RELIABILITY CONSIDERATIONS FOR STEEL FRAMES DESIGNED WITH ADVANCED ANALYSIS S.G. Buonopane 1, B.W. Schafer 2 & T. Igusa 3

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1 RELIABILITY CONSIDERATIONS FOR STEEL FRAMES DESIGNED WITH ADVANCED ANALYSIS S.G. Buonopane 1, B.W. Schafer 2 & T. Igusa 3 INTRODUCTION The design of steel structures by advanced analysis is emerging as a practical design tool with the availability of non-linear analysis software, and U.S. technical committees are presently considering the adoption of advanced analysis techniques in upcoming specifications. Advanced analysis captures important non-linear structural phenomena; foremost are geometric second-order effects, frame stability and material yielding. The existing load and resistance factor U.S. design (LRFD) specifications (AISC 1999) approximate such effects through moment amplification factors, effective lengths and alignment charts. While the current approximate methods work well for a large class of steel structures, for others their application can be ambiguous and the results over-conservative. Advanced analysis directly captures system behavior and can therefore simplify the design process by eliminating the need for individual member checks. By more faithfully modeling important structural phenomena, advanced analysis provides predictions of frame strength with greater accuracy than LRFD provisions. The improved accuracy of advanced analysis can result in more efficient structures with acceptable reliability. LRFD specifications enforce a target reliability through the use of load and resistance factors. Existing proposals for design by advanced analysis have used the load and resistance factors from the existing specifications with no explicit probabilistic justification. Strength 1 Graduate Research Asst., 2 Asst. Prof., and 3 Prof., Dept. of Civil Engrg., Johns Hopkins Univ., Baltimore, MD 21218, USA

2 predictions by advanced analysis may have different means and variances than those by LRFD, thereby requiring a different value of φ to maintain the desired target reliability. Existing advanced analysis design proposals Ziemian et al. (1992a,b) analyzed a series of two-bay, two-story planar frames and a 22-story, three-dimensional frame and showed that design by advanced analysis could save about 12% steel by weight compared to design by the 1986 LRFD specifications. These analyses captured discrete plastic hinging and geometric non-linearities. Resistance factors were incorporated by scaling the yield surface. A successful design required the total load at plastic collapse (frame strength) to equal or exceed the total factored design load. Galambos (1988) considers methods of incorporating system reliability into the design process, which remains a fundamental distinction between system-based advanced analysis design techniques and the current member-based LRFD methods. Chen and Kim (1997) also provide guidelines for design with advanced analysis and present several modeling approaches (e.g. notional load, reduced tangent modulus, semi-rigid connections). No resistance factor is used, and again the design condition requires the frame strength to exceed the factored loads. Additional research on advanced analysis has focused on the development of analysis techniques and tools for frames which exhibit complex non-linear behavior. Galambos (1998) summarizes various analysis and design techniques. While general guidelines are available for the design of steel frames by advanced analysis, no previous research has directly compared the structural reliability of frames designed by advanced analysis to those designed by LRFD methods. STRUCURAL RELIABILITY For a structure with random strength (R) subjected to random load (Q), the probability of failure is

3 ( r, q) f ( r) f ( q dr dq Pf = I R Q ) (1) where f R and f Q are probability density functions (PDFs) of strength and load, and I ( r, q) is an indicator function which takes the value 1 when the failure condition is satisfied and 0 otherwise. Typically, Eq. 1 cannot be solved in closed-form, but is instead estimated using analytical or numerical techniques. First-order LRFD format The LRFD format enforces a target reliability through the load (γ) and resistance factors (φ) in the design equation γ iqi φrn (2) i The development of the LRFD code is presented in detail in Ravindra and Galambos (1978) and Ellingwood et al. (1980). Using the failure condition ln( R / Q) 0, and assuming independent normal distributions and small variances for R and Q, results in the first-order reliability index 2 2 ( R Q ) V + V β = ln (3) FO m m R Q where the subscript m indicates a mean value; V R =coefficient of variation (COV) of R; and V Q =COV of Q. For a given target reliability, β t, the resistance factor is φ = R R exp 0. 55β V (4) ( ) ( ) m n t R where R m =mean of true resistance; and R n =nominal resistance or code strength. Monte Carlo and Importance sampling The probability of failure may be estimated through direct Monte Carlo sampling, thereby avoiding some of the assumptions of the first-order method. However, for small P f, a prohibitively large number of samples may be required to achieve an estimate with sufficient accuracy. Importance sampling can produce a satisfactory estimate with fewer samples than naïve Monte Carlo sampling. Since the integral to be

4 estimated here has only two dimensions, importance sampling is an appropriate technique. Its application in higher dimensional spaces is more difficult and may require additional refinements (Melchers 1999). Eq. 1 may be rewritten as P I( x, y) fr( x) fq( y) = hxy ( x, y dx dy (5) h ( x, y) f ) XY This integral may be estimated by importance sampling as N I( xi, yi ) f R ( xi ) f Q ( yi ) Pˆ 1 = f N (6) i= hxy ( xi, y 1 i ) where the samples (x i,y i ) are drawn from the importance sampling density h XY. For this application we perform importance sampling over the load dimension only, h XY =f Q*. A normal distribution is selected for Q * with a mean equal to the average of the means of R and Q, and a COV equal to that of Q. Figure 1 compares estimates of the probability of failure by direct Monte Carlo sampling and importance sampling for an increasing number of samples. P f x SP50LA Direct Monte Carlo Importance Sampling Number of samples Figure 1. Estimates of probability of failure by direct Monte Carlo and importance sampling for frame SP50LA.

5 FRAME STRUCTURES The steel frames analyzed are based on those of Ziemian (1990) typical of low-rise industrial structures. Figure 2 shows the geometry, support conditions and loads for the 16 frames discussed in this paper. The frames are labeled with the following nomenclature: S, U: symmetric or unsymmetric geometry; P, F: pinned or fixed base; 50: 50 ksi (345 MPa) nominal yield strength steel; H, L: heavy or light gravity load; A, E: member sizes determined by advanced analysis or elastic LRFD. Member sizes for frames UP50HA and UP50HE are given in Table 1; the member sizes of all frames are listed in Ziemian (1990). The yield strengths of beams and columns, and gravity loads are modeled as random variables. Other potential random effects, such as residual L: kn/m H: kn/m 6.10 m 4.57 m B3 B4 C4 C5 C6 L: kn/m H: kn/m B1 B2 C1 C2 C3 U: 6.10 m U: m S: m S: m Figure 2. Dimensions and loads of frames. P F

6 Table 1. Member sizes for frames UP50HE and UP50HA. UP50HE UP50HA Member SI U.S. SI U.S. C1 W310x28.3 W12x19 W310x21 W12x14 C2 W360x196 W14x132 W360x147 W14x99 C3 W360x162 W14x109 W360x122 W14x82 C4 W250x17.9 W10x12 W250x17.9 W10x12 C5 W360x162 W14x109 W360x162 W14x109 C6 W360x162 W14x109 W360x162 W14x109 B1 W690x125 W27x84 W690x125 W27x84 B2 W920x201 W36x135 W920x201 W36x135 B3 W460x60 W18x40 W460x60 W18x40 B4 W690x140 W27x94 W690x140 W27x94 stresses and geometric imperfections, are not considered in the current analyses. Random yield strengths Based on Galambos and Ravindra (1978), the yield strengths are modeled as a normal distribution F y ~N(1.05F yn, 0.10) where the first parameter is the mean; the second, COV. More recent data reported in FEMA (2000) indicates a slightly smaller mean of 1.04F yn and COV of For a given frame analysis, random simulations were performed in which all members had uncorrelated F y and perfectly correlated F y. Random gravity loads The design of these frames is controlled by gravity loading, therefore only the load combination 1.2Dn Ln is considered (Ziemian et al. 1992a). Dead and live loads are assumed to be equal. The total nominal gravity load is Q n =1021 kn for the light load case and Q n =3327 kn for the heavy. The dead loads are normally distributed D~N(1.05D n, 0.10). The live loads follow an extreme type I distribution L~ExI(L n, 0.10) (Ellingwood et al. 1980). The total random gravity load, Q, is the sum of four random variables, dead and live load on each of two stories. The distribution of Q cannot be expressed in closed form but its PDF can be

7 computed through numerical integration. After normalizing by the total design load Q n, both the light and heavy load cases have a mean of and COV of Analysis details Structural analyses were performed with OpenSees (McKenna & Fenves 2001), including geometric non-linear effects and an elasticperfectly-plastic material model. Displacement-based beam-column elements with a cubic shape function were used. Columns were subdivided into 4 elements, and beams, 8. Cross-section yielding and axial-moment interaction was captured with a fiber element model, integrated at 4 points along the element length. All members have their webs in the plane of the frame, and out-of-plane behavior was restrained. Uniform gravity loads were applied as equal concentrated loads at all 9 nodes along the beam lengths. Symmetric frames were given an initial out-of-plumb imperfection of 1/400th of the building height for numerical stability. No initial imperfection was given to the unsymmetric frames. SIMULATION RESULTS For each frame, 10,000 non-linear structural analyses were performed with random yield strengths. To determine the distribution of frame strength, the frames were loaded with an increasing gravity load until plastic collapse. For each simulation, the applied load at the occurrence of the first plastic hinge was also recorded. A plastic hinge was assumed to have occurred when the moment-curvature slope was reduced to 15% of its elastic value. The first plastic hinge strength provides an analog to the member-based LRFD design criteria. Figure 3 shows the histogram of the frame strength and first plastic hinge strength, both normalized by the design load Q n, for UP50HE with uncorrelated F y. Figure 4 shows the strength histograms for UP50HE with correlated F y. Both Figures 3 and 4 include the PDF of the normalized load. The mean and COV of the sampled strengths for all frames are given in Table 2 for uncorrelated F y, and Table 3 for correlated F y.

8 Frames with correlated F y have a slightly larger mean strength but also a significantly larger COV than the frames with uncorrelated F y. For a given frame, the failure mode of each sample with uncorrelated F y depends on the spatial distribution of the random F y, and the effect of a single weak member may be offset by a strong member elsewhere in the frame. In contrast, for a given frame with correlated F y, all samples exhibit the same failure mode Q mean=1.927 cov= Normalized Frame Strength Q mean=1.521 cov= Normalized Strength at 1st Plastic Hinge Figure 3. Normalized strength distributions for frame UP50HE with uncorrelated F y.

9 Those frames designed by advanced analysis have smaller member sizes in some locations, and therefore have smaller mean strengths than the corresponding elastically designed frame. The advanced analysis mean frame strengths range from about 80-90% of mean strengths of the LRFD designed frames; the first plastic hinge strengths range from about 70-80%. The COVs of the first plastic hinge strength are typically greater than those of frame strength. Also the design method of the frame does not greatly effect the COV of either strength measure Q mean=1.969 cov= Normalized Frame Strength Q mean=1.618 cov= Normalized Strength at 1st Plastic Hinge Figure 4. Normalized strength distributions for frame UP50HE with correlated F y.

10 Figure 5 plots frame strength against first plastic hinge strength for two of the frames. For UP50HA no correlation exists. For UP50LE, a band of correlated samples is apparent at the upper left of the plot; it is likely that this subset of samples has a common failure mode. Other frames exhibited various correlation structures between the two extremes of Figure 5. This lack of consistent correlation between these two performance measures suggests that there is no simple means of relating the member-based failure criterion to the system-based criterion. Table 2. Parameters of strength distributions for uncorrelated F y. Frame Frame strength 1st plastic hinge Mean COV Mean COV UP50HA UP50LA UF50HA UF50LA SP50HA SP50LA SF50HA SF50LA UP50HE UP50LE UF50HE UF50LE SP50HE SP50LE SF50HE SF50LE

11 Strength at 1st Plastic Hinge Table 3. Parameters of strength distributions for correlated Fy. Frame Frame strength 1st plastic hinge Mean COV Mean COV UP50HA UP50LA UF50HA UF50LA SP50HA SP50LA SF50HA SF50LA UP50HE UP50LE UF50HE UF50LE SP50HE SP50LE SF50HE SF50LE UP50HA Frame Strength 1 UP50LE Frame Strength Figure 5. Correlation of frame strength and first plastic hinge strength.

12 Reliability results Tables 4 and 5 list the reliabilities of all 16 frames for both frame strength and first plastic hinge strength for uncorrelated and correlated F y. The reliability is expressed as a probability of failure and a reliability index, β = Φ -1 (P f ) where Φ is the standard normal cumulative distribution function. The reliability by sampling is also compared to the first order reliability, β FO, computed from Eq. 3. The first-order approximation generally overestimates β compared to the sampling estimate, since the lower tail of the strength distributions have a higher probability mass than the normal distribution assumed for β FO. The frames designed by advanced analysis have a lower reliability index than those designed by LRFD. Table 4. Frame reliability for uncorrelated F y. Frame strength 1st plastic hinge Frame P f 10-6 β β FO P f 10-3 β β FO UP50HA UP50LA UF50HA UF50LA SP50HA SP50LA SF50HA SF50LA UP50HE UP50LE UF50HE UF50LE SP50HE SP50LE 0.00 > SF50HE SF50LE

13 Table 5. Frame reliability for correlated F y. Frame strength 1st plastic hinge Frame P f 10-6 β β FO P f 10-3 β β FO UP50HA UP50LA UF50HA UF50LA SP50HA SP50LA SF50HA SF50LA UP50HE UP50LE UF50HE UF50LE SP50HE SP50LE SF50HE SF50LE For the LRFD designed frames, β FO for first plastic hinge is greater than 3 for all frames. The target β for steel beam-columns in LRFD is approximately 3, and thus these analyses demonstrate that the LRFD code is effective in achieving this goal. For these frames, the sampling estimate of β is also generally near 3 as well. Examining the effect of correlation on reliability, reveals that β based on frame strength and uncorrelated F y is greater than the β for correlated F y. Whereas for first plastic hinge strength, β for correlated F y is greater than that for uncorrelated F y. Correlation among member yield strengths tends to decrease the probability of occurrence of the first plastic hinge but to increase the probability of overall frame failure. For the frames designed by advanced analysis, the reliability indices based on frame strength are above 3 and thus could be considered

14 acceptable designs. The values of β for first plastic hinge are generally between 1 and 2, corresponding to probabilities of occurrence of up to 14%. This result suggests that the occurrence of a plastic hinge in a frame designed by advanced analysis and subjected to nominal gravity loads may not be an infrequent event. In the selection of member sizes for these frames, Ziemian et al. (1992a) prohibited the formation of plastic hinges under service loads. However, this deterministic design requirement does not necessarily result in a zero probability of occurrence of plastic hinging when random loads and strengths are included in the analysis. The occurrence of a plastic hinge may affect the serviceability performance, even though it does not compromise overall strength and safety. This observation highlights the importance of serviceability issues in frames designed by advanced analyses. Elastic LRFD design controls the occurrence of the first plastic hinge, while advanced analysis may allow plastic hinging at service load levels in cases where it does not compromise frame strength. The existence of plastic hinges may also impact structural behavior for extreme load events, such as seismic, since the structure cannot be assumed to begin in an elastic, undamaged state. Resistance factors for advanced analysis The general expression for the resistance factor has been given in Eq. 4. The distribution of true resistance is ideally based on experimental data, and related to nominal resistance by R = Rn PMF (7) with mean and COV given by R = R P M F, m n m m m R 2 P 2 M 2 F V = V + V + V (8) where P, M and F are the random variables of professional, material and fabrication factors, respectively; the subscript m indicates a mean value and V R, V P, V M and V F are COVs of the random variables. The values used in the LRFD specifications were determined by experimental data, analytical models and professional judgment. For steel beam-columns, the LRFD values are P m =1.02, M m =1.05, F m =1.00, V P =0.10, V M =0.10, V F =0.05, V R =0.15 (Bjorhovde et al. 1978). Using

15 these values and β t =3.00 in Eqs. 4 and 8, results in φ =0.84. The value of φ =0.90 used in LRFD for beams corresponds to β t =2.10. The goal of advanced analysis methods is to provide predictions of strength which are closer to the true strength than existing elastic-based code methods. Since experimental data are not available for the frames analyzed here, we assume the limiting case that the distribution of true strength is exactly predicted by the advanced analysis with random material properties. Practical design with advanced analysis will be based on a single analysis with nominal yield strengths, resulting in the n nominal strength R. We relate the deterministic nominal strength to the probability distribution of strength by R = Rn B F (9) where B is the random variable bias factor and F is the LRFD fabrication factor, retained since our analyses do not consider random geometric properties. This equation is analogous to Eq. 7. Without test data for frame strength it is not possible determine individual bias factors equivalent to P, M and F. Table 6 gives values of nominal strengths and mean bias factors for frame strength and first plastic hinge for both uncorrelated and correlated F y. The mean bias factor, B m, is the mean strength from Table 1 or 2 divided by R n. The bias factors fall within the range of 0.93 to 1.08, and the mean of the bias factors is close to 1.0 indicating that R n is a good predictor of R m with little bias. For comparison, the combined bias factor assumed in LRFD for beam-columns is Also important is the observation that the bias factors appear independent of the method by which the frame was designed. Tables 7 and 8 present values of the variance of the advanced analysis strength prediction (including V F =0.05) and values of φ for both β t =3.00 and The COVs of the strengths are in the range of 0.07 to 0.10, somewhat less than the LRFD value of For a target reliability of 3.00 on frame strength, the values of φ range from 0.86 to 0.91 for both correlated and uncorrelated F y. For first plastic hinge strength, the values of φ for β =3.00 range from 0.80 to For a

16 Table 6. Nominal strengths and bias factors. Frame Frame strength 1st plastic hinge Uncorr. Corr. F y Uncorr. Corr. F y R n B m B m R n B m B m UP50HA UP50LA UF50HA UF50LA SP50HA SP50LA SF50HA SF50LA UP50HE UP50LE UF50HE UF50LE SP50HE SP50LE SF50HE SF50LE min mean max target reliability of 2.10, the values of φ are slightly higher, 0.84 to 0.95, including the cases of frame strength and correlated and uncorrelated F y. The resistance factor is affected by the mean bias and the variance of the strength distribution (Eq. 4). For the frames analyzed here, both the typical COV and bias factor of strength was less than that assumed by LRFD. These differences offset one another, resulting in values of φ which are approximately equal to current LRFD values. The

17 Table 7. COVs and resistance factors for uncorrelated F y. Frame Frame strength 1st plastic hinge φ for φ for φ for φ for V R V β t =3.00 β t =2.10 R β t =3.00 β t =2.10 UP50HA UP50LA UF50HA UF50LA SP50HA SP50LA SF50HA SF50LA UP50HE UP50LE UF50HE UF50LE SP50HE SP50LE SF50HE SF50LE min mean max incorporation of other random variables (e.g. residual stress, imperfections), as well as professional judgment, might justify larger COVs of strength, in which case smaller values of φ would be necessary to maintain the same target reliability. CONCLUSIONS Advanced analysis is emerging as the next-generation design tool for steel structures. This paper summarized research into the probabilistic character of design by advanced analysis, due to randomness in structural properties and loads. Based on a series of 16 steel frames

18 Table 8. COVs and resistance factors for correlated F y. Frame Frame strength 1st plastic hinge φ for φ for φ for φ for V R V β t =3.00 β t =2.10 R β t =3.00 β t =2.10 UP50HA UP50LA UF50HA UF50LA SP50HA SP50LA SF50HA SF50LA UP50HE UP50LE UF50HE UF50LE SP50HE SP50LE SF50HE SF50LE min mean max with random yield strengths and random applied gravity loads, nonlinear structural analysis simulations were performed to calculate distributions of frame and first plastic hinge strengths. Frames designed by advanced analysis had a smaller mean strength than those designed by LRFD, since they contain smaller member sizes. However, the calculated reliability indices of the frames designed by advanced analysis were still above 3.0 based on the failure condition of frame strength. Because advanced analysis primarily controls frame strength, some frames exhibit non-negligible probabilities of plastic

19 hinging under service load conditions. Even a deterministic design requirement which prohibits plastic hinging at nominal loads does not necessarily preclude the occurrence of plastic hinging when random loads and strengths are included in the analysis. Occurrence of such hinges may require greater attention to serviceability criteria such as deflection and drift, as well as consideration of a frame s initial state when subjected to extreme load events. The resistance factors determined from these simulations are generally in the range of 0.80 to 0.90 for a target reliability of 3.00 and 0.85 to 0.95 for a target reliability of 2.10, suggesting that current resistance factors may be acceptable for design with advanced analysis. However, these values depend on the variability of the strength distributions, which may increase as additional random effects are introduced into the analysis. Since advanced analysis is predicated on system behavior, the probabilistic results are dependent on the peculiarities of a given structure s behavior. The results presented here are based on a group of sixteen, closely-related steel frames, and the conclusions may not be representative of other structures. Without requiring explicit probabilistic analysis, one of the greatest challenges of design by advanced analysis may be the selection of design coefficients and resistance factors which are applicable to a wide range of structural systems and behaviors. ACKNOWLEDGEMENTS This research has been supported in part by a National Science Foundation Graduate Research Fellowship and Grant No. DMI REFERENCES American Institute of Steel Construction (AISC) Load and resistance factor design specification for structural steel buildings, 2nd ed. Chicago, Ill.: AISC. Bjorhovde, R., Galambos, T.V. & Ravindra, M.K LRFD criteria

20 for steel beam-columns. Journal of the Structural Division 104(9): Chen, W.F. & Kim, S-E LRFD steel design using advanced analysis. Boca Raton, Fla.: CRC Press. Ellingwood, B., Galambos, T.V., MacGregor, J.G., & Cornell, C.A Development of a probability based load criterion for American national standard A58. National Bureau of Standards Special Publication No. 577, Washington, D.C.:U.S. Dept. of Commerce. Federal Emergency Management Agency (FEMA) State of the art report on base metals and fracture. FEMA-355A. Washington, DC. Galambos, T.V Guide to stability design criteria for metal structures, 5th ed. New York: John Wiley & Sons. Galambos, T.V Reliability of Structural Steel Systems. Structural Engineering Report No Univ. of Minnesota, Minneapolis, Minn. Galambos, T.V. & Ravindra, M.K Properties of steel for use in LRFD. Journal of the Structural Division 104(9): McKenna, F. & Fenves, G.L The OpenSees command language manual, version 1.2. Pacific Earthquake Engineering Research Center, Univ. of California at Berkeley. edu. Melchers, R.E Structural reliability, 2nd ed. Chichester, England: John Wiley & Sons. Ravindra, M.K. & Galambos, T.V Load and resistance factor design for steel. Journal of the Structural Division 104(9): Ziemian, R.D Advanced methods of inelastic analysis in the limit states design of steel structures. PhD. dissertation. Cornell Univ. Ithaca, NY. Ziemian, R.D. McGuire, W. & Deierlein, G.G. 1992a. Inelastic limit states design. part I: planar frame studies. Journal of Structural Engineering, 118(9): Ziemian, R.D. McGuire, W. & Deierlein, G.G. 1992b. Inelastic limit states design. part II: three-dimensional frame study. Journal of Structural Engineering, 118(9):