You'll see expansion of the wood until either the steel frame ruptures (may or may not, depending on its structural inegrity & mass) or the wood fibers crush (as in the eye of an axe), and you'll see contraction such that the wood draws away from the metal frame, leaving a gap around the rim (as in the eye of a loose-handled axe). It's possible, if that wood panel is tightly bonded to the steel frame, that it might crack somewhere in the middle, too - I constrained a piece of oak not much different in size, once, and was rewarded by a good 1/8" crack for my efforts.
For choice of woods... there are large charts all over the Web listing various woods & their expansion/contraction radially and circumferentially (usually called "tangentially"), movement in those two directions being different in nearly all woods.
Certain mahoganies are highly prized for their extremely low movment - that makes them exceptionally well-suited for such as casting.
The better you can seal that wood, the less will be (because the slower will be) its humidity-related movement. If the steel-framed wood is exposed to extremes of temperature, there will be some difference in thermal expansion coefficients, too... but unless you're talking about something LARGE it's going to be pretty negligible.
If ya really gotta' constrain it, if ya can't come up with any way around it, then try & use a low-movement wood and seal the SNOT out of it, every way you can. Hardening oil, followed by shellac after the oil is fully cured, followed by (at least) several coats of lacquer.
Just how heavy a frame are we talking about? A steel frame can't be built that allows NO "give", but if it's built really heavily it may allow very little.
Don't underestimate the power of expanding wood - that expansion was used to split large pieces of granite off quarry walls; they'd drill holes, drive dry wooden wedges into the holes, and pour water on the wedges. SHOOM.