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I'm in the process of making a work bench, mostly following the Nicholson style, and with a vise like this one.

The only difference is that I'm going to be using hard wood for the vise chop, and I want to put dog holes in the top of it.

Because of cost, I'll probably have to laminate 2 pieces of wood together to form the chop, rather than using a single thick piece.

My question is this: given that I want to put dog holes (3/4") in the chop, how thick should I make it? I was thinking 2" - is that enough?

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    Remember that the thicker the chop, the less space there is between the jaws... – keshlam Jan 9 '17 at 22:48
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Because of cost, I'll probably have to laminate 2 pieces of wood together to form the chop, rather than using a single thick piece.

This can be stronger and more stable than one solid piece so it's often a good way to go anyway. If the piece or pieces you get are flat-sawn, alternate the grain as in the vice bottom-left of the image in this Answer.

My question is this: given that I want to put dog holes (3/4") in the chop, how thick should I make it? I was thinking 2" - is that enough?

Yes 2" would be more than sufficient. You can install dogs in vice linings much thinner than this but for the style of face vice you're making thick chops are usual.

Lining
Since you are going with hardwood you should give thought to lining the chop with leather to help prevent marring of workpieces in softer woods.

Leather linings substantially increase grip at more modest clamping pressures anyway, reducing wear and tear on the mechanism. Not a big deal here but an important consideration with cheaper modern all-metal vices that aren't made as robustly as their vintage counterparts.

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As you tighten the vise your dog will place all the pressure from tightening the vise on the outerlayer of wood in the chop. I would recommend at least one inch of hardwood onm the outside face of the chop. The inside face need not be as thick since it will not be compressed during tightening, so about 1/2" should do it. All told this would result in 2 1/4" minimum chop. I would align my pieces with the grain traveling parallel with the vise travel and use plenty of glue when assembling the chop.

The vise installation in your video shows the chop installed along the front of the bench. When planning my own bench I had both a front leg vise and a traveling wagon vise that extends side to side. The wagon vise had dogs that extend along the entire front of the vise. This allows me to mount short or long boards and easily plane them along their entire length, so it gets plenty of use. I chose not to put dogs in the leg vise since any board mounted using the dogs would extend from front to back of the bench and is not convenient for most prep work including planing.

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