RELIABILITY ANALYSIS OF TYPICAL COLD-FORMED STEEL BEAMS SECTIONS

Transcription

1 RELIABILITY ANALYSIS OF TYPICAL COLD-FORMED STEEL BEAMS SECTIONS Raylza Santos da Silva Campos André Luis Riqueira Brandão Universidade Federal de Itajubá Rua Irmã Ivone Drumond, 200, Distrito Industrial II, , Minas Gerais, Itabira, Brazil Marcílio Sousa da Rocha Freitas Programa de Pós-Graduação em Engenharia Civil, Universidade Federal de Ouro Preto, Campus Universitário Morro do Cruzeiro, , Ouro Preto - MG, Brasil, Abstract.This paper presents a reliability analysis o cold-ormed steel beams, based on the FORM and MC simulation and using data obtained rom experimental tests perormed at the Johns Hopkins University, Baltimore (Schaer, 2003; 2006). This paper approached two sets o lexural tests, one reerring to the ailure by local buckling, and another or distortional buckling. A ailure unction is assembled to obtain, by taking into account the statistical parameters o the material (M), abrication (F), and proessional actors (P). The material and abrication actors were taken rom AISI standard. The proessional actor was determined by comparing the tested ailure loads and the predicted ultimate loads calculated rom the selected design provisions. The goal o the paper is the assessment o reliability index or two nominal live-to-dead load ratios, as well as to compare the value ound considering the load combinations or ultimate limit states. Keywords: Reliability, Steel Beams, Buckling, Cold-ormed stee.

2 Reliability Analysis o Cold-ormed Steel Members Subject to Bending Moment 1 INTRODUCTION In the mid 1970s, there were signiicant changes in international construction standards, recommending probabilistic delineation in the design o structures. The reliability theory has been incorporated in the assessment o structural saety projects in construction and shipbuilding, becoming known rom the 70s, as Structural Reliability (Ang and Cornell, 1974). The reliability theory carries as undamental role, in the construction branch o saety and perormance analysis, a realistic description o a project template or structural analysis. The statistical nature o materials actions and properties has been the cause o study in dierent countries. Such study has shown that the uncertainty o the applied orces and structural resistance generate uncertainty in structural perormance, such uncertainties can be analyzed through the reliability theory application. In this context, structural reliability presents itsel as a valuable tool or the structure saety analysis providing a more accurate measure o the degree o security, since this is based on determining ailure probabilities (P ) or reliability indices (). In building construction, cold-ormed steel elements are mainly used as structural members, loors, walls, diaphragms and roos. The main advantage is lightweight which can be translated in reduced costs by using less steel or structural elements. The most recent edition o the brazilian speciicacion or the design o cold-ormed steel structural members was published in 2010 (ABNT NBR 14762, 2010) and is mainly based in the AISI speciication valuable at the time. This study uses experimental data rom Yu e Schaer (2003, 2006), who tested cold-ormed steel members subjected to bending moment. These members had lipped C- and Z-sections, and were conditioned to ailure by local or distortional buckling. The comparison between experimental and theoretical results provided by Yu e Schaer (2003, 2006), deined as proessional actor (model error), was used as a random variable in the structural reliability analysis o this work. 2 STRUCTURAL RELIABILITY Structural reliability analysis is based on the existence o a ailure unction or limit state design G(X), wherein X=(X 1, X 2,..., X n ) represents the set o random variables involved in the analysis, in other words, all those with some statistical inormation regarding it. The ailure unction G(X) must be deined so that the limit G(X)=0 separates the ailure domain (G(X)<0) and the saety domain (G(X)>0) (Hsiao, 1989). Structural reliability analysis is based on the existence o a ailure unction or limit state design G(X), wherein X=(X 1, X 2,..., X n ) represents the set o random variables involved in the analysis, in other words, all those with some statistical inormation regarding it. The ailure unction G(X) must be deined so that the limit G(X)=0 separates the ailure domain (G(X)<0) and the saety domain (G(X)>0) (Hsiao, 1989). So structural reliability must seek the probability o ailure events, i.e., what is the ailure unction probability o taking values belonging to the ailure domain. This probability is known as ailure probability and described by: P P G( X) 0 (1)

3 Raylza Santos da Silva Campos, André Luis Riqueira Brandão, Marcílio Sousa da Rocha Freitas Knowing that X (X) represents the joint probability density unction (pd) o all X variables involved in the analysis, ailure probability can be described by the integral: P ( X) dx (2) X G( X) 0 Thus, reliability is deined as C 1 P (3) The integration o Eq. (2) can be illustrated by Fig. 1 or a case o two random variables. The igure illustrates the joint probability density unction and its level curves projected on X (X) surace on the X 1 X 2 plan. All the points contained in the same contour curve have same value as (X), in other words, same probability density (Hsiao, 1989). X (X, X ) X 1 2 X 2 G(X, X )>0 1 2 região segura G(X, X )<0 1 2 região de alha X 2 G(X, X )=0 1 2 X 1 X 1 Figure 1. Failure probability integral in the basic variables space. In structural analysis, it is possible to deine the ailure unction G(X), as being: G(X ) Z R Q (4) where R e Q represent random variables o resistance and load eects, respectively. It is assumed that the probability density unction and cumulative density unction o R and Q are known (Hsiao, 1989). Thereby, probabilities associated with the events can be deined by: P P( R Q) P( G( X) 0) ( X) dx (5) X G( X) 0 Assuming that R and Q are statistically independent, the probability o ailure can be represented as: qdrdq F q q P R( r) Q R Q dq (6) where Q (q) is the probability density unction o Q variable and F R (q) is cumulative distribution unction o variable R.

4 Reliability Analysis o Cold-ormed Steel Members Subject to Bending Moment The large number o random variables involved in many practical problems makes it quite diicult to obtain the joint probability density unction and the integration o the equation is nearly always ineasible. Alternatively, the structural saety is measured as a unction o the reliability index β, deined as the shortest distance rom the origin o standardized or reduced variables space to ailure surace (Cornell, 1969). The reliability index is undoubted important rom the point o view o design saety because such index is a relative saety measure, i.e., when comparing two or more projects, the one that presents the biggest β is more reliable (AISI S100, 2007). 2.1 FORM method FORM method is based on the transormation o basic variables X, whether or not correlated, in statistically independent standard variables U, called reduced variables, as well as the ault unction that can be deined in the reduced variables space. There are several ways to turn the random variables X into random standard variables U and statistically independent. The methodology that uses structural reliability the most bases itsel on the transormation o correlated standard variables into statistically independent standard variables. This transormation is known as Nata transormation. Another method step consists in the ailure surace approximation G(U)=0, unction deined in the reduced variables space, by a linear surace (irst order o Taylor expansion) * * * * on the point with the shortest distance to the origin, identiied by U U 1, U2,..., Un. This is the design point in the reduced variables space, that is, the highest local density probability point. One o the most commonly used algorithms to obtain the design point is the one developed by Hasoer and Lind (1974) and improved by Rackwitz and Fiessier (1978). This algorithm is commonly reerred to as HLRF. The distance between design point and origin is called reliability index. Ergo, reliability * index probability can be measured by vector U length, i.e., * U (7) where. represents magnitude o a vector. Being, * U a where a is the unit vector, perpendicular to ailure surace, on the design point. Ditlevsen e Madsen (1996) demonstrated, using standard distribution properties, that ailure probability is given by: P (9) where (.) represents cumulative standard distribution. 2.2 FOSM method Random variables probability distribution inormation is not considered in the analytical method FOSM (Hsiao, 1989). The linearization o the ailure unction G(X) is achieved (8)

5 Raylza Santos da Silva Campos, André Luis Riqueira Brandão, Marcílio Sousa da Rocha Freitas through the average o random variables, using mean and variance, in other words second order statistical moments. Equations (10) and (11) o mean and standard deviation, respectively, were obtained assuming that random variable are independent among them and trucating Taylor expansion in linear terms. G( X ) G X, X,..., Xn ) (10) ( 1 2 P G X ) n ( 1 i1 2 2 g X X i Failure probability can be determined through Eq. (12). g (12) g The relation is known as Cornell reliability index (1969). g 2.3 Monte Carlo simulation (MCS) g From a set o n random variables X={x 1,x 2,...,x n }, these being individually characterized by their marginal probability density unction and their marginal cumulative distribution unction F Xi (x i ), ailure probability, associated with a limit state design G(X), which deines a ailure region, can be calculated by: P ( X ) dx I[ G( X )] ( X dx (13) ) G( X ) 0 where x (X) is the probability density unction o random variables and I[g(X)] is a indicator unction deined by: 1 se G( X ) 0 0 se G( X ) 0 I [ g( X )] (14) Indicator unction enables the calculation o integral rom Eq. (13) on the entire domain. The average indicator unction value represents this equation s result. Thereore, ailure probability can be estimated through Eq. (15). ^ n 1 P I[ G( x j )] (15) ns j1 where ns is the number o simulations, x j is the j-th sample vector simulated containing n n variables and I [ G( )] represents the sum o simulation numbers ocurred in the ailure j1 x j region (n). Hence, Eq. (15) can be rewritten as illustrated below: ^ P n ns The variance, or small values o ailure probability, is expressed as: (11) (16)

6 Reliability Analysis o Cold-ormed Steel Members Subject to Bending Moment ^ Var( P ^ ^ (1 P ) ) P (17) ns Structures ailure probability is generally low, order rom 10-3 to 10-5, and as its variance is expressed inversely proportional to the total number o simulations (ns), ns value must be elevated in order to obtain acceptable approximations o P (Pulido et al., 1992). Once the estimated ailure probability is calculated by simulation, using Eq. (16), β reliability index is obtained by the expression: ^ 1 (1 P ) (18) where -1 (.) is the inverse o cumulative standard probability density unction. 3 BEAMS SUBMITTED TO BENDING MOMENT NBR (2010), based on limit state design, establishes basic requirements that must be met in the design o cold-ormed steel, made out o sheet metal or carbon strip steel or lowalloy steel, intended or building structures. This standard oresees three design methods: eective width method (EWM), eective section method (ESM) and direct strength method (DSM). In this paper, it was adopted the DSM, which considers the geometric properties o the gross section and overall analysis o elastic stability, which identiies, or this case, all modes o buckling and their critical eorts. In case o steel bar submitted to bending moment, basic conditions are that solicited bending moment (M Sd ) must be smaller than or equal to resistance bending moment (M Rd ). Thereore resistance bending moment is calculated according to Eq. (19) at the beginning o eective section low, and according to Eq. (20) due to lateral torsional buckling (FLT): We y M Rd (19) M Rd FLTWc, e y (20) where, W e eective section elastic resistance modulus in relation to extreme iber that reaches low, W c,e eective section elastic resistance modulus in relation to external compressed iber and is resistance actor. Cross section geometrical properties (W e and W c,e ) can be calculated based on EWM or ESM. Formulation o DSM considers strength elastic modulus o gross section (W) modulus in relation to extreme iber that reaches low (AISI, 2007; NBR 14762, 2010). AISI (2007) speciies a resistance actor = 0.90 or LRFD (Load and Resistance Factor Design). ABNT NBR (2010) uses equivalent value to AISI or LRFD, in other words, resistance actor = 1.10 ( is the inverse o ). In case o LSD (Limit States Design), resistance actor speciied by american standard is = Direct strength method Determination o axial orces or bending moments associated with local, distortional and global buckling involves initial step o DSM. DSM requires numerical techniques such as

7 Raylza Santos da Silva Campos, André Luis Riqueira Brandão, Marcílio Sousa da Rocha Freitas inite element method or inite strip method (FSM), in order to perorm a linear stability analysis o open sections and thin-walled proiles. FSM employs strip elements transversely along longitudinal direction, being the length o these strips assumed to be equal to the length o a hal wave o buckling. It is noteworthy that the FSM model used results in ewer degrees o reedom than a inite element mesh, which greatly acilitates the entry and processing o data. CUFSM (Cornell University Finite Strip Method) program developed by Schaer (2001) is appropriated to cold-ormed steel design analysis. Such sotware is available or ree on Cornell University website, and employs the semi-analytical inite strip method to provide solutions or the cross-section stability o cold-ormed steel members. Figure 2 illustrates CUFSM program use, or one o the member subject to bending, experimental program Yu and Schaer (2003) constant. It can be seen that the irst branch o the graph corresponds to the local buckling plate and the second branch corresponds to distortional mode. Third brand corresponds to global buckling (lateral torsional buckling). Critical values (minimum values) o local elastic buckling bending moments and distortional are employed in obtaining nominal bending moment strength (M Rk ) Figure 2. Distortional Mode. Source: Author.

8 Reliability Analysis o Cold-ormed Steel Members Subject to Bending Moment 4 PROCEDURES FOR RELIABILITY ANALYSIS 4.1 Resistance and load statistics Ravindra and Galambos (1978) describe resistance o a structural element as shown in Eq. (21): R R PMF (21) n where R n is strutuctural element nominal resistance, P is proessional actor (model error), M is material actor and F is abrication actor. Considered dimensionless random variables, P, M and F relect model, material and geometric properties uncertainties, respectively (Hsiao, 1989). Assuming that P, M and F variables are uncorrelated and using irst order probabilistic theory, resistance mean (R m ) can be obtained through Eq. (22). R R ( P M Fm ) (22) m where, n m m P m é is ratio mean between experimental and theoretical resistance, calculated according to a given model, using material and geometric properties; M m is the ratio mean between material mechanical resistance obtained rom testing and speciied minimum value; F m is ratio mean between geometric property measured and speciied value (nominal). V R The variation coeicient o variable R is equal to V V V (23) where, 2 P 2 M 2 F V P is measurement error model variation coeicient; V M is material actor variation coeicient; V F is manuacturing actor variation coeicient. Hence, P m, M m, F m, V P, V M and V F needed statistical data to determine resistance statistical properties R m e V R. Mean and standard deviation rom M and F random variables can be obtained rom NBR (2010) and AISI (2007) standards. Probability distribution unctions were assumed to be lognormal (LN). Statistical data obtainment o random variable P will be presented in subsection 5.1. Most combinations o actions involving gravitational actions is represented by the sum o dead load (D) plus live load (L). Combinations o gravitational actions drive projects in many practical situations and are particularly important. Dead loads present little variability over structure useul lie. Table 1 displays this study s statistical properties or dead and live load, proposed by Galambos et al. (1982).

9 Raylza Santos da Silva Campos, André Luis Riqueira Brandão, Marcílio Sousa da Rocha Freitas Table 1. Random variables statistics o dead and live loads (Galambos et al., 1982) Type o load Mean Value Coeicient o Variation Type o Distribution (pd) Dead load D m = 1.05D n V D = 0.10 Normal Live load L m = L n V L = 0.25 Extreme Type I 4.2 Failure unction domain Equation (24) correlates nominal resistance (R n ) with nominal loads ollowing limit state design used by ABNT NBR (2010) standard. Rn c( ) DDn LLn (24) where is resistance actor, D n and D n are the nominal values o the dead and live load, D and L are dead and live load actors, respectively, and c is deterministic coeicient that relates load intensities to the loads eects. AISI (2007) calibration data can be observed in Table 2, which are parameters used in reliability analysis. LSD (Limit States Design) with load combination 1.25D n +1.5L n is used in both brazilian and american standard. Brandão and Freitas (2013) considered a cross between load combinations and nominal dead-to-live load ratio (D n /L n ) by inexistence, in NBR (2010), o speciic procedures deinition and speciic reliability analysis parameters. Table 2. Calibration data (AISI-S100, 2007). LRFD LSD D L 1.2D n +1.6L n 1.25D n +1.5L n D n L n D n /L n 1/5 1/3 o P 6.21 x x 10-3 Equation (25) represents ailure unction, in which R, D and L variables were remade based on their nominal values, by using limit state design deined according to Eq. (24), nominal resistance and D n /L n ratio. It is noteworthy that FORM and MCS reliability methods use probability distribution unctions in addition to mean and standard deviation (Brandão and Freitas, 2015). G(.) R c( D L) (25)

10 Reliability Analysis o Cold-ormed Steel Members Subject to Bending Moment 5 RESULTS 5.1 Resistance and load statistics The experiment conducted by Yu and Schaer (2003, 2006), who tested cold-ormed steel members subjected to bending moment, was speciically developed to enable local or distortional buckling occurrence. In Yu and Schaer (2003), a trapezoidal tile panel connected to the tested beams upper lange restricted global and distortional buckling (Fig. 3). It is noteworthy that theoretical analysis pointed out to distortional buckling, i not considered the lock imposed by placing the tile panel. In the experiment conducted by Yu and Schaer (2006), the objective was to study beams subjected to distortional buckling. In this study were considered Yu and Schaer (2003, 2006) data with lipped C- and Z-sections. Measurement error model was calculated by Yu e Schaer (2003, 2006), rom the ratio between experimental resistance bending moment (M test ) and theoretical resistance bending moment, to each specimen. Theoretical resistance bending moment reer to resistance bending moment characteristic value associated with local buckling (M Rl ) or distortional buckling (M Rdist ). Table 3 presents Yu e Schaer (2003, 2006) test data, organized or this study s reliability analysis. Each data group presented in Table 3 was analyzed through Minitab 16 computer program in order to obtain statistical parameters o measurement model error random variable (P). Thus, it was easible to describe this variable through mean (P m ), variation coeicient (V P ) and probability distribution unction (pd) that better adjusted to data set. Figure 3. Overall view o test conducted by Yu and Schaer (2003)

11 Raylza Santos da Silva Campos, André Luis Riqueira Brandão, Marcílio Sousa da Rocha Freitas Table 3. Experimental groups. Source: Yu and Schaer (2003, 2006) Nomenclature Failure mode Number o tests V P P m Type o Distribution (pd) C-L Local Lognormal C-L Local Lognormal C-D Distortional Lognormal C-D Distortional Lognormal All data Local and Distortional Lognormal 5.2 Reliability index calculation Reliability indices, presented in Table 4, were obtained by using reliability methods FORM, FOSM and MCS, with 100,000 iterations, to two load combinations and two relations D n /L n. Table 4. Reliability indices obtained by section type and ailure mode. Case P Statistics Ratio 1.2D n + 1.6L n 1.25D n + 1.5L n P m V P pd D n /L n FORM FOSM MCS FORM FOSM MCS C-L LN 1/ / Z-L LN 1/ / C-D LN 1/ / Z-D LN 1/ / All LN 1/ data 1/

12 Reliability Analysis o Cold-ormed Steel Members Subject to Bending Moment Results obtainted, via FORM method, compared to the ones by MCS conirm analytical method accuracy or the perormance unction used in this work. Despite FOSM method s simplicity and reduced accuracy, it proceeded with this analysis or comparative purposes, since the calibration coeicients o the American standard AISI (2007) were calibrated with such methodology. With American standard calibration data (combination 1.2D n + 1.6L n and ratio D n /L n o 1/5), reliability indices were always higher than the target reliability index ( o =2.5). American standard presents certain conservatism since the coeicient considering dead load D n is inerior and the coeicient considering live load L n is superior when compared to the value adopted by Brazilian standard. Figure 4. Comparison o reliability indices obtained rom FORM method to each load combination. Figue 4 shows a comparison between reliability indices obtained by FORM method according to load combinations and D n /L n ratio. It has been veriied that results rom Brazilian standard combination 1.25D n + 1.5L n are not satisactory when compared to target reliability index o 2.5, with exception o C-L case (lipped channel section submitted to local instability). 6 CONCLUSIONS It was presented in this paper a procedure or analysis o cold-ormed steel design submitted to bending moment. Direct strength method (MRD) was used or analysis, in order to obtain bending moment resistance capacity, o sections deined rom experimental data set. It was veriied that in the gravitational load combination analysis rom Brazilian standard NBR (2010), reliability indices demonstrated small deviation in relation to target reliability index ( o =2.5). Using load combination rom American standard AISI S100 (2007) or LRFD, reliability indices were greater than the target reliability index. Despite design criteria similarity or bending deined by Brazilian and American standards, it can be said that they do not have the same saety level due to dierent load combinations. American model was more conservative, according to values used as load weighting coeicients. Thereore, it is suggested to review standards committees, a clear deinition o target reliability indices ( o ) and o nominal load ratio (D n /L n ), in order to veriy resistance actor rom Brazilian standard NBR (2010) or bending. In this case, it is recommended FORM reliability method.

13 Raylza Santos da Silva Campos, André Luis Riqueira Brandão, Marcílio Sousa da Rocha Freitas ACKNOWLEDGEMENTS The authors thank CNPq, FAPEMIG and Uniei or sponsoring this study. REFERENCES ABNT NBR 14762, Dimensionamento De Estruturas De Aço Constituídas Por Peris Formados A Frio. Rio de Janeiro. AISI S100, Speciication For The Design O Cold-Formed Steel Structural Members. Alves, A. R.; Brandão, A. L. R.; Freitas, M. S. R., Coniabilidade De Barras Em Peris Formados A Frio Submetidos À Força Axial De Compressão Via Método Form. Iberian Latin-American Congress On Computational Methods In Engineering. Abmec, Rio de Janeiro, Rio de Janeiro, Brasil. Ang, A. H-A. E Cornell, C. A., Reliability Bases O Structural Saety And Design. Journal O The Structural Division, ASCE. Vol. 100, Número 9, Pp Ang, A. H-S. E Tang, W. H., Probability Concepts In Engineering Planning And Design Decision, Risk and Reliability. Vol. Ii, John Wiley & Sons, 562p, EUA. Brandão, A. L. R.; Freitas, M.S. R., Calibração De Coeiciente De Ponderação Da Resistência Em Ligações Soldadas De Peris Formados A Frio. Iberian Latin-American Congress On Computational Methods In Engineering. Abmec, Pirenópolis, Goiás, Brasil. Cornell, A. C., A probability based structural code. ACI Journal, vol. 66, 12, Freitas, M. S. R., Combinação de modelos probabilísticos e possibilísticos para a análise de coniabilidade estrutural. Tese de Doutorado, Universidade Federal do Rio de Janeiro (COPPE). Galambos, T.V., Ellingwood, B., MACGREGOR, J.G., Cornell, C.A., Probability based load criteria: assessment o current design practice. Journal o the Structural Division,Vol. 108, n. ST5, pp Javaroni, C.E., Estruturas De Aço: Dimensionamento De Peris Formados A Frio. 1 a Ed. Elsevier Editora Ltda. Kiureghian, A.D., Analysis O Structural Reliability Under Parameter Uncertainties. Probabilistic Engineering Mechanics, Vol. 23, No. 4, Pp Pulido, J. E., Jacobs, T. L., Prates de Lima, E. C., Structural reliability using Monte arlo simulation with variance reduction techniques on elastic-plastic structures. Computer and Structures, p Ravindra, M. K.; Galambos, T. V., Load And Resistance Factor Design For Steel. Journal o the Structural Divison, 104, ST9, Schaer, B. W. (2001). Cusm 2.5 Sotware. User Manual And Tutorials. Disponível No Site Acesso em 15 de março de Yu, C., Schaer, B. W., Local Buckling Tests On Cold-Formed Steel Beams. In: Journal O Structural Engineering /(ASCE) (2003)129:12(1596), Yu, C. and Schaer, B., Distortional Buckling Tests on Cold-Formed Steel Beams. In: Journal O Structural Engineering /(ASCE) (2006)132:4(515),