IntroductionBuckling-restrained braces (BRBs) are seismic devices composed of a tapered, energy-dissipating steel core encased in an axially decoupled restrainer, which is separated by a debonding interface. The debonding interface serves two purposes: to create a gap that accommodates compressive Poisson expansion of the yielded core and reduce friction at wavecrests that form as the core undergoes small-amplitude higher-mode buckling during compression half-cycles. These design objectives may be met using the debonding interfaces classified in Fig. 1. These include a thin antifriction material that undergoes plastic deformation with an internal slip plane (Interface type I), a thick antifriction material that compresses elastically and slides against the core plate (Interface type II), or a hard antifriction material that is offset by compressible filler or removable spacers and slides against the core plate (Interface type III). Bare steel–steel or steel–mortar interfaces with an air gap are also feasible (Interface type IV), but at the cost of a higher friction coefficient, while lubricants requiring maintenance are undesirable given that most applications have a multidecade design life and the BRB or interface is often inaccessible.Selecting an appropriate type of debonding interface depends on the desired performance, restrainer composition, and fabrication method. In a typical construction of mortar-filled steel-tube BRBs, which are widely used in practice, the debonding material is first attached to the core plate before placing the core in the steel tube and casting the mortar, which imposes a small hydrostatic pressure on the order of 0.05–0.5 MPa and bonds to the debonding material. For this type of BRB, good performance has been achieved using a soft, thin film of butyl, chloroprene, or silicon rubber (Chen et al. 2016; Guo et al. 2017; Iwata et al. 2000; Tsai et al. 2014). However, despite extensive device-level testing (Black et al. 2004; Tsai et al. 2014), there are few to no direct friction data for these Interface type I materials under the unique sliding conditions experienced in BRBs.The primary use case for the friction coefficient in BRBs is to numerically simulate the effects of higher-mode buckling of the core plate using nonlinear finite-element software, such as LS-Dyna, Abaqus, and Ansys. However, owing to a lack of direct friction data, current practice is to adopt a Coulomb friction model calibrated against peak BRB forces (Avci-Karatas et al. 2019; Budaházy and Dunai 2015; Chen et al. 2016; Guo et al. 2017; Stratan et al. 2020). These constant friction coefficients have varied from μ=0.03 to 0.5, but μ=0.1 is the most common estimate. Unfortunately, this trial-and-error procedure is potentially error prone because friction is a second-order force in BRBs and the friction coefficient may vary with the core geometry. Indirect calibration may mask compensating errors that affect key failure modes, and so a direct experimental basis for the friction coefficient is preferred.Direct experimental data are particularly important for polymers [e.g., polytetrafluoroethylene (PTFE), ultra-high-molecular-weight polyethylene (UHMWPE), rubber], because the friction coefficient tends to decrease with pressure and temperature (ambient and cyclic heating) but increase with velocity (Stachowiak and Batchelor 2014). Adhesion results in significant material transfer to the countersurface (Stachowiak and Batchelor 2014), while viscoelasticity may result in a greater response when loaded from an at-rest state (Kasai and Nishizawa 2010; Kumar et al. 2015). Dynamic breakaway effects are uniquely important for BRBs because the core straightens and contracts during tension half-cycles, causing contact to recur each compression half-cycle (one-way sliding), which differs from conventional reciprocating motion with continuous contact (two-way sliding). These dependencies are depicted schematically in Fig. 2.The objective of this study is to experimentally quantify the friction coefficient of a soft, thin polymer of Interface type I (Fig. 1), at full scale and with realistic sliding conditions. First, the plausible range of bearing pressures, slip velocities, and slip displacements are developed from a preliminary parametric numerical study. Next, the test rig is introduced, and results are presented from a comprehensive program of one-way and two-way sliding tests. These data are used to develop a nonlinear pressure–velocity–displacement–dependent friction model. Finally, implications for modeling higher-mode buckling and friction, qualification, and testing of BRBs are discussed.Pressure, Velocity, and Distance CharacterizationNo codified test protocol currently exists to systematically obtain the friction coefficient for BRB debonding interfaces, and the contact and slip demands that may plausibly be encountered in practice are not well understood. Therefore, a parametric study (Sitler and Takeuchi 2021) of 576 Abaqus/Explicit 2017 models (Smith 2017) was postprocessed to characterize the maximum and minimum bearing pressures, slip velocities, and slip distances and narrow the test specification.Two simplifications were adopted. First, because the proposed nonlinear friction model was not known a priori, constant-friction Coulomb models were adopted, following current practice. Additional validation models were later analyzed using the proposed nonlinear friction model, but they produced only minor changes. This confirmed that while many BRB response parameters and failure modes may be sensitive to friction, the contact and slip demands are not. Secondly, half-length two-dimensional (2D) models were adopted to improve runtime. These conservatively capture the demands at the plate edges, as three-dimensional (3D) anticlastic plate deformation reduces contact over the middle of the rectangular core plates (Sitler and Takeuchi 2021). BRB failure modes, such as global buckling and bulging, were not modeled to avoid arbitrarily limiting the analysis by a design-specific capacity.A wide range of core properties were investigated in the preliminary numerical study, including different core steel grades (LY100, LY225, SN400B, SN490B, SA440B, SA700), yield lengths (Lp=2, 6, 10, 14 m), thicknesses (tc=16, 25, 40 mm), debonding gaps ratios (sw/tc=0.01, 0.03, 0.05, 0.1) and friction coefficients (μ=0.1, 0.3). The subsequent validation study reanalyzed the SN400B, Lp = 2, 6 or 10 m and tc=16, 25 or 40 mm models using the nonlinear friction model with sw=0.5, 1.0 or 2.0 mm. Each model consisted of a fine mesh (tc/5×tc/5) of reduced integration shell elements (CPS4R) over a depth tc and length Lp/2. The mesh was offset by debonding gaps (sw, per face) to upper and lower analytical rigid surfaces representing a stiff, elastic restrainer. Hard friction contact was assigned but validated against a tabular hardening contact, which was also used for the nonlinear friction models. Half the BRB axial displacement (δ/2) was applied at the free end, while the opposite end was restrained to represent a midspan shear key. The protocol consisted of two cycles each at an average axial strain of ε¯=δ/Lp=0.5%, 1.0%, 1.5%, 2.0%, 2.5%, and 3.0%, which was applied sinusoidally at a 2 s period for the velocity-dependent models. The constitutive material models and further details are provided in Sitler and Takeuchi (2021).Bearing PressureThe bearing pressure (σN) was calculated by averaging the Abaqus field output CPRESS for the nodes in contact at each wavecrest, and the mean is shown in Fig. 3 at three strain levels. The bearing pressure was primarily determined by the debonding gap ratio (sw/tc) and was not meaningfully correlated with the other dimensions, location along the yield length, nor friction model. The low yield point (LY100, LY225) and mild steels (SN400B, SN490B) stabilized to similar bearing pressures at large strains (ε¯>2%), while the high-strength steels (SA440B, SA700) with yield strengths exceeding fy>400  MPa produced higher bearing pressures. This was due to the mechanics of higher-mode buckling and material properties, which are explained as follows.Normal force is generated at the higher-mode buckling wavecrests during compression half-cycles due to the core axial force acting at an incline determined by the sinusoidal wavelength and wave height (Fig. 3). Both are smallest at the core ends and gradually increase along the yield length as friction sheds force from core to the restrainer, which reduces the compressive stress (σc) and strain (εc). The wave height equals the total as-built debonding gap (2sw) less Poisson expansion of the core (−νp|εc|tc, where νp=0.5 is the plastic Poisson ratio), while the sinusoidal wavelength (lp,w) is a function of the material-specific axial stress (σc) and tangent stiffness (Et) and may be estimated as lp,w/tc≈2π√(Et/3σc) (Sitler and Takeuchi 2021). The lower tangent stiffnesses of the high-strength (SA440B, SA700) and low-yield-point (LY100, LY225) steels result in shorter wavelengths than SN400B and SN490B, while the axial stress increases with the steel grade. This suggests that the bearing pressure will be substantially higher for SA400B and SA700, but almost identical for LY100, LY225, SN400B, and SN490B, which is consistent with the numerical results.Based on this understanding, an analytical estimate of the wavecrest bearing pressure (σN) is given by Eq. (1), where the compressive stresses (σc), wave heights (2sw−νp|εc|tc), and sinusoidal wavelengths (lp,w) may be obtained by cycling the engineering stress-strain curve to εc=−ε¯. The contact length (lc) was complex because highly concentrated initial contact tended to be alleviated as the wavecrests flattened, until new waves eventually formed via snap-through buckling. Nevertheless, the total contact length was about lc≈0.9tc for fully developed wavecrests, but about half this length at initial contact (Fig. 3). Both the average and peak bearing pressure may be obtained as(1) σN≈2σc·tc0.5lp,w·(2swtc−νp|εc|)·tclcGiven that the debonding gap ratio (sw/tc) was the dominant parameter at large strains for most steel grades, Eq. (1) may be further simplified to Eq. (2), where the pressure concentration factor (αP) may be taken as αP=1 for fully developed wavecrests with lc=0.9tc and αP=2 at initial contact. Eq. (2) may be used by engineers to estimate the wavecrest bearing pressure (2) σN≈αP·50  MPa·10swtc(for  fy<400  MPa, ε¯>2%)As the most common debonding gap ratios (0.0130  MPa. The one- and two-way friction coefficients converged for quasi-static and large-amplitude (±40  mm) sliding, while the velocity dependency peaked at Vmax=30  mm/s, with higher velocities producing a beneficial cyclic heating effect, which is conservatively neglected.DiscussionFriction Coefficient for Use in Numerical SimulationThe nonlinear friction model presented in Eqs. (9)–(13) was implemented as Abaqus/Explicit 2017 user subroutines (Sitler 2021). Given the complex dependencies and variable slip along the core yield length, simplified friction models indirectly calibrated against the BRB force-displacement hysteresis may be inaccurate when the core geometry or loading protocol significantly differs from the test specimen. Specifically, a larger friction coefficient is expected for thicker cores with smaller debonding gap ratios (sw/tc), as these produce lower bearing pressures, and for dynamic loading tests. Similarly, the greater cumulative slip displacement of longer cores increases the wear effect, which may be artificially suppressed in the small-scale specimens often used in academic studies.Nevertheless, in some cases it may be reasonable to simplify the nonlinear friction model, specifically when the cumulative friction effects are small or to validate a complex model. In these cases, the small amplitude (k1=1.8), high pressure (σN,0=30  MPa, kP=0.9), slow speed (Vmax<5  mm/s, kV=0.7), and final cycle (∑δslip=0.3  m, tpoly=1  mm, μW=0.02) friction coefficient of μ=0.09 is nearly equal to the μ=0.1 used by many researchers and may be a reasonable starting point. However, in general, experimentally validated friction models that include the unique dynamic breakaway and wear effects are recommended, such as the proposed nonlinear friction model.Quasi-Static Qualification TestingDynamic testing is not currently required by the AISC 341-16 specification owing to cost and test facility limitations (§Comm.K3.4, AISC 2016), while dynamic device-level tests have produced mixed results (Hasegawa et al. 1999; Lanning et al. 2016; Qu et al. 2020). However, the argument for incorporating loading rate effects fundamentally differs for the steel core, which is discussed in AISC 341-16, and the debonding material, which is not. Dynamic loading increases both the BRB forces and yield strengths of the force-controlled members, minimizing the change in the demand-to-capacity ratio. However, velocity-dependent friction exclusively affects demand, specifically by increasing the friction component (βF) of the compressive adjustment factor (β).This is an important distinction because the 2016 edition of AISC 341 §K3.8.d increased the acceptance criteria to β<1.5, such that friction may now constitute up to approximately 40% of the quasi-static BRB compressive force, significantly increasing the potential influence of the friction coefficient on the overall behavior. Because dynamic testing introduces its own challenges, a practical approach may be to (1) obtain β from full-length quasi-static tests, (2) estimate the friction component as βF≈β/(1+2ε¯), (3) scale the quasi-static βF to an average dynamic value (βF,dyn) using Eq. (14), and (4) recombine with the Poisson component to produce a design value of βdyn=βF,dyn×(1+2ε¯). This effectively reimposes the previous acceptance criteria of β<1.3 for quasi-static tests, assuming Cslow=0.6/k1≈0.45, while increasing the compression design forces by up to 20%. The unadjusted β test value may still be used for full-length dynamic testing, velocity-insensitive debonding materials (e.g., Interface type IV) or if friction is negligible (e.g., β<1.1), otherwise, the following correction applies: (14) βF,dyn=1+(βF−1)·∫0VstrokekV(V)Cslow≈1+βF−1CslowFinally, dynamic tests with peak stroke velocities exceeding about 100  mm/s should produce similar friction velocity dependency due to the low saturation velocity of kV. Cyclic heating may partially mitigate this adverse effect, but there are currently no comparative temperature data for full-scale BRBs, a prerequisite to take advantage of beneficial temperature dependency.References AISC. 2016. Seismic provisions for structural steel buildings. AISC 341-16. Chicago: AISC. Avci-Karatas, C., O. C. Celik, and S. 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