I've got some large lillies in quite shady parts of my garden in tubs and then flower very well. How about a clematis montana - mine is on a north wall and is fabulous when it flowers.

Thank you Cammomile for the suggestion of a clematis. I know they like their feet in the shade but I assumed the head must be in the sun. i will look into it. Thankyou.

Euonymus fortunii will climb if planted against a wall - takes a while to get going but tough as old boots and doesn't mind the shade.And it keeps its leaves in winter. You could put bulbs under it, and a bit of begonia semeperflorens in summer?
If you want a clem montana will grow ever so much bigger than your trellis quite quickly so will take a lot of keeping in check. One of the spring flowering clems alpinus might fit better?

You're right about clematis liking their heads in sun, John - if the area gets any sun at all at any point in the year, Akebia quinata might be worth trying, or Passiflora. Otherwise you're stuck with Hydrangea anomala, but that's self clinging and will stick to a fence or wall, doesn't really need the trellis. Garrya elliptica isn't a climber, but prefers to be planted against a wall or fence for protection and can be pruned flat against it. Doesn't mind shade, but can be a bit susceptible at the roots to frost - if you want that one, choose the variety 'James Roof' - it has longer catkins.