modeling of steel moment frames for seismic loads -...

Journal of Constructional Steel Research 58 (2002) 529–564www.elsevier.com/locate/jcsr

Modeling of steel moment frames for seismicloads

Douglas A. Foutcha,∗, Seung-Yul Yunb

a University of Illinois at Urbana-Champaign, 3129 Newmark Lab, 205 N. Mathews, Urbana,IL 61801, USA

b University of Illinois at Urbana-Champaign, 3113 Newmark Lab, 205 N. Mathews, Urbana,IL 61801, USA

Received 23 April 2001; received in revised form 15 August 2001; accepted 7 September 2001

Abstract

Simple elastic models based on centerline dimensions of beams and columns are widelyused for the design of steel moment resisting frames. However, for the performance predictionand evaluation of these structures, different nonlinear models are being used to better simulatetheir true behavior. Simple nonlinear modeling methods widely used as well as those withmore detailed modeling representations are investigated and compared.

A 9-story building and a 20-story building were designed for this study according to the1997 NEHRP provisions. Different models for these structures were developed and analyzedstatically and dynamically. The models investigated involved the use of centerline dimensionsof elements or clear length dimensions, nonlinear springs for the beam connections, and linearor nonlinear springs for the panel zones. A second group of models also incorporated thefracturing behavior of beam connections to simulate the pre-Northridge connection behavior.Two suites of ground motions were used for the dynamic analysis: typical California and nearfault ground motions. The differences in structural responses among different models for bothsuites of motions are investigated.

According to static pushover analyses with roof displacement controlled, the benefit of theincrease in capacity that results from the detailed models is consistently observed for both the9-story and 20-story buildings. When the models were excited by different ground motionsfrom each suite, the median responses of the more detailed models showed an increase incapacity and a decrease in demand as expected. However, due to the randomness inherent inthe ground motions, variations were also observed. Overall, the model which incorporatesclear length dimensions between beams and columns, panel zones and an equivalent gravity

∗ Corresponding author.E-mail address: [email protected] (D.A. Foutch).

0143-974X/02/$ - see front matter 2002 Published by Elsevier Science Ltd.PII: S0143 -974X(01)00078-5

530 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) 529–564

bay without composite action from the slab seems to be a practical model with appropriateaccuracy. 2002 Published by Elsevier Science Ltd.

Keywords: Seismic analysis; Steel moment frames; Steel buildings; Earthquake response; Non-linearanalysis; Elastic analysis

1. Introduction

The engineer’s ability to model buildings has increased quickly over the past sev-eral years with the development of advanced analysis programs and the competitionamong software developers. In fact, our ability to model structural behavior probablyexceeds our ability to fully understand the observed behavior.

The first structural analysis programs that were developed in the early 1960s couldhandle only linear prismatic beam and column members with fully restrained orpinned joints and centerline dimensions. Programs in use today have a number ofelements that model material and geometric nonlinearities, rigid or partiallyrestrained connections, and flexible foundations and diaphragms. This paper willcover commonly used modeling procedures for steel moment frames.

A word of caution is required. Although the modeling procedures described hereinare quite detailed and match measured behavior very well, it must be rememberedthat this is still greatly simplified from the case of a real building which has cladding,partitions, mechanical equipment, stairways and many other discounted attributes. Areal building might have irregularities and flexible foundations that are importantbut not included here. It must be remembered that the calculations that follow areonly estimates of actual behavior.

A 9-story and a 20-story building were designed in accordance with the 1997NEHRP provision for this study. Different models for those structures weredeveloped and analyzed statically as well as dynamically. Two suites of groundmotions were used for the dynamic analyses: typical California and near fault groundmotions. The comparisons of computed structural responses for the different modelsare investigated.

2. Design of 9-story and 20-story buildings according to the 1997 NEHRPprovisions

The plan and elevation views of the buildings are given in Fig. 1. The buildingswere designed for a site in downtown Los Angeles where SS is 1.61g and S1 is 1.15g.The perimeters of the buildings were designed as special moment frames so theresponse reduction factor of R=8 was used. According to the 1997 NEHRP pro-visions, the base shears corresponding to the 9-story and 20-story building were 300and 244 kips, respectively. The approximate period equation prescribed in the pro-vision was used to check for strength as well as drift requirements. Drift requirementsgoverned the design for both of the buildings. The section members assigned for

531D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) 529–564

Fig. 1. Plan view and elevation view of 9-story and 20-story building.

each of the buildings are listed in Table 1. Box sections were used for the cornercolumns of the 20-story frame since they needed to resist bi-axial bending fromlateral loadings. Doubler plates were inserted at the middle story panel zones of theinterior columns to satisfy the shear requirement as shown in the table.

The new element in the DRAIN-2DX program developed by Foutch and Shi [1]was used to model the nonlinear behavior of the beam connections as well as panelzones. Detailed descriptions of the nonlinear springs used for the beams and panelzones will follow in a later section.

Six different models of the buildings were investigated. The first model used cent-erline dimensions with nonlinear springs for yielding of beams as well as a leaningcolumn attached to the moment resisting frame to correctly account for the P��effect for the building. This model is denoted as M1-WO. The next three modelsused clear length dimensions with nonlinear springs to model the panel zones aswell the beams. The first model of the three is similar to M1-WO but used the clear


Table 1Sections assigned for the 9-story and 20-story buildings

Story w14 Story w24Columns Doubler plate Beam Columns Doubler plate Beam

Exterior Interior Exterior Interior Exterior Interior Exterior Interior

9 w14×342 w14×398 0 0 w21×62 20 15×15×0.5 w24×207 0 0 w18×468 w14×342 w14×398 0 0 w27×94 19 15×15×0.5 w24×207 0 0 w24×557 w14×398 w14×455 0 0 w33×118 18 15×15×0.75 w24×250 0 0 w27×846 w14×398 w14×455 0 0 w33×118 17 15×15×0.75 w24×250 0 0.125 w30×1085 w14×455 w14×550 0 0 w36×150 16 15×15×1.0 w24×279 0 0 w30×1084 w14×455 w14×550 0 0 w36×150 15 15×15×1.0 w24×279 0 0.25 w33×1183 w14×550 w14×550 0 0 w36×150 14 15×15×1.0 w24×279 0 0.375 w33×1182 w14×550 w14×550 0 0 w40×183 13 15×15×1.0 w24×335 0 0.125 w33×1181 w14×550 w14×605 0 0 w40×183 12 15×15×1.0 w24×335 0 0.25 w36×1350 w14×550 w14×605 0 0 w40×183 11 15×15×1.0 w24×335 0 0.25 w36×135

10 15×15×1.25 w24×408 0 0 w36×1359 15×15×1.25 w24×408 0 0 w36×1358 15×15×1.25 w24×408 0 0 w36×1357 15×15×1.25 w24×408 0 0 w36×1356 15×15×1.25 w24×408 0 0.125 w36×1355 15×15×1.25 w24×408 0 0.125 w36×1354 15×15×2.0 w24×492 0 0 w36×1353 15×15×2.0 w24×492 0 0 w33×1182 15×15×2.0 w24×492 0 0 w33×1181 15×15×2.0 w24×492 0 0 w33×1180 15×15×2.0 w24×492 0 0 w33×118�1 15×15×2.0 w24×492 0 0 w14×22

length of beams and columns with the flexibility of the panel zones modeled intothe joint. This is denoted as M2-WO. The second of the three includes one bay ofthe frame model that represents all of the interior gravity columns but with simpleconnection properties assumed for the beam springs and is denoted as M2-SC. Thelast of the three is identical to the second model, but with resistance from the com-posite slab on top of the beam in the gravity frames modeled into the beam springs,and this is referred to as M2-Comp. The last two models of the six models areidentical to the M2-WO and M2-SC but fracturing behavior of the beam connectionsis incorporated into the models. For those connections, when the plastic moment isreached, the strength of the beam connection drops down to 10% of the plasticmoment capacity. The periods of each model for the 9-story and 20-story momentframes are listed in Table 2. The model with the equivalent gravity bay frame withrotational resistance from the slab is the stiffest since the contribution from the conti-nuity of the interior columns and rotational strength of the beam connection isincluded. It is interesting to note that M2-WO is stiffer than M1-WO. This is dueto the fact that the M2 model uses clear lengths of the beams and columns that makethis structure stiffer even though a flexible panel zone is also included. When clearlength models and centerline models are pushed statically using displacement control,the demands for elements for the clear length model will be larger. The natural


Table 21st and 2nd mode of each model

M1 M2WO WO SC Comp WO-frac SC-frac Comp-fr

9-storyT1, (s) 2.49 2.38 2.35 2.30 2.38 2.35 2.30T2, (s) 0.89 0.85 0.83 0.81 0.85 0.83 0.8120-storyT1, (s) 3.77 3.48 3.45 3.41 3.47 3.45 3.41T2, (s) 1.30 1.20 1.20 1.19 1.20 1.20 1.19

periods for these models are remarkably similar and will respond about the samefor response to dynamic motions if linear elastic behavior is assumed.

First, drift demands using static pushover analyses and then dynamic analysesusing both suites of ground motions were investigated. Finally, dynamic driftcapacities of the models were calculated using Increment Dynamic Analysis (IDA)which will be described later in this paper.

3. Ground motions

Two different suites of accelerograms were used for the study. The first suite ofaccelerograms represents the typical ground motions for the LA site. The secondsuite represents near fault ground motions. Each of the typical ground motions inthe first suite was scaled in a least square manner to match the 2% in 50-year hazardspectra of the site at periods of 1.0, 2.0 s and 4.0 s. The descriptions of the groundmotions with their scaling factors are given in Table 3. The scale factors range from1.72 to 1.87. A different scaling method was used for the second suite of groundmotions since those ground motions were generated specifically to represent the 2%

Table 3Description of typical ground motions

Name Ground motion name Scale factor used

EQ01 Taft (1952) 1.72EQ02 Castica (1971) 1.87EQ03 Imperial Valley (1979) 1.83EQ04 Pacoima Dame (1971) 1.83EQ05 Northridge (1994) 1.85EQ06 El Centro (1940) 1.85EQ07 San Fernando (1971) 1.39EQ08 Mammoth Lakes (1980) 1.82EQ09 Morgan Hill (1984) 1.87EQ10 North Palm Spring (1986) 1.81


Table 4Description of near fault ground motions

Name Ground motion name Scale factor used

LF21 Kobe (1995) 0.65LF23 Loma Prieta (1989) 0.65LF25 Northridge (1994) 0.65LF27 Northridge (1994) 0.65LF29 Tabas (1974) 0.65LF31 Elysian Park (Simulated) 0.65LF33 Elysian Park (Simulated) 0.65LF35 Elysian Park (Simulated) 0.65LF37 Palos Verdes (Simulated) 0.65LF39 Palos Verdes (Simulated) 0.65

in 50-year hazard level. They are the normal component of the LA 2% in 50-yearhazard level ground motions developed by Somerville et al. [2] for the SAC PhaseII project. The ground motions were scaled to minimize the error for the medianresponse of the ground motions. The scaling factor for this suite of ground motionscame out to be 0.65. The descriptions of the ground motions with their scaling factorsare given in Table 4. The scaled response spectra of the both suites of ground motionsare shown in Figs. 2 and 3. It was interesting to notice that the spectral accelerationsin the short period range (less than 1.0 s) for the typical ground motions were highcompared to those for the near fault motions whereas some of the near fault motionspossess bumps in the period region of 0.7–1.8 s. Therefore, the effects of highermodes for the typical motions and the pulse motion for the near fault motions shouldbe examined for the calculated responses. Because of the large spectral accelerations

Fig. 2. Scaled response spectra for EQ01–EQ10 and their median spectra.


Fig. 3. Scaled response spectra for LF21, LF23, LF25, ..., LF39 and their median spectra.

at longer periods, it should be expected that the near-fault motions on average wouldaffect the 9-story and 20-story buildings more than for shorter structures.

4. Description of systems

4.1. Linear centerline models

When designing new buildings or evaluating existing or damaged buildings twoacceptance criteria must be checked: member strength and building stiffness (drift).For new steel moment frame buildings the drift limitation always governs in highseismic regions.

Research done by Krawinkler [3] has shown that a linear elastic model usingcenterline dimensions is acceptable for design of special moment frames. The beammoments may be checked at the location in the beam where it intersects the columnflange. Even though this model gives adequate results for design, it will not alwaysgive good estimates of the distribution of shears, moments and axial forces through-out the building under dynamic loads. The panel zones must be modeled explicitlyfor frames with weak panel zones.

4.2. Elastic models with panel zones included

The next increase in reality is to include the panel zone behavior in the model.The panel zone is the region in the column web defined by the extension of thebeam flange lines into the column as shown in Fig. 4. The simplest way to modelthe panel zone for linear analysis is referred to as the scissors model also shown inFig. 5. The beams and column are modeled with a rigid link through the panel zoneregion and a hinge in the beam is placed at the intersection of the beam and column


Fig. 4. Definition of panel zone.

Fig. 5. Scissors model for panel zone modeling.

centerlines. A rotational spring with stiffness kθ is then used to tie the beam andcolumn together. The rigid links stiffen the structure but the panel zone spring addsflexibility. The net result is that this building model is usually stiffer than the center-line model. Since it is stiffer it will help in satisfying the drift design criteria. It willalso give better estimates of shears, moments and axial forces in the members. Mostfinite element programs currently used by engineers for seismic analysis have thisfeature. The equations for determining the stiffness of the panel zone spring arebased on the yield properties of the panel zone. The yielding property of the panelzone is


gy �Fy

�3G� qy (1)

My � Vy·db � 0.55Fydct·db (2)

where,

Fy=the yield strength of the panel zoneG=the shear modulus=E /2·(1 � n)dc=depth of columnt=thickness of panel zone which is the thickness of the web of the columnplus the thickness of the doubler plates if they are utilized.db=depth of beamν=Poisson’s ratio=0.3

So, the stiffness of the panel becomes

Kq �My

qy

(3)

4.3. Nonlinear centerline models

Models that allow yielding in the beams and columns are much more realisticthan linear models. Although nonlinear models are not required for design of newbuildings, they are very useful for evaluating existing and damaged buildings [4].Most commonly used programs model this behavior by including a nonlinear flexuralspring at the ends of elastic beam and column members. The springs should beassigned a very high stiffness compared to that of the beam or column. However,the spring yields at the plastic moment capacity of the member. The correct structurestiffness is maintained because it comes from the actual members rather than fromthe spring. This model is shown schematically in Fig. 6 and is referred to in thispaper as M1-WO.

The spring is rigid until the plastic moment of the member is reached. After yield-ing a post-yield stiffness is assigned to the spring that represents the strain hardeningbehavior of the member. A strain hardening coefficient, α, is assigned to the springafter yielding. A value of α equal to 0.03 is a reasonable choice. The spring behaviorand member plus spring behavior are shown in Fig. 6. The value of α equal to 0.03is a good choice for calculating story drift angles out to about 3–4%. After this,local flange buckling will begin to occur that causes α to gradually decrease to zeroand then it can become negative with larger drifts. Most programs will not allow anegative value of α. For calculating building behavior beyond 4%, it is best to choosea strain-hardening factor of zero.

For performance evaluation, the expected values of the yield strengths of the steelsshould be used. Expected yield strengths of commonly used steels are given in Table5 [5,6].


Fig. 6. Centerline model with nonlinear elements.

Table 5Expected and lower bound material properties for structural steel of various grades [5,6] 12

Yield strength Tensile strengthMaterial Year of Lower bound Expected Lower bound Expectedspecification construction

ASTM, A36 1961–1990Group 1 41 51 60 70Group 2 39 47 58 67Group 3 36 46 58 68Group 4 34 44 60 71Group 5 39 47 68 80ASTM, A572 1961–Group 1 47 58 62 75Group 2 48 58 64 75Group 3 50 57 67 77Group 4 49 57 70 81Group 5 50 55 79 84A36 and dual 1990–1999grade 50Group 1 48 55 66 73Group 2 48 58 67 75Group 3 52 57 72 76Group 4 50 54 71 76

1 Lower bound values for material are mean�2 standard deviation values from statistical data. Expectedvalues for material are mean values from statistical data.

2 For wide flange shapes, indicated values are representative of material extracted from the web of thesection. For flange, reduce indicated values by 5%.


4.4. Nonlinear models with panel zones

Most of the pioneering work on nonlinear panel zone modeling has been perfor-med by Krawinkler. His state-of-the-art report [3] provides a good discussion of thistopic and includes references to his earlier work [7,8]. Two methods of modelingthe nonlinear behavior of frames with yielding beams, columns and panel zones areavailable. One procedure is based on the scissors model shown in Fig. 5. The panelzone springs as well as the springs at the ends of the members are nonlinear. Thebehavior of the member spring is exactly the same as described in the previoussection. The panel zone spring is assigned a stiffness of

Kq �My

qy(4)

where

My � Vy·db � 0.55Fydct·db (5)

qy � gy �Fy

�3G(6)

In most cases, panel zones have a steeper post yield stiffness. Therefore, a valueof α equal to 0.06 is a reasonable value to use.

A better model is shown in Fig. 7. This model holds the full dimension of thepanel zone with rigid links and controls the deformation of the panel zone using twobilinear springs that simulate a tri-linear behavior. With this, the large strength differ-ence between the real behavior and the model is reduced.

The first slope post yield is steep and represents the behavior between the timethat yielding is initiated and the full plastic capacity is reached. After the plastic

Fig. 7. Panel zone modeling.


capacity is reached a small slope (2%) or zero slope may be used. This is shown inFig. 8.

Since yielding in the beams, columns and panel zones is represented well by thismodel, the actual distribution of yielding throughout the structure will be representedwell. For design of new special moment frames, the panel zones yield first. But,because of the steep strain-hardening slope for the panel zones, the beams will yieldshortly thereafter. This model is referred to as M2-WO.

Fig. 1 shows the 20-story building that was designed according to the 1997NEHRP Provisions. This building will be used for comparing the different modelsdescribed here. The results of pushover analyses for the buildings with the specifieddistribution of lateral forces required for new design in the 1997 NEHRP provisionsare shown in Fig. 9. ‘M1’ in the figure is the modeling case with the centerlinedimensions, whereas, ‘M2’ is for the model based on clear lengths plus panel zones.M2 also includes the modeling of the panel zones. The panel zone is modeled withtri-linear model spring and the full dimension of the member for the analysis. Ascan be seen, the M2 model is initially a little stiffer than the M1 model. The M1model with P�� gives the lowest strength. Care should be taken when plotting theroof drift ratio versus the total base shear. The roof drift ratio can be misleadingbecause it is incapable of capturing the local drift concentration. A good exampleof this case can be seen for this building that is pushed to about 4% of global driftwith P�� effects. The concentration of plastic deformations around the 3rd levelwas the controlling factor. Fig. 10 shows the plot of global roof drift ratio, top storydrift ratio, and 3rd level story drift ratio versus total base shear. Global roof driftratio is defined as the roof displacement divided by the total height of the building.Top story drift ratio is the story drift divided by the height of the story. The globaldrift ratio shows the averaged drift ratio over the whole height. This pushover plotreaches a peak at a drift of about 0.02 and rapidly has an increasing negative slope

Fig. 8. Panel zone load–deformation behavior.


Fig. 9. Comparison of modeling for 1997 NEHRP 20-story building.

Fig. 10. Comparison between global drift ratio vs story drift ratios for 20-story building.

after that. When each story drift ratio is plotted the 3rd level concentration of plasticdeformation is very noticeable as shown in Fig. 10. Note, however, that this doesnot reach a peak until almost 0.04 drift and then slowly becomes negative. The topstory drift actually remains elastic throughout the entire loading sequence. A plot ofdisplaced shapes of the building with increasing roof displacement is shown in Fig.11. The story level where the tangential slope is small indicates a large change indrift ratio. The concentration of plastic deformation can clearly be seen in Fig. 12where the story drift ratio for each story level with increasing lateral load is plotted.These results indicate that any nonlinear static procedure that relies on global roofdrift for a static pushover analysis is highly questionable.


Fig. 11. Displaced shape from static pushover analysis for 20-story building.

Fig. 12. Story drift ratio from static pushover analysis for 20-story building.

5. Description of components

5.1. Nonlinear springs for beams, columns, and panel zones

5.1.1. Reduced beam section connectionFor new buildings, reduced beam sections that are also referred to as dog-bone

members were used for the analysis. They exhibit very good hysteretic behaviorwith stable loops and good energy dissipation. Tests were performed by Venti and


Engelhardt [9]. A typical case of the hysteretic behavior is shown in Fig. 13. Thistest used a w14×398 column member and w36×150 beam section. Both membershave a nominal yield strength of 50 ksi. A model for the analysis using the DRAIN-2DX program is shown in Fig. 13. The expected yield strength of 57.6 ksi was usedin the modeling. The behavior of the member was modeled using a tri-linear model.The model simulated the specimen behavior very well. The ratio between the beamplastic moments to the first yielding point as well as the second moment value werecalculated and used for determining the yielding properties of the other membersizes. Seventy-four percent of the plastic moment of the beam was used as the firstyield moment for both positive and negative moments. For the second yield momentvalue, factors of 132% of the first yielding moment for the positive side and 120%of that for the negative side of the connection were used. The rotational value forthe second yielding moment of 0.03 radians for the positive side and 0.017 radiansfor the negative side were used for the protocol model. The rotational values thatare proportional to the plastic section modulus were assigned for the other beamsections. The strength degradation ratio that is the drop of the strength at each newplastic excursion was assigned a value of 0.83. This value was fixed for all membersizes although in reality, there would be variations from member to member. Thedrift demand is not significantly affected by the choice of this ratio. Differences indrift demand calculations would not vary by more than 2 or 3% because of thisdifference. An illustration of the yielding values for the protocol member (w36×150)and the 6th level beam in the 9-story building (w33×118) are shown in Fig. 14. Theplastic moments for the members are 33,750 (k-in) for w36×150 and 23,904 (k-in)for the w33×118 member.

5.1.2. Fracturing beam connectionFracturing beam connections were incorporated into the model to simulate the

behavior observed for the pre-Northridge buildings. The measured and the modeledhysteresis behavior of the connection are shown in Fig. 15. The new element in

Fig. 13. Measured [9] and model [4] of moment–rotation behavior of RBS connection.


Fig. 14. Illustration of yielding values for w36×150 (protocol) and w33×118 (6th and 7th level of 9-story building).

Fig. 15. Measured [13] and model [4] of moment–rotation behavior of fracturing connection.

DRAIN-2DX developed by Foutch and Shi [1] was used to model the strength dropin the connection after fracture. In the positive rotation case, the strength was mod-eled to drop to 10% of the original strength of the connection when the plasticmoment was reached just like the measured response. In the negative rotation side,the loss of strength at about 0.04 radians is observed for the measured behavior. Fornegative moment the crack in the bottom flange closes so typical bilinear-type ofbehavior occurs out to about 0.04 radians when the top flange fractures. However,due to limitations of the element, a gradual decrease in strength was modeled into theconnection. Therefore, the connection arrives at about zero strength at 0.04 radians tosimulate the fracture of the connection. Significant increase in demand as well asdecrease in capacity of structural response is expected.


Fig. 16. Illustration of simple connection in gravity frames.

5.1.3. Simple connections in gravity framesThe gravity frames are usually thought of as frames with no resistance to the

lateral load since the beam flanges are not connected to the column flanges. Theframe is sometimes modeled with pinned connections to capture the P-Delta effectdue to additional gravity load from the interior frames. However, according to theexperimental results from Liu and Astaneh-Asl [10] the resistance not only existsbut sometimes is significant due to the additional resistance occurring when a com-pression force in the composite floor slab is connected by a tension force in theshear tab. Additional resistance is encountered when the flanges of the beam comein contact with the column. An illustration of the connection is shown in Fig. 16.Fig. 17 shows a typical case where the shear tab with concrete slab on top of thebeam resists lateral load for many cycles of motion. This is a case with w18×35beam connected to the w14×90 column with shear tab and concrete on top. Minimumreinforcement was used for the slabs. The moment–rotation behavior of the connec-tion was modeled with a nonlinear spring that drops in strength at specified rotations.

Fig. 17. Measured [10] and model [4] of moment–rotation behavior of simple beam in gravity frame.


The model of the connection is shown in Fig. 18. A portion of the beam stiffnesswas used for the stiffness of the connection since it will not be like the rigid cases.The proportion was determined to be 25% of the stiffness of the beam. Also, theconnections cannot be expected to develop the full plastic moment capacity. Themaximum moment for the positive moment was taken as 38% of Mp and that forthe negative side as 11% because these values resulted in good matches betweenexperiment and analysis. The fact that the positive side develops higher moment isattributed to the compressive resistance of concrete slab on top of the girder bearingagainst the column. The tensile strength of the slab cannot be expected to help muchsince minimum reinforcement is used. The rotation at which the strength drops isassigned a value of 0.045 radians for the positive side and 0.05 radians for thenegative side of the connection. The drop in strength was assigned a value of 53%for the positive and 89% for the negative side. Those rotational values for the othersections were calculated using the disproportional value to the depth of the beams.Again gradual degradation of strength was modeled using 0.97 as the strength degra-dation factor. As will be seen later in this paper, the resistance from the gravityframe is significant. However, most of the contribution is not from the compositeconnection but from the flexural resistance from the continuous columns acting inconjunction with the rigid floor slabs. According to the report by Yun and Foutch[11], the differences in responses between the models with the simple connectionare negligible as long as the continuity of the gravity frame columns is modeled.This is due to the fact that the connections lose strength at very early stages of theground motions leaving only the columns to resist the lateral load. Fig. 18 showsan illustration of the yielding properties of a protocol connection and the connectionfrom a typical floor of the 9-story building.

5.2. Other modeling attributes

Another feature that should be included for analysis of tall buildings or shorterbuildings taken out to large drifts is the P�� effect. When the structure is displaced

Fig. 18. Illustration of yielding properties for w18×35 (protocol) and w16×26 beams in gravity frames(typical beam for 9-story building).


Fig. 19. Modeling interior columns for P�� effect only.

laterally the gravity forces acting through the displacement causes additional over-turning moments to develop in the structure. For a perimeter frame building this canbe a very significant effect since the perimeter frames must carry the overturningmoments of the entire building including the gravity frames.

One way to do this is to provide a dummy column in the model that carries thegravity loads in the building not directly carried by the moment frame. The columnis connected to the moment frame using rigid links with hinges at each end as shownin Fig. 19. The columns are hinged at both top and bottom. By doing this, onlyadditional overturning moment from the lateral displacement will be induced. Thecolumns will not help carry any of the lateral loads since they are pinned. Howeverin reality, the interior columns do help the moment frames since the columns arenot connected with a hinge and some resistance exists for the shear tab connectionin the beams due to the slab on top. An additional bay that has the equivalent proper-ties for all of the interior frames can be used as shown in Fig. 20. The columns and

Fig. 20. Modeling interior columns for P�� effect and resistance from the equivalent interior bay.


beams will have the equivalent stiffness and strength for the corresponding storiesof the interior gravity frames. The beam springs used for the gravity frame have thehysteresis behavior described in Fig. 17. The contribution of the equivalent gravitybay comes from both the flexural resistance of the columns and well as those fromthe beam springs used. However, since the strengths of the beam springs are verysmall compared to the moment frame springs, most of them will yield at a very earlystage of the excitation. Modeling parameters for these gravity frames connectionsare given in Yun and Foutch [11].

6. Static analysis

Static pushover analyses using both 9-story and 20-story building models wereperformed using the lateral force distribution calculated from the 1997 NEHRP Pro-visions. The approximate period from the provisions was used to obtain the totalbase shear for each building. The lateral force distribution coefficient is defined in1997 NEHRP Provision as

Cvx �Wx·hx

k

�wi·hik

(7)

Models were pushed to 5% global drift ratio with roof displacement controlled.For the models with fracturing connections, the structures were unable to sustain 5%global drift ratio so the analysis had to be stopped earlier. The static pushover plotsfor both of the buildings are shown in Figs. 21 and 22. The model with the equivalentgravity bay and slab effect shows the highest capacity and that with centerline modelexhibiting the lowest capacity for both 9-story and 20-story buildings. The strengthdifference between the two is 18% for the 9-story building and 25% for the 20-story

Fig. 21. Result from static pushover analysis for the 9-story building models.


Fig. 22. Result from static pushover analysis for the 20-story building models.

building which is significant. Little difference in strength is observed among themodels with fracturing beam connections. This is due to the fact that the resistancelost when reaching the plastic moment of the beam connection is large. So whenone joint fractures, the other joints have to share the additional force and this causesthe other joints to fracture almost simultaneously. The difference in behavior betweenthose with connections that fracture and those that do not is substantial. This is astrong indication that models used to evaluate existing buildings with pre-Northridgeconnections must include the effects of fracture if the results are to be meaningful.This can be done more simply than indicated here [4]. The percent of lateral resist-ance from the equivalent gravity bay is very small compared to the main lateralresisting frame. The higher effect of P�� is very noticeable for the 20-story framefor which the load–deformation response becomes negative at a much smaller drift.

It should be pointed out that the response in the negative tangential stiffness regionis not realistic. This part of the curve exists because a displacement controlled staticpushover method was used. Therefore if collapse is defined as the drift angle at zerotangential slope, the drift capacity of the 9-story and 20-story models would be about3.5 and 1.7%, respectively. As will be verified later in the dynamic analysis section,the capacity obtained from static pushover is very conservative. The reduction inlateral force due to elongation of fundamental period and cyclic nature of the groundmotions will help the structure to sustain larger drifts in dynamic response.

7. Dynamic analysis

Nonlinear dynamic analyses using both suites of scaled ground motions describedearlier have been used for drift demand as well as drift capacity calculations.


7.1. Dynamic drift demands

The median of the maximum inter-story drifts from each scaled ground motionwas calculated for all of the models described. All of the maximum drift values aswell as the median responses for the 9-story and 20-story frames are listed in Tables6 and 7, respectively. Results for the typical ground motions are listed on the leftwhereas those for the near fault ground motions are listed on the right side of thetable. The maximum drift demands for each earthquake are plotted in Figs. 23 and 24.

The median drifts for the 9-story models without fracture are all about 0.032 forthe ordinary motions and 0.043 for the near-fault motions. The drift demands forthe M2 model with composite action in the gravity frames were about 10% smallerthan the other three models in both cases, but this is not significant. The mediandrift demands for the frames with connection fracture were significantly larger thanfor frames without. The median values were about 0.056 for the frames with fracturedconnections for both sets of ground motions. The frames with fracturing connectionscollapsed during two of the ordinary and three of the near fault motions.

The median drift demands for the 20-story building models without fracture wereslightly smaller than for the 9-story buildings for both sets of ground motions. How-ever, the accuracy is overstated in these figures so one could say that the driftdemands averaged 0.03 for all 9- and 20-story buildings without fracture. There wasa very significant difference between the 20-story models with fracturing connec-tions. Model M2-WO-fr model collapsed during eight ordinary ground motions andsix near fault motions, but M2-SC-fr did not collapse for any of the accelerograms.This implies that the gravity frames should be included in models if pre-Northridgeconnections are used in the buildings.

Figs. 25 and 26 show maximum inter-story drifts over the height of the buildingfor the M1-WO and the M2-Comp model excited by near fault ground motions.Although median drifts are similar, the M2-Comp model had a smaller standarddeviation over the height of the frame. In addition, the M2-Comp 20-story modelsustained significantly smaller maximum drift demands for the one or two extremeground motions.

The effect of incorporating the contribution from the equivalent gravity bay ismore noticeable for the 20-story frame models due to larger P�� effects. The modelwith fracturing beam connections but without the equivalent gravity bay collapsedfor many ground motions of both suits as mentioned above. However, only oneground motion induced collapse for the model with fracturing beam connections andthe equivalent gravity bay.

7.2. Dynamic drift capacities

7.2.1. Incremental Dynamic Analysis (IDA)The Incremental Dynamic Analysis (IDA) procedure was used to determine the

capacity of the frames. Median responses were calculated using a similar procedureas for the statistical drift demand calculations. It is important to note that the analyti-cal model used for determining the global drift demand reproduces the major features

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Table 6Drift demands for the 9-story building for ordinary ground motions (left) and near fault ground motions (right)

M1-WO M2-WO M2-SC M2-Comp M2-WO- M2-SC-fr M1-WO M2-WO M2-SC M2-Comp M2-WO-fr M2-SC-frfr

Eq. 1 0.0507 0.0430 0.0430 0.0379 0.1037 collapse If21 0.0498 0.0542 0.0537 0.0524 0.0539 0.0570Eq. 2 0.0292 0.0350 0.0389 0.0318 0.0436 0.0407 If23 0.0277 0.0284 0.0286 0.0286 0.0379 0.0365Eq. 3 0.0443 0.0445 0.0514 0.0509 0.0539 0.0629 If25 0.0359 0.0361 0.0358 0.0325 0.0400 0.0383Eq. 4 0.0268 0.0264 0.0266 0.0255 0.0286 0.0300 If27 0.0390 0.0411 0.0393 0.0376 0.0377 0.0443Eq. 5 0.0240 0.0221 0.0267 0.0260 0.0466 0.0469 If29 0.0289 0.0300 0.0301 0.0245 0.0268 0.0347Eq. 6 0.0279 0.0287 0.0330 0.0293 0.0719 0.0556 If31 0.0331 0.0346 0.0346 0.0252 0.0540 0.0630Eq. 7 0.0303 0.0306 0.0290 0.0281 0.0878 collapse If33 0.0440 0.0450 0.0446 0.0448 0.0480 0.0658Eq. 8 0.0402 0.0461 0.0431 0.0308 0.0605 0.0604 If35 0.0685 0.0596 0.0553 0.0529 collapse 0.3258Eq. 9 0.0191 0.0207 0.0234 0.0217 0.0327 0.0345 If37 0.0759 0.0689 0.0672 0.0615 collapse collapseEq. 10 0.0233 0.0230 0.0244 0.0258 0.0299 0.0386 If39 0.0345 0.0356 0.0359 0.0361 0.0406 0.0442δ 0.3135 0.2946 0.2773 0.2657 0.4322 0.2935 δ 0.3644 0.3036 0.2885 0.3147 0.2852 0.3544Xm 0.0301 0.0307 0.0327 0.0297 0.0509 0.0469 Xm 0.0409 0.0414 0.0408 0.0377 0.0406 0.0442

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Table 7Drift demands for the 20-story building for ordinary ground motions (left) and near fault ground motions (right)


Eq. 1 0.0379 0.0258 0.0273 0.0261 collapse 0.0383 If21 0.0282 0.0257 0.0259 0.0257 collapse 0.0483Eq. 2 0.0341 0.0277 0.0274 0.0262 collapse 0.0312 If23 0.0241 0.0239 0.0217 0.0215 0.0321 0.0289Eq. 3 0.0372 0.0401 0.0412 0.0421 collapse 0.0352 If25 0.0278 0.0276 0.0283 0.0266 0.0341 0.0352Eq. 4 0.0188 0.0167 0.0168 0.0167 3.0468 0.0245 If27 0.0211 0.0185 0.0171 0.0169 collapse 0.0188Eq. 5 0.0218 0.0255 0.0292 0.0280 0.0260 0.0287 If29 0.0211 0.0164 0.0188 0.0198 collapse 0.0224Eq. 6 0.0175 0.0165 0.0165 0.0161 2.2844 0.0262 If31 0.0209 0.0193 0.0189 0.0193 0.0239 0.0236Eq. 7 0.0752 0.0577 0.0433 0.0369 collapse 0.0467 If33 0.0357 0.0300 0.0269 0.0262 0.0427 0.0367Eq. 8 0.0311 0.0368 0.0428 0.0454 collapse 0.0322 If35 0.0865 0.0859 0.0724 0.0711 21.5200 collapseEq. 9 0.0211 0.0194 0.0213 0.0198 0.0248 0.0265 If37 0.0395 0.0384 0.0358 0.0353 collapse 0.0393Eq. 10 0.0160 0.0180 0.0156 0.0156 collapse 0.0181 If39 0.0223 0.0199 0.0180 0.0177 collapse 0.0243δ 0.5276 0.4395 0.3766 0.3853 collapse 0.2557 δ 0.5624 0.6099 0.5408 0.5346 collapse 0.3074Xm 0.0270 0.0258 0.0262 0.0253 collapse 0.0298 Xm 0.0279 0.0254 0.0245 0.0243 collapse 0.0289


Fig. 23. Drift demands for the 9-story building for ordinary ground motions (left) and near fault groundmotions (right).

Fig. 24. Drift demands for the 20-story building for ordinary ground motions (left) and near fault groundmotions (right).

Fig. 25. Drift demands for the 9-story M1-WO (left) and M2-Comp (right) excited by near fault groundmotions (right).


Fig. 26. Drift demands for the of 20-story M1-WO (left) and M2-Comp (right) excited by near faultground motions (right).

of the measured response such as sudden loss of strength or pinching. This means thatthe measured hysteresis behavior must be modeled as closely as possible. Modelingrequirements are given in an earlier part of this paper. It should be noted that basedon the SAC research the connection that reaches a plastic rotation of 0.03 withoutsignificant loss of strength and 0.05 without complete loss of strength should havea median global drift capacity of 0.09 or greater for both 9- and 20-story modelsfor L.A.-type ground motions. This can be thought of as the lower bound behaviorof a connection that satisfies the AISC test protocol. Including the gravity columnsin the model helps to stabilize the building at large drifts. If the computer programis capable of handling complex moment–rotation behavior, the moment developedin gravity frames through the columns composite beam action can be included.

The global stability limit is determined using the Incremental Dynamic Analysis(IDA) technique developed by Cornell and his associates [12, 15]. The procedurethat was used to perform this analysis is as follows:

1. Choose a suite of ten to twenty accelerograms representative of the site and hazardlevel. The SAC project developed typical accelerograms for Los Angeles, Seattleand Boston sites [2]. These might be appropriate for similar sites.

2. Perform an elastic time history analysis of the building for one of the accelerog-rams. Plot the point on a graph whose vertical axis is the spectral ordinate forthe accelerogram at the first period of the building and the horizontal axis is themaximum calculated drift at any story. Draw a straight line from the origin ofthe axis to this point. The slope of this line is referred to as the elastic slope forthe accelerogram. Calculate the elastic slope for the rest of the accelerogramsusing the same procedure and then calculate the median slope. The slope of thismedian line is referred to as the elastic slope, Se (see Fig. 27).

3. Perform a nonlinear time history analysis of the building subjected to one of theaccelerograms. Plot this point of maximum drift on the graph. Call this point �1.

4. Increase the amplitude of the accelerogram and repeat step 3. This may be doneby multiplying the accelerogram by a constant that increases the spectral ordinatesof the accelerogram by 0.1g. Plot this point as �2. Draw a straight line between


points �1 and �2. If the slope of this line is less than 0.2 Se then �1 is the globaldrift limit. This can be thought of as the point at which the inelastic drifts areincreasing at 5 times the rate of elastic drifts.

5. Repeat step 4 until the straight-line slope between consecutive points �i and �i+1,is less than 0.2 Se. When this condition is reached, �i is the global drift capacityfor this accelerogram.

6. Choose another accelerogram and repeat steps 3 through 5. Do this for each acce-lerogram. The median capacity for global collapse is the median value of thecalculated set of drift limits. An example for one accelerogram for an L.A. sitefor a 20-story weak-column OMF building is shown in Fig. 27. The open circlesrepresent the IDA calculations for an accelerogram where the 0.2 Se slope determ-ined the capacity. The point �7 would be considered to be the drift capacity.

For the SAC project, the upper bound on the drift capacity was assumed to be0.10. It was believed that the analytical results for drift greater than 0.10 wouldnot be reliable [14]. The issue of the safety of the occupants was paramount inthis design.

7.2.2. Calculated capacityThe IDAs were performed according to the procedures described in the previous

section. The median drifts as well as the spectral acceleration capacities were calcu-lated for each suit of ground motions described earlier. A strain-hardening ratio of0.03 was used for all of the analyses in this study. The increment of ground motionintensity used was 0.2g for the cases without fracturing connections and 0.1g for themodels with fracturing connections since those models are expected to collapse atearlier intensities of ground motions. However, the 20-story model with fracturingconnections without the equivalent gravity bay collapsed even at 0.1g for two ordi-nary ground motions and for one near fault ground motion. The lateral force that 9-story and 20-story building was designed for is 0.066g and 0.042g, respectively. A

Fig. 27. Definition of collapse from IDA analyses for two ground motions.


smaller increment of 0.02g was used for some IDA analyses. It was found that this20-story model reaches incipient collapse at 0.08g that is only a little greater thanthe design level. The ground motion increment must be small enough so that driftincrement is relatively small for each step. The values given above should be con-sidered as upper bounds. The use of a larger increment would usually result in asmaller drift capacity and larger variation of the capacity. Therefore, it would giveconservative results.

The individual drift capacities for each 9-story model along with the median valuesare given in Table 8. The capacities for the 20-story buildings are given in Table 9.Plots of the drift capacities are shown in Figs. 28 and 29.

For the 9-story models without fracture, the median capacities for the ordinaryCalifornia ground motions were all about the same. For the near-fault motions, themodel with the gravity frames had greater capacities. The drift capacities for the 9-story models with fracturing connections had much smaller capacities than the mod-els without fracture. For instance the capacities for M2-SC and M2-SC-fr were 0.13and 0.08, respectively, for the standard motions and 0.18 and 0.06, respectively, forthe near fault motions. There was not a significant difference between the two modelswith fractures.

The differences in capacity among the various models were much greater for the20-story models when subjected to the ordinary accelerograms. Models M1-WO andM2-WO had drift capacities of 0.07 and 0.05, respectively, while models M2-SCand M2-Comp had capacities of 0.09 and 0.10, respectively. For the near-faultmotions, the four models without fracture had comparable capacities of about 0.10.Again, the capacities for the model with fracture were significantly smaller than forthose without fracture. The capacities for M2-WO-fr and M2-SC-fr were 0.02 and0.05, respectively, for the ordinary motions and 0.03 and 0.05 for the near-faultmotions. In addition, the capacities for the fracturing models without the gravity baywere significantly smaller than those with the gravity bay.

It should be pointed out once again that this is a numerical exercise where relativecapacities are compared. One should not expect that a real building would be ableto resist drift levels of 0.20 without collapse. This is why the SAC project placedan upper limit of drift capacity of 0.10.

The SAC project computed uncertainties and confidence levels in terms of storydrifts. This was chosen because it is a quantity that is calculated by the designer asa regular part of the design process. Another quantity that could be used is thecollapse limit based on spectral amplitude. These are also a natural result of the IDAanalyses. The median strengths for the 0- and 20-story buildings are given in Tables10 and 11 respectively. The spectral acceleration capacities for each ground motionsare shown in Figs. 30 and 31.

The spectral acceleration capacities vary greatly for the ordinary ground motionsbut very little for the near-fault motions. Spectral acceleration capacities were sug-gested by some researchers because it was believed that there would be less scatter.This is clearly not so. The same conclusions that were drawn from the drift capacitieswould also be derived from the spectral acceleration capacities.

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Table 8Drift capacities for the 9-story building for ordinary ground motions (left) and near fault ground motions (right)

M1-WO M2-WO M2-SC M2-Comp M2-WO-fr M2-SC-fr M1-WO M2-WO M2-SC M2- M2-WO-fr M2-SC-frComp

Eq. 1 0.0590 0.0555 0.0542 0.1049 0.0351 0.0655 If21 0.1531 0.1908 0.2315 0.2086 0.0545 0.0722Eq. 2 0.1043 0.1009 0.1561 0.1235 0.0664 0.0803 If23 0.1936 0.2180 0.2341 0.1941 0.0625 0.0426Eq. 3 0.1410 0.1630 0.1332 0.0858 0.0638 0.0836 If25 0.0884 0.1477 0.1329 0.1271 0.0547 0.0490Eq. 4 0.2240 0.2051 0.2313 0.2583 0.0680 0.0744 If27 0.1662 0.1701 0.1558 0.1935 0.0621 0.0866Eq. 5 0.1236 0.1081 0.1156 0.1334 0.0736 0.0838 If29 0.1161 0.1406 0.1957 0.1666 0.0697 0.0310Eq. 6 0.0516 0.0638 0.0843 0.0531 0.0665 0.0953 If31 0.1192 0.1498 0.2317 0.1857 0.0873 0.1069Eq. 7 0.0526 0.0775 0.0842 0.0759 0.0444 0.0385 If33 0.1431 0.1685 0.1191 0.1506 0.0700 0.0777Eq. 8 0.1312 0.1289 0.1466 0.1314 0.0637 0.1082 If35 0.1525 0.1152 0.1549 0.1368 0.0788 0.0709Eq. 9 0.1870 0.1983 0.2052 0.1779 0.0994 0.1208 If37 0.2111 0.2159 0.1405 0.2492 0.0387 0.0389Eq. 10 0.1324 0.1650 0.1590 0.1638 0.0561 0.0731 If39 0.2014 0.1936 0.2569 0.2262 0.0609 0.0593δ 0.4486 0.4108 0.3876 0.4306 0.2651 0.2718 δ 0.2538 0.1946 0.2670 0.2100 0.2105 0.3617Xm 0.1091 0.1164 0.1271 0.1192 0.0615 0.0794 Xm 0.1496 0.1678 0.1788 0.1798 0.0625 0.0595

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Table 9Drift capacities for the 20-story building for ordinary ground motions (left) and near fault ground motions (right)




Fig. 28. Drift capacities for the 9-story building for ordinary ground motions (left) and near fault groundmotions (right).

Fig. 29. Drift capacities for the 20-story building for ordinary ground motions (left) and near fault groundmotions (right).

8. Summary and conclusions

Six models of a 9-story building and six of a 20-story building were developedfor this study. The two buildings were designed to comply with the 1997 NEHRPprovisions. Four of the models for each building had ductile connections modeledafter test results for a reduced beam section connection. The other two models foreach building had connections that could fracture in a similar fashion to a fullywelded pre-Northridge connection. The effects of gravity frames were included inthree models for each building. Each model was subjected to a suite of 10 standardCalifornia accelerograms and one of 10 near fault motions.

Results of the study suggest the following conclusions:

1. A bare-frame model using centerline dimensions is more flexible and weaker thanall other models. This model is conservative for use in design of new buildings butis not recommended for performance evaluation of existing or damaged buildings.

2. Models which include the interior gravity frames had smaller drift demands andgreater drift capacities than those without. The effects of including composite

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Table 10Spectral acceleration capacities for the 9-story building for ordinary ground motions (left) and near fault ground motions (right)



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Table 11Spectral acceleration capacities for the 20-story building for ordinary ground motions (left) and near fault ground motions (right)




Fig. 30. Spectral acceleration capacities for the 9-story building for ordinary ground motions (left) andnear fault ground motions (right).

Fig. 31. Spectral acceleration capacities for the 20-story building for ordinary ground motions (left) andnear fault ground motions (right).

action in the gravity frame model were not significant. Therefore, it is rec-ommended that the columns in the gravity frames be included for evaluation andperformance prediction of steel moment frame buildings.

3. The frame models that had fracturing connections had significantly larger driftdemands and smaller drift capacities than those with ductile connections. This istrue for both static and dynamic analyses.

4. The effects of the gravity frames had significantly greater effect for frames withpre-Northridge connections. In many cases they made the difference between col-lapse or survival. It is highly recommended that the gravity frames be includedin models used for evaluation of existing or damaged pre-Northridge buildings.

5. Several of the models with pre-Northridge connections collapsed for some acceler-ograms that represented the 2% in 50 year hazard level. So, it should not besurprising if some existing steel moment frame buildings collapse if subjected tocomparable ground motions.


Acknowledgements

This research was funded by the Federal Emergency Management Agency througha grant to the SAC Joint Venture. The SAC Joint Venture is composed of the Struc-tural Engineers Association of California (SEAOC), the Applied Technology Council(ATC) and the California Universities for Research in Earthquake Engineering(CUREE). This support is gratefully acknowledged. Any results, findings and con-clusions are solely those of the authors and do not necessarily represent those ofthe sponsors.

References

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[13] Goel SC, Stojadinovic B. Test results of Pre-Northridge, simulated filed welding, weld fracture,small plastic rotation. SAC test results. Ann Arbor, MI: University of Michigan, 1999.

[14] Barsom JM. Failure analysis of welded beam to column connections. SAC Report No. SAC/BD-99/23, 1999.

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modeling of steel moment frames for seismic loads -...

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