Hi Jerry. This is quite a complex subject, but I can point you to a Rail Safety and Standards Board document that covers the subject. It is probably far more than you want and I suggest searching for the word radius within the document. Possibly table G 1 on page 15 will answer your question. Please remember that this relates to current legislation and may not be applicable to anything vintage you might be thinking of modelling.
Like all official document, it is not an easy read.
This is the URL for the document.
Hope this helps.