4. МЕЂУНАРОДНА КОНФЕРЕНЦИЈА Савремена достигнућа у грађевинарству 22. април Суботица, СРБИЈА

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1 4. МЕЂУНАРОДНА КОНФЕРЕНЦИЈА Савремена достигнућа у грађевинарству. април 016. Суботица, СРБИЈА ARAMETRIC STABILITY ANALYSIS OF STEEL FRAME STRUCTURES Stanko Ćorić 1 UDK: DOI: /konferencijaGFS Summary: In this paper it is presented parametric stabiity anaysis of frame structures with accuracy assessment of soutions given in codes for design of stee structures. Method for cacuating the critica oad and the effective bucking ength of the frame structures that is based on the goba stabiity anaysis is formuated. According to resuts of anaysis of the whoe structure, the critica oad and effective bucking ength for each coumn member can be obtained. The numerica anaysis in this paper is based on the cacuation of the critica oad in easo-pastic domain. Keywords: energy efficiency of buidings, renovation, mesurements 1. INTRODUCTION Research of the stabiity of inear structures, starting from the first Euer s investigations unti recenty, was mainy based on soving the differentia equation of bucking according to the second order theory. In order to find simpe soutions for the more compex structures, the researchers performed some approximations in their cacuation. It means that isoated members with different boundary conditions were anayzed. Numerica expressions and graphica representation of the obtained resuts for the critica can be found, for exampe in [1], []. Such approximations were used to formuate procedures for cacuations of muti- frames. The most used methods for this type of cacuation are Sope defection method and Stiffness distribution method, and they can be found in the iterature, for exampe [1], [3]. The above procedures for cacuation of one-bay muti-storey frames can be appied for more compex muti-bay frames. Namey, in this case, the cacuation of muti-bay framework can be reduced to cacuation of equivaent one-bay frame, as it is shown in [4], [5]. Sometimes, the framework because of its irreguarity can not be reduced to an equivaent singe-bay frame in order to cacuate their critica oad. In that case, some other procedure, such as energy method [6], are suggested. Theoretica approach based on the cacuation of isoated member was appied in the nationa and European reguation for the stabiity of frame structures [7] - [9]. However, the appication of these codes shows that such cacuation, in some cases, may ead to the substantia errors, because obtained resuts are approximate [10]. Therefore, in recent years, considerabe effort has been made in order to improve these approximate 1 Stanko Ćorić, dip.inž. građ., Facuty of Civi Engineering, University of Begrade, Buevar kraja Aeksandra 73, Begrade, Serbia, te: , e mai: cstanko@grf.bg.ac.rs ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (016) 33

2 4 th INTERNATIONAL CONFERENCE Contemporary achievements in civi engineering. Apri 016. Subotica, SERBIA cacuation methods. So, for exampe, the objective of the anaysis [11], [1] is to propose improved input parameters for the determination of the effective bucking ength coefficient of coumns in muti- frames. In this anaysis, the finite eement method, as the most effective method for numerica anaysis of stabiity of frame structures is appied. Appication of finite eement method in the stabiity anaysis of the inear frames was investigated by many authors, for exampe [13], [14]. The aim of this paper is to formuate the exact matrix stabiity anaysis. So, the interpoation functions shoud be derived from the soution of the differentia equation of bending according to the second order theory. Aso, when the bucking of structure occurs in pastic domain, constant moduus of easticity E shoud be repaced with the tangent moduus Et. More detais about this approach can be found in [15].. DETERMINATION OF THE EFFECTIVE BUCKLING LENGTH OF THE COMRESSED MEMBERS Cacuation based on the theory of eastic stabiity is widey appied in engineering practice. It is because it can be assumed that buiding structures generay have eastic behavior when they are subjected to standard expoitation oad, even in the case when the critica oad is reached. Namey, in order to design structures with high senderness, designers often make ta structures with ow stiffness, so bucking occurs in the eastic domain. Therefore it is reasonabe that this type of eastic cacuation is the basis of the reguations for the stabiity anaysis of frame structures. Such cacuation procedure is defined through the determination of the effective bucking ength of frame coumns. From the physica point of view, effective bucking ength is a ength of the equivaent member with constant cross-section that is pined at the both ends and is subjected to the compressive axia force. Critica force is defined by: cr EI (1) and it is equa to the critica force of the anayzed member with arbitrary characteristics. From the mathematica (geometrica) point of view, effective bucking ength is a distance between infection points of the bended member. The effective bucking ength is given as the product of the coumn's effective ength factor β and the geometric ength of the coumn : i () rocedure for cacuation of the effective bucking ength for the members in stee frame structures is given in nationa JUS standards [9] and European reguations EC3 [7], [8]. On the basis of such obtained bucking engths, the cacuation based on the bucking curves is appied for the axiay compressed members. 34 CONFERENCE ROCEEDINGS INTERNATIONAL CONFERENCE (016)

3 4. МЕЂУНАРОДНА КОНФЕРЕНЦИЈА Савремена достигнућа у грађевинарству. април 016. Суботица, СРБИЈА Effective bucking engths for the members with different boundary conditions are shown in Figure 1. According to obtained bucking engths, critica oad is derived from the equation (1). Figure 1. Effective bucking engths in function of the boundary conditions This approach to cacuate the critica oad, when the effective bucking ength is obtained, is appied aso to the coumns of the frame structures. For exampe, anaysis of the whoe frame structure (Figure a) is performed using the equation (1), and critica oad and effective bucking engths shoud be defined for each coumn individuay. The reason for this is the fact that unti recenty it was considered that goba stabiity anaysis of frame structures is too compicated for engineering practice. That is the reason why the codes for the stabiity anaysis of pane frame coumns [7] - [9], use simpified static scheme, as presented in Figure b. racticay, this means that codes consider ony coumns which are isoated from the frame structure. These isoated coumns are supported ony by the adjacent coumns and beams. Basicay, presence of the other structura eements connected to the considered one is introduced by the corresponding boundary conditions. a) b) Figure. Simpified static scheme for the stabiity anaysis according to the codes ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (016) 35

4 4 th INTERNATIONAL CONFERENCE Contemporary achievements in civi engineering. Apri 016. Subotica, SERBIA This approximate, simpified cacuation has its advantages because resuts can be obtained very easiy and they are shown by adequate diagrams and approximate formuas. However, very important question is how these approximate soutions are correct and how they can be used for the various exampes in engineering practice. Aso, there is important question shoud such soutions sti be used, due to the fact that fast deveopment of computing possibiities is in the progress and the programs for stabiity anaysis of frame structures become generay avaiabe. The methodoogy of cacuation that is used in this anaysis is reated to the goba stabiity anaysis of frame structures. This means that critica oad for the whoe structure ( cr,g,) shoud be cacuated first. When this critica oad is cacuated, the critica oad for each coumn ( cr) can be obtained. Based on these resuts, effective ength factor of individua coumns is given by: EI (3) cr In the case of bucking in the pastic domain, the expression for the coefficient β has the same form as (3), but critica oad ( cr) is reated to the critica force that is computed in the ineastic anaysis of the frame ( cr,ine). Aso, moduus of easticity E is no onger constant, but it is stress dependent and it is repaced by the tangent moduus E t: Et I (4) cr,ine In this anaysis it is used an empirica reationship between the two modui [15] in a form: E t 4 E 1 (5) y y 3. NUMERICAL EXAMLES The determination of the bucking ength of frame structures is presented in this section. The cacuation is performed using the corresponding computer program ALIN deveoped in the C++ programming anguage [16], [17]. Obtained resuts are compared with the resuts from the European EC3 and nationa JUS standards [7] - [9]. A six- three-bay pane frame, camped at the base, is first considered in this anaysis (Figure 3a). Graphica presentation of the resuts for the coefficient β is shown in Figure 3b. These resuts are obtained using the code ALIN, as we as European and nationa codes for stee structures. It can be seen that in this case there is a quite we coinciding of the resuts. 36 CONFERENCE ROCEEDINGS INTERNATIONAL CONFERENCE (016)

5 4. МЕЂУНАРОДНА КОНФЕРЕНЦИЈА Савремена достигнућа у грађевинарству. април 016. Суботица, СРБИЈА coumns- I girders- I/ а) b) effective ength factor Figure 3 a) Six- sway frame with constant axia force in the coumns b) The vaues of the coefficient β for the anayzed frame ALIN ec3, outer jus, outer ec3, inner jus, inner A anaysed six- three-bay frame with the oad that is exerted on each coumn at each in the frame is shown in Figure 4а). Resuts for the effective bucking ength coefficient are given in Figure 4b. effective ength factor 5.0 ALIN ec3, outer 4.5 jus, outer ec3, inner jus, inner coumns- I girders- I/ а) b) Figure 4 а) Six- sway frame with oad at each b) The vaues of the coefficient β for the anayzed frame This graph shows that errors in the appication of reguations increase when compared to the previous exampe. One of the main reasons for this is that the cacuations that are based on the isoated treatment of considered compressed eements (as in the ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (016) 37

6 4 th INTERNATIONAL CONFERENCE Contemporary achievements in civi engineering. Apri 016. Subotica, SERBIA reguations) do not take into consideration the axia force vaue in the eement. They ony consider stiffness of the other structura eements connected to the considered one. Thus, in anayzed numerica exampe obtained errors are arger than in the previous case. It is cear that that these differences in the determination of the bucking ength affect the cacuation of the bucking resistance of axiay compressed members. Athough the camped coumns are more present in the engineering practice, this paper aso considers the frames that are pinned at the base. The characteristics and the geometry of the frames are assumed the same as in the previous numerica exampes, except for the boundary conditions at the bottom. The externa oads are exerted on each coumn at each in the frame. The moduus of easticity for the bucking anaysis in the eastic range is adopted to be constant (E). In the ineastic range, tangent moduus aproach is appied. Comparative review of the resuts for the coefficient β for the camped and hinged frame is given herein. Resuts obtained using the code ALIN and Eurocode 3 are presented in Figure 5. effective ength factor ALIN, camped frame ALIN, pined frame effective ength factor а) b) ec3-outer, camped ec3-inner, camped ec3-outer, pinned ec3-inner, pinned Figure 5 Coefficient β for the coumns of the frames which are camped and pinned at the base a) obtained using the program ALIN; b) using the EC3 standard Regarding the fact that hinged frames have ower stiffness when compared to camped frames, it is cear that smaer critica oad is obtained for them. Thus their effective bucking ength coefficient has a higher vaue for a foors. Aso, it is cear that, by appying the standards, the resuts for anayzed frames are different ony at the first foor. Negection of the stabiity treatment of the structure as a whoe, is just one of the main reasons for the errors that are obtained using the standards. The six- three-bay pane frame, camped at the base, with the oad on each coumn at each in the frame is aso anayzed. In order to make comparison of the resuts simper, a data is given in the parametric form. Considered frame has the same characteristics as the frame given in Figure 4а. In this paper are presented the comparative resuts for the effective bucking ength coefficient for the considered sway and non-sway frames (Figure 6). Since the non-sway frame (considered in this numerica exampe) has six times higher critica force with respect to the same sway frame (given in Figure 4) these resuts for the coefficient β are expected. 38 CONFERENCE ROCEEDINGS INTERNATIONAL CONFERENCE (016)

7 effective ength factor 4. МЕЂУНАРОДНА КОНФЕРЕНЦИЈА Савремена достигнућа у грађевинарству. април 016. Суботица, СРБИЈА 4.5 sway frames non-sway frames Figure 6 Comparison of the vaues β for the six- sway and non-sway frames 4. CONCLUSIONS In this paper it was investigated the accuracy of the soutions given in JUS and EC3 standards in detais, reated to the effective bucking ength determination. Muti foor sway and non-sway frames were anayzed. The effect of the various parameters to the critica oad vaue, such as different supports, oads, etc. was investigated. Appying the proposed method it was shown that substantia errors can be made when the approximate soutions from the codes are used. That means that there is a need for the innovation of these standards in the part where the effective ength of frame coumns is considered. It can be done, for exampe, in the way as it is proposed herein. It shoud be emphasized that some steps have aready been taken in EC3 standards, in the part reated to the compex deformabe structures which require the cacuation according to second order theory, however, no more detais are given in EC3. REFERENCES [1] Aen, H.G., Buson,.S.: Backgroung to bucking, McGraw Hi, U.K., [] European Convention for Constructiona Steework (ECCS): European recommendations for stee structures, [3] Horne M.R., Merchant W.:The stabiity of frames, ergamon, Oxford, [4] White D., Hajjar J.:Bucking Modes and Stabiity Design of Stee Frames: a Unified Approach, Journa of Constructiona Stee Research, 1997., Vo.4, Iss.3, pp [5] White D., Hajjar J.: Accuracy and simpicity of aternative procedures for stabiity design of stee frames, Journa of Constructiona Stee Research, 1997., Vo.4, ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (016) 39

8 4 th INTERNATIONAL CONFERENCE Contemporary achievements in civi engineering. Apri 016. Subotica, SERBIA Iss.3, pp [6] Horne M.R.: An approximate method for cacuating the eastic critica oads of muti-storey pane frames, Structura Engineer, 1975.,Vo.56, Iss.6, pp [7] Eurocode 3, Design of Stee Structures - ENV : 199, European Committee for Standardization, [8] Eurocode 3, Design of Stee Structures - ENV : 005, European Committee for Standardization, 006. [9] Buđevac D., Мarković Z., Bogavac D., Тоšić D.: Stee Structures, Facuty of Civi Engineering University of Begrade, (in Serbian) [10] Ćorić S.: Contribution to the Effective Length Stabiity Anaysis of Frame Structures, master thesis, Facuty of Civi Engineering University of Begrade, 006. (in Serbian) [11] Mageirou G.: Contribution to the design of muti- stee frames against fexura bucking, Doctora Thesis, Nationa Technica University of Athens, Greece, 001. [1] Gantes C., Mageirou G.: Improves stiffness distribution factors for evauation of effective bucking engths in muti- sway frames, Engineering Structures, 005, Vo.7, Iss.7, pp [13] Gaagher R.H.,: Finite Eement Anaysis, rentice-ha, Inc. New Jersey, [14] Bathe K.J.: Finite Eement rocedures in Engineering Anaysis, rentice-ha, 198. [15] McGuire W., Gaagher R., Ziemain R., Matrix Structura Anaysis, nd edition, John Wiey and Sons, New York, USA, 000. [16] Ćorić S.: Noninear stabiity anaysis of the frame structures, Doctora dissertation, Facuty of civi engineering, University of Begrade, 013. [17] Ćorić S., Brčić S.: Easto-pastic stabiity cacuation of the frame structures using the code ALIN, Internationa Conference Contemporary Achievements in Civi Engineering, pp , 4-5. Apri 014, Subotica, Serbia ПАРАМЕТАРСКА АНАЛИЗА СТАБИЛНОСТИ ЧЕЛИЧНИХ ОКВИРНИХ НОСАЧА Резиме: У овом раду је приказана параметарска анализа стабилности оквирних носача са оценом тачности решења датих у стандардима за прорачун челичних оквирних конструкција. Формулисана је метода за прорачун критичне силе, односно дужине извијања оквирних конструкција, а која се заснива на анализи глобалне стабилности конструкције. Тако срачуната сила омогућује да се даље срачунају и критичне силе, односно дужине извијања појединих притиснутих штапова у конструкцији. Нумеричка анализа у овом раду се заснива на прорачуну критичног оптерећења у еласто-пластичној области. 330 CONFERENCE ROCEEDINGS INTERNATIONAL CONFERENCE (016)