6 erive the shape of a tree The uniform stress hypothesis is

6 erive the shape of a tree The \"uniform stress hypothesis\" is a popular basis to describe the shape and growth of trees I, Of course stems grow \"up\" in a quest for light That phenomenon is called \"phototropism Stems also grow out, or thicken. in or der to hold themselves up. The d(z) essential idea behind the uniform stress hypothesis is that a stem will below some threshold value, which is the same all along the trunk. In this problem, we are going to use the uniform stress hypothesis to de- Figure 8: rive the shape of a tree trunk 6.1 shape based on static loading 6.1.1 Static loads Let Wc denote the weight of the crown of the tree, the weight density of the tree, z the height of any section of tree, h the height of the trunk, d(z) 2r (z) the trunk diameter at height z see the figure. 1.) Show that the axial load as a function of z is: (15) 6.1.2 The uniform stress hypothesis Now assume that 32(z) amax (16) wind Here, max is a species de pendent constant 2.) Determine r(z) so that (16) is true. Hint: try differentiating the equation P A. Figure 9: Wind loading 6.2 shape based on dynamic or live loading The function r(z) derived in section 6.1.2 would describe the shape of a tree trunk if tree growth were regulated by static stress in the wood. There are physiological reasons to believe, however,

Solution

1.Elementary theory
Tree stems may be considered to be cantilever beams fixed at a point above the region
of butt swell. Elementary theory states that the longitudinal stress, (3, in beams of
circular cross section with bending moment, M, is given by:
Sigma=d^3/32
where d is the beam diameter (provided that beam diameter is
much less than beam length, and the deflection is small. The height-diameter profile
of tree stems having uniformly distributed stress (0) below and within the crown can
be calculated using this elementary theory, in which the horizontal (wind) force, F,
is distributed within the crown in a simple way.
Consider the two simple cases in Figure 1. Equation 1 implies that, for stress to be
uniformly distributed along the stem, the cube of the diameter at any height must be
proportional to the distance from that height (h) to the center of pressure of the
horizontal wind force on the crown (II,). That is, to give uniform stress along the
0.0 0.2 0.4 0.6 0.8 1.0
Relative Diameter
Figure 1. Theoretical (lines) and calculated (points) height-diameter profiles of stems with uniform stress
along their length. (a) A uniform distribution of wind force, and (b) a triangular distribution of wind
force. The theoretical relationships (lines) were based on text Equations 2 and 4 for case (a), and
Equations 2 and 6 for case (b). The points were calculated independently using the transfer matrix
method. H = tree height; H, = length of crown, h = height, h, = height within crown, H, = height to the
center of pressure from the wind force, F, acting at point.

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