Solving Lateral Beam Buckling Problems by Means of Solid Finite Elements and Nonlinear Computational Methods

Transcription

1 International Journal of athematical and Computational ethods Solving Lateral Beam Buckling Problems b eans of Solid Finite Elements and Nonlinear Computational ethods JAN VALEŠ ZDENĚK KALA JOSEF ARTINÁSEK Department of Structural echanics Brno Universit of Technolog Facult of Civil Engineering Veveří 33/ Brno CZECH REPUBLIC vales.j@fce.vutbr.c Abstract: The present paper deals with the comparison of values of resistance of a hot-rolled steel beam with initial imperfections. Ver detailed 3D calculation models of beams were eated in the computer programme Anss appling the finite elements SOLID85. Eight nodes having three degrees of freedom at each node define the element SOLID85. Resistances were computed b application of geometricall and material nonlinear methods to the selected degree of non-dimensional slenderness depending on nominal length of the beam. Beams are subjected to bending and their total and elastic resistance are compared with analtical empirical and standard tpes of resistance. Ke-Words: Lateral Beam Buckling Resistance Imperfections Hot-rolled Beam Anss Introduction The stiffness of steel beams subjected to bending is substantiall higher in the plane associated with bending about their major principal axis than in the plane associated with bending about their minor principal axis []. The static resistance is the ucial quantit influencing the safet and reliabilit of steel beams. Initial imperfections differ according to the manufacturing tpe of the beam []. Initial imperfections of steel plated girder resistance arise during the welding process in particular [3]. On the other side initial imperfections of hot-rolled steel beams are often modelled so that the originate in the first eigenmode of lateral beam buckling [4]. The imperfections of a hot-rolled steel I-beam loaded b equal end bending moments of opposite sense consist of initial axis curvature v 0 and initial oss-section rotation φ 0 [5]. The imperfection modelled according to the first eigenmode of lateral beam buckling assumes that v 0 and φ 0 are dependent functionall and have the correlation [6]. From the point of view of design reliabilit it is conservative because it leads to the most rapid deease of resistance. In general reliabilit analses are important parts of multi-objective optimiation of economic aspects of building engineering and mechanical one [7-9]. Large attention is to be paid to computational models of supporting elements in particular. The presented paper is concentrated on finite element modelling and nonlinear analses of resistance of a hot-rolled steel beam I00. Computational odel The computational model represents a hot-rolled beam I00 steel grade 35. The oss-section geometr was simplified according to Fig.b) so that it could be possible to define it b four quantities h b t and t. h h R b t t R t 3 t4 b/4 h t a) b) Fig. : Profile I00: a) real b) idealied The initial geometrical imperfection of beams is designed according to the first eigenmode of buckling at the stabilit loss b lateral beam buckling. It consists of initial displacement of axis v 0 and of initial rotation of oss-sections φ 0. These imperfections are considered to be affine to the final shape as the functions sinus v 0 being the curvature b t b/4 ISSN: X 03 Volume 06

2 International Journal of athematical and Computational ethods of the beam axis in the direction of major axis i.e. in plane x and φ 0 rotation of oss-sections along the beam length see Fig.. sin π x v0 = av0 0 0 L sin π x ϕ = aϕ L φ 0 = v 0 () opposite sense. For such a loading case the relation for can be derived according to [] in the form π π EI = +. (5) L ω EI GI t GI tl The bending moment on the edge oss-section of 3D model was eated as a pair of forces in its nodes. The forces act during loading perpendicularl to the end oss-section and so their arms r i remain constant see Fig.3a). Boundar conditions are set so that the edge oss-sections can warp see Fig.3b). The support in direction of axis x u x = 0 is introduced at one end onl. x Pressure + u =0 Fig. : Initial imperfections If the beam is curved according to the first eigenmode thus it holds for amplitudes a v0 and aφ0 a vo e0 = h π EI + L a π EI = () ϕ 0 av0 L where e 0 is the amplitude of one half-wave of the sine function relating to the upper edge of the osssection and is designed as L/000 [0] h is the oss-section height E is the Young s modulus of elasticit I is the inertia moment to the axis L is the beam length and is the elastic itical moment at lateral beam buckling. The elastic behaviour of beams can be analsed using two differential equations []: v + ( ϕ + ϕ0 ) = EI 0 x 3 ϕ ϕ v v0 EIω GI 0 3 t + + = x x x x (3) (4) where G is the shear modulus I ω is the warping constant and I t is the torsion constant.. Loading and Boundar Conditions The beam is considered as simpl supported and loaded at both ends b equal bending moment of r i Pressure - x u =0 u =0 u x =0 a) b) Fig. 3: a) Loading of edge oss-sections b) boundar conditions. Finite Element odel Computational models were eated in the programme Anss using 3D elements SOLID85. SOLID85 is an 8-node element which can be used for 3D modelling of solid structures and has plasticit hperelasticit stress stiffening eep large deflection and large strain capabilities. It is defined b eight nodes having three degrees of freedom at each node: translations in the nodal x and directions. The element was set to be a homogeneous structural solid element. The enhanced strain formulation was considered. The enhanced strain formulation prevents shear locking in bending-dominated problems and volumetric locking in nearl incompressible cases. The formulation introduces certain number of internal (and inaccessible) degrees of freedom to overcome shear locking and an additional internal degree of freedom for volumetric locking (except for the case of plane stress in -D elements). All internal degrees of freedom are introduced automaticall at the element level and condensed out during the solution phase of the analsis []. x ISSN: X 04 Volume 06

3 International Journal of athematical and Computational ethods An example of the computational model is presented in Fig.4 and Fig.5. Initial imperfections are illustrated here in magnified scale. Fig. 4: Computational model in Anss - axonometr D + D + D 4 5 R = 4 W ( ) ( ) 4D + D + D + 4D D D W where D = f WW D = W + P av0 W D3 = W P av0 W D4 = P I a v 0 D5 = D EI P = π L. (6) W is the oss-section module to axis and W is the oss-section module to axis. Fig. 5: Computational model in Anss views.. aterial Properties An elastic-plastic stress-strain diagram without hardening according to the standard ENV :99 is used for the computation. The value of ield strength f is considered b nominal value 35 Pa. σ f dσ dε =E Fig. 6: Bilinear stress-strain diagram 3 Load-Carring Capacit The value of elastic resistance R of the beam can be derived on behalf of equations (3) and (4). R represents the bending moment at which the maximum value of the von ises stress corresponds to ield strength f of the steel and is given b the relation [4]: ε Table : Cross-section characteristics Characteristic Smbol Value Cross-section height h 0.00 m Cross-section width b m Web thickness t m Flange thickness at quarter of the width t m Second moment of area about axis I.35E-6 m 4 Second moment of area about axis I.88E-6 m 4 Torsion constant I t.87e-7 m 4 Warping constant I ω.07e-8 m 6 Section modulus about axis W.35E-5 m 3 Section modulus about axis W.639E-5 m 3 Plastic section modulus about axis W pl 4.684E-5 m 3 To calculate the plastic resistance plr it is possible to appl b means of (6) the empirical relation according to [3] where W = ( ) W α + α (7) pl pl R R R 4 α = 4. (8) + λ ISSN: X 05 Volume 06

4 International Journal of athematical and Computational ethods λ is the non-dimensional slenderness at lateral beam buckling according to the Eurocode 3 W pl λ = (9) f where W pl is the plastic oss-section module to axis. Cross-section characteristics of the idealied profile I00 according to Fig.b) are given in Table. 3. Resistance According to Eurocode 3 The design resistance moment of the beam at lateral beam buckling brd of a horiontall not supported beam is determined from the relation f = χ W (0) γ b Rd The oss-section I00 is the oss-section class and therefore the oss-section module W can be determined as W = W pl. For the partial resistance factor of oss-section when evaluating the stabilit γ holds γ =.0. Reduction factor for lateraltorsional buckling χ for the appropriate slenderness χ in which λ ma be determined from = Φ + Φ λ ( ) () Φ = α λ 0. + λ. () The curve of lateral beam buckling b can be used for the oss-section I00. The value of imperfection factor for lateral-torsional buckling is α = Resistance of the Computational odel The computational model in the programme Anss is loaded ineasingl and calculated in geometricall nonlinear wa b the Newton- Raphson method. The plastic resistance planss is defined as the maximum value of bending moment when the determinant of the stiffness matrix is non-ero and the calculation converges. The elastic resistance RAnss is given b reaching of presibed stress (of ield strength f ) in an point of the beam. With regard to the plane smmetr of the beam along the plane passing through its centre and in parallel with the reaching the ield strength takes place in one of oss-section tops in the middle of span. The linear regression was carried out to accuratel quantif the elastic resistance for a narrow set of data including the value of acting moment and corresponding value of the von ises stress near the ield strength. As the basic linear regression model the polnomial of the seventh degree was applied where the absolute error was still negligible. 4 Comparison of Resistances The values of analticall computed resistances according to (6) (7) and (0) are depicted b the curve in Fig.7. The diagram is completed b the Euler hperbola according to (5) and the values of resistance d given b the relation = f W (3) d pl λ The range of non-dimensional slenderness is from 0 to.. According to (9) and (5) nondimensional slenderness is in the present problem dependent onl on the beam length L. This is limited to the length of metres b the manufacturer of hot-rolled profiles. The relation for the dependence of length on non-dimensional slenderness can be thus derived in the form: L = in which Load-Carring Capacit [knm] λ I 0.5 π ( EI ) ω 0.5 ( Q ) 0.5 λ 0.5 I GIt λ t 4 n pl Q = I G I + f W I ω d brd R plr (4) RAnss planss Nondimensional slenderness λ [-] Fig. 7: Resistance vs. non-dimensional slenderness ISSN: X 06 Volume 06

5 International Journal of athematical and Computational ethods Due to the non-linear relation between λ and L the resistance is laid out also to the beam length see. Fig.8. Load-Carring Capacit [knm] d brd R plr RAnss planss Span Length [ m] even for the lowest considered values of slenderness the oss-section of the beam need not plasticie full at reaching the total resistance. Such a case of stress course σ x is observed e.g. for the slenderness λ = 0.6 presented in Fig.0. The von ises stress which decides on the resulting value of resistance is depicted in Fig. and Fig.. σ x.35e+09.83e+09.3e e+08.6e E E E E E+09 Fig. 8: Diagram resistance vs. span length 4. Stress of the Beam under Limit State The elastic resistance is given b reaching the ield strength f in an point of the beam without occurrence of plasticiation of the oss-section. The course of stress σ x in the span middle is illustrated in Fig.9a). It is evident from the diagram in Fig.7 that the absolute difference of resistance planss and RAnss is ineasing with deeasing value of slenderness. There takes place the use of plastic reserve of the oss-section. For the slenderness approximatel higher that.4 this difference is negligible..35e+09.83e+09.3e e+08.6e E E E E E+09 Fig. 0: Stress course in web σ x for slenderness λ = 0.6 at reaching the total resistance.35e+09.09e+09.83e+09.57e+09.3e+09.04e e+08.5e+08.6e+08 0 Fig. : Course of the von ises stress σ v in the beam for λ = 0. 6 at reaching the total resistance σ x σ x a) b) Fig. 9: Stress course in web σ x at reaching : a) elastic resistance b) plastic resistance It can be noticed that the plastic resistance of the oss-section subjected to bending is an important part of numerous optimiation analses [4]. The oss-section can theoreticall plasticie totall according to Fig.9b). However the realit is so that Fig. : Course of the von ises σ v in the osssection in the span middle for slenderness λ = 0. 6 at reaching the total resistance ISSN: X 07 Volume 06

6 International Journal of athematical and Computational ethods 5 Conclusion It is clear from Fig.7 that with ineasing slenderness the values of analticall and b programme computed resistances approach the Euller hperbola and the problem becomes a stabilit problem. In case of elastic resistance the values from computational model RAnss full agree with the analtical R. When comparing the values of standard resistance brd with total computed planss there is obtained good agreement approximatel fromλ 0.7. For the lower nondimensional slenderness the standard resistance is b 6 % higher. The empirical total resistance plr according to (7) gives the values lower than brd and planss and thus it is rather safe assessment of total resistance based on the elastic resistance R. Although it is well known that the influence of residual stress deeases the resistance of hot-rolled struts under compressions [5] it was not considered in the present problem. It remains however the topic of future studies. Acknowledgement The article was elaborated within the framework of project GAČR S and project No. LO408 AdAs UP. References: [] T. V. Galambos Stabilit Design Criteria for etal Structures John Wile & Sons 998 p.9. [] Z. Kala J. Kala Resistance of Plate Girders Under Combined Bending and Shear In Proc. of the 3rd WSEAS Int. Conf. on Engineering echanics Structures Engineering Geolog (EESEG 0) Corfu Island (Greece) 00 pp [3] Z. Kala J. Kala Resistance of Thin-walled Plate Girders under Combined Bending and Shear WSEAS Transactions on Applied and Theoretical echanics Vol.5 No.4 00 pp [4] Z. Kala Elastic Lateral-torsional Buckling of Simpl Supported Hot-rolled Steel I-beams with Random Imperfections Procedia Engineering Vol pp [5] Z. Kala Sensitivit and Reliabilit Analses of Lateral-torsional Buckling Resistance of Steel Beams Archives of Civil and echanical Engineering Vol.5 No.4 05 pp [6] J. Valeš The Influence of Correlation Between Initial Axis Curvature and Cross-section Rotation on the Beam Static Resistance AIP Conference Proceedings Vol Article number doi:0.063/.493. [7] J. Antucheviciene Z. Kala. arouk E.R. Vaidogas Solving Civil Engineering Problems b eans of Fu and Stochastic CD ethods: Current State and Future Research athematical Problems in Engineering Vol Article number [8] J. Antucheviciene Z. Kala. arouk E.R. Vaidogas Decision aking ethods and Applications in Civil Engineering athematical Problems in Engineering Vol Article number [9] S. Chakrabort J. Antucheviciene E. K. Zavadskas Applications of Waspas ethod as a ulti-iteria Decision-making Tool Economic Computation and Economic Cbernetics Studies and Research Vol. 49 No. 05 pp. -7. [0] Z. Kala Stochastic Analsis of the Lateral- Torsional Buckling Resistance of Steel Beams with Random Imperfections AIP Conference Proceedings Vol pp [] TRAHAIR N. S. The behaviour and design of steel structures John Wile and Sons Ltd. 977 p.30. [] ANSYS Element Reference Release. ANSYS Inc [3] Z. Kala Reliabilit Analsis of the Lateral Torsional Buckling Resistance and the Ultimate Limit State of Steel Beams Journal of Civil Engineering and anagement Vol. No.7 05 pp [4] J. Atkočiūnas T. Ulitinas S. Kalanta G. Blaževičius An Extended Shakedown Theor on an Elastic plastic Spherical Shell Engineering Structures Vol.0 05 pp [5] Z. Kala J. Kala Variance-based Sensitivit Analsis of Stabilit Problems of Steel Structures In Proc. of the Int. Conf. on odelling and Simulation (S'0) Edited b Štemberk P. Prague (Cech Republic) 00 pp ISSN: X 08 Volume 06