Open the sofa bed and pull up the mattress. Set aside the mattress for the duration of your repair. Measure the frame of the sofa bed.
Divide your plywood into three pieces which equal the measurement of the frame of the entire sofa bed. Cut the plywood so you have three pieces that will sit horizontally on your bed frame. The pieces should fit snugly down into the frame and should not have room to move around.
Drill three evenly spaced holes along the one edge of two of your pieces of plywood. Drill three holes along both edges of your third piece of plywood that line up with the holes in the other two pieces of plywood. The third piece will be going in the middle of the other two.
Lay the pieces of wood on the floor so that the piece with holes on each side is in the middle. Lace your wire through each set of holes to connect your wood pieces together. This will allow your frame to fold easily. You should leave at least one inch of slack in the wire to give the pieces enough room to fold over each other.
Trim any wires that are left sticking out and lay the plywood onto the flat sofa frame. Place your mattress back on top of your newly reinforced frame and test it out by laying on the mattress. Fold up the sofa bed to make sure that you have left enough space between the pieces of plywood for the bed to fold properly.