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

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: 1)

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 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 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 1993-1-1:2005) and is in accordance with the standard recommendations (EN 1993-1-1:2005).

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.

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 λ¯ \bar \lambda (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 1993-1-1:2005).

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) λ¯=AfyNcr=Afyαcr,NNEd=1αcr,NUk,N \bar \lambda = \sqrt {{{A{f_y}} \over {{N_{cr}}}}} = \sqrt {{{A{f_y}} \over {{\alpha _{cr,N}}{N_{Ed}}}}} = \sqrt {{1 \over {{\alpha _{cr,N}}{U_{k,N}}}}} 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 cross-section characteristics and U_k,N is the level of utilisation of the cross section’s characteristic strength, calculated from the formula: (2) Uk,N=NEdNRk≤1.0 {U_{k,N}} = {{{N_{Ed}}} \over {{N_{Rk}}}} \le 1.0

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 cross-section 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: (3) Ub,N=γM1Uk,Nχ≤1.0 {U_{b,N}} = {{{\gamma _{M1}}{U_{k,N}}} \over \chi} \le 1.0

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.

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 χ_LT reduction factor for lateral torsional buckling for the general case (sect. 6.3.2.2 [EN 1993-1-1:2005]) is determined on the basis of relative slenderness λ¯LT {\bar \lambda _{LT}} (6.56) (EN 1993-1-1:2005), which depends on M_cr (elastic critical moment for lateral torsional buckling based on gross cross-sectional properties).

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.

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) λ¯LT=WyfyMcr=Wyfyαcr,MMy,Ed=1αcr,MUk,M {\bar \lambda _{LT}} = \sqrt {{{{W_y}{f_y}} \over {{M_{cr}}}}} = \sqrt {{{{W_y}{f_y}} \over {{\alpha _{cr,M}}{M_{y,Ed}}}}} = \sqrt {{1 \over {{\alpha _{cr,M}}{U_{k,M}}}}} 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: (5) Uk,M=My,EdMy,Rk≤1.0 {U_{k,M}} = {{{M_{y,Ed}}} \over {{M_{y,Rk}}}} \le 1.0

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: (6) Ub,M=γM1Uk,MχLT≤1.0 {U_{b,M}} = {{{\gamma _{M1}}{U_{k,M}}} \over {{\chi _{LT}}}} \le 1.0

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.

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. 6.3.4 (1) [EN 1993-1-1:2005]),

Under the above standard assumptions, the structural stability condition (6.63) (EN 1993-1-1:2005) takes the form: (7) χopαult,kγM1≥1.0 {{{\chi _{op}}{\alpha _{ult,k}}} \over {{\gamma _{M1}}}} \ge 1.0 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 λ¯op {\bar \lambda _{op}} (6.64) (EN 1993-1-1:2005) corresponding to lateral and lateral torsional buckling at (8) λ¯op=αult,kαcr,op {\bar \lambda _{op}} = \sqrt {{{{\alpha _{ult,k}}} \over {{\alpha _{cr,op}}}}} 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 λ¯op {\bar \lambda _{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 cross-section strength condition: (9) 1αult,k=NEdNRk+My,EdMy,Rk {1 \over {{\alpha _{ult,k}}}} = {{{N_{Ed}}} \over {{N_{Rk}}}} + {{{M_{y,Ed}}} \over {{M_{y,Rk}}}} and then the stability condition (6.65) (EN 1993-1-1:2005) assumes the form: (10) NEdNRk/γM1+My,EdMy,Rk/γM1≤χop {{{N_{Ed}}} \over {{N_{Rk}}/{\gamma _{M1}}}} + {{{M_{y,Ed}}} \over {{M_{y,Rk}}/{\gamma _{M1}}}} \le {\chi _{op}}

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) λ¯op=αult,kαcr,op=1αcr,NMUk,NM {\bar \lambda _{op}} = \sqrt {{{{\alpha _{ult,k}}} \over {{\alpha _{cr,op}}}}} = \sqrt {{1 \over {{\alpha _{cr,NM}}{U_{k,NM}}}}} 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) Uk,NM=NEdNRk+My,EdMy,Rk≤1.0 {U_{k,NM}} = {{{N_{Ed}}} \over {{N_{Rk}}}} + {{{M_{y,Ed}}} \over {{M_{y,Rk}}}} \le 1.0

Resistances N_Rk and M_Rk are determined according to the cross-section class, taking into account the plastic resistance of cross sections of class 1, 2 or effective resistance of cross sections of class 4 (Boissonnade et al., 2006; Sedlacek & Naumes, 2009).

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) Uk,NM,el=σN+σMyfy≤1.0 {U_{k,NM,el}} = {{{\sigma _N} + {\sigma _{My}}} \over {{f_y}}} \le 1.0 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: (14) Ub,NM,1=γM1Uk,NMmin(χNM;χLT,NM)≤1.0 {U_{b,NM,1}} = {{{\gamma _{M1}}{U_{k,NM}}} \over {{min}({\chi _{NM}};{\chi _{LT,NM}})}} \le 1.0 or (15) Ub,NM,2=γM1Uk,NMχNMUk,NUk,NM+γM1Uk,NMχLT,NMUk,MUk,NM==Ub,NM,χUk,NUk,NM+Ub,NM,χLTUk,MUk,NM≤1,0 \matrix{{{U_{b,NM,2}} = {{{\gamma _{M1}}{U_{k,NM}}} \over {{\chi _{NM}}}}{{{U_{k,N}}} \over {{U_{k,NM}}}} + {{{\gamma _{M1}}{U_{k,NM}}} \over {{\chi _{LT,NM}}}}{{{U_{k,M}}} \over {{U_{k,NM}}}} =} \cr {= {U_{b,NM,\chi}}{{{U_{k,N}}} \over {{U_{k,NM}}}} + {U_{b,NM,\chi LT}}{{{U_{k,M}}} \over {{U_{k,NM}}}} \le 1,0} \cr} where χ_NM and χ_LT,NM are the buckling and lateral torsional buckling reduction factors, respectively, determined for the proper buckling curve, at cross-section strength utilisation U_k,NM and α_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 U_b,NM values if the cross section’s buckling curve and lateral torsional curve are identical (χ_NM=χ_LT,NM), and the level of utilisation of the cross-section capacity is determined consistently from the following equations: U_k,N Eq. (2), U_k,M Eq. (5), U_k,NM Eq. (12). In the other cases, Eq. (14) yields higher U_b,NM values. Of course, if the element is protected against strong or weak axis flexural buckling, torsional buckling, flexural–torsional 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).

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

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: ‘first-order 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. (12) and U_b,NM Eq. (15) coherently reduce their values to U_k,N Eq. (2), U_b,N Eq. (3) or U_k,M Eq. (5), and U_b,M Eq. (6), respectively. Because of assumption 5, such a coherent transition would not be possible in the standard’s general method.

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: 1)

Determine the cross section’s strength utilisation: U_k,N Eq. (2), U_k,M Eq. (5), U_k,NM Eq. (12). In the light of the formulas proposed above, the most stressed cross section is representative of the member’s stability.

In the graphs (Figs 1–6), the critical load amplifier (α_cr) values are marked on the horizonal axis while the member strength utilisation (U_k Eqs (2, 5, 12)) values determined for the characteristic strength of the cross section are marked on the vertical axis.

The graphs shown in Figs 1–6 were drawn for γ_M1=1.0. In the case of other values of coefficient γ_M1 read off the graph, one should multiply the value of U_b Eqs (3, 6, 14, 15) by γ_M1 and check if it is lower than 1.

Fig. 1 shows the structure resistance utilisation curves U_b, respectively, for the buckling curves a₀, 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: (16) αlim=1Ub {\alpha _{lim}} = {1 \over {{U_b}}}

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) αcr≥25Uk {\alpha _{cr}} \ge {{25} \over {{U_k}}}

A graphical interpretation of Eq. (17) is shown in Fig. 1 where the buckling curves a₀, 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 λ¯≤0.2 \bar \lambda \le 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).

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.

Figure 7

Data:

Steel: S235, f_y=235 MPa, E=210 GPa

Height of the frame: h_c=4.0 m

Span of the frame: L_b=4.0 m

Compression force in the column: N_Ed=1700 kN

Column and girder cross section: HEA 300, h=290 mm, b_f=300 mm, A=112.5 cm², I_y=18.260 cm⁴

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

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.