It is very important to be aware of the direction or vector of the force which is applied to the tree . Forces which act in a perpendicular ( at 90 °) direction to the axis of the lever arm apply significantly more bending moment than forces which act at an acute angle ( less than 90 °) to the axis .
To calculate the actual bending moment applied at a point , the angle between the lever arm and the applied force must be considered .
The formula to calculate bending moment at a point is :
M = rFsin∅
Where :
M
r
F
Ø is the bending moment applied at the axis , is the distance from the axis to the force , is the magnitude of force being applied , and is the angle between the force and the arm .
The three examples below show the use of a pulling rope to assist with felling a tree . In each case , the pulling rope is pulled with the same force , roughly equivalent to a 100kg person applying their full weight onto the rope ( 1kN ).
Assuming that in every case above , the length of the stem is 10 m and the force applied is 1000 N ( ≈100 kg ):
M = rFsin∅
M = 10 x 1000 sin30 °
M = 10,000 ( 0.5 )
M = rFsin∅
M = 10 x 1000 sin45 °
M = 10,000 ( 0.707 )
M = rFsin∅
M = 10 x 1000 sin90 °
M = 10,000 ( 1 )
M = 5,000 Nm M = 7,071 Nm M = 10,000 Nm
This calculation is provided to illustrate the use of the formula , and also to show that the angle between a force and a lever arm has a significant impact on the bending moment applied by that force .
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