4.5.
Step-by-step procedure of proposed method................................................ 74
5.
VERIFICATION AND DISCUSSION ................................................................ 79
6.
CONCLUSION AND SUGGESTION ................................................................ 91 6.1.
Conclusion.................................................................................................... 91
6.2.
Suggestion .................................................................................................... 92
REFERENCE.............................................................................................................. 93 APPENDIX ................................................................................................................. 95 A.1.
MATLAB Code for extended N2 method for this research ......................... 95
LIST OF TABLES Table 2.1. Damping modification factor
(Ou, 2012)............................................12
Table 2.2. Structural behavior type (Ou, 2012).........................................................12 Table 2.3. Near source factors (Ou, 2012) ................................................................13 Table 2.4. Seismic source type (Ou, 2012) ...............................................................14 Table 2.5. Minimum allowable value for
and
........................................15
Table 2.6. Drift limits (Ou, 2012)..............................................................................17 Table 2.7. Values for modification factor C0 (FEMA-356, 2000) .............................19 Table 2.8. Values for effective mass factor Cm (FEMA-356, 2000)..........................20 Table 2.9. Values for modification factor
(FEMA-356, 2000) ...........................21
Table 2.10. Comparison result of each method ...........................................................34 Table 2.11. Difference action in Basic N2, FEMA 356 AND ATC-40 .......................35 Table 3.1. Details of members for each building ......................................................41 Table 3.2. Natural period of mode n of the building .................................................41 Table 3.3. Effective mass factor in x-direction about mode n ...................................42 Table 3.4. List of earthquake ground motion ............................................................44 Table 5.1. Calculation sheet for defining target top displacement for 14-storey building with 10% eccentricity. ................................................................80 Table 5.2. The displacement error resulted from basic N2, extended N2, and proposed method at center of mass ..........................................................82 Table 5.3. The coefficient of torsion error resulted from basic N2, extended N2, and proposed method................................................................................84 Table 5.4. The inter-storey drift error resulted from basic N2, extended N2, and proposed method at center of mass ..........................................................85 Table 5.5. The displacement error resulted from basic N2, extended N2, and proposed method at flexible edge.............................................................88 Table 5.6. The inter-storey drift error resulted from basic N2, extended N2, and proposed method at flexible edge.............................................................90
LIST OF FIGURES Figure 2.1. Elastic acceleration spectra: (a) Sa-T format; (b) AD format ...................8 Figure 2.2. Pushover analysis of a building (Ou, 2012) ..............................................9
−
Figure 2.3. Pushover curve (capacity curve) (Ou, 2012) ............................................9 Figure 2.4. Capacity curve: (a)
format; (b) Sa-Sd format (Ou, 2012) ..........10
Figure 2.5. Derivation of damping (Ou, 2012)..........................................................11
Figure 2.6. Damping modification factor (ATC-40, 1996) .......................................12 Figure 2.7. Reduction of 5% damped spectrum by
and
.........................15
Figure 2.8. Procedure to determine performance point (Ou, 2012) ..........................16 Figure 2.9. The acceptance criteria for performance objectives (FEMA-356, 2000) .......................................................................................................17
Figure 2.10 Idealized Force-Displacement Curves (FEMA-356, 2000) ....................18 Figure 2.11. Figure 2.12. Figure 2.13.
1 values (FEMA-356, 2000) ................................................................20 2 from Table 2.9 (FEMA-356, 2000) ..................................................21 2 from nonlinear response history analysis (FEMA-356, 2000) ..........22
Figure 2.14. EPP and SD hysteretic models ................................................................22 Figure 2.15.
3 values (FEMA-356, 2000)................................................................23
Figure 2.16. Building data and Elastic acceleration spectra (Fajfar, 2000).................24 Figure 2.17. Elastic and inelastic response spectra for constant ductility (Fajfar, 1999) .......................................................................................................25 Figure 2.18. Idealized bilinear capacity curve with zero post-yielding stiffness and transformation from base shear-displacement format to Sa-Sd
∗
format (Fajfar, 1999) ...............................................................................27
∗ ≥
Figure 2.19. Determination of displacement demand,
: (a)
<
; (b)
(Fajfar, 2000) ...........................................................................27
Figure 2.20. Simple version of ductility factor (Fajfar, 1999) ....................................29 Figure 2.21. The plan view of the building model ......................................................30 Figure 2.22. Performance point of ATC-40 .................................................................31 Figure 2.23. Defining the target displacement by FEMA 356 ....................................32 Figure 2.24. Target displacement by basic N2 method ...............................................34 Figure 2.25. The displacement shape of each method ................................................35
Figure 2.26. Example of correction factor for higher mode effect in elevation, C E (Kreslin & Fajfar, 2011)..........................................................................36 Figure 2.27. Example of correction factor for higher mode effect in plan, C T (Kreslin & Fajfar, 2011)..........................................................................37 Figure 2.28. Response spectra: (a) Original response spectra from the ground motions; (b) Compatible response spectra from the compatible ground motions .......................................................................................37 Figure 3.1. Plan view and elevation view: (a) 2-storey (b) 8-storey; (c) 14-storey; (d) 20-storey............................................................................................40 Figure 3.2. Response spectra: (a) Original response spectra from the ground motions; (b) Compatible response spectra from the compatible ground motions .......................................................................................43
Figure 4.1. Displacement result for 2-storey building: (a) 0% eccentricity 0.1g
( = 0.46 ) ; (b) 0% eccentricity 0.4g ( = 1.86 ) ; (c) 5% eccentricity 0.1g
= 0.48 ; (d) 5% eccentricity 0.4g ( = 1.904 );
(e) 10% eccentricity 0.1g ( = 0.5 ) ; (f) 10% eccentricity 0.4g
( = 1.99 ) ; (g) 15% eccentricity 0.1g ( = 0.52 ) ; (h) 15% eccentricity 0.4g ( = 2.08 ) .................................................................46 Figure 4.2. Displacement result for 2-storey building: (a) 0% eccentricity 0.6g
( = 2.78 ); (b) 0% eccentricity 1g ( = 4.64 ); (c) 5% eccentricity 0.6g
= 2.86 ; (d) 5% eccentricity 1g ( = 4.76 ) ; (e) 10%
eccentricity 0.6g ( = 2.98 ); (f) 10% eccentricity 1g ( = 4.96 ); (g) 15% eccentricity 0.6g ( = 3.13 ); (h) 15% eccentricity 1g
( = 5.21 ) .............................................................................................47 Figure 4.3. Displacement result for 8-storey building: (a) 0% eccentricity 0.1g
( = 0.85 ) ; (b) 0% eccentricity 0.4g ( = 3.4 ) ; (c) 5% eccentricity 0.1g
= 0.85 ; (d) 5% eccentricity 0.4g ( = 3.4 ); (e)
10% eccentricity 0.1g ( = 0.84 ) ; (f) 10% eccentricity 0.4g
( = 3.36 ) ; (g) 15% eccentricity 0.1g ( = 0.83 ) ; (h) 15% eccentricity 0.4g ( = 3.31 ) .................................................................48 Figure 4.4. Displacement result for 8-storey building: (a) 0% eccentricity 0.6g
( = 5.1 ); (b) 5% eccentricity 0.6g
= 5.1 ; (c) 10% eccentricity
0.6g ( = 5.04 ); (d) 15% eccentricity 0.6g ( = 4.96 ) ....................49
Figure 4.5. Displacement result for 20-storey building: (a) 0% eccentricity 0.1g
( = 0.38 ) ; (b) 0% eccentricity 0.4g ( = 1.51 ) ; (c) 5% eccentricity 0.1g
= 0.38 ; (d) 5% eccentricity 0.4g ( = 1.51 );
(e) 10% eccentricity 0.1g ( = 0.38 ); (f) 10% eccentricity 0.4g
( = 1.5 ) ; (g) 15% eccentricity 0.1g ( = 0.37 ) ; (h) 15% eccentricity 0.4g ( = 1.48 ) .................................................................50 Figure 4.6. Displacement result for 20-storey building: (a) 0% eccentricity 0.6g
( = 2.27 ); (b) 0% eccentricity 1g ( = 3.79); (c) 5% eccentricity = 2.27 ; (d) 5% eccentricity 1g ( = 3.78 ) ; (e) 10%
0.6g
eccentricity 0.6g ( = 2.25 ); (f) 10% eccentricity 1g ( = 3.75 ); (g) 15% eccentricity 0.6g ( = 2.22 ); (h) 15% eccentricity 1g
( = 3.71 ) .............................................................................................51 Figure 4.7. Displacement result for 20-storey building: (a) 0% eccentricity 1.4g
(μ = 5.3 ); (b) 5% eccentricity 1.4g μ = 5.29 ; (c) 10% eccentricity 1.4g (μ = 5.25 ); (d) 15% eccentricity 1.4g (μ = 5.19 ) .....................51
Figure 4.8. Coefficient of torsion result for 2-storey building: (a) 5% eccentricity 0.1g
= 0.48 ; (b) 5% eccentricity 0.4g ( = 1.904 ); (c) 10%
eccentricity 0.1g ( = 0.5 ); (d) 10% eccentricity 0.4g ( = 1.99 );
(e) 15% eccentricity 0.1g ( = 0.52 ); (f) 15% eccentricity 0.4g
( = 2.08 ) .............................................................................................53
Figure 4.9. Coefficient of torsion result for 2-storey building: (a) 5% eccentricity 0.6g
= 2.86 ; (b) 5% eccentricity 1g ( = 4.76 ) ; (c) 10%
eccentricity 0.6g ( = 2.98 ); (d) 10% eccentricity 1g ( = 4.96 );
(e) 15% eccentricity 0.6g ( = 3.13 ) ; (f) 15% eccentricity 1g
( = 5.21 ) .............................................................................................53
Figure 4.10. Coefficient of torsion result for 8-storey building: (a) 5% eccentricity 0.1g
= 0.85 ; (b) 5% eccentricity 0.4g ( = 3.4 ) ; (c) 10%
eccentricity 0.1g ( = 0.84 ) ; (d) 10% eccentricity 0.4g ( =
3.36 ); (e) 15% eccentricity 0.1g ( = 0.83 ); (f) 15% eccentricity 0.4g ( = 3.31 ) ....................................................................................54 Figure 4.11. Coefficient of torsion result for 8-storey building: (a) 5% eccentricity
0.6g
= 5.1 ; (b) 10% eccentricity 0.6g ( = 5.04 ) ; (c) 15%
eccentricity 0.6g ( = 4.96 ) .................................................................55
Figure 4.12. Coefficient of torsion result for 20-storey building: (a) 5% eccentricity 0.1g
= 0.38 ; (b) 5% eccentricity 0.4g ( = 1.51 );
(c) 10% eccentricity 0.1g ( = 0.38 ); (d) 10% eccentricity 0.4g
( = 1.5 ) ; (e) 15% eccentricity 0.1g ( = 0.37 ) ; (f) 15% eccentricity 0.4g ( = 1.48 ) .................................................................56
Figure 4.13. Coefficient of torsion result for 20-storey building: (a) 5% eccentricity 0.6g ( = 2.27); (b) 5% eccentricity 1g ( = 3.78 );
(c) 10% eccentricity 0.6g ( = 2.25 ) ; (d) 10% eccentricity 1g
( = 3.75 ) ; (e) 15% eccentricity 0.6g ( = 2.22 ) ; (f) 15% eccentricity 1g ( = 3.71 ) ....................................................................57
Figure 4.14. Coefficient of torsion result for 20-storey building: (a) 5% eccentricity 1.4g ( = 5.29) ; (b) 10% eccentricity 1.4g ( = 5.25 ); (c) 15% eccentricity 1.4g ( = 5.19 )....................................................57
Figure 4.15. Drift result for 2-storey building: (a) 0% eccentricity 0.1g
( = 0.46 ) ; (b) 0% eccentricity 0.4g ( = 1.86 ) ; (c) 5% eccentricity 0.1g
= 0.48 ; (d) 5% eccentricity 0.4g ( = 1.904 );
(e) 10% eccentricity 0.1g ( = 0.5 ) ; (f) 10% eccentricity 0.4g
( = 1.99 ) ; (g) 15% eccentricity 0.1g ( = 0.52 ) ; (h) 15% eccentricity 0.4g ( = 2.08 ) .................................................................58 Figure 4.16. Drift result for 2-storey building: (a) 0% eccentricity 0.6g
( = 2.78 ); (b) 0% eccentricity 1g ( = 4.64 ); (c) 5% eccentricity 0.6g ( = 2.86 ) ; (d) 5% eccentricity 1g ( = 4.76 ) ; (e) 10% eccentricity 0.6g ( = 2.98 ); (f) 10% eccentricity 1g ( = 4.96 ); (g) 15% eccentricity 0.6g ( = 3.13 ) ; (h) 15% eccentricity 1g
( = 5.21 ) ..............................................................................................59 Figure 4.17. Drift result for 8-storey building: (a) 0% eccentricity 0.1g
( = 0.85 ); (b) 0% eccentricity 0.4g ( = 3.4 ); (c) 5% eccentricity 0.1g
= 0.85 ; (d) 5% eccentricity 0.4g ( = 3.4 ) ; (e) 10%
eccentricity 0.1g ( = 0.84 ); (f) 10% eccentricity 0.4g ( = 3.36 ); (g) 15% eccentricity 0.1g ( = 0.83 ); (h) 15% eccentricity 0.4g
( = 3.31 ) ..............................................................................................60 Figure 4.18. Drift result for 8-storey building: (a) 0% eccentricity 0.6g ( = 5.1 ); (b) 5% eccentricity 0.6g (
= 5.1 ) ; (c) 10% eccentricity 0.6g
( = 5.04 ); (d) 15% eccentricity 0.6g ( = 4.96 ) ...............................61 Figure 4.19. Drift result for 20-storey building: (a) 0% eccentricity 0.1g
( = 0.38 ) ; (b) 0% eccentricity 0.4g ( = 1.51 ) ; (c) 5% eccentricity 0.1g ( = 0.38) ; (d) 5% eccentricity 0.4g ( = 1.51 ); (e) 10% eccentricity 0.1g ( = 0.38 ); (f) 10% eccentricity 0.4g
( = 1.5 ) ; (g) 15% eccentricity 0.1g ( = 0.37 ) ; (h) 15% eccentricity 0.4g ( = 1.48 ) .................................................................62 Figure 4.20. Drift result for 20-storey building: (a) 0% eccentricity 0.6g
( = 2.27 ); (b) 0% eccentricity 1g ( = 3.79); (c) 5% eccentricity 0.6g ( = 2.27 ) ; (d) 5% eccentricity 1g ( = 3.78 ) ; (e) 10% eccentricity 0.6g ( = 2.25 ); (f) 10% eccentricity 1g ( = 3.75 ); (g) 15% eccentricity 0.6g ( = 2.22 ) ; (h) 15% eccentricity 1g
( = 3.71 ) ..............................................................................................63 Figure 4.21. Drift result for 20-storey building: (a) 0% eccentricity 1.4g
( = 5.3 ) ; (b) 5% eccentricity 1.4g ( = 5.29 ); (c) 10% eccentricity 1.4g ( = 5.25 ); (d) 15% eccentricity 1.4g ( = 5.19 ) ..64 Figure 4.22. Linear assumed displacement vs proposed displacement shape in 20-storey building 5% eccentricity: (a) load pattern; (b) assumed displacement shape .................................................................................66 Figure 4.23. Reduction factor to the target top displacement considering the torsion resulted by NRHA ......................................................................68 Figure 4.24. The multiplier or weight factor of CT value resulted from NRHA: (a) CTt PO multiplier; (b) CTtRSA multiplier................................................69 Figure 4.25. The proposed multiplier of PO and RSA to calculate the final
≈
coefficient of torsion ...............................................................................70 Figure 4.26. The “whip effect” on chi-chi earthquake: 20-storey 1.4g μ
5............71
Figure 4.27. The max storey displacement versus the max drift displacement in 0% eccentricity building: (a) 2-storey 0.4g; (b) 8-storey 0.4g; (c) 20-storey 0.4g .........................................................................................71
Figure 4.28. The value of max drift displacement divided by the max storey displacement in the top floor: (a) NRHA result; (b) proposed γr . ..........73 Figure 4.29. The value of decomposed max drift displacement divided by the max storey displacement: (a) 2-storey by ground motion; (b) 2-storey by proposed coefficient,γ j ;(c) 8-storey by ground motion; (d) 8-storey by proposed coefficient, γ j ;(e) 20-storey by ground motion; (f) 20-storey by proposed coefficient,γ j .......................................................74 Figure 4.30. Elastic acceleration spectra: (a) Sa-T format; (b) AD format .................75
Figure 4.31. Flowchart of the proposed method .........................................................78
∅
Figure 5.1. (a) Normalized lateral force pattern shape
; (b) assumed displacement
.................................................................................................80
Figure 5.2. Graphical way to obtain the target top displacement of SDOF of 14-storey
building,
10%
eccentricity,
0.6g
resulted
by
1 st
modification of proposed method. ..........................................................80 Figure 5.3. Displacement result at center of mass for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g ........................................................................82 Figure 5.4. Coefficient of torsion result for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g..............................................................................83 Figure 5.5.
Inter-storey
drift
result
at
center
of
mass
for
14-storey
10%-eccentricity: (a) pga=0.6g; (b) pga=1g ...........................................85 Figure 5.6. Displacement result at flexible edge for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g ........................................................................87 Figure 5.7. Inter-storey
drift
result
at
flexible
edge
for
14-storey
10%-eccentricity: (a) pga=0.6g; (b) pga=1g ...........................................89
Figure A.1. Extended N2 result from the MATLAB code .......................................105
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1. INTRODUCTION 1.1.Background and Research Motivation
Nowadays, seismic design criteria tend to shift from the force-based procedure to performance-based procedure both for design and evaluation purpose. Pushover analysis becomes very well known method in determining seismic demand because of its simplicity and accuracy for short and symmetric or 1st mode dominant building. Pushover analysis estimates seismic demands directly from the earthquake design spectrum and capacity curve, excluding the complications to select and to scale the ground motions. In the other side, Nonlinear Response History Analysis (NRHA) typically demands high computational resources. For tall and asymmetric building, traditional pushover analysis cannot come up with higher mode effect both in elevation and in plan (torsional effect), respectively. Moreover, traditional pushover (ATC-40 and FEMA 356) requires an iteration process to get the performance point or target displacement. A lot of researches have been done to make the traditional pushover can give better result and retain its simplicity. (Chopra & Goel, 2002, 2004; Chopra, Goel, & Chintanapakdee, 2004; Fajfar, 1999, 2000; Kreslin & Fajfar, 2011; Kunnath, 2004; ̆ ić & Fajfar, 2005; R. Rofooei, K. Attari, Rasekh, & Shodja, 2006; Reyes & Marus Chopra, 2011; Rofooei, Attari, Rasekh, & Shodja, 2007). The previous methods are time consuming and there is an assumption that higher mode effect will remain in elastic behavior. In the other side, N2 method has a simple way (no need iteration) to get the target displacement which uses the inelastic response spectra and the capacity curve to get the target displacement (Fajfar, 1999, 2000). The extended N2 method has the same way as the basic N2 to calculate the target displacement. The extension is the usage of response spectrum analysis (RSA) as the correction to the seismic demands which assume that the higher mode effect will keep in the elastic behavior (Kreslin & ̆ ić & Fajfar, 2005). Though N2 method has its simplicity and has Fajfar, 2011; Marus been extended, but it usually still conservative in determining the coefficient of torsion in large earthquake and in some cases result unconservatism in determining
1
the drift. By explanation explained above, a proposed method is introduced to come up with the higher mode effect problems using the pushover analysis. The proposed method also uses the inelastic response spectra like the N2 method does to keep the simplicity, i.e. no need iteration in obtaining the target displacement.Additional 4 modifications are made to improve the PO such the seismic demands will approach the real behavior. The modifications are made to consider the higher mode effect and based on the behavior of the real building. NRHA is assumed to be similar with the real behavior of the building. Three different buildings which are 2-storey, 8-storey and 20-storey reinforced concrete frame building with 0%, 5%, 10%, and 15% eccentricity and with several pga value are taken as the source of database for finding the behavior. By these modifications, there is no need to assume that the higher mode effect will keep in elastic behavior. A 14-storey reinforced concrete frame building with NRHA, basic N2, extended N2, and the proposed method is established to check the competence of the proposed method. Several seismic demands are taken into account, which are displacement and drift at center of mass and at the flexible edge
1.2.Objectives and scopes
The objective of this research is to find the simplest and the most accurate method to approach the real behavior. This objective is accomplished by several small modifications: 1. New lateral load pattern which takes account the contribution of higher mode effect in elevation 2. Adjusted target displacement which considers the higher mode effect in plan (torsional effect) based on the real behavior 3. Adjusted coefficient of torsion based on the real behavior 4. Adjusted inter-storey displacement to calculate inter-storey drift based on the real behavior Three different buildings which are 2-storey, 8-storey and 20-storey reinforced concrete frame building with 0%, 5%, 10%, and 15% eccentricity and with several pga value of ground motions are taken as the source of database for finding the real
2
behavior.
1.3. Outline
This thesis is divided into 6 chapters as follows:
Chapter 1 gives general introduction about this research, including background and research motivation, objectives and scopes, and outline.
Chapter 2 describes briefly literatures which are related to this research.
Chapter 3 shows buildings example and ground motions which are used in analytical study and verification study.
Chapter 4 describes the result of NRHA, basic N2, and extended N2, including the modifications that should be applied based on the analytical study and step by step procedure.
Chapter 5 contains verification of proposed method in a 14-storey building and the comparison with the NRHA result, basic N2, and extended N2 method.
Chapter 6 describes conclusions and suggestions of this study.
3
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4
2. LITERATURE REVIEW 2.1. Previous research
There are a lot of researches done to modify the pushover such approach the NRHA result. Some of them are described as follows:
2.1.1. Modal Pushover Analysis (MPA)
Modal Pushover Analysis (MPA) combines the pushover analysis with uncouple modal response history analysis. Several dominant modes are used separately as the lateral load for the pushover analysis with target displacement defined from the uncouple response history analysis result to an inelastic SDOF system (Chopra & Goel, 2002, 2004). This method gives good result compare with NRHA, but it is a time consuming method because need to run the uncouple modal response history analysis in order to get the target displacement for each mode. The short description of this method is described as follows:
− ϕ −
The governing differential equations of the response of MDOF system subjected
( ) are presented in Eqs.(2.1) to (2.3).
to horizontal ground motion
+
+
=
=
=
=1
=
;
(2.1) (2.2)
=1
=
;
=
where is lateral floor displacement relative to the ground;
(2.3)
, , are the mass,
damping, and stiffness matrices, respectively; is influence vector of each element which equal to 1;
is the structural natural vibration of nth mode.
For „exact‟ NRHA, because history of the displacement controls the next displacement, thus the relation between the lateral force the lateral displacements
at the N floor levels and
are not single-valued. Therefore, Eq.(2.1) becomes:
+
+
,
=
(2.4)
By neglecting the coupling of the N equations, Eqs.(2.5) to (2.7) will become the governing equation of uncoupled modal response history analysis ( UMRHA). 5
− − +
+
+2
=
,
+
,
=
(2.5)
=
(2.6)
=
,
(2.7)
2.1.2. Modified Modal Pushover Analysis (MMPA)
Modified Modal Pushover Analysis (MMPA) takes account the higher mode effect by assuming that the behaviour of higher mode will remain in elastic state. By this assumption, clasical modal analysis for linear system is used to take account the higher mode effect. Therefore no need to perform the pushover analysis for higher mode. NRHA is used to get the target displacement of the inelastic first mode which in turn will be used in the pushover analysis. This method is simpler than MPA, but will result larger error in larger degree of inelastic action(Chopra et al., 2004)
2.1.3. Practical Modal Pushover Analysis (PMPA)
Practical Modal Pushover Analysis (PMPA) is similar with the MMPA with additional simplification to determine the target displacement of inelastic first mode. The target displacement of inelastic first mode is obtained by multiplying the median target displacement of linear system with inelastic deformation ratio (Reyes & Chopra, 2011). This method has good prediction of the seismic demand and will be similar as RSA result in linear system, and no need to run the NRHA, but still need to select the ground motions and run many of linear dynamic analysis to get the median elastic target displacement.
2.1.4. Method of Modal Combination (MMC)
Method of Modal Combination (MMC) tries to combine several modes by adding or reducing the contribution of different mode in determining the lateral forces. This method will have so many alternative lateral force patterns and will consume much time (Kunnath, 2004).
6
2.1.5. Adaptive Pushover (APO)
Adaptive Pushover (APO) tries to change the load pattern in every step by following the displacement pattern from the previous step. The load pattern will be changed as many as the step required in pushover analysis. This cause the method also becomes time consuming (Rofooei et al., 2007).
2.1.6. Dynamic pushover with SRM load pattern
Combination of effective modal load pattern and NRHA in SDOF system to get the target displacement (Dynamic pushover with SRM load pattern) tries to take account the higher mode effect in the loading pattern by including the higher mode with effective modal mass as the multiplication factor, and use the NRHA to get the target displacement (R. Rofooei et al., 2006)
2.2. ATC-40
ATC-40 (ATC-40, 1996) uses peak roof displacement of the building to determine the performance of the building subjected to earthquake ground motion. Combination between capacity curve and demand response spectrum with some iteration process are used to get the performance point or target displacement. ATC-40, called Capacity Spectrum method, requires AD format for both of capacity curve and demand spectra. The steps of ATC-40 are described as follows: 1. Convert a demand response spectrum found in the building codes from the standard
S a
(Spectra Acceleration) - T (Period) format to AD format as shown
in Figure 2.1and Eq. (2.8)
2
=
4
7
2
(2.8)
1.2
1.2
) g ( n o i t0.8 a r e l e c c a l a r0.4 t c e p S
) g ( n o i t0.8 a r e l e c c a l a r0.4 t c e p S
0
0
1
2 3 Period (s)
4
0
5
0
20 40 60 80 Spectral displacement (cm)
(a)
(b)
Figure 2.1. Elastic acceleration spectra: (a) Sa-T format; (b) AD format
2. Perform pushover analysis and generate the relationship between roof displacement
and base shear
V
(pushover capacity curve). This is
illustrated in Figure 2.2 and Figure 2.3. The lateral story forces applied to the structure is in proportion to the product of the mass and first mode shape which is described in Eq. (2.9). Alternatively, Eq.(2.10) can be used to determine the lateral story force pattern. Gravity loads should be included in this analysis. Subsequently, convert the base shear-displacement format to the
ϕ ϕ ϕ ϕ
Sa-Sd format.
=
(2.9)
=1
=
(2.10)
=1
where and
is the lateral force at level x;
or
is the weight at level x or j;
or is the displacement at level x or j corresponding to first mode
shape of the structure;
or
is the height from the base to level x or j; For
structures having a period of 0.5 seconds or less, k = 1; For structures having a period of 2.5 seconds or more, k = 2; For structures having a period between 0.5 and 2.5 seconds, k can be determined by linear interpolation between 1 and 2 or may be taken equal to 2.
8
F 4
Level 4
F 3
Level 3
F 2
Level 2
F 1
Level 1
Figure 2.2. Pushover analysis of a building (Ou, 2012)
200
) f t ( r a e h s e s a B
160 120 80 40 0 0
10 20 30 Roof displacement (cm)
40
Figure 2.3. Pushover curve (capacity curve) (Ou, 2012)
(a) Convert points
(roof displacement) into
S d
Eq.(2.11) is used to convert the roof displacement coordinate.
into the
Γ ∙ Γ ∙∙ =
1
S d
(2.11)
,1
where 1 is modal participation factor for the first mode and is defined by Eq. (2.12)
1
=
=1 =1
(b) Convert points
V
(base shear) into
1 2 1
S a
Eqs.(2.13) and (2.14) is used to convert the base shear coordinate.
9
(2.12)
V
into the
S a
∙∙ =
(2.13)
1
2
1
=1
=
=1
) f t ( r a e h s e s a B
1 2 1
200
0.5
160
0.4
120 80
) 0.3 g ( a S
40
0.1
(2.14)
0.2
0
0 0
10 20 30 Roof displacement (cm) (a)
Figure 2.4. Capacity curve: (a)
40
−
0
5
10 15 Sd (cm)
20
(b)
format; (b) Sa-Sd format (Ou, 2012)
3. Estimation of damping and reduction of 5 percent damped response spectrum The damping that occurs when an earthquake ground motion shakes the structure to the inelastic range can be defined as a combination of hysteretic damping and viscous damping that is natural in the structure. Hysteretic damping is defined as the area inside the loops that are formed when the base shear is plotted in opposition to the structure displacement as
−
shown in Figure 2.5.
The equivalent viscous damping, displacement of
, correlated with a maximum
can be estimated from the Eqs. (2.15) to (2.18)
=
0
=
0
+ 0.05
1
4
=
(2.16)
0
= 4( 0
(2.15)
)
(2.17)
/2
(2.18)
where 0 is hysteretic damping represented as equivalent viscous damping;
and 0.05 is 5% viscous damping which is natural in the structure (assumed to be constant);
is energy dissipated by damping; and
10
0
is maximum strain
25
energy as shown in Figure2.5.
Figure 2.5. Derivation of damping (Ou, 2012)
Figure 2.5 shows an idealized hysteresis loop which is reasonable for a ductile detailed building subjected to relatively short duration ground motion (not enough cycles to extensively degrade elements) and with equivalent viscous damping less than approximately 30%. For non ductile buildings, calculation of the equivalent viscous damping using Eq. (2.15)and the idealized hysteresis loop in Figure 2.5will overestimate the realistic value of
damping. In order to consider imperfect hysteresis loops (loops reduced in area), a damping modification factor, , is used in Eq. (2.19)
The
=
0
+ 0.05
(2.19)
-factor is listed in Table 2.1 and Table 2.2. Moreover, it is shown in
Figure 2.6.The
-factor depends on the structural behavior of the building, i.e.
the quality of the seismic resisting system and the duration of ground shaking.
For easiness, ATC-40 defines three categories of structural behavior. Structural behavior Type A represents stable, reasonably full hysteresis loops
11
most similar to Figure 2.5, and is assigned a
of 1.0 (except at higher
damping values). Type B represents a moderate reduction of area and is assigned a basic
of 2/3 ( is also reduced at higher values of
to be
consistent with the Type A relationships). Type C represents poor hysteretic
behavior with a substantial reduction of loop area (severely pinched) and is assigned a
of 1/3.
Table 2.1. Damping modification factor Structural behavior type
0
16.25
16.25
Type A
(Ou, 2012)
1.0 1.13-
0 . 5 1 a y d
d yap
p
a p d p
25
25
Type B
0.67 0.845-
0 . 4 4 6 a y d
p
d yap
a p d p
Type C
Any value
0.33
Figure 2.6. Damping modification factor (ATC-40, 1996)
Table 2.2. Structural behavior type (Ou, 2012) Shaking duration(a) Short Long
Essentially new building(b) Type A Type B 12
Average existing building(c) Type B Type C
Poor existing building(d) Type C Type C
(a) Shaking duration
Sites with a near-source factor,
N
1.2 (see Table 2.3 and Table 2.4),
may be assumed to have short-duration ground shaking. (sites near a seismic source (fault), a relatively short duration of very strong shaking would be expected )
Sites located in seismic zone 3 should be assumed to have long duration ground shaking. (sites far from fault rupture, a much longer duration of ground shaking would be expected at the level of response described by the site spectrum. Although, ground shaking is not as strong the previous case, longer duration of shaking increases the potential for degradation of the structural system)
Sites located in seismic zone 4 (with a near-source factor of N <1.2) should be assumed to have long-period ground shaking.
Long duration ground shaking should be assumed for soft soil sites.
(b) Essentially new building: buildings whose primary elements make up an essentially new lateral system and little strength or stiffness is contributed by noncomplying elements. (c) Average existing building: buildings whose primary elements are combinations of existing and new elements, or better than average existing systems. (d) Poor existing building: buildings whose primary elements make up noncomplying lateral force systems with poor or unreliable hysteretic behavior.
Table 2.3. Near source factors (Ou, 2012) Seismic source type A B C
Closest distance to known seismic sources
5km
2km
10km
15km
N A
N V
N A
N V
N A
N V
N A
N V
1.5 1.3 1.0
2.0 1.6 1.0
1.2 1.0 1.0
1.6 1.2 1.0
1.0 1.0 1.0
1.2 1.0 1.0
1.0 1.0 1.0
1.0 1.0 1.0
13
Table 2.4. Seismic source type (Ou, 2012) Seismic source definition Maximum Slip rate, SR moment (mm/year) magnitude
Seismic source type
Seismic source description
A
Faults that are capable of producing large magnitude events and Which have a high rate of seismic activity
B
All faults other than types A and C
C
Faults that are not capable of producing large magnitude earthquakes and that have a relatively low rate of seismic activity
M 7.0
SR 5
Not applicable
Not applicable
M 6.5
SR<2
− −
Taking the damping into account, the response spectrum is reduced by reduction factors (Note: input
and
=10 for 10% damping)
=
=
Both
which are described in Eqs. (2.20) and (2.21)
and
1
1
=
=
3.21
0.68ln (
)
2.12
2.31
0.41ln (
)
(2.20) (2.21)
1.65
must be greater than or equal to allowable values in
Table 2.5. The elastic response spectrum (5% damped) is thus reduced to a response spectrum with damping values greater than 5% critically damped (See Figure 2.7).
14
1.6
S D S
1.2
Elastic response spectrum, 5% damped
) g ( n o i t a r e l e c c a l a r t c e p S
S D S S R A 0.8
S D 1 T
0.4
S D 1 S R V T
Elastic response spectrum, 15% damped
0 0
1
2 Period (s)
3
Figure 2.7. Reduction of 5% damped spectrum by
Table 2.5. Minimum allowable value for Structural behavior type Type A Type B Type C
S R A
4
and
and
S R V
0.33 0.44 0.56
0.50 0.56 0.67
4. Performance point determination The procedure for obtaining the performance point is described as follows: (a) Draw the capacity curve and 5% damped ADRS response spectrum in the same figure. (b) Define a trial performance point displacement approximation.
,
. This is done using the equal
(c) A bilinear approximation of the capacity curve is made such that the area
under the capacity curve and the bilinear representation is the same. (d) The spectral reduction factors
and
are computed and the
demand spectrum is reduced. The reduced demand spectrum is plotted together with the capacity spectrum. i. If the reduced demand spectrum intersects the capacity spectrum at
or if the intersection point
15
,
is within 5% of S dp i , then this point
≤
represents the performance point.
ii. If the intersection point does not lie within acceptable tolerance ( 5% of
) then select another point and repeat steps (c) to (d). The
intersection point obtained in Step (d) can be used as the starting point for the next iteration.
Figure 2.8. Procedure to determine performance point (Ou, 2012)
5. Check for response limits Right after performance point
,
is obtained, the base shear ( V p ) and
roof displacement ( p ) at the performance point are calculated by Eqs. (2.13) and (2.11), respectively. Then, check the global building response and local element response. The global building responses consist of checking whether the lateral force resistance has degraded by more than 20% of the peak resistance and the lateral drift limits should satisfy the limits given in the Table 2.6. The local element response, identify the critical components and check as detailed in Chapter 11 of ATC-40
16
Table 2.6. Drift limits (Ou, 2012) Inter-story drift limit
Immediate occupancy
Maximum total drift
0.01
Performance level Damage Life safety control 0.01-0.02
0.33
0.02
Maximum 0.005 0.005-0.015 No limit inelastic drift where V is the total calculated lateral shear force in story i and i
Structural stability V i P i
No limit P i
is the total
gravity load (i.e. dead plus likely live load) at st ory i.
2.3. FEMA 356
FEMA 356 (FEMA-356, 2000) is firstly established for retrofitting of existing structures, but the procedures are equally applicable for new design. The owner decides the expected performance of the building, and then the engineer designs a new or retrofit structure to achieve the performance objective. The performance objectives are divided into 4 categories which are operational performance, immediate occupancy (IO), life safety (LS), and collapse prevention (CP). Each performance is divided into 2 parts which are primary and secondary member. The limitation of each category is drawn in Figure 2.9.
Figure 2.9.The acceptance criteria for performance objectives (FEMA-356, 2000) The most common lateral load patterns are inverted triangular, uniform, or first mode load pattern. The capacity curve resulted from the pushover analysis is transformed into an idealized force-displacement curve explained as follows (see Figure 2.10): 1. The first linear line of the idealized force-displacement curve begins from the origin.
17
2. The second linear line ends at the calculated target displacement. 3. The meeting point of the two linear line defines effective lateral stiffness (
K e
),effective yield strength (
stiffness(
K e
V y
), and effective positive post-yield
).The intersection point is determined by satisfying following
constraints: The effective stiffness,
K e
, is defined such the first line passes through the
calculated curve at a point where the base shear is 60% of the effective yield strength or 60% of intersection point. The areas below the idealized bilinear curve should be approximately equal
with the areas below the actual capacity curve with the same ending point. The effective yield strength should not be taken as greater than the maximum
base shear force at any point along the actual curve.
(a) Positive post-yield slop
(b) Negative post-yield slop
Figure 2.10. Idealized Force-Displacement Curves (FEMA-356, 2000)
FEMA 356 or coefficient method modifies the linear elastic response of the equivalent SDOF system by giving a series of coefficients C0 toC3 to generate an estimation of maximum global displacement (elastic and inelastic), which is termed as
target displacement. The top target displacement c an be calculated by Eq. (2.22) 2
=
where: 0
0 1 2 3
4
2
(2.22)
is modification factor to transform the spectral displacement of an equivalent SDOF system to the roof displacement of the building MDOF system. It can be
18
calculated from
The first modal participation factor
The modal participation factor at the level of the control node calculated using a shape vector corresponding to the deflected shape of the building at the target displacement.
The appropriate value from Table 2.7.
Table 2.7. Values for modification factor C 0 (FEMA-356, 2000)
1
is modification factor to adjust the displacements calculated for the linear elastic response to the expected maximum displacements of an inelastic SDOF oscillator
≥ −
with EPP hysteretic properties
1
1
=
1
1+
(2.23)
<
=
where
/
(2.24)
is characteristic period of the response spectrum, defined as the
transition period from the constant-acceleration segment to the constant-velocity segment of the spectrum; strength demand; capacity curve;
is ratio of elastic strength demand to reduced elastic
is yield strength calculated using the idealized bilinear
is effective seismic weight, consist of total dead load and
some portions of other gravity loads as calculated in Section 3.3.1.3.1 of FEMA 356;
is effective mass factor to account for higher mode mass participation
effects as obtained from Table 2.8.
19
Table 2.8.Values for effective mass factor C m (FEMA-356, 2000)
The
1
≥
should not greater than the values given in Section 3.3.1.3.1 (Linear
Static Procedure, LSP section) nor less than 1. Values of are
1
=
1 1.5
1 for
for
(2.25)
the intermediate values of
Figure 2.11(a) shows an example of values of and periods
Section3.3.1.3.1
< 0.1
with linear interpolation used to calculate
various
1 in
1
T e
.
according to Eq. (2.23)for
equal to 0.4 seconds. Limitation on values
1
according to Eq. (2.25)is also shown in Figure 2.11(a). Figure 2.11(b) shows values of of
1.
1 from
It can be seen that Eq. (2.23) basically captures the behavior of
and
nonlinear response history analysis, representing true values
.
1
versus
Eq. (2.25)
(a)
1
from Eq. (2.23)
Figure 2.11.
(b)
1 values
20
1
from nonlinear response history analysis
(FEMA-356, 2000)
2
is modification factor to take into account the effect of pinched hysteretic shape,
stiffness degradation, and strength deterioration on the maximum displacement response. Values of
2 depends
on framing systems and structural performance
levels (i.e., immediate occupancy, life safety, and collapse prevention) and listed
in Table 2.9.Figure 2.12shows an example of values of values of
2 from
2.
Figure 2.13shows the
nonlinear response history analysis for ratios of maximum
displacements of SD (stiffness degrading) systems to corresponding EPP (elastic perfectly plastic) systems. Hysteretic behaviors of these two systems are shown in Figure 2.14.
Table 2.9. Values for modification factor
2
(FEMA-356, 2000)
Note: T is fundamental period of the building determined using Eigen value analysis or empirical equations (Section 3.3.1.2 of FEMA 356).
Figure 2.12.
2
from Table 2.9 (FEMA-356, 2000)
21
Figure 2.13.
2
from nonlinear response history analysis (FEMA-356, 2000)
Figure 2.14. EPP and SD hysteretic models
3
is modification factor to take into account the dynamic P-Δ effects.
−
3
is equal to
1 for buildings with positive post-yield stiffness. For buildings with negative post-yield stiffness, values of
where
3
3
arecalculated using Eq. (2.26)
1
=1+
3/2
is post-yield stiffness.Figure2.15 shows the examples of
22
3
(2.26) values.
(a) An Example of
3
by Eq. (2.27)
Figure 2.15.
3
(b) Values of
3 from
nonlinear
response history analysis
values (FEMA-356, 2000)
is response spectrum acceleration at the effective fundamental period of the building.
is gravitational acceleration. is effective fundamental period of the building which is defined in Eq. (2.27)
(2.27)
=
where
analysis;
is elastic fundamental period (in seconds) calculated by elastic dynamic is elastic lateral stiffness of the building; and
stiffness of the building as defined previously.
is effective lateral
2.4. Basic N2 method
N2 method is a variant of capacity spectrum method established in Europe by Peter Fajfar. The name N2 consists of N and 2 which is defined as nonlinear analysis and two mathematical models, respectively. The conventional capacity spectrum design uses elastic spectra with equivalent damping and period. The N2 method uses an inelastic spectrum which is obtained by applying a reduction factor to typical smooth elastic design spectra. The N2 method is proposed to eliminate 2 drawbacks of conventional capacity
23
spectrum ( FEMA 356 and ATC-40) which are the need of iteration process and the usage of equivalent damping and period. By N2 method, the iteration process will be eliminated and no need to use the equivalent damping and period. These proceed is done because there is no physical principle that justifies the existence of a stable relationship between the hysteretic energy dissipation and equivalent viscous damping, and the period associated with the intersection of the capacity curve with the highly damped spectrum may have no relation with the dynamic response of the inelastic system (Fajfar, 1999). The inelastic spectra are based on statistical analyses, in which near-fault and impulsive type of ground motion has not been included. Care should be taken in long period range (actual displacements are typically constant) and in very long period range (where the spectral displacements decrease to the level of the peak ground displacement). The steps of N2 method is described as follows: 1. Building data and Elastic acceleration spectra a.
Structure data: mass, height, and properties of the section
b.
Elastic acceleration spectra Sae
Figure 2.16. Building data and Elastic acceleration spectra (Fajfar, 2000) 2. Convert the seismic demand into AD format and get inelastic spectra a.
Elastic spectra
2
=
4
2
(2.28)
Where S ae and S de are the value of elastic acceleration and elastic displacement spectrum, respectively corresponding to the period T and fixed viscous damping ratio (usually 5%). b.
Inelastic spectrum for constant ductility Inelastic spectrum is defined by Eqs. (2.29) and (2.30) 24
=
2
where
=
=
4
(2.29)
2
=
2
4
(2.30)
2
is inelastic acceleration; R μis the reduction factor due to ductility;
is inelastic displacement; μis the ductility factor (ratio between the
− ≥
maximum displacement and the yield displacement)
=
1
+1
<
(2.31)
=
(2.32)
Figure 2.17. Elastic and inelastic response spectra for constant ductility (Fajfar, 1999)
At the longer periods, the displacement spectrum is typically constant so the acceleration spectrum will decreases with the square of period T. Long period is depending on the earthquake and site characteristics (can be start at 2 seconds). In very long period range, spectral displacements decrease to the value of the peak ground displacement
3. Pushover analysis
∅ ∅ ∅
a.
Assume displacement shape
b.
Determine lateral load distribution by Eq. (2.33)
where
=
;
is the lateral load in level i;
displacement shape at level i. c.
=
is mass at level i;
∅
(2.33) is assumed
Run the pushover analysis and determine the base shear (V ) – top 25
displacement ( Dt ) relationship
4. Equivalent SDOF and capacity diagram a.
Determine mass m* by Eq. (2.34)
The
∅
∗ ∅ ∅∅ =
value should be constant and does not change during the structural
response to ground motion.
is normalized (the value of top is 1) and any
reasonable shape can be used for shape can be assumed b.
(2.34)
. As special case, the elastic first mode
∗∗ ΓΓ ∗ ∅ ∅ Γ ∅∅ ∅ ∅
Transform MDOF quantities (Q) to SDOF (Q*)
=
1
=
is equivalent to
1 in
/
(2.35)
= /
(2.36)
=
2
=
2
(2.37)
capacity spectrum method (ATC-40) , and to C 0 in
the displacement coefficient method (FEMA 356).The initial stiffness both
of equivalent SDOF system and MDOF is the same.N2 method requires the post-yield stiffness is equal to zero, because the reduction factor
is
defined as the ratio of the required elastic strength to the yield strength.The strain hardening is incorporated in the demand spectra. The moderate strain hardening does not have a significant influence on displacement demand. Thus, the N2 method approximately apply for systems with zero or small strain-hardening
∗ ∗
∗ ∗ ∗∗ ∗ ∗ ∗
c.
Determine an approximate elasto-plastic force-displacement relationship
d.
Determine strength respectively,
,
,and period
by Eqs. (2.35), (2.36), and (2.38),
=2
e.
(2.38)
Determine capacity diagram (Acceleration versus displacement)
=
Step 4b.until 4e. are drawn in Figure 2.18. 26
(2.39)
Figure 2.18. Idealized bilinear capacity curve with zero post-yielding stiffness and transformation from base shear-displacement format to Sa-Sd format (Fajfar, 1999)
− ∗ ∗∗ ≥ ∗ − ∗ ∗∗ ≥
5. Seismic demand for the equivalent SDOF model a.
Determine the reduction factor
(Vidic et al. 1994) based on the elastic
period of the idealized capacity curve.
=
=
1
+1
(2.40) (2.41)
<
=
b.
=
Determine the displacement demand
=
1
+1
(2.42)
(refer to Figure 2.19)
<
(2.43)
=
(2.44)
Note: all steps in the procedure can be performed numerically without using the graph
∗
(a)
<
∗ ≥ ∗ ∗ ≥ (b)
Figure 2.19. Determination of displacement demand, (Fajfar, 2000)
27
: (a)
<
; (b)
6. Global seismic demand for the MDOF model Transform SDOF displacement demand to the top displacement of the MDOF model by Eq. (2.45)
Γ =
(2.45)
7. Local seismic demand for the MDOF model a.
Perform pushover analysis of MDOF model up to the top displacement (or to an amplified value of
). Pushover result from result in step 3
should be used only until target top displacement b.
in step 6.
Determine local quantities (eg. Story drifts, rotation θ) corresponding to
8. Performance evaluation (Damage analysis)
.
Compare local and global seismic demands in step 7 with the capacities for the relevant performance level. The basic N2 method is limited to the planar or symmetric structure. When the higher mode effects are significant, the basic N2 will not accurate again. The inelastic spectra are based on the equal displacement rule (medium- and long-period range) which yields too small inelastic displacements in the case of near-fault ground motions, hysteretic loops with significant pinching or significant stiffness/or strength deterioration, and for systems with low strength (yield strength to required elastic strength ratio less than 0.2). Equal displacement rule also will not have satisfied result for soft soil conditions In the case of short period structures, inelastic displacements are larger than the elastic ones, so R μ is smaller than μ. Transition period or characteristic period Tc will decrease and increase with a decreasing and increasing ductility factor, respectively. Equations (2.46) to (2.48) and Figure 2.20 are simple version formula by Vidic et al (1994). Conservative result (higher seismic demand) are obtained for short-period structures in the case of low ductility demand ( μ<4). Non conservative results happen for higher ductility demand.
28
Figure 2.20. Simple version of ductility factor (Fajfar, 1999)
− ≤≥ =
1
+1
<
(2.46)
0
=
0
= 0.65
0.3
(2.47) (2.48)
In the short period structures, the inelastic displacement is more sensitive due to the changing of structural parameters than in the medium- and long-period ranges. Consequently, estimation of inelastic displacement are less accurate in the short-period range, but it is still can be achieved since the absolute values of displacements in the short-period region are small and typically they do not control the design.
2.5. Comparison between ATC-40, FEMA 356, and Basic N2 method
Eight storey RC frame building is chosen as a building model to compare the available nonlinear static procedure. The methods compared are ATC-40, FEMA 356, and basic N2 method. The displacement is used as the parameter to be compared. The plan view of the building is drawn in Figure 2.21. The storey height of the building is 5 m for first and second floor, and 3.1 m for third floor and above. The slab thickness is 20 cm. Total weight and height of the building is 3639 ton and 28.6 m, respectively. The concrete strength and steel yield strength is 25 MPa and 500 MPa, respectively. The size of the column and beam is 60x60 cm 2and 40x60 cm2, respectively. The longitudinal reinforcement steel for the column is 8Ø25, and 4Ø20 for each top and bottom of the beam. The shear strength is design over the flexural strength such the failure is not happen in shear failure. Fixed-base condition is used and P- Δ effect is
29
ignored. The elastic flexural and shear stiffness properties of cracked elements are assumed to be one-half of the uncracked one. Material nonlinearity is modeled by plastic hinges at both ends of every planar element which are beams and columns. Bi-linear moment-rotation relationship without strain hardening is used in defining the plastic hinges. Unlimited ductility is assumed.
Figure 2.21. The plan view of the building model The displacement shape of the building is assumed. The assumption used in this
∅
building is a linear inverted triangle: = [0.175; 0.350; 0.458; 0.566; 0.675; 0.783; 0.892; 1]
The corresponding lateral load pattern can be defined by the Eq. (2.33): T
= [0.184; 0.367; 0.458; 0.566; 0.675; 0.783; 0.892; 1]
For comparison purpose, all of the methods use the same lateral load pattern and capacity curve. ATC-40
The capacity curve resulted is converted to Sa-Sd format and drawn in the same plot with the 5% demand spectra. A trial performance point
1 ,
1 which
is
0.092g, 29.07 cm is defined by equal displacement rule. A bilinear curve then is drawn such the area under the capacity curve and the bilinear representation is the same.
29.07
From
the
Figure
2.22,
= 0.092 ;
= 8.66
;
= 0.092 ;
=
. The building is assumed to be type B. Then the hysteretic damping is
calculated based on Eqs. (2.16) to (2.18).
30
− 0
=
2
= 44.89%
1
Demand spectra
0.9 0.8
First iteration performance point
0.7
Reduced demand spectra
0.6
Initial performance point
) g ( a 0.5 S
0.4 0.3
Idealized capacity curve
Capacity curve
0.2
Say1 0.1
Sap1
Sdy1
0 0
2
4
6
8
Sdp1
10
12
14
16
18
20
22
24
26
28
Sdp2 30
32
34
Sd (cm)
Figure 2.22. Performance point of ATC-40 Using Table 2.1., the damping modification factor effective viscous damping is
− − −− ≥ ≥ calculated.
= 0.845
0.446(
)
= 0.53
Then, the effective damping is obtained using, Eq. (2.19):
=
0
+ 0.05 = 28.8%
The damping will reduce the response spectrum. The reduction factor is defined by Eqs. (2.20) and (2.21) and governed by Table 2.5
=
=
1
1
=
=
3.21
0.68 ln 2.12
2.31
= 0.44
0.44 (
)
0.41 ln
= 0.565 0.56 ( ) 1.65 Plot the reduce demand spectrum into the graph and the intersection between this reduced demand spectrum with the capacity curve will result new performance point. The iteration is done by repeat the calculation until the new performance point has less than 5% error compare with the previous performance point.
31
The last performance point is:
Γ
= 0.092 = 31
= 0.097 = 9.23
The target displacement of SDOF is equal to
= 31
The target displacement of MDOF is equal to
31 = 43.08
1
×
,1
×
= 1.3898 × 1 ×
FEMA 356 4000
Ki
Final iteration of bilinear curve
3500
Bilinear curve
3000 ) N2500 k ( r a e2000 h s e s a1500 B
Capacity curve
1000 500 0 0
10
20 30 Top displacement (cm)
40
50
Figure 2.23. Defining the target displacement by FEMA 356 After plotting the capacity curve, an initial target displacement is defined, for example 5% of the total height. Then, the bilinear approximation with a rule explained in FEMA 356 section is done. From the Figure 2.23:
0.6
0.6
= 1622 = 7.2
= 2703 = 12
= 225.19
/
= 225.47
/
=
= 1.96
32
−≈ 1
=
= 0.3 (
=
0.017
>
)
/
0
From the FEMA 356 section, the value of the coefficient is:
= 1.3 Table 2.7.
0
1
= 1 Eqs.(2.23) to (2.25) 2
= 1 Table 2.9
3
= 1 Eq. (2.26)
The target displacement is calculated based on Eq. (2.22) 2
=
0 1 2 3
4
2
= 1.3 × 1.0 × 1.0 × 1.0 × 0.3 ×
Then, set
= 37.23
1.972 4
2
× 981 = 37.23
as the target displacement for the next iteration until
the error produced between the new target displacement and the previous one is less than 5%. The final target displacement is
= 38.08
Basic N2 method
Figure 2.24draw the process to obtain the target displacement. Modal
∗ Γ ∅ ∗ ∗∗ ∗ ∗∗ ∗ →
participation factor is calculated using Eq. (2.37):
=
2
= 1.3898
Define the bilinear curve. This can be done by follow the FEMA 356 rule with post yielding stiffness is equal to zero. Then, transform to the SDOF and obtain:
= 11.97
=
= 2696
= 8.62
=
=2
= 1940
= 1.966 >
33
=
;
=
= 0.3
=
= 29.32
Say = 0.09g
=
=
= 3.4
And the target displacement of MDOF is:
=
= 40.75 cm
1.2
1
0.8
Elastic demand spectra Inelastic demand spectra
) g ( 0.6 a S
Ideali zed response spectra
0.4
T*=1.966 s
Sae=0.31 R μ=3.4
0.2
Say=0.09
Sde=29.32
0 0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 Sd (cm)
Figure 2.24. Target displacement by basic N2 method The comparison result and the difference action of each method is listed in Table 2.10and Table 2.11., respectively.
Table 2.10. Comparison result of each method
Triangular displacement form C0 or
K e (kN/cm) α Dt (cm)
Basic N2
FEMA-356 ATC-40
1.3898
1.3
1.3898
225.15
225.47
225.19
0
-0.00007
-0.027
40.75
38.08
43.08
Figure 2.25. shows the displacement shape of each method. The basic N2 result values in between the ATC-40 and FEMA 356.
34
Figure 2.25. The displacement shape of each method
Table 2.11. Difference action in Basic N2, FEMA 356 AND ATC-40 Basic N2 method
Pushover analysis
Transformation from the MDOF to SDOF system
FEMA 356
Any desired lateral force distribution can be obtained by assuming appropriate displacement shape
Several different lateral load patterns are suggested
( same with FEMA 273 if same displacement shape is used)
any reasonable primary slope can be used Bilinear idealization
ATC-40
1 (restricted only in
C
0
first mode shape)
Has a rule for model the bilinear idealization
No idealization of the pushover curve is made (just bilinear curve with the same area)
The difference in secondary slope has no practical consequences
represents C
0
μ /R represents
Determination of the displacement demand (target displacement)
μ
C
1
no C and C (assume 2
3
equal to one or can be considered by multiply the displacement demand or dividing the reduction factor by appropriate modification factor)
Elastic displacement times by 4 coefficient (C ,C ,C ,C ) 0
1
2
3
Determined from equivalent elastic spectra by using equivalent damping and period to consider the inelastic behaviour of the structure
Another comparison between several nonlinear static pushover shows the same trend with the above example (Causevic & Mitrovic, 2011)
35
2.6. Extended N2 method
Extended N2 method combines the result of basic N2 method and RSA. RSA is linear dynamic analysis or response spectrum analysis. The result of RSA should be adjusted such the top displacement of the building will have same value with that of basic N2 method. The larger value between basic N2 method and RSA will be used as the final result. The purpose of including the RSA is to take account the higher mode effect of the building which is assumed to be keep in elastic behavior. Correction factor for higher mode effect both in elevation (C E ) and in plan (C T) are taken into account and drawn in Figure 2.26 and Figure 2.27, respectively.
Figure 2.26. Example of correction factor for higher mode effect in elevation, C E (Kreslin & Fajfar, 2011)
36
Figure 2.27. Example of correction factor for higher mode effect in plan, C T (Kreslin & Fajfar, 2011)
2.7. Compatible ground motion matching a spectrum
The response spectra from each ground motions are homogenized in amplitude to be similar with a target response spectrum. The purpose is to make the response in linear dynamic is the same for all spectra, such only the effect of nonlinearity of the structure will make different. Figure 2.28 shows the original response spectra resulted from original earthquake ground motion and the modified response spectra resulted from compatible earthquake ground motion. 1.6 ) g ( 1.2 n o i t a r e l e c0.8 c a d n u 0.4 o r G
0
1.2
Earthquake ground motions
) g 1 ( n o i t0.8 a r e l e c0.6 c a l a r0.4 t c e p 0.2 S
Eurocode 8 0.4g
0
1
2 3 Period (s)
4
0
5
(a)
0
1
2 3 Period (s)
4
5
(b)
Figure 2.28.Response spectra: (a) Original response spectra from the ground motions; (b) Compatible response spectra from the compatible ground motions
The scaling and matching process is done by RspMatch2005 proposed by Hancock (Hancock et al., 2006). Several artificial compatible ground motions with the
37
same amplitude are used and applied to the building to find the inelastic response of the building.
38
3. BUILDING
EXAMPLE
AND
GROUND
MOTION 3.1. Buildings Example
The building example for the analytical study is consisting of 3 kinds of buildings which are 2-storey, 8-storey and 20-storey building. While, the building example for verification purpose is a 14-storey reinforced concrete frame building. The building plan view and the elevation view are shown in Figure 3.1 and the details of members for each building are shown in Table 3.1. For 2-storey building, the storey height is 3.5 m both for 1 st and 2ndstorey. The slab thickness is 15 cm. Total weight of the building is 305 ton. For 8-storey building, the storey height is 5 m for first and second floor, and 3.1 m for above second floor. The slab thickness is 20 cm. Total weight of the building is 3639 ton. For 14-storey building, the storey height is 5 m for first and second floor, and 3.5 m for above second floor. The slab thickness is 20 cm. Total weight of the building is 8990 ton. For 20-storey building, the storey height is 3.5m for all storey. The slab thickness is 20 cm. Total weight of the building is 13710 ton. These building s‟ natural period is in velocity constant which is in between the 0.6s (transitional period from acceleration constant to velocity constant) and 3s (displacement is assumed to be constant when the period larger than 3 s).
(a)
39
(b)
(c)
(d) Figure 3.1.Plan view and elevation view: (a) 2-storey (b) 8-storey; (c) 14-storey;
(d) 20-storey
40
Table 3.1. Details of members for each building
Column details
No. of storey
Level
2 8 14
20
Size (cm2)
(MPa)
1-2
25 x 25
1-8
Beam details
′
′
Steel
Size (cm2)
(MPa)
Top steel
Bottom steel
25
8Ø16
20 x 30
25
3Ø16
3Ø16
60 x 60
25
8Ø25
40 x 60
25
4Ø20
4Ø20
1-10
120 x 120
30
32Ø25
11-14
100 x 100
30
20Ø28
60 x 80
30
8Ø25
8Ø25
1-10
150 x 150
35
32Ø32
11-18
120 x 120
30
32Ø25
60 x 80
30
8Ø25
8Ø25
19-20
100 x 100
30
20Ø28
For 2-,8-, and 20-storey building, there are four eccentricities which are 0%, 5%, 10%, and 15%, while for 14-storey building, there is only one eccentricity which is 10%. All steel has yield stress equal to 500 MPa and ultimate stress equal to 625 MPa. Fixed-base condition is used and P-Δ effect is ignored. The elastic flexural and shear stiffness properties of cracked elements are assumed to be one-half of the uncracked one. Table 3.2 and Table 3.3 describe the dynamic properties of buildings. Material nonlinearity is modeled by plastic hinges at both ends of every planar element which are beams and columns. Bi-linear moment-rotation relationship without strain hardening is used in defining the plastic hinges. Unlimited ductility is assumed. PERFORM-3D V4.0.4 (CSI, 2008) program is used in this research to do all types of analysis which are pushover analysis, response spectrum analysis, and nonlinear response history analysis. MATLAB (The MathWorks, 2009) is used to make the calculation faster.
Table 3.2. Natural period of mode n of the building Tn (seconds) 2-storey
Mode 0% ecc
5% ecc
10% 15% ecc ecc
8-storey 0% ecc
5% ecc
14storey
20-storey
10% 15% ecc ecc
10% ecc
1
0.92 0.94 1.00 1.09 2.14 2.18 2.29 2.45
1.98
2.51 2.53 2.61 2.74
2
0.92 0.92 0.92 0.92 2.14 2.14 2.14 2.14
1.91
2.51 2.51 2.51 2.51
3
0.30 0.72 0.71 0.69 0.60 1.58 1.57 1.56
1.22
0.83 1.60 1.62 1.65
4
0.30 0.31 0.33 0.35 0.60 0.62 0.65 0.69
0.59
0.83 0.84 0.87 0.92
5
0.30 0.30 0.30 0.32 0.60 0.60 0.60
0.57
0.45 0.83 0.83 0.83
6
0.24 0.23 0.23 0.32 0.45 0.44 0.44
0.38
0.45 0.56 0.57 0.57
41
0% ecc
5% ecc
10% 15% ecc ecc
7
0.22 0.32 0.34 0.37
0.30
0.29 0.45 0.47 0.50
8
0.22 0.32 0.32 0.32
0.29
0.29 0.45 0.45 0.45
9
0.16 0.24 0.24 0.26
0.20
0.20 0.31 0.31 0.33
10
0.16 0.23 0.24 0.23
0.18
0.20 0.29 0.31 0.31
11
0.11 0.22 0.22 0.22
0.17
0.15 0.29 0.29 0.29
12
0.11 0.17 0.17 0.18
0.12
0.15 0.21 0.21 0.23
13
0.08 0.16 0.16 0.16
0.12
0.11 0.20 0.21 0.21
14
0.08 0.16 0.16 0.16
0.11
0.11 0.20 0.20 0.20
15
0.07 0.12 0.12 0.13
0.09
0.09 0.15 0.16 0.17
16
0.08
0.09 0.15 0.15 0.15
17
0.08
0.07 0.14 0.14 0.14
18
0.06
0.07 0.11 0.12 0.13
19
0.06
0.06 0.11 0.11 0.11
20
0.06
0.06 0.11 0.11 0.11
21
0.05
0.05 0.09 0.09 0.10
22
0.05
0.05 0.09 0.09 0.09
23
0.04
0.04 0.08 0.08 0.08
24
0.04
0.04 0.07 0.08 0.08
Table 3.3. Effective mass factor in x-direction about mode n Effective mass factor in x-direction (%) 2-storey
Mode 0% ecc 1
0
5% ecc
10% 15% ecc ecc
14store y
8-storey 0% ecc
5% ecc
10% 15% 10% ecc ecc ecc
20-storey 0% ecc
5% ecc
10% 15% ecc ecc
42.48 37.01 32.98 90.23 43.3 39.4 35.53 39.04 73.92 36.45 35.01 32.93
2
91.89 45.95 45.95 45.95
0
3
5.72 3.46 8.93 12.96 5.11 1.81 5.72 9.58 1.9 1 11.05 0.51 1.95 4.05
4
2.76 3.73 3.23 2.88
0
45.11 45.11 45.11 40.96
2.45 2.23 2.01 4.47
5
4.05 4.05 4.05 2.44 2.56 2.56 2.56
6
0.33 0.82 1.18
7
0
0.1
4.8
1.86 1.17 1.06 0.96
1.5
0
1.22 1.22 1.22 1.65
9
0
0.05
11
0.5
0.32 0.89 0.46 0
36.96 36.96 36.96
0
5.41 5.11 4.72
0
5.52 5.52 5.52
0.32 0.53 0.34 4.65 0.11
8
10
0
0.4
0.77
1.48 2.27 2.13 1.97 0
2.33 2.33 2.33
0.68 0.13 1.79 0.08 0.57 1.26 0.3
0.89
0
1.25 0.82
0.2
0.93 0.93 0.93 0.98 0.25 1.31 1.31 1.31
12
0.04 0.04 0.12 0.11 0.08 1.23 0.05 0.82 0.73
13
0.01 0.16 0.15 0.22 0.58 0.03 0.85
14
0
0.16 0.16 0.16 0.64
15
0
0.01 0.01 0.01
16
0.4 0.08
42
0
0.1
0.21
0.89 0.89 0.89
0.84 0.65 0.61 0.56 0
0.67 0.67 0.67
17
0.46 0.56 0.02 0.08 0.16
18
0.28
19
0.33 0.52 0.45 0.45 0.45
20
0.06
21
0.11 0.11 0.41 0.37 0.34
22
0.13 0.26 0.42 0.42 0.42
23
0.04
0
0.02 0.06 0.22
24
0.04
0
0.27 0.24
0
0
0.43
0.4
0.36
0.02 0.07 0.12
0.1
3.2. Ground motion
Various types of short duration of ground motions with wide range variation of pga (0.15g-0.84g) are chosen. Table 3.4 displays the earthquakes that are chosen in this research. The records are selected from the PEER ground motion database (PEER). A target response spectrum chosen for benchmark is Eurocode 8 (EC8) response spectrum. Each ground motion is changed to be compatible to the target response spectrum. Refer Figure 3.2 for the original response spectra from the ground motions and the compatible response spectra from the artificial compatible ground motions. Several pga, i.e. 0.1g, 0.4g, 0.6g, 1g, 1.4g of target response spectra are
σ
chosen to represent small and large earthquake. The average value from all of max value of ground motions response with its ± standard deviation
is used as the
reference for verifying the accurac y of extended N2 and proposed method.
1.6 ) g ( 1.2 n o i t a r e l e c0.8 c a d n u 0.4 o r G
0
1.2
Earthquake ground motions
) g 1 ( n o i t0.8 a r e l e c0.6 c a l a r0.4 t c e p 0.2 S
Eurocode 8 0.4g
0
1
2 3 Period (s)
4
0
5
(a)
0
1
2 3 Period (s)
4
5
(b)
Figure 3.2. Response spectra: (a) Original response spectra from the ground motions; (b) Compatible response spectra from the compatible ground motions 43
Table 3.4. List of earthquake ground motion No.
Earthquake
Station
Site Condition (USGS)
1
1979 Imperial Valley 6
El Centro Array #6
C
140
0.376
0.631
0.269
2
1989 Loma Prieta
Los Gatos -Lexington Dam
A
0
0.442
0.844
0.147
3
1992 Landers
Yermo
C
0
0.151
0.290
0.228
4
1994 Northridge
Sylmar
C
0
0.843
1.289
0.326
5
1995 Kobe
Takatori
D
0
0.611
1.271
0.358
6
1999 Chi-Chi
TCU074
C
0
0.597
0.733
0.204
7
1990 Upland
Pomona
C
0
0.1860
0.104
0.011
44
Orientation (degree)
PGA (g)
PGV (m/s)
PGD (m)
4. NRHA STUDY
RESULT
AND
ANALYTICAL
4.1. Maximum displacement result
Each building is given several pga of earthquake to study the real behavior of that building in different degree of inelasticity whose values are from less than 1 to 5. The displacement described in this research is the displacement at the center of mass. The displacement results of 2-storey buildings with pga equal to 0.1g and 0.4g are described in Figure 4.1. The displacement results of 2-storey buildings with pga equal to 0.6g and 1g are described in Figure 4.2.The displacement results of 8-storey buildings with pga equal to 0.1g and 0.4g are described in Figure 4.3.The displacement results of 8-storey buildings with pga equal to 0.6g are described in Figure 4.4.The displacement results of 20-storey buildings with pga equal to 0.1g and 0.4g are described in Figure 4.5.The displacement results of 20-storey buildings with pga equal to 0.6g and 1g are described in Figure 4.6.The displacement results of 20-storey buildings with pga equal to 1.4g are described in Figure 4.7.
800
800
average
average
700
Ave+stdev
700
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
extended N2
100
Ave+stdev Ave-stdev extended N2
100
0
0 0
1
2
3
4
5
0
5
Displacement (cm)
(a) 800
average
20
average
700
Ave+stdev
) 600 m c ( t 500 h g i e 400 h y e 300 r o t S 200
Ave-stdev
Ave+stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
15
(b)
800 700
10
Displacement (cm)
Ave-stdev extended N2
extended N2
100
100
0
0 0
1
2
3
4
0
5
Displacement (cm)
5
10
Displacement (cm)
(c)
(d)
45
15
20
800
800
average
average
700
Ave+stdev
700
) 600
Ave-stdev
) 600 m c ( t 500 h g i e400 h y e r300 o t S200
m c ( t 500 h g i e 400 h y e 300 r o t S 200
extended N2
Ave+stdev Ave-stdev extended N2
100
100
0
0 0
1
2
3
4
0
5
5
10
(e) 800
average
average 700
Ave+stdev
)600 m c ( t500 h g i e400 h y e300 r o t S200
20
(f)
800
700
15
Displacement (cm)
Displacement (cm)
Ave+stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev extended N2
Ave-stdev extended N2
100
100
0
0 0
1
2
3
4
0
5
5
10
15
20
Displacement (cm)
Displacement (cm)
(g) (h) Figure 4.1. Displacement result for 2-storey building: (a) 0% eccentricity 0.1g
( = 0.46 ); (b) 0% eccentricity 0.4g ( = 1.86 ); (c) 5% eccentricity 0.1g = 0.48 ; (d) 5% eccentricity 0.4g ( = 1.904 ); (e) 10% eccentricity 0.1g
( = 0.5 ); (f) 10% eccentricity 0.4g ( = 1.99 ); (g) 15% eccentricity 0.1g ( = 0.52 ); (h) 15% eccentricity 0.4g ( = 2.08 ) 800
800
average
average
700
700
Ave+stdev
)600 m c ( t500 h g i e400 h y e300 r o t S200
Ave+stdev
)600 m c ( t500 h g i e400 h y e300 r o t S200
Ave-stdev extended N2
100
Ave-stdev extended N2
100
0 0
5
10
15
20
25
0
30
0
Displacement (cm)
10
(a) 800
20
30
40
50
Displacement (cm)
(b) 800
average
average
700
Ave+stdev
700
Ave+stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev
) 600 m c ( t 500 h g i e400 h y e r300 o t S200
Ave-stdev
extended N2
100
extended N2
100
0
0
0
5
10
15
20
25
30
0
Displacement (cm)
10
20
30
Displacement (cm)
46
40
50
(c)
(d)
800
800 average
700
average 700
Ave+stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave+stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev extended N2
Ave-stdev extended N2
100
100
0
0 0
5
10
15
20
25
0
30
10
20
Displacement (cm)
40
50
40
50
(f)
(e) 800
30
Displacement (cm)
800
average
average
700
Ave+stdev
700
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
extended N2
100
Ave+stdev Ave-stdev extended N2
100
0
0 0
5
10
15
20
25
30
0
10
20
Displacement (cm)
30
Displacement (cm)
(g) (h) Figure 4.2. Displacement result for 2-storey building: (a) 0% eccentricity 0.6g
( = 2.78 ); (b) 0% eccentricity 1g ( = 4.64 ); (c) 5% eccentricity 0.6g = 2.86 ; (d) 5% eccentricity 1g ( = 4.76 ); (e) 10% eccentricity 0.6g
( = 2.98 ); (f) 10% eccentricity 1g ( = 4.96 ); (g) 15% eccentricity 0.6g ( = 3.13 ); (h) 15% eccentricity 1g ( = 5.21 ) 3500
3500
average
average
3000
3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t 1000 S
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t S1000
Ave-stdev extended N2
Ave-stdev extended N2
500
500
0
0 0
2
4
6
8
10
0
12
10
Displacement (cm)
(a) 3500
Ave+stdev
) m2500 c ( t h 2000 g i e h y 1500 e r o t S 1000
Ave-stdev
30
40
50
40
50
(b) 3500
average
3000
20
Displacement (cm)
average
3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t S1000
extended N2
Ave-stdev extended N2
500
500
0
0 0
2
4
6
Displacement (cm)
8
10
0
12
10
20
30
Displacement (cm)
47
(c)
(d)
3500
3500
average 3000
average 3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t 1000 S
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o1000 t S
Ave-stdev extended N2
Ave-stdev extended N2
500
500
0
0 0
2
4
6
8
10
0
12
10
20
30
40
50
Displacement (cm)
Displacement (cm)
(e)
(f)
3500
3500 average
average 3000
3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t S1000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t 1000 S
Ave-stdev extended N2
Ave-stdev extended N2
500
500
0
0 0
2
4
6
8
10
0
12
10
20
30
40
50
Displacement (cm)
Displacement (cm)
(h) (g) Figure 4.3. Displacement result for 8-storey building: (a) 0% eccentricity 0.1g
( = 0.85 ); (b) 0% eccentricity 0.4g ( = 3.4 ); (c) 5% eccentricity 0.1g = 0.85 ; (d) 5% eccentricity 0.4g ( = 3.4 ); (e) 10% eccentricity 0.1g
( = 0.84 ); (f) 10% eccentricity 0.4g ( = 3.36 ); (g) 15% eccentricity 0.1g ( = 0.83 ); (h) 15% eccentricity 0.4g ( = 3.31 ) 3500
3500
average
average 3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t 1000 S
Ave-stdev extended N2
500
3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t S1000
Ave-stdev extended N2
500
0
0 0
10
20
30
40
50
60
70
0
Displacement (cm)
10
20
30
40
Displacement (cm)
(a)
(b)
48
50
60
70
3500
3500 average
average 3000
3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t S1000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t 1000 S
Ave-stdev extended N2
Ave-stdev extended N2
500
500
0
0 0
10
20
30
40
50
60
0
70
10
20
30
Displacement (cm)
40
50
60
70
Displacement (cm)
(c) (d) Figure 4.4. Displacement result for 8-storey building: (a) 0% eccentricity 0.6g
( = 5.1 ); (b) 5% eccentricity 0.6g
= 5.1 ; (c) 10% eccentricity 0.6g
( = 5.04 ); (d) 15% eccentricity 0.6g ( = 4.96 )
8000
8000
average 7000 ) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
average
Ave+stdev
7000
Ave+stdev
Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev
extended N2
1000
extended N2
1000 0
0 0
3
6
9
12
0
15
10
Displacement (cm)
20
30
(a) 8000
60
50
60
50
60
8000 average 7000
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
50
(b)
average
7000
40
Displacement (cm)
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev extended N2
Ave-stdev extended N2
1000
1000
0
0 0
3
6
9
12
0
15
10
20
Displacement (cm)
(c)
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
40
(d)
8000 7000
30
Displacement (cm)
average
8000
Ave+stdev
7000
Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e r3000 o t S2000
average
extended N2
1000
Ave+stdev Ave-stdev extended N2
1000
0
0 0
3
6
9
12
15
0
10
20
30
Displacement (cm)
Displacement (cm)
(e)
(f)
49
40
8000
8000
average
average 7000
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev extended N2
7000
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev extended N2
1000
1000
0
0 0
3
6
9
12
0
15
10
20
Displacement (cm)
30
40
50
60
Displacement (cm)
(g) (h) Figure 4.5. Displacement result for 20-storey building: (a) 0% eccentricity 0.1g
( = 0.38 ); (b) 0% eccentricity 0.4g ( = 1.51 ); (c) 5% eccentricity 0.1g = 0.38 ; (d) 5% eccentricity 0.4g ( = 1.51 ); (e) 10% eccentricity 0.1g
( = 0.38 ); (f) 10% eccentricity 0.4g ( = 1.5 ); (g) 15% eccentricity 0.1g ( = 0.37 ); (h) 15% eccentricity 0.4g ( = 1.48 ) 8000
8000
average
average 7000
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev extended N2
7000
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev extended N2
1000
1000 0
0 0
20
40
60
80
100
0
30
(a) 8000
average
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
90
120
150
(b)
8000 7000
60
Displacement (cm)
Displacement (cm)
average
Ave+stdev
7000
Ave+stdev
Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev
extended N2
extended N2
1000
1000
0
0 0
20
40
60
80
0
100
30
(c) 8000
60
90
120
150
Displacement (cm)
Displacement (cm)
(d) 8000
average
average
7000
Ave+stdev
7000
) 6000 m c ( t 5000 h g i e 4000 h y e 3000 r o t S 2000
Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
extended N2
1000
Ave+stdev Ave-stdev extended N2
1000
0 0
20
40
60
80
0
100
0
Displacement (cm)
30
60
90
Displacement (cm)
50
120
150
(e)
(f) 8000
8000
average
average
7000
Ave+stdev
7000
) 6000 m c ( t 5000 h g i e 4000 h y e 3000 r o t S 2000
Ave-stdev
)6000 m c ( t5000 h g i e4000 h y e3000 r o t S2000
extended N2
1000
Ave+stdev Ave-stdev extended N2
1000
0 0
20
40
60
80
0
100
0
30
60
Displacement (cm)
90
120
150
Displacement (cm)
(g) (h) Figure 4.6. Displacement result for 20-storey building: (a) 0% eccentricity 0.6g
( = 2.27 ); (b) 0% eccentricity 1g ( = 3.79); (c) 5% eccentricity 0.6g = 2.27 ; (d) 5% eccentricity 1g ( = 3.78 ); (e) 10% eccentricity 0.6g
( = 2.25 ); (f) 10% eccentricity 1g ( = 3.75 ); (g) 15% eccentricity 0.6g ( = 2.22 ); (h) 15% eccentricity 1g ( = 3.71 )
8000
8000
average
average 7000
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev extended N2
7000
Ave+stdev
) 6000 m c ( t 5000 h g i e 4000 h y e r 3000 o t S2000
Ave-stdev extended N2
1000
1000
0
0 0
50
100
150
200
0
250
50
(a)
200
250
8000
average ) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
150
(b)
8000 7000
100
Displacement (cm)
Displacement (cm)
average
Ave+stdev
7000
Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
extended N2
1000
Ave+stdev Ave-stdev extended N2
1000
0
0 0
50
100
150
200
250
0
50
Displacement (cm)
100
150
200
250
Displacement (cm)
(c) (d) Figure 4.7. Displacement result for 20-storey building: (a) 0% eccentricity 1.4g
μ
μ
( = 5.3 ); (b) 5% eccentricity 1.4g
μ μ
= 5.29 ; (c) 10% eccentricity 1.4g
( = 5.25 ); (d) 15% eccentricity 1.4g ( = 5.19 )
51
4.2. Coefficient of torsion result
Coefficient of torsion represents the effect of higher mode effect in plan which is equal to the displacement at the flexible edge divided by the displacement at the center of mass. The displacement at the stiff edge is assumed to be the same as at the center of mass. In the symmetrical building which has 0% eccentricity will have the coefficient of torsion value equal to 1. The coefficient of torsion results of 2-storey buildings with pga equal to 0.1g and 0.4g are described in Figure 4.8. The coefficient of torsion results of 2-storey buildings with pga equal to 0.6g and 1g are described in Figure 4.9. The coefficient of torsion results of 8-storey buildings with pga equal to 0.1g and 0.4g are described in Figure 4.10. The coefficient of torsion results of 8-storey buildings with pga equal to 0.6g are described in Figure 4.11. The coefficient of torsion results of 20-storey buildings with pga equal to 0.1g and 0.4g are described in Figure 4.12. The coefficient of torsion results of 20-storey buildings with pga equal to 0.6g and 1g are described in Figure 4.13. The coefficient of torsion results of 20-storey buildings with pga equal to 1.4g are described in Figure 4.14. 800
800
700
700
Ave+stdev
600
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev
average
) m500 c ( t h 400 g i e h 300 y e r o 200 t S
average Ave+stdev
Ave-stdev extended N2
100
100
ba sic N2
0 1
extended N2 basic N2
0
1.05
1.1
1.15
1.2
1.25
1.3
1
1.05
1.1
Coefficient of Torsion
1.15
1.2
1.25
1.3
1.35
Coefficient of Torsion
(a)
(b) 800
800 average 700
Ave+stdev
) 600 m c ( t 500 h g i e 400 h y e 300 r o t S 200
Ave-stdev
700 ) 600 m c ( t 500 h g i e400 h y e300 r o t S200
extended N2 basic N2
100
average Ave+stdev Ave-stdev extended N2
100
0
basic N2
0 1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1
1.1
1.2
1.3
Coefficient of Torsion
Coefficient of Torsion
(c)
(d)
52
1.4
1.5
800
800
average
average
700
Ave+stdev
700
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev
) 600 m c ( t 500 h g i e 400 h y e 300 r o t S200
extended N2 basic N2
Ave+stdev Ave-stdev extended N2 basic N2
100
100
0
0 1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1
1.4
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
Coefficient of Torsion
Coefficient of Torsion
(e) (f) Figure 4.8. Coefficient of torsion result for 2-storey building: (a) 5% eccentricity
0.1g
= 0.48 ; (b) 5% eccentricity 0.4g ( = 1.904 ); (c) 10% eccentricity
0.1g ( = 0.5 ); (d) 10% eccentricity 0.4g ( = 1.99 ); (e) 15% eccentricity 0.1g ( = 0.52 ); (f) 15% eccentricity 0.4g ( = 2.08 )
800
800
700
700
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
average Ave+stdev Ave-stdev extended N2
100
1
Ave+stdev Ave-stdev extended N2
100
basic N2
0
average
basic N2
0
1.05
1.1
1.15
1.2
1.25
1
1.3
1.05
1.1
Coefficient of Torsion
(a) 800
700
700
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
average Ave+stdev Ave-stdev basic N2 1
1.1
1.15
1.3
1.35
Ave+stdev Ave-stdev extended N2 basic N2
0
1.05
1.25
average
100
extended N2
0
1.2
(b)
800
100
1.15
Coefficient of Torsion
1.2
1.25
1.3
1
1.35
1.05
1.1
1.15
1.2
1.25
1.3
1.35
Coefficient of Torsion
Coefficient of Torsion
(c)
(d)
800
800
700
700
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
average Ave+stdev Ave-stdev
100
basic N2 1
Ave+stdev Ave-stdev
100
extended N2
0
average
extended N2 basic N2
0 1.1
1.2
1.3
1.4
1
Coefficient of Torsion
1.05
1.1
1.15
1.2
1.25
1.3
1.35
Coefficient of Torsion
(e) (f) Figure 4.9. Coefficient of torsion result for 2-storey building: (a) 5% eccentricity 53
0.6g
= 2.86 ; (b) 5% eccentricity 1g ( = 4.76 ); (c) 10% eccentricity 0.6g
( = 2.98 ); (d) 10% eccentricity 1g ( = 4.96 ); (e) 15% eccentricity 0.6g ( = 3.13 ); (f) 15% eccentricity 1g ( = 5.21 )
3500
3500
average 3000
) 2500 m c ( t h2000 g i e h y1500 e r o t S1000
Ave+stdev
3000
Ave-stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t 1000 S
extended N2 basic N2
average Ave+stdev Ave-stdev extended N2
500
500
ba sic N2
0
0 1
1.05
1.1
1.15
1.2
1.25
1
1.3
1.05
1.1
(a) 3500
average
) 2500 m c ( t h2000 g i e h y1500 e r o t S1000
1.2
(b)
3500 3000
1.15
Coefficient of Torsion
Coefficient of Torsion
average
Ave+stdev
3000
Ave+stdev
Ave-stdev
) m2500 c ( t h 2000 g i e h y 1500 e r o 1000 t S
Ave-stdev
extended N2 basic N2
extended N2 basic N2
500
500
0
0 1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1
1.4
1.05
1.1
(c) 3500
1.25
1.3
1.3
1.35
average
average
) 2500 m c ( t h2000 g i e h y1500 e r o t S1000
1.2
(d)
3500 3000
1.15
Coefficient of Torsion
Coefficient of Torsion
Ave+stdev
3000
Ave+stdev
Ave-stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t 1000 S
Ave-stdev
extended N2 basic N2
extended N2 basic N2
500
500
0
0 1
1.1
1.2
1.3
1.4
1
1.5
1.05
1.1
1.15
1.2
1.25
Coefficient of Torsion
Coefficient of Torsion
(e) (f) Figure 4.10. Coefficient of torsion result for 8-storey building: (a) 5% eccentricity
0.1g
= 0.85 ; (b) 5% eccentricity 0.4g ( = 3.4 ); (c) 10% eccentricity 0.1g
( = 0.84 ); (d) 10% eccentricity 0.4g ( = 3.36 ); (e) 15% eccentricity 0.1g ( = 0.83 ); (f) 15% eccentricity 0.4g ( = 3.31 )
54
3500
3500
average Ave+stdev
3000
Ave-stdev
) m2500 c ( t h 2000 g i e h y 1500 e r o t S 1000
average
3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t 1000 S
extended N2 basic N2
500
Ave-stdev extended N2 basic N2
500
0
0 1
1.05
1.1
1.15
1.2
1
1.05
1.1
Coefficient of Torsion
1.15
1.2
1.25
1.3
Coefficient of Torsion
(a)
(b) 3500
average
3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o1000 t S
Ave-stdev extended N2 basic N2
500 0 1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
Coefficient of Torsion
(c) Figure 4.11. Coefficient of torsion result for 8-storey building: (a) 5% eccentricity
0.6g
= 5.1 ; (b) 10% eccentricity 0.6g ( = 5.04 ); ); (c) 15% eccentrici eccentricity ty 0.6g
8000 7000
( = 4.96 4.96 )
average
8000
Ave+stdev
7000
average
Ave-stdev
6000
) m c5000 ( t h g i e4000 h y e r3000 o t S
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
extended N2 basic N2
2000
Ave-stdev extended N2 basic N2
1000
1000
0
0 1
1 .02
1.0 4
1 .0 6
1 .0 8
1.1
1 .12
1
1 .1 4
1.05
1.1
1.25
average
Ave+stdev
7000
Ave+stdev
Ave-stdev
6000
) m c5000 ( t h g i e4000 h y e r3000 o t S
1.3
8000
average
7000
1.2
(b)
(a) 8000
1.15
Coefficient of Torsion
Coefficient of Torsion
Ave-stdev
6000
extended N2
) m c5000 ( t h g i e4000 h y e r3000 o t S
basic N2
2000
extended N2 basic N2
2000
1000
1000
0 1
1 .0 5
1 .1
1 .1 5
1 .2
0
1 .2 5
1
Coefficient of Torsion
1 .0 5
1 .1
1 .15
1 .2
1 .25
Coefficient of Torsion
(c)
(d)
55
1 .3
1 .35
1 .4
8000
8000
average
average
7000
Ave+stdev
6000
extended N2
7000
Ave+stdev
Ave-stdev
) m c5000 ( t h g i e4000 h y e r3000 o t S
Ave-stdev
6000
) m c5000 ( t h g i e4000 h y e r3000 o t S
basic N2
2000
2000
1000
1000
extended N2 basic N2
0
0 1
1 .0 5
1 .1
1.1 5
1 .2
1 .2 5
1 .3
1
1 .3 5
1 .05
1 .1
1 .15
1 .2
1 .2 5
1 .3
1 .35
1 .4
Coefficient of Torsion
Coefficient of Torsion
(e) (f) Figure 4.12. Coefficient of torsion result for 20-storey building: (a) 5%
eccentricity 0.1g
= 0.38 .38 ; (b) 5% eccentricity eccentricity 0.4g 0.4g ( = 1.51 ); (c) (c) 10% 10%
eccentricity 0.1g ( = 0.38 ); (d) (d) 10% 10% eccentricit eccentricity y 0.4g 0.4g ( = 1.5 ); ); (e) 15% 15% eccentricity 0.1g ( = 0.37 ); (f) (f) 15% 15% eccentricit eccentricity y 0.4g 0.4g ( = 1.48 1.48 )
8000
8000
average
average
7000
7000
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev
Ave+stdev Ave-stdev
) 6000 m c ( 5000 t h g i 4000 e h y e 3000 r o t S2000
extended N2 basic N2
extended N2 basic N2
1000
1000
0
0 1
1.05
1.1
1.15
1.2
1.25
1.3
1
1.35
1.05
1.1
1.15
1.2
1.25
Coefficient of Torsion
Coefficient of Torsion
(a)
(b) 8000
8000
average
average
7000 ) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave+stdev
7000
Ave-stdev
) 6000 m c ( 5000 t h g i e 4000 h y e 3000 r o t S 2000
extended N2 basic N2
Ave+stdev Ave-stdev extended N2 basic N2
1000
1000
0
0 1
1.1
1.2
1.3
1
1.4
1.05
1.1
1.15
1.2
1.25
Coefficient of Torsion
Coefficient of Torsion
(c)
(d)
8000
8000
average
7000
Ave+stdev Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
average
7000
Ave+stdev Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
extended N2 basic N2
extended N2 basic N2
1000
1000
0
0 1
1.1
1.2
1.3
1
1.4
Coefficient of Torsion
1.05
1.1
1.15
Coefficient of Torsion
(f)
(e)
56
1.2
1.25
Figure 4.13. Coefficient of torsion result for 20-storey building: (a) 5% eccentricity 0.6g
= 2.27 ; (b) 5% eccentricity 1g ( = 3.78 ); (c) (c) 10% 10%
eccentricity 0.6g ( = 2.25 ); (d) (d) 10% 10% eccentricit eccentricity y 1g 1g ( = 3.75 ); (e) (e) 15% 15% eccentricity 0.6g ( = 2.22 ); (f) (f) 15% 15% eccentricit eccentricity y 1g 1g ( = 3.71 3.71 ) 8000
average
7000
Ave+stdev
) 6000 m c ( t 5000 h g i e 4000 h y e 3000 r o t S 2000
Ave-stdev
8000
average
7000
Ave+stdev Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
extended N2 basic N2
1000
extended N2 basic N2
1000
0
0 1
1.0 2
1.0 4
1.06
1 .08
1.1
1.1 2
1
1.05
1.1
1.15
1.2
1.25
Coefficient of Torsion
Coefficient of Torsion
(a)
(b) 8000
average
7000
Ave+stdev Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
extended N2 basic N2
1000 0 1
1.05
1.1
1.15
1.2
1.25
Coefficient of Torsion
(c) Figure 4.14. Coefficient of torsion result for 20-storey building: (a) 5%
eccentricity 1.4g
= 5.29 .29 ; (b) 10% eccentricity 1.4g ( = 5.25 ); (c) (c) 15% 15% eccentricity 1.4g ( = 5.19 5.19 )
4.3. Maximum inter-storey drift result result
The drift described in this research is the maximum inter storey drift at the center of mass. The Drift results of 2-storey buildings with pga equal to 0.1g and 0.4g are described in Figure 4.15. The Drift results of 2-stor ey buildings with pga equal to 0.6g and 1g are described in Figure 4.16. The Drift results of 8-storey buildings with pga equal to 0.1g and 0.4g are described in Figure 4.17. The Drift results of 8-storey buildings with pga equal to 0.6g are described in Figure 4.18. The Drift results of 20-storey buildings with pga equal to 0.1g and 0.4g are described in Figure 4.19. The Drift results of 20-storey buildings with pga equal to 0.6g and 1g are described in Figure 4.20. The Drift results of 20-storey buildings with pga equal to 1.4g are described in Figure 4.21. 57
800
800
average
700
Ave+stdev
)600 m c ( t500 h g i e400 h y e300 r o t S200
Ave-stdev extended N2 basic N2
average
700
Ave+stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev extended N2 basic N2
100
100
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.7
1
Drift (%)
(a)
(b) 800
800
700
Ave+stdev
Ave-stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
basic N2
100
4
3
4
Ave+stdev Ave-stdev
) 600 m c ( t 500 h g i e 400 h y e 300 r o t S 200
extended N2
3
average
average
700
2
Drift (%)
extended N2 basic N2
100
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.7
0
1
2
Drift (%)
Drift (%)
(c)
(d) 800
800
average
average 700
700
Ave+stdev
) 600 m c ( t 500 h g i e400 h y e r300 o t S200
Ave+stdev
) 600 m c ( t 500 h g i e 400 h y e 300 r o t S200
Ave-stdev extended N2
basic N2
Ave-stdev extended N2 ba sic N2
100
100
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.7
1
2
Drift (%)
(e) 800
3
4
Drift (%)
(f) 800
average
average
700
Ave+stdev
700
Ave+stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev
extended N2 basic N2
100
extended N2 basic N2
100
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.5
1
Drift (%)
1.5
2
2.5
3
3.5
4
Displacement (cm)
(g) (h) Figure 4.15. Drift result for 2-storey building: (a) 0% eccentricity 0.1g ( =
0.46 ); (b) 0% eccentricity 0.4g ( = 1.86 ); (c) 5% eccentricity 0.1g
= 0.48 ;
(d) 5% eccentricity 0.4g ( = 1.904 ); (e) 10% eccentricity 0.1g ( = 0.5 ); (f)
10% eccentricity 0.4g ( = 1.99 ); (g) 15% eccentricity 0.1g ( = 0.52 ); (h) 15% eccentricity 0.4g ( = 2.08 )
58
800
800
average
700
Ave+stdev
700
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
extended N2 basic N2
average Ave+stdev Ave-stdev extended N2 basic N2
100
100 0
0 0
1
2
3
4
5
0
6
2
4
6
8
10
12
Drift (%)
Drift (%)
(a)
(b) 800
800
average
average
Ave+stdev
700
Ave+stdev
Ave-stdev
Ave-stdev
200
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
100
100
700 600
) m c ( 500 t h g i e400 h y e r 300 o t S
extended N2 basic N2
0
extended N2 basic N2
0 0
1
2
3
4
5
6
0
7
2
4
6
8
10
12
Drift (%)
Drift (%)
(c)
(d)
800
800
average
average
700
Ave+stdev
700
Ave+stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev
) 600 m c ( t 500 h g i e400 h y e300 r o t S200
Ave-stdev
extended N2 basic N2
100
extended N2 basic N2
100
0
0 0
1
2
3
4
5
6
7
0
2
4
6
8
10
12
Drift (%)
Drift (%)
(e)
(f)
800
800
average
700
Ave+stdev
700
)600 m c ( t500 h g i e400 h y e300 r o t S200
Ave-stdev
)600 m c ( t500 h g i e400 h y e300 r o t S200
extended N2 basic N2
100
average Ave+stdev Ave-stdev extended N2 basic N2
100
0
0 0
1
2
3
4
5
6
7
0
Drift (%)
2
(g)
4
6
Drift (%)
8
10
12
(h)
Figure 4.16. Drift result for 2-storey building: (a) 0% eccentricity 0.6g ( = 2.78 ); (b) 0% eccentricity 1g ( = 4.64 ); (c) 5% eccentricity 0.6g
= 2.86 ;
(d) 5% eccentricity 1g ( = 4.76 ); (e) 10% eccentricity 0.6g ( = 2.98 ); (f)
10% eccentricity 1g ( = 4.96 ); (g) 15% eccentricity 0.6g ( = 3.13 ); (h) 15% eccentricity 1g ( = 5.21 ) 59
3500
3500
average
average 3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o1000 t S
Ave-stdev extended N2 basic N2
500
3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t S1000
Ave-stdev extended N2 basic N2
500
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
1
2
3
4
5
Drift (%)
Drift (%)
(a)
(b)
3500
3500
average
average
3000
Ave+stdev
3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t S1000
Ave-stdev
) m2500 c ( t h 2000 g i e h y 1500 e r o t S 1000
Ave-stdev
extended N2 basic N2
500
extended N2 basic N2
500
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
1
2
Drift (%)
3
4
5
Drift (%)
(c)
(d)
3500
average
3000
3500
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o1000 t S
average
3000
Ave-stdev
Ave+stdev
) m2500 c ( t h2000 g i e h y 1500 e r o 1000 t S
extended N2 basic N2
500
Ave-stdev extended N2 basic N2
500
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
1
2
Drift (%)
3
4
5
Drift (%)
(e)
(f)
3500
average
3500
3000
Ave+stdev
3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t 1000 S
Ave-stdev
) m2500 c ( t h 2000 g i e h y 1500 e r o 1000 t S
Ave-stdev
extended N2 basic N2
500
average
extended N2 basic N2
500
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
1
Drift (%)
2
3
4
5
Drift (%)
(g) (h) Figure 4.17. Drift result for 8-storey building: (a) 0% eccentricity 0.1g ( =
0.85 ); (b) 0% eccentricity 0.4g ( = 3.4 ); (c) 5% eccentricity 0.1g
= 0.85 ;
(d) 5% eccentricity 0.4g ( = 3.4 ); (e) 10% eccentricity 0.1g ( = 0.84 ); (f)
10% eccentricity 0.4g ( = 3.36 ); (g) 15% eccentricity 0.1g ( = 0.83 ); (h) 15% eccentricity 0.4g ( = 3.31 )
60
3500
3500
average
average
3000
Ave+stdev
3000
Ave+stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t S1000
Ave-stdev
) 2500 m c ( t h2000 g i e h y1500 e r o t S1000
Ave-stdev
extended N2 basic N2
extended N2 basic N2
500
500 0
0 0
1
2
3
4
5
6
7
0
1
2
Drift (%)
3
4
5
6
7
Drift (%)
(a)
(b)
3500
3500
average
average
3000
Ave+stdev
3000
) 2500 m c ( t h2000 g i e h y1500 e r o t 1000 S
Ave-stdev
) 2500 m c ( t h2000 g i e h y1500 e r o1000 t S
extended N2 basic N2
Ave+stdev Ave-stdev extended N2 basic N2
500
500
0
0 0
1
2
3
4
5
6
0
7
1
2
3
4
5
6
7
Drift (%)
Drift (%)
(c) (d) Figure 4.18. Drift result for 8-storey building: (a) 0% eccentricity 0.6g ( = 5.1 );
(b) 5% eccentricity 0.6g
= 5.1 ; (c) 10% eccentricity 0.6g ( = 5.04 ); (d)
15% eccentricity 0.6g ( = 4.96 ) 8000
8000
7000
7000
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
average Ave+stdev Ave-stdev extended N2 basic N2
average Ave+stdev Ave-stdev extended N2 basic N2
1000
1000
0
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.35
0.2
0.4
0.6
0.8
1
1.2
1.4
1
1.2
1.4
Drift (%)
Drift (%)
(a)
(b)
8000
8000
7000
7000
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
average Ave+stdev Ave-stdev extended N2
average Ave+stdev Ave-stdev extended N2
basic N2
basic N2 1000
1000
0
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.35
Drift (%)
0.2
0.4
0.6
0.8
Drift (%)
(c)
(d)
61
8000
8000
7000
7000
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev
average Ave+stdev Ave-stdev extended N2 basic N2
1000
average
extended N2 basic N2
1000
0
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.2
0.4
Drift (%)
0.6
0.8
1
1.2
1.4
1
1.2
1.4
Drift (%)
(e)
(f)
8000
8000
7000
7000
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
average Ave+stdev Ave-stdev extended N2 basic N2
1000
average Ave+stdev Ave-stdev extended N2 basic N2
1000
0
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.2
0.4
Drift (%)
0.6
0.8
Drift (%)
(g) (h) Figure 4.19. Drift result for 20-storey building: (a) 0% eccentricity 0.1g
( = 0.38 ); (b) 0% eccentricity 0.4g ( = 1.51 ); (c) 5% eccentricity 0.1g = 0.38 ; (d) 5% eccentricity 0.4g ( = 1.51 ); (e) 10% eccentricity 0.1g
( = 0.38 ); (f) 10% eccentricity 0.4g ( = 1.5 ); (g) 15% eccentricity 0.1g ( = 0.37 ); (h) 15% eccentricity 0.4g ( = 1.48 )
8000
8000
7000
7000
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
average Ave+stdev Ave-stdev extended N2 basic N2
average Ave+stdev Ave-stdev extended N2
1000
1000
basic N2
0
0 0
0.5
1
1.5
2
0
0.5
1
1.5
2
2.5
3
3.5
Drift (%)
Drift (%)
(a)
(b)
8000 7000
average
8000
Ave+stdev
7000
Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
average Ave+stdev Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
extended N2 basic N2
1000
extended N2 basic N2
1000
0
0 0
0.5
1
1.5
2
0
Drift (%)
(c)
0.5
1
1.5
(d)
62
2
Drift (%)
2.5
3
3.5
8000 7000
average
8000
Ave+stdev
7000
average Ave+stdev
Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
extended N2 basic N2
1000
extended N2 basic N2
1000
0
0 0
0.5
1
1.5
2
0
Drift (%)
0.5
1
(e)
1.5
2
2.5
Drift (%)
3
3.5
(f)
8000
8000
average
7000
Ave+stdev Ave-stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
extended N2 basic N2
average
7000
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev
1000
extended N2 basic N2
1000
0
0 0
0.5
1
1.5
2
0
0.5
1
1.5
Drift (%)
2
2.5
3
3.5
Drift (%)
(g) (h) Figure 4.20. Drift result for 20-storey building: (a) 0% eccentricity 0.6g
( = 2.27 ); (b) 0% eccentricity 1g ( = 3.79); (c) 5% eccentricity 0.6g = 2.27 ; (d) 5% eccentricity 1g ( = 3.78 ); (e) 10% eccentricity 0.6g
( = 2.25 ); (f) 10% eccentricity 1g ( = 3.75 ); (g) 15% eccentricity 0.6g ( = 2.22 ); (h) 15% eccentricity 1g ( = 3.71 )
8000
8000
average
7000
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev
average
7000
Ave+stdev
) 6000 m c ( t 5000 h g i e 4000 h y e 3000 r o t S 2000
extended N2 basic N2
1000
Ave-stdev
extended N2 ba sic N2
1000
0
0 0
1
2
3
4
5
0
Drift (%)
1
(a)
2
Drift (%)
3
4
5
(b)
8000
8000
average
average
7000
Ave+stdev
7000
Ave+stdev
) 6000 m c ( t 5000 h g i e4000 h y e3000 r o t S2000
Ave-stdev
) 6000 m c ( t 5000 h g i e 4000 h y e 3000 r o t S 2000
Ave-stdev
extended N2 basic N2
extended N2 basic N2
1000
1000
0
0 0
1
2
3
4
0
5
Drift (%)
1
2
3
Drift (%)
(c)
(d)
63
4
5
Figure 4.21. Drift result for 20-storey building: (a) 0% eccentricity 1.4g
( = 5.3 ); (b) 5% eccentricity 1.4g
= 5.29 ; (c) 10% eccentricity 1.4g
( = 5.25 ); (d) 15% eccentricity 1.4g ( = 5.19 )
4.4. Analytical study
In the conventional pushover analysis, there are two important steps to do, i.e. how to determine the lateral load pattern and how to use response spectrum and pushover curve to determine the target displacement. Extended N2 method uses the inelastic design spectra to obtain the target displacement. Therefore, this method does not need any iteration process. Moreover, Extended N2 method combines the basic N2 with linear dynamic analysis called response spectrum analysis (RSA) to take account the higher mode effect. This method is done by taking the larger value of the seismic demand, i.e displacement, drift, and coefficient of torsion for each degree of freedom which is obtained in basic N2 and RSA. The RSA need to be normalized until the roof target displacement of RSA is the same as the basic N2. The main idea of the extended N2 method is by assuming that the behavior of higher mode effects will be in elastic behavior. Basic N2 will capture nonlinearity for the first mode since the load pattern is based on linear assumed displacement shape. RSA will capture the elastic higher mode effect. In the other side, the proposed method will consider that inelastic behavior of the higher mode effect could occur especially when the earthquake becomes so intense. The proposed method also uses the inelastic response spectra like the N2 method does to keep the simplicity in obtaining the target displacement with 4 modifications based on the real behavior (NRHA result). The idea of the proposed method is include the higher mode effect in elevation by using the lateral load pattern described in 1st modification. This first modification modifies the basic N2 method by including the higher mode contribution. The result will get nearer to the NRHA result rather than the basic N2. The next modification is including the higher mode effect in torsion by using reduction factor to the target displacement described in 2nd modification. By these two modifications, the target displacement could be defined more accurate than the conventional pushover with triangular assumed displacement shape.
64
The next two modifications (3 rd and 4th modification) try to capture the higher mode effect and dynamic response. Since the PO result is nonlinear static analysis, therefore the target displacement should be enlarged to adjust the static result become dynamic result. The 3rd and 4th modification describes how to deal with the coefficient of torsion and the drift, respectively. st
nd
1 and 2 modification:
From Figure 4.1 to Figure 4.7, it is obtained that the target displacement resulted by extended N2 method still too conservative. This is shown by the result which greater than the value of average plus one times standard deviation from NRHA. The reason of this phenomenon is because this research only uses linear assumed displacement shape or inverted triangular for basic N2 that will ignore the contribution of higher mode in elevation. Moreover, in the same ductility, the error of the displacement increases as the eccentricity increases. It is caused by the same target top displacement result produced by pushover analysis. In the other hand, The NRHA generates smaller target top displacement in larger eccentricity as the effect of higher mode effect in plan. This caused larger difference between the result of pushover analysis and NRHA in larger eccentricity. Above explanation requires an improvement to take the higher mode effects both in elevation and in plan into account. Higher mode effect in elevation is solved by using combination of modal load pattern based on Eq. (4.1). Effective mass factor is used to represent the contribution of each mode. Figure 4.22shows the difference between the linear assumed displacement shape and displacement shape resulted from proposed load pattern.
∙ ∙Г ∙ ∙ Г ∙∙ ∙∙ =
(
)2
(4.1)
=1
=
=1 =1
=
=1
2
2
2
=1
65
(4.2)
(4.3)
where
∙∅ Г
(4.4)
=
is horizontal loading at floor j;
normalized mode shape in mode i and floor j; mode shape i; mode shape i;
is total mass in floor j;
is
is the modal participation factor for
is the spectral acceleration of the response spectra in the period of
∅ ∙ ∙Г ∙
is the effective mass factor for mode shape i which equal to
effective modal mass of mode i divided by total mass;
is mode shape in mode i
and floor j; n is number of mode; k is number of floor;
is assumed displacement
shape in floor j. As shown in Eq.(4.1),
represents the equivalent
static force at floor j. This force from each mode is combined using SRSS rule.
8000
8000
proposed load pattern
proposed load pattern 7000
7000
basic N2 method
)6000 m c ( t5000 h g i e4000 h y e3000 r o t S2000
)6000 m c ( t5000 h g i e4000 h y e3000 r o t S2000
1000
1000
factor and
basic N2 method
0
0 0
0.2
0.4
0.6
0.8
1
1.2
0
Normalized load pattern
0.2
0.4
0.6
0.8
1
1.2
Normalized assumed displaecment shape
(a)
(b)
Figure 4.22. Linear assumed displacement vs proposed displacement shape in 20-storey building 5% eccentricity: (a) load pattern; (b) as sumed displacement shape
Compared to the basic N2, the proposed load pattern will give better results, since we consider the contribution of higher mode. Applying the proposed load pattern to the building will result smaller target top displacement such the level of conservatism will decrease. This equation is different with the other methods mentioned in literature review. Effective mass factor is included in the equation as weight factor to represent the contribution of each mode. By using this new approximation in determining the lateral force pattern, the participation weight of different modes can be considered. The number of modes which should be taken into account is when the total of effective mass factor of those modes is larger than 90%. Higher mode effect in plan (torsional effect) has not been include in the above
66
modification, thus second modification should be added to take the torsional effect into account. In the FEMA-356 and ATC-40, some iteration steps that require much time are needed to obtain the target displacement. N2 method has no iteration while using the inelastic response spectra to define the target displacement. Moreover, the target displacement resulted by N2 is similar with FEMA-356 and ATC-40as explained in the literature review and in another paper (Causevic & Mitrovic, 2011). Therefore, this research adopts the N2 method (Fajfar, 2000) in using the inelastic response spectra to get the target displacement. In torsional dominant building, basic PO will result similar value of target displacement as in symmetric building. In the other hand, The NRHA produces smaller target displacement in larger eccentricity as described previously from Figure 4.1 to Figure 4.7. Therefore, a modification is made based on the NRHA result in
2-,8-,and 20- storey building with 0%, 5%, 10%, 15% of eccentricity. Several pga are chosen to make the level of inelasticity,
, vary from less than 1 to 5. The reduction
factor of maximum top displacement resulted by NRHA for all of the investigated building is shown in Figure 4.23. From the real behavior (NRHA), actually the intensity does not affect so much in the reducing factor. The main factor is the eccentricity itself. When the eccentricity higher, the reduction factor also larger. Torsional mode effective mass factor is used to represent the eccentricit y and torsional effect that contribute to the building. The second modification to come up the torsional effect is described in Eq.(4.5).
∙ − n
dtt = dtti
1
0.8789
(4.5)
i=1
where dtt is final target displacement at top floor; dttiPO is initial target displacement at top floor resulted from the PO analysis in 1 st modification;
− 1
0.8789
target
n i=1
is regression line to determine the reduction factor for
displacement
x-direction;
The value of
represented
by
torsional
effective
mass
factor
in
is the effective mass factor of torsional mode in x-dir at mode-i.
n i=1
becomes larger in larger eccentricity.
represent the degree of torsional effect.
67
is used to
Figure 4.23. Reduction factor to the target top displacement considering the torsion resulted by NRHA
By two modifications explained above, the modified load pattern and target displacement, the final target displacement will have better accuracy compare with the NRHA.
rd
3 modification:
Higher mode effect in plan, torsional effect, is represented by coefficient of torsion (CT) which equal to the comparison value between the displacement at the flexible edge and displacement at center of mass (CM) in the same level of storey. The extended N2 assume that the higher mode effect will keep in elastic, thus use the RSA to capture the higher mode effect. By this assumption, the result from extended N2 method will be more conservative in larger earthquake (higher degree of inelasticity) since the large earthquake produce smaller seismic demand. This is because in large earthquake, the real CT value will be decreased due to the inelas ticity of the structure, while the CT value from the RSA will keep in large value. This cause the RSA tend to give conservative result especiall y in large earthquake. In the other hand, the pushover result (basic N2) usually tends to underestimate the CT value especially in small earthquake. A study to the real behavior of the top floor of the building through NRHA is done to understand the contribution of pushover and RSA using the inelasticity of the structure as the parameter. The result of the study is drawn in Figure 4.24. Based on the real behavior that the contribution of RSA should be decreased and contribution of PO should be increased while the 68
degree of inelasticity,
, increases, the proposed method want to combine the result
from the PO result and RSA, such the conservatism caused by the assumption that
higher mode will keep in elastic behavior will be reduced. The weight factor that based on the degree of inelasticity,
,
is used to define the participation of the PO
and the RSA. Eq. (4.6) is used to get the weight factor both for PO and RSA, and the final CT value of the system. The multiplier or the weight factor both for PO and RSA is drawn in Figure 4.25.
∙ − ∙ 2
CTj final =
+ 0.9
2
CTj PO + 1
+ 0.9
CTj RSA
(4.6)
where CTjfinal is the final coefficient of torsion in floor j; CTj PO is the coefficient of torsion in floor j resulted from PO; CTj RSA is coefficient of torsion in floor j resulted from RSA. Note that the PO in this formula should be defined as the basic N2 with two modifications described previously.
(a)
(b)
Figure 4.24. The multiplier or weight factor of CT value resulted from NRHA: (a)
CTt PO multiplier; (b) CTt RSA multiplier
69
Figure 4.25. The proposed multiplier of PO and RSA to calculate the final coefficient of torsion By this third modification, the assumption used in extended N2 that the structure will have elastic behavior of higher mode is eliminated and take into account the inelastic higher mode effect by giving the weight factor both for PO and RSA.
th
4 modification:
Maximum inter-storey drift resulted from NRHA is not directly derived from the maximum storey displacement. This is because the max inter-storey drift in NRHA possibly happens not in every maximum storey displacement. These higher mode and dynamic effect, called “whip effect”, is explained in Figure 4.26. It is shown that in lower level of the structure, maximum inter-storey drift happen in the similar value as the maximum displacement in corresponding level occur. In upper level of the building, the maximum inter-storey drift happens in much smaller value compare with the maximum displacement of corresponding level. When the max drift resulted from NRHA (Figure 4.26) is decomposed to the displacement in each storey level, called max drift displacement, the result shows that max drift displacement gives larger value in upper storey level, while the lower storey is keep the same. This phenomenon is explained in Figure 4.27.
70
Figure 4.26. The “whip effect” on chi-chi earthquake: 20-storey 1.4g
(a)
μ ≈ 5
(b)
(c) Figure 4.27. The max storey displacement versus the max drift displacement in 0% eccentricity building: (a) 2-storey 0.4g; (b) 8-storey 0.4g; (c) 20-store y 0.4g 71
From this phenomenon, the target displacement from the PO result actually needs to be modified for calculating the inter-storey drift purpose. Figure 4.28 and Figure 4.29shows the multiplication factor of target displacement in top floor and the shape of the multiplication factor in each floor, respectively, resulted both from the ground motion and the proposed coefficient. The proposed coefficient which considers the height of the building, the ductility (degree of inelasticity), and the fundamental period of the building is described in Eqs. (4.7) to (4.10). Eq. (4.10) describes the contribution of the stiffness of the building which is represented by the natural period of the building. When the natural period is larger, the stiffness is smaller, and the effect of higher mode effect is larger, hence the multiplication of the top floor becomes larger. Figure 4.28 also explains that when the ductility increases until the value is equal to 3, the multiplier of the top floor also increases. This is because the structure has its both elastic behavior and inelastic behavior. Then, when the ductility becomes so large, which is assumed to be larger than 3, the effect of the inelastic behavior will control the behavior such the multiplication value becomes smaller. The parabolic approach is made to catch that behavior resulted by NRHA and concluded in Eq. (4.9). The value 1.03 shows that the peak multiplication value is increase by 3% from the linear state of the structure. The value of 3 describe that the peak value is happen in ductility equal to 3. Eq. (4.8) is made based on the Figure 4.29where in the lower level of the structure (assumed equal to 25% of the total height), the multiplication factor will equal to 1, and will increase linearly to the top of the building to the value obtained in (4.9).
∙γ γ − ∙ γ − γ ≥ γ γ ∙γ − μ − ∙γ μ≥ γ ∙ − dmaxdrift
j
=
hj
0.25ht
r
= 1.03
T
where dmaxdrift
j is
1
r
0.75ht
−
j
= dtj
+1
;
3
T
133.33
= 0.01
1+
(4.7)
j
j
1 and
1
=1
(4.8)
2
;
1
(4.9)
T
+1
(4.10)
the dummy displacement in floor- j which is used to calculate
the max inter-storey drift; dmax
−
j is
the target displacement in floor- j resulted from 72
the 1st and 2nd modification presented previously;
γ
γ
j
is the multiplication factor
which include the effect of storey height, ductility, and period of the structure; hj is
γ
the height of the floor; ht is the total height; building; period
T
μ
r
is multiplication factor at roof of the
is the stiffness effect of the structure represented by fundamental
of the structure;
is the ductility;
is the change point from the
acceleration constant to the velocity constant in demand spectra.
(b)
(a)
γ
Figure 4.28. The value of max drift displacement divided by the max storey displacement in the top floor: (a) NRHA result; (b) proposed
(a)
(b)
73
r
(c)
(d)
(e)
(f)
Figure 4.29. The value of decomposed max drift displacement divided by the max
γ γ
storey displacement: (a) 2-storey by ground motion; (b) 2-storey by proposed coefficient, j ;(c) 8-storey by ground motion; (d) 8-storey by proposed coefficient,
j ;(e)
20-storey by ground motion; (f) 20-storey by proposed coefficient,
γ
j
4.5. Step-by-step procedure of proposed method
In this section, step-by-step of proposed method will be described. The whole procedure is similar with the basic N2 method and additional four modifications.
Figure 4.31shows the flowchart of the proposed method. 1. a. Calculate mass of each floor center of the building.
and be placed at x% of eccentricity from the
This step is to make the building example become has an eccentricity. x value can
74
be 5, 10, or any value desired. In asymmetric building, this step can be ignored since the building already has its eccentricity.
b. Choose the elastic acceleration spectra and change to AD format. This research takes EC8 with pga equal to 0.4g and 0.1g as the elastic acceleration spectra with 5% damping (see Figure 4.30). The conversion from Sa-T format to Sa-Sd format can be accomplished using Eq. (4.11).
Sde = where Sde and Sae
π
T2 4
2
(4.11)
Sae
are elastic spectral displacement and
elastic
spectral
acceleration, respectively.
1.2
1.2
) g ( n o i t 0.8 a r e l e c c a l a r0.4 t c e p S
) g ( n o i t 0.8 a r e l e c c a l a r0.4 t c e p S
0
0
1
2 3 Period (s)
4
0
5
(a)
0
20 40 60 80 Spectral displacement (cm)
(b)
Figure 4.30. Elastic acceleration spectra: (a) Sa-T format; (b) AD format
2. Run the Pushover analysis Before running the pushover analysis, spatial distribution of lateral force should be determined. As outlined in theory and assumptions chapter, lateral force distribution can be obtained by several steps. The first step is get several dominant mode shape of the building and normalized it. The second step is calculating the modal participation factor and effective mass factor described in Eqs. (4.2) and (4.3), respectively. The third step is obtaining the spectral acceleration in corresponding period of each mode shape and the mass participation factor for each mode. The last step is by following Eq. (4.1), the lateral force distribution is
75
defined and the applied lateral force to the building is normalized value so that at the top of the building equal to one.
3. Transform the MDOF capacity curve to SDOF capacity curve The real MDOF capacity curve system is modified to bilinear system. The FEMA approach where the intersection point between the real MDOF capacity
α
curve and the bilinear curve is 60% of the yield point of the bilinear curve is used with
equal to zero. Eqs.(4.12) to (4.16) are used to transform the MDOF
bilinear curve to SDOF bilinear curve and transform from the base shear-top
Г ∗ ∙∅ Г ∙∅ ∗ ∗∗ ∗ ∗ ∗
displacement format to Sa-Sd format.
=
=
(4.12)
=1
(4.13)
2
=1
(4.14)
=2
where
could be replaced with
=
(4.15)
=
(4.16)
,
,
, or
which are defined as
Г∅ ∗ ∙∅
ultimate base shear, yield base shear, target displacement, or yield displacement, respectively; star rank is used to distinguish the SDOF from MDOF system;
is
defined as modal participation factor for assumed displacement shape;
is
defined as assumed displacement shape at floor j (note that the used assumed displacement shape has been explained at theory and assumptions section); defined as equivalent mass of SDOF system which equal to
is
.
4. Get inelastic acceleration spectra From step 1b and by using Eqs.(4.17) to (4.20), inelastic acceleration spectra can be obtained. This step of defining the inelastic acceleration spectra is the same with N2 method.
76
Sae
μμ μ
Sa = Sd =
μ μ
R =
where
μ
R =
μ − μ 1
Sde
R
T Tc
(4.17)
R
(4.18)
+1
T < TC
(4.19)
T
(4.20)
≥
TC
is displacement ductility factor obtained by dividing the target
displacement with yield displacement; R
μ
is reduction factor due to ductility; TC
is the characteristic period of the ground motion defined as transition period between constant acceleration and constant velocity in the response, i.e. 0.6 second. 5. Seismic demands a. Target displacement at center of mass Target displacement for SDOF system can be obtained by using Eqs. (4.21) and (4.22). These two equations are resulted from combination of Eqs. (4.17) to (4.20). Eq. (4.12) should be used to convert the target displacement from SDOF system to MDOF system. After obtaining the target displacement of MDOF, it should be noted to modify the target displacement to take the torsional effect into account described in second modification section by Eq. (4.5).
∗ μ μ − ∗ ∗ ∗ ∗≥
dtti =
Sde R
1+ R
dtti = Sde T
1
TC
T
TC
T < TC
(4.21) (4.22)
b. Coefficient of torsion
Torsional effect is discussed in term of coefficient of torsion (CT) which is defined as the comparison between the displacement value in the flexible edge and the displacement value in the center of mass. The stiff edge will have smaller displacement value compare with that of in the center of mass, therefore in this research for conservatism purpose, the displacement value in stiff edge is assumed to be the same as that of in the center of mass. Then, the term CT in this research is only for comparison between the displacement value in the flexible edge and the displacement value in the center of mass.
77
Extended N2 method uses the maximum value of CT between RSA and the basic N2 result as it assumes that the higher mode effect will keep in elastic behavior. In the other hand, the proposed method try to include the inelastic behavior of the higher mode by using weight factor to get the combination between PO and RSA described by Eq. (4.6) c. Inter-storey drift at center of mass In this proposed method, as explained in theory and assumption that the maximum inter-storey drift resulted from NRHA is not directly derived from the maximum storey displacement, thus the max drift displacement gives larger value in upper floor of the building than the original max storey displacement. The multiplier of the max storey displacement for calculating storey drift is described in Eqs. (4.7) to (4.10) Eqs. (4.1) to (4.4) F j & j
Run Pushover
Capacity curve
Eqs. (4.12) to (4.13)
Eq. (4.15)
MDOF to SDOF
F-D to SaSd
Eqs. (4.17) to (4.20) Start
m j,, Sae & Sde
Run RSA
Sa & Sd
Linear response
CTRSA
d*tti
Eqs. (4.21) to (4.22)
dtti
Eqs. (4.12) to (4.13)
Eq. (4.5) Eq. (4.6)
CTfinal
CTPO
End
Figure 4.31. Flowchart of the proposed method
78
Eqs. (4.7) to (4.10)
dtt
dmaxdrift-j
End
End
5. VERIFICATION AND DISCUSSION The proposed method is verified by a 14-storey building with 10% eccentricity. The ground motions attached are0.6g and 1g to represent the medium and intense ground motion. This ground motion is chosen to show the effect of nonlinearity towards the seismic demands. The specification of the building and ground motion has been described in Chapter 3. Both extended N2 and proposed method are examined with the NRHA. When the result is between the average value plus one standard deviation and average value minus one standard deviation, it is categorized as zero error. This represents that the degree of conservatism is still reasonable. When the result is larger than the average value plus one standard deviation or smaller than average value minus one standard deviation, the error is calculated. The calculation of error is described in Eqs. (5.1) and (5.2).
%error = %error =
σσ− ∙ −σ−σ− ∙ m ean +
Y
(mean + )
(mean
(mean
)
Y
)
σ −σ
100%
for Y > mean +
(5.1)
100%
for Y < mean
(5.2)
where Y is the seismic demand (displacement, drift, or coefficient of torsion) resulted
σ
both from extended N2 method or proposed method; mean is the average seismic demand from all of the maximum value of the ground motions ‟ response;
is the
standard deviation. Based on the Eqs. (5.1) and (5.2), the negative error represents the conservative result, while the positive error represents the unconservative result. The lateral force patterns and assumed displacement shape for 14-storey building both in extended N2 and proposed method are described in Figure 5.1. The difference of the lateral load pattern becomes clearer when the contribution of higher mode in elevation is larger. Or in the other word, the difference between extended N2 method and the proposed method will be clearer in high rise building. The target displacement can be determined either by calculation sheet in Table 5.1or by graphical way in Figure 5.2. In Table 5.1, It can be concluded that target top displacement resulted by extended N2 method is larger than the proposed method. This occurrence happens especially when the higher mode effect becomes dominant.
79
∅
(a)
Figure 5.1. (a) Normalized lateral force pattern
(b) ; (b) assumed displacement shape
Figure 5.2.Graphical way to obtain the target top displacement of SDOF of 14-storey building, 10% eccentricity, 0.6g resulted by 1 st modification of proposed method.
Table 5.1. Calculation sheet for defining target top displacement for 14-storey building with 10% eccentricity. Basic N2 and Extended N2 method
Proposed method
m (kg sec 2 /cm)
5026
5716
ɼ
1.46
1.32
Dy (cm)
25.97
25.97
Fy = Fu (kN)
15432
15432
Note
Bilinear approximation of MDOF and SDOF
∗
∙
80
∗
Dy (cm)
17.83
19.61
Fy = Fu (kN)
10597
11655
T (s)
1.81
1.93
Say (g)
0.219
0.212
yield pga (g)
-
0.273
-
0.033
Sde (cm)
67.5
71.95
Sae (g)
0.829
0.777
R
3.784
3.669
dtti (cm)
67.5
71.95
dtti (cm)
98.3
95.32
dtt (cm)
98.3
92.52
Sde (cm)
40.48
43.17
Sae (g)
0.497
0.466
R
2.27
2.2
dtti (cm)
40.48
43.17
dtti (cm)
58.98
57.19
dtt (cm)
58.98
55.54
∗ ∗ ∗
n
i=1
Calculate target displacement in 1g of pga
Calculate target displacement in 0.6g of pga
μ∗ μ∗
The displacement result at the center of mass, coefficient of torsion result, and inter-storey drift result at the center of mass are graphed in Figure 5.3, Figure 5.4, and Figure 5.5, respectively. In those figure, the basic N2 method, extended N2 method, proposed method, and the ground motions are compared. The ground motions results result s are summarized become three value which are average from all of the maximum value of each ground motions, and plus minus one times standard deviation. From Table 5.2, the maximum absolute error of displacement induced by both basic N2 and extended N2 is 5.06% and 6.07% for pga equal to 0.6 and 1g, respectively. The error is induced from the linear assumption of the displacement shape and the ignorance of the reduction effect in target displacement in CM caused by torsion (described previously in 1 st and 2nd modification). The basic N2 and extended N2 have the same value, because pushover result dominates the target displacement instead of RSA result. In the other hand, the maximum absolute error of displacement induced by proposed method is 0.47% and 0.39% for pga equal to 0.6 and 1g, respectively respec tively.. The small error (less than 5% for displacement) resulted from proposed method shows that by 1st and 2nd modifications, the proposed method can capture the effect of the 81
higher mode effect both in elevation and in plan toward the target displacement.
5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0
Average Ave+Stdev Ave-Stdev Basic N2 Extended N2 Proposed method
0
15 30 45 Displacement (cm)
60
(a)
(b) Figure 5.3.Displacement result at center of mass for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g
Table 5.2. The displacement error resulted from basic N2, extended N2, and proposed method at center of mass Error percentage (%) Height (cm)
0.6g ( =2.2) =2.2) 1g ( =3.7) =3.7) Floor level basic N2 Extended Proposed basic N2 Extended Proposed N2 method method method N2 method method method
5200
14
-0.85
-0.85
0.00
-4.81
-4.81
0.00
4850
13
-1.12
-1.12
0.00
-5.08
-5.08
0.00
82
4500
12
-1.48
-1.48
0.00
-5.54
-5.54
0.00
4150
11
-1.98
-1.98
0.00
-6.00
-6.00
0.00
3800
10
-2.73
-2.73
0.00
-6.07
-6.07
-0.25
3450
9
-3.67
-3.67
0.00
-5.85
-5.85
-0.39
3100
8
-4.58
-4.58
0.00
-5.36
-5.36
-0.28
2750
7
-5.02
-5.02
-0.16
-4.52
-4.52
0.00
2400
6
-5.06
-5.06
-0.47
-3.39
-3.39
0.00
2050
5
-4.55
-4.55
-0.21
-2.20
-2.20
0.00
1700
4
-3.94
-3.94
0.00
-1.06
-1.06
0.00
1350
3
-3.28
-3.28
0.00
-0.20
-0.20
0.00
1000
2
-2.60
-2.60
0.00
0.00
0.00
0.00
500
1
-1.56
-1.56
0.00
0.00
0.00
0.00
5500
Average Ave+Stdev Ave-Stdev Basic N2 Extended N2 Proposed method
5000 4500 4000 3500 3000 2500 2000 1500 1000 500 1.04 1.08 1.12 1.16 1.2 1.24 Coefficient of torsion, CT
(a) 5500
Average Ave+Stdev Ave-Stdev Basic N2 Extended N2 Proposed method
5000 4500 4000 3500 3000 2500 2000 1500 1000 500 1.04
1.08 1.12 1.16 Coefficient of torsion, CT
1.2
(b) Figure 5.4.Coefficient of torsion result for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g
83
Table 5.3. The coefficient of torsion error resulted from basic N2, extended N2, and proposed method Error percentage (%) 0.6g ( =2.2) 1g ( =3.7) Height Floor (cm) level basic N2 Extended Proposed basic N2 Extended Proposed method N2 method method method N2 method method 5200
14
3.65
0.00
0.00
1.94
-4.96
0.00
4850
13
4.01
0.00
0.00
1.62
-4.94
0.00
4500
12
4.32
0.00
0.11
1.44
-4.92
0.00
4150
11
3.98
0.00
0.00
1.37
-4.99
0.00
3800
10
3.87
0.00
0.00
1.48
-5.59
0.00
3450
9
3.89
0.00
0.00
1.61
-6.09
0.00
3100
8
3.99
0.00
0.00
1.82
-6.13
0.00
2750
7
3.87
0.00
0.00
2.06
-6.14
0.00
2400
6
3.66
0.00
0.00
2.40
-6.20
0.00
2050
5
3.24
0.00
0.00
2.69
-6.13
0.00
1700
4
3.18
0.00
0.00
2.77
-5.69
0.00
1350
3
3.18
0.00
0.00
2.76
-5.20
0.00
1000
2
3.19
0.00
0.00
2.88
-5.07
0.00
500
1
3.24
0.00
0.00
3.15
-4.75
0.00
5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0
Average Ave+Stdev Ave-Stdev Basic N2 Extended N2 Proposed method
0
0.4
0.8 1.2 Drift (%)
1.6
(a)
84
2
5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0
Average Ave+Stdev Ave-Stdev Basic N2 Extended N2 Proposed method
0
1
2
3
Drift (%)
(b) Figure 5.5.Inter-storey drift result at center of mass for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g
Table 5.4. The inter-storey drift error resulted from basic N2, extended N2, and proposed method at center of mass Error percentage (%) Height (cm)
Floor level
0.6g ( =2.2)
1g ( =3.7)
basic N2 Extended Proposed method N2 method method
basic N2 Extended Proposed method N2 method method
5200-4850 14-13
33.97
-7.46
0.00
30.37
-25.34
-0.41
4850-4500 13-12
29.88
-10.79
0.00
24.13
-29.65
0.00
4500-4150 12-11
18.83
-13.31
0.00
8.99
-33.08
-2.18
4150-3800 11-10
0.39
-5.96
0.00
0.00
-8.34
0.00
3800-3450
10-9
0.00
0.00
0.00
0.00
-4.55
0.00
3450-3100
9-8
0.00
0.00
0.00
0.00
-5.39
-3.43
3100-2750
8-7
0.00
0.00
0.00
-0.85
-6.62
-7.15
2750-2400
7-6
0.00
0.00
0.00
-4.72
-7.35
-9.64
2400-2050
6-5
0.00
0.00
-0.66
-6.09
-6.25
-9.58
2050-1700
5-4
-2.25
-2.25
-4.07
-5.20
-5.20
-7.12
1700-1350
4-3
-5.13
-5.13
-5.37
-2.91
-2.91
-3.28
1350-1000
3-2
-5.04
-5.04
-1.31
-1.74
-1.74
0.00
1000-500
2-1
-3.52
-3.52
0.00
-0.60
-0.60
0.00
500-0
1-0
-1.56
-1.56
0.00
0.00
0.00
0.00
From Figure 5.4, the coefficient of torsion resulted by NRHA is smaller when the inelastic degree is larger. This indicates that the extended N2 method will result larger 85
degree of conservatism in larger ductility. The value of coefficient of torsion in extended N2 method is relatively the same in all intensity because the RSA result which has the same result in arbitrary pga usually takes control. In the other hand, by rd
giving weight factor described in 3 modification, the proposed method reduces the level of conservatism. From Table 5.3, basic N2 usually underestimate the result, while the extended N2 overestimate it. Both extended N2 and proposed method give very good result in pga equal to 0.6g, which are 0% and 0.1% error for extended N2 and proposed method, respectively. When the pga equal to 1g, the maximum error of CT generated by basic N2 and extended N2 method is 3.15% and 6.2%, respectively. In the other hand the proposed method still gives good result which is zero error. This coefficient of torsion actually is not the final result of seismic demand. The coefficient of torsion should be multiplied by the displacement and drift at center of mass to define the displacement and drift at flexible edge. The comparison of displacement and drift at flexible edge are described in Figure 5.6 and Figure 5.7, respectively. Moreover, the comparisons of those errors are described in Table 5.5 and Table 5.6. From Figure 5.5, it is drawn that all of the method gives small error in the middle of the building. The proposed method becomes better than basic N2 and extended N2 method in the upper level of the building. From Table 5.4, the maximum absolute error of inter-storey drift produced by basic N2 method is 33.97% and 30.37% for pga equal to 0.6 and 1g, respectively. The maximum absolute error of inter-storey drift induced by extended N2 is 13.31% and 33.08% for pga equal to 0.6 and 1g, respectively. The maximum absolute error of inter-storey drift produced by proposed method is 5.37% and 9.64% for pga equal to 0.6 and 1g, respectively. The error produced by extended N2 method becomes larger in larger inelastic degree of a structure. This is caused by the assumption that the higher mode effect keep in elastic state. In the other hand, because the proposed method is made based on the real behavior of the building, the error generated is below 10% which is very small.
86
5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0
Average Ave+Stdev Ave-Stdev Basic N2 Extended N2 Proposed method
0
15 30 45 60 Displacement (cm)
75
(a)
(b) Figure 5.6.Displacement result at flexible edge for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g
From Figure 5.6 and Table 5.5, it is shown that the error from basic N2 is reduced compare with the error produced in center of mass, while the extended N2 method produce larger error in flexible edge compare with the error produced in center of mass. It is because the coefficient of torsion (CT) value resulted by basic N2 is unconservative and the displacement at the CM is overestimated, hence make the result in flexible edge eventually become better. In the other side, both CT value and displacement at CM resulted by the extended N2 are overestimated, hence make the result in flexible edge become much more overestimated. The proposed method gives better result both in center of mass and in flexible edge because the CT value and displacement at center of mass resulted by proposed method have no error and small 87
error, respectively. From Table 5.5, the maximum absolute error of displacement at flexible edge produced by basic N2 method is 0% and 2.13% for pga equal to 0.6 and 1g, respectively. The maximum absolute error of displacement at flexible edge induced by extended N2 is 8.12% and 13.51% for pga equal to 0.6 and 1g, respectively. The maximum absolute error of displacement at flexible edge produced by proposed method is 0% and 0.83% for pga equal to 0.6 and 1g, respectively.
Table 5.5. The displacement error resulted from basic N2, extended N2, and proposed method at flexible edge Error percentage (%) Height (cm)
0.6g ( =2.2) 1g ( =3.7) Floor level basic N2 Extended Proposed basic N2 Extended Proposed N2 N2 method method method method method method
5200
14
0.00
-4.13
0.00
-0.32
-11.40
0.00
4850
13
0.00
-4.38
0.00
-0.56
-11.68
0.00
4500
12
0.00
-4.66
0.00
-1.03
-12.18
0.00
4150
11
0.00
-5.08
0.00
-1.58
-12.76
0.00
3800
10
0.00
-5.81
0.00
-2.02
-13.30
-0.33
3450
9
0.00
-6.71
0.00
-2.13
-13.51
-0.83
3100
8
0.00
-7.34
0.00
-1.63
-13.05
-0.72
2750
7
0.00
-7.79
0.00
-0.81
-12.25
-0.26
2400
6
0.00
-8.10
0.00
0.00
-11.19
0.00
2050
5
0.00
-8.12
0.00
0.00
-10.05
0.00
1700
4
0.00
-7.56
0.00
0.00
-8.96
0.00
1350
3
0.00
-6.76
0.00
0.00
-8.21
0.00
1000
2
0.00
-5.91
0.00
0.00
-7.53
0.00
500
1
0.00
-4.56
0.00
0.00
-6.21
0.00
88
5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0
Average Ave+Stdev Ave-Stdev Basic N2 Extended N2 Proposed method
0
0.4
0.8 1.2 1.6 Drift (%)
2
2.4
(a) 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0
Average Ave+Stdev Ave-Stdev Basic N2 Extended N2 Proposed method
0
1
2 Drift (%)
3
4
(b) Figure 5.7.Inter-storey drift result at flexible edge for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g
From Figure 5.7, it is drawn that all of the method gives small error in the middle of the building. The proposed method becomes better than basic N2 and extended N2 method in the upper level of the building. From Table 5.6, the maximum absolute error of inter-storey drift produced by basic N2 method is 38.51% and 39.57% for pga equal to 0.6 and 1g, respectively. The maximum absolute error of inter-storey drift induced by extended N2 is 14.88% and 33.96% for pga equal to 0.6 and 1g, respectively. The maximum absolute error of inter-storey drift produced by proposed method is 5.71% and 10.73% for pga equal to 0.6 and 1g, respectively. The basic N2 produces large error both in pga equal to 0.6g and 1g. This phenomenon is explained th
in the 4 modification. The error produced by extended N2 method becomes larger in 89
larger inelastic degree of a structure. This is caused by the assumption that the higher mode effect keep in elastic state. In the other hand, because the proposed method is made based on the real behavior of the building, the maximum error generated is only 10% which is very small. Table 5.6. The inter-storey drift error resulted from basic N2, extended N2, and proposed method at flexible edge Error percentage (%) Height (cm)
0.6g ( =2.2) 1g ( =3.7) Floor level basic N2 Extended Proposed basic N2 Extended Proposed method N2 method method method N2 method method
5200-4850
14-13
38.51
-9.09
0.00
39.57
-20.96
0.00
4850-4500
13-12
34.96
-12.39
1.33
32.73
-27.16
0.00
4500-4150
12-11
24.52
-14.88
0.00
18.82
-33.96
0.00
4150-3800
11-10
5.07
-7.04
0.00
0.00
-11.62
0.00
3800-3450
10-9
0.00
0.00
0.00
0.00
-9.61
0.00
3450-3100
9-8
0.00
0.00
0.00
0.00
-11.21
-2.22
3100-2750
8-7
0.00
0.00
0.00
0.00
-13.34
-6.63
2750-2400
7-6
0.00
-1.07
0.00
-0.78
-15.04
-9.92
2400-2050
6-5
0.00
-3.64
-1.70
-2.87
-14.82
-10.73
2050-1700
5-4
0.00
-7.53
-4.99
-2.28
-14.13
-8.58
1700-1350
4-3
-0.68
-9.98
-5.71
-0.04
-11.77
-4.73
1350-1000
3-2
-0.68
-10.01
-1.74
0.00
-10.68
0.00
1000-500
2-1
0.00
-8.26
0.00
0.00
-9.68
0.00
500-0
1-0
0.00
-4.56
0.00
0.00
-6.21
0.00
90
6. CONCLUSION AND SUGGESTION 6.1. Conclusion
Several main conclusions of this research are gathered as follows: 1. Compare with the conventional pushover, the proposed method does not need the iteration process. This is because the proposed method adopts the N2 method that use the inelastic spectra thus does not require the iteration process. 2. In the first modification, the effective mass factor (EMF) included in determining the lateral load pattern effectively contributes to give a better result of target displacement, particularly in high rise symmetric buildings. 3. In the second modification, the reduction factor toward the displacement which includes the torsional effective mass factor (TEMF) successfully gives a better result of the target displacement, particularly in the torsional dominated building. 4. By combining the 1 st and 2nd modification, an asymmetric high rise building can be well predicted in target displacement.
5. In the third modification, the ductility or degree of inelasticity, , is used as the weight factor towards the coefficient of torsion of the Pushover (PO) result and Response Spectra Analysis (RSA) result. Pushover (PO) result represents the nonlinear behavior, while Response Spectra Analysis (RSA) result represents the linear behavior. This third modification gives better result of the Coefficient of Torsion (CT) in reducing the level of conservatism caused by the elastic behavior assumption of the higher modes. 6. In the fourth modification, the multiplication factor towards the target displacement to get the maximum inter-storey drift can predict the actual inter-storey drift. This factor allows nonlinear behavior of higher modes. 7. The proposed method eliminates the assumption that the higher mode effects keep in the elastic behavior and hence reduces the level of either conservatism or unconservatism resulted by the elastic higher mode assumption.
91
6.2.
Suggestion
Based on this research, several suggestions to the future studies are proposed as follows: 1. Since this research only focuses on medium and long period structures, the future studies can observe the validity of this proposed method in short and very long period structures. 2. Include the effect of the strength and stiffness degradation by using the appropriate hysteretic models such as the Takeda model.
92
REFERENCE ATC-40. (1996). Seismic Evaluation and Retrofit of Concrete Buildings . Washington, DC: Applied Technology Council. Causevic, M., & Mitrovic, S. (2011). Comparison between non-linear dynamic and static seismic analysis of structures according to European and US provisions. Bulletin
of
Earthquake
Engineering,
9(2),
467-489.
doi:
10.1007/s10518-010-9199-1 Chopra, A. K., & Goel, R. K. (2002). A modal pushover analysis procedure for estimating seismic demands for buildings. Earthquake Engineering and Structural Dynamics, 31(3), 561-582. doi: 10.1002/eqe.144
Chopra, A. K., & Goel, R. K. (2004). A modal pushover analysis procedure to estimate seismic demands for unsymmetric-plan buildings. Earthquake Engineering & Structural Dynamics, 33(8), 903-927. doi: 10.1002/eqe.380
Chopra, A. K., Goel, R. K., & Chintanapakdee, C. (2004). Evaluation of a Modified MPA Procedure Assuming Higher Modes as Elastic to Estimate Seismic Demands. Earthquake Spectra, 20(3), 757-778. CSI. (2008). PERFORM-3D
Nonlinear Analysis and Performance Assessment 3D
Structures (Version 4.0.4). Berkeley, California: Computers and Structures, Inc. Fajfar, P. (1999). Capacity spectrum method based on inelastic demand spectra. Earthquake Engineering & Structural Dynamics, 28 (9), 979-993. doi:
10.1002/(sici)1096-9845(199909)28:9<979::aid-eqe850>3.0.co;2-1 Fajfar, P. (2000). A Nonlinear Analysis Method for Performance-Based Seismic Design. Earthquake Spectra, 16 (3), 573-592. FEMA-356. (2000). Pre standard and Commentary for the Seismic Rehabilitation of Buildings-FEMA 356 . Washington, DC: Federal Emergency Management
Agency. Hancock, J., Jennie Watson-Lamprey, Abrahamson, N. A., Bommer, J. J., Markatis, A., Mccoy, E., & Mendis, R. (2006). An Improved Method of Matching Response Spectra of Recorded Earthquake Ground Motion using Wavelets. Journal of Earthquake Engineering, 10(1), 1-23.
Kreslin, M., & Fajfar, P. (2011). The extended N2 method considering higher mode 93
effects in both plan and elevation. Bulletin of Earthquake Engineering, 10 (2), 695-715. doi: 10.1007/s10518-011-9319-6 Kunnath, S. K. (2004). Identification of modal combinations for nonlinear static analysis of building structures. Computer-Aided Civil and Infrastructure Engineering, 19(4), 246-259. doi: 10.1111/j.1467-8667.2004.00352.x ̆ ić, D., & Fajfar, P. (2005). On the inelastic seismic response of asymmetric Marus
buildings under bi-axial excitation. Earthquake Engineering & Structural Dynamics, 34(8), 943-963. doi: 10.1002/eqe.463
Ou, Y.-C. (2012). Seismic Resistant Design. Lecture note. Construction Engineering. National Taiwan University of Science and Technology. R. Rofooei, F., K. Attari, N., Rasekh, A., & Shodja, A. H. (2006). Comparison of Static and Dynamic Pushover Analysis in Assessment of the Target Displacement. [Research Paper]. International Journal of Civil Engineering, 4(3), 212-225.
Reyes, J. C., & Chopra, A. K. (2011). Three-dimensional modal pushover analysis of buildings subjected to two components of ground motion, including its evaluation for tall buildings. Earthquake Engineering & Structural Dynamics, 40(7), 789-806. doi: 10.1002/eqe.1060
Rofooei, F. R., Attari, A., Rasekh, A., & Shodja, A. H. (2007). Adaptive Pushover Analysis. Asian Journal of Civil Engineering (Building and Housing), 8 (2007), 343-358. The MathWorks, I. (2009). MATLAB The Language of Technical Computing (Version 7.8.0.347 (R2009a)): The MathWorks, Inc.
94
APPENDIX A.1.
MATLAB Code for extended N2 method for this research
%% input by user n=101; aaa=linspace(0,1,n)'; for zzz=1:n pga=aaa(zzz); coef1=2.5; coef2=0.4; coef3=0.6; Sds=coef1*pga; % Sd1=pga/coef2*coef3; T0=0.1; Ts=Sd1/Sds; TL=3; Tc=0.6; %% Pick-up the data from txt file load 'building data.txt'; load 'PO output.txt'; load 'RSA result.txt'; load 'PO displacement history.txt'; stheight=building_data(:,1); height=cumsum(stheight); ads=building_data(:,2); mass=building_data(:,3); drift=PO_output(:,1); bshear=PO_output(:,2); cmRSA=RSA_result(:,1); f1RSA=RSA_result(:,2); f4RSA=RSA_result(:,3); %% Lateral force distribution & modal participation factor [a b]=size(mass); lfd=zeros(a,b); for i=1:a lfd(i,b)=ads(i,b).*mass(i,b)./mass(a,b); end Ln=mass.*ads; Mstar=mass.*ads.^2; MPF=sum(Ln)/sum(Mstar); %% Target displacement displ=drift*height(a); bshear=bshear*9.8145/1000; % convert kg to KN
95
maxdispl=max(displ); [c d]=size(bshear); area_po=zeros(c,d); area_po(1,1)=0; for i=2:c area_po(i,d)=area_po(i-1,d)+.5*(displ(i,d)-displ(i-1,d))*(bshear(i,d)-bshear(i-1,d))+(d ispl(i,d)-displ(i-1,d))*min(bshear(i,d),bshear(i-1,d)); end max_area_po=max(area_po); x_seed=rand(1,1)*maxdispl; x1=x_seed; diffx_60=1; while diffx_60>10^-10 y1=(max_area_po)/(maxdispl-.5*x1); x_60=0.6*x1; y_60=0.6*y1; cc=y_60-bshear; [e f]=size(cc); dd=zeros(e-1,1); for i=1:e-1 dd=cc(i,f)*cc(i+1,f); end [dd_sort ind]=sort(dd,'ascend'); ind1=ind(1,1); ind2=ind(1,1)+1; x_60_new=interpolate(bshear(ind1),displ(ind1),bshear(ind2),displ(ind2),y_60); diffx_60=abs(x_60-x_60_new); x1=1/0.6*x_60_new; end Dtstar=maxdispl/MPF; Dystar=x1/MPF; Fystar=y1/MPF; Fustar=Fystar; Tstar=2*pi*(sum(Ln)*Dystar*981.45/100/(Fystar*1000))^.5; Say=Fystar*1000/sum(Ln)/9.8145/981.45; if Tstar > 0 && Tstar < T0 Sae=(Sds-pga)/T0*Tstar+pga; elseif Tstar >= T0 && Tstar <= Ts Sae=Sds; elseif Tstar > Ts && Tstar <= TL Sae=Sd1/Tstar; else Sae=Sd1*TL/Tstar^2; end Sde=Tstar^2/4/pi^2*Sae*981.45; Rm=Sae/Say; 96
if Tstar < Tc Sd = Sde/Rm*(1+(Rm-1)*Tc/Tstar); else Sd=Sde; end [g h]=size(aaa); Dt(zzz,1)=MPF*Sd; end X1 = displ; Y1 = bshear; X2 = Dt; Y2 = aaa; %obtain yield disp and pga elastic yield_displ=x1; %or displ(2,1) --> need check ii=1; while yield_displ>X2(ii,1) ii=ii+1; end pga_elastic = interpolate(X2(ii-1,1),Y2(ii-1,1),X2(ii,1),Y2(ii,1),yield_displ); X1mod=max(X2); ii=1; while X1mod>X1(ii,1) && X1(ii,1)
'XColor',,'k' 'XColor' 'k',,'YColor' 'YColor',,'k' 'k'); ); hl2 = line(X2,Y2,'Color' line(X2,Y2, 'Color',,'k' 'k',,'Parent' 'Parent',AX(2)); ,AX(2)); set(gca,'FontName' set(gca,'FontName',,'Timesnewroman' 'Timesnewroman',,'fontSize' 'fontSize',14) ,14) set(get(AX(1),'Ylabel' set(get(AX(1),'Ylabel'), ),'String' 'String',,'Base Shear (kN)',,'FontName' (kN)' 'FontName',,'Timesnewroman' 'Timesnewroman',,'fontSize' 'fontSize',14) ,14) set(get(AX(2),'Ylabel' set(get(AX(2),'Ylabel'), ),'String' 'String',,'pga (g)', (g)','FontName' 'FontName',,'Timesnewroman' 'Timesnewroman',,'fontSize' 'fontSize',14) ,14) set(get(AX(1),'Xlabel' set(get(AX(1),'Xlabel'), ),'String' 'String',,'Target top displacement (cm)',,'FontName' (cm)' 'FontName',,'Timesnewroman' 'Timesnewroman',,'fontSize' 'fontSize',14) ,14) set(get(AX(2),'Xlabel' set(get(AX(2),'Xlabel'), ),'String' 'String',,'Target top displacement (cm)',,'FontName' (cm)' 'FontName',,'Timesnewroman' 'Timesnewroman',,'fontSize' 'fontSize',14) ,14) grid on %% Torsional Effect & Drift & Displacement CTf1=flipud(f1RSA)./flipud(cmRSA); [a b]=size(CTf1); for i=1:a i=1:a if CTf1(i,1)<1 CTf1(i,1)<1 CTf1(i,1)=1; end end CTf1(1,1)=CTf1(2,1); CTf4=flipud(f4RSA)./flipud(cmRSA); [a b]=size(CTf4); for i=1:a i=1:a if CTf4(i,1)<1 CTf4(i,1)<1 CTf4(i,1)=1; end end CTf4(1,1)=CTf4(2,1); pga_i=input('pga= pga_i=input('pga= '); '); Sds=coef1*pga_i; Sd1=pga_i/coef2*coef3; Ts=Sd1/Sds; Dt_i=convert( pga_i, Sds, Sd1, T0, Ts, TL, Tc ); [a b]=size(PO_displacement_history); displmatrix=[PO_displacement_history(:,b),PO_displacement_history(:,2:b-1)]; [c d]=size(displmatrix); iii=1; while Dt_i while Dt_i > displmatrix(iii,d) iii=iii+1; end xx1=displmatrix(iii-1,d); xx2=displmatrix(iii,d);
98
storeydispl=zeros(d,1); for i=1:d i=1:d storeydispl(i,1)=interpolate(xx1,displmatrix(iii-1,i),xx2,displmatrix(iii,i),Dt_i); end figure; plot(storeydispl,height) set(gca,'FontName' set(gca,'FontName',,'Timesnewroman' 'Timesnewroman',,'fontSize' 'fontSize',14) ,14) title('Center title('Center of mass') mass') xlabel('Displacement xlabel('Displacement (cm)') (cm)') ylabel('Height ylabel('Height (cm)') (cm)') hold on normRSA=flipud(cmRSA*max(storeydispl)/max(cmRSA)); p=plot(normRSA,height); p=plot(normRSA,heigh t); set(p,'Color' set(p,'Color',,'red' 'red',,'LineWidth' 'LineWidth',2, ,2,'LineStyle' 'LineStyle',,'--' '--')) legend('basic legend('basic N2', N2','extended N2', N2','Location' 'Location',,'best' 'best')) grid on [a b]=size(height); c=1+(a-2)*2+1; height2=zeros(c,1); height2(1,1)=height(1,1); height2(c,1)=height(a,1); j=2; for i=2:c-1 i=2:c-1 height2(i,1)=height(floor(j),1); j=j+0.5; end drift1=zeros(a-1,1); for i=1:a-1 i=1:a-1 drift1(i,1)=(storeydispl(i+1,1)-storeydispl(i,1))/(height(i+1,1)-height(i,1)); end j=1; for i=1:c i=1:c driftN2(i,1)=drift1(floor(j),1); j=j+0.5; end drift3=zeros(a-1,1); for i=1:a-1 i=1:a-1 drift3(i,1)=(normRSA(i+1,1)-normRSA(i,1))/(height(i+1,1)-height(i,1)); end j=1; for i=1:c i=1:c driftnormRSA(i,1)=drift3(floor(j),1); j=j+0.5; end figure; 99
set(gca,'FontName' set(gca,'FontName',,'Timesnewroman' 'Timesnewroman',,'fontSize' 'fontSize',14) ,14) plot(driftN2,height2) title('Center title('Center of mass') mass') xlabel('Drift xlabel('Drift (%)') (%)') ylabel('Height ylabel('Height (cm)') (cm)') hold on p=plot(driftnormRSA,height2); p=plot(driftnormRSA,height2 ); set(p,'Color' set(p,'Color',,'red' 'red',,'LineWidth' 'LineWidth',2, ,2,'LineStyle' 'LineStyle',,'--' '--')) legend('basic legend('basic N2', N2','extendedN2' 'extendedN2',,'Location' 'Location',,'best' 'best')) grid on CEdrift=driftnormRSA./driftN2; [a b]=size(CEdrift); for i=1:a i=1:a if CEdrift(i,1)<1 CEdrift(i,1)<1 CEdrift(i,1)=1; end end CEdispl=normRSA./storeydispl; [a b]=size(CEdispl); for i=1:a i=1:a if CEdispl(i,1)<1 CEdispl(i,1)<1 CEdispl(i,1)=1; end end CEdispl(1,1)=1; hold off figure; plot([CEdrift],[height2]) set(gca,'FontName' set(gca,'FontName',,'Timesnewroman' 'Timesnewroman',,'fontSize' 'fontSize',14) ,14) hold on title('Coefficient title('Coefficient of higher mode') mode' ) xlabel('CE' xlabel('CE')) ylabel('Height ylabel('Height (cm)') (cm)') p=plot([CEdispl],[height]); set(p,'Color' set(p,'Color',,'red' 'red',,'LineWidth' 'LineWidth',2, ,2,'LineStyle' 'LineStyle',,'--' '--')) legend('CE legend('CE drift', drift','CE displacement', displacement','Location' 'Location',,'best' 'best')) grid on extN2displ_sted=storeydispl.*CTf1.*CEdispl; extN2displ_cm=storeydispl.*CEdispl; extN2displ_fled=storeydispl.*CTf4.*CEdispl; figure; plot([extN2displ_sted],[height]) set(gca,'FontName' set(gca,'FontName',,'Timesnewroman' 'Timesnewroman',,'fontSize' 'fontSize',14) ,14) hold on title('Stiff title('Stiff edge') edge') 100
xlabel('Displacement (cm)') ylabel('Height (cm)') p=plot(storeydispl,height); set(p,'Color','red','LineWidth',2,'LineStyle','--') legend('Extended N2 method','basic N2 method','Location','best') grid on figure; plot([extN2displ_cm],[height]) set(gca,'FontName','Timesnewroman','fontSize',14) hold on title('Center of Mass') xlabel('Displacement (cm)') ylabel('Height (cm)') p=plot(storeydispl,height); set(p,'Color','red','LineWidth',2,'LineStyle','--') legend('Extended N2 method','basic N2 method','Location','best') grid on figure; plot([extN2displ_fled],[height]) set(gca,'FontName','Timesnewroman','fontSize',14) hold on title('Flexible edge') xlabel('Displacement (cm)') ylabel('Height (cm)') p=plot(storeydispl,height); set(p,'Color','red','LineWidth',2,'LineStyle','--') legend('Extended N2 method','basic N2 method','Location','best') grid on [a b]=size(CTf1); CTf1_2=zeros((a-2)*2+2,1); CTf1_2(1,1)=CTf1(1,1); CTf1_2((a-2)*2+2,1)=CTf1(a,1); j=2; for i=2:(a-2)*2+1 CTf1_2(i,1)=CTf1(floor(j),1) j=j+0.5; end [a b]=size(CTf4); CTf4_2=zeros((a-2)*2+2,1); CTf4_2(1,1)=CTf4(1,1); CTf4_2((a-2)*2+2,1)=CTf4(a,1); j=2; for i=2:(a-2)*2+1 CTf4_2(i,1)=CTf4(floor(j),1) j=j+0.5; 101
end extN2drift_sted=driftN2.*CTf1_2.*CEdrift; extN2drift_cm=driftN2.*CEdrift; extN2drift_fled=driftN2.*CTf4_2.*CEdrift; figure; plot([extN2drift_sted],[height2]) set(gca,'FontName','Timesnewroman','fontSize',14) hold on title('Stiff edge') xlabel('Drift (%)') ylabel('Height (cm)') p=plot(driftN2,height2); set(p,'Color','red','LineWidth',2,'LineStyle','--') legend('Extended N2 method','basic N2 method','Location','best') grid on figure; plot([extN2drift_cm],[height2]) set(gca,'FontName','Timesnewroman','fontSize',14) hold on title('Center of Mass') xlabel('Drift (%)') ylabel('Height (cm)') p=plot(driftN2,height2); set(p,'Color','red','LineWidth',2,'LineStyle','--') legend('Extended N2 method','basic N2 method','Location','best') grid on figure; plot([extN2drift_fled],[height2]) set(gca,'FontName','Timesnewroman','fontSize',14) hold on title('Flexible edge') xlabel('Drift (%)') ylabel('Height (cm)') p=plot(driftN2,height2); set(p,'Color','red','LineWidth',2,'LineStyle','--') legend('Extended N2 method','basic N2 method','Location','best') grid on The MATLAB code above is supported by interpolate.m which has content as follows: function [ y ] = interpolate( x1, y1, x2, y2, x ) y = y1+(y2-y1)/(x2-x1)*(x-x1); end
102
The input example of the above MATLAB program in 8-storey 15% eccentricity with pga equal to 0.4g is described as follows: building data.txt 0 0 500 0.234911551 500 0.548076012 310 0.678669741 310 0.781939308 310 0.865439429 310 0.929550794 310 0.973937645 310 1
0 488.0269003 474.6284899 461.2304871 461.2300795 461.2304871 461.2300795 461.2296719 439.3698798
PO output.txt 0 0 2.74E-03 174354 2.87E-03 181911.5 2.99E-03 189126.6 3.12E-03 195925.4 … 4.98E-02 276013.5 4.99E-02 276042.6 5.00E-02 276053.7 5.01E-02 276079.6 RSA result.txt 27.718 25.323 26.96 24.627 25.701 23.474 23.925 21.85 21.651 19.77 18.848 17.207 15.277 13.942 6.5953 6.0126 0 0
29.849 29.036 27.683 25.773 23.326 20.309 16.467 7.1148 0
PO displacement history.txt 0 0 2.74E-03 0 2.87E-03 0 2.99E-03 0 3.12E-03 0
0
0
0
0
0
0
0
0
1.8459
4.3037
5.328
6.1382
6.793
7.2959
7.6438
7.8477
1.937
4.5033
5.5723
6.418
7.1014
7.6263
7.9895
8.2025
2.0368
4.709
5.8211
6.7007
7.4115
7.9576
8.3355
8.5573
2.1414
4.9219
6.0754
6.9872
7.724
8.2901
8.6819
8.9121
103
…. 4.98E-02 0 4.99E-02 0 5.00E-02 0 5.01E-02 0
69.949
135.37
137.52
139.03
140.22
141.15
141.83
142.29
70.131
135.72
137.87
139.38
140.57
141.5
142.18
142.64
70.313
136.08
138.22
139.73
140.92
141.86
142.54
142.99
70.494
136.43
138.58
140.09
141.28
142.21
142.89
143.35
The figures resulted from the above MATLAB program in 8-storey 15% eccentricity with pga equal to 0.4g are drawn as follows:
(a)
(b)
(c)
(d)
104
(e)
(f)
(g)
(h)
(i) (j) Figure A.1. Extended N2 result from the MATLAB code
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Profile
Name
Yusak Oktavianus ( 蔡優光)
Gender Place/Date of birth Nationality Email
Male Jember / October 16, 1988 Indonesia
[email protected] [email protected] Academic profile
2006-2010
2010-2012
Bachelor Degree Civil Engineering Department Petra Christian University Surabaya, Indonesia Master Degree Construction Engineering Department National Taiwan University of Science and Technology (Taiwan Tech) Taipei, Taiwan