridge in Southern Italy in order to investigate topographic effects. First, the analyses were carried out on a simplified model by employing simple artificial signals, in order to get preliminary physical insights into the two-dimensional phenomena involved. Then, numerical analyses were carried out on a more realistic heterogeneous subsoil model developed on the basis of geotechnical and geophysical investigations. Real accelerograms were selected for these analyses. Particular attention was devoted to separating topographic from stratigraphic amplification. Finally, the topographic amplification factors were compared with literature data and Eurocode 8 recommendations.     

TORSIONAL EFFECTS IN THE PUSHOVER-BASED SEISMIC ANALYSIS OF BUILDINGS

The general trends of the inelastic behaviour of plan-asymmetric structures have been studied. Systems with structural elements in both orthogonal directions and bi-axial eccentricity were subjected to bi-directional excitation. Test examples include idealised single-storey and multi-storey models, and a three-storey building, for which test results are available. The response in terms of displacements was determined by nonlinear dynamic analyses. The main findings, limited to fairly regular and simple investigated buildings, are: (a) The amplification of displacements determined by elastic dynamic analysis can be used as a rough, and in the majority of cases conservative estimate in the inelastic range, (b) Any favourable torsional effect on the stiff side, which may arise from elastic analysis, may disappear in the inelastic range. These findings can be utilised in the approximate pushover-based seismic analysis of asymmetric buildings, e.g. in the N2 method. It is proposed that the results obtained by pushover analysis of a 3D structural model be combined with the results of a linear dynamic (spectral) analysis. The former results control the target displacements and the distribution of deformations along the height of the building, whereas the latter results define the torsional amplifications. The proposed approach is partly illustrated and evaluated by test examples.     

Cumulative Ductility Spectra for Seismic Design of Ductile Structures Subjected to Long Duration Motions: Concept and Theoretical Background

A seismic design procedure that does not take into account the maximum and cumulative plastic deformation demands that a structure will likely undergo during severe ground motion could lead to unreliable performance. Damage models that quantify the severity of repeated plastic cycling through plastic energy are simple tools that can be used for practical seismic design. The concept of constant cumulative ductility strength spectra, developed from one such model, is a useful tool for performance-based seismic design. Particularly, constant cumulative ductility strength spectra can be used to identify cases in which low-cycle fatigue may become a design issue, and provides quantitative means to estimate the design lateral strength that should be provided to a structure to adequately control its cumulative plastic deformation demands during seismic response. Design expressions can be offered to estimate the strength reduction factors associated to the practical use of constant cumulative ductility strength spectra.     

Constant Ductility Energy Factors for the Near-Fault Pulse-Like Ground Motions

This study is focused on the constant ductility energy factors for bilinear system under the near-fault pulse-like ground motions. The variation of energy factors is studied in consideration of the earthquake magnitude, rupture distance, damping ratios, and post-yield stiffness ratios. The results indicate that the near-fault pulse-like ground motions would increase the energy dissipation of structures. The energy factors are significantly influenced by the earthquake magnitude. The damping ratios have more obvious influences on the energy factors than the post-yield stiffness ratios. A predictive model is proposed for the application of constant ductility energy factors for near-fault pulse-like ground motions.     

Robust Period-Independent Ground Motion Selection and Scaling for Effective Seismic Design and Assessment

A period-independent approach for the selection and scaling of ground motion records aimed at reducing demand variability is proposed for seismic response history analysis. The same set of scaled records can be used to study various structures at the same site regardless of their dynamic characteristics. The statistical robustness of the proposed and current approaches is compared through nonlinear inelastic dynamic analyses performed on single-degree-of-freedom systems and multi-story braced frames. The proposed approach leads to consistent response predictions with a limited number of records. This is advantageous for day-to-day structural design or assessment against code hazard-based seismic demand levels.     

AN ALGORITHM FOR COMPUTING ISODUCTILE RESPONSE SPECTRA

The computation of constant ductility (or isoductile) response spectra for single-degree-of-freedom systems can require numerous individual response history analyses. Recognising that the same ductility response may be obtained for different strength oscillators of a given period, greater computational effort is required to reduce the possibility that a desired solution is not overlooked. Even a single solution may not exist if a local discontinuity in the strength-ductility relationship coincides with the desired value of ductility. This paper describes a two-phase algorithm to identify the highest strength solution for which the corresponding ductility equals (or does not exceed) the desired ductility. The first phase adopts a “check-reject” approach to reject intervals of strength where the possibility of unidentified higher-strength solutions is considered to be remote, thereby narrowing the strength interval in which the solution will be found. The second phase identifies a solution within this interval as rapidly as possible using a bisection approach. The algorithm is implemented in the USEE software program. The efficiency and accuracy of the algorithm are demonstrated by comparison to results obtained with other software programs.