If you plan to drive your vintage sports car regularly, OEM style bias plies are a foolish affectation. Upgrade to vintage looking radials, they will still offer less performance than conventional performance rubber, but at least they won't be dangerous.
Yep, so step one is to go to your tire shop and have them remove these tires, have them give them back to you, and get a quality set of tires that will be used for the duration of testing. Modernize where you can and make it as safe as you can.
I have a '68 Camaro built in the late 70's that I am trying to make period-correct with 10 inch Cragar SS's and N50's, but the only ones I can find are new Pro Trac or Mickey Thompson brand bias plys. I'd love for a modern radial to be made that has vintage looks but with the better handling characteristics. I never plan on using my car in the rain, but with my kids in the car, I want it to be at least safe for normal driving to and from car shows and cruises.
Not much available at Tire Rack for the Vette, just some T-rated all-season radials. Coker tire just has bias ply tires. A significant upgrade to the tires will require new wheels, it seems.
It isn't easy to have a period correct show type classic car that you drive everyday. Time to decide what gets upgraded to make it usable. That will be one of the interesting learning processes to watch unfold. I've got an attic full of original parts off my old car so that someday if someone wants to they can go back to stock but for me I'm driving it so I want it to be comfortable, reliable and fun over factory stock.
While I'm no fan of the 30-series tires on huge wheels, a good set of performance 50-series tires on, say, some Torque Thrust wheels would work for me, along with a modern power brake system, as has been mentioned. Keep all the old parts, put the tires in sealed bags to reduce ozone cracking, and have fun!
As I mentioned in another comment on another post, one of the real criticisms of this generation of Corvette was that it was almost impossible to fit anything larger than a 70-series tire under it, if you were going with an original diameter tire. Maybe a 215/70-15 - that will match the rolling diameter of the OEM 7.75-15s that are on it now - for those 15X6 wheels.
Once my dad bought a '69 Firebird in a crisis moment, and it was shod with bias plys. Driving the car was terrifying. Probably the worst was how the car would try and hurl itself into adjacent lanes because of truck ruts on asphalt highways...likely you'll be spared that behaviour in SoCal. But once we put on a set of BFG Radial T/A's, the car settled down a fair bit, although braking and offramps were still an adventure.
I drove a 65 Mustang in the rain once on bias-plys. Truly terrifying and dangerous. Anyone who drives a car with bias-plys on public roads in the wet is taking a chance with all our lives and property...upgrade.
I remember driving on bias ply tires in my very first car ('71 Oldsmobile), and the tires seemed perfectly fine. Radials are better, of course. It's just a matter of what you are used to.
Bias weren't great, but they weren't a death trap, either. I used them for 10 years, no problem. I bet some folks' bad experiences were either with worn ones (they wore out much quicker) or old ones with hard rubber.
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I've got an attic full of original parts off my old car so that someday if someone wants to they can go back to stock but for me I'm driving it so I want it to be comfortable, reliable and fun over factory stock.