Reliability Analysis of a Steel Frame M. Sýkora

Transcription

1 Acta Polytechnica Vol. 4 No. 4/00 Reliability Analysis of a Steel Frae M. Sýkora A steel frae with haunches is designed according to Eurocodes. The frae is exposed to self-weight, snow, and wind actions. Lateral-torsional buckling appears to represent the ost critical criterion, which is considered as a basis for the liit state function. In the reliability analysis, the probabilistic odels proposed by the Joint oittee for Structural Safety (JSS) are used for basic variables. The uncertainty odel coefficients take into account the inaccuracy of the resistance odel for the haunched girder and the inaccuracy of the action effect odel. The tie invariant reliability analysis is based on Turkstra s rule for cobinations of snow and wind actions. The tie variant analysis describes snow and wind actions by jup processes with interittencies. Assuing a 50-year lifetie, the obtained values of the reliability index vary within the range fro 3.95 up to The cross-profile IPE 330 designed according to Eurocodes sees to be adequate. It appears that the tie invariant reliability analysis based on Turkstra s rule provides considerably lower values of than those obtained by the tie variant analysis. Keywords: steel frae, lateral-torsional buckling, reliability, jup processes. 1 Introduction Application of the partial factor ethod introduced in operational European standards for structural design often leads to unequal reliability of structures or structural ebers ade of different building aterials and exposed to different cobinations of actions. Well-balanced structural reliability can be achieved using design procedures based on probabilistic ethods. This approach to the verification of structural reliability is allowed in the fundaental European docuent on structural design EN 1990 Basis of Structural Design [1]. At present, the basic principles and data for the design and verification of structural ebers using probabilistic ethods are partly provided in the technical literature and also in recent ISO and EN standards. Detailed guidelines can be found in the JSS working aterials []. It is expected that probabilistic design will becoe a practical design tool. Unfortunately, ipleentation of the design is liited by lack of required input data. The reliability analysis presented in this paper provides reliability verification of a steel frae designed according to recoendations given in the Eurocodes [1,3]. The reliability index as a basic indicator of the level of reliability, is deterined using both tie invariant and tie variant analysis provided by the software product OMREL [4]. The basic variables are described using probabilistic odels recoended by JSS []. The subitted analysis indicates possible procedures for ipleenting probabilistic ethods of structural reliability in the design of civil engineering structures. Deterinistic design.1 Geoetry The portal frae analysed in this study is a double-pinned frae stiffened by haunches in the frae corners as indicated in Fig. 1. The span of the frae is The height of the structure is 7.6. The slope of the roof is approxiately 15. The axiu loading width is The cross-section of the frae consists of the rolled I-profile IPE 330. In the location of the haunches, a T-section of variable height (10 80 ) is welded on it (see Fig. 1). The axiu section height is 610 in the frae corner. The lengths of the haunches are.0 and.8, respectively. 15 IPE 330. Effects of actions The frae is exposed to the self-weight of the load bearing girders and the roof, snow, and wind action. The effect of the iposed action and theral actions is negligible. The action effects of the actions considered in the analysis consist of an axial force N and bending oent M.Inthedesigncalculation, the axial force and bending oent are represented by the design values N d and M d. The cobination of actions is deterined considering expression (6.10b), given in EN 1990 [1]. If the snow load is the leading variable action, then it follows that: Nd GNfrae, k Nroof, k (1) N N M IPE 330 Fig. 1: Geoetry of the frae [] A-A Q snow, k 0w wind, k h = 610 Md G Mfrae, k Mroof, k () Q snow, k 0w Mwind, k where 0.85 is the reduction factor for peranent actions, G 1.35 is the partial factor for peranent actions, Q 1.5 is the partial factor for variable actions, and 0,w 0.6 is the factor for the cobination value of the wind action. N frae,k is the characteristic value of the axial force due to the self-weight of the frae (the rolled sections) estiated as 0.49 kn/. In the location of the haunches, it ranges fro 0.49 to 0.76 kn/. N roof,k is the characteristic value of the axial force due to the self-weight of the roof structure. The load, including the secondary longitudinal girders, is estiated as 0.15 kn/. N snow,k is the characteristic value of the A A zech Technical University Publishing House 7

2 Acta Polytechnica Vol. 4 No. 4/00 axial force due to snow action. The characteristic value of the snow load s k deterined according to [5] is given as: sk 1 et sg,k (3) where is the load shape coefficient considered for a unifor snow load covering a whole roof area and for a roof slope about 15. Both the exposure coefficient e and the heat coefficient t are chosen equal to 1 and the characteristic value of the snow load on the ground at the weather station is taken as s g,k 1.33 kn/ considering a given site locality (approxiately corresponding to region III for the zech Republic). N wind,k is the characteristic value of the axial force due to wind action. Following the recoendations provided in [6], the characteristic value of the wind pressure w k is given as: wk cpqp z (4) where c p is the pressure coefficient dependent on the building geoetry and the size of the loaded area (here, the loaded area is assued to be larger than 10 ). It describes the outside pressure and suction cobined with either the inside suction or the inside overpressure. In this case, ore unfavourable effects are caused by a cobination of outside pressure and inside suction. The peak velocity pressure can be written as: 1 q p z c g z c r z v b (5) where c g (z).4 is the gust factor specified for the height of the structure z 7,5 and for the terrain category II open terrain with isolated obstacles [6]. The roughness factor c r (z 7,5 ) 0.95 is also defined for the terrain category II. The air density is taken as 1.5 kg/ 3. The reference wind speed v b is 6 /s. It results q p (z) 0.9 kn/. The design values of the bending oents are derived fro the sae assuptions as for the design values of the axial forces..3 Structural analysis The internal forces were deterined using the deforation ethod. The structure has been odelled as a double-pinned frae. To odel the real behaviour of the frae, the haunches of the girder were divided into 6 parts, each having a constant height corresponding to the iddle cross-section of the relevant part. It appears that the shear does not affect the bending capacity and need not be taken into account. The structure is classified as a sway frae and consequently the sway oents caused by the wind action are increased by the factor k 1.8. The buckling length of the colun with respect to axis y L y 1.48 is taken as.6-ultiple of the length of the colunfollowingtheapproxiateprocedureforswayfraes shownin[3].thebucklinglengthl z with respect to axis z is chosen as., which is the distance between the stays for lateral buckling restraint. As for the diaphrag bea, L y is half of the bea span. L z is again.. Each of the cross-sections within the haunch is checked against buckling without lateral-torsional buckling and buckling with lateral-torsional buckling. The design criterion for buckling without lateral-torsional buckling sees to yield the ost critical criterion for checking of the colun. The ost critical criterion for the diaphrag bea is the criterion for buckling with lateral-torsional buckling. It appears that the critical cross-sections within the colun and diaphrag bea are just at the origin of the haunches. The design criterion for buckling without lateral-torsional buckling is expressed as: N k M Sd y y Sd,y A f 1 (6) y,k fy,k Wp1, y M1 M1 where, in the critical cross-section of the colun, N Sd 115kNisthedesignvalueoftheaxialforcedueto the actions, 0.63 is the buckling coefficient (the lower of the values y 0.63 and z 0.80), A is the area of the relevant cross-section (A IPE ), f yk 75 MPa is the characteristic value of the yield strength of the steel S75, M1 1.1 is the partial factor for the aterial property, k y 1.09 is the oent aplification factor, y 0.95 is the equivalent unifor oent factor, M Sd,y 13 kn is the design value of the bending oent due to the actions and W pl,y is the plastic sectional odulus ( W pl,y,ipe ). The design criterion for buckling with lateral-torsional buckling is given as: N M Sd Sd,y A f 1 (7) y,k fy,k z LTWp1,y M1 M1 where, in the critical cross-section of the bea, N Sd is 91 kn, z is 0.80, M Sd,y is 161 kn, and LT 0.89 is the buckling coefficient of lateral-torsional buckling. The Eurocodes [3] do not provide a procedure for deterining the critical bending oent at the liit of the lateral-torsional buckling M cr of the haunched girder, which is required for calculation of LT. Therefore, the critical bending oent M cr is approxiately calculated neglecting the effect of the haunch. It is assued that the I-section alone without the haunch resists lateral-torsional buckling. onsidering the criterion for buckling, the ratio between the design action effect and the design resistance for the critical cross-section of the colun at the origin of the haunch is 0.8. For the critical cross-section of the bea at the origin of the haunch, the ratio is 0.97 taking into account the criterion for buckling with lateral-torsional buckling. Thus, this cross- -section is also the ost critical one within the whole structure and for this reason its reliability is verified in the following analysis. 3 Liit state function As entioned above, the reliability analysis concentrates on the critical cross-section of the bea at the origin of the haunch. The liit state function is derived fro the design criterion for lateral-torsional buckling (7). In addition, the uncertainty odel coefficients are used to take into account the inaccuracy of the resistance odel for the haunched girder and the inaccuracy of the action effect odel. The liit state function reads as: N M gx 1 EN S EM S, y Af W f 0 (8) RNz y RMLT p1, y y 8 zech Technical University Publishing House

3 Acta Polytechnica Vol. 4 No. 4/00 where EN is the coefficient of the odel uncertainties for axial force and EM for bending oent, RN is the coefficient of the odel uncertainties for axial force resistance and RM for bending oent resistance. Utilizing the results of the structural analysis, the internal forces in the critical cross-section can be siply written as: NS 67. Bgroof gfrae 647. Bs 54. Bw (9) MS,y 837. Bgroof gfrae 809. Bs 137. Bw (10) where B 6.48 is the loading width, g roof is the self-weight of the roof [kn/ ], g frae is the self-weight of the load bearing girders [kn/], s is the snow load [kn/ ]andw is the wind action [kn/ ]. The liit state function given by equation (8) is applied in the following reliability analysis considering appropriate probabilistic odels for the basic rando variables described below. 4 Theoretical odels for basic variables 4.1 Basic variables Probabilistic odels for basic variables are used in accordance with the odels proposed by the Joint oittee for Structural Safety (JSS). The sectional area A,the plastic sectional odulus W pl,y, the loading width B and the span of the girder L are assued to be deterinistic values (D), while the others are considered as rando variables. The statistical properties of the rando variables are described by the noral distribution (N), lognoral distribution (LN) and the Gubel distribution (G) indicated by the oent characteristics (the ean and standard deviation ) [,7] as listed in Tab. 1. The skewnesses areiplicitlygivenbythetypeof distributions as: N 0, LN 3V X + V X 3,and G The statistical paraeters used for the yield strength are estiated assuing: fy fy, k fy (11) Table 1: Statistical properties of basic variables Var. types Sybol X Nae of basic variable Dist. Di. Paraeters X k X X /X k X V X MP* f y Yield strength LN MPa A Sectional area D Geoetric data Model uncertainties Actions W pl,y Plastic sectional odulus D B Loading width D L Girder span D no no z Buckling coefficient N oefficient of lateralbuckling torsional N LT EN Axial force action effect N EM Bending action effect N RN Axial force resistance N RM Bending resistance N g frae Self-weight due to girders N kn/ g roof Self-weight due to roof N kn/ s year extrees of snow G kn/ s 1 Annual extrees of snow G kn/ w year extrees of wind G kn/ w 1 Annual extrees of wind G kn/ *MP = aterial properties X k is the characteristic value of the variable, X is the ean, X is the standard deviation and V X is the coefficient of variation. zech Technical University Publishing House 9

4 Acta Polytechnica Vol. 4 No. 4/00 Table : Statistical properties of the basic variables used to derive coefficients z and LT Var. types Sybol X Nae of basic variable Dist. Di. Paraeters X k X X /X k X V X MP f yp Yield strength (S35) LN MPa oefficients Iperfection coefficient N Loading and end restraint N factor Loading and end restraint N factor 1 fy 008. fy (1) The paraeters of the buckling coefficient z are derived fro the recoendation indicated in EN 1993 [3] taking into account the rando variability of the yield strength f y, coparative yield strength f yp and iperfection coefficient. Siilarly, the statistical paraeters of the coefficient of lateral-torsional buckling LT are derived. In addition, the factors depending on loading and end restraint conditions 1 and are also considered as rando variables. The paraeters applied in the analysis are listed in Tab.. The coefficients of the odel uncertainties cover the iprecision and incopleteness of the theoretical odels for the frae with the haunches. Their statistical paraeters are assued as in the JSS Probabilistic Model ode []. The probabilisticodelsfortheself-weightactions(g frae and g roof )are considered as in [7]. 4. Snow load As for the snow load s, the statistical paraeters are derived considering equation (3). The characteristic value of the snow load on the ground s g,k is assued to have the probability p 0.0 to be exceeded by annual extrees. Assuing a Gubel distribution and the coefficient of variation V sg, [8], the ean of the annual extrees sg, kn/ canbeobtainedfrothefollowingequation for a fractile of the Gubel distribution: sg,k sg, ln ln 1 p wsg,1 (13) The standard deviation of the annual extrees is then sg 0.33 kn/. For tie invariant analysis, the paraeters of the 50-year extrees ust be deterined. The standard deviation of the 50-year extrees is equal to the standard deviation of the annual extrees for the Gubel distribution. The ean of the N-year extrees can be derived fro the annual extree paraeters as: sg,n sg, lnn sg (14) For N 50, the ean of the 50-year extrees is sg, kn/. The statistical paraeters of the other variables are used in accordance with JSS Probabilistic Model ode []. The statistical paraeters of the annual extrees and 50-year extrees of the snow load (denoted as s 1 and s 50 in Table 1) result fro equation (3) using the statistical odels for rando variables shown in Table Wind action The statistical paraeters of wind pressure w are deterined assuing that: w c p c g c r q q b (15) Table 3: Variables used in calculating the paraeters of the snow load s 1 e t s g Var. types Sybol X Nae of basic variable Dist. Di. Paraeters X k X X /X k X V X oef. 1 e Shape and exposure coef. N t Heat coefficient D Actions s g,1 s g,50 Annual extrees of snow load on ground 50-year extrees of snow load on ground G kn/ G kn/ zech Technical University Publishing House

5 Acta Polytechnica Vol. 4 No. 4/00 Table 4: Variables used in calculating the paraeters of the wind action w c p c g c r q q b Var. types Sybol X Nae of basic variable Dist. Di. Paraeters X k X X / X k X V X oef. c p c g c r q Pressure coefficient N - no no 1 0.1no 0.1 Gust factor N Roughness factor N Model coefficient N Actions Annual extrees of basic wind pressure q b,1 50-year extrees of basic wind pressure q b,50 G kn/ G kn/ where q is the odel coefficient describing the ratio between the expected and coputed value of the basic wind pressure q b, which can be written as: qb 1 v b (16) The characteristic value q b 0.4 kn/ is defined to have the probability p 0.0 to be exceeded by the annual extrees. The coefficient of variation of the annual extrees of the reference wind speed v b is V vb,1 0. [8]. Supposing that the annual extrees of the reference wind speed can be odelled by a Gubel distribution, the coefficient of variation of the annual extrees of the basic wind pressure V qb, results fro: Vvb, Vvb, 1 Vqb, 1 (17) 1 Vvb, 1 The ean of the annual extrees of the basic wind pressure qb,1 0.0 kn/, the ean of the 50-year extrees qb, kn/ and the standard deviation qb kn/ can be derived identically as for the statistical paraeters of the extrees of the snow load on the ground (13,14). The statistical paraeters of the other variables used in calculating the statistical paraeters of the wind pressure w are taken in accordance with the JSS Probabilistic Model ode [] as listed in Table 4. 5 Reliability analysis liatic actions due to snow and wind are coplex tie- -variant quantities that significantly coplicate the reliability analysis. Two different approxiations for describing the are considered in the following analysis. Firstly, Turkstra s rule is accepted in conjunction with tie invariant analysis. Secondly, the Ferry Borges-astanheta odel (FB) is applied together with tie variant reliability analysis. The variable actions due to snow and wind are assued to be uncorrelated. The software product OMREL [4] has been applied in both types of analysis. 5.1 Tie invariant analysis In accordance with Turkstra s rule, the leading action is described by its lifetie (assued as 50 years) extree while the accopanying action is considered by its point-in-tie value (approxiated by annual extrees). In the following analysis, each cliatic action, the snow and the wind action, is considered to be either a leading or an accopanying action w w k = 0.9 w d = 1.38 The probability densities of the 50-year extrees of the snow load s 50 (considered as the leading variable action) and the annual extrees of the wind pressure w 1 are shown in Fig.. The characteristic value of the wind pressure w being the 98-percentage fractile of Gubel s distribution is denoted as w k,andw d denotes the design value. 5. Tie variant analysis The tie variant reliability analysis is based on the FB odel for the snow and wind actions. Both the cliatic s 50 w, s [kn/ ] Fig. : The probability densities of the 50-year extrees of the snow load s 50 and the annual extrees of wind pressure w 1 zech Technical University Publishing House 31

6 Acta Polytechnica Vol. 4 No. 4/00 actions are described by jup processes without interittencies (actions soeties take a zero value), which approxiate their real variation in tie by rectangular wave renewal functions. Each jup process with interittencies is characterized by the jup rate (the average nuber of agnitude changes of the square waves in a reference tie T ref )andbythe interarrival-duration intensity (the product of the arrival rate and the ean duration with respect to a reference tie T ref ). As for the snow load, it is assued that it takes its extree five ties a year. onsidering the reference tie T ref 1year, Snow load s reliability index than those obtained by the tie variant analysis. The tie variant analysis predicts the interval at which the reliability index can be expected (a higher value of corresponds to the lower bound of a failure probability while a lower value of corresponds to the upper bound of a failure probability). The results obtained by the tie variant analysis areorefavourableandindicatethateventhesallerprofile LOWER BOUND (p f ) TIME VARIANT ANALYSIS UPPER BOUND (p f ) TARGET VALUE OF ~14 ~14 ~14 ~14 ~14 ~ 73 days ~ 73 days ~ 73 days ~ 73 days ~ 73 days 365 days = 1 year t day IPE 300 TIME INVARIANT ANALYSIS TURKSTRA S RULE W pl,y [ ] IPE 330 Fig. 3: The approxiation of the snow load used in the tie variant analysis the arrival rate of on-ties is therefore s 5. The ean duration (the tie during which the structure is loaded by the extree snow load) is supposed to be about 14 days. The interarrival-duration intensity is thus s 5 14/ The possible approxiation of the snow load during the reference tie T ref 1yearisshowninFig.3. Windstors are expected to appear ten ties a year ( w 10) and the ean duration of the stor is estiated as 8 hours. The interarrival-duration intensity is then w 10 8/(4 365) Fig. 4: Reliability index as a function of the plastic sectional odulus W pl,y IPE 300 ight be acceptable. The expected values of the reliability index foripe300arewithintherangefro3.57(the upper bound) to 4.84 (the lower bound) for IPE 330 fro 4.49 up to 5.56 as listed in Table 5. Fig. 4 shows the reliability index deterined by both the analyses as a function of the plastic sectional odulus W pl,y Results of reliability analysis 5. LOWER BOUND (p f ) According to the results of the tie invariant analysis, the reliability of a structure of the IPE 300 profile sees to be rather low ( is less than 3), while the cross-profile IPE 330 sees to be acceptable. For the higher profile the resulting reliability index 3.95 corresponds well to the recoended value 3.8 [1], as shown in Table 5. Nevertheless, it should be entioned that the tie invariant analysis based on Turkstra s rule provides considerably lower values for UPPER BOUND (p f ) Fig. 5: Reliability index as a function of the jup rate of the snow load s s Table 5: Results of reliability analysis Analysis Used load odels Reliability index IPE 300 IPE 330 Tie invariant s 50 + w s 1 + w Tie variant Jup processes with interittencies zech Technical University Publishing House

7 Acta Polytechnica Vol. 4 No. 4/ Effect of input data on resulting reliability index The odel paraeters and required for the tie variant analysis are very difficult to specify. Nevertheless, paraetric studies indicate that uncertainty in and have an insignificant effect on the resulting reliability. For exaple, ifthevalueofthejuprateofthesnowload s increases fro 1 to 5 (i.e. if the snow load takes its extree lasting 14 days five ties a year, which is not real), the upper bound of decreases approxiately by 0.4 for the cross-profile IPE330, as shown in Fig. 5. Paraetric studies of the jup rate of the wind action w and of the interarrival-duration intensities of the two cliatic actions s and w provide siilar results. 5.5 Probabilistic optiisation Probabilistic optiisation is based on iniisation of a siplified objective function expressed as the su of the initial, arginal and expected alfunction cost. The decisive paraeter is the sectional area A. The total cost tot can be expressed as: tot 0 A f pf A (18) where 0 denotes the initial cost independent of paraeter A and failure probability p f. The arginal cost is the product of theunitcostofthesectionalarea and the sectional area A. The expected alfunction cost is the product of failure probability p f and alfunction cost f when failure occurs. For probabilistic optiisation, equation (18) ay be adapted as: tot 0 tot A f p f A (19) Therelativeincreentofthetotalcost tot is dependent only on the decisive paraeter A andontheratio f /. hoosing various values of this ratio, different cross-sections see to be adequate according to the results of the probabilistic optiisation shown in Fig. 6. The arrows point to the inia of the relative increent tot for assued ratios f /. The dot-and-dash curve shows the resulting reliability index dependent on sectional area A assuing Turkstra s rule (the 50-year extrees of the snow load and the annual extrees of the wind action). The horizontal dashed line arks the target value of ( t 3.8). Obviously with the increasing cost ratio f / the optiu cross-sectional area A increases. Profile IPE 330 sees to be optial for f / = To get credible results of the optiisation, it is necessary to deterine the values of and f exactly. 6 onclusions The structural analysis of the frae shows that lateral-torsional buckling represents the ost critical design criterion and indicates that the snow load is the leading variable action. onsidering a 50-year lifetie, the reliability index for IPE 330 varies within the range fro 3.95 up to According to the results of the reliability analysis, cross-profile IPE 330 designed using Eurocodes sees to be adequate. The tie invariant analysis based on Turkstra s rule provides considerably lower values of than those obtained by the tie variant analysis. It sees that Turkstra s rule leads to a rather conservative reliability level for a cobination of variable actions having significant interittencies. The great differences between the lower and upper bounds are ost likely caused by the considerable interittencies of the two variable actions. The odel paraeters required for tie variant analysis are, however, very difficult to estiate. Nevertheless paraetric studies indicate that this uncertainty has an insignificant effect on the resulting reliability. Structural analysis of a bea with haunches exposed to lateral-torsional buckling is a very coplicated task. It is foreseen that ore precise results ay be obtained by an analysis based on the odel using the Finite Eleent Method. Total ost tot 7500 f f 10 6 f IPE 300 f 10 5 IPE Fig. 6: Relative increent of total cost tot as a function of sectional area A using Turkstra s rule (s 50 + w 1 ) A[ ].4 zech Technical University Publishing House 33

8 Acta Polytechnica Vol. 4 No. 4/00 Acknowledgeent This research has been conducted at the Klokner Institute, zech Technical University in Prague, zech Republic, as a part of research project TU Basis for Probabilistic Design of Structural Eleents in Accordance with Eurocodes. References [1] EN 1990 Basis of Structural Design. Brussels: European oittee for Standardisation, 00. [] JSS Probabilistic Model ode. Joint oittee for Structural Safety, 001 ( [3] pren Design of Steel Structures. Brussels: European oittee for Standardisation, 001. [4] RP Reliability onsulting Progras: Strurel: AStructural Reliability Analysis Progra Syste. OMREL & SYS- REL User's Manual. Munich: RP onsult, [5] pren Actions on Structures Snow Loads. Final draft. Brussels: European oittee for Standardisation, 001. [6] pren Actions on Structures Wind Actions. Final draft. Brussels: European oittee for Standardisation, 001. [7] Holický M.: Reliability Based alibration of Eurocodes onsidering Steel oponent. JSS Workshop on Reliability Based ode alibration, Zürich: Joint oittee for Structural Safety, 001. [8] Tichý, M.: Zatížení stavebních konstrukcí. (Actions on Structures). Technický průvodce 45, Praha: SNTL, Ing. Miroslav Sýkora phone: fax: e-ail: sykora@klok.cvut.cz zech Technical University in Prague Klokner Institute Šolínova Praha 6, zech Republic 34 zech Technical University Publishing House