A METHOD FOR SEISMIC DESIGN OF RC FRAME BUILDINGS USING FUNDAMENTAL MODE AND PLASTIC ROTATION CAPACITY

A seismic design method is proposed for RC frame buildings, with focus on two of the seven virtues of earthquake resistant buildings, namely deformation capacity and desirable collapse mechanism. Fundamental lateral translation mode of the building and plastic rotation capacity of beams are included as input to estimate lateral force demand. Guidelines are provided to proportion beam and column cross-sections through: (a) closed-form expressions of flexural rigidities to maximize participation of the fundamental mode, and (b) relative achievable plastic rotation capacity using current design and detailing practice. This method is seen to surpass two prominent displacement-based design methods reported in literature. Results of nonlinear static pushover and nonlinear time history analyses of buildings of three different heights designed by this and the said two methods are used to make a case for the proposed method; the proposed method is able to control plastic rotation demand in beams and provide at least 20% more lateral deformation capacity than the said methods.


INTRODUCTION
The seven virtues of earthquake resistant buildings (ERBs) are ( Figure 1): (1) regular structural configuration, (2) at least a minimum lateral stiffness, (3) sufficient lateral strength, (4) good overall lateral ductility, (5) large overall lateral deformability, (6) desirable collapse mechanism, and (7) large energy dissipation capacity. In the traditional force-based earthquake resistant design of RC buildings, most design codes have provisions to meet directly the first three virtues and the fourth through prescriptive ductile detailing. But, the last three virtues are not in direct focus in current force based design practice. Because earthquake ground shaking imposes lateral displacement demand on structures and inputs energy to them at their base, the last three virtues are essential. Eventually, design codes should guide designers to meet these three virtues also. Studies should be undertaken and design methods suggested towards achieving this intent. Literature indicates that many studies have attempted this [1][2][3][4][5][6][7][8][9][10]. Two prominent studies, whose variants have been adopted in various other studies, are: (1) Direct Displacement Based Design (DDBD) [11][12][13], and (2) Performance-based Plastic Design (PBPD) [14]. Of these two methods, the latter has attempted to bring in the last three virtues, though in a simple way. The important merits and limitations of these two methods are summarized in Table 1.
Further, it is customary in the seismic coefficient method of the traditional force based design to consider the fundamental lateral translational mode to be the dominant mode. If this assumption can be realised through appropriate proportioning of lateral stiffness and associated lateral strength of buildings along their height, the method can be used readily by practising structural engineers. Thus, improved performance of frame buildings can be achieved, if buildings have: (a) rotation demands at plastic hinges less than those which can be provided practically, and (b) overall lateral deformation capacity more than that imposed by earthquake shaking. The former is contingent on the latter.
Attempts were made to improve design methods to address these factors. Members were designed to sustain combined effects of fundamental and higher modes of oscillation [13,24]. But, till date, no method explicitly proportions the member sizes and strengths to make the fundamental mode of oscillation become the dominant mode, with at least 80% mass (say) participating in just the fundamental mode alone. While codes specify limits on maximum slenderness l/d of beams, the values specified (20)(21)(22)(23)(24)(25)(26) makes beams too flexible to sustain good inelastic action [25]. Limiting lateral displacement in each storey under service loads and designing members by capacity design are practiced routinely now. But, low column-to-beam flexural strength ratios (~1.4) are recommended in design codes [26,27], even though higher values (2.2-2.8) are recommended in literature [28,29].
Further, in the recent past, plastic rotation capacity pbc was recommended as a design input in a design method [14,30], but adequate provisions to limit the plastic rotation demand pbd was not integrated into the design method. Consequently, results of response history analyses show that the actual pbd demands are much higher than pbc [14].
pbd can be limited to pbc, if more beams are made to participate in the collapse mechanism. More beams participate in the response, if relative stiffness and strength of beams and columns are proportioned appropriately, thereby reducing the possibility of concentration of plastic actions in limited beams and columns. Furthermore, if the fundamental mode shape can be related to pbd and if pbd is ensured to be less than a fraction of pbc during design stage, most beams can utilise fully the pbc; in turn, this will help maximise the lateral deformation capacity of buildings. Hence, quantifying the available pbc is the first step. Typically, RC beams designed and detailed by current seismic design codes have pbc in the range 0.015-0.030 rad [31][32][33]. These values are small owing to many factors, like: (1) large flexural rigidity of beams owing to heavy gravity loads, and (2) stiffness degradation and strength deterioration of RC beams under reverse cyclic response during strong earthquake shaking [34]. Until such time,pbc is increased through new design and/or detailing strategies, it is prudent to have design guidelines such that pbd is restricted to withinpbc. Thus, design methods should formally recognise the limited pbc made available when members are designed and detailed by the current methods.
Design methods are available, which use pbc of beams as design input to estimate the lateral force demand of the building [10,14,30]. But, specific design guidelines are not available to ensure that pbd does not exceed pbc of beams.
A single method is not available yet, which: (1) proportions stiffness and strength of members, to maximize the contribution of the fundamental mode, and (2) uses the limited pbc available in beams as design input and ensures that pbd does not exceed it during strong earthquake shaking. To address these challenges and the overcome limitations in a holistic way, three actions are required in design, namely: (1) proportion member sizes considering a single mode, the fundamental lateral mode, reducing effects of higher modes; (2) design all beams along the height of the building so that they have near uniform pbd, and (3) include pbc as a design input so as to ensure that pbd is less than pbc. Plastic rotation at beam ends is related to design lateral force. Focus is indirectly on 5 th Virtue of ERBs, i.e., overall lateral deformability.

(b) Shortcoming
Many other factors also affect inter-storey drift, e.g., cracking, and not just geometric dimensions of members.
Parameter not easy to control at beam ends along the entire height of the building. Effects considered implicitly by an empirical distribution of lateral force demand along height.

(b) Shortcoming
No attempt is made to enhance the contribution of first mode.
No attempt is made to enhance the contribution of first mode.
But, even when a single mode, namely the fundamental lateral mode, is made to dominate, buildings can deform in shear, linear or flexure type lateral profiles. If they deform in linear mode, pbd is uniform along its height, thereby improving its lateral deformation capacity. But, to make buildings have large modal mass in this mode, buildings should deform in shear mode; in such a case, pbd is unduly large in few storeys near the base of buildings. This paper presents an analytical method that balances these competing requirements by identifying a fundamental mode shape {}1 of the building, which has large modal mass participation and which gives near-uniform pbd in beams along the height. Also, the method uses the limited pbc available in beams as design input.

PROPOSED METHOD OF DESIGN
The

Step 1: Choose Fundamental Mode Shape and Proportion Stiffness of Members
The sub-steps involved in proportioning of members of buildings are: Step 1a: Select regular grid in plan and elevation of the building.
Step 1d: Identify flexible lower storeys (i.e., Ki/Ki+1 <1) from Eq.(3) using i from Step 1c: where mi is the seismic mass lumped at floor i and Ki the lateral translational stiffness of storey i. If more than 0.2N storeys of the N-storey building have flexible lower storeys (i.e., Ki/Ki+1 <1), then select smaller M1 * . By sacrificing some M1 * , the number of storeys with flexible lower storeys reduces ( Figure 4). Thus, choice of M1 * also controls stiffness proportioning.

Figure 4: Influence of mode shape parameter  on number of flexible lower storeys in a 12-storey building (with equal storey height and storey mass) and fundamental modal mass M1 * .
Step 1e: Choose sizes of columns and beams in the top storey based on gravity load considerations, and thereby their gross moments of inertia IcN and IbN, respectively. When doing so, keep l/d ratio of members in the range 10-14; this reduces design iterations and leads to plastic rotation capacity pbc in the practical range.
Step 1f: Estimate required gross moments of inertia Ici and Ibi of columns and beams, respectively, in each lower storey i, starting from the (N-1) th storey and going downwards, using Eq.(4) and Eq.(5) as shown below: where A1, A2 and A3 are given by: Step 1g: Since member sizes are rounded off in Step 1f, the dynamic characteristics of the building change slightly. Hence, update T1 and {}1 of the building using modal analysis with new member sizes as determined in Step 1f. And, estimate Ki of each storey i using Eq.(9) [36].
where i, the mode shape coefficients corresponding to lateral translation degree of freedom, is obtained from modal analysis and not from Eq. (2), and 1 = 2/T1. Then, estimate the initial lateral translational stiffness Kinitial of the building as: where the effective height h1 * of the building is given by in which hi+1 and hi are heights from the base of the building to floors i and i+1 between which h1 * is located (Figure 4), hj is height j from base of the building to floor j) and {}1 the fundamental lateral translational mode shape: (13) Step 1h: Examine adequacy of sizes chosen of members by checking if the drift demand is within the allowable limits under the design seismic lateral force Hdesign given in the seismic design code. If the lateral drift is more, then increase IcN of columns and/or IbN of beams in the top storey in Step 1e until the lateral drift limit is less than the allowable limit. Hdesign estimated in this step is NOT used in the strength design of the building.
Step 2: Estimate Lateral Force Demand Assume that: (a) lateral force-displacement response is elasticperfectly plastic ( Step 2a: Estimate the Elastic Maximum Lateral Force He as: where Z is the Zone Factor, I the importance factor, (Sa/g)1 the spectral acceleration at T1 (corresponding to Maximum Considered Earthquake (MCE)) and W the total seismic weight of the building [37][38].
Step 2b: Estimate the Elastic Lateral Displacement Demand e as: Using Equal Displacement Rule [39], the inelastic lateral displacement demand d of a flexible building is given as: Step 2c: Estimate the Plastic Lateral Displacement Capacity pc as: where for flexible building (T1 > 0.5s) h1 * is as per Eq. (11) and Lb * the distance between plastic hinges in beam, and Lb centerline length of beam bay. Choosing a safety factor  of 2.0 for plastic rotation capacity pbc of beams in flexible buildings and of 1.5 in stiff buildings, the Design Lateral Plastic Displacement Capacitypdes is:  is calibrated using results of the time history analysis of 12 buildings (of 4-, 8-and 12-storeys) subjected to a suite of 30 ground motions.
Step 2d: Estimate the Yield Displacement Capacity y of flexible buildings (T1 > 0.5s) as: of stiff buildings (T1 < 0.5s) using work balance equation as: Step 2e: Estimate the Overstrength Lateral Force Demand HΩ as: Estimate the Design Lateral Force Demand HD as where Ω (=1/0.9) is the overstrength factor (in which 0.9 is the resistance factor).
Step 2f: Ensure y obtained is within the limits where y=fy/Es and (Lb/d)avg are yield strain of flexural reinforcement in beams and average (Lb/d) ratios of all beams in the building, respectively [13]. To minimize the number of iterations needed to match EIeff assumed (Step 1) and EIeff estimated (Step 4), ensure that y is within the specified limit. If not, change (Lb/d) ratio of beams and choose a new IbN in Step 1e.

Step 3: Proportioning Member Strengths
Step 3a: Perform linear elastic structural analysis, and obtain: (a) flexural demands where  is capacity reduction factor as defined in seismic design code [26].
Step 4: Updating Member Stiffness Step 4a: Re-evaluate EIeff of members as My/y, where My -y curve is obtained using characteristic  curves of concrete and reinforcing steel [40]. Compare these values with EIeff taken in Step 1 of beams and columns as 0.35EIgross and 0.5EIgross, respectively.
Step 4b: If EIeff is away by more than 10% of that considered in Step 1, repeat the analysis and building redesigned. Experiences from design of buildings indicate that one iteration is sufficient if: (a) yield displacement y from Eq. (19) is within the limits given in Eq. (23), and (b) e/y is in the range 1.5-2.5.
Step 5: Detailing Members Step 5a: Detail all members as per ductile detailing requirements given in design code.

Details of Study Buildings
Three RC buildings of 4-, 8-and 12-storeys are considered as study buildings whose details are available in literature ( Figure 6) [14,41]. Buildings are designed by the PD method and two other state-of-the-art design methods, namely Direct Displacement Based Design (DDBD) [8,13] and Performance Based Plastic Design (PBPD) methods [30]. The inputs and assumptions made in design are listed in Tables 2 to 4 Crosssectional details of members, ratios of effective to gross rigidities of members and reinforcement provided in members of buildings designed using PD and DDBD methods are available in Annex A; the same for buildings designed using PBPD method is available in literature [14]. Step 3: Proportioning Member Strength

3.1
Beams are designed using b  in the range of 0.5-1.0 to ensure that they yield and to control whiplash effect.
All columns are designed using c  of 0.5 to ensure that they do not yield. (2) Design lateral force estimated using yield displacement and d  .

Proportioning Member Strength
3.1 (1) Flexural demand is estimated in beams using equilibrium-based analysis considering lateral loads alone.
(2) Columns are designed for combined actions of flexural demand (estimated using lateral loads alone) and axial demand (estimated from gravity loads alone).
(3) Capacity protection factor used to ensure columns do not yield [8]. (1) All beams yield, and sustain nearly same plastic rotation demand.
(2) Inter-storey drift demand is uniform along the height.

Proportioning Member Strength
3.1 (1) Flexural demand in beams estimated considering lateral loads alone.
(2) Flexural demand on columns estimated by the Principle of Virtual Work using the free-body diagram of columns subjected to external lateral forces and moments, and to internal overstrength-based plastic moment hinges developed in the beams framing into columns.

Modeling Details
Typical 2D interior frames oriented along X-direction ( Figure  6) are considered to assess seismic performance of the study buildings. Commercially available Perform  [43].
The reduction in stiffness is as per literature [44]. Shear failure of members is precluded through capacity design and detailing. Beam-column joints are considered to be stiff and strong. Rayleigh damping of 5% between 0.9T1 and 0.25T1 (as recommended in the manual of Perform 3D) is used.

Methods of Analyses
The dynamic characteristics of the buildings are estimated using modal analysis of buildings. Performances of designed buildings are assessed by both Nonlinear Static (NSA) and Nonlinear Time History Analyses (NTHA). NSA is used to obtain the lateral force-displacement response of buildings and the lateral displacement capacity; when reporting the lateral force-displacement response, the lateral displacement at the effective height h1 * of the building is used, to ensure consistency between the lateral force-displacement curve considered in the analysis and design stages. The performance point of a building is obtained using equivalent linearization procedure [45]. In NTHA, each building is subjected to a suite of 30 ground motions (Table 5) [46][47][48], which are selected to have significant randomness (i.e., coefficient of variation) in their characteristics [48] (Table 6). Further, 30 ground motions are selected to limit epistemic uncertainty related to selection of ground motions. Ground motions are scaled using spectral scaling method to ensure the buildings are subjected to the MCE level of earthquake shaking. P- effects are considered in both NSA and NTHA.
During NSA and NTHA, stated 'failure' of buildings denotes at least one structural element reaching any one of the limit states: (1) exhausting plastic rotation capacity pbc of beams, and (2) reaching ultimate compressive strain cu of confined concrete in columns. Exhausting pbc of beams may not lead to collapse of buildings, but only may lead to disruption in the gravity load path resulting in increased demand in few columns. In contrast, crushing failure of columns by reaching cu of confined concrete can lead to local failure, and even, global failure of buildings. Thus, the said limit states are considered to assess the guaranteed capacity of buildings. Further, average and maximum estimates of plastic rotation demand and inter-storey drift demand are obtained using NTHA results of 27 of the 30 ground motions; 3 outlier data points on the higher side are ignored. Ignoring outliers is acceptable as the general acceptance criteria used in design codes allow failure of certain percentile of samples (e.g., definitions of minimum specified loads and material strengths).
For interpreting results of NTHA, the uniformity in the variation of responses (such as plastic rotation demand and inter-storey drift demand) is quantified along the building height; data of N-2 responses is used to estimate their CoV (N is the number of storeys in the building) . Effectively, CoV is estimated without considering responses of the first and top storeys of a building, because responses of these storeys are significantly influenced by either the fixity of columns at the base or discontinuity of members at the roof level [49].   Figure 7 shows lateral force-displacement curves of the study buildings obtained from NSA, and Table 7  (3) Buildings designed by PD method have highest lateral drift capacity (at least 20% more), because both stiffness and strength are proportioned explicitly; buildings designed by PBPD method have lowest lateral drift capacity. (4) Total energy stored in the buildings (estimated as area under the lateral force-displacement curve) ( Table 7) is highest in buildings designed by PD and DDBD methods in 8-and 12-storey buildings, respectively; buildings designed by PBPD method have lowest total energy.

Performance of Buildings
Acceptability of the design of a building is examined by the number of ground motions that the building withstands without exceeding pbc of beams (= 0.03 rads). Table 8 lists the number of instances when pbc is exceeded in buildings when resisting MCE level earthquake shaking; it is estimated using the counted statistics method (as in [50]). Also, it provides results from NTHA along with the number of ground motions that cause yielding of columns. And, Figure 8 shows number ground motions that cause yielding of members designed by the three methods. The salient observations are: (1) Buildings designed by PD and DDBD methods withstand about 90% of ground motions (i.e., at least 27 of 30 ground motions) without exceeding pbc of beams; those designed by PBPD method withstand only ~70% of ground motions, respectively. (2) All 30 ground motions result in yielding of columns in 12storey buildings designed by DDBD method, because design of columns is based on axial demand from gravity load analysis, and flexural demand from lateral load analysis. Also, the method uses a capacity protection factor to prevent yielding of columns. Notwithstanding this, the method underestimates demand on columns in exterior bays (where axial demand on columns changes significantly due to overturning action under earthquake shaking); consequently, exterior columns sustain significant yielding in 12-storey buildings. Thus, the design of columns is inadequate to prevent yielding of columns. This observation is consistent with results present in literature [51].

Plastic Rotation Demand
The PD and PBPD methods use available plastic rotation capacity pbc of beams as input to estimate design lateral force of buildings (Tables 2 and 4). The PD method uses a safety factor to decide the design capacity pbdes (= pbc/) from the available capacity pbc. In contrast, PBPD method uses the available capacity pbc as the design capacity pbdes. Table 9 shows these values along with the maximum pbd,max and average pbd,avg of absolute plastic rotation demands in beams when resisting at least 27 (of the 30) ground motions without exceeding pbc. In addition, the PD and PBPD methods assume all beams to undergo nearly similar pbd under severe ground shaking. To examine the validity of this assumption, the average pbd,avg,storey and maximum pbd,max,storey plastic rotation demands are examined on beams in each storey ( Figure 9); CoVs of these values are listed in Table 9. The salient observations on plastic rotation demands in buildings designed by the three methods are: (1) DDBD method: The plastic rotation demands in beams in 12-storey buildings show the smallest pbd,max. Also, these values are: (a) less than the available pbc of 0.03 rads, and (b) near uniform along the height, even though significant yielding of columns is observed (Figure 9).
(2) PBPD method: The plastic rotation demand pbd,max in beams exceeds pbc in 5-9 ground motions, because this method uses available pbc itself as the design value. This highlights the need to use a safety factor of plastic rotation capacity in design to limit the plastic rotation demand in beams. Also, beams in 8-storey building have the largest pbd,max. In most storeys of 8-storey and 12-storey buildings, pbd,max,storey exceed the available pbc of 0.03 rad. Also, pbd,max is concentrated in the first few storeys.
Thus, the plastic actions (and hence damage) are localised in buildings designed by PBPD method. Thus, pbc of beams is exceeded and the assumptions that plastic rotation demand is uniform along the height is violated.
(3) PD method: It uses a safety factor  on available plastic rotation capacity pbc to estimate lateral force demand on buildings. Hence, the plastic rotation demand in beams is less than the available pbc; this is not observed in any other method. Further, plastic rotation demands in beams are almost uniform along the height. Thus, as in buildings designed by the DDBD method, damage in building design using PD method is well distributed along the building height.  (Table   10). The salient observations on the maximum inter-storey drift demands in buildings designed by the three methods are:

Figure 9: Variation in plastic rotation in beams along the height of 4-, 8-and 12-storey buildings designed by the three methods: (a) average of maximum rotation, and (b) absolute maximum rotation.
(1) DDBD method: d assumed in design and d,max demand from analysis differ by less than 16%. Further, avg is less than the critical inter-storey drift assumed in design to estimate lateral force demand obtained from NTHA, but max is more, but almost uniform along the height (Table   10).
(2) PBPD method: d and d,max differ by up to +11%. Also, inter-storey drift demand is concentrated in the first few storeys of 12-storey building. Thus, the inter-storey drift demand assumed in design does not match with that obtained from NTHA.
(3) PD method: d and d,max differ by less than 16%. Also, the inter-storey drift demand (max andavg) is nearly uniform along the height; this is reflected by the CoV values also (Table 10).

Bill of Quantities
The sizes of members of buildings designed by the PBPD and PD methods do not differ much. Hence, the amounts of concrete in buildings designed by these methods are comparable (Table 11). In contrast, buildings designed by the DDBD method require the larger amount of concrete. On the other hand, the longitudinal reinforcement is least in buildings designed by the PBPD method; this is expected because these buildings have the least lateral strengths (Figure 7). Smaller lateral force reduces the longitudinal reinforcement, but imposes unduly large plastic rotation demands in beams. Buildings designed by the PD and DDBD methods have the larger strength, and require the larger reinforcement ( Figure  7). Higher design acceleration coefficient increases the range of elastic response, and thereby reduces the inelastic plastic rotation demand on beams. It is desirable to limit this plastic rotation demand on beams, even though the longitudinal reinforcement is larger.

Summary
The results suggest that buildings designed by PD method demonstrate the best seismic performance and those designed by the PBPD method the worst. The overall ratings of the three methods are presented in Table 12 based on different considerations. The PD method provides an acceptable building using: (a) the properties of the first translational mode alone, and (b) the maximum plastic rotation capacity of beams, such that plastic rotation demand in beams are limited to levels within practically achievable values and are uniform along the height; the maximum plastic rotation demand (averaged over all beams in a storey) is 77%-84% of the design values in 8-and 12-storey buildings.

CONCLUSIONS
The salient conclusions of this study are: 1. A new method is proposed for seismic design of moment frame low-rise buildings. The design method considers: (a) A single mode, namely the fundamental lateral translational mode, and (b) Plastic rotation capacity of beams as a design input, with a safety factor of 2.0 on available plastic rotation capacity in them. The resulting building possesses good seismic performancedesirable mechanism and large deformability. Further, results of the numerical study highlight the efficacy of the proposed design method in limiting: (a) the contribution of higher modes of oscillation in the seismic response of buildings and (b) the plastic rotation demand pbd is successfully restricted to within to practically achievable pbc and available in members, which is considered as design input. 2. Based on numerical study presented, the relative performances of buildings designed by the proposed and two other methods show that the Proposed Design method is: (a) better than the DDBD, and (b) significantly better than the PBPD method in controlling seismic behavior of buildings. This method is not applicable to tall RC MF buildings, because it is difficult to make first mode dominate in such buildings.