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.     

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.     

Nonlinear Static Methods vs. Experimental Shaking Table Test Results

Three different Nonlinear Static Methods (NSM's), based on pushover analysis, are applied to a 3-story, 2-bay, RC frame. They are (i) the Capacity Spectrum Method (CSM), described in ATC-40, (ii) the Displacement Coefficient Method (DCM), presented in FEMA-273 and further developed in FEMA 356, and (iii) the N2 Method, implemented in the Eurocode 8. Pushover analyses are conducted with DRAIN-3DX by using four different lateral force distributions, according to the acceleration profile assumed along the height of the structure: uniform, triangular, modal-proportional, and multimodal fully adaptive. In the numerical model, RC members are modeled as fiber elements.

The numerical predictions of each method are compared to the experimental results of the shaking table tests carried out on two similar 1:3.3-scale structural models, with and without infilled masonry panels, respectively. The comparison is made in terms of maximum story displacements, interstory drifts, and shear forces. All the NSM's are found to predict with adequate accuracy the maximum seismic response of the structure, provided that the associated parameters are properly estimated. The lateral load pattern, instead, is found to little affect the accuracy of the results for the three-story model considered, even if collapse occurs with a soft story mechanism.     

Direct Displacement-Based Design of Glulam Timber Frame Buildings

We introduce a direct Displacement-Based Design methodology for glued laminated timber portal frames with moment-resisting doweled joints. We propose practical expressions to estimate ultimate target displacement and equivalent viscous damping, and we demonstrate that these expressions provide prior values that are close to those obtained a posteriori using a more refined model. Applied to case studies, the method yields base-shear forces lower than those obtained using the force-based approach of Eurocode 8. This is due to the high dissipation capacity of the specific connection technology, which apparently is conservatively accounted for in the q-factor of Eurocode 8.     

A Static Nonlinear Procedure for the Evaluation of the Elastic Buckling of Anchored Steel Tanks Due to Earthquakes

Ground-supported steel tanks experienced extensive damage in past earthquakes. The failure of tanks in earthquakes may cause severe environmental damage and economic losses. This study deals with the evaluation of the elastic buckling of above-ground steel tanks anchored to the foundation due to seismic shaking. The proposed nonlinear static procedure is based on the capacity spectrum method (CSM) utilized for the seismic evaluation of buildings. Different from the standard CSM, the results are not the base shear and the maximum displacement of a characteristic point of the structure but the minimum value of the horizontal peak ground acceleration (PGA) that produces buckling in the tank shell. Three detailed finite element models of tank-liquid systems with height to diameter ratios H/D of 0.40, 0.63, and 0.95 are used to verify the methodology. The 1997 UBC design spectrum and response spectra of records of the 1986 El Salvador and 1966 Parkfield earthquakes are used as seismic demand. The estimates of the PGA for the occurrence of first elastic buckling obtained with the proposed nonlinear static procedure were quite accurate compared with those calculated with more elaborate dynamic buckling studies. For all the cases considered, the proposed methodology yielded slightly smaller values of the critical PGA for the first elastic buckling compared to the dynamic buckling results.     

AN EVALUATION OF THE STRONG GROUND MOTION RECORDED DURING THE MAY 1, 2003 BINGÖL TURKEY,EARTHQUAKE

An important record of ground motion from a M6.4 earthquake occurring on May 1, 2003, at epicentral and fault distances of about 12 and 9 km, respectively, was obtained at a station near the city of Bingöl, Turkey. The maximum peak ground values of 0.55 g and 36 cm/s are among the largest ground-motion amplitudes recorded in Turkey. From simulations and comparisons with ground motions from other earthquakes of comparable magnitude, we conclude that the ground motion over a range of frequencies is unusually high. Site response may be responsible for the elevated ground motion, as suggested from analysis of numerous aftershock recordings from the same station. The mainshock motions have some interesting seismological features, including ramps between the P-and S-wave that are probably due to near- and intermediate-field elastic motions and strong polarisation oriented at about 39 degrees to the fault (and therefore not in the fault-normal direction). Simulations of motions from an extended rupture explain these features. The N10E component shows a high-amplitude spectral acceleration at a period of 0.15 seconds resulting in a site specific design spectrum that significantly overestimates the actual strength and displacement demands of the record. The pulse signal in the N10E component affects the inelastic spectral displacement and increases the inelastic displacement demand with respect to elastic demand for very long periods.     

Inelastic Demand Spectra of Bi-Linear Models for Capacity Spectrum Method and Response of Simple Structures under Near-Source Records

A very useful tool for the preliminary design of structures is the elastic demand spectrum that can be used in the capacity spectrum method. A pseudo-acceleration relationship has to be assumed when constructing a demand spectrum. This assumption results in large errors for long period structures with large damping ratios and the conventional demand spectra require a substitute elastic structure. In the present study, the conventional demand spectra are extended to bi-linear models. Pseudo-acceleration is still assumed but results in acceptably small errors, when a constant viscous damping coefficient for a single-degree-of-freedom (SDF) structure is calculated from the tangent stiffness and the damping ratio is set at 5% in both elastic and yield phases. For nonlinear structures, tangent stiffness dependency of damping force could be acceptable because energy absorption is primarily the result of structural nonlinear deformation. To extend the conventional demand spectra to a bi-linear model, effective period calculated from the secant stiffness has to be used. The use of effective period introduces no approximation because the peak displacement of the SDF structure is computed from nonlinear analysis in the time domain. The method presented in this study is also valid if damping coefficient proportional to initial elastic spectra is used. In this case, the pseudo-acceleration is defined as the base shear coefficient that is required to produce the peak displacement of the SDF structure in a static manner. We present demand spectra of bi-linear models for a number of near-source records from large earthquakes, and spectral ratios of two horizontal components. The effects of different types of ground motion on the response reduction factor due to inelastic deformation are investigated.     

SOIL DAMPING FORMULATION IN NONLINEAR TIME DOMAIN SITE RESPONSE ANALYSIS

Nonlinear time domain site response analysis is used to capture the soil hysteretic response and nonlinearity due to medium and large ground motions. Soil damping is captured primarily through the hysteretic energy dissipating response. Viscous damp-ing, using the Rayleigh damping formulation, is often added to represent damping at very small strains where many soil models are primarily linear. The Rayleigh damping formulation results in frequency dependent damping, in contrast to experiments that show that the damping of soil is mostly frequency independent. Artificially high damp-ing is introduced outside a limited frequency range that filters high frequency ground motion. The extended Rayleigh damping formulation is introduced to reduce the over-damping at high frequencies. The formulation reduces the filtering of high frequency motion content when examining the motion Fourier spectrum. With appropriate choice of frequency range, both formulations provide a similar response when represented by the 5% damped elastic response spectrum.

The proposed formulations used in non-linear site response analysis show that the equivalent linear frequency domain solution commonly used to approximate non-linear site response underestimates surface ground motion within a period range relevant to engineering applications. A new guideline is provided for the use of the proposed formulations in non-linear site response analysis.     

Inelastic Displacement Demand of Strength-Degraded Structures

This article presents findings from parametric studies involving nonlinear time-history analyses of inelastic systems with and without strength degradation. Results showed that estimates based on the equal-displacement and equal-energy propositions can be exceeded significantly by the inelastic displacement demands in the acceleration and velocity-sensitive regions of the response spectrum. The displacement demand behaviour is sensitive to the strength degradation and the frequency properties of the ground shaking. With a modest strength reduction factor of 2, the inelastic displacement demand would typically be constrained by the Peak Displacement Demand as indicated on the elastic displacement response spectrum for 5% damping.     

Time-Domain Spectral Matching of Earthquake Ground Motions using Broyden Updating

Time-domain spectral matching of an earthquake ground motion consists of iteratively adding sets of wavelets to an acceleration history until the resulting response spectrum sufficiently matches a target spectrum. The spectral matching procedure is at its core a nonlinear problem because the addition of a wavelet often causes shifting in the time of peak response or creation of a larger second peak at a different time. A modification to existing time-domain spectral matching algorithms is proposed using Broyden updating for solving the set of nonlinear equations. Three wavelet bases are evaluated and the corrected tapered cosine wavelet is selected. The proposed algorithm is then tested and compared with other methods that are commonly used for spectral matching. The results show that the proposed algorithm is able to match the target spectrum while reasonably preserving the spectral nonstationarity, energy development, and the frequency content of the original time histories.     

Floor Spectra of Inelastic RC Frame Buildings Considering Ground Motion Characteristics

A range of reinforced concrete frame buildings with different levels of inelasticity as well as periods of vibration is analyzed to study the floor response. The derived floor acceleration response spectra are normalized by peak ground acceleration, peak floor acceleration, and ground response spectrum. The normalization with respect to ground response spectrum leads to the lowest coefficients of variation. Based on this observation as well as previous studies, an amplification function is proposed that can be used to develop design floor spectra from the ground motion spectrum, considering the building’s dynamic characteristics and level of inelasticity.     

Estimation of the Displacement of Viscously Damped Pinching Hysteretic Structures Subjected to Near-fault Ground Motions

To fulfill a displacement-based design or response prediction for nonlinear structures, the concept of equivalent linearization is usually applied, and the key issue is to derive the equivalent parameters considering the characteristics of hysteretic model, ductility level, and input ground motions. Pinching hysteretic structures subjected to dynamic loading exhibit hysteresis with degraded stiffness and strength and thus reduced energy dissipation. In case of excitation of near-fault earthquake ground motions, the energy dissipation is further limited due to the short duration of vibration. In order to improve the energy dissipation capability, viscous-type dampers have been advantageously incorporated into these types of structures. Against the viscously damped pinching hysteretic structure under the excitation of near-fault ground motions, this study aims to develop a seismic response estimation method using an equivalent linearization technique. The energy dissipation of various hysteretic cycles, including stationary hysteretic cycle, amplitude expansion cycle, and amplitude reduction cycle, is investigated, and empirical formulas for the equivalent damping ratio is proposed. A damping modification factor that accounts for the near-fault effect is introduced and expanded to ensure its applicability to structures with damping ratios less than 5%. An approach for estimating the maximum displacement of a viscously damped pinching hysteretic structure, in which the pinching hysteretic effect of a structure and the near-fault effect of ground motions are considered, is developed. A time history analysis of an extensive range of structural parameters is performed. The results confirm that the proposed approach can be applied to estimate the maximum displacement of a viscously damped pinching hysteretic structure that is subjected to near-fault ground motions.     

Nonlinear Static Procedure for Multi-Story Asymmetric Frame Buildings Considering Bi-Directional Excitation

The present article focuses on a nonlinear static procedure (NSP) for a multi-story asymmetric frame building with regular elevation subjected to bi-directional ground motion. In this procedure, two simplified models—an equivalent single-story model and an equivalent single-degree-of-freedom (SDOF) model—are used to predict the peak response of multi-story asymmetric buildings. The peak response is predicted through pushover analysis of an equivalent single-story model considering the effect of bi-directional excitations and an estimation of the nonlinear response of equivalent SDOF models. The predicted results are compared with the nonlinear dynamic analysis results, and satisfactory predictions can be obtained by the proposed procedure.     

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.     

Equivalent Damping in Support of Direct Displacement-Based Design

The concept of equivalent linearization of nonlinear system response as applied to direct displacement-based design is evaluated. Until now, Jacobsen's equivalent damping approach combined with the secant stiffness method has been adopted for the linearization process in direct displacement-based design. Four types of hysteretic models and a catalog of 100 ground motion records were considered. The evaluation process revealed significant errors in approximating maximum inelastic displacements due to overestimation of the equivalent damping values in the intermediate to long period range. Conversely, underestimation of the equivalent damping led to overestimation of displacements in the short period range, in particular for effective periods less than 0.4 seconds. The scatter in the results ranged between 20% and 40% as a function of ductility. New equivalent damping relations for four structural systems, based upon nonlinear system ductility and maximum displacement, are proposed. The accuracy of the new equivalent damping relations is assessed, yielding a significant reduction of the error in predicting inelastic displacements. Minimal improvement in the scatter of the results was achieved, however. While many significant studies have been conducted on equivalent damping over the last 40 years, this study has the following specific aims: (1) identify the scatter associated with Jacobsen's equivalent damping combined with the secant stiffness as utilized in Direct Displacement-Based Design; and (2) improve the accuracy of the Direct Displacement-Based Design approach by providing alternative equivalent damping expressions.     

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.     

Design of Passive Viscous Fluid Control Systems for Nonlinear Structures Based on Active Control

A practical procedure is developed for the design of passive control systems using viscous fluid dampers for nonlinear structures. The design methodology takes advantage of the modification of the damping, strength, and stiffness properties of the structure to achieve the desired relative displacement and absolute acceleration response. For this purpose, a study of poles in the complex plane is used to determine the required changes in the dynamic properties of nonlinear structures. Furthermore, a relatively simple relation between the ductility demands of highly damped single- and multiple-degree-of-freedom (SDF and MDF respectively) systems is established to reduce the computational burden of the proposed design method.     

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.     

Seismic Performance of Tuned Liquid Column Dampers for Structural Control Using Real-Time Hybrid Simulation

This article presents real-time hybrid simulation (RTHS) in a single-degree-of-freedom (SDOF) steel frame incorporated with tuned liquid column damper (TLCD). The SDOF steel frame is numerically simulated, and the TLCD alone is physically experimented on a shaking table. The delay-dependent stability of RTHS system for TLCD investigation is first assessed; and the delay-dependent accuracy is verified by comparing the responses obtained through the RTHS, the conventional shaking table test, and an analytical solution. Then, RTHSs are carried out to evaluate the effects of mass ratio, structural damping ratio, structural stiffness, and peak ground acceleration on the reduction effectiveness of STLCD. The nonlinear behavior of the STLCD is experimentally captured. Finally, the structural responses under STLCD and multiple TLCDs (MTLCD) control are compared. It is found that the performance of STLCD strongly depends on structural parameters and properties of earthquakes; both MTLCD and STLCD induce approximately the same response reductions, and the former can enhance the control performance in certain cases. These results presented here may contribute to improve the design and application of TLCDs in practical engineering.