Modeling of steel moment frames for seismic loads

Transcription

1 Journal of Constructional Steel Research 58 (2002) Modeling of steel moment frames for seismic loads Douglas A. Foutch a,, Seung-Yul Yun b 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 widely used for the design of steel moment resisting frames. However, for the performance prediction and evaluation of these structures, different nonlinear models are being used to better simulate their true behavior. Simple nonlinear modeling methods widely used as well as those with more detailed modeling representations are investigated and compared. A 9-story building and a 20-story building were designed for this study according to the 1997 NEHRP provisions. Different models for these structures were developed and analyzed statically and dynamically. The models investigated involved the use of centerline dimensions of elements or clear length dimensions, nonlinear springs for the beam connections, and linear or nonlinear springs for the panel zones. A second group of models also incorporated the fracturing 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 near fault ground motions. The differences in structural responses among different models for both suites of motions are investigated. According to static pushover analyses with roof displacement controlled, the benefit of the increase in capacity that results from the detailed models is consistently observed for both the 9-story and 20-story buildings. When the models were excited by different ground motions from each suite, the median responses of the more detailed models showed an increase in capacity and a decrease in demand as expected. However, due to the randomness inherent in the ground motions, variations were also observed. Overall, the model which incorporates clear length dimensions between beams and columns, panel zones and an equivalent gravity Corresponding author. address: d-foutch@uiuc.edu (D.A. Foutch) X/02/$ - see front matter 2002 Published by Elsevier Science Ltd. PII: S X(01)

2 530 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) bay without composite action from the slab seems to be a practical model with appropriate accuracy Published by Elsevier Science Ltd. Keywords: Seismic analysis; Steel moment frames; Steel buildings; Earthquake response; Non-linear analysis; Elastic analysis 1. Introduction The engineer s ability to model buildings has increased quickly over the past several years with the development of advanced analysis programs and the competition among software developers. In fact, our ability to model structural behavior probably exceeds our ability to fully understand the observed behavior. The first structural analysis programs that were developed in the early 1960s could handle only linear prismatic beam and column members with fully restrained or pinned joints and centerline dimensions. Programs in use today have a number of elements that model material and geometric nonlinearities, rigid or partially restrained connections, and flexible foundations and diaphragms. This paper will cover commonly used modeling procedures for steel moment frames. A word of caution is required. Although the modeling procedures described herein are quite detailed and match measured behavior very well, it must be remembered that this is still greatly simplified from the case of a real building which has cladding, partitions, mechanical equipment, stairways and many other discounted attributes. A real building might have irregularities and flexible foundations that are important but not included here. It must be remembered that the calculations that follow are only estimates of actual behavior. A 9-story and a 20-story building were designed in accordance with the 1997 NEHRP provision for this study. Different models for those structures were developed and analyzed statically as well as dynamically. Two suites of ground motions were used for the dynamic analyses: typical California and near fault ground motions. The comparisons of computed structural responses for the different models are investigated. 2. Design of 9-story and 20-story buildings according to the 1997 NEHRP provisions The plan and elevation views of the buildings are given in Fig. 1. The buildings were designed for a site in downtown Los Angeles where S S is 1.61g and S 1 is 1.15g. The perimeters of the buildings were designed as special moment frames so the response reduction factor of R=8 was used. According to the 1997 NEHRP provisions, the base shears corresponding to the 9-story and 20-story building were 300 and 244 kips, respectively. The approximate period equation prescribed in the provision was used to check for strength as well as drift requirements. Drift requirements governed the design for both of the buildings. The section members assigned for

3 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) 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 corner columns of the 20-story frame since they needed to resist bi-axial bending from lateral loadings. Doubler plates were inserted at the middle story panel zones of the interior 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 panel zones. Detailed descriptions of the nonlinear springs used for the beams and panel zones will follow in a later section. Six different models of the buildings were investigated. The first model used centerline dimensions with nonlinear springs for yielding of beams as well as a leaning column 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 models used clear length dimensions with nonlinear springs to model the panel zones as well the beams. The first model of the three is similar to M1-WO but used the clear

4 532 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Table 1 Sections assigned for the 9-story and 20-story buildings Story w14 Story w24 Columns Doubler plate Beam Columns Doubler plate Beam Exterior Interior Exterior Interior Exterior Interior Exterior Interior 9 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w14 22 length of beams and columns with the flexibility of the panel zones modeled into the joint. This is denoted as M2-WO. The second of the three includes one bay of the frame model that represents all of the interior gravity columns but with simple connection properties assumed for the beam springs and is denoted as M2-SC. The last of the three is identical to the second model, but with resistance from the composite 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 are identical to the M2-WO and M2-SC but fracturing behavior of the beam connections is incorporated into the models. For those connections, when the plastic moment is reached, the strength of the beam connection drops down to 10% of the plastic moment capacity. The periods of each model for the 9-story and 20-story moment frames are listed in Table 2. The model with the equivalent gravity bay frame with rotational resistance from the slab is the stiffest since the contribution from the continuity of the interior columns and rotational strength of the beam connection is included. It is interesting to note that M2-WO is stiffer than M1-WO. This is due to the fact that the M2 model uses clear lengths of the beams and columns that make this structure stiffer even though a flexible panel zone is also included. When clear length models and centerline models are pushed statically using displacement control, the demands for elements for the clear length model will be larger. The natural

5 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Table 2 1st and 2nd mode of each model M1 M2 WO WO SC Comp WO-frac SC-frac Comp-fr 9-story T 1, (s) T 2, (s) story T 1, (s) T 2, (s) periods for these models are remarkably similar and will respond about the same for response to dynamic motions if linear elastic behavior is assumed. First, drift demands using static pushover analyses and then dynamic analyses using both suites of ground motions were investigated. Finally, dynamic drift capacities 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 of accelerograms represents the typical ground motions for the LA site. The second suite represents near fault ground motions. Each of the typical ground motions in the first suite was scaled in a least square manner to match the 2% in 50-year hazard spectra of the site at periods of 1.0, 2.0 s and 4.0 s. The descriptions of the ground motions with their scaling factors are given in Table 3. The scale factors range from 1.72 to A different scaling method was used for the second suite of ground motions since those ground motions were generated specifically to represent the 2% Table 3 Description of typical ground motions Name Ground motion name Scale factor used EQ01 Taft (1952) 1.72 EQ02 Castica (1971) 1.87 EQ03 Imperial Valley (1979) 1.83 EQ04 Pacoima Dame (1971) 1.83 EQ05 Northridge (1994) 1.85 EQ06 El Centro (1940) 1.85 EQ07 San Fernando (1971) 1.39 EQ08 Mammoth Lakes (1980) 1.82 EQ09 Morgan Hill (1984) 1.87 EQ10 North Palm Spring (1986) 1.81

6 534 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Table 4 Description of near fault ground motions Name Ground motion name Scale factor used LF21 Kobe (1995) 0.65 LF23 Loma Prieta (1989) 0.65 LF25 Northridge (1994) 0.65 LF27 Northridge (1994) 0.65 LF29 Tabas (1974) 0.65 LF31 Elysian Park (Simulated) 0.65 LF33 Elysian Park (Simulated) 0.65 LF35 Elysian Park (Simulated) 0.65 LF37 Palos Verdes (Simulated) 0.65 LF39 Palos Verdes (Simulated) 0.65 in 50-year hazard level. They are the normal component of the LA 2% in 50-year hazard level ground motions developed by Somerville et al. [2] for the SAC Phase II project. The ground motions were scaled to minimize the error for the median response of the ground motions. The scaling factor for this suite of ground motions came out to be The descriptions of the ground motions with their scaling factors are given in Table 4. The scaled response spectra of the both suites of ground motions are shown in Figs. 2 and 3. It was interesting to notice that the spectral accelerations in the short period range (less than 1.0 s) for the typical ground motions were high compared to those for the near fault motions whereas some of the near fault motions possess bumps in the period region of s. Therefore, the effects of higher modes for the typical motions and the pulse motion for the near fault motions should be examined for the calculated responses. Because of the large spectral accelerations Fig. 2. Scaled response spectra for EQ01 EQ10 and their median spectra.

7 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) 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 would affect 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 two acceptance criteria must be checked: member strength and building stiffness (drift). For new steel moment frame buildings the drift limitation always governs in high seismic regions. Research done by Krawinkler [3] has shown that a linear elastic model using centerline dimensions is acceptable for design of special moment frames. The beam moments may be checked at the location in the beam where it intersects the column flange. Even though this model gives adequate results for design, it will not always give good estimates of the distribution of shears, moments and axial forces throughout the building under dynamic loads. The panel zones must be modeled explicitly for frames with weak panel zones 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 the beam flange lines into the column as shown in Fig. 4. The simplest way to model the panel zone for linear analysis is referred to as the scissors model also shown in Fig. 5. The beams and column are modeled with a rigid link through the panel zone region and a hinge in the beam is placed at the intersection of the beam and column

8 536 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Fig. 4. Definition of panel zone. Fig. 5. Scissors model for panel zone modeling. centerlines. A rotational spring with stiffness k θ is then used to tie the beam and column together. The rigid links stiffen the structure but the panel zone spring adds flexibility. The net result is that this building model is usually stiffer than the centerline model. Since it is stiffer it will help in satisfying the drift design criteria. It will also give better estimates of shears, moments and axial forces in the members. Most finite element programs currently used by engineers for seismic analysis have this feature. The equations for determining the stiffness of the panel zone spring are based on the yield properties of the panel zone. The yielding property of the panel zone is

9 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) g y F y 3G q y (1) M y V y d b 0.55F y d c t d b (2) where, F y =the yield strength of the panel zone G=the shear modulus=e/2 (1 n) d c =depth of column t=thickness of panel zone which is the thickness of the web of the column plus the thickness of the doubler plates if they are utilized. d b =depth of beam ν=poisson s ratio=0.3 So, the stiffness of the panel becomes K q M y q y (3) 4.3. Nonlinear centerline models Models that allow yielding in the beams and columns are much more realistic than linear models. Although nonlinear models are not required for design of new buildings, they are very useful for evaluating existing and damaged buildings [4]. Most commonly used programs model this behavior by including a nonlinear flexural spring at the ends of elastic beam and column members. The springs should be assigned 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 structure stiffness is maintained because it comes from the actual members rather than from the spring. This model is shown schematically in Fig. 6 and is referred to in this paper as M1-WO. The spring is rigid until the plastic moment of the member is reached. After yielding a post-yield stiffness is assigned to the spring that represents the strain hardening behavior of the member. A strain hardening coefficient, α, is assigned to the spring after yielding. A value of α equal to 0.03 is a reasonable choice. The spring behavior and member plus spring behavior are shown in Fig. 6. The value of α equal to 0.03 is 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 zero and then it can become negative with larger drifts. Most programs will not allow a negative value of α. For calculating building behavior beyond 4%, it is best to choose a strain-hardening factor of zero. For performance evaluation, the expected values of the yield strengths of the steels should be used. Expected yield strengths of commonly used steels are given in Table 5 [5,6].

10 538 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Fig. 6. Centerline model with nonlinear elements. Table 5 Expected and lower bound material properties for structural steel of various grades [5,6] 12 Yield strength Tensile strength Material Year of Lower bound Expected Lower bound Expected specification construction ASTM, A Group Group Group Group Group ASTM, A Group Group Group Group Group A36 and dual grade 50 Group Group Group Group Lower bound values for material are mean 2 standard deviation values from statistical data. Expected values for material are mean values from statistical data. 2 For wide flange shapes, indicated values are representative of material extracted from the web of the section. For flange, reduce indicated values by 5%.

11 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Nonlinear models with panel zones Most of the pioneering work on nonlinear panel zone modeling has been performed by Krawinkler. His state-of-the-art report [3] provides a good discussion of this topic and includes references to his earlier work [7,8]. Two methods of modeling the nonlinear behavior of frames with yielding beams, columns and panel zones are available. One procedure is based on the scissors model shown in Fig. 5. The panel zone springs as well as the springs at the ends of the members are nonlinear. The behavior of the member spring is exactly the same as described in the previous section. The panel zone spring is assigned a stiffness of K q M y (4) q y where M y V y d b 0.55F y d c t d b (5) q y g y F y 3G (6) In most cases, panel zones have a steeper post yield stiffness. Therefore, a value of 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 the panel zone with rigid links and controls the deformation of the panel zone using two bilinear springs that simulate a tri-linear behavior. With this, the large strength difference between the real behavior and the model is reduced. The first slope post yield is steep and represents the behavior between the time that yielding is initiated and the full plastic capacity is reached. After the plastic Fig. 7. Panel zone modeling.

12 540 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) capacity is reached a small slope (2%) or zero slope may be used. This is shown in Fig. 8. Since yielding in the beams, columns and panel zones is represented well by this model, the actual distribution of yielding throughout the structure will be represented well. 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 yield shortly thereafter. This model is referred to as M2-WO. Fig. 1 shows the 20-story building that was designed according to the 1997 NEHRP Provisions. This building will be used for comparing the different models described here. The results of pushover analyses for the buildings with the specified distribution of lateral forces required for new design in the 1997 NEHRP provisions are shown in Fig. 9. M1 in the figure is the modeling case with the centerline dimensions, 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 with tri-linear model spring and the full dimension of the member for the analysis. As can be seen, the M2 model is initially a little stiffer than the M1 model. The M1 model with P gives the lowest strength. Care should be taken when plotting the roof drift ratio versus the total base shear. The roof drift ratio can be misleading because it is incapable of capturing the local drift concentration. A good example of this case can be seen for this building that is pushed to about 4% of global drift with P effects. The concentration of plastic deformations around the 3rd level was the controlling factor. Fig. 10 shows the plot of global roof drift ratio, top story drift ratio, and 3rd level story drift ratio versus total base shear. Global roof drift ratio 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 global drift ratio shows the averaged drift ratio over the whole height. This pushover plot reaches a peak at a drift of about 0.02 and rapidly has an increasing negative slope Fig. 8. Panel zone load deformation behavior.

13 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) 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 plastic deformation is very noticeable as shown in Fig. 10. Note, however, that this does not reach a peak until almost 0.04 drift and then slowly becomes negative. The top story drift actually remains elastic throughout the entire loading sequence. A plot of displaced 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 in drift ratio. The concentration of plastic deformation can clearly be seen in Fig. 12 where 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 roof drift for a static pushover analysis is highly questionable.

14 542 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) 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 Reduced beam section connection For new buildings, reduced beam sections that are also referred to as dog-bone members were used for the analysis. They exhibit very good hysteretic behavior with stable loops and good energy dissipation. Tests were performed by Venti and

15 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Engelhardt [9]. A typical case of the hysteretic behavior is shown in Fig. 13. This test used a w column member and w beam section. Both members have 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 used in 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 beam plastic moments to the first yielding point as well as the second moment value were calculated and used for determining the yielding properties of the other member sizes. Seventy-four percent of the plastic moment of the beam was used as the first yield moment for both positive and negative moments. For the second yield moment value, 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 for the second yielding moment of 0.03 radians for the positive side and radians for the negative side were used for the protocol model. The rotational values that are proportional to the plastic section modulus were assigned for the other beam sections. The strength degradation ratio that is the drop of the strength at each new plastic excursion was assigned a value of This value was fixed for all member sizes although in reality, there would be variations from member to member. The drift demand is not significantly affected by the choice of this ratio. Differences in drift demand calculations would not vary by more than 2 or 3% because of this difference. 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. The plastic moments for the members are 33,750 (k-in) for w and 23,904 (k-in) for the w member Fracturing beam connection Fracturing beam connections were incorporated into the model to simulate the behavior observed for the pre-northridge buildings. The measured and the modeled hysteresis 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.

16 544 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Fig. 14. Illustration of yielding values for w (protocol) and w (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 drop in the connection after fracture. In the positive rotation case, the strength was modeled to drop to 10% of the original strength of the connection when the plastic moment 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. For negative moment the crack in the bottom flange closes so typical bilinear-type of behavior 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 the connection. Therefore, the connection arrives at about zero strength at 0.04 radians to simulate the fracture of the connection. Significant increase in demand as well as decrease in capacity of structural response is expected.

17 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Fig. 16. Illustration of simple connection in gravity frames Simple connections in gravity frames The 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. The frame is sometimes modeled with pinned connections to capture the P-Delta effect due to additional gravity load from the interior frames. However, according to the experimental results from Liu and Astaneh-Asl [10] the resistance not only exists but sometimes is significant due to the additional resistance occurring when a compression force in the composite floor slab is connected by a tension force in the shear tab. Additional resistance is encountered when the flanges of the beam come in 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 the beam resists lateral load for many cycles of motion. This is a case with w18 35 beam connected to the w14 90 column with shear tab and concrete on top. Minimum reinforcement was used for the slabs. The moment rotation behavior of the connection 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.

18 546 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) The model of the connection is shown in Fig. 18. A portion of the beam stiffness was 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, the connections cannot be expected to develop the full plastic moment capacity. The maximum moment for the positive moment was taken as 38% of M p and that for the negative side as 11% because these values resulted in good matches between experiment and analysis. The fact that the positive side develops higher moment is attributed to the compressive resistance of concrete slab on top of the girder bearing against the column. The tensile strength of the slab cannot be expected to help much since minimum reinforcement is used. The rotation at which the strength drops is assigned a value of radians for the positive side and 0.05 radians for the negative 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 other sections 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 degradation factor. As will be seen later in this paper, the resistance from the gravity frame is significant. However, most of the contribution is not from the composite connection but from the flexural resistance from the continuous columns acting in conjunction with the rigid floor slabs. According to the report by Yun and Foutch [11], the differences in responses between the models with the simple connection are 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 the ground motions leaving only the columns to resist the lateral load. Fig. 18 shows an illustration of the yielding properties of a protocol connection and the connection from a typical floor of the 9-story building Other modeling attributes Another feature that should be included for analysis of tall buildings or shorter buildings 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).

19 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Fig. 19. Modeling interior columns for P effect only. laterally the gravity forces acting through the displacement causes additional overturning moments to develop in the structure. For a perimeter frame building this can be a very significant effect since the perimeter frames must carry the overturning moments of the entire building including the gravity frames. One way to do this is to provide a dummy column in the model that carries the gravity loads in the building not directly carried by the moment frame. The column is connected to the moment frame using rigid links with hinges at each end as shown in Fig. 19. The columns are hinged at both top and bottom. By doing this, only additional overturning moment from the lateral displacement will be induced. The columns will not help carry any of the lateral loads since they are pinned. However in reality, the interior columns do help the moment frames since the columns are not connected with a hinge and some resistance exists for the shear tab connection in the beams due to the slab on top. An additional bay that has the equivalent properties 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.

20 548 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) beams will have the equivalent stiffness and strength for the corresponding stories of the interior gravity frames. The beam springs used for the gravity frame have the hysteresis behavior described in Fig. 17. The contribution of the equivalent gravity bay comes from both the flexural resistance of the columns and well as those from the beam springs used. However, since the strengths of the beam springs are very small compared to the moment frame springs, most of them will yield at a very early stage of the excitation. Modeling parameters for these gravity frames connections are given in Yun and Foutch [11]. 6. Static analysis Static pushover analyses using both 9-story and 20-story building models were performed using the lateral force distribution calculated from the 1997 NEHRP Provisions. The approximate period from the provisions was used to obtain the total base shear for each building. The lateral force distribution coefficient is defined in 1997 NEHRP Provision as C vx W k x h x (7) k w i h i 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 plots for both of the buildings are shown in Figs. 21 and 22. The model with the equivalent gravity bay and slab effect shows the highest capacity and that with centerline model exhibiting the lowest capacity for both 9-story and 20-story buildings. The strength difference 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.

21 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Fig. 22. Result from static pushover analysis for the 20-story building models. building which is significant. Little difference in strength is observed among the models with fracturing beam connections. This is due to the fact that the resistance lost when reaching the plastic moment of the beam connection is large. So when one joint fractures, the other joints have to share the additional force and this causes the other joints to fracture almost simultaneously. The difference in behavior between those with connections that fracture and those that do not is substantial. This is a strong indication that models used to evaluate existing buildings with pre-northridge connections 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 resistance from the equivalent gravity bay is very small compared to the main lateral resisting frame. The higher effect of P is very noticeable for the 20-story frame for 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 region is not realistic. This part of the curve exists because a displacement controlled static pushover method was used. Therefore if collapse is defined as the drift angle at zero tangential slope, the drift capacity of the 9-story and 20-story models would be about 3.5 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 in lateral force due to elongation of fundamental period and cyclic nature of the ground motions will help the structure to sustain larger drifts in dynamic response. 7. Dynamic analysis Nonlinear dynamic analyses using both suites of scaled ground motions described earlier have been used for drift demand as well as drift capacity calculations.

22 550 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Dynamic drift demands The median of the maximum inter-story drifts from each scaled ground motion was calculated for all of the models described. All of the maximum drift values as well as the median responses for the 9-story and 20-story frames are listed in Tables 6 and 7, respectively. Results for the typical ground motions are listed on the left whereas those for the near fault ground motions are listed on the right side of the table. 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 for the ordinary motions and for the near-fault motions. The drift demands for the M2 model with composite action in the gravity frames were about 10% smaller than the other three models in both cases, but this is not significant. The median drift demands for the frames with connection fracture were significantly larger than for frames without. The median values were about for the frames with fractured connections for both sets of ground motions. The frames with fracturing connections collapsed during two of the ordinary and three of the near fault motions. The median drift demands for the 20-story building models without fracture were slightly smaller than for the 9-story buildings for both sets of ground motions. However, the accuracy is overstated in these figures so one could say that the drift demands averaged 0.03 for all 9- and 20-story buildings without fracture. There was a very significant difference between the 20-story models with fracturing connections. Model M2-WO-fr model collapsed during eight ordinary ground motions and six 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-northridge connections are used in the buildings. Figs. 25 and 26 show maximum inter-story drifts over the height of the building for 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 standard deviation over the height of the frame. In addition, the M2-Comp 20-story model sustained significantly smaller maximum drift demands for the one or two extreme ground motions. The effect of incorporating the contribution from the equivalent gravity bay is more noticeable for the 20-story frame models due to larger P effects. The model with fracturing beam connections but without the equivalent gravity bay collapsed for many ground motions of both suits as mentioned above. However, only one ground motion induced collapse for the model with fracturing beam connections and the equivalent gravity bay Dynamic drift capacities 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 procedure as for the statistical drift demand calculations. It is important to note that the analytical model used for determining the global drift demand reproduces the major features

23 Table 6 Drift 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-fr fr Eq collapse If Eq If Eq If Eq If Eq If Eq If Eq collapse If Eq If collapse Eq If collapse collapse Eq If δ δ Xm Xm D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002)

24 Table 7 Drift demands for the 20-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-fr fr Eq collapse If collapse Eq collapse If Eq collapse If Eq If collapse Eq If collapse Eq If Eq collapse If Eq collapse If collapse Eq If collapse Eq collapse If collapse δ collapse δ collapse Xm collapse Xm collapse D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002)

25 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Fig. 23. Drift demands for the 9-story building for ordinary ground motions (left) and near fault ground motions (right). Fig. 24. Drift demands for the 20-story building for ordinary ground motions (left) and near fault ground motions (right). Fig. 25. Drift demands for the 9-story M1-WO (left) and M2-Comp (right) excited by near fault ground motions (right).

26 554 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) Fig. 26. Drift demands for the of 20-story M1-WO (left) and M2-Comp (right) excited by near fault ground motions (right). of the measured response such as sudden loss of strength or pinching. This means that the measured hysteresis behavior must be modeled as closely as possible. Modeling requirements are given in an earlier part of this paper. It should be noted that based on the SAC research the connection that reaches a plastic rotation of 0.03 without significant loss of strength and 0.05 without complete loss of strength should have a median global drift capacity of 0.09 or greater for both 9- and 20-story models for L.A.-type ground motions. This can be thought of as the lower bound behavior of a connection that satisfies the AISC test protocol. Including the gravity columns in the model helps to stabilize the building at large drifts. If the computer program is capable of handling complex moment rotation behavior, the moment developed in 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 procedure that was used to perform this analysis is as follows: 1. Choose a suite of ten to twenty accelerograms representative of the site and hazard level. The SAC project developed typical accelerograms for Los Angeles, Seattle and Boston sites [2]. These might be appropriate for similar sites. 2. Perform an elastic time history analysis of the building for one of the accelerograms. Plot the point on a graph whose vertical axis is the spectral ordinate for the accelerogram at the first period of the building and the horizontal axis is the maximum calculated drift at any story. Draw a straight line from the origin of the axis to this point. The slope of this line is referred to as the elastic slope for the accelerogram. Calculate the elastic slope for the rest of the accelerograms using the same procedure and then calculate the median slope. The slope of this median line is referred to as the elastic slope, S e (see Fig. 27). 3. Perform a nonlinear time history analysis of the building subjected to one of the accelerograms. Plot this point of maximum drift on the graph. Call this point Increase the amplitude of the accelerogram and repeat step 3. This may be done by multiplying the accelerogram by a constant that increases the spectral ordinates of the accelerogram by 0.1g. Plot this point as 2. Draw a straight line between

27 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) points 1 and 2. If the slope of this line is less than 0.2 S e then 1 is the global drift limit. This can be thought of as the point at which the inelastic drifts are increasing 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 S e. When this condition is reached, i is the global drift capacity for this accelerogram. 6. Choose another accelerogram and repeat steps 3 through 5. Do this for each accelerogram. The median capacity for global collapse is the median value of the calculated set of drift limits. An example for one accelerogram for an L.A. site for a 20-story weak-column OMF building is shown in Fig. 27. The open circles represent the IDA calculations for an accelerogram where the 0.2 S e slope determined 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 be It was believed that the analytical results for drift greater than 0.10 would not be reliable [14]. The issue of the safety of the occupants was paramount in this design Calculated capacity The IDAs were performed according to the procedures described in the previous section. The median drifts as well as the spectral acceleration capacities were calculated for each suit of ground motions described earlier. A strain-hardening ratio of 0.03 was used for all of the analyses in this study. The increment of ground motion intensity used was 0.2g for the cases without fracturing connections and 0.1g for the models with fracturing connections since those models are expected to collapse at earlier intensities of ground motions. However, the 20-story model with fracturing connections without the equivalent gravity bay collapsed even at 0.1g for two ordinary 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.

28 556 D.A. Foutch, S.-Y. Yun / Journal of Constructional Steel Research 58 (2002) smaller increment of 0.02g was used for some IDA analyses. It was found that this 20-story model reaches incipient collapse at 0.08g that is only a little greater than the design level. The ground motion increment must be small enough so that drift increment is relatively small for each step. The values given above should be considered as upper bounds. The use of a larger increment would usually result in a smaller drift capacity and larger variation of the capacity. Therefore, it would give conservative results. The individual drift capacities for each 9-story model along with the median values are 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 ordinary California ground motions were all about the same. For the near-fault motions, the model with the gravity frames had greater capacities. The drift capacities for the 9- story models with fracturing connections had much smaller capacities than the models without fracture. For instance the capacities for M2-SC and M2-SC-fr were 0.13 and 0.08, respectively, for the standard motions and 0.18 and 0.06, respectively, for the near fault motions. There was not a significant difference between the two models with fractures. The differences in capacity among the various models were much greater for the 20-story models when subjected to the ordinary accelerograms. Models M1-WO and M2-WO had drift capacities of 0.07 and 0.05, respectively, while models M2-SC and M2-Comp had capacities of 0.09 and 0.10, respectively. For the near-fault motions, the four models without fracture had comparable capacities of about Again, the capacities for the model with fracture were significantly smaller than for those without fracture. The capacities for M2-WO-fr and M2-SC-fr were 0.02 and 0.05, respectively, for the ordinary motions and 0.03 and 0.05 for the near-fault motions. In addition, the capacities for the fracturing models without the gravity bay were significantly smaller than those with the gravity bay. It should be pointed out once again that this is a numerical exercise where relative capacities are compared. One should not expect that a real building would be able to resist drift levels of 0.20 without collapse. This is why the SAC project placed an upper limit of drift capacity of The SAC project computed uncertainties and confidence levels in terms of story drifts. This was chosen because it is a quantity that is calculated by the designer as a regular part of the design process. Another quantity that could be used is the collapse limit based on spectral amplitude. These are also a natural result of the IDA analyses. The median strengths for the 0- and 20-story buildings are given in Tables 10 and 11 respectively. The spectral acceleration capacities for each ground motions are shown in Figs. 30 and 31. The spectral acceleration capacities vary greatly for the ordinary ground motions but very little for the near-fault motions. Spectral acceleration capacities were suggested 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 capacities would also be derived from the spectral acceleration capacities.