Destructively Assessed Property Relationships in Reliability Theory with an Application in Relating Structural Properties of Dimension Lumber

We present a novel approach for predicting one lumber strength property from another, each being measured destructively. The objective is to reduce the cost of lumber monitoring programs, by measuring one of the properties and predicting the other. To reach the objective, we review single proof load design (SPLD) proposed to assess dependence between two jointly normally distributed random variables X and Y that cannot be observed simultaneously. The SPLD tests specimens in one mode X up to a determined load (proof loading), and the survivors are tested to failure in a second mode Y. To resolve the near non-identifiability of parameters in the SPLD approach, we redesign the SPLD for improving the estimation performance. The new design assigns specimens to one of the two groups: SPLD and shoulder. The shoulder group tests specimens to failure in the Y mode. The SPLD with a shoulder approach results in a more accurate and precise estimate. To quantify damage caused by proof loading lumber specimens, we use the maximum likelihood method to estimate theoretical quantiles of the strength distribution using a sample from a SPLD with a shoulder experiment with a low proof load level. The comparison between those estimated theoretical quantiles and empirical quantiles of the proof load survivors reveals the damage at higher proof load levels. Using our experimental data on manufactured lumber, we find low and high proof load levels leave survivors undamaged, but intermediate load levels do damage weaker survivors. We generalize the SPLD with a shoulder approach to incorporate proof load damage, and thus finally provide a method for estimating the X-Y dependence when damage occurs. This generalized approach is applied on our experimental data to estimate the relationship between bending and log--transformed tension. The high correlation found in our application suggests that if there is a need to verify the two strength properties, only one of them needs to be measured and the other is predicted from the measured strength.