t/R = 0.3 | 70% hollow or 30% solid?

Recently, we caught a podcast where a tree was declared 'safe' if it's less than 30% hollow. We think they meant 70% hollow. Either way, this isn't right for several reasons.

The heart of the confusion is the t/R = 0.3 fallacy. t/R = 0.3 is when a residual wall thickness (t) is 30% of the stem radius (R). It's often cited as a failure threshold. It's not. The Why t/R Ratios Aren't Very Helpful pdf explains why in detail.

In short, one reason is because of a geometric property called section modulus. Wind load and material properties remaining equal, if you double the diameter you increase the load bearing capacity of a tree by 8 times.

To add to the confusion, t/R 0.3 is often referred to as 70% hollow. In fact, a 0.3 t/R ratio is only 50% hollow.  70% is the radius, which is one dimension. t/R 0.3 is the area, which is two dimensions.

This graph from Paul Muir shows the relationship of central hollowing on:

A = Cross Sectional Area
Z = Section Modulus

t/R = 0.3
A = 49% loss of cross sectional area
Z = 24% reduction in load bearing capacity

To make matters worse. A tree with a t/R ratio of 0.3 can have a very high likelihood of failure, or it can have a very low likelihood of failure.

If all that wasn't enough, it's seldom that where decay is of concern we're dealing with a cross sectional area of a tree that's a circle.

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