Collapse Prediction

8/2/2019 Collapse Prediction

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COMPDYN 2009

ECCOMAS Thematic Conference on

Computational Methods in Structural Dynamics and Earthquake EngineeringM. Papadrakakis, N.D. Lagaros, M. Fragiadakis (eds.)

Rhodes, Greece, 2224 June 2009

COLLAPSE ASSESSMENT OF A 4-STORY STEEL MOMENT

RESISTING FRAME

Dimitrios G. Lignos1, Helmut Krawinkler

1, and Andrew S. Whittaker

2

1Stanford University

Stanford, CA 94305-4020

[email protected], [email protected]

2State University of New York at Buffalo (SUNY)

Buffalo, NY, 14260, [email protected]

Keywords: Sidesway collapse, collapse prediction, deterioration modeling, P-Delta effects.

Abstract. Although building seismic codes and standards of practice assume that the prob-

ability of collapse is low in very rare earthquake shaking, the likelihood of collapse in such

shaking is almost never checked by analysis.

This paper discusses analytical modeling of component behavior and structure response from

the onset of inelastic deformations to lateral deformations at which a structure becomes dy-

namically unstable. In this context dynamic instability implies loss of vertical load carryingcapacity due to global P- effects and component deterioration. The basis of analytical mod-

eling for collapse prediction is two series of component tests at Stanford University and

earthquake-simulator tests on two models of a four-story steel moment frame tested at the

NEES facility at the University at Buffalo.

The first series of component experiments served to calibrate deterioration behavior based on

cyclic tests with symmetric deformation histories. The shaking table tests served to provide a

comprehensive set of data on inelastic response through collapse, including reliable data on

the actual deformation histories to which critical components were subjected. In the second

series of component tests, which was performed after the shaking table tests, advantage was

taken of the deformation histories experiences by structural components during the shaking

table tests. This led to improved analytical modeling of deterioration and an accurate predic-

tion of inelastic response all the way to collapse. The focus of the paper is on the process of

model development and calibration, pre-shaking table test response prediction, model im-

provements through post-shaking table test component experiments, and post-earthquake

simulator test prediction of inelastic response to collapse.


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Dimitrios G. Lignos, Helmut Krawinkler and Andrew S. Whittaker

2

1 INTRODUCTION

Only recently has significant progress been made in our ability to predict collapse and to

develop engineering measures that provide consistent collapse protection. The main obstacleshave been our inability to predict deterioration properties of structural components and the

lack of physical experiments that could validate and improve our analytical tools. This paper

focuses on component deterioration modeling improvements by taking advantage of (1) recentearthquake simulator tests through collapse of a scale model of a 4-story steel moment resist-ing frame and (2) various cyclic and monotonic tests of components of the scale models prior

and after the completion of the earthquake simulator (shake table) tests to collapse.

2 DETERIORATION MODELING

Experimental studies have shown that the hysteretic behavior of a structural component isdependent upon numerous structural parameters that greatly affect the deformation and energy

dissipation characteristics, leading to the development of a wide range of more versatile dete-rioration models. In the early 1960s and 1970s Hisada et al. [1], Clough and Johnson [2], Ma-

hin and Bertero [3], Takeda et al. [4] developed analytical models in which stiffness and

strength of a structural component is updated in each excursion based on the maximum de-formation experienced in previous excursions. Song and Pincheira [5] developed a model thatincorporates a strength and post-capping behavior, but does not incorporate cyclic strength

deterioration. Foliente [6] presented a summary of the widely known Bouc-Wen model [7, 8].

In the same paper Foliente presents the main modifications of the model by others (Baber and

Noori [9], Casciati [10], Reinhorn et al. [11]) to incorporate component deterioration.

Sivaselvan and Reinhorn [12], based on earlier models by Iwan [13] and Mostaghel [14], de-

veloped a versatile smooth hysteretic model with stiffness and strength degradation and

pinching characteristics, derived from inelastic material behavior.

Lignos [15] modified the deterioration model by Ibarra et al. [16] (Ibarra-Krawinkler (IK)

model) to address asymmetric component hysteretic behavior including different rates of cy-

clic deterioration in the two loading directions, residual strength and incorporation of an ulti-

mate deformation u at which the strength of a component drops to zero. This model is usedfor the purpose of the research presented in this paper. The phenomenological IK model is

based on a backbone curve that defines a reference envelope for the behavior of a structural

component and establishes strength and deformation bounds, and a set of rules that define thebasic characteristics of the hysteretic behavior between the bounds or backbone curve. Up to

four modes of cyclic deterioration are defined with respect to the backbone curve based on thehysteretic response of the component. The cyclic deterioration rates are controlled by a rule

developed by Rahnama and Krawinkler [17] that is based on the hysteretic energy dissipatedwhen the component is subjected to cyclic loading. The main assumption is that every com-

ponent has a reference hysteretic energy dissipation capacity tE , regardless of the loading his-

tory applied to the component. The reference hysteretic energy dissipation capacity in Lignos

[15] is expressed as a multiple of y pM ,

t p yE M or t yE M (1)

wherep is the reference cumulative deformation capacity, and p , and yM are the

plastic rotation capacity and effective yield strength of the component, respectively. Figure 1

illustrates the monotonic backbone curve of the modified IK model together using a simulated

bilinear cyclic response.


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3

Chord Rotation

MomentM

Initial Backbone

p

pcMc

My

u

r

c

y

Post -CappingStrength Det.

Basic Strength Det.

Unlod.Stiff. Det.

Mr= M

y

Mr= M

y

My

Mc

Ke

Effective yield strength and rotation (

yM - y )

Effective stiffness e y yK M

Capping strength and associated rotation for monotonic loading (c c

M )

Plastic rotation capacity for monotonic loadingp

Post-capping rotation capacity pc

Residual strength r yM M

Ultimate rotation capacityu

as shown in the figure

Figure 1. Modified Ibarra Krawinkler (IK) deterioration model. Backbone curve, basic modes of cyclic dete-

rioration and associated definitions

The basic deterioration rule by Rahnama and Krawinkler [17] has been modified for the

case of asymmetric hysteretic response to consider different rates of cyclic deterioration in

positive and negative loading directions.

/ /

, , , 1

1

c

is c k i i

t j

j

ED

E E

(2)

Where/

, , ,s c k i = parameter defining the deterioration in excursion i, strength

/

,s i , post-

capping strength /,c i

and unloading stiffness deterioration /,k i , Ei =hysteretic energy dissi-

pated in excursion i and/D = parameter defining the decrease in rate of cyclic deterioration

in the positive or negative loading direction (e.g. in the case of a composite beam the slab de-

celerates the deterioration in the positive direction).D can only be/

1D . When the rate of


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cyclic deterioration is the same in both loading directions then / 1D and the cyclic dete-rioration rule is essentially the same as the one included in the original Ibarra-Krawinkler

model. The deteriorated yield strength/

iF , deteriorated post-capping strength ,

/

ref iF and de-

teriorated unloading stiffness ,u iK per excursion i are given by the set of the following equa-

tions,

/ /

, 1(1 )i s i iF F

(3)

,

/ / /

, , 11 c iref i ref iF F

(4)

, 1

/ / /

, ,(1 )

u iu i k iK K

(5)

An example of the modified IK deterioration model for the effect of asymmetric behavior due

to composite action is shown in Figure 2 for the same steel beam with/without composite ac-tion.

-0.1 -0.05 0 0.05 0.1-800

-600

-400

-200

0

200

400

600

800

Ke

= 45000

My+ = 552M

y

-= -514

p+

= 0.030

p-

= 0.030

pc+

= 0.250

pc-

= 0.200

s = 1.0

c= 1.0

a = 1.0k = 1.0M

c/M

y+

= 1.05

Mc/M

y

-= 1.05

k = 0.00

Chord Rotation (rad)

Moment(kN-m)

E-Defense-Beam-MomentRotation

-0.1 -0.05 0 0.05 0.1

-800

-600

-400

-200

0

200

400

600

800

Ke

= 100000

My

+= 650

My-

= -485

p

+= 0.030

p

-= 0.015

pc

+= 0.250

pc-

= 0.250

s

= 1.0

c

= 1.0

a = 1.0

k= 0.5

Mc/M

y

+= 1.20

Mc/M

y

-= 1.05

k = 0.30


Moment(kN-m)

E-Defense-BeamC-MomentRotation

(a) (b)Figure 2. Calibration of modified Ibarra Krawinkler (IK) deterioration model for asymmetric hysteretic re-

sponse (data from E-Defense [18])

All deterioration model modifications were based on a recently developed database by

Lignos and Krawinkler [19] on deterioration properties of structural components. The modi-fied IK deterioration model has been implemented in a single degree of freedom (SDOF) dy-

namic analysis program (SNAP) and a multi degree of freedom (MDOF) dynamic analysisplatform (DRAIN 2DX [20]).

3 PROTOTYPE AND MODEL STRUCTURE FOR EXPERIMENTAL STUDIES

In order to validate analytical modeling capabilities for the collapse prediction effort a co-

ordinated analytical and experimental program is conducted in which a 4-story steel momentresisting frame is designed based on recent code requirements [21, 22] and serves as a proto-

type for analytical predictions and experimental validation to collapse. The structural systemis a special moment resisting frame (SMRF) with reduced beam sections (RBS) designed per

FEMA350 [23]. A detailed description of the building is presented in Lignos [15]. Two 1:8

scale model frames, whose properties represent those of the prototype structure, are tested on

the earthquake simulator of the Network for Earthquake Engineering Simulation (NEES) fa-

cility at the State University of New York at Buffalo (SUNY-UB).


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3.1 Scale model frames for earthquake simulator collapse tests

The scale of the two model frames whose properties represent those of the prototype frame

discussed earlier was dictated by the size of the earthquake simulator facility at SUNY-UBand fabrication constraints. The two test specimens that were tested through collapse were

nominally identical and represent only half of the 4-story prototype building due to cost and

the payload capacity of the earthquake simulator. At a 1:8 model scale, the total weight of halfof the structure is approximately 40 kips based on similitude rules described by Moncarz andKrawinkler [24].

In an analysis model, gravity can be placed on a leaning P column (Gupta and Krawin-kler [25]). In the physical model tested on the SUNY-Buffalo earthquake simulator, two sub-

structures were used to achieve the same goal: (1) the scale model of the SMRF (denoted as

the model frame in Figure 3) and (2) a mass simulator (see Figure 3). Both sub-structures are

connected with rigid links at each floor level to transfer the P effect from the masssimulator to the test frame. Each link is equipped with a load cell to measure story forces and

an articulated joint (hinge) at each end. Figure 3 illustrates both sub-structures after erection

on the earthquake simulator. Details about design, erection process are presented in Lignos

[15].

Figure 3. Four-story scale model and mass simulator on the the SUNY Buffalo NEES earthquake simulator

The model frame consists of elastic aluminum beam and column elements and elastic

joints that are connected by plastic hinge (lumped plasticity) elements. The mechanical prop-

erties of the elastic elements are selected to correctly simulate scaled element stiffness. A

plastic hinge element consists of (1) two steel flange plates that are tuned such that plastichinging at the end of beams and columns is realistically represented at the model scale and (2)

a spherical hinge that transfers shear. Spacer plates are used to permit adjustment of the buck-ling length of the flange plates, that is, to control basic strength and cyclic deterioration of

components. The final geometry and flange plate dimensions are the product of an extensivecomponent testing program discussed next. Figure 4a is a photograph of a typical plastic

hinge element and Figure 4b shows one of the flange plates of the plastic hinge element afterbuckling. A series of fifty component tests were undertaken in advance of the earthquake


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simulator tests to identify the configuration dimensions and geometry of flange plates for each

plastic hinge element (see Figure 4c) and most importantly to replicate the hysteretic behavior

of the prototype connections (see Figure 4d). In the following sections emphasis is put on de-

terioration modeling characteristics of each component of the model frames.

(a) plastic hinge element (b) flange plate after buckling

-0.1 -0.05 0 0.05 0.1-2300

-1150

0

1150

2300


Moment(kN-m)

(c) control parameters (d) typical behavior of prototype RBS connection

Figure 4. Component testing program; (a) typical plastic hinge element (b) flange plate after buckling (c) geo-metric parameters to be identified; (d) typical targeted hysteretic behavior of a beam with RBS similar to the

prototype moment resisting frame (data from Uang et al. [26])

4 EXPERIMENTAL-ANALYTICAL COMPONENT PROGRAM

To identify the deterioration parameters of the plastic hinge elements, a series of monotonic

and cyclic tests are conducted with single and double flange plate configurations at Stanford

University. The experimental setup used for this component testing program is shown in Fig-

ure 5a and consists of the test frame, data acquisition system (DAQ) a hydraulic actuator,

which is controlled from a computer unit and a recorder unit. The basic component used in

this testing program is shown schematically in Figure 5b and it consists of the aluminum parts

1-1, 3-4 and the plastic hinge element discussed previously (part 1-2). total is the total deflec-

tion of the cantilever component due to bending of all its parts shown in Figure 5b.


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(a) experimental setup for component tests

(b) basic cantilever component for pre-Buffalo experimental program

Figure 5. Experimental setup for component tests and basic cantilever component

For all the cyclic tests a standard symmetric loading protocol was used. Figure 6a shows atypical cyclic test with double plates. From this figure it can be seen that the behavior of the

specimen is pinched after deformations larger than 0.03rad. The pinching is clearly noticeable

in the execution of cyclic tests with a symmetric loading protocol. This pinching effect is a

weakness in the model simulation of prototype RBS behavior because pinching in actual RBS

hysteretic diagrams is barely noticeable unless if the RBS connection is bolted in the web areawith the column face (i.e. slip in the bolts causes pinching in the hysteretic behavior). Most of

the pinching effect in the hysteretic response of the model connection is attributable to the ab-

sence of the web in the model plastic hinge element. In these elements flange plate buckling is

not restrained by a web, and during subsequent load reversal the flange will straighten out at a

much reduced axial stiffness before picking up full tensile resistance, which causes pinching

behavior more characteristic for axially loaded steel members.


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-0.1 -0.05 0 0.05 0.1-3.50

-1.75

0 .00

1.75

3.50


Moment(kN

-m)

(a) calibration with modified IK model

-0.1 -0.05 0 0.05 0.1-2.5

-1.25

0

1.25

2.5

1.50"

(rad)

Moment(kN-m)

Exp. Data

ABAQUS

(b) calibration with ABAQUS model (moment versus rotation over 1.5 plastic hinge length)

Figure 6. Hysteretic behavior of various configurations together with calibration of analytical models; (a) plastic

hinge element with two flange plates with calibrated IK deterioration model; (b) plastic hinge element with oneflange plate together calibrated ABAQUS model with combined isotropic and kinematic hardening.

The modified Ibarra-Krawinkler model is calibrated based on monotonic and cyclic re-

sponse curve of the same subassembly. The bilinear hysteretic response of the modified

Ibarra-Krawinkler model is used to calibrate the cyclic deterioration parameters for eachcomponent, ignoring the pinching effect that is evident in all symmetric cyclic loading tests.

Overall, the hysteretic behavior of the plastic hinge elements is captured fairly well even if thepinching effect is not simulated since emphasis is put on strength and stiffness deterioration.

Using a more refined continuous finite element model in ABAQUS [26] that includes com-bined isotropic and kinematic hardening the hysteretic behavior of these joints can be mod-

eled accurately (see Figure 6b, which shows a single flange plate configuration test in whichthe pinching effect addressed previously is more evident). But the use of continuum models is

computationally expensive for analytical collapse simulations of the entire test frame and

required large computational capabilities (parallel computing). On the other hard, first yield-


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0.00

0.50

1.00

1.50

2.00

2.50

0 0.05 0.1 0.15 0.2 0.25Roof Drift (/H)

GroundMotionM

ultiplier

Experimenta Data

Pre-Buffalo Prediction

SLE

DLE

MCE

CLE

CLEF

(a) Frame 1

0.00

0.50

1.00

1.50

2.00

2.50

-0.3 -0.2 -0.2 -0.1 -0.1 0 0.05 0.1 0.15 0.2 0.25Roof Drift (/H)

GroundMotionMultiplier

Experimental Data

Pre-Buffalo PredictionSLE (CP)

DLE (CP)

MCE (LL)

CLE (CP)

CLEF(CP)

(b) Frame 2

Figure 7. IDAs of pre-Buffalo analytical predictions together with experimental data

For Frame 2 the objective was to investigate the effect of cumulative damage prior to col-lapse; thus we used the Chile 1985 Llolleo record for the MCE level after using the CP mo-

tion for SLE and DLE intensities. Figure 7b shows the physical IDA obtained from theexperimental data and pre-test analytical predictions. The analytical prediction indicates that

Frame 2 would be close to collapse at the MCE level. However the Llolleo record was notreproduced successfully in the earthquake simulator test. Thus we switched back to CP re-

cord. During CLE Frame 2 drifted in the opposite direction of Frame 1, and it collapsed inthis direction at the CLEF level ground motion (2.2 times Canoga Park) (see Figure 7b). The

roof drift response of both frames is shown in Figure 8, in which we show only the response

during the large amplitude motions of the full test series from SLE to CLEF. Each test is sepa-

rated with a dashed line from the other.


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After the completion of the two scale model test frames at the NEES facility at Buffalo a

comprehensive data set was assembled that includes information regarding the performance of

both test frames through collapse. A total of 314 channels were used for the instrumentation

of Frame 1 and 247 channels were used for Frame 2. The data is uploaded, curated and publi-

cally available through the NEES repository (http://central.nees.org).

-0.05

0.00

0.05

0.10

0.15

0.20

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Time (sec)

RoofDrift(

r/H)

SLE DLE MCE CLE CLE

(a) Frame 1

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Time (sec)

RoofDrift(r/H)

SLE DLE MCE CLE CLEF

(b) Frame 2

Figure 8. Roof drift ratio history for Frames 1 and 2 at various ground motion intensities

6 POST-BUFFALO COLLAPSE PREDICTIONS

The question to be answered in this section is what are the required analytical modeling re-finements that should be considered in order to explain the difference between pre-Buffalo

response predictions and earthquake simulator tests for various ground motion intensities. The

main issue is to investigate the effect of deterioration modeling parameters for critical plastic

hinges of the test frames on their collapse capacity since the initial collapse predictions were

based on calibrations of critical components of the test frames from symmetric cyclic loading

histories as presented in Section 4 of this paper.


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6.1 Post Buffalo component tests

In the earthquake simulator tests the rotation histories of all the plastic hinge elements of

both test frames are recorded, but the associated moments are unknown or have to be deducedfrom strain measurements or analytical models. In the elastic response regime the strain

moment calibrations provide good data on moments (see Lignos [15]), but in the inelastic re-

gime the strain - moment calibrations become strongly history dependent.

0 0.1 0.2 0.3 0.4-4.6

-2.3

0

2.3

4.6

1.5''

(rad)

Moment(kN-m)

Experimental Data

Monotonic BackboneAnalytical Prediction

(a) moment-rotation diagram obtained from

component test and model re-calibration

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 20 40 60

1

.5''

Buffalo Test

Post-Buffalo Test

(b) rotation history of plastic hinge element at base of exterior column of Frame 1

Figure 9. Post-Buffalo component test using the earthquake simulator test rotation history

After the completion of the earthquake simulator tests a series of component tests are per-

formed for selected critical plastic hinge locations, in which the recorded rotation histories areused to derive the tip displacement histories of component tests (see Figure 5b). The compo-

nent specimen type shown in Figure 5b and nominally identical plastic hinge elements withthe ones in the earthquake simulator test frames are employed. The earthquake simulator rota-

tion history over 1.5 length as deduced from clip gage extensometers (see Figure 4a) of aplastic hinge element is used as the basis for the deduction of the moment rotation diagrams

of critical plastic hinge locations.

To transform the input rotation history into a tip displacement history for the actuator, the

contributions of all the parts of the components that remain elastic (see Figure 5b, Parts 1-1


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and 3-4) need to be predicted. Hence an estimate of the moment at the plastic hinge element

of interest is needed. For this reason a mathematical model that simulates hysteretic behavior

based on the modified IK model is developed. Using the predicted stiffness and deterioration

parameters from the preshaking table component tests summarized in Table 1 and the rota-

tion1.5

as an input loading history, the moment needed to estimate elastic contributions is

back-calculated analytically. Finally the input rotation history of the selected plastic hingeelement is transformed into a tip displacement history for the component test. Figure 9a

shows the deduced moment rotation relationship for the exterior column base of Frame 1 to-gether with the calibrated IK deterioration model and Figure 9b illustrates the comparison of

measured plastic rotation over 1.5 length of the same plastic hinge element between the re-

corded response during the earthquake simulator tests and the response after repeating the his-

tories at Stanford laboratory. The two measurements are almost identical indicating

confidence on the results. The modeling parameters obtained from the component test series

discussed in this section and focused on four critical locations of the test frames are summa-

rized in Table 2. Due to the effect of loading history on deterioration parameters of compo-

nentspc

values between component tests after and prior to the earthquake simulator tests are

noticeably different. The post-Buffalo analytical response predictions are based on deteriora-tion parameters summarized in Table 2.

Location eK (kN-m/rad) c yF F p (rad) pc (rad)

C1S1B 2904 1.10 0.050 2.0 1.30

C1S1T 2331 1.10 0.050 2.0 1.30

F2B1R 1469 1.10 0.050 1.6 1.80

C1S3T 1265 1.08 0.055 2.4 1.00

Table 2. Component modeling parameters for postBuffalo collapse prediction

(a) CLE (b) ICL

Figure 10. Moment equilibrium of exterior subassembly during CLE and incipient collapse level (ICL)

An assessment of the results from the component tests discussed in this section allows usto illustrate the effect of component deterioration at critical plastic hinge locations. Figure 10

shows the moment equilibrium for during CLE and and CLEF ground motion intensities at


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selected times after the beginning of each motion. For CLEF the incipient collapse level (ICL)

is selected. This figure together with Figure 9a illustrate that the reduction in moments for the

plastic hinge locations at the base column and first floor beam from CLE to ICL is due to

strength deterioration. Based on Figure 10 it is evident that the point of inflection for the col-

umns does not stay constant but its location depends on the loading history of the components.

6.2 Post Buffalo response predictions

The purpose of the post-Buffalo response predictions described in this section is to investi-gate if we can predict the seismic behavior of the two model frames through collapse more

accurately after modifying the analytical model based on information that became available

from earthquake simulator tests and post-Buffalo component tests.

For the post-Buffalo response predictions the recorded earthquake simulator motions are

used rather than the actual ground motions. After comparing the input motions and recorded

motions in the frequency domain, the differences are small except for the input error made in

reproducing the Llolleo MCE motion for Frame 2.

During the earthquake simulator tests both frames exhibited considerable friction damping

attributed primarily to the spherical hinges of the mass simulator gravity links shown in Fig-

ure 3. Thus for the post-Buffalo analytical response predictions of both frames a friction ele-ment was inserted per story at each end of the gravity links (P-Delta column) in each story,

that were initially modeled as rigid elements with hinged ends (see Lignos [15]). After the

DLE shaking, the effect of friction has very little effect on the response as seen from Figure

11 that shows the DLE roof displacement response for both test frames. Friction damping isonly important for the analytical model to accurately reproduce the residual deformations in

each story up to DLE. It was found that friction damping does not affect the collapse intensityof the selected motion for both frames.

Comparison of Roof Displacements

Between Frames #1 and #2, DLE

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Time (sec)

r(

in)

Frame#1

Frame#2

Figure 11. Comparison of roof displacement history between Frames 1 and 2 at DLE shaking

Tables 1 and 2 show that the values of plastic rotation capacity (p

) for the component

tests prior and after the earthquake simulator tests are essentially identical. The rates of cyclicdeterioration ( ) of the beams based on the post-Buffalo component tests are slightly higherthan for the pre-Buffalo component tests. Past studies by Ibarra and Krawinkler [29] haveshown that variations of this parameter of the magnitude seen here do not have a significant

effect on the collapse capacity of deteriorating structural systems.


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From Tables 1 and 2 the calibrated values of post-capping rotation ( pc ) based on post-

Buffalo component tests are clearly larger than the calibrated values from pre-Buffalo com-

ponent test series. A smaller pc value (i.e. the post-capping slope of the deterioration model

becomes steeper) accelerates the P effect on the collapse capacity of a frame structure.That is the structure deflects more due to P and collapse occurs earlier. Using the deterio-ration parameters of Table 2 the predicted physical IDA curves are plotted for both frames

in Figure 12 together with the experimental data, indicating that the global response of bothscale models is predicted well. We draw the following conclusions from this figure:

o Accurate prediction of response of frame structures near collapse is sensitive to theaccurate representation of deterioration parameters of critical components.

o The use of the modified IK model presented in Section 2 can capture the global re-sponse near collapse provided that accurate modeling parameters can be obtained.

0.00

0.50

1.00

1.50

2.00

2.50

0 0.05 0.1 0.15 0.2 0.25Roof Drift (/H)


Post-Buffalo Prediction

Experimental Data

DLE

SLE

MCE

CLE

CLEF

(a) Frame 1

0.00

0.50

1.00

1.50

2.00

2.50

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05Roof Drift (/H)


Experimental Data

Post-Buffalo Prediction

SLE (CP)

DLE (CP)

MCE (LL)

CLE (CP)CLEF(CP)

(b) Frame 2

Figure 12. IDAs for both scale models based on post-Buffalo predictions and experimental data


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To illustrate the effect of cumulative damage in both frames after CLE we show the dashed

dotted black line in both Figures 12a and 12b. This line indicates that both frames would

have collapsed at a lower level of intensity than was executed in the CLEF tests. The slope of

the dashed dotted line for Frame 1 is much steeper than the one for Frame 2, illustrating that

due to much larger residual deformation in Frame 1verus Frame 2, a much lower intensity is

needed to collapse Frame 1. This can be seen in Figures 13a and 13b, which show the residualdrift ratio profiles along the height of the two frames together with analytical simulations us-

ing the post-Buffalo refined analytical model. The analytical predictions for all levels of

ground motion intensity prior to collapse indicate that the differences with the measured data

are not large. For DLE level of shaking the residual drift for both frames is about the same.

The reason that the analytically predicted residual drifts for both frames are slightly overesti-mated compared to the experimental data for the upper stories is the presence of friction

damping.

1.00

2.00

3.00

4.00

5.00

0.00 0.03 0.06 0.09 0.12 0.15SDRr (rad)

Floor

DLEMCECLEDLE-Pr.MCE-Pr.CLE-Pr.

Roof

(a) Frame 1

1.00

2.00

3.00

4.00

5.00

-0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04

SDRr (rad)

Floor

DLE

MCECLEDLE-Pr.

MCE-Pr.CLE-Pr.

Roof

(b) Frame 2

Figure 13. Residual story drift after various ground motion intensities for Frames 1 and 2


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6.3 Predicted story drift ratio histories and peak values

Figure 14 shows a time window of the story drift histories obtained from the CLE earth-

quake simulator tests from post-Buffalo analytical simulations for Frame 1. The responsesfrom the post-Buffalo analytical simulations are also shown. As seen from this figure, the

fourth story drift histories are overestimated by the analytical model but in the first three sto-

ries, which are part of the collapse mechanism of Frame 1, the analytical prediction is not farfrom the experimental data considering the simplifications of the mathematical model dis-cussed previously.

0 5 10 15 200

0.075

0.15

Time, t (sec)

SDR1

(rad)

SDR1

SDR1

Analytical

0 5 10 15 200

0.075

0.15

SDR2

(rad)

SDR2

SDR2

Analytical

0 5 10 15 200

0.075

0.15

SDR3

(ra

d)

SDR3

SDR3

Analytical

0 5 10 15 200

0.05

0.100

SDR4

(rad)

SDR4

SDR4

Analytical

Figure 14. Story drift ratio histories of Frame 1 based on analytical post-Buffalo predictions and experimentaldata for CLE

The peak story drift ratios (SDRs) recorded at various levels of ground motion intensity are

presented in Figure 15 for both frames. In the same figure we have superimposed the analyti-

cally predicted peak story drift ratios based on the postBuffalo analytical model discussedearlier. The agreement with the measured data is fairly good considering the simplifications of

the analytical model. The peak response for both frames is underestimated at DLE due to

modeling of friction damping discussed earlier in this paper.


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1.00

2.00

3.00

4.00

5.00

0.00 0.05 0.10 0.15 0.20

SDR (rad)

Floor

SLE

DLE

MCE

CLE

SLE-Pr

DLE-Pr

MCE-Pr

CLE-Pr

Roof

(a) Frame 1

1.00

2.00

3.00

4.00

5.00

-0.20 -0.15 -0.10 -0.05 0.00 0.05

SDR (rad)

Floor

SLE

DLE

MCE

CLE

SLE-Pr

DLE-Pr

MCE-Pr

CLE-Pr

Roof

(b) Frame 2

Figure 15. Peak story drift ratios at various ground motion intensities for both frames

6.4 Predicted base shear 1st story drift

The model frame with mass simulator tested on the Buffalo earthquake simulator allows

the assessment of shear force histories up to collapse including the effect of P since the

rigid links used to connect the model frame and mass simulator are instrumented and act asload cells. Of particular interest is the base shear versus first story drift relationship up to col-

lapse. The purpose of this section is to compare the base shear 1st

story drift ratio to collapse

using the post-Buffalo analytical model with the base shear 1st

story drift relationship as

measured experimentally. Figure 16 shows this diagram for Frame 1 based on the analytical

response predictions and experimental data during CLE and CLEF levels of intensity. Base

shear 1A

V is computed after summing up the acceleration histories per floor times the mass of

the individual floors and is normalized with respect to the total scaled weight (40kips). It can


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be seen that the dynamic response is captured fairly well analytically including the dynamic

shear amplification after 0.12rad due to ground motion reversal.

To guarantee stability of the analytical model for large deformations near collapse (SDR >

12%) a very small time step ( dt = 0.0005sec) is used for the implicit analysis solver. Up to

CLE level of intensity the employed time step was an order of magnitude larger ( dt =

0.005sec), indicating that predicting dynamic response near collapse can be computationallyexpensive.

0 0.1 0.2

0

0.25

0.50

SDR1 (rad)

V/W

-0.25

-0.500 0.1 0.2

0

0.25

0.50

SDR1 (rad)

V/W

-0.25

-0.50

Pushover P-

V1A

V1Pred.

Push.

Figure 16. Base shear 1st

story IDR of Frame 1

(a) predicted collapse mechanism atIDR1 = 25% (b) collapse mechanism after CLEF

Figure 17. Collapse mechanism of model frame 1

The pushover curve (base shear versus 1st

story drift) up to collapse, based on first mode

lateral load pattern, is superimposed in Figure 16 (in a pushover curve collapse is associated

with the drift ratio at which base shear V/Wbecomes zero). As seen from Figure 16, elastic

stiffness and yield base shear are captured fairly well based on pushover analysis. The col-

lapse mechanism is also predicted accurately as seen from Figure 17, which shows the pre-

dicted collapse mechanism at about 25% first story drift (see Figure 17a) together with aphoto of the final collapse mechanism of Frame 1 (see Figure 17b). Coming back to Figure 16,


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the pushover curve is not able to simulate the dynamic amplification of the base shear due to

ground motion reversals, particularly near collapse and the maximum dynamic story shear

force demand is about 50% larger than predicted by the pushover. Hence nonlinear dynamic

analysis is the recommended to be used in order to predict analytically the behavior of a struc-

tural system near collapse.

7 SUMMARY AND CONCLUSIONS

This paper presents analytical model modifications and refinements to simulate the dy-namic behavior through collapse of two scale models of a 4-story steel moment resisting

frame. Both model frames were tested on the earthquake simulator of the Buffalo NEES facil-

ity. The comprehensive set of experimental data is available through the NEES repository

(http://nees.central.org). Deterioration model modifications and calibration of critical compo-

nents used for preBuffalo and postBuffalo response predictions is based on two component

test programs. The main findings from the research project are:

Deterioration models used for prediction of dynamic response near collapse shouldbe calibrated with loading protocols that include relatively small inelastic cycles

followed by unidirectional loading to extreme deformations.

The use of continuum models represents better the hysteretic response of compo-nents compared to phenomenological models, in terms of capturing details that af-

fect the behavior of a component in early cycles (cyclic hardening, Bauschinger

effect, pinching effect). But the behavior of a frame structure can be predicted

fairly well up to collapse without using a computationally expensive analytical

model, as long as the deterioration characteristics of critical components near col-

lapse are accurately represented in a phenomenological model.

Accurate modeling of the post-capping rotation capacity pc of structural compo-

nents is essential for prediction of collapse capacity of frame structures.

Nonlinear response-history analysis with direct integration may require a verysmall analysis step to capture the dynamic behavior of frame structures at large de-

formations near collapse. Pushover analysis provides valuable information regarding elastic response, yield

base shear and global behavior of a frame structure. The collapse mechanism of a

first mode dominated structure can also be predicted accurately from a pushoveranalysis. However, dynamic shear amplification due to ground motion reversals

cannot be simulated with the use of nonlinear static analysis.

ACKNOWLEDGMENTS

This study is based on work supported by the United States National Science Foundation(NSF) under Grant No. CMS-0421551 within the George E. Brown, Jr. Network for Earth-

quake Engineering Simulation Consortium Operations. The financial support of NSF is grate-fully acknowledged. The authors also thank REU students Mathew Alborn, Melissa Norlund

and Karhim Chiew for their invaluable assistance during the earthquake simulator collapsetest series. The successful execution of the earthquake-simulator testing program would not

have been possible without the guidance and skilled participation of the laboratory technicalstaff at the SUNY-Buffalo NEES facility. Any opinions, findings, and conclusions or recom-

mendations expressed in this paper are those of the authors and do not necessarily reflect the

views of NSF.


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