Pierre-
I'm not a stuctural engineer, but I've designed and built a lot of things like this, and tend to go overboard. I think your design is entirely satisfactory. My main concern would have been if you were using U-shaped brackets at the bottom of each 4 X 4 rather than planting them in concrete. I also think that the 2 X 4's will be fine; there may be a trace of sag, over a 9' span, but not noticeable. As long as you are using rot-resistant wood like cedar or redwood, and treated posts, this structure will last many years.

For context, let me tell you that I just moved from a house that had a 20' X 20' car port which was supported on one side by the house and on the opposite side by just 3 pairs of 2 X 6's. Those pairs of 2 X 6's each supported a 20' beam that was 6 X 12, a flat roof of 2 X 6 decking, a torch down roof over that, and 2" of water when I neglected to clean the roof drains. That carport was built in 1952 and was still solid. I don't think the weight of those vines will do a thing to your pergola.

Yeah, I have a water tank stand that is less engineered than that! You can't avoid sag, only choose an amount you feel acceptable.

Thanks for the feedback! I haven't built too many things like this before so your posts help.

Pierre

There is nothing wrong with the design, but remember that a 2x4 will bow in time UNDER ITS OWN WEIGHT.

All boards have a "crown" or a bow, some more than others.

If you fell that sagging will be a problem, when you purchase the lumber look for rafters with a slight crown in them and make sure to put the crown facing upwards.

This is one instance where you may want warped boards to make up for natural sagging.

Make sure that the warp is in one direction, you don't want boards that are twisted.

You should however pick some straight 2x6s for the joists, at such a short span, sagging will not be a problem.

You may also consider using 4x6s for your joists.
It is very easy to notch both the 4x4 and the 4x6 and have a "post and beam" look instead of 2 2x6s sandwiched on each side of the 4x4 uprights.

I'm a newbie to gardening, but have a few decades of building houses under my belt. ;)

This forum has helped me in more ways than I can count, will be happy to give you detailed instructions on the post and beam method if you choose that method.

I have 4m spans of 1x3 green hardwood that hold up my greenhouse roof and they haven't bowed. On the other hand, I've seen 2m spans with 2x4s that have sagged 3cm in the middle. I think it depends very much on the wood! Softwoods are terrible with bowing, and this modern trend towards treated softwood means that people build what look to me like ancient egyptian architecture - big columns, fat roof beams and a close feeling. I'd rather a gothic cathedral with light and airy spaces filled with light!

(I also think people tend to underestimate the strength of wood)