Farklı zemin parametrelerinin istatistiksel özellikleri ve regrasyon analizi

Abstract

Geoteknik mühendisliğinde karşılaşılan belirsizlik lerden dolayı semin parametrel erindeki değişkenliğin sonucu olarak deterministlk yöntemler yetersiz kalmakta ve bu nedenle olasılık yöntemlerinin kullanılması daha gerçekçi olmaktadır. Bu çalışmada, istatistiksel metodlar kullanılarak istanbul Teknik üniversitesi Zemin Mekaniği Laboratu- varında günümüze kadar yapılmış olan konsolidasyon deney sonuçları değerlendirilmiştir. Herbir değişkenin istatistiksel parametreleri oluşturulan tüm alt gruplar için hesaplanarak bilinen dağılım şekillerine uygunluk ları, Kolmogorov-Smirnov uygunluk testi kullanılarak belirlenmiş ve istatistiksel anlamlılıkları incelenmiştir. Sonuç olarak semin özelliklerinin çoğunun lognormal, normal, basılarının ise beta ve gama dağılımlarına uydu ğu gözlenmiştir. Sıkışma indisinin, daha kolay bir şekilde elde edilen boşluk oranı, likit limit ve su muhtevası ile korelasyonu araştırılmış ve çeşitli regresyon eşitlikleri büyük korelasyonlarla bulunmuştur. Doğrusal regresyon analisinin yanında y = ax J şeklinde doğrusal olmayan bası dönüşümlerin yüksek korelasyonlar gösterdiği görülmüştür.

In this study, statistical analyses are conducted on 800 laboratory test results is obtained for various soils from different parts of Turkey. In order to investigate the variations of index and engineering properties of different soils, statistical parameters have been calculated and histograms have been plotted for each parameters. Statistical distribution functions of each parameters have been determined and their statistical significance have been tested by Kolmogorov-Smi rnov Goodness of fit method. It is observed that most of the variations can be modelled in terms of either the lognormal or normal distribution and some with beta and gamma distributions. The regression analysis method of statistic is examined for compression index to express in terms of other readily obtained index properties. It is observed that the nonlineer simple regression analyses have higher coefficient of correlations than simple lineer regression analyses. Statistical and probabilistic methods are being used at different levels in the different areas of civil engineering. Eventhough the classical statistical and probabilistic theories are well established, their application to civil engineering problems are recent. Design decisions in geotechnical engineering are usually based on uncertain experimental data, and the results obtained by use of deterministic analyses may be misleading since they promote a false sense of accuracy. The uncertainties encountered in soil engineering include (a) uncertainty in the soil parameters due to variations in actual material properties, specimen geometry, and testing procedures, as well as limited spatial sampling, (b) uncertainties in the magnitude and distributions of imposed loads, and (c) uncertainty in the mathematical models used in modelling random nature of natural processes involved in deposition of soil and systematic or physical changes which are often identified in conventional exploration techniques. Furthermore, because of the complexity of soils, limita tions in the laboratory measurement of properties relevant to a given design problem. The uncertainties associated with the imposed loads arise mainly from the randomness of the value assigned to dead and live loads; in fact, the live loads vary both spatially and temporally. Finally, most of the analytical techniques used in the field of soil engineering are based on various assumptions that introduce some uncertainty in the design. Due to random deviations from homogeneity within a soil formation deterministic procedures are not entirely satisfactory for some soil parameters, that are needed for design purposes, and hence, the problem should be approached on a probabilistic basic. This has been well recognized virtually by all soil engineers, and the use of engineering judgement has served to subjectively take this situation into account. However, the application of probability theory to a given problem should not be used to supplement proper soils investigation, laboratory tests, and experience; rather, it should be used to provide a framework within which a practicing engineer can accumulate, organize and evaluate past experience. The definition of soils on a probabilistic basics is not an easy task due to the fact that a correct formulation of heterogenity in soils will introduce an endless list of cases. The complex behaviour of soils requires information on many different parameters in order to study performance under different conditions. The application of probabilistic methods to geotechnical engineering has been concern of several investigators during the several decade. Distributions and statistical analysis of parameters describing natural soils are given by Peck (1940), Thornburn and Larsen (1959), Hammit (1966), Ungar (1967), Lumb (1966, 1970), Borus and Rev (1970), Schultze (1971), Fredlund and Dahlman (1971), Holtz and Krizek (1971), Ladd, Moh and Gifford (1971), Paal (1974), Azzouz (1974), Corotis et al. (1975), Rethatti (1983) and Lav (1986). Those investigators reported that most of the soil properties follow normal or lognormal distribution. For example, according to Lumb (1966, 1970). Liquid limit, plastic limit, plasticity index, liquidity index, cohesion, compression index, angle of internal friction, void ratio and soil strength follow normal probabilty distributions, while the coefficient consolidation appears lognormal. vli The determination of soil properties needed for design purposes often requires long, expensive, and cumbersome tests; therefore, different researches have used statistics in the form of regression analyses of one soil property with respect to others. For example, statistical correlations and regression analyses of compression index with more readily determined properties, liquid limit, void ratio and water content are pro posed by Skempton, 1944; Peck and Reed, 1954; Nishida, 1956; Hough, 1957; Cossolino, 1961; Sowers, 1970; Azzouz, 1974; Lav, 1986. Some typical uses of statical analysis in soils engineering include; 1) Data evaluation and reduction 2) Search for inconsistent and /or redundant data. 3) Tests for significance, both statistical and from the soil mechanics point of view. 4> Prediction of certain engineering properties from classification of index data. 5) Aid in establishing design parameters based on a rational definition of safety factor. Randomness in a parameter means that more than one outcome is possible; in other words the actual outcome is (to some degree) unpredictable. The possible outcomes are usually a range of measured or observed values; moreover, within this range certain values may occur more frequently than others. Mathematical representation of a random variable is thus a primary task is any probabilistic formulation. In a probabilistic design procedure, a prediction of the future is made using information from the past, including experiences and judgement whenever possible. Thus, it is necessary to collect all relevant information from the past for this purpose. A typical flow chart of the steps involved is shown in Fig. 1. The information collected will constitute the sample space. The randomness characteristics can be described graphically in the form of a histogram or frequency diagram. For a more general representation of the randomness, the frequency diagram can be fitted to some theoretical probability density functions fx(x). By integrating the probability density function thus obtained, a probability distribution function, Fx(x) can be obtained. However, to describe the probability density function uniquely, some parameters of the distribution need to be estimated. The estimation of these parameters, which are called statistics, is itself a major part of the uncertainty analysis. When the randomness in the load and the resistance parameters are modelled in terms of distributions and the corresponding statistics, the risk viii involved in the design can be estimated. In this way, the risk associated with each design alternative can be evaluated. The information on risk and the corresponding consequences in case of failure can be combined using the decision analysis framework to obtain the so-called "best" solution to the problem under consideration. CONSEQUENCE Figure 1. Steps in a Probabilistic Study The second chapter of this study, contains an information about previous researches on the use of statistical methods in geotechnics and creating a data base. In this study by using all of the relevant experimental data which were obtained in the Soil Mechanics Laboratory of Istanbul Technical University, an attempt is made to create a data bank for Turkey or some regions, such as Istanbul, which has great amount of experimental data. Furthermore, the places which have a swelling problem in Turkey and their maximum swelling pressures are presented. The third chapter is be devoted to regression analyses of statistics. In this chapter, the theoretical information about regression and correlation analyses are summarised and the regression and correlation analyses ix of compressibility index with respect to readily determinated void ratio, water content and liquid limit are examined and the results are presented. In the fourth chapter, statistical variations of some general soil parameters taken into consideration, were investigated and statistical parameters were calculated and their distribution functions were determined. The applicability of some well-known distribution functions were tested by Kolmogonov-Smirnov Goodness of fit Test. Finally, the conclusions deduced are summarized in chapter