slide 4_static nonlinear analysis
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STATIC NONLINEAR ANALYSISANALYSIS
Advanced Earthquake EngineeringCIVIL-706
Instructor:Amin KARBASSI, PhD
By the end of today’s course…
You will be able to answer:
• What are NSA advantages • What are NSA advantages
over other structural
analysis methods?
• How to perform an NSA?
• What are the key elements
Static nonlinear analysis Advanced Earthquake Engineering CIVIL-706
• What are the key elements
when performing NSA?
Earthquake Engineering Assessment
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FE
MA
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Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
So
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FE
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Earthquake Engineering Assessment
Action
StructureStatic Dynamic
LinearEquivalent Force
method
Response
Spectrum method
Advanced Earthquake Engineering CIVIL-706
method Spectrum method
Non-linear PushoverNon-linear
Dynamic
Static nonlinear analysis
Nonlinear time history analysis
• Advantage: considering the complexity of the
dynamic load + the dynamic behavior of structuredynamic load + the dynamic behavior of structure
• Disadvantage: time consuming
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Nonlinear static procedure
• How does it work? Capacity vs. Demand
Sa
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Sd
Nonlinear static procedure
• Necessary elements?
– Acceleration-displacement response spectrum (ADRS)– Acceleration-displacement response spectrum (ADRS)
Sa
T(sec.)
V (force)
Δ (disp.)
Sa
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Sa
Sd
Nonlinear static procedure
• Developing capacity curves:
– Displacement-based methods– Displacement-based methods
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Nonlinear static procedure
• Assumptions
– Response: maximum displacement– Response: maximum displacement
– Deformation: “most often” following the first mode
• Benefits
– Displacement-based analysis
– Nonlinear behavior of the structure
• Drawbacks
Advanced Earthquake Engineering CIVIL-706
• Drawbacks
– Simplified approach (static)
– Cyclic response resembled to monotonic response
– Damping difficult to represent
Static nonlinear analysis
Content
• Pushover technique – capacity curve
• Seismic performance – damage• Seismic performance – damage
• Acceleration-Displacement Response Spectrum
(ADRS)
• Inelastic response spectrum and Performance Point
• Large-scale vulnerability assessment
Advanced Earthquake Engineering CIVIL-706
• Large-scale vulnerability assessment
Static nonlinear analysis
Content
• Pushover technique – capacity curve
• Seismic performance – damage• Seismic performance – damage
• Acceleration-Displacement Response Spectrum
(ADRS)
• Inelastic response spectrum and Performance Point
• Large-scale vulnerability assessment
Advanced Earthquake Engineering CIVIL-706
• Large-scale vulnerability assessment
Static nonlinear analysis
Capacity Curve
• Pushover: to estimate the capacity curve
Structural model
(FEM, AEM, etc.)
Advanced Earthquake Engineering CIVIL-706
• Objectives:
– Estimate the maximum horizontal displacement
– Estimate the (global) ductility of the structure
Static nonlinear analysis
(FEM, AEM, etc.)
Capacity Curve
• Load pattern: distribution of forces should represent
the dynamic behavior the dynamic behavior
– Triangular (if building regular enough)
– Following the first mode
– Other simplified distributions
depending on the building
Advanced Earthquake Engineering CIVIL-706
Base Shear
Static nonlinear analysis
Capacity Curve
• Load pattern: IDARC practice
– Effect of lateral load pattern– Effect of lateral load pattern
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Capacity Curve
• Force/Displacement to spectral coordinates
Equivalent SDOF
V
Δ
Sa
Sd
V
Equivalent SDOF
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
1a
VWS
α=
1 1,( )droof
SPF
∆=×Φ
1
1,
1
(mod.1)
(mod.1)
(mod.1)roof
mass participation factor
roof level amplitude
PF Participation Factor
α =Φ =
=
Content
• Pushover technique – capacity curve
• Seismic performance – damage• Seismic performance – damage
• Acceleration-Displacement Response Spectrum
(ADRS)
• Inelastic response spectrum and Performance Point
• Large-scale vulnerability assessment
Advanced Earthquake Engineering CIVIL-706
• Large-scale vulnerability assessment
Static nonlinear analysis
Seismic Performance/Damage
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Seismic Performance/Damage
• Damage-based design/assessment:
– Related to social and economical costs of – Related to social and economical costs of
damage/mitigating measures
• Performance-based design/assessment:
– Related to life safety, reduce injuries, etc.
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
FEMA 356 Performance levels
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
EMS98 damage grades
FEMA 356 Performance levels
Example of damage grade estimation
RC building with FE modeling
– Pushover on the FE model: damage grade– Pushover on the FE model: damage grade
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Unreinforced masonry (URM)
Example
URM building in Yverdon-les-Bains
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Unreinforced masonry (URM)
Example
Architectural plan
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Unreinforced masonry (URM)
Example
Simplification for the modeling
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Unreinforced masonry (URM)
Key-parameter: Frame effect (coupling of horizontal/vertical elements)horizontal/vertical elements)
– flexible slab (w/o frame effect)
– Infinitely stiff lintels
0
2
3 totalh h= ×
Ze
ro m
om
en
t
Advanced Earthquake Engineering CIVIL-706
– Infinitely stiff lintels
(total frame effect)
Static nonlinear analysis
0
1
2 storeyh h= ×
Unreinforced masonry (URM)
Example
Extreme case studies:
0
26.7
3 totalh h m= × =
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
0
11.25
2 storeyh h m= × =
Unreinforced masonry (URM)
Example
Lateral strength – transverse direction
Lw (m) Nd (kN)w/o frame effect
(kN)
Complete frame
effect (kN)
2.0 180 22 96
5.0 360 117 216
6.0 1520 367 624
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
6.0 1520 367 624
total 690 1440
Too conservative Too optimistic
Unreinforced masonry (URM)
Example
Lateral strength – transverse direction
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Content
• Pushover technique – capacity curve
• Seismic performance – damage• Seismic performance – damage
• Acceleration-Displacement Response Spectrum
(ADRS)
• Inelastic response spectrum and Performance Point
• Large-scale vulnerability assessment
Advanced Earthquake Engineering CIVIL-706
• Large-scale vulnerability assessment
Static nonlinear analysis
Reminder: Response Spectrum
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Reminder: Response Spectrum
2a
d
SS
ω≈
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Acceleration-Displacement Response
Spectrum (ADRS)
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Acceleration-Displacement Response
Spectrum (ADRS)
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Content
• Pushover technique – capacity curve
• Seismic performance – damage• Seismic performance – damage
• Acceleration-Displacement Response Spectrum
(ADRS)
• Inelastic response spectrum and Performance Point
• Large-scale vulnerability assessment
Advanced Earthquake Engineering CIVIL-706
• Large-scale vulnerability assessment
Static nonlinear analysis
Inelastic Response Spectrum
Best linear SDOF which represents the nonlinear
structure? (examples covered here)structure? (examples covered here)
• Equivalent linearization (FEMA 440)
• Improved Spectrum Method of ATC40
• N2 method (equal displacement rule, EC8 approach)
Advanced Earthquake Engineering CIVIL-706
• N2 method (equal displacement rule, EC8 approach)
Static nonlinear analysis
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)
Advanced Earthquake Engineering CIVIL-706
ATC-40 (1996)
• Improved later in FEMA 440 document (2005)
Static nonlinear analysis
Equivalent Linearization Method
Optimal equivalent linear parameters (i.e.,
effective period, Teff, and effective damping, effective period, Teff, and effective damping,
βeff) are determined through a statistical
analysis that minimizes, in a rigorous manner,
the extreme occurrences of the difference (i.e.,
error) between the maximum response of an
actual inelastic system and its equivalent
Advanced Earthquake Engineering CIVIL-706
error) between the maximum response of an
actual inelastic system and its equivalent
linear counterpart.
Static nonlinear analysis
“FEMA 440”
Equivalent Linearization
Basic equations…
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Equivalent Linearization-
Performance Point
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Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
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Equivalent Linearization
• Advantages:
– Linear computation– Linear computation
– Use of pushover analysis
• Drawbacks:
– Value of damping (through iterations)
Advanced Earthquake Engineering CIVIL-706
– Value of damping (through iterations)
– Not always conservative
Static nonlinear analysis
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)
Advanced Earthquake Engineering CIVIL-706
(ATC-40)
• Based on inelastic spectra rather than elastic
spectra
Static nonlinear analysis
N2 Method Procedure (1)
Required tools:
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Structural model (MDOF) Elastic response spectrum
N2 Method Procedure (2)
,aea d de
SS S S
R Rµ µ
µ= =Reduction factorR Rµ µ
( 1) 1 CC
C
TR T T
T
R T T
µ
µ
µ
µ
= − + <
= ≥
Reduction factor
“Vidic et al. 1992”
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Inelastic response spectrum (ADRS)
N2 Method Procedure (3)-
Performance Point
:Cif T T≥
Basic equations…
:Cif T T≥
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
N2 Method Procedure (3)-
Performance Point
d deS S=d deS S=
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
N2 Method Procedure (3)-
Performance Point
:Cif T T<
Basic equations…
:Cif T T<
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
N2 Method Procedure (3)-
Performance Point
1 ( 1)de Cd
S TS Rµ
= + − *1 ( 1)dS R
R Tµµ
= + −
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Content
• Pushover technique – capacity curve
• Seismic performance – damage• Seismic performance – damage
• Acceleration-Displacement Response Spectrum
(ADRS)
• Inelastic response spectrum and Performance Point
• Large-scale vulnerability assessment
Advanced Earthquake Engineering CIVIL-706
• Large-scale vulnerability assessment
Static nonlinear analysis
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
Advanced Earthquake Engineering CIVIL-706
• Vulnerability assessment of each class including
variability (probabilistic assessment)
Fragility functions
Static nonlinear analysis
Large-scale vulnerability assessment
Empirical methods (e.g. EMS98, GNDT, Risk-UE LM1,
Vulneralp) based on damage surveysVulneralp) based on damage surveys
• Relationship between ground motion (intensity)
and distribution of observed damage grades for
structures with a given vulnerability index
Advanced Earthquake Engineering CIVIL-706
Damage Probability Matrix
Static nonlinear analysis
Empirical Methods
EMS98, GNDT, Risk-UE, Vulneralp) based on damage
surveyssurveys
• Relationship between ground motion (intensity)
and distribution of observed damage grades for
structures with a given vulnerability index
Advanced Earthquake Engineering CIVIL-706
Damage Probability Matrix
Static nonlinear analysis
Empirical Methods
Empirical methods example
European Macroseismic Scale 98European Macroseismic Scale 98
• Calculate Vulnerability index
Advanced Earthquake Engineering CIVIL-706
• Calculate Vulnerability index
from vulnerability class
• Structural parameters is only
the structure class
Static nonlinear analysis
Empirical Methods
EMS 98 (Empirical method):
So
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Ris
k-U
E 2
00
3
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
So
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Ris
k
Empirical Methods
Example of EMS98 application: (Feriche et al. 2009)
b. Damage
grade from
intensity VI
a. Vulnerability
class in
downtown
Malaga
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Empirical Methods
• Application in large-scale assessment (engineering offices, insurance companies, etc.), real-time offices, insurance companies, etc.), real-time computations (WAPMERR) or risk-based seismic assessment in codes (SIA 2018)
BUT
• Not compatible with modern hazard assessment (site effect, etc.)
Only valid for large number of buildings (classification)
Advanced Earthquake Engineering CIVIL-706
• Only valid for large number of buildings (classification)
• No surveys available for areas with moderate seismic hazard
Static nonlinear analysis
Mechanical Methods
• AKA as predicted methods (e.g., HAZUS, RISK-UE LM2)
• Based on computation (generally nonlinear static)• Based on computation (generally nonlinear static)
• Relation between ground motion (PGA, spectral
ordinate) and expected distribution of damage grade
Fragility curves
• Takes variability into account
Advanced Earthquake Engineering CIVIL-706
• Takes variability into account
Static nonlinear analysis
Mechanical Methods
Fragility curves
So
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HA
ZU
S 2
00
3
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
So
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HA
ZU
S 2
00
3
Mechanical Methods
RISK-UE LM2 method:
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Mechanical Methods
• Compatible with modern hazard definitions
• Independent of observed damage data (only verification)• 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)
• Crude approximation in computations
Advanced Earthquake Engineering CIVIL-706
• Crude approximation in computations
• Fragility curves only available for certain building classes
• Difficulties in estimating variability
Static nonlinear analysis
Mechanical Methods
Example of RISK-UE application:
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Probability of D4 and D5 in case of a 1887-type
scenario earthquake at 30 km (deterministic)
Mechanical Methods
Example of RISK-UE application:
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
Probability of D4 and D5 for the 475-year return
period probabilistic scenario (probabilistic)
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