A deductive design method to simplify ASCE 41-17 nonlinear static procedure preserving the adjusted collapse margin ratio of steel moment frames

The present study offers a deductive design procedure based on designs that have been optimally screened in accordance with ASCE 41-17's nonlinear static procedure (NSP). The subsequent phase inferred a linear static methodology from the design philosophy underlying the NSP of ASCE 41-17. The deduced design procedure is applied to provide a set of optimal 6-story SMRF designs under two load patterns: the code-based and the first-mode shape patterns. The optimal designs by the proposed deduced linear procedure are compared with those optimized under ASCE 41-17's NSP using FEMA p695 metrics. Consequently, the adjusted collapse margin ratio is evaluated for both design approaches using incremental dynamic analyses. A linearization algorithm is suggested for idealizing pushover curves. The evaluation results demonstrate that the deduced linear static design produces designs that are comparable to those of the ASCE 41-17's NSP without necessitating a sophisticated inelastic analysis. Using the code-based lateral load pattern, the linear static design resulted in a greater margin of safety against collapse and heavier designs with respect to applying the first-mode shape load pattern. The deduced design procedure yields an idea that supports a seismic design philosophy to prevent conflict between design and retrofit standards.


Introduction
Developing seismic design, evaluation, and retrofit standards is one of the most important issues facing the building construction industry in countries prone to earthquakes, such as Iran. The methods for improving existing standards or proposing new standards are especially important. In addition, this cannot be achieved without understanding the fundamental ideas upon which the standards should be based. Developing methods for evaluating and updating available seismic design criteria is a crucial field of research in earthquake engineering.
Developing novel design approaches via structural optimization is an emerging field of study that allows for the establishment of more resilient design methods or improvement of existing design methods (Hajirasouliha et al., 2011;Kaveh et al., 2010;Shahrouzi & Farah-Abadi, 2018). The behavior of steel moment-resisting frames (SMRFs) has already been studied under seismic loadings (Movahed et al., 2014;Ouazir et al., 2018). Nabati and Gholizadeh (2020) optimized SMRF's for various strong column-weak beam ratios, indicating that optimization will produce superior designs when the ratio is limited to one. Milany and Gholizadeh (2021) compared optimally designed SMRFs with fixed-base and flexible-base models, declaring that considering soil-structure interaction reduces optimal structural weight and increases drift. Talebpour et al. (2021) developed a reliability-based constraint handling method to account for fire effects during optimization. Shahrouzi and Rahemi (2014) treated a problem in which the structural sections and the design load pattern are simultaneously optimized. Ganjavi and Hao (2013) developed the optimal seismic design load pattern for elastic shear buildings. Applying the load pattern in seismic design can improve performance by resulting in uniform damage distribution across the structure. Shahrouzi and Ojani (2018) optimized mid-rise building frames by linear design under three patterns of load distribution and compared them in terms of the formation of plastic hinges under earthquake time-history records.
Structural optimization techniques can help us better understand the behavior of structures under seismic loads Kaveh & Hassani, 2009;Kaveh & Nasrollahi, 2014). Hosseini Lavassani et al. (2022) investigated optimal linguistic design variables of fuzzy logic controllers to explain the behavior of a fifteen-story structure under pulse-type earthquake records. Ebadijalal and Shahrouzi (2022) studied the optimal structural weight with respect to the collapse margin ratio. The authors concluded that there is no direct and strong relationship between seismic safety and the optimal weight of steel momentresisting frames. Topology optimization of braced frames, is another issue addressed in the context of performancebased seismic design; acting as a guide to choose the proper location for steel braces (Ghasemiazar & Gholizadeh, 2022;Gholizadeh & Ebadijalal, 2018).
Another intriguing research topic is the relationship between structural optimization and seismic evaluation. Ebadijalal and Shahrouzi (2022) demonstrated that metaheuristic algorithms can generate sample structures for evaluation purposes. Ghasemiazar and Gholizadeh (2022) evaluated the seismic safety of several chevron-braced frames that were topologically optimized. Their assessment of seismic collapse safety results indicates that all optimal designs are acceptable.
Performance-based engineering design is focused on comprehensibility, simplicity, straightforwardness, flexibility, economy, and consistency in design, assessment, and retrofitting. Due to the complexity of seismic design, existing building codes have failed to meet some of the goals (Freeman et al., 2004). Comprehensibility can be achieved through clear provisions and strong theoretical foundations. Seismic design provisions can also conflict with seismic assessment code requirements, resulting in inconsistencies between standards.
Designing building structures to behave elastically during strong ground motion is not economically possible. It has long been assumed that some structural elements will experience after-yield deflections and sustain damage. According to this viewpoint, structural engineers are pursuing displacementbased design paradigms rather than force-based (or strengthbased) ones. However, the validity of any design approach can be determined by its capability to produce satisfactory results.
Based on the inevitable damage caused by powerful earthquakes, structural elements can be classified as either damageable or elastic (indestructible). Damaged elements are ductile, and their failure reduces the input energy of an earthquake. When a strong ground motion is likely to destroy an element, the question arises whether it is necessary to check it for strength-based or displacement-based design requirements under earthquake loads or to provide practical seismic detailing requirements to resist ground motion cycles in a ductile manner.
Consider SMRF beams that were originally designed for gravity loads, as a concrete example. Is it necessary to design the beams directly under seismic loads, whether based on strength or displacement? Steel-braced frames may experience a similar issue with their braces. Should braces be designed under seismic loads, or is it sufficient to have seismic details that allow for ductile buckling? In conventional design approaches, it is believed that such elements should be designed under seismic load and with proper seismic detailing in order to provide maximum energy dissipation.
The next part of lateral load-resisting systems that must operate securely during intense ground motion consists of elements with high sensitivity to damage. The columns in a steel frame are part of the group whose loss can cause the structure's global instability. Designing such elements to withstand relatively powerful motions may be prudent without sustaining damage. This is only possible with the seismic strength-based design requirements. Ebadijalal and Shahrouzi (2022) demonstrated that the nonlinear static procedure (NSP) of (ASCE 41-17 2017) is remarkably similar to a force-based design method in which ductile elements do not need to be designed directly under seismic loads and indestructible elements are elastic until the collapse prevention (CP) level. The procedure assumes that columns remain undamaged while moment frames do not require direct seismic design for beams.
To clarify, CP and LS (Life Safety) performance levels are checked against the maximum considered earthquake (MCE) and design-based earthquake (DBE or DE) hazard levels (where Sa MCE = 1.5*Sa DE (ASCE 7-16 2016)). The columns in the standard satisfy the strength-based design requirements regardless of whether the flexural action is force-control or displacement-control. Complying with a strength-based or force-based design requirement of NSP results in elastic columns up to the corresponding target displacements for the selected hazard levels.
The design method for damagable elements is different, and the SMRFs' beams will not be directly designed to withstand seismic loads. Investigating a beam with a W-section profile in a SMRF can demonstrate it. Acceptable plastic rotations at LS and CP levels are on the order of 10θ y , according to Table 9.7.1 of ASCE 41-17 (ASCE 41-17 2017). θ y is at least 0.005 radians for a typical W-section profile, so that the plastic rotation acceptance criteria will be approximately 0.05 radians (or higher). The acceptance value is (at least) two times greater than the demand for plastic rotation at the MCE hazard level, and beams will never have encountered such extensive plastic rotation at the LS or CP performance levels. In addition, there are no drift requirements in the standard, and the design procedure yields structures with an inter-story drift of less than 2.5% under consistent ground motions scaled to the MCE level. Nonetheless, more than double this drift (at least 0.05 rad or more) may cause dynamic instability in SMRFs.
In order to transform the time-consuming NSP of (ASCE 41-17 2017) into a linear static design method with the same level of seismic safety, the seismic forces in the undamaged members (i.e., columns) should be estimated at the MCE hazard level (or cap point of the pushover curve, as there is a slight difference between the cap and MCE base shears, especially in the mid-rise structures). The forces can be calculated by multiplying the seismic forces by an overstrength factor that turns the base shear of the equivalent static method into the actual base shear at the cap point of the pushover curve. According to the study by Ebadijalal and Shahrouzi (2022), a mid-rise (about 5-to 10-story) SMRF can be considered to have a constant overstrength factor of 3.5.
Assuming an overstrength factor of 3.5 in the present study, at least 30 optimal results will be found using the deduced linear design procedure. They will be compared with other optimal designs obtained by ASCE 41-17 NSP. Separate elastic designs for code-based and first-mode load patterns are developed. For each optimal design, an adjusted collapse margin ratio (ACMR) is calculated to assess both design approaches (under the applied load patterns) based on the FEMA p695 evaluation procedure (FEMA-P695, 2009). The proposed method of deduced linear design is validated through comparison with the ASCE 41-17 nonlinear static procedure (ASCE 41-17 2017).

The formulation of the problem
The structural weight as the raw cost function, is combined here with the optimization constraints in the form of the following pseudo-objective exterior penalty function: where the constant r j denotes the jth penalty factor, W(X) and g j (X) are the structural weight and the jth constraint for the vector of discrete design variables or X, respectively. In the extended form, however, the optimization problem can be stated as: where A j stands for the cross-sectional area of the jth member with the length L j , and ne is the number of structural elements, while ρ stands for the weight of the unit volume. The design methods in the present work have some general and specific constraints. The next section discusses such behavioral constraints in detail.

General constraints
Both the ASCE 41-17 nonlinear static procedure (NSP) (ASCE 41-17 2017) and the deduced linear static procedure check each candidate solution for gravity constraints, strong column-weak beam requirements at each joint, and suppression of higher mode effects. These are known as general constraints that apply to both design methods. Every structural element should be checked under a gravity load combination of 1.2D + 1.6L based on AISC 360-16 (ANSI/ AISC 360-16 2016), as follows: where D and L stand for dead and live loads, respectively. P c and M c are the available axial and flexural strengths, respectively. The corresponding required strengths are denoted by P r and M r , respectively (ANSI/AISC 360-16 2016). To prevent columns from forming plastic hinges before the formation of plastic hinges in beams, the strong column-weak beam constraint is checked for each beam-to-column joint as follows (ANSI/AISC 341-16 2016): where M pc and M pb are the plastic moment strengths corresponding to the columns and beams that are connected to a joint, respectively. To ensure higher mode effects are negligible, the following constraints are checked for all stories (ASCE 41-17 2017): where V 90% stands for the story shear resulting from the response spectrum analysis considering at least 90% mass participation. The fundamental mode of vibration contributes to the story shear by V 1 .

The nonlinear static procedure's specific constraints
The real response of a typical structure to a strong ground motion is a time-varying inelastic response that produces structural damage. The responses can be represented by performing a nonlinear time history (NTH) analysis on a nonlinear inelastic model. Current codified analysis approaches try to mimic and estimate enveloped responses during the NTH analyses. Nonlinear static (pushover) analysis as an intermediate analysis approach is a relatively simpler procedure to identify the behavior of structures. It relies on the assumption that the structural responses are vertically distributed in proportion to a fixed specific pattern (usually the first mode shape) (Krawinkler & Seneviratna, 1998). The assumption is used to relate the response of a multi-degreeof-freedom (MDOF) system to a single-degree-of-freedom (SDOF) system. In other words, the nonlinear displacement of the building's roof (target displacement) relates to the nonlinear displacement of the equivalent SDOF.
Some constraints of the NSP proposed by (ASCE 41-17 2017) should be checked at the target displacement. In the conventional performance-based design, LS and CP performance levels are considered against MCE and two-thirds of MCE as the hazard levels. The target displacement at each selected hazard level during the optimization process can be calculated through the displacement modification procedure of (ASCE 41- 17 2017): where δ t is target displacement and C 0 is the first modal mass participation factor of the structure multiplied by the ordinate of the first mode shape at the control node (or Γ i ϕ roof,1 ). Sa denotes the spectral acceleration corresponding to the selected hazard level. C 1 and C 2 factors can be calculated as follows (ASCE 41-17 2017): where the factor a is 60 for the site class D and T e is the effective fundamental period, which can be calculated by: T i and K i stand for the elastic fundamental period and the elastic lateral stiffness of the structure, respectively. K e is the initial slope of an idealized pushover curve (or V y /Δ y ) that its calculation will be discussed in the next section. μ strength as the demand strength per the effective lateral strength of the idealized pushover curve (Vy), is calculated by (ASCE 41-17 2017): C m is the first mode effective mass participation factor, determined according to Table 7-4 of ASCE 41-17; based on the type of the structural system, number of stories, and the structure's fundamental period. Because of the higher mode effects, the factor usually reduces the target displacements. To limit the potential of dynamic instability, ASCE 41-17 sets the following constraint: where μ max is given by: Δ d and Δ y are the cap and yield displacements in the bilinearized pushover curve, and the parameter h can be determined by: Here, α e is the effective negative post-cap slope ratio of the idealized pushover curve, derived as follows (ASCE 41- 17 2017): where λ is considered 0.8 to represent near-field effects in the study, and α P-Δ denotes the increase in the negative slope ratio of the idealized pushover curve caused by P-Δ effects.
Plastic rotations in beams and columns should be compared to each performance level's plastic rotation acceptance criteria using the following constraint: where θ and θ all are the plastic rotation at a given hazard level and the allowable plastic rotation of beams or columns at the corresponding performance level.
In addition, a strength-based constraint for the columns is checked at each target displacement as follows: MU and PU are the columns' flexural moment and axial force at the selected hazard level, while PCE and MPE are the expected compression and flexural strength capacity of the element. The next section provided an idealization algorithm for the first time to extract important parameters of the pushover curve.

Pushover curve idealization algorithm
In the nonlinear static procedure (NSP), an original inelastic MDOF system is transformed into an equivalent SDOF system to estimate its maximum displacement demand. To obtain the characteristics of the equivalent system, the pushover curve must be idealized. The pushover curve is usually idealized using visual techniques which are inaccurate and cannot be used in automatic iterative design procedures (like structural optimization). According to (ASCE 41-17 2017), the pushover curve should be idealized using three line segments. In Fig. 1, the starting and ending points of each segment line are identified. The peak point of the idealized pushover curve (∆ d , V d ) coincides with the cap point of the actual curve. The yield point (∆ y , V y ) is commonly defined so that the areas under the actual and idealized curves from the origin to the peak point are equal. Based on the assumption, the locus of the yield points is a line and can be formulated as follows: Eq. 17 denotes the area under the curve before the cap point. Due to the fact that the abscissa and the ordinate are simultaneously unknown, another criterion is necessary to determine the effective yield point (∆ y , V y ). Assuming equal initial stiffness for both curves (K i = K e ), the following could be obtained: where K i and K e are the initial tangents of the actual and idealized pushover curves, respectively. Likewise, the effective yield strength could be assumed as the maximum base shear capacity of the structure (V y = V d ). According to (ASCE 41-17 2017), there is another criterion by which the idealized pushover curve must intersect the actual curve at the 0.6V y strength. The pseudo-code for the ASCE 41-17 bi-linearization method is presented in Fig. 2. It is also programmed here in the Python-3 language (Van Rossum & Drake, 2009) using the Numpy library (Harris et al., 2020) as given in the Appendix. The peak point of the pushover curve is denoted by Cap in the code.
The starting point of the third segment line is the cap point of the pushover curve (∆ d , V d ). According to ASCE 41-17, the next point of the segment is the 0.6V y base shear level on the pushover curve.

The deduced linear static design's specific constraints
It has been exhibited that the ASCE 41-17 complex nonlinear static procedure can be simplified into a strength-based design method where no displacement-based constraints are active for SMRF, while columns' strength-based constraints are the most important requirements . Hence, one can use an elastic method with similar results if the columns' forces in the ASCE 41-17 NSP can be estimated by a linear static design method. The forcebased response quantities usually show low dispersion in the nonlinear static analysis, so a code-based prescriptive base shear can be factored to estimate the column forces. In prescriptive seismic design codes like (ASCE 7-16 2016), such a scale has been introduced as the overstrength factor (Ω 0 ). The overstrength factor can be defined by dividing the maximum base shear of the pushover curve by the design code base shear. Low-rise buildings with a fundamental period in a high spectral acceleration region of the design spectrum can show a large overstrength factor (approximately 6), while mid-rise buildings (6-to 9-story SMRFs) have a relatively constant overstrength factor of 3.5 . By designing columns under a seismic load combination of ΩE + D + 0.25L (where E, D, and L are abbreviations for earthquake, dead, and live loads, respectively, while Ω stands for overstrength factor) in a typical linear static design method, they will be elastic until the CP performance level in NSP of ASCE 41-17. In other words, columns are treated in the design as undamageable elements, while beams are damageable and do not require direct force-based or displacementbased design checks (providing adequate seismic detailing for their construction).
Step-by-step explanations of the deduced design procedure will be given below.
A codified seismic demand in the form of the base shear equation is considered in the deduced design. The design base shear is calculated as follows due to ASCE 7-16: W is the effective seismic weight determined by the D + 0.25L combination, while C s is the seismic response coefficient defined according to ASCE 7-16 (ASCE 7-16 2016) as: where S DS and S D1 are the design spectral response acceleration parameters in the short period range and at a 1-s period, respectively. S 1 and S S are the mapped maximum considered (MCR) earthquake spectral response acceleration parameters at a 1-s period and short periods, respectively. T 1 is the fundamental period of the structure. R and I e are the response modification factor (behavior factor) and the importance factor that are considered 8 and 1, respectively. The amount of R in the method doesn't matter, since a change in the R amount will result in a change in omega as well.
The present work offers a deduced design procedure (DDP) that utilizes a pattern x for lateral loading. At any level x, base-shear V is distributed among stories to form a lateral design load For the first case (DDP-LP1), such a design load pattern is taken as the fundamental mode shape of the structure: where ϕ x1 represents the first mode component in the xth story. The mode shape should be normalized so that the sum of its components equals one.
The second case (DDP-LP2) considers a code-based vertical distribution according to the ASCE 7-16 provisions: where h and H are the heights of the xth story from the base level and the roof, respectively. k is considered to estimate the envelope of inertial forces based on the results of the nonlinear time history analyses as follows (ASCE 7-16 2016): Beams, as damageable elements, have no direct seismic design under the applied loads, while columns should be checked according to the following constraint: M r and P r are the required flexural and axial strengths, respectively. P c and M c are the corresponding expected capacities based on (ANSI/AISC 360-16 2016).

Adjusted collapse margin ratio
The seismic design process will be entrusted to a wellknown metaheuristic algorithm with a random nature. Sample structures can be produced using a metaheuristic that is devoid of the designers' biases. It is possible to evaluate and compare different design methods by conducting incremental dynamic analysis (IDA) on such sample structures .
As part of its evaluation procedure, FEMA P-695 uses a performance metric called adjusted collapse margin ratio (ACMR) to quantify structural safety against collapse based on collapse fragility curves obtained from IDA results. In order to develop the lognormal collapse fragility curves, a median and a standard deviation should be calculated. The median is derived by fitting a lognormal distribution to the collapsed intensities of the IDA curves, while the lognormal standard deviation should represent the total system uncertainty. Assuming a good design, good test data, and good model quality based on FEMA-P695 (2009), a 0.525 total system collapse uncertainty, β TOT , is considered for both NSP and the proposed design method. Initially, the collapse margin ratio (CMR) must be calculated. Then, it is adjusted for spectral shape effects (FEMA-P695, 2009). CMR is defined by: where Sa 50% and Sa MCE are the spectral accelerations corresponding to the collapse fragility curve's 50 percent collapse probability and the spectral acceleration at MCE level, respectively. Sa MCE is usually calculated by the median response spectrum of earthquake records that are used in IDA. The median spectrum should be consistent with the MCE design spectrum. For far-field records of FEMA p695, the median response spectrum complies with the ASCE 7-16 design spectrum outside of the constant spectral acceleration region. In the present work, Sa MCE is directly calculated based on the design spectrum of ASCE 7-16. A spectral shape factor (SSF) should be applied to CMR to account for spectral shape effects, as follows: For a valid design method, the individual and average ACMRs of the designed structures should be above the FEMA p695 acceptance values. Assuming β TOT = 0.525, ACMR20% = 1.56 and ACMR10% = 1.96 are considered as the respective acceptance values for the individual and average ACMRs.

Numerical simulation
In order to compare the results of the proposed linear static design method with the nonlinear static (NSP) method in ASCE 41-17, designs of a 6-story 3-bay steel moment-resisting frame in Fig. 3 are screened here by optimization. Consequently, sample structures are provided for the assessment procedure. Figure 3 also shows the geometrical and grouping details of the case study. The beam's length is 5 m, and the column's height is 3 m. On all floors, the beams are loaded with 24,516.6 N/m of dead load and 9806.7 N/m of live load. The yield stress and elasticity modulus are 235 MPa and 210 GPa, respectively. S s and S 1 are 1.5 and 0.6, respectively. All columns are fixed at the base. Covariance matrix adaptation in evolution strategies (CMA-ES) (Igel et al., 2007) is selected for the current optimization task, as a well-known optimization algorithm (Kaveh et al., 2012(Kaveh et al., , 2015.
Available standard sections for beams and columns are selected from the W-section profiles in Table 1 during optimization. The plastic moment and plastic rotation capacities are used to select and sort the beam sections.
As previously discussed, there are two different design approaches used for optimization. The first approach is based on the requirements of ASCE 41-17 for NSP, while the second design is a deduced linear static design method based on the philosophy behind the previous design method. In both cases, all candidate solutions satisfy Eq. 3 through Eq. 5. Additional seismic requirements of Eqs. 11, 15, and 16 should be met for NSP, while the constraint of Eq. 26 is the only constraint that will be satisfied for columns during the deduced method.
It is emphasized that nonlinear modeling, pushover analysis, and pushover curve idealization should be used in the first design method based on ASCE 41-17. The constraints Using each design method, optimization is done at least thirty times to provide optimal designs as the sample structures to be assessed in the seismic evaluation and comparison. However, more trail runs are implemented for NSP due to its complexity. ASCE 41-17 plastic hinges are considered for optimization. The near-fault ground motions of FEMA P695 (FEMA-P695, 2009) are employed for the incremental dynamic analysis (IDA). In nonlinear dynamic analyses, plastic hinge models explicitly simulate cyclic deterioration for the beams (Lignos & Helmut, 2011) and columns (Lignos et al., 2019). Dynamic instability is defined as reaching the maximum inter-story drift of 0.15 or reducing the slope of an IDA curve to 20% of the initial median slope of all records. The adjusted collapse margin ratio (ACMR) is derived for each optimal design based on collapse fragility analysis. More details about nonlinear modeling and structural properties can be found in .

Optimal design by the nonlinear static procedure
A number of optimal designs based on ASCE 41-17 NSP are listed in Table 2, where they are identified from D1 to D6. The displacement-based constraints of Eqs. 11 and 15 are inactive. Additionally, the higher mode constraint is inactive in such optimal designs (Eq. 5). The overstrength factors and ACMRs are provided for each of the optimum results. The overstrength factors are almost the same for the welloptimized solutions. Also, individual ACMR values are well above ACMR20% = 1.56.
A blue line illustrates the pushover curve of the best design (D1) in Fig. 4 that is derived using ASCE41-17 plastic hinge models. The dashed black line represents the idealized pushover curve. LS and CP target displacements are marked with vertical dashed lines on the pushover curve, which are 28.6 cm and 42.8 cm, respectively. Additionally,   Fig. 4 illustrates the base shear in the equivalent static method of ASCE 7-16 (16.9 tonf) and the base shear in the response spectrum analysis considering 90% modes (13.4 tonf), by two horizontal dashed lines. The black line in Fig. 4 is the pushover curve of a linear static model without any modification in stiffness. The red pushover curve corresponds to the distributed plasticity model. Bilinear steel material is assigned to beam and column fiber sections. A nonlinear force-based beam-column element can be used with the fiber section to model distributed plasticity in OpenSees (McKenna, 2011). The constitutive law should be bilinear with a pure strain hardening slope of 0.003 of the elastic modulus in the model. However, 0.006 (number of stories per 1000 for mid-rise SMRFs) usually leads to more consistent pushover curves (as shown in Fig. 4), because the concentrated plasticity model of ASCE 41-17 lacks axial-flexural interaction for columns. Previous texts have repeatedly employed incorrect strain hardening of 0.03 for steel material, which can result in a steep after-yield pushover curve.
The ACMR values for all optimal results are calculated and depicted against the optimal structural weights in Fig. 5. The average of the ACMRs is 3.03, which is quite above the acceptance limit of ACMR10% = 1.96. There is no strong relationship between ACMR and structural weight. As a result, having a heavier structure does not guarantee better seismic safety .
The demand-to-capacity ratio of columns is the only active constraint in the nonlinear static procedure of ASCE 41-17. The method is inherently a strength-based design method in which displacement-based constraints are never active. So, the inter-story drift requirement is not checked by the design standard. It is assumed that if strength-based design requirements are met, drift will be within acceptable limits.
In Fig. 6, the median drift profiles are shown for six optimal designs of Table 2 at two hazard levels: maximum considered earthquakes (MCE) and design-based earthquakes (DBE). According to the figure, a maximum inter-story drift of slightly more than 1.5% and 2.0% can be achieved for the DBE and MCE hazard levels by satisfying the strengthbased constraints of NSP. The maximum drift values are significantly less than the drift accepted for CP and LS performance levels (5% and 2.5% due to FEMA356 at CP and LS, respectively). The design procedure results in maximum inter-story drift between the upper stories.
In addition, the median maximum displacements of the roof story under the suite of ground motion records are approximately 1.5 times larger than the target displacements calculated by Eq. 6. Since column forces cannot be increased significantly, the shortcoming is not critical in NSP. The convergence histories of the optimization algorithm for solving the six-story structure under NSP of ASCE 41-17 are shown in Fig. 7, where the average convergence history of independent optimization runs is compared with the convergence histories of the worst and best design results.

Optimal design by the proposed linear static procedure
In Table 3, six optimal solutions for the deduced linear design method are derived by using the load pattern of the first mode. The overstrength factors and ACMRs are provided for each design, where reported overstrength   Table 3 are satisfactory. It confirms that the overstrength factors obtained for the linear design method are similar to those derived by NSP in the previous section. The results of applying the code-based load pattern in the deduced linear static design are reported in Table 4. It can be noticed that the best optimal results with the first-load pattern are lighter than the code-based load pattern. Although an overstrength factor of 3.50 was assumed in the design, the optimal results exhibit an overstrength of about 3.37.
The ACMR values for at least 30 optimal results are derived with each load pattern and depicted against the optimal structural weights in Fig. 8a and b. Averages of ACMRs for the first mode and code-based load patterns are respectively 3.16 and 3.30, showing a great margin from the acceptance limit of ACMR 10% = 1.96. While the worst and best ACMRs for the code-based load pattern are better than those for the first-mode load pattern, structures designed with the first-mode load pattern tend to be lighter. Figure 9a and b compare the convergence history of the optimization algorithm in solving the six-story SMRF under the deduced design procedure by the first mode (DDP-LP1) and code-based load patterns (DDP-LP2). Figure 9 shows less distance between the average convergence history and those of the best and the worst designs with respect to Fig. 7. Consequently, optimal design is made simpler by the proposed deduced procedure, neglecting the complex performance-based design requirements of ASCE 41-17.

Discussion
To develop more comprehensible, consistent, and flexible seismic design, evaluation, and retrofit standards, this section explains a seismic design philosophy. For the purpose of establishing the idea, two major arguments are presented.  D1  D2  D3  D4  D5  D6   B1  W12X22  W14X22  W12X22  W12X22  W12X22  W14X22  B2  W12X22  W14X22  W12X22  W12X22  W12X22  W12X22  B3  W12X22  W12X22  W12X22  W12X22  W12X22  W12X22  C1  W14X90  W14X90  W14X90  W14X90  W14X90  W14X90  C2  W14X48  W14X43  W14X43  W14X43  W14X53  W14X43  C3  W14X30  W14X30  W14X34  W14X30  W14X34  W14X38  C4  W14X99  W14X99  W14X99  W14X99  W14X99  W14X99  C5  W14X53  W14X61  W14X61  W14X61  W14X53  W14X61  C6  W14X38  W14X38  W14X38  W14X43  W14X38  First, it is important to acknowledge that valid seismic design is not limited to a single design method. Among these linear static designs, the proposed procedure in current paper is the only one with a zero beam load and a maximum column load level. It explains the overstrength factor in prescriptive designs and how it can be derived. The proposed approach can be considered a particular method in a set of acceptable linear design procedures in which the seismic design loads in beams and columns can be calculated with different seismic load intensities. In an ideal seismic design, force amplification factors (overstrength factors) can be separately derived for beams and columns. As it is already denoted by ordinary, intermediate, and special momentresisting frames in some prescriptive design codes, e.g. (ANSI/AISC 341-16 2016), in which beams and columns should be designed for different levels of seismic design loads and ductile details.
The second is that seismic evaluation and retrofit standards must comply with the seismic design standards as well as their predecessors in order to avoid conflicts (Searer et al., 2008). Some designs obtained by the linear design methods may be condemned by the available seismic evaluation methods (like NSP of ASCE 41-17). However, they can be considered acceptable for different combinations of overstrength factors of beams and columns. The design criteria of nonlinear static procedures (in seismic evaluation and retrofit standards) must be compatible with each of the aforementioned elastic design methods; otherwise, a conflict may arise.
Both issues can be addressed by a coprehensive seismic assessement and retrofit standard. Assuming the displacement-based requirements will be met for all pairs of overstrengths, Fig. 10 depicts a schematic curve that can distingiush between feasible and infeasible design approaches. Decision for retrofitting or demolishing of a structure in the infeasible region is necessary, while structures outside the area can be considered acceptable with varying levels of structural safety and damage. Developing such assessment curves for each structural system can present an evaluation and retrofit standard that is non-restrictive and flexible. Accepting that there are multiple justifiable design procedures, the methodology allows engineers to make better engineering judgments during the seismic assessment, which is impossible under ASCE 41-17 and its predecessors. In addition, it does not bring about conflict between the seismic design and the seismic evaluation requirements.

Conclusion
This paper compared the nonlinear static procedure of ASCE 41-17 with a linear static design based on optimal results. The linear static design was inferred from the design philosophy behind NSP in ASCE 41-17. In the methodology, beams and columns are designed to be either damageable or undamaged elements of a moment frame. The seismic design method considers columns as indestructible elements that should remain elastic until a certain hazard level (MCE level in the study). The requirement cannot be met unless a strength-based constraint is satisfied in columns. As beam elements are allowed to be damaged during strong ground motions, there is no need to design them with direct displacement-based or force-based requirements, and ductile detailing (such as section compactness) is considered sufficient to absorb seismic energy. Based on numerical simulations of the sample moment frames screened by optimal design, a number of concluding remarks can be derived as: • ASCE 41-17 nonlinear static procedure (NSP) is inherently a strength-based design method. It is while the proposed deduced design procedure can shortcut such a complex design procedure; by which a comparable margin of safety was provided against collapse for the 6-story sample frames. • Two design load patterns were compared in the optimal design due to the deduced procedure. The codebased load pattern (DDP-LP2) typically results in better ACMRs for the treated example, while the first-mode load pattern tends to result in lighter structures. The code-based load pattern is recommended if a higher ACMR is desired, as in FEMA-356. • The constraint of higher modes effect remained inactive for all 6-story SMRF models of the present study. • The SMRFs can be categorized into four groups based on the overstrength factor and the constraint of higher mode effects. There is a considerable overstrength factor (over 4.0) for the frames in the first group. Other groups had an overstrength factor of about 3.5, while the higher mode effects differ. The constraint of higher mode effects is always inactive for the second group, while it can be active or inactive for the third (design can have dominant higher mode effects or not). In the last group, higher mode effects cannot be suppressed. • In low-rise structures of group 1 designed due to ASCE 41-17 NSP, the overstrength factor will increase. Designed elastically with an overstrength of 3.5, the group has a less safety margin from collapse. Author contributions M. Shahrouzi: supervision, methodology, conceptualization, visualization, writing -review and editing. M. Ebadijalal: software, programming, conceptualization, visualization, writing -original draft.
Funding No funding was received to assist with the preparation of this manuscript.

Declarations
Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest.