Weld thickness Flange width Web thickness Modulus of elasticity of steel Compressive yield stress of a flange (web) Tensile yield stress of a flange (web) Ultimate tensile strength Web depth Effective depth of a web for compressed cross-section Ultimate (failure) load of a test beam Applied force that corresponds to My Ultimate (failure) bending moment
201
202 D. Beg, L. Hladnik
t Elastic resistance moment Flange thickness Elastic section modulus of the full cross-section
O~f
O~ w
%
Flange stiffness Web stiffness (235/fy) °'5 P d P y = M d M y ~ non-dimensional ultimate carrying capacity of the cross-section
1 INTRODUCTION
Because of the favorable ratio between high load capacity and price, micro- alloyed high strength steel is being used increasingly in the construction of steel structures. Some research reports I predict even a possibility of the appli- cation of plastic analysis of such structures. In any case, an interesting question is how to determine more precisely the slenderness limit between slender cross-sections (Class 4) and semicompact cross-sections (Class 3). Because elements made of high strength steel are exploited in the presence of very high strains and stresses, demands for compactness of cross-sections are very strict and more exact knowledge of those criteria can contribute to more econ- omical design.
Previous research in connection with the local stability and the compactness limit for the third class cross-sections of high strength steel 2--8 applies in gen- eral to stub column tests of columns with box, cruciform and I cross-sections.
Results of Rasmussen & Hancock, 8 for example, show that the slenderness which demarcates a slender and semicompact flange of compressed cross- sections is the same for mild structural steel and high strength steel and amounts to (dOle = 15 [e = (235/fy)°'5], in accordance with provisions of EVN 1993-1-1 (EUROCODE 3---EC39).
For I beams in bending it is expected that the limiting slenderness will be slightly higher, especially when more compact webs are used. In order to take into consideration this effect in the analysis of the cross-section local stability, it is necessary to consider the interaction between the stability of the web and the flange. Technical regulations or standards, with the exception of Japan, ~° presently do not take this interaction into account.
In the paper, an experimental and numerical analysis of the local stability of welded I beams made of micro-alloyed high strength steel with a yield stress around 800 MPa was presented.
Ten beams loaded in bending with different slenderness of flanges were tested up to failure load. The influence of web slenderness was also analysed
S,Ienderness limit of Class 3 I cross-sections made of high strength steel 203
using additional numerical analysis. Based on the obtained results, the limit between sle;nder and semicompact cross-sections was constructed at which the interaction between flange and web was taken into consideration in a way similar to that suggested by Kato. n
2 EXPERIMENTAL ANALYSIS
2.1 Description of test beams and testing equipment
Ten welded steel beams with I c ros s - se~ t ion 12,13 w e r e tested. They were made of fine grahaed micro-alloyedhigh strength steel NIONICRAL 70 with the nominal yield stress fy = 700 MPa, a product of the Steel-Mill Jesenice-Acroni in Slovenia, which corresponds to StE 690. The MAG welding process using the flux-cored electrode FILTUB 32B (1.4 mm diam., fy = 690 MPa) and pre- heating to 150-175°C was applied. The chosen nominal dimensions and static scheme of tested beams are illustrated in Fig. 1. The test beams were loaded with concentrate forces at two points 1.2 m apart so that in the span between these two points the bending moment was constant. Arrangement of lateral and torsional supports assured safety from lateral buckling, except for the beam with the narrowest flange where numerical analysis predicted approxi- mately simultaneous occurrence of the local buckling of compressed flange and lateral buckling of the middle part of the beam. The test beams have been divided into five groups from A to E with two beams of equal nominal dimensions. Individual groups differ only in the width or slenderness of flange. They were chosen in the area of slender and semicompact flanges. In all cases the webs meet the requirements for the first class of compactness.
The actual cross-section dimensions are illustrated in Table 1 and present the average,, value of three measurements made along the middle part of the beams.
Introduction of forces has been accomplished with a pair of hydraulic actu- ators. The :input force was measured using load cells. With the help of dis- placement l~ransducers and strain gauges the characteristic vertical and lateral displacement of a cross-section, as well as the deformation of a cross-section in the middle of a span and at the point of force application were measured (Fig. 2).
2.2 Mechanical properties of material, geometrical imperfections and residual stresses
The basic mechanical properties were defined using the standard tensile test (proportional test specimens) and compression test (small cylindrical
204 D. Beg, L. Hladnik
X- lateral supports~ P ~P
100 1800 1200 1800 ! I I I
test I beam I b (mm)
A 1300 B ] 270 c 1250 D 1220
, ~ E 1200
~a=5
o
100
Fig. 1. Static scheme and nominal dimensions (mm) of test beams.
Cross-section in the mid-span
Cross-section at the aplication point o f the force
compression s d s9 s]o LC 1(2)
o,I, flange DT 3 s6 [ sl2
I tensile ss [ std ~ DT 6 flange ~ ~ " ~1s(2) ~ s74o)
10 10 10 | 10
H H H DTI(2) H
DT - displacement transducer LC - load cell Si - strain gauge i
Fig. 2. Measuring points.
specimens), and are given in Table 1. Yield stresses were defined at the perma- nent deformation of 0.2%.
The greatest geometrical imperfections of the cross-section (Fig. 3) in the middle of the span are collected in Table 2 and in general do not surpass ordinary allowed tolerance. 14
The arrangement of compressive residual stresses was also measured in flanges of test beams from groups B and D (Fig. 4). The average value of compressive residual stresses for cross-section B amounted to 73 MPa (0.09fy), and for cross-section D it was 123 MPa (0.14fy), which is in accord-
Slenderness limit of Class 3 I cross-sections made of high strength steel
TABLE 1 Actual Dimensions (mm) and Yield Stresses (MPa) of Test Beams
ance with l~he resul ts o f our p rev ious research, 15 where the ave rage compress -
ive residual s tresses in f langes amoun ted to app rox ima te ly 100 M P a (0 . 12 -
0 .18fy) .
2.3 Test results
All 10 b e a m s were loaded to the fai lure load. Ul t imate toad factors are s u m m a - r ized in T,tble 3. F igure 5 presents the non-d imens iona l load-de f l ec t ion diag- rams. Py is; the force that causes at m id - span the e m e r g e n c e o f yie ld m o m e n t (M = My = uz ¢rl~gea ,, el7 y¢ j, and Wy is the per ta in ing deflection. The d iagrams show
m m m m 220 M P a t 1 i0 i~lo ' / /270 14a °'°' t 3o
-2011.0 1
D - ' - ou t s ide
i n s i d e
average
B
50.13 M P a o .c ~ -
6O
-100.0
.200.0
°300.0
o0 I MPa o.o
-100.0, * / ' " ~ " ~
Fig. 4. Measured residual stresses.
m m I10 190 270
that with increasing slenderness of flanges due to local buckling of compressed flanges (Fig. 6), the non-dimensional carrying capacity and ductility of beams decrease.
Figure7 illustrates non-dimensional ultimate carrying capacity ~xp. = P J P y = M J M y of all 10 test beams depending on the slenderness of flanges
(blt)/e. To calculate the coefficient e, the yield stress of flanges f~y~g= from Table 1 has been used. For reasons of comparison, ultimate carrying capacities are given which were obtained by previous numerical analysis 12.13 (Section 3).
Slenderness limit of Class 3 I cross-sections made of high strength steel 207
Fig. 5. Measured deflection beneath the load point.
The experimental results correlate very well with those obtained through numerical analysis.
From the diagram on Fig. 7 a limit slenderness of flanges (bl t ) l~40, which divides slender and semicompact welded I cross-sections from high strength steel, loaded in bending, can be obtained. The determined limit is much more
208 D. Beg, L. Hladnik
Fig. 6. Test beam D1 after the failure - - the local buckling of compressed flange.
favorable than the limit for flanges (b/ t ) /~30, in accordance with EC 3. The average slenderness of webs in test samples amounted to (hw/d)/e~40.
The collapse of tested beams resulted from the local buckling of compressed flange (Fig. 6) with the exception of test beams E1 and E2, where apart from local buckling, also the lateral buckling of beams 12A3 influenced the ultimate carrying capacity. This was confirmed by the measured local deflections of compressed flange and lateral movements of compessed and tension flange, as well as measured deformations in the middle of the span. Figure 8 illustrates the measured local deflections of the outer edge of compressed flange and lateral displacements of compressed and tension flange of test beam A2.
3 NUMERICAL ANALYSIS
With the experimental analysis on 10 beams, the effect of flange slenderness on the carrying capacity of the beams was defined. In order to determine the effect of other parameters, in the first place the slenderness of the web, nonlin- ear numerical analysis was used (computer program U L N A S 16 - - shells finite elements).
The calculation considers the nominal dimensions and the static scheme
Slenderness limit of Class 3 I cross-sections made of high strength steel 209
Fig. 7. Non-.dimensional ultimate carrying capacity of test beams y. in relation to the slender- ness of flange (b/t)/e.
from Fig. 1 and initial geometrical imperfection, material and residual stresses from Fig. 9. Residual stresses were taken from the results of our previous research work. 15 First the analysis of the influence of flange slenderness on limit carrying capacity was done. The results are illustrated in Fig. 7. An agreement with the experimental results is good. Then the effect of web slen- derness on limit carrying capacity was analysed. Support beams from groups B, D and E were calculated at five different web depths (hw = 220, 350, 500, 700, 900 rnm). Other dimensions remained unchanged. Webs with depths of 700 and 990 mm are classified according to EC 3 as slender (hwld > 124 e). The obtained ultimate load factors are illustrated in Fig. 10 depending on web slenderness.
The results show that when web slenderness is increased, the limit carrying capacity is decreased. Cross-section D with the flange (b/t)/e = 33.6 can be classified among semicompact cross-sections up to web slenderness of (hw/d)le = 82, but the cross-section with more slender web must be classified as slender. Stability interaction between the flange and the web is therefore present after all and influences the ultimate carrying capacity of a cross- section.
2 1 0 D. Beg, L. Hladnik
DT6 4OO.OO(; 350.00
300.00
250.00 P (IN)
200.00
150.00 I
100.00
50.00
[ n t u a ~ J . v v
-0.005 0.000
DT5 :: w=DT3 -DT4
0.005 0 .010 0 .015 0.020 0 .025 0.030
UlrrSf6~ w/L (L=I20 cm)
Fig. 8. The local deflection of compressed flange and lateral displacement of compressed and tension flange of test beam A2.
4 THE SLENDERNESS LIMIT BETWEEN SLENDER AND SEMICOMPACT I SECTIONS IN BENDING
The relation between ultimate load factor and flange or web slenderness may be defined as suggested by Kato H for ultimate carrying capacity obtained by stub column tests:
The coefficients A, B, and C were determined using linear regression analy- sis from the results of the numerical analysis presented in Table 4. We arrived at
Fig. 9. Geometrical imperfections and residual stressses considered in numerical analysis.
eqn (2) from the results of nonlinear numerical analysis. The correlation factor amounts to 0.993 and confirms the suitability of eqn (2) for the range 0 .8<%<1.1.
Equation (2) was verified by experimental results (Fig. 12). Our test results are marked with black circles. Omitted, however, have been the results for the beams of group E where apart from the local buckling of compressed flange the lateral buckling of the beams influenced the ultimate carrying capacity. The correlation coefficient amounts to 0.963. McDermott's exper- imental results 5 are marked with black triangles and refer to the determination of bending ductility of cross-sections. Among others he also tested five rolled I sections loaded in bending and made of high strength steel with an average yield stress of 857 MPa. The collapse of these beams occurred because of the local buckling of compressed flange.
212 D, Beg, L. Hladnik
1111iljm"
1.1 ~ _
l -
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1 0 ~ , J i I i i !
20 40 60 80 100 120 140 160 180
E (b/~200/12, d=10)
D (b/t=220/12, d=10)
A B (b/t=270/12, d=10)
. . . . slenderness limit for the web from Class 3 according to EC 3
0aw/d)lc Fig. 10. The influence of web slenderness (h./d)/E on ultimate carrying capacity of a cross-
section.
TABLE 4 The Results of Numerical Analysis (fy = 790 MPa)
Slenderness limit of Class 3 I cross-sections made of high strength steel 213
1.2
1.1
l Tu~naly t.
0.9
O.S
/
0.7
0.7
/
/
/ /
/
/
/
a"
¢¢ • numerical
analysis (t/d=l.2)
i : ; I i
0.8 0.9 1 1.1 1.2
~u hum-
Fig. 11. Correlation between analytical (eqn 2) and numerical (Table 4) ultimate load factors.
]L .3
)L.2 t 1.1
1.0
0.9
O.S ~uanalyt" 0.7
0.6
0.5
0.4
(3,.3
(L2
/
/ /
/
/ /
/
~, ," • Beg, Hladnik A, (t/d 1 2)
/
/
/
°'le ~ • McDermot t ~ Q ( t /d=l .2-1.5)
/ ~ o Rasmussen, ," Hancock
/
" (t/d--1.O) /
, O ," * Davids ¢ $
," Hancock /
, o (t /d=l O) / •
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
~u exP"
Fig. 12. Correlation between analytical (eqn 2) and experimental (Table 3) ultimate load factors.
214 D. Beg, L. Hladnik
Equation (2) can be modified in such a way that will also be valid for I sections loaded only in compression. 13 When calculating the stiffness of the web aw, instead of using an actual depth of a web hw, the effective depth of a web hwe is used. The effective depth hw, for compressed cross-sections is defined under the condition that the elastic critical stress of the local buckling is equal in both cases (Ot'~r~SSu~ = tr~n'Ung). Based on these suppositions hw, is defined with the expression (3). K,~ is the coefficient of the local buckling. 9
/ --~-~nding ~34. 9 hwo = hw = - - " hw = (3 )
Modified equations were verified using the experiment results of Ras- mussen s and Davids. 7 Both authors performed stub column tests of welded I cross-sections at various values of bit and h,dd. With Rasmussen's tests the yield stress was at 740 MPa and with David's test 411 MPa. In Fig. 12, circles and rhombi illustrate the deviation of their experimentally determined ultimate carrying capacities ~xp. from the ultimate carrying capacities y~yt. calculated by using eqn (2) and effective depth hwe.
It should be mentioned that in our research, the ratio of flange thickness and web thickness amounts to 1.2. With McDermott, Rasmussen and Davids, this ratio was within the range 1-1.5.
If 7u=l is applied in eqn 2, the expression representing the limit between slender and semicompact welded I cross-sections made of high strength steel is obtained, taking into account the interaction between the flange and web. The limits for cross-sections loaded in bending or compression obtained in such manner are illustrated in Fig. 13 in comparison with limits according to EC 3. The white circle illustrates our experimentally obtained limit between slender and semicompact I cross-sections loaded in bending (Fig. 7). A slightly more conservative position of our numerically obtained limit regarding the experiment is most probably the consequence of the strain hardening of the existing material and more favorable distribution of the local geometrical imperfections along the flange of test beams in comparison with the distri- bution applied in the numerical analysis. 13
For purposes of design a simplified limit between slender and semi-compact welded I sections in bending made of high strength steel illustrated in Fig. 13 can be proposed. As long as the web is very compact (hJd)/E<40, the limit slenderness between slender and semicompact flange amounts to (blt)l~=- 40>30 which is more than according to EC 3. At the web which is according to EC 3 on the limit between slender and semi-compact, the corresponding flange slenderness amounts to 30, which is also in accordance with EC 3 where no flange-web interaction is considered. To extend the suggested simplified
Slenderness limit of Class 3 I cross-sections made of high strength steel 215
40
35
30
25 (bit)Is
20
15
I0
5
0
our tost result ( t /d=l .2 ) - - ~ - - o - - . ~ simplified slenderness limit
EC 3 \x EC 3 " " ' , , / pure 'x pure x x compression, '~ bending x
cross-section in t are compression Eq. (2) ; P.=Py, hwe=2.444hw
t I I
| i I ] *, I ~ I I ~ ~,
20 40 60 80 1 O0 120 140 160 180 200
0add) /~
Fig. 13. The limit between slender (Class 4) and semicompact (Class 3) welded cross-sections made of high strength steel taking into account the interaction between flange and web.
limit to slender webs with (hJd)l~> 124, additional experimental and numeri- cal analysis would be needed.
5 CONCLUSION
Results of experimental and numerical analysis of local stability of welded beams with I cross-sections made of high strength steel loaded in bending illustrate that the ultimate carrying capacity of cross-sections is greatly influ- enced by the flange and web interaction. According to the obtained results and method proposed by Kato, H the analytical expression for demarcation between shmder and semicompact I cross-sections taking into consideration the interaction between flange and web was derived.
The achieved expression enables that, for more compact webs, more slender flanges are used in the third class of compactness than anticipated in EVN 1993-1-1. 9 The obtained expression has been verified by experimental results available from other authors and the compatibility was high.
In order to establish the obtained slenderness limit for Class 3 cross-sections as a design rule further research work is needed. In the first place the influence of the flange to web thickness ratio has to be analysed.
216 D. Beg, L. Hladnik
We analysed I cross-sections made of high strength steel, but to a high degree, these results also relate to cross-sections made of mild structural steel. The essential difference with regard to local stability is higher intensity of compressive residual stresses in flanges and thus relatively lower carrying capacity when mild structural steel is used. The numerical analysis we carried out in relation to this proves that the difference in carrying capacity of a cross- section in non-dimensional form is small and does not exceed 3%.
ACKNOWLEDGEMENTS
The work presented in the article originated within the project 'Development and introduction of micro-alloyed high strength steel' which was financed by Steel-Mill Jesenice-Acroni and the Ministry for Science and Technology of the Republic of Slovenia.
REFERENCES
1. Spangemacher, R. & Sedlacek, G., Zum Nachweis ausreichender Rotationsf~tig- keit von Fliel3gelenken bei der Anwendung des Fliel3gelenkverfahrens. Stahlbau 61 (1992) H.11, s.329-339.
2. Nishino, F., Ueda, Y. & Tall, L., Experimental Investigation of the Buckling of Plates with Residual Stresses. ASTM Special Technical Publication NO419, American Society for Testing, PA, 1967, pp. 12-30.
3. Nishino, F., Tall, L. & Okumura, T., Residual stresses and torsional buckling strength of H and cruciform columns. Jap. Soc. Civil Engrs Trans., 160 (1968) 75-87.
4~ McDermott, J. F., Local plastic buckling of A514 steel members. J. Struct. Div. Proc. ASCE, 95 ST9 (1969) 1837-1849.
5. McDermott, J. F., Plastic bending of A514 steel beams. J. Struct. Div. Proc. ASCE, 95 No. ST9 (1969) 1851-1870.
6. Usami, T. & Fukumoto, Y., Local and overall buckling of welded box columns. J. Struct. Div. Proc. ASCE, 108 No. ST3 (1982) 525-542.
7. Davids, A. J. & Hancock G. J., Compression tests of short welded 1-sections. J. Struct. Engng, 112 (1986) 960-975.
8. Rasmusen, K. J. R. & Hancock, G. J., Plate slenderness limits for high strength steel sections. J. Construct. Steel Res. 23 (1992) 73-96.
9. Eurocode No. 3 (ENV 1993-1-1:1992), Design of Steel Structures, Part I - - Gen- eral Rules and Rules for Buildings, 1992.
10. Beedle, L. S. (ed.), Stability of Metal Structures, A World View (2nd edn). Struc- tural Stability Research Council, Lehigh University, 1991.
11. Kato, B., Deformation capacity of steel structures. J. Construct. Steel Res. 17, (1990) 33-94.
12. Beg, D. & Hladnik, L., Development and Introduction of Fine-grained Micro- alloyed High Strength Steel: The Influence of Flange Slenderness on Bending
Slenderness limit of Class 3 I cross-sections made of high strength steel 217
Carrying Capacity of Welded 1-cross-sections Made of Micro-alloyed High Strength'. Steel. Report on Research Project No. B-649, Faculty of Architecture, Civil Engineering and Geodesy, Ljubljana, August 1994 (original in Slovene).
13. Hladnik, L., Stability of Steel Structures Made of Micro-alloyed High Strength Steel. Iv[.Sc. Thesis, Faculty of Architecture, Civil Engineering and Geodesy, Department of Civil Engineering and Geodesy, Ljubljana, October 1994 (original in Slovene).
14. SIA 161, Stahlbauten, Schweizericher lngenieur und Architekten, Zurich, 1979. 15. Beg, D. & Hladnik, L., Eigenspannungen bei geschweil3ten I-profilen aus hoch-
festen St~thlen. Stahlbau 5 (1994), H.5, s.134-138. 16. Trueb, U., Stability Problems of Elastic-Plastic Plates and Shells by Finite
Elements. Ph.D. thesis, Faculty of Engineering, University of London, 1987.
