TH BHAVIOR OF CCNTRICALLY BRACD FRAMS WITH SHORT LINKS Helmuth Köber 1, erban Dima 2 ABSTRACT This paper represents a more detailed completion of a previous work The Behavior of Different Bracing Systems of Multistoried ccentrically Braced Frames. The paper is intended to illustrate some features of different bracing systems used for eccentrically braced frames located in seismic areas. Non-linear static and non-linear dynamic analysis was performed for each of the different braced frames. The maximal values for the displacements, the base shear force, the plastic hinge rotations, the bending moments and axial forces in different structural elements were compared. The history of the formation of plastic hinges was observed. 1. INTRODUCTION A ten-story structure was considered, having the eccentrically braced frames placed as shown in the following figure: ccentrically braced frames Figure 1: Location of the eccentrically braced frames The structure has two spans and six bays of 6.6m. The eccentrically braced frames were configured according to appropriate Romanian codes. The eccentrically braced frames were designed in six different configurations (using six different bracing systems). ach solution used short links with a unique length of 1.2m. The six geometry types are presented in figure 2. 1 Assistant Professor, Steel Department, Technical University of Civil ngineering, Bucharest 2 Professor, Steel Department, Technical University of Civil ngineering, Bucharest
K Frame DC Frame DM Frame V Frame Z Frame Y Frame Figure2: Types of braces geometry The different braced frames were sized so that they could have very close eigenvalues. The values of the first three eigenperiods of the six analyzed braced frames are given in the next table: Table 1: igenperiods igenperiod K Frame DC Frame DM Frame V Frame Z Frame Y Frame T 1 (s) 1.125 1.111 1.067 1.104 1.109 1.112 T 2 (s) 0.405 0.401 0.366 0.385 0.406 0.390 T 3 (s) 0.226 0.227 0.207 0.223 0.242 0.227 2. STATIC NON-LINAR ANALYSIS: To establish the successive occurrence of plastic deformations in different structural elements, a static non-linear analysis was performed for each of the analyzed eccentrically braced frames. The calculation consists of a biographical analysis under constant gravity loads and monotonically increased horizontal loads distributed on the height of the frame according to the first (fundamental) vibration mode. Table 2 contains some results of these analyses (S is the value of the base shear force and < represents the corresponding value of the horizontal displacement of the last floor). Table 2: Static non-linear analysis results Frame Occurrence of the first plastic hinge Occurrence of the first plastic hinge outside the links Previous stadium to plastic failure mechanism S (kn) $ (mm) S (kn) $ (mm) S (kn) $ (mm) K 1317.6 52.029 2415.5 250.909 2703.8 1856.039 DC 1304.1 51.850 2527.3 248.552 3480.6 1432.375 DM 1318.0 51.767 2860.7 217.404 3520.7 1319.182 V 1311.7 51.857 3204.6 265.357 3405.0 1608.640 Z 1312.4 51.803 2903.8 271.128 3284.5 1542.133 Y 1302.0 51.912 2581.6 190.341 3836.1 1829.671
3. INLASTIC DYNAMIC ANALYSIS ach of the six eccentrically braced frames was analyzed under the same base excitation. The excitation was taken from the Vrancea 04.03.1977 earthquake (the most severe earthquake in Romania in the last sixty years, with a magnitude of 7.2 on the Richter scale) and it consisted of the N-S component acceleration record. The peak acceleration for this record is approximately 0.2 times the acceleration of gravity. The first 20 seconds of this record were used since this period contains all the high acceleration peaks and nearly the whole inelastic activity is expected to occur during this period. The entire analysis was performed with a 0.01 second time step. The results with a larger time step (0.012 seconds) were sufficiently similar to the results with a smaller time step (0.0075 and 0.005 seconds). No damping was considered. Although the six frames were sized so that they could have very close eigenvalues, their behavior was quite different. Table 3 contains the greatest values of the base shear force (S min and S max ) and of the horizontal displacement of the last floor (< min and < max ) recorded during the analysis. S max is the greatest value of the base shear force and < max is the greatest value of the horizontal displacement of the top floor for one sense of motion, whereas S min and < min are the greatest recorded values for the other sense of motion. Table 3: Non-linear dynamic analysis results Frame S min (kn) S max (kn) min (m) max (m) K -2304 2573-0.0451 0.3202 DC -2507 2873-0.0993 0.2987 DM -3556 3393-0.1577 0.2333 V -3232 3189-0.1570 0.2652 Z -3004 3181-0.1171 0.2823 Y -3251 3502-0.2243 0.2503 4. BHAVIOR OF K FRAM xcept for some small deformations in the central column and in some braces from the upper stories no inelastic deformations were observed outside the potential plastic zones (links and the base ends of the first floor columns and braces). During the analysis plastic deformations were registered in almost all links. These deformations are greater in the first seven stories. The largest recorded plastic hinge rotation was of 0.08345 radians and the largest cumulated plastic hinge rotation was of 0.12241 radians. Beam segments outside the links remained elastic during the analysis although they have smaller cross-sections than similar members of the other analysed frames. Generally the greatest values of the forces were reached at the link end (not at the column end) of the beam segment. Small inelastic deformations could be observed in some braces at different moments during the analysis. In most cases the inelastic deformations were accidental (the plastic hinge rotation was equal to the value of the cumulated plastic hinge rotation, = ). The inelastic deformations appeared at the lower end of the first floor braces (in potential plastic zones) and in some braces of the upper stories. xcept for the second story central column, no inelastic deformations were registered in the columns outside the potential plastic zones near the base. All these inelastic
deformations were small and appeared accidentally. The greatest values of the bending moments on the columns (1411.5 knm in the marginal column and 569.7 knm in the central column) were much smaller than the bending moments that appeared in the columns of the other frames. This could be explained in part by the fact that links are not located near the columns in the K frame. 5. BHAVIOR OF DC FRAM Plastic deformations could be noticed in almost all links. Compared to the values registered in the K frame, the deformations of the links were smaller in the lower stories and greater in the upper half of the frame. The largest recorded plastic hinge rotation was of 0.07902 radians and the largest cumulated plastic hinge rotation was of 0.12199 radians. The beam segments outside the links had an elastic behavior during the analysis. Most of the braces suffered inelastic deformations at different moments during the analysis. Most of them were accidental (= ). These quite small inelastic deformations had comparable values along the height of the frame. xcept for a marginal column at the fourth story, no inelastic deformations were recorded in the columns outside the potential plastic zones near the column base. All plastic deformations in the columns were accidental. The values of the bending moments reached in the columns were much greater than those recorded in the columns of the K frame. Table 4: Greatest bending moments, axial forces and plastic hinge rotations in the marginal columns Frame Quantity K DC DM V Z Y Bending moment (knm) 1411.5 2052.76 2332.5 2261.83 2033.48 2439.42 Axial force (kn) 5123.94 5613.01 6318.65 5871.06 5589.25 6511.61 0.00623 0.00209 0.00599 0.00140 0.00314 0.00326 0.00623 0.00209 0.00599 0.00140 0.00314 0.00426 Table 5: Greatest bending moments, axial forces and plastic hinge rotations in the central columns Frame Quantity K DC DM V Z Y Bending moment (knm) 569.7 1142.61 1114.62 1052.53 979.95 1307.36 Axial force (kn) 2462.24 1421.66 3164.89 1810.31 1829.61 2155.22 0.00548 0.00525 0.00151 0.00548 0.00593 0.00293 Considering the fact that most of the braces suffered inelastic deformations at different moments during the analysis, it can be appreciated that the K frame had a more favorable behavior than the DC frame. 6. BHAVIOR OF DM FRAM During the analysis plastic deformations were registered in all links. Compared to the values reached in the DC frame, the deformations of the links were smaller except for the first two stories. The largest recorded plastic hinge rotation was of 0.06109 radians; the largest cumulated plastic hinge rotation was of 0.11673rad. No inelastic deformations were observed in any beam segments outside the links. The bending moments and the shear forces were greater than those reached in the DC frame but smaller than the values observed in the beams segments of the K frame. The recorded axial forces in the beam segments had the greatest values of all analysed frames.
Plastic deformations appeared in almost all braces at different moments during the analysis (Fig.3). The deformations in the braces were bigger than those reached in the DC frame. The plastic rotations were not accidental especially in the higher stories. This less favorable behavior can be explained through the great axial forces that occurred in the braces. These axial forces were the greatest that appeared in the frames with the braces connected directly to the columns. 7. BHAVIOR OF V FRAM xcept for a single brace from the second story, no inelastic deformations were observed outside the potential plastic zones. Plastic deformations appeared in all links during the analysis. The recorded plastic hinge rotations were much smaller than those registered in the K-, DC- or DM frame. The greater number of links in the V frame can explain this basically. The greatest registered plastic hinge rotation was of 0.00387radians and the greatest cumulated plastic hinge rotation was of 0.0457radians. A more uniform vertical distribution of the plastic deformations in the links could be observed (compared to the other frames). The greatest cumulated plastic hinge rotations in the links from the stories 2-7 had comparable values. Table 6: Greatest plastic hinge rotations in the links (radians) Story Frame Rotation 1 2 4 5 7 9 0.02738 0.06606 0.08345 0.06466 0.03887 0.00994 K DC DM V Z Y 0.03436 0.12241 0.10277 0.06988 0.05977 0.00994 0.01368 0.04849 0.07902 0.07498 0.03441 0.01005 0.02413 0.10725 0.10899 0.12199 0.05239 0.01991 0.02939 0.06019 0.04981 0.02375 0.02135 0.01592 0.04851 0.11673 0.07912 0.03240 0.03216 0.01961 0.00724 0.02323 0.03870 0.03767 0.02709 0.01783 0.01432 0.04432 0.04570 0.04022 0.04154 0.01895 0.06182 0.07185 0.07041 0.05971 0.03863 0.01613 0.06522 0.07899 0.07041 0.06409 0.05707 0.01651 0.00812 0.02921 0.03993 0.02244 0.01164 0.00916 0.02264 0.08201 0.09844 0.05416 0.04098 0.03740 The beam segments outside the links remained in the elastic range. The beam segments outside the links carry together with the braces a great part of the gravitational loads from the floors. That is why the values of the bending moments and shear forces are much greater than those that appeared in the beam segments of the K-, DC- or DM frame.
K Frame DC Frame DM Frame V Frame Y Frame Z Frame Figure 3: Plastic hinges at 6.38 seconds from the start of the dynamic analysis The braces had an elastic behavior during the analysis except for a single brace from the second story, where small inelastic deformations appeared (= = 0.00043 radians). The columns also remained in the elastic range except for the potential plastic zones from the bottom of the marginal columns where small inelastic deformations could be observed (= = 0.00140 radians). The values and the vertical distribution of the bending moments in the marginal columns are similar to the DM frame, whereas those in the central column are similar to the DC frame. The main disadvantage of this frame compared to the frames that had the braces connected to the columns (K-, DC- or DM frame) consists in the greater relative vertical deformations suffered by the floors (Fig.4). The parts of the floors supported by the braces suffered larger vertical deformations than the parts of the floors supported by the columns. This can be explained by the greater stiffness of the columns compared to that of the beam-
braces system when subjected to vertical loads. The greatest remaining link axis rotations at the end of the analysis are shown in the following table: Table 6: Remaining link axis rotations (the values are measured in radians) Frame Floor K DC DM V Z Y 2 0.00941 0.00188 0.00343 0.00441 0.04565 0.00352 5 0.01184 0.00129 0.00269 0.01025 0.03919 0.00901 7 0.00693 0.00212 0.00094 0.01424 0.02978 0.00038 9 0.00279 0.00617 0.01370 0.01884 0.00324 0.00117 K Frame DC Frame DM Frame V Frame Y Frame Z Frame Figure 4: Deformed shapes of the frames at the end of the analysis (20.08 seconds)
8. BHAVIOR OF Z FRAM During the analysis plastic deformations were recorded in almost all links. These deformations were greater than those reached in the V frame, but smaller than those registered in the K-, DC- or DM-frame. The largest recorded plastic hinge rotation was of 0.07185 radians and the greatest cumulated plastic hinge rotation was of 0.07899rad. The vertical distribution of the plastic link deformations was not as uniform as in the V frame. The greatest hinge rotations appeared in the first five stories. The beam segments outside the links had an elastic behavior during the analysis. The beam segments outside the links carry together with the braces (like in the V frame) a great part of the gravitational loads from the floors. Therefore the recorded values of the shear forces and bending moments are much greater than those that appeared in the beam segments of the K-, DC- or DM frame. Compared to the V frame the bending moments were smaller and the axial forces were greater. Most of the braces suffered plastic deformations at different moments during the analysis. All inelastic deformations were accidental and they were greater than those registered in the braces from the DC- and DM frame. xcept for some small plastic deformations recorded in the central column (at the stories 4, 5 and 7) no other inelastic deformations could be observed outside the potential plastic zones from the bottom of the columns. The greatest values of the bending moments in the columns were slightly smaller than those registered in the DC frame. The relative vertical deformations of the floors were larger than those observed in the V frame because the system formed by the braces and beam segments in the Z frame has a smaller stiffness under vertical loads than the one in the V frame (Fig.4). Table 7: Greatest plastic hinge rotations in the braces (radians) Story Frame Rotation 1 2 4 7 9 0.00124 0.00028 0.00133 K DC DM V Z Y 0.00124 0.00028 0.00148 0.00092 0.00151 0.00143 0.00090 0.00206 0.00092 0.00151 0.00143 0.00090 0.00355 0.00391 0.00187 0.00307 0.00285 0.00363 0.00391 0.00195 0.00341 0.00473 0.00982 0.00043 0.00043 0.00499 0.00629 0.00209 0.00159 0.00252 0.00499 0.00629 0.00209 0.00159 0.00252 0.00086 0.00513 0.00369 0.00086 0.00680 0.00465
Figure 5: Dissipated energies (Set 1 = energy dissipated through plastic deformations in the links, Set 2= energy dissipated through plastic deformations in the columns, braces and beam segments outside the links) 9. BHAVIOR OF Y FRAM Plastic deformations appeared in almost all links during the analysis. These deformations were smaller than those registered in the DC- and DM frame, but greater than those recorded in the V frame. The largest registered plastic hinge rotation in a link was of 0.03993 radians and the greatest cumulated plastic hinge rotation was of 0.09844 radians.
Large inelastic deformations could be observed at the ends of the beams of the first five stories. Only the beams of the upper five stories had an elastic behavior. During the analysis inelastic deformations could be observed in many braces, especially in the lower stories. Generally these deformations were not accidental (< ). Inelastic deformations were registered in the columns from the first five stories of the frame. These deformations were larger in the central column than in the marginal columns. The recorded bending moments in the columns had the greatest values of all analyzed frames. Considering the fact, that many beams, braces and columns from the first five stories of the Y frame suffered inelastic deformations during the analysis, the behavior of this frame can be appreciated to be the poorest of all analyzed frames (Fig.5). This behavior can be explained by the fact that a favorable global collapse mechanism was difficult to size by design. As long as the plastic deformations remained only in the links the Y frame had a favorable, predictable behavior. When the loading level increases and plastic deformations appeared in other members (beams, braces or columns) the behavior of the frame was difficult to control. The influence of the plastic hinges that appeared in the columns, braces and beams (respective beam segments outside the links) can be observed from the graphics in figure5. 10. CONCLUSIONS The behavior of eccentrically braced frames during strong earthquakes (quakes which are generating plastic deformations in the frames), is conditioned more on the dynamic characteristics of the structures at the time when the greatest seismic impulses are registered, than on the eigenvalues of the frames (established in the elastic range of behavior). The main disadvantage of the eccentrically braced frames, which hadn t the braces directly attached to the columns, consists in the greater remaining relative vertical deformations registered by the floors of the V- and Z frame. The values of the plastic hinge rotations recorded in the links of the V- and Z frame were smaller than those registered in the other eccentrically braced frames. The placing of the links near the columns leads to greater values of the bending moments in the columns. The bracing configuration of the DC frame is more favorable than that of the DM frame, because it leads to smaller axial forces in the braces and columns. The absence of a favorable global collapse mechanism, conducts after the plastification of the most dissipative members to an unpredictable behavior of the Y frame. 11. RFRNCS: [1] rdbebenbemessung von Stahlbetonhochbauten, Thomas Paulay, Hugo Bachmann, Konrad Moser, 1990 [2] Normativ pentru proiectarea antiseimic& a construc'iilor de locuin'e, social-culturale, agrozootehnice )i industriale indicativ P100-92, 1992 [3] Seismic provisions for structural steel buildings, LRFD-AISC, 1997 [4] Auslegung von Bauwerken gegen rdbeben, urocode 8,1998 [5] Cod de proiectare seismic& P100-2004 (Propunere normative- redactarea 4) [6] The Behavior of Different Bracing Systems of Multistoried ccentrically Braced Frames, Helmuth Köber, Bogdan tefpnescu - The 10 th International Conference on Metal Structures TimiRoara 2003