we just had a deck built on top of an existing concrete patio structure with a trellis to block out some of the morning sun.
the deck is *free standing* in that it is not attached to the house. it uses 4 4x4 posts (~11 foot above the ground and 18" below ground in concrete) to support the trellis. my friend noticed that the structure easily moved when you lean against any of the 4x4s. the height of the trellis is about 9 feet above the finished deck floor.
Not sure where you live, but 18" below ground isn't much for a deck that tall. 24" to 36" would have give much better support for the deck itself.
As for lateral movement in the trellis, and the deck itself for that matter, I'd think you could take some of that by putting in some diagonal bracing. Of course with the trellis, if the flex is front to back you're probably out of luck unless you can figure a way to brace in this direction as well.
Those dimensions are somewhat large for not having bracing at all. Was the deck professionally installed? If so, get them back to fix the problem.
It might be possible to make the joints more rigid by adding some hefty steel angle connections. I'm not clear enou,gh on the connections that exist to prescribe the locations and arrangements. Investigate the direction of movement at each joint - starting below the deck - and add connectors. At some point in that process the system will be rigid enough to suit you. This is similar to your idea of knee braces, and less visible. That's a nice looking trellis and it would be a pity to change the appearance more than needed.
Consider bracing of 4x4's. Looking at the trellis head on, the 4x4 would come out from each post 3ft (which the double beam would sandwich), and down
3ft on the post. These would be a 45 degree cut against the posts. You could carriage bolt for the sandwich. For the post fastening, you could carriage bolt also through recessing the end/washer/nut. (The 3 ft. might be a little extreme, but maybe not). I don't believe this would ruin the look of your trellis, I had done something similar to enhance the appearance.
On the sides for back and forth movement, hard to tell from the pictures, but I'm sure you could do similar with the post and main joist (2x8's).
Nice looking trellis, one factor I would add to enhance the appearance (wouldn't help stability in anyway). Cut from 2x4 of same material. 16" long pieces with a 45 degree bevel on one end only, to go on the front and back of posts under the beams, the bevel faces the ground. On the side of the posts, you cut longer pieces to slide up between the beams, beveled one end to face ground and the bevel meet symmetrical with the already installed decorative pieces. This gives the posts a beefier look, very pleasing to the eye and helps to rid of the chicken leg look.
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