Mécanique de l'arbre sur pied : modélisation d'une structure en croissance soumise à des chargements permanents et évolutifs. 1. Analyse des contraintes de support

Une méthode d'analyse des contraintes mécaniques supportées par le bois de l'arbre sur pied est proposée, qui tient compte de la croissance secondaire. Elle est appliquée à l'étude des contraintes longitudinales engendrées par le poids propre supporté par la tige, chargement permanent à l'échelle de la croissance du tronc. Les distributions d'efforts ainsi calculées sont tout à fait inattendues : d'une part, les niveaux de contraintes sont très faibles partout à la périphérie de la tige, là où le bois est très jeune donc sollicité depuis peu de temps; d'autre part, la localisation à l'intérieur de la tige et le niveau de chargement des parties les plus tendues ou comprimées ne dépend pas seulement de l'état actuel observable mais de toute l'histoire de l'arbre. Les notions intuitives de «face tendue ou comprimée» sont donc à reconsidérer. L'illustration de ces conclusions est faite à travers plusieurs situations numériques réalistes : cas de l'arbre vertical parfaitement symétrique, d'un houppier qui s'excentre dans un plan fixe plus ou moins vite ou qui se redresse, d'un houppier qui s'excentre en changeant de direction.Mechanics of standing trees: modelling a growing structure submitted to continuous and fluctuating loads. 1. Analysis of support stresses. A general analysis of mechanical stresses which develop in stems as the tree grows in weight and volume is presented and applied to the study of the distribution of longitudinal stresses due to the self weight supported. Compressive and bending loads, the main loads due to weight supported, are analysed using simple concepts of beam theory. The effect of radial growth is taken into account. Compared to the classical distribution of stresses in a non-growing initially straight cantilever beam and fully loaded at a given moment in time, the stress patterns so calculated are totally unconventional: on the one hand, stress values are very low everywhere at the stem surface where young wood has been loaded for a short time; on the other hand, the positions and values of maximal tensile or compressive stresses depend not only on the actual state but on the entire history of the tree. The intuitive concepts of "tensile or compressive face" must be reconsidered. These conclusions are shown by several realistic numerical simulations: in the case of the symmetrical straight tree (fig 3), near the pith where the wood is older, compressive stresses can be 3-6 times greater than the uniform stresses calculated from the standard distribution of the whole weight on the final cross-section. In the case of the tree which offsets its crown eccentrically in a fixed direction (fig 4), the greater stress is not at the surface: the more recent the offset is, the nearer to the surface is the position of greater stress. The case of the tree which straightens its eccentric crown in a fixed direction (fig 5), clearly shows that a tree which is straight at the present time can undergo quite high tensile or compressive stresses Inside. In the case of the tree with an eccentric twisting crown (fig 6), the position of greater tensile or compressive stresses is not in the axis of the bending observed at present, but depends on the history of twisting