econ41
Senior Member
Another way to approach it is to take a simple example and analyse the vectors.
IF we have a rigid beam loaded with 2W at the centre there will be a downward force of W on each end support. And no pull in force.
Replace it with a rope. Still holding the same 2W load. Depending on the length of rope the support to weight - both sides will be a straight line (zero weight rope) and WHAT EVER the length of rope will determine the angle of each side. BUT the downwards force vector on each support will still be W
Now if the rope sags so that each end slopes downwards at 45 degrees:
1) The downwards force vector on each support will still be W;
2) The horizontal force vector will also be W - inwards; AND
3) The tension in the rope will be 1.414W
Pure geometry of the vectors. And at 45 degrees inwards pull = half the centrally applied download.
Now shorten the rope so that it slopes at 10 degrees below horizontal:
1) The downwards force vector on each support will still be W;
2) The horizontal force vector will now be 5.67 W - inwards; AND
3) The tension in the rope will be 5.76 W
Shorten again to 5 degrees slope:
1) The downwards force vector on each support will still be W;
2) The horizontal force vector will now be 11.43 W - inwards; AND
3) The tension in the rope will be 11.47W
At 2 degrees:
1) The downwards force vector on each support will still be W;
2) The horizontal force vector will now be 28.63W - inwards; AND
3) The tension in the rope will be 28.65W
At 1 degree - very close to horizontal:
1) The downwards force vector on each support will still be W;
2) The horizontal force vector will now be 57.29W - inwards; AND
3) The tension in the rope will be 57.29W
All that rope tension from a 2W load at the centre. ~30 times multiplier.
And a direct consequence of the vector geometry. Which shows why:
Oystein's hammocks needed a stronger rope; AND
Why JS Orling can see his anchor chain move the boat.
PLUS my exploitation of child labour worked - and my grandson could bend in that "perimeter column" by pushing the taut rope sideways.
AND why the natural - original - unaugmented weight of ONE WTC floor could exert a large inwards pull once the joists got hot enough to act like flexible ropes.
IF we have a rigid beam loaded with 2W at the centre there will be a downward force of W on each end support. And no pull in force.
Replace it with a rope. Still holding the same 2W load. Depending on the length of rope the support to weight - both sides will be a straight line (zero weight rope) and WHAT EVER the length of rope will determine the angle of each side. BUT the downwards force vector on each support will still be W
Now if the rope sags so that each end slopes downwards at 45 degrees:
1) The downwards force vector on each support will still be W;
2) The horizontal force vector will also be W - inwards; AND
3) The tension in the rope will be 1.414W
Pure geometry of the vectors. And at 45 degrees inwards pull = half the centrally applied download.
Now shorten the rope so that it slopes at 10 degrees below horizontal:
1) The downwards force vector on each support will still be W;
2) The horizontal force vector will now be 5.67 W - inwards; AND
3) The tension in the rope will be 5.76 W
Shorten again to 5 degrees slope:
1) The downwards force vector on each support will still be W;
2) The horizontal force vector will now be 11.43 W - inwards; AND
3) The tension in the rope will be 11.47W
At 2 degrees:
1) The downwards force vector on each support will still be W;
2) The horizontal force vector will now be 28.63W - inwards; AND
3) The tension in the rope will be 28.65W
At 1 degree - very close to horizontal:
1) The downwards force vector on each support will still be W;
2) The horizontal force vector will now be 57.29W - inwards; AND
3) The tension in the rope will be 57.29W
All that rope tension from a 2W load at the centre. ~30 times multiplier.
And a direct consequence of the vector geometry. Which shows why:
Oystein's hammocks needed a stronger rope; AND
Why JS Orling can see his anchor chain move the boat.
PLUS my exploitation of child labour worked - and my grandson could bend in that "perimeter column" by pushing the taut rope sideways.
AND why the natural - original - unaugmented weight of ONE WTC floor could exert a large inwards pull once the joists got hot enough to act like flexible ropes.
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