ON THE DESIGN CURVES FOR BUCKLING PROBLEMS

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1 EUROSTEEL 008, 3-5 September 008, Gra, Austria O THE DESIG CURVES FOR BUCKLIG PROBLES Jósef Salai a, Ferenc Papp b a KÉSZ Ltd., Budapest, Hungar b Budapest Universit of technolog and Economics, Department of Structural Engineering, Budapest, Hungar ITRODUCTIO The paper deals with the design curves for the member buckling problems included in man modern standards as the basis for the stabilit design of steel structures. In the Eurocode 3 [] member buckling curves are used for the two basic cases: the flexural buckling of columns and the lateral-torsional buckling of beams (B). These two buckling curves are of quite high concern and necessar for the standard design of general member stabilit, therefore the accurate theoretical and experimental verification of them is ver important. The column buckling has been researched comprehensivel for man ears as the basic and simplest case of stabilit problems. The research included a huge number of experimental tests, complete theoretical investigations using both analtical and numerical models, and deep probabilistic examinations []. As a main result the multiple column curves concept has been developed, and as the basic design model the Arton- Perr formula (APF) was adopted. This formula has the following main advantages: clear mechanical background, simplicit and flexibilit; however it is important to note that it is connected directl with the flexural buckling phenomena. The B is far more complicated to handle, the experimental research can usuall run into modelling difficulties, the analtical desiption is limited for the most basic cases; nowadas the most appropriate analsis is the numerical simulation [3]. Exploiting the flexibilit of the APF this form was chosen as the design curve for B adjusting the design parameters carefull so as to be in accordance with the results of the numerical simulations [3]. Although such a wa the B curves can be considered adequatel verified, the theoretical background of the direct connection between the APF and B is not clarified so far. The most important drawback of this is connected with the imperfection factor as the main parameter of the APF: the correct form of the generalied imperfection factor is unknown and the appropriate equivalent geometric imperfections can not be defined. The paper shows a possible wa to connect directl the B and APF b generaliing the APF using a special rule for the definition of the imperfection factor. COLU BUCKLIG The APF was applied originall for geometricall imperfect columns loaded b uniform compression, where the load carring capacit is corresponds to the onset of ielding at the most compressed fibre [4]. It is important to note that this formula neglects the influence of other imperfections (residual stress, eccentricit) and partial plasticit; however these additional effects can be efficientl modelled through the generalied imperfection factor. The maximum total second order lateral displacement of such a simpl supported prismatic member assuming sinusoidal imperfections writes: v v = () 0. where v 0 is the midspan amplitude of the half-sine wave,. is the elastic itical buckling load about the minor axis and is the actual compressive force. At the midspan oss-section the most compressed fibre should reach the ield stress (first ield iterion): A v + = f () where A is the oss-sectional area, is the elastic sectional modulus and f is the ield stress. Substituting Eq. () into Eq. () the original form of the APF can be written:

2 ( σ σ )( σ ) σ σ η (3) b f b = where σ is the elastic itical compressive stress, σ b is the actual compressive stress, and A η = v 0 is the generalied imperfection factor. Eq. (3) can also be written b the standard notations: b χ + χ η = 0 + (4) λ λ λ Af where λ is the well-known slenderness ( λ = ), χ is the buckling reduction factor ( χ = ).. Af The solution of Eq. (4) ields the standard form of the column buckling curve. GEERALIZATIO OF THE AYRTO-PERRY FORULA The derivation of the column buckling curve has distinct steps, which can be generalied in order to be applicable for other buckling problems, in our case for B. The final aim is alwas the determination of the generalied form of Eq. (4), which ields the reduction factor (χ gen ) in terms of the slenderness and imperfection factor. The general form of the slenderness writes: α λ gen = (5) α where the α and α are the first order load amplifiers corresponding to the first ield of the oss section and the elastic itical load respectivel. This definition is somewhat similar to the one applied in the general method for stabilit check in EC3 [], but is not exactl the same, because the final solution corresponds still to the first ield iterion. The steps of the derivation of APF for general stabilit problems are the following:. Determination of the one parameter load and displacement and corresponding geometrical imperfection variables, and defining the governing differential equation(s);. Appling suitable initial geometric imperfections; 3. Determination of the second order elastic displacements in terms of the load parameter, elastic itical loads (amplification factors) and initial imperfections; 4. Definition of the first ield iterion based on second order section forces which depend on the second order elastic displacements; 5. From the first ield iterion writing the normall quadratic equation for the reduction factor in terms of the generalied slenderness and imperfection, and the solution is the appropriate buckling curve. It is ver important to understand, that the ucial point of the general process is step, since the proper definition of the shape of imperfections has a great impact on the main equation derived from the first ield iterion (step 4). The main principle of this paper is that if the shape of the initial geometric imperfection is taken exactl the same as the buckling shape (or first eigenshape) of the perfect, elastic member then the equation of step 5 is alwas has the same form as Eq. (4), the onl deviations between the different buckling problems are the conete definition of slenderness (Eq. (4)) and imperfection factor. In the paper this theorem will be proved for B of beams showing that the usual simplifications in the shape of initial imperfection do not lead to the form of Eq. (4), therefore the APF can not be constructed. 3 LATERAL-TORSIOAL BUCKLIG Consider the basic case for B, which is a simpl supported prismatic beam loaded b uniform bending moment about its major axis. In this case the one parameter load is the bending moment,

3 w( θ( v( Fig.. Displacements of the oss-sections in B the displacements are the two deflections w(, v( and section rotation θ( depending from the axis x through the centreline of the member (Fig. ), and the well-known governing sstem of partial differential equations is the following: EI EI EI w( = v( + 3 θ ( GI SV ( θ ( + θ ( ) 0 = 0 θ ( v( v0( + + = 0 where v 0 ( and θ 0 ( are the initial imperfections. Since the first equation does not influence the buckling it will be omitted henceforward. Step is now the selection of the suitable shape for the initial imperfections. Omitting the initial imperfections from Eqs. (7-8) the solution gives the buckling shape of the perfect member for which the following relationship holds: (6-8) vb ( = θb ( (9) where the b subsipt refers to the buckling shape, and is the elastic itical bending moment. This is the ke relationship which should be applied to the shape of initial imperfections further on. In step 3 solving Eqs. (7-8) with the general imperfection terms ields the total second order elastic displacements, which takes the following form at the midspan oss-section: v = θ ( / ).. v0 θ 0 This is a general relationship containing a quadratic amplification factor and showing that both initial imperfections influence both displacements. At this point it is a usual simplification to neglect one of the two imperfection terms [5, 6], however following this wa the quadratic amplification factor still remains, and the APF in the form of Eq. (4) can not be achieved, since this main equation will be cubic instead of quadratic with an untreatabl complex solution. So appling the principle introduced in the second section, and accordingl substituting the relationship Eq. (9) into Eq. (0) one obtains the following form: (0)

4 v = θ / 0 0 v0 θ 0 That is the ucial step desibed in the second section which is resulted in a far simpler relationship and has the same form as Eq. (4) obtained for flexural buckling with the same linear amplification factor. The other main conclusion is that if the direction of the vector of initial imperfections is the same as the one of the vector of the buckling displacements then the certain displacements depend on the corresponding imperfection onl. oving on to step 4 the first ield iterion in the midspan oss-section is the following [7]: + ( v + wθ ) GI SV θ θ + = where, and w are the elastic major axis, minor axis and warping sectional moduli respectivel, and w is the deflection about the major axis (Fig. ) which can be expressed at L midspan as w =. On the left side of Eq. () the second and third terms are the normal EI stresses due to the second order bimoment and minor axis bending respectivel. In the second order bimoment the advantageous effect of the Saint-Venant torsional rigidit, and the additional mixed term due to the simultaneous deflection (w) and twist (θ) are considered. Substituting Eq. () into f Eq. () and appling λ = and χ = the quadratic APF for B writes: f f () () χ.. = χ η (3) λ λ λ where the three additional terms are the new generalied imperfection factor, the Saint-Venant torsional rigidit factor and the deflection factor (appling the relationship Eq. (9)): η... = v0 + θ0 = v0 + GI SV. GI SV = + θ0 = + v0 f f w = θ0. π w = v0.. π (4-6) Using these expressions the buckling curve for B can be written as the solution of Eq. (3) in the well-known form of the EC3 []: χ φ = φ + φ.. [ + η + λ ] = 0.5. λ (7-8) This is the fundamental form of the APF based B curve belonging to the first ield iterion of Eq. () and the speciall shaped initial geometric imperfection defined b Eq. (9). 4 DISCUSSIO In order to reveal the most important features and characteristics of the obtained B curve keeping in mind that it is onl the basic form belonging to the first ield iterion let examine deeper the main peculiarities of Eqs. (7-8) which are the effects of the special imperfection factors of Eqs. (4-6). In the recent form of EC3 [] multiple buckling curves are used for B,

5 and the distinction is made upon the section tpe (I-shaped or not), fabrication process (hot-rolled or welded) and in the case of I profiles the height-width ratio (h/b). In this paper we deal with I- shaped profiles and since the effect of fabrication through the residual stresses has been omitted onl the influence of the geometrical parameters is investigated further. It is found, that it is not solel the h/b ratio what affects significantl the shape of the B curves, therefore the examinations are carried out for six different hot-rolled sections (HEAA300, HEA300, HEB300, HEAA900, HEA900, HEB900) in order to show the other tpes of influential parameters. In Figs. -6 the solid lines represent the 300 sections, the dashed lines represent the 900 sections, and the thickness of the lines show the heaviness of the proper section (thin HEAA, thick HEB). All the results are calculated considering a v 0 =L/000 initial out-of-straightness and a S35 material. In Fig. the buckling curves and the non-dimensional Euler curve, in Figs. 3-6 the imperfection factors of Eqs. (4-6) are plotted for the six oss-sections against the non-dimensional slenderness. Fig.. The B buckling curves From Fig. it can be first noticed that there is onl a little difference between the curves which correspond to higher h/b ratio (~3, dashed lines), and generall these curves take higher values at higher slenderness (λ < ) and lower values at lower slenderness then the curves correspond to lower h/b ratio (~, solid lines). It is important here to note that the EC3 distinguish the curves belonging to different h/b ratios onl b the generalied imperfection factor (here η ) and the factor remains the same for all the cases, this approach does not ield the above difference between the curves. On the other hand there is significant difference between the solid curves at medium slenderness; the curve of the HEB300 section takes higher value with up to 0% then curve of HEAA300 section. This is mainl because of the fact that more compact and heavier profiles have considerabl smaller η values while the advantageous effect of their Saint-Venant torsional rigidit ( ) is more significant as it is seen in Figs This observation suggests an additional distinction between the heav and light sections b for example the b/t f ratio. It is also interesting to remark the effect of the two values in Figs Generall the deflection factor. has a dominant influence in case of wider flange profiles (solid lines); while for the higher sections this effect is negligible. It is the result of the much greater deflection along with almost the same minor axis inertia, and that is the reason for the difference between the solid and dashed curves at higher slenderness. This phenomenon suggests that the factor in the design B curves of the EC3 should also depend on the h/b ratio.

6 Fig. 3. The generalied imperfection factors Fig. 4. The Saint-Venant torsional rigidit factors Fig. 5. The deflection factors Fig. 6. The combined effect of.. 5 COCLUSIO The main point of the paper was to clarif the mechanical basis of the design curve used for lateraltorsional buckling in the EC3. This aim has been achieved b generaliing the original form of the Arton-Perr formula the basis for flexural buckling design curve introducing a specific definition of the initial geometric imperfection. B this approach the direct connection between the generalied Arton-Perr formula and the lateral-torsional buckling has been established and the correct meaning of the special imperfection factors has been deduced. Examining these factors the specialties of the formula have been revealed and suggestions for improvements have been made. REFERECES [] European Standard, EuroCode 3, Design of Steel Structures Part-: General rules and rules for buildings, E 993--, 005 [] Galambos T. V., Guide to stabilit design of metal structures, ile, 995 [3] Greiner R., Salgeber G., Ofner R., ew lateral-torsional buckling curves κ - numerical simulations and design formulae, ECCS TC8 Report o. TC , 000 [4] Arton. E., Perr J., On Struts, The Engineer, 886 [5] Kaim P., Spatial buckling behaviour of steel members under bending and compression, PhD Dissertation, TU Gra, 004 [6] Boissonnade., Villette., ueau J.P., About amplification factors for lateral-torsional buckling and torsional buckling, In: Festschrift Richard Greiner, TU Gra, 00 [7] Papp F., Computer Aided Design of Steel Beam-Column Structures, PhD Dissertation, Budapest Universit of Technolog and Economics, Edinburgh, 994