Modeling of Dynamic Behavior and Estimation of Damage Incurred by Self-Centering Rocking Walls

Transcription

1 Journal of Rehabilitation in Civil Engineering 4-2 (2016) journal homepage: Modeling of Dynami Behavior and Estimation of Damage Inurred by Self-Centering Roking Walls A. Jafari 1, M.R. Ghasemi 2*, H. Akbarzadeh Bengar 3, B. Hassani 4 1. Ph.D. Candidate, Department of Civil Engineering, University of Sistan and Baluhestan, Zahedan, Iran. 2. Professor, Department of Civil Engineering, University of Sistan and Baluhestan, Zahedan, Iran. 3. Assistant Professor, Department of Civil Engineering, University of Mazandaran, Babolsar, Iran. 4. Professor, Department of Mehanial Engineering, Ferdowsi University of Mashhad, Mashhad, Iran. Corresponding author: mrghasemi@hamoon.usb.a.ir ARTICLE INFO Artile history: Reeived: 15 February 2017 Aepted: 03 April 2017 Keywords: Self-Centering roking walls, Damage index, Post-Tensioning tendons, Modeling of nonlinear behavior. ABSTRACT Self-entering roking walls are known as viable alternatives to typial shear walls, as they provide a number of solutions for eliminating seismi flaws of onventional designs. These roking walls have a generally positive impat on the seismi behavior of strutural systems, but their design makes them suseptible to onrete rushing around their base, whih an lead to signifiantly adverse effets on their seismi performane. This paper first models the dynami behavior of these walls under yli loading and then uses a new approah to estimate the extent and quality of damage inurred by the wall at element level. The damage index used for this purpose ats as a quantitative sale measuring the quality of damage inurred by the onrete and therefore gauging the status of the wall. This paper uses the PERFORM 3D software for the proedure of modeling and damage estimation. To assess the auray of the modeling tehnique, results of numerial analyses are ompared with the results of a full-sale load test. The quantitated damage inurred by the wall is then plotted for its surfae and these damages are then ompared with the atual results obtained from the test. The results indiate that the tehnique used by this paper to model the dynami behavior of these walls an aurately simulate their behavior. Also, the damage index used in this paper provides an adequately aurate estimate of the damages inurred by this type of walls. 1. Introdution A major objetive of ivil engineering is to reate a design that would prevent or limit the extent of earthquake-indued damage in reinfored onrete strutures. RC strutural walls have always played an effetive role in preventing the ollapse of strutures under severe earthquakes and limiting the extent of damages aused by subsequent seismi

2 94 A. Jafari et al./ Journal of Rehabilitation in Civil Engineering 4-2 (2016) effets. [1] A multitude of tests arried out on RC strutural walls built with onventional steel reinforement have reported their adequate flexural dutility [2], even in the event of sustaining onsiderable damages in the plasti zone of their base or under the risk of shear slip. [3] Meanwhile, the earthquakeindued large defletions highlight the need for adequate apaity of highly dutile walls, whih an be provided by extensive inelasti deformation in the plasti zone of the wall. Plasti rotation of the base of the wall aompanied with a managed yield of reinforement bars an be used to provide adequate strutural dutility and energy dissipation under yli load and dampen earthquake-indued inertial fore. However, this approah will result in severe damage to the base of the wall and permanent deformation of struture, whih will require diffiult and ostly repairs or demolition of damaged setions. Furthermore, preventing the loss of life is no longer the sole purpose of earthquake-resistant designs, as designers are urrently required to make sure of struture s post-earthquake servieability with minimum repair and renovation. This has enouraged the researhers to searh for novel seismi designs that not only would ensure maximum safety of residents but also would minimize the damages inurred by the struture itself. Many researhers [4], [5] have suggested that reinfored onrete roking walls an at as an alternative to onventional shear walls for preventing or limiting earthquake-indued strutural damages. The major advantage of roking systems is their ability to remain stable under massive displaements without inurring a severe damage. This feature makes these systems an attrative design option for those buildings that need to maintain ontinuous servie in postearthquake onditions. Roking walls are strutural walls that are reinfored by unbonded post-tensioned ables. These ables allow the wall to move with the load and then return to the enter after the end of eah loading yle. Appliation of roking walls is not limited to new strutures, sine they an also be used to retrofit onventional RC walls exhibiting an inadequate seismi resistane or the struture of already damaged buildings. [6] The first researh on roking systems was onduted by Housner [7], who studied the response of a rigid roking blok under free osillation. Meek [8] studied the effet of strutural flexibility of a roking system inluding a foundation. Aslam et al [9] proposed an RC roking system in whih unbonded ables apply a prestressing fore. These researhers found that the roking strength of a rigid struture depends on how it is anhored to the ground and how the prestressing fores are applied to its anhoring elements. The use of unbonded post-tensioned ables in beam-to-olumn onnetions was first proposed by Priestley and Tao [10], and good performane of this design was later proved by Priestley and MRae [11]. This approah allowed the designers to avoid plasti deformation of the walls and strengthen the ritial regions of frames and walls by a roking omponent [11], [12], [13]. Kurama [12] studied the performane of preast walls that were posttensioned with unbonded ables in beam-toolumn onnetions and provided a method for seismi design of these walls. This was followed by the work of Mander and Cheng [14], who developed a damage avoidane design. Their design sought to inorporate various behavioral aspets of roking strutures suh as strutural flexibility and prestressing. The self-entering mehanism of roking walls auses their apaity urve to exhibit a bilinear elasti behavior under semi-stati loading. This in turn auses the amount of energy dissipated in a yle of roking system to be muh less than the energy dissipated in an elastoplasti yle of

3 A. Jafari et al./ Journal of Rehabilitation in Civil Engineering 4-2 (2016) a onventional plasti RC hinge. To overome this flaw, researhers have tested and suggested a number of damper tools to aelerate the energy dissipations of roking systems [15], [4] or proposing the hybrid system like the preast roking walls with end olumns (PreWEC) with employing the O-Connetor for improving the energy dissipating of the roking wall system [16]. An experimental study onduted by Boroshek and Yanez [17] showed that roking mehanism has a generally positive impat on the displaement of the entire struture. They reported that due to the more stable yli behavior of roking RC strutures in large displaements, they suffer muh lesser earthquake-indued damage than onventional RC strutures. They added that using post-tensioned ables in plae of onventional reinforement bars an allow the designer to ontrol these strutures and avoid any damage through plasti bukling of rebars. As previously mentioned, the onventional strutural walls are highly suseptible to plasti bukling of rebars, whih is the most probable form of dutile behavior in the plasti zone of base of the walls, and this may lead to rushing of onrete and failure of reinforement bars in those regions [18], [19], [20]. Preti and Giuriani [21] studied the slip strength of selfentering roking walls subjeted to lateral loads and reported that the weakness of these walls is the low slip strength of their base, whih an be easily addressed by adding a shear key at those setions, strengthening them against shear failures. Preti and Meda [22] studied the retrofitting of damaged monolithi roking walls using high strength fiber onrete, and reported that after retrofitting, roking walls exhibited a good and safe behavior against exerted loads and are therefore a viable option for reduing the repair osts and extending the servieability of struture. Yoopraserthai et al [23] investigated the appliation of BRB for improving the energy dissipation of preast onrete roking wall (PCRW). These authors stated that the stable elastoplasti BRB supplies vibration energy to the nonlinear elasti responses of the PCRW. Their finding onfirms that the PCRW-BRB system is a suitable alternative for use in seismi-resistant strutures. To address the absene of exat professional odes for design and analysis of roking walls, Hasanli et al [24] reviewed and analyzed the existing experimental data and developed a set of parametri formulas for the foredisplaement behavior of these walls. These authors stated that the developed formulas allow the designer to gain a better understanding about the behavior of roking walls and so allows the proess to be arried out with a lesser degree of unertainty. Henry et al [25] were studied the urrent methods that are used to ensure that reentering is ahieved during the design of self-entering onrete systems to find the flaws in these urrent proedures. They performed some time-history analyses and it was onluded that due to dynami shake-down the residual drifts at the onlusion of the ground motion were signifiantly less than the maximum possible residual drifts that were observed from the yli hysteresis response, and were below aeptable residual drift performane limits established for seismi resilient strutures. They were reommended a residual drift ratio that an be implemented during the design proess to ensure that residual drift performane targets are ahieved for self-entering onrete wall systems. Self-entering roking walls have a positive impat on the struture s seismi behavior, but their unique design makes them vulnerable to a number of fators. These strutural walls are vulnerable to the damage

4 96 A. Jafari et al./ Journal of Rehabilitation in Civil Engineering 4-2 (2016) around their base and toe (at the point where they are onneted to the foundation), so their behavior highly depends on to the extent of damage inurred by their base. To investigate the behavior of these walls, this study first uses the results of a loading test onduted on a full-sale monolithi selfentering roking wall to develop a model for its nonlinear behavior, and then assesses the validity of the developed model. After assessing the auray of the model, the extent and quality of damage inurred by wall base is alulated by a novel damage index. This paper uses this index, whih an express important aspets of onrete s nonlinear behavior, to obtain the element level of the onrete damage index and then it plotted for its surfae. The quality and extent of damage inurred by the wall during a desired load yle will be obtained by alulating this index for the wall elements using the quantitative sale orresponding to the qualitative damage. To gauge the auray of the developed damage index, the alulated values will be ompared with the extent of damage aused in the atual test, and the estimated quality of the inurred damage will be disussed. 2. Damage Estimation Finding a damage index that an properly and quantitatively expresses the damage inurred by a struture is an objetive that has long preoupied the minds of many researhers. This index must provide a quantitative measure of damages inurred by the struture under desired loading, and subsequently allow a onlusive deision to be made about the damaged struture. Damage indies are often ategorized into two levels: element-level (loal) and struture-level (global). These indies are often expressed as a value between 1 (representing the diminished strength of the struture) and zero (representing the perfet health of the struture). Some of the damage indies use numbers suh as 0.7 or 1.27, instead of 1, to express the diminished strength of the struture. Table 1 lists some of the most widely used struture-level damage indies [26], [27], [28] along with their values and orresponding levels of damage. Most element-level indies have a umulative nature and reflet the dependene of damage to both amplitude and number of loading yles. The main disadvantages of most element-level damage indies inlude the diffiulty to find appropriate oeffiients for different strutural members, the lak of a preise alibration proess for varying degrees of damage inurred by different members, and the diffiulty of defining the impat of damage inurred by individual members on the instability of the entire struture. Considering the unique nature and behavior of self-entering roking walls, using the onventional damage indies to estimate the damage inurred by these walls often leads to an inaurate assessment. The heavy duty post-tensioning ables used in these walls, whih are the soure of their self-entering nature, ause them the exhibit a minimal residual displaement under most loading onditions. On the other hand, ables often behave elastially and it is only the wall toe that exhibits a plasti behavior, so nonlinear energy dissipation of these systems is muh lower than that of onventional shear walls. This is also refleted in the differene between the hysteresis urves of these roking walls and ordinary shear walls, as their hysteresis urve is flag-shaped and has an area muh smaller than that of ordinary shear walls. Therefore, those damage indies that are based on energy loss will not be able to return aurate results for these roking

5 A. Jafari et al./ Journal of Rehabilitation in Civil Engineering 4-2 (2016) walls. This highlights the importane of using an index that would be able to aurately measure the damage inurred by the base of the wall. This study uses damage Damage state index developed by Kim et al. [29] to estimate the damage inurred by the selfentering roking walls. Table 1. Correlations of the damage index with the damage state Stone and Taylor DI [26] Minimum value of damage index Williams et al. DI [27] Hindi and Sexsmith DI [28] Damage that an be repaired Damage that annot be repaired Collapse This index has been developed through parametri studies onduted via finite element analysis. In this index, the value 0.1 represents the onset of damage or the presene of negligible damage; the value 0.4 represents signifiant (not easily repairable) damage, for example, signifiant rushing of onrete over due to bending or shear raking after the yield of longitudinal rebars; the value 0.75 represents the point of failure (rupture of longitudinal rebars or rushing of onrete); and the value 1.00 represents failure of most rebars and strutural ollapse. This damage index, whih has been developed speifially for RC strutures, an be alulated in two modes: ompressive and tensile. The ompressive damage index only represents the severity of damage in onrete setions and the tensile damage index onsiders only the damages inurred by rebars of RC struture. The roking walls studied in this paper lak any longitudinal rebars, so the extent of damage is assessed only by ompressive damage index. 3. Nonlinear modeling of the selfentering roking wall Preti et al [30] have onduted a full-sale test on a monolithi self-entering roking wall. This wall was post-tensioned by unbonded tendons. Preti et al [30] used two walls for this test (Fig. 1-a), the reation wall whih was a highly post-tensioned shear wall, and the self-entering roking wall, ating as the test subjet. The tested wall was designed as part of a hypothetial 5-storey building (Fig. 1-b) and was subjeted to a load proportionate to this assumption. Height of the tested wall was onsidered to be h w = 10 m to ensure that the shear span indued at the highest level by the horizontal fore would be equal to that in the supposed building. Horizontal load was applied at the height of h f = 8.80 m (from foundation) (Fig. 1-) whih was about one meter lower than the shear span of the supposed building. The tested wall had a uniform mm 2 ross-setion throughout its height. As Fig. 1- shows, instead of onventional longitudinal rebars, the wall had only 8 unbonded steel tendons. The tendons, was employed for wall post-tensioning, were enased in sheaths to prevent any interation with onrete. Considering the absene of longitudinal rebars, 8 Teflon-oated metal sheaths were installed in the bottom 4.00 meters of the wall to at as shear keys and prevent slip shear around the base. The unbonded posttensioned steel tendons in the bottom 4 meters of the wall passed through these sheaths. [30]

6 98 A. Jafari et al./ Journal of Rehabilitation in Civil Engineering 4-2 (2016) (a) (b) () Fig. 1 Full sale experimental roking wall (a), Full sale test wall (b), and position of the posttensioned unbonded tendons and dowels () [30] Transverse reinforement of the wall had two different layouts. As Fig. 2 shows, in the bottom 4.00 meters of the wall, transverse reinforement had a heavier and denser layout. Cross-setion of the wall was subjeted to an axial fore of F v = 2500 kn applied through 8 tendons, eah omposed of three 0.6 inhes diameter steel strands, reating a post-tensioning stress of 700 MPa. It should be noted that 1000 kn of this axial fore was equivalent to the gravitational fores exerted on the wall and the remaining 1500 kn was to subjet the tendons to prestressing. The reiproating horizontal fore F h was exerted by a hydrauli pump to a point loated at the height of 8.80 meters. Fig. 2 ross-setion of the wall, layout of transverse reinforement in both setions, and position of the tendons and metal sheaths [30] In this study, nonlinear behavior of selfentering roking wall is modeled with PERFORM 3D (Version 4.0.0). Considering the aspet ratio of tested wall, it an be regarded as a slender wall and its nonlinear behavior an be onsidered to be ontrolled by flexure; so this behavior is modeled by shear wall element provided in the mentioned software. [31] Shear wall elements are multilayer shell finite elements whose rosssetions are omposed of fibers. To model the behavior of the wall, the stress-strain

7 A. Jafari et al./ Journal of Rehabilitation in Civil Engineering 4-2 (2016) urves of fibers of eah element are assumed similar to that of materials used in the wall (steel reinforement or onrete). To model the axial-bending behavior of the wall, ross setion of eah element is developed by defining an adequate layout of steel and onrete fibers in one or several layers. After that, the behavior of the wall is modeled by defining a layer with stritly shear behavior and then forming a parallel attahment between this layer and the axial-bending layer Wall elements Considering the absene of longitudinal rebars, the behavior of tested roking wall is modeled by using a single layer for axialbending behavior and a single layer for shear behavior. Due to lak of any interation between post-tensioning tendons and onrete, behavior of these tendons is modeled separately from the wall elements. Considering the absene of onsisteny in irregular meshing and onnetion of two differently-sized elements, the wall is meshed as shown in Fig. 3-a. Tendons are onneted to the two ends of the wall (base and top), so transverse meshing at the tendon-wall onnetions is inevitable. In the remaining two ends, at left and right, the wall is divided into two equal parts. In the longitudinal diretion, height of the elements is seleted aording to their aspet ratio, geometry and struture of the wall, and loading onditions. In those setions where the steel sheaths are used as a shear key (heights of lower than 4.00 meters), height of the elements is seleted to be 50m to ensure a near-1 aspet ratio for most of these elements. In the rest of the wall, height of the elements is seleted to be 1 meter to ensure that the lowest aspet ratio would be about 0.3. Due to appliation of onentrated horizontal loading to the top of the wall, this setion is modeled by 80 and 120-m high elements. (a) (b) Fig. 3 Multilayer finite element model of the wall (sorted by the type of onrete fiber) (a), and uniaxial stress-strain urve of onfined and unonfined onrete fibers (b) [32] 3.2. Axial-bending and shear layers of wall elements The behavior of onrete setions is modeled by the elements shown in Fig. 3-a; these

8 100 A. Jafari et al./ Journal of Rehabilitation in Civil Engineering 4-2 (2016) elements onsist of two layers: axial-bending layer and shear layer. Those regions that have a high potential for raking or rushing, like edges, are modeled by relatively smaller fibers. The smaller area of these fibers ensures their quiker raking or rushing with the inrease of fore, whih leads to adequately aurate modeling of these regions as well as adequate shift of onrete s neutral axis. As Fig. 2 shows, the entire area of ross-setion (A-A), whih pertains to the lower 4 meters, inludes onfinement reinforement and so exhibits a behavior similar to that of onfined onrete. In ross setion (B-B), whih pertains to rest of the wall, only two edges of the wall have onfinement reinforement; so in this ross setion, onrete of those two edges are assumed to be onfined and that of other setions are modeled with unonfined status. In Fig. 3-a, those elements wherein onrete fiber of axial-bending layer is onfined are marked with hathing. Properties of axialbending layer of onrete fibers are defined by Kappos equations [32]. Kappos suggested stress-strain relations for onfined and unonfined onretes (Fig. 3- b): 2 f 2 o o o f 1 u( ) 0.25 f o o Eq. (1) Eq. (2) Aording to Eq. (1), Eq. (2) and Fig. 3-b, the uniaxial stress-strain urve is omprised of two parts: i) The asending part (Eq. (1)): in this part, the stress is inreased up to the maximum onfined onrete strength (f ), orresponding to the strain of ε o, defined as: f o k f k 2 o f y b k 1 a( w ) f Eq. (3) where, k is the onfining index, f y is the yield strength of reinforement; and, a and b are experimental oeffiients whih depend on the hoop reinforement layout (Fig. 3-b). ii) The desending part (strain softening): in this part (Eq. (2)), the stress is linearly dereased with respet to the dereasing rate, u: u 0.75 w b s w 0.5 f f k f k o Eq. (4) where f, ρ w, b, and S w are the maximum unonfined onrete strength, the volumetri ratio of hoop reinforement, the width of the onfined ore, and the hoop spaing, respetively. In this paper, tensile strength of onrete is not inorporated into the model. Here, the onrete fiber of axial-bending layer extends only along vertial diretion (along the height); in other diretions (along the width and the out-of-plane axis) the wall is assumed to exhibit an elasti behavior. Given the details of the ross setion of the wall, its shear behavior an be partitioned into 4 regions shown in Fig. 4-a. The shear behavior of the wall is modeled without using fiber and only through defining a suitable shear stress-strain urve and seleting a fitting ross-setion for shear layer.

9 A. Jafari et al./ Journal of Rehabilitation in Civil Engineering 4-2 (2016) (a) (b) Fig. 4 Shear layers of multilayer finite elements of the wall (sorted by type) (a) and shear stress-strain urve of shear layers [31] (b) Aording to Fig. 2 and Fig. 4-a, the shear layer types 1 and 3 are defined based on shear strength of the wall and onfinement reinforements, the shear layer type 2 is defined based on shear strength of onrete, steel sheaths, and onfinement reinforements, and the shear layer type 4 is defined only by the shear strength of onrete. The behavior of shear layer of the wall elements is assumed to be non-linear and independent of axial fore and its properties are defined based on Esfandiari equations [31]. The proposed shear stress-strain urve (Fig. 4-b) an be used to model the behavior of shear layer of element, but this requires the introdution of a few parameters. The first part of this shear stressstrain urve exhibits a slope of G = 0.4 E until reahing the raking strength (f r ) and its orresponding strain In the seond part, shear strength of the setion equals its nominal shear apaity (V n ), whih is alulated using the reommendations of ASCE41-06 [32]. Based on reommendations of Esfandiari, the shear strain orresponding to this shear strength is set to 0.004, and the maximum shear strain in all shear layers is assumed to be Modeling of unbonded posttensioning tendons The post-tensioning tendons, whih are extended over the wall s entire height, are seured within sheaths and have no interation with onrete. These tendons are modeled by bar-shaped elements made of, tension only inelasti steel materials. The eight unbonded post-tensioning tendons that have been used in the wall are modeled with four bar elements, eah having an area twie the area of eah tendon. As Fig. 5-a shows, tendons are fixed at the bottom by 4 hinged supports and their other ends have a hinged onnetion to the nodes of onrete elements situated on top setion of the wall. The mentioned hinged supports are defined slightly away from the base of the wall to prevent an interation between the support of tendons and those of the wall. The four barshaped elements defined over the wall s entire length are fully separate from the wall and have no onnetion with wall elements exept at the top of the wall.

10 102 A. Jafari et al./ Journal of Rehabilitation in Civil Engineering 4-2 (2016) Modeling of wall supports (a) (b) Fig. 5 The bar-shaped elements used for modeling of unbonded post-tensioning tendons plus supports and onnetions (a) and tensile stress-strain urve of tendons s [35] (b) The stress-strain urve of tendons, whih is shown in Fig. 5-b, is tri-linear and has a slight strain hardening after yield. This urve is defined based on material properties provided in [30] as well as Walsh and Kurama reommendations [35]. After modeling the wall, the post-tensioning fore of tendons (exerted on the wall) is modeled by alulating the strain equivalent to the stress indued by appliation of tensile fore F v = 2500 kn on 8 tendons, and then inorporating this strain into the initial tensile strain of tendons. Wall supports must be modeled by an element that would be able to properly exhibit its roking property. Aording to the result of the test onduted in [30], after the exertion of lateral fore, one side the studied wall easily detahes from its support, ausing the other side to be pushed further against the support. The roking property is simulated with the help of nonlinear elasti gap-hook bar elements shown in Fig. 6. These elements have positive and negative reation displaement (gap) and stiffness. To form the roking behavior, positive stiffness of these elements are assumed to be very small (10 N/m) and their negative stiffness are assumed to be very large ( N/m). Meanwhile, positive gap of these elements is assumed to be relatively large (0.3 m) and their negative gap is assumed to be zero. Based on these properties, when the system is subjeted to tensile fore, these elements easily detah from the support and move upward, but when it is subjeted to ompressive fore, the wall support exhibits a high stiffness and ats like a rigid support. One end of these elements is onneted to the nodes of the wall elements and the other end is onneted to the hinged supports. A roller support is used on one side of the wall to prevent the lateral fore from triggering a lateral movement. Fig. 6 The non-linear elasti gap-hook bar elements and the supports used for modeling of wall support elements

11 A. Jafari et al./ Journal of Rehabilitation in Civil Engineering 4-2 (2016) Validity assessment 4.1. Validation of the model of selfentering roking wall To assess the validity of developed model, the experimentally attained lateral load drift urve provided by [30] is ompared with the simulated urve obtained by the developed model. The lateral load test onduted in [30] has been performed in two modes: with frition and without frition; but the present paper only disusses the results obtained in the presene of frition. In the first part of the mentioned test (with frition), whih assessed the behavior of self-entering roking walls subjeted to quasistati horizontal yli loading, seven yles of yli loading, as shown in Fig. 7-a, were applied to the wall. To plot the lateral load drift urve of the model, after applying the gravitational load and post tensioning fore of tendons, we apply and inrease the horizontal lateral load to the designated point suh that it would reate relative displaements similar to those plotted in Fig. 7-a. Next, the base shear and relative displaement indued by seven yles of this lateral fore is measured and plotted as lateral load drift urve. Comparing the lateral load drift urve obtained from the test data with the one obtained by the model (Fig. 7-b) demonstrates a good agreement between atual behavior of the wall and simulations of the model, and therefore onfirms its validity and preision. (a) (b) Fig. 7 Loading history in the test [30] (a) and the urves of lateral load - drift obtained by the test and model (b) 4.2. Validation of damage estimation As previously mentioned, the damage inurred by the wall is assessed by the damage index proposed by Kim et al [29]. In this damage index, the value 0.75 represents the ompressive failure (rushing) of onrete. This ompressive failure ours when prinipal ompressive strain of onrete element reahes ε u (failure riterion), whih an be alulated by Eq. (5) This damage index assigns the value 0.4 to irreparable damage (rushing of a signifiant portion of onrete over), i.e. when prinipal ompressive strain of onrete element reahes up to the value of ompressive strain at maximum ompressive stress of the onrete. 1.4s f yh sm u Eq. (5) f In Eq. (5), f ' is the onfined ompressive strength of the onrete, ρ s is the volume

12 104 A. Jafari et al./ Journal of Rehabilitation in Civil Engineering 4-2 (2016) ratio of transverse onfinement reinforement, f yh is the yield stress of transverse onfinement reinforement, and ε sm is the strain orresponding to the maximum tensile stress of steel. Kim et al. have developed Eq. (6) for measuring the ompressive damage index of onrete: (2 ) u s D. I. Compressive 1 ftg 2 AD C u 2 Eq. (6) ftg Eq. (7) AD C 1 N 2 f Eq. (8) In Eq. (6), ftg is the onrete fatigue parameter, whih must be alulated by Eq. (7), ε u denotes the ultimate ompressive strain of onrete, and ε s is the priniple ompressive strain of onrete at the desired step of analysis. In Eq. (7), AD is a parameter representing the fatigue-indued aggregate (umulative) damage inurred by onrete, whih must be alulated by Eq. (8), N 2f is the number of full yles of loading before failure, whih must be alulated by Eq. (9). In Eq. (9), k is the onrete s oeffiient of variation, whih must be alulated by Eq. (10) and Eq. (11). N log k 2 f max< 0. 7εo, 2 1 ( εo εmin ) ( εo ε 1 2 β ( εo εmin ) εmax 0. 7εo, 0. 09εu 1 max min εu 0.7εo u min k 2.0 s k s k f f o max ) 2 Eq. (9) Eq. (10) Eq. (11) In Eq. (11), f ' and f o ' are the onfined and unonfined ompressive strength of onrete. In Eq. (9), ε o denotes the strain orresponding to the maximum unonfined ompressive strength of onrete, ε min and ε max are minimum and maximum values of strain in a single yle of loading, and β is a material-based onstant, whih is for onrete. The next step is to evaluate the damages inurred by the proposed nonlinear model. Considering the high onentration of damage in the base of the wall, this stage of work employs smaller elements, espeially in the lower setions. The elements used for the lateral ends of the wall in its bottom 1.40 meters have a dimension of mm 2. To ahieve a uniform meshing, other elements of the wall must inevitably have the same pattern, i.e. to shrink in both diretions. After re-meshing the wall, the model is analyzed by fore-ontrolled nonlinear stati analyses. After applying the lateral loads equivalent to those shown in Fig. 7-a, the damage index is alulated for the end of yle at 2% drift. To alulate the damage index, one numerial analyses are finished, first the parameter N 2f is alulated for eah loading yle and then parameters AD and ftg pertaining to eah wall element and eah yle are obtained. The parameter AD is alulated umulatively over onseutive yles, and finally damage index of eah element at eah loading yle is alulated by the use of parameters ε u (ultimate ompressive strain of onrete) and ε s (prinipal ompressive strain of onrete). Shear strain and ompressive axial strain of elements at eah step of yle are used to alulate their prinipal ompressive strain ε s, and the peak value of this parameter in eah yle is used to alulate the damage index of elements. After alulating the damage index of eah individual yle, the highest value among all yles is seleted as the damage index of the element. Fig. 8-a and Fig. 8-b show the damaged wall after repairs, and the ontour of damage index aross its surfae.

13 A. Jafari et al./ Journal of Rehabilitation in Civil Engineering 4-2 (2016) (a) (b) Fig. 8 Details of the damage sustained by the base of the tested wall as a result of lateral loading (after repairs) [30] (a) and the ontour of damage index aross the surfae of the wall after yli loading leading to relative lateral displaement of 2% Fig. 8-a shows that onrete of base of the wall has been damaged up to a height of about 60 m, and the first 20 m has been damage to the extent that the rushed onrete need to be replaed. The extent of damage has dereased with height, but the damage inurred by onrete over up to a height of about 60 m is well evident. On the other hand, the visible depth of damage has been about 40 m, and this has also dereased with height. Aording to definitions of damage index used in this paper, the undamaged setions of the wall have a damage index of less than 0.1 (In Fig. 8-b) these regions are olored gray. The onset of formation of minute raks is equivalent to a damage index of between 0.2 and 0.3, and the points having a damage index of 0.4 to 0.7 represent, respetively, the damaged (dislodged) onrete over and the onset of rushing. Any point whose damage index is higher than 0.7 has ertainly been subjeted to onrete rushing onditions and will no longer have its former strength. Those points on the tested wall that have a damage index of between 0.1 and 0.4 have exhibited no tangible damage, but it should be noted that this range of damage index represents a transition from undamaged ondition of onrete to start of damage and raking. Comparing the Fig. 8-a and Fig. 8-b indiates that this damage index an predit the damage inurred by the wall with great preision. 5. Conlusions The following are the onlusions obtained through numerial analyses and omparison of the obtained results with the results of a full-sale lateral loading test. This paper used multilayer shell finite elements omposed of fiber setions to model the nonlinear behavior of selfentering roking walls. This behavior was modeled by defining two layers of elements: axial-bending layer and shear layer. Properties of axial-bending layer were defined by defining its ross setion using onrete fibers. Properties of shear layer were defined based on onrete s shear stress-strain urve and shear-resistant ross setion of the wall. The unbonded post tensioning tendons were modeled by bar-shaped elements with onstitutive mehanial properties similar to those of atual tendons. In the end, the roking behavior of the wall support was simulated with the help of bar-shaped gap-hook elements. As stated in the previous setion, using this method for modeling the behavior of this type of roking wall leads to a set of results that are aeptably onsistent with its atual behavior observed in the test. In this paper, the damage inurred by the self-entering roking walls was

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