Stability and Ductility of Steel Structures

Transcription

1 Stability and Ductility of Steel Structures Professor Ottó HALÁSZ Memorial Session 2002, BUDAPEST edited by M. IVANYI AKADÉMIAI KIADÓ, BUDAPEST

2 COLLOQUIUM PATRON Ákos Detreköi, Rector, Budapest University of Technology and Economics SCIENTIFIC COMMITTEE János Szabó, Hungary (Honorary Chairman) Miklós Iványi, Hungary (Chairman) Miroslav Škaloud, Czech republic (Vice-Chairman) Zoltán Agócs, Slovakia Zsolt Gáspár, Hungary Federico M. Mazzolani, Italy Lambis Baniotopoulos, Greece Victor Gioncu, Romania Janusz W. Murzewski, Poland Eduardo M. Batista, Brazil Nikola Hajdin, Yugoslavia Jean-Pierre Muzeau, France Ronaldo C. Battista, Brazil Gregory Ji. Hancock, Australia David A. Nethercot, United Kingdom Reidar Bjorhovde, United States Nestor R. Iwankiw, United States Takeo Nishiwaki, Japan Darko Beg, Slovenia Pavol Juhas, Slovakia Hartmut Pasternak, Germany Dinar R. Z. Camotim, Portugal Aarne Jutila, Finland Jacques Rondal, Belgium Zbigniew Cywinsky, Poland Ulrike H. Kuhlmann, Germany Donald R. Sherman, United States Dan Dubina, Romania Shigeru Kuranishi, Japan Jiri Studnicka, Czech republic John Ermopoulos, Greece Audronis K. Kvedaras, Lithuania Tibor Tarnai, Hungary Józscf Farkas, Hungary Joachim Lindner, Germany Tsutomu Usami, Japan Yuhshi Fukumoto, Japan Le-Wu Lu, United States Riccardo Zandonini, Italy Theodore V. Galambos, United States René Maquoi, Belgium Abdul H. Zureick, United States ORGANIZING COMMITTEE György Farkas, BUTE (Honorary Chairman) Miklós Iványi, Bute (Chairman) Gyula Czeglédi(Hung. Acad. Sci.) László Horváth, (BUTE) István Szatmári (BUTE) Sándor Fernezelyi (BUTE) Károly Jármai (Univ. Miskole) Géza Varga (BUTE) Attila Fülöp (BUTE) Gábor Medved József Varga László Hegedüs (BUTE) Ferenc Papp (BUTE) József Vörös (MÁV Co. Ltd.) M. Iványi (ed.), Budapest, 2002 The articles are published as received. The publishers take no responsibility for their content or style. ISBN Printed in Hungary by Akadémiai Nyomda, Martonvásár

3 Stability and Ductility of Steel Structures (SDSS 2002) M. Iványi, editor 2002, Akadémiai Kiadó, Budapest RESPECTING THE INFLUENCE OF GEOMETRICAL AND MATERIAL IMPERFECTIONS OF STEEL BEAM WHEN CALCULATING THEIR LOAD-CARRYING CAPACITY Zdenìk Kala and Jiøí Kala Faculty of Civil Engineering, Brno, University of Technology, Department of Structure Mechanics, Veveøí 95, Brno, Czech Republic ABSTRACT In the presented paper, a rolled steel beam from the profile IPE 240 with slenderness = 0.6 is being studied. One of the essential quantities which can be of significance for evaluation of a structure in a decisive manner, is the yield strength of the material. As the yield strength is, by its nature, a random quantity, its variability also influences the variability of the beam load-carrying capacity. The mutual correlation among individual elements of cross-section was considered. The correlation was introduced by using the so-called correlation lengths. The bar was modelled by applying the programme system ANSYS and by using the finite element SHELL 43. In addition to yield strength, also the randomness of the other imperfections (geometrical, structural) was considered. The problem is discussed in connection with design reliability conditions of the Eurocode 1, too. KEYWORDS Yield strength, random fields, simulation methods, Latin Hypercube method, imperfections, lateral buckling. INTRODUCTION Real beams are quite far from theoretical simplification to the bar made of homogeneous isotropic material. Within the framework of the modern conception of stability theory, it is necessary to take the presence of structural imperfections into account. Among the major quantities largely influencing the load-bearing capacity is the yield strength. Experimental results of yield strength tests of materials carried out on larger number of beams and sheets are given in ( Rozlívka & Dvoøáèek & Fajkus 1999, Mrázik & Sadovský 1992). Real yield strength characteristic values of Czech structural steels differ from nominal values guaranteed in technical standards of these steels very significantly, see Fig. 1. The real yield strength characteristic value of evaluated set of steels S235 is about 251 MPa, so that the probability of the not

4 104 reaching the guaranteed value 235 MPa is remarkably lower than 5 %. However, keeping the characteristic values itself may not guarantee the design reliability, as this depends on a series of imperfections. For the stochastic analysis, it is necessary, indeed, to have much information on statistic characteristics of these imperfections (Daddi & Mazzolani 1974, Fukumoto & Kajita & Aoki 1976, Koteš Absolute frequency Vièan & Slavík 2001, & Soares 1988). By means of nonlinear methods, the influence of the imperfections MPa Steel S235 mentioned can be taken into consideration in computation models, and by the same, an idea of loadcarrying capacity variability can be obtained MPa 60 In the paper presented, we will demonstrate that in the case of yield strength of rolled IPE profiles, it is not sufficient to know only statistical characteristics of Yield strength [MPa] samples taken from one cross-section point. According Figure 1: Histogram of Yield Stength to experimental results (Daddi & Mazzolani 1974), the yield strength is in flanges lower than in web due to non-uniform cross-section cooling down. The maximum deviation, 26 %, was found in beams HE200. When examining the web, this aspect cannot be omitted, when considering the significance of flanges for the profile resistance. The yield strength statistic characteristics often represent the input data for further research work. The yield strength histograms of sheets, shaped bars, etc. are known in detail. For rolled profiles, material tests are carried out based on samples taken from one third of a flange. In this paper, we will show that the rolled profile yield strength description by only one histogram is insufficient. We should know the same histograms also for the web, for the edges of flanges, etc. The histogram of each part of cross-section would represent one random quantity (of one cross-section point). However, the most important task would be determination of the yield strength corresponding values of individual cross-section parts. THE PROBLEM OFYIELD STRENGTH VARIABILITYALONG CROSS-SECTION The yield strength is being determined by means of material i tests on short bodies obtained by cutting up the cross-sections of a short beam (Daddi & Mazzolani 1974). Let us carry out an intellectual experiment: a short beam will be cut up into j? individual segments determined before, see Fig. 2. By means of material test, we will then determine yield strength for each segment. It can be expected that, in general, yield strength of each cross-section segment will be different. Further, it can be anticipated that the yield strength values of neighbouring segments will show strong correlation. By evaluating on larger number of bodies (cross-sections) of two neighbouring Figure 2: Segments of Cross Sections segments, we will obtain the values the correlation of which will be higher than that of more distant segments. The correlation decrease among the segment i, the neighbouring segments i + 1, i + 2 etc. can be described by the auto-correlation function. The determination of this auto-correlation function based on the results of real material tests would be very valuable but at the same time, also highly demanding both from the economic and time aspects. Therefore the study of these problems by using of simulation methods is advantageous. In this connection, several fundamental questions arise: 1) How the yield strength variability along cross-section influences the load-bearing capacity? 2) How does this influence change with the beam slenderness? 3) How is the real correlation function among individual cross-section segments? Note: The last question can be answered only by means of experiments.

5 Only the decreasing character of the auto-correlation function can be stated with certainty. In case we would obtain a sufficient number of experimental data from material tests, this function could be determined based on results of measurements. The majority of real random phenomena occurring in the nature can be described by a correlation function which decreases approximately exponentially. In our studies, we will, according to our previous experience, consider the auto-correlation function in the form: i, j / L c cor i, j e, (1) where Lcor is the positive parameter called correlation length, is the standard deviation of the random field, and i, j =xj -xi is the distance between two points xi and xj. Besides the correlation function (1), sometimes the so-called Gaussian correlation function (2) is also used, for which the relation /L is considered in square (Novák & Lawanwisut & Bucher 2000). ij cor / L 2 2 i, j cor i, j e, In our study, we applied the function (1). The correlation coefficient be obtained by: ij (2) of correlation matrix can i, j ci, j ci, i c j, j, (3) Individual values of yield strength ones in discretisation points are then computed by the appropriate routines of software package STATREL, and by our own programmes, as well. 1 Correlation function Probability Function of Yield Point i j Yield point of segments i j Figure 3: Yield point probability function of segments i, j Further on, we will try to answer the question no. 1 see above.

6 106 RANDOM FIELDS The beam with cross-section IPE 240 of a S235 steel was solved for several variants. The crosssection was divided into 36 segments, see Fig. 1, i.e., both flanges and web were divided into 12 segments. As for the first variant, we introduced yield strength as an independent random quantity for each of the 36 segments, i.e., Lcr=0. For the other variants, the correlation was considered between the yield strength values of individual segments. The one-dimensional correlation function (1) was assumed. The random field was introduced for yield strength of segments of one half of flange and of the adjacent half of web. So the problem was defined by one-dimensional correlation function (1) with variable parameter Lcr. For the other segments, the symmetrical yield strength was introduced (like for the biaxially symmetrical cross-section). In general, 4 variants were solved: Variant A) Yield strength values of segments are statistically independent (hypothetically Lcor=0). Variant B) Yield strength values are correlated according to (1) with correlation length Lcor=90 mm. Variant C) Yield strength values are considered to be correlated according to (1) with Lcor=180 mm. Variant D) Yield strength is considered to be constant along the cross-section ( L = mm) Yield Strength [MPa] Variant A, Lcor= 0 mm Variant B, Lcor= 90 mm Variant C, Lcor= 180 mm Variant D, L = For the segments of flange, we considered the yield strength to be a random quantity with mean value mfy=285.7 MPa, and with standard deviation Sfy=26.76 MPa ( Rozlívka & Dvoøáèek & Fajkus 1999). According to Daddi & Mazzolani (1974), the average yield strength of the web is higher. In comparison with the flange, it is due to quicker cooling down. Therefore, a higher average value mfy=305,7 MPa was considered. The standard deviation was considered, both for web and flange, to have the value Sfy=26.76 MPa. The realizations of random yield strength of individual segments were determined by the Latin Hypercube Sampling method. 200 simulation runs of the method mentioned were used. After statistic evaluation of all the 200 simulated yield strength realizations of each cross-section segment, the presumed statistic characteristics mfy, Sfy, can be quantified back, yield strength of individual segments being correlated according to the variantsa D. RANDOM IMPERFECTIONS OF THE BEAM cor Web Flange Segments 1-12 Figure 4a: The yield strength course along the segments 1-12 We considered the other random quantities to be statistically independent (constant in each realization on the beam). The initial curvature of the beam with length L=1.9 m was introduced in form of sinusoids both for initial deflection in the primary bending plane in direction of the axis y and cor Segments of Flange Segments of Web Figure 4b: Random field

7 for lateral buckling, in direction of the axis z. In both cases, we considered the amplitudes of initial deflection to be random quantities, uniformly distributed in the interval 0 - L/1000, i.e., 0 mm mm. For the Young s modulus E, we considered the mean value me =210 GPa. The standard deviation was introduced, according to two independent examinations (Fukumoto & Kajita & Aoki 1976, Soares 1988), as the value SE=12.6 GPa. An ideal elastic-plastic material was considered. The residual stress was introduced, its mean value being mrs=60 MPa, and its standard deviation, Srs=20 MPa; their distribution both in the web and flanges was triangular (Daddi & Mazzolani 1974). For geometrical cross-section characteristics, we supposed that nominal values of dimensions, were the mean values mx. The Gaussian cut off distribution of probabilities was considered. The restriction of function distribution was carried out for the limited interval m - 4 SX, m + 4 SX. Standard deviations SX were determined from Tolerance Standard EN 10034:1993, based on the rule, 2 SX. All input random quantities are given in Tab. 1 synoptically. Random variables TABLE 1 INPUT RANDOM QUANTITIES Dimension Probability Distribution Function Mean X Standard deviation Flange yield strength f y Normal MPa 26.8 MPa Web yield strength f y Normal MPa 26.8 MPa Young s modulus E Normal 210 GPa 12.6 GPa Cross-sectional depth h Normal 240 mm 1.5 mm Cross-sectional width b Normal 120 mm 1.5 mm Web thickness t 1 Normal 6.2 mm mm Flange thickness t 2 Normal 9.8 mm 0.75 mm Upper flange displacement k 1 Normal 0 mm 0.09 mm Lower flange displacement k 2 Normal 0 mm.09 mm Initial web deflection f Normal 0 mm 0.75 mm Initial web throwing out of straightness m Normal 0 mm 1.25 mm Residual stress rs Normal 60 MPa 20 MPa Amplitude of initial column Curvature e 0 Rectangular in the interval <0 mm; 1.9 mm> X 107 NON-LINEAR COMPUTATION MODEL The load-bearing capacity of a beam IPE 240 with length L = 1.9 m was analyzed. The beam was meshed in the programme ANSYS. The four-node thin-walled element SHELL 43 was applied. The symmetry was used with regard to the very exacting character of the problem solved. In the bar half in the symmetry plane, we supposed the shift prevention in all cross-section nodes in direction of axis X,

8 108 and rotation around axes Y and Z. On the second edge of the bar half solved, we prevented the shifts of nodes in direction of the axis Y on the flange of profile IPE240. On the lower flange of that edge, we prevented the shifts in the direction of the axis Z. The upper flange was left free. Within the framework of each run of the LHS method, the load-bearing capacity was solved by non-linear computation by means of the programme system ANSYS. A very detailed FEM model was used. In general, the beam was modelled of 540 elements, type SHELL 43, i.e., the thin-walled effect was taken into account. Also the influence of local imperfections which can contribute to the load-bearing capacity loss, was thereby taken into consideration. In geometrically and physically non-linear FEM solution, the Euler method was applied based on proportional loading in combination with the Newton-Raphson method. We determined the loadbearing capacity as the loading constant at which the matrix determinant of tangential stiffness K t of the structure will approach zero with certain accuracy. As we required the determination of loadbearing capacity with accuracy 0.1 %, it was necessary to use, with Euler method, automatic control of the loading run. For the steel S235, we supposed bilinear kinematic material strengthening. Further on, we also supposed that the initial steel plastification occurs when main stress exceeds yield strength. DESIGN RELIABILITY CONDITIONS According to the partial reliability coefficients method, the design values are defined, for all the quantities considered, as uncertain ones (basic quantities). The design is considered to be convenient when limit states are not reached by using the design values in computation models. It can be written in symbols: E d R d, (4) where Ed design loading effect, R design load-bearing capacity. d For the so-called basic reliability level ( = 3.8) the design values Ed, Rd correspond to quantiles with probability 0.90 and/or According to EC1, the design load-bearing capacity can be determined as 0.1 quantile of appropriate distribution type. Given the numerically very exacting computation model, the load-bearing capacity was assessed for 200 runs of the LHS method, for example, Novák & Lawanwisut & Bucher (2000), Kala (2001). The LHS method gives very good results for the first two moments even at a relatively low number of simulation runs. We determined the design load-bearing capacity value, based both on normal distribution of probabilities, and on the lognormal one. SOLUTION The results of load-bearing capacity analysis of a bar bending stressed and solved with the influence of lateral buckling are given in Tab. 2. The results are clearly plotted in Fig. 5. It is evident that with increasing correlation length the design load-bearing capacity determined according to EC1 decreases. The load-bearing capacity was determined both for normal and lognormal distributions. With increasing correlation length, standard deviation of load-bearing capacity increases. Mean value shows no sensitivity to the correlation length change, see Fig. 5. In the problem solved, we considered the auto-correlation function in the form (1). The variants A, D are limit cases of the given physical phenomenon, and thus they are not connected with the

9 application of function (1) or (2). The difference between the load-bearing capacities of variants A, D is approximately 10 %. TABLE 2 STATISTIC CHARACTERISTICS OF RANDOM LOAD-CARRYING CAPACITY In comparison with the course presented in Fig. 5, a rapider decrease of design load-bearing capacity for smaller correlation lengths (and rapider increase of standard deviation) could be expected if the function (2) were applied. M [knm] Load-Carrying Capacity Correlation Variants Arithmetic mean Standard deviation Design loadbearing capacity by EC1 L cor (0.1% quantile Gauss distribution) (0.1% quantile lognormal dist.) =0mm NO A 83.8 knm 6.9 knm 62.5 knm 58.0 knm 57.1 knm 56.1 knm 64.8 knm 61.4 knm 60.9 knm 60.1 knm 90mm EC3 EC1- lognormal distribution Standard deviation YES B C D EC knm 84.7 knm 84.8 knm knm 8.9 knm 9.3 knm 180mm Arithmetic mean EC1-normal distribution Mean & St. Deviation Quantile Estimation Eurocode 3 0 Variant A Variant B Variant C Variant D Figure 5: Characteristics of load-carrying capacity The determination of real correlation length of the function (1) or (2) would be possible only based on experimental data. According to our study for the beam with slenderness = 0.6 we can obtain, by taking into account or by omitting the yield strength variability along the cross-section, a difference up to 10 % among the design load-bearing capacity values. For the bar with slenderness = 1, such a high influence was not proved, for example, Kala (2001). With increasing slenderness, since the stability phenomena start prevailing, i.e., at the loss of both lateral buckling and load-bearing capacity, the stress values in the most stressed points of the beam are lower than those of the yield strength; bending and torsion stiffness is more decisive. Similar results were obtained for steel frames, too, where the yield strength variability influence along stressed beams was studied, for example, Kala (2001). The auto-correlation function (2) was applied. It was proved, identically as in the present paper, that with increasing correlation length the standard deviation of load-bearing capacity increases, i.e., the design load-bearing capacity decreases. In the future, the study of these problems for a large number of frames is planned knm ( a = 0.89) 109

10 110 CONCLUSION The results of our studies show that more attention should be drawn to the variability of material characteristics along cross-section. The determination of yield strength of samples taken from only one cross-section point in insufficient, as thereby, it is not possible to determine how is the variability of yield strength along cross-section. Therefore, we should take several samples both from flange and web of a cross section in each rolled cross-section. According to Daddi & Mazzolani (1974), yield strength on flanges was in average by up to 26 % lower than that on the web. Further research work should prove whether it is sufficient to assess the yield strength statistic characteristics of flange and web of a rolled profile separately. We should concentrate, above all, on scattering of yield strength taken from different cross-section points. The best basis for further theoretical studies were thus represented by complete information on yield strength variability of samples taken from different cross-section points. As the complete information, we could understand the following: point from which the sample was taken, mean value, standard deviation, correlation with the other points. The measurement of real correlation lengths of real cross-sections could be one among particular aims. The present paper was elaborated under the project No and No D022 and within the research project MSM REFERENCES ANSYS manual rev. 5.7, SAS IP, Daddi, I., Mazzolani, F., M. (1974). Détermination expérimentale des imperfections structurales des profilés en acier, Construction métallique n.1, pp Fukumoto, Y., Kajita, N.,Aoki, T. (1976). Evaluation of Column Curves Based on Probabilistic Concept, In: Proc. of Int. Conference on Stability, Prelim. Rep., Publ. by Gakujutsu Bunken Fukyu Kai, Tokyo, pp Kala, Z. (2001). Respecting The Influence of Geometrical and Material Imperfections of Steel Frames When Calculating Their Load-Carrying Capacity, Proc. of the IInd Int. Scientific Conference Quality and Reliability in Building Industry, Levoèa, pp , ISBN Kala, Z., Kala, J. (2001). The influence of yield point along cross section of a rolled beam on its loadcarrying capacity when considering lateral beam buckling, Proc. of the Conference in Jahodná (SR), pp , ISBN (in Czech). Koteš, P., Vièan, J., Slavík, J. (2001). Influence of Reinforcement Corrosion on Reliability of Existing Concrete Structures, Communications 4/2001, pp , ISSN Rozlívka, L., Dvoøáèek, P., Fajkus, M. (1999). Dimension deviations of welded steel beams and their influence on design strength of structural steels, Building Review, 8-2/99, pp (in Czech). Soares, G. C. (1988). Uncertainty Modeling in Plate Buckling, Structural Safety, 5, pp Mrázik, A., Sadovský, Z. (1992). Register of statistic information s about yield strength, material strength and tensility, Part 4 period , ÚSTARCH SAV, Bratislava. (in Slovak) Novák, D., Lawanwisut, W., Bucher, C. (2000). Simulation of random fields based on orthogonal transformation of covariance matrix and Latin Hypercube Sampling. In: Proc. of Int. Conference on Monte Carlo Simulation, Monte Carlo, pp

11 This book presents the general reports and the presentations of the International Colloquium on Stability and Ductility of Steel Structures, held in Budapest, Hungary on September 26-28, On hundred papers have been contributed by Europian, American, Japanese and Australian engineers and researchers concerned with a wide range of applications of stability and ductility problems. The aim of the Colloquium was to bring together specialist in the stability and ductility of steel structures, as well as to provide a forum for the exchange of information regarding the theoretical foundations and experimental investigations and methods, and to promote further development. Highlights of volume: Design theories and the present phase of creating new standards and codes. Theoretical and experimental problems of buckling of columns, lateral buckling of beams, and lateral-torsional buckling of beam-columns. The stability of steel frames and the effects of semi-rigid connections. Numerous new results achieved in the field of plate and box girders, including those of experimental and theoretical studies on fatigue induced web breathing. Experimental tests and theories of cold-formed and thin-walled steel structures. New methodology to analyse the problem of seismic stability of steel frames. This edition providing an international approach to the stability and ductility of steel structures, will be a useful addition to research conducted in this field. It is very important to emphasies that this Colloquium was devoted to the memory of the late professor Ottó Halász. AKADÉMIAI KIADÓ BUDAPEST