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UBC Theses and Dissertations

Statistical models for the effect of length on the strength of lumber Williamson, Justin Andrew


This thesis concerns the problem of modelling the effect of length on the strength of lumber. The length effect manifests as a shift to lower values of the strength distribution from shorter to longer boards. This effect is also noticeable for other materials. It is usually attributed to a statistical effect: a longer member has a greater chance of having a serious defect. This effect is not currently allowed for in design codes, which results in boards of different length having different reliability. A method is needed to find adjustment factors that will provide equal reliability. It has been modelled with the Weibull model, which significantly over-predicts the length effect. It is deduced that the over-prediction is due to the dependence between elements of the board, which is not allowed by the Weibull model. The presence of this dependence has been found experimentally. Two classes of alternative model are developed. One requires input of the magnitude of the dependence and the strength distribution. The other requires only the strength distribution and gives an estimate of the dependence. A simulation study of 100,000 boards shows with considerable certainty that both classes of model work better than the currently accepted models on data which contains dependence. Real lumber strength data from bending and tension tests of 3000 boards from 4 sources are used to validate the proposed models. Using a wide variety of measures it is shown with a high degree of certainty that the proposed models are superior to the existing models. Theory is developed which shows that this model-based approach has much less uncertainty than a purely data-based approach, and this is confirmed from the data. Machine graded lumber may be better fitted by a model developed from the Gumbel distribution. It is shown that equal reliability adjustment factors can be obtained from the model parameters very easily. The average factor proposed to be applied to design strength for halving the length is 1.14. It appears that this factor is lower for high grade lumber. The tools used for the length effect in single members have been extended to computing the reliability of weakest link structures. This is a useful class of structures, and this method appears to be superior to existing methods. This can be used to show that the different length adjustments make a considerable difference to the calculated reliability of structural systems such as trusses. Two truss types were considered and found to have effective lengths of up to 7.3m, for a β = 3. These required a (downward) adjustment to the single member strength of up to 24%.

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