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:
xxc
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.