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STATIC NONLINEAR
ANALYSIS (2/2)
Advanced Earthquake Engineering
CIVIL-706
Instructor:
Lorenzo DIANA, PhD
1
Advanced Earthquake Engineering CIVIL-706
Content
• Capacity curve
• Seismic damage
• Acceleration-Displacement Response Spectrum (ADRS)
• Performance Point
• Large-scale vulnerability assessment
• N2 method reliability and optimization
• Method result confrontation
Static nonlinear analysis 2
Advanced Earthquake Engineering CIVIL-706
Determination of the performance point
Two of the main static non linear methods to determine the performance point (examples covered here)
• Equivalent linearization (FEMA 440) • Improved Spectrum Method of ATC40
• N2 method (equal displacement rule, EC8 approach)
Static nonlinear analysis 3
Advanced Earthquake Engineering CIVIL-706
Equivalent Linearization Method
A little bit of background…
• First introduced in 1970 in a pilot project as a
rapid evaluation tool (Freeman et al. 1975)
• Basis of the simplified analysis methodology in ATC-40 (1996)
• Improved later in FEMA 440 document (2005)
Static nonlinear analysis 4
Advanced Earthquake Engineering CIVIL-706
Equivalent Linearization Method
Based on equivalent linearization.
The displacement demand of a non-linear SDOF system is estimated from the displacement demand of a linear-elastic SDOF system. The elastic SDOF system, referred to as an equivalent system, has a period and a damping ratio larger than those of the initial non-linear system (ATC, 2005).
Static nonlinear analysis
A version of the Capacity-Spectrum Method (CSM) is proposed by ATC (Applied Technology Council, US). This version is based on equivalent linearization. Therefore, the displacement demand of a non-linear SDOF system is estimated from the displacement demand of a linear-elastic SDOF system. The elastic SDOF system, referred to as an equivalent system, has a period and a damping ratio larger than those of the initial non-linear system (ATC, 2005).
5
Advanced Earthquake Engineering CIVIL-706
Equivalent Linearization
Static nonlinear analysis
Basic equations…
6
Advanced Earthquake Engineering CIVIL-706
Equivalent Linearization- Performance Point
Static nonlinear analysis
Sou
rce:
FEM
A 4
40
𝑆𝑑 = 𝑆𝑑 𝑇𝑒𝑞; ζ𝑒𝑞 = 𝑆𝑑 𝑇𝑒𝑞; ζ=5% ∙ 𝜂 = 𝑆𝑎 𝑇𝑒𝑞; ζ=5% ∙𝑇𝑒𝑞2
4𝜋2∙ 𝜂
7
Advanced Earthquake Engineering CIVIL-706
Equivalent Linearization- Performance Point
Static nonlinear analysis 8
𝑆𝑑 = 𝑆𝑑 𝑇𝑒𝑞; ζ𝑒𝑞 = 𝑆𝑑 𝑇𝑒𝑞; ζ=5% ∙ 𝜂 = 𝑆𝑎 𝑇𝑒𝑞; ζ=5% ∙𝑇𝑒𝑞2
4𝜋2∙ 𝜂
𝑆𝑑 𝑇𝑒𝑞; ζ𝑒𝑞 Spectral displacement of the equivalent system
𝑇𝑒𝑞 Equivalent period of vibration
ζ𝑒𝑞 Equivalent viscous damping ratio
𝑆𝑑 𝑇𝑒𝑞; ζ=5% Displacement demand of the linear system with 5%-damping elastic
ratio 𝜂 Reduction factor depending from the damping modification factor
𝜂 =1
0.5+10ζ𝑒𝑞
S_d (T_eq;ξ_eq )
𝜂 =1
0.5 + 10 𝜉𝑒𝑞
Advanced Earthquake Engineering CIVIL-706
Equivalent Linearization- Performance Point
Static nonlinear analysis 9
𝑆𝑑 = 𝑆𝑑 𝑇𝑒𝑞; ζ𝑒𝑞 = 𝑆𝑑 𝑇𝑒𝑞; ζ=5% ∙ 𝜂 = 𝑆𝑎 𝑇𝑒𝑞; ζ=5% ∙𝑇𝑒𝑞2
4𝜋2∙ 𝜂
The equivalent period and the equivalent damping ratio are functions of the strength reduction factor of the non-linear SDOF system and, respectively, of the initial period of vibration and of the damping ratio. The various equivalent linear methods differ from each other mainly for functions used to compute 𝑇𝑒𝑞 and ζ𝑒𝑞.
In their work (2008), Lin & Miranda give the equivalent period and the equivalent damping ratio as follows:
𝑇𝑒𝑞 = 1 +𝑚1
𝑇2∙ 𝑅𝜇
1.8 − 1 ∙ 𝑇
ζ𝑒𝑞 = ζ=5% +𝑛1𝑇𝑛2
∙ 𝑅𝜇 − 1
Advanced Earthquake Engineering CIVIL-706
Equivalent Linearization- Performance Point
Static nonlinear analysis 10
𝑆𝑑 = 𝑆𝑑 𝑇𝑒𝑞; ζ𝑒𝑞 = 𝑆𝑑 𝑇𝑒𝑞; ζ=5% ∙ 𝜂 = 𝑆𝑎 𝑇𝑒𝑞; ζ=5% ∙𝑇𝑒𝑞2
4𝜋2∙ 𝜂
𝑇𝑒𝑞 = 1 +𝑚1
𝑇2∙ 𝑅𝜇
1.8 − 1 ∙ 𝑇
ζ𝑒𝑞 = ζ=5% +𝑛1𝑇𝑛2
∙ 𝑅𝜇 − 1
Advanced Earthquake Engineering CIVIL-706
Equivalent Linearization
• Advantages:
– Linear computation
– Use of pushover analysis
• Drawbacks:
– Value of damping
– Not always conservative
Static nonlinear analysis 11
Advanced Earthquake Engineering CIVIL-706
N2 Method
A little bit of background…
• Started in the mid 1980’s (Fajfar and Fischinger 1987, 1989)
• A variant of the Capacity Spectrum Method (ATC-40)
• Based on inelastic spectra rather than elastic spectra
Static nonlinear analysis 12
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis 13
N2 Method Procedure
R
Reduction factor
Ductility factor
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis 14
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis 15
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis 16
N2 Method Procedure
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis 17
N2 Method Procedure
Advanced Earthquake Engineering CIVIL-706
Content
• Capacity curve
• Seismic damage
• Acceleration-Displacement Response Spectrum (ADRS)
• Performance Point
• Large-scale vulnerability assessment
• N2 method reliability and optimization
• Method result confrontation
Static nonlinear analysis 18
Advanced Earthquake Engineering CIVIL-706
Large-scale vulnerability assessment
Steps for large-scale vulnerability assessment:
• Typology of the building stock with structural characteristics
• Distribution of building classes in the area under study
• Vulnerability assessment of each class including variability (probabilistic assessment)
Fragility functions
Static nonlinear analysis 19
Advanced Earthquake Engineering CIVIL-706
Large-scale vulnerability assessment
Steps for large-scale vulnerability assessment:
• Typology of the building stock with structural characteristics
• Distribution of building classes in the area under study
• Vulnerability assessment of each class including variability (probabilistic assessment)
Fragility functions
Static nonlinear analysis 20
Advanced Earthquake Engineering CIVIL-706
Large-scale vulnerability assessment
Empirical methods (e.g. EMS98, GNDT, Risk-UE LM1, Vulneralp) based on damage surveys
• Relationship between intensity and distribution of observed damage grades for structures with a given vulnerability index
Damage Probability Matrix
Static nonlinear analysis 21
Advanced Earthquake Engineering CIVIL-706
Empirical Methods
Empirical methods example
European Macroseismic Scale 98
• Calculate Vulnerability index from vulnerability class
• Structural parameters is only the structure class
Static nonlinear analysis 22
Advanced Earthquake Engineering CIVIL-706
Empirical Methods
EMS 98 (Empirical method):
Static nonlinear analysis
Sou
rce:
Ris
k-U
E 2
00
3
23
Advanced Earthquake Engineering CIVIL-706
Empirical Methods
Static nonlinear analysis 24
Risk-UE LM1 method:
Advanced Earthquake Engineering CIVIL-706
Empirical Methods Risk-UE LM1 method:
Static nonlinear analysis 25
Mean damage grade
Advanced Earthquake Engineering CIVIL-706
Empirical Methods
Static nonlinear analysis
Vulnerability class in downtown Malaga (EMS98)
26
Mean damage of downtown Malaga with I=VI (LM1)
Advanced Earthquake Engineering CIVIL-706
Risk-UE LM1 method : (Rachel de Blaireville, PdM 2016)
Yverdon-les-Bains
Static nonlinear analysis 27
I=VI I=VII
Advanced Earthquake Engineering CIVIL-706
Empirical Methods
Risk-UE LM1 method:
Static nonlinear analysis
Sou
rce:
Ris
k-U
E 2
00
3
28
Advanced Earthquake Engineering CIVIL-706
Mechanical Methods
• AKA as predicted methods (e.g., HAZUS, RISK-UE LM2)
• Based on computation (generally nonlinear static)
• Relation between ground motion and expected distribution of damage grade
Fragility curves
Static nonlinear analysis 29
Advanced Earthquake Engineering CIVIL-706
Mechanical Methods
RISK-UE LM2 method:
Static nonlinear analysis 30
Advanced Earthquake Engineering CIVIL-706
Mechanical Methods RISK-UE LM2 method:
Definition of capacity curves for building typologies
Static nonlinear analysis 31
Advanced Earthquake Engineering CIVIL-706
Mechanical Methods RISK-UE LM2 method:
Definition of capacity curves for building typologies
Definition of damage limit states.
Static nonlinear analysis 32
Advanced Earthquake Engineering CIVIL-706
Mechanical Methods
RISK-UE LM2 method:
Static nonlinear analysis 33
Advanced Earthquake Engineering CIVIL-706
RISK-UE LM2 method:
Determination of the performance point
Static nonlinear analysis 34
DG2
DG3
DG4
Advanced Earthquake Engineering CIVIL-706
Mechanical Methods
RISK-UE LM2 method:
Static nonlinear analysis 35
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis 36
Risk-UE LM2 method : (Rachel de Blaireville, PdM 2016)
Yverdon-les-Bains
DG0
DG1
DG2
DG3
DG4 NO
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis 37
Risk-UE LM2 method : (Lestuzzi et al., 2016)
Method LM2: results for the city of Sion
DG0
DG1
DG2
DG3
DG4
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis 38
DG0
DG1
DG2
DG3
DG4
Risk-UE LM2 method : (Lestuzzi et al., 2016)
Method LM2: results for the city of Martigny
Advanced Earthquake Engineering CIVIL-706
Mechanical Methods
RISK-UE LM2 method:
Static nonlinear analysis 39
Advanced Earthquake Engineering CIVIL-706
Mechanical Methods
Capacity Curve and Fragility curves
Static nonlinear analysis
Sou
rce:
HA
ZUS
20
03
40
Advanced Earthquake Engineering CIVIL-706
Mechanical Methods
• Independent of observed damage data (only verification)
• Applicable to single buildings (capacity curves)
BUT
• Difficulties to build accurate models for existing buildings (lack of information)
• Capacity curves available only for certain building classes
• Difficulties in estimating variability
Static nonlinear analysis 41
Advanced Earthquake Engineering CIVIL-706
Mechanical Methods
Example of RISK-UE application:
Static nonlinear analysis
Probability of D4 and D5
42
Advanced Earthquake Engineering CIVIL-706
Content
• Capacity curve
• Seismic damage
• Acceleration-Displacement Response Spectrum (ADRS)
• Performance Point
• Large-scale vulnerability assessment
• N2 method reliability and optimization
• Methods results comparison
Static nonlinear analysis 43
Advanced Earthquake Engineering CIVIL-706
N2 method reliability
Michel C, Lestuzzi P, Lacave C (2014)
• N2 method may lead to overestimation or underestimation of the displacement prediction
• Pointed out the lack of accuracy of the N2 method on the plateau interval
• Important consequences in the results of mechanical model-based assessment of seismic vulnerability at the urban scale
Static nonlinear analysis 44
Advanced Earthquake Engineering CIVIL-706
N2 method reliability
Methodology
• the methodology used in this study consists in the computation of the non-linear responses of SDOF systems subjected to earthquake records (CONSIDERED AS TRUE VALUES) and in the assessment of the difference between the obtained peak displacement demands and those predicted by N2 method.
Static nonlinear analysis 45
%𝛥𝑇𝑖 =𝐷𝐷𝑁2𝑇𝑖
− 𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖
%.
Advanced Earthquake Engineering CIVIL-706
N2 method reliability
Example of the method applied
Static nonlinear analysis 46
N2 method prediction NLTHA mean prediction (+ and – standard deviation)
R = Sae / Say
R
R
Advanced Earthquake Engineering CIVIL-706
N2 method reliability
Non Linear time history analysis (NLTHA)
• The non-linear structural behaviour is described by the modified Takeda hysteretic model. For each EC8 soil class, a set of 12 recordings selected from a database, such as the European Strong Motion Database, and slightly modified to match the corresponding response spectrum, are first developed.
Static nonlinear analysis 47
Advanced Earthquake Engineering CIVIL-706
N2 method optimization
Selection and modification of 12 earthquakes
Static nonlinear analysis 48
Advanced Earthquake Engineering CIVIL-706
N2 method reliability
Methodology
• The comparison is performed for the response spectrum of EC8 different soil classes. The SDOF are based on different periods (these corresponding to plateau) and different strength reduction factors (R = [1.5 5.0] and R = [1.5 9.0] with intervals of 0.5)
• The displacement demand derived by non-linear time-history analysis is calculated as the average of the displacements obtained by the 12 seismic recordings selected and is considered as the true value of displacement demand.
Static nonlinear analysis 49
%𝛥𝑇𝑖 =𝐷𝐷𝑁2𝑇𝑖
− 𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖
%.
The N2 method and the NLTHA are compared in the following sections with respect to displacement demand determination, to point out the differences between them. The comparison is performed for the response spectrum of EC8 type 1 and different soil classes. The SDOF are based on different periods (these corresponding to plateau) and different strength reduction factors (R = [1.5 5.0] with intervals of 0.5). The displacement demand derived by non-linear time-history analysis is calculated as the average of the displacements obtained by the 12 seismic recordings selected and is considered as the true value of displacement demand. The difference between the displacement demand provided by the N2 method and time-history analysis is expressed as a percentage of the NLTHA displacement:
EC8 type 1 SOIL class B, Ag= 1.00 m/s2 Tb = 0.15 s Tc = 0.40 s Sea,max = 2.5 m/s2 S = 1 R = 5.0
T N2 [m]
(displacement)
% ΔTi (DDN2 Ti - DDNLTHA Ti) /
DDNLTHA Ti
NLTHA (displacement)
R
- σ average + σ
pla
teau
0.150 0.0049 -18.0% 0.0047 0.0060 0.0073 5.0
0.167 0.0055 -21.2% 0.0054 0.0070 0.0086 5.0
0.182 0.0060 -24.4% 0.0063 0.0080 0.0096 5.0
0.200 0.0067 -23.3% 0.0068 0.0087 0.0106 5.0
0.222 0.0075 -24.3% 0.0078 0.0099 0.0120 5.0
0.250 0.0086 -21.3% 0.0087 0.0109 0.0130 5.0
0.267 0.0092 -21.2% 0.0093 0.0117 0.0141 5.0
0.286 0.0099 -19.6% 0.0097 0.0124 0.0150 5.0
0.308 0.0108 -17.7% 0.0105 0.0131 0.0157 5.0
0.333 0.0118 -15.2% 0.0113 0.0140 0.0166 5.0
0.364 0.0131 -10.2% 0.0118 0.0146 0.0173 5.0
0.400 0.0146 -5.9% 0.0125 0.0155 0.0186 5.0
0.444 0.0165 -3.7% 0.0140 0.0172 0.0203 5.0
0.500 0.0190 3.3% 0.0153 0.0184 0.0216 5.0
T>Tc
0.571 0.0217 4.5% 0.0160 0.0208 0.0256 5.0
0.667 0.0254 2.5% 0.0204 0.0247 0.0291 5.0
0.800 0.0304 1.6% 0.0242 0.0300 0.0358 5.0
1.000 0.0380 1.2% 0.0310 0.0376 0.0442 5.0
% |Δ|(R) 16.4%
Advanced Earthquake Engineering CIVIL-706
N2 method reliability
Methodology
• The difference between the displacement demand provided by the N2 method and time-history analysis is expressed as a percentage of the NLTHA displacement:
Static nonlinear analysis 50
%𝛥𝑇𝑖 =𝐷𝐷𝑁2𝑇𝑖
− 𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖
%.
The N2 method and the NLTHA are compared in the following sections with respect to displacement demand determination, to point out the differences between them. The comparison is performed for the response spectrum of EC8 type 1 and different soil classes. The SDOF are based on different periods (these corresponding to plateau) and different strength reduction factors (R = [1.5 5.0] with intervals of 0.5). The displacement demand derived by non-linear time-history analysis is calculated as the average of the displacements obtained by the 12 seismic recordings selected and is considered as the true value of displacement demand. The difference between the displacement demand provided by the N2 method and time-history analysis is expressed as a percentage of the NLTHA displacement:
EC8 type 1 SOIL class B, Ag= 1.00 m/s2 Tb = 0.15 s Tc = 0.40 s Sea,max = 2.5 m/s2 S = 1 R = 5.0
T N2 [m]
(displacement)
% ΔTi (DDN2 Ti - DDNLTHA Ti) /
DDNLTHA Ti
NLTHA (displacement)
R
- σ average + σ
pla
teau
0.150 0.0049 -18.0% 0.0047 0.0060 0.0073 5.0
0.167 0.0055 -21.2% 0.0054 0.0070 0.0086 5.0
0.182 0.0060 -24.4% 0.0063 0.0080 0.0096 5.0
0.200 0.0067 -23.3% 0.0068 0.0087 0.0106 5.0
0.222 0.0075 -24.3% 0.0078 0.0099 0.0120 5.0
0.250 0.0086 -21.3% 0.0087 0.0109 0.0130 5.0
0.267 0.0092 -21.2% 0.0093 0.0117 0.0141 5.0
0.286 0.0099 -19.6% 0.0097 0.0124 0.0150 5.0
0.308 0.0108 -17.7% 0.0105 0.0131 0.0157 5.0
0.333 0.0118 -15.2% 0.0113 0.0140 0.0166 5.0
0.364 0.0131 -10.2% 0.0118 0.0146 0.0173 5.0
0.400 0.0146 -5.9% 0.0125 0.0155 0.0186 5.0
0.444 0.0165 -3.7% 0.0140 0.0172 0.0203 5.0
0.500 0.0190 3.3% 0.0153 0.0184 0.0216 5.0
T>Tc
0.571 0.0217 4.5% 0.0160 0.0208 0.0256 5.0
0.667 0.0254 2.5% 0.0204 0.0247 0.0291 5.0
0.800 0.0304 1.6% 0.0242 0.0300 0.0358 5.0
1.000 0.0380 1.2% 0.0310 0.0376 0.0442 5.0
% |Δ|(R) 16.4%
Advanced Earthquake Engineering CIVIL-706
N2 method reliability
Example of the method applied
Static nonlinear analysis 51
N2 method prediction NLTHA mean prediction (+ and – standard deviation)
R = Sae / Say
R
R
Advanced Earthquake Engineering CIVIL-706
N2 method reliability
Static nonlinear analysis 52
%𝛥𝑇𝑖 =𝐷𝐷𝑁2𝑇𝑖
− 𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖
%.
The N2 method and the NLTHA are compared in the following sections with respect to displacement demand determination, to point out the differences between them. The comparison is performed for the response spectrum of EC8 type 1 and different soil classes. The SDOF are based on different periods (these corresponding to plateau) and different strength reduction factors (R = [1.5 5.0] with intervals of 0.5). The displacement demand derived by non-linear time-history analysis is calculated as the average of the displacements obtained by the 12 seismic recordings selected and is considered as the true value of displacement demand. The difference between the displacement demand provided by the N2 method and time-history analysis is expressed as a percentage of the NLTHA displacement:
EC8 type 1 SOIL class B, Ag= 1.00 m/s2 Tb = 0.15 s Tc = 0.40 s Sea,max = 2.5 m/s2 S = 1 R = 5.0
T N2 [m]
(displacement)
% ΔTi (DDN2 Ti - DDNLTHA Ti) /
DDNLTHA Ti
NLTHA (displacement)
R
- σ average + σ p
late
au
0.150 0.0049 -18.0% 0.0047 0.0060 0.0073 5.0
0.167 0.0055 -21.2% 0.0054 0.0070 0.0086 5.0
0.182 0.0060 -24.4% 0.0063 0.0080 0.0096 5.0
0.200 0.0067 -23.3% 0.0068 0.0087 0.0106 5.0
0.222 0.0075 -24.3% 0.0078 0.0099 0.0120 5.0
0.250 0.0086 -21.3% 0.0087 0.0109 0.0130 5.0
0.267 0.0092 -21.2% 0.0093 0.0117 0.0141 5.0
0.286 0.0099 -19.6% 0.0097 0.0124 0.0150 5.0
0.308 0.0108 -17.7% 0.0105 0.0131 0.0157 5.0
0.333 0.0118 -15.2% 0.0113 0.0140 0.0166 5.0
0.364 0.0131 -10.2% 0.0118 0.0146 0.0173 5.0
0.400 0.0146 -5.9% 0.0125 0.0155 0.0186 5.0
0.444 0.0165 -3.7% 0.0140 0.0172 0.0203 5.0
0.500 0.0190 3.3% 0.0153 0.0184 0.0216 5.0
T>Tc
0.571 0.0217 4.5% 0.0160 0.0208 0.0256 5.0
0.667 0.0254 2.5% 0.0204 0.0247 0.0291 5.0
0.800 0.0304 1.6% 0.0242 0.0300 0.0358 5.0
1.000 0.0380 1.2% 0.0310 0.0376 0.0442 5.0
% |Δ|(R) 16.4%
Advanced Earthquake Engineering CIVIL-706
N2 method reliability
Static nonlinear analysis 53
%𝛥𝑇𝑖 =𝐷𝐷𝑁2𝑇𝑖
− 𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖
%.
The N2 method and the NLTHA are compared in the following sections with respect to displacement demand determination, to point out the differences between them. The comparison is performed for the response spectrum of EC8 type 1 and different soil classes. The SDOF are based on different periods (these corresponding to plateau) and different strength reduction factors (R = [1.5 5.0] with intervals of 0.5). The displacement demand derived by non-linear time-history analysis is calculated as the average of the displacements obtained by the 12 seismic recordings selected and is considered as the true value of displacement demand. The difference between the displacement demand provided by the N2 method and time-history analysis is expressed as a percentage of the NLTHA displacement:
Soil B – Difference in percentage (vertical axis) between NLTHA average displacement and N2 method (thin dashed black line). The two mirrored thick black lines show the standard deviation of the 12 seismic recordings
ADD COMMENTS
Advanced Earthquake Engineering CIVIL-706
N2 method reliability
Static nonlinear analysis 54
%𝛥𝑇𝑖 =𝐷𝐷𝑁2𝑇𝑖
− 𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖
%.
The N2 method and the NLTHA are compared in the following sections with respect to displacement demand determination, to point out the differences between them. The comparison is performed for the response spectrum of EC8 type 1 and different soil classes. The SDOF are based on different periods (these corresponding to plateau) and different strength reduction factors (R = [1.5 5.0] with intervals of 0.5). The displacement demand derived by non-linear time-history analysis is calculated as the average of the displacements obtained by the 12 seismic recordings selected and is considered as the true value of displacement demand. The difference between the displacement demand provided by the N2 method and time-history analysis is expressed as a percentage of the NLTHA displacement:
EC8 type 1 SOIL class B, Ag= 1.00 m/s2 Tb = 0.15 s Tc = 0.40 s Sea,max = 2.5 m/s2 S = 1 R = 5.0
T N2 [m]
(displacement)
% ΔTi (DDN2 Ti - DDNLTHA Ti) /
DDNLTHA Ti
NLTHA (displacement)
R
- σ average + σ p
late
au
0.150 0.0049 -18.0% 0.0047 0.0060 0.0073 5.0
0.167 0.0055 -21.2% 0.0054 0.0070 0.0086 5.0
0.182 0.0060 -24.4% 0.0063 0.0080 0.0096 5.0
0.200 0.0067 -23.3% 0.0068 0.0087 0.0106 5.0
0.222 0.0075 -24.3% 0.0078 0.0099 0.0120 5.0
0.250 0.0086 -21.3% 0.0087 0.0109 0.0130 5.0
0.267 0.0092 -21.2% 0.0093 0.0117 0.0141 5.0
0.286 0.0099 -19.6% 0.0097 0.0124 0.0150 5.0
0.308 0.0108 -17.7% 0.0105 0.0131 0.0157 5.0
0.333 0.0118 -15.2% 0.0113 0.0140 0.0166 5.0
0.364 0.0131 -10.2% 0.0118 0.0146 0.0173 5.0
0.400 0.0146 -5.9% 0.0125 0.0155 0.0186 5.0
0.444 0.0165 -3.7% 0.0140 0.0172 0.0203 5.0
0.500 0.0190 3.3% 0.0153 0.0184 0.0216 5.0
T>Tc
0.571 0.0217 4.5% 0.0160 0.0208 0.0256 5.0
0.667 0.0254 2.5% 0.0204 0.0247 0.0291 5.0
0.800 0.0304 1.6% 0.0242 0.0300 0.0358 5.0
1.000 0.0380 1.2% 0.0310 0.0376 0.0442 5.0
% |Δ|(R) 16.4%
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N2 method reliability
Global discrepancy for soil classes
• The global variance in the displacement demand determination for the plateau set of values of the periods has been carried out for each EC8 soil class. This value has been provided for each strength reduction factor (R) in the studied interval. It is expressed as the average of the absolute value of the single per cent differences for each period corresponding to the plateau:
Static nonlinear analysis 55
% 𝛥 𝑅 =1
𝑡 % 𝛥 𝑇𝑖
𝑡
𝑖=1
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N2 method reliability
Static nonlinear analysis 56
Per cent variance in absolute value expressed for each strength reduction factor (R)
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N2 method reliability
Global discrepancy for soil classes
Static nonlinear analysis 57
17.90%
SOIL A SOIL B SOIL C SOIL D
15.72% 16.26% 16.39% 23.24%
SOIL A SOIL B SOIL C SOIL D
R = 1.5 22.07% 30.90% 30.87% 41.71%
R = 2.0 18.03% 29.14% 32.67% 45.31%
R = 2.5 6.03% 16.47% 19.00% 31.86%
R = 3.0 5.55% 7.72% 7.88% 14.61%
R = 3.5 12.50% 6.07% 3.86% 7.13%
R = 4.0 17.59% 9.82% 8.98% 10.76%
R = 4.5 20.98% 13.57% 12.75% 15.27%
R = 5.0 22.98% 16.43% 15.11% 19.23%
Formula N2
𝑆𝑑 =𝑆𝑑𝑒𝑅
∙ 𝑅 − 1 ∙𝑇 𝑇+ 1
% 𝑅
%
% 𝑇 𝑇
% 𝛥 =1
𝑟 % 𝛥 𝑅𝑘
𝑟
𝑘=1
% 𝛥 𝑅 =1
𝑡 % 𝛥 𝑇𝑖
𝑡
𝑖=1
%𝛥𝑇𝑖 =𝐷𝐷𝑁2𝑇𝑖
− 𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖
%.
% 𝛥 𝑇 𝑇 =1
𝑠 % 𝛥 𝑘
𝑠
𝑘=1
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N2 method optimization
Optimization approach for new formulas
Two main optimizations:
• 1 variable optimization: α1 = 1; α2 = 1; the formula has been optimized with respect only to β;
• 3 variable optimization: α1 , α2 and β have been considered as optimization variables.
Static nonlinear analysis 58
𝑆𝑑 =𝑆𝑑𝑒𝑅𝛼1
∙𝑅
𝛼2 − 1
𝛽
∙𝑇𝐶𝑇+ 1
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N2 method optimization
Optimization approach for new formulas (Manno A)
Definition of the single error
Definition of the global error function
Minimization objective
Static nonlinear analysis 59
𝑆𝑑 =𝑆𝑑𝑒𝑅𝛼1
∙𝑅
𝛼2 − 1
𝛽
∙𝑇𝐶𝑇+ 1
𝑒 𝑠𝑅𝑟𝑇𝑡 𝛼1, 𝛼2, 𝛽 = |𝐷𝐹 𝑠𝑅𝑟𝑇𝑡 𝛼1, 𝛼2, 𝛽 − 𝐷𝑆 𝑠𝑅𝑟𝑇𝑡|
| 𝐷𝑆 𝑠𝑅𝑟𝑇𝑡|
𝐸 𝛼1, 𝛼2, 𝛽 = 𝑒𝑇
𝑡=1𝑅𝑟=1 𝑠𝑅𝑟𝑇𝑡
𝛼1, 𝛼2, 𝛽 𝑠=1
𝑆 ∗ 𝑅 ∗ 𝑇
min𝛼1,𝛼2,𝛽∈ℜ
𝐸 𝛼1, 𝛼2, 𝛽
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N2 method optimization
1 variable optimization
Static nonlinear analysis 60
𝑆𝑑 =𝑆𝑑𝑒𝑅
∙ 𝑅 − 1 1.13 ∙𝑇𝐶𝑇+ 1
17.90%
SOIL A SOIL B SOIL C SOIL D
15.72% 16.26% 16.39% 23.24%
SOIL A SOIL B SOIL C SOIL D
R = 1.5 22.07% 30.90% 30.87% 41.71%
R = 2.0 18.03% 29.14% 32.67% 45.31%
R = 2.5 6.03% 16.47% 19.00% 31.86%
R = 3.0 5.55% 7.72% 7.88% 14.61%
R = 3.5 12.50% 6.07% 3.86% 7.13%
R = 4.0 17.59% 9.82% 8.98% 10.76%
R = 4.5 20.98% 13.57% 12.75% 15.27%
R = 5.0 22.98% 16.43% 15.11% 19.23%
Formula N2
𝑆𝑑 =𝑆𝑑𝑒𝑅
∙ 𝑅 − 1 ∙𝑇 𝑇+ 1
% 𝑅
%
% 𝑇 𝑇
β* 1.13
15.78%
SOIL A SOIL B SOIL C SOIL D
10.65% 14.99% 15.79% 21.67%
SOIL A SOIL B SOIL C SOIL D
R = 1.5 17.29% 25.33% 25.37% 35.15%
R = 2.0 18.03% 29.14% 32.67% 45.31%
R = 2.5 9.30% 21.11% 23.72% 37.35%
R = 3.0 3.49% 13.86% 15.85% 22.17%
R = 3.5 5.13% 8.92% 8.71% 10.98%
R = 4.0 8.96% 6.76% 5.34% 6.26%
R = 4.5 10.93% 6.79% 6.37% 7.13%
R = 5.0 12.07% 8.04% 8.30% 9.02%
Formula Optimization (exp β)
% 𝑇 𝑇
%
% 𝑅
𝑆𝑑 =𝑆𝑑𝑒𝑅
∙ 𝑅 − 1 𝛽 ∙𝑇 𝑇+ 1
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N2 method optimization
1 variable optimization
OPTexp1.13 (R=[1.5 5.0])
OPTexp1.16 (R=[1.5 9.0])
Static nonlinear analysis 61
𝑆𝑑 =𝑆𝑑𝑒𝑅
∙ 𝑅 − 1 1.13 ∙𝑇𝐶𝑇+ 1
𝑆𝑑 =𝑆𝑑𝑒𝑅
∙ 𝑅 − 1 1.16 ∙𝑇𝐶𝑇+ 1
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N2 method optimization
3 variable optimization
Static nonlinear analysis 62
𝑆𝑑 =𝑆𝑑𝑒𝑅
1.48
∙𝑅
1.45− 1
1.35
∙𝑇𝐶𝑇+ 1
α1* 1.48
α2* 1.45
β* 1.35
6.90%
SOIL A SOIL B SOIL C SOIL D
7.97% 6.91% 5.24% 7.46%
SOIL A SOIL B SOIL C SOIL D
R = 1.5 3.01% 1.59% 3.96% 3.23%
R = 2.0 4.71% 7.27% 4.02% 7.04%
R = 2.5 7.34% 7.30% 3.16% 11.25%
R = 3.0 9.10% 6.96% 2.95% 8.67%
R = 3.5 9.46% 7.24% 4.11% 7.08%
R = 4.0 10.04% 7.95% 5.81% 7.21%
R = 4.5 9.92% 8.29% 7.80% 7.43%
R = 5.0 10.22% 8.69% 10.12% 7.81%
for R<1.45;
Sd=Sd (opt1.13)
Condition
Formula optimization 3 variables (α1;α2;β)
𝑆𝑑 =𝑆𝑑𝑒𝑅 1
∙𝑅
2− 1
𝛽
∙𝑇 𝑇+ 1
% 𝑅
% 𝑇 𝑇
%
17.90%
SOIL A SOIL B SOIL C SOIL D
15.72% 16.26% 16.39% 23.24%
SOIL A SOIL B SOIL C SOIL D
R = 1.5 22.07% 30.90% 30.87% 41.71%
R = 2.0 18.03% 29.14% 32.67% 45.31%
R = 2.5 6.03% 16.47% 19.00% 31.86%
R = 3.0 5.55% 7.72% 7.88% 14.61%
R = 3.5 12.50% 6.07% 3.86% 7.13%
R = 4.0 17.59% 9.82% 8.98% 10.76%
R = 4.5 20.98% 13.57% 12.75% 15.27%
R = 5.0 22.98% 16.43% 15.11% 19.23%
Formula N2
𝑆𝑑 =𝑆𝑑𝑒𝑅
∙ 𝑅 − 1 ∙𝑇 𝑇+ 1
% 𝑅
%
% 𝑇 𝑇
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N2 method optimization
1 variable optimization
OPT3varA ( R=[1.5 5.0] )
OPT3varB ( R=[1.5 9.0] )
Static nonlinear analysis 63
𝑆𝑑 =𝑆𝑑𝑒𝑅
1.48
∙𝑅
1.45− 1
1.35
∙𝑇𝐶𝑇+ 1
𝑆𝑑 =𝑆𝑑𝑒𝑅
1.38
∙𝑅
1.38− 1
1.28
∙𝑇𝐶𝑇+ 1
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N2 method optimization
Static nonlinear analysis 64
Soil B – Difference in percentage (vertical axis) between NLTHA average displacement and the optimized formulas. The two mirrored thick black lines show the standard deviation of the 12 seismic recordings
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N2 method optimization
Static nonlinear analysis 65
Soil D – Difference in percentage (vertical axis) between NLTHA average displacement and the optimized formulas. The two mirrored thick black lines show the standard deviation of the 12 seismic recordings
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N2 method optimization
Static nonlinear analysis 66
Trend of the N2 formula (black line) and the optimized formulas in the average discrepancy for the different R-values with the NLTHA results
R=1.5:5.0 R=1.5:9.0
N2 17.90% 20.80%
OPT exp1.13 15.78% 14.34%
OPT exp1.16 16.04% 13.86%
OPT 3varA 6.91% 9.56%
OPT 3varB 7.29% 8.19%
OPT 3varC 6.97% 9.44%
% 𝑇 𝑇
Total average discrepancies for N2 formula and for optimized formulas
Advanced Earthquake Engineering CIVIL-706
Content
• Capacity curve
• Seismic damage
• Acceleration-Displacement Response Spectrum (ADRS)
• Performance Point
• Large-scale vulnerability assessment
• N2 method reliability and optimization
• Methods results comparison
Static nonlinear analysis 67
Simplified seismic demand determination confrontation
• N2 method
• Lin&Miranda
• N2 optimization
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Methods results comparison
Static nonlinear analysis 68
𝑆𝑑 =
𝑆𝑑𝑒𝑅𝜇
∙ 𝑅𝜇 − 1 ∙𝑇𝐶𝑇+ 1 𝑇 < 𝑇𝐶 𝑎𝑛𝑑 𝑅𝜇 > 1
𝑆𝑑 = 𝑆𝑑𝑒 𝑇 ≥ 𝑇𝐶 𝑜𝑟 𝑅𝜇 ≤ 1
𝑆𝑑 = 𝑆𝑎 𝑇𝑒𝑞; 𝜉=5% ∙𝑇𝑒𝑞2
4𝜋2∙ 𝜂 2
Sd = SdeRμ1.48
∙Rμ
1.45− 1
1.35
∙TCT+ 1 𝑇 < TC 𝑎𝑛𝑑 Rμ > 1
Sd = Sde 𝑇 ≥ TC 𝑜𝑟 Rμ ≤ 1
6
Test cities: SION and MARTIGNY
The accuracy of damage prediction linked to the investigated displacement demand determination methods is tested on two typical Swiss cities (Sion and Martigny) in moderate seismicity area.
These tests provide realistic building stock distributions and seismic conditions (microzone studies available).
Tests are related to the damage distribution and have been carried out on the EC8 soil class C and on microzones.
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Methods results comparison
Static nonlinear analysis 69
Test cities: SION and MARTIGNY
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Methods results comparison
Static nonlinear analysis 70
Introduction specific typologies for Swiss buildings
A1 unreinforced masonry (URM) buildings with a basement floor in reinforced concrete (RC)
A2 mixed URM-RC buildings
B2 buildings with RC pillars in the base floor
C buildings with RC shear walls
D2 buildings with URM shear walls
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Methods results comparison
Static nonlinear analysis 71
Distribution
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Methods results comparison
Static nonlinear analysis 72
Sion Martigny
Distribution
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Methods results comparison
Static nonlinear analysis 73
Sion Martigny
Non linear time history analysis
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Methods results comparison
Static nonlinear analysis 74
Methodology
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Methods results comparison
Static nonlinear analysis 75
N2 method Methodology
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Methods results comparison
Static nonlinear analysis 76
Lin&Miranda Methodology
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Methods results comparison
Static nonlinear analysis 77
N2 optimization Methodology
Performance point: SION and MARTIGNY – soil class C
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Methods results comparison
Static nonlinear analysis 78
Performance point: SION and MARTIGNY – soil class C
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Methods results comparison
Static nonlinear analysis 79
3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9
A1 114% 60% 33% 12% 2% 0% 4% A1 26% -1% -3% -4% 2% 3% 15% A1 97% -2% -1% -4% -4% -5% 16%
A2 52% 42% 44% 33% 16% 9% 5% A2 -1% -2% 5% 5% 0% 2% 7% A2 43% 34% -1% 0% -6% -6% -4%
C 14% 3% 1% 11% 16% 11% 15% C -3% 5% 9% 17% 20% 15% 18% C -4% -4% 3% 20% 16% 16% 15%
D2 39% 20% 5% -2% -1% 1% - D2 -4% -3% 0% 8% 11% 11% - D2 2% -1% -4% -3% 4% 6% -
N2 method Lin&Miranda N2 OPT
Sion - EC8 Soil Class C Sion - EC8 Soil Class C Sion - EC8 Soil Class C
3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9
A1 125% 51% 19% 6% -4% - - A1 24% -2% -6% 0% 6% - - A1 15% 2% -2% -2% -4% - -
A2 52% 48% 28% 11% 4% 4% - A2 -1% 5% 2% -1% 2% 11% - A2 43% 1% -1% -6% -6% -2% -
C - 9% 8% 8% 9% 12% - C - -1% 8% 12% 10% 11% - C - -9% -4% 1% 7% 12% -
D2 39% 13% 1% -3% 2% 0% - D2 -4% -5% 2% 12% 9% 15% - D2 2% -3% -3% 0% 5% 13% -
N2 method Lin&Miranda N2 OPT
Martigny - EC8 Soil Class C Martigny - EC8 Soil Class C Martigny - EC8 Soil Class C
Per cent differences of performance points prediction of the three methods analysed with the NLTHA prediction.
Performance point: SION and MARTIGNY – microzones
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Methods results comparison
Static nonlinear analysis 80
Performance point: SION and MARTIGNY – microzones
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Methods results comparison
Static nonlinear analysis 81
3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9
A1 - - 93% 59% 50% 13% - A1 - - -7% 0% -14% -10% - A1 - - 0% 11% -6% -3% -
A2 60% 50% 54% 51% - 26% 23% A2 2% 1% 0% 1% - -4% 0% A2 47% 37% 45% -2% - -5% -1%
C 48% - 10% -1% -1% 6% 0% C 0% - 2% 3% 15% 4% 8% C 7% - 1% -3% 3% 0% -1%
D2 69% 45% 15% 7% 5% 5% - D2 -7% -8% -16% -11% -2% 0% - D2 3% 3% -7% - - 0% -
3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9
A1 - 85% 44% 16% -14% -15% -24% A1 - -2% -11% -16% -28% -25% 17% A1 - 3% 2% -5% -22% -21% -16%
A2 80% 59% 57% 43% - 2% -7% A2 3% -3% -1% -3% - -18% -19% A2 65% 49% -3% -1% - -17% -19%
C 20% -11% -17% -18% -17% -15% -17% C -14% -23% -14% -6% -4% 11% 2% C -5% -20% -18% -14% -7% -14% -11%
D2 60% 34% -12% -17% - -20% - D2 -7% -9% -29% -23% - -14% - D2 10% 5% -22% -20% - -18% -
3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9
A1 198% 82% 43% 10% 0% - - A1 27% -9% -12% -18% -11% - - A1 23% 15% 13% -1% -1% - -
A2 105% 88% 54% 43% 17% - - A2 4% 0% -9% -6% -14% - - A2 8% 2% 1% 6% -4% - -
C 15% 5% 8% 10% 15% 14% 11% C -16% -6% 18% 39% 65% 32% 45% C 0% 3% 14% 25% 37% 24% 27%
D2 46% 21% 1% 0% 4% - - D2 -14% -16% -15% -1% 20% - - D2 14% 5% -2% 5% 16% - -
Sion - MA3 Sion - MA3Sion - MA3
N2 method Lin&Miranda N2 OPT
Sion - MA1
Sion - MA2
Sion - MA1 Sion - MA1
Sion - MA2 Sion - MA2
Summary of the results obtained for the microzones of Sion. On the left, the per cent differences between N2 method displacement demand and the NLTHA average value; on the middle the Lin & Miranda method results; on the right the N2 OPT method. Values outside of one standard deviation are bold face
Per cent differences of performance points prediction of the three methods analysed with the NLTHA prediction (Sion microzone MA1, MA2, MA3)
Performance point: SION and MARTIGNY – microzones
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Methods results comparison
Static nonlinear analysis 82
Performance point: SION and MARTIGNY – microzones
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Methods results comparison
Static nonlinear analysis 83
3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9
A1 97% 46% 16% -2% -1% - - A1 9% -6% -9% -8% 9% - - A1 1% -1% -6% -11% -2% - -
A2 - 45% - 3% 4% - - A2 - 3% - -8% 2% - - A2 - -1% - -15% -7% - -
C - 7% - - 13% 6% - C - -3% - - 14% 6% - C - -13% - - 10% 6% -
D2 40% 4% -4% 3% 1% 1% - D2 -3% -12% -3% 21% 35% 36% - D2 3% -12% -10% 5% 9% 9% -
3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9
A1 45% 10% -4% -10% -2% - - A1 -5% -2% 12% 29% 70% - - A1 -4% -6% -5% -3% 12% - -
A2 35% 14% 7% -1% 1% 4% - A2 0% 2% 14% 11% 9% 10% - A2 -6% -6% -1% -2% 5% 4% -
C - 5% 7% 6% 8% 7% - C - 12% 9% 8% 8% 6% - C - 1% 6% 6% 7% 6% -
D2 4% -5% -8% -1% -1% 0% - D2 1% 16% 9% 15% 12% 10% - D2 -6% -3% 3% 15% -1% 0% -
Martigny - MM2 Martigny - MM2 Martigny - MM2
Martigny - MM3 Martigny - MM3 Martigny - MM3
N2 method Lin&Miranda N2 OPT
Per cent differences of performance points prediction of the three methods analysed with the NLTHA prediction (Martigny microzone MM2, MM3).
Damage distribution
Mechanical methods for damage distribution (Lagormarsino and Giovinazzi, 2006) produce a lognormal cumulative probability function. In this study a different procedure has been employed related to a particular damage probability matrix. This matrix is utilized in macroseismic vulnerability methods (Lagomarsino and Giovinazzi, 2006) and is achieved by the probability mass function (PMF) of the binomial distribution.
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Methods results comparison
Static nonlinear analysis 84
𝑝 𝑠𝑡𝑜𝑟,𝑘 =5!
𝑘! 5 − 𝑘 !∙𝜇𝑑5
𝑘
∙ 1 −𝜇𝑑5
5−𝑘
Damage distribution
Definition of average damage grade:
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Methods results comparison
Static nonlinear analysis 85
𝜇𝑑 =
0 + 𝑆𝑑 − 0
𝑆𝑑,1 − 0 , 𝑆𝑑 ≤ 𝑆𝑑,1
1 + 𝑆𝑑 − 𝑆𝑑,1𝑆𝑑,2 − 𝑆𝑑1
, 𝑆𝑑,1 < 𝑆𝑑 ≤ 𝑆𝑑,2
2 + 𝑆𝑑 − 𝑆𝑑,2𝑆𝑑,3 − 𝑆𝑑,2
, 𝑆𝑑,2 < 𝑆𝑑 ≤ 𝑆𝑑,3
3 + 𝑆𝑑 − 𝑆𝑑,3𝑆𝑑,4 − 𝑆𝑑,3
, 𝑆𝑑,3 < 𝑆𝑑 ≤ 𝑆𝑑,4
4 + 𝑆𝑑 − 𝑆𝑑,42 ∙ 𝑆𝑑,4
, 𝑆𝑑 > 𝑆𝑑,4
Damage distribution: SION and MARTIGNY – soil class C
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Methods results comparison
Static nonlinear analysis 86
damage
degree0 1 2 3 4 5
damage
degree0 1 2 3 4 5
damage
degree0 1 2 3 4 5
N2 47.63 150.06 217.73 189.92 101.02 25.65 L&M 57.95 174.25 228.77 172.33 79.57 19.13 N2 OPT 56.16 171.31 229.96 176.24 80.31 18.03
NLTHA 62.64 178.64 226.61 168.63 77.56 17.93 NLTHA 62.64 178.64 226.61 168.63 77.56 17.93 NLTHA 62.64 178.64 226.61 168.63 77.56 17.93
diff. -15.01 -28.58 -8.88 21.29 23.46 7.72 diff. -4.69 -4.39 2.16 3.70 2.01 1.20 diff. -6.48 -7.33 3.35 7.61 2.75 0.10
7.72DIFF.
TOT15.01 28.58 8.88 21.29 23.46
Sion - typ. A1+A2+C+D2 (total number of buildings = 732)
EC8 Soil class C (total number of buildings = 732)
DIFF.
TOT4.69 4.39 2.16 3.70
Total buildings damage distribution difference
N2 method damage distribution analysis Lin&Miranda damage distribution analysis N2 OPT damage distribution analysis
Sion - typ. A1+A2+C+D2 (total number of buildings = 732) Sion - typ. A1+A2+C+D2 (total number of buildings = 732)
EC8 Soil class C (total number of buildings = 732) EC8 Soil class C (total number of buildings = 732)
2.01 1.20DIFF.
TOT6.48 7.33 3.35 7.61 2.75 0.10
18.15 Total buildings damage distribution difference 27.63
Per cent damage distribution difference 2.48% Per cent damage distribution difference 3.77%
Total buildings damage distribution difference
Per cent damage distribution difference
104.93
14.33%
damage
degree0 1 2 3 4 5
damage
degree0 1 2 3 4 5
damage
degree0 1 2 3 4 5
N2 12.73 40.44 76.12 102.73 86.22 32.75 L&M 16.09 49.97 81.40 92.99 76.15 34.39 N2 OPT 16.58 50.80 84.18 96.73 74.04 28.67
NLTHA 17.16 50.83 82.90 96.67 75.11 28.33 NLTHA 17.16 50.83 82.90 96.67 75.11 28.33 NLTHA 17.16 50.83 82.90 96.67 75.11 28.33
diff. -4.43 -10.39 -6.77 6.06 11.11 4.43 diff. -1.07 -0.86 -1.49 -3.68 1.04 6.06 diff. -0.58 -0.03 1.28 0.06 -1.07 0.34
N2 method damage distribution analysis Lin&Miranda damage distribution analysis N2 OPT damage distribution analysis
Martigny - typ. A1+A2+C+D2 (total of analysed buildings = 351) Martigny - typ. A1+A2+C+D2 (total of analysed buildings = 351) Martigny - typ. A1+A2+C+D2 (total of analysed buildings = 351)
EC8 Soil class C (total number of buildings = 351) EC8 Soil class C (total number of buildings = 351) EC8 Soil class C (total number of buildings = 351)
DIFF.
TOT4.43 10.39 6.77 6.06 11.11 4.43
DIFF.
TOT1.07 0.86 1.49 3.68 1.04 6.06
DIFF.
TOT0.58 0.03 1.28
Per cent damage distribution difference 12.30% Per cent damage distribution difference 4.05% Per cent damage distribution difference 0.96%
0.06 1.07 0.34
Total buildings damage distribution difference 43.18 Total buildings damage distribution difference 14.21 Total buildings damage distribution difference 3.36
Number of buildings and relative damage grade for the city of Sion and Martigny, evaluated for Soil class C and with the three methods
Damage distribution: SION and MARTIGNY – soil class C
Advanced Earthquake Engineering CIVIL-706
Methods results comparison
Static nonlinear analysis 87
0.00
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125.00
150.00
175.00
200.00
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250.00
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Damage grade
N2 Method Lin&Miranda N2 OPT NLTHA
Damage distribution of Sion (soil C)
Martigny
Damage distribution: SION and MARTIGNY – soil class C
Advanced Earthquake Engineering CIVIL-706
Methods results comparison
Static nonlinear analysis 88
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110.00
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Damage grade
N2 Method Lin&Miranda N2 OPT NLTHA
Damage distribution of Martigny (soil C)
Damage distribution: SION – microzone
Advanced Earthquake Engineering CIVIL-706
Methods results comparison
Static nonlinear analysis 89
damage
grade0 1 2 3 4 5
damage
grade0 1 2 3 4 5
damage
grade0 1 2 3 4 5
N2 7.87 24.65 35.55 31.21 17.11 4.60 L&M 11.99 32.69 38.71 25.67 9.98 1.96 N2 OPT 10.13 29.97 38.76 28.03 11.72 2.39
NLTHA 12.18 32.44 38.06 25.65 10.47 2.20 NLTHA 12.18 32.44 38.06 25.65 10.47 2.20 NLTHA 12.18 32.44 38.06 25.65 10.47 2.20
diff. -4.31 -7.79 -2.51 5.56 6.64 2.41 diff. -0.19 0.25 0.65 0.02 -0.49 -0.24 diff. -2.05 -2.47 0.70 2.38 1.25 0.19
damage
grade0 1 2 3 4 5
damage
grade0 1 2 3 4 5
damage
grade0 1 2 3 4 5
N2 8.72 30.58 49.52 49.95 33.40 11.84 L&M 12.42 40.05 56.22 45.44 23.20 6.68 N2 OPT 11.45 37.82 54.78 46.55 25.56 7.84
NLTHA 10.77 36.24 52.23 44.38 27.95 12.42 NLTHA 10.77 36.24 52.23 44.38 27.95 12.42 NLTHA 10.77 36.24 52.23 44.38 27.95 12.42
diff. -2.05 -5.66 -2.71 5.57 5.45 -0.58 diff. 1.65 3.81 3.99 1.06 -4.75 -5.74 diff. 0.68 1.58 2.55 2.17 -2.39 -4.58
damage
grade0 1 2 3 4 5
damage
grade0 1 2 3 4 5
damage
grade0 1 2 3 4 5
N2 12.24 51.32 98.05 118.03 99.71 47.66 L&M 17.70 72.17 124.70 119.90 70.13 22.38 N2 OPT 16.28 66.33 115.60 117.00 79.54 32.23
NLTHA 19.67 73.33 118.30 112.40 73.60 29.67 NLTHA 19.67 73.33 118.30 112.40 73.60 29.67 NLTHA 19.67 73.33 118.30 112.40 73.60 29.67
diff. -7.43 -22.01 -20.25 5.63 26.11 17.99 diff. -1.97 -1.16 6.40 7.50 -3.47 -7.29 diff. -3.39 -7.00 -2.70 4.60 5.94 2.56
Microzone MA2 (total number of buildings = 184) Microzone MA2 (total number of buildings = 184)
Microzone MA3 (total number of buildings = 427) Microzone MA3 (total number of buildings = 427)
Microzone MA1 (total number of buildings = 121) Microzone MA1 (total number of buildings = 121)Microzone MA1 (total number of buildings = 121)
Sion - typ. A1+A2+C+D2 (total number of buildings = 732)
9.15 9.58 7.34TOT
DIFF.3.81 5.22 11.04 8.58
Microzone MA2 (total number of buildings = 184)
Microzone MA3 (total number of buildings = 427)
Total buildings damage distribution difference
Per cent damage distribution difference
150.66
20.58%
MA3 damage distribution difference (23.28%) 99.42
8.71 13.27
Sion - typ. A1+A2+C+D2 (total number of buildings = 732)
N2 method damage distribution analysis
Sion - typ. A1+A2+C+D2 (total number of buildings = 732)
N2 OPT damage distribution analysisLin&Miranda damage distribution analysis
TOT
DIFF.13.80 35.46 25.47
MA1 damage distribution difference (24.15%) 29.22
Total buildings damage distribution difference 49.19
Per cent damage distribution difference 6.72%
MA3 damage distribution difference (6.13%) 26.19
MA2 damage distribution difference (7.58%) 13.95
MA1 damage distribution difference (7.47%) 9.04
Total buildings damage distribution difference 50.63
Per cent damage distribution difference 6.92%
MA1 damage distribution difference (1.52%) 1.84
MA2 damage distribution difference (11.41%) 21.00
MA3 damage distribution difference (6.51%) 27.79
MA2 damage distribution difference (11.97%) 22.03
16.75 38.20 20.98TOT
DIFF.6.12 11.05 5.95
Number of buildings and relative damage grade for the city of Sion, evaluated for microzone MA1, MA2 and MA3, and with the three methods
Damage distribution: SION – microzone
Advanced Earthquake Engineering CIVIL-706
Methods results comparison
Static nonlinear analysis 90
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Damage grade
N2 Method Lin&Miranda N2 OPT NLTHA
Damage distribution of Sion (microzones MA1+MA2+MA3)
Damage distribution of Sion (microzones MA1+MA2+MA3)
Damage distribution: MARTIGNY – microzone
Advanced Earthquake Engineering CIVIL-706
Methods results comparison
Static nonlinear analysis 91
damage
grade0 1 2 3 4 5
damage
grade0 1 2 3 4 5
damage
grade0 1 2 3 4 5
N2 3.24 9.81 18.60 26.40 23.08 8.87 L&M 3.80 11.43 19.51 24.57 21.25 9.44 N2 OPT 3.95 11.89 20.63 25.61 20.32 7.61
NLTHA 3.90 10.99 18.97 25.41 22.07 8.66 NLTHA 3.90 10.99 18.97 25.41 22.07 8.66 NLTHA 3.90 10.99 18.97 25.41 22.07 8.66
diff. -0.66 -1.18 -0.37 0.99 1.01 0.20 diff. -0.11 0.43 0.55 -0.84 -0.81 0.78 diff. 0.04 0.90 1.66 0.20 -1.74 -1.06
damage
grade0 1 2 3 4 5
damage
grade0 1 2 3 4 5
damage
grade0 1 2 3 4 5
N2 7.39 22.58 43.05 64.64 63.15 28.19 L&M 7.30 20.99 35.67 54.70 67.62 42.73 N2 OPT 7.91 23.76 42.09 60.26 62.38 32.60
NLTHA 8.12 23.41 41.83 61.98 63.31 30.35 NLTHA 8.12 23.41 41.83 61.98 63.31 30.35 NLTHA 8.12 23.41 41.83 61.98 63.31 30.35
diff. -0.73 -0.82 1.22 2.65 -0.16 -2.16 diff. -0.82 -2.42 -6.16 -7.29 4.31 12.38 diff. -0.21 0.35 0.26 -1.72 -0.94 2.25
Martigny - typ. A1+A2+C+D2 (total of analysed buildings = 319) Martigny - typ. A1+A2+C+D2 (total of analysed buildings = 319) Martigny - typ. A1+A2+C+D2 (total of analysed buildings = 319)
N2 method damage distribution analysis Lin&Miranda damage distribution analysis N2 OPT damage distribution analysis
Microzone MM2 (total number of buildings = 90) Microzone MM2 (total number of buildings = 90) Microzone MM2 (total number of buildings = 90)
MM2 damage distribution difference (4.91%) 4.42 MM2 damage distribution difference (3.91%) 3.52 MM2 damage distribution difference (6.23%) 5.60
2.36TOT
DIFF.0.93
Microzone MM3 (total number of buildings = 229) Microzone MM3 (total number of buildings = 229) Microzone MM3 (total number of buildings = 229)
MM3 damage distribution difference (3.38%) 7.74 MM3 damage distribution difference (14.57%) 33.37 MM3 damage distribution difference (2.50%) 5.73
1.92 2.68 3.31
Total buildings damage distribution difference 12.16 Total buildings damage distribution difference 36.89 Total buildings damage distribution difference 11.33
2.85 6.71 8.13 5.12 13.16TOT
DIFF.0.25 1.25 1.92
TOT
DIFF.1.39 2.00 1.59 3.65 1.17
Per cent damage distribution difference 3.81% Per cent damage distribution difference 11.56% Per cent damage distribution difference 3.55%
Number of buildings and relative damage grade for the city of Martigny, evaluated for microzone MM2 and MM3, and with the three methods
Damage distribution: MARTIGNY – microzone
Advanced Earthquake Engineering CIVIL-706
Methods results comparison
Static nonlinear analysis 92
Damage distribution for Martigny (microzones MM2 and MM3)
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N2 Method Lin&Miranda N2 OPT NLTHA
Conclusion
Advanced Earthquake Engineering CIVIL-706
Methods results comparison
Static nonlinear analysis 93
MA1 24.15% MA2 11.97% MA3 23.28% MA1 1.52% MA2 11.41% MA3 6.51% MA1 7.47% MA2 7.58% MA3 6.13%
MM1 - MM2 4.91% MM3 3.38% MM1 - MM2 3.91% MM3 14.57% MM1 - MM2 6.23% MM3 2.50%
SION - Microzones SION - Microzones SION - Microzones
Damage distribution difference 20.58% Damage distribution difference 6.92% Damage distribution difference 6.72%
MARTIGNY - Microzones MARTIGNY - Microzones
MARTIGNY - EC8 Soil class C MARTIGNY - EC8 Soil class C MARTIGNY - EC8 Soil class C
Damage distribution difference 12.30% Damage distribution difference 4.05% Damage distribution difference 0.96%
Damage distribution difference 3.81% Damage distribution difference 11.56% Damage distribution difference 3.55%
Lin&Miranda
SION - EC8 Soil class C
Damage distribution difference 2.48%
N2 OPT
SION - EC8 Soil class C
Damage distribution difference 3.77%Damage distribution difference 14.33%
MARTIGNY - Microzones
N2 method
SION - EC8 Soil class C
Summary - Differences between the three methods and the NLTHA damage distribution for the city of Sion and Martigny obtained on EC8 soil class C and on the microzones
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