NUMERICAL ANALYSIS OF FLEXURAL BUCKLING RESISTANCE OF NON-UNIFORM COMPRESSION

Konstruktivni elementi sa neuniformnom promenom poprečnih preseka imaju značajnu primenu u zgradarstvu, najčešće kod stubova okvirnih nosača velikih raspona, ili kod stubova industrijskih hala koje su opremljene kranovima velike nosivosti. Promena poprečnog preseka, koja prati neuniformnu raspodelu presečnih sila duž elementa, može da bude linearna duž elementa, ili skokovita u određenom broju diskretnih tačaka elementa. Ovakvim konstruktivnim rešenjem postiže se značajna ušteda u količini čeličnog materijala i ceni konstrucije. Proračun nosivosti neuniformnih elemenata na fleksiono izvijanje kompleksan je i zahtevan sa stanovišta svakodnevne inženjerske prakse. Izraz za elastičnu kritičnu silu izvijanja nije obuhvaćen osnovnim Ojlerovim slučajevima izvijanja. Vrednost elastične kritične sile izvijanja treba da se odredi vodeći računa o tačnoj raspodeli geometrijskih


INTRODUCTION
Structural members with non-uniform change in cross sections have a significant role in constructions, usually as columns of large-scale structural frames, or as columns in industrial halls that are equipped with cranes of high load carrying capacity.The crosssectional change, which follows the non-uniform distribution of the internal forces along the member, can be linear along the member, or stepped in a certain number of discrete points.This structural solution achieves significant savings in the amount of steel material and in the costs of construction.
Analysis of flexural buckling resistance of nonuniform compression members is complex and demanding from a standpoint of everyday engineering practices.
Ovaj rad prikazuje rezultate parametarske numeričke analize neuniformnih, obostrano zglobno oslonjenih i konzolnih elemenata, koja je sprovedena u softveru Abaqus [7].Neuniformnost se ogleda u promeni poprečnog preseka kroz dva segmenta i stepenastoj promeni aksijalne sile pritiska.Variran je odnos momenta inercije poprečnih preseka gornjeg i donjeg critical elastic bucklingload should be determined taking into account the exact distribution of geometric characteristics along the length of the member, boundary conditions, and the existence of possible eccentricity in the position of the system axes of the adjacent segments in members with a stepped change in cross section.The critical elastic buckling load can be determined using theoretical analyses that require the solution of nonlinear differential equations, or using one of the advanced numerical analysis methods.In simpler cases, simplified calculation procedures can be used in which the value of the elastic buckling force is determined by modifying the basic Eulerian expressions for buckling of uniform elements with an equivalent moment of inertia or an equivalent effective length, both of which take into account the geometric non-uniformity of the analysed member.
The position of the critical cross-section in which the ultimate flexural buckling load is achieved is unclearly defined.This is especially pronounced in the case of members with linear change of cross-section, where finding the solution asks for iterative calculations.In general, it can be at the cross-section in which normal stress reaches the maximum value.
For practical engineering applications, the flexural buckling resistance of non-uniform members should be based on the use of global second-order analysis, taking into account the influence of geometric, material and structural imperfections on the behaviour of the structure in the state of ultimate load capacity.Second-order influences can be included in design in one of the two ways [1], [2]:  Indirectly, by determining the value of the critical buckling load using approximate methods or linear-elastic analysis for members without imperfections, after which the general method of calculating the flexural buckling resistance, which is valid for uniform elements, should be applied in accordance with SRPS EN 1993-1-1 [3].
 Directly, by applying a second order theory with the equivalent initial global and local imperfections of members.The shape of imperfection should correspond to the fundamental form of elastic selfbuckling in the appropriate plane of buckling, and the size of the imperfection should be in accordance with the recommendations given in SRPS EN 1993-1-1 / NA [4].By this procedure, the calculation of the flexural buckling resistance of members is reduced to the calculation of the load bearing capacity of the critical cross-section for the compressive axial force and bending moment of second order.
Finite element method should be mentioned as an effective method in analysis of different stability problems that include geometric and material nonlinearities.Except for being the basis of advanced software packages (RSTAB, Sofistik, DIANA FEA...), it is also the basis of individual scientific programs developed using different programming languages.As an example, program ALIN [5], [6] solves complex problems of static and dynamic analysis, linear analysis of buckling through which the critical load can be determined both in the elastic and in the non-linear stress region.

OPŠTA METODA PRORAČUNA PREMA SRPS EN 1993-1-1
Opšta metoda proračuna nosivosti elemenata na izvijanje data u poglavlju 6.3.1,SRPS EN 1993-1-1 [3] odnosi se na pritisnute elemente konstantnog poprečnog preseka.Opšti format kontrole nosivosti na izvijanje prema ovoj metodi, podrazumeva zadovoljenje uslova da je odnos proračunske vrednosti sile pritiska N Ed i proračunske nosivosti elementa na izvijanje N b,Rd manji ili jednak od jedan.Proračunska nosivost elementa na izvijanje N b,Rd predstavlja proizvod bezdimenzionalnog koeficijenta izvijanja χ i nosivosti poprečnog preseka koja odgovara naponu na granici razvlačenja Af y , a koji je redukovan parcijalnim koeficijentom sigurnosti γ M1 .Bezdimenzionalni koeficijenat izvijanja χ zavisi od relativne vitkosti elementa  i koeficijenta imperfekcije α kojim su obuhvaćeni uticaji nesavršenosti realnih elemenata.Relativna vitkost za fleksiono izvijanje u slučaju klase poprečnog preseka 1, 2 i 3 može da se odredi prema opštem izrazu: The non-uniformity is reflected in the change of crosssection through two segments and the stepped change in compressive axial force.The ratio of the moment of inertia of cross-sections of the upper and lower segments and the ratio of axial forces in the segments is varied.The length of the members is 10 m, the height of the upper segment is 4 m, and the height of the lower segment is 6 m.For the hinged elements, the crosssection of the lower segment is HEA 300 while the cross-section of the upper segment varies in the range: HEA 160, HEA 180, HEA 200, HEA 220 and HEA 240.For the cantilever members, the cross section of the lower segment of the column is HEB 450 while the cross-section of the upper segment varies in the range: HEB 220, HEB 240, HEB 260, HEB 280 and HEB 300.The analysed range of force ratio at the ends of the FE model, P 1 /(P 1 +P 2 ) is from 0.05 to 0.50 with a step of 0.05.The total number of FE models analysed is 100.The purpose of this paper is to determine, on the basis of the given parameters and the results of the linear analysis of buckling, the range of effective length coefficients for the major axis of the individual segments of analysed non-uniform elements.The analysis of the flexural stability of the members for the minor axis of inertia has not been analysed in this paper, given the practical importance of determining the value of the buckling coefficient for the major axis, which cannot be uniquely determined as it is in the case of buckling for the minor axis of cross-section.Also, the paper assesses the accuracy of the design values for flexural buckling resistance according to the general method [3] by comparing them with the results of nonlinear numerical analysis.

GENERAL DESIGN METHOD ACCORDING TO SRPS EN 1993-1-1
The general design method for buckling resistance of members given in Chapter 6.3.1,SRPS EN 1993-1-1 [3] refers to compressed members with uniform crosssection.The general format of buckling resistance design according to this method implies that the ratio of the design value of compressive load N Ed and design value of buckling resistance N b,Rd is less than or equal to one.The design value of buckling resistance N b,Rd represents the product of non-dimensional buckling coefficient χ and the cross-sectional load carrying capacity correspondent to the yield stress Af y , which is reduced by the partial safety factorγ M1 .The nondimensional coefficient of buckling χ depends on the relative slenderness of the member  and the coefficient of imperfection α, which includes the influence of imperfections of real members.The relative slenderness for flexural buckling in the case of the crosssection classes 1, 2 and 3 can be determined according to the general expression: gde su: N cr Ojlerova kritična sila za fleksiono izvijanje, E modul elastičnosti, I momenat inercije poprečnog where: N cr is Euler critical flexural buckling load, Eis modulus of elasticity, Iis moment of inertia of the cross-preseka u ravni izvijanja a L cr dužina izvijanja elementa.
FE modeli su realizovani sa solid elementima i wedge mrežom konačnih elemenata dimenzija 15 mm.U section in the buckling plane and L cr is effective length of member.
The imperfections of real elements reduce their buckling resistance.Eurocode 3 [3] defines the analytical relation between the relative slenderness of the member and the non-dimensional coefficient of buckling φ with a family of buckling curves.The basis for the mathematical interpretation of these curves is the Ayrton-Perry formula, in which the influence of the imperfections of real elements is taken into account using the coefficient of imperfection α.The value of this coefficient depends on the shape of the cross-section, the relevant plane of buckling, the type of the production process (hot, cold rolled or welded), as well as the quality of the steel material.
The general design method can also be applied in the flexural buckling resistance design of the compressed non-uniform members with a modification that refers to the determination of the critical buckling force value N cr for which the expression (2) cannot be used.Modern design methods involve determining the critical elastic buckling loadusing a linear-elastic analysis in the appropriate software, or using approximate, simplified methods, given in the relevant literature [1], for example in the function of the equivalent moment of inertia.

NUMERICAL MODELS DESCRIPTION
Two different types of analysis were undertaken for each numerical model: linear-elastic analysis and nonlinear static analysis of flexural buckling for major axis, using the Riks solver.Linear-elastic analysis, based on the problem of bifurcation stability, gives the critical elastic buckling load of an ideal real member without imperfection.The response of the member is accompanied by negligibly small lateral deformations, and when the load reaches the ultimate critical value, a sudden buckling followed by large deformations occurs.Since the modulus of elasticity in the linear-elastic region is constant, the value of the critical buckling load depends exclusively on the slenderness ofmembers and the boundary conditions at the ends of the members.On the other hand, material and geometric non-linearity as well as structural imperfections of real structural members limit their flexural buckling resistance.In these cases, the linear-elastic analysis is used to determine the basic modes of buckling, and its results are used for the interpretation of initial geometric imperfections in later stages of analysis of the stability problems of real members.In Abaqus [7] there are several different numerical methods for solving nonlinear static problems.The Riks method, or the arc-length method, is the basic and most widely used method for analysing the behaviour of compressed membersin different modes of buckling.The requirementfor its application is that loaddisplacement curve is smooth and without branching.The accuracy of the critical load value depends greatly on the size of the initial geometric imperfections of the real member.If these deviations are idealised, or negligible, the initial, ascending part of the curve is steep with a sudden transition to an unstable equilibrium state, when the application of the Riks method can lead to a divergence incalculations.FE models are developed using "solid" elements and
"wedge" finite element mesh with a size of 15 mm.In centroids of cross-sections, at both ends of the FE model, reference points are defined and they are coupledwiththe cross-sections at the endsusing the "kinematic coupling type" option.Depending on the type of the analysed FE model, the reference points are assigned appropriate boundary conditions that simulate an idle hinge in the case of simply supported beam, or a clamp in the case of cantilever.Concentrated compressive forces in the direction of the longitudinal axis of the FE model are applied at the centroid of the upper cross section, and at the stepped change in the cross section, respectively.The nominal values of the stress-strain curve obtained by experimental testing of the specimens taken from the final hot-rolled profile HEB 260 with the quality of steel material S275 [8] were adopted to describe the mechanical properties of materials for all FE models.The experimental values were transformed into the actual values of the stressstrain.In the analysis, the usual values of the elastic constants of steel material E = 210000 N/mm 2 andv = 0,3were adopted.The residual stresses resulting from the manufacturing process were not modelled.Design code SRPS EN 1993-1-5 [9] in Appendix C suggests that geometric imperfections can be based on the buckling modein the relevant plane and allows for the reduction of manufacturing geometric tolerances of 80% in the interpretation of initial imperfections.However, since the residual stresses are not included in the analysis of the static response of the member, the allowable manufacturing tolerance has been adopted for the amplitude of the initial imperfection, and it includes deviation of the member axis from the vertical plane in the full amount of L/750 in accordance with SRPS EN 1090-2 [10].The normalized coordinate values of the deformed model for the critical buckling mode are scaled and assigned with the "Imperfection" command in the "Edit keyword" option for each individual model.The influences of local imperfections of the cross-section are disregarded in this analysis.
Figures 1 and 2 show the distribution of Mises stresses at the ultimate load state of the analysed nonuniform simply supported member due to the flexural buckling for major axis.The failure of the member is determined by the plasticization of the critical crosssection in the lower and upper segment, respectively.The numerical values of the ultimate flexural buckling for the major axis N b,FEA are normalized with the value of the force in which the cross-section is plasticized Af y for each individual segment and compared with the results of the general design method according to SRPS EN 1993-1-1 [3] through adequate choosing of the curves for the corresponding cross-sections.These values are graphically presented in Figure 3 in the case of simply supported FE member, that is, in Figure 4 in the case of cantilevermembers.Taking into account the shape of the cross section of the segment, the design ultimate strength of simply supported members is determined by choosing the curve b.In the case of cantilever members, the design ultimate strength of the lower segments are calculated for the curve a, that is, for the curve b in the case of the upper segments.Na osnovu zadatih ulaznih parametara koji se ogledaju u odnosu momenata inercije gornjeg i donjeg segmenta I 2 /I 1 i odnosa aksijalnih sila u segmentima analiziranih FE modela P 1 /(P 1 +P 2 ) s jedne strane i dobijenih vrednosti kritičnih sila izvijanja N cr s druge strane, definisane su vrednosti koeficijenata izvijanja β pojedinačnih segmenata koje su prikazane u tabelama 1 i 2. Nomogrami prezentovani na slici 5a i 5b za slučaj obostrano zglobno oslonjenih elemenata, odnosno konzolnih elemenata omogućavaju grafičko određivanje koeficijenta izvijanja β.Na apscisi nomograma dat je odnos aksijalnih sila u segmentima.Crne linije na nomogramu definišu vrednosti koeficijenta dužine izvijanja za gornji segment koje se očitavaju na levoj ordinati, dok crvene linije na nomogramu definišu vrednosti koeficijenta dužine izvijanja za donji segment koje se očitavaju na desnoj ordinati.
Based on the given input parameters that represent the relation between the moment of inertia of the upper and lower segment I 2 /I 1 and the ratio of axial forces in the segments of the analysed FE models P 1 /(P 1 +P 2 ) on one hand, and the obtained values of the critical buckling loads N cr on the other, the values of the buckling coefficients β of single segments are defined and shown in Tables 1 and 2. The nomograms presented in Figures 5a and 5b in the case of simply supported or cantilever members allow the graphical determination of the bucklingcoefficient β.The abscissa of the nomogram gives the ratio of axial forces in the segments.The black lines in the nomogram define the values of the effective length coefficients for the upper segment andare read on the left ordinate, while the red lines on the nomogram definethe values of theeffective length coefficient for the lower segment and are read on the right ordinate.
The effective length coefficient for the lower segment of the simply supported members is in the range of 1.4 to 3.1, and for the cantilever member from 2.1 to 3.6.The effective length coefficients have a higher value for greater values of the force at the top to the force at the segment change ratio.The value of the coefficient reduces with the increase of the ratio of the stiffness of upper and lower segments.
The value of the effective length coefficient for the upper segment of simply supported members is in the range of 2 to 6.1, and for the cantilever members from 2.4 to 7.8.The effective length coefficient decreases with the increase in the ratio of the force at the top and force at the segment change, and increases with the increase in the ratio of the stiffness of the upper and lower segments.Slika 5. Koeficijent dužine izvijanja gornjeg i donjeg segmenta u funkciji odnosa momenta inercije I 2 /I 1 i odnosa sila na krajevima elementa P 1 /(P 1 +P 2 ) Figure 5. Effective length coefficient of upper and lower segments in function of moments of inertia ratio I 2 /I 1 and axial forces ratioP 1 /(P 1 +P 2 )

Konzolni stub
The verification was carried out by comparison of the values of the flexural buckling critical force obtained in the commercial software Autodesk Robot 2012 and those obtained from the nomograms shown in this paper.In order to determine the value of the flexural buckling critical force using the nomograms given in Figure 5 it is necessary to determine the ratio of the moment of inertia of the upper and lower segments and the ratio of the axial forces in the upper and lower segments.

CONCLUSIONS
The method that indirectly takes into account the effects of the second order with the predetermination of the flexural buckling critical force of a non-uniform members provides a good prediction of its flexural buckling resistance.
Based on the results of the parametric numerical analysis that included 100 FE models, graphic nomograms and tabular modules were defined for determining effective buckling length coefficients of nonuniform members' segments.The analysis deals with the most common types of columns: simply supported and cantilever.
Developed nomograms and tabular modules are easy to apply in everyday engineering practice, as shown in numerical examples.The deviations between the critical force values obtained by linear-elastic analysis and from the nomograms or tables are acceptable from accuracy and reliability standpoint.
The parametric analysis shown in this paper should be further developed taking into account different heights of segments in order to define the nomograms that would have extensive application in the structural engineering practices.

Aljosa FILIPOVIC Jelena DOBRIC Milan SPREMIC Zlatko MARKOVIC Nina GLUHOVIC
This paper presents parametric linear-elastic analysis of flexural buckling of idealised non-uniform member and nonlinear analysis of flexural buckling with equivalent imperfections, using software package Abaqus.The analysis includes hinged and cantileverstepped members, where the stiffness ratio of the upper and lower segments and the ratio of the values of axial forces at the top and at the change in the cross-section are varied.The aim of this paper is to define graphic and table models for determining effective lengths coefficients of non-uniform members' segments based on a relevant and reliable database that was obtained using the Finite element method.In addition, the reliability of the method for calculating the flexural stability of the compressed columns according to EC3 was evaluated in which the critical load value was determined in the previous step using the Elastic buckling analysis of idealised elements.

Table 3
gives values of flexural buckling critical force values obtained from software modelling and graphically determined using nomograms and their comparison.