The posts will support a lot of weight. But I assume since it's 20' long that means one post on each end and one in the middle. That means a 10' span, and you will probably have sagging of the joists running across the tops of the posts, unless they are sized large enough.

The posts are not as important as the beams joining them. What is the plan there? Two 10' 2x8 nailed together to cover each span would work. Simpson tie plates at the post to hold the beams together and to the post. A plate on each side would be best.

Look at putting the posts on top of concrete pillars in the ground. You can use a post base to attach the post to the concrete base. Even with pressure treated wood in a dry climate, posts cemented in soil are not as durable as putting them above ground. Any additional concrete used should be covered by having to use shorter posts.

I used 6" x 6" which is over kill, but I wanted the look of mass. I used 2 - 2" x 10"s, but I cut away part of the 10". Pictures here:

Here is a link that might be useful: My Garden

i purchased an aluminum pergola and want to attach to concrete patio. the holes are 1/4" in the base. i bought some 1/4x 2 3/4 concrete tapcon anchors. is this all i need? i want to make sure that this thing will stand up to our high desert winds.

You don't say how big/heavy this pergola is, what its design is, or how many holes there are per attachment point. However, I suspect that the Tapcon's will be adequate as long as you use the special bit that Tapcon sells for that screw, and you use a hammer drill so that you put the holes in cleanly (use of a regular drill is sometimes slow and results in the hole becoming oversize).

My pergola is L shaped 16 ft across the beam and 7 feet of that has 16' joists, last 9 is 12' joists. So the beams span 16' but the joists only span 10. I used 4x4 posts buried 2-1/2' with 160# of 5500psi concrete and sandwiched them with 2x8 carriage bolted together. The height from ground to lower beams is roughly 11'.
On top of the 2x8 joists, I ran 2x2 strips for decoration and to tie the joists on top too.
I don't believe I will have any problems; just wondering if I should re-enforce the posts somehow. Thoughts?

Warren-
That's a complex design and it would really be hard to answer that from the info given. I don't know if I can give you a good answer even with more info, but a photo of your design would help. For one thing, you don't say how many posts you have supporting the structure, and what their spacing is. For another thing, you don't say if this is free-standing or is attached to a structure.

My main concern with the info you provided is that I would have planted the posts deeper and used more concrete. That's a pretty tall load supported by posts planted 2-1/2' deep. I'm not going to say it will be a problem, but it is often a good idea to over-engineer on these structures.

Finally, it's always a good idea to start a new thread with a new issue, rather than piggy-backing on an old one. You might get more replies that way.