NSCC2009 Reliability of steel flexural members accordig to EC i serviceability limit state D. Hofi & A. Mårtesso Divisio of Structural Egieerig, Lud Uiversity, Lud, Swede ABSTRACT: To achieve a relatively cosistet probability of failure for structural elemets, most desig codes apply reliability based code calibratio process. Such approaches commoly focus o the ultimate stregth of the structural members, which is related to the ultimate limit state (ULS). I the desig of steel beams the performace of the structural elemets is ofte limited by the serviceability requiremets, which are related to the serviceability limit state (SLS) usig differet load combiatios tha those applied i the ultimate limit state. The curret study aims to ivestigate the reliability for serviceability desig for flexural steel members accordig to the specificatios of the Eurocode. Secod-order reliability method (SORM) is applied to determie the reliability idex for differet load ratios. 1 INTRODUCTION 1.1 Serviceability Ultimate failure of structural elemets is relatively rare. However, serviceability o-compliaces occur more frequetly. The mai type of serviceability o-compliaces are excessive floor ad roof deflectios which may cause for example (Stewart 1996): cracig ad local crushig of structural ad o-structural masory walls; cracig of reiforced cocrete floor beams ad slabs; gaps below partitios; oticeable dishig of floors; doors out of square; filig cabiets ad dess tilted; cracig of plastered ceiligs; damage to services; podig ad water/moisture peetratio (i case of roof); or simply be aesthetically aoyig or give the feelig of beig usafe. To achieve a relatively cosistet probability of failure for structural elemets, most desig codes such as Eurocode apply reliability based code calibratio process. Such approaches commoly focus o the ultimate stregth of the structural members. Therefore the characteristic values of actios 446
ad the combiatio factors which are used i serviceability limit states (SLS) as well are maily developed ad optimized for the ultimate limit state (ULS). The preset paper aims to ivestigate the reliability of Eurocode specificatios for serviceability, cosiderig a steel structural member subjected to bedig. 2 DESIGN ACCORDING TO EUROCODE 2.1 Deflectio limits The vertical deflectios of horizotal structural elemets should be limited to avoid deformatios that affect appearace/comfort/fuctioig of the structure or that cause damage to fiishes or ostructural members. Figure 1. Defiitio of vertical deflectios. The defiitio of vertical deflectios is show i Figure 1, where w c is the precamber i the uloaded structural member; w 1 is the iitial part of the deflectio uder permaet loads of the relevat combiatio of actios; w 2 is the log-term part of the deflectio uder permaet loads; w 3 is the additioal part of the deflectio due to the variable actios of the relevat combiatio of actios; w tot is the total deflectio as sum of w 1, w 2, w 3 ; w max is the remaiig total deflectio taig ito accout the precamber. If the fuctioig or damage of the structure or to fiishes, or o-structural members (e.g. partitio walls, claddigs) is beig cosidered, the verificatio for deflectio should tae accout of those effects of permaet ad variable actios that occur after executio of the member or fiish cocered, see EN 1990:2002 A1.4.3 (3) (CEN 2002). j 1 G + Q + ψ Q (1), j,1 i 1 0, i, i where G,j deotes the characteristic value of the jth permaet actio, Q,1 is the characteristic value of the leadig variable actio, Q,i is the characteristic value of the ith variable actio ad ψ 0,i is the factor for combiatio value of a ith variable actio. If the appearace of the structure is beig cosidered, the quasi-permaet combiatio should be used, see EN1990:2002 A1.4.3 (4), j 1 G + ψ Q (2), j i 1 2, i, i where ψ 2,i is the factor for quasi-permaet value of the ith variable actio. I case of steel structures EN 1993-1-1:2005 (CEN 2005) states that the limits for vertical deflectios accordig to Figure 1 should be specified for each project ad agreed with the cliet ad otes that the Natioal Aexes may specify these limits. 2.2 Natioal Aexes The Natioal Aexes (NA) of EN 1993 may specify the limits of deflectios. The values give i the NAs are oly suggested values, there are ot obligatory rules give. These prescriptios are quite varyig ad ca be differet depedig o the fuctio (e.g. accessible/o-accessible roof, floor etc.), the importace (mai girder, purli), the type of the carried material (plaster, brittle fiish, o-brittle fiish) or other coditios of the ivestigated elemet. 447
The deflectio limit values for a geeral steel beam ot carryig brittle fiish or havig some special requiremet are summarized i Table 1. These values are valid for the characteristic combiatio of actios. Table 1. Deflectio limits for a geeral floor beam ot carryig brittle fiish w max w 3 UK - L/200 Demar - L/400 Filad L/400 - Greece L/250 L/300 Spai - L/300 Hugary L/250 L/300 2.3 Load represetatio I the preset study maximum deflectio is calculated from the characteristic load combiatio give i Equatio 1. Cosiderig a desig situatio where the leadig variable actio is the floor live load ad the secod variable actio is the wid load this formula simplifies to: = + +ψ 0 (3) q G Q, where q represets the omial load, G is the characteristic value of the permaet actio, Q is the characteristic value of the live load, is the characteristic value of the wid load ad ψ 0,w is its combiatio factor. To ivestigate the effect of the variable actios o the reliability, the followig expressios ca be implemeted accordig to Gulvaessia & Holicy (2002): Q + = G + Q + χ (4) = (5) Q where χ represets the ratio of the variable loads to the total load (load factor) ad deotes the ratio of the wid actio to the live load (variable load factor). Usig the above equatios the characteristic values ca be expressed as: q G = (6) 1+ χψ 0, 1+ (1 + )(1 χ) χg Q = (7) ( 1+ )(1 χ) = Q (8) These represetatios of loads will be used i the followig. 2.4 Target reliabilities The target reliability for irreversible serviceability limit states i EN 1990 Aex C is set to be 1.5 for a 50 years referece period ad 2.9 for a 1 year referece period for reliability class 2 (RC2) structural members. Class RC2 ca be associated with cosequece class 2, which covers structures with medium cosequece for loss of huma life, ecoomic, social or evirometal cosequeces cosiderable, e.g. residetial ad office buildigs, public buildigs where cosequeces of failure are medium. 448
3 RELIABILITY ANALYSIS 3.1 Serviceability limit state The serviceability limit state defies what costitutes a serviceability failure. I the preset paper serviceability o-compliace is deemed to occur whe a deflectio exceeds a allowable deflectio limit as a result of flexure. Sice the deflectio model of the structural steel member is based o elastic behavior, the maximum deflectio of a simply supported steel beam ca be expressed as: 5 ql 4 δ max = (9) 384 EI where δ is the midspa deflectio of the beam, q is the uiformly distributed load, L is the spa, E is the modulus of elasticity ad I is the secod momet of iertia. If the beam is well desiged the stiffess of the sectio is determied i such a way, that the calculated deflectio should be equal to the allowable deflectio limit (if the serviceability limit state is critical). Thus the omial value of the deflectio is equal to deflectio limit give by the code (subscript meas omial values): 4 5 ql δ limit = δ = (10) 384 EI The limit state fuctio for a member satisfyig the deflectio limit ca be expressed as: 4 5 ql δ max EI qei g( X ) = 1 = 1 384 = 1 4 δ limit 5 q L qei 384 E I I the above equatio the omial values are determiistic values give by codes ad stadards, however the actual values ca be represeted as stochastic radom variables. Therefore there is a certai probability of serviceability o-compliace, which is accordig to the above equatio theoretically idepedet from the deflectio limit, assumig that the deflectio limit itself is a determiistic variable. To cosider the effect of model ucertaities the limit state fuctio for the reliability aalysis should be writte i the followig form: θ δ g( X ) θ δ R limit (11) E max = 1 (12) where θ E is the coefficiet expressig the ucertaity of the actio effect ad θ R is the ucertaity of the resistace model. 3.2 Model of basic variables The probabilistic models of basic variables for time ivariat aalysis are give i Table 2. Table 2. The simplified probabilistic models of basic variables for time ivariat aalysis Descriptio X Distributio μ X COV(X) Youg s modulus Momet of Iertia E I Normal Normal E I 0.04 0.03 Dead load Live load 50 years Live load 5 years id load 50 years id load 1 year G Q Q Normal Gumbel Gumbel Gumbel Gumbel G 0.6Q 0.2Q 0.7 0.5 0.10 0.35 1.10 0.25 0.40 Actio effect Resistace factor θ E θ R Logormal Logormal 1 1 0.10 0.05 449
Colum X cotais the radom variables ad μ x the mea value of the radom variable X. The parameters have bee tae from JCSS Probabilistic Model Code (JCSS 2001). Note that there are o factors give taig ito accout the model ucertaities i case of deflectio, therefore θ E for momets i frames ad θ R for bedig momet capacity have bee cosidered. The applied load model was the same as the model used by Gulvaessia & Holicy (2002) which ca be used for geeral purposes, however whe the reliability of differet type of structural members uder particular coditios is assessed, the proposed models may be modified. The load combiatios are modeled usig Turstra s combiatio rule to model load combiatios, which states that the maxi-mum value of sum of two idepedet radom processes occurs whe oe of the processes has its maximum value. Usig this rule the variable load i our case is give as: apt Q max + Pmax = max (13) apt Q + max where P max deotes the maximum value of the variable actios, Q max, Q apt ad max, apt are the 50- years maximum ad the arbitrary-poit-i-time value of the live load ad the wid load respectively. 4 RESULTS 4.1 Remaiig total deflectio, w max A parametric study has bee carried out ivestigatig the effect of chagig ratio of the variable loads to total load, χ ad the ratio of the wid actio to the live load,. The computatios have bee made usig Secod Order Reliability Method (SORM) with the structural reliability software COMREL 8.10 (RCP 2008). The combiatio factor of the secod variable load (wid actio) ψ 0, was set to 0.6 accordig to EN 1990 Aex A1.2.2. Figure 2 ad 3 preset the reliability idex β ad probability of serviceability o-compliace P f versus load ratio χ respectively. The momet of iertia has bee calculated from the criterio w max L/250 from the characteristic combiatio. It should be oted that the value of the deflectio limit does ot ifluece the serviceability ocompliace. Figure 2. Reliability idex β versus load ratio χ (w max L/250 characteristic combiatio). 450
Figure 3. Probability of failure P f versus load ratio χ (w max L/250 characteristic combiatio). From the figures it ca be see that reliability is ot cosistet. I case of relatively low variable loads (or high value of self-weight) the reliability is very low ad therefore the probability of serviceability o-compliace is high. For the first sight it ca be surprisig that whe the beam is loaded oly by permaet actios ad desiged for that situatio oly the reliability idex is almost equal to zero which meas that the probability of serviceability o-compliace is 50%. However it ca be theoretically prove that if the deflectio depeds oly o G, E ad I ad all of them are ormal radom variables β will be equal to zero. The obtaied values here are slightly higher, sice actio effect ad resistace factor have bee tae ito accout. It should be metioed, that from a practical poit of view this case is ot importat, sice structures are desiged to carry loads. The target reliability β=1.5 give i EN 1993-1-1 Aex C is idicated i Figure 2. I most of the cases the reliability is below that value, however the actual values of β are ot of great importace, sice the applied model cotais some simplificatios. A more importat issue is the remarable differeces of β for low ad high values of the load factor χ. It ca also be observed that icreasig the secod variable actio (i this case the wid) the reliability decreases ad so the probability of o-compliace icreases, however this effect is ot so sigificat. 4.2 Additioal part of the deflectios, w 3 I case of icremetal deflectio of the beam the load factor χ (the variable loads to total load) obviously has o effect o the reliability, sice it is calculated oly from the variable actios. Figure 4 shows the effect of varyig the variable load factor (the ratio of the wid actio to the live load) o the reliability idex β. The required momet of iertia i this case has bee calculated from the criterio w 3 L/300 from the characteristic combiatio of the variable loads. The target reliability idex β=1.5 give i EC3 Aex C is idicated i this figure too. It ca be see that the values are, i most of the cases, over the target reliability they fall below the lie oly for sigificat secod variable load. For the icremetal deflectio criteria the reliability ca be cosidered as cosistet (stadard deviatio of the results is 0.07). 451
Figure 4. Reliability idex β versus variable load ratio (w 3 L/300 characteristic combiatio). 4.3 Compariso with ULS I Figure 5 the differet behavior of β is illustrated i ultimate ad serviceability limit state whe the load ratio is chagig (i both cases =0.5). I case of ULS we obtai higher reliability values i the low variable load regio (χ<0.5) ad less reliability whe the variable loads are higher (χ>0.5). This is cotradictory to the results of SLS, which suggests that serviceability problems occur more liely to structures where the self-weight is more relevat. Figure 5. Compariso of β with varyig load factor i case of ULS ad SLS. 452
5 CONCLUSIONS A secod order reliability aalysis has bee carried out to ivestigate the cosistecy of the probability of serviceability o-compliace accordig to Eurocode i case of a simply supported steel beam subjected to uiform loadig. The reliability related to the total deflectio has bee foud to be ot cosistet i the serviceability limit state. I case of low variable loads the reliability seemed to be low ad was geerally below the target value (β=1.5) give by the code. By icreasig the secod variable actio the reliability decreased. The reliability i case of the icremetal deflectio was close to the target reliability ad it was relatively cosistet. Although the actual value of the deflectio limit does ot ifluece the reliability of the serviceability limit state, it is importat, sice this limit will decide the relatio to the ultimate limit state ad therefore ifluece the desig. Ufortuately the origi of these values is ot clear ad they are ot harmoized amog the differet coutries, therefore a itesive ivestigatio o this topic is essetial. REFERENCES CEN (2002) EN 1990 Eurocode Basis of structural desig. CEN (2005) EN 1993-1-1 Eurocode 3 Desig of steel structures: Geeral rules ad rules for buildigs. Gulvaessia, H & Holicy, M (2002) Reliability based calibratio of Eurocodes cosiderig a steel member. JCSS orshop o Reliability Based Code calibratio. JCSS (2001) Probabilistic Model Code. Stewart, MG (1996) Optimizatio of serviceability load combiatios for structural steel beam desig. Structural Safety 18(2): 225-238. RCP (2008) COMREL Versio 8.10. ww.strurel.de. 453