Collapse-based seismic design method

Abstract

Continuing the structural analysis until the model reaches its ultimate capacity using an incremental nonlinear time history analysis can incorporate all the behavioral complexities of the structure. This approach ensures that all factors influencing the seismic behavior of structures, which can be simulated in numerical modeling, are fully considered in the analysis. This technique addresses many shortcomings in seismic analysis and structural design, using simple criteria for the accurate design of members. With this perspective, this paper explores the collapse-based seismic design method and provides an approach to determine the ultimate tolerable seismic intensity for a structure, or an appropriate collapse seismic intensity. It also establishes design criteria based on the maximum capacity of structural members under maximum loading and utilizes the endurance time method to perform incremental nonlinear time history analysis quickly and easily. Evaluations show that models designed using this method achieve the desired performance level at different seismic hazard levels, provide sufficient seismic capacity, meet all minimum requirements of the seismic code and standards, and are optimized. Therefore, the procedure presented in this paper is introduced for the analysis, design, evaluation, and retrofitting of structures with simple, accurate, and uniform principles.

Keywords

Collapse-based design (CBD)endurance time method (ETM)moment resisting frame (MRF)concentrically braced frame (CBF)collapse seismic intensity of structures collapse spectrum

Introduction

In contemporary seismic engineering, seismic codes and structural analysis and design standards are evolving to incorporate advanced structural systems and methodologies, aiming to effectively analyze and control structures through equivalent analysis techniques (ANSI/AISC 360-22, 2022; ANSI/AISC 341-22, 2022;ACI Committee 318, 2019; ASCE/SEI 7-22, 2022; EN 1991-1-1:2002, 2002; EN 1992-1-1:2004, 2004; EN 1993-1-1:2005, 2005; EN 1998-1:2004, 2004). These guidelines represent the new generation of earthquake-resistant design codes, which are being developed to increase the resilience of structures. While there is no limit on the use of advanced methods, seismic analysis and design guidelines strive to prepare structures for proper behavior in future seismic events by equating the real effects of earthquakes on structures. Equating earthquake effects on structures involves applying hypothetical equivalent loads in specific locations on structures to simplify the analysis process. Although this practice has significantly simplified the analysis, it has led to the creation of numerous formulas, parameters, factors, and concepts that are not only error-prone and vary for different structures, but also diverge from the real nature of the behavior of structures under seismic events. Today, although the seismic design of structures is considered one of the most complex engineering sciences, the approaches inevitably used in contemporary practice have increased this complexity. The situation has now changed significantly, and continuing with the old principles only slows progress and keeps the seismic design of structures complicated and imprecise. Persisting on this path widens the gap between equalization and reality, but ultimately, real simulation will replace these simplified approaches.

These approaches were reasonable in the past due to limitations and the lack of necessary tools. However, despite progress, persisting with these outdated processes only serves to maintain complexity in analysis and design. The most realistic method for seismic analysis of structures is nonlinear time history analysis. In contrast, methods like equivalent static, response spectrum, and pushover are equivalent analysis methods, each attempting to simulate real earthquake effects to some extent. These methods were developed when nonlinear time history analysis for structural design was not feasible. If such analysis could now be conducted easily and quickly, there would be no need for equivalent analysis methods. Several factors contribute to the continued usefulness of equivalent analysis methods for designing structures. These include the complexity of selecting and adjusting ground motions, the variability of structural behavior to ground motions, the necessity of using multiple ground motion records, the time-consuming nature of analysis, the volume of data and results, the complexity of interpreting analysis results, and the absence of a suitable design method. Despite being the most efficient tools currently available for seismic design of structures, relying solely on previous principles prolongs the analysis and design process, making it complex and dependent on sub-parameters.

For more accurate analysis and design of structures, the nonlinear behavior of structural members is now modeled, with significant efforts directed toward developing methods that design structures based on their plastic behavior. To achieve this, methods such as performance-based seismic design (Leelataviwat et al., 1999; Priestley, 2000), performance-based plastic design (Goel et al., 2010; Sahoo and Chao, 2010), direct displacement-based seismic design (Chopra and Goel, 2001; Priestley and Kowalsky, 2000), and various earthquake-resistant design concepts (Gholizadeh and Moghadas, 2014; Grigorian et al., 2023; Moehle, 1992) have been introduced to replace the traditional force-based seismic design approach. Table 1 shows various methods of analysis, design, and seismic assessment of structures across different categories.

Table 1.

Seismic Analysis and Structural Design Methods.

Analysis methods	Design and seismic assessment methods	Categories
Equivalent static analysis	Force-based seismic design method	FBD
Response spectrum analysis
Linear time history analysis
Nonlinear static pushover analysis	Performance-based seismic design method	PBD
Nonlinear time history analysis	Performance-based plastic design method
	Displacement-based seismic design method
	Energy-based seismic design method
Nonlinear time history analysis	Seismic performance assessment	Seismic assessment methods
Incremental dynamic analysis	Probabilistic assessment of vulnerability
Incremental dynamic analysis	Seismic reliability assessment
Endurance time analysis	Seismic risk assessment
Endurance time analysis	Seismic resilience assessment
Endurance time analysis	Collapse-based seismic design method	CBD

Novel approaches to addressing shortcomings

Studies indicate that employing new methods and modeling the plastic behavior of structural members can significantly enhance the seismic performance of structures. However, most methods still adhere to traditional approaches and lack the capacity to be used independently for the analysis and design of new buildings. This limitation arises because new methods cannot encompass all the details and influencing factors. They often equate seismic effects and perform incomplete analyses, necessitating the use of additional parameters and seismic provisions to cover all effective factors comprehensively.

For example, if seismic provisions like the strong column-weak beam or capacity design concept are not applied to a model using a force-based design method, and the structure is designed solely based on performance-based design principles at the target displacement, the columns will remain completely elastic at this displacement, corresponding to the DBE (Design Basis Earthquake) or MCE (Maximum Considered Earthquake) hazard levels. However, the columns and other non-ductile lateral load carrying members will still lack the required capacity. This is because applying these seismic provisions ensures that the non-ductile lateral load carrying members remain elastic even beyond the DBE and MCE hazard levels, allowing the ductile members to provide the necessary ductility through their plastic behavior. Therefore, these seismic provisions must either be applied to the model using a force-based design method, or similar provisions must be defined for the performance-based seismic design method. In both cases, the analysis and design process remains complicated. However, if the analysis continues until collapse and the structural members are designed to withstand an appropriate collapse seismic intensity, these seismic provisions and similar requirements can be eliminated. This allows all structural members to be designed using simple and uniform principles, as demonstrated in the models designed in this article.

Seismic performance factors, such as the response modification coefficient (R), overstrength factor (Ω₀), and deflection amplification factor (C_d), are all determined based on the ultimate capacity of the structure at the point of collapse. If the analysis is carried out only up to the target displacement or the DBE or MCE hazard levels, these factors cannot fully influence the analysis. Buildings are typically designed for life safety performance at the seismic intensity of the DBE and for collapse prevention performance at the MCE. However, all design control parameters are adjusted to ensure that structural systems achieve a specific ultimate seismic capacity. Therefore, although analyzing the structure up to the target displacement is more appropriate than a linear analysis, it remains an incomplete approach that introduces problems in the design process. This has necessitated the definition of supplementary concepts, formulas, parameters, and seismic provisions for resolution. If the analysis of the model continues until collapse, these factors, along with issues such as the P-delta effect, strain-hardening, and stiffness deterioration, can directly, accurately, and completely influence the model’s analysis (Deyanova et al., 2023; Jouneghani and Haghollahi, 2020a; FEMA P440A, 2009). Consequently, certain factors, formulas, parameters, and seismic provisions can be eliminated from the analysis and design process.

In designing structures, the objective is to select the optimal section for each structural member. However, in performance-based design methods like ASCE/SEI 41-17, 2017 and similar approaches, the defined life safety performance level for ductile members is quite broad. At the target displacement, various sections can be assigned to a ductile member while still exhibiting plastic behavior within the life safety performance level. This makes it unclear how to determine the best and most suitable section, leading to a qualitative approach in structural member design. Evaluations using incremental dynamic analysis (IDA) have shown that assigning the smallest acceptable section in such scenarios results in insufficient seismic capacity, failing to meet minimum seismic code requirements. This highlights the necessity, when designing new structures using performance-based methods, to ensure that the designed model complies with the seismic requirements stipulated by the seismic code and standards. The new methods may contain numerous undisclosed weaknesses. However, in the method outlined in this paper, the structure is subjected to the maximum seismic intensity, and each structural member is designed based on the ratio of applied loads to their maximum plastic capacity. Accordingly, the criteria for designing structural members are clear, enabling the determination of the best section for each member with high precision.

In structural design, the target displacement varies with the model’s stiffness (ASCE/SEI 41-17, 2017). While the target displacement refers to the roof displacement, interstory drift is more critical for designing structures. FEMA 356 (2000) specifies the appropriate interstory drift value for each structural system at different seismic hazard levels. However, even models of the same structural system have different stiffness and periods, which directly affect the target drift value (ASCE/SEI 41-17, 2017). FEMA P58-1, 2012 provides a more accurate yet complex method to determine the target drift value. Nonetheless, this value is always an estimate, and estimates inherently carry errors. On the other hand, an interstory drift ratio of 2% or 2.5%, considered the target drift ratio in some new methods, may be suitable for plastic design of moment resisting frame systems (Noruzvand et al., 2021), but it is not appropriate for braced frames (Jaberi et al., 2024) or other structural systems (Jouneghani et al., 2021). Due to the variability and dependence of physical displacements on many factors, using this parameter as a design criterion is problematic. Despite the challenges, current analysis and design approaches necessitate considering this factor. However, the method presented in this paper allows for designing structures without relying on physical displacements as a primary criterion while still effectively managing them. Furthermore, after analyzing the model, the collapse displacement value and the displacements corresponding to each seismic hazard level are identified. The maximum displacement and drift at which the model can maintain stability can also be easily determined.

Differences in structural analysis and design methods

As illustrated in Figure 1, the force-based design (FBD) method involves designing structures based on the elastic behavior of structural members under reduced seismic intensity, adjusted by the response modification coefficient. This approach requires numerous parameters and seismic provisions to accurately adjust the plastic capacity of the structure. In contrast, the performance-based design (PBD) method aims to enhance analysis and design accuracy while minimizing estimations by modeling the plastic behavior of structural members. The structure is then designed to meet a specific performance level under the target seismic intensity. The collapse-based design (CBD) method focuses on realistic structure modeling and seismic loading simulations to simplify and improve the accuracy of the analysis and design process. This method involves performing incremental nonlinear time history analysis up to the appropriate collapse seismic intensity of the structure. All structural members are then designed based on their maximum capacity under the target seismic intensity. This method can operate independently of seismic codes and many provisions by utilizing plastic behavior modeling, quick and simple analysis with the endurance time method (ETM), and designing structural members based on the ratio of applied loads to maximum capacity.

Figure 1.

Seismic loading levels in various design methods.

Additionally, there are seismic assessment methods, which, while not designated as specific design methods, can enhance the seismic capacity of models and focus on the plastic behavior of structures. These methods evaluate the vulnerability and damage of the structure under various seismic hazard levels probabilistically or quantify the model’s maximum capacity (Baker, 2015; Hassanzadeh and Gholizadeh, 2019; Mahsuli and Haukaas, 2013; Mirfarhadi et al., 2021; Salimbahrami and Gholhaki, 2018; Zareian and Krawinkler, 2007). By using seismic assessment methods, damage levels and structural weaknesses or strengths are identified, which can then inform design refinements or seismic evaluation objectives. Furthermore, new methodologies for the plastic design of earthquake-resilient structures are rapidly developing (Grigorian et al., 2019). Expectations for predetermined performance objectives have increased, leading to the proposal of design methods that enable rapid return to occupancy after a severe earthquake and the reuse of buildings (Jaberi and Asghari, 2020a, 2020b; Shoeibi et al., 2017). Effective techniques to prevent progressive collapse have also been introduced (Brunesi et al., 2016; Ravasini et al., 2021). Emphasis is increasingly placed on designing structures to meet the resilience needs of structures and infrastructure during seismic events (Dong et al., 2022).

Necessity of structural analysis until collapse

The performance-based design method is more accurate than the force-based design method because it accounts for the nonlinear behavior of structural members and designs for higher seismic hazard levels. Similarly, the collapse-based design method achieves greater accuracy by designing structural members to withstand ultimate seismic intensity based on their maximum capacity. The key is to create a scenario where all factors influencing the seismic behavior of the structure are completely and directly considered in the model analysis. When designing a structure for DBE or MCE hazard levels, numerous parameters and factors may not be fully accounted for in the analysis. This necessitates additional controls and criteria to ensure the required seismic capacity of the model and to fully incorporate the effects of critical factors. Analyzing the structure only up to the target displacement or seismic intensity at DBE and MCE levels is incomplete, and designing the structure based on this limited analysis increases reliance on supplementary parameters. However, if the structure is analyzed using nonlinear time history analysis by the endurance time method until it reaches an appropriate collapse level, all factors affecting the seismic behavior of the structure can be directly and accurately included in the model analysis. This approach makes certain formulas, factors, parameters, and seismic provisions unnecessary, allowing for the design of structures using simpler and clearer principles. In such a scenario, structural irregularities, system indeterminacy, overstrength, the ability to provide the required ductility, and the ability to maintain stability during lateral displacements, as well as specific behavioral complexities, are all directly and realistically considered in the nonlinear time history analysis. The stability of the structure after this analysis indicates that the designed model can withstand all the numerically modeled factors influencing seismic behavior. Meanwhile, other factors not included in the numerical modeling can still be addressed through seismic provisions.

Using the endurance time method for simple and rapid nonlinear time history analysis

As mentioned, seven factors generally hinder the efficiency of nonlinear time history analysis for designing structures. However, advances in software have made this analysis faster and easier to perform, and this development will continue. And this paper presents a suitable design method using nonlinear time history analysis at the collapse level of structures. Issues such as the complexity of selecting and adjusting ground motions, the variability of structural behavior under different ground motions, the need for multiple ground motion records, the time-consuming nature of the analysis, the large volume of data and results, and the complexity of interpreting these results can be effectively addressed using the endurance time method.

Endurance time method is a novel approach for conducting nonlinear time history analysis, enabling structural analysis through incremental artificial ground motion (Estekanchi et al., 2008). This method not only mitigates the primary drawbacks of time history analysis but also offers many practical advantages. In endurance time method, incremental artificial ground motion is generated to simulate a series of ground motion records or specific earthquakes (Estekanchi et al., 2020). As depicted in Figure 2, the results of nonlinear time history analysis using ETM ground motion closely resemble the median outcomes of nonlinear time history analysis using the target ground motion record set. Therefore, the endurance time method facilitates nonlinear time history analysis using incremental artificial ground motion (Mohsenian et al., 2022). Typically, in nonlinear time history analysis, the median results are considered the primary outcomes due to the variability of analysis results under different ground motion records (Cavalieri et al., 2023). ETM ground motion can quickly and accurately obtain these median results, making it a viable substitute for real ground motion record sets (Hariri-Ardebili et al., 2014). Comparisons of analysis outcomes using ETM-generated artificial ground motion with the median results from nonlinear time history analysis using a real ground motion record set have demonstrated that ETM can reliably estimate the median results of time history a`25 nalysis (Jaberi and Asghari, 2022).

Figure 2.

Comparison of the results of endurance time analysis with the median results of nonlinear time history analysis at different seismic intensities. (Jaberi and Asghari, 2022).

The response spectrum of an ETM ground motion can perfectly match the design spectrum or the median response spectrum of a set of ground motion records (Valamanesh and Estekanchi, 2013, 2014). As shown in Figure 3, the ground motions in the endurance time method are created incrementally, with their response spectrum increasing during the ground motion while maintaining a form similar to the target spectrum. The incremental nature of the ground motion and the uniform increase of the response spectrum over time allow for obtaining the median results of incremental dynamic analysis using an incremental nonlinear time history analysis (Estekanchi and Basim, 2011). Additionally, this method enables the evaluation of structural behavior at different seismic hazard levels, as a specific seismic intensity is applied to the structure at every second of the artificial ground motion (Ahmadie Amiri et al., 2022). This incremental approach allows for a detailed assessment of the structure’s performance under varying levels of seismic intensity. Accordingly, the endurance time method is utilized as a fast and straightforward time history analysis technique.

Figure 3.

The scaled artificial incremental ground motion of the endurance time method and its acceleration response spectrum at different seismic hazard levels.

Studies on this method demonstrate its accuracy and efficiency, enabling the rapid attainment of median results from nonlinear time history analyses. Currently, advanced procedures for generating ETM ground motions are available (Mashayekhi et al., 2019a; 2019b), and these generated ground motions are accessible (Estekanchi, n.d.). Although generating ETM ground motions is somewhat intricate and requires a deep understanding of creating incremental artificial ground motions and earthquake characteristics, using these generated ground motions is straightforward. It’s important to note that there’s no need to produce a distinct ground motion for each structure. Similar to the design spectrum, ETM ground motions are generated for a region, allowing all structures within that region to be analyzed using the same ground motion. In this approach, employing an ETM ground motion instead of a set of ground motion records is adequate. The resulting analysis can be considered as providing median outcomes comparable to those obtained using a set of ground motion records, and other analysis steps are similar to traditional nonlinear time history analysis.

Given that the response spectrum of the generated ground motion is akin to the design spectrum or the median response spectrum of a set of ground motion records, scaling is straightforward. By directly scaling the acceleration response spectrum of the ground motion to the design spectrum at each seismic hazard level, analysis results for that level can be obtained. Figure 3 illustrates how the ground motion can be scaled, determining various seismic intensities based on the design spectrum of seismic hazard levels throughout the duration of the ETM ground motion.

Using a single ground motion for the analysis and design of structures might initially seem to contradict seismic codes. However, the endurance time method can effectively adjust the structure’s behavior based on the analysis results from a set of ground motion records. In any case, evaluating the efficacy of designed models is straightforward, and verification can be conducted using one of the endorsed methods or procedures outlined in seismic codes.

Collapse-based seismic design method

Introduction

The collapse-based design method represents the third generation of seismic analysis and design techniques, following the force-based and performance-based design methods. The aim of this method is to better capture the true nature of seismic behavior in structures and provide a more realistic analysis and design process. It seeks to simplify the analysis and design process, facilitating advancements in seismic design science. As new analysis and design software strive to model structures with greater detail, offering a more realistic method can help keep pace with these developments. The collapse-based design method follows a straightforward yet accurate process based on two principles:

⁃ Analyzing the structure under the ultimate seismic intensity (referred to as collapse seismic intensity).

⁃ Utilizing the maximum capacity of structural members against the applied loads at the target seismic intensity.

In this method, the goal is for the structure to remain stable under the target seismic intensity, equivalent to an appropriate collapse intensity for the structure. To achieve optimal design, the maximum plastic capacity of structural members is utilized against loading. Although the likelihood of such a severe earthquake may seem low, accurately and straightforwardly designing structural members necessitates conducting nonlinear time history analysis up to the ultimate capacity of the structure. By employing this method, the structure’s stability against a severe earthquake can be more confidently assured.

To design a structure using the collapse-based design method, at least one ETM ground motion is created for each seismic zone, with a response spectrum that completely matches the design spectrum. Then, the appropriate collapse seismic intensity of the model is determined using equation (1) to establish the target seismic intensity. Subsequently, the nonlinear time history analysis of the model is performed by multiplying a calculated scale factor in the ETM ground motion. Following the analysis, the loads on any structural member must not exceed its maximum plastic capacity. Ensuring that the plastic behavior of all structural members does not exceed their capacity curve maintains the structure’s stability and enables it to withstand applied loads.

In this method, many traditional concepts and parameters can be discarded, including the response modification coefficient, overstrength factor, deflection amplification factor, ductility factor, lateral force distribution over height, target displacement, structural performance levels, base shear, and structure stiffness. Furthermore, none of the formulas, parameters, factors, or controls from seismic codes are utilized, except for the design spectrum and the fundamental period of the structures, which are employed to adjust the appropriate seismic intensity.

Target seismic intensity (the intensity at which the structure collapses)

Seismic hazard levels are typically defined by seismic intensities with a probability of exceedance in 50 years and an associated return period. For instance, seismic hazard levels with 2%, 10%, 20%, and 50% probability of exceedance in 50 years are established, each corresponding to a specific return period. In most seismic codes, buildings are designed for the design basis earthquake (DBE), which has a 10% probability of exceedance in 50 years and a return period of 475 years. Buildings at this seismic intensity should meet the life safety performance level. In the event of the maximum considered earthquake (MCE), which has a 2% probability of exceedance in 50 years and a return period of 2475 years, the building would likely fall into the collapse prevention performance level. However, these estimates of future seismic behavior are tentative. To ensure accuracy, the model must be precisely analyzed at both DBE and MCE hazard levels. While seismic codes typically require buildings to be designed for the DBE, this involves considering the elastic behavior of structural members under seismic intensity reduced by the response modification coefficient. To achieve the design performance objectives, seismic performance factors and various seismic provisions are applied.

Incremental dynamic analysis (IDA) is a method used to determine the collapse seismic intensity of structures (Vamvatsikos and Cornell, 2002). In this method, ground motion records are applied incrementally, starting from low intensities and increasing until the structure collapses. This process helps determine the collapse seismic intensity of the structure. However, each ground motion record induces collapse at a different seismic intensity, leading to variation in the collapse intensity for a structure subjected to different ground motion records. As shown in Figure 4, when structural collapse criteria are applied and a 10% interstory drift ratio is considered as the ultimate drift limit, all 44 ground motion records at different intensities caused the model to collapse. For this model, the smallest collapse intensity is 0.55 g, and the largest is 2.83 g. Thus, collapse intensity varies for each structure and each earthquake. Due to these variations in collapse seismic intensities, the median collapse intensity ( ${\hat{S}}_{CT}$ ) is considered. This value is determined based on the collapse fragility curve and represents the seismic intensity at which 50% of the ground motion records caused the structure to collapse.

Figure 4.

Combining the fundamental concepts of the structures’ behavior under incremental dynamic seismic loading to achieve an appropriate collapse seismic intensity.

In the IDA method based on FEMA P695 (2009), the S_MT factor represents the seismic intensity of the maximum considered earthquake. According to the design performance objectives, structures must be prevented from collapsing at the maximum considered earthquake hazard level. Therefore, if the designed model has sufficient seismic capacity, most or all of the ground motion records at larger spectral accelerations than S_MT can cause the model to collapse. One of the most important results in the IDA method is the collapse margin ratio (CMR). This parameter indicates the seismic capacity of the structure, which is equal to the ratio of the median collapse intensity ( ${\hat{S}}_{CT}$ ) to the seismic intensity of the maximum considered earthquake (S_MT). All factors affecting the structural seismic capacity are incorporated into the collapse margin ratio value. Since building frames are designed for a specific earthquake intensity, it is expected that their seismic capacity ratio, despite different geometric and loading conditions, structural systems, and seismic zones, remains consistent. This consistency makes CMR a comparable parameter for expressing the seismic capacity ratio. Essentially, a structure with a larger CMR factor has a greater seismic capacity relative to its design seismic intensity. It should be noted that the CMR factor is the ratio of the median collapse intensity ( ${\hat{S}}_{CT}$ ) to the seismic intensity of the maximum considered earthquake (S_MT). Therefore, by multiplying the seismic intensity of the MCE hazard level by an appropriate collapse margin ratio, the median collapse intensity can be determined, utilizing the concept of the IDA method.

In structural and earthquake engineering, due to earthquake uncertainties and the varying behavior of structures under different ground motions, the median result and data are typically considered a suitable quantitative value. According to the IDA method concept, the seismic capacity of each structure (collapse margin ratio factor) equals the ratio of the median collapse intensity to the seismic intensity at which the collapse of the structure must be prevented (spectral acceleration value at the fundamental period of the structure on the MCE spectrum, denoted as S_MT). Since the seismic capacity is presumed to be uniform for all buildings with a similar seismic importance factor, the MCE spectrum can be augmented using an appropriate collapse margin ratio to depict the collapse spectrum at the median collapse intensity.

The design spectrum concept has always been suitable and effective for designing structures, as it is determined for each zone based on the soil type and seismic risk level. Also, the design spectrum can still be used to determine the collapse seismic intensity for each structure. In the collapse-based design method, an artificial incremental ground motion of the endurance time method is used, which directly achieves the median of the analysis results using a set of real ground motion records. The median collapse intensity is determined by multiplying the MCE spectrum by an appropriate collapse margin ratio. Accordingly, as shown in Figure 5, both the appropriate design seismic intensity and the seismic intensity applied to the structure using the scaled ETM ground motion are adjusted using a collapse spectrum at the median collapse intensity.

Figure 5.

Scaling the ETM ground motion based on the acceleration response spectrum to the median collapse intensity.

Furthermore, a crucial aspect in the design of structures using nonlinear time history analysis concerns the disparity between the fundamental period (seismic code period) and the numerical modeling period (first mode period). This disparity directly impacts the seismic intensity exerted on the structure. During structural analysis for design purposes, seismic intensity is adjusted based on the seismic code, using the fundamental period. However, in nonlinear time history analysis, the analysis relies on the period of the numerical model modes. When there is a significant difference between the fundamental period and the first mode period of the model, and the first mode period surpasses the fundamental period, a lower seismic intensity is applied to the structure. This intensity could fall below the minimum seismic loading outlined in the seismic code. Thus, it is imperative to increase the seismic intensity applied to the structure, based on the ratio of spectral acceleration at the fundamental period to the spectral acceleration at the first mode period of the model. As shown in Figure 5, endurance time method ground motions are typically generated based on the design spectrum at the DBE hazard level. In the collapse-based design method, the generated or selected ground motion must possess a response spectrum that closely matches the design spectrum to apply the appropriate and reliable seismic intensity to the structure at all stages. Once a suitable ETM ground motion is created or selected, it is then multiplied by the SF_TOT scale factor to produce the target seismic intensity. The SF_TOT scale factor is calculated using equation (1).

{S F}_{T O T} = {S F}_{S A R} * {S F}_{D S R} * {S F}_{C M R}

(1)

To determine the appropriate collapse seismic intensity for each structure, three factors are employed to enhance the intensity of the ETM ground motion. Firstly, the scale factor of spectral acceleration ratio (SF_SAR) is used to adjust the seismic intensity of the first mode period of the model to match the seismic intensity of the fundamental period of the structure according to the seismic code. Secondly, the scale factor of design spectrum ratio (SF_DSR) is applied to elevate the seismic intensity from the DBE level to the MCE level. Thirdly, the scale factor of collapse margin ratio (SF_CMR) is used to raise the seismic intensity from the MCE level to an appropriate intermediate collapse seismic intensity level. Accordingly, after choosing or generating a suitable ETM ground motion and determining the scale factors necessary to achieve the appropriate collapse intensity for the model within the ground motion, nonlinear time history analysis of the model is conducted using the scaled ETM ground motion.

Scale factor of spectral acceleration ratio (SF_SAR)

To determine the seismic intensity for designing buildings, seismic codes typically use the fundamental period. However, in the collapse-based design method, nonlinear time history analysis is employed, which operates based on the modes of the numerical model. This necessitates adjusting the design seismic intensity according to the seismic code, taking into account the disparity between the structure’s fundamental period (T) and the first mode period of the model (T₁). This discrepancy directly affects the seismic intensity applied to the model. To address this, the ratio of the spectral acceleration at the fundamental period (Sa_FP) to the spectral acceleration at the first mode period of the model (Sa_FMP) is used as a modification factor. When the first mode period of the model is shorter than the fundamental period, the first mode period is used for the design. Consequently, the scale factor of the spectral acceleration ratio (SF_SAR) cannot be less than 1, and this value is calculated using equation (2).

{S F}_{S A R} = ({S a}_{F P} / {S a}_{F M P}) \geq 1

(2)

Scale factor of design spectrum ratio (SF_DSR)

To determine the collapse seismic intensity, the seismic intensity of the maximum considered earthquake is increased using an appropriate collapse margin ratio. However, the ETM ground motions are typically generated based on the seismic intensity of the design basis earthquake. Therefore, it is necessary to increase their seismic intensity to the MCE level. This is achieved by multiplying the ETM ground motion by the SF_DSR scale factor, which adjusts the seismic intensity from the DBE to the MCE hazard level. The SF_DSR scale factor is defined as the ratio of the MCE to DBE spectrum and is calculated using equation (3). In most seismic codes, this value is typically 1.5.

{SF}_{DSR} = {MCE}_{spectrum} / {DBE}_{spectrum}

(3)

Scale factor of collapse margin ratio (SF_CMR)

The appropriate value of the collapse margin ratio can be determined through various techniques, including engineering judgment, statistical studies of approved models, or by following the method provided in FEMA P695. Incremental dynamic analysis of building frames has revealed that well-designed models typically exhibit a CMR ranging from 1.5 to 2.5 (Asghari and Saharkhizan, 2019; Jaberi et al., 2024; Jouneghani and Haghollahi, 2020b). Given that a similar seismic capacity ratio is generally expected for all structural systems, a CMR of two is considered optimal and acceptable. Nevertheless, additional research could help establish a more precise value for the appropriate collapse margin ratio.

Design criteria of structural members in collapse-based design method

There are various methods for numerically modeling the plastic behavior of structural members. As shown in Figure 6, these methods can be divided into two categories: concentrated plasticity and distributed plasticity. In the category of concentrated plasticity, the plastic behavior of structural members is modeled based on force-displacement using plastic hinges or nonlinear springs. In the category of distributed plasticity, it is based on stress-strain using finite length hinge zones, fiber sections, and finite elements. In the collapse-based design method, the objective is to ensure that at the target seismic intensity, the plastic behavior of any structural member does not exceed its ultimate capacity limit. Therefore, after analyzing the model, the plastic behavior of each structural member and its ultimate capacity must be specified. By determining the plastic behavior of structural members and the ultimate limit of their capacity, each member can be designed based on the ratio of the applied loads to the maximum considered capacity. This ratio must always be less than or equal to one for each member. Accordingly, in the collapse-based design method, the primary design criterion is to calculate the ratio of loads to capacity for each structural member. Based on equation (4), the optimal section for each member will be the one where the ratio of its maximum plastic behavior to its ultimate plastic capacity is less than or equal to one. For this purpose, the plastic behavior of the members can be modeled using any of the mentioned methods, and the following two methods can be used to determine the capacity of the structural members.

S u i t a b l e s e c t i o n f o r e a c h m e m b e r \Rightarrow (a p p l i e d l o a d s / u l t i m a t e c a p a c i t y l i m i t) \leq 1

(4)

Figure 6.

Numerical modeling techniques of the plastic behavior of structural members. (Roohi et al., 2021).

First method: drawing capacity curve

For each structural member in various situations and under different loading conditions, the capacity curve can be drawn as a backbone curve based on the formulas presented in ASCE 41-17. As shown in Figure 7, as long as the plastic behavior of a structural member does not exceed its capacity curve, that member has the required capacity to withstand the applied loads. However, if the member’s plastic behavior exceeds its capacity curve, it indicates insufficient capacity, necessitating an increase in its section. Also, formulas in ASCE 41-17 and other similar sources can be used to determine the maximum elastic capacity of members. This method is very common in modeling the plastic behavior of members using plastic hinges, although it can be applied to all methods for modeling plastic behavior and drawing capacity curve (backbone curve). Software such as ETABS: CSI, 2023, SAP2000: CSI, 2023, and Perform 3D: CSI, 2023 can automatically draw the member capacity curve and display its plastic behavior, which is highly efficient for use in the collapse-based design method.

Figure 7.

Drawing of the plastic behavior of ductile members along the capacity curve under the effect of scaled ETM ground motion.

Second method: defining ultimate capacity limits as acceptance criteria

In collapse-based design, only the maximum load and capacity values are important. Although the first method involves drawing the member’s capacity curve, only its maximum value serves as a controlling criterion. Hence, in the second method, which is more suitable for modeling the plastic behavior of members in the form of distributed plasticity, the maximum elastic and plastic capacities are determined based on the material strain. For example, to determine the maximum elastic capacity of a member, the yield strain (elongation) of the material is defined as the criterion for exceeding the elastic phase. According to Hooke’s law, the yield strain (ε_y) of materials is equal to the yield stress (σ_y) divided by the modulus of elasticity (E), as calculated from equation (5). In the modeling of plastic behavior based on stress-strain, which is very common in the use of fiber sections, reaching the yield strain of a member exactly means reaching its ultimate elastic capacity.

ε_{y} = σ_{y} / E

(5)

The failure strain of materials is also defined as the ultimate limit of plastic capacity, though this criterion lacks the required accuracy. However, considering that modeling plastic behavior using distributed plasticity methods necessitates calibrating the material properties with an experimental model, it is possible to ensure the accuracy of failure modeling to some extent. As shown in Figure 8, the ultimate capacity limit of ductile members in this method must be considered the most important control criterion, based on appropriate experimental models such as maximum rotation, maximum deformation, or maximum displacement. Therefore, in the second method, the material properties are first calibrated using a suitable experimental model (Sumner and Murray, 2002; D’Aniello et al., 2013; Wijesundara et al., 2014; Fell et al., 2009). Then, the yield strain of the material is defined as an acceptance criterion for exceeding the elastic phase, and the ultimate limit of capacity is considered as maximum rotation, maximum deformation, or maximum displacement based on experimental models.

Figure 8.

Calibrating the plastic behavior of ductile members based on experimental models and determining the ultimate capacity limit.

The behavior of each structural member is crucial for the design of structures. In general, structural members can be divided into two categories: ductile and non-ductile. Ductile members are those that behave plastically during an earthquake, providing the inelastic capacity and ductility of the structure through their plastic behavior. The inelastic capacity of a structure is derived from the plastic behavior of these members. While the plastic behavior of ductile members does not threaten the stability of the structure, their failure will trigger the collapse mechanism. Unlike ductile members, non-ductile members are used solely to maintain the stability of the structure. Usually, these members lack the ability to contribute useful inelastic capacity and are designed to allow ductile members to reach their maximum capacity. The plastic behavior of non-ductile members typically jeopardizes the stability of the structure, and their failure can render the structure unstable. For instance, beams in moment resisting frames, braces in concentrically braced frames, links in eccentrically braced frames, and shear walls in shear wall systems are ductile, while other structural members are non-ductile.

Design approaches differ for frames with special, intermediate, and ordinary ductility. This paper focuses on special structural systems. Seismic provisions for frames with special ductility are established to ensure high ductility within the system (Asghari and Jaberi, 2018, Asghari et al., 2019). To achieve high ductility, ductile members must use their maximum capacity and exhibit perfect plastic behavior (Tajik et al., 2024). Meanwhile, the behavior of non-ductile members also plays a significant role. Non-ductile members should facilitate conditions for ductile members to attain their maximum plastic capacity. While it may seem that non-ductile members can absorb and dissipate energy through partial plastic behavior, thereby enhancing the system’s plastic capacity, this minimal energy absorption not only proves ineffective but also disrupts the functioning of ductile members, potentially jeopardizing structural stability. Recognizing the insignificant plastic capacity of non-ductile members and its potential to swiftly lead to structural instability, it is imperative to delay the plastic behavior of non-ductile members through special seismic provisions until ductile members reach their maximum capacity. Consequently, each seismic provision for special frames aims to enable ductile members to reach their maximum capacity, thereby enhancing the ductility of the lateral load carrying system.

In the collapse-based design method, the sole criterion is the precise design of structural members, one by one. Design criteria can be developed within this method, and as initial principles for designing special structural systems, ductile members are designed based on their maximum plastic capacity, while non-ductile members are designed based on their maximum elastic capacity. Based on this, the sections designed using the collapse-based design method are compared with those designed using the force-based design method for the studied models, as presented in Table 2. Consequently, after analyzing the structure using scaled ETM ground motion, it is sufficient to satisfy the following five conditions for design:

⁃ The plastic behavior of any ductile member must not exceed its ultimate capacity limit.

⁃ The behavior of any non-ductile member should not exceed the elastic phase.

⁃ The structure must maintain stability.

⁃ Structural members under the effect of gravity loads must have the required capacity based on the relevant standards.

⁃ To the extent possible, assign the smallest and most optimal section to each member.

Table 2.

Sections of Structural Members of the Designed Models.

System	MRF				CBF
Method	FBD		CBD		FBD			CBD			Simple frame
Element	Column	Beam	Column	Beam	Column	Beam	Brace	Column	Beam	Brace	Column	Beam
Story	W14*	W24*	W14*	W24*	W14*	W18*	HSS	W14*	W18*	HSS	W14*	W18*
6	283	76	283	62	132	97	10*0.312	68	97	7.500*0.312	68	97
5	342	94	370	84	132	106	10*0.312	132	97	9.625*0.375	82	86
4	398	131	426	94	176	86	10*0.500	159	86	9.625*0.375	82	86
3	426	146	426	131	176	106	10*0.500	233	97	10.00*0.500	132	86
2	426	146	455	131	283	86	10*0.625	283	86	10.00*0.625	132	86
1	426	146	455	131	283	106	10*0.625	283	97	12.75*0.500	132	86

Exception: In moment resisting frame systems or when the connections of the columns to the foundation are fixed, the plastic behavior at the base of the first-floor columns is inevitable at high seismic intensities. Thus, the plastic behavior of the columns fixed to the foundation is not limited to the elastic phase. However, it is better to control their plastic behavior level to prevent initiating the collapse mechanism in the structure.

Principles and methodologies of structural collapse in seismic design

Determining and calculating the collapse of structures is highly complex and challenging, arguably the most intricate aspect of seismic analysis (Moehle and Deierlein, 2004). Despite these complexities, the collapse-based design method relies on simple and rational principles to identify structural collapse. While distinguishing structural collapse considering the physical behavior of the structure is challenging, comparing the capacity of each structural member to the loads acting upon them makes it easier to ascertain stability. In the CBD method, the validity of model analysis results hinges on ensuring that the loads acting on any element do not surpass its ultimate capacity. Exceeding the ultimate capacity limit of any main structural member is deemed collapse. Although this marks the initiation of the collapse mechanism, since all members are designed to reach their ultimate capacity at the target seismic intensity, the structure loses seismic capacity after the first member fails, indicating structural collapse. Thus, identifying collapse in this method signifies the emergence of the first threat to structural stability, which, according to other approaches, is not far from overall structural failure. Ensuring structural stability guarantees the accuracy of the performed analysis and demonstrates the maximum seismic capacity of the structure against applied seismic loads at collapse intensity. While the focus is on maintaining stability at collapse seismic intensity, utilizing the maximum capacity of all structural members against seismic loads indicates the ultimate seismic capacity of the designed model, even a slight increase in seismic load will render the structure unstable. Despite the complexity and uncertainties surrounding the investigation of structural collapse, maintaining stability under ultimate loading and utilizing the maximum capacity of structural members in an incremental time history analysis is straightforward and rational.

Assessing seismic behavior in designed models

In this section, a new rapid method for evaluating the seismic behavior of designed models and quantifying the maximum seismic capacity is presented. By employing a rapid and robust analysis technique for collapse-based design, it has become possible to comprehensively assess the behavior of the model after each analysis, in addition to the design phase. This technique combines the concept of incremental dynamic analysis with approaches presented in FEMA P695 and utilizes the remarkable capabilities of the endurance time method. This enables the quantification of the seismic capacity of the structure based on the method outlined in FEMA P695 through an incremental nonlinear time history analysis using artificial ETM ground motion, as well as the evaluation of the building’s performance level at various seismic hazard levels.

As illustrated in Figure 9, this method involves analyzing the model until collapse using scaled ETM ground motion. After this analysis, the interstory drift ratios for each story can be plotted against ground motion time. As explained in Section 1.4, this method employs nonlinear dynamic analysis, and since seismic intensity increases over the duration of the ground motion, this trend can be represented in terms of spectral acceleration. Determining the spectral acceleration applied to the model during ETM ground motion is straightforward. This is done by plotting the acceleration response spectrum at each step of the ground motion and reading the spectral acceleration at the first mode period of the model. By obtaining the values of spectral acceleration and interstory drift ratios for each story at each step, Figure 9 can be constructed. Using the concept of the design spectrum, the intensity corresponding to each seismic hazard level can then be determined in terms of spectral acceleration. This allows for the evaluation of the building’s performance levels at various seismic intensities and different hazard levels.

Figure 9.

Plotting interstory drift ratios corresponding to spectral acceleration applied to the model and monitoring the structure’s behavior.

The value of the scale factor that causes the collapse of the model is calculated using equation (1) for models designed with the method presented in this paper (CBD method). However, for other models with different and uncertain maximum seismic capacities, the appropriate scale factor for collapse should be determined through trial and error. In the IDA method, since the collapse seismic intensity under each ground motion varies, a collapse fragility curve is drawn. The intensity at which 50% of the ground motions cause the structure to collapse is considered the median collapse intensity ( ${\hat{S}}_{CT}$ ). In this method, an incremental artificial ground motion is utilized to obtain results similar to the median results of a set of ground motion records. Hence, there is no need to draw a collapse fragility curve, and the collapse seismic intensity of the model under this ground motion can be considered equal to the ${\hat{S}}_{CT}$ factor. As depicted in Figure 10, the collapse seismic intensity of the model is equal to the spectral acceleration on the acceleration response spectrum of the scaled ETM ground motion at the first mode period of the model, and the value of the S_MT factor is equal to the spectral acceleration on the design spectrum of the maximum considered earthquake hazard level at the fundamental period of the structure (seismic code period).

Figure 10.

The ratio of the collapse intensity to the MCE level.

Finally, by dividing the median collapse intensity ${\hat{S}}_{CT}$ ) by the seismic intensity at which the collapse of the model must be prevented (S_MT), the collapse margin ratio (CMR) of the model is calculated. The CMR is a valuable measure indicating the maximum seismic capacity of the model. Moreover, calculating this factor can expedite the seismic evaluation process outlined in FEMA P695. Consequently, as shown in Figure 11, a dynamic capacity curve can be drawn for the model using endurance time analysis and quantified using FEMA P695 principles. This curve can be considered equivalent to the median of the IDA curves. By employing this technique, it is possible to document the seismic behavior of the model from the onset of lateral load application to collapse and scrutinize the details using the unique results of a nonlinear incremental time history analysis with high accuracy and capabilities. Although this section primarily discusses the quantification of maximum seismic capacity and the evaluation of the building’s performance level at different seismic hazard levels based on interstory drift values, the results can be used from any perspective and for any purpose. Additionally, this technique can be an efficient method for the seismic retrofitting of existing buildings.

Figure 11.

Dynamic seismic capacity curve of the model.

Prototype models

In this research, two 6-story, two-dimensional models of special steel moment resisting frame (6-S MRF) and special concentrically braced frame (6-S CBF) are considered as prototype models. The gravity loads, seismic mass, and geometric conditions of the models are based on SAC model (FEMA 355-C, 2000). As displayed in Figures 12 and 17, in this modeling, lateral load carrying bays are placed on the perimeter of the structure, while the other columns and beams resist only gravity loads. According to the geometry and loading conditions of the SAC building, all four lateral load carrying frames on the perimeter of the structure are identical. These frames have four bays with a length of 9.14 m and a story height of 3.96 m. The seismic mass of each floor is 956 tons, and for the roof, it is 1035 tons. The gravity loads of each bay are placed in their actual locations, and the loads of the other bays are modeled using the P-Delta leaning column. The connection of the columns to the foundation is fixed for the 6-S MRF model and pinned for the 6-S CBF model. In the 6-S MRF model, the behavior of the panel zone is modeled, and in both models, rigid end offsets are considered for the members. It is assumed that the models are located in a high seismic risk zone on site class D soil. The values of the spectrum parameters S_MS and S_M1 for the maximum considered earthquake (MCE) are 1.5 g and 0.9 g, respectively. For the design basis earthquake (DBE), the values of S_DS and S_D1 are 1 g and 0.6 g, respectively. The steel material used is ASTM A992 with a yield stress of 345 MPa. The models are designed once using the force-based design method and once using the collapse-based design method, with the sections presented in Table 2. In the force-based design method, the models are analyzed using the equivalent static method based on ASCE7-22 and designed using AISC 360-16 and AISC 341-16. In the collapse-based design method, the models are analyzed and designed using the presented method.

Figure 12.

6-Story model of the special steel moment resisting frame system.

Seismic design and evaluation of prototype models

Moment resisting frame system

This section deals with the design of prototype models using the collapse-based seismic design method. For better comprehension, sections designed using the force-based design method, according to the seismic code and standards, are also presented in Table 2, allowing for comparison with sections designed using the CBD method. Comparing sections designed with the FBD and CBD methods can provide valuable insights into these approaches. To implement the presented method on prototype models, initial assumptions are made for each structural member, and the analysis and design process is iterated until the target design objectives are met.

At least one endurance time method ground motion is generated or selected, with a response spectrum that aligns with the design spectrum. This ground motion is then scaled by the SF_TOT scale factor to achieve the target seismic intensity. To calculate the total scale factor, the fundamental period based on the seismic code for the 6-S MRF model is 1.28 s, while the first mode period of the model is currently 1.88 s. According to the design spectrum, the spectral acceleration at the fundamental period of the structure is 0.47 g, and at the first mode period, it is 0.32 g. Therefore, based on equation (6), the seismic intensity modification factor for numerical model design using time history analysis is 1.47. As presented in Section 2.2.2, the ratio of MCE to DBE spectral acceleration in this research is 1.5. And the appropriate collapse margin ratio for buildings is considered to be 2. Eventually, the total scale factor required to reach the target seismic intensity of the 6-S MRF model is 4.41, based on equation (7). By multiplying this factor with the ETM ground motion, an incremental nonlinear time history analysis is performed until a seismic intensity proportional to the collapse intensity of the 6-S MRF model is reached.

{SF}_{DSR} = 2, {SF}_{CMR} = 1.5, {SF}_{SAR} = 0.47 g / 0.32 g = 1.47

(6)

{SF}_{TOT} = {SF}_{SAR} * {SF}_{DSR} * {SF}_{CMR} = 2 * 1.5 * 1.47 = 4.41

(7)

After analyzing the model, two conditions for the design must be met. Firstly, the plastic behavior of any ductile structural member must not exceed the backbone curve. Secondly, the behavior of non-ductile members should remain in the elastic phase while assigning the smallest possible section to the members. In moment resisting frame systems, beams are ductile members, and columns are non-ductile members. It is worth noting that in this system, plastic behavior at the base of the first-story columns is inevitable at high seismic intensities, so this case is considered an exception. By implementing these two principles and choosing the smallest possible sections, the most suitable sections with the necessary and sufficient capacity are assigned to each member. As shown in Figure 13, the section is selected for each of the beams so that after the analysis of the model under the target intensity, the plastic behavior of none of them exceeds its capacity curve, and they are the smallest sections that meet the required conditions.

Figure 13.

The plastic behavior of beams along their capacity curve under the scaled ETM ground motion loading.

In this design (compared to the model designed by the FBD method), it was possible to reduce the section of all beams by only one level. Moreover, as shown in Figure 14(a), the behavior of all the columns is maintained in the elastic phase. To achieve this, it was necessary to increase the sections of the columns in the first, second, fourth, and fifth stories by one level. As presented in Table 2, the difference in the section of members between the FBD and CBD methods is very small, and these models have almost the same weight of materials used. The difference between these two methods is that in the CBD method, there is no estimation, and the analysis continues until the beams reach their maximum plastic capacity. In contrast, in the FBD method, the beams are designed based on elastic behavior using seismic performance factors. Furthermore, in the FBD method, maintaining the columns in the elastic phase is achieved using the strong column-weak beam seismic provision. However, in the CBD method, because the effects of the beams are directly applied to the columns upon reaching their maximum capacity, and the columns are subjected to the loads caused by the plastic behavior of the beams, the columns are designed correctly even without using this seismic provision.

Figure 14.

The plastic hinges in the frames and the spectral accelerations corresponding to the period of the models.

In this two-dimensional modeling, the weight of the steel used in the model designed using the CBD method is 2% lighter. However, the model still satisfies all the minimum requirements of the seismic code, despite not applying the principles of the seismic code explicitly. It is worth mentioning that if the section of any designed member is reduced by even one level, its behavior under the target seismic intensity will exceed its ultimate capacity limit. Therefore, it can be concluded that the smallest section with the necessary and sufficient capacity has been assigned to the members.

As the concepts of the CBD method are described in detail, maintaining the stability of the model under the target seismic intensity, equivalent to the collapse seismic intensity for the model, means that the model has the necessary and sufficient capacity. Utilizing the maximum capacity of the structural members against the applied loads leads to optimal design. Maintaining stability in this method is contingent on the ductile members not exceeding the ultimate plastic capacity limit and the non-ductile members remaining within the elastic phase. Thus, the performed analysis and the stability of the structure are both valid.

In this method, aiming for the highest level of seismic loading enables the evaluation of the structure’s performance level at all seismic hazard levels and verifies whether the structure’s performance is satisfactory at the target hazard level. For this purpose, based on the description in Section 2.5 and as depicted in Figure 15, the interstory drift ratios versus the spectral acceleration applied to the model are plotted, and the corresponding spectral acceleration can be determined to represent each seismic hazard level. As illustrated in Figure 14(c), the spectral acceleration values corresponding to the DBE and MCE hazard levels, based on the design spectrum and the first mode period of the model, are 0.32 g and 0.48 g, respectively. By identifying these seismic intensities on Figure 15, the maximum interstory drift ratio corresponding to these spectral accelerations can be determined and assessed. Accordingly, the maximum interstory drift ratio for the 6-S MRF model at the DBE and MCE hazard levels is 1.46% and 2.6%, respectively, representing the life safety and collapse prevention performance levels based on values provided in most sources (FEMA 356, 2000). Here, the analysis results up to 4.7 s are equivalent to the model analysis results at the DBE hazard level, and similarly, the model analysis results up to 6.1 s are equivalent to the model analysis at the MCE hazard level. At the S_MT, collapse must be strictly avoided, and at the ${\hat{S}}_{CT}$ , the model reaches the specified collapse seismic intensity. This evaluation demonstrates that as a structural system capable of maintaining stability under large displacements, the special steel moment resisting frame system can uphold its stability up to a 10% interstory drift ratio. In this model, the 10% interstory drift ratio is the displacement at which the beams, as ductile members of the system, have reached their maximum plastic capacity. This interstory drift and its corresponding displacement signify the collapse displacement of the model.

Figure 15.

Plotting the interstory drift ratios corresponding to the spectral acceleration applied to the 6-S MRF model.

As described, the collapse margin ratio is the best measure to quantify the seismic capacity of the model. It equals the ratio of the median collapse intensity to the seismic intensity at which the collapse of the model must be prevented. As shown in Figure 16, for the 6-S MRF model, the median value of the collapse seismic intensity is equal to the spectral acceleration at the first mode period of the model on the acceleration response spectrum of the scaled ETM ground motion, which is 1.57 g. And the seismic intensity that must prevent the collapse of this model equals the spectral acceleration at the fundamental period of the structure on the MCE spectrum, which is 0.7 g. By calculating the ratio of these two values, the CMR factor for the 6-S MRF model is 2.24. Based on these results, it can be concluded that the model designed using the CBD method has the required seismic capacity and is at the desired performance level at different seismic hazard levels. It is also completely optimal in terms of the materials used in the structure.

Figure 16.

Calculation of the collapse margin ratio factor for 6-S MRF models.

Figure 17.

6-Story model of the special concentrically braced frame system.

Concentrically braced frame system

Currently, individual principles and criteria are used for the design of each structural system, derived from extensive evaluations and experiences regarding the seismic behavior of those systems. However, using the CBD method, all structural systems are designed using the same principles and criteria. Therefore, this section deals with the design of the CBF system based on the principles of the CBD method. For this purpose, the fundamental period of the 6-S CBF model is 0.73 s based on the seismic code, and the first mode period of the numerical model is 0.85 s. Based on the design spectrum, the spectral acceleration values at the fundamental period and the first mode period are 0.812 g and 0.706 g, respectively. Therefore, according to equation (2), the value of the spectral acceleration modification factor is 1.15. Likewise, based on the descriptions in Sections 2.2.2 and 2.2.3, the values of the SF_DSR and SF_CMR factors are 1.5 and 2, respectively. Finally, the total scale factor is 3.45 based on equation (1).

To create the target seismic intensity for designing the 6-S CBF model, the calculated total scale factor is multiplied by the ETM ground motion, and the model is analyzed using the scaled ETM ground motion. After analyzing the 6-S CBF model, the plastic behavior of any brace as a ductile member in the CBF system must not exceed its capacity curve, and the behavior of other non-ductile members (columns and beams) should remain in the elastic phase. The sections designed using the CBD method for the 6-S CBF model are presented in Table 2, which are comparable to the sections designed using the FBD method. In this design, the sections of the braces in the first, fourth, and fifth stories and the sections of the columns in the third, fourth, and sixth stories differ with very slight variations. The weight of the designed models is almost the same, with a 1% difference, and the model designed using the CBD method satisfies all the minimum requirements of the seismic code.

As shown in Figure 18, in the CBF system under the target seismic intensity, only the braces should exhibit plastic behavior, while all the columns and beams should remain elastic. However, these sections have been chosen in such a way that reducing any of the sections by even one level would fail this condition. For the 6-S CBF model, the capacity design based on AISC 341-22 is conducted using the FBD method, but such a provision is not used in the model design using the CBD method. In the incremental nonlinear time history analysis, which continues up to the model’s maximum capacity, the columns and beams are directly affected by the additional loads resulting from the plastic behavior of the braces, and no additional provision is needed to cover this issue.

Figure 18.

The plastic behavior of braces along their capacity curve under the scaled ETM ground motion loading.

In Figure 19, the values of interstory drift ratios are plotted against the spectral acceleration applied to the model. The first mode period of the model is 0.85 s, and based on the design spectrum, the spectral acceleration at the DBE and MCE hazard levels for the 6-S CBF model is 0.706 g and 1.06 g, respectively. As shown in Figure 19, the maximum value of interstory drift ratios corresponding to the DBE and MCE seismic hazard levels is 0.85% and 1.41%, respectively. In this evaluation, the results of model analysis up to 5.07 s are equivalent to the results of model analysis at the DBE level, and up to 7.6 s are equivalent to model analysis at the MCE level. Finally, the 6-S CBF model collapsed under a spectral acceleration of 2.787 g at an interstory drift ratio of 2.13%. Meanwhile, the similar moment resisting frame model collapses at a 10% interstory drift ratio and under a spectral acceleration of 1.57 g. The main seismic parameters of these two models are presented in Table 3, showing significant differences in the seismic behavior of these systems. However, the maximum seismic capacity of both models, based on CMR factors, is the same and proportional to the target design hazard level.

Figure 19.

Plotting the interstory drift ratios corresponding to the spectral acceleration applied to the 6-S CBF model.

Table 3.

Important Seismic Parameters in Designing Structures Using CBD Method.

Frame	Period (s)		Scale factor	S_a (g)			Max drift ratio (%)			Capacity ratio			Seismic code requirements
Frame	T	T₁	Scale factor	DBE	MCE	Collapse	DBE	MCE	Collapse	${\hat{S}}_{CT}$	S_MT	CMR	Seismic code requirements
MRF	1.28	1.88	4.41	0.32	0.48	1.57	1.46	2.60	10.0	1.57	0.70	2.24	Satisfied
CBF	0.73	0.85	3.45	0.70	1.00	2.78	0.85	1.41	2.13	2.78	1.22	2.27	Satisfied

As shown in Figure 20, the collapse seismic intensity for the 6-S CBF model is 2.787 g, which is equal to the spectral acceleration at the first mode period on the acceleration response spectrum of the scaled ETM ground motion. The seismic intensity at which the collapse of the model must be prevented is 1.225 g, determined based on the spectral acceleration at the fundamental period of the structure on the MCE spectrum. By dividing the collapse intensity by the intensity that must prevent the model from collapsing, the collapse margin ratio is calculated as 2.27. Therefore, the CBF system is well-designed according to the principles of the CBD method. The results of the analysis and evaluation of the designed model show that this model has sufficient seismic capacity and exhibits the desired performance level at different seismic hazard levels. It is designed optimally with high reliability.

Figure 20.

Calculation of the collapse margin ratio factor for 6-S CBF models.

Capabilities and weaknesses of collapse-based seismic design method

Capabilities and advantages

⁃ Accurate seismic analysis and design of structures using nonlinear time history analysis.

⁃ Optimal design and utilization of the maximum capacity of structural members.

⁃ Ability to prepare artificial ground motion for structure design based on seismic intensity and soil type of the zone (provision of ETM ground motions along with design spectrum).

⁃ Assessment of structure vulnerability from lateral load initiation to collapse.

⁃ Evaluation of collapse mechanism occurrence and postponement until target seismic intensity.

⁃ Suitable and accurate method for seismic evaluation and retrofitting of existing buildings.

⁃ Quick assessment of seismic capacity of existing buildings and comparison with required capacity.

⁃ Fast simulation of incremental dynamic analysis based on the FEMA P695 method and calculation of the maximum seismic capacity of the model after each analysis.

⁃ Accurate design of connections based on the maximum loads they will be subjected to.

⁃ More accurate design of foundation based on maximum loads at the base of columns.

⁃ Independence from seismic performance factors like response modification coefficient (R), overstrength factor (Ω₀), and deflection amplification factor (C_d), and other estimator parameters in the force-based design method.

⁃ Ability to define simple acceptance criteria for member design, such as maximum capacity, rotation, displacement, or deformation.

⁃ Design of structural members with high reliability in seismic behavior during earthquakes.

⁃ Evaluation of seismic behavior at different hazard levels and collapse scenarios.

⁃ No need for special procedures to design complex systems, dual systems, fuse systems, and other innovative systems (Jaberi, 2023).

⁃ Adjustment of maximum seismic capacity of structures with different structural systems to specific and reasonable ultimate capacity levels.

⁃ Reduction in error percentage due to non-utilization of estimator parameters.

⁃ Independence from other design methods and standalone usability.

⁃ Alignment with growth trends and recent developments in seismic analysis and design methods, compatible with various software such as ETABS, SAP2000, Perform 3D, SeismoStruct: Seismosoft, 2023, OpenSees (2023), etc.

Weaknesses and disadvantages

⁃ Requirement for knowledge of nonlinear time history analysis.

⁃ Need for accurate modeling of the nonlinear behavior of ductile members.

⁃ Sensitivity to the precise definition of effective factors in collapse, such as P-delta effect, stiffness deterioration effect, and strain hardening.

⁃ Requirement for accurate and sufficient information on the collapse criteria of structural members.

⁃ Dependency on the use of artificial incremental ground motion perfectly matches with the design spectrum.

Conclusion

This article discusses the fundamental principles and processes of seismic design using the collapse-based design method, specifically applied to the special steel moment resisting frame system and the special concentrically braced frame system. It identifies significant shortcomings in the current seismic analysis and design processes and proposes a new technique to address these issues using the collapse-based design method. The research points out that deviations from the true nature of seismic loading are the primary source of these shortcomings. To address this, the study introduces a technique that aligns more closely with the actual seismic behavior of structures during earthquakes. Instead of relying on the qualitative performance levels of IO, LS, and CP, the research advocates using the ultimate capacity limit of structural members as a clearer and simpler criterion. By designing structural members individually based on loads derived from incremental nonlinear time history analysis, the need to control physical displacements of the structure during design is eliminated. Consequently, the interstory drift ratios and displacements are maintained within favorable limits. The study further explains that after analyzing the structure, the collapse displacement and corresponding interstory drifts for each story can be calculated. Despite the significant behavioral differences among various structural systems, the uniform principles can be applied to analyze, design, evaluate, and retrofit them. This approach simplifies the design process by accommodating the complexities of seismic behavior through robust analysis. By applying the maximum seismic load according to the expected capacity of the structure and utilizing the maximum capacity of structural members, the accuracy of the design is enhanced. This method enables the easy identification and assignment of the most optimal sections that provide the required capacity.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

CRediT authorship contribution statement

Vahid Jaberi: Writing – original draft, Writing – review & editing, Visualization, Supervision, Project administration.

ORCID iD

Vahid Jaberi

Data availability statement

No data was used for the research described in the article.*

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Collapse-based seismic design method

Abstract

Keywords

Introduction

Novel approaches to addressing shortcomings

Differences in structural analysis and design methods

Necessity of structural analysis until collapse

Using the endurance time method for simple and rapid nonlinear time history analysis

Collapse-based seismic design method

Introduction

Target seismic intensity (the intensity at which the structure collapses)

Scale factor of spectral acceleration ratio (SFSAR)

Scale factor of design spectrum ratio (SFDSR)

Scale factor of collapse margin ratio (SFCMR)

Design criteria of structural members in collapse-based design method

First method: drawing capacity curve

Second method: defining ultimate capacity limits as acceptance criteria

Principles and methodologies of structural collapse in seismic design

Assessing seismic behavior in designed models

Prototype models

Seismic design and evaluation of prototype models

Moment resisting frame system

Concentrically braced frame system

Capabilities and weaknesses of collapse-based seismic design method

Capabilities and advantages

Weaknesses and disadvantages

Conclusion

Footnotes

Declaration of conflicting interests

Funding

CRediT authorship contribution statement

ORCID iD

Data availability statement

References

Scale factor of spectral acceleration ratio (SF_SAR)

Scale factor of design spectrum ratio (SF_DSR)

Scale factor of collapse margin ratio (SF_CMR)