Deterioration Modeling of Steel Components in Support of Collapse Prediction of Steel Moment Frames under Earthquake Loading

Cumulative distribution functions (CDFs) for ${\theta }_p$, ${\theta }_{pc}$, and $\Lambda$ as obtained from the four data sets are shown in Fig. 3. Each plot shows CDFs for other-than-RBS and RBS sections. The CDFs reveal general statistical characteristics but do not display dependencies on individual properties. This information is relevant for detailed studies concerned with quantifying modeling uncertainties and their effect on the collapse capacity of structural systems subjected to earthquake excitations; so far, a systematic collection of experimental data that could be used to document statistical information (median and standard deviation) on deterioration parameters of components was not available. The lognormally distributed CDFs for the four data sets shown in Fig. 3 are comparable, but, in general, the median value of the modeling parameters for beams with other-than-RBS connections is smaller than that for beams with RBS connections. The dispersion is larger for other-than-RBS beams compared with beams with RBS, partially because this set includes experimental data from different connection types.

An increase in beam depth $d$ is associated with a clear decrease in modeling parameters. This is supported by Fig. 4(a), which shows data and a linear regression line for the precapping plastic rotation ${\theta }_p$ for data set 1 (full data set for other-than-RBS beams). This data set includes beams with a depth varying from 102 to 914 mm (4 to 36 in.). Others (FEMA 2000a, b) have pointed out the strong dependence of plastic rotation capacity on beam depth. This strong dependence is driven in part by the incorporation of small sections in the database and is not confirmed for the range of primary interest for tall buildings [$d\ge 533\mathrm{mm}$ (21 in.)] on the basis of Fig. 4(b).

On the basis of simple curvature analysis with disregard of local instabilities, ${\theta }_p$ of a given beam section is perceived to be linearly proportional to the beam shear span $L$ (distance from plastic hinge location to point of inflection). This perception is supported by the plot in Fig. 5(a), which shows the dependence of ${\theta }_p$ on $L/d$ for the full other-than-RBS data set [beams with 102 mm (4 in.) $\le d\le 914\mathrm{mm}$ (36 in.)]. But the strong dependence on $L/d$ is not evident when only beams of depth $\ge 533\mathrm{mm}$ (21 in.) are considered [see Fig. 5(b)]. The reason is that most deep beams are susceptible to a predominance of web buckling and lateral torsional buckling, and both of these susceptibilities increase with a decrease in the moment gradient (more uniform moment, as implied by an increase in the $L/d$ ratio). This phenomenon offsets much of the curvature integration effect for a larger plastic hinge length. On the basis of this information, it is concluded that, for beams with depth $\ge 533\mathrm{mm}$ (21 in.), a description of beam plastic deformation capacity in terms of a ductility ratio ${\theta }_p/{\theta }_y$ is often misleading because ${\theta }_y$ increases linearly with $L$ (for a given beam section) but ${\theta }_p$ does not. Assume two cantilever beams made of the same W section. One beam has length $L$ and the other $L/2$. The yield rotation ${\theta }_y$ for the first beam with length $L$ will be $M_y/6EI/L$ and for the second beam with length $L/2$ will be $M_y/(6EI/L/2)$, i.e., ${\theta }_y$ linearly increases with length. But the experimental data for set with $d\ge 533\mathrm{mm}$ show that ${\theta }_p$ does not depend strongly on $L/d$ [see Fig. 5(b)]. In other words, the ratio ${\theta }_p/{\theta }_y$ depends strongly on beam span $L$. Similar observations are made for the parameters ${\theta }_{pc}$ and $\Lambda$.

This ratio is associated with sensitivity to lateral torsional buckling. The parameter $L_b$ is defined in this case as the distance from the column face to the nearest lateral brace, and $r_y$ is the radius of gyration about the $y$-axis of the beam. The American Institute of Steel Construction (2005) requires that this ratio be less than $2,500/F_y$. Results from the steel beam database indicate that ${\theta }_p$ is somewhat but not greatly affected by $L_b/r_y$, provided that the ratio is close to or smaller than the value required by seismic codes. A decrease of $L_b/r_y$ to 50% of the value required by AISC (2005) increases ${\theta }_p$, on average, by 2.5 and 10% for other-than-RBS beams and beams with RBS, respectively. Providing lateral bracing close to the RBS portion of a beam decreases the rate of cyclic deterioration, because twisting of the RBS region is delayed. Uang et al. (2000) reached the same conclusion for beams with RBS.

When the effect of the $b_f/2t_f$ ratio on ${\theta }_p$ is viewed in isolation, a small $b_f/2t_f$ ratio has a negligible effect on ${\theta }_p$. For most of the deeper beams in the database, a small $b_f/2t_f$ implies a narrow wide-flange beam with small radius of gyration $r_y$ and large fillet-to-fillet web depth over web thickness ratio $h/t_w$, both of which have a detrimental effect on ${\theta }_p$ because (1) a larger $h/t_w$ ratio makes a beam more susceptible to web local buckling, and (2) a small $r_y$ makes a beam more susceptible to lateral torsional buckling. In contrast, the data show a clear benefit of a smaller $b_f/2t_f$ ratio for the parameters ${\theta }_{pc}$ and $\Lambda$, since a beam with a smaller $b_f/2t_f$ ratio does not develop a large flange local buckle, i.e., postcapping strength deterioration and cyclic deterioration occur at a slower rate.

This geometric parameter is found to be very important for all three modeling parameters (see Fig. 6). The reason is that a beam with a large $h/t_w$ ratio is more susceptible to web local buckling. This triggers flange local buckling and, at larger inelastic cycles, also triggers lateral torsional buckling (Lay 1965; Lay and Galambos 1966) Fig. 6 indicates also that the trends for all three modeling parameters are similar for RBS and other-than-RBS sections.

For the full data set for other-than-RBS beams (data set 1), the equation for ${\theta }_p$ obtained from multivariate regression analysis using 107 specimens is

The large values of regression coefficients for web depth over thickness ratio $h/t_w$, beam depth $d$, and span-to-depth ratio $L/d$ confirm trends pointed out previously. Fig. 7(a) shows data points for predictions obtained from Eq. (7) plotted against the data points obtained from experimental results on the basis of the calibration process described previously in this paper.

For the data set of beams with $d\ge 533\mathrm{mm}$ (21 in.), the regressed equation for precapping plastic rotation ${\theta }_p$, on the basis of 78 specimens, is given by

In Eq. (8), the effects of $d$ and $L/d$ on ${\theta }_p$ are not as significant as in Eq. (7) for the entire range of data, as concluded from the trends plots discussed in the previous section of this paper.

On the basis of 72 test specimens with beams with RBS with $d\ge 533\mathrm{mm}$ (21 in.) the regressed equation for precapping plastic rotation ${\theta }_p$ is given by

Eq. (9) indicates that the effect of $h/t_w$ and $d$ dominates on plastic rotation capacity ${\theta }_p$ of beams with RBS. Uang and Fan (1999) came to similar conclusions regarding the effect of $h/t_w$ on ${\theta }_p$, on the basis of a data set of 55 RBS specimens and using the difference between the rotations at 80% of the ultimate strength and at yield strength as a definition of plastic rotation capacity.

For the development of predictive equations for ${\theta }_{pc}$, only specimens with clear indication of postcapping behavior are considered from the W-section database. For other-than-RBS beams, 104 specimens were used. The empirical equation for ${\theta }_{pc}$, obtained from multivariate regression analysis of the full set of other-than-RBS beams, is given by

Predicted versus calibrated ${\theta }_{pc}$ values for the total range of data set 1 are presented in Fig. 7(b).

After eliminating specimens with $d<533\mathrm{mm}$ (21 in.) (data set 3), the proposed empirical equation for ${\theta }_{pc}$, on the basis of 72 specimens, is given by

The regression equation for ${\theta }_{pc}$ for beams with RBS, on the basis of 61 specimens, is

Patterns reflected in Eqs. (10) - (12) agree with the ones from previous studies by Axhag (1995) and White and Barth (1998). These researchers proposed empirical equations for predicting the descending slope of the moment-rotation curve of beams and concluded that flange and web local buckling are the primary contributors to the descending slope of the beams.

As discussed previously, the reference cumulative plastic rotation $\Lambda$ is a parameter that defines the rate of cyclic deterioration. The specimens considered for the development of predictive equations for $\Lambda$ are the ones that fail in a ductile manner and for which cyclic deterioration is clearly observed. All modes of cyclic deterioration are assumed to be defined by the same $\Lambda$. The exponent $c$ of Eq. (3) is kept equal to 1.0 for the sake of simplicity.

Eq. (13) is the best-fit multivariate regression equation for predicting the cumulative rotation capacity $\Lambda$ for the full set of other-than-RBS beams on the basis of 85 specimens with clear indication of cyclic deterioration:

Eq. (13) indicates that the geometric parameter $d$, $L/d$, and $L_b/r_y$ become statistically insignificant. The reason why the $L_b/r_y$ ratio has a small effect on $\Lambda$ is that all the specimens included in the multivariate regression analysis satisfy the AISC (2005) lateral bracing requirements. The small effect of $L_b/r_y$ was also pointed out by Roeder (2002).

For the data set of beams with nominal depth larger than 533 mm (21 in.) the following equation is derived to predict $\Lambda$ (66 specimens showed clear indication of cyclic deterioration):

The proposed equation for predicting the cumulative rotation capacity $\Lambda$ for beams with RBS, on the basis of 55 specimens, is

Uang et al. (2000) have shown that beams with RBS are susceptible to twisting at the RBS region because of the reduced flanges, and additional lateral bracing reduces the rate of strength deterioration at large deformation levels because it reduces the lateral buckling amplitude near the RBS location. Roeder (2002) came to the same conclusion. This is reflected in the exponent of the $L_b/r_y$ term in Eq. (15).

Experimental data with the following range of parameters are used in deriving Eqs. (7) - (15):The specimens included in the steel database were fabricated from three main types of steel material: A36, A572, Grade 50, and A992, Grade 50. The yield strength values reported in this paper are the ones obtained from actual coupon tests conducted by the experimentalists.

The range of validity of the regression equations is only as good as the experimental data allows it to be. The data do not include heavy W14 sections (heavier than $\mathrm{W}14\times 370$) and heavy (heavier than $\mathrm{W}36\times 150$) and deep (e.g., deeper than W36) beam sections. The predictions from the regression equations have been compared with data from the only series of experiments found in the literature on heavy W14 sections (Newell and Uang 2006) and have been found to provide reasonably close values of experimentally obtained modeling parameters. Until more tests on columns become available, the preceding equations provide the best estimates that can be offered for columns.

Tables 1 and 2 summarize the variation of deterioration parameters for other-than RBS beams and beams with RBS, respectively, for a range of sections (W21 to W36) that satisfy seismic compactness criteria. The range of deterioration parameter values is also reflected in the cumulative distribution functions of these parameters presented in Fig. 3. Sections whose geometric or material properties are slightly outside the range of properties, on which the predictive equations are based, are noted.

As noted previously, the modified IK deterioration model does not account for cyclic hardening, but the effect of isotropic hardening is incorporated approximately by increasing the yield moment (bending strength) to an effective value $M_y$ that accounts for isotropic hardening on average. The effective yield strength typically is larger by a small amount than the predicted bending strength $M_{y,p}$, which is defined as the plastic section modulus $Z$ times the measured material yield strength obtained from coupon tests. Table 3 summarizes the mean and standard deviation of $M_y/M_{y,p}$ ratios for other-than-RBS beams and beams with RBS. For the latter, $M_{y,p}$ is defined on the basis of the reduced section properties.

Options exist for more refined modeling that account explicitly for combined isotropic and kinematic hardening (e.g., Sivaselvan and Reinhorn 2000; Jin and El-Tawil 2003). Such options were not incorporated to keep the model relatively simple for engineering implementation.

Postyield hardening, and subsequently $M_c$, is described by the ratio of the maximum moment on the backbone curve shown in Fig. 1(a) over the effective yield bending strength, $M_y$, discussed previously. The $M_c/M_y$ and ${\theta }_c/{\theta }_y$ ratios define the strain hardening stiffness of the backbone curve shown in Fig. 1(a). This stiffness is important because of its effect on the $P-\Delta$ stability of a structural system (Medina and Krawinkler 2003). Table 3 summarizes statistics (mean and standard deviation) of $M_c/M_y$ for RBS and other-than-RBS connections on the basis of information extracted from the database of steel components. In general, $M_c/M_y$ is a more stable parameter to describe postyield strength increase than the traditional strain hardening ratio because the latter depends strongly on yield rotation, which, in turn, depends strongly on the beam span selected in the experiment, i.e., on the moment gradient. Experimental data used in this study have shown that the strength increase beyond yielding is much less sensitive to the moment gradient than the yield rotation, which is linearly proportional to the beam span.

Low-cycle fatigue experimental studies (Krawinkler et al. 1983; Ricles et al. 2004) indicate four ranges of cyclic deterioration. The first range has negligible deterioration in which local instabilities have not yet occurred or are insignificant. The second range involves an almost constant rate of cyclic deterioration attributable to continuous growth of local buckles. In the third range, deterioration proceeds at a very slow rate because of the stabilization in buckle size; this range is associated with the residual strength of a steel component. These three ranges are followed by a range of very rapid deterioration, which is caused by crack propagation at local buckles (ductile tearing). From the data sets for W-sections, a residual strength ratio $\kappa =M_r/M_y$ of approximately 0.4 is suggested for sets 3 and 4. This value is based on a relatively small set of data points from which an estimate of $\kappa$ could be made with confidence. To assess residual strength more reliably, more experiments with very large deformation cycles need to be conducted.

At very large inelastic rotations, cracks may develop in the steel base material close to the apex of the most severe local buckle, and rapid crack propagation will then occur, followed by ductile tearing and essentially complete loss of strength [see Ricles et al. (2004) for illustrations and end of last loading cycle of the experimental data shown in Fig. 2(b)]. The modified IK deterioration model captures this failure mode with the ultimate rotation capacity ${\theta }_u$. This rotation depends on the loading history and may be very large for cases in which only a few very large cycles are executed (e.g., near-fault loading history or ratcheting type of global behavior) as discussed in Uang et al. (2000) and Lignos and Krawinkler (2009). Estimates of ${\theta }_u$ are made in this paper only for experiments with stepwise increasing cycles of the type required in the AISC (2005) seismic specifications. For other-than-RBS beams, an estimate of ${\theta }_u$ is 0.05 to 0.06 rad on the basis of available data from various researchers (Allen et al. 1996; Ricles et al. 2000). For beams with RBS, an estimate of ${\theta }_u$ is 0.06 to 0.07 rad (Engelhardt et al. 2000; Ricles et al. 2004). For monotonic type of loading ${\theta }_u$ is on the order of three times as large as the ${\theta }_u$ values reported in the preceding sections for symmetric cyclic loading protocols. Ductile tearing is not found to be critical in analytical studies in which the collapse capacity of a steel moment-resisting frame has been evaluated, because steel frame structures usually approach their collapse capacity before ductile tearing occurs (Ibarra and Krawinkler 2005; Lignos and Krawinkler 2009; NIST 2010; Lignos et al. 2010, 2011).