تجزیه و تحلیل حساسیت از مشکلات ثبات فریم های فولادی هواپیما

The objective of the paper is to analyse the influence of initial imperfections on the load-carrying capacity of a single storey steel plane frame comprised of two columns loaded in compression. The influence of the variance of initial imperfections on the variance of the load-carrying capacity was calculated by means of Sobol’ sensitivity analysis. Monte Carlo based procedures were used for computing full sets of first order and second order sensitivity indices of the model. The geometrical nonlinear finite element solution, which provides numerical results per run, was employed. The mutual dependence of sensitivity indices and column non-dimensional slenderness is analysed. The derivation of the statistical characteristics of system imperfections of the initial inclination of columns is described in the introduction of the present work. Material and geometrical characteristics of hot-rolled IPE members were considered to be random quantities with histograms obtained from experiments. The Sobol sensitivity analysis is used to identify the crucial input random imperfections and their higher order interaction effects.

The reliability analysis of structural systems in the limit state methods is aimed at an assessment of safety and serviceability. Steel structures are composed of thin members and hence the problem of stability can prove to be one of the most important constituents of the safety. The stability loss is caused by the change of geometry of steel structures or structural components. Thus, to assess the structural stability, the equilibrium equations must be described under deformed geometry. This implies that the consideration of geometrical nonlinearity is inevitable for ultimate limit state analysis. The behaviour of compressed members in the loading process leading to the ultimate limit state is influenced by initial imperfections generally divided into three categories: geometrical imperfections, material imperfections and structural imperfections [1] and [2]. The limit state theory for individual struts has been worked out and corroborated with experiments. However, isolated struts occur rarely in real structures. Generally, each structure is a system of members, which mutually influence each other by their behaviour. This interaction is most significant in structures with rigid joints (frame systems). On the contrary, this mutual interaction is small in truss structures and is generally neglected. Within the division of structures into members and frame systems, we can accept the further division of initial imperfections into local (member) and global (frame systems) ones [1]. The local imperfections include: initial straightness deviation of member axis, deviation from the theoretical layout of the hot-rolled cross section, load eccentricity, dispersion of the mechanical material properties, residual stress, etc. Global imperfections include initial inclination of any column in systems and imperfections in the realization of joints, connections, anchorage and other structural details, which are apparent in comparison with the theoretical assumptions introduced in the solution of idealized system. One of the challenging issues in modern civil engineering analysis is the typically large number of random quantities defining the input and system parameters [3]. Most building structures are atypical and hence a higher number of measurements are conceivable from the statistical point of view just for local imperfections of mass produced members. The basic indicators of production quality include the yield strength, tensile strength, ductility and geometrical characteristics of hot-rolled IPE profiles which have been under long term statistical evaluation within the framework of non-commercially aimed research programmes, see e.g. [4] and [6]. Relatively sufficient statistical information on material and geometrical characteristics of mass produced members of steel structures is available in comparison to other building structures. The frame depicted in Fig. 1 represents a typical stability problem of a system comprised of more members. The fundamental question in terms of safety of a structure is how significant is the effect of inevitable initial imperfections on the load-carrying capacity. An approach to make such problems tractable is to identify the most important sources of uncertainty and to focus attention primarily on those uncertainties of the input space. Such a method using the Sobol decomposition [7], global sensitivity analysis method, is proposed here. The Sobol decomposition is used to decompose the variance of the load-carrying capacity into contributions of the individual input variables (initial imperfections). Full-size image (11 K) Fig. 1. IPE-section symmetric portal frame. Figure options The Sobol sensitivity analysis quantifies the relative importance of input imperfections in determining the value of load-carrying capacity. The crucial imperfections, which should be paid greater attention both in the modeling phase and in the interpretation of model results, are identified using sensitivity analysis. One of the advantages of the Sobol sensitivity analysis is that it enables the identification of interaction effects between input quantities on the monitored output. With the development of new concepts of the reliability analysis, these procedures can contribute to a qualitative improvement of the reliability analysis of structures.

The sensitivity analysis was used to detect and rank those imperfections that need to be measured with greater accuracy in order to reduce the variance of load-carrying capacity of the steel plane frame. System imperfections e1 and e2 have a dominant effect on the load-carrying capacity of frequently used column lengths with non-dimensional slenderness close to one. The sensitivity analysis results have shown that interactions between imperfections e1 and e2 have approximately twice the effect on the variance of the load-carrying capacity than when considered individually, see Fig. 7. The main advantage of sensitivity analysis is that it provides quantified evaluation of the influence of individual imperfections and their interaction on the ultimate limit state, and results may also be used for probabilistic assessments of reliability and calibration methods and approaches, respectively methods of the verification of tolerance limits Δa, Δb and other imperfections. The sensitivity analysis results illustrate that the statistical characteristic of system imperfections e1 and e2 should be determined with increased accuracy; however, it is difficult or practically impossible under hard service conditions. The second greatest interaction effect of the second order S 6,18=0.08 was obtained between the yield strength of the left and right columns for View the MathML sourceλ¯=0, see Fig. 8. However, the value of this interaction effect is, in comparison with the main effect S6 or S18, approximately a quarter S6,18≈0.25⋅S6=0.25⋅S18 and hence of little significance. Let us note that S6,18 is approximately equal to the main effect of flange thickness of the left or right column S6,18≈S4=S16. The most important characteristics, checked on mass produced steel IPE members, are the yield strength and geometrical characteristics of the cross-section. For the practice, we can recommend the thorough measurement and check of yield strength and flange thickness, the variances of which have an influence on the variances of the load-carrying capacity of compressed columns and may also be of significance under other types of strain and loading. Let us note that the mean value of flange thickness should be equal to the nominal value. In the case of the yield strength, the member reliability may be increased by decreasing the variance or by increasing the mean value. The cross-section height of columns, flange width of columns and cross-beam characteristics were identified as the factors which, if left free to vary over their range of uncertainty, make no significant contribution to the variance of the load-carrying capacity. The identified imperfections can be fixed at any given value within their range of variation without affecting the load-carrying capacity. This analysis can be performed on groups of factors, especially for large models, to identify non-influential subset of imperfections. The load-carrying capacity of very slender members is very sensitive to Young’s modulus, the variance of which cannot be influenced in production. The second most significant variable is the flange thickness the variance of which can be favourably influenced in production. No significant higher order interaction effects were found for the hereby analysed problem for high slenderness values. The initial imperfections are among the basic and frequently most important input data of theoretical studies and hence they should be paid great attention. In this regard, it is generally necessary to point out the publications dedicated to research problems in the field of stability problems [25]. For the load-carrying capacity problems, it is also necessary to study the influence of interaction amongst imperfections formally identical to the buckling modes which cause the instability [26], [27] and [28], the influence of load action eccentricity [29] and that of joint stiffness [30]. Variance-based methods can be beneficial when studying the first and the higher order interaction effects of imperfections on limit states and reliability of structures.