A Procedure for Assessing Low-Cycle Fatigue Life of Buckling-Restrained Braces

The concept of buckling-restrained brace frames (BRBFs) was originally developed in Japan (Uang et al. 2004; Xie 2005; Takeuchi and Wada 2017) and has been a popular seismic force-resisting system in the US since it was first introduced to the AISC Seismic Provisions and ASCE 7 in 2005. Extensive research including component and system-level testing has demonstrated that BRBFs are a reliable seismic force-resisting system (Clark et al. 2000; Iwata et al. 2003; Sabelli et al. 2003; Uang et al. 2004; Tremblay et al. 2006; Fahnestock et al. 2007; Tsai et al. 2014). As a result, BRBFs have gained wide acceptance for new construction or seismic retrofit of buildings and other structures in high seismic regions in the world.

BRBFs are expected to provide significant inelastic deformation capacity primarily through buckling-restrained braces (BRBs). Under seismic loading, the steel core of a BRB yields in tension and buckles plastically in a high-mode buckling pattern in compression. These behaviors contribute to stable hysteric energy dissipation during cyclic loading. Although results from numerous testing showed that a BRB can easily meet the seismic demand from a significant seismic event, the question pertaining to the remaining fatigue life of the BRB for future earthquakes remains to be answered. It is highly desirable that a low-cycle fatigue (LCF) damage assessment procedure be established such that an engineer can make a decision if BRBs need to be replaced after an earthquake.

To date, there have been limited experimental studies on the LCF of BRBs. In Japan, Maeda et al. (1998) conducted fatigue tests on 75-mm-long flat core plates made of low-yield-strength steels. Nakagomi et al. (2000) performed experiments on all-steel BRBs in which the steel cores were restrained by a square tube. Their test results showed that the Coffin-Manson relation (Coffin 1954; Manson 1953), which is widely used to describe the LCF in terms of a strain-life relationship for metallic materials, can be applied to steel core plates. Nakamura et al. (2000) conducted tests on 11 2,000-mm-long BRBs with a steel core made of low-yield and mild-strength steels (LYP100, LYP235, or SN400B), which had a measured yield stress of 95, 230, and 260 MPa, respectively. The lengths of the yielding steel core ranged from 470 to 1,180 mm, the yield axial forces ranged from 320 to 693 kN, and the imposed core strain amplitudes ranged from $\pm 0.08\%$ to $\pm 2.25\%$. Test results showed that similar fatigue properties could be found from BRBs with various steel grades. Usami et al. (2011) and Wang et al. (2012) tested nine 2,015-mm-long all-steel BRBs with steel cores that were made of SM400 steel with measured yield stresses of 271 and 291 MPa and had a yielding steel core length of 1,375 mm. The imposed core strain amplitudes ranged from $\pm 1$ to $\pm 4\%$. Test results showed that overall strain-life relationship of theses all-steel BRBs was similar to but differed slightly from that observed by Nakamura et al. (2000). A comparison of the Coffin-Manson fatigue curves obtained from the BRB tests and material tests showed that the fatigue life of a BRB was shorter than that of the steel core base metal. This is the result of the high-mode buckling that takes place during the compression excursion; the additional flexural strains consume fatigue life of the steel core.

In addition to the Coffin-Manson LCF models that were calibrated based on the results of BRB fatigue tests in Japan, researchers also developed cumulative deformation capacity models for BRBs. Takeuchi et al. (2008) developed a model to estimate the cumulative deformation capacity of BRBs by decomposing the hysteretic loop into the skeleton and Bauschinger parts. It is shown through a validation with experimental data that their proposed model predicts the fatigue damage of BRBs better than a conventional approach that assesses cumulative damage by using Miner’s method (Miner 1945). Based on the results of 76 BRB tests, Andrews et al. (2009) developed empirical models for the total and remaining ductility capacities of BRBs by using a probabilistic modeling framework. Matsui and Takeuchi (2012) proposed a BRB cumulative deformation capacity assessment model by applying the available LCF models of steel core metallic materials to the strain concentration zone in the BRB. Which considers the effect of high-mode buckling on the core plate. Wei et al. (2019) investigated the thermal-induced LCF of BRBs in end diaphragm systems of bridge structures by using three LCF models for BRBs that were calibrated on the basis of fatigue test data in Japan.

The fatigue performance of BRBs is related to the local bucking amplitude of steel core, which in turn is a function of the details of the BRB design (e.g., the gap between the steel core and the buckling-restraining mechanism). The BRB specimens used in previous fatigue tests in Japan were about 2-m long and had an actual yield strength smaller than 700 kN, which falls in the lower-end region of BRB capacity in real applications. Also, a significant portion of the tested BRBs were made from low-yield-strength steel (e.g., LY100 and LY220 with a nominal yield stress of 100 and 220 MPa, respectively), which is not commonly used in the U.S. Therefore, a fatigue test program on representative full-scale BRBs commonly used in the US was initiated. A total of 18 BRB specimens were tested. The overall length of the BRB specimens was about 5,400 mm and the actual yield strength ranged from 1,110 to 3,330 kN. Symmetric, constant-amplitude fatigue tests with an imposed strain amplitude ranging from $\pm 0.25$ to $\pm 3\%$ were conducted on 11 specimens. These 11 tests provided a database to develop a standard LCF model in a form equivalent to the Coffin-Manson-Basquin relation (Morrow 1945) and an alternative model. A procedure using the proposed LCF models together with the rainflow cycle counting method (Matsuishi and Endo 1968) and Miner’s damage rule (Miner 1954) was then developed to assess the low-cycle fatigue life for BRBs subjected to random cyclic loading histories. Five variable-amplitude fatigue tests, including simulated earthquake loading tests, were performed to examine the accuracy of the proposed procedure. In addition, two asymmetric, constant-amplitude cyclic tests were conducted to investigate the mean strain effect on the LCF performance of BRBs.

Three sets of nominally identical BRB specimens for a total of 18 BRB specimens were tested in the Seismic Response Modification Device (SRMD) Test Facility at the University of California, San Diego, with the test setup as shown in Fig. 1. Table 1 depicts the test matrix. The steel cores and steel casing were manufactured from A36 and A500 Gr. B steel, respectively. The three sets (Series A, B, and C) had incrementally larger core cross-sectional areas, $A_{sc}$, with an actual yield strength, $P_{ya}=A_{sc}{F}_{ya}$, of 1,110, 2,220, and 3,330 kN, respectively, where the measured yield stress, $F_{ya}$, of the A36 steel cores in Series A, B, and C specimens were 303, 307, and 299 MPa, respectively. Fig. 2 shows the specimen geometries, indicating the overall brace length and the length of yielding steel core, $L_y$.

The fatigue tests were conducted in a displacement-control mode and were divided into two groups: “constant-amplitude” and “variable-amplitude” cyclic tests. The constant-amplitude tests were further categorized into two types: tests with a zero-mean strain and tests with non-zero-mean strains. There were 11 specimens tested with symmetric strain cycles to fracture [Fig. 3(a)]. Within each set, three specimens were tested with a constant amplitude of $\pm 0.25$, $\pm 0.75$, and $\pm 2.0\%$ core strains, respectively. The core strain, $\epsilon$, was defined as ${\mathrm{\Delta }}_b/L_y$ for simplicity, where ${\mathrm{\Delta }}_b$ was the brace axial deformation. The three core strain amplitudes corresponded to axial deformations of $1.67{\mathrm{\Delta }}_{by}$, $5.00{\mathrm{\Delta }}_{by}$, and $13.3{\mathrm{\Delta }}_{by}$, where ${\mathrm{\Delta }}_{by}=P_{ya}{L}_y/E{A}_{sc}$ was the brace axial deformation at first yield.

Note that the brace deformation ${\mathrm{\Delta }}_b$ was measured between the measuring points (marked as + in Fig. 2) outside the casing (also see Fig. 1). Because ${\mathrm{\Delta }}_b$ contains the elastic strains outside the yield zone, the core strain determined by ${\mathrm{\Delta }}_b/L_y$ would slightly overestimate the actual core strain. Based on the geometry of the BRBs tested in this research, core strains thus calculated would overestimate the actual core strain by about 5% when the braces are deformed in the elastic range. However, the overestimation would reduce to about 2.5% and 0.5% when the core strain reaches 0.25% and 2%, respectively. To apply a fatigue model to BRBs in real buildings, it is inevitable that the BRB axial deformation be measured in a way similar to that used in this test program. Therefore, such slight overestimation of the strains is acceptable.

Two additional specimens in the B series were tested with higher strains. Specimen B5 was tested with a constant amplitude of $\pm 3.0\%$ for a strain range of 6% and Specimen B6 was tested with $\pm 2.5\%$ cycles for a strain range of 5%. To evaluate the mean strain effect, Specimens B7 and B8 were tested with the same strain range as B6 but with a mean strain of $+1$ and $-1\%$, respectively. That is, B7 was tested with the strain amplitudes at $-1.5\%$ and $+3.5\%$, and B8 was tested with the strain amplitudes at $-3.5\%$ and $+1.5\%$.

In the variable-amplitude test group, two specimens were subjected to symmetric cycles, i.e., with a zero-mean strain but with variable strain amplitudes. Specimen A5 was tested with repeats of a symmetric, increasing (SI) cyclic loading sequence [Fig. 3(c)], which was adapted from the loading protocol of the BRB qualification test prescribed in the AISC Seismic Provisions (AISC 2016), until fracture. Specimen C5 was subjected to a two-stage constant-amplitude (2SCA) test with symmetric cycles [Fig. 3(d)]. It was first subjected to a total of 1,100 cycles at $\pm 0.25\%$ strain to consume about a half of the fatigue life before $\pm 0.75\%$ strain cycles were applied until fracture.

Simulated earthquake loading tests were conducted for three specimens, one for each of the three series of specimens. These loading sequences were designated with EQ. Nonlinear time-history analyses were performed on an example four-story BRBF in the AISC Seismic Design Manual (AISC 2018) to generate the BRB core strain time histories. The nonlinear structural analysis software PISA3D (Lin et al. 2009) was used for the analyses. For scaling of the selected ground motions, the response spectrum of the maximum considered earthquake (MCE) based on $S_{DS}=1.0$ and $S_{D1}=0.6$ was first constructed (ASCE 2016). The analytical fundamental period, $T_1$, for the BRBF model was 0.819 s.

Specimen A4 was subjected to the earthquake loading sequence EQ1 [Fig. 3(e)], which was repeated until the BRB fractured. This earthquake suite was composed of three parts. The first part was generated from a time-history analysis with an input motion from the 1989 Loma Prieta earthquake record (Gilroy Array #6 Station) scaled to the 120% MCE level at $T_1$. The simulated core strain time history of a BRB in the first story is shown in the first part of Fig. 3(e). The second part of the response was based on the 1999 Hector Mine earthquake (Hector Station) scaled to the MCE level. Note that this part had a large residual strain. Reversing the sign of the strain time-history of this part constituted the third part such that the combined time history of the three parts had no residual strain.

Specimen B4 was tested with the loading sequence EQ2 [Fig. 3(f)] repeatedly until fracture. The first part of this earthquake suite was generated from a near-fault ground motion record (Sylmar-Converter Station) from the 1994 Northridge earthquake scaled to the MCE level. Reversing the sign of the first part constituted the second part, resulting in a zero residual strain.

Specimen C4 was tested with the loading sequence EQ3 [Fig. 3(g)] repeated until fracture. This earthquake suite was composed of six parts of MCE level earthquake responses. The first two parts were composed of two repeated core strain responses obtained from the analysis with a ground motion record (Gilroy Array #6 Station) from the 1989 Loma Prieta earthquake. Part 3 was the response from the 2016 Amberley earthquake in New Zealand (Kekerengu Valley Road Station); reversing the sign of this part constituted Part 4. Similarly, Parts 5 and 6 were the response from the 1985 Michoacán earthquake in Mexico (La Union, Guerrero Array Station).

Table 2 summarizes results of the constant-amplitude tests. The BRB fatigue lives under the $\pm 0.25\%$, $\pm 0.75\%$, and $\pm 2.0\%$ strain cycles were on the order of 2,000, 200, and 20 cycles, respectively. Based on the limited test data, the BRBs survived 11 and 9 completed cycles for the $\pm 2.5\%$ and $\pm 3.0\%$ strain amplitudes, respectively.

Fig. 4 shows the measured axial deformation time-history a sample specimen (Specimen B1) from a symmetric constant-amplitude test. For a BRB like this one with a high force requirement ($P_{ya}=\mathrm{2,220}\mathrm{kN}$), it was not possible to complete all the cycles required to fracture the core (more than 2,300 cycles) in one test run due to hydraulic limitations of the test facility. Therefore, this test was completed in 12 separate test runs (the duration between each test run was about 20 min). This example response shows that the test facility was not able to reproduce faithfully the target amplitude. This was accounted for by using the actual averaged strain amplitudes recorded to develop low-cycle fatigue models. Fig. 5 shows sample hysteretic responses of the B Series specimens tested with constant-amplitude cycles and a zero-mean strain.

For specimens subjected to constant-amplitude tests, the normalized total inelastic axial deformation for the $i$th cycle, ${\mu }_i$, with a deformation level greater than the yield deformation ${\mathrm{\Delta }}_{by}$ is given bywhere ${\mathrm{\Delta }}_i^{+}$ and ${\mathrm{\Delta }}_i^{-}$ = peak tension and compression deformations for the $i$th cycle, respectively. The cumulative inelastic deformation ($CID$) is then computed as the sum of the normalized inelastic axial deformation of each cycle up to fracture

See Table 2 for the calculated $CID$ values. For BRB qualification tests, AISC Seismic Provisions (AISC 2016) require tested BRBs to achieve a $CID$ of at least 200. All the constant-amplitude test specimens achieved a $CID$ greater than this requirement. The $CID$ of the $\pm 0.25\%$ strain specimens reached from 5,600 to 6,800. The $\pm 0.75\%$ strain specimens withstood a $CID$ ranging from 2,900 to 4,000. The specimens tested with $\pm 2.0\%$ strains sustained a $CID$ between 800 and 1,500. When the strain amplitude was increased to $\pm 2.5\%$ and $\pm 3.0\%$, the BRB still achieved a $CID$ of 694 and 687, respectively. It is apparent that the larger the imposed strain amplitude, the smaller the $CID$ the BRBs will likely achieve.

In addition, Table 2 lists the averaged compressive overstrength factor, $\beta$, for the constant-amplitude tests with a zero-mean strain. The $\beta$ is defined as $P_{\max }/T_{\max }$, where $P_{\max }$ and $T_{\max }$ are the brace axial forces at the peak compression and tension displacements in each cycle, respectively. Test results showed that the average $\beta$ value was less than the AISC limiting value of 1.5 for all specimens.

To show the mean strain effect, Fig. 6 shows the test results of the three Series B specimens tested with the same strain range (5%) but with different mean strains. Specimen B6, which had a zero-mean strain (i.e., the strain amplitude ranged from $–2.5\%$ to $+2.5\%$), withstood 11 completed cycles and ruptured just before the positive tensile strain excursion in the 12th cycle was reached. Specimen B7 was tested with a mean strain of $+1.0\%$ (i.e., the strain amplitude ranged from $–1.5\%$ to $+3.5\%$) and survived 14 completed cycles. Specimen B8 was loaded in a manner similar to that of Specimen B7 but with a mean strain of $–1.0\%$ (i.e., the strain amplitude ranged from $–3.5\%$ to $+1.5\%$). The specimen ruptured before completing 10 cycles. Based on this limited database, a positive (tensile) mean strain would increase the BRB fatigue life, but a negative (compressive) mean strain would decrease the fatigue life. These results suggest that compression strains are more detrimental to BRB fatigue life than tension strains. The higher localized strains that result from the high-mode buckling of steel core in compression are believed to be the primary culprit. Also, the compressive strains tend to oblate microvoids, which could exacerbate the ductile fracture process (Kanvinde and Deierlein 2007).

Fig. 7 shows the core strain time-histories and hysteretic responses of variable-amplitude tests. Table 3 summarizes the test results. Specimen A5 completed six test runs of the adapted AISC protocol [Fig. 3(c)] and fractured during the first 2.0% cycle in the seventh test run [Fig. 7(a)]; the value of $CID$ reached 1,580 at rupture. Specimen C5 first completed 1,100 cycles at $\pm 0.25\%$ strain amplitude, followed by another 113 cycles of $\pm 0.75\%$ strain cycles before fracture [Fig. 7(b)]. The specimen developed a $CID$ value of 5,033. Figs. 7(c–e) show the responses of the three specimens tested with simulated earthquake responses. Specimens A4, B4, and C4 generated $CID$ values of 1,584, 1,592, and 998, respectively. It is noted that each of these three specimens survived the simulated MCE response more than 10 times.

To provide a tool that allows the designer to assess the remaining life of buckling-retrained braces (BRBs) and to determine if these braces need to be replaced after a seismic event, a set of cyclic tests to failure was performed. These tests include a total of 18 full-scale braces tested at the SRMD facility at the University of California San Diego. Eleven of these specimens were tested with constant strain amplitudes with a zero-mean strain, and the test data were used to establish one standard [Eq. (9)] and one alternative [Eq. (11)] fatigue model. In addition to the mean fatigue life, prediction intervals with a confidence level of 95% are also developed [Eqs. (18) and (19)]. Together with the rainflow counting method and the Miner’s damage rule, a procedure to assess the fatigue damage of BRB is proposed. The accuracy of the proposed procedure is verified not only by these 11 tests but also two constant-amplitude tests with a non-zero-mean strain, and five variable-amplitude tests. The following conclusions can be made:

It should be noted that the proposed fatigue models are based on one type of commercially available BRBs with A36 steel core. Because the fatigue performance of BRBs would be affected by parameters such as steel grade of the core plate, details of the buckling-restraining mechanism, stiffness of the concrete grout, and gap size in the debonding zone, different BRB fatigue models would need to be developed when the aforementioned parameters deviate significantly from the ones investigated in this research.

Some or all data, models, or code generated or used during the study are proprietary or confidential in nature and may only be provided by CoreBrace, LLC, with restrictions. However, some data could be shared with reviewers on a limited basis as needed.

The authors would like thank Messrs. D. Innamorato and E. Stovin, staff members of Seismic Response Modification Device (SRMD) Test Facility at the University of California, San Diego, for their technical assistance towards the completion of this test program.