Where do you live? Zone 6B goes across the entire US.

If you have bermuda now, then St Augustine might work. Also there are some seldom used varieties of zoysia that do well in the shade. I believe they are the Manila varieties or hybrids thereof. Otherwise you would be looking at ground covers that are not turf. Do you need to walk on the area?

Can't imagine bermuda grass being at all suitable for a zone 6b climate! It is typically not very tolerant of subzero temperatures, which are not at all uncommon in any zones under 7.

An overhead structure is going to limit the use of most turf grasses. That is pretty heavy shade compared to the mostly partial shade conditions of underplanting tree canopies and very few grasses will tolerate no direct sunlight at all. You would probably be better served with either a solid surface (gravel, pavers, concrete, hardwood mulches) or a shade and drought tolerant groundcover.

We live in NE Ok. and Bermuda does very well here!!!

Yes it will be walked on.

It gets afternoon and evening sun from south and west..

Don't really want solid surface b/c of the heat here!!

Thanks .

Check with your local nurseries for the availability of zoysia. Also ask them if it is suitable for shady areas. The company selling Shadow Turf has discontinued their grass production in favor of crop hybrids. I have pictures of it growing very densely under an English walnut tree with a low canopy. The tree allowed no direct sunlight all day long, and the grass thrived. So I know there are varieties of zoysia that will work.