The extended N2 method was applied to analyze two reinforced concrete buildings with asymmetric setbacks. The extended N2 method accounts for higher mode effects by combining results from a pushover analysis and an elastic modal analysis. It was applied to a 5-story and 9-story building. For the 5-story building, the extended N2 method provided results close to the displacement coefficient method in FEMA 356. Correction factors from the modal analysis were less than 1, so the basic N2 method results were used. This shows the extended N2 method can accurately analyze buildings with vertical irregularities like asymmetric setbacks.

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APPLICATION OF EXTENDED N2 METHOD TO
REINFORCED CONCRETE FRAMES WITH
ASYMMETRIC SETBACKS
Dini Devassy Menachery1
, Manjula N.K.2
1
Post Graduate Student, Sree Narayana Gurukulam College of Engineering, Kadayiruppu, India
2
Research Scholar, NIT Calicut, India
ABSTRACT
N2 method is a simplified nonlinear static analysis procedure used for the seismic evaluation and design of
structures. Recent improvements of this procedure, like the extended N2 method, made the application of it to irregular
structures possible. Asymmetric setback is a common form of vertical irregularity found in multi-storied building frames.
The Extended N2 method proposed by Prof. Peter Fajfar is based on the assumption that the structure remains in the
elastic range when vibrating in higher modes. The seismic demand in terms of displacements and storey drifts can be
obtained by enveloping the results of basic pushover analysis and the results of standard elastic modal analysis. The
proposed procedure was applied to two reinforced concrete buildings. Different ground motion records were used for the
time history analysis. The results obtained from extended N2 method were compared with the time history analysis and
displacement coefficient method in FEMA 356, and it was identified that the method predict displacement and drift
demands of the frames types considered with reasonable accuracy.
Keywords: Extended N2 method, Nonlinear static analysis, Plastic mechanism, Pushover analysis, Response spectrum.
1: INTRODUCTION
1.1 General
Seismic performance of a structure is strongly dependent on the damage suffered by itunder the design
earthquake. Since damage in structures results from inelasticdisplacement, assessment of the performance of an existing
or new structure requiresinelastic structural analysis to determine the magnitude of demand parameters in thestructures.
These demand parameters are used to compare the performance against theacceptance criteria.
Global displacement (roof level displacement) and inter-storey drift are the major demand parameters which can
be predicted and controlled in performance based design.In principle, the non-linear time-history analysis is the correct
approach. However, such an approach, for the time being, is not practical for everyday design use. It requires additional
input data (time-histories of ground motions and detailed hysteretic behaviour of structural members) which cannot be
reliably predicted. Non-linear dynamic analysis is, at present, appropriate for research and for design of important
structures. It represents a long-term trend. On the other hand, the methods applied in the great majority of existing
building codes are based on the assumption of linear elastic structural behavior and do not provide information about real
strength, ductility and energy dissipation. They also fail to predict expected damage in quantitative terms. For the time
being, the most rational analysis and performance evaluation methods for practical applications, which should be used in
addition to the established elastic procedures, seem to be simplified inelastic procedures, which combine the non-linear
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static (pushover) analysis of a relatively simple multi degree-of- freedom (MDOF) mathematical model and the response
spectrum analysis of an equivalent single-degree-of-freedom (SDOF) model. This is termed as N2 method. They can be
used for a variety of purposes such as design verification for new buildings and bridges, damage assessment for existing
structures, determination of basic structural characteristics in direct displacement based design, and rapid evaluation of
global structural response to seismic ground motion of different intensities. In recent years, a breakthrough of these
procedures has been observed. They have been implemented into the modern guidelines and codes (Euro code 8
standard- Annex B of part 1).
The basic assumption used in pushover-based methods is that the structure vibrates predominantly in a single
mode. This assumption is not always fulfilled, especially in the case of high-risebuildings and/or torsionally flexible
plan-asymmetric buildings. A lot of work has been done worldwide to take into account the influence of higher modes in
elevation and in plan. Development of the extended N2 method is applicable to structures with important higher mode
effects and the basic idea was to keep the procedure as simple as possible. The higher mode problem was approached by
using correction factors based on the results of elastic modal analysis. The same idea can be used in any pushover-based
approach. In the current study, the extended N2 method which takes into account higher mode effects in elevation is
presented. First, the basic idea of N2 method is explained followed by extended N2 method. The computational
procedure is then summarized, and applied to two test building.
1.2: Need for the Study
N2 method is a simplified analysis procedure used for the seismic evaluation and design of structures. Recent
improvements of this procedure, like the extended N2 method, made the application of it to irregular structures possible.
Asymmetric setback is a common form of vertical irregularity found in multi-storeyed building frames. Application of
the Extended N2 method for such structures, if found successful, will reduce the computational efforts greatly, as
Nonlinear time history analysis is the current requirement for such structures as per different design codes.
2. THE PROPOSED APPROACH- EXTENDED N2 METHOD
The basic N2 method is, like other pushover-based methods, based on the assumption that the structure vibrates
predominantly in a single mode. If higher modes of vibration are important, either in plan or in elevation, some
corrections have to be applied to the basic procedure. The extension of the N2 method to plan-asymmetric buildings,
where torsional influences are important, was made by assuming that the torsional influences in the inelastic range are
the same as in the elastic range. The torsional influences are determined by the standard elastic modal analysis. They are
applied in terms of correction factors, which are used for the adjustment of results obtained by the usual pushover
analysis. Practically the same idea has been used for the extension of the N2 methodto medium- and high-rise buildings,
where higher mode effects are important along the elevation ofthe structure. It is assumed that the structure remains in
the elastic range when vibrating in higher modes, and that the seismic demands can be estimated as an envelope of
demands determined by a pushover analysis, which does not take into account the higher mode effects, and normalized
demands determined by an elastic modal analysis, which includes higher mode effects. Typically, the pushover analysis
controls the behavior of those parts of the structure where the major plastic deformations occur, whereas the elastic
analysis determines seismic demand at those parts where the higher mode effects are important. Due to the similarity of
the approaches, basically the same procedure as in the case of torsion can be applied. The influence of higher modes is
determined by standard elastic modal analysis, and used for the adjustment of the results obtained by the usual pushover
analysis.The proposed procedures (for taking into account higher mode effects in plan and in elevation) are consistent
and compatible. Both effects can be considered simultaneously bytwo sets of correction factors.
In order to predict the structural response for a building with a non-negligible effect of higher modes along the
elevation, the following procedure can be applied:
1. Perform the basic N2 analysis as explained in the previous section and target roof displacement is found out
2. Perform the standard elastic modal analysis of the MDOF model considering all relevantmodes. Determine
storey drifts for each storey. Normalize the results in such a way that thetop displacement is equal to the target
top displacement.
3. Determine the envelope of the results obtained in Steps 1 and 2.
(3a) For each storey, determine the correction factors CHM , which are defined as the ratio between the results
obtained by elastic modal analysis (Step 2) and the results obtainedby pushover analysis (Step 1). If the ratio is
larger than 1.0, the correction factor CHM is equal to this ratio, otherwise it amounts to 1.0. Note that the
correction factors fordisplacement are small and can be neglected in most practical applications. The correction
factors for storey drifts are important.
(3b)The resulting storey drifts (and displacements, if necessary) are obtained by multiplying the results
determined in Step 1 with the corresponding correction factors CHM.

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4. Determine other local quantities. The resulting correction factors for storey drifts CHMapplyto all local
deformation quantities (e.g. rotations). Correction factors CHM for storey driftsalso apply to internal member
forces, provided that the resulting internal forces do not exceedthe load-bearing capacity of the structural
member.
3. DESCRIPTION OF TEST BUILDING AND MATHEMATICAL MODELLING
3.1. Introduction
Two test examples, G+5storey and G+9storey reinforced concrete buildings having vertical setbacks are used
for the study. The structures designed for two different seismic zones. G+5storey building belongs to seismic zone III
and G+9storey building belongs to seismic zone IV.
Structures are modeled and designed using the software SAP2000. For the analysisthree-dimensional structural
model is chosen. The plastic rotational hinges at both ends of every line element (beam, column) are used. Five per cent
damping is assumed.
3.2. Application of Extended N2 method to G+5storeyReinforced Concrete building
Application of extended N2 method to G+5storey reinforced concrete building is described based on the
procedure described above and the results obtained are compared with the FEMA 356 values. The 3D model of the
structure considered is shown in Fig.1.Height of each storey is 5m.The structure belongs to seismic zone III (zone factor
Z=0.16).
Fig.1: 3D model of G+5storey RC building
The storey masses from bottom to the top amounts to
[m1, m2, m3, m4, m5, m6] = [848.04, 858.04, 618.28, 594.12, 347.28, 205.88]kN-S2
/m.
Linear displacement shape obtained from modal analysis is
Φ T
=[Φ1, Φ2, Φ3, Φ4, Φ5, Φ6] = [0.115, 0.324, 0.552, 0.769, 0.922, 1]
The MDOF system is transformed to an equivalent SDOF system using Eqs.
∆∗ = ∆τ/ Γ (1)
Φ∗ = ς/ Γ (2)
Mass of equivalent SDOF system, m* = ∑mi Φi = 1724.080 kN-S2
/m (3)
Transformation factor, Γ= m*/∑mi Φi2
= 1.509 (4)
With this force pattern, push over analysis were performed in both X and Y directions. Loading was applied
with +ve and– signs. The base shear Vstop displacement Dt relationship obtained from SAP 2000 in X- positive direction
is shown in Fig.2.
Fig.3defines both the V–Dt relationship for the MDOF system, and the force F* -displacement D* relation for
the equivalent SDOF system.The scale of the axes, however, is different for the MDOF and SDOF systems. The
factorbetween the two scales is equal to Γ. Bilinear idealization of the pushover curve is shown in Fig.3. Iteration was
used for determination of the bilinear idealization of pushover curves.

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Fig.2: Push over curve in X- positive direction
From Fig.3Assume dm*= 0.3m
The yield strength and displacement amount tofy*= 2250 kN and dy*= 0.22m.
The elastic period is T*= = 2.57 s (5)
From response spectrum shown in Fig.4, accelerationSa/g = 0.4 and time period,
Tc = 0.5331
The period of the system T* is larger than Tc.
Thus the equal displacement rule applies: µ = Rµ ,Sd = Sde.
The seismic demand for the equivalent SDOF system is found out using the formulae,
Sde = Se(T*)[ T*/ ]2
(6)
Se(T*) will obtained from response spectrum for corresponding value of time period T* and the value is Se(T*) = 0.853.
Sd = Sde= 0.143m
This Sdvalue is then used as the new approximation for target displacement. Iteration continued until similar
displacements obtains. Table 1 shows the calculation involved in finding out the target displacement of SDOF system

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Fig.3: Bilinear idealization of push over curve in X- positive direction
Fig.4: Response Spectrum

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Table1: Calculation of target displacement of SDOF system in X-direction
X-direction (+ve) X-direction (-ve)
Trial 1 Trial 2 Trial 3 Trial 1 Trial 2 Trial 3
Target displacement
assumed, dm*(m)
0.3 0.143 0.13 0.3 0.139 0.136
Yield strength Fy*(kN) 2250 1700 1500 2250 1500 1500
Yield displacement
dy*(m)
0.22 0.143 0.13 0.22 0.139 0.136
Elastic Time period,
T*= 2π √((m)*dy*)/Fy* (sec)
2.57 2.39 2.43 2.57 2.51 2.51
Elastic accleration,
Se(T*) m/sec2
0.853 0.9005 0.8834 0.833 0.853 0.853
Target displacement
of SDOF(m), Sd
0.143 0.13 0.132 0.139 0.136 0.136
The seismic demand for the equivalent SDOF system in X direction obtained from Table1 is 0.136m (higher
value of +ve and –ve direction).
In the next step the displacement demand of the equivalent SDOF system is transformed back to the top
displacement of the MDOF system (using Eq.1)
Dt = 0.205m
Similarly pushover analysis is done Y- direction also.
Table 2 shows the displacements obtained at various floors by N2 method.
In extended N2 method correction factors are applied which are based on the ratio between the normalized
results obtained by elastic modal analysis and the results obtained by push over analysis. Here the correction factor
obtained is less than one. So the target displacement need not be multiplied with correction factors.
Inference:
Target displacementobtained from FEMA 356 in X- direction = 0.188m
Target displacementobtained from FEMA 356 in Y- direction = 0.184m
Since there is no correction factor required, the results of N2 method and Extended N2 method are the same.
The results of the N2 method were within the range of results obtained from FEMA 356 or fairly close to the test results.
Time period obtained for first three modes from software is given in the Table 3,results are similar to those obtained by
N2 method.
Table 2: Displacements obtained by N2 method
Displacements obtained by N2 method
Floor X-direction in cm Y- direction in cm
1 1.44 1.98
2 3.98 5.56
3 9.56 9.92
4 13.33 14.35
5 19.95 17.73
6 20.5 20.5

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Table 3: Time Period of first three modes
Mode Period (Sec)
1 2.648037
2 2.514996
3 1.889716
3.3. Application of Extended N2 method to G+9Storey Reinforced Concrete building
Application ofextended N2 method to G+9Storey Reinforced concretebuilding is described based on the
procedure described above and the results obtained are compared with nonlinear dynamic analysis. The 3D model of the
structure considered is shown in Fig.5. Height of each storey is 3.5m.The structure belongs to seismic zone IV( zone
factor Z=0.24).
Fig.5: 3D model of G+9 storey RC building
The storey masses from bottom to the top amounts to
[m1, m2, m3, m4, m5, m6, m7, m8, m9, m10] = [1920.52, 1887.21, 1278.78, 1272.38, 974.58, 971.43, 673.63, 670.48,
372.68, 366.38]kN-S2
/m.
Linear displacement shape obtained from modal analysis is
Φ T
= [Φ1, Φ2, Φ3, Φ4, Φ5, Φ6, Φ7, Φ8, Φ9, Φ10] = [0.064206, 0.16397, 0.291203, 0.429896, 0.567577, 0.692787,
0.800725, 0.88844, 0.954837, 1]
The MDOF system is transformed to an equivalent SDOF system using Eqs.(1).
Mass of equivalent SDOF system, m* = ∑mi Φi = 4435.576 kN-S2
/m
Transformation factor, Γ = m*/∑mi Φi2
= 1.556
With this force pattern, push over analysis were performed in both X and Y directions. Loading was applied
with +ve and – signs. The base shear V–top displacement Dt relationship obtained from SAP 2000 in X- positive
direction is shown in Fig.6.
Fig.7defines both the V –D t relationship for the MDOF system, and the force F*- displacement D* relation for
the equivalent SDOF system.The scale of the axes, however, is different for the MDOF and SDOF systems. The
factorbetween the two scales is equal to Γ. Bilinear idealization of the pushover curve is shown in Fig.7. Iteration was
used for determination of the bilinear idealization of pushover curves.

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Fig.6: Push over curve in X- positive direction
From Fig.7 Assume dm*= 0.3m
The yield strength and displacement amount tofy*= 4800 kN and dy*= 0.1m.
The elastic period is T* = 1.91s(Eq.5).
Fig.7: Bilinear idealization of push over curve in X- positive direction
From response spectrum, Acceleration Sa/g = 0.6 and time period, Tc = 0.5331

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The period of the system T* is larger than Tc. Thus the equal displacement rule applies,
µ = Rµ ,Sd = Sde.
The seismic demand for the equivalent SDOF system is found out using the formulae,
Sde = Se(T*)[ T*/ ]2
Se(T*) will obtained from response spectrum for corresponding value of time period T* and the value is Se(T*) = 1.68.
Sd = Sde= 0.155m
This Sdvalue is then used as the new approximation for target displacement. Iteration continued until similar
displacements obtains. Table4 shows the calculation involved in finding out the target displacement of SDOF system in
X-direction.
The seismic demand for the equivalent SDOF system in X direction obtained is 0.156m (higher value of +ve
and –ve direction).
In the next step the displacement demand of the equivalent SDOF system is transformed back to the top
displacement of the MDOF system (using Eq.1)
Dt = 0.242m
Table 4: Calculation of target displacement of SDOF system in X-direction
X-direction (+ve) X-direction (-ve)
Trial 1 Trial 2 Trial 3 Trial 1 Trial 2 Trial 3
Target displacement
assumed, dm*(m)
0.3 0.155 0.156 0.2 0.143 0.149
Yield strength Fy*(kN) 4800 4700 4700 5000 4900 4900
Yield displacement dy*(m) 0.1 0.1 0.1 0.1 0.095 0.1
Elastic Time period ,
T*= 2π √((m)*dy*)/Fy* (sec)
1.91 1.93 1.93 1.87 1.84 1.89
Elastic accleration,
Se(T*) m/sec2 1.68 1.663 1.663 1.612 1.743 1.698
Target displacement
of SDOF(m), Sd
0.155 0.156 0.156 0.143 0.149 0.152
Similarly pushover analysis is done Y- direction also.
Displacements Obtained at various floors by N2 method is given in Table5 and Inter storey drift is shown in Table 6.
Table 5: Displacements Obtained by N2 method
Displacements Obtained at various floors by N2 method
Storey
Displacement
(X-direction) in mm
Displacement
(Y-direction) in mm
10 0.24 0.264
9 0.2036 0.2558
8 0.1695 0.2184
7 0.1394 0.1977
6 0.111 0.1485
5 0.0874 0.1179
4 0.0638 0.0707
3 0.0423 0.0469
2 0.023 0.014
1 0.009 0.056

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Table 6: Inter Storey Drift
Inter Storey Drift
Storey Drift in X direction Drift in Y direction
9 &10 0.009742857 0.010685714
8&9 0.0086 0.005914286
7&8 0.008114286 0.014057143
6&7 0.006742857 0.008742857
5&6 0.006742857 0.013485714
4&5 0.006142857 0.0068
3&4 0.005514286 0.0094
2&3 0.004 0.012
1&2 0.002571429 0.016
Elastic modal analysis of the MDOF model was performed, considering the first three modes. Their
effectivemodalmassescontribute98%ofthetotalmass.TheSRSSruleforthecombinationof different modeswasapplied.
Displacements and storey drifts were determined and are shown in Table 7.
Table 7: Inter Storey Drift by Modal Analysis
Inter Storey Drift by Modal Analysis
Storey Drift in X direction Drift in Y direction
9 &10 0.00176846 0.000224214
8&9 0.002016678 0.000257008
7&8 0.0015465 0.000625204
6&7 0.001093343 5.60596E-05
5&6 0.000759428 0.001025099
4&5 0.000940758 0.000629244
3&4 0.001241681 0.001161497
2&3 0.001458921 0.000764556
1&2 0.001238861 0.000711759
Determination of correction factors
IntheextendedN2method,theresultsofpushoveranalysisaremultipliedbycorrection factors CHM,which depend on
the elevation of the structure.Different values of CHMapply to the displacements and storey drifts. The correction factors
are based on the ratio between the normalized results obtained by elastic modal analysis and the results obtained by
pushover analysis. If this ratio is larger than 1.0, then the correction CHM is equal to this ratio, otherwise it amounts
to1.0.The correction factors are in the range between 1.0 and 1.8. The correction factors obtained are given in the Table
8. Storey drifts obtained by N2 method, Extended N2 method and time history analysis are shown on Table 9 and Fig. 8
Table 8: Correction Factors for Storey Drift
Correction Factors for Storey Drift
Storey Correction factor
9 &10 1
8&9 1.217273833
7&8 1.057522429
6&7 0.792399804
5&6 0.662339309
4&5 0.820487556
3&4 1.188715224
2&3 1.555895923
1&2 1.821380533

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Table 9: Comparison of Storey Drift
Storey Drift obtained
Storey N2 method
Extended N2
method
Time history
analysis
9 &10 0.0104 0.0104 0.009842
8&9 0.009742857 0.011859725 0.009339
7&8 0.0086 0.009094693 0.008157
6&7 0.008114286 0.008114286 0.008014
5&6 0.006742857 0.006742857 0.01
4&5 0.006742857 0.006742857 0.006
3&4 0.006142857 0.007302108 0.007617
2&3 0.005514286 0.008579655 0.006787
1&2 0.004 0.007285522 0.0049
Fig.8: Storey Drifts obtained by N2, Extended N2 method and Time history analysis
Inference:
Storey drifts obtained by extended N2 method are compared with storey drifts obtained by time history analysis
shown in Fig.9. In the upper stories values obtained by extended N2 method are higher than time history values.
In asymmetric setback plane i.e. in X-direction, drift values are more dependent on higher modes or extended
N2 method can safely predict the drift demands. In Y-direction extended n2 method is overestimating the drift demands.
We can say that extended N2 method can be used for asymmetric plane performance evaluation.
From Fig. 5.19, it can be seen that at mid height the drift values obtained extended N2 method is less compared to time
history values due to the reason that the correction factors are obtained from elastic modal analysis which does not
consider the non-linear effects due to higher modes.
Extended N2 method is based on displacement controlled pushover analysis. In medium rise or higher storied
buildings push over results obtained from generally applied load pattern cannot predict the base shear demands
accurately due to the higher mode effects
CONCLUSIONS
Extended N2 method is a simple, rational analysis performance evaluation method for practical applications. It
has proven to better results for displacement and drift demands of asymmetric buildings. In this study buildings with
setbacks on one plane was chosen and the predictions of extended N2 method are evaluated with respect to the time

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history analysis. It could be seen that the predictions from this method are safer for the plane of asymmetry and for the
symmetric plane the performance evaluation can be done even with the basic N2 method.
This study focus on the application of extended N2 method for 3D building frames. Parametric study can be
conducted on set of buildings with varying irregularities. Soil structure interaction effects can be incorporated for further
studies.
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