Civil Engineering Dimension, Vol. 10, No. 1, March 2008, 14-27 ISSN 1410-9530 Civil Engineering Dimension ISSN 1410-9530 print © 2008 Petra Christian University http://puslit.petra.ac.id/journals/civil Spline Nonparametric Regression Analysis of Stress-Strain Curve of Confined Concrete Tavio 1 , I Nyoman Budiantara 2 , and Benny Kusuma 3 Abstract: Due to enormous uncertainties in confinement models associated with the maximum compressive strength and ductility of concrete confined by rectilinear ties, the implementation of spline nonparametric regression analysis is proposed herein as an alternative approach. The statistical evaluation is carried out based on 128 large-scale column specimens of either normal- or high-strength concrete tested under uniaxial compression. The main advantage of this kind of analysis is that it can be applied when the trend of relation between predictor and response variables are not obvious. The error in the analysis can, therefore, be minimized so that it does not depend on the assumption of a particular shape of the curve. This provides higher flexibility in the application. The results of the statistical analysis indicates that the stress-strain curves of confined concrete obtained from the spline nonparametric regression analysis proves to be in good agreement with the experimental curves available in literatures. Keywords: confinement model, ductility, spline nonparametric regression analysis, stress- strain curves. Introduction The regression analysis has been playing an impor- tant role in the theory of approximation and statistics for many years. It is used to evaluate the effects of the independent to the dependent variables by observing the trend of relation between the two types of variables. The evaluation can be carried out using two approaches, i.e. the parametric approach, which is frequently used in practice, and the nonparametric approach. The use of piecewise polynomial has been increa- singly popular due to its flexible nature. It is effective in handling a function or a set of data comprising the local natures [1,2]. One of the essential piecewise polynomial is a spline polynomial. The application of the spline nonparametric regression analysis has been widely found in many fields, such as in economics [3] and medicine [4]. However, the imple- mentation of this type of analysis in civil engineer- ing, particularly in reinforced concrete area, is still seldom and not popular. 1 Department of Civil Engineering, Sepuluh Nopember Institute of Technology (ITS), Surabaya, Indonesia E-mail: tavio@its.ac.id 2 Department of Statistics, Sepuluh Nopember Institute of Techno- logy (ITS), Surabaya, Indonesia 3 Ph.D Candidate, Department of Civil Engineering, Sepuluh Nopember Institute of Technology, Surabaya (ITS), Indonesia Note: Discussion is expected before June, 1st 2008, and will be published in the “Civil Engineering Dimension” volume 10, number 2, September 2008. Received 7 March 2007; revised 14 May 2007; accepted 27 June 2007. The spline polynomial has many beneficial statistical properties for use in the regression analysis [1,5]. It is a piecewise polynomial which has segmental properties. The segmental properties of the spline polynomial give higher flexibility in the application compared to the ordinary polynomial. This allows the spline polynomial effectively adapting itself with the local characteristics of a function or a set of data. To date, the regression analysis, which is frequently used by the researchers to obtain the stress-strain curve of confined concrete, is the parametric regres- sion analysis, mostly linear or quadratic. Since it is very simple, the resulting stress-strain curves of confined concrete still have appreciable discrepancies with the experimental curves, particularly in term of ductility along the descending branch. This causes very significant uncertainties in the confinement model of concrete under uniaxial compression. Problems of confined concrete have long been recog- nized and investigated both experimentally and analytically in the past [6]. Several confinement models have been developed to predict the stress- strain curve of normal- as well as high-strength concrete. All proposed models are based on previous experimental work where several parameters have been derived from the parametric statistical process- ing of the results. However, the evaluation of these models against other models was based on limited number of specimens and model parameters. Fur- thermore, the confinement models for predicting the stress-strain curve of confined concrete proposed by various researchers are applicable only for normal- or high-strength concrete. Hence, it is deemed 14