Method N2-According to Fajfar

Full text

Proc. of the Second Intl. Conf. on Advances in Civil, Structural and Construction Engineering - CSCE 2015
Copyright © Institute of Research Engineers and Doctors, USA .All rights reserved.
ISBN: 978-1-63248-042-2 doi: 10.15224/ 978-1-63248-042-2-95
83
METHOD N2 – ACCORDING TO FAJFAR
Arton Dautaj, Naser Kabashi and Hajdar Sadiku
Abstract: A relatively simple nonlinear method for the seismic
performance evaluation of structures (the N2 method) is
presented. The method combines the nonlinear static
(pushover) analysis and the response spectrum approach. The
method yields results of reasonable accuracy if the structure
oscillates predominantly in the first mode. In the paper the
method is formulated in the acceleration – displacement
format. This versions combine the advantages of the visual
representation of CSM developed by Freeman. By reversing
the analysis process, the method can be used as a tool for the
implementation of the direct displacement-based design
approach.
Keywords: ”Pushover analysis”, performance evaluation,
inelastic behavior, ductility etc.
Introduction
For the rational design of buildings from seismic
operations, design methods must meet the following
requirements:
a) adequately respond to requests of stiffness,
strengthness and ductility during an expected earthquake[5],
and
b) not be complicated[5].
The methods applied in the various codes ( analysis
of equivalent lateral force and modal spectral analysis) are
based on the assumption of linear elastic behavior of the
structure and as such, regardless of the application to them
of various modifier and corrective factors, fail to satisfy the
first request adequately.
On the other hand, nonlinear dynamical analysis of
the system with many degrees of freedom is relatively
complicated and, as such, are not very suitable for everyday
design.
________________________________________________
Arton Dautaj¹
Naser Kabashi¹
Hajdar Sadiku¹
¹Civil Engineering and Architectural Faculty, University “Hasan Prishtina”
of Prishtina
Republic Of Kosova,
Method N2 (N2 designation relates to the fact that it
requires the application of a nonlinear method - "Nonlinear"
- and the construction of two models) meets the two
requirements above
N2 method in Europe is developed in Slovenia, in the
University of Ljubljana and is based on the so-called model-
Q, filed by Saiidi and Sozen [23] for simple nonlinear
analysis of systems with multi degrees of freedom. Later,
this method has been developed through different stages and
for different systems, while now has been included in the
final draft of the EC 8 [3]
N2 method provides results with sufficient accuracy
and can be used for systems where seismic response is
dominated by the contribution of the first form of vibration.
Initially N2 method is presented for "regular" systems
(Fajfar & Fischinger, 1987 ECEE [4], 1989 WCEE [5]). As
basic proposals of this method are the use of two different
mathematical models and the application of three main steps
in the analysis. In the first step is determined the stiffness
(rigidity), strengthens and ductility. For this applies
nonlinear static analysis of the system with many degrees of
freedom (MDOF) by the action of a form of distribution of
loads that monotonically increasing.
In the second step is defined equivalent system with
single degree of freedom. Nonlinear characteristics of
equivalent system are based on the relation base shear force
-displacement on the roof designated by nonlinear static
analysis in the first step. While on the third step of the
method N2 from nonlinear dynamic analysis of the
equivalent system with single degree of freedom is
determined the maximum displacement (and corresponding
demand on ductility). The third step, in a simple form can be
performed using inelastic spectra.
As mentioned above, the use of inelastic spectra in
the third step can simplify a lot the analysis and make it
highly suitable for daily practice project.
After processing of the first ideas, the method has
already found numerous applications of seismic analysis and
anti-seismic design (reinforced concrete buildings
considering the remaining damages, according Fajfar and
Gaspersic [16], bridges, asymmetric buildings; spatial
buildings; and, finally assessment of performance).
Actually, the N2 method is formulated in the format AD
[14,17], acceleration-displacement.

Proc. of the Second Intl. Conf. on Advances in Civil, Structural and Construction Engineering - CSCE 2015
Copyright © Institute of Research Engineers and Doctors, USA .All rights reserved.
ISBN: 978-1-63248-042-2 doi: 10.15224/ 978-1-63248-042-2-95
84
The following is given recently developed version of
the method N2.
1. Summary of N2 method
I. Data
a) Structure
b) Moment-curved geometry relationship
c) Elastic spectra of accelerations
II. Seismic demands on format AD
a) Define elastic spectra in AD format
aede S
T
S2
2
4


b) Define the inelastic spectra for constant ductility

R
S
Sae
a
,
dedS
R
S



1.8
T=Tc=0.5s
44.76 Tc
R
T


=1
=2
=3
Sa(g)
Sd(cm)
1.8
T=Tc=0.5s
44.76 Tc
R
T


=1
=2
=3
Sa(g)
Sd(cm)
III. ''Pushover'' analysis
a) Assume the form of displacement
b) Determine the vertical distribution of lateral force
 
 
 
 MP
,
iii pmP
c) Define the relationship "based shear force -
displacement on the roof".
IV. Equivalent model with single
degree of freedom.
a) Define the mass
*
m
i
n
ii
mm  
1
*
b) Transform quantities (Q) of system with multi
degrees of freedom in quantities (
*
Q
) of the system with
single degree of freedom

Proc. of the Second Intl. Conf. on Advances in Civil, Structural and Construction Engineering - CSCE 2015
Copyright © Institute of Research Engineers and Doctors, USA .All rights reserved.
ISBN: 978-1-63248-042-2 doi: 10.15224/ 978-1-63248-042-2-95
85

Q
Q*
,



 n
iii
m
m
1
2
*
c) Determine the approximate relationship elastic-
plastic force-displacement
d) Determine the strength
*
y
F
, yield displacement
y
d
and period
*
T
.
*
**
*2
y
y
F
dm
T


,
*** yy dFk 
e) Determine the diagram of capacity (acceleration
versus displacement)
*
*
m
F
Sa
Md*
Mm*
Fb
F*
Dd*
Yy
F*
y
Dd*
V. Seismic demand for the model
with single degree of freedom.
a) Define reducing factor

R
ay
ae
S
S
R

b) Define displacement demand
 





 *
*11 T
T
R
R
S
Sd Cde
d


,
C
TT 
*
dedSSd 
*
,
C
TT 
*
T=Tc
T<Tc
Sd(cm)
44.7
Sa(g)
Sae
Sd(cm)
44.7
T=0.72S
T=Tc
Sa(g)
1.8
*
*
m
Fy*
*
m
Fy
M=3 M=3
M=1
Dd* Dd* Dd*
Dd* Dd*
Dd*=
T=Tc
T<Tc
Sd(cm)
44.7
Sa(g)
Sae
Sd(cm)
44.7
T=0.72S
T=Tc
Sa(g)
1.8
*
*
m
Fy*
*
m
Fy
M=3 M=3
M=1
Dd* Dd* Dd*
Dd* Dd*
Dd*=
VI. Global seismic demand for the
model with many degrees of freedom.
a) Transform the displacement demand of the system
with single degree of freedom to the top
displacement of multi degrees of freedom
model.
*
ddt
VII. Local seismic demand
a) Apply the analysis "pushover" in the
model with multi degrees of freedom ( MDOF) until
reaching the displacement in
t
d
b) Determine local quantities (relative
displacement of floors, rotation of joints
etc.), corresponding to dt
Fb
dt

dt=d*

Proc. of the Second Intl. Conf. on Advances in Civil, Structural and Construction Engineering - CSCE 2015
Copyright © Institute of Research Engineers and Doctors, USA .All rights reserved.
ISBN: 978-1-63248-042-2 doi: 10.15224/ 978-1-63248-042-2-95
86
VIII. Performance Assessment
a) Compare local and global seismic demand with
capacity to the required level of performance.
2 . APPLICATION OF THE
METHOD N2 IN DESIGN
With the inversion of procedure of method N2 used
for assessment of the performance, can be developed a
design methodology based directly on the displacement.
Practically will be done this way:
1st Step.
For the required performance (defined) is given the
displacement
 
nt
d
, which represents the displacement on
the roof of the system with many degrees of freedom.
2nd Step.
Determining the displacement of the system with
single degree of freedom

 t
dt d
Sd **
4.15
where
123

 nn
or
2
1ii
n
i
*
m
m

 

,
where n-number of floors,
i
n
ii
mm  
1
*
,
i

- linear form of vibration (first form of
vibration).
3rd Step.
Determining the ductility or stiffness (rigidity) of the
structure. To define the ductility, initially should be
determined the yield displacement for the system.

yt
yt
d
d*
4.16
Ductility will be
*
*
yt
t
d
d


4.17
4th Step.
For the defined quantity
*
t
d
appreciate
ae
S
and
*
T
.
Further apply the following expressions:


R
for
C
TT 
*
, 4.18
 
11 *
C
T
T
R


for
C
TT 
*
4.19
T=Tc
T<Tc
Sd(cm)
44.7
Sa(g)
Sae
Sd(cm)
44.7
T=0.72S
T=Tc
Sa(g)
1.8
*
*
m
Fy*
*
m
Fy
M=3 M=3
M=1
Dd* Dd* Dd*
Dd* Dd*
Dd*=
T=Tc
T<Tc
Sd(cm)
44.7
Sa(g)
Sae
Sd(cm)
44.7
T=0.72S
T=Tc
Sa(g)
1.8
*
*
m
Fy*
*
m
Fy
M=3 M=3
M=1
Dd* Dd* Dd*
Dd* Dd*
Dd*=

Proc. of the Second Intl. Conf. on Advances in Civil, Structural and Construction Engineering - CSCE 2015
Copyright © Institute of Research Engineers and Doctors, USA .All rights reserved.
ISBN: 978-1-63248-042-2 doi: 10.15224/ 978-1-63248-042-2-95
87

R
S
Sae
ay 
4.20
Thus, base shear force of the system with single
degree of freedom will be:
*
y
*
*
y
ay F
m
F
S
4.21
5th Step.
Determine the base shear force for the system with
multi degree of freedom.
 *
yb FF
4.22
6th Step.
The distribution in the height of the structure of
horizontal load adapting assumed displacement profile:
b
n
iii
ii
iF
m
m
F




1
4.23
Example:
According to the Fajfar method N2 to evaluate the structure
for seismic demand by EC-8,B,ag=0.3g ,04g,0.5g and 0.6g
.
From the "pushover" analysis is defined the
relationship base shear force – displacement on the roof.
Fb=670 kN,
cmdyt 13
Fig.1,Displacement demand for ag=0.6g,B,EC8,v.2002
Fig.2,Displacement demand for 0.3g,B,EC8,v2002
For four cases of seismic demand, the results are presented
below in tabular form.
0.3g
0.4g
0.5g
0.6g
Fb(kN)
670
670
670
670
yt
d
(cm)
13
13
13
13
*
m
(ton)
141
141
141
141
*
y
F
(kN)
523
523
523
523
*
yt
d
(cm)
10.15
10.15
10.15
10.15
*
T
(s)
1.04
1.04
1.04
1.04
*
k
(kN/m)
5153.8
5153.8
5153.8
5153.8
ay
S
 
2
sm
0.378g
0.378g
0.378g
0.378g
ae
S
 
2
sm
0.432g
0.57g
0.72
0.86

R
1.14
1.53
1.91
2.29
*
t
d
(cm)
11.65
15.53
19.42
23.3
t
d
(cm)
14.93
19.91
24.89
29.86

Proc. of the Second Intl. Conf. on Advances in Civil, Structural and Construction Engineering - CSCE 2015
Copyright © Institute of Research Engineers and Doctors, USA .All rights reserved.
ISBN: 978-1-63248-042-2 doi: 10.15224/ 978-1-63248-042-2-95
88
According to the DDBD N2method -
Fajfar,
For the defined performance dt =15 cm and seismic
demand, ag=0.3g,0.4g,0.5g and 0.6g to be determined the
base shear force
28.1
141
*m
ton
cmdt71.11
*
cmdyt 4.141200
50
600
002.05.0 
cm13%10 
cmdyt 15.10
*
15.1

From the design spectra can be obtained (read):
gSae 43.0
and
04.1
*T
g
R
S
Sae
ay 37.0

,
kNFy28.518
*
kNFb12.664
kNF15.328
1
kNF39.234
2
kNF57.101
3
For four cases of seismic demand, the results are presented below in
tabular form
0.3g
0.4g
0.5g
0.6g
*
m
(ton)
141
141
141
141

1.28
1.28
1.28
1.28
t
d
(cm)
15
15
15
15
*
t
d
(cm)
11.71
11.71
11.71
11.71
yt
d
(cm)
13
13
13
13
*
yt
d
(cm)
10.15
10.15
10.15
10.15

1.15
1.15
1.15
1.15
ae
S
 
2
sm
0.43g
0.76g
1.195
1.49g
*
T
(s)
1.04
0.78
0.627
0.52

R
1.15
1.15
1.15
1.15
ay
S
 
2
sm
0.37g
0.66g
1.035g
1.49g
*
y
F
(kN)
518.28
921.37
1439.64
2073.09
Fb(kN)
664.12
1180.65
1844.78
2656.48
Conlusion
N2 method can be used both for the seismic performance
evaluation of newly designed or existing structures.
Furthermore, by reversing the analysis process, the method
can be used as a tool for the implementation of direct
displacement-based design approach, in which design starts
from a predetermined target displacement. The limitations of
the methods based on pushover analysis should be
recognized. A detailed discussion of pushover analysis can
be found in (Krawinkler and Seneviratna)

Proc. of the Second Intl. Conf. on Advances in Civil, Structural and Construction Engineering - CSCE 2015
Copyright © Institute of Research Engineers and Doctors, USA .All rights reserved.
ISBN: 978-1-63248-042-2 doi: 10.15224/ 978-1-63248-042-2-95
89
Literature
1. A.S.Elnashai.”DO WE REALLY NEED INELASTIC DYNAMICS
ANALYSIS”.Journal of Earthquake Engineering,Vol.6,Special
issue 1(2002), pp 123-130.
2. A.D.Dautaj.”VEÇORITË E APLIKIMIT TË METODIKAVE
TË PROJEKTIMIT BAZUAR NË ZHVENDOSJE PËR
STRUKTURAT BETONARME NË ZONAT SIZMIKE”.Tema
e Magjistratures,Prishtinë,2005.
3. “Eurocode 8: Design of structures for earthquake
resistance”Part 1: General rules, seismic actions and rules for
buildings.DRAFT No 6,Version for translation, January
2003,CEN.
4. P.Fajfar and M.Fishinger,” Non-linear seismic analysis of RC
buildings: implications of a case study”,European Earthquake
Engineering,1,31-43.(1987).
5. P.Fajfar and M.Fishinger,”N2-a method for non-linear seismic
analysis of regular buildings”,Proc.9th World Conf. Earthquake
Engineering,Vol.V,
Tokyo,Kyoto,1988,Maruzen,Tokyo,1989,pp.277-287.
6. P.Fajfar.”Elastic and inelastic design spectra”.In 10th
Eur.Conf.Earthquake Engineering,(ed.G.Duma),
Vienna,1994,Balkema,Rotterdam,Vol.2,pp.1169-1178.
7. M.Fischinger, P.Fajfar and T.Vidic.”Factors contributing to the
response reduction.” Proc.5th U.S. Conference on Earthquake
Engineering,Vol.1.Chicago,IL,1994,pp 97-106.
8. P.Fajfar and P.Gaspersic,”The N2 method for the seismic
damage analysis for RC buildings”,Earthquake Engineering and
Structural Dynamics,25,23-67(1996).
9. P.Fajfar and H.Krawinkler(1992 ),”Nonlinear Seismic Analysis
and Design of Reinforced Concrete Buildings”,Elsevier Applied
Science,New York,1992.
10. P.Fajfar,”Equivalent ductility factors taking into account low-
cycle fatigue”, Earthquake Engineering and Structural
Dynamics,21,837-848(1992).
11. P.Fajfar.(1996): “Design spectra for new generation of codes.In
11th World Conference on Earthquake
Engineering,Acapulco,28-28 June 1996,CD-ROM 2127.
12. P.Fajfar.”Trends in seismic design and performance evaluation
approaches”.In 11th European Conference on Earthquake
Engineering,Paris,6-11 September
1998,Balkema,Rotterdam,Invited Lectures,237-249.
13. P.Fajfar and H.Krawinkler (eds), Seismic Design Methodology
for the Next Generation of codes”,Balkema ,Rotterdam,1997.
14. P.Fajfar,1999,Capacity spectrum method based on inelastic
demand spectra”, Earthquake Engineering and Structural
Dynamics,28,979-993.
15. P.Fajfar and D.Drobnic,1998,”Nonlinear seismic analysis of
ELSA buildings”,Proceedings of the 11th European Conference
on Earthquake Engineering,Paris,CD-ROM,Balkema,
Rotterdam.
16. P.Fajfar ,P.GaspersicD.Drobnic,1997,”A simplified nonlinear
method for seismic damage analysis of structures”, in P.Fajfar
and H.Krawinkler(eds),Seismic Design Methodology for the
Next Generation of codes,Balkema ,Rotterdam,1997,pp.183-
194.
17. P.Fajfar,” A nonlinear analysis method for performance-based
seismic design”Earthquake Spectra 2000,16(3):573-592.
18. P.Fajfar.”A Practical Nonlinear Method for Seismic
Performance Evaluation.” Proceedings of the 2000 Structures
Congress&Exposition,ed.M.Elgaaly,May 8-
10,2000,Philadelphia,Pennsylvania.
19. S.A.Freeman,”Development and use of capacity spectrum
method”.Proc.of 6th U.S.National Conference on Earthquake
Engineering,Sttle,CD-ROM,Oakland:EERI,1998.
20. S.A.Freeman,”The capacity Spectrum Method as a Tool for
Seismic Design”,Proc.11th European Conference on Earthquake
Engineering,Paris,CD-ROM,A.A.Balkema,Roterdam,1998.
21. H.Krawinkler and G.D.P.K.Seneviratna.”Pros and Cons of
Pushover analysis for seismic performance
evaluation”Engineering Structures.20,452-464(1998).
22. S.R.Lawson and H.Krawinkler. ”Nolinear static pushover
analysis-Why,when and how?”Proc.5th U.S. Conference on
Earthquake Engineering, Vol.1.Chicago,IL,1994,pp 283-292.
23. M.Saiidi and M.A.Sozen.”Simple nonlinear seismic analysis of
R/C structures”.Journal of the Structural
Division,ASCE,1981;107(ST5);937-951.