Sensitivity Analysis of Steel Structures with Imperfections

Z. Kala1*
1Brno University of Technology, Department of Structural Mechanics, Czech Republic
kala.z@fce.vutbr.cz

Sensitivity analysis methods in application to the analysis of the influence of imperfections of steel structures on their reliability are presented in the paper. The aim of sensitivity analysis is the assessment of the influence of input random variables on the variability of output random variables. In regard to the limit states of structures, the load carrying capacity (or deflection at selected section) is frequently considered as the output variable. In mass produced products (e.g. steel hot–rolled profiles, etc.) it is possible to according to the relative sensitivity determine those variables that the monitored output (e.g. load carrying capacity) is especially sensitive to. It is then possible to concentrate control activity on these variables with aim at securing sufficient stability in statistical parameters or at lowering their random variability.
Sensitivity analysis can be generally divided into two groups:
a) Deterministic sensitivity analysis (design sensitivity) is relatively well known, frequently used during structural design. The most frequent type of sensitivity analysis is a parametric study utilizing a computational model, which accompanies the design process [1]. This study may be simply realized as a sequence of calculations with gradually varying values of a certain input parameter Xi in each calculation step j(j = 1, 2,…,K) in a certain real range. The influence of parameter Xi on the response Yj can be observed through the comparison of the calculative results Yj (the set of the structure response). Certain more advanced computational programmes include such an option and everything is performed automatically. Quantified data on sensitivity is however not obtained.
 b) Stochastic sensitivity analysis offers more complex and quantified information on the influence of parameters. It is however necessary to utilize more sophisticated numerical methods. Different procedures of stochastic sensitivity analysis are frequently implemented in reliability based software. Input parameters are considered as random variables, described by their probability distributions with given statistical parameters: mean value, standard deviation, respectively by their skewness and kurtosis. The procedure for the determination of sensitivity is to a certain degree similar to the deterministic sensitivity analysis. The parameter is also gradually changed, however, within the framework of the applied simulation technique. The influence of this change on the output variable is observed. The change in input variable is performed with respect to its distribution (the frequency of its occurrence), i.e., other valuable information, which is neglected in the deterministic sensitivity analysis, is utilized. The sensitivity analysis is a suitable supplement of every level of reliability analysis. It is usually performed in connection to utilized reliability analysis.
It would be very valuable to perform the stochastic sensitivity analysis utilizing experimental results. This is however practically impossible especially due to economic restraints. Only certain characteristics of beams can be reliably measured by non–destructive methods. It is better to use numerical sensitivity analysis. Particularly due to large diversity of initial imperfections and possibilities of their mutual combination it is more suitable to utilise the so–called numerical sensitivity analysis. During which it is possible to utilize valuable results of long term research e.g. [4, 5].
The subject of the article is the numerical stochastic sensitivity analysis of the influence of imperfections on the load carrying capacity of steel structures. Imperfections are generally random variables, only very little statistical information of these variables from measurements on real structures is available. Typical examples include system imperfections of plane frames; see Fig. 1 and Fig. 2. The loading type of steel frames in Fig. 1 predicts large influence of system imperfections on the load carrying capacity. System imperfections are depicted by a dashed line in Fig. 1. The influence of boundary conditions on the sensitivity of the load carrying capacity to the imperfection is studied. Boundary conditions were considered as hinged and clamped end in the numerical study. These represent limit variants of reality. Additionally considered imperfections include strut imperfections of columns, geometric deviations of profile dimensions, material and geometric characteristics.
The output load carrying capacity was evaluated by means of the geometric non–linear solution of the finite element method. Input random imperfections were considered according to results of experimental research [2] and [3]. The random load carrying capacity was evaluated utilizing the Latin Hypercube Sampling method, which is an improved version of the Monte Carlo method. 800 simulation runs were utilized. The sensitivity analysis was performed in the form of Spearman’s rank–order correlation coefficient. A frame of height 6m and span 6m with cross beam IPE 360 and columns IPE 270 was solved.

Fig. 1: System imperfections of steel plane frames

The influence of input variables on the load carrying capacity was evaluated by means of sensitivity analysis. Sensitivity analyses results illustrate varying sensitivity of the load carrying capacity to the system imperfections of both frames depicted in Fig. 1. The variability of the load carrying capacity is more sensitive to the variability of system imperfections in the left frame. Conversely, the dominant variables of the right frame are the yield strength and residual stress. Relatively high correlation between the load carrying capacity and the flange thickness of the columns was observed in practically all of the analysed problems.
Statistical characteristics of the dominant input variables should be determined and checked with increased care [2]. Input random variables can generally be divided into two basic groups – imperfections, whose statistical characteristics can be positively influenced in production (yield strength, residual stress, geometrical characteristics), and imperfections, which are not sensitive to changes in the technology of production (e.g. variability of Young’s modulus E) [2]. The first group of variables may be further subdivided into two subgroups: (i) variables for which both the mean value and the standard deviation can be changed by improving the quality of production, e.g. the yield strength; (ii) variables whose mean value cannot be significantly changed because it should approximately correspond to the nominal value (geometric characteristics of profile dimensions) [2].

Acknowledgement:
The present paper was elaborated under the project GAČR 103/07/1067, junior research project KJ201720602 of Czech Academy of Science, Research Centre Project CIDEAS 1M68407700001(1M0579) and project GAČR 103/05/2059.

References

[1] Balut N, Moldovan A: Sensitivity of steel structures to different categories of imperfections, In: Proceedings of SDSS´99, Elsevier Science 1999.

[2] Kala Z: Sensitivity Analysis of the Stability Problems of Thin–Walled Structures. Journal of Constructional Steel Research 61(3), 415–422, (2005) ISSN 0143–974X. 

[3] Kala Z: Auxiliary Sensitivity Analysis Applied to Stability Problems of Steel Frame Structures, Journal of Engineering Mechanic 12(1), Brno: VUT, 2005, ISSN 1210 2717.

[4] Kala Z, Kala J, Škaloud M, Teplý B: The Stress State in the Crack–Prone Areas of „Breathing“ Thin Walled Girders – a Sensitivity Analysis, In: Proc. of the Fourth International Conference on Coupled Instabilities in Metal Structures, Rome (Italy) 2004, pp.265–276.

[5] Kala Z, Kala J, Škaloud M, Teplý B: Sensitivity Analysis of the Initial Imperfections Influence on the Stress State in the Crack Prone Areas of Breathing Webs, In: Proc. of 4th European Conference on Steel and Composite Structures, Maastricht (Netherlands), 2005, pp.1.7–47 to 1.7–54, ISBN 3–86130–812–6.

[6] Kala Z, Kala J, Škaloud M, Teplý B: Sensitivity Analysis of the State of Stress in the Crack–Prone Areas of Breathing Webs, In: Proc. of Int. Conf. Lightweight Structures in Civil Engineering, Warsaw, 2005, pp. 219–228, ISBN 83–908867–9–0.