# Building a workbench, considering building one.

• posted on February 16, 2014, 8:10 pm
See:

It's all in the engineering.
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Jeff

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• posted on February 17, 2014, 2:52 am
On 2/16/2014 12:10 PM, woodchucker wrote:

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 or Fy*H - Fsc*W = 0 or 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.
Dan

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• posted on February 17, 2014, 3:52 am
On 2/16/2014 9:52 PM, Dan Coby wrote:

Well not exactly the yellow grip tite is the stretcher. But to represent 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.

You are taking this too literally, it's to show the benefit of a middle stretcher vs a bottom stretcher.

That really is not supported in reality. I can take a bench with the same size stretcher and move it from the middle to the bottom and cause the bench to rack. All things being equal.

Certainly it is. I agree that wide stretchers are necessary, but if I 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.

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Jeff

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• posted on February 17, 2014, 4:17 am
On 2/16/2014 10:52 PM, woodchucker wrote:

I'll build a small mortise and tenon setup (2) to show middle pressure vs end pressure.
This is about leverage and lever arms. A longer lever has more ability to break the joint down, that a shorter one.
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Jeff

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• posted on February 17, 2014, 5:01 am
On 2/16/2014 8:17 PM, woodchucker wrote:

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.
Dan

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• posted on February 17, 2014, 4:51 am
On 2/16/2014 7:52 PM, woodchucker wrote:

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.
Dan

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• posted on February 18, 2014, 3:46 am
On 2/16/2014 11:51 PM, Dan Coby wrote:

Ok, added another video.
http://youtu.be/F1MlotN9ssY

http://youtu.be/F1MlotN9ssY
you try it.
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Jeff

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• posted on February 18, 2014, 5:15 am
On 2/17/2014 7:46 PM, woodchucker wrote:

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 bench.
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.
Dan

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• posted on February 18, 2014, 5:44 am
On 2/18/2014 12:15 AM, Dan Coby wrote:

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.
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Jeff

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• posted on February 18, 2014, 6:52 am
On 2/17/2014 9:44 PM, woodchucker wrote:

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 joint correctly.

> looking to minimize the leverage of the top or the floor. putting it

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 leg.
Dan