The stability solution with buckling length is based on the calculation of the critical load action F_cr at which the system (without imperfections) loses its stability (Table 1). Axial forces in beams intended for buckling with a displacement of joints and their respective buckling lengths correspond to the critical load action.

The design load-carrying capacity F_max can be determined for steel S235 according to Eurocode 3 [2]. The standard provides the calculation method of the buckling coefficient using buckling lengths L_cr1 and L_cr2. Eurocode 3 [2] further gives procedures for the computation of the design load-carrying capacity and for the assessment of the steel plane frame at the ultimate limit state. Nevertheless, it permits simplifications, which lead to larger or smaller inaccuracies in the dimensioning of beams. It is a paradox of buckling lengths that extremely high buckling lengths are obtained for columns with small axial force (see L_cr2=∞ of the right column for δ=0).

Eurocode 3 [2] makes it possible to compute the design load-carrying capacity with the use of the geometrically nonlinear solution with consideration to the influence of all imperfections. The imperfections include system imperfections (out-of-plumb) and bow (out-of-straightness) imperfections of the columns under compression (Fig. 2).

Fig. (2). The combination of imperfections.

The load-carrying capacity was determined using the Euler method in combination with the Newton-Raphson method. The bow imperfection was introduced in the shape of a half-wave of the sine function with a so-called equivalent amplitude e₀ = 9.33 mm [2]. The out-of-plumb imperfections were introduced as 1:200 of the height of the frame [2, 25]. The out-of-plumb or out-of-vertical imperfections can be applied in all horizontal directions but were considered in only one direction [2]. The frame can be solved using eight imperfections of frames (Fig. 2). Eight variants of imperfections yield different values of the design load-carrying capacity, the maximum and minimum values being interesting (Table 2).

For more complicated systems with more beams, it is not possible to study all combinations of bow and out-of-plumb imperfections, and differences between the minimum and maximum load-carrying capacities can be more significant.

According to EN1990 [14], one of the possibilities for determining the design load-carrying capacity is the application of the statistical solution with consideration to the random influence of all imperfections. Stochastic models of the geometrical imperfections of the frame in Fig. (1) were published in [26, 27]. It was assumed that the initial bow curvature of the column in the shape of a one-half wave of the sine function with e₀ would have a Gauss probability density function with zero mean value and standard deviation S_e0=1.5 mm. This probabilistic model is based on the rule 2S_x, i.e, that 95% random realizations lie within the interval ±L/1000=0.003 m. Out-of-plumb imperfections of the columns are introduced with Gauss pdf with zero mean value; the standard deviation was introduced using the method described in [27]. The principle of forming stochastic combinations (random realizations) of initial imperfections is explained in detail in [27] (Fig. 3).

Fig. (3). The random realizations of initial imperfections [27].

The statistical characteristics of imperfections of cross-sectional dimensions were introduced according to the results of experimental research [17, 18]. The yield strength of steel grade S235 was introduced as a random variable with normal probability distribution with mean value m_fy=297.3 MPa and standard deviation S_fy=16.8 MPa [17]. The residual stress was not considered. An ideally elastic-plastic material without hardening was considered. The influence of deviations of physical-mechanical characteristics was taken into account by the variability of Young’s modulus E, the mean value was considered as m_E=210 GPa and standard deviation S_E =10.5 GPa [28]. The residual stress was not considered.

The geometrically and materially nonlinear solution was chosen for the computational model [17]. Statistical analysis was performed for two hundred thousand runs of the Monte Carlo method. F due to which the yield strength is reached in the most stressed point of the web of the left or the right column was calculated in each run of the Monte Carlo method. The random variability of the load-carrying capacity is due to the random variability of initial imperfections. For design reliability index β_d = 3.8, the design load-carrying capacity can be computed as 0.1percentile [14]. Reliability index β has a target value of 3.8 provided that we consider the ultimate limit state for common design situations within the reference period of 50 years [29]. The design reliability index β_d = 3.8 can be used for medium consequences of loss of human life, economic, social or environmental consequences. The one hundredth lowest load-carrying capacity from the statistical set of sampling (runs) of load-carrying capacities sorted in ascending order can be considered as the estimate of the design value. The design load-carrying capacity computed as 0.1 percentile [14] is an alternative to the approaches of the standard [2].

A graphical summarization of the design values of the load-carrying capacity is shown in Fig. (4) and Table 3. It is evident that the safest design having the lowest load-carrying capacity is reached using the geometrically nonlinear solution given in Chapter 2.2. This design also contains certain subjective uncertainty, which is introduced by the engineer when choosing the shape of the imperfection from Fig. (2). Selection of the shape of the imperfection gives a certain freedom in design in which the engineer may reflect the effort for the lightest structure (economic aspect) or the effort for a robust structure (safety aspect). In the case of complex steel structures with an equivalent number of columns and beams, the differences between the load carrying capacities due to the subjective selection of the shape of the imperfection may be even more significant, thus, design using the geometrically nonlinear solution must be approached with the utmost caution and responsibility.

Fig. (4). Comparison of three approaches of the computation of load-carrying capacity.

The stochastic solution leads to approximately similar results as the stability solution with buckling length Fig. (4). The stability solution with buckling length is well established in engineering practice and in the case of the presented analysis is shown to be in good agreement with the design reliability conditions of the Standard EN1990 [14]. In the stochastic solution, refinement of the shape of the initial curvature of the column can be discussed, for e.g. using random fields [11]. Modelling using random fields can refine the load-carrying capacity if the non-dimensional slenderness of the columns is close to one. However, knowledge of the correlation length is necessary. Moreover, greater influence can be expected from the out-of-plumb imperfections [27]. One way to evaluate the influence of initial imperfections is to apply sensitivity analysis methods [30, 31].