Photo by Artan Sadiku on Unsplash.
Some trees are genetically programmed to grow taller than others. California redwood, Douglas-fir, Sitka spruce in North America; a number of Dipterocarp species (“Philippine” mahoganies) in Southeast Asia; and several species of eucalypts in Australia grow to heights exceeding 165′ (50 meters). Do these and other tall trees have special properties that give them mechanical advantages? If so, would these properties produce superior masts and spars for boats?
Why some trees have evolved to grow so much taller than others has fascinated me for more than two decades. Certainly, the environment in which trees find themselves plays a role. The two tallest trees in the world, California redwood (Sequoia sempervirens) and Australia’s mountain ash (Eucalyptus regnans), can reach maximum heights of greater than 328′ (100 meters). They grow in very similar latitudes (35° to 40° North and South) and are blessed with winter rains, summer fogs, and ambient temperatures in the Goldilocks range—not too warm, not too cold.
Trees are unsupported columns with crowns that are free to sway in the wind. So, are tall trees stronger than shorter trees? We know that wood density is correlated with many strength properties, yet tall trees do not, necessarily, have denser and stronger wood. Redwood and western red cedar have rather low-density, weak woods. So, some other mechanical properties must be critical for preventing buckling in long, slender columns.
To get to the root of this question, we must go back in time to the 18th century and to the Swiss mathematician Leonhard Euler (pronounced oyler). Born in 1707, Euler enrolled in the University of Basel at the tender age of 13 and received a Master of Philosophy by age 16. In his “spare” time, he was tutored by a family neighbor, Johann Bernoulli, who just happened to be Europe’s foremost mathematician at the time. Euler’s father urged Leonhard to become a pastor, but fortunately for us Bernoulli’s influence prevailed and in 1726 Euler completed his PhD dissertation.
At age 19, Euler sought a position at the University of Basel but was rejected. Disheartened, in 1727 he entered the competition for the Paris Academy Prize. That year, the question to be solved was: “What is the best way to place masts on a ship?” Unfortunately, Euler only took second prize, while Pierre Bouguer, who later became known as the father of naval architecture, won.
Just as his future was not looking very auspicious, Euler’s old friend, Bernoulli, came to his rescue. Bernoulli’s two sons had posts at the Imperial Russian Academy. When one son died suddenly of appendicitis, the brother recommended Euler to fill a position, and this led to a professorship in physics by 1731. Turmoil in Russia led to Euler taking a post at the Berlin Academy in 1740; but despite his great contributions to that Academy’s prestige, he fell out of favor in the Court of Frederick the Great and eventually returned to Russia, where he lived the rest of his life.
Plagued with failing eyesight since his youth, Euler became completely blind by 1766. Yet this had little effect on his prodigious academic productivity—publishing one mathematical paper per week in 1775. It is tempting to think Euler may have been recollecting the “lost” Paris Academy Prize concerning ship masts, when, in 1757, he published his now-famous critical buckling formula for an ideal column. The unusual feature of this theory is that unlike virtually any other mechanical formula, it does not depend on stress/strain (strength) relationships.
The basic formula is F = π2 EI/(KL)2, where F = critical force (maximum vertical load), E = modulus of elasticity, I = area moment of inertia, L = unsupported column length, and K = column-effective length factor [for a column with one end fixed and one end free to move laterally, K = 2.0]. Area moment of inertia (I) is related to the geometry of the column and is adjusted in trees by diameter to height scaling.
This leaves E (written as MOE by wood scientists) and L as variables. What Euler is saying is that in order to avoid critical buckling, the modulus of elasticity (a measure of stiffness) must be greater for longer columns.
In 2006, I presented a paper at the 5th Plant Biomechanics Conference in Stockholm, Sweden. Scouring wood science databanks, I accumulated mechanical properties for 169 tree species from Latin America, Southeast Asia, Australia, and North America. I found that within each geographic region, tall tree species of similar densities had MOE values significantly higher than short tree species of similar densities (measured as basic specific gravity, Gb).
How does this apply to the building of masts and spars? First, what I found applies to a species, not individual trees. This strongly suggests genetic control of stiffness. So, you don’t need to know if the wood you are using came from a tall or less-tall tree; you simply need to know whether the species has the potential to grow taller than other trees within its native region. (Maximum tree height can be obtained from tree identification books and on the Internet.)
Notes: Boatbuilding uses from Bootle, K.R. 1983. Wood in Australia, McGraw-Hill, Sydney. Gb = basic specific gravity. MOE = modulus of elasticity.
How might you make use of this information? Let’s say you have sailed from Seattle, Washington, to Sydney, Australia, and along the way one of your Sitka-spruce masts was snapped off in a storm. You find a Sydney boatbuilder who says he can make a replacement mast for you and has access to either hoop pine (Araucaria cunninghamia) or King William pine (Athrotaxus selaginoides). Both are commonly used boatbuilding woods but King William pine has decay resistance, while hoop pine lacks it. Yet, this local builder strongly recommends hoop pine for your mast replacement. Do you go with his recommendation?
The accompanying table lists some properties of six Australian boatbuilding woods. The first three are tall trees with an average MOE/Gb of 25.0. The last three trees are comparatively short and have an average MOE/Gb of only 15.2. Although the three short tree species have higher decay resistance, they are generally not used for masts or spars (see the last column in the table). For comparison, Sitka spruce trees can grow to 197′ (60 meters) in height, lack decay resistance, but have an average MOE/Gb of 22.9. Thus, it appears that Euler’s theory for ideal, long, slender columns is reflected in nature by trees that have evolved to reach greater maximum heights. And by trial and error boatbuilders have recognized that the wood from these taller tree species produces superior masts and spars.
Dr. Richard Jagels is an emeritus professor of forest biology at the University of Maine, Orono. Please send correspondence to Dr. Jagels by mail to the care of WoodenBoat, or via e-mail to Senior Editor Tom Jackson, tom@woodenboat.com.