ARISTOTLE UNIVERSITY OF THESSALONIKI CIVIL ENGINEERING DEPARTMENTLABORATORY OF REINFORCED CONCRETE

COMPILED

GREGORY G. PENELISGREGORY G. PENELIS

ANDREAS J. KAPPOSANDREAS J. KAPPOS

3D PUSHOVER ANALYSIS:

THE ISSUE OF TORSION

12th European Conference on Earthquake EngineeringLONDON – SEPTEMBER 2002

INTRODUCTION

Torsional strain is often observed on damaged Torsional strain is often observed on damaged buildings after earthquakesbuildings after earthquakes

This effect is more transparent in the nonlinear This effect is more transparent in the nonlinear response of stuctures (I.e. severe damage)response of stuctures (I.e. severe damage)

The nonlinear analysis of buildings is gradually The nonlinear analysis of buildings is gradually being introduced in codes and guidelinesbeing introduced in codes and guidelines ( (ATC-ATC-40, FEMA 273 & 356, HAZUS, RISK-UE etc40, FEMA 273 & 356, HAZUS, RISK-UE etc)- )- mainly by utilising the more perceptible by the mainly by utilising the more perceptible by the practicing engineer practicing engineer PUSHOVER ANALYSISPUSHOVER ANALYSIS..

INELASTIC TORSION TO DATE: STATE OF THE ART

Two “categories” of reports: Two “categories” of reports:

(Α) (Α) The theoretical study of inelastic torsionThe theoretical study of inelastic torsion

(Β) (Β) The design of torsionally restrained new buildingsThe design of torsionally restrained new buildings

From these:From these:

The static eccentricity is modified as the elastic center CR The static eccentricity is modified as the elastic center CR shifts towards the center of shear CS. (PAULAY)shifts towards the center of shear CS. (PAULAY)..

The limit surfaceThe limit surface BST (BASE SHEAR TORSIONBST (BASE SHEAR TORSION) ) defined defined by triads of points corresponding to different failure by triads of points corresponding to different failure mechanismsmechanisms ( (Chopra)Chopra)..

From the state of the art the issue of From the state of the art the issue of nonconvergence between static nonlinear analysis nonconvergence between static nonlinear analysis and dynamic nonlinear analysis is obvious.and dynamic nonlinear analysis is obvious.- All approaches seem to be case sensitive to the All approaches seem to be case sensitive to the excitationexcitation- The modal loads (elastic) seem to be the load The modal loads (elastic) seem to be the load vector approximating the dynamic nonlinear vector approximating the dynamic nonlinear analysis betteranalysis better

SCOPE OF WORK

The primary results of a 3D static nonlinear The primary results of a 3D static nonlinear analysis methodology for the assessment of the analysis methodology for the assessment of the vulnerability of structures which converges with vulnerability of structures which converges with the results of 3D dynamic nonlinear analysis.the results of 3D dynamic nonlinear analysis.

Α) Α) Definition of an appropriate load vector for the Definition of an appropriate load vector for the static nonlinear analysisstatic nonlinear analysis

Β) Β) Definition of the equivalent single dof oscillator Definition of the equivalent single dof oscillator for the spectral assessment of the vulnerability for the spectral assessment of the vulnerability under a specific excitation. under a specific excitation.

CC) ) The introduction of the excitationThe introduction of the excitation..

PRINCIPLES OF THE METHODOLOGY

Α) Α) LOAD VECTORLOAD VECTOR: : One that causes the same One that causes the same displacement and torque on a structure using static displacement and torque on a structure using static linear analysis as the ones calculated by elastic spectral linear analysis as the ones calculated by elastic spectral dynamic analysis (icluding all important modes)dynamic analysis (icluding all important modes).. A A kind of modal loads…kind of modal loads…

Β) Β) EQUIVALEN SDOF OSCILATOR:EQUIVALEN SDOF OSCILATOR: ((For translation & For translation & torquetorque) ) The methodology of Saidi& Sozen (1981) The methodology of Saidi& Sozen (1981) which defined the sdof oscillator for translation was which defined the sdof oscillator for translation was modified to take into account the torsional effect. modified to take into account the torsional effect.

CC) ) SPECTRASPECTRA: : Mean normalised inelastic acceleration-Mean normalised inelastic acceleration-displacement spectra (ADRS) displacement spectra (ADRS)

ONE STOREY BUILDING (1)

1)1) Selection of accelerogramsSelection of accelerograms (3-5) (3-5) which are normalisedwhich are normalised ((acc.acc. Pga orPga or Ι) Ι)

2)2) Calculation of the mean elastic spectra of the selected Calculation of the mean elastic spectra of the selected accelerograms and execution of spectral dynamic accelerograms and execution of spectral dynamic analysis in order to define the elastic translation and analysis in order to define the elastic translation and rotation of the center of mass. rotation of the center of mass.

3)3) The displacement vector of step 2 is used as a constraint The displacement vector of step 2 is used as a constraint in order to calculate the corresponding load vector. in order to calculate the corresponding load vector.

4)4) Calculation of the modification factors for the sdof Calculation of the modification factors for the sdof oscillator.oscillator...

ONE STOREY BUILDING (2)ψψδδ = P = P

11/M/M11 (1)(1)

ψψΜ Μ = -1= -1

(2)(2)

cc1 1 = (m = (m u uy2y22 2 + J+ J

m m θ θz2z222) / m ) / m uuy2y2 (3)(3)

cc2 2 = (u= (uy2 y2 ψ ψδ δ + + ψψM M θ θz2z2 )/ )/ ψψδδ (4)(4)

mm* * = m= muuy2y2 (5)(5)

WhereWhereψψδδ, ψ, ψ

ΜΜ:: parameters related to the modal loads parameters related to the modal loads,,

PP1, 1, MM11 : : the load vector defined by step 3 the load vector defined by step 3

cc11, , cc22:: parameters for the tranformation of a mdof to a parameters for the tranformation of a mdof to a

sdof systemsdof system,,

In general parameterIn general parameter c c11 corresponds to displacements and corresponds to displacements and

parameterparameter c c22 to loadingto loading..

υy1 static elastic P1 normalization uy2 = 1

θz1 Analysis with constraint M1 of displacement vector θz2 =θz1/uy1

Μέσο φάσμα επιταχύνσεων

0

2

4

6

8

10

12

0 1 2 3

Τ(sec)

Acc(m/sec

2)

Elastic

CM

Excitation

Translation – Rotation:υy1,θz1

ONE STOREY BUILDING (3)5)5) Pushover analysis with the load vector at Center of Mass Pushover analysis with the load vector at Center of Mass (P (P

11, ,

MM11)). The . The P-δP-δ curve of the multi dof curve of the multi dof -> -> single dof usingsingle dof using cc11, c, c2 2

ΡΡ* * = c = c22 p/m* p/m* δδ** = c = c

1 1 uuyy (6)(6)

6)6) For the selected accelerograms the mean inelastic normalised For the selected accelerograms the mean inelastic normalised spectra (A-D) are calculated. The demand is defined for several spectra (A-D) are calculated. The demand is defined for several ductilities ductilities ((I.e.I.e. FajfarFajfar--DolsekDolsek, 2000), 2000)

7)7) The The P-δP-δ curve of the sdof is plotted on the demand spectraand the curve of the sdof is plotted on the demand spectraand the performace point is defined. This is the target displacement of performace point is defined. This is the target displacement of the sdof -> the sdof -> uu**

targtarg. .

8)8) The target displacement of the mdof is calculatedThe target displacement of the mdof is calculated

uutarg targ = u= u**targtarg / c / c

11 (7)(7)

and the target rotationand the target rotation (R (Rtargtarg) ) as it is defined by the pushover as it is defined by the pushover

analysisanalysis ((P-P-θθ curve) of the mdof for the target dispacement curve) of the mdof for the target dispacement u utargtarg

RESULTS - COMPARISON

Α) Α) Comparison of theComparison of the P-P-δδ andand Ρ-θ Ρ-θ curves curves of the pushover analysis of the pushover analysis ((stepssteps 1-3 &5) 1-3 &5) with the corresponding dynamic envelope with the corresponding dynamic envelope

Β) Β) Calculation of the target displacement and rotation using pushover Calculation of the target displacement and rotation using pushover analysis with inelastic spectra and comparison with the results of analysis with inelastic spectra and comparison with the results of nonlinear time history analysis. nonlinear time history analysis.

Torsionally Unrestrained Torsionally Restrained

Α) P-δ and Ρ-θ curves The dynamic envelope is calculated for the 1st set The dynamic envelope is calculated for the 1st set

of 4 accellerograms usingof 4 accellerograms using: :

T.URT.UR :40 :40 time history nonlinear analysistime history nonlinear analysis

T.RT.R.. : 80 : 80 time history nonlinear analysistime history nonlinear analysis

1)Lp-Tre s ure Is l-Tr . -κανονικοποιημένο

-6

-4

-2

0

2

4

0 5 10 15 20

2)LP-Lick -lab-tr -κανονικοποιημένο

-6

-4

-2

0

2

4

6

0 5 10 15 20

3)Northr idge -Ne w hall Fire Station-L-κανονικοποιημένο

-6

-4

-2

0

2

4

6

0 5 10 15 20

4)Kobe -HYOGO KEN - l-κανονικοποιημένο

-6

-4

-2

0

2

4

0 5 10 15 20 25 30

TORSIONALLY RESTRAINEDTORSIONALLY RESTRAINED

TORSIONALLY UNRESTRAINEDTORSIONALLY UNRESTRAINEDPolynomial fit to 40 inelastic time history dynamic analyses and

comparison w ith the pushover curve

P-δ curve

0

500

1000

1500

2000

2500

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014

δ (mm)

P (

Um

ax)(

kN

)

LP-Tres.Isl. LP-LickLab No-New h

Kob-Hyog.Ken Pushover Polynomial 3rd

Polynomial fit to 40 inelastic time history dynamic analyses and comparison with the pushover curve

P-θ curve

0

500

1000

1500

2000

2500

0.0E+00 2.0E-04 4.0E-04 6.0E-04 8.0E-04 1.0E-03 1.2E-03

θ (rad)

P (

Um

ax)(

kN

)

LP-Tres.Isl. LP-LickLab No-New h

Kob-Hyog.Ken Pushover Polynomial 3rd

Polynomial fit to 80 inelastic time history dynamic analyses and comparison with the pushover curve

P-δ curve (T.R.)

0

1000

2000

3000

4000

5000

6000

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

δ (mm)

P (

Um

ax)(

kN

)

LP-Treas.Isl. LP-LickLab No-New h

Kob-Hyog.Ken Pushover polynomial 6th

Polynomial fit to 80 inelastic time history dynamic analyses and comparison with the pushover curve

P-θ curve (T.R.)

0

1000

2000

3000

4000

5000

6000

0.0E+00 2.0E-03 4.0E-03 6.0E-03 8.0E-03 1.0E-02

θ (rad)

P (

Um

ax)(

kN)

LP-Treas.Isl. LP-LickLab No-New h

Kob-Hyog.Ken Pushover log f it

W1W2

Torsionally Unrestrained Building (T.U)

Τ3

Τ3

W1W2

Torsionally Restrained Building (T.R)

Β) TARGET DISPLACEMENT & ROTATION

TheThe 4 4 selectedselected accelerogramsaccelerograms scaled to scaled to pgapga = = 0.40.4gg 6% 6% deviation in displacementdeviation in displacement and and 2% 2% in rotation for in rotation for

the torsionally unrestrained building. the torsionally unrestrained building. 3.7% deviation in displacement3.7% deviation in displacement and 6.8% in and 6.8% in

rotation for the torsionally restrained building. rotation for the torsionally restrained building.

CONCLUSIONS - COMMENTS TheThe Ρ-δ Ρ-δ andand Ρ-θ Ρ-θ curves of the pushover analysis curves of the pushover analysis

approximate the dynamic envelopeapproximate the dynamic envelope The target displacement and rotation are The target displacement and rotation are

accurately calculated for the one storey buildingaccurately calculated for the one storey building The implementation for multi storey buildings is The implementation for multi storey buildings is

yet to comeyet to come

Problems - ObservationsProblems - Observations

Α) Α) Adaptive pushover analysisAdaptive pushover analysisChange in Change in Κ -> [Φ] -> [Κ -> [Φ] -> [V, T]V, T]

Μέσο Ικανοτικό Φάσμα ομαλοποιημένο

0

2

4

6

8

10

12

0 0.01 0.02 0.03 0.04

Επιτ

. (m

/sec

2 )

Elastic

Pushover

Πολυωνυμική(duct.= 2)

Πολυωνυμική(duct.= 1.5)

Πολυωνυμική(duct.=1.75)

Πολυωνυμική(1.9)

Mean inelastic normalised spectra

Β) Β) Mean inelastic normalised spectra / Highly Mean inelastic normalised spectra / Highly damped spectradamped spectra

Μέσο Ικανοτικό Φάσμα χωρίς ομαλοποίηση

0

2

4

6

8

10

12

0 0.01 0.02 0.03 0.04

Disp (m)

Acc

(m

/se

c2)

Elastic

duct.= 1.5

duct.= 2

duct.= 4

Mean inelastic spectra without normalisation

Name Country Date Depth mb ML Ms MoSource Mechanism

aftershock of Friuli earthquake Italy 9/15/76 15 km 5.7 6.2 6.06 6.3e+017 Nm oblique

GazliUzbekistan 5/17/76 13 km 6.2 6.4 7.04 1.8e+019 Nm thrust

Tabas Iran 9/16/78 5 km 6.4 7.33 1.3e+020 Nm thrust

MEAN SMOOTHENED ADRS SPECTRA

0

2

4

6

8

10

12

0 0.05 0.1 0.15 0.2

Elastic

Poly. (duct.= 2)

Poly. (duct.=1.5)

Poly.(duct.=1.75)

Poly. (duct.=1.1)

CC) ) Inconsistency of t-h inelastic analysis?Inconsistency of t-h inelastic analysis?

Torsionally Unrestrained 1 storey building65 Timehistory analysis

P-δ curve

0

1000

2000

3000

4000

5000

6000

7000

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Δ

V

maxV

maxD

Torsionally Unrestrained 1 storey building65 Timehistory analysis

P-θ curve

0

1000

2000

3000

4000

5000

6000

7000

0 0.002 0.004 0.006 0.008 0.01 0.012

Θ

V

maxV

maxd

Dynamic EnvelopeDynamic Envelope

MaxV -> disp & rot

Maxdisp -> V & rot