Probably not enough sun. Most plants require close to 6hrs, I have some raspberries and blueberries getting about 5 hrs of direct sun and they don't do as well as others getting more...

If conditions other than sunlight (soil, water, fertilizer, etc.) are favorable, you could get some fruit. Raspberries, unlike most blackberries, need insect pollination, so if those canes are kind of hidden away, insects might not notice the blossoms. Berry plants largely in shade should not be watered overhead, takes too long for foliage or fruit to dry out.

Any chance of a picture of the situation? How will the patio be used?

This isn't what you were looking for, but creeping wintergreen, Gaultheria procumbens, is a cool plant to grow in that condition and it can be used for teas. It is evergreen and a nice ornamental appeal, plus it is an Appalachian native. You might be able to have a whole mint garden there.