Sep 20, 2007 - Ross" examined a set of n= 1 1 samples (each test is regarded as one sample), ...... John Wiley and Sons, New York, 1973. 15 Tukey, J. W. Exploratory data anlysis, Addisun-Wesley, Reading,. MA, 1977. 16 Bezdek, J. C.
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Structural failure determination with fuzzy sets a
b
James C. Bezdek, Norah T. Grimball , James M. Carsonf & Timothy J. Ross a
c
Computer Science Department, University of South Carolina, Columbia, SC, 29208, USA
b
New Mexico Engineering Research Institute, The University of New Mexico, Albuquerque, NM, 87131, USA c
Air Force Weapons Laboratory, Kirtland AFBNM, 87117, USA
Version of record first published: 20 Sep 2007
To cite this article: James C. Bezdek, Norah T. Grimball, James M. Carsonf & Timothy J. Ross (1986): Structural failure determination with fuzzy sets, Civil Engineering Systems, 3:2, 82-92 To link to this article: http://dx.doi.org/10.1080/02630258608970430
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Structural failure determination with fuzzy sets
Downloaded by [The University Of Melbourne Libraries] at 05:59 03 July 2012
James
C. Bezdek,
Norah
T. G r i m b a l l * , James M. Cat-son? a n d T i m o t h y J . Ross**
This paper has four objectives. Firstly, modifications are recommended and presented to a well known set of US Army test data representing eleven tests that involve destruction of buried concrete box structures by ground transmitted shock waves. The resultant database is available for failure mode analysis. Secondly. a new approach forfailure mode analysis based on pattern recognition techniques to the structural engineering community is introduced. Thirdly, a comparison is presented of the results of preprocessing. feature extraction, and cluster analysis for failure modes to several previous studies. The objective here is to inform the structural engineering community about data interpretations that can be inferred using this approach: the authors d o not claim to have a conclusive or definitive classification of failure modes in these data Finally, the importance of a prior! decisions about data analysis during the planning stages of physical experiments is emphasized by pointing out the many statistical and other uncertainities that are associated with the data being discussed. Keywords: damage assessment, failure mode analysis, feature extraction, fuzzy cmeans, pattern recognition
Given data from various sensors that measure parameters of failure in structures due to shock or blast, what can be learned by analysis of the data using pattern recognition techniques? The analysis of vulnerability of structures is a complex problem involving many components and various uncertainties. Typically, structural failure analysis includes the identification of failure modes, which in turn may point to the basic causes of structural failure. There are several uncertainties associated with t he analysis of structural failure, not the least of which is the limitation imposed by the lack of an adequate amount of data. Failure analysis is also limited by the lack of knowledge about, understanding of and vagueness concerning the nature of failure and the geometric and material weaknesses that cause it. A third source of uncertainty is due to the (incorrect) placement and type(s) of sensors that are sometimes utilized for the physical measurements, (Indeed, sensors capable of unequivocally measuring the phenomena of interest may not exist). Finally, one is never certain that the features derived from sensor signals are the 'best' information contained by the waveforms for investigating the problem at hand. Some of the uncertainties discussed above can be treated with classical statistical theories. Specifically, when data can be assigned labels which are crisp (nonfuzzy, hard), and when the mode of failure seems plausibly modelled as a stochastic process, the probabilistic approach has been utilized succe~sfully'-~. In most cases, however, particularly those based on sparse information, damage assessment based on statistical methodology has been unsuccessful. In this latter instance, uncertainties
* Computer Science Depanrnenr, University of South Carolina, Columbia, SC 29208, USA t New hlexico Engineering Research Institute, The University of New Mexico. Albuquerque. NM 87131. USA ** Air Force Weapons Laboratory, Kinland AFB, Nhl 87117, USA (Received M a y 1986)
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Civ. Engng Syst. 1986. Vol 3. J u n e
may be better treated by fuzzy set modcls, because these models seem more consistent with the types of uncertainties encountered4-'. As the information becomes more clearly defined, fuzzy descriptions can becomc more precise. Another benefit of fuzzy set theory is that it allows quantification and use of subjective information and judgemental opinion in the analysis process. The attcntion in this paper is directed towards buried structures of reinforced concrete (RC) such as those described by Kiger and Slawson" and Ross and Krawinkler' '. Structures of this type are presumed to fail under impulsive loading in possibly three ways. These modes of failure will hereafter be designated as: a type F - flexure a type FS - flexure-shear
type DS-direct
shear
Unfortunately, data sets available for structural damage analysis are far from optimal. The extremely high cost of conducting even one test of structural failure usually means that there are a small number of tests in a data set. Compounding this problem is the fact that there are many measurements for each sample. For example, Ross" examined a set of n = 1 1 samples (each test is regarded as one sample), described at length in the section on damage assessment with fuzzy c-means algorithms, each of which possesses up to 24 sensor waveforms. Since each waveform can be used to generate various waveform characteristics called features, the potential number of features one might use for analysis of the tests is far in excess of the basic number of samples. To this date only fairly simple characteristics of the measured waveforms have been used. Ross and Krawinkler proposed" a method for predicting the mode of failure based on various physical parameters of the structure under test; and then attempted to substantiate their predictive model OZ63-0Z57/86/0208Z-l1SO3.00
0 1986 Buttenvonh & Co. (Publishes) Ltd
Structural failure determination with fuzzy sets: J. C. Bezdek er at
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by making a posteriori assessments of failure mode based on visual inspection of outputs from three interface pressure sensors and post test photographs showing the failure surface. Their analysis of data was conhed to visual comparisons of pairs of waveforms; the numerical features used implicitly were peak pressure, rise time and peak pressure decay. Carson has illustrated the potential of using more sophisticated features1'; we will amplify on this point below. A further complication is that, in many instances, one has only a vague idea of the actual failure mode experienced by the structure. Thus, the analysis of structural failure using pattern recognitions techniques must account for both statistically unqualified data and vaguely understood failure modes. In preparation for examining several specific data analysis techniques, a brief overview of a typical pattern recognition system (PRS) is provided in the next section.
Pattern recognition systems Pattern recognition is most aptly defined as a search for structure in data. In the statistical approach to numerical pattern recognition, which is treated thoroughly by Devijver and Kitler13, each input observation is represented as a multi-dimensional data vector (feature vector) where each component is called a feature. The purpose of the pattern recognition system is to assign each imput to one of c possible pattern classes. Presumably, different input observations should be assigned to the same class if they have similar features, and to different classes if they have dissimilar features. Statistical PRSs rest on mathematical models; it is crucial that the measure of mathematical similarity used to match feature vectors with classes assess a property shared by physically similar components of the process generating the data. The components of a typical PRS are illustrated in Fig 1. The data used to design a pattern recognition system is usually divided into two categories: design (or training) data and test data. Design data are used to establish the algorithmic parameters of the pattern recognition system. The design samples may be labelled, (the class to which each observation belongs is known), or unlabelled, (the class to which each data sample belongs is unknown). Test data are labelled samples used t o test its overall performance. Throughout this paper the following
I
Feature nomination
selection
Classifier design
Feature extraction
Cluster analysis
I
I
Cluster validitv
Fig 1 Pattern recognition systems
X = {x1,x2,x.) =data
x,€RP kth sample in X xk@ =jth measured feature of r,
n = number of samples in data p = number of original (nominated) features s = number of selected or extracted features c = number
of clusters
Feature analysis
Feature analysis refers to methods for conditioning the raw data so that the information which is most relevant for classification and interpretation is enhanced and represented by a mimimal number of features. Feature analysis consists of three components: nomination, selection, and extraction. Feature nomination (FN)refers to the process of proposing the original p features; it is usually done by workers close to the physical process, and may be heavily influenced by physical constraints, c.g. what can be measured by a particular sensor. In the following discussion, the nominated features correspond to simple characteristics of the various sensors which are represented by digitization of the sensor records. Feature selection (FS) refers to choosing the 'best' subset of s features ( s < p ) from the originals. Feature extraction (FE) denotes transforming the original pdimensional feature space into an s-dimensional space in some manner that 'best' preserves or enhances the information available in the original space. This is usually accomplished mathematically by means of some linear combination of the initial measurements. Another method of feature extraction that lies closer to thc expertise of the structural engineer is heuristic nomination and/or extraction. In other words. the process being examined may suggest choices for analytic features, c.g. slopes (of rise and decay), areas (impulse or energy) during rise and decay or even transformations of the measurements waveform. Implicit in both FS and F E is a means for evaluating feature sets chosen by a particular procedure. The usual benchmark of feature quality is the empirical error rate achieved by a classifier on labelled t e s t data. A second method of assessing feature quality is to refer algorithmic interpretations of the data to domain experts: do the computed results make sense'? This latter test is less esoteric than mathematical criteria, but very important-an attempt will be made to apply it to the results presented.
)
Design data
w
notation will be used:
I
Classifier design
Classifier design refers to partitioning feature space into c regions, one for each sub
