Incorporation of Torsional & Higher-Mode Responses in Displacement-Based Seismic Design of Asymmetric RC Frame Buildings

Featured Application: This article proposes a displacement-based seismic design procedure for asymmetric RC frame buildings. Abstract: Direct Displacement-Based Design (DDBD) is currently a widely used displacement-based seismic design method. DDBD accounts for the torsional response of Reinforced Concrete (RC) frame buildings by using semi-empirical equations formulated for wall-type buildings. Higher-mode responses are incorporated by using equations obtained from only a few parametric studies of regular planar frames. In this paper, there is an attempt to eliminate torsional responses by proportioning frames’ secant stiffnesses so that the centers of rigidity and supported mass (the mass on and above each story) coincide. Once the torsional rotations are signiﬁcantly reduced and only translational motions are achieved, higher-mode responses are included using a technique developed by the authors in their recent paper. The efﬁciency of the proposed design procedure in fulﬁlling the intended performance objective is checked by two plan-asymmetric 20-story RC frame building cases. Case-I has the same-plan conﬁguration while Case-II has a different-plan conﬁguration along the height. Both cases have different bay widths in orthogonal directions. Veriﬁcation of the case studies by Nonlinear Time History Analysis (NTHA) has shown that the proposed method results in designs that satisfy the performance objective with reasonable accuracy without redesigning members. It is believed that a step forward is undertaken toward rendering design veriﬁcation by NTHA less necessary, thereby saving computational resources and effort.


Introduction
A reliable Performance-Based Seismic Design (PBSD) uses a probabilistic design approach to achieve performance objectives (Bertero and Bertero [1], Collins and Stojadinovic [2], and Castaldo et al. [3]). However, PBSD is implemented for routine design using simplified design methods like the displacement-based design and force-based design. This is due to the complex nature of the probabilistic design approach and scarcity of data on experiments, fragility curves, cost functions, damage states, and so on. Currently, the displacement-based design is found to be a better choice than force-based design since problems with force-based design have been clearly identified by Priestley [4], and damage is correlated with displacements (strains) than forces (stresses). Various displacement-based design methods have been developed, which can be categorized into two types (Bertero and Bertero [1]). The first type, discussed in Mohele [5] and Applied Technology Council (2) Scaling the moment of inertia reduction factors so that the allowable story displacements, drifts, and plastic hinge rotations are not exceeded. (3) Determining elastic higher-mode force effects that are combined with DDBD-obtained force effects. The buildings' stiffness modified according to tasks (1) and (2) is used for modal analysis. Design can then be performed for the modal superposed lateral force effects, P-∆ forces, and gravity loading.
Before applying the proposed procedure, member dimensions are assumed and gravity loads are determined. A given performance objective is chosen, which the building must fulfill under specified seismic hazard levels anticipated at the building site. With the above data obtained, the steps to be followed are given below. A flowchart is provided to show a brief outline of the steps as given below ( Figure 1). In this study, uni-directional ground motions in the y-direction (Figure 2) are applied and responses checked. Locate the center of mass along the building's height.
Distribute the story secant stiffness of the frames so that eccentricity b/n the centers of story's secant stiffness and supported mass is zero.
Perform response spectrum analysis to find firstmode center of mass displacements at story i, ui.
Scale ui, based on allowable drift to get i th story center of mass design displacement, ∆i.
Find the equivalent SDOF's parameters and lateral loads using equations of DDBD.
Analyze the structure for the lateral loads, P-∆ forces, and seismic weights to obtain i th story displacement, Δi. Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 28 Accordingly, γj,k values are chosen so that ex,k will be zero. To minimize strength difference between frames and result in insignificant torsional rotation, m − 2 frames are assigned arbitrary γj,k values that range from 0.7 to 1.2. Thus, torsion will not impose high displacement demand to frames far from the supported mass center compared to those near it. One-frame relative stiffness value for story k, γ1,k, is calculated by setting Equation (5) to zero and, can have a value other than a value between 0.7 to 1.2.
If the origin is taken from the x-coordinate of the center of supported mass, ex,k will be equal to xcr,k. Setting xcr,k to zero as in Equation (5) and letting ' j x be the x-coordinate from the center of supported mass to the frame j, γ1,k can be determined.
For frames closer to the center of supported mass, lower values of the range can be used and vice versa. When a negative value is obtained in calculating the γ1,k, the range can be adjusted until a positive value is found. The relative stiffness values can vary along the height following the variation of the center of supported mass location. In that case, average relative stiffness values can be used for values within a ten percent variation in a given consecutive story range.
Typically, some frames have fewer columns and beams than other frames in a given story. For example, in Figure 2, frame A has two columns and one beam while frame J has three columns and two beams. In this case, an adjustment is needed.
The k-th-story's lateral stiffness for the j-th frame and the reference frame can be equal if factors ηbk and ηck are used for the j-th frame as follows:  (Step 1) Consider the plan view of a given building at the k-th story as shown in Figure 2. Using Equation (1), calculate the x-coordinate's location of the k-th story center of supported mass, x cm,k .
where m i is the mass at story i, x m,i , the x-coordinate of the center of mass at story i, and n represents the total number of stories. ( Step 2) Equation (2) is developed by Smith et al. [36] to determine the lateral stiffness of a given frame at story k, K k .
where E c is the young modulus of the concrete material, s k is the k-th-story height, and l b is the beam's span length. b and c represent the number of beams and columns in story k, respectively. I gb and I gc are the gross moment of inertia for beams and columns, respectively. To determine the x-coordinate of the center of rigidity at story k of a building in nonlinear range, x cr,k , Equation (3) is used.
where K j,k = γ j,k K ref,k . K ref,k is the secant lateral stiffness of a reference frame at story k. A reference frame is preferably chosen arbitrarily, which rises to the top of the building roof. Figure 2 shows that frame A is a reference frame while its elevation view is depicted in Figure 3. The k-th story secant stiffness for the reference frame is determined by substituting I gb with α b I gb and I gc with α c I gc (for the first-story column only). α b and α c are the moment of inertia reduction factors for beams and columns, respectively. x j is the x-coordinate for frame j. γ j,k is a factor assigned to the j-th frame to relate the frame to the reference frame secant stiffness at story k. γ ref,k , is clearly 1, and m stands for the number of frames at story k.
To eliminate the out-of-plane torsional floor rotation at story k, the in-plane stiffness eccentricity at story k, e x,k , given by Equation (4) below, is set to zero.
Accordingly, γ j,k values are chosen so that e x,k will be zero. To minimize strength difference between frames and result in insignificant torsional rotation, m − 2 frames are assigned arbitrary γ j,k values that range from 0.7 to 1.2. Thus, torsion will not impose high displacement demand to frames Appl. Sci. 2019, 9, 1095 6 of 28 far from the supported mass center compared to those near it. One-frame relative stiffness value for story k, γ 1,k , is calculated by setting Equation (5) to zero and, can have a value other than a value between 0.7 to 1.2.
If the origin is taken from the x-coordinate of the center of supported mass, e x,k will be equal to x cr,k . Setting x cr,k to zero as in Equation (5) and letting x j be the x-coordinate from the center of supported mass to the frame j, γ 1,k can be determined.
For frames closer to the center of supported mass, lower values of the range can be used and vice versa. When a negative value is obtained in calculating the γ 1,k , the range can be adjusted until a positive value is found. The relative stiffness values can vary along the height following the variation of the center of supported mass location. In that case, average relative stiffness values can be used for values within a ten percent variation in a given consecutive story range.
Typically, some frames have fewer columns and beams than other frames in a given story. For example, in Figure 2, frame A has two columns and one beam while frame J has three columns and two beams. In this case, an adjustment is needed. Accordingly, γj,k values are chosen so that ex,k will be zero. To minimize strength difference between frames and result in insignificant torsional rotation, m − 2 frames are assigned arbitrary γj,k values that range from 0.7 to 1.2. Thus, torsion will not impose high displacement demand to frames far from the supported mass center compared to those near it. One-frame relative stiffness value for story k, γ1,k, is calculated by setting Equation (5) to zero and, can have a value other than a value between 0.7 to 1.2.
If the origin is taken from the x-coordinate of the center of supported mass, ex,k will be equal to xcr,k. Setting xcr,k to zero as in Equation (5) and letting ' j x be the x-coordinate from the center of supported mass to the frame j, γ1,k can be determined.
For frames closer to the center of supported mass, lower values of the range can be used and vice versa. When a negative value is obtained in calculating the γ1,k, the range can be adjusted until a positive value is found. The relative stiffness values can vary along the height following the variation of the center of supported mass location. In that case, average relative stiffness values can be used for values within a ten percent variation in a given consecutive story range.  The k-th-story's lateral stiffness for the j-th frame and the reference frame can be equal if factors η bk and η ck are used for the j-th frame as follows: where I gb,ref and I gc,ref are the gross moment of inertia for beams and columns of the reference frame at story k, respectively. I gb,j and I gc,j are the gross moment of inertia for beams and columns of the j-th frame at story k, respectively. l b,ref and l b,j are the beam span for the reference and j-th frame at story k, respectively. b ref and c ref represent the number of beams and columns in story k for the reference frame, respectively. b j and c j represent the number of beams and columns in story k for the j-th frame, respectively. If E c , ref = E c , j , then Keeping the equality of the left and right sides of Equation (7) and after simple algebraic manipulation, Determine the factors γ, η b and η c at all stories for all frames in the y-direction. Frames in the x-direction are unchanged. There is no need to calculate the actual value of the story's secant stiffness for any of the elements. The factors α b and α c along with γ, η b and η c can be input as section property modifiers in the analysis software, in this case, SAP 2000 V. 18. Steps 1 and 2 significantly reduce torsional responses and result in predominantly translational motion of the plan-asymmetric buildings. The next steps to be followed are almost the same as the Steps 2 to 9 of the proposed method developed by Abebe and Lee [35] that considers higher-mode effects in planar RC frames. The only difference is that instead of using design displacement at each story of the frames, the design displacement at the center of mass location of a floor is used. The steps are reproduced here for convenience. (Step 3) Conduct a modal analysis to calculate the first-mode displacement, u i , at the center of mass of the story i in the y-direction using Equation (10).
where φ i is the first-mode shape at the center of mass of i-th story in y-direction. Γ 1 is the corresponding first-mode modal participation factor. D 1 is the first-mode spectral displacement for the mean displacement response spectrum of the selected ground motions obtained using an elastic damping ratio (i.e., 5 percent). (Step 4) Determine design displacement, ∆ i , at the center of mass of story i. ∆ i is found by scaling u i so that the allowable story drift, θ, is not exceeded. ∆ i is given as Equation (11).
where u j and u j−1 are the first-mode displacements at the center of mass location of story j and j − 1, respectively. s j is the j-th story height. j = 1, 2, 3, . . . , n. n stands for the number of stories. (Step 5) Calculate the effective mass, m eff , effective height, h eff , design displacement at effective height, ∆ d , average yield rotation, θ y,avg , yield displacement at effective height, ∆ y , ductility at ∆ d , µ, equivalent viscous damping ratio, ξ eq , effective period, T eff , base shear for the first-mode, V b , and lateral force at story i-th, F i , using equations of DDBD as provided by Priestley et al. [37]. These equations are also provided in Appendix A below. Using SeismoSpect 2016 [38], the displacement response spectra for the equivalent viscous damping ratios can be plotted.
(Step 6) Find δ i , the output displacement at the center of mass in the y-direction obtained using a linear analysis of the structural model modified as Step 2. The loading applied includes F i , the seismic weight, and the additional lateral force due to P-∆, Q i . MacGregor et al. [39] derived Q i as follows: where P i is the sum of story seismic weights at the i-th level and above i. s i is the i-th story height. (Step 7) Update the structural model's secant stiffness given in Step 2 by setting a new moment of inertia reduction factor for beams, α b , and for columns, α c , as in Equation (13) . . , n. n is the number of stories The factor multiplying α b and α c in Equation (13) is found to yield a reasonable estimate of design story displacements. It is based on the linear force-displacement relation, whereby keeping the force equal, desired displacements can be obtained by scaling the stiffness. It is known that multiplying a stiffness matrix by a factor does not change the first-mode shapes, φ i , and first-mode modal participation factor, Γ 1 . Though the first-mode spectral displacement, D 1 , changes as the first-mode period varies, it cancels out from Equation (11). Hence, the design displacements, ∆ i , and the story lateral forces, F i , remain unaltered. If so, and for multi-story frames, [K final ] is approximated as in Equation (15) where i = 1, 2, 3, . . . , n. n is the number of stories. [K initial ] is the secant stiffness matrix of the frames with I gb and I gc modified by α b , α c , γ, η b and η c , respectively. [K final ] will then be approximated as the final secant stiffness matrix with I gb and I gc modified by α b and α c , γ, η b and η c , respectively. (Step 8) Calculate the force effects by linear analysis applying F i on the structural model modified in Step 7.
(Step 9) Determine the elastic higher-mode force effects for the modes contributing to at least 90 percent of the total mass. Using the SRSS method, combine force effects from Step 8 and the elastic higher-mode force effects. SRSS is used since the modal frequencies are well separated, i.e., the adjacent frequencies have a ratio of greater than 2 or less than 0.5. In addition, translational motion of each frame is expected which allows the use of SRSS modal combination. Superpose the force effects from the seismic weight, P-∆ forces, and the modal-combined lateral force effects to get design force effects. SAP2000 V.18 has been used for the purpose of structural analysis.
(Step 10) Employ capacity design to achieve the beam-sway mechanism. For a plastic hinge location, the expected material strength is used while the strength reduction factor is set as unity (Priestley et al. [37]). At other column locations, design moments, M D , are calculated as follows: where ϕ s is the strength reduction factor, ϕ o is an over-strength factor and M A is the applied moment.

Numerical Examples
To investigate the performance of the proposed design procedure, two plan-asymmetric 20-story RC frame buildings are chosen. Ali et al. [40] stated that moment-resisting frames will be ineffective in resisting seismic loading above the 20-story limit. Hence, the case studies are imposed on this recommendation. Case-I has same-plan configuration while Case-II has different-plan configuration along the height. Both cases have different bay widths in orthogonal directions as shown in Figures 4 and 5. Columns and beams have dimensions of 800 mm × 800 mm and 800 mm × 300 mm, respectively, for all stories for the frames in the y-direction. In the x-direction, the beam dimensions are 500 mm × 300 mm for all stories. All stories are 3.5 m high. The seismic load acting on all beams is 15 kN/m, excluding the frame members' weight. A characteristic cylinder strength, f c , of 30 MPa for the concrete, a characteristic yield strength of 413.7 MPa, and an ultimate strength of 620.5 MPa for the rebar are used. The characteristic concrete and rebar strengths are factored by 1.3 and 1.1, respectively, to obtain the corresponding expected material strengths.

Numerical Examples
To investigate the performance of the proposed design procedure, two plan-asymmetric 20-story RC frame buildings are chosen. Ali et al. [40] stated that moment-resisting frames will be ineffective in resisting seismic loading above the 20-story limit. Hence, the case studies are imposed on this recommendation. Case-I has same-plan configuration while Case-II has different-plan configuration along the height. Both cases have different bay widths in orthogonal directions as shown in Figures  4 and 5. Columns and beams have dimensions of 800 mm × 800 mm and 800 mm × 300 mm, respectively, for all stories for the frames in the y-direction. In the x-direction, the beam dimensions are 500 mm × 300 mm for all stories. All stories are 3.5 m high. The seismic load acting on all beams is 15 kN/m, excluding the frame members' weight. A characteristic cylinder strength,  In this study, it is desired that a basic safety objective must be fulfilled according to the Federal Emergency Management Agency (FEMA-356) standard [41]. Therefore, the buildings when subjected to Design Basis Earthquake (DBE) and Maximum Considered Earthquake (MCE) hazard levels, the Life Safety (LS) and Collapse Prevention (CP) performance levels were not exceeded, respectively. To achieve the performance levels LS and CP, the mean NTHA story drifts shall not be higher than 2 percent and 4 percent, respectively (Table C1- The case studies are located near San Andreas Fault having coordinates of 37° N, −122° W. Class C (very dense soil and soft rock) is assumed for the soil type. The U.S. seismic design map is utilized to obtain the site's design spectral accelerations, Sd1 and Sds, and long period, TL. These values are input in the PEER NGA-West 2 ground motion database [42] to generate the acceleration response spectrum for DBE. The acceleration response spectrum for MCE is 1.5 times the acceleration response spectrum for DBE. Again, the PEER NGA-West 2 ground motion database is used to select and scale eleven ground motions (Tables 1 and 2) to match the 5-percent damping DBE and MCE acceleration response spectra (Figure 6a,b). Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 28

Numerical Examples
To investigate the performance of the proposed design procedure, two plan-asymmetric 20-story RC frame buildings are chosen. Ali et al. [40] stated that moment-resisting frames will be ineffective in resisting seismic loading above the 20-story limit. Hence, the case studies are imposed on this recommendation. Case-I has same-plan configuration while Case-II has different-plan configuration along the height. Both cases have different bay widths in orthogonal directions as shown in Figures  4 and 5. Columns and beams have dimensions of 800 mm × 800 mm and 800 mm × 300 mm, respectively, for all stories for the frames in the y-direction. In the x-direction, the beam dimensions are 500 mm × 300 mm for all stories. All stories are 3.5 m high. The seismic load acting on all beams is 15 kN/m, excluding the frame members' weight. A characteristic cylinder strength,  In this study, it is desired that a basic safety objective must be fulfilled according to the Federal Emergency Management Agency (FEMA-356) standard [41]. Therefore, the buildings when subjected to Design Basis Earthquake (DBE) and Maximum Considered Earthquake (MCE) hazard levels, the Life Safety (LS) and Collapse Prevention (CP) performance levels were not exceeded, respectively. To achieve the performance levels LS and CP, the mean NTHA story drifts shall not be higher than 2 percent and 4 percent, respectively (Table C1- The case studies are located near San Andreas Fault having coordinates of 37° N, −122° W. Class C (very dense soil and soft rock) is assumed for the soil type. The U.S. seismic design map is utilized to obtain the site's design spectral accelerations, Sd1 and Sds, and long period, TL. These values are input in the PEER NGA-West 2 ground motion database [42] to generate the acceleration response spectrum for DBE. The acceleration response spectrum for MCE is 1.5 times the acceleration response spectrum for DBE. Again, the PEER NGA-West 2 ground motion database is used to select and scale eleven ground motions (Tables 1 and 2) to match the 5-percent damping DBE and MCE acceleration response spectra (Figure 6a,b). In this study, it is desired that a basic safety objective must be fulfilled according to the Federal Emergency Management Agency (FEMA-356) standard [41]. Therefore, the buildings when subjected to Design Basis Earthquake (DBE) and Maximum Considered Earthquake (MCE) hazard levels, the Life Safety (LS) and Collapse Prevention (CP) performance levels were not exceeded, respectively. To achieve the performance levels LS and CP, the mean NTHA story drifts shall not be higher than 2 percent and 4 percent, respectively (Table C1- The case studies are located near San Andreas Fault having coordinates of 37 • N, −122 • W. Class C (very dense soil and soft rock) is assumed for the soil type. The U.S. seismic design map is utilized to obtain the site's design spectral accelerations, S d1 and S ds , and long period, T L . These values are input in the PEER NGA-West 2 ground motion database [42] to generate the acceleration response spectrum for DBE. The acceleration response spectrum for MCE is 1.5 times the acceleration response spectrum for DBE. Again, the PEER NGA-West 2 ground motion database is used to select and scale eleven ground motions (Tables 1 and 2) to match the 5-percent damping DBE and MCE acceleration response spectra (Figure 6a,b). no variation of γ along the height. Frame A is taken as a reference frame, and the relative stiffness values for frames B, C, and D are assigned arbitrarily ( Table 3). The relative stiffness value for frame E is calculated using Equation (5). Correspondingly, the moment of inertia reduction factors, αb and αc, for the frames used are shown in Table 4. From Step 9, the new moment of inertia reduction factors are 0.44 times the initial moment of inertia reduction factors. CP performance level governs for this case.  To define the secant stiffness of the case studies, the moment of inertia reduction factors, α b for beams and α c for first-story columns, of the reference frames need to be assumed. Kowalsky [43] and Kappos et al.
[44] adopted 10 percent of the uncracked section stiffness to represent the secant stiffness of RC bridge piers as a start for their iterative design methods. Cheng [34] determined the member stiffness matrix with plastic hinges at one or both ends. Accordingly, for members with hinges at both ends, the stiffness of that member will be the stiffness based on the gross moment of inertia of members times the ratio of post-yield stiffness to the elastic stiffness of the plastic hinge. For members with only one hinge, no such simplification has been made. Thus, if that ratio is assumed as 0.1 for members with two hinges at member ends, the moment of inertia reduction factor, α b , will be also 0.1. The values of the reduction factors are chosen arbitrarily as a starting value as it does not affect the values for α c and α b . Therefore, α c is assigned 0.3 since only one plastic hinge is expected and α b as 0.1 or, in some cases, 0.05. For the reference frames of the case studies, α c is set as 0.3, whereas α b for the lower ten story beams is 0.1 and for the top 10 stories 0.05. A lower value of α b at the top stories is chosen because seismic demand reduces as the height increases.
Steps 3 to 5 need to be performed for both LS and CP levels. When using Step 5, the peak value of the mean displacement response spectra for the equivalent viscous damping may be less than the initial design displacement at effective height, ∆ d,initial . In that case, a method discussed in Section 3.4.6(b) of Priestley et al. [37] is employed. Hence, lower final displacement, ∆ d,final , at reduced equivalent damping ratio is found. To attain ∆ d,final , the design story displacements need to be factored by the ratio ∆ d,final /∆ d,initial . The DDBD base shear, V b , for the two performance levels is compared, and the maximum is taken as a governing design base shear.
When the value of P-∆ force at story i, Q i is found to be negative, it is not added to the lateral force F i and, hence, only F i is taken as the design force at that story. Column design moments, M D , are amplified by the ratio ϕ o /ϕ s = 1.2 using ϕ o = 1.1 and ϕ s = 0.9.
For Case-I, since the center of story mass and the supported mass do not vary per story, there is no variation of γ along the height. Frame A is taken as a reference frame, and the relative stiffness values for frames B, C, and D are assigned arbitrarily ( Table 3). The relative stiffness value for frame E is calculated using Equation (5). Correspondingly, the moment of inertia reduction factors, α b and α c , for the frames used are shown in Table 4. From Step 9, the new moment of inertia reduction factors are 0.44 times the initial moment of inertia reduction factors. CP performance level governs for this case. Table 3. Case-I's relative stiffness values.
a γ E is calculated using Equation (5) by setting e x,k to zero. For Case-II, the relative stiffness values are varied following center of supported mass variation along the height to yield e x,k equal to zero. Frame C is chosen as the reference frame ( Table 5). The choice of the reference frame is arbitrary as long as the frame rises to the top-story level. The reference frame C is assigned α b equal to 0.05 and 0.1 for the top ten stories and bottom ten stories, respectively. α c is set as 0.3. Since some frames have fewer columns and beams than others in a given story, modification per Step 2 is necessary ( Table 6). α b and α c for frames other than the reference frame are multiples of those α b and α c values for frame C times the factors of the frames given in Tables 5 and 6. From Step 7, the new moment of inertia reduction factors, α b and α c , are 0.432 times the initial moment of inertia reduction factors. LS performance level governs for this case.

Verification of the Proposed Method by NTHA
For the NTHA of case studies, the flexural reinforcements are first calculated with the Section Designer module of SAP2000 V.18 for the design actions obtained from Step 10 of Section 2. It is required to account for the difference in rebar amount for the first-story columns' bottom end (plastic hinge) and top end. Hence, a section representing the plastic hinge is incorporated at the bottom end of the column. The section will have a rebar amount provided for the plastic hinge and a length, L p determined by Priestley et al. [37] as shown in Equation (17).
where k = 0.2(f u /f y − 1) ≤ 0.08. L c is the length from the plastic hinge center to the point of contra flexure in the member, and the strain penetration length L sp = 0.022f ye d bl . f u and f y are the ultimate and yield strength of the rebar, respectively. f ye is the expected yield strength of the rebar in MPa, and d bl is the diameter of the rebar in meters. Nonlinear analysis for the effect of seismic weight is performed to be continued by NTHA for the ground motions. The analyses are conducted by PERFORM-3D V.7 with FEMA 356 hinges assigned to the members' ends. The floor slabs are modeled as a rigid diaphragm. The cyclic strength and stiffness degradation of the plastic hinges in PERFORM-3D is accounted for by specifying energy degradation factors. Ghodsi and Ruiz [45] provide reasonable values for the energy degradation factors. Rayleigh damping is used to model the elastic damping. A damping ratio of 5 percent is assumed at periods 1.5 and 0.1 times the fundamental period in the y-direction to calculate mass and stiffness matrix coefficients. A P-∆ option is included in the analyses.

Case-I Building
From NTHA outputs, it is observed that the allowable story drift of the governing performance level (i.e., 4 percent) is not exceeded for all ground motions (Figure 7). Design drifts determined from the design displacements are not exceeded when compared to the mean NTHA drifts. Mean NTHA drifts are low compared to design drifts though the mean plus one standard deviation and design drifts are approximately equal (Figure 7). The NTHA drifts tend to follow design drift profile; thus, the method produces a design that reflects the design drift ( Figure 7). It is observed that NTHA drifts are nearly the same for all frames. The fact that design drifts are being exceeded for some ground motions is acceptable as long as the mean NTHA drifts from the eleven ground motions did not exceed the allowable drifts. Since the design by the proposed method is performed using the design response spectrum which corresponds to the mean response spectrum of the eleven ground motions, it is expected that some ground motions result in the design drifts being exceeded. The NTHA displacements are below the design displacements, except for one ground motion. Mean plus one standard deviation NTHA displacements are closer to the design displacements at the lower-half stories compared to the top-half stories (Figure 8). Mean NTHA displacements and drifts are almost identical for all frames (Figure 9). For example, the difference in 20th-story mean NTHA displacements at the extreme frames A and E is 0.35 percent. This signifies that the proposed method works well for plan-asymmetric buildings when the plan configuration is identical throughout the height. will be conservative.

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Negative and positive moment capacities of beams, though different values, are the same on a given bay. Thus, some beams will have higher strength than needed.

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The choice of amplification factor (ϕ o /ϕs) applied to columns forces affect the design.

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The empirical equations of yield drift and equivalent viscous damping ratio induce errors.    (e) (f) To check whether allowable hinge rotations for the CP performance level are exceeded, usage ratio graphs of Perform-3D V.7 are used. Usage ratio graphs depict the ratio of hinge rotation demand of the ground motions to the allowable hinge rotation through time. An average of 0.8 is achieved for the beams' usage ratio as shown in Figure 10a-c. For the columns' usage ratio an average of 0.2 is obtained, which is predominantly from hinge rotations at the base of the building (Figure 10d-f). It can be said that from the hinge status point of view, the procedure produces economical design and a desirable collapse mechanism (i.e., a beam-sway mechanism.)

Case-II Building
The code specified drift for the governing performance level (i.e., 2 percent) and design drifts are not exceeded for any of NTHA drifts ( Figure 11). The result shows the method is conservative in attaining the design drifts for the LS performance level. This is because secant stiffness of the frames obtained based on the 2-percent drift is higher than that from 4-percent drift. As a result, elastic higher-mode periods are obtained that produce significant elastic force effects since periods are on the rising and constant plateau of the design acceleration response spectrum. Thus, design for these force effects gives the frames higher strength than required. NTHA drifts decrease as the story increases although this is not the case for the design drift profile ( Figure 11). Less discrepancy is obtained between NTHA and design displacements as well as NTHA and design drifts at the lower stories (Figures 11 and 12). NTHA displacements and drifts of the frames show less identical response compared to the Case-I building. The difference of the mean story displacements between the frames at the extreme ends of the buildings (for example, at the 5th story for frame A and E) is 6 percent, at the 13th story for frame A and D 10 percent, and at the 20th story for frame A and C 11 percent (Figure 13a).  (e) (f)  (e) (f)  (e) (f)  Beam usage ratios for the frames range from 0.4 to 0.45 with an average of 0.43 (Figure 14a-e), while for the columns the usage ratios vary between 0.02 and 0.033, and the mean is 0.03 (Figure 14f-j). The ranges show that similar seismic demands are imposed on each frame, which indicates that uneven responses of frames due to floor rotations (global torsion) are insignificant. Conservatism for the ranges can be minimized by redesigning members, but since the aim is to check the performance of the procedure without redesigning members, it can be inferred that the safety is ensured using the proposed design procedure.
In general, low mean NTHA outputs are produced compared to the design outputs. Factors affecting the outputs include the following:

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To reduce computational effort and time, the design should not be done for each member. Members having similar analysis outputs must be grouped in one category, and design has to be performed for the maximum output from the category. Consequently, some members' design will be conservative. • Negative and positive moment capacities of beams, though different values, are the same on a given bay. Thus, some beams will have higher strength than needed.

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The choice of amplification factor (ϕ o /ϕ s ) applied to columns forces affect the design.

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The empirical equations of yield drift and equivalent viscous damping ratio induce errors.  (e) Case-II frame E-beams'usage ratio; (f) Case-II frame A-columns' usage ratio; (g) Case-II frame B-columns' usage ratio; (h) Case-II frame C-columns' usage ratio; (i) Case-II frame D-columns' usage ratio; (j) Case-II frame E-columns' usage ratio.

Conclusions
A performance-based seismic design procedure is proposed to account for torsional and highermode responses in the DDBD of plan-asymmetric RC frame buildings. In the procedure, a technique is developed to minimize torsional response by assigning secant stiffness of the frames that eliminate the eccentricity between the centers of secant-stiffness rigidity and supported mass (the mass on and above that story). To define the design story displacements, the procedure employs a modal analysis and scales the mode shapes so that allowable story drift is not exceeded. In such a way, the effect of stiffness and mass irregularity along the height are considered. The P-Δ effect is included using a non-parametric equation, which is applicable for general cases.
The procedure does not exceed the allowable story drifts of the intended performance levels though the mean NTHA displacements and drifts are low compared to the design displacements and drifts. It greatly reduces the floor rotations of the building with mass and stiffness irregularity along the height. Translational and rotational coupling has become insignificant, and approximately only translational motion is achieved when subjected to the ground motions. Allowable hinge rotations are compared with the corresponding mean NTHA outputs by usage ratios. It was observed that the mean NTHA outputs of usage ratios are within a tolerable limit, and uniform responses for each frame are obtained. The target failure mechanism (i.e., the beam-sway mechanism) is also achieved. Usage ratio for the members of Case-II. (a) Case-II frame A-beams'usage ratio; (b) Case-II frame B-beams' usage ratio; (c) Case-II frame C-beams' usage ratio; (d) Case-II frame D-beams' usage ratio; (e) Case-II frame E-beams'usage ratio; (f) Case-II frame A-columns' usage ratio; (g) Case-II frame B-columns' usage ratio; (h) Case-II frame C-columns' usage ratio; (i) Case-II frame D-columns' usage ratio; (j) Case-II frame E-columns' usage ratio.

Conclusions
A performance-based seismic design procedure is proposed to account for torsional and higher-mode responses in the DDBD of plan-asymmetric RC frame buildings. In the procedure, a technique is developed to minimize torsional response by assigning secant stiffness of the frames that eliminate the eccentricity between the centers of secant-stiffness rigidity and supported mass (the mass on and above that story). To define the design story displacements, the procedure employs a modal analysis and scales the mode shapes so that allowable story drift is not exceeded. In such a way, the effect of stiffness and mass irregularity along the height are considered. The P-∆ effect is included using a non-parametric equation, which is applicable for general cases.
The procedure does not exceed the allowable story drifts of the intended performance levels though the mean NTHA displacements and drifts are low compared to the design displacements and drifts. It greatly reduces the floor rotations of the building with mass and stiffness irregularity along the height. Translational and rotational coupling has become insignificant, and approximately only translational motion is achieved when subjected to the ground motions. Allowable hinge rotations are compared with the corresponding mean NTHA outputs by usage ratios. It was observed that the mean NTHA outputs of usage ratios are within a tolerable limit, and uniform responses for each frame are obtained. The target failure mechanism (i.e., the beam-sway mechanism) is also achieved.
At the current stage, the proposed design method has been tested for moment-resisting RC frame buildings subjected to uni-directional ground motions. The proposed procedure is found to produce designs that achieve the performance objective without redesigning members. It is hoped that upon using the procedure, a step forward is undertaken toward rendering NTHA less necessary, thereby saving computational resources and effort.
Future research should consider the effect of bi-directional ground motions on the proposed method. The applicability of the method to buildings made of steel, precast, prestressed, composite, FRP reinforced, etc., should be checked. It should also be investigated for sufficient samples of lateral-load resisting systems other than rigid-frame buildings such as braced, wall, and wall-frame buildings. Figure A1. Representation of the multi-story frame by a SDOF system. Let design displacement, height to story i and mass at story i be given as Δd, hi and mi, respectively, and the effective height, Heff, be given as Design displacement for the SDOF system, Δd, is where lb, db and εy are beam span length, beam depth and yield strain, respectively.
For more than one bay on a given story, average yield drift ratio θy,avg is obtained using Equation (A5) based on the rationale that equal moment capacities are assigned to each beam but not necessarily equal negative and positive moments as discussed by Priestley et al. [36]. where n is the number of bays.
Yield displacement, Δy, at the effective height is found as  Figure A1. Representation of the multi-story frame by a SDOF system.
Let design displacement, height to story i and mass at story i be given as ∆ i , h i and m i , respectively, and the effective height, H eff , be given as Design displacement for the SDOF system, ∆ d , is Effective mass, m eff , is given as For RC frames, yield drift ratio θ y is where l b , d b and ε y are beam span length, beam depth and yield strain, respectively. For more than one bay on a given story, average yield drift ratio θ y,avg is obtained using Equation (A5) based on the rationale that equal moment capacities are assigned to each beam but not necessarily equal negative and positive moments as discussed by Priestley et al. [37].
where n is the number of bays. Yield displacement, ∆ y , at the effective height is found as ∆ y = θ y,avg H e f f (A6) Ductility, µ, at the design displacement is For RC frames, Equation (A8) is used to calculate the equivalent viscous damping ratio, ξ eq ξ eq = 0.05 + 0.565 µ − 1 µπ (A8) Using the displacement response spectrum for ξ eq and ∆ d , the effective period, T eff , can be obtained. The effective stiffness, K eff , and the base shear, V b , can then be determined as shown in Equations (A9) and (A10), respectively.
K e f f = 4π 2 m e f f T 2 e f f (A9) The base shear is distributed up the height of the building as a set of equivalent lateral forces, F i , using Equation (A11).