Direct Displacement-based Design Procedure Research Articles

Displacement-based design procedure for the seismic retrofit of existing buildings with self-centering dissipative braces

Existing framed buildings, especially those designed according to outdated seismic codes, often offer inadequate seismic protection levels due to material degradation, low amount of reinforcement and poor construction details. Among the possible retrofit strategies, elasto-plastic dissipative braces (EPDBs) can be installed to provide supplemental stiffness and energy dissipation in the main resisting frame. Since they are based on the plastic deformation of sacrificial steel components, EPDBs are inherently affected by scarce recentering capability, i.e., presence of residual deformation at the end of the mainshock that could impair their effectiveness for aftershock events. As demonstrated in recent seismic sequences, there might be not enough time to restore (e.g., though hydraulic actuators and a reaction frame) the undeformed configuration of the structure before a subsequent shock. This issue can be overcome through self-centering dissipative braces (SCDBs), i.e., braces equipped with dampers characterized by a “flag-shaped” hysteretic behavior, which inherently facilitates the immediate re-centering of the frame. Despite this evident benefit, there is still lack of practical design procedures for the seismic retrofit of buildings through SCDBs. To fill this gap, a direct displacement-based design procedure for SCDBs tailored to building frames is here conceived and illustrated. The proposed method aims at achieving simultaneously two performance requirements: (i) a target maximum roof displacement under the ultimate-limit-state seismic event; (ii) the absence of significant residual inter-storey drifts at the end of the shaking. The implementation of the procedure is described by two applications to a steel and to a reinforced concrete case-study building. Nonlinear time-history analyses are carried out to prove the effectiveness of the proposed method, along with a comparison of the seismic performance offered by SCDBs versus traditional EPDBs implemented in the same case-study buildings.

Calibration of Equivalent Viscous Damping Expressions for Displacement-Based Design and Application to RC Frames

Equivalent viscous damping plays a central role in displacement-based design procedures. In this paper, approaches for estimating the equivalent viscous damping of RC frame buildings are proposed. At first, the analytical formulation of Blandon and Priestley was analysed, and then a calibration of the coefficients of this formulation was performed. Compared with the work of Blandon and Priestley, a larger set of synthetic accelerograms, related to different types of soil and different intensities, and a wider range of the effective periods were considered. In particular, two different sets of parameters are proposed: the first is usable in the case of spectra obtained numerically (approach 1), and the second is usable in the case of code-based spectra and damping modification factor (approach 2). To test the performed calibration and to compare the considered formulations (i.e., the proposed and literature equations), the direct displacement-based design procedure has been applied to three case studies of reinforced concrete frame structures, and then pushover and nonlinear time-history analyses have been performed. The results show that the use of the calibrated parameters (for both the considered approaches) has determined more conservative results, in terms of design base shear and maximum drift from NLTH. Moreover, the average displacement profiles and the inter-storey drifts obtained from time-history analyses for the frames designed with the calibrated parameters match better the design profile.

Direct displacement‑based seismic design for a retrofitted RC frame by hinged wall with BRBs at base

A hinged wall (HW) with buckling-restrained braces (BRBs) at the base (HWBB) has been proposed to enhance the seismic performance of a reinforced concrete frame (RCF). In this study, a new direct displacement-based design procedure based on the bottom shear method (DDBDP-BSM) is proposed for low- and multi-rise RCFs with an HWBB (RCF-HWBB). First, the theoretical relationship between the basic period target and displacement target of the RCF-HWBB was deduced based on the DDBDP-BSM and performance targets. Then, a method to determine the HW configuration and BRB sectional area was proposed. Finally, the optimal selection of earthquake intensity at BRB yielding and the applicability of the DDBDP-BSM on a high-rise RCF-HWBB were investigated through a comparison between the design results and the time history analysis results.

Hybrid self-centering dual rocking core system for seismic resilience by controlling both structural and nonstructural damage

This study proposes a novel hybrid self-centering dual rocking core (HSDRC) system with structural and nonstructural damage control functions for the building structure’s seismic resilience. The rigid rocking cores (RCs), viscous dampers (VDs), and shear friction spring dampers (SFDs) are introduced in HSDRC systems. The rocking cores with pinned bases are adopted for promoting uniform inter-story drifts over the building height. The SFDs are introduced to dissipate hysteretic energy and bring self-centering capacity by eliminating residual deformation. The VD’s velocity-proportional damping can help HSDRC systems control the nonstructural damage and building content’s damage sensitive to absolute floor accelerations. The expected behavior of HSDRC systems is first described. A direct displacement-based design (DDBD) method is following proposed to achieve the desired nonlinear behavior of HSDRC. Following the developed DDBD procedure, six HSDRC systems are designed. For highlighting the advantages of HSDRC systems compared to the existing self-centering energy-absorbing rocking core (SEDRC) system, two SEDRC systems are also designed. Nonlinear dynamics were included for investigating the performance of the designed systems subjected to seismic events. The numerical results demonstrate that the designed HSDRC systems show the expected nonlinear behavior and achieve the design target, confirming the developed DDBD method’s effectiveness. Compared to SEDRC systems, the newly proposed HSDRC systems show smaller absolute floor acceleration responses while obtaining the comparative capacity in controlling both peak and residual inters-tory drifts, indicating the proposed HSDRC systems can be a promising seismic resilient structural system for both structural and nonstructural damage control.

Optimum displacement profile for the direct displacement-based design of steel moment-resisting frames

The direct displacement-based design (DDBD) procedure is well established for designing reinforced concrete and steel moment-resisting frames (SMRFs). However, a limited number of researches is available on optimum DDBD of SMRFs. Furthermore, the nonlinear time-history analysis response of structures designed based on this procedure is inconsistent with its initial assumptions. Design displacement profile is one of the most essential and influential parameters in the DDBD method because it can impress other design parameters. In this paper, an optimum displacement profile is proposed to improve the results of the DDBD procedure via using the particle swarm optimization (PSO) algorithm. In this regard, nine SMRFs with a different number of stories have been designed using the DDBD procedure. Nonlinear time history analyses are performed for these frames using OpenSees software. The models are subjected to a set of 20 ground motion records. Then, PSO algorithm has been used to optimize the displacement profile of the DDBD procedure to achieve uniform drift distribution along with the height of the frames. Finally, based on regression analysis, the optimum design displacement profile for SMRFs with a different number of stories is formulated. The proposed equations show an average reduction of about 20% and 40% in steel usage and design base shear of the frames, respectively, while they have uniform drift along their heights.

Improvements in the direct displacement-based design procedure for mid-rise steel MRFs equipped with viscous dampers

This paper aims to propose straightforward modifications in the direct displacement-based design (DDBD) procedure to reach an optimal design for mid-rise steel moment-resisting frames (MRFs) equipped with linear fluid viscous dampers (FVDs). The modifications include using an alternative equivalent damping as well as applying the ratio of pseudo-spectral velocity to spectral velocity. To compare the proposed method with respect to the conventional DDBD procedure, the seismic performance of a set of low to mid-rise structures consisting of 3-, 6-, 9- and 12-story steel MRFs is investigated. This is carried out via nonlinear time-history analyses by using a set of design spectrum compatible ground motion records at two seismic hazard levels, i.e., the design earthquake (DE) and maximum considered earthquake (MCE). Furthermore, two different patterns, i.e., uniform and non-uniform, for distribution of damping coefficients at the height of the structures are used. Results show that the proposed method can meet the performance targets defined at the DE and MCE hazard levels. Moreover, the peak roof displacement of the structures that is estimated by the proposed method is acceptably the same as captured via the nonlinear analyses. Finally, implementing the proposed method leads to a reduction in the weight of the steel used in the MRFs in comparison with the structures designed by the conventional DDBD procedure and smaller damper forces, i.e., lighter damper constants, can also be achieved.

Applicability of the direct displacement-based design procedure to concentrically braced frames with setbacks

ABSTRACT The objective of this paper is to investigate and develop seismic design guidelines for multi-storey vertical irregular concentrically braced frames (CBFs). The work develops a direct displacement-based design (DDBD) procedure for irregular CBFs associated with steps in building plan area. In this procedure, design displacements considered are decided upon the code and material drift limits, then the strength required to achieve this displacement is calculated and finally all structural elements are designed. A case study of a 12-storey CBF structure with vertical irregularity is designed using the developed DDBD procedure. The configuration of the vertical irregularity assessed is in the form of setbacks up the vertical axis of the building where the frames have more bays at the base of the building than at the top. Non-linear time history analysis (NLTHA) using seven different accelerograms with displacement response spectra matching the design displacement spectrum are used to record the behaviour of the irregular CBF structure when subjected to real earthquakes. It is found that the design displacements and storey drifts from the DDBD procedure for the case study matched relatively well with those recorded through the NLTHA analyses and a new DDBD procedure for CBFs with vertical irregularity is validated.

Seismic performance of self-centering steel-timber hybrid shear wall structures

This paper presents the seismic performance assessment of self-centering steel-timber hybrid shear wall (SC-STHSW) structures. The hybrid structure introduces extra re-centering action to itself through the combination of the post-tensioned steel frame (PTSF) with the light-framed wood shear wall. Slip friction dampers (SFDs) are used as connectors between the frame and the wood wall. An OpenSees model for the SC-STHSW was established and validated versus experimental results. Then, a 9-story SC-STHSW structure was designed with the direct displacement-based design procedure, and the design was checked through nonlinear time history analysis. Based on the designed 9-story SC-STHSW structure, the influence of a key parameter (i.e., self-centering ratio α E ) on the structure's dynamic performance was investigated by considering different hazard levels. The dynamic response of three 9-story SC-STHSW structures with different α E values was compared in terms of maximum inter-story drift (MaxISD) and maximum residual inter-story drift (MaxRISD). Meanwhile, the fragility curve of a 9-story conventional steel-timber hybrid shear wall (STHSW) structure without self-centering capability was calculated based on the cloud analysis, and the fragility curves were compared with those of the 9-story SC-STHSW. It was found that the increase of α E was not only beneficial to decrease the system's MaxRISD but also positive in enhancing system's control over its MaxISD. The lower limit of α E was suggested to be 0.6 when the system was used in high seismic regions. The 9-story SC-STHSW ( α E = 0.6) was comparable to the 9-story STHSW system in regards to the control of MaxISD, which illustrated the advancement of the new system since, at the same time, it had better re-centering ability to decrease post-earthquake repair costs. • A self-centering steel-timber hybrid shear wall structure was designed through the direct displacement-based design. • The influences of α E on the new structure’s seismic performance were explored. • Based on cloud analysis, the seismic fragility of the new structure was compared with the other structure.

Self-centering companion spines with friction spring dampers: Validation test and direct displacement-based design

This paper aims to propose self-centering companion spines with friction spring dampers (denoted as SCS) for obtaining better post-earthquake resilience in steel building structures. Two typical configurations of SCSs in buildings were first introduced. The SCS mainly consists of two rigid spines (RSs) and many friction spring dampers (FSDs). The two RSs have adequate strength and stiffness to create uniform inter-story drift distribution in SCS. The FSDs are the lateral force-resistant and energy-absorbing components and provide self-centering capacity for SCS. Some validation tests were further performed to investigate the hysteretic behavior of the developed SCS. Two different loading protocols were adopted in the experiments to test the stability of SCS subjected to different loading histories. 6% was set as the maximum loading roof drift to indicate the superior deformability of SCS. Moreover, three repeated loadings were performed for each loading protocol to study the behavior of SCS under aftershocks or multiple earthquakes. The experimental results show that the developed SCS can obtain stable and convincing self-centering hysteretic responses under repeated loading with different loading protocols, and no strength or stiffness deterioration can be found. These observations confirm that SCS is fully recoverable after earthquakes and has the potential to be used as a resilient seismic-resistant system in the building located in seismic regions. Lastly, the direct displacement-based design (DDBD) method was developed for SCS. A three-story SCS was designed on the basis of the proposed DDBD method. Nonlinear static and dynamic analyses were performed to validate the efficiency of the proposed DDBD procedure. The numerical results show that the designed SCS can show the anticipated nonlinear behavior and obtain the expected performance target.

Equivalent Viscous Damping of Cross-Laminated Timber Structural Archetypes

This paper attempts to formulate an expression for the estimation of equivalent viscous damping (EVD) for cross-laminated timber (CLT) structural archetypes as a function of ductility. The paper aims at contributing toward the direct displacement-based design procedure (DDBD) of CLT structures, which makes use of equivalent viscous damping and secant stiffness estimations as proxies for the estimation of the nonlinear behavior of structures. The available hysteretic models of CLT structures are very elaborate, and the inelastic time-domain simulations using hysteretic models are not easily manageable. Hence, the adoption of equivalent viscous damping is a crucial feature of any DDBD method. In this paper, the authors estimate the EVD of a set of 16 typologies of CLT buildings with an increasing slenderness ratio. The minimum of the squared error between the maximum drift of the elastic time-history and the hysteretic response gives the EVD. The extended energy-dependent generalized Bouc Wen (EEGBW) model, calibrated on experimental data, is used to predict the inelastic seismic response of the single CLT shear wall. The correlation between EVD and ductility is compared to existing EVD formulations referring to concrete and steel structures.

Efficient direct displacement-based seismic design approach for structures with viscoelastic dampers

Viscoelastic damper (VED) is one of most effective structural control techniques, because it provides both additional stiffness and damping to the structures. Current seismic standards implement force-based methods for the design of VEDs, which generally require extensive iterative calculations. It also causes inconsistent risk degree and economic efficiency because it is displacements, rather than forces, that are directly related to structural damage. In the past decades, researchers have developed a direct displacement-based design (DDBD) methodology. The DDBD approach begins with selecting a target displacement and estimating of the equivalent damping ratio, and then calculate the seismic demand of the structure using a substitutive equivalent elastic system. However, because of the strong nonlinearity of the VED devices, it is difficult to appropriately proportion the key parameters (i.e., damping, period) of the VED controlled structure system, which makes the DDBD method ineffective. To solve this problem, this paper proposed an efficient DDBD approach that is in accordance with the Chinese seismic design principles. The relationship between the equivalent damping and structure period of the VED-structure system is established, so the proposed approach requires minimum iterative calculations and the efficiency improves significantly. A step-by-step description of the approach is presented, in which the two key VED parameters, namely the strain amplitude and its additional damping ratio contribution to the structure, are determined by the target displacements. The approach is further examined and verified by a 24 story building. Nonlinear history analysis results confirm that, by using the proposed DDBD procedure, the designed target story drift is achieved with no iterative calculations and the structural responses of the VED-structure system satisfy the code requirements.

Peak Velocity of Viscous Dampers for Direct Displacement Based Design of RC Moment Resisting Frames

ABSTRACT Previous researchers have extended the direct displacement-based design (DDBD) procedure, to design reinforced concrete moment-resisting frame (RC-MRF) buildings with linear viscous dampers (LVDs). In those studies, the viscous damper forces are found to be larger than the design viscous damper forces because the actual relative structural velocity is different from the pseudo-velocity and also due to higher-mode effects. In the present study, a damper velocity correction factor is introduced along the height of the RC-MRF with LVDs, to calculate the corrected velocity of the LVD from the design velocity of the LVD, when the DDBD procedure is used. The damper velocity correction factor is proposed using the results of nonlinear response history analysis (NLRHA) for a set of RC-MRFs with LVDs, which are designed using the DDBD procedure. The DDBD procedure is then used to design RC-MRFs with nonlinear viscous dampers (NLVDs) making use of the proposed damper velocity correction factor, by calculating the NLVD characteristics using the equal energy (EE) dissipation approach or the equal power (EP) consumption approach. The accuracy of the proposed damper velocity correction factor is evaluated by carrying out NLRHA of two sets of RC-MRFs with NLVDs, one set where the damper velocity correction factor is not used, and the other set where it is used. When the damper velocity correction factor is used; the design LVD forces either agree with NLRHA results or are slightly more conservative, the design NLVD forces are close to the NLRHA results, and the maximum value of peak inter-story drift ratio (IDR) along the height of the RC-MRF from NLRHA is close to the design IDR limit. There is no significant difference in the peak story shear forces whether the damper velocity correction factor is used or not.

Seismic performance of energy-dissipating post-tensioned CLT shear wall structures II: Dynamic analysis and dissipater comparison

The work described herein was conducted within the performance-based seismic assessment framework on both post-tensioned (PT) CLT shear wall structures with UFP dissipaters (PT-Strs-U) or with friction dissipaters (PT-Strs-F). In this paper (Part II of the work), a more extensive study was conducted for the PT-Strs-U and PT-Strs-F, dealing with structural design method, structural modeling, seismic performance assessment, and post-earthquake loss assessment. In the previous study (Part I of the work), both the design capacities of the PT CLT shear walls and the corresponding parameters of the incorporated dissipaters were provided. Using these data, a direct displacement-based design (DDD) procedure was applied for designing both one 8-storey PT-Str-U and one 8-storey PT-Str-F. A simplified structural model was also developed in OpenSees for both structures. Pushover and time-history dynamic analysis were conducted to calibrate the calculated structural performance objectives within the design targets of the DDD procedure. Finally, based on the calibrated structural models, seismic performance of the PT-Str-U and of the PT-Str-F was assessed using the fragility curves. The post-earthquake loss assessment was also conducted afterwards in terms of the inter-storey drift, absolute floor acceleration, and shear deformation of the dissipaters. Overall, the maximum inter-storey drift of the PT-Str-U is similar to that of the PT-Str-F under the earthquakes. In most cases, the friction dissipaters perform better than the UFP dissipaters for mitigating the absolute response accelerations of the structures. When taking the total post-earthquake loss of the structures into account, the friction dissipaters are recommended, which can result in a less post-earthquake loss for the energy-dissipating PT CLT shear wall structure, compared to the UFP dissipaters.

Direct Displacement Based Design of Reinforced Concrete Elevated Water Tanks Frame Staging

Elevated water tanks supported by the reinforced concrete (RC) Staging are classified as inverted pendulum structures. These are considered as structures of high post-earthquake importance and should remain functional after the seismic events. National codes of various countries recommend Force-Based Design (FBD) procedure for water tank staging, which does not ensure nonlinear performance level for a given hazard. Therefore, it becomes necessary to design these structures with a performance-based design approach like Direct Displacement-Based Design (DDBD). Many design engineers consider that the behavior of frame staging of the elevated water tank is similar to the building's frame and generally adopt the same design principles for both types of structures. However, the seismic behavior of the building frame is significantly different from frame staging due to the absence of diaphragm action at the bracing level and concentrated mass at the top level only. Therefore, it may not be rational to utilize the same DDBD procedure of the building's frame for the design of frame staging of the elevated water tanks. The present study proposes some modification in existing DDBD procedure (used for the design of frame building) based on the nonlinear time history analysis of twenty meters high RC frame staging with four different configurations. The modifications are proposed in terms of inelastic displacement profile, design displacement, effective height, and effective mass calculation. Further, the performance of the same RC frame staging designed using the proposed DDBD procedure has been assessed using nonlinear static and dynamic analyses to verify the suitability of proposed modifications.

Optimal DDBD procedure for designing steel structures with nonlinear fluid viscous dampers

This paper presents an optimal direct displacement-based design (DDBD) procedure for designing steel moment-resisting frames equipped with nonlinear fluid viscous dampers (FVDs). The design method is based on defining an optimization problem to determine the optimal distribution of FVDs along the height of the structure to achieve a desired added damping ratio by FVDs with the minimization of total damping coefficient of dampers. The optimization problem is solved using the distributed genetic algorithm (DGA) to determine the damping coefficient of each FVD. For numerical illustration, the proposed optimal DDBD procedure has been used to design four moment-resisting steel frames of 4, 8, 12, and 20-storey equipped with linear and nonlinear FVDs. The effectiveness of the proposed method has then been evaluated by performing nonlinear time history analysis on the considered frames under 20 earthquake records. Moreover, the effect of the displacement profile on the efficiency of the proposed design method has been discussed. The results of extensive numerical simulations show that using the proposed method for optimal distribution of the dampers in DDBD procedure has been considerably effective in reducing the total damping coefficient and maximum force of the dampers while satisfying the expected performance criteria. It has also been found that designing nonlinear FVDs based on the proposed optimal DDBD method have resulted in a significant reduction in the maximum damper force compared to linear FVDs.

A direct displacement-based design procedure for base-isolated building structures with lead rubber bearings (LRBs)

This paper proposes a direct-displacement based design (DDBD) procedure, to make base-isolated building structure with lead-rubber bearing (LRBs) satisfy the performance objectives prescribed by displacement thresholds. To this end, the DDBD for regular building structures is incorporated with the two-degree-of-freedom (2DOF) model of base-isolated building structures. Fundamentals of the standard DDBD procedure is briefed at first. The equivalent linearized 2DOF model of a base-isolated building structure is then detailed. Modal analysis and response spectrum analysis of this 2DOF model has been carried out, which suggests a simplistic and readily-invertible closed-form relationship from structural characteristics to seismic displacements. Given the displacement requirements, the relationship facilitates the solution of the structural parameters of the base-isolated building structure. Following the basic steps of DDBD but utilizing this 2DOF model as the substitute for the original nonlinear structural model, an alternative DDBD procedure is then developed and detailed. This DDBD procedure is verified by numerical examples, in which the nonlinear time-history analysis of the designed structures yields seismic displacement results matching the preset target thresholds well. The design solutions of the proposed procedure are also compared with those given by an existing one. Numerical results indicate that the proposed DDBD procedure is reliable, straightforward, and convenient for the seismic design of base-isolated building structures with LRBs.

Seismic performance assessment of conventional CLT shear wall structures and post-tensioned CLT shear wall structures

This paper presents the performance-based seismic assessment framework on conventional CLT shear wall structures (C-CLTStrs) and post-tensioned CLT shear wall structures (PT-CLTStrs). Numerical models of both the conventional CLT shear walls and the PT CLT shear walls were developed and calibrated with experimental results. A direct displacement-based design (DDD) procedure was then developed and demonstrated by the design examples of one 8-storey C-CLTStr and a set of 8-, 12-, and 16-storey PT-CLTStrs. Corresponding simplified structural models were developed. A series of pushover and time-history dynamic analysis was conducted afterwards to calibrate the calculated structural performance objectives with the design targets of the DDD procedure. Finally, using 50 mainshock (MS) and 50 mainshock-aftershock (MSAS) earthquake records, probability seismic demanding analysis (PSDA) were conducted to evaluate two engineering demanding parameters of the structures, namely the maximum inter-storey drift (MaxISDR) and the residual inter-storey drift (ResISDR). The MaxISDR limit states of the PT-CLTStrs are recommended at 0.7%, 1.4%, and 2.2% for the immediate occupancy (IO), life safety (LS), and collapse prevention (CP) hazard levels, respectively. The MaxISDR of the PT-CLTStrs is almost twice that of the C-CLTStrs, while the ResISDR is at most 50% of that of the C-CLTStrs. The damage probability of the 8-storey PT-CLTStr indicated by ResISDR is not significantly sensitive to the consideration of aftershocks seismic input, whereas, when aftershocks are considered, up to a 7.0% difference under the LS hazard level is caused in the damage probability of the 8-storey C-CLTStr indicated by ResISDR.

Structural Design and Modelling Method for the Post-tensioned CLT Shear Wall Structures

This paper presents the direct displacement-based design (DDD) procedure, structural modelling method, and structural performance calibration for post-tensioned CLT shear wall structures (PT-CLTstrs). Numerical models of the post-tensioned (PT) CLT shear walls were developed and calibrated with the experimental results. Based on the developed shear wall models, parametric analysis were conducted to investigate the lateral performance influencing factors. Then, a DDD procedure was developed and demonstrated by the design examples of a set of 8-, 12-, and 16-storey PT-CLTStrs. The corresponding simplified structural models were developed, and then a series of pushover and time-history dynamic analysis were conducted to calibrate the calculated structural performance objectives with the design targets of the DDD procedure. Finally, the empirical cumulative distribution functions (CDFs) of the maximum inter-storey drift (MaxISDR) were constructed. It is found that when the width of the PT CLT shear walls increases from 1.8 m to 3.0 m, the base shear at the drift of 2.0 % increases by twice accordingly. When the diameter of the PT strand increases from 15.2 mm to 34.6 mm, the base shear at the drift of 2.0 % increases by up to five times. Additionally, the MaxISDR limitation of the PT-CLTStrs is recommended as 2.2 % under the collapse prevention (CP) hazard level. The study results can serve as guidelines for the development of engineering design methods for the PT-CLTStrs.

DDBD assessment of steel CBFs using full scale shake table tests with realistic connections

A direct displacement based design (DDBD) methodology for concentrically braced frames (CBFs) is validated using a combination of experimental and numerical data taking into account the behaviour of brace-beam-column connections. Twelve full-scale shake table tests with four different brace cross-section sizes, two different gusset plate design methodologies and two different brace connection configurations are used to evaluate the design procedure at three different performance levels (continued operation, life safety and collapse/near collapse) through successive scaling of test excitations.It is found that the DDBD procedure can predict the behaviour of the shake table CBFs for the three levels of excitation. Accurate results have been found for life safety design where the mean ratio and coefficient of variation of base shear (Fb) calculated using the DDBD method to the values found from shake table tests are 0.96 and 0.17, respectively. Furthermore, the mean ratio and coefficient of variation of brace area (Ab) calculated from the DDBD method to the values used in shake table tests are 1 and 0.08, respectively.The design procedure was also verified using nonlinear time history analysis (NLTHA) using a selection of earthquake records with displacement spectra compatible with those of the exact accelerograms recorded during the shake table tests. At the life safety performance level, the mean peak displacement demand and base shear values obtained from NLTHA were observed to be 82% and 91%, respectively, of the design values determined by the DDBD methodology.

An investigation of P-delta effect in conventional seismic design and direct displacement-based design using elasto-plastic SDOF systems

The seismic responses of structures are always influenced by P-Δ effects. The significance of this effect may be negligible when the structure responds elastically but is very important when the structure responds into inelastic range. The P-Δ effect usually increases the displacement response of structures. It may even cause dynamic instability when the structure subjected to severe earthquake ground motions. In conventional seismic design codes, the P-Δ effect usually leads to an increase in design base shear. The same approach is used in direct displacement-based design. Various researchers have proposed different relationships for base shear increase which can be used in different seismic design approaches such as force-based design and performance-based design. In this paper, the proposed expressions are reviewed extensively and their adequacy is evaluated by detail. Then two main purposes are pursued: (1) the development of new expressions for strength amplification factor due to P-Δ effect for current force-based design seismic codes; and (2) the modification of equivalent viscous damping or the required additional base shear considering the P-Δ effect in direct displacement-based design procedure. For this purpose, a new algorithm based upon elasto-plastic SDOF system analyses is presented. The algorithm is implemented 102,500 times overall for different periods, ductility levels, stability indices and different earthquake ground motions that each implementation needs a large amount of trial and error process in linear and nonlinear SDOF systems. The results seem to present a good development of the P-Δ effect relations for the seismic design procedures.