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Cope-and-stick joinery is used in cabinet doors more than any other application.
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Cope-and-stick is a modern, machine-cut joint for right-angle flat frames intended to house a panel. It's the linchpin of present-day frame-and-panel constructions. Think doors. But also think casework.
Back in the 18th century, the maker of a basic frame-and-panel door had to chop out four mortises, saw four tenons, and plow a panel groove and a decorative profile in each frame member. Still not done, he'd have to trim away both the groove and the profile from the mortise shoulders, and miter the profiles in the bargain, just to assemble the frame and seat the joints.
The door would be strong, provided the maker fit everything just right. But it sure was a lot of work.

Nowadays, the frame for a basic door is completed with two cuts — the cope cut on the ends of the rails and the sticking cut made along each edge of both stiles and rails. The sticking cut plows the panel groove and the profile simultaneously. The cope cut forms a stub tenon on the end of the rail, along with the "negative" of the profile. The cope fits quite snugly into the sticking, and you have a strong right-angle joint with what appears to be a miter joint between the profile on the rail and the one on the stile.
Had I used traditional mortise-and-tenon joinery to construct the side assemblies of this chest of drawers, the wide rails would have required multiple tenons to prevent them from splitting. Using cope-and-stick joinery saved a lot of work.

When the original of this chest was built, the maker had to cut and fit more than two dozen mortise-and-tenon joints to assemble the front, end, and back frames. Router-cut cope-and-stick joinery simplified the construction of this new version of the original without sacrificing strength or durability.
Cope-and-stick joinery is cut and fit quickly and easily. You need a bit or pair of bits and a mid-power, table-mounted router to drive them. With a very little practice, you can set up and cut the joinery for a door in less than 15 minutes.

Cut quickly with special router bits (or shaper cutters), the cope-and-stick joint is the standard for cabinet doors. Properly fitted and glued, it is a strong, enduring joint, despite what Old World traditionalists often claim.
Preparing the Stock
By industry convention, cope-and-stick bits are designed for 3/4"-thick stock. Because this stock thickness is standard in most areas of the United States and Canada, you shouldn't have problems if you buy dressed stock.
The usual caveats about your stock apply: Use defect-free, straight-grained lumber.
I know, I know. We usually take such provisos with a grain of salt, and you can get away with using slightly bowed stock for a frame-and-panel unit so long as it isn't a door. If the wood in a frame-and-panel unit is bowed, the unit will be bowed. If the unit is a structural part of the case, it will be anchored to other elements that will pull it into line and hold it there. But if it is a door, it won't hang flat, and that you won't be able to conceal.
Dress the chosen stock. You ought to prep extra stock, because you need several pieces for testing the setups. To me, the important thing is to plane all the stock to a consistent thickness. I achieve consistency by planing all of it at the same time.
Rip the stock to width, then crosscut the parts to length. When you cut the rails, you have to account for the profile width. For example, if you are making an 18"-wide door and using 1 3/4"-wide stiles, the distance between the stiles is 14-1/2". But the rails must be long enough to overlap the sticking profile. If it's 3/8" wide, then you need to add 3/4" to the length of the rails (3/8" for each stile, or twice the width of the profile). While this may be awkward to verbalize, it is quite easy to visualize; see the drawing.
Cope-and-Stick Bits
Cope-and-stick frame joinery can be cut with any of three styles of router bits.
Two-Bit Set
Most common is the two-bit set. One is the sticking bit, which cuts the groove and profile simultaneously. The other is the cope bit, which forms the stub tenon and the cope.
Typically, the profile cutter is an integral part of the shank, while the bearing and the slot cutter are separate parts, secured on an arbor projecting from the bit. The stick cutter has a bearing mounted on the tip, so it can be used to make curved cuts, as is required to make arched rails. The cope bit has the bearing mounted between the profile cutter and the groover. Usually, the bits come all set up to produce a properly fitted joint. It'll be a slip fit, not particularly tight. The logic is that glue will swell the wood and, when swollen with glue, the fit will be just right.
A few makers include two or three shims with the bits, to give you a little control over the fit. Most often, they become useful after the bits have been resharpened a time or two. You use them to alter the thickness of the tenon. To tighten the joint's fit, you add a shim or two between the two cutters on the cope bit's arbor, thus increasing the tenon thickness. To loosen the fit, you remove shims from that location.

The two-bit set is the most common way to cut cope-and-stick joinery. The decorative profile and the panel groove, collectively known as the sticking, are formed in one pass using the sticking bit (resting on the tabletop). The cope cutter — in the router collet —machines the stub tenon and profile cope on the ends of the rails and muntins.
The overall advantage of the two-bit set is that you don't have to break down the bit to switch from one cut to the other. For those who do lots of frame-and-panel work, this means that two routers can be set up, one for each bit. This makes the switch from sticking to coping even easier.
Reversible Assembly
A less expensive — and also less convenient — bit is the reversible assembly. You get an arbor with an integral shank, a bearing, and separate profile and slot cutters. Instead of swapping bits to switch from sticking to coping, you switch the positions of the cutters on the arbor. To make the cope cut, you put the slot cutter on the bottom, then the bearing, and finally the profile cutter. For the sticking cut, the profile cutter is on the bottom, topped by the slot cutter and the bearing. This may not be the right match for you if you're at all random or the least bit fumble-fingered.
This kind of bit harbors a couple of "gotcha!"s. One is in the swapping of cutters. Don't turn either one over. The router only spins one direction, and when it is mounted in a table, you see the bit turn counterclockwise. If you inadvertently flip the cutters over (which is easiest to do with the slotter; the profile cutter always faces up), the cutting tips will be facing the wrong way, and the bit won't cut. (Smoke? Yes, but not cut.) The second gotcha! is the workpiece orientation. With a matched set, the bits are configured so the workpiece orientation is the same for both cope and sticking cuts (usually face down). With a reversible assembly, you cut the sticking with the face down, but you do the copes with the faces up.

The reversible bit produces both the sticking and cope cuts. The appearance of the cuts and the strength of the joint equal those produced by two-bit sets. It's generally less expensive, but somewhat more onerous to use.

The reversible bit's onerousness stems from the need to rearrange the cutters on the arbor when alternating between cope cuts and sticking cuts. The typical assembly consists of an arbor (in the router collet here), a profile cutter (top left), a slotter (bottom left), some shims (on the slotter), a pilot bearing and a nut. You supply the wrench.

Set up for a cope cut. Drop the slotter onto the arbor (making sure the cutting tips are oriented to cut on a counterclockwise rotation). Then add the bearing, shims as necessary, the profile cutter, and the nut to secure everything.

For a sticking cut, drop the profile cutter onto the arbor first. Then add shims as necessary, and the slot cutter. The bearing and nut go on last.
Stacked Bit
The third style is the stacked bit. This is a single bit with both cope and stick cutters on the same body. The least expensive versions have the cutting edges laid out in as compact a way as possible, so you have that face-up/face-down gotcha to look out for. This version has a single bearing.
Top o' the line versions eliminate the gotcha, but to do it, they have to make the bit considerably larger, and they have two bearings: one above the cutters, one below. With some brands, you orient the work face down; with others, face up.
For the most part, cope-and-stick cutters, of whatever style, are 1-1/2" in diameter. Those high-end stackers are massive bits, however, as much as 2" in diameter and 4" tall, overall.
Any of these three types will cut the frame joinery just fine. You should be able to choose from several profiles, typically an ogee profile, a quarter-round profile, or a bead. Some makers have an even grander assortment.

The stacked cope-and-stick bit tends to be bulky, because it has both cutting profiles and two bearings on a single shank. Nevertheless, it can be powered by a midsized router, running flat out. It produces the same cuts as matched pairs, but with a little less setup, since you merely raise or lower the router to switch from one cut to the other.
Make a Setup Block
If you just purchased bits to do cope-and-stick joinery, I think you should spend a little time getting familiar with it. Take as much time as you need to make both cope and sticking cuts. Here's your goal: a setup block with an edge sticked and an end coped. With a setup block, you take the guesswork out of setting up your cutters. Not only are bit elevations easy to set without a lot of test cuts and fine-tuning, but it also puts you in control of the appearance of the finished joints.
From an aesthetic viewpoint, the sticking cut is paramount, because it establishes the appearance of the joinery. But operationally, the sticking usually is cut after the rails have been coped. Unfortunately, doing the copes first pretty much consigns you to accepting whatever appearance you get. The two cuts do have to match.
A setup block solves the problem.

A setup block can take the worry out of making the critical bit height settings, especially for those who don't do the the job frequently. For my pair of Amana bits, which aren't supplied with setup blocks, I made one that's essentially a short piece of a rail. The end is coped and the edge is sticked.
Make sticking cuts on sample stock, experimenting with the bit elevation. When you have a setting that's optimum to your eye, mark it and set it aside.
Switch cutters and rout some copes, fitting them to your sample with the optimum sticking profile. When you've got the perfect fit, cope one end of the piece with the optimum sticking cut. Now you have one piece of wood with the optimum profiles, perfectly matched to each other.
Trim the block to a tidy size for storage with your bits. Next project, you can set the cope cutter quickly using the setup block, knowing you won't be disappointed by the appearance of your finished assemblies.
Cutting the Joinery
The usual routine is to cope the rail ends first, then stick all the stiles and rails.
Cope the Rail Ends
Before doing any setup or cuts, reflect on the fact that the cope cut is cross-grain. That means you need to back up the work to prevent splinters from being torn from the back edge by the cutter. Depending on the size and number of rails, I'll usually gang them up and feed the lot of them past the cutter, pushing them along the fence with a square scrap. The pusher acts as a backup, preventing the splintering.
Some woodworkers prefer to use a more formal guide, like the coping sled, and to cope the rails one at a time. No shame in that at all. I bring this up now because the sled does impact the bit height setting: you have to accommodate the sled base thickness.
The first setup task, of course, is the bit. Place the cope cutter in the router's collet. To do this, you'll probably shift the fence back from the bit opening in the tabletop.
Once the bit is secured in the collet, establish a height setting using your setup block. You can tuck it into the bit and adjust the bit up and down. If you are using a coping sled, you must, of course, set the block on the sled when gauging the bit elevation.

Setting the height of the coping bit is easy if you have a setup block. You want to be able to slide the block on and off the cutter. If the cutter pinches the block against the tabletop, it's too low, and, of course, if it lifts the block off the tabletop, it's too high.
Set the fence next, positioning it with the face tangent to the pilot bearing.

The face of the fence must be tangent to the bit's pilot bearing. Slide a straightedge back and forth across the fence and bearing. You want the straightedge to graze the bearing without turning it.
Of course, you need to make a test cut. Fit the test piece to the setup block's sticked edge. If necessary, adjust the bit up or down to get the surfaces flush.
With your setup tested, set hold-downs. I use a featherboard or two to keep the work tight to the tabletop and ensure the cut is consistent from rail to rail. Position the featherboards just fore and aft of the bit, where you need the pressure. Then you are ready to cut.

Make a test cut in scrap material of exactly the same thickness as the rail-and-stile stock. Fit the test cut to your setup block and assess the fit. The faces of the two parts should be flush, and the profiles should fit tight together. If need be, make an adjustment to the bit height or the fence position, then make another test cut.
Pay attention when you turn the rails to cope the second end. You want to turn them, not flip them. Mark the face that's supposed to be up as you make the cope cut. Before you cut, look for the mark.

Once the setup is dialed in, rout the copes on the ends of the rails. I don't usually use a coping sled. Ganging up a brace of rails and a longish backup helps keep the rail-ends square against the fence and expedites the work to boot. The tandem featherboards, positioned so the direction pressure is not on the bit but on either side of it, help keep the work against the tabletop.
Rout the Sticking
Chuck the sticking bit in the router collet, and adjust its height. To do this, you'll have to shift the fence out of the way. Make an initial adjustment using your setup block. After the fence is reset, you can make a test cut. Fit your test piece to one of the coped rails; chances are, if you set the bit carefully with the setup block, you'll be dead-on. But if some adjustment is necessary, make it and run a new test piece across the bit. Keep adjusting and testing until you have the fit you want dialed in.

Here's one of two practical ways to set the height of the sticking bit. Use a coped rail and adjust the bit's slotter to align with the stub tenon.

Another practical way to adjust the sticking bit height is with a setup block. Here I'm using a plastic block provided with the bits by the manufacturer. I hunker down so I can sight across the tabletop to the bit. From this position, it's convenient to reach under the table to adjust the router.
Position the fence, setting it so the facing is tangent to the bit's pilot bearing, and make a quick test cut that you can fit to your setup block on an already coped rail.

Make a test cut before setting the featherboards. Use a short piece of the working stock for this.

Check the fit of this test cut to the cope on a rail. (Here, I'm using the cope cut in the setup block.) You want to see that the faces are flush and that the sticking cut is full-depth, as is the case here.
Before actually routing the workpieces, make a zero clearance fence opening for the bit. This will minimize chipping along the edge (though it may not completely eliminate it).
Chipping is a fugitive phenomenon. I've done long stretches of sticking without seeing any more than scattered and inconsequential chipping. But occasionally — always when it's most consequential, of course — milling more than a few inches without chipping seems impossible.
Chipping along the edge of the profile is usually a result of susceptible wood colliding with a delicate profile and an aggressive feed. By susceptible wood, I mean particular boards with gnarly, swirling, undulating grain that runs off the edge being worked. Or even boards with fairly straight grain that happens to run out the working edge. It's just prone to chipping.
The best thing you can do to minimize chipping is to provide support for the wood right up to the cutting edges of the bit. Craftsmen who use hand planes control surface chipping by closing down the tool's throat, crowding the blade and leaving the barest slit for the wood shaving. This is the same idea: the zero-clearance opening. But you'll want that zero clearance on the infeed side; on the outfeed side, tight clearance between cutter and fence face is less important.
If your router table fence has split facings, you can slide the infeed face onto the spinning bit to create the zero clearance.
A universal alternative is to clamp a thin facing to the fence, with just a small notch or hole for the fence. I use 1/4" MDF, plywood, or hardboard. Drill or saw a hole for the pilot bearing and arbor of the bit — a starting hole — and cut through with the bit. You do have to reset the fence so its new face is tangent to the bearing.

Use a thin hardboard or plywood facing for the fence, and cut a zero-clearance opening for the bit. Cut an initial opening for the bearing and arbor. Clamp the outfeed side against the fence, but flex the rest to hold it clear of the bit. Switch on the router and roll the facing onto the fence, while the bit cuts through the edges of the opening.

The zero-clearance facing.

Position the fence tangent to the bit's pilot bearing with a straightedge or small rule. Shift the fence until the rule is just clear of the bearing.
Set your hold-downs after you've completely tweaked the setup. Use a featherboard or two to keep the work tight to the tabletop. Position the featherboards just fore and aft of the bit, where you need the pressure.

With the fence and bit all set, clamp featherboards to the fence. I use a tandem featherboard, which is simply two separate featherboards joined in parallel with a cleat. The boards are positioned with the bit in the gap between them. I use a scrap of 3/4" plywood as a spacer to set the boards because it's about 1/32" thinner than my frame stock. That translates into firm but not excessive featherboard pressure.

Routing the sticking is a straightforward enterprise, at this point. While you can feed each stick through the cut with your hands, most woodworkers are more comfortable using a pusher. Mine has a V-groove in the bottom edge and a heel at the end to catch the end of the workpiece. Pushing thus forces the workpiece against the fence as well as through the cut. The featherboards keep the work against the tabletop and shield the bit as well.

Once the rails are coped and all the parts are sticked, the frame is ready for the panels.
Making a Coping Sled
Prefer to guide your cope cuts with a device more "formal" than a scrap block of wood? What you need is a coping sled.
This simple sled is used in conjunction with the fence. That is, there's no slide for a miter gauge slot. Instead, the sled is pushed against the fence at the same time it is advanced along the fence.
The construction is evident in the drawing.
A couple of construction notes: I recommend the base be 1/2" material, because it provides a better backing for the toggle clamp. I know I have a propensity to want that clamp to snap, making me feel it is holding the work tight. Maybe you will, too. But the workpiece in turn is being pressed with great force against the base. If the base is thin and bendy, it'll distort and you won't get consistent, accurate cuts with the sled.
The handgrip can be a block of wood, a fat dowel, a turned shape, or, as shown, a copy of a hand plane's tote. I cocked the tote at an angle, so when you push, you are applying pressure toward the fence as well as along it.
The sled's backup fence will be cut by the cope bit the first time you use it. On the plus side, this allows you to use the sled to set the cope bit height.
On the negative side, it behooves you to make a different sled for each different cope-and-stick set you use; otherwise, you'll get tearout as the bit exits the work and cuts into the sled. And the sticking cut will remove that tearout only at one end of the rail.
An alternative is to use a separate backup strip between the sled's fence and the rail. Hold it in place with carpet tape.

A coping sled holds the rail securely for the cut and provides some backup against tearout. It gives you a comfortable, sure grip on the work and keeps your hand out of the bit's way. A tradeoff is that the bit has to be extended an extra 1/2" above the tabletop, which isn't a simple matter on every router table.