Innovative Look at the ‘General Method’ of Assessing Buckling Resistance of Steel Structures

: Stability checking is an essential element of the dimensioning of steel frame structures. One of the stability checking methods allowed by EN 1993-1-1 is the so-called general method of assessing structural stability, based on buckling curves and relative structure slenderness op usually determined through numerical analyses. But this method is not widely used because of the limited computing capabilities of the engineering programs dedicated to static load analyses and difficulties in interpreting the results of the computations. The commonly used computer programs enable one to determine the shape of buckling and critical load amplifier α cr , from which one cannot directly determine the risk of buckling of a real structure. This paper presents a modified and innovative approach to the general method of assessing structural stability, which uses only three parameters, that is, the type of cross section, cross-section strength utilisation and α cr , to determine a member’s/structure’s bearing capacity mobilisation from the stability condition. The problem solution is presented in the form of simple formulas and legible diagrams. Finally, synthetic conclusions are formulated.


Introduction
In order to evaluate the ultimate limit state (ULS) of steel members in compression or/and bending, it is necessary to determine their buckling resistance ( The classical method 1 is most commonly used by designers owing to its simplicity, reliability and computation speed. Methods 3 and 4 are rarely used because of the required long computing time, the difficult choice of proper imperfections for a large number of bars and load combinations and the fact that they have not been widely implemented in engineering computer programs. Also, method 2 is not commonly used due to insufficient knowledge as to its proper use, difficulties in interpreting its results and the lack of software that would enable a comprehensive analysis of structures in different computing systems. Currently, in engineering programs (e.g. RFEM by Dlubal, SCIA Engineer by Nemetschek, Advance Design by Graitec, Idea StatiCa Member by IDEA StatiCa, AxisVM by Inter-CAD), there are additional functions that try to fill this gap. The Hungarian ConSteel software seems to be the most advanced in this matter.
The method 2, 3, 4 of the calculation of steel structures was implemented to ConSteel software in cooperation with Szalai and Papp (2011), Szalai (2017Szalai ( , 2018, Szalai and Papp (2019) and Papp et al. (2019). The aim of this study was to simplify the general method of stability assessment (sect. 6.3.4 [EN 19936.3.4 [EN -1-1:2005) and popularise it in the design of steel structures. Moreover, this paper provides an answer to the question regarding the safe value of α cr (the critical load amplifier for elastic buckling) yielded by numerical computations. Moreover, the paper presents an innovative modification of the general stability assessment method, resulting in its unification and simplification. It should be noted that the proposed modification does not affect the standard level of safety of the structure being designed (EN 1993(EN -1-1:2005 and is in accordance with the standard recommendations (EN 1993(EN -1-1:2005. Since denotations consistent with the nomenclature of EN 1993-1-1 (EN 1993-1-1:2005) are used in this paper, some of the symbols are not explained here as they are defined in the standard.

Assessment of Buckling Resistance of Compressed Members
A well-known condition, proposed in EN 1993-1-1:2005, for the ULS of a compressed member is described by equation (6.46) (EN 1993-1-1:2005), where the χ buckling coefficient is determined on the basis of the relative slenderness (6.49) (EN 1993-1-1:2005), which depends on N cr (elastic critical force for the relevant buckling mode based on the gross cross-sectional properties).
The critical force N cr is very often determined using computer programs. However, such programs usually do not specify critical force values for the bars, providing only the critical load amplifier α cr,N and the shape of the buckling of the whole analysed structure. On the other hand, the computer programs used should be able to determine N cr or α cr,N in accordance with the requirements of the standard (EN 1993(EN -1-1:2005. In order to facilitate interpretation of the results of numerical structural buckling computations and to reduce possible errors, it is proposed to modify the method of checking the stability of compressed members presented in EN 1993-1-1:2005. For this purpose, the relative slenderness of the structure under compression (EN 1993-1-1:2005) is modified as follows: (1) where α cr,N is the critical load amplifier for elastic (flexural weak or strong axis, torsional or flexural-torsional) buckling of the member, determined for the gross crosssection characteristics and U k,N is the level of utilisation of the cross section's characteristic strength, calculated from the formula: The cross section's characteristic compressive strength N Rk (the strength with no coefficient γ M0 taken into account) is determined in accordance with the procedures specified in the standard EN 1993-1-1:2005, depending on the crosssection class. When γ M0 =1.0, then N Rk =N c,Rd .
Moreover, the load capacity condition for the compressed member is written as follows: The proposed procedure for evaluating the stability of a member in compression is valid for structures/members satisfying the requirements of the standard EN 1993-1-1:2005 and for any form of buckling (flexural, torsional and flexural-torsional). The key to a correct assessment of the stability of a member is to find a buckling form with a minimal critical load value.
For the so-defined condition in Eq. (3), both buckling reduction factor χ and member resistance utilisation U b,N depend on only three variables: the kind of cross section (and hence, the associated buckling curve), cross-section strength utilisation U k,N (2) and the critical load amplifier α cr,N . Thanks to this modification, easy-to-interpret nomographs or a spreadsheet for the quick assessment of the stability of a member can be developed. The critical moment M cr is very often determined using computer programs. However, such programs usually do not specify critical moment values for the beams, providing only the critical load amplifier and the form of the buckling of the whole analysed structure. On the other hand, the computer programs used should be able to determine M cr or α cr,M in accordance with the requirements of the standard EN 1993-1-1:2005.

Assessment of Flexural Buckling Resistance of Members
In order to facilitate interpretation of the results of numerical structural stability computations with regard to lateral torsional buckling and to reduce the possible errors, it is proposed to modify the method of checking the stability of beams in bending. For this purpose, the relative slenderness (6.56) (EN 1993-1-1:2005) of the structure under bending only is modified as follows: (4) where α cr,M is the critical load amplifier for the elastic lateral torsional buckling of the member, determined for the gross cross-section characteristics, and U k,M is the level of utilisation of the cross section's characteristic strength, calculated from the formula: The cross section's characteristic resistance to bending M y,Rk (the resistance with no coefficient γ M0 taken into account) is determined in accordance with the procedures specified in the standard EN 1993-1-1:2005, depending on the cross-section class. When γ M0 =1.0, then M y,Rk =M c,Rd .
Moreover, the load capacity condition of the member in bending is written as follows: For the so-defined buckling condition in Eq. (6), both the reduction factor for lateral torsional buckling χ LT and member resistance utilisation U b,M depend only on three variables: the kind of cross section (and hence, the associated buckling curve), cross-section strength utilisation U k,M and critical load amplifier α cr,M . Thanks to this modification, easy-to-interpret nomographs or a spreadsheet for the quick assessment of the stability of a member can be developed.

Assessment of Stability of Members Being Simultaneously in Compression and In-Plane Bending
It is proposed to modify the general method described in sect. 6.3.4 of standard EN 1993-1-1:2005 to assess the stability of members being simultaneously in compression (N Ed ) and in-plane bending relative to the axis of greater stiffness (M y , Ed ). According to the standard recommendations, the general method of evaluating structural buckling resistance can be applied to the following: 1) single members or built-up (sect. 6 Under the above standard assumptions, the structural stability condition (6.63) (EN 1993-1-1:2005) takes the form: where α ult,k is the minimal design load amplifier at which the critical cross section reaches the characteristic strength in plain strain conditions with proper geometric imperfections taken into account and χ op is the reduction factor for non-dimensional slenderness op (6.64) (EN 1993-1-1:2005) corresponding to lateral and lateral torsional buckling at (8) where α cr,op is the minimum amplifier for the in-plane design loads to reach the elastic critical resistance of the structural component with regards to lateral or lateral torsional buckling without accounting for the in-plane flexural buckling. Reduction factor χ op can be evaluated on the basis of relative slenderness op as the lower of the factors χ or χ LT determined from the standard buckling curves. Load amplifier α ult,k is calculated from the characteristic crosssection strength condition: (9) and then the stability condition (6.65) (EN 1993-1-1:2005) assumes the form: (10) In this paper, a modification of the formula of general method is proposed. For this purpose, the relative slenderness of members simultaneously compressed and bent is defined as follows: (11) where α cr,NM is the critical load amplifier for the elastic buckling of a member being simultaneously in compression and mono-axial bending, determined for the gross cross-section characteristics. As opposed to its counterpart (α cr,op ) in the standard EN 1993-1-1:2005, this coefficient takes into account the effects of in-system plane and out-of-system plane buckling. U k,NM is the level of characteristic cross-section strength utilisation calculated from the formula: (12) Resistances If resistance M Rk is determined only in the elastic state (e.g. for cross-section class 3), then the level of characteristic cross-section strength utilisation can be calculated from the relation: (13) where σ N denotes the normal stresses in the cross section, produced by compressive force N Ed , and σ My denotes the normal stresses in the cross section, produced by bending moment M y,Ed .
Stresses above can be determined according to the first-or second-order analysis as defined in sect. 5.2.1 (3) (EN 1993-1-1:2005). If the structure has a small deformation, second-order analysis gives the same internal forces and stresses as first-order analysis.
The values of U k,NM determined according to Eqs (12) and (13) are identical if the characteristic bending resistance of the cross section M y,Rk determined on the basis of the elastic bending section modulus W y,el is inserted into Eq. (12) as for the section of class 3.
In the modified method, the load capacity condition for a member being simultaneously in compression and in-plane bending can be defined in two ways: Of course, if the element is protected against strong or weak axis flexural buckling, torsional buckling, flexuraltorsional buckling or lateral torsional buckling, the appropriate coefficients χ NM =1 or χ LT,NM =1 should be adopted in Eq. (15). On the basis of the presented nomographs (Figs 1-6), one can easily determine structural stability from Eq. (14). When structural buckling resistance is evaluated according to Eq. (15), one can determine U b,NM,χ and U b,NM,χLT using nomographs (Figs 1-6) and then, they should be multiplied like in Eq. (15).
It should be noted that the general method (EN 1993-1-1:2005) can be used for analysing the buckling resistance of a structure not only out of the system's plane, but also in its plane for the critical factor α cr,NM ≥10 (but the cross sections of the structure's components should be so positioned that the plane of greater stiffness and the bending plane coincide with the system's plane). Moreover, Polish Annexe to EN 1993-1-1:2005 states thus: 'firstorder analysis without taking into account geometrical imperfections can be used for non-sway systems as well as for single-storey sway systems'. It means that for typical single-story steel frame systems, general method could be used for the assessment of in-plane buckling resistance. In the proposed method, when only a compressive force or a bending moment acts, parameters U k,NM Eq. In the case of difficulties in interpreting the obtained form of a member's buckling, a safe estimate of its stability will be made assuming the more disadvantageous buckling curve (out-of-system plane buckling).

Results and Discussion
Thanks to the presented unified approach to the assessment of the stability of members in compression or/ and bending, the following simple procedure for stability assessment can be proposed:  Fig. 1 shows the structure resistance utilisation curves U b , respectively, for the buckling curves a 0 , a, b, c, d, which in a given group (U b =1.0;0.8;0.6, etc.) show the same utilisation level (each point on the curve in a given group represents the same utilisation).
The cases in which coordinates α cr and U k indicate the points below curve U b =1.0 are regarded as safe states of the member/structure (Figs 1-6). The points situated above this curve represent the states in which the member loses its stability.
Moreover, it follows from the graphs (Figs 1-6) that when α cr ≤1.0, the real member buckles, regardless of the level of cross-section strength utilisation U k .
A great advantage of this method is that the value by which the load of a member/structure can be increased so that the member/structure reaches the ULS can be easily determined from the general stability condition. The load increasing multiplier can be calculated from the formula: Moreover, the analyses showed that the buckling reduction factor χ or the lateral torsional reduction factor χ LT reaches the value of 1.0 (no reduction in member resistance with regard to buckling) when (17) A graphical interpretation of Eq. (17) is shown in Fig.  1 where the buckling curves a 0 , a, b, c, d of the ULS (U b =1.0) converge in one point α cr =25, U k =1.0. In this case, the member's resistance with buckling taken into account reaches the cross section's resistance and further increasing α cr no longer results in increased resistance of the member or the structure. This finding explains the standard's statement (EN 1993-1-1:2005) in section 6.3.1.2: 'For slenderness ≤0.2 or for N Ed ⁄N cr ≤0.04, the buckling effects may be ignored and only cross sectional checks apply', since it was shown above that for U k =1.0, 1⁄α cr =N Ed ⁄N cr =1⁄(25=0.04).
In the case of lower cross-section strength utilisation levels (U k ≤1.0), the reduction factors χ and LT reach the value of 1.0 at higher values of α cr , for example, for U k =0.2, α cr =25⁄0.2=125.
By analysing the properties of the obtained boundary surfaces (Figs 1-6), it can be concluded that for any utilisation level of a structure element Ub,i<1 resulting from the parameters U k,i , α cr,i , it is easy to determine the parameters for the limit state U b,lim =1 and they are equal to U k,lim =U k,i ⁄U b,i , α cr,lim =α cr,i U b,i (applies to sections 1, 2, 3).

Workout Example
Determine the utilisation of load-bearing capacity of frame columns in Fig. 7a. The frame is restrained against out-ofplane buckling and lateral torsional buckling. Verification is carried out by the classical method and by the general method using the nomographs proposed in this paper.        There is satisfactory convergence of calculations between the two methods, which is adequate in practical design.

Conclusions
The analyses conducted allowed for the formulation of the following conclusions regarding the proposed modified method of assessing the ultimate load-bearing capacity of a structural member from the stability loss condition: 1) This method can be applied to arbitrarily supported members being in compression and bending relative to the axis of greater stiffness, for any load distribution. The method can also be used to calculate members with a cross section variable along their length, but the results obtained are conservative. 2) The method eliminates the necessity to determine the member buckling length coefficient and to choose between a sway system and a non-sway system. The method enables a smooth transition between the two kinds of systems.
3) The so-formulated method is easy in practical use and time-efficient (short computation time). By means of the nomographs (Figs 1-6), one can quickly check the load-bearing capacity utilisation of even complex structures. Consequently, errors due to an incorrect interpretation of the results of numerical analyses are reduced. The method can be used as a tool for quick verification of structural design. 4) The method eliminates the need to separately check the different buckling forms (flexural, torsional, flexural-torsional and lateral torsional). Maximum member resistance utilisation U b is determined on the basis of the minimal value of α cr . Therefore, it is essential that the software used for structural computations is able to reliably determine not only the flexural modes of buckling, but also the torsional and flexural-torsional ones. In the case of beam finite elements, this means the seventh degree of freedom. 5) In the case of compression-only elements, the proposed nomographs (Figs 1-6 In the general method, calculations are conducted for the cross section with maximum utilisation of load-bearing capacity. According to more accurate calculation with first mode of buckling shape adopted as equivalent geometrical imperfection of member, the representative cross section is less utilised. Despite the fact that the method is conservative and overestimates utilisation of the structure load-bearing capacity, it can be useful in detecting the so-called 'thick' design errors or for quick selection of cross section of elements. 8) As opposed to geometrically and material-wise nonlinear analyses covering geometric imperfections, especially arduous in the case of models with a large number of bars and load combinations, the proposed method is easily implementable in computing programs and fast in computation. 9) The application of the general method with the use of the proposed procedure and nomographs is intuitive for the user. The nomograms clearly show the degree of utilisation of the resistance of the cross section, the degree of utilisation of the element-bearing capacity, taking into account the global stability, and allow for a quick assessment of the limit load of the structure.