You correctly point out that using a wide stretcher will make for
a stiffer bench. If the stretcher is rigidly connected to the leg,
then its contribution to the stiffness of the bench is proportional
to the square of its height. So doubling the height of the stretcher
will quadruple its effect on the bench's stiffness. However other
things like the size of the upper apron, legs, top, and the rigidity
of the joints also affect stiffness. Diagonal braces can also have
a major impact.
However there are a few problems with your analysis when it comes
to the location of the stretcher.
If I am interpreting your video correctly, the magnet represents the
stretcher and the bar of the magnetic base represents the leg of the
bench. The joint between the stretcher and leg is free to pivot.
Real benches are not normally built with joints that pivot. The
rigidity of many bench designs comes from the rigidity of the joints
between the legs, aprons, top, and stretchers. If none of these joints
were rigid then the bench would simply fall down due to gravity. You
can build a bench with flexible joints but you need to have some form
of diagonal bracing to convert some of the parallelograms into triangles.
Lets do a simple analysis of a bench leg. To make things very simple,
I am going to have a bench with only a single leg and without a top or
an apron. Obviously this is not very representative of real benches.
However this is somewhat equivalent to what you were showing in your
video. I am also going to replace the stretcher with just a couple
of forces. When the leg tries to rack (twist) it will apply a force
at the top of the stretcher which is trying to push the stretcher to
the right and a force at the bottom of the stretcher which is trying
to pull the stretcher to the left. The stretcher pushes back on the
leg with equal and opposite forces so the stretcher will push back to
the left at the top and pull to the right at the bottom. With a rigid
joint between the stretcher and leg, the top of the stretcher will be
in compression and the bottom will be in tension as the stretcher tries
to keep the bench from racking. (In a real joint the forces will be
distributed across the joint and vary smoothly from compression at the
top to tension at the bottom.)
Fy ----> |---|
| | |
| | |
| | |
| | |
| | |
| |<--- Fsc |
| | | |
| | W H
| |---> Fst | |
| | | |
| | | |
| | | |
| | S |
| | | |
| | | |
|---| <---- Ff | |
In this diagram:
Fy is the force you are applying to bench while working
Ff is the friction force at the bottom of the leg at the floor
Fsc is the force due to trying to compress the top of the stretcher
Fst is the force due to applying tension to the bottom of the stretcher
S is the height of the bottom of the stretcher
W is the width of the stretcher
H is the height of the top of the bench
If we apply Newton's second law to the stretcher we see:
Fsc - Fst = 0
This means that Fsc = Fst. I.e. there is as much tension force at the
bottom of the stretcher as compression at the top. If these forces did
not balance the the stretcher would start to accelerate.
If we apply Newton's second law to the leg we see:
Fy - Fsc + Fst - Ff = 0
Since Fsc = Fst we get Fy - Ff = 0
Which means Fy = Ff
This is what we would expect. There needs to be enough friction at the
floor to keep the bench from moving.
Now lets look at the torques being applied to the leg. Please
remember that a torque is determined by both the force being applied
and the distance (moment arm) from the reference point. Lets use
the floor as our reference point. This gives us:
Fy*H - Fsc*(S + W) + Fst*S - Ff*0 = 0
Substituting Fst = Fsc we get:
Fy*H - Fsc*S - Fsc*W + Fsc*S = 0
Fy*H - Fsc*W = 0
FY*H = Fsc*W
This last equation tells us that the torque applied by Fy*H is equal
to the counter torque being applied by the stretcher Fsc*W.
PLEASE NOTE THAT THE HEIGHT OF THE STRETCHER ABOVE THE FLOOR DOES NOT
APPEAR IN THE FINAL EQUATION. The height of the stretcher dropped out
of the equation because the tension at the bottom of the stretcher is
balanced by the compression at the top of the stretcher.
A final note: The width of the stretcher does determine the magnitude
of the tension/compression in the stretcher needed to balance the torque
due to the forces being applied to the bench. Since I simplified the
analysis to just two forces in the stretcher, the magnitude drops
inversely with the width instead of the square of the width.
Your bench is stiff due to your building it with wide stretchers and
rigid joints and not due to the stretchers being higher above the floor.
the lever principle simply, they are not connected. The point is that
even w/o locking them together solidly, even if you had loose bolts the
middle would offer less leverage and the bottom, would be too much leverage.
stretcher vs a bottom stretcher.
same size stretcher and move it from the middle to the bottom and cause
the bench to rack. All things being equal.
put that wide stretcher at the bottom I have more leverage and therefore
that joint will fail eventually. It will cause compression of the fibers
and the socket will widen if I were to put that stretcher at the very
bottom like I have seen on benches.
See my comment in another post about there being two lever arms to consider.
The first is the lever arm between the top and the stretcher. The second is
the lever arm between the bottom and the stretcher. As you make one shorter,
the other is getting longer. The resulting torques remain constant.
Another way to look at the leverage issue. There are forces from both the
your efforts at the top of the bench and counter balancing reaction force
from the floor being applied to the bottom of the leg. As you move the
location of the stretcher up, you are shortening the lever arm between the
top of the bench and the stretcher. However at the same time, you are
lengthening the lever arm between the force reaction at the bottom of the
leg and stretcher. The decrease in the top torque is balanced by an
increase in the bottom torque. The result is the the torque applied to
the stretcher/leg joint is constant.
If the stretcher were attached to something fixed like a wall then I
would agree with what you have been showing. Indeed if you are fastening
your bench to a wall then you should fasten it near the top of the
However the stretcher is not attached to a fixed object. It is simply
attached to another leg of the bench. That leg can also move and flex.
Let me give you another example. Take your argument to the extreme and
move the stretcher all the way to the top of the bench. If I understand
your arguments then since this would produce a near zero lever arm then
the bench would be extremely rigid.
However this is basically the same situation as most tables with an apron
around the top. (The stretcher in this case is the same as the apron.)
However to make a table rigid, table makers have to go to great lengths to
make the leg/apron joint very strong and rigid. The problem is that the
table leg makes a very long lever arm connected to the bottom of the
apron/stretcher. You have simply shortened one lever arm and lengthened
another when you move the stretcher.
A table is not meant to overcome racking forces. A workbench is. The
floor is now the top when you put the apron at the top. You are looking
to minimize the leverage of the top or the floor. putting it closer to
the middle does this.
I don't understand why you say if it is fixed to a wall.
I'm at a loss to understand that.
Both benches and table have to overcome racking forces. Take a look at
the joints between a table leg and table top and apron sometime. These
can be massive on a large heavy table. Check out any book on table
construction and you will see the emphasis that is placed on making this
To keep the bench from racking, you have to counteract the torque created
when you push on the top. This applied force also creates an equivalent
force at the floor. Both of those forces are trying to twist the joint
between the stretcher and the leg. You are not including this second
force in your demonstrations or in your arguments.
Moving the stretcher up or down simply increases the torque from one
force while decreasing the other.
When the stretcher is at the bottom then:
torque = F*L + F*0 = F*L
Where F is the force applied and L is the length of leg.
When the stretcher is at the top then:
torque = F*0 + F*L = F*L
When the stretcher is in the middle then:
torque = F*L/2 + F*L/2 = F*L
Please note that the result is the same in each case.
I am trying to say that your analysis would be correct if the other end of
your stretcher were connected to a fixed object instead of another bench
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