Survival Analysis of Tensile Strength Variation and Simulated Length–Size Effect along Oak Boards

Abstract

Experiments and a simulation of the effect of localized tensile strength in European white oak boards and predictions of the effect of the board’s length on global tensile strength are reported. Furthermore, the cross-correlation of local strength with a local modulus of elasticity (MOE) is investigated. This knowledge is a fundamental prerequisite for stochastic simulations of glued laminated timber with adequate representation of the lamination effect. The sample consisted of 47 boards, each subdivided virtually into 15 consecutive cells 100 mm in length and for which local MOE and its autocorrelation had been previously determined. The remnants of the first, that is, global, failure in the axial tension tests were loaded to rupture in secondary loadings up to four times. This procedure, resulting in a total of 102 primary and secondary strengths, was enabled by the rather blunt failure planes presented by the investigated oak boards. The censored localized strength data, including the lower bounds of the unbroken virtual cells, were analyzed using survival analysis with a modified likelihood function (LHF) applied to six different strength distribution models: (1) four censored parametric distributions (two-parameter Weibull, beta, grafted Weibull–Gaussian, and Duxbury–Leath–Beale); and (2) two censored regression models (Weibull and beta) with shape and scale parameters depending on global MOE. A detailed analysis of the lower tail of the experimental distribution evidenced the quasibrittle nature of the material, thus supporting the use of the Weibull–Gaussian distribution. The simulation of the tensile strength variation along the board’s length was conducted by an adapted vector autoregressive (VAR) model. Here, the localized MOEs were cross-correlated with the obtained cell-related strength distributions employing Gaussian normalization of both distributions, followed by mapping the normalized strength values into the LHF-derived strength distributions. The length effect of tensile strength was analyzed with virtual boards of different lengths with 50,000 simulations per board length and a strength distribution model. The obtained length effect exponent of a simple power law, usually applied in timber engineering, was roughly equal to 0.33 for the mean strength level for all models and grade combinations. At the 5%-quantile level, the calculated exponents varied and were grade-dependent and model-dependent from 0.14 to 0.34. The simulated length effects were similar to those of softwoods at the 5%-quantile level but markedly higher at the mean level, which is well explained by the finite weakest link theory by considering the more general size dependency that governs quasibrittle materials.