THE CALIBRATION OF THE STEEL BEAMS UNDER SNOW LOADS IN CROATIA Marta Sulyok-Selimbegović University of Zagreb, Faculty of Architecture, Zagreb, Croatia The beam design requirements for limit state lateral torsional buckling are analysed and criteria for two codes (AISC and ECCS) are compared. As the differences in the design requirements are substantial, the comparision of the achieved reliability indeces is made in order to find out the model, which one is closer to the target reliability level. For this purpose the four example is done concerning with the various observation data of snow loads on four locations in Croatia. The characteristic values from the actual snow loads on the roof constructions are derived and the laterally unsupported rolled steel beams are designed with various unbraced lengths due to the ultimate moments capacity requirements for the testing models. The statistical parametars of the experimental results are evaluated and FOSM method is used for the procedure of the calibration. As the results varries with the applied snow loads and the slenderness ratios, for two examined models of the designed rolled beams, which are compared, it is necessary to achieve target values of reliability by correction of the model and resistance factors. THEORETICAL MODELS FOR LATERAL TORSIONAL BUCKLING These divergences are due to the different perceptions of the effects of initial imperfections. For specifications ECCS and AISC many variabilities arise especially in the inelastic range and for the beams under moment gradient. The following general treatments are used for beam design rules: Use of the columns formula with the “equivalent” slenderness parameter : M MP M E Mp is the plastic moment of the cross section ME is the elastic lateral torsional buckling moment M f M is the buckling coefficient is the coefficient of initial imperfection for rolled (0.21) and welled (0.49) beams, such as speciffied in EC3. Use of the analytically exact lateral-torsional buckling solutions for the cross section, loading and end condition, can be empirically modified to account for bucking in the inelastic range, such as in AISC Specification, with the linear interaction in the inelastic range, which can accommodate idealized conservative simplifications. The most general equation is the one adopted in Western Europe: 1 M M 1 u 2n p M 1 n , with the exponent “n” which takes on values 2.5 for rolled and 2.0 for welded beams. The variation of the buckling strength with M for the various values of “n” and ““ is shown in Fig. 1. Figure 1. Buckling curves by ECCS with imperfection coefficients and sistem factor n=2.5 Generality of the LRFD citeria of the AISC is comprised in the elastic solution which is expressed in the form: ME= CbMr , Mr is the buckling solution for the case of uniform bending Cb is the coefficient for the effect of loading The inelastic buckling solution is approximated by a straight line (Fig. 2). The ultimate moment is determined as follows: M M for L L u p p M C M M M u b p p L L M L L p r r p M M for L L where L 1.76 E u E r p p F y (1) In these equations are: E = modulus of elasticity Fy = yield stress Mr = S(Fy - Fr ), yield moment Mp = maximum moment capacity S = elastic section modulus Fr = maximum compresive residual stress Lr = unbraced length corresponding to Mr Lp = spacing of the braces necessary to reach Mp without rotation capacity for L L p Figure 2. Variation of Mu with slenderness parameters by AISC Specification As ECCS curve for n=2.50 provides a reasonable mean strength over short and medium slenderness range for rolled beams, the curve for n=1.50 forms a lower bound for the test points with the (m-2s) strength curve shown on Fig. 3. On Fig. 4 same is valid for the welded beams with different system factors “n”, which are 2.0 and 1.50 respectively. In order to determine the best fit of the assumed implicit function to the experimental data, such as selected 324 rolled beams and 132 welded beams, the mean values of nondimensional strength coefficients (i), as well as 5% fractiles (m-1.64s) is evaluated by the method of least squares (Eq. 2) in the nonlinear regression analysis. In this problem, as nonlinearity is encountered, the higherorder equations with one independent variable should be tried to fit data with the correlation coefficient near 1. 1 M M min and ,where M M 1 1 2 n N i 1 2n i u p p E i ……. ......................................................................................... (2) The results for the rolled beams from the sellected tests data are n=2.64, which is concerning mean values, and n=1.425 for 5% fractiles. For the welded beams, on which the same evaluation is done, the results for mean value is n=2.44 and 5% fractile n=1.095 . In order to derive the probabilistic evaluation of lateral-torsional buckling strength of ECCS and AISC design formula comparing them with test results, the realised indeces of reliability is performed on the following examples. Figure 3. Experimental results and lateral-torsional buckling curves for rolled beams Figure 4. Experimental results and lateral-torsional buckling curves for welded beams THE EVALUATION OF THE RELIABILITY INDECES FOR THE ROLLED BEAMS UNDER SNOW LOADS The snow load is taken as dominant load during 30 years of measurements of metheorological data with the characteristical values as 95% fractile with the return period of 30 years. The experimental results are selected for the needed slenderness ratios of the rolled beams designed by the theoretical models of ECCS and AISC Specifications, and the proposed fractile curve with system factor n=1.425 . The statistical evaluation of the snow loads data The snow measurements, which are converted to the snow loads on the flat roofs, are analysed and compared with extreme probability distribution function type I of Gumbel as it is shown on Fig. 5 and Eq.3. F x exp exp a x x 0 . .................. (3) . where mod is: xxc a mean value of the distribution: x standard deviation of the distribution: a 0 6 Function of the extreme probability distribution during the period of n years: 1 F x F x exp exp a x x ln n a . . n n … 0 (4) where the mean value for return period of n years X n is: 6 x x 1 V lnn ............................. (5) n 0 0 The characteristic value for the load during the period of n years x k,n , with the probability p that it will not be exceeded is: . x k ,n x 1 lnn ln ln p .................................. (6) a x x 1 0.4501V 0.7801V lnn ln ln p (7) k ,n 0 0 0 The general expression for the snow loads on the flat roofs with probability density function of Gumbel, is defined as: y lnn ln ln p s 0.8 x N 0n 0 N y i 0 0 (8) N are mean value and standard deviation of the calculated snow loads. N N On Fig.5 the histograms and extreme type I distribution function is shown for the measure data of snow loads in continental parts of Croatia, which are compared with theoretical frequencies on Fig.6 and 7. frequency 25 GUMBEL TYPE I 20 Xm0 = 50 kp/m2 X = 60,91 kp/m2 s = 25,74 kp/m2 X+s = 86,65 kp/m2 15 20 5 kp/m2 50 100 150 200 Xi, Xi, s - MEASURED DATA FOR SNOW LOADS kp/m 2 Figure 5. Histogram and distribution function for snow load 112 94 78 66 56 48 35 30 17 FT 0,01 0,1 1 10 50 80 90 99 99,9 99,99% 96,66 Figure 6. Comparison of measured data for snow in Varaždin on probability paper with Henry-diagram kp/m 2 204 200 180 168 143 133 126 97 92 88 80 70 57 48 FT 0,010,1 1 10 50 6,66 26,66 56,66 80 90 99 73,33 86,66 84,1 99,9 99,99% 96,66 93,33 Figure 7. Comparison of measured data for snow in Ogulin on probability paper with Henry-diagram The statistical parameters for the load and the resistance variable The ultimate lateral-torsional buckling resistances of the rolled beams with sections I 200x100x8x5.5 are designed by the above stated models, under the characteristic values of snow loads. The first example is on the location in Zagreb with the statistical parametar for the snow loads: q 0 0.35kN /m2 ; 0=0.21 kN/m2 ; V0 =0.60 q p ,n q0 (1 4.52V0 ) 1.30kN / m 2 ............. characteristic values q n q 0 (1 2.652V0 ) 0.907kN / m 2 ........mean values for n years The evaluated girder is from the group of the tested beams with the variable strength shown on Fig. 5, designed by the values of the loads qp,n , with ultimate strength Mu . The second example is on the location in Varaždin (Fig.6) with the statistical parameters for the snow loads: q 0 0.51kN / m 2 ; 0 0.28kN / m 2 ;V0 0.55 q np 0.51(1 4.52V0 ) 1.78kN / m 2 q n 0.51(1 2.652V0 ) 1.254kN / m 2 ;Vn 0.223 The third example is for location in Ogulin (Fig.7) with the parameters of the snow loads: q 0 0.929 kN m2 0 0.51 kN m2 V0 0.55 q np 3.20 kN m2 q n 2.26 kN m2 Vn 0.225 The fourth example is for location in Slavonski Brod with the characteristics of snow load: q 0 0.490 kN m 2 0 0.244 kN m 2 V0 0.490 q np 2.960 kN m 2 q n 2.120 kN m 2 Vn 0.113 The evaluation of the reliability indeces for the calibration of the rolled beams The reliability index is derived from the equation of the ultimate limit state with two basic variables, which are statitically independent, g(x) =R-Q, with the probability of the failure (pf): p P( R Q) F ( x) f ( x)dx .. (9) f R Q FR - cumulative distribution function of resistance R fQ - probability density function of load Q The basic variable is not distibutated by normal probability distribution function, so Rackwitz & Fiesslermethod is used with the transformation of the basic variable into the equivalant of normal distribution and the N N parameters x , x, under the circumstances, that the cumulative distribution and probability density functions are the same, as for basic and aproximated variables, in the reper points on the ulimate limit state plane: g ( x1* , x2* ,....., xn* ) 0 . The equivalent mean value and standard deviation of basic variable is: x x F x N 1 * i N xi i * N i xi F x 1 * i f x i * i i ......................................... (10) F,f - distribution and density function of basic variable xi , - cumulative distribution and density function of standard normal variable. The iterative procedure for approaching to minimum value of , is obtained by the equation system: g x g x N xi i ................................................... (11) i m i 1 N2 xi i x x * i i i N xi g x , x ,.....,x 0 * * * 1 2 n .................................................. (12) The partial derivations g x are evaluated for x * i , i and i of the basic variables xi . After the convergation of this algorythm, reliability index * is evaluated, and approximate value of the probability of failure p 1 f * . THE RESULTS OF THE CALIBRATION FOR THE ROLLED BEAMS Tab. 1. Ultimate strength and calibration for the snow load in Zagreb Theoretical results Mteo [kNm] for models: AISC (m-2s) Mexp 0.605 ECCS curve [kNm] 21.63 21.22 19.22 57.45 M max 5.01 4.91 4.45 1.58 Mmax 3.8 3.9 4.20 Table 2. Calibration for the snow load data in Varaždin Theoretical results Mteo [kNm] for models: ECCS AISC (m-2s) Mexp 0.77 curve [kNm] 20.51 19.75 17.17 46.30 M max 4.51 4.40 3.83 2.77 Mmax 3.20 3.50 4.20 Table 3. Calibration for the snow load data in Ogulin Theoretical results Mteo [kNm] for models: ECCS AISC (m-2s) Mexp 0.918 curve [kNm] 18.40 18.10 15.07 47.27 M max 4.14 4.07 3.40 5.69 Mmax 3.21 3.10 3.81 Table 4. Calibration for the snow load data in Slavonski Brod Theoretical results Mteo [kNm] for models: ECCS AISC (m-2s) Mexp 1.17 curve [kNm] 14.51 14.88 11.93 40.53 M max 1.64 1.69 1.35 3.40 Mmax 5.18 4.80 6.20 CONCLUSION The analysis of laterally unsupported steel beams for various design models is obtained, by which the ultimate limit state of the lateral torsional buckling strength is evaluated for the purpose of the calibration of the rolled beams under the snow loads from the measured data in Croatia. The results of the calibration varies with applied snow loads and slenderness ratio for three examined designed models: For ECCS criteria they are in the range from realised reliability indices = 3.20 to 5.19. For AISC Specifications indices are lower, such as =3.10 to 4.80. For the model of proposed system factor “n” is quite on the target safety side with =4.20 and 6.20. As the calibration is performed with the designed model by global and constant safety factor, the differences are the result of the basic formulations of the buckling curves. It is evident that there is no necessity to change the system factor “n” of buckling curves, but to correct the evaluation model by model and resistance factors with the target reliability level in order to achieve uniform reliability with the proposed loads factors, concerning the applied loads in certain cases.