STATIC NONLINEAR ANALYSIS (2/2) Advanced Earthquake ...€¦ · ANALYSIS (2/2) Advanced Earthquake Engineering CIVIL-706 Instructor: Lorenzo DIANA, PhD 1 . Advanced Earthquake Engineering

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


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)



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)



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


Equivalent Linearization


Basic equations…

6


Equivalent Linearization- Performance Point


Sou

rce:

FEM

A 4

40

𝑆𝑑 = 𝑆𝑑 𝑇𝑒𝑞; ζ𝑒𝑞 = 𝑆𝑑 𝑇𝑒𝑞; ζ=5% ∙ 𝜂 = 𝑆𝑎 𝑇𝑒𝑞; ζ=5% ∙𝑇𝑒𝑞2

4𝜋2∙ 𝜂

7





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





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





4𝜋2∙ 𝜂

𝑇𝑒𝑞 = 1 +𝑚1

𝑇2∙ 𝑅𝜇

1.8 − 1 ∙ 𝑇

ζ𝑒𝑞 = ζ=5% +𝑛1𝑇𝑛2

∙ 𝑅𝜇 − 1


Equivalent Linearization

• Advantages:

– Linear computation

– Use of pushover analysis

• Drawbacks:

– Value of damping

– Not always conservative



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


Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis 13

N2 Method Procedure

R

Reduction factor

Ductility factor




N2 Method Procedure


N2 Method Procedure


Content

• Capacity curve

• Seismic damage





• Method result confrontation



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




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




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



Empirical Methods

Empirical methods example

European Macroseismic Scale 98

• Calculate Vulnerability index from vulnerability class

• Structural parameters is only the structure class



Empirical Methods

EMS 98 (Empirical method):


Sou

rce:

Ris

k-U

E 2

00

3

23


Empirical Methods


Risk-UE LM1 method:


Empirical Methods Risk-UE LM1 method:


Mean damage grade


Empirical Methods


Vulnerability class in downtown Malaga (EMS98)

26

Mean damage of downtown Malaga with I=VI (LM1)


Risk-UE LM1 method : (Rachel de Blaireville, PdM 2016)

Yverdon-les-Bains


I=VI I=VII


Empirical Methods

Risk-UE LM1 method:


Sou

rce:

Ris

k-U

E 2

00

3

28


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



Mechanical Methods

RISK-UE LM2 method:



Mechanical Methods RISK-UE LM2 method:

Definition of capacity curves for building typologies



Mechanical Methods RISK-UE LM2 method:

Definition of capacity curves for building typologies

Definition of damage limit states.



Mechanical Methods

RISK-UE LM2 method:



RISK-UE LM2 method:

Determination of the performance point


DG2

DG3

DG4


Mechanical Methods

RISK-UE LM2 method:



Risk-UE LM2 method : (Rachel de Blaireville, PdM 2016)

Yverdon-les-Bains

DG0

DG1

DG2

DG3

DG4 NO


Risk-UE LM2 method : (Lestuzzi et al., 2016)

Method LM2: results for the city of Sion

DG0

DG1

DG2

DG3

DG4


DG0

DG1

DG2

DG3

DG4

Risk-UE LM2 method : (Lestuzzi et al., 2016)

Method LM2: results for the city of Martigny


Mechanical Methods

RISK-UE LM2 method:



Mechanical Methods

Capacity Curve and Fragility curves


Sou

rce:

HA

ZUS

20

03

40


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



Mechanical Methods

Example of RISK-UE application:


Probability of D4 and D5

42


Content

• Capacity curve

• Seismic damage





• Methods results comparison



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




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.


%𝛥𝑇𝑖 =𝐷𝐷𝑁2𝑇𝑖

− 𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖𝐷𝐷𝑁𝐿𝑇𝐻𝐴𝑇𝑖

%.



Example of the method applied


N2 method prediction NLTHA mean prediction (+ and – standard deviation)

R = Sae / Say

R

R



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.



N2 method optimization

Selection and modification of 12 earthquakes




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.




%.

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%



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:




%.



T N2 [m]

(displacement)


DDNLTHA Ti


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%



Example of the method applied


N2 method prediction NLTHA mean prediction (+ and – standard deviation)

R = Sae / Say

R

R






%.



T N2 [m]

(displacement)


DDNLTHA Ti


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%






%.


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






%.



T N2 [m]

(displacement)


DDNLTHA Ti


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%



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:


% 𝛥 𝑅 =1

𝑡 % 𝛥 𝑇𝑖

𝑡

𝑖=1




Per cent variance in absolute value expressed for each strength reduction factor (R)



Global discrepancy for soil classes


17.90%

SOIL A SOIL B SOIL C SOIL D

15.72% 16.26% 16.39% 23.24%


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



%.

% 𝛥 𝑇 𝑇 =1

𝑠 % 𝛥 𝑘

𝑠

𝑘=1



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.


𝑆𝑑 =𝑆𝑑𝑒𝑅𝛼1

∙𝑅

𝛼2 − 1

𝛽

∙𝑇𝐶𝑇+ 1



Optimization approach for new formulas (Manno A)

Definition of the single error

Definition of the global error function

Minimization objective


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



1 variable optimization



∙ 𝑅 − 1 1.13 ∙𝑇𝐶𝑇+ 1

17.90%


15.72% 16.26% 16.39% 23.24%


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


∙ 𝑅 − 1 ∙𝑇 𝑇+ 1

% 𝑅

%

% 𝑇 𝑇

β* 1.13

15.78%


10.65% 14.99% 15.79% 21.67%


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


∙ 𝑅 − 1 𝛽 ∙𝑇 𝑇+ 1




OPTexp1.13 (R=[1.5 5.0])

OPTexp1.16 (R=[1.5 9.0])



∙ 𝑅 − 1 1.13 ∙𝑇𝐶𝑇+ 1


∙ 𝑅 − 1 1.16 ∙𝑇𝐶𝑇+ 1






1.48

∙𝑅

1.45− 1

1.35

∙𝑇𝐶𝑇+ 1

α1* 1.48

α2* 1.45

β* 1.35

6.90%


7.97% 6.91% 5.24% 7.46%


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%


15.72% 16.26% 16.39% 23.24%


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


∙ 𝑅 − 1 ∙𝑇 𝑇+ 1

% 𝑅

%

% 𝑇 𝑇




OPT3varA ( R=[1.5 5.0] )

OPT3varB ( R=[1.5 9.0] )



1.48

∙𝑅

1.45− 1

1.35

∙𝑇𝐶𝑇+ 1


1.38

∙𝑅

1.38− 1

1.28

∙𝑇𝐶𝑇+ 1




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




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




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


Content

• Capacity curve

• Seismic damage





• Methods results comparison


Simplified seismic demand determination confrontation

• N2 method

• Lin&Miranda

• N2 optimization


Methods results comparison


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.




Test cities: SION and MARTIGNY




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




Distribution




Sion Martigny

Distribution




Sion Martigny

Non linear time history analysis




Methodology




N2 method Methodology




Lin&Miranda Methodology




N2 optimization Methodology

Performance point: SION and MARTIGNY – soil class C




Performance point: SION and MARTIGNY – 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 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% -


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








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


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)









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


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.




𝑝 𝑠𝑡𝑜𝑟,𝑘 =5!

𝑘! 5 − 𝑘 !∙𝜇𝑑5

𝑘

∙ 1 −𝜇𝑑5

5−𝑘

Damage distribution

Definition of average damage grade:




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




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%


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


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


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





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N2 Method Lin&Miranda N2 OPT NLTHA

Damage distribution of Sion (soil C)

Martigny





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Damage distribution of Martigny (soil C)

Damage distribution: SION – microzone




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


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


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


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)


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)


Per cent damage distribution difference

150.66

20.58%

MA3 damage distribution difference (23.28%) 99.42

8.71 13.27


N2 method damage distribution analysis


N2 OPT damage distribution analysisLin&Miranda damage distribution analysis

TOT

DIFF.13.80 35.46 25.47


Total buildings damage distribution difference 49.19

Per cent damage distribution difference 6.72%




Total buildings damage distribution difference 50.63

Per cent damage distribution difference 6.92%





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




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Damage distribution of Sion (microzones MA1+MA2+MA3)

Damage distribution of Sion (microzones MA1+MA2+MA3)

Damage distribution: MARTIGNY – microzone




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


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


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)


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




Damage distribution for Martigny (microzones MM2 and MM3)

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Conclusion




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



Lin&Miranda

SION - EC8 Soil class C

Damage distribution difference 2.48%

N2 OPT


Damage distribution difference 3.77%Damage distribution difference 14.33%

MARTIGNY - Microzones

N2 method


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

Documents

STATIC NONLINEAR ANALYSIS (2/2) Advanced Earthquake ...€¦ · ANALYSIS (2/2) Advanced Earthquake Engineering CIVIL-706 Instructor: Lorenzo DIANA, PhD 1 . Advanced Earthquake Engineering