تجزیه و تحلیل حساسیت از فریم های فولادی هواپیما با نقص اولیه

The article presents the sensitivity and statistical analyses of the load-carrying capacity of a steel portal frame. It elaborates a typical stability problem of a system comprising two single-storey columns loaded in compression. The elements of this system mutually influence each other, and this fact, in conjunction with the random imperfections, influences the load-carrying capacity variance. This mutual interaction is analysed using the Sobol’ sensitivity analysis. The Sobol’ sensitivity analysis is applied to identify the dominant input random imperfections and their higher order interaction effects on the load-carrying capacity. Majority of imperfections were considered according to the results of experimental research. Realizations of initial imperfections were simulated applying the Latin Hypercube Sampling method. The geometrical nonlinear solution providing numerical result per run was employed. The frame was meshed using beam elements. The columns of the plane frame are considered with two variants of boundary conditions. The dependence between mean and design load-carrying capacities and column non-dimensional slenderness is analysed.

Along with the progress of structural design theories and the technological advancement of steelworks production, more and more large-scale and high-rise steel bar structures are implemented in modern structures. The issue of stability of these structures becomes more apparent due to the utilization of more slender members. The frame stability requires that all structural members and connections of the frame have adequate strength to resist the applied loads where static equilibrium is satisfied on the deformed geometry of the structure. In order to determine the load-carrying capacity of an actual structure, it is necessary to take into consideration initial imperfections and to consider the geometrically nonlinear analysis. In general, all imperfections are of random character. The reliability of steel structures depends on the variance of input imperfections which influences the evaluation of limit states of building structures. The attainment of limit states is generally a random phenomenon, which is examined in the field of reliability using probabilistic theories and mathematical computation models. One of the most important characteristics occurring in probabilistic methods of reliability assessment of steel structures is the variance of the load-carrying capacity which is primarily given by the quality of production. Basic indicators of production quality include the yield strength, tensile strength, ductility and geometrical characteristics of cross sections; see, e.g., [1] and [2]. Relatively sufficient statistical information is provided for the material and geometrical characteristics of mass produced hot-rolled members of steel structures in comparison to other building branches. Scarcely measurable imperfections of steel plane frames include the inevitable initial crookedness of bar members (bow imperfections) and out-of-plumb inclinations of the columns (sway imperfections) in the same frame [3] and [4]. Some measurements have been made in connection with testing programmes [5], but very little data is available. The frames depicted in Fig. 1 and Fig. 2 represent a typical stability problem of a structural system consisting of more members. The frames are typical lean-on systems which are characterized by the structural members tied or linked together in such a way that buckling of the column would require adjacent members to buckle with the same lateral displacement. The imperfection interaction effects can have a significant influence on the overall performance of the frames. The steel frame depicted in Fig. 1 has rotation and translation fixed boundary conditions of both column ends. The steel plane frame in Fig. 2 is similar to that in Fig. 1 with the exception that there is no rotation restrain at the column ends. The rotation fixed and rotation free conditions represent the two limits of real anchorage in practice. Let us denote the frame in Fig. 1 as Frame 1 and the frame in Fig. 2 as Frame 2. Full-size image (21 K) Fig. 1. Frame 1, rotation and translation fixed boundary conditions. Figure options Full-size image (20 K) Fig. 2. Frame 2, rotation free and translation fixed boundary conditions. Figure options In the presented paper, the effects of input imperfections on the load-carrying capacities of Frames 1 and 2 are evaluated by means of sensitivity analysis. The lean-on imperfect system (left column leans on right column) requires the utilization of sensitivity analysis which enables the evaluation of the influence of individual imperfections on the load-carrying capacity as well as of their higher order interaction effects. An outline of sensitivity analysis methods with examples of their application in a number of scientific fields is listed, e.g., in [6]. With regard to the random character of initial imperfections, the influence of imperfections on the load-carrying capacity of the frame systems will be studied applying the Sobol’ sensitivity analysis [7], [8] and [9]. One of the advantages of Sobol’ sensitivity analysis is that it enables the identification of interaction effects among input quantities on the monitored output. The effects of the dominant imperfections, which have the greatest influence on the load-carrying capacities, will be described. Design load-carrying capacities evaluated statistically according to EN1990 [10] and according to the partial factor method of EUROCODE 3 [11] will be compared later on in the article. Obtained results will be discussed in connection with the results of Sobol’ sensitivity analysis.

The sensitivity analyses of Frames 1 and 2 provide sensitivity information concerning the load-carrying capacity as influenced by initial imperfections. The paper was aimed at the comparison of the influence of individual initial imperfections on the load-carrying capacity of Frames 1 and 2. Results of the sensitivity analyses of the load-carrying capacities of both frames show that the influence of initial bow imperfections δ1,δ2δ1,δ2, compared to the influence of the initial sway imperfections Θ1,Θ2Θ1,Θ2, is very small; see Fig. 7 and Fig. 8. This conclusion is valid for both the main effects as well as the second-order interaction effects. As the non-dimensional slenderness approaches one, the second-order interaction effects between Θ1Θ1 and Θ2Θ2 become significant and the main effect is of secondary importance. Imperfections Θ1,Θ2Θ1,Θ2 may generate, as a result of their mutual interactions, extreme values of the load-carrying capacity. This is important for the analysis of reliability and economy of structural design. Variability of imperfections Θ1,Θ2Θ1,Θ2 significantly contributes to the output variability, and thus, additional research may be recommended in order to strengthen their knowledge base. However, under heavy service conditions, this is difficult or practically impossible. Higher order interaction effects were obtained for Frame 2 for View the MathML sourceλ¯=0.93. Mainly imperfections Θ1,Θ2Θ1,Θ2 are involved in interactions with other variables; see Fig. 9. The total effect index STiSTi is a summarized sensitivity measure which includes the interaction effects of any order. Imperfections that interact with other imperfections with main effect close to zero are worth noticing. Change in such imperfection does not cause any significant change of the load-carrying capacity, if not accompanied by additional changes of one or more significantly interacting imperfections. For example, the main effects of imperfections δ1,δ2δ1,δ2 of Frame 2 are practically equal to zero but the total effect indices ST7ST7 and ST20ST20 are the second most significant ones; see Fig. 9. Let us note that the total effect index is derived from a notion of Sobol’ which involved the problem of “freezing” the unimportant factors to their midpoint [8]. Sensitivity indices S7=S20≈0S7=S20≈0 represent a necessary but insufficient condition for fixing imperfections δ1,δ2δ1,δ2 of Frame 2. Results depicted in Fig. 9 show that all column imperfections have total effect indices greater than zero, and thus, they cannot be fixed at any value within its range of uncertainty without greater or lesser effect on the value of the variance of the load-carrying capacity. For View the MathML sourceλ¯=0, the first-order sensitivity indices of yield strength S6=S19=0.32S6=S19=0.32 are cardinal; see Fig. 7 and Fig. 8. The interaction effect of the second order S6,19=0.08S6,19=0.08 also exists between the yield strengths of the left and the right columns. The sensitivity indices of flange thickness S4=S17=0.06S4=S17=0.06 are the third most important ones among all. Results of the sensitivity and statistical analyses for View the MathML sourceλ¯=0 are the same for both frames and are practically valid for columns under tension. Young’s modulus S5=S18=0.31S5=S18=0.31 and flange thickness S4=S17=0.15S4=S17=0.15 are the dominant variables for View the MathML sourceλ¯→∞. Higher order interactions are relatively small. For high slenderness, the load-carrying capacity in limit approaches the Euler’s critical force (buckling load), and is thus sensitive to variables preventing buckling. From the point of view of production technology of hot-rolled steel members, we can strive for decrease in the variability of flange thickness, however, the variability of Young’s modulus cannot be significantly influenced in practice. Results of sensitivity analysis of both frames differ, for View the MathML sourceλ¯≈1.0, most significantly; see Fig. 7 and Fig. 8. Sensitivity indices pertinent to flange thickness S4,S17S4,S17 are small for all analysed slenderness. This may even be the second most significant for high slenderness. Let us compare the hereby presented results with the results of sensitivity analyses of the strut published in [13]. In the case of the strut, the load-carrying capacity is not significantly influenced by the higher order interactions between initial imperfections. The evident discrepancies between the mean and design load-carrying capacities are depicted in Fig. 11 and Fig. 12. Discrepancies between the design load-carrying capacities evaluated according to EUROCODE 3 and EN1990 (0.1 percentile) are relatively small. The 0.1 percentile yields greater values within the interval λ∈(0.5,1.0)λ∈(0.5,1.0); this may be due to the fact that residual stresses were neglected. The 0.1 percentile plots of Frames 1 and 2 differ slightly, however, we cannot conclude that discrepancies for other frame types may not be greater. The increase in the values obtained from the 0.1 percentile can be achieved by decreasing the standard deviation of input imperfections. The influences of individual variables (and their interactions) were quantified applying the tools of sensitivity analysis. The sensitivity analysis was used to determine where additional information on imperfections (obtained perhaps from measurement) would be most beneficial in terms of uncertainty reduction in probabilistic model results of the frames. By decreasing the standard deviation of the dominant input imperfections, we can significantly increase the reliability of newly designed structures. In practice, sensitivity analysis provides a basis utilizable in production and in the operating of control activities which can thereby be concentrated on the most important input variables with the greatest effect on the load-carrying capacity. The Sobol’ sensitivity is generally suitable for the analysis of majority of stability problems of steel structures with imperfections. Sampling based methods may be applied for the analysis of the effect of local plated and global bar geometrical imperfections on the ultimate limit state of thin-walled structures [29]. Some researchers [30] introduced multiple local modes into the numerical model to consider possible additional interactions among the global mode and multiple local modes [31]. The Sobol’ sensitivity analysis can quantify the interaction effect amongst imperfections formally identical to the buckling modes which cause instability. The solution should be based on measurements of real imperfections; see, e.g., [32].