Blowing in the WindContinued By making some assumptions about the density of crowns

Chapter 3, Problem 10

(choose chapter or problem)

Blowing in the Wind-Continued  By making some assumptions about the density of crown's foliage, the total force F on the tree can be calculated using a formula from physics: \(F=\rho A v^{2} / 6\) where \(\rho \approx 1.225 \mathrm{~kg} / \mathrm{m}^{3}\) is the density of air, v is the wind speed in m/s, and A is the cross-sectional area of the tree's crown. If we assume that the crown in roughly cylindrical, then its cross section is a rectangle of area A=(2R)(L/2)=RL, where R is the average radius of the cylinder. Then the total force per unit length is then \(w_{0}=F /(L / 2)=0.408 R v^{2}\).

The cross-sectional moment of inertia for a uniform cylindrical beam is \(I=\frac{1}{4} \pi r^{4}\), where r is the radius of the cylinder (tree trunk).

By your answer to part (c) in Problem 9 and the explanations above, the amount that a loblolly will bend depends on each of the parameters in the following table.

                 

(a) Mathematically show how each parameter affects the bending, and explain in physical terms why this makes sense. (For example, a large value of E results in less bending since E appears in the denominator of y(L). This means that hard wood like oak bends less than a soft wood such as palm.)

(b) Graph y(x) for 40 m/s winds for an “average” tree by choosing average values of each parameter (for example, r  0.2, R  3.5, and so on). Graph y(x) for 60 m/s winds for a “tall” (but otherwise average) pine.

(c) Recall that in the derivation of the beam equation on page 166 it was assumed that the deflection of the beam was small. What is the largest possible value of y(L) that is predicted by the model if all parameters are chosen from the given table? Is this prediction realistic, or is the mathematical model no longer valid for parameters in this range?

(d) The beam equation always predicts that a beam will bend, even if the load and flexural rigidity reflect an elephant standing on a toothpick! Different methods are used by engineers to predict when and where a beam will break. In particular, a beam subject to a load will break at the location where the stress function \(y^{\prime \prime}(x) / I(x)\) reaches a maximum. Differentiate the function in part (b) of Problem 9 to obtain \(E I y^{\prime \prime}\), and use this to obtain the stress \(y^{\prime \prime}(x) / I(x)\).

(e) Real pine tree trunks are not uniformly wide; they taper as they approach the top of the tree. Substitute r(x)=0.2-x/(15L) into the equation for I and then use a graphing utility to graph the resulting stress as a function of height for an average loblolly. Where does the maximum stress occur? Does this location depend on the speed of the wind? On the radius of the crown? On the height of the pine tree? Compare the model to observed data from Hurricane Hugo.

(f) A mathematical model is sensitive to an assumption if small changes in the assumption lead to widely different predictions for the model. Repeat part(e) using r(x)=0.2-x/(20L) and r(x)=0.2-x/(10L) as formulas that describe the radius of a pine tree trunk that tapers. Is our model sensitive to our choice for these formulas?