The calculation of bending moment provided on the previous page can be combined with our previous calculation of the resultant force applied to a pulley to quickly analyse and compare three different methods of installing a lowering pulley at the end of a 5m branch with 45 ° inclination .
In the first example , the pulley is secured to the branch end .
The loaded rope running through the pulley is secured to an anchor such that the two sides of the rope are parallel .
In the second example , the loaded rope running through the pulley is secured to an anchor fixed to the tree , such that the fall side of the rope runs parallel to the branch .
In the final example , the loaded rope running through the pulley is secured to an anchor fixed to the top of the tree , such that the resulting force on the pulley acts directly down the branch as compression .
M = r ( 2F ! cos / θ 2 1 ) sin∅
M = 5 ( 2000 ) sin45 ° M = 7 . 1 kNm
M = r ( 2F ! cos / θ 2 1 ) sin∅
M = 5 ( 1850 ) sin22.5 ° M = 3 . 5 kNm
M = r ( 2F ! cos / θ 2 1 ) sin∅
M = 5 ( 1,410 ) sin0 ° M = 0 kNm
Where :
M is the bending moment applied at the axis , r is the distance from the axis to the force ,
F is the magnitude of force being applied , a is the angle between the two legs of the rope , and Ø is the angle between the resultant force and the lever arm of the branch .
This example clearly shows how a small investment of planning when installing tree anchors and rope systems can significantly reduce the magnitude of force and almost eliminate bending moment .
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