As arborists and storm responders, the precision and planning involved in tree felling are paramount to ensure safety and success. Understanding the mathematical principles behind calculating face cut depth and wedge height plays a crucial role in tipping a back-leaning tree over its center effectively and safely.
When dealing with back-leaning trees, achieving the right balance between the face cut depth and wedge height is key. Mathematical formulas offer a systematic approach to calculating these parameters, allowing arborists to anticipate and control the direction and extent of canopy movement before any cuts are made. In this post we will discuss how canopy movement is a function of wedge height, tree diameter, and fulcrum distance, as well as how to calculate each parameter as a part of the tree felling plan. For the sake of safe tree felling practice, this discussion will be based on a hinge length that is 80% of the overall tree diameter measured at breast height (4.5 feet off the ground).
Calculation Disclaimer: Formulas assume the tree in question is perfectly round and with a uniform canopy as to easily determine back lean. Calculations are approximate and will vary slightly in real world applications.
Formulas for Calculating Canopy Movement, Hinge Length, Face Cut Depth and Fulcrum Distance
Canopy Movement Formula
The Canopy Movement Formula calculates the canopy movement “CM” as a function of wedge height “Hw”, tree height “Ht”, and Fulcrum Distance “Fd”. Canopy movement refers to the distance the canopy of a tree will advance in the direction of lay when a felling wedge is utilized in the back cut. Fulcrum Distance is constrained by the distance measured from the front of the hinge to the back of the tree where the back cut will be initiated. Deriving this distance will be analyzed in a later formula.
Example Problem:
“Determine the canopy movement, in feet, of an 85 foot tree with a uniform canopy when a 1″ tall wedge is driven completely into the back cut. The diameter at breast height (DBH) is measured to be 24 inches and the fulcrum distance is measured to be approximately 19 inches.”
We can use the formula above to calculate the canopy movement of the tree in question. It is important to convert all known variables into the same unit of measurement. We will convert all variables into inches in order to reduce the complexity of the calculation process, and then convert back to feet for the final result.
80% Hinge Length Formula
This formula is used to determine the optimal hinge length “Hl” of a tree with a given diameter “Dt”. In most accepted publications, the optimal hinge length is 80%, or .8 of the tree’s diameter measure at breast height (DBH).
Example Problem:
“Calculate the optimal hinge length of a tree with a DBH of 17 inches.”
Central Hinge Angle Formula
The Central Hinge Angle Formula is used to determine the angle that intersects the front face of the hinge along the circumference of the trunk originating from the center of the trunk, and is necessary for determining the face cut depth. Again, “Hl” denotes the optimum hinge angle and “r” the radius of the trunk.
Example Problem:
“Calculate the Central Hinge Angle of a tree with the optimal hinge length of 13.6 inches and a DBH of 17 inches.”
IMPORTANT: When using the “80% Rule” for hinge length the Central Hinge Angle will always be 106.26 degrees.
Face Cut Depth Formula
The Face Cut Depth Formula is used to determine the distance from the front of a tree to the front face of the hinge. This distance represents the face cut depth and is calculated using a circle chord equation. The hinge length “Hl” and the central hinge angle “theta” must be determined before calculating the face cut depth. The face cut depth will ultimately dictate the Fulcrum Distance, thus directly influencing the Canopy Movement Formula discussed at the beginning of this post.
Example Problem:
“What is the proper face cut depth for a tree that has an optimum hinge length of 13.6 inches when the central hinge angle has been calculated to be 106.26 degrees?”
Fulcrum Distance Formula
The Fulcrum Distance Formula is the final piece in calculating the canopy movement as it relates to wedge height and tree height. Displacing a tree canopy past its tipping point “hinges” on the distance from the front of the hinge to the back of the tree. Once the face cut depth “FC” has been determined, the Fulcrum Distance “Fd” is simply the diameter “D” minus the face cut depth.
Example Problem:
“A tree has been measured to have a DBH of 17 inches. A face cut has been properly sawed at a depth of 3.4 inches. What is the Fulcrum Distance?”
Real World Application
Thank you for sticking around this long! There may be a lot of information to digest, but it is important to understand the mathematical basis behind certain tree felling plans and techniques. Let’s bring everything together we’ve discussed and see how this might apply to a real world situation.
Example Problem:
“You are tasked with felling a tree with a uniform canopy that has been subjected to strong winds due to a microburst. The visual tree assessment shows no signs of wood fiber fracture or signifigant root crown disturbance, and therefore, standard felling methods are deemed appropriate. You measure the tree to be approximately 82 feet tall with a DBH of 23 inches, and roughly 5 feet of back lean. Determine if one plastic felling wedge 8 inches long and 1 inch high is enough to tip the tree over center using the ‘80%’ hinge length rule.”
We need to sort our known values and indentify the variables that need to be solved for. The goal is to determine if a single 1 inch tall felling wedge is able to tip this specific back leaning tree over center.
We will now calculate hinge length “Hl”, face cut depth “FC”, and fulcrum distance “Fd” in order to solve for canopy movement “CM”. The canpoy movement must be greater than 5 feet in order to overcome the existing back lean with the single felling wedge.
The task at hand is to determine if one plastic felling wedge measuring 1 inch tall could correct 5 feet of back lean in a tree that is 82 feet tall and measuring 23 inch at DBH. The canopy was calculated to advance 54.67 inches, or 4.5 feet. In this case a single 1 inch felling wedge would not be enough to send this particular tree over center into the direction of lay. Knowing that more than a single wedge is needed before the felling operation can begin could help develop a more effective plan such as using a rope, a steeper slope wedge, or multiple wedges. These are things to consider especially in dealing with trees that are near targets.
Conclusion
It is no myth that developing a proper felling plan is a key factor in a successful and safe tree removal. Understanding the relationship between a tree’s dimensions and the capability of the tools at hand can help prevent unwanted results during a tree felling operation. There are several key ideas shown in this exercise, and we encourage you to follow the examples as well as change the dimensions of the tree at hand in order to become more familiar with the following concepts:
- Canopy movement is determined by the tree’s height, the effective wedge height, and the distance from the front of the hinge to the back of the tree (Fulcrum Distance).
- When using the “80%” rule, the hinge length will always equal the fulcrum distance. This allows quick calculation of the fulcrum distance, canopy movement, and face cut depth.
