I would like to cover the ceiling a porch with something. What is
there now are 2X4 appearance "beams" spaced 16 inches apart. I like
the look of bead board, but am thinking that the 1/4" thickness isn't
substantial enough. It may end up sagging.

I don't think 1/4" plywood will sag with 16" supports, but I've been
wrong before.

So what I thought I might do is cover the area with plywood first,
then attach the bead board to the plywood.

If you want extra support, your choices are sheet goods and blocking.
Blocking will be cheaper, but more labor.

Or, attach (glue?) the
bead board to the plywood first then install over the ceiling.

The construction adhesive is unneeded, but won't hurt. Don't use _only_
adhesive. You need nails.

Good idea, bad idea?

My suggestion: Paint both sides of the 1/4" plywood, then nail it to
the joists. Fill the joints and nails, then touch up the paint.

--
Steve Bell
New Life Home Improvement
Arlington, TX USA