LIGHTWEIGHT STRUCTURES in CIVIL ENGINEERING - CONTEMPORARY PROBLEMS

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1 ITERATIOAL SEMIAR Organized by Polish Chapter o International Association or Shell and Spatial Strctres LIGHTWEIGHT STRUCTURES in CIVIL EGIEERIG - COTEMPORARY PROBLEMS STOCHASTIC CORROSIO EFFECTS O RELIABILITY OF THE STEEL BEAMS WITH I PROFILES M. Kamiński, I. Kraze Chair o Steel Strctres, Department o Strctral Mechanics Faclty o Civil Engineering, Architectre and Environmental Engineering Al. Politechniki 6, 90-9 Łódź, POLAD, tel/a Marcin.Kaminski@p.lodz.pl ABSTRACT: The main aim o this paper is to develop a probabilistic method or analysis o the delection or steel I beams eposed to corrosion and the corresponding reliability indices. This approach is dal in the contet o an application o the analytical and, independently, the pertrbation-based techniqe to determine basic random characteristics o the delection lctations. To get the most similar models to the real corrosion eects we created the new, redced cross-sectional areas or ten varios typical I proiles. Compter simlation is based on the 3D Finite Element Method models provided in the compter program ROBOT sing the shell inite elements. One type o the I proile consists o the several models created in a similar way (the same length, restraint bondary conditions and the same eternal orces), the only dierence is in a depth o possible destrction. Instead o the Direct Dierentiation Methods we employ the Response Fnction Method, where the basic polynomial response nction is based on the FEM tests and determined throgh the nweighted least sqares method. It is embedded into the stochastic generalized pertrbation scheme, where the polynomial eqation o delection is sbjected to the seqential analytical symbolic dierentiation to ind the probabilistic moments o the imm beam delection. Stochastic redction o cross-sectional area is assmed as the linear time series with the Gassian coeicients having niqely deined irst two probabilistic moments coming rom the previos eperimental tests. This assmption was adopted to simlate nprotected corrosive steel beam eposed to atmospheric corrosion; the basic random variable o the problem is the depth o steel corrosion destrction. Thereore, inal reliability inde determined is inlenced by the eploitation time o the strctres analyzed to assess the time-dependent drability o the steel beams applied in dierent environments (rban and marine at least). Or method is based on the statements o Erocode 3 and Erocode 0 and is developed or a direct application in the reliability o steel strctres sing the compter algebra systems like MAPLE, v. 3, or instance and any FEM engineering system. Keywords: steel strctres, delection, corrosion, reliability analysis, stochastic pertrbation techniqe. ITRODUCTIO Corrosion phenomenon is a common problem o steel strctres. It cases disintegration o material in de to chemical reactions with its srrondings which has a large eect on parameters o strctral steel. Figre : Corrosion chemical reactions with the srronding environment. As a reslt o oidation, the rst is ormed. It cases loss o steel crosssection reslting in a strength redction and increasing delection, or instance. This paper ocses on the changes o delection as a reslt o atmosphere corrosion. All calclations are based on Erocode 0 [] and Erocode 3 []. The approach is dal in the contet o application. One way is simlation and the other, independently, is pertrbation techniqe. In both cases we calclated the reliability inde as a nction o the delection or the most poplar ten I proiles eposed to increasing corrosion. Moreover, the reliability inde is determined by time nction, which is assmed as a linear to model steel strctres as nprotected corrosive. According to Robert E Melchers`s paper [3], we assmed that annl increase in the depth o corrosion, or steel strctres in inland atmospheric conditions, is 0,05mm. The depth o corrosion destrction is the only random variable in or simlations. These simlations are based on three-dimensional models. For each type o ten I proiles are made several cross-sections. ew redced models are created by redcing bottom part o cross-section []. All the models simlate an increasing corrosion - the later years the deeper loss o area. Calclations are made in Finite Element Method analyst program. Reslts are the basis to determinate the main nction o delection. The inal orm o the ormla is bilt sing the compter algebra system MAPLE [5] jst as the vale o reliability inde. However, pertrbation techniqe [6] is based on mathematical eqation o delection and all calclated sing compter algebra system. In this aspect, loss o steel cross-section is hidden in moment o area. Moment o area is deined or new lat igres, where the depth o loss is a random vale. The vale o reliability inde is given sing the compter algebra system.

2 . THEORETICAL MODELIG OF CORROSIO PROCESS Corrosion eect is modelled as a loss o cross-section area in the bottom part o beam (Fig. ). It creates ll 3D model o destrction, where only one dimension, the depth, is variable. The others are assmed as d d 0,035mm () w h where d w is width, and d h height o the corrosion loss. The analytical ormla known rom the strength o materials and, separately, FEM-based reslts are contrasted here since the realistic thinwalled character o this proile is inclded in the second approach only. At the same time, the engineers reqently apply the classical soltion, so that the dierence in probabilistic and stochastic contet shold be known, especially in the view o the reliability analysis. The stochastic analytical pertrbation techniqe ses mathematical epression or a delection (Eqn. ) or simply spported niormly loaded Eler beam, which is given by 5qL () 38EI where q is load, L - the length o the beam and traditionally, E stands or the elasticity modls. The moment o area is calclated rom the deinition I A y da (3) Figre : The loss o cross-sectional area simlated the corrosion destrction. In simlation aspect, or a single type o the I proile, several models are made. The mechanism o the corrosion represented by the certain geometrical parameters relects the eperimental observations, where a lower part o the proile is highly eposed to the atmospheric corrosion, while pper hal remains the same. and inally takes the orm 3 3 ( b t ) h t w ( dwdd ) I bt ( ) 3 3 hw dd ( twhw ) ( dddh ) dd w d( ) 3 hw dh ( r r ) hw r dddh( ) r r( ) r 8 ( r ) hw r ( ) ( r ) () 3. ELEMETS OF STATISTICAL AALYSIS Several elements o statistical analysis are sed to carry ot this calclation. Starting with the epected vale which is deined by statistical estimation as () i E. (5) i Then, the nbiased estimator or the variance is applied in the orm Figre 3: The eample o the FEM model. Using FEM analyst program, ROBOT, are created similar, or every I beams, 3D static models (Fig.3). Modelled beams are single-span, simply spported and niormly loaded. Finite element mesh density is assmed as 0 cm 0 cm, which totally gives abot 3,000 elements, depending on the actal I proile. We se the -noded linear elastic and isotropic shell inite elements to realistically model the thin-walled behavior o this proile. The reslts o stochastic delection are the basis to bild p the appropriate symbolic least sqares method approimation. () i Var E, (6) i which enables to determine the standard deviation as its sqare root Var The ratio o the standard deviation to the epected vale is called the coeicient o variation. (7) E. (8) Generally, the mth central probabilistic moment is estimated sing the ollowing ormla: m ( i) m E, (9) i Figre : The eample o redced cross-sectional area. which practically is most reqently sed to approimate third and orth central probabilistic moments. Althogh the vales o the latter eqal 0 and 3, respectively, or the Gassian variables, real comptations prove that one needs the very large poplations to obtain those reslts with the satisactory precision. The asymmetry coeicient and krtosis are additionally deined as

3 3 3 3 (0). PERTURBATIO METHOD AD RELIABILITY AALYSIS It is given the random variable b b( ) and its probability density nction as pb (). Farther, the epected vales as well as its central mth probabilistic moments are deined as and E b b 0 b pbdb, m ] m b b E[ b pb db. () () It is known that, stochastic pertrbation is based on Taylor series. All the inpt variables are epanded abot the additional vales sing the parameter 0. Is random qantity b e, the ormla is given by 0 where b b b n 0 n e n e e n n! b, n b (3) is the irst variation o b arond its epected 0 vale b. Then, coming back to the problem, the vale o random delection is epanded sing ormla (3) and inserted to the deinition () E b ( b) p b db n 0 n n n! b n pbdb n b Finally, the epected vale is calclated as o E () b b b 6 6! ( b) 6! 6 6( b)... b b Similar ormla describes the 6 th order epressions o the variance 0 Var b b b b ! b 3! b b b ! b E[ ] p( b) db 5 b () (5) (6) Frther we se the probabilistic moments compted above or a determination o the reliability inde o the steel beams analyzed according to the First Order Reliability Method (FORM). We employ or this prpose the general ormla describing the limit nction as a simple combination o the allowable (R) and imm (E) displacements as g R E (7) Usally both R and E are some combinations o the inpt random variable(s), whose nctions in the worst case are determined via the Response Fnction Method. Once we assme that the limit nction g is o the Gassian natre (which not always is realistic), then the reliability inde takes the ollowing orm: Eg [ ] (8) [ g] where β is this inde, E[g] stands or the epected vale o the limit nction analyzed (displacements, stresses, temperatres, atige cycles etc.), whereas σ(g) is the standard deviation o this nction. Taking into accont a physical meaning o or case one can rewrite this inde as where E[ ] dop [ ] dop E[ ] E[ ] dop Var( ) Var( ) dop (9) dop, denoting the allowable and imm vales take the ollowing orms: l dop 300 (0) 5 ql 38 E J () 5. COMPUTER SIMULATIO The calclations are condcted sing two dierent methods, the analytical approach as well as the pertrbation-based, whose nmerical comparison is presented in the net section. To begin with compter simlations, it is based on ten dierent I beams sed most reqently in civil engineering. For all o them we created new, redced cross-sectional areas simlating the intermediate and inal eect o atmospheric corrosion. All the calclations are carried ot in commercial Finite Element Method analysis system ROBOT. The reslts o compter simlations are the basis o bilding the main nction, the ormla o delection cased increasing o corrosion. The analytical response nction is created in the compter algebra system MAPLE sing the least sqare method procedre and it is approimated by a polynomial nction o the lowest possible order (to optimize approimation procedre and minimize nmerical error). However, the analytical generalized stochastic pertrbation analysis is based on mathematical eqation o the delection (). In this case, the redced cross-section is hidden in cross-sectional area and its inertia moment, c. (). In both cases, the coeicient o variation o the corrosion depth adjacent to the atmospheric corrosion is assmed as 0,5. Material parameters are o corse eqal to E=0,0 GPa and G=80,0 GPa as well as ν=0.30. All reslts o comptational analysis are provided in the orm o epected vales and variances o the displacements combined into the reliability indices they are all marked in the vertical ais and presented as the nctions o the eploitation time. The time scale is adeqate to the irst three years o an eploitation to check how ast the nprotected steel proiles really corrode. Additionally we have plotted 3D graphs o the variances or a delection and reliability indices as the nctions o time and the inpt coeicient o variation o the corrosion depth. In this case the coeicient is contained in the interval [0;0,3], whereas time lctations are considered or the irst ive (or iteen) years o the beams eploitation. The irst general observation qite consistent with the engineering intition is that the epected vales o the imm delection increase together with the eploitation time. O corse, the smaller I proile, the larger initial delection and the larger its rther corrosion driven deormation. Comparison o Figre 5 and 8 shows clearly that the epectations determined sing the analytical and the FEM-based eperiments eqal to each other, so practically no dierence is noticed. The variances given in Figs. 6 and 9 accordingly, also increase together with the eploitation time, bt this change is apparently nonlinear and conve, especially or smaller proiles. ow we notice the dierences in variances rom analytical and nmerical methods, especially or the larger proiles, which inlence at most the reliability indices given in Figs. 7 and 0.

4 Figre 5: Epected vales o the delection, 0 th order pertrbation techniqe. Figre 8: Epected vales o the delection, 0 th order pertrbation or compter simlations. Figre 6: Variances o the delection, 0 th order pertrbation techniqe. Figre 9: Variances o the delection, 0 th order pertrbation or compter simlations. Figre 7: Reliability inde, 0 th order pertrbation techniqe. Figre 0: Reliability inde, 0 th order pertrbation or compter simlations.

5 Figre : 3D variances o the delection, 0 th order pertrbation techniqe. Figre : 3D reliability inde, 0 th order pertrbation techniqe. Figre 3: 3D variances o the delection, 0 th order pertrbation or compter simlations. One realizes rom the reliability indices collected on those graphs that analytical method retrns overestimated reslts o this inde, having even the vales 5% higher than or the FEM analysis especially or the largest I proile. Taking into accont the limit vales o the reliability indices given in Erocode 0 we can distingish in-between the proiles conorming the designing rles and those, who cannot be sed in the view o the time-dependent reliability or the given period o eploitation. The beams having the negative reliability inde are atomatically irrelevant in rther reliability-based considerations. Decisively more interesting reslts are contained in Figs., 3 as well as and, where we have the variances and the reliability indices as the nctions o the two inpt parameters. It is apparent that or the coeicient o variation close to 0 or problem becomes deterministic and then variances eqal 0 in any time o the eploitation, which is consistent with the mathematical interpretation. As it is epected, the largest lctations o the reslting variances are noticed or the imm inpt coeicient o variation. Mathematical interpretation o the eqation (9) lead to the conclsion that the reliability inde mst behave somewhat in the opposite way it has very stable reslts or a wide interval o larger vales o the inpt coeicient α, while ehibits enormos variations close to the deterministic problem (when the denominator in (9) becomes closer to 0). Then, or inal indices have the variability closer to that given in the previos, D graphs being positive at the beginning o the corrosion process (positive veriication) and negative at the end o the time scale (negative veriication). Larger random lctations o the corrosion depth may make all those I proiles inadeqate to the given geometrical beam parameters and the eternal load applied, so that an application o the larger proiles wold be necessary. 6. COCLUDIG REMARKS Compter analysis condcted and presented in this paper show that the stochastic lctations o the steel I beams delection depends on increasing corrosion, what conirm both mathematical assmptions and, irst o all, engineering intition and eperience. It is a direct aect o loss o cross-sectional area inclded into its total cross-section area and its inertia moment I, which reslts in a decrease in strength o the steel beam or traditional and shell Finite Element Method model. Analytical relation is initially provided by the well-known delection ormla transormed net sing the generalized stochastic pertrbation techniqe. Frther, analyzing reslts o reliability inde or all ten I beams, it may be noticed that the higher I beam, the slower increase o delection and its probabilistic moments. The major conclsion conirms similarity in the reslts o both methods in the case o epected vales, whereas some dierences are noticed or the variances and even more apparent in the case o the reliability inde. Frther works need to ocs on the elastoplastic analysis o the I beams, non-symmetric thin-walled proiles as well as the higher order reliability methods. 7. REFERECES [] P-E 990:00. Erocode 0. Principles o Strctral Design (in Polish). [] DD EV Erocode 3. Design o Steel Strctres, 00. [3] R.E. Melchers, Advances in mathematical-probabilistic modelling o the atmospheric corrosion o strctral steels in ocean environments, Glasgow, Jly 006. [] J.W. van de Lindt, T.M. Ahlborn, Development o Steel Beam End Deterioration Gidelines, Janary 005. [5] MAPLE v.3 - User Manal, Maplesot, [6] M. Kamiński, Comptational Mechanics o Composite Materials. Springer-Verlag, London, 005. Figre : 3D reliability inde, 0 th order pertrbation or compter simlations.