slide 4_static nonlinear analysis

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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Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis

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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


Nonlinear static procedure

• How does it work? Capacity vs. Demand

Sa


Sd


• Necessary elements?

– Acceleration-displacement response spectrum (ADRS)– Acceleration-displacement response spectrum (ADRS)

Sa

T(sec.)

V (force)

Δ (disp.)

Sa


Sa

Sd


• Developing capacity curves:

– Displacement-based methods– Displacement-based methods



• Assumptions

– Response: maximum displacement– Response: maximum displacement

– Deformation: “most often” following the first mode

• Benefits

– Displacement-based analysis

– Nonlinear behavior of the structure

• Drawbacks


• Drawbacks

– Simplified approach (static)

– Cyclic response resembled to monotonic response

– Damping difficult to represent


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




Capacity Curve

• Pushover: to estimate the capacity curve

Structural model

(FEM, AEM, etc.)


• Objectives:

– Estimate the maximum horizontal displacement

– Estimate the (global) ductility of the structure


(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


Base Shear


Capacity Curve

• Load pattern: IDARC practice

– Effect of lateral load pattern– Effect of lateral load pattern


Capacity Curve

• Force/Displacement to spectral coordinates

Equivalent SDOF

V

Δ

Sa

Sd

V

Equivalent SDOF


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




(ADRS)






Seismic Performance/Damage


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.


FEMA 356 Performance levels


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


Unreinforced masonry (URM)

Example

URM building in Yverdon-les-Bains



Example

Architectural plan



Example

Simplification for the modeling



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


– Infinitely stiff lintels

(total frame effect)


0

1

2 storeyh h= ×


Example

Extreme case studies:

0

26.7

3 totalh h m= × =


0

11.25

2 storeyh h m= × =


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


6.0 1520 367 624

total 690 1440

Too conservative Too optimistic


Example

Lateral strength – transverse direction


Content




(ADRS)






Reminder: Response Spectrum


Reminder: Response Spectrum

2a

d

SS

ω≈


Acceleration-Displacement Response

Spectrum (ADRS)


Content




(ADRS)






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)


• 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)


ATC-40 (1996)

• Improved later in FEMA 440 document (2005)


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


error) between the maximum response of an

actual inelastic system and its equivalent

linear counterpart.


“FEMA 440”

Equivalent Linearization

Basic equations…


Equivalent Linearization-

Performance Point

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Equivalent Linearization

• Advantages:

– Linear computation– Linear computation

– Use of pushover analysis

• Drawbacks:

– Value of damping (through iterations)


– Value of damping (through iterations)

– 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)


(ATC-40)

• Based on inelastic spectra rather than elastic

spectra


N2 Method Procedure (1)

Required tools:


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”


Inelastic response spectrum (ADRS)

N2 Method Procedure (3)-

Performance Point

:Cif T T≥

Basic equations…

:Cif T T≥



Performance Point

d deS S=d deS S=



Performance Point

:Cif T T<

Basic equations…

:Cif T T<



Performance Point

1 ( 1)de Cd

S TS Rµ

= + − *1 ( 1)dS R

R Tµµ

= + −


Content




(ADRS)






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


• Vulnerability assessment of each class including

variability (probabilistic assessment)

Fragility functions


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


Damage Probability Matrix


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


Damage Probability Matrix


Empirical Methods

Empirical methods example

European Macroseismic Scale 98European Macroseismic Scale 98

• Calculate Vulnerability index


• Calculate Vulnerability index

from vulnerability class

• Structural parameters is only

the structure class


Empirical Methods

EMS 98 (Empirical method):

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Empirical Methods

Example of EMS98 application: (Feriche et al. 2009)

b. Damage

grade from

intensity VI

a. Vulnerability

class in

downtown

Malaga


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)


• Only valid for large number of buildings (classification)

• No surveys available for areas with moderate seismic hazard


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


• Takes variability into account


Mechanical Methods

Fragility curves

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Mechanical Methods

RISK-UE LM2 method:


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


• Crude approximation in computations

• Fragility curves only available for certain building classes

• Difficulties in estimating variability


Mechanical Methods

Example of RISK-UE application:


Probability of D4 and D5 in case of a 1887-type

scenario earthquake at 30 km (deterministic)

Mechanical Methods

Example of RISK-UE application:


Probability of D4 and D5 for the 475-year return

period probabilistic scenario (probabilistic)

slide 4_static nonlinear analysis

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