Ultra-Low Cycle Fatigue Fracture Life of a Type of Buckling Restrained Brace

Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42

journal homepage: http://civiljournal.semnan.ac.ir/

N. Hoveidae* *Assistant Professor, Civil Engineering department, Azarbaijan Shahid Madani University, Tabriz, Iran

ARTICLE INFO ABSTRACT Article history: Received: 05 May 2017 Buckling restrained braced frames (BRBFs) are considered as Accepted: 05 September 2017 popular seismic-resistant structural systems. A BRB sustains large plastic deformations without brace buckling. The core of a Keywords: buckling restrained brace is prone to fatigue fracture under cyclic Buckling Restrained Brace, loading. The earthquake induced fracture type of the core plate in a Ultra-Low Cycle Fatigue, buckling restrained brace can be categorized as ultra-low cycle Cyclic Void Growth Model, fatigue fracture. This paper investigates the ultra-low cycle fatigue Finite Element Analysis. fracture life of a type of composite buckling restrained brace previously tested. The newly developed cyclic void growth model was adopted to theoretically predict the fracture and crack initiation in the core. In addition, the Coffin-Manson fatigue damage model was applied to estimate the fracture life of the brace. A FEM model of the BRB developed in ABAQUS was used to evaluate the fatigue life. The analysis results showed that the cyclic void growth model is capable to nearly predict the fracture life of the core in buckling restrained brace.

[2], Qiang [3], Watanebe et al. [4], 1. Introduction Tremblay et al. [5], Usami et al. [6,7], Hoveidae et al. [8-10], Chou et al. [11], Recently, Buckling Restrained Braced Eryasar et al. [12]. Fig. 1 illustrates the Frames (BRBFs) are considered as popular typical BRB detail. The encasing member seismic-resisting structural systems. A in a BRB inhibits the brace global buckling BRBF is distinguished from ordinary and minimizes the core local buckling. buckling type brace, where the its buckling A variety of arrangements have been is inhibited by a restraining member. There proposed for BRBs. Hoveidae et al. [10] are different details for BRBs including proposed a novel all-steel BRB in which a all-steel and composite BRBs. The cyclic shorter core plate was used and then the response of a BRB is more stable. BRBs seismic response of SCBRB was evaluated are famous for their higher ductility through nonlinear time history and finite capacity. The ductility in a BRB is element analyses. Bazzaz et al. [13] produced by plastic deformation of core proposed a new type of non-buckling steel member without significant local and ring dissipater in off-center and centric global buckling of the brace. Many bracing systems in order to enhance the researches have been conducted on seismic ductility of braced system. Moreover, response of BRBs which can be found in Bazzaz et al. [14] investigated the behavior the works by Black et al. [1], Inou et al.

DOI: 10.22075/JRCE.2017.11262.1185 N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42 30 of off-center bracing system with ductile factor of 8.68 was achieved. Furthermore, element. Andalib et al. [15] analytically Bazzaz et al. [16] proposed a new bracing and experimentally studied the usage of system using circular element (circular steel rings made of steel pipes as an energy dissipater). The analytical results and dissipater at the intersection of braces. comparison between plots of these two These studies showed that the brace with models showed that the first model has the steel ring exhibits a steady and wide higher performance than the others. hysteresis curve and a tensile ductility

Fig. 1. Typical configuration of a BRB Ozcelik et al. [17] investigated the buckling-restrained brace to develop the response of BRBs with different types of high-performance BRB used in bridge restraining members. It was found that the engineering were conducted. According to energy dissipation capacity of BRBs the mentioned experimental results, the considerably depends on compression BRBs with stoppers possess a higher low- strength adjustment factor, β, and strain cycle fatigue performance than those hardening adjustment factor, ω. without stoppers. Razavi et al. [18] proposed reduced length Yan-Lin et al. [20] proposed core-separated buckling restrained brace (RLBRB), in BRBs and theoretically and experimentally which a shorter core was sandwiched investigated the behavior of the brace. It between a restraining member. The test was found that the proposed detail results indicated that the reduction in BRB improves flexural rigidity of the restraining core length and consequently the increase system. in strain of core up to amplitudes of 4-5% Most of research areas in the literature enhance the risk of low-cycle fatigue focuses on seismic response of BRBs and failure. The low cycle fatigue response of the lack of studies on low cycle fatigue the core plate was examined by Coffin- response of BRBs is evident. Manson fatigue damage criteria as well. Wang et al. [19] surveyed the low cycle Typically, a core plate in a BRB is made fatigue behavior of all-steel buckling up of a ductile steel rectangular plate. The restrained braces. Experimental and core plate is normally designed according numerical studies on the effect of stoppers to code-based forces. In general, the limit on the low-cycle fatigue performance of state of a BRB is the core fracture at mid- 31 N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42 length or at the core ends close to well-known Coffin–Manson approach transition zones, depending on core details. (Pereira et al. [21]). If a stopper is provided on the core plate to prevent the slippage of restraining Kanvinde and Deierlein [31] proposed member, the core plate tensile fracture is another micro-mechanical based ULCF likely to take place at a region near to the model based on the cyclic behavior of stopper [11]. However, in some cases, micro-voids, which enters the stress especially for the BRBs without stopper, triaxiality effects into material degradation. the fracture tends to attend the core ends The model was called Cyclic Void Growth [18]. The fracture of the core plate in a Model (CVGM). Based on this model, BRB can be classified as a low cycle void growth and shrinkage lead to ductile fatigue or ultra-low cycle fatigue fracture fracture initiation during cyclic loading of problem, depending on the loading history the material. This model was verified for applied. several steel types, geometries and loading histories. Despite the low cycle fatigue Normally, the fracture of steel material response of BRBs is investigated in some under extreme loads occurs at small prior works, which can be found in the number of cycles (less than 100 cycles). literature, the ultra-low cycle fatigue This fatigue regime is called ultra-low response is not deliberated meticulously. In cycle fatigue (ULCF) [21]. The steel this paper, the CVGM is applied to a BRB damage induced by earthquake can be previously tested by Chou et al. [11] in classified as ultra-low cycle fatigue order to predict the ultra-low cycle fatigue (Shimada et al. [22], Kuroda [23], Nip et life. In addition, the fracture life of the core al. [24]), which is categorized by large plate is examined by the well-known plastic strains and few cycles to fracture Coffin-Manson damage rule and compared (Zhou et al. [25]). The ULCF fracture is with that evaluated by CVGM model. often the governing limit state in steel structures subjected to severe earthquakes. 2. CVGM Formulation The extremely random loading histories of ULCF associated with very few cycles Based on the researches by Kanvinde and make them difficult to adapt to techniques Deierlein [31], the two key procedures to developed for high and low cycle fatigue, capture in ULCF regime are void growth such as rain-flow cycle counting method “demand,” including the effects of void (Downing et al. [26]) and strain-life growth and shrinkage/collapse, and approaches proposed by Manson [27] and degraded void growth “capacity,” related Coffin [28]. to cyclic strain concentrations of the inter void ligament material. The CVGM model A number of fatigue damage models, proposed by Kanvinde and Deierlein proposed by various authors are available improves the concepts described by Rice in the literature. ULCF prediction models and Tracey [32], Hancock and Mackenzie are usually categorized into two coupled [33] for monotonic loading, and Ristinmaa and uncoupled models. An example of [34] and Skallerud and Zhang [35] for coupled plasticity-damage models was cyclic loading. The fundamental proposed by Lemaitre [29]. An exponential mechanisms of low cycle fatigue fracture ULCF damage rule was proposed by Xue involve cyclic void growth, collapse, and [30] which is more accurate for life distortion. Fig. 2 represents the ductile prediction in ULCF regime, compared to fracture mechanism in metals based on N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42 32

CVGM model. In addition, a fractograph of ULCF is displayed in Fig. 3 [36].

Fig. 2. Micromechanism of ULCF Fig.3. Fractograph of ULCF

First, this paper aims to briefly review the In order to show the fracture through a components of CVGM model. The void void growth index, the calculations above growth rate can be designated by the next can be simplified. equation for a single spherical void, (Rice (VGI ), which is compared to its and Tracey, [32]): monotonic critical value, VGI critical can be defined dR monotonic Cexp(1.5 T ) d  p (1)  p R R VGIexp(1.5 T ) d  VGIcritical  ln( ) critical / C monotonic0 p monotonic monotonic where R is the average void radius; T R0 represents the stress triaxiality which is the as below: ratio of mean stress to effective stress and (5)

C is a material parameter. In addition, d  p The relations above state the void growth denotes the incremental equivalent plastic model and can be used for the damage strain as defined in Eq. (2). detection under monotonic loadings. However, for cyclic loading, Kanvinde and 2 Deierlein suggested that the fracture dd d  pp (2) pij ij3 initiates when Eq. (6) is satisfied over the By integrating Eq. (2), the void radius can characteristic length l * [37]: be stated as: critical VGIcyclic VGI cyclic (6) R  p critical ln C exp(1.5 T ) d  (3) Where VGI cyclic is the critical cyclic void R 0 p 0 growth index. Considering void shrinkage where R denotes the initial void size. As 0 during compressive (negative) triaxialities, the void growth is considered as the Eq. (5) has been upgraded to the following controlling parameter of the damage, form: fracture is triggered when the void ratio reaches a critical value, i.e.: VGI e1.5TT d e 1.5 d (7) cyclic p p Tensile cycles Compressive cycles critical critical R  p It is supposed that the critical cyclic void ln C exp(1.5 T ) d  (4) 0 p growth damage index can be assessed from R0 monotonic its monotonic counterpart as follows: 33 N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42

criticalcritical benchmark. A number of equations can be VGIVGIcyclicmonotonicp.exp( ) (8) found in the literature which tries to The stress and strain histories obtained predict the fracture life of BRBs [19]. In from finite element analysis can be used to this paper the equation proposed by calculate VGI cyclic demand. The parameter Nakamura et al. [40] as the most  is a constant which shows the material conservative equation is selected to damageability. estimate the fracture life of the BRB specimen throughout the loading history. The proper calibration of VGI critical and monotonic Whenever the damage index ( DI ) reaches  parameters ascertains the accuracy of to unity, Low cycle fatigue fracture CVGM model. Lacking the data for the triggers. Damage in each loading phase is characteristic length of the ULCF, estimated by dividing the number of cycles Kanvinde and Deierlein [31] adopted those at that constant amplitude ( ni ) by the of monotonic ductile fracture. number of constant amplitude cycles at

As discussed earlier, based on CVGM that amplitude ( N fi ) necessary to cause model, during ULCF in steel material, fracture. The damage accumulation is damage initiates when Eq. (6) is met over based on Miner's rule. The fracture of the an element with the characteristic length of component subjected to different levels of l * . Thereafter, when cracking occurs in an strain demand will occur when the index element, adjacent elements become more DI reaches to 1 as follows: vulnerable to damage initiation and the n process of cracking expedites for them as a  i 1 (10) redistribution in stress and strain happens N fi in the vicinity of cracking zone [38]. N n DI  i (11)   1 3. Coffin-Manson Damage Model i 1 ()i m  0 A well-known relation for estimating the Based on the equation proposed by low cycle fatigue fracture life of materials Nakamura et al. [45] the parameters proposed by Coffin [30] and Manson [29] and  0 are set to -0.490 and 0.2048, can be expressed as follows: respectively.  N m (9) if0 4. Assessment of ULCF in BRBs Eq. (9) is represented by a linear relation in a bi-logarithm diagram, where  and N i f As mentioned previously, this paper aims are uniaxial plastic strain amplitude and to address the ULCF response of buckling the number of cycles to failure, restrained braces through the newly respectively. 0 is the fatigue ductility developed CVGM model. By knowing the coefficient and m is the fatigue ductility predicted fracture time by CVGM, the exponent. Some authors such as Tateishi et damage index estimated by Coffin-Manson al. [39] have shown that the Coffin– rule is also calculated at threshold of Manson relation over-predicts ULCF. fracture and the ability of this method to However, this issue is going to be capture the fracture in ULCF domain is examined in this paper by selecting the evaluated. For this purpose, a sample fracture life predicted by CVGM as a composite buckling restrained brace N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42 34 recently tested by Chou et al. [11] is 150 mm and 22 mm, respectively. ASTM considered. In order to implement the A36 steel with a nominal yield strength of CVGM method for fracture prediction of 250 MPa was specified for the channels as the core plate in the BRB specimen, restraining members, and ASTM A572 ABAQUS [41] general purpose finite Grade50 steel was specified for the core, element software is employed. A Fortran side, and face plates. The specified 28-days UVARM subroutine is developed to concrete strength was 35 MPa. Table 2 calculate the damage indices at each summarizes the material properties used in integration point. Fig. 4 represents the specimen 1. The test set-up of the BRB characteristics of the BRB specimen tested conducted by Chou et al. [11] is indicated by Chou et al. [11]. Table 1 summarizes in Fig. 5. member size of the BRB specimen. Core plate width, bc , and thickness, t c , were

Fig. 4. Details and dimensions of the sample BRB

Table 1. Characteristics of the sample BRB (Specimen-1) Channel and Bolt spacing Specimen Core plate face No. of Bolts (mm) plates(mm) b (mm) t (mm) L (mm) 150x75x6.5x10 1 c c y 32 186 150 22 2800 270x12

Table 2. Material properties of the sample BRB (Specimen-1) Concrete Specimen Core plate (A572 Gr 50) Channel Face plate strength (MPa) F (MPa) F (MPa) F (MPa) F (MPa) F (MPa) F (MPa) 1 y u y u y u 57 367 525 274 425 441 565 35 N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42

Fig. 5. Test set-up of the sample BRB (Chou et al. 2010) Poisson ratio of steel material were set to 5. Numerical Modeling of the BRB 203 GPa and 0.3, respectively. A combined Specimen isotropic/Kinematic hardening model represented in Fig. 6a, was used for the A three dimensional finite element model steel material in order to accurately capture of the specimen-1 has been considered to the cyclic response. The initial kinematic assess the applicability of the numerical hardening modulus C and the rate factor γ approach in this paper. The detail and were set to 2 GPa and 12, respectively. The dimensions of BRB model in numerical calibration of hardening parameters analysis corresponds to the BRB detail signified in Fig. 6b, was conducted via represented in Fig. 4. The connection comparison between the hysteretic curves portion of the braced was eliminated in the obtained from the previous test by FEM model since the brace member was Tremblay et al. [5] and the FEM analysis loaded axially. Material nonlinearity with conducted by the author. For isotropic the Von-Mises yielding criterion was Q  160 hardening,  MPa and a rate factor considered in the steel core and restraining b  5 members. The elastic modulus and the of were adopted

. a) b) Fig. 6. a) 3D representation of the hardening in the nonlinear isotropic/kinematic model; 6b) Material Calibration for A572-Grade50 steel; Concrete infill as a part of restraining modeled with eight-node solid elements member was modeled with an elastic (C3D8R). A fine mesh pattern was material with a Young modulus of 25 GPa introduced in the core plate in order to and a Poisson ratio of 0.2. Other parts of accurately capture the strain history. the brace including core plate, restraining However, coarser mesh was assigned to channels, and the filler plates were the other parts of the brace because they N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42 36 were expected to generally remain elastic. strongly dependent on key parameters of A contact interaction was used to capture material specially the amount. the interface between the core plate and the Since it is extremely time-consuming to restraining member during loading. A consider the characteristic length l * for all friction coefficient of 0.1 was implemented of the elements in finite element model, to simulate interface between the core plate only the critical zone in which the fracture and the encasing which is similar to the is observed in experimental test is modeled value adopted in the FEM analysis by with finer mesh. Fig. 7 shows the mesh Chou et al. [11]. An initial geometric generation at the critical zone of the core imperfection was introduced in the model plate. As observed in the test, the core of based on the data extracted from the first BRB specimen 1 was fractured at the buckling model. The tie interaction was middle section during the cyclic loading. used to model the welding connection of The BRB specimen in the test was the brace components. The brace was positioned at an inclination of  50 . pinned at one core end and the axial However, it is modeled horizontally in the displacement history was applied at the finite element program and the other core end. Nonlinear quasi static corresponding horizontal displacement analysis including both material and history is applied at the brace end. The geometric nonlinearity and initial and standard and fatigue loading protocols maximum increment size of 0.25 was suggested by AISC seismic provisions [43] conducted in ABAQUS 6.13. Full Newton were applied on BRB model. Standard method was assumed as the solution loading regime was defined at levels technique. corresponding to core strains of 0.33, 0.52, In this paper, material parameters of 1.05, 1.58 and 2.1%. After the standard CVGM model for the Grade50 steel core loading, the BRB specimen is subjected to VGI critical  including monotonic and are assumed constant-amplitude large deformation as 1.13 and 1.18 and the characteristic demands which is called fatigue test at a length l * was set to 0.18mm, as proposed core plate strain of 1.6%, up to failure. previously in a paper by Kanvinde and This type of loading history with large Deierlein [42]. It should be noted that, in strain amplitudes and limited number of the absence of enough laboratory test cycles can be classified in ultra-low cycle equipment, it may be possible to use the fatigue domain. Fig. 8 shows the loading CVGM parameters based on the values protocol applied at the end of BRB model reported in the literature, provided that the as was used in the test. In addition, Figs. 9a specification and mechanical properties of and 9b illustrate the finite element model the selected steel material in the analysis of the entire BRB and also the restraining and the one formerly calibrated through member, respectively. Moreover, the finite notched bar tests, closely match together. element model of the core plate and its The ULCF prediction by CVGM model is mesh generation is illustrated in Fig. 10.

Critical zone

Fig. 7. FE Mesh generation of the BRB core at critical zone 37 N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42

Fig. 8. Applied loading protocol in the BRB test and FEM modeling

a) b) Fig. 9. a) FE model of the sample BRB, b) FE model of the restraining parts of BRB core including the channels, filler plates, face plates, and the concrete infill

Fig. 10. FE model and mesh generation of the BRB core

Fig. 11. Comparison of test and FEM hysteretic curves N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42 38

VGIcyclic 6. Finite Element Analysis Results (i.e. critical ). As shown in Fig. 11b, the VGIcylic The BRB model was subjected to a cyclic damage index evaluated by CVGM at the displacement history at the end and the mid-length of the core and near to the hysteretic response of the brace was stopper has the maximum value. The captured. Fig. 11 displays the hysteretic fracture is supposed to initiate when the response of the BRB model during damage index reach to 1. Therefore, the standard loading protocol and it is FEM model could properly predict the compared to the hysteric curve obtained fracture site. Moreover, the evolution of from the test. As shown in Fig. 11, the CVGM demand and capacity over the hysteretic response of the BRB member in characteristic length in the core and at the the test and the FEM model closely match fracture zone is illustrated in Fig. 13. As together and then, the FEM model could shown in Fig. 13, the quantity of VGI cyclic properly capture the BRB behavior under capacity decreases based on the prescribed condition and loading history. accumulation of plastic strain at the As mentioned before, the CVGM is beginning of each tensile excursion of critical applied to validate the fracture in the BRB loading and the value of VGI cyclic increases specimen. Based on test results, the BRB and decreases based upon the sign of member tolerated large plastic triaxiality. The intersection point of the deformations during standard loading and VGI critical and VGI predicts the failure. held a stable hysteretic behavior up to core cyclic cyclic strain of 2.1%. After the standard loading As revealed in Fig. 13, the intersection protocol, the brace was subjected to the point is close to analysis time 125s, which closely coincides with the beginning of the fatigue loading sequence with a constant th core strain of 1.6% up to failure. The test tensile sequence at 22 cycle of the results showed that the core plate in the imposed loading history. Therefore, the BRB specimen was fractured at the mid- fracture time closely meet the fracture in section during the low cycle fatigue test the test and the CVGM could properly and the fracture occurred at the beginning envisage the failure. Hence, in terms of the of 22th cycle of low cycle fatigue phase fracture site and the time, the FEM model because of the crack initiation and and also CVGM could successfully predict evolution. Remarkably, the CVGM the failure of the BRB in ultra-low cycle implemented on the FEM model of the fatigue regime. For comparison, the fatigue BRB specimen is also able to predict the fracture life of the core plate in the BRB crack initiation and the fracture of the core was also estimated by Coffin-Manson plate at the same loading sequence and also relation based on the core strain history the same site. The finite element analysis during the FEM analysis. The results showed that the core fracture starts at the showed that at the threshold of fracture mid-length close to the core stopper. The predicted by CVGM, the Coffin-Manson fracture points in the test and also the FEM damage index is just about 0.62 which is model are close together as shown in Figs. remarkably far from 1 (i.e. failure). 12a and 12b. The contours in Fig. 12b Therefore, the Coffin-Manson damage display the UVARM6 field-output which criterion seems to overestimate the fracture corresponds to the CVGM damage index life of the core plate in the BRB. Such a result can be found in the literature, emphasizing that Coffin-Manson damage 39 N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42 rule can better capture the fracture in low should be distinguished from ultra-low to high cycle damage regimes which cycle fatigue regime.

Fig. 12. Critical section of the BRB core in the; a) test; b) FEM model

Fig. 13. Comparison of CVGM demand and capacity in FE analysis this paper, the Finite element model of the 7. Conclusions BRB specimen was developed in ABAQUS together with a Fortran In this paper, cyclic void growth model subroutine to predict the onset of fatigue (CVGM) as a micromechanical-based fracture in the core. It is observed that the fracture model is implemented to validate CVGM model is able to successfully the ultra-low cycle fatigue fracture life of a predict the fracture initiation in terms of buckling restrained brace. The assumed location and time (loading sequence) in the BRB specimen was previously core plate and the results closely meet experimentally examined and the fracture those observed in the test. As a result, it was observed after 21 cycles in the low can be concluded that the cyclic void cycle fatigue loading sequence which growth model acts as an acceptable model corresponded to the core strain of 1.6%. In for predicting crack initiation and the N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42 40 failure of BRBs (core plate) in ultra-low Experimental Structural Engineering, cycle fatigue regimes. In addition, the San Francisco. results indicated that Coffin-Manson [8] Hoveidae, N., Rafezy, B. (2012). “Overall damage rule cannot properly predict the buckling behavior of all-steel buckling restrained braces”. Journal of fracture in ultra-low cycle fatigue regime constructional steel researches, Vol. and is likely to overestimate the fracture (79),151-158. life. More experimental tests together with [9] Hoveidae, N., Rafezy, B. (2015). “Local FEM analyses are required to validate the Buckling Behavior of Core Plate in All- CVGM model for ULCF damage Steel Buckling Restrained Braces”. prediction of BRBs with different International journal of steel structures, Vol.15(2), 249-260. arrangements and details. [10] Hoveidae, N., Tremblay, R., Rafezy, B., Davaran A. (2015). “Numerical REFERENCES investigation of seismic behavior of short-core all-steel buckling restrained [1] Black, CJ., Makris, N., Aiken, ID. (2002). braces”. Journal of constructional steel “Component Testing, stability analysis, researches, Vol. (114) 89–99. and characterization of buckling [11] Chou, C., Chen, S. (2010). “Sub- restrained braced braces”. Report No. assemblage tests and finite element PEER 2002/08, Univ. of California, analyses of sandwiched buckling- Berkeley, CA. restrained braces”. Engineering [2] Inoue, K., Sawaizumi, S., Higashibata, Y. structures, Vol. 32, Issue 8, Pages 2108– (2001). “Stiffening requirements for 2121. unbonded braces encased in concrete [12] Eryasar, M., Topkaya, C. (2010). “An panels”. ASCE Journal of Structural experimental study on steel-encased Engineering, Vol.127(6),712-9. buckling restrained brace hysteretic [3] Qiang, X. (2005). “State of the art of damper”. Journal of Earthquake buckling-restrained braces in Asia”. Engineering Structural Dynamics, Vol. Journal of Constructional Steel 39, 561-81. Researches, Vol. 61,727-48. [13] Bazzaz, M., Andalib, Z., Kheyroddin, A. [4] Watanabe, N., Kat, M., Usami, T., Kasai, and Kafi, M.A. (2015). “Numerical A. (2003). “Experimental study on Comparison of the Seismic Performance cyclic elasto-plastic behavior of of Steel Rings in Off-centre Bracing buckling-restraining braces”. JSCE System and Diagonal Bracing System”, Journal of Earthquake Engineering, Vol. Journal of Steel and Composite 27 [Paper No. 133]. Structures, Vol. 19, No. 4, 917-937. [5] Tremblay, R., Bolduc, P., Neville, R., [14] Bazzaz, M., Kheyroddin, A., Kafi, M.A., Devall, R. (2006). “Seismic testing and Andalib, Z. and Esmaeili, H. (2014). performance of buckling restrained “Seismic Performance of Off-centre bracing systems”. Canadian Journal of Braced Frame with Circular Element in Civil Engineering, Vol. 33(1),183-98. Optimum Place”, International Journal [6] Usami, T. (2006). “Guidelines for seismic of Steel Structures, Vol. 14, No 2, 293- and damage control design of steel 304. bridges”. Edited by Japan Society of [15] Andalib, Z., Kafi, M.A., Kheyroddin, A. Steel Construction; Gihodo-Shuppan, and Bazzaz, M. (2014). “Experimental Tokyo [in Japanese]. Investigation of the Ductility and [7] Usami, T., Ge, H., Luo, X. (2009). Performance of Steel Rings Constructed “Experimental and analytical study on from Plates”, Journal of Constructional high-performance buckling restrained steel research, Vol. 103, 77-88. brace dampers for bridge engineering”. [16] Bazzaz, M., Andalib, Z., Kafi, M.A. and Proceeding of 3rd International Kheyroddin, A. (2015). “Evaluating the Conference on Advances in Performance of OBS-C-O in Steel 41 N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42

Frames under Monotonic Load”, Journal International Journal of Fatigue Vol. 48, of Earthquakes and Structures, Vol. 8, 90–100. No.3, 697-710. [26] Downing, SD., Socie, DF. (1982). [17] Ozcelik, R., Dikiciasik, Y., Erdil, E. “Simple rainflow counting algorithms”. (2017). “The development of the International Journal of Fatigue, Vol. buckling restrained braces with new end 4(1), 31–40. restrains”. Journal of Constructional [27] Manson, SS. (1954). “Behavior of Steel Research, Vol. 138, 208-220. materials under conditions of thermal [18] Razavi, A., Mirghaderi, R., Hosseini, A. stress”. National Advisory Commission (2014). “Experimental and numerical on Aeronautics, Report 1170. Cleveland: developing of reduced length buckling- Lewis Flight Propulsion Laboratory. restrained braces”. Engineering [28] Coffin, Jr. (1954). “A study of the effects Structures, Vol. (77), 143–160. of cyclic thermal stresses on a ductile [19] Wang, C., Usami, T., Funayama, J. metal”. Trans ASME, Vol. 76(6), 931– (2012). “Improving Low-Cycle Fatigue 50. Performance of High-Performance [29] Lemaitré, J., Chaboche, J-L. (1990). Buckling-Restrained Braces by Toe- Mechanics of solid materials. Finished Method”. Journal of Cambridge, UK: Cambridge University Earthquake Engineering, Vol. 16,8, Press. 1248-1268. [30] Xue, L. (2007). “A unified expression for [20] Yan-Lin, G., Jing, T., Bo-Hao, Z., Bo-Li, low cycle fatigue and extremely low- Z., Yong-Lin, P. (2017). “Theoretical cycle fatigue and its implication for and experimental investigation of core- monotonic loading”. International separated buckling-restrained braces”. Journal of Fatigue, Vol. 30,1691–8. Journal of Constructional Steel [31] Kanvinde, A., Deierlein, G. (2007). Research, Vol. 135, 137-149. “Cyclic void growth model to assess [21] Pereira, J., Jesus, A., Xavier, J., ductile fracture initiation in structural Fernandes, A. (2014). “Ultra-low-cycle steels due to ultra-low cycle fatigue”. fatigue behavior of a structural steel”. Journal of Structural Engineering, Vol. Engineering Structures, Vol. (60), 214– 133(6),701–12. 222. [32] Rice, J. R., Tracey, D. M. (1969). “On the [22] Shimada, K., Komotori, J., Shimizu, M. ductile enlargement of voids in triaxial (1987). “The applicability of the stress fields”. J. Mech. Phys. Solids, Manson-Coffin law and Miner’s law to Vol. 35, 201–217. extremely low cycle fatigue”. J Jpn Soc [33] Hancock, J. W., Mackenzie, A. C. (1976). Mech Eng, Vol. 53(491),1178–85. “On the mechanics of ductile failure in [23] Kuroda, M. (2002). “Extremely low cycle high-strength steel subjected to fatigue life prediction based on a new multiaxial stress states”. J. Mech. Phys. cumulative fatigue damage model”. Solids, Vol. 24,147–169. International Journal of Fatigue, Vol. [34] Ristinmaa, M. (1997). “Void growth in 24(6), 699–703. cyclic loaded porous plastic solid”. [24] Nip, KH., Gardner, L., Davies, CM. Mech. Mater., Vol. 26(4), 227–245. (2010). “Extremely low cycle fatigue [35] Skallerud, B., and Zhang, Z. L. (2001). tests on structural carbon steel and “On numerical analysis of damage stainless steel”. Journal of evolution in cyclic elastic–plastic crack Constructional Steel Researches, Vol. growth problems”. Fatigue Fract. Eng. 66(1), 96–110. Mater. Struct., Vol. 24(1), 81–86. [25] Zhou, H., Wang, Y., Shi, y., Xiong, J., [36] Deierlein, G., Kanvinde, A., Myers, A., Yang, L. (2013). “Extremely low cycle Fell, B. (2011). “LOCAL CYCLIC fatigue prediction of steel beam-to- VOID GROWTH CRITERIA FOR column connection by using a micro- DUCTILE FRACTURE INITIATION mechanics based fracture model”. IN STEEL STRUCTURES UNDER LARGE-SCALE PLASTICITY”. N. Hoveidae/ Journal of Rehabilitation in Civil Engineering 6-2 (2018) 29-42 42

Proceeding of EUROSTEEL2011, Budapest, Hungary. [37] Amiri, H., Aghakoochak, A., Shahbeuk, S., Engelhardt, M. (2013). “Finite element simulation of ultra-low cycle fatigue cracking in steel structures”. Journal of Constructional Steel Research, Vol. 89, 175–184. [38] Afzalan, M., Ghasemieh, M. (2015). “Finite element modeling of failure in steel moment connection subjected to ultra-low cycle fatigue loading”. ACEM15, Korea. [39] Tateishi, K., Hanji, T. (2004). “Low cycle fatigue strength of butt-welded steel joint by means of new testing system with image technique”. International Journal of Fatigue, Vol. 26,1349–56. [40] Nakamura, H., Maeda, Y. & Sasaki, T. (2000). “Fatigue Properties of Practical- Scale Unbonded Braces”, Nippon Steel corporations. [41] ABAQUS (2013). Standard user’s manual version 6.13. Providence, RI: Hibbitt, Karlsson & Sorensen Inc. [42] Kanvinde, A., Deierlein, G. (2006). “The void growth model and the stress modified critical strain model to predict ductile fracture in structural steels”. Journal of Structural Engineering, Vol.132(12),1907–18. [43] AISC (American Institute of Steel Construction), 2010. Seismic provisions for structural steel buildings, Chicago, IL.