BEAM-COLUMN RESISTANCE ACCORDING TO SLOVAK STANDARD AND EUROCODES BEAM-COLUMN RESISTANCE ACCORDING TO SLOVAK STANDARD AND EUROCODES

In the case of the torsionally restrained beam-column along its length, i.e. the member whose effects of lateral-torsional buckling are eliminated by the relevant supporting, and considering the initial deflection in the direction of the y-y axis only, the system of differential equations (1) can be rewritten into the following elementary pair of independent differential equations, describing the buckling resistance of beam-column respecting the above mentioned cross-sectional shape and load is the Young’s modulus of elasticity, is the shear modulus, is the second moment of area about the y-y or z-z axis, respectively, is the St. Venant torsional constant, The article deals with the verification of resistance of torsionally restrained hot-rolled beam-columns of H or I cross-sectional shape and also of the perpendicular hollow cross-section subjected to the biaxial bending and axial compression. Results of analysis of the beam-column resistances according to methods of former STN 73 1401 [1] and the current STN EN 1993-1-1 [2] are compared with results of numerical approaches. The observed members are subjected to the normal force and to the transverse loads uniformly distributed along the length of both axes. The cross-sections of HEB 300, IPE 300 and RHS 300x200x10 were chosen to compare obtained results.


Introduction
Nowadays, the development of new materials creates a competitive environment for the use of steel as a structural material in constructions. High-strength concrete and reinforced concrete cross-sections of high quality compete with steel with their dimensions. Therefore, many specialists are searching to improve the accuracy of design formulas for steel members, especially for members subjected to the combined actions. The beam-column subjected to combination of bending moments and axial compression is an example of this problem. Therefore, the possibility of using cross-sectional plastic reserve or improving and more specifying the design approaches to verification of beam-column resistance becomes more important. If the standard prescriptions do not meet criteria of the optimal structural design, then the determination of the structural resistance and its verification need necessarily to be more precise and often complex based on the numerical calculations using FEM models.

Analytical approach
Stability analysis of a beam-column has been done in the past using solution of the relevant homogeneous differential equations. The buckling resistance of initially deflected beam-column with double symmetric constant cross-section, subjected to combined constant axial compression and biaxial bending due to transverse load uniformly distributed along the length of the both axes, could be described by the system of inhomogeneous differential equations in accordance with [3] EI w N w w @ @ @ @ @ (1) In the case of the torsionally restrained beam-column along its length, i.e. the member whose effects of lateral-torsional buckling are eliminated by the relevant supporting, and considering the initial deflection in the direction of the y-y axis only, the system of differential equations (1) can be rewritten into the following elementary pair of independent differential equations, describing the buckling resistance of beam-column respecting the above mentioned cross-sectional shape and load where E is the Young's modulus of elasticity, G is the shear modulus, I y , I z is the second moment of area about the y-y or z-z axis, respectively, I t is the St. Venant torsional constant, The equations become even more inhomogeneous when the initial imperfections are implemented into the system. The member initial deflection of v 0 in the direction of the y-y axis was considered to simulate the shape of the member equivalent geometrical imperfection described by means of the sinusoidal function as follows / sin v e x L ,z 0 0 r =^h (4) where the amplitude e 0,z is defined according to the standard [2] in the following form and the imperfection factor α l depends on the cross-sectional shape.
Using the differential expression for bending moment, the equations (3) can be rewritten into the differential form [4] as follows The final solutions of the above described differential equations can be obtained using homogenous boundary condition, w = v = 0 for x = 0 and for x = L, in the form as follows I w is the warping torsional constant, N is the normal compression force, M y , M z are the bending moments about the y-y and z-z axis, respectively, induced by transverse loads, i s is the cross-sectional polar radius of gyration around to the centre of shear, q y , q z are the transverse uniformly distributed loads in the direction of the y-y or z-z axis, respectively, v o is the initial deflection of a member in the direction of the y-y axis, v is the deflection increment of a member in the direction of the y-y axis, w is the deflection increment of a member in the direction of the z-z axis, w o is the initial deflection of a member in the direction of the z-z axis, z g,y , z g,z are the distances of the applied transversal load from the centre of shear measured in the direction of the y-y or z-z axis, respectively, θ o is the angle of the initial cross-sectional rotation of a member about the x-x axis, θ is the angle increment of cross-sectional rotation of a member about the x-x axis.
Bending moments M y and M z depend on the type of transverse load and usually have the non-constant shape. According to Fig.  1, the bending moments at the point C at a distance of x from A (see Fig. 1 Table 1. The comparison is presented in the form of ratio of results of numerical calculations to results obtained using derived analytical solutions. The comparison presented in Table 1 is statistically evaluated to determine the error of the numerical approach.
Statistically evaluated comparisons of analytical calculations with results of numerical ones The comparison presented in Table 1 shows a very good accordance insomuch that above described numerical model can be applied for parametric numerical study of the beam-column resistance determination. It was necessary to evaluate a large number of combinations of normal forces and transverse loads to create compact resistance surface of the observed members. Therefore, the following input values were taken into account for numerical calculations in frame of parametric study: where W el,y , W el,z are the elastic sectional modulus about the y-y or z-z axis, respectively.

Numerical approach
Due to the complexity of analytical solutions of system of inhomogeneous differential equations according to (1), the numerical analyses are used to obtain results. Numerical models are usually created using computer software based on FEM. In this case the Ansys-Workbench was used. Numerical model developed in this environment consists of one dimensional finite element BEAM 188 (see Fig. 2). The element is based on Timoshenko's beam theory including shear-deformation effects. The unrestrained warping of cross-section or restrained one respectively can be taken into account. This element is a linear, quadratic, or cubic two-node beam element in 3D. BEAM188 has six or seven degrees of freedom at each node. These include translations in the x, y, and z directions and rotations about the x, y, and z directions. The seventh degree of freedom -warping magnitude, is optional. This element is well-suitable for linear large rotation and/or large deflection nonlinear applications.
The sinus function was chosen in accordance with [2] to simulate the initial shape of bow imperfection of the member in the direction of the y-y axis. Its amplitude was taken in compliance with recommendation for flexural buckling in [2]. In the case of applied numerical model, the boundary conditions Rot,x = 0 were considered, allowing for torsionally restrained member with disabled rotation about the x-axis.
Ideal elastic-plastic material model was chosen. Material characteristics were used according to standard recommendations [2]. STN EN 1993-1-1 comparisons. In accordance with this standard, the spatial display creates approximately "planar resistance surface" connecting the borderline cases lying on the axes of the graph. Resulting isoclines of methods A and B show that the standard [2] uses the plastic reserve of members more accurately. Surface resistance of these approaches is already convex describing more realistic beam-column resistance. Both methods for assessing the combination of biaxial bending and axial compression according to [2] describe the resistance of member with the closed crosssection of RHS 300x200x10 better. In the cases of the higher levels of normal forces, standardized approaches show reserve of resistance compared to results of numerical calculations. If members are subjected to the smaller bending moment M y , the method A provides higher levels of the member resistance for non-dimensional slenderness z m greater than 1.0 than numerical calculations for open cross-sections IPE 300 and HEB 300. Assessments of members with torsional restraints according to method A are less suitable for members with non-dimensional M z,Ed /M z,Rd are shown on the horizontal and vertical graph axes. Characteristic values of resistances without partial safety factor γ M1 for material were considered as the standardized member resistances for all normative approaches to compare the obtained results of numerical calculations. Graphs are processed for non dimensional slenderness z m =1.0; 1.5 and 2.0. The investigated member is subjected to the axial compression and bending due to the uniformly distributed transverse load for both axes of symmetry of the cross-section so that the shape of the bending moment is parabolic. Calculations were considered with torsionally restrained cross-section HEB 300 and IPE 300, i.e. the effect of lateral-torsional buckling was neglected. The cross-section of RHS 300x200x10 is not susceptible to torsional deformations.

Conclusions
Conservative approach to the resistance verification according to the standard [1] is evident from all the result's slenderness z m greater than 1.0. The recommendation to use the method B in Slovakia seems to be correct from this point of view [8].