I've read on all of these ships rather extensively, and would like to offer up the following comments
Design & armor:
First off, the Bismarck wasn't really a new design, but rather an enlarged, up-armored and engined Baden of WWI vintage. She was designed and built to fight at shorter ranges in the foggy waters of the baltic and north seas, and her armor layout reflected this, and the German experiences in battle in WWI. Basically, her armor and guns were laid out in such a way to withstand heavy, short-ranged bombardment from multiple capital ships. Everything had some amount of armor on it, even the AA guns and extreme bow and stern (Though the stern proved eventually to be structurally weak regardless.)This layout was vindicated at her final battle wherein she absorbed 70-80 close range hits from guns of 14" and above, and according to some sources more than 600 cruiser and destroyer gun hits before succumbing to the effects of the scuttle order.
This close-and-heavy philosophy is part of the reason she shipped her main armamant in 4 double turrets, rather than three triples like American, Italian, and Japenese contemporaries, or in quads like the French BC's and BB's. I guess their thinking behind this was that if one or two main turrets were to be knocked out, there would still be good turrets to fire. This also was somewhat vindicated in her last battle when her two forward turrets were knocked out but the two aft continued to fire and do some splinter damage to Nelson (Rodney? Both? I forget). Unfortunately for the Germans, she scored not one hit that day, despite her above-average shooting against Hood and PoW. The British Vanguard was also laid out like this, but by the time she was completed, her layout was obsolete as per the KGV class and the earlier Nelsons.
The Iowas were designed as fast carrier escorts to be able to maintain pace with the fast American carriers, which themselves been designed from the outset as fast battlecruisers (Lex, Sara). Hence their long range, powerful engines, and very high speed. Their armor didn't suffer for this however because of the "All-or-nothing concept", meaning that spaces that absolutely needed armor were protected to the max, (Conning tower, engines, turrets, magazines, ect.) were armored to withstand the effect of it's own guns, while spaces that didn't need to be heavily armored, weren't. Their design was also more modern than either the Bis or the Yamato, and in conjunction with their intended use in the vast and relatively unpopulated waters of the pacific, they were afforded greater protection against plunging fire that is more common in long-range fights. Plunging fire is shells coming it at a ballistic trajectory, rather than a direct (non-arcing) trajectory.
Their guns used the newer 3x3 layout (2 triple turrets forward, 1 triple turret aft) because the designers realized this provided 2 distinct advantages over the Bismarck's 4x2 layout: Firstly that 9 guns in 3 turrets weighed as much or less than 8 guns in 4 turrets, and secondly that three turrets made for a shorter citadel than 4 turrets. (Armored box that contains the turrets, magazines, machinery, and C&C spaces. The real important stuff)
The Yamatos were designed to be world-beaters: Able to take anything the enemy gave and to be able to dish it out half again; Able to destroy with crushing 18" guns anything that couldn't flee quickly enough. Their enourmous size and girth provided the designers with 2 very real advantages: Firstly to have the room and weight to hang an absurd amount of armor off the thing, and secondly to have the size to be able to absorb terrible amounts of damage. These two combined can clearly be seen in the final battles of both ships. Both required many, many hits from aerial bombs and torpedoes to go down, though sadly, the battleship duel for which these monsters had been designed never occored because of the Americans overwhelming air superiority in the pacific at the stage in the war when the two ships of this class went down.
However the 2 afforementioned advantages were also among this classes greatest drawbacks: Huge size and heavy armor made the ships slower than any of their contemporaries by at least 3 knots, and less manuverable, and it made them very hard to miss targets when overwhelmed by planes. (Or destroyers and torpedo cruisers maybe?) It was also a handicap that the immense amount of armor fitted to these ships was vastly inferior in quality (Stopping power) to the American and German armors of the time.
Like Rocky Balboa, these ships were slow, but with an iron chin. But unlike Rocky, they didn't win in the end
Guns and control:
The Bismarcks were armed with the excellent SK-C/34 380mm (14.96") gun. It was accurate, long-ranged (High elevation), and hard-hitting. It also fired very rapidly (Between 3 and 3.3 rounds per minute) for a battleship gun. The power of this weapon fell off rapidly at extreme ranges, I guess emphasizing the Germans' intent to fight at shorter ranges.
The Bismarcks fire control consisted of world-renowned Zeiss range-finders, in widths from 10.5m to 7m, with smaller devices serving the secondary weapons. The radar as fitted in 1940 was the rather lacking FuMO 23, which was sometimes disabled by the recoil of the main guns. It was reported to be effective when it worked. I'm sure Tirpitz had a more effective radar, but she's outside the scope of our discussion here.
The Iowas were armed with 9 highly effective 16"/50 cal guns. Accurate, long ranged, and with the best ammo and propellant (At least till 1989). From all I've read, these guns fired about 2 rounds per minute, and that the armor-piercing variety of shell was heavier per inch of diameter than anything else.
Her fire-control was excellent, clearly the best out of the three. In 1944, one of the Iowas was able to retain a gun lock on a target at night, in heavy weather, at high speed, while going in a figure-8. This was a feat that no other US BB had done before (And probably no brit, definitely no German, and no Japenese). That's all I need to say about American FC.
The Yamatos had 9 massive 460mm guns, originally developed as the "406mm special" or some such to avoid the washington treaty limits. Clearly the heavyweight in both shell weight and in destructive effect when accurate, they were also the slowest of the group: between 1 rpm and 1.5 rpm. These weapons also apparently had an interesting secondary role: AA. Some Admiral or Captain shot these monsters at incoming aircraft, though I doubt they were effective in this role.
I have no details on Yamato's fire control, but I feel safe in saying that it was inferior to both the American and German systems operational in this discussion, by both day or night. Perhaps only due to poor crew quality, or poor equipment, or some combination thereof. I have no proof that Japenese FC sucked, so feel free to flame me on that assumption if I need it.
After action report:
Iowa wins 1 on 1 against Bismarck and 1 on 1 against Yamato.
Iowas potent combo of great speed, adequate armor, highly-effective guns, and excellent fire control lay waste to the Japenese giant by day and especially at night. Despite her great size and huge guns, the Yamato simply isn't in the same league as Iowa. She might have a better shot against a SoDak or NC.
And against Bismarck, Iowa uses her 2-3 knot speed advantage to regulate the distance of the fight, while letting her better fire control wear down the German ship. Though on the pursuit, Bismarck can field 1 more gun, her overall armament is inferior, as is her fire control, but not by such a degree as above. I call this fight as Bismarck sunk, and Iowa moderately-to heavily damaged.
In the hypothetical engagement of Yamato by Bismarck, I see the fight turning out pretty much as above. German radar and better rangefinders / directors will give the Bis first blood, and with her ability to put at least half again the number of rounds in the air per unit of time, Bis will hit first, and hit hard. That is, until, she shoots out her own radar. Then I see The Germans doing one of two things: Closing quickly to her designed battle range and pummeling the Yamato with salvoes only 20 seconds apart and dealing with the occasional impact of an 18.1" shell or three, or keeping the range long and still using superior FC but less effective shells to slowly wear the giant down.
I'd go for the close-and-pummel route myself. :nutkick
Cheers