You can do the job without disassembling the chair; all you need is a scarf joint.
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The chair you see above suffered from a common problem: a broken stretcher rail. If the chair's joints were loose, we would have disassembled it, and made a solid replacement stretcher. This chair's joints were sound, however, so we wanted to repair the stretcher without taking the chair apart. Here's how we did just that.

Match the wood

First, decide what species of wood you need for the new stretcher. Although the seat and back of our chair are oak, the stretchers are maple. We measured the diameter at 58 ", as shown in the photo, below, and bought a standard maple dowel as our workpiece.

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Fixing a round stretcher rail isn't as difficult as it might look. Start by measuring the diameter and length of the piece.

Remove the old stretcher, measure the space between the legs, and add the depths of the tenon holes. Subtract 18 " to make installation easier. Cut the dowel to that length.

Next, install a 14 " core-box bit in your table-mounted router, and use a tenon on the old stretcher as a gauge to set the bit's height. The setup shown in the photo below serves as a quick way to shape the new tenons. Set the router-table fence to act as a stop that determines the length of the tenon. Secure an auxiliary fence to your miter gauge, locate the gauge so that your dowel is centered on the bit, and clamp it to the table.

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Push the stock all the way to the fence, and rotate it once counterclockwise. Then nibble away the rest of the waste.

Cut the scarf joint

After making the tenons, go to the bandsaw, and cut a long, sharply angled scarf joint across the dowel. Ours was a 15° cut 214 " long. Cut as straight as possible, making sure not to rotate the dowel, and then smooth the sawn surfaces on 150-grit sandpaper, as shown in the photo below.

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Place self-adhesive sandpaper on a flat surface, such as your workbench, and carefully smooth the scarf joint.

Now, coat the tenons with yellow glue, insert them into the holes, apply glue to the scarf joint, and fit it together, as shown in the opening photo, above. Wrap it with masking tape to serve as a clamp. After the glue dries, sand the joint.

All that remains is to match the replacement piece to the chair's finish. Clean the chair, then use a dowel scrap to test stains until you find the color you want.