AP02_04.vp 1 Introduction Application of the partial factor method introduced in operational European standards for structural design often leads to unequal reliability of structures or structural mem- bers made of different building materials and exposed to different combinations of actions. Well-balanced structural reliability can be achieved using design procedures based on probabilistic methods. This approach to the verification of structural reliability is allowed in the fundamental European document on structural design EN 1990 Basis of Structural Design [1]. At present, the basic principles and data for the design and verification of structural members using probabilistic methods are partly provided in the technical literature and also in recent ISO and EN standards. Detailed guidelines can be found in the JCSS working materials [2]. It is expected that probabilistic design will become a practical design tool. Unfortunately, implementation of the design is limited by lack of required input data. The reliability analysis presented in this paper provides reliability verification of a steel frame designed according to recommendations given in the Eurocodes [1,3]. The reliabil- ity index �� as a basic indicator of the level of reliability, is determined using both time invariant and time variant analy- sis provided by the software product COMREL [4]. The basic variables are described using probabilistic models recom- mended by JCSS [2]. The submitted analysis indicates possible procedures for implementing probabilistic methods of structural reliability in the design of civil engineering structures. 2 Deterministic design 2.1 Geometry The portal frame analysed in this study is a double-pinned frame stiffened by haunches in the frame corners as indicated in Fig. 1. The span of the frame is 17.71 m. The height of the structure is 7.26 m. The slope of the roof is approximately 15°. The maximum loading width is 6.48 m. The cross-sec- tion of the frame consists of the rolled I-profile IPE 330. In the location of the haunches, a T-section of variable height (10–280 mm) is welded on it (see Fig. 1). The maximum section height is 610 mm in the frame corner. The lengths of the haunches are 2.0 and 2.8 m, respectively. 2.2 Effects of actions The frame is exposed to the self-weight of the load bear- ing girders and the roof, snow, and wind action. The effect of the imposed action and thermal actions is negligible. The ac- tion effects of the actions considered in the analysis consist of an axial force N and bending moment M. In the design calcu- lation, the axial force and bending moment are represented by the design values Nd and Md. The combination of actions is determined considering expression (6.10b), given in EN 1990 [1]. If the snow load is the leading variable action, then it follows that: � � � � N N N N N d G frame, k roof, k Q snow, k 0 w wind, k � � � � � � � � � � (1) � � � � M M M M M d G frame, k roof, k Q snow, k 0 w wind, k � � � � � � � � � � (2) where � � 0.85 is the reduction factor for permanent actions, �G � 1.35 is the partial factor for permanent actions, �Q � 1.5 is the partial factor for variable actions, and �0,w � 0.6 is the fac- tor for the combination value of the wind action. Nframe,k is the characteristic value of the axial force due to the self-weight of the frame (the rolled sections) estimated as 0.49 kN/m. In the location of the haunches, it ranges from 0.49 to 0.76 kN/m. Nroof,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 esti- mated as 0.15 kN/m2. Nsnow,k is the characteristic value of the © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 27 Acta Polytechnica Vol. 42 No. 4/2002 Reliability Analysis of a Steel Frame M. Sýkora A steel frame with haunches is designed according to Eurocodes. The frame is exposed to self-weight, snow, and wind actions. Lateral-torsional buckling appears to represent the most critical criterion, which is considered as a basis for the limit state function. In the reliability analysis, the probabilistic models proposed by the Joint Committee for Structural Safety (JCSS) are used for basic variables. The uncertainty model coefficients take into account the inaccuracy of the resistance model for the haunched girder and the inaccuracy of the action effect model. The time invariant reliability analysis is based on Turkstra’s rule for combinations of snow and wind actions. The time variant analysis describes snow and wind actions by jump processes with intermittencies. Assuming a 50-year lifetime, the obtained values of the reliability index � vary within the range from 3.95 up to 5.56. The cross-profile IPE 330 designed according to Eurocodes seems to be adequate. It appears that the time invariant reliability analysis based on Turkstra’s rule provides considerably lower values of � than those obtained by the time variant analysis. Keywords: steel frame, lateral-torsional buckling, reliability, jump processes. IPE 330 h = 610mm 15° IPE 330 A - A´ A A ´ IPE 330 h = 610mm 15° IPE 330 A - A´ A A ´ A A ´ Fig. 1: Geometry of the frame [m] axial force due to snow action. The characteristic value of the snow load sk determined according to [5] is given as: s C C sk e t g, k� �1 (3) where �1 � 0.8 is the load shape coefficient considered for a uniform snow load covering a whole roof area and for a roof slope about 15°. Both the exposure coefficient Ce and the heat coefficient Ct are chosen equal to 1 and the characteristic value of the snow load on the ground at the weather station is taken as sg,k � 1.33 kN/m 2 considering a given site locality (ap- proximately corresponding to region III for the Czech Re- public). Nwind,k is the characteristic value of the axial force due to wind action. Following the recommendations provided in [6], the characteristic value of the wind pressure wk is given as: � �w c q zk p p� (4) where cp is the pressure coefficient dependent on the build- ing geometry and the size of the loaded area (here, the loaded area is assumed to be larger than 10 m2). It describes the out- side pressure and suction combined with either the inside suc- tion or the inside overpressure. In this case, more unfavourable effects are caused by a combination of outside pressure and inside suction. The peak velocity pressure can be written as: � � � � � �q z c z c z vbp g r� 2 21 2 � (5) where cg(z) � 2.4 m is the gust factor specified for the height of the structure z � 7,5 m and for the terrain category II – open terrain with isolated obstacles [6]. The roughness factor cr (z � 7,5 m) � 0.95 is also defined for the terrain category II. The air density � is taken as 1.25 kg/m3. The reference wind speed vb is 26 m/s. It results qp(z) � 0.92 kN/m 2. The design values of the bending moments are derived from the same assumptions as for the design values of the axial forces. 2.3 Structural analysis The internal forces were determined using the de- formation method. The structure has been modelled as a double-pinned frame. To model the real behaviour of the frame, the haunches of the girder were divided into 6 parts, each having a constant height corresponding to the middle 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 frame and consequently the sway moments caused by the wind action are increased by the factor k � 1.28. The buckling length of the column with respect to axis y Ly � 12.48 m is taken as 2.6-multiple of the length of the col- umn following the approximate procedure for sway frames shown in [3]. The buckling length Lz with respect to axis z is chosen as 2.2 m, which is the distance between the stays for lateral buckling restraint. As for the diaphragm beam, Ly � 8.855 m is half of the beam span. Lz is again 2.2 m. Each of the cross-sections within the haunch is checked against buckling without lateral-torsional buckling and buck- ling with lateral-torsional buckling. The design criterion for buckling without lateral-torsional buckling seems to yield the most critical criterion for checking of the column. The most critical criterion for the diaphragm beam is the criterion for buckling with lateral-torsional buckling. It appears that the critical cross-sections within the column and diaphragm beam are just at the origin of the haunches. The design criterion for buckling without lateral-torsional buckling is expressed as: N A f C M W f Sd y, k M1 y my Sd, y p1, y y, k M1 k � � � �1 (6) where, in the critical cross-section of the column, NSd � 115 kN is the design value of the axial force due to 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 (AIPE330 � 6261 mm 2), fyk � 275 MPa is the characteristic value of the yield strength of the steel S275, �M1 � 1.1 is the partial factor for the material property, ky � 1.09 is the moment amplification factor, Cmy � 0.95 is the equivalent uniform moment factor, MSd,y � 132 kNm is the design value of the bending moment due to the actions and Wpl,y is the plastic sectional modulus ( Wpl, y, IPE330 � 804�10 3 mm3). The design criterion for buckling with lateral-torsional buckling is given as: N A f M W f Sd z y, k M1 Sd, y LT p1, y y, k M1 � � � �1 (7) where, in the critical cross-section of the beam, NSd is 91 kN, �z is 0.80, MSd,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 deter- mining the critical bending moment at the limit of the lat- eral-torsional buckling Mcr of the haunched girder, which is required for calculation of �LT. Therefore, the critical bending moment Mcr is approximately calculated neglecting the effect of the haunch. It is assumed that the I-section alone without the haunch resists lateral-torsional buckling. Considering the criterion for buckling, the ratio between the design action effect and the design resistance for the criti- cal cross-section of the column at the origin of the haunch is 0.8. For the critical cross-section of the beam 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 most critical one within the whole structure and for this reason its reliability is verified in the following analysis. 3 Limit state function As mentioned above, the reliability analysis concentrates on the critical cross-section of the beam at the origin of the haunch. The limit state function is derived from the design criterion for lateral-torsional buckling (7). In addition, the uncertainty model coefficients are used to take into account the inaccuracy of the resistance model for the haunched girder and the inaccuracy of the action effect model. The limit state function reads as: � �g X N Af M W f � � � � � � � � �1 0 � � � � EN S RN z y EM S, y RM LT p1, y y (8) 28 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 42 No. 4/2002 where EN is the coefficient of the model uncertainties for ax- ial force and EM for bending moment, RN is the coefficient of the model uncertainties for axial force resistance and RM for bending moment resistance. Utilizing the results of the structural analysis, the internal forces in the critical cross-sec- tion can be simply written as: � �N Bg g Bs BwS roof frame�� � � �6 7 6 47 2 54. . . (9) � �M Bg g Bs BwS, y roof frame� � � �8 37 8 09 13 27. . . (10) where B � 6.48 m is the loading width, groof is the self-weight of the roof [kN/m2], gframe is the self-weight of the load bearing girders [kN/m], s is the snow load [kN/m2] and w is the wind action [kN/m2]. The limit state function given by equation (8) is applied in the following reliability analysis considering appropriate probabilistic models for the basic random variables described below. 4 Theoretical models for basic variables 4.1 Basic variables Probabilistic models for basic variables are used in accor- dance with the models proposed by the Joint Committee for Structural Safety (JCSS). The sectional area A, the plastic sec- tional modulus Wpl,y, the loading width B and the span of the girder L are assumed to be deterministic values (D), while the others are considered as random variables. The statistical properties of the random variables are described by the nor- mal distribution (N), lognormal distribution (LN) and the Gumbel distribution (G) indicated by the moment character- istics (the mean � and standard deviation ) [2,7] as listed in Tab. 1. The skewnesses � are implicitly given by the type of distributions as: �N � 0, �LN � 3VX + VX 3, and �G � 1.14. The statistical parameters used for the yield strength are estimated assuming: � fy y, k fy� �f 2 (11) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 29 Acta Polytechnica Vol. 42 No. 4/2002 ParametersVar. types Symbol X Name of basic variable Dist. Dim. Xk �X �X/Xk X VX MP* fy Yield strength LN MPa 275 327 1.19 26.2 0.08 A Sectional area D mm2 6261 6261 1.00 - - Wpl,y Plastic sectional modulus D mm 3 804000 804000 1.00 - - B Loading width D m 6.48 6.48 1.00 - - L Girder span D m nom nom 1.00 - - z Buckling coefficient N - 0.64 0.67 1.04 0.04 0.06 LT Coefficient of lateral - bucklingtorsional N - 0.79 0.82 1.03 0.04 0.05 �EN Axial force action effect N - - 1 - 0.05 0.05 �EM Bending action effect N - - 1 - 0.1 0.1 �RN Axial force resistance N - - 1.1 - 0.07 0.07 �RM Bending resistance N - - 1.1 - 0.07 0.07 gframe Self-weight due to girders N kN/m 0.49 0.49 1 0.03 0.05 groof Self-weight due to roof N kN/m 2 kN/m2 kN/m2 kN/m2 kN/m2 0.15 0.15 1 0.02 0.1 s 50 50-year extremes of snow G 1.06 1.18 1.11 0.32 0.27 s 1 Annual extremes of snow G 1.06 0.38 0.36 0.27 0.72 w50 50-year extremes of wind G 0.92 0.64 0.7 0.21 0.33 w1 Annual extremes of wind G 0.92 0.28 0.3 0.14 0.52 G eo m et ri c d at a M od el u n ce rt ai n ti es A ct io n s Table 1: Statistical properties of basic variables *MP = material properties Xk is the characteristic value of the variable, �X is the mean, X is the standard deviation and VX is the coefficient of variation. �fy fy� 0 08. (12) The parameters of the buckling coefficient �z are derived from the recommendation indicated in EN 1993 [3] taking into account the random variability of the yield strength fy, comparative yield strength fyp and imperfection coefficient �. Similarly, the statistical parameters of the coefficient of lat- eral-torsional buckling �LT are derived. In addition, the fac- tors depending on loading and end restraint conditions C1 and C2 are also considered as random variables. The parame- ters applied in the analysis are listed in Tab. 2. The coeffi- cients of the model uncertainties cover the imprecision and incompleteness of the theoretical models for the frame with the haunches. Their statistical parameters are assumed as in the JCSS Probabilistic Model Code [2]. The probabilis- tic models for the self-weight actions (gframe and groof) are considered as in [7]. 4.2 Snow load As for the snow load s, the statistical parameters are derived considering equation (3). The characteristic value of the snow load on the ground sg,k is assumed to have the probability p � 0.02 to be exceeded by annual extremes. Assuming a Gumbel distribution and the coefficient of varia- tion Vsg, 1 � 0.7 [8], the mean of the annual extremes �sg,1 � 0.47 kN/m 2 can be obtained from the following equa- tion for a fractile of the Gumbel distribution: � �� �� � � sg,1 g,k sg,1 � � � � � s p w1 0 45 0 78 1. . ln ln (13) The standard deviation of the annual extremes is then sg � 0.33 kN/m 2. For time invariant analysis, the parameters of the 50-year extremes must be determined. The standard deviation of the 50-year extremes is equal to the standard deviation of the annual extremes for the Gumbel distribution. The mean of the N-year extremes can be derived from the annual extreme parameters as: � �� � sg,N sg,1 sg� � 0 78. ln N (14) For N � 50, the mean of the 50-year extremes is �sg,50 � 1.48 kN/m 2. The statistical parameters of the other variables are used in accordance with JCSS Probabilistic Model Code [2]. The statistical parameters of the annual extremes and 50-year extremes of the snow load (denoted as s1 and s50 in Table 1) result from equation (3) using the statistical models for random variables shown in Table 3. 4.3 Wind action The statistical parameters of wind pressure w are deter- mined assuming that: w c c c m q� p g r q b 2 (15) 30 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 42 No. 4/2002 ParametersVar. types Symbol X Name of basic variable Dist. Dim. Xk �X �X/Xk X VX MP fyp Yield strength (S235) LN MPa 235 280 1.19 22.4 0.08 � Imperfection coefficient N - 0.32 0.275 0.028 0.028 0.1 C 1 Loading and end restraint factor N - 1.59 1.90 1.19 0.19 0.1 C 2 Loading and end restraint factor N - 0.78 0.67 0.86 0.067 0.1C oe ff ic ie n ts Table 2: Statistical properties of the basic variables used to derive coefficients �z and �LT s � �1 Ce C t sg ParametersVar. types Symbol X Name of basic variable Dist. Dim. Xk �X �X/Xk X VX �1C e Shape and exposure coef. N - 0.8 0.8 1 0.12 0.15 C t Heat coefficient D - 1 1 1 - - s g,1 Annual extremes of snow load on ground G kN/m2 1.33 0.47 0.35 0.33 0.7 s g,50 50-year extremes of snow load on ground G kN/m2 1.33 1.48 1.11 0.33 0.22 C oe f. A ct io n s Table 3: Variables used in calculating the parameters of the snow load where mq is the model coefficient describing the ratio between the expected and computed value of the basic wind pressure qb, which can be written as: q vb b 2� 1 2 � (16) The characteristic value qb � 0.42 kN/m 2 is defined to have the probability p � 0.02 to be exceeded by the annual ex- tremes. The coefficient of variation of the annual extremes of the reference wind speed vb is Vvb,1 � 0.2 [8]. Supposing that the annual extremes of the reference wind speed can be mod- elled by a Gumbel distribution, the coefficient of variation of the annual extremes of the basic wind pressure Vqb,1 � 0.43 results from: V V V V qb vb vb vb , , , , . 1 1 1 1 2 2 1 114 1 � � � (17) The mean of the annual extremes of the basic wind pressure �qb,1 � 0.20 kN/m 2, the mean of the 50-year extremes �qb,50 � 0.46 kN/m 2 and the standard deviation qb � 0.085 kN/m 2 can be derived identically as for the statisti- cal parameters of the extremes of the snow load on the ground (13,14). The statistical parameters of the other vari- ables used in calculating the statistical parameters of the wind pressure w are taken in accordance with the JCSS Probabilistic Model Code [2] as listed in Table 4. 5 Reliability analysis Climatic actions due to snow and wind are complex time- -variant quantities that significantly complicate the reliability analysis. Two different approximations for describing them are considered in the following analysis. Firstly, Turkstra’s rule is accepted in conjunction with time invariant analysis. Sec- ondly, the Ferry Borges-Castanheta model (FBC) is applied together with time variant reliability analysis. The variable actions due to snow and wind are assumed to be uncorrelated. The software product COMREL [4] has been applied in both types of analysis. 5.1 Time invariant analysis In accordance with Turkstra’s rule, the leading action is described by its lifetime (assumed as 50 years) extreme while the accompanying action is considered by its point-in-time value (approximated by annual extremes). In the following analysis, each climatic action, the snow and the wind action, is considered to be either a leading or an accompanying action. The probability densities of the 50-year extremes of the snow load s50 (considered as the leading variable action) and the annual extremes of the wind pressure w1 are shown in Fig. 2. The characteristic value of the wind pressure w being the 98-percentage fractile of Gumbel’s distribution is denoted as wk, and wd denotes the design value. 5.2 Time variant analysis The time variant reliability analysis is based on the FBC model for the snow and wind actions. Both the climatic © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 31 Acta Polytechnica Vol. 42 No. 4/2002 w � cp cg cr2 mq qb Symbol X Xk �X �X Xk X VX c p c g c r 2 mq q b,1 2 q b,50 2 ParametersVar. types Name of basic variable Dist. Dim. / Pressure coefficient N - nom nom 1 0.1nom 0.1 Gust factor N - 2.4 2.4 1 0.24 0.1 Roughness factor N - 0.91 0.73 0.8 0.073 0.1 Model coefficient N - 1 0.8 0.8 0.16 0.2 Annual extremes of basic wind pressure G kN/m 0.42 0.20 0.44 0.085 0.43 50-year extremes of basic wind pressure G kN/m 0.42 0.46 1.10 0.085 0.18 C oe f. A ct io n s Table 4: Variables used in calculating the parameters of the wind action � w1 s50 0 1 2 3 4 0 0.5 1 1.5 wk = 0.92 wd = 1.38 w s, [kN/m ] 2 Fig. 2: The probability densities of the 50-year extremes of the snow load s50 and the annual extremes of wind pressure w1 actions are described by jump processes without intermit- tencies (actions sometimes take a zero value), which approxi- mate their real variation in time by rectangular wave renewal functions. Each jump process with intermittencies is characterized by the jump rate � (the average number of magnitude changes of the square waves in a reference time Tref) and by the interarrival-duration intensity � (the product of the arrival rate � and the mean duration with respect to a reference time Tref). As for the snow load, it is assumed that it takes its extreme five times a year. Considering the reference time Tref � 1 year, the arrival rate of on-times is therefore �s � 5. The mean dura- tion (the time during which the structure is loaded by the ex- treme snow load) is supposed to be about 14 days. The interarrival-duration intensity is thus �s � 5 × 14/365 � 0.19. The possible approximation of the snow load during the reference time Tref � 1 year is shown in Fig. 3. Windstorms are expected to appear ten times a year (�w � 10) and the mean duration of the storm is estimated as 8 hours. The interarrival-duration intensity is then �w � 10 × 8/(24 × 365) � 0.009. 5.3 Results of reliability analysis According to the results of the time invariant analysis, the reliability of a structure of the IPE 300 profile seems to be rather low (� is less than 3), while the cross-profile IPE 330 seems to be acceptable. For the higher profile the resulting reliability index � � 3.95 corresponds well to the recommend- ed value � � 3.8 [1], as shown in Table 5. Nevertheless, it should be mentioned that the time invariant analysis based on Turkstra’s rule provides considerably lower values for reliability index � than those obtained by the time variant analysis. The time variant analysis predicts the interval at which the reliability index � can be expected (a higher value of � corre- sponds 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 time variant analysis are more favourable and indicate that even the smaller profile IPE 300 might be acceptable. The expected values of the reli- ability index � for IPE 300 are within the range from 3.57 (the upper bound) to 4.84 (the lower bound)� for IPE 330 from 4.49 up to 5.56 as listed in Table 5. Fig. 4 shows the reliability index � determined by both the analyses as a function of the plastic sectional modulus Wpl,y. 32 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 42 No. 4/2002 t day~14 ~14 ~14 ~14 ~14 365 days = 1 year ~ 73 days ~ 73 days ~ 73 days ~ 73 days ~ 73 days Snow load s Fig. 3: The approximation of the snow load used in the time vari- ant analysis Analysis Used load models Reliability index � IPE 300 IPE 330 Time invariant s50 + w1 2.87 3.95 s1 + w50 2.97 3.97 Time variant Jump processes with intermittencies 3.57–4.84 4.49–5.56 Table 5: Results of reliability analysis � 600 650 700 750 800 850 900 2.6 3.0 3.4 3.8 4.2 4.6 5.0 Wpl,y [10 3 mm3] IP E 3 3 0 IP E 3 0 0 LOWER BOUND (pf) TIME VARIANT ANALYSIS UPPER BOUND (pf) TIME INVARIANT ANALYSIS TURKSTRA‘S RULE TARGET VALUE OF � � 600 650 700 750 800 850 900 2.6 3.0 3.4 3.8 4.2 4.6 5.0 Wpl,y [10 3 mm3] IP E 3 3 0 IP E 3 0 0 LOWER BOUND (pf) TIME VARIANT ANALYSIS UPPER BOUND (pf) TIME INVARIANT ANALYSIS TURKSTRA‘S RULE TARGET VALUE OF � Fig. 4: Reliability index � as a function of the plastic sectional modulus Wpl,y 1 2 3 4 5 6 7 8 9 10 4.4 4.8 5.2 5.6 6.0 � �s LOWER BOUND (pf) UPPER BOUND (pf) 1 2 3 4 5 6 7 8 9 10 4.4 4.8 5.2 5.6 6.0 � �s LOWER BOUND (pf) UPPER BOUND (pf) Fig. 5: Reliability index � as a function of the jump rate of the snow load �s 5.4 Effect of input data on resulting reliability index � The model parameters � and � required for the time variant analysis are very difficult to specify. Nevertheless, parametric studies indicate that uncertainty in � and � have an insignificant effect on the resulting reliability. For example, if the value of the jump rate of the snow load �s increases from 1 to 5 (i.e. if the snow load takes its extreme lasting 14 days five times a year, which is not real), the upper bound of � decreases approximately by 0.4 for the cross-profile IPE330, as shown in Fig. 5. Parametric studies of the jump rate of the wind action �w and of the interarrival-duration intensities of the two cli- matic actions �s and �w provide similar results. 5.5 Probabilistic optimisation Probabilistic optimisation is based on minimisation of a simplified objective function expressed as the sum of the initial, marginal and expected malfunction cost. The decisive parameter is the sectional area A. The total cost Ctot can be expressed as: � �C C C A C p Atot m f f� � �0 (18) where C0 denotes the initial cost independent of parameter A and failure probability pf. The marginal cost is the product of the unit cost of the sectional area Cm and the sectional area A. The expected malfunction cost is the product of failure prob- ability pf and malfunction cost Cf when failure occurs. For probabilistic optimisation, equation (18) may be adapted as: � � C C C C A C C p Atot m tot f m f � � � � �0 (19) The relative increment of the total cost Ctot is dependent only on the decisive parameter A and on the ratio Cf/ Cm. Choosing various values of this ratio, different cross-sections seem to be adequate according to the results of the probabilis- tic optimisation shown in Fig. 6. The arrows point to the minima of the relative increment �Ctot for assumed ratios Cf/ Cm. The dot-and-dash curve shows the resulting reliability index � dependent on sectional area A assuming Turkstra’s rule (the 50-year extremes of the snow load and the annual extremes of the wind action). The hori- zontal dashed line marks the target value of � (�t � 3.8). Obviously with the increasing cost ratio Cf/ Cm the opti- mum cross-sectional area A increases. Profile IPE 330 seems to be optimal for Cf/ Cm = 5 × 10 6. To get credible results of the optimisation, it is necessary to determine the values of Cm and Cf exactly. 6 Conclusions The structural analysis of the frame shows that lateral-tor- sional buckling represents the most critical design criterion and indicates that the snow load is the leading variable action. Considering a 50-year lifetime, the reliability index � for IPE 330 varies within the range from 3.95 up to 5.56. According to the results of the reliability analysis, cross-profile IPE 330 designed using Eurocodes seems to be adequate. The time invariant analysis based on Turkstra’s rule pro- vides considerably lower values of � than those obtained by the time variant analysis. It seems that Turkstra’s rule leads to a rather conservative reliability level for a combination of vari- able actions having significant intermittencies. The great dif- ferences between the lower and upper bounds are most likely caused by the considerable intermittencies of the two variable actions. The model parameters required for time variant analysis are, however, very difficult to estimate. Nevertheless parametric studies indicate that this uncertainty has an insig- nificant effect on the resulting reliability. Structural analysis of a beam with haunches exposed to lateral-torsional buckling is a very complicated task. It is fore- seen that more precise results may be obtained by an analysis based on the model using the Finite Element Method. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 33 Acta Polytechnica Vol. 42 No. 4/2002 � IP E 3 0 0 IP E 3 3 0 � IP E 3 0 0 IP E 3 3 0 � IP E 3 0 0 IP E 3 3 0 � IP E 3 0 0 IP E 3 3 0 IP E 3 3 0 � IP E 3 0 0 IP E 3 3 0 � IP E 3 0 0 IP E 3 3 0 IP E 3 3 0 � IP E 3 0 0 A[mm ] 2 IP E 3 3 0 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 � IP E 3 0 0 5000 5250 5500 5750 6000 6250 6500 5000 5500 6000 6500 7000 7500 totTotal Cost C´ IP E 3 3 0 IP E 3 3 0 C 6 m f 10.5� C C 5 m f 10.5� C 6 10 C m f � C 5 10 C m f � C Fig. 6: Relative increment of total cost C tot as a function of sectional area A using Turkstra’s rule (s50 + w1) Acknowledgement This research has been conducted at the Klokner Institute, Czech Technical University in Prague, Czech Republic, as a part of research project CTU0214417 “Basis for Probabilistic Design of Structural Elements in Accordance with Eurocodes”. References [1] EN 1990 Basis of Structural Design. Brussels: European Committee for Standardisation, 2002. [2] JCSS Probabilistic Model Code. Joint Committee for Struc- tural Safety, 2001 (http://www.jcss.ethz.ch\). [3] prEN 1993-1-1 Design of Steel Structures. Brussels: Euro- pean Committee for Standardisation, 2001. [4] RCP Reliability Consulting Programs: Strurel: A Struc- tural Reliability Analysis Program System. COMREL & SYS- REL User's Manual. Munich: RCP Consult, 1995. [5] prEN 1991-1-3 Actions on Structures – Snow Loads. Final draft. Brussels: European Committee for Standardisa- tion, 2001. [6] prEN 1991-1-4 Actions on Structures – Wind Actions. Final draft. Brussels: European Committee for Standardisa- tion, 2001. [7] Holický M.: Reliability Based Calibration of Eurocodes Con- sidering Steel Component. JCSS Workshop on Reliability Based Code Calibration, Zürich: Joint Committee for Structural Safety, 2001. [8] Tichý, M.: Zatížení stavebních konstrukcí. (Actions on Structures). Technický průvodce 45, Praha: SNTL, 1987. Ing. Miroslav Sýkora phone: +420 224 353 850 fax: +420 224 355 232 e-mail: sykora@klok.cvut.cz Czech Technical University in Prague Klokner Institute Šolínova 7 166 08 Praha 6, Czech Republic 34 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 42 No. 4/2002