Reliability of the Weibull analysis of the strength of construction materials

Abstract

The breaking force during transversal loading of fibre-cement corrugated roofing sheets was measured on several test samples from a serial production. The results were statistically analyzed assuming the 2-parameter Weibull statistics. In addition, Monte Carlo statistical simulations were made by using a computerised built-in random-number generator. While smaller sample data groups, mostly containing up to 50 samples, were studied in the literature, we extended their size up to 400 samples. We showed that some trends in the evaluation of statistical parameters which hold for smaller data groups, apply well to larger data groups. In particular, we confirmed that the statistical distribution of the Weibull parameters obtained from repeated Monte Carlo simulations is log-normal. Furthermore, we considered the influence of the measurement uncertainty on the statistical parameters.

Appendix: The Fisher matrix and the parameter confidence bounds

First the logarithmic likelihood function Λ is defined which can also be used for the estimation of the Weibull parameters according to the maximum likelihood method:

$$ \Uplambda (m,F_0 ) = \sum\limits_{i = 1}^N {\ln p(F_i ,m,F_0 )} , $$

where N is the number of measurements, p is the distribution function (1a), F _i is the i-th breaking force, and the parameters m and F ₀ can still be varied. The Fisher information matrix F is defined by the second derivatives of the function Λ with respect to Weibull parameters:

$$ \underline{\underline F} = \left[ {\begin{array}{*{20}c} {\frac{{\partial ^2 \Uplambda }} {{\partial m^2 }}} & {\frac{{\partial ^2 \Uplambda }} {{\partial m\partial F_0 }}} \\ {\frac{{\partial ^2 \Uplambda }} {{\partial m\partial F_0 }}} & {\frac{{\partial ^2 \Uplambda }} {{\partial F_0 ^2 }}} \\ \end{array} } \right] $$

When the estimated Weibull parameters (obtained, for instance, by the linear regression method) are inserted into the Fisher matrix which is afterwards inverted the covariance matrix C is obtained which consists of variances and covariance of the Weibull parameters:

$$ \underline{\underline C} = \left[ {\begin{array}{*{20}c} {Var(m)} & {Cov(m,F_0 )} \\ {Cov(m,F_0 )} & {Var(F_0 )} \\ \end{array} } \right] = \underline{\underline F} ^{ - 1} $$

Finally the variances of the parameters are used to obtain the corresponding “half-widths” w in Eq. 5, for instance for the parameter m:

$$ w = \frac{{\sqrt {Var(m)} }} {m}, $$

from which the confidence bounds can be calculated as described above: $m_{{\hbox{UB}}} ,m_{{\hbox{LB}}} = m \cdot \exp ( \pm w \cdot erf^{ - 1} (\rm CL))$, and similarly for F ₀.