Bivariate Normal Parameter Estimation under Destructive Testing

Many materials used in construction and other applications can be characterized by two or more important physical strength properties. In assessing the acceptability of the materials the individual strength means and standard deviations are clearly important. In addition, the correlation between the various strength properties is also very important. For physical structures subject to a variety of stresses, large correlations between strength modes have the effect of increasing the variability of a structure's load carrying capacity, thus making it less reliable. In many applications, however, the strength of an item can only be determined through destructive testing. Lumber, for example, has a number of physical properties such as bending strength, tensile strength, shear strength, and compression strength, that can only be determined destructively. As a result, one is able to ascertain only the breaking strength in a single mode for each unit. In such situations, the correlations among the various strength properties cannot be measured directly and must be approximated. A number of past studies, Amorim (1982), Evans, Johnson and Green (1984), Green, Evans and Johnson (1984) have addressed the problem of estimating the correlation between two destructively determined variables by using a combination of proof-loading and precise measurement. Proof-loading involves stressing units only up to a prescribed (proof) load, thereby breaking only the weaker members of a population. This way, although some units break before the proof-load is reached, others survive and can be subjected to further testing in other strength modes. This paper proposes four simple procedures which use only proof loading to estimate the correlation between two variables that can individually only be measured destructively. In Section 1, we describe Procedure I which is very simple, but requires prior knowledge of the individual means and variances. Section 2 presents Procedure II that requires a slightly more complicated testing procedure, but is more flexible since it does not require as much prior information. More details on Procedure I and II can be found in the upcoming paper, Steiner and Wesolowsky (1995). Extensions of this work that yield estimates for all five bivariate normal parameters are discussed in Section 3. Estimating all parameters from one experiment is highly desirable when prior information about individual means and standard deviations is scant and there is a need to minimize testing requirements. Finally, Section 4 summarizes the results and provides practitioners guidance in choosing the most appropriate estimation procedure. One should note that all procedures based on proofloading, including all methods discussed in this paper, implicitly assume that survivors of the proof-load are not damaged. Experimental studies by Madsen (1976), and Strickler et al. (1970) suggest that this may be a reasonable assumption regarding the static strength of lumber, although a few pieces whose strength is only slightly greater than the proof-load stress will likely be weakened. Similarly, according to cumulative damage theory, Gerhards (1979, p. 139), “the theoretical results suggest that some percentage of the population will fail during the proof-load, a very small additional percentage will be weakened, but the remainder will have residual strength virtually equal to original strength.”