Alternative approach to description of the BRB backbone curve

With my participation an alternative Eurocode-conform bilinear representation of the BRB force-displacement envelope has been developed. Parameters of the proposed bilinear curve are determined with the so-called Strength Adjustment Method (SAM). The SAM is based on regulations in AISC 341-10; it uses variables that can directly be applied by engineers in their calculations.

The yielding point of the BRB is described by the material overstrength factor:

γ_m,ov=𝑅y,a

𝑅_y,c=𝑓y,a𝐴y

𝑓_y𝐴_y =𝑓y,a

𝑓_y (6)

where γm,ov is the material overstrength factor; Ry,a is the actual yield capacity of the device;

Ry,c is the nominal capacity; fy,a is the actual yield strength (preferably from material tests); fy

is the characteristic yield strength of the BRB material; Ay is the cross-sectional area of the steel core in the yielding zone. According to experimental results [Z12, Z13, Z14] the initial stiffness of BRBs is in good agreement with that of the steel core, thus the stiffness of structural steel can be applied in BRB design calculations. Therefore, given γm,ov the yield force and displacement of any BRB made of the corresponding material can be calculated.

BRB hardening is divided into a symmetric and an asymmetric component. It is described by two strain dependent ratios. The ωε strain hardening adjustment factor represents the symmetric hardening component, while the βε compression strength adjustment factor describes the additional asymmetric hardening under compression:

ω_ϵ=𝑉_ϵ,T⁄𝑅_y,a (7)

β_ϵ=𝑉_ϵ,C⁄𝑉_ϵ,T (8)

where Vε,T and Vε,C are the load bearing capacities corresponding to ε strain under tension and compression respectively. (A device with identical behavior under compression and tension is characterized by βε=1.0 for all ε strains, while ωε=1.0 corresponds to a device with no inelastic hardening capability.) Generally both ωε and βε are nonlinear functions of BRB strain.

Therefore, a bilinear force-displacement relationship can only be defined by selecting a pre- defined ε level as a reference point for design and testing. This εref reference strain level is proposed to be device specific. The two factors and the bilinear relationship they define are shown in Fig. 19. A straightforward way to represent these data is in a normalized force – yielding zone strain plane. An advantage of this normalized representation is that it makes the results of different specimens directly comparable.

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Fig. 19 Proposed alternative bilinear relationship based on the Strength Adjustment Method

I evaluated parameters of both the TBC and the SAM from experimental test data. Besides the results from BME, 14 tests from the University of Utah [9] and the University of California, San Diego [5] have also been included in the evaluation with the permission of Star Seismic LLC. This extension of the set of results provides data about the full spectrum of BRBs in terms of both length and capacity. Therefore, the evaluated backbone is expected to be more general and widely applicable.

Evaluated variables for individual specimens for both approaches are presented in Appendix A2. Because the TBC is defined in a specimen-dependent manner, the individual TBC curves cannot be used to provide a general result applicable to all BRBs. Therefore, the SAM-based general curve is used to describe BRB behavior in both subsequent research and practical applications. A reference strain level of 2% is proposed because BRBs in buildings designed as per EC8 regulations shall not have to accommodate larger design deformations under the design seismic excitation. Table 2 shows the evaluated parameters for SAM based on the mean of experimental results. The individual backbone curves and the mean are displayed in Fig. 20. An expected deviation limit from the mean is also assigned to each value that can be used in design and quality control. The limit for the overstrength factor is based on the results of material tests. Deviation limits for the hardening ratios are defined to be in line with the limits for other NLD properties in EN 15129 6.2.

The proposed SAM is not limited to BRBs, but intended for a more general application for all nonlinear displacement dependent devices (NLDs) [Z3]. It is considered a more advantageous approach to the backbone curve of NLDs than the TBC in EN 15129. SSE BRBs described with SAM were approved for the CE marking by Austrian Standards plus GmbH using the characteristics in Table 2.

Table 2 Expected characteristics of Buckling Restrained Braces based on test results

nominal BRB capacity (Ry,n) [kN] 100 - 5000

material overstrength factor (γov,m) 1.13 ±8%

strain hardening adjustment factor at εref = 2% (ω2%) 1.57 ±15%

compression strength adjustment factor at εref = 2% (β2%) 1.18 ±15%

total hardening adjustment under compression at εref = 2% (ω2%β2%) 1.85 ±15%

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Fig. 20 Bilinear representations of BRB behavior from the 21 test results (black) and the proposed ideal backbone (red) with its acceptable deviation (dashed red)

2.5.4 Quality control

Definition of a backbone curve is not only beneficial for practicing engineers, but also useful for quality control purposes. The objective of quality control for NLDs is typically to ensure stable cyclic behavior without excessive increase of internal forces to limit the necessary overstrength applied at adjacent elastic members. Note that the actual value of strain- dependent hardening is not a concern, because it can be taken into consideration with a sufficiently high overstrength factor. Cyclic hardening⁶ on the other hand can have detrimental consequences on structural performance if not addressed appropriately during design.

EN 15129 includes specifications on quality control. An alternative procedure has been developed at BME that uses the bilinear backbone curve based on the SAM approach. Its advantages over the standardized approach are briefly explained in Appendix A3.

Based on the evaluation I performed on the behavior of 20 BRB specimens (see Appendix A2 for detailed results) I proposed the following formulae to regulate the quality of a set of BRB elements:

Κ_ω=�ωϵ_ref,𝑖− ωref�

ω_ref ≤0.15 𝑖> 1 (9)

Κ_ωβ =�ωϵ_ref,iβϵ_ref,𝑖− ωrefβref�

ω_refβ_ref ≤0.15 𝑖 > 1 (10) where ωref and βref are the expected BRB hardening rates, such as the values in Table 2; ωεref,i

and βεref,i are the hardening rates of the tested specimen in the i^th load cycle at εref strain level.

6 cyclic hardening refers to the increase of material strength in consecutive load cycles at the same strain level

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BRB