### Structural Behavior of Timber Beams with a Modulus of Elasticity Represented by a Random Field

#### Abstract

A stochastic model of the structural behavior of a Eucalyptus grandis timber beam is herein presented. Three strength classes obtained according to the the criterion of the Argentinean standard IRAM 9662-2:2006 are studied. The Modulus of Elasticity (MOE) is considered as a random field.

The MOE uncertainties are quantified in the beam deflection and the random field is given by a gamma probability density function (PDF) and an exponential correlation function. Experimental data obtained from bending tests performed with 349 sawn beams of Argentinean Eucalyptus grandis are employed to find the PDF parameters. Statistical results of the deflection are reported for the case of pure bending of a simply supported beam. Focus is made on the central deflection. The methodology was previously verified comparing with some results published by other authors. The influence of the correlation length is assessed for limiting and intermediate cases. The problem is solved by a discretization with a finite element model and the Nataf transform is employed in order to generate and simulate the random field.

Additionally, the random field is discretized by means of the Midpoint Method. Numerical results are obtained by means of Monte Carlo simulations and previously a convergence study was carried out to determine the number of realizations that gives a reasonable accuracy. The PDF and cumulative distribution function (CDF) of the central deflection allows obtaining statistical estimates. As a particular case, the homogeneous model with the MOE assumed as a random variable is studied with some detail. The propagation is made analytically and the explicit PDF of the central deflection is obtained. Since wood is a material with very variable mechanical properties, assuming the MOE as a random field contributes to attain a more realistic structural model. In particular, the study of the correlation length variability also provides a larger amount of information about the range of the response. It should be noted that within the standard code in use, the MOE is eventually modeled as a random variable and not as a random field. It is shown that the MOE modeled as a random variable constitutes a more conservative approach.

The MOE uncertainties are quantified in the beam deflection and the random field is given by a gamma probability density function (PDF) and an exponential correlation function. Experimental data obtained from bending tests performed with 349 sawn beams of Argentinean Eucalyptus grandis are employed to find the PDF parameters. Statistical results of the deflection are reported for the case of pure bending of a simply supported beam. Focus is made on the central deflection. The methodology was previously verified comparing with some results published by other authors. The influence of the correlation length is assessed for limiting and intermediate cases. The problem is solved by a discretization with a finite element model and the Nataf transform is employed in order to generate and simulate the random field.

Additionally, the random field is discretized by means of the Midpoint Method. Numerical results are obtained by means of Monte Carlo simulations and previously a convergence study was carried out to determine the number of realizations that gives a reasonable accuracy. The PDF and cumulative distribution function (CDF) of the central deflection allows obtaining statistical estimates. As a particular case, the homogeneous model with the MOE assumed as a random variable is studied with some detail. The propagation is made analytically and the explicit PDF of the central deflection is obtained. Since wood is a material with very variable mechanical properties, assuming the MOE as a random field contributes to attain a more realistic structural model. In particular, the study of the correlation length variability also provides a larger amount of information about the range of the response. It should be noted that within the standard code in use, the MOE is eventually modeled as a random variable and not as a random field. It is shown that the MOE modeled as a random variable constitutes a more conservative approach.

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Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**