1. Introduction
Determining the structural response modification factor (R) is crucial for establishing prescribed seismic design actions for buildings. Notably, R values vary across different countries’ codes and standards, enabling designers to employ a linear elastic force-based design methodology that accounts for nonlinear behavior and deformation constraints. Furthermore, the response modification factor is essential for developing equal-ductility inelastic response spectra within the context of performance-based seismic design.
In the seismic design approach adopted in China in 1978 [1], seismic design forces were determined by applying a seismic influence coefficient, which is the reciprocal of the response modification factor (1/R), to the elastic earthquake forces corresponding to the design seismic intensity. However, the Code for Seismic Design of Buildings (GBJ 11-89) [2], introduced in 1989, abandoned the concept of a response modification factor. Instead, seismic design forces were established directly using the elastic design response spectrum for minor earthquakes. This method determines seismic design forces through a singular structural response modification factor, aiming to reduce the elastic response at the design seismic intensity without considering the diverse inelastic energy dissipation capacities of different structures. To introduce performance-based seismic design, the General Rule for Performance-based Seismic Design of Buildings (CECS 160:2004) [3] provided values for structural seismic influence coefficients and displacement amplification factors for 25 structural systems. These values were derived from the Code for Seismic Design of Industrial and Civil Buildings (TJ11-78) [1] and data from buildings that experienced seismic ground motions. However, this approach lacks sufficient theoretical research and experimental data to substantiate its validity. Consequently, numerous researchers in China have reevaluated the current seismic design methodology. From the authors’ perspective, quantifying and evaluating R would enhance the scientific, rational, and economical aspects of seismic design while facilitating the implementation of performance-based seismic design (PBSD) in China.
Seismic design of structures has traditionally relied on elastic analysis as the primary tool. However, recent earthquake events have revealed that elastic analysis alone is insufficient to capture the actual behavior of structures. Consequently, there has been a fundamental shift in seismic design provisions within design and material codes, such as ASCE, UBC, and NEHRP codes, following the Northridge earthquake in 1994 [4,5,6]. The prevalent approach, known as the equivalent static method, used in many seismic design codes, involves the utilization of the response modification factor (R), also referred to as the force reduction factor. Table 1 lists the response modification factor for different structural types in Chinese and American standards. Essentially, the design loads are determined by dividing the earthquake loads by the R factor, causing the structure to enter the inelastic range. Consequently, the structure must undergo significant inelastic deformations to dissipate the earthquake’s energy. The ability of a structure to endure earthquake loads is directly linked to its capacity to deform within the inelastic range, commonly referred to as its ductility capacity. For systems displaying idealized bilinear behavior (as depicted in Figure 1), structural ductility, denoted as μ, is defined as the ratio of the maximum displacement to the displacement at the yielding point. Structures with higher values of the force reduction factor R require greater ductility capacity μ. Thus, the factors R and μ are interconnected and play a pivotal role in the energy dissipation mechanism of structures.
The response modification factor (R) and displacement amplification factor (Cd) can be determined using the structural response curve shown in Figure 1 (Uang, 1991) [7]. This figure shows the relationship between the roof displacements of the structure and the base shear. The roof displacements are plotted on the horizontal axis, while the base shear is represented on the vertical axis. The line OA illustrates the structure’s performance curve under perfect elastic conditions. However, when the structure encounters earthquake excitation, its actual elastic response is depicted by the curve ODEF. The anticipated response in an ideal scenario is represented by the idealized performance curve labeled as ODCF. The actual nonlinear response of the structure can be approximated by a bilinear elastic–plastic relationship, transitioning from curve ODEF to ODCF in Figure 1. The initial linear response of the structure is represented by the bilinear line ODCF, with the area under this line corresponding to the area under the actual curve.
The response modification factor R is defined as follows:
The ductility reduction factor, represented as Rμ, is essential for assessing the reduction in strength caused by inelastic behavior. The definition of Rμ involves the ratio of the maximum seismic demand for elastic response (Ve) to the base shear at the maximum inelastic displacement (Vy), which corresponds to the structure’s yield force. In this context, another crucial parameter is denoted as RΩ, the overstrength factor, which indicates the ratio of Vy to the design base shear (Vd). Following this, the displacement amplification factor (Cd) is calculated using the provided equation.
In seismic design considerations, two key parameters related to structural displacement stand out: Δmax, which represents the maximum displacement during moderate or severe seismic ground motion, and Δd, the designated roof displacement. Δmax indicates the largest displacement magnitude a structure may undergo during an earthquake, while Δd represents the intended displacement level that the structure is designed to withstand. Δmax and Δd, these parameters, play a crucial role in the comprehensive evaluation of a structure’s seismic performance. They are frequently combined with other metrics to assess structural integrity and safety under seismic loads.
Four Y-shaped eccentrically braced steel frames were designed using high-strength steel combinations for multiple high-rise buildings in compliance with China’s seismic code. The response modification factor for these frames, determined through the Incremental Dynamic Analysis method, was found to be 4.889 [8]. A method to calculate the elastic stiffness and bearing capacity of K-type, D-type, and V-type eccentrically braced steel frames was proposed. Using the virtual work principle and the ideal failure mode of these frames, the yield and ultimate bearing capacities were determined based on frame drift angle and verified through experimental results [9]. Based on previous studies, pushover analysis and nonlinear dynamic analysis of 4-, 8-, 12-, and 16-story eccentrically braced high-strength steel frame structures with different bracing forms were conducted using performance-based seismic design methods. The results indicated that D-shaped eccentrically braced steel frames had the greatest stiffness and bearing capacity [10]. The seismic performance of three chevron-braced steel frames (CBSFs) of 3-, 6-, and 9-story Performance-based Plastic Design (PBPD) designs were compared, leading to the proposal of a PBPD framework for CBSFs considering mainshock–aftershock sequences [11]. Nonlinear time history analyses and performance-based seismic design were conducted on steel MRFs, EBFs, and BRBFs. A simple expression linking the peak seismic layer velocity to the peak ground velocity was provided, with its efficiency demonstrated through numerical examples [12]. Additionally, steel plate shear wall structures, consisting of 4, 8, and 12 stories, were developed following the Code for Seismic Design of Buildings (GB 50011-2010) [13]. The response modification factors for both single-frame planes and integral steel shear wall structures were obtained under uniform and inverted triangle loads using the pushover analytical method in Midas/Gen. A recommended response modification factor of 3.25 was proposed for steel plate shear wall structures with a maximum of 12 floors [14].
Furthermore, a high-strength steel fabricated framed-tube structure with replaceable shear links (HSS-SFT) was introduced. To investigate the response modification factor of HSS-SFTs, eight models with ideal yield mechanisms were established. Pushover analyses were conducted using the stepwise lateral force adjustment method. Based on the analysis results, a recommended response modification factor (R) of 3.65 is suggested for HSS-SFT designs under seismic action [15].
To assess the response modification factors of buckling restrained braced frames (BRBFs) used in the rehabilitation of steel frames, static nonlinear analyses were conducted on building models with single and double bracing bays, multi-floors, and various brace configurations including chevron V and inverted V. The response modification factors for different types of BRBFs with a single bracing bay ranged from 7 to 16, whereas for double bracing bays, the range extended from 8 to 22 [16].
Additionally, linear time history analyses were performed using multiple earthquake records on 4-, 8-, 12-, and 16-story steel frame buildings with different concentrically braced systems, namely X brace, V brace, and inverted V brace. The results showed an increase in the response modification factor with the number of stories for all systems except the inverted V-braced system [17].
Furthermore, the research group conducted studies on the response modification factors of eccentrically braced structures. A K-type eccentrically braced structure with varying numbers of stories (5, 10, 15, and 20 stories) and link lengths (900, 1000, 1100, and 1200 mm) was designed using the performance-based seismic design method. Static nonlinear pushover analysis and Incremental Dynamic Analysis (IDA) were carried out. The average R value, obtained through both methods of analysis, was determined to be 7.194 for the 20-story structure. Additionally, a linear relationship between the response modification factor (R) and the first period (T1) of structures with fewer than 20 stories is established as R = 4.275T1 + 4.1274 [18].
In summary, our research delivers key insights into the seismic performance and response modification factors of various structural systems, greatly aiding in the seismic design and rehabilitation of high-rise buildings. Eccentrically braced steel frames (EBFs) integrate the advantages of both moment-resisting and concentrically braced steel frames, providing superior ductility and energy dissipation. Illustrated in Figure 2 are four common EBF configurations. One key characteristic of these frames is a component known as a “fuse”, which is intended to experience extensive plastic deformation, keeping the remaining structural elements elastic. Utilizing high-strength steel in this design enables the use of smaller beam and column cross-sections due to its enhanced elastic strength. Our earlier studies concentrated on identifying the response modification factors for high-strength steel frames equipped with Y-type and K-type braces, employing both Incremental Dynamic Analysis (IDA) and pushover analysis methods. Our research team thoroughly examined the parameters R and Cd for high-strength steel frames with D-type braces using an enhanced pushover analysis method. In this current study, we utilized the IDA method to determine the response modification factor specifically for high-strength steel frames with D-type braces (D-HSS-EBFs).
2. Structural Design and Finite Element Model
2.1. The Design Principle
According to the Code for Seismic Design of Buildings, Figure 3 illustrates the plan layout of the D-HSS-EBF. Designed to endure a seismic acceleration of 0.30 g, this prototype structure corresponds to a peak ground acceleration of 0.3 g, with a 10% chance of exceedance over 50 years. The building stands 3 m tall with columns 6 m apart. It features a plane dimension of 18 m × 30 m, comprising 3 spans along the Y direction and 5 spans along the X direction.
The frame column utilizes a square steel tube, while welded H-shaped sections form the other components. Constructed from 120 mm thick cast-in-place C30 concrete, the floor is designed to handle several load considerations: a floor load of 5.0 kN/m2, a live load of 2.0 kN/m2, a roof load of 5.5 kN/m2, a roof live load of 2.0 kN/m2, and a basic wind pressure of 0.35 kN/m2.
A schematic diagram of the frame structure after calculation is presented in Figure 4. As depicted in Figure 5, the steel stress–strain curve in the SAP2000 model remains at its default value [19].
A standardized naming system is employed to classify prototype structures, indicating steel materials and their properties. For instance, the designation “D1-8-0.9” represents an 8-story D-type eccentrically braced frame incorporating Q345 and Q460 steel, with a link length of 0.9 m. Using the same pattern, the structure combining Q345 and Q690 steel follows suit, yet the prefix “D2” denotes a distinct steel blend. Hence, the prototype structure, composed of Q345 and Q690 steel, comprises 8 layers and boasts a link length of 0.9 m, denoted as “D2-8-0.9”.
In accordance with the basic principles of performance-based seismic design (PBDS), as outlined in the academic work of Li and Tian [20], the design approach for D-HSS-EBFs is established. A detailed depiction of the design process is provided in the design flow diagram, illustrated in Figure 6. As specified in the Load Code for the Design of Building Structures (GB50009-2012) [21], the fundamental period of high-rise steel structures is T = 0.1N, with N indicating the number of stories. In the case of an eight-story D-HSS-EBF, this period calculates to 0.8 s. Table 2 provides a detailed summary of the PBSD design parameters relevant to D-HSS-EBF.
2.2. Shear Link Model
The performance of eccentrically braced structures during earthquakes relies primarily on the shear yield behavior of the links, which is essential for both energy dissipation and maintaining structural integrity. The specimen and finite element model are illustrated in Figure 7. To accurately describe this linkage behavior, the analytical framework incorporates the ShearV2 plastic hinge model introduced by Li and Gao [22]. The SAP2000 software was used to determine the force–displacement correction parameters for the model, with the shear plastic hinge reflecting the force–deformation relationship as depicted in Figure 8. To accurately capture the nonlinear behavior of the shear link, the analysis incorporated empirical data from Tables 5 and 6 of FEMA-356 [22]. The link’s ultimate shear force (Vu) was set at 1.5 times the yield shear force (Vp). Deformation standards for immediate occupancy (ΔIO), life safety (ΔLS), and collapse prevention (ΔCP) were evaluated based on the parameters in Tables 5 and 6 of FEMA-356. Figure 8 illustrates this analysis.
Ref. Richards [23] drew from the experimental studies by Popov and Ramadan [24,25] to propose a shear–deformation relationship for the shear link, overlooking the intricate interplay between bending moment and shear force. The pertinent test results are presented in Figure 8b.
A spring model, integrating four nodes, was devised to simulate the diverse behaviors of the shear link, shown in Figure 9. This model ensures elasticity is maintained under various loading conditions, preventing yielding at its ends. The two external nodes mirror the arrangement of the two internal nodes, ensuring a distance of zero between each external node and its corresponding internal node. At the link’s end, the critical moment hinge emerges, while the shear hinge extends across its web, constrained to primary and secondary nodes. The length of this pivotal structural unit, denoted “e”, precisely matches that of the shear link. By confining the inelastic response to the ends of the link, this arrangement ensures that the rest of the link stays elastic, thereby maintaining structural stability.
An amplification factor λ is introduced to address the enhancement effect of flange shear on the shear resistance of the plastic link. This factor seamlessly integrates into the shear control formula, resulting in a refined expression, as follows:
Equations (3) and (4) distinctly establish the fundamental parameters needed for assessing the shear of a plastic link. The width of the flange is represented by bf in these equations, while the cross-sectional area of an individual flange is denoted by Af. The connection length is signified by e, whereas the thickness and area of the web are, respectively, indicated by tw and Aw. The noticeable aspect is that within the SAP2000 software, the plastic hinges display a clear lack of elasticity, confining all plastic deformation exclusively to the hinge region. Meanwhile, the remaining structure consistently demonstrates its inherent elastic response.
Point B in Figure 8 vividly illustrates the yield of the hinge, showing a critical bearing capacity (VB = 1.1αVP). Subsequently, at point C, the ultimate bearing capacity of the hinge is marked, with a gradual decrease in capacity beyond point C (VC = 1.5αVP), and the associated angle θC = 6.12αfy/G. Point D signifies the residual strength of the hinge, set at 0.6 for beams with a flange height-to-thickness ratio h0/tw ≤ 418/fy1/2, and at 0.2 for those with h0/tw ≥ 640/fy1/2. For other flange ratios, the values are linearly interpolated between 0.2 and 0.6. The angle θD matches θC. Point E marks the hinge’s total failure. The specific parameters related to these points are listed in Table 3.
In this research, each link is designed to undergo shear yield across the entire cross-sectional area simultaneously. For each link, three shear hinges are strategically positioned at the two ends and the center of the link.
2.3. Plastic Hinge of Non-Yielding Member
The model for plastic hinges in frame beams and columns incorporates the default M3 and P-M-M plastic hinges, positioned at both ends of the structural elements. Specifying these plastic hinge models allows for determining the plastic moment bearing capacity of the beams and columns.
D-HSS-EBFs use MCE for design bending moment, with Z representing section modulus and Fye indicating steel yield strength. Columns bear axial force P and have compressive capacity Pye. The midpoint support’s axial force is termed P hinge, and SAP2000 software automatically assigns plastic hinge characteristics.
3. IDA Method
3.1. The IDA Analysis Procedure
The primary distinction between the Incremental Dynamic Analysis (IDA) method and pushover analysis in the computation of structural performance coefficients lies in the derivation of the structural performance curve. The steps involved in acquiring structural performance coefficients through IDA analysis are outlined below:
(1) Selection of Ground Motion Records: Initiate the process by judiciously selecting a set of ground motion records, typically ranging from 10 to 20 records.
(2) Scaling of Ground Motion Records: Proceed by amplifying the chosen ground motion records into multiple sets. This amplification involves applying appropriate scaling factors to each record, yielding a series of ground motion sets, with each set corresponding to distinct maximum acceleration levels, as specified in Table 4.
(3) Time History Analysis: Conduct time history analyses of the structure employing the various sets of ground motion records. During this phase, capture and record pertinent data points characterizing the base shear and roof displacement responses for each ground motion record. Subsequently, we employed Origin (2021) software to conduct a data fitting process, resulting in the derivation of the structural performance curve.
(4) Target Displacement Determination: Employ the spectral method to ascertain the target displacement point within the structural system.
(5) Performance Coefficient Computation: Utilize the formulas delineated in Formulas (7)–(10) to compute a spectrum of performance coefficients for the structure, reflecting its behavior under different seismic hazard levels.
In summary, the IDA method for determining structural performance coefficients involves a systematic procedure encompassing ground motion record selection, record scaling, time history analysis, structural performance curve derivation, target displacement determination, and subsequent computation of performance coefficients. This approach provides a comprehensive insight into the structure’s behavior across varying seismic intensities.
3.2. The Calculation of Factors by IDA
The specific steps for obtaining response modification factors through the capacity spectrum method, as shown in Figure 10 within the IDA framework are elucidated as follows:
Nonlinear Dynamic Analysis: Perform nonlinear dynamic analyses under various ground motions, resulting in envelope curves for roof displacement and base shear force. Similar curves are obtained for other ground motion groups.
Curve Fitting: Fit all the acquired Δ-V envelope curves to generate the IDA response curve, which bears resemblance to the pushover curve.
Yield Point Determination: Determine the yield shear force (Vy) and yield displacement (Δy) by introducing bilinear characteristics into the fitted IDA response curve.
Spectrum Transformation: Convert the fitted IDA response curve into the spectrum acceleration–displacement curve (Sai − Sdi).
Capacity Point Determination: Plot the response curve obtained in step 4 alongside the plastic–elastic demand spectrum for different ground motions within the same coordinate system. Iteratively identify the capacity points for different ground motions.
Factor Calculation: Utilize the provided Formulas (7) and (10) to calculate the response modification factor (R) and displacement amplification factor (Cd) for the structure.
Figure 10.The IDA method.
Figure 10.The IDA method.
In summary, the IDA methodology involves a comprehensive sequence of steps encompassing structural modeling, ground motion selection and amplification, dynamic analysis, curve fitting, and subsequent factor computation. This rigorous approach offers valuable insights into the structural behavior under varying seismic conditions, facilitating the assessment of performance and safety.
4. Calculation Results for Each Case
4.1. IDA Method Process
The results of the Incremental Dynamic Analysis (IDA) for a single ground motion record, specifically the RSN184 seismic record, are outlined below. This analysis involves conducting elastic–plastic time history analyses while adjusting the peak ground acceleration of the seismic motion. The base shear and roof displacement values for each prototype structure are meticulously calculated and presented in Table 5. Additionally, the IDA analysis produces envelope points for each load case and presents pushover curves using both inverted triangle and uniform loading patterns, as depicted in Figure 11. These results provide a comprehensive understanding of the structural response under varying seismic conditions, enabling a robust assessment of structural performance and behavior.
Upon examination of Table 5, it becomes evident that the enveloped values of base shear and roof displacement for the structure exhibit a gradual increase with the progressive augmentation of acceleration amplitude. However, it is noteworthy that an intriguing observation arises when the peak acceleration is elevated from 1310 cm/s2 to 1420 cm/s2: both the base shear and roof displacement experience a decrease. This phenomenon can be primarily attributed to a critical structural condition. Specifically, when the peak acceleration is adjusted to 1420 cm/s2, the inter-story displacement of the fifth floor attains a significant value of 65.589 mm, corresponding to an inter-story drift angle of 2.19%. Importantly, this drift angle exceeds the specified limit of 2% for elastic–plastic inter-story drift angle. Consequently, the structure reaches its ultimate state, and the analysis can be rightfully terminated at this point. This observation underscores the importance of adherence to specified performance criteria in assessing the structural response under varying seismic conditions.
From Figure 11, it can be observed that the general trend of the results obtained through the Incremental Dynamic Analysis (IDA) for the single ground motion record in model D1-12-0.9 aligns consistently with the observed variations in the pushover analysis curve.
The IDA analysis results for model D1-12-0.9 under the influence of 10 seismic ground motions are graphically depicted in Figure 12. An examination of this graph reveals a notable trend: most data points closely align with the uniformly distributed loading pushover curve. This suggests that for model D1-12-0.9, the performance curve derived through the uniformly distributed loading mode in the pushover analysis more accurately represents the actual structural response during seismic events. This finding underscores the importance of selecting an appropriate loading mode when conducting structural assessments in seismic scenarios.
4.2. Calculation of Structural Performance Factors
(1) Determination of Significant Yielding Point
Utilizing the nonlinear least squares method, a polynomial fitting procedure is applied to the IDA analysis data presented in Figure 13. The resultant performance curve equation for case D1-12-0.9 is expressed as follows:
Here, ‘x’ denotes the roof displacement of the structure, while ‘y’ represents the base shear of the structure. By substituting the maximum roof displacement value, Δm = 417.78 mm, into the equation, the maximum yield shear force, Vm, is computed as 5893.6 kN. Additionally, the structural elastic stiffness, Ke, is determined as the average value obtained from the time history analysis conducted under various seismic ground motions.
Using Matlab (2018a) software, the integral of the equation is solved to determine the area enclosed by the performance curve and the horizontal axis. Subsequently, based on the formulas provided in this chapter, the significant yielding point for the given case is calculated.
It is noteworthy that there are discrepancies between the yielding points obtained from pushover analysis for the same case. Vy and Δy were determined as 3784.9 kN and 191.73 mm, respectively, using the pushover analysis method. The primary reason for this disparity can be attributed to the differences in the initial stiffness and yield shear force values derived from the IDA analysis of the structural performance curve, which are relatively larger than those obtained from the pushover analysis. Furthermore, it is essential to acknowledge that the IDA fitting results exhibit a certain degree of dispersion, with an R-squared value (R2) of 0.865.
(2) Determination of Structural Target Displacement
Using the formulas outlined in this study ((11) and (12)), the fitted performance curve is transformed into the capacity spectrum for the first three modes. Additionally, a demand spectrum curve is established to represent seismic excitations acting upon the structure. Through a series of iterative calculations, a ductility demand of μ = 1.945 and a structural target displacement of Δ = 340.5 mm are determined. The intersections between the capacity spectrum and the demand spectrum for each mode are visually depicted in Figure 14. These intersections provide critical insights into the structural response under seismic loading conditions and inform the assessment of structural performance and safety.
(3) Calculation of Performance Factors for Case D1-12-0.9.
In the case of minor seismic excitations, specific structural parameters are determined as follows: the elastic design point (Vd) is computed as 927.83 kN, and the associated displacement (Δd) is found to be 37.01 mm. Furthermore, the initial stiffness of the structure (Ke) is established at 24.552 kN/mm, while the yield shear force (Vy) is calculated to be 4238.76 kN. Additionally, the displacement demand (Δ) is determined to be 340.5 mm.
By applying the formulas denoted as (7) to (10), the ensuing performance factors for case D1-12-0.9 are meticulously computed and derived.
4.3. Summary of Calculation Results
4.3.1. IDA Fitting Curves for Each Case
In accordance with the IDA analysis process outlined in the preceding section, the fitting performance curves for all cases subjected to multiple seismic waves are presented in Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20, and information on the seismic records are as Appendix A. These figures also incorporate the pushover curves for both uniformly distributed and inverted triangular loading modes. Notably, for the majority of cases, the initial stiffness of the structure, as determined from the IDA fitting curves, falls within the range defined by the inverted triangular loading and uniformly distributed loading modes. Furthermore, the structural capacity derived from the IDA fitting curves shows closer alignment with the results obtained under uniformly distributed loading conditions. This observation highlights the importance of employing the IDA approach to comprehensively assess structural behavior under diverse seismic scenarios, facilitating a more accurate representation of structural capacity.
4.3.2. Initial Stiffness and Significant Yielding Point for Each Case
Derived from the IDA analysis, coupled with the obtained performance curves for each case, the initial stiffness and significant yielding point of the structures are meticulously determined. A comprehensive summary of these specific results is provided in Table 6.
Upon examining the table, a clear trend emerges: as the number of structural stories increases, the initial stiffness (Ke) of the structure gradually decreases, while the ultimate bearing capacity (Vm) shows an incremental rise. Additionally, it is noteworthy that for cases with the same number of stories, the Q690 composite case demonstrates lower initial stiffness and ultimate bearing capacity compared to the Q460 composite case. This observation highlights the significant influence of structural configuration and material properties on the dynamic response and ultimate performance of the analyzed structures.
4.3.3. Seismic Performance Points for Each Case
Utilizing the capacity spectrum method, capacity spectrum curves for the first three modes of each case are systematically plotted on the same coordinate system alongside the demand spectrum curve representative of moderate seismic excitations. Through an iterative computational process, the ductility demand (μ) and the seismic performance point for moderate seismic excitations are determined for each individual structure. These specific results are meticulously illustrated in Figure 21, Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26.
4.3.4. Displacement Demand (Δ) for Each Case under Moderate Seismic Level
Drawing from the spectral displacements corresponding to the performance points derived from the demand spectrum and capacity spectrum for moderate seismic excitations, the vertex displacements Δi (i = 1, 2, 3) are systematically computed using the formula (12). Subsequently, the structural displacement demand Δ is determined by taking the square root of the sum of squares of the vertex displacements. A comprehensive summary of these specific results is presented in Table 7.
4.3.5. Elastic Base Shear (Ve) for Each Model under Moderate Seismic Level
The calculation of elastic base shear (Ve) for each case, under conditions of moderate seismic excitations, is predicated on the initial stiffness (Ke) of the structure and the displacement demand (Δ). Detailed outcomes of these calculations are meticulously presented in Table 8.
4.3.6. Summary of Performance Factors for Each Case
Drawing upon the calculated results derived from the IDA analysis method elucidated earlier, comprehensive performance factors for each individual case can be ascertained. A comprehensive compilation of these specific results is succinctly summarized in Table 9.
5. The Impact of Design Parameters on R and Cd
Focusing on the number of floors (N), link length (e), and alterations in steel strength of non-dissipative frames, this section examines their influence on critical structural indicators such as the response modification factor (R), displacement amplification factor (Cd), and structural overstrength factor (RΩ), all derived through Incremental Dynamic Analysis (IDA) methods. Compared to the data from the pushover analysis, it is noteworthy that the structure influence coefficient and displacement amplification factor obtained through IDA analysis are relatively smaller. This disparity can be attributed to the higher elastic stiffness of the fitted performance curve in IDA analysis, which generally surpasses the initial stiffness derived from the distributed lateral force adjustment method used in the pushover analysis. Additionally, the discrete nature of the IDA analysis method introduces certain distinctions in the structural performance factors obtained, highlighting the importance of considering the specific analytical method employed when evaluating the impact of design parameters on key structural response metrics.
5.1. Impact of Design Parameters on Response Modification Factor, R
(1) Influence of Story Number N
Figure 27 reveals that the response modification factor, as obtained from IDA analysis, does not exhibit a discernible regular pattern with respect to the story number. Specifically, as the story number increases from 8 to 12, there is a decrease in the R value for the Q460 combination case, while conversely, there is an upward trend in the R value for the Q690 combination case. However, the alteration in the R value is relatively minor when the story number increases from 12 to 16.
(2) Influence of Link Length e
In Figure 27, it is apparent that the response modification factor for most structural configurations tends to decline with an increase in the link length. Nevertheless, this decrease is not particularly substantial. Moreover, the correlation between the response modification factor and the link length is relatively modest, notably in the case of the 16-story configuration with the Q460 model and the 8-story configuration with the Q690 model.
(3) Influence of Variation in Steel Strength
A comparison of the response modification factor between two distinct high-strength steel combinations reveals nuanced variations. Specifically, for the 8-story configuration with the Q690 steel models, the response modification factor is marginally lower than that of its counterpart with the Q460 models. Conversely, for the 12-story and 16-story configurations with equivalent link lengths, the response modification factor for the Q690 high-strength steel surpasses the results obtained for the Q460 high-strength steel.
5.2. Impact of Design Parameters on Displacement Amplification Factor Cd
(1) Influence of Story Number N
Figure 28 reveals that the displacement amplification factor tends to increase with a rise in the story number, notably in the scenario with a link length of 900 mm. The most significant surge in the displacement amplification factor is observed during the transition from 12 stories to 16 stories. However, for other link lengths, the relationship between the displacement amplification factor and the story number is not as pronounced.
(2) Influence of Link Length e
From Figure 28, it becomes evident that the displacement amplification factor generally diminishes with an increase in the link length across various structural configurations. Notably, in the 12-story model with the Q460 high-strength steel and the 8-story model with the Q690 high-strength steel, the displacement amplification factor exhibits relatively stable changes with a weak correlation to the number of floors.
(3) Influence of Variation in Steel Strength
Figure 28 highlights that in specific cases, such as the 12-story configuration with link lengths of 1000 mm or 1100 mm, and the 16-story configuration with a link length of 900 mm, the disparities in displacement amplification factor Cd between the two steel strength combinations are marginal. However, in the remaining scenarios, the displacement amplification factor for the Q460 high-strength steel models slightly exceeds the results observed for the Q690 high-strength steel models.
5.3. Impact of Design Parameters on Structure Overstrength Factor RΩ
(1) Influence of Story Number N
Figure 29 reveals noteworthy trends in the structure overstrength factor RΩ. Specifically, in cases involving the Q460 high-strength steel combination, a substantial decline in RΩ is evident as the number of stories increases from 8 to 12 for link lengths of 900 mm and 1000 mm. Moreover, in the scenario featuring a link length of 1100 mm, there is a more pronounced decrease in RΩ when transitioning from 12 to 16 stories. In contrast, for the Q690 high-strength steel combination cases, the RΩ values for the link length of 900 mm exhibit an increase as the number of stories rises, while other cases showcase varying degrees of decrease.
(2) Influence of Link Length e
Figure 29 illustrates that, for structures with 8 stories, the structure overstrength factor RΩ typically increases with a growing number of stories. In the case of 12-story structures, there is a slight reduction in RΩ as the link length extends from 900 mm to 1000 mm, while an increase is noted when the link length is further increased to 1100 mm. This increase is particularly pronounced in the Q460 high-strength steel combination cases. Meanwhile, in the context of 16-story structures, RΩ exhibits a decreasing trend as the link length increases.
(3) Influence of Variation in Steel Strength
Figure 29 demonstrates that the influence of steel strength variation on the structure overstrength factor RΩ, as gleaned from IDA analysis, lacks a distinct regular pattern. The average RΩ values for the two different steel combinations, with the same number of stories, are as follows: 5.952, 5.496, 5.001, 5.337, 4.351, and 5.223.
6. Conclusions
In this study, the Incremental Dynamic Analysis (IDA) method was utilized to calculate key structural metrics for 18 structural models. These metrics include the response modification factor (R), displacement amplification factor (Cd), and structure overstrength factor (RΩ). The analysis focused on the impact of three critical design parameters: the number of floors, the link lengths, and variations in steel combinations. The conclusions are as follows:
(1) The response modification factor (R) and structure overstrength factor (RΩ) obtained through Incremental Dynamic Analysis (IDA) show only minor discrepancies compared to the results from pushover analysis. However, the displacement amplification factor (Cd) exhibits significant differences between the two methods. These differences are primarily due to variations in the initial stiffness of the structural performance curves derived from the two methods of analysis.
(2) The displacement amplification factor (Cd) derived from Incremental Dynamic Analysis (IDA) showed little variation with the number of stories, with no significant effect observed. However, the impact of the link length (e) on the Cd value exhibits irregularity in IDA analysis. Moreover, for various high-strength steel composite models, the Cd value obtained from IDA analysis for the Q690 high-strength steel composite model shows inconsistent results.
(3) The influence of the story number on the response modification factor (R) is minimal, according to IDA analysis. Similarly, a clear relationship between the R value and the link length is not evident in the 16-story cases, which is consistent with findings from the pushover analysis. In the other cases, a general trend is observed where the R value decreases as the link length increases. Notably, for the 12-story cases, both methods of analysis yield congruent results, demonstrating that the structure influence coefficient is higher for the Q690 high-strength steel combination compared to the Q460 high-strength steel combination.
(4) For cases with a link length of 900 mm, the displacement amplification factor (Cd) from IDA analysis shows an increasing trend as the story number rises, though this effect is less pronounced in other cases. Comparing the two methods of analysis, pushover analysis demonstrates a more substantial increase in Cd with an increase in the story number. Additionally, the agreement between the two methods on the influence of link length and different high-strength steel combinations on Cd is relatively limited.
(5) For cases with a link length of 1100 mm, both methods of analysis show a significant decrease in the structure overstrength factor (RΩ) as the number of floors increases. This decrease is particularly pronounced. Additionally, in the 8-story and 12-story cases, an increase in link length results in an improvement in RΩ, displaying a consistent trend in both methods. Notably, pushover analysis yields a higher structure overstrength factor for the Q690 high-strength steel combination compared to the Q460 high-strength steel combination, while the trend obtained from IDA analysis is less pronounced.
These findings provide valuable insights into the impact of critical design parameters on key structural response metrics, clarifying the complexities involved in assessing structural behavior under seismic loading conditions.
Author Contributions
Methodology, Y.M.; Software, J.Y.; Validation, Y.M.; Investigation, J.Y.; Data curation, J.Y.; Writing—original draft, Y.M.; Writing—review & editing, X.M. All authors have read and agreed to the published version of the manuscript.
Funding
This study was supported by the Ningxia Natural Science Foundation (Grant No. 2021AAC03189, Grant No. 2023AAC03317) and the National Natural Science Foundation of China (Grant No. 52368011). The financial support is greatly appreciated.
Data Availability Statement
Data are contained within the article.
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A
Table A1.Seismic records for model D1-8-0.9.
Table A1.Seismic records for model D1-8-0.9.
| Seismic Wave Number | Seismic Wave | Seismic Wave Station | PGA/g | Earthquake Degree |
|---|---|---|---|---|
| RSN9 | Borrego | El Centro Array #9 | 0.066 | 6.5 |
| RSN161 | Imperial Valley-06 | Brawley Airport | 0.163 | 6.53 |
| RSN764 | Loma Prieta | Gilroy—Historic Bldg. | 0.285 | 6.93 |
| RSN879 | Landers | SCE 24 Lucerne | 0.727 | 7.28 |
| RSN1489 | Chi-Chi Taiwan | TCU049 | 0.279 | 7.62 |
| RSN1602 | Duzce Turkey | Bolu | 0.739 | 7.14 |
| RSN3746 | Cape Mendocino | Centerville Beach Naval Fac | 0.318 | 7.01 |
| RSN4040 | Bam Iran | Bam | 0.808 | 6.6 |
| RSN6927 | Darfield New Zealand | LINC | 0.461 | 7 |
| RSN6960 | Darfield New Zealand | Riccarton High School | 0.19 | 7 |
Table A2.Seismic records for models D1-8-1.0, D1-8-1.1.
Table A2.Seismic records for models D1-8-1.0, D1-8-1.1.
| Seismic Wave Number | Seismic Wave | Seismic Wave Station | PGA/g | Earthquake Degree |
|---|---|---|---|---|
| RSN161 | Imperial Valley-06 | Brawley Airport | 0.163 | 6.53 |
| RSN173 | Imperial Valley-06 | El Centro Array #10 | 0.11 | 6.53 |
| RSN764 | Loma Prieta | Gilroy—Historic Bldg. | 0.285 | 6.93 |
| RSN802 | Loma Prieta | Saratoga—Aloha Ave | 0.513 | 6.93 |
| RSN879 | Landers | SCE 24 Lucerne | 0.727 | 7.28 |
| RSN1085 | Northridge-01 | Sylmar—Converter Sta East | 0.853 | 6.69 |
| RSN1489 | Chi-Chi Taiwan | TCU049 | 0.279 | 7.62 |
| RSN1602 | Duzce Turkey | Bolu | 0.739 | 7.14 |
| RSN3746 | Cape Mendocino | Centerville Beach Naval Fac | 0.318 | 7.01 |
| RSN4040 | Bam Iran | Bam | 0.808 | 6.6 |
Table A3.Seismic records for model D1-12-0.9.
Table A3.Seismic records for model D1-12-0.9.
| Seismic Wave Number | Seismic Wave | Seismic Wave Station | PGA/g | Earthquake Degree |
|---|---|---|---|---|
| RSN163 | Imperial Valley | Calipatria Fire Station | 0.129 | 6.53 |
| RSN176 | Imperial Valley | El Centro Array #13 | 0.049 | 6.53 |
| RSN184 | Imperial Valley-06 | El Centro Differential Array | 0.353 | 6.53 |
| RSN802 | Loma Prieta | Saratoga—Aloha Ave | 0.513 | 6.93 |
| RSN879 | Landers | SCE 24 Lucerne | 0.727 | 7.28 |
| RSN1085 | Northridge-01 | Sylmar—Converter Sta East | 0.853 | 6.69 |
| RSN1602 | Duzce Turkey | Bolu | 0.739 | 7.14 |
| RSN4228 | Niigata Japan | NIGH11 | 0.599 | 6.63 |
| RSN6927 | Darfield New Zealand | LINC | 0.461 | 7 |
| RSN6933 | Darfield New Zealand | MAYC | 0.068 | 7 |
Table A4.Seismic records for model D1-12-1.0.
Table A4.Seismic records for model D1-12-1.0.
| Seismic Wave Number | Seismic Wave | Seismic Wave Station | PGA/g | Earthquake Degree |
|---|---|---|---|---|
| RSN171 | Imperial Valley-06 | El Centro—Meloland Geot.Array | 0.317 | 6.53 |
| RSN184 | Imperial Valley-06 | El Centro Differential Array | 0.353 | 6.53 |
| RSN802 | Loma Prieta | Saratoga—Aloha Ave | 0.513 | 6.93 |
| RSN1085 | Northridge-01 | Sylmar—Converter Sta East | 0.853 | 6.69 |
| RSN1148 | Kocaeli Turkey | KOERI 99999 Arcelik | 0.22 | 7.51 |
| RSN1493 | Chi-Chi Taiwan | TCU053 | 0.229 | 7.62 |
| RSN1602 | Duzce Turkey | Bolu | 0.739 | 7.14 |
| RSN4228 | Niigata Japan | NIGH11 | 0.599 | 6.63 |
| RSN5783 | Wate Japan | Semine Kurihara City | 0.142 | 6.9 |
| RSN6927 | Darfield New Zealand | LINC | 0.461 | 7 |
Table A5.Seismic records for model D1-12-1.1.
Table A5.Seismic records for model D1-12-1.1.
| Seismic Wave Number | Seismic Wave | Seismic Wave Station | PGA/g | Earthquake Degree |
|---|---|---|---|---|
| RSN171 | Imperial Valley-06 | El Centro—Meloland Geot.Array | 0.317 | 6.53 |
| RSN184 | Imperial Valley-06 | El Centro Differential Array | 0.353 | 6.53 |
| RSN802 | Loma Prieta | Saratoga—Aloha Ave | 0.513 | 6.93 |
| RSN1004 | Northridge-01 | LA—Sepulveda VA Hospital | 0.753 | 6.69 |
| RSN1085 | Northridge-01 | Sylmar—Converter Sta East | 0.853 | 6.69 |
| RSN1119 | Kobe_ Japan | Takarazuka | 0.697 | 6.9 |
| RSN1165 | Kocaeli, Turkey | ERD 99999 Izmit | 0.22 | 7.51 |
| RSN1493 | Chi-Chi Taiwan | TCU053 | 0.229 | 7.62 |
| RSN1602 | Duzce Turkey | Bolu | 0.739 | 7.14 |
| RSN3746 | Cape Mendocino | Centerville Beach Naval Fac | 0.318 | 7.01 |
Table A6.Seismic records for models D1-16-0.9, D1-16-1.0, D1-16-1.1.
Table A6.Seismic records for models D1-16-0.9, D1-16-1.0, D1-16-1.1.
| Seismic Wave Number | Seismic Wave | Seismic Wave Station | PGA/g | Earthquake Degree |
|---|---|---|---|---|
| RSN171 | Imperial Valley-06 | El Centro—Meloland Geot.Array | 0.317 | 6.53 |
| RSN173 | Imperial Valley-06 | El Centro Array #10 | 0.11 | 6.53 |
| RSN184 | Imperial Valley-06 | El Centro Differential Array | 0.353 | 6.53 |
| RSN185 | Imperial Valley-06 | Holtville Post Office | 0.258 | 6.53 |
| RSN767 | Loma Prieta | Gilroy Array #3 | 0.342 | 6.93 |
| RSN879 | Landers | SCE 24 Lucerne | 0.727 | 7.28 |
| RSN1004 | Northridge-01 | LA—Sepulveda VA Hospital | 0.753 | 6.69 |
| RSN1085 | Northridge-01 | Sylmar—Converter Sta East | 0.853 | 6.69 |
| RSN1489 | Chi-Chi Taiwan | TCU049 | 0.279 | 7.62 |
| RSN1493 | Chi-Chi Taiwan | TCU053 | 0.229 | 7.62 |
Table A7.Seismic records for models D2-8-0.9, D2-8-1.0, D2-8-1.1.
Table A7.Seismic records for models D2-8-0.9, D2-8-1.0, D2-8-1.1.
| Seismic Wave Number | Seismic Wave | Seismic Wave Station | PGA/g | Earthquake Degree |
|---|---|---|---|---|
| RSN161 | Imperial Valley-06 | Brawley Airport | 0.163 | 6.53 |
| RSN764 | Loma Prieta | Gilroy—Historic Bldg. | 0.285 | 6.93 |
| RSN802 | Loma Prieta | Saratoga—Aloha Ave | 0.513 | 6.93 |
| RSN879 | Landers | SCE 24 Lucerne | 0.727 | 7.28 |
| RSN1489 | Chi-Chi Taiwan | TCU049 | 0.279 | 7.62 |
| RSN1602 | Duzce Turkey | Bolu | 0.739 | 7.14 |
| RSN3746 | Cape Mendocino | Centerville Beach Naval Fac | 0.318 | 7.01 |
| RSN4040 | Bam Iran | Bam | 0.808 | 6.6 |
| RSN6927 | Darfield New Zealand | LINC | 0.461 | 7 |
| RSN6960 | Darfield New Zealand | Riccarton High School | 0.19 | 7 |
Table A8.Seismic records for models D2-12-0.9, D2-12-1.0, D2-12-1.1.
Table A8.Seismic records for models D2-12-0.9, D2-12-1.0, D2-12-1.1.
| Seismic Wave Number | Seismic Wave | Seismic Wave Station | PGA/g | Earthquake Degree |
|---|---|---|---|---|
| RSN171 | Imperial Valley-06 | El Centro—Meloland Geot.Array | 0.317 | 6.53 |
| RSN173 | Imperial Valley-06 | El Centro Array #10 | 0.11 | 6.53 |
| RSN802 | Loma Prieta | Saratoga—Aloha Ave | 0.513 | 6.93 |
| RSN879 | Landers | SCE 24 Lucerne | 0.727 | 7.28 |
| RSN1086 | Northridge-01 | Sylmar—Olive View Med FF | 0.605 | 6.69 |
| RSN1119 | Kobe_ Japan | Takarazuka | 0.697 | 6.9 |
| RSN1165 | Kocaeli, Turkey | ERD 99999 Izmit | 0.22 | 7.51 |
| RSN1602 | Duzce Turkey | Bolu | 0.739 | 7.14 |
| RSN4228 | Niigata Japan | NIGH11 | 0.599 | 6.63 |
| RSN6927 | Darfield New Zealand | LINC | 0.461 | 7 |
Table A9.Seismic records for model D2-16-0.9.
Table A9.Seismic records for model D2-16-0.9.
| Seismic Wave Number | Seismic Wave | Seismic Wave Station | PGA/g | Earthquake Degree |
|---|---|---|---|---|
| RSN173 | Imperial Valley-06 | El Centro Array #10 | 0.11 | 6.53 |
| RSN184 | Imperial Valley-06 | El Centro Differential Array | 0.353 | 6.53 |
| RSN185 | Imperial Valley-06 | Holtville Post Office | 0.258 | 6.53 |
| RSN767 | Loma Prieta | Gilroy Array #3 | 0.342 | 6.93 |
| RSN802 | Loma Prieta | Saratoga—Aloha Ave | 0.513 | 6.93 |
| RSN879 | Landers | SCE 24 Lucerne | 0.727 | 7.28 |
| RSN1004 | Northridge-01 | LA—Sepulveda VA Hospital | 0.753 | 6.69 |
| RSN1602 | Duzce Turkey | Bolu | 0.739 | 7.14 |
| RSN4228 | Niigata Japan | NIGH11 | 0.599 | 6.63 |
| RSN6897 | Darfield New Zealand | DSLC | 0.257 | 7 |
Table A10.Seismic records for models D2-16-1.0, D2-16-1.1.
Table A10.Seismic records for models D2-16-1.0, D2-16-1.1.
| Seismic Wave Number | Seismic Wave | Seismic Wave Station | PGA/g | Earthquake Degree |
|---|---|---|---|---|
| RSN173 | Imperial Valley-06 | El Centro Array #10 | 0.11 | 6.53 |
| RSN185 | Imperial Valley-06 | Holtville Post Office | 0.258 | 6.53 |
| RSN828 | Cape Mendocino | Petrolia | 0.591 | 7.01 |
| RSN879 | Landers | SCE 24 Lucerne | 0.727 | 7.28 |
| RSN1086 | Northridge-01 | Sylmar—Olive View Med FF | 0.605 | 6.69 |
| RSN1493 | Chi-Chi Taiwan | TCU053 | 0.229 | 7.62 |
| RSN1602 | Duzce Turkey | Bolu | 0.739 | 7.14 |
| RSN4228 | Niigata Japan | NIGH11 | 0.599 | 6.63 |
| RSN6897 | Darfield New Zealand | DSLC | 0.257 | 7 |
| RSN6906 | Darfield New Zealand | GDLC | 0.765 | 7 |
References
- TJ11-78; Code for Seismic Design of Industrial and Civil Buildings. China Architecture Industry Press: Beijing, China, 1978. (In Chinese)
- GBJ 11-89; Code for Seismic Design of Buildings. China Architecture Industry Press: Beijing, China, 1989. (In Chinese)
- CECS 160:2004; General Rule for Performance-Based Seismic Design of Buildings. China Plan Press: Beijing, China, 2004. (In Chinese)
- International Council of Building Officials. 1997 Uniform Building Code; International of Building Officials: Whittier, CA, USA, 1997. [Google Scholar]
- ASCE7-05; Minimum Design Loads for Building and Other Structures. American Society of Civil Engineers: Reston, VA, USA, 2006.
- Building Seismic Safety Council (BSSC). NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures. In National Earthquake Hazard Reduction Program (NEHRP); Building Seismic Safety Council (BSSC): Washington, DC, USA, 2000. [Google Scholar]
- Uang, C.M. Establishing R (or Rw) and Cd factors for building seismic provisions. J. Struct. Eng. 1991, 117, 19–28. [Google Scholar] [CrossRef]
- Su, M.Z.; Liu, Y.L. Study on structural behavior factor of high-strength steel composite Y-type eccentrically braced frame based on incremental dynamic analysis. J. Xi’an Univ. Archit. Technol. (Nat. Sci. Ed.) 2014, 46, 635–642. [Google Scholar]
- Xu, J.; Wang, S.L.; Mohamed, H.S. Seismic performance of shear energy dissipation beams in D-shaped eccentrically braced steel frames. J. Constr. Steel Res. 2021, 180, 106584. [Google Scholar] [CrossRef]
- Li, S.; Wang, Z.; Guo, H. Seismic performance of high strength steel frames with variable eccentric braces based on PBSD method. Earthq. Struct. 2020, 18, 527–542. [Google Scholar]
- Mohammadgholipour, A.; Billah, M.A. Performance-based plastic design and seismic fragility assessment for chevron braced steel frames considering aftershock effects. Soil Dyn. Earthq. Eng. 2024, 178, 108440. [Google Scholar] [CrossRef]
- Kalapodis, N.A.; Muho, E.V.; Papagiannopoulos, G.A. Integration of peak seismic floor velocities and accelerations into the performance-based design of steel structures. Soil Dyn. Earthq. Eng. 2022, 154, 107160. [Google Scholar] [CrossRef]
- GB50011-2010; Code for seismic design of buildings. China Architecture Industry Press: Beijing, China, 2010. (In Chinese)
- Zheng, L.; Qin, C. Research on the structural influencing coefficient of steel plate shear wall. Ind. Constr. 2020, 50, 154–161. [Google Scholar]
- Cheng, Q.Q.; Su, M.Z. Response modification factor of high-strength steel fabricated framed-tube with replaceable shear links. Eng. Mech. 2020, 37, 145–158. [Google Scholar]
- Mussa, M.; Mahdi, Z. Determination of the Response Modification Factors of Buckling Restrained Braced Frames. Procedia Eng. 2013, 54, 222–231. [Google Scholar]
- Yousef, H.; Al, Q.; Besan, Y.A. Evaluating seismic response modification factor of steel frames with different bracing systems. Int. J. Earthq. Impact Eng. 2018, 2, 203–225. [Google Scholar]
- Li, S.; Liang, W.G. Response modification factor and displacement amplification factor of K-shaped eccentrically braced high-strength steel frames. J. Asian Archit. Build. Eng. 2022, 21, 2399–2425. [Google Scholar] [CrossRef]
- SAP2000; Integrated Software for Structural Analysis & Design, Version 14.2.4—Analysis Reference Manual. Computers and Structures, Inc. (CSI): Berkeley, CA, USA, 2010.
- Li, S.; Tian, J.B. Response on the seismic behavior of Y-type eccentrically braced frames combined with high strength steels based on PBSD method. Ind. Constr. 2018, 48, 140–150. [Google Scholar]
- GB50009-2012; Load Code for Design of Building Structures. China Architecture Industry Press: Beijing, China, 2012. (In Chinese)
- Li, S.; Gao, Y.T. Story shear distribution of high strength steel frame with D-eccentric braces based on multi-seismic-hazard levels. Structures 2023, 48, 616–641. [Google Scholar]
- Richards, P.W. Cyclic Stability and Capacity Design of Steel Eccentrically Braced Frames; University of California: San Diego, CA, USA, 2004. [Google Scholar]
- Rides, J.M.; Popov, E.P. Inelastic Link Element for EBF Seismic Analysis. J. Struct. Eng. 1994, 120, 441–463. [Google Scholar] [CrossRef]
- Ramadan, T.; Ghobarah, A. Analytical model for shear-link behavior. J. Struct. Eng. 1995, 121, 1574–1580. [Google Scholar] [CrossRef]
Figure 1.Performance curve of the structure.
Figure 1.Performance curve of the structure.
Figure 2.Types of eccentrically braced frames.
Figure 2.Types of eccentrically braced frames.
Figure 3.D-8-0.9 structural plan layout.
Figure 3.D-8-0.9 structural plan layout.
Figure 4.Computing frame element.
Figure 4.Computing frame element.
Figure 5.Stress–strain curves of steel.
Figure 5.Stress–strain curves of steel.
Figure 6.Flow chart of PBSD method.
Figure 6.Flow chart of PBSD method.
Figure 7.Comparison diagram of test specimen and finite element model.
Figure 7.Comparison diagram of test specimen and finite element model.
Figure 8.Generalized force–deformation relation for shear link (FEMA-356).
Figure 8.Generalized force–deformation relation for shear link (FEMA-356).
Figure 9.Shear link springs model.
Figure 9.Shear link springs model.
Figure 11.IDA analysis results of example D1-12-0.9 under seismic wave RSN184.
Figure 11.IDA analysis results of example D1-12-0.9 under seismic wave RSN184.
Figure 12.IDA fitting curve of D1-12-0.9.
Figure 12.IDA fitting curve of D1-12-0.9.
Figure 13.Example D1-12-0.9 of IDA calculation data.
Figure 13.Example D1-12-0.9 of IDA calculation data.
Figure 14.Performance point of model D1-12-0.9 under design earthquake.
Figure 14.Performance point of model D1-12-0.9 under design earthquake.
Figure 15.Structure–performance curve of the 8 stories (D1).
Figure 15.Structure–performance curve of the 8 stories (D1).
Figure 16.Structure–performance curve of the 12 stories (D1).
Figure 16.Structure–performance curve of the 12 stories (D1).
Figure 17.Structure–performance curve of the 16 stories (D1).
Figure 17.Structure–performance curve of the 16 stories (D1).
Figure 18.Structure–performance curve of the 8 stories (D2).
Figure 18.Structure–performance curve of the 8 stories (D2).
Figure 19.Structure–performance curve of the 12 stories (D2).
Figure 19.Structure–performance curve of the 12 stories (D2).
Figure 20.Structure–performance curve of the 16 stories (D2).
Figure 20.Structure–performance curve of the 16 stories (D2).
Figure 21.Seismic performance points of the 8 stories (D1).
Figure 21.Seismic performance points of the 8 stories (D1).
Figure 22.Seismic performance points of the 12 stories (D1).
Figure 22.Seismic performance points of the 12 stories (D1).
Figure 23.Seismic performance points of the 16 stories (D1).
Figure 23.Seismic performance points of the 16 stories (D1).
Figure 24.Seismic performance points of the 8 stories (D2).
Figure 24.Seismic performance points of the 8 stories (D2).
Figure 25.Seismic performance points of the 12 stories (D2).
Figure 25.Seismic performance points of the 12 stories (D2).
Figure 26.Seismic performance points of the 16 stories (D2).
Figure 26.Seismic performance points of the 16 stories (D2).
Figure 27.The influence of design parameters on R. (a) Example of Q460 high-strength steel combination calculation. (b) Example of Q690 high-strength steel combination calculation.
Figure 27.The influence of design parameters on R. (a) Example of Q460 high-strength steel combination calculation. (b) Example of Q690 high-strength steel combination calculation.
Figure 28.The influence of design parameters on Cd. (a) Example of Q460 high-strength steel combination calculation. (b) Example of Q690 high-strength steel combination calculation.
Figure 28.The influence of design parameters on Cd. (a) Example of Q460 high-strength steel combination calculation. (b) Example of Q690 high-strength steel combination calculation.
Figure 29.The influence of design parameters on RΩ. (a) Example of Q460 high-strength steel combination calculation. (b) Example of Q690 high-strength steel combination calculation.
Figure 29.The influence of design parameters on RΩ. (a) Example of Q460 high-strength steel combination calculation. (b) Example of Q690 high-strength steel combination calculation.
Table 1.Seismic response modification factor in individual national specifications.
Table 1.Seismic response modification factor in individual national specifications.
| Canonical Name | Structure Type | Recommended Value |
|---|---|---|
| Code for Seismic Design of Industrial and Civil Buildings (TJ11-78) | Steel frame structure | 4 |
| General Rule for Performance-based Seismic Design of Buildings (CECS 160:2004) | Steel frame structure | 4 |
| Steel eccentrically braced frame | 3.33 | |
| Steel concentrically braced frame | 3.70 | |
| Code for Seismic Design of Buildings (GBJ 11-89) | Steel eccentrically braced frame | 2.8125 |
| Uniform Building Code 1997 of America (UBC97) | Steel eccentrically braced frame | 7.0 |
| Steel ordinary braced frame | 5.6 | |
| American Society of Civil Engineers (ASCE7-05) | Steel eccentrically braced frame (Beam column rigid connection) | 8 |
| Steel eccentrically braced frame (Hinge joint of beam and column) | 7 | |
| National Earthquake Hazard Reduction Program (NEHRP 2000) | Steel eccentrically braced frame | 8 |
Table 2.The relationship of ductility reduction factor and period.
Table 2.The relationship of ductility reduction factor and period.
| Design Parameters | Rare Earthquake |
|---|---|
| Seismic influence coefficient α (m/s2) | 0.643 |
| Sa (m/s2) | 1.537 |
| Fundamental period T/s | 0.8 s |
| Yield drift θy | 0.5156% |
| Target drift θu | 1.7156% |
| Plastic drift θp | 1.2% |
| Ductility μs | 3.33 |
| Ductility reduction factor Rμ | 3.33 |
| Energy correction factor γ | 0.51 |
| V/G | 0.39 |
| Design base shear V/kN | 2861.43 |
Table 3.Parameter for shear–displacement relation of link.
Table 3.Parameter for shear–displacement relation of link.
| VB | VB1 | VC | K0 | K1 | K2 | K3 |
|---|---|---|---|---|---|---|
| 1.1VP | 1.3VP | 1.5VP | 2GAw/e | 0.03K0 | 0.015K0 | 0.002K0 |
Table 4.Maximum acceleration value for time history analysis under four-level ground motions.
Table 4.Maximum acceleration value for time history analysis under four-level ground motions.
| Earthquake Hazard Level | 6 Degrees | 7 Degrees | 8 Degrees | 9 Degrees |
|---|---|---|---|---|
| Frequent earthquakes | 18 | 35(55) | 70(110) | 140 |
| Fortification earthquakes | 49 | 98(147) | 196(294) | 392 |
| Rare earthquake | 125 | 220(310) | 400(510) | 620 |
| Extremely rare earthquake | 147 | 294(441) | 588(882) | 1176 |
Table 5.The results of model D1-12-0.9 under ground motion RSN184.
Table 5.The results of model D1-12-0.9 under ground motion RSN184.
| Number | Acceleration (cm/s2) | Roof Displacement Δ (mm) | Base Shear V (kN) |
|---|---|---|---|
| 1 | 110 (Frequency earthquake) | 28.503 | 684.572 |
| 2 | 170 | 43.528 | 1193.839 |
| 3 | 230 | 58.553 | 1615.191 |
| 4 | 294 (Moderate earthquake) | 74.582 | 2064.633 |
| 5 | 344 | 87.102 | 2415.764 |
| 6 | 394 | 99.623 | 2766.893 |
| 7 | 444 | 112.144 | 3118.025 |
| 8 | 510 (High earthquake) | 128.672 | 3581.38 |
| 9 | 580 | 146.322 | 3892.323 |
| 10 | 690 | 175.751 | 4420.492 |
| 11 | 800 | 209.99 | 4931.886 |
| 12 | 882 (Extreme earthquake) | 238.591 | 5129.263 |
| 13 | 980 | 275.121 | 5155.457 |
| 14 | 1090 | 320.271 | 5387.287 |
| 15 | 1200 | 367.952 | 5408.085 |
| 16 | 1310 | 417.785 | 5509.309 |
| 17 | 1420 | 270.218 | 5235.821 |
Table 6.The initial stiffness and significant yield points.
Table 6.The initial stiffness and significant yield points.
| Model | Δm (mm) | Vm (kN) | Ke (kN/mm) | A (kN·mm) | Vy (kN) | Δy (mm) |
|---|---|---|---|---|---|---|
| D1-8-0.9 | 430.667 | 4936.789 | 31.4 | 1,672,809 | 4459.219 | 141.981 |
| D1-8-1.0 | 335.947 | 5195.302 | 34.4 | 1,316,090 | 4786.129 | 138.789 |
| D1-8-1.1 | 270.211 | 6043.783 | 42.8 | 1,178,524 | 5608.73 | 130.975 |
| D1-12-0.9 | 417.785 | 5893.627 | 24.2 | 1,601,000 | 4238.761 | 174.96 |
| D1-12-1.0 | 505.353 | 6065.311 | 22.4 | 1,968,500 | 3703.977 | 164.862 |
| D1-12-1.1 | 603.736 | 5819.289 | 24.0 | 2,830,006 | 5937.389 | 247.095 |
| D1-16-0.9 | 697.921 | 6921.277 | 16.4 | 3,098,830 | 4932.992 | 299.898 |
| D1-16-1.0 | 909.117 | 7410.585 | 16.1 | 4,432,884 | 4738.037 | 294.005 |
| D1-16-1.1 | 772.672 | 7518.736 | 16.5 | 3,603,942 | 4398.911 | 266.075 |
| D2-8-0.9 | 346.949 | 4301.339 | 25.6 | 1,062,025 | 3530.434 | 137.904 |
| D2-8-1.0 | 339.558 | 4622.684 | 24.7 | 1,071,201 | 3747.746 | 151.393 |
| D2-8-1.1 | 320.256 | 5543.01 | 28.6 | 1,194,575 | 4834.807 | 168.574 |
| D2-12-0.9 | 549.343 | 5566.875 | 20.3 | 2,194,523 | 4834.114 | 237.954 |
| D2-12-1.0 | 485.217 | 4859.031 | 19.2 | 1,646,965 | 4011.697 | 207.921 |
| D2-12-1.1 | 536.004 | 5997.204 | 20.0 | 2,146,260 | 4557.528 | 227.582 |
| D2-16-0.9 | 1062.703 | 7370.32 | 14.1 | 5,457,170 | 5673.852 | 399.947 |
| D2-16-1.0 | 811.687 | 6180.02 | 15.3 | 3,484,873 | 4766.574 | 309.944 |
| D2-16-1.1 | 781.336 | 6001.567 | 14.8 | 3,238,867 | 4754.159 | 320.933 |
Table 7.Seismic displacement requirements in each example.
Table 7.Seismic displacement requirements in each example.
| Model | Sd1 | Sd2 | Sd3 | Δ1 | Δ2 | Δ3 | Δ |
|---|---|---|---|---|---|---|---|
| D1-8-0.9 | 164.557 | 74.853 | 65.984 | 229.523 | 41.942 | 20.219 | 234.198 |
| D1-8-1.0 | 149.164 | 71.572 | 61.810 | 208.367 | 41.749 | 20.645 | 213.509 |
| D1-8-1.1 | 125.95 | 68.88 | 58.96 | 177.191 | 40.671 | 18.649 | 182.753 |
| D1-12-0.9 | 228.648 | 82.159 | 61.373 | 329.293 | 53.463 | 25.596 | 334.586 |
| D1-12-1.0 | 231.879 | 85.481 | 72.583 | 332.494 | 53.135 | 27.044 | 337.797 |
| D1-12-1.1 | 223.11 | 82.941 | 48.777 | 321.919 | 53.903 | 17.998 | 326.897 |
| D1-16-0.9 | 388.069 | 125.779 | 108.095 | 571.235 | 88.159 | 44.572 | 579.714 |
| D1-16-1.0 | 390.299 | 124.764 | 101.019 | 574.533 | 88.834 | 43.621 | 582.995 |
| D1-16-1.1 | 355.598 | 124.532 | 87.967 | 525.528 | 89.948 | 40.725 | 534.723 |
| D2-8-0.9 | 176.417 | 72.814 | 65.689 | 250.446 | 44.911 | 18.997 | 255.149 |
| D2-8-1.0 | 173.917 | 75.126 | 66.941 | 249.411 | 50.554 | 22.602 | 255.485 |
| D2-8-1.1 | 162.906 | 79.087 | 67.906 | 234.392 | 53.416 | 21.334 | 241.346 |
| D2-12-0.9 | 276.216 | 104.061 | 70.611 | 409.138 | 74.309 | 27.791 | 416.759 |
| D2-12-1.0 | 278.392 | 100.371 | 68.594 | 407.008 | 67.193 | 25.017 | 413.275 |
| D2-12-1.1 | 264.846 | 108.341 | 89.743 | 392.209 | 77.868 | 34.257 | 401.329 |
| D2-16-0.9 | 421.119 | 176.422 | 79.601 | 676.545 | 130.288 | 32.495 | 689.742 |
| D2-16-1.0 | 405.683 | 143.051 | 97.358 | 607.085 | 106.063 | 41.380 | 617.669 |
| D2-16-1.1 | 415.609 | 149.441 | 105.557 | 621.239 | 109.786 | 43.475 | 632.361 |
Table 8.The seismic elastic base shear in each example.
Table 8.The seismic elastic base shear in each example.
| Model | D1-8-0.9 | D1-8-1.0 | D1-8-1.1 | D1-12-0.9 | D1-12-1.0 | D1-12-1.1 |
|---|---|---|---|---|---|---|
| Ve (KN) | 7355.49 | 7362.83 | 7825.97 | 8105.99 | 7589.34 | 7854.94 |
| Model | D1-16-0.9 | D1-16-1.0 | D1-16-1.1 | D2-8-0.9 | D2-8-1.0 | D2-8-1.1 |
| Ve (KN) | 9535.63 | 9395.24 | 8840.37 | 6531.95 | 6324.52 | 6921.96 |
| Model | D2-12-0.9 | D2-12-1.0 | D2-12-1.1 | D2-16-0.9 | D2-16-1.0 | D2-16-1.1 |
| Ve (KN) | 8466.58 | 7973.85 | 8036.94 | 9785.02 | 9499.01 | 9367.52 |
Table 9.Summary table of performance coefficients of each calculation example.
Table 9.Summary table of performance coefficients of each calculation example.
| Model | D1-8-0.9 | D1-8-1.0 | D1-8-1.1 | D1-12-0.9 | D1-12-1.0 | D1-12-1.1 |
|---|---|---|---|---|---|---|
| R | 9.229 | 9.147 | 8.815 | 8.736 | 8.378 | 8.398 |
| Cd | 9.325 | 8.775 | 8.261 | 9.040 | 9.09 | 8.895 |
| Rμ | 1.649 | 1.538 | 1.395 | 1.912 | 2.048 | 1.322 |
| RΩ | 5.595 | 5.946 | 6.317 | 4.568 | 4.089 | 6.348 |
| Model | D1-16-0.9 | D1-16-1.0 | D1-16-1.1 | D2-8-0.9 | D2-8-1.0 | D2-8-1.1 |
| R | 8.766 | 8.838 | 8.162 | 9.055 | 8.866 | 9.079 |
| Cd | 9.784 | 9.596 | 8.872 | 8.699 | 8.564 | 8.864 |
| Rμ | 1.933 | 1.982 | 2.009 | 1.850 | 1.687 | 1.431 |
| RΩ | 4.535 | 4.457 | 4.061 | 4.894 | 5.254 | 6.341 |
| Model | D2-12-0.9 | D2-12-1.0 | D2-12-1.1 | D2-16-0.9 | D2-16-1.0 | D2-16-1.1 |
| R | 10.049 | 9.660 | 9.551 | 9.916 | 9.830 | 9.831 |
| Cd | 9.281 | 9.067 | 8.865 | 9.780 | 8.570 | 8.640 |
| Rμ | 1.751 | 1.987 | 1.763 | 1.724 | 1.992 | 1.970 |
| RΩ | 5.737 | 4.86 | 5.416 | 5.749 | 4.933 | 4.989 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
