Numerical model

Although results in Chapter 3 confirm the superior accuracy of the developed Steel04-based BRB model, the computational performance of this solution admittedly lags behind that of a simpler bilinear approach. The additional computational time required for the analysis is

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justified if the difference in element behavior accuracy is also apparent in the global behavior and the evaluation results. This is investigated by comparing the results of three models:

- M1 is the Steel04 based complex approach from Chapter 3

- M2 is a bilinear kinematic hardening approach with 4% hardening rate. The hardening rate was calibrated to the experimental results in Chapter 1 to give a good approximation of the envelope BRB response. This model is a good representation of BRB modeling in typical BRBF-related studies such as [66,17].

- M3 is a bilinear kinematic hardening model, but with a significantly lower hardening rate of 0.5% under tension and 2.5% under compression. These rates were selected to match the kinematic hardening of the calibrated Steel04 model. This approach was suggested based on results of uniaxial numerical analysis in [Z3]. The rationale for the selected hardening rates is the accurate modeling of the kinematic hardening of the device.

Stress-strain response of the three models under a typical load protocol with gradually increasing amplitudes is displayed in Fig. 67. Median IDA curves with ± 1 standard deviations from response history analyses, calculated lower and upper bound fragility curves and collapse probabilities are shown in Fig. 69. The M2 model clearly overestimated the performance by a large margin (31% in median collapse intensity and more than 50% in collapse probability), thus its application leads to highly unconservative results. Fig. 68 compares the stress-strain response of the BRBs on the ground floor of the sample structure with the three models at Sa(T1) = 0.5 g intensity level. The figure is representative of observed responses on other stories, other seismic intensities and under other ground motion records.

Note that BRBs with M2 experience unrealistic axial forces because of their unrealistically high post-yield stiffness. This additional stiffness is the reason behind the increased performance, because it effectively limits horizontal displacements and the influence of the P- Δ effect. Nevertheless it also introduces higher internal forces in the columns and eventually leads to incorrect expectations of premature column failure.

Fig. 67 The stress-strain response of the advanced Fig. 68 Stress strain response of the materials from Fig.

nonlinear and two bilinear approaches are compared 67 under dynamic loading.

under typical quasi-static loading (M1 – gray, M2- red, M3 – blue)

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Fig. 69 Median drift curves (left) and fragility curves (right) of the same structure with three different settings for BRB material (M1 – gray, M2 – red, M3 – blue)

Because a typical earthquake at the design seismicity does not induce excessive isotropic hardening, the approximate response of M3 in absence of isotropic effects is still a considerably better solution than M2. More importantly, M3 provides conservative results, thus its use involves less risk. However, the inferior nature of the model needs to be recognized by its users and considered during evaluation with special emphasis on structural overstrength, because column forces are typically underestimated with this model.

I believe that the increase in accuracy and the customizable behavior of M1 justifies its application in the presented research.

BRB hardening rate and material overstrength

The effect of different BRB model parameters is tested in this analysis. Besides the calibrated results, three additional configurations for inelastic hardening were prescribed according to Table 7. They represent the lower and higher bounds of experimental result and an additional test with highly asymmetric hardening. Numerical analysis results in Fig. 70 confirm that the inelastic stiffness has significant influence on BRBF performance. The values selected during calibration provide results below the mean of lower and higher bounds, thus they are considered to be on the conservative side. Note that the asymmetric setup did not lead to performance reduction, which implies that BRBs with high hardening rates under compression are still acceptable for seismic design.

The influence of material overstrength is examined through the test of two additional values:

1.10 and 1.20 are selected as the lower and upper bounds from experimental results. Results in Fig. 71 show that a higher γov has a minor and trivial influence on structural behavior. The larger its value the better the frame performs in general. Note that this only holds within the range of reasonable overstrength values for BRBs. Very high overstrength would lead to excessive BRB hardening and premature column failure.

Table 7 Hardening characteristics of the investigated BRB models

hardening type low normal high asymmetric

ρiso 1.3 1.4 1.55 1.4

bt 0.3 % 0.4 % 0.6 % 0.3 %

bc 2.0 % 2.5 % 4.0 % 4.0 %

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biso 0.012 0.015 0.018 0.012

b_iso,c 0.008 0.013 0.015 0.015

Fig. 70 Median drift curves (left) and fragility curves (right) of four structures modeled with different BRB hardening characteristics(normal – gray; high– blue; low – red, asymmetric – green)

Fig. 71 Median drift curves (left) and fragility curves (right) of four structures modeled with different BRB material overstrengths (1.13 – gray; 1.10– blue; 1.20 – red)

Material model for steel columns

The application of a more complex bilinear model with isotropic hardening and low cycle fatigue consideration as explained in section 4.3 is justified by the following analysis. The applied model (M1) is compared to a simple linear elastic (M2) and a bilinear model with a strict elastic behavior by limiting plastic strain at 0.5% (M3). Fig. 72 displays the performance of the sample structure with the three different models, while Fig. 73 shows the force and moment response at the bottom column of the braced frame over time. M2 clearly overestimates structural performance, because column failure is not represented in this model. Note how the accurate yielding of columns at their plastic capacity affects the overall structural response. Also note that the 0.5% strain cap leads to failure of the first model because of column failure.

Stability failure is an important concern in the design of columns that shall be addressed in the analysis. The plots in Fig. 73 highlight that base columns are typically subjected to excessive bending and their axial loading does not reach their flexural buckling resistance. Therefore, application of initial imperfections (a measure that significantly increases computation time) on the model to properly simulate the buckling effect is not recommended. Columns on higher levels experience axial forces closer to their capacity, but the results imply that in these type of 77

structures the base column is expected to fail before flexural buckling of the upper members.

The base column is also susceptible to lateral torsional buckling, although note that its wide flanges and short length lead to sufficient resistance against this failure mode. The typical value of the χLT reduction factor that shall be applied at the bending resistance is in the range of 0.98 – 0.995. This is taken into account by reducing the yield strength of the material for the columns by a factor of 0.98.

Fig. 72 Median drift curves (left) and fragility curves (right) of three structures modeled with different inelastic properties for columns (M1 – gray; M2– blue; M3 – red)

Fig. 73 Influence of the column material model selection on the utilization (i.e. demand / capacity) in the bottom outer column of the braced frame under seismic excitation.

4.4.2 Design procedure