On modelling of the buckling resistance of welded I-section columns

This paper analyzes the influence of geometrical and material imperfections on the buckling resistance of welded I-section columns subjected to axial compression through numerical and analytical models. The paper is divided into two parts. The first part recalls a FEM parametric study of members under compression taking into account different slenderness ratios, as well as different amplitudes of initial crookedness and different values of postwelding residual stresses. The formulation of analytical approach is the main issue of the second part of the paper. Analytical formulation of the buckling resistance is based on a statistical hypothesis of the minima value approach, called the Marchant-Rankine'sMurzewski approach (M-R-M). Calibration of imperfection factors included in the analytical formulation is made using the results of FEM simulations performed in the first stage of research investigations.


Introduction
Recent design strategies are based on advanced analysis applying the stiffness degradation concept to account for the combined effects of plasticity and imperfections on the nonlinear behaviour of structures. Many standards, including Eurocode 3 [1,2], permit the use of non-linear finite element method (FEM) simulations. Geometrically nonlinear refined plastic hinge analysis of GMNIA type (geometrically and materially nonlinear analysis with imperfections) is the method accurate enough in the resistance assessment for engineering practice.
The main aim of the paper is to assess the influence of geometrical and material imperfections on the buckling resistance of welded I-section columns subjected to axial compression. To achieve this goal the authors divided the paper into two parts. In the first part the results of a FEM parametric study (with the use of GMNIA) of compressed members were recalled [3]. The presented outcomes form the basis for the analytical formulation of the buckling resistance of welded I-section columns which is the subject of the second part of the paper. The analytical approach is based on a statistical hypothesis of the minima value approach [4,5], called the Marchant-Rankine's-Murzewski approach (M-R-M). Calibration of the imperfection factors contained in the analytical formulation is performed with the use of the results of FEM simulations recalled in the first stage of research investigations.

Effect of I-section postwelding residual stresses on the equivalent stress-strain model
Let us consider a perfectly straight column made of steel grade S355 with an elastic -plastic model of parent steel material, and with the same stress strain characteristics for tension and compression (Fig. 1a). A low value of the hardening modulus of Ehar=E/1000 is adopted. The initial self-equilibrated postwelding stresses distribution is approximated by adopting a standard residual stress block postulated in [3] (Fig. 1b). The buckling direction is about z-z axis. Table 1 presents the parameters αf tf and αw tw (the dimensions of respectively flange and web parts under tension) for the discrete values of ψten equal to 1.0 for the common steel grade S355 and ψcom from the range of (0.1, 0.5) with the interval of 0.1.   When an I-section column is subjected to axial compression, firstly the column behaves elastically so that the first piece of multi-linear equivalent σeffεeff relationship is σeff=Eεeff. The first yielding of compression zones starts when NEd=(1-ψcom)Npl,Rk that corresponds to the reduced yield stress σy,red=(1-ψcom)fy. The column tangent stiffness is then suddenly degraded from its elastic value of EIz to its tangent value of ETIz.
The stress-strain relationship can be therefore formulated deterministically as a minimum of three variables being the functions of ε: σ = min (E ε, σH,eff + ET ε, σy,red + Ehar ε) (1) in which σH,eff is the value of initial postwelding stress (see Fig. 2a) and the initial stress value of rigid-plastic-strain-hardening part of the parent steel stress-strain relationship is expressed as: The results of finite element simulations of the stub test representing the equivalent stress-strain model (σeff -εeff model) for a considered steel grade of S355 are given in Fig.  2a, together with the estimates of ET and σH,eff values (σH,eff,0.1 is for ψcom = 0.1, σH,eff,0.2 is for ψcom = 0.2, etc.). The parameters of the equivalent stress-strain characteristic are given in Table 2.
It can be seen that the intersection of the Euler's curve for the residual stress free perfectly straight column and that of the perfectly straight column affected by welding residual stresses travels towards larger values of the slenderness ratio when ψcom increases. The slenderness ratio corresponding to the intersection point is greater than unity and calculated as follows: The slenderness given by Eqn. (5) is tabulated in Table 3, where ξH,eff, ξE,eff from Table 2.
While the members are affected by residual stresses in a lower extent, ξH,eff → 1 and ξE,eff → 0, so the modified slenderness The real members are affected by material and geometric imperfections. The global effect of imperfections and their interaction during the progressive yielding under applied axial loads on the buckling resistance may be modeled using the semi-probabilistic minima value approach in which the variables Ncr, Ncr,eff and Ncr,har of Eqn. (3) are considered independent random variables following a Weibull distribution. Such a statistical hypothesis of the minima value approach, resulting from the randomization of variables used in Eqn. (3) is called the Merchant-Rankine's-Murzewski approach (M-R-M approach [5]). In the case study presented herein, such approach leads to a three dimensional Weibull minima distribution of the buckling resistance. As a result, the following equation is obtained for the reduction factor χz of an imperfect column subjected to residual stresses and initial bow deformations: in which the n is the imperfection factor being an inversion of the equivalent Weibull coefficient of variation u, i.e. n=1/u. The u parameter plays the same role as the imperfection factor α in the Eurocode 3 [1]. While the values of u and α are higher, the resistance reduction factor χz is lower and vice versa.
The results of finite element simulations are presented in [3] for imperfect columns subjected to the residual stress block presented in the previous sections (for five different   . Similar simulations of residual stress influence on the flexural buckling of welded I-girders were presented also in [6,7]. The FEM calculations of compressed elements with the use of concept of equivalent geometric imperfections were described in [8][9][10][11]. the convergence of the NMinimize method was very good allowing for finding the global minimum of the goal function inside the domain. The results are presented in Table 4. The results presented in Table 4 prove that the imperfection parameter nopt decreases with the increase of both imperfection factors, 0 e and ψcom (or increase with the equivalent coefficient of variation uopt = 1/nopt). Applying the results given in Table 4    The authors presented the issue of buckling resistance modelling of welded I-section columns in a complementary way. The results of FEM calculations were recalled herein and were used to develop the buckling curve analytical formulation. The application of statistical hypothesis of the minima value approach (M-R-M approach) gave a good fit of the analytical solutions and the FEM calculations outcomes. The results of linear and nonlinear approximations were described in detail and summarized in particular subsections. One could observe a higher imperfection factor 0 e triggers a better fit of the analytical solutions and FEM results. The adoption of the very small value of the dimensionless geometric amplitude 0 e equal to 1/10000 leads to the analysis of a practically perfect element. In this case the compliance of results was not satisfactory. The perfect match was observed in case of the value 0 e equal to 1/750. The results presented in the paper lead to the conclusion that the calculations should be also executed for other types of steel, especially high strength steels. Presently, the authors perform simulations of elements made of steel grade S 690 to check the applicability of M-R-M approach not only in case of common used steels.