Slides NSP DHL

8/20/2019 Slides NSP DHL

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Dominik H. Lang, Dr.‐Ing. Dr.philos.Senior Advisor, Project Manager

NORSAR, Kjeller, Norway

Indo ‐Norwegian Training Programme on

Nonlinear Modelling and Seismic Response Evaluation of StructuresDecember 14 ‐16, 2014 – Continuing Education Center, IIT Roorkee

Seismic response estimation using non ‐linear static methods


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Table of contents

IntroductionMethods of structural analysisNon ‐linear static (pushover) method (NSP)

Performance estimationStructural capacity vs. seismic demand

Methodologies and procedures for performance estimationCapacity Spectrum Methods (CSM)

Conventional CSM (ATC–40)

MADRS method (FEMA 440) ‐ Procedures A, B, CDisplacement Coefficient Methods (DCM)

DCM (FEMA 273, FEMA 356)

Improved DCM (FEMA 440)N2 method (Fajfar 2002; EN 1998)


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Introduction – Methods of analysis

various structural analysis methods, both elastic (linear) and inelastic (non ‐linear) are available:

linear (elastic)

methods

non ‐linear (inelastic) methods

• lateral force method

•

modal response spectrum

• static ('pushover')

analysis (NSP)

• dynamic time history

analysis (NLTHA)• Incremental Dynamic

Analysis (IDA)

dynamic methods

(NDP)

(quasi) staticmethod

the NSP can be considered as a simplified non ‐linear analysis method in which the dynamic load is substituted by a gradually increasing (static) lateral load

NOTE: A nice overview of the various analysis methods is provided by FEMA 356 (FEMA 2000), Chapter 3.3 !!!


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spatial (3D) model

planar (2D) model

lateral force modal RS NSP NDPlinear methods non ‐linear methods

Introduction – Methods of analysis

the decision which structural analysis method to choose strongly depends on:calculation effort

uncertainties in results

calculation effort

calculation effort


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Purpose of the NSP:• to verify or revise the overstrength ratio values α u/ α 1• to estimate the plastic mechanisms and the distribution of damage•

to assess the structural performance of existing or retrofitted buildings • as an alternative to the design based on linear ‐elastic analysis methods,

which use behavior factor q

a good alternative to NDP as these are considered as overly complex and the length of time required to perform a computation process might be significant

• NDP requires defining a complete hysteretic behavior of the materials• NDP requires a set of ground motion records to perform dynamic response

history analysis

Nonlinear static ‘pushover’ method (NSP)


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serves as a good alternative to the NDP due to its simplicity, ease in capturing global structural behavior, and its ability to identify weak links in the structural system

a lateral force distribution is applied which is gradually increased until the structure becomes unstable or until a predetermined limit is reached

V d

B a s e

S h e a r

, V

Roof displacement, d


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F 3

F 2

F 1

V

Type 2:F x = [w x h x / ∑ w x h x ]·V

Type 3: F x = [w x Φ x / ∑ w x Φ x ]·V Type 5: including higher modes

n th mode 1st modeType 4: same as Type 3 until first yielding; the mode shapes are adapted after first yielding

1st mode

F 3

F 2

F 1

Type 1:F x = [w x / ∑ w x ]·V

F 3

F 2

F 1

Minimum required load patterns

that need to be considered acc. to EN 1998 ‐1


types of lateral force distribution to the building:


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main component of the NSP is the elaboration of the capacity (‘pushover’) curve

capacity curve represents the lateral displacement of any structural point (e.g.

roof displacement d ) versus total base shear V

V d

B a s e

S h e a r

, V



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B a s e s

h e a r

, V


analysis segments

yield point for element(s)

V 1

V u

V 1 ‐ base shear at first yield

V u ‐ base shear at ultimate point


capacity (‘pushover’) curve:

V u / V 1 = u / 1


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step ‐by‐step procedure to determine structural capacity: Generation of the structural model

• check if structural and geotechnical features of the foundation have to be included

δ i Θi

Θi δ i

Definition of nonlinear hinge properties(i.e., moment ‐rotation or force ‐deformation relationships) for the yielding elements

(beams, columns, and struts)

Definition of load cases• compilation of permanent and live loads• definition of certain combinations of permanent,

live and seismic loads ( code provisions)(G + Ψ

2 Q)

Calculation of story masses• permanent and participating live loads

Gk + ∑i (Ψ Ei Q Ki )with: Ψ Ei = Φ Ψ 2i

Ψ2i

= 0.3 : for residential buildings

Run of static analysis (prerequisite for the definition of nonlinear hinge properties), modal analysis, and Pushover analysis


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Create a non ‐linear computer model of the building:


i j

plastic hinge region

l pl l pl

l pl = 0.5 ·d – d

with: d ‐ depth of section

Moment

Curvature


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Apply gravity loads and conduct static analysis:


G + ΨEi Q

B a s e

S h e a r

, V

Roof Displacement, d


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Apply lateral loads to the structure in proportion to the selected load pattern:


B a s e

S h e a r

, V


d V


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Calculate member forces under the applied lateral load and gravity load combination – check for yielding in the members:


B a s e

S h e a r

, V


d V

Moment

Curvature


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Record the base shear V and roof displacement d :


B a s e

S h e a r

, V


d V

Moment

Curvature


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– n: Repeat Steps to for increments of lateral load:


B a s e

S h e a r

, V


d

Moment

Curvature


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B a s e

S h e a r

, V


d

Moment

Curvature


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B a s e

S h e a r

, V


d

Moment

Curvature


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B a s e

S h e a r

, V


d

Moment

Curvature


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Performance estimation

the estimation of structural performance through simplified nonlinear analysis methods requires procedures that combine

S p e c t r a

l a c c e

l e r a t i o n

Spectral displacement

estimation of the building’s maximum displacement under a given seismic load

structural capacity seismic demand


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PBSD: structural damage is mainly determined by lateral displacementin order to analytically predict the structural damage that a building of a given capacity will produce under a given seismic impact, different methods are

available (NORSAR 2009):• Capacity Spectrum Methods (CSM) (ATC–40, 1996 ; FEMA 440, 2005 ), • Collapse ‐based methods (CBM) (e.g. FaMIVE, D’Ayala and Speranza, 2002 ; VULNUS,

Bernardini et al., 1990 ; Cosenza et al., 2005 ),• Displacement ‐based methods (DBM) (e.g. DBELA, Crowley et al., 2004 ; Miranda,

1999 ),• Displacement coefficient methods (DCM) (FEMA 273, 1997a ; FEMA 356, 2000 ;

FEMA 440, 2005 ),• Incremental dynamic analysis (IDA) (e.g. Shome and Cornell, 1999 ; Vamvastsikos

and Cornell, 2002 )

CSM and DCM received greatest attention to date, mainly because: •

published as various FEMA provisions, and • in case of CSM: established the basis for the HAZUS methodology (FEMA, 2003 )


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to be able to correlate the response spectrum with the capacity curve, both need to be converted from their individual domains into a common domain, i.e. spectral acceleration–spectral displacement domain (Sa –Sd)

Seismic

demand

(response

spectrum):

S p e c t r a

l a c c

e l e r a t i o n

S a

Period T [s]

S p e c t r a

l a c c e

l e r a t i o n

S a

Spectral displacement Sd

d d a ST SS

2

2 2

T 1T 1

T 2T 2

T 3 T 3

f


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Structural capacity (capacity curve):

B a s e s

h e a r V

Roof displacement

S p e c t r a

l a c c

e l e r a t i o n

S a


equivalent SDOFstructural model (MDOF)

T n = T elastic T >>

P f i i


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with: V ‐ base shearW ‐ building weight (masses from dead and live loads)

α 1 ‐ modal mass coefficient 1st mode shape

δ roof ‐ roof displacement

γ 1 ‐ modal participation factor 1st mode shape

Φ roof,1 ‐ amplitude of 1st mode shape at roof level

a

d

S

S2T

n

1 j

21, j j

n

1 j1, j j

1

m

m

n

1 j

21, j j

n

1 j j

2n

1 j1, j j

1

mm

m

with:

with:

Φ roof,1Φ 3,1

Φ 2,1

Φ 1,1


Structural capacity (capacity curve):Conversion of V –δ capacity curve into Sa –Sd format (ADRS)

1a W/VS

1,roof 1

roof dS

C i S M h d


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Capacity Spectrum Methods

History :

roots can be traced to John A. Blume’s Reserve Energy Technique (RET) (Blumeet al. 1961 )

first introduced in the 1970s as a rapid evaluation procedure in a pilot project for assessing seismic vulnerability of buildings at the Puget Sound Naval Shipyard (Freeman et al. 1975 )

in the 1980s CSM was used to find correlation between seismic ground motion and building performance (ATC 1982), as well as development into a design verification procedure for the Tri‐services Army, Navy and Air Force (Freeman et al. 1984 ; Army 1986 )

in the 1990s under the direction of the State of California, the ATC was assigned the task of developing guidelines for the seismic evaluation and retrofit of concrete structures which lead to ATC–40 (ATC 1996)

CSM P i i l th d l


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CSM – Principal methodology

ATC 40 th d l g


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ATC–40 methodology

’Conventional’ Capacity Spectrum Method (ATC–40, FEMA 273 ):

the location of the performance point must satisfy two conditions:• the point must lie on the capacity curve• the point must lie on a spectral demand curve, reduced from the elastic, 5%

damped design spectrum that represents the nonlinear demand at the same structural displacement

iterative procedure to determine the location of the performance point trial ‐and ‐error search for satisfaction of the two criteria

ATC–40 provides three alternate procedures for this iterative process:• Procedure A: purely formula ‐based (analytical) method; easy to understand

• Procedure B: relative direct solution with little iteration; less transparent than A

• Procedure C: pure graphical method; most convenient for hand analysis; least transparent

ATC 40 methodolog


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Procedure A: construct bilinear representation of capacity spectrum:• required to estimate effective damping and reduction of spectral demand• most cumbersome step of the whole procedure, as it differs for various shapes of

capacity curves (natural or bilinear)• choose trial performance point d pi / a pi and construct bilinear representation

target: area A1 area A2 so that equal areas under capacity curve and its bilinear representation

ATC 40 methodology


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construct bilinear representation of capacity spectrum:• choose trial performance point d pi / a pi

equal displacement approximation can be used as first estimate of d pi / a pi

S p e c t r a

l a c c e

l e r a t i o n

S a


Ki

d inelastic = d pi

d elastic

elastic demand spectrum (5%)

d y

a y a pi

ATC 40 methodology


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construct bilinear representation of capacity spectrum:• construct bilinear representation

target: area A1 area A2

S p e c t r a

l a c c e

l e r a t

i o n

S a


Ki

d inelastic = d pi

d elastic

elastic demand spectrum (5%)

a pi

d y

a y Area A1 Area A2

ATC 40 methodology


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determination of effective viscous damping eff :

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50

pi pi

pi y pi y eff

d a

ad d a )(.

Sa

Sd

A1

A1

A2A2

A3

A3

d y d pi

a y

a pi

κ – degradation factor (damping

modification factor) that defines the effective amount of hysteretic damping as a function of earthquake duration and energy ‐

absorption capacity of the structure during cyclic earthquake load (see HAZUS, Table 5.18 in FEMA 2003 )

0 – hysteretic damping represented as

equivalent viscous damping

in [%]

ATC 40 methodology


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determination of effective viscous damping eff :

1.00Type A

• values of , (SR A )min and (SRV )min are given in Tables 8.1 and 8.2 of ATC–40• dependent on structural behavior types:

• qualitative function of :

Shakingduration

Essentially new building

Average existing building

Poor existing building

short Type A Type B Type C

long Type B Type C Type C

0

Type B

Type C

ATC 40 methodology


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compute spectral reduction factors SR A and SRV :

eff

min)(.

)ln(.. A

eff A SRSR

122

680213

min)(.

)ln(..V

eff V SRSR

651

410312

• qualitative function of eff

:

0

Type A

Type BType C

5%

Structural behavior type (SRA)min (SRV)min

Type A 0.33 0.50

Type B 0.44 0.56

Type C 0.56 0.67

• minimum allowable values for reduction factors (SRA)min and ( SRV)min :



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compute reduced (inelastic) demand spectrum:

elastic (5%)

S p e c t r a

l a c c e

l e r a t i o n

S a


2.5 CA

CV /T

(2.5 CA) SR

A

(CV /T) SRV

CAinelastic (>5%)



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ATC 40 methodology

check if demand spectrum intersects the capacity spectrum at point (d pi, a pi):‐ if yes → (d pi, a pi) is the performance point

‐ if not → select a new (d pi, a pi) and repeat from step

elastic (5%)

S p e c t r a

l a c c e

l e r a t i o n

S a


inelastic (>5%)

d pi

a pi

a i

d i

acceptable, if

0.95 d pi d i 1.05 d pi



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ATC 40 methodology

Procedure B (in short):• simplified assumptions (that may not always be valid!!!):

1. initial slope of the bilinearized capacity curve Ke remains constant2. yield point a y, d y remains constant3. post ‐yield slope Ks remains constant

this procedure does not make sense!!

Procedure C (in short):• graphical solution to be used by hand• often the performance point is found at first try

this procedure can only be drawn by hand and seems to be very “homemade”!



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ATC 40 methodology

’Conventional’ Capacity Spectrum Method (ATC–40, FEMA 273 ):Disadvantages of CSM:• Iterative procedure is unnecessarily complex for the intended use (Freeman

2004 )• Inelastic reduction factors and hence damping values appear to most

researchers to be too large, i.e. non ‐conservative, underestimating the

seismic demand and thus predicting lower damageupdated procedure called MADRS (Modified Acceleration ‐Displacement Response Spectrum) has been introduced in FEMA‐440 (FEMA 2005 )

MADRS is based on a similar theoretical background but it has some advantages over CSM: • more accurate displacement demand estimates•

it provides the engineer with a visualization tool• not iterative computationally less expensive

Displacement Coefficient Method (DCM)


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Displacement Coefficient

Method

(FEMA 273, 1996 ; FEMA 356, 2000 ):

direct numerical process for calculating displacement demand

no need of transferring the capacity spectrum into spectral coordinated

limited to buildings that are regular and do not have adverse torsional or multimode behaviors

first presented by FEMA 273 (1996)

slightly modified (especially in terms of terminology) in ATC–40 (1996)• ‘target displacement’ refers to ‘performance point’• ‘performance objective of collapse prevention’ refers to ‘structural stability’



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construct a bilinear representation of the capacity curve:

B a s e s

h e a r V

Roof displacement

Ki

Ks

• draw a post ‐elastic stiffness Ks to represent an average stiffness of this branch where the structure’s strength is leveling off

Ke

• draw the effective elastic stiffness Ke by constructing a secant line

Vy

0.6 Vy

y t

target: intersection point with capacity curve 0.6 V y iterative process!!!

Note: This bilinear curve is

generally different from those constructed for the CSM !!!



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p ( )

calculate effective fundamental period T e :

e

iie K

KTT

with: T i – elastic fundamental period (in [s])

K i – elastic lateral stiffness

K e – effective lateral stiffness

Note: K i may be equal to K e!!

in [s]

calculate target (roof!) displacement t :

2

2

ea3210t 4T

SCCCC in [m]

C0 – modification factor to relate spectral displacement and likely roof displacement (first modal participation factor at roof level; tabular value dep. on story number)

C1 – modification factor to relate expected maximum inelastic displacement to displacement calculated for linear elastic response

C2 – modification factor to represent the effect of hysteresis shape on the maximum displacement response

C3 – modification factor to represent increased displacements due to second ‐order effects

with: Sa – spectral acceleration at the effective fundamental period of the building (in [m/ s2])

Improved Displ. Coeff. Method (I–DCM)


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p p ( )

y

ye A

D2T in [s]

Improved Displacement

Coefficient

Method

(FEMA 440, 2005 ):

the DCM modifies the displacement demand of the equivalent linear single degree of freedom (SDOF) system by multiplying it by a series of coefficients in

order to generate an estimate of the maximum displacement demand of the nonlinear oscillator

the process begins with the generation of the capacity curve of the nonlinear oscillator; the effective period T

e of the system is then computed as:

Sa

Sd Dy

Ay



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p p ( )

2e

el,ael,d 2T

SS

compute acceleration demand of the equivalent linear SDOF system Sa,el :

S p e c t r a

l a c c e

l e r a t i o n

S a

Period T

T e

Sa,el

derive peak elastic spectral displacement demand Sd,el :



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p p ( )

compute target displacement t :

el,d21t SCC in [m]

C1 – modification factor to relate the expected maximum displacement demand of a nonlinear oscillator with elastic ‐perfectly ‐plastic (EPP) hysteretic properties to the peak displacement demand of the linear oscillator

C2 – modification factor to represent the effect of pinched hysteretic shape and

stiffness degradation on the maximum displacement response

coefficients C 1 and C 2 can be approximatively derived (FEMA 440, 2005 ):

2e

1 Ta1R

1C

2

e2 T

1R800

11C

a – equation constant dep. on soil class: a = 130 for SC B

a = 90 for SC Ca = 60 for SC D

R – ratio of elastic strength demand to the calculated strength capacity, i.e. R = Sa,el / Ay

elastic, T e inelastic, EPP

MADRS method (FEMA 440, 2005)


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MADRS – Improved Nonlinear Static Procedures (Procedures A, B, C)

more accurate estimates of displacements than CSM when compared to time ‐

history analysis

However: The principals of these improved procedures are similar to

the 'ATC–40 CSM' !

provides the engineer with a visualization tool

by facilitating a direct graphical comparison of capacity and demand

( )

Modified Acceleration

Displacement

Response

Spectrum

Method :



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Modified Acceleration

Displacement

Response

Spectrum

Method :

the ATC–40 CSM uses the secant period as the effective linear period in determining the maximum displacement (performance point)

•

this assumption results in the maximum displacement occurring at the intersection of the capacity curve for the structure with the demand curve for the effective damping

• however, it has been shown in several studies that this method cannot be used with a non ‐IBC response spectrum and that it does not provide an accurate performance point in some cases

improvements of the method have been published in FEMA 440 (2005)• visualization tool by facilitating a direct graphical comparison of capacity and demand

• very effective solution strategies for equivalent linearization that rely on a modified ADRS demand curve (MADRS) that intersects the capacity curve at the maximum displacement

However: Results of MADRS may not be reliable for extremely high ductility values, e.g., greater than 10 to 12!!!



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select ground motion demand spectrum for an initial damping i (usually 5%)

design spectrum

S p e c t r a

l a c c e

l e r a t i o n

S a

Period T

deterministic spectrum hazard spectrum

if applicable: modify demand spectrum for SSI effects (reduction in spectral ordinates and modification of system damping i to 0

convert demand spectrum into Sa–Sd domain ( initial ADRS demand)




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S p e c t r a l a c c e

l e r a t i o n

S a


generate capacity curve for the structure to be analyzed in the ADRS formatselect initial performance point (a pi , d pi ) equal ‐displacement approximation

d pi

a pi

develop bilinear representation of capacity curve following ATC–40 procedure

d y

a y

so far the procedure (more or less) follows ATC–40!!!



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determine effective damping eff : a) compute post yield stiffness and ductility for the bilinear representation

of the capacity curve:

y

y

ypi

ypi

d

a

ddaa

y

pi

d

d

– post yield (post elastic) stiffness

– displacement ductility

b) compute effective period of the structure T eff for various ranges of :

• for < 4.0 :• for 4.0 6.5:

•

for > 6.5 :

032eff T1)1(H)1(GT 0eff T1)1(JIT

0eff T11)2(L1)1(

KT

G, H, I, J, K – coefficients dependent on and hysteresis type (tabular values)



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determine effective damping eff (cont’d) :c) compute eff using the previous derived parameters:

• for < 4.0 :

• for 4.0 6.5:

• for > 6.5 :

032

eff )1(B)1(A

0eff )1(DC

0

2

0

eff 2eff T

T)1(F

1)1(FE

A, B, C, D, E, F – coefficients dependent on and hysteresis type (tabular values)

Note: Both set of coefficients (A–F and G–K) have been optimized for model oscillators and not actual building models. Approximate coefficients independent

of hysteretic model type or are provided in FEMA 440, Ch. 6.2.1 and 6.2.2.



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adjustment of the initial ADRS to effective damping eff and determination of target displacement d pi :

FEMA 440 provides three possible procedures for this (iterative) process:•

Procedure A: Direct Iteration, trial ‐and ‐error• Procedure B: Intersection with MADRS• Procedure C: MADRS Locus of Possible Performance Points

all 3 procedures can be conducted by hand



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l e r a t i o n

S a

Procedure A: Direct Iteration


d pi

a pi


intital ADRS

A.1 adjust the initial ADRS to effective damping eff

ADRS, eff ( , CS)



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l e r a t i o n

S a



d pi

a pi


intital ADRSADRS, eff ( , CS)

A.2 determine estimated maximum displacement d i by intersecting the effective period T eff with the adjusted ADRS ( eff )

d i

T eff ( , CS)



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l e r a t i o n

S a



d pi

a pi

adjustment of the initial ADRS to effective damping eff

and determination of target displacement d pi :

intital ADRS

A.2 the estimated maximum acceleration a i is that corresponding to d i on the capacity curve

ADRS, eff ( , CS)

d i

T eff ( , CS)

a i



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l e r a t i o n

S a



d pi

a pi




d i

T eff ( , CS)

a i

A.3 compare estimated maximum displacement d i with the initially assumed d pi if within acceptable limits, performance point is found !!



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l e r a t i o n

S a



d pi

a pi




d i

T eff ( , CS)

a i

A.3 compare estimated maximum displacement d i with the initially assumed d pi if not, repeat iteration with new d pi (from step )



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l e r a t i o n

S a

Procedure B: Intersection with MADRS


d pi

a pi



intital ADRS

B.1 adjust the initial ADRS to effective damping eff

ADRS, eff ( , CS)



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l e r a t i o n

S a



d pi

a pi



intital ADRS

B.2 multiply (only!) acceleration ordinates of the adjusted ADRS with modification factor M

ADRS, eff ( , CS)

0sec T)1(1T

2

sec

eff

TT

M

MADRS, eff ( , CS, M)

y

y0 a

d2T

d y

a y



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l e r a t i o n

S a



d pi

a pi



intital ADRS

B.3 determine estimated maximum displacement d i and acceleration a i by intersecting the MADRS with the capacity curve

ADRS, eff ( , CS)MADRS, eff ( , CS, M)

d i

a i



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l e r a t i o n

S a



d pi

a pi



intital ADRS

B.4 compare estimated maximum displacement d i with the initially assumed d pi if within acceptable limits, performance point is found !!


d i

a i



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l e r a t i o n

S a



d pi

a pi



intital ADRS

B.4 compare estimated maximum displacement d i with the initially assumed d pi if not, repeat iteration with new d pi (from step )


d i

a i



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l e r a t i o n

S a

Procedure C: MADRS Locus of Possible Performance Points


d y

a y



intital ADRS, 0 ( = 1)

C.1 the initial ADRS is considered to correspond to ductility = 1.0



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l e r a t i o n

S a






C.2 construct a suite of MADRS for various ductility factors = 2, 3, ....10(?); this requires the calculation of d pi, , T eff , T sec , M, and eff

MADRS ( = 2)MADRS ( = 3)MADRS ( = 4)MADRS ( = 5)MADRS ( = ...)



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l e r a t i o n

S a






C.3 draw radial lines for secand periods T sec for each ductility factor

MADRS ( = 2)MADRS ( = 3)MADRS ( = 4)MADRS ( = 5)MADRS ( = ...)

T 0 = T sec, =1 T sec, =2T sec, =3

T sec, =4T sec, =5T sec, =...

C.4 mark intersections of T sec and MADRS for each ductility factor


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N2 method


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N – non ‐linear , 2 – two mathematical models (1 SDOF and 1 MDOF model)

procedure is specifically applicable only for structures that are characterized by

bilinear elasto ‐perfectly plastic capacity curves extended version of the method, which can be applied for asymmetric structures as well

process consists in obtaining the appropriate inelastic response spectrum for which the extracted performance point (defined as point of intersection of idealized SDOF system based capacity curve with the inelastic demand spectrum) reaches, or is close to, the displacement capacity at the considered damage state threshold

N2 method based on the work by Fajfar (2002) and recommended by EN 1998 :

Fajfar, P. (2002). Structural analysis in earthquake engineering ‐ a break ‐through of simplified nonlinear methods. Twelfth European Conference on Earthquake Engineering , London, paper 843.

N2 method


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N2 method :

elastic ADRS

inelastic ADRS

(

=

3)

T C

generate inelastic demand spectrum in ADRS for ductility factor :

)T(R

)T(S

)T(S el,a

inelastic,a C

C

C

TTfor

TTfor1

T

T)1(

)T(R

)T(R

)T(S

)T(S el,d

inelastic,d

N2 method


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elastic ADRS

inelastic ADRS ( = 3)

T C

overlay capacity spectrum (with elastic period T *):

Case 1: T * TC (medium ‐ and long ‐period structures)

ductility demand = reduction factor Rinelastic displacement demand = elastic displ. demand:

T *

)T(SS *el,d*

inel,d

*

inel,dS

)T(S *

el,d)T(S)T(S *

ae*

a

ayS

N2 method :

N2 method


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elastic ADRS


T C


Case 2: T * < TC (short ‐period structures)

if Sa,y Sa,el (T*) the response is elastic:

T *

)T(SS *

el,d*

inel,d

ayS)T(S)T(S *ae

*a

*

inel,dS

N2 method :

N2 method

h d


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elastic ADRS


T C


Case 2: T * < TC (short ‐period structures)

if Sa,y < Sa,el (T*

) the response is non ‐linear :

T *

)T(STT

)1R(1R

)T(SS *el,d*

C*

el,d*inel,d

ayS

)T(S)T(S *ae*

a

*

inel,dS)T(S *

de

N2 method :


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ReferencesApplied Technology Council (1996) Seismic evaluation and retrofit of concrete buildings


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Applied Technology Council (1996). Seismic evaluation and retrofit of concrete buildings , Report No. ATC–40, Redwood City, CA.

Fajfar, P. (2002). Structural analysis in earthquake engineering ‐ a break ‐through of simplified nonlinear methods. Twelfth European Conference on Earthquake Engineering , London, paper 843.

Federal Emergency Management Agency (1997). NEHRP Guidelines for the Seismic Rehabilitation of Buildings, FEMA 273 , Washington, D.C., October 1997.

Federal Emergency Management Agency (2000). Prestandard and Commentary for Seismic Rehabilitation of Buildings, FEMA 356 , Prepared by the American Society of Civil Engineers for the Federal Emergency Management Agency, Washington, D.C.

Federal Emergency Management Agency (2003). HAZUS‐MH MR4 Technical Manual. Washington, D.C.

Federal Emergency Management Agency (2005). Improvement of Nonlinear Static Seismic Analysis Procedures, FEMA 440, Prepared by Applied Technology Council (ATC‐55 Project), Washington, D.C., United States, June 2005.

Krawinkler, H., and Seneviratna, G.D.P.K. (1998). Pros and cons of a pushover analysis for seismic performance evaluation, Engineering Structures 20 : 452–464.

NORSAR (2009). GEM Memorandum on GEM1, September 2009, (Authors: J. Crempien, D.H. Lang, E. Erduran, C.D. Lindholm), 12 pp.

ReferencesD’Ayala D and Speranza E (2002) An integrated procedure for the assessment of seismic


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D Ayala, D., and Speranza, E. (2002). An integrated procedure for the assessment of seismic vulnerability of historic buildings, Proceedings of the 12th European Conference on Earthquake Engineering, London, U.K., Paper no. 561.

Bernardini, A., Gori, R. and Modena, C. (1990). Application of coupled analytical models and experimental knowledge to seismic vulnerability analyses of masonry buildings, in:

Engineering Damage Evaluation and Vulnerability Analysis of Building Structures, A. Koridze (ed.), Omega Scientific, Oxon, U.K.

Cosenza, E., Manfredi, G., Polese, M. and Verderame, G.M. (2005). A multi ‐level approach to the capacity assessment of existing RC buildings, Journal of Earthquake Engineering 9(1):

1–22.Crowley, H., Pinho, R., and Bommer, J.J. (2004). A probabilistic displacement ‐based

vulnerability assessment procedure for earthquake loss estimation, Bulletin of Earthquake Engineering 2: 173–219.

Miranda, E. (1999). Approximate seismic lateral deformation demands in multistory buildings, Journal of Structural Engineering 125 (4): 417–425.

Shome, N., and Cornell A. (1999). Probabilistic Seismic Demand Analysis of Nonlinear Structures, Report No.RMS‐35, Department of Civil Engineering, Stanford University.

Vamvastsikos, D. and Cornell, A.C. (2002). Incremental dynamic analysis, Earthquake Engineering and Structural Dynamics 31 (3): 491–514.

Contact


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Dominik H. Lang Dipl.‐ing., Dr.‐Ing. (PhD), Dr.philos.Senior Advisor, Project ManagerNORSAR2027 Kjeller, NorwayPhone: (47) 988 42 924 (Norway)

(91) 97 60 21 46 80 (India)email: [email protected]: http://www.norsar.no

http://selena.sourceforge.net

Associate Editor

EERI/IAEE's World Housing EncyclopediaEarthquake Engineering Research InstituteOakland, CA http://www.eeri.org

http://www.world ‐housing.net

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