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 (Boissonnade et al., 2006; Sedlacek & Naumes, 2009; Davison & Owens, 2012; Stachura & Giżejowski, 2015; Hajdu & Papp, 2018). According to EN 1993-1-1:2005, one can use one of the following four methods to assess the ULS of a structure or its members with regard to global buckling:

The classical method based on buckling lengths, buckling curves and the standard ULS criteria for compressed and flexural members (sect. 6.3.1-6.3.3 [EN 1993-1-1:2005])

The general method of evaluating stability, using buckling curves and global relative structural slenderness

A geometrically non-linear elastic analysis (GNA) of a structure with initial bow imperfections of bars and sway imperfections of frames, which can be replaced with equivalent forces (sect. 5.3.2 [EN 1993-1-1:2005])

A GNA of a structure with initial equivalent imperfections corresponding to the scaled shape of elastic buckling mode (sect. 5.3.2 (11) [EN 1993-1-1:2005])

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 (2017, 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 1993-1-1:2005]) and popularise it in the design of steel structures. Moreover, this paper provides an answer to the question regarding the safe value of _{cr}

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.

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
_{cr}

The critical force _{cr}_{cr,N}_{cr}_{cr,N}

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:
_{cr,N}_{k,N}

The cross section’s characteristic compressive strength _{Rk}_{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 _{M0}=1.0, then _{Rk}_{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 _{b,N}_{k,N}_{cr,N}

A well-known condition, proposed in EN 1993-1-1:2005, for the ULS of lateral torsional buckling of a member in bending has the form (6.54) (EN 1993-1-1:2005), where the _{LT}_{cr}

The critical moment _{cr}_{cr}_{cr,M}

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:
_{cr,M}_{k,M}

The cross section’s characteristic resistance to bending _{y,Rk}_{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 _{M0} =1.0, then _{y,Rk}_{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 _{LT}_{b,M}_{k,M}_{cr,M}

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 (_{Ed}_{y,Ed}

single members or built-up (sect. 6.3.4 (1) [EN 1993-1-1:2005]),

members with uniform cross section or not (sect. 6.3.4 (1) [EN 1993-1-1:2005]),

members with complex support conditions (sect. 6.3.4 (1) [EN 1993-1-1:2005]),

members subjected to compression and/or mono-axial bending in the system plane (sect. 6.3.4 (1) [EN 1993-1-1:2005]) and

members losing elastic stability only out of the system plane (see the standard’s definition of parameters _{cr,op}_{op}

Under the above standard assumptions, the structural stability condition (6.63) (EN 1993-1-1:2005) takes the form:
_{ult,k}_{op}_{cr,op}

Reduction factor _{op}_{LT}_{ult,k}

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:
_{cr,NM}_{cr,op}_{k,NM}

Resistances _{Rk}_{Rk}

If resistance _{Rk}_{N}_{Ed}_{My}_{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 _{k,NM}_{y,Rk}_{y,el}

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:
_{NM}_{LT,NM}_{k,NM}_{cr,NM}

Eq. (15) enables linear interpolation (recommended in sect. 6.3.4 (4) b) [EN 1993-1-1:2005]) between the buckling and lateral torsional buckling reduction factors in the case where they are determined from various buckling curves. Eqs (14) and (15) yield identical _{b,NM}_{NM}_{LT,NM}_{k,N}_{k,M}_{k,NM}_{b,NM}_{NM}_{LT,NM}_{b,NM,χ}_{b,NM,χLT}

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 _{cr,NM}_{k,NM}_{b,NM}_{k,N}_{b,N}_{k,M}_{b,M}

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).

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:

Determine the cross section’s strength utilisation: _{k,N}_{k,M}_{k,NM}

Determine the minimal critical load amplifier for the elastic buckling (_{cr,N}_{cr,M}_{cr,NM}

Identify the member’s buckling curve/curves (_{0},

Determine the level of utilisation of the structure’s resistance from the stability condition: _{b}

Repeat steps 1–4 for all the load combinations. Each time, resistance utilisation _{b}

Generate an envelope of structure resistance utilisation _{b}

In the graphs (Figs 1–6), the critical load amplifier (_{cr}_{k}

The graphs shown in Figs 1–6 were drawn for _{M1}=1.0. In the case of other values of coefficient _{M1} read off the graph, one should multiply the value of _{b}_{M1} and check if it is lower than 1.

Fig. 1 shows the structure resistance utilisation curves _{b}_{0}, _{b}

The cases in which coordinates _{cr}_{k}_{b}

Moreover, it follows from the graphs (Figs 1–6) that when _{cr}_{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 _{LT}

A graphical interpretation of Eq. (17) is shown in Fig. 1 where the buckling curves _{0}, _{b}_{cr}_{k}_{cr}

This finding explains the standard’s statement (EN 1993-1-1:2005) in section 6.3.1.2: ‘For slenderness
_{Ed}_{cr}_{k}_{cr}_{Ed}_{cr}

In the case of lower cross-section strength utilisation levels (_{k}_{LT}_{cr}_{k}_{cr}

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 _{k,i}_{cr,i}_{b,lim}_{k,lim}_{k,i}_{b,i}_{cr,lim}_{cr,i} U_{b,i}

Determine the utilisation of load-bearing capacity of frame columns in Fig. 7a. The frame is restrained against out-of-plane 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.

Data:

Steel: S235, _{y}

Height of the frame: _{c}

Span of the frame: _{b}

Compression force in the column: _{Ed}

Column and girder cross section: HEA 300, _{f}^{2}, _{y}^{4}

Static calculations and stability analysis were performed in the software Consteel v.14.

Buckling length of a column was determined according to Boissonnade et al. (2006).

Distributor factor in the bottom node of the column

_{1} = ∞ – rigid support

Distributor factor in the top node of the column

Buckling length

Critical force of buckling

Buckling reduction factor

^{2})=0.6077

Buckling resistance of compression member

_{cr,N}

_{b,N}

There is satisfactory convergence of calculations between the two methods, which is adequate in practical design.

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:

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.

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.

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.

The method eliminates the need to separately check the different buckling forms (flexural, torsional, flexural–torsional and lateral torsional). Maximum member resistance utilisation _{b}_{cr}

In the case of compression-only elements, the proposed nomographs (Figs 1–6) provide the same results of the assessment of the ultimate load-bearing capacity of the elements as the classical approach presented in sect. 6.3.1 (EN 1993-1-1:2005). This method enables a quick and easy assessment of the load-bearing capacity of columns with any support conditions, any axial force distribution and variable cross section along the length.

In the case of bending-only elements, the proposed nomographs (Figs 1–6) provide the same results of the assessment of the ultimate load-bearing capacity of the elements as the approach presented in sect. 6.3.2.2 (EN 1993-1-1:2005) (lateral torsional buckling curves – general case). This method enables a quick and easy assessment of utilisation of the load-bearing capacity of single-span and multi-span beams with any support conditions, any bending moment distribution and a variable cross section along the length. Of course, one can use other lateral torsional buckling curves, for example, the ones presented in sect. 6.3.2.3 (EN 1993-1-1:2005), and calculate

In the case of elements being simultaneously in bending and axial compression, the estimates of their ultimate load capacity according to the proposed modification are the same as the estimates presented in sect. 6.3.4 (4) a) and b) (EN 1993-1-1:2005). The precision of general method mainly depends on which procedure (sect. 6.3.2.2 [EN 1993-1-1:2005] or sect. 6.3.2.3 [EN 1993-1-1:2005]) was chosen to calculate lateral–torsional reduction factor. In Bijlaard et al. (2010), it was shown that the general method is conservative. 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.

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.

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.

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