Attaching posts to concrete

I'm about to repair a wood fence on the side of my yard. It's rotted over the years, especially the posts and bottom rail which are sitting directly on top of the concrete pad. I'm going to replace the posts, rails and fencing. The existing posts were attached to the concrete pad using L-brackets and have lasted 20+ years. For the new posts, I was going to attach them to the concrete using a post base and anchor... the kind Simpson StrongTie make (perhaps model ABA44Z). However, the sign at the Home Depot said "not recommended for non top-supported installations (like fencing)".
So, I'm a bit confused. How can that not be strong enough, yet some lousy L-brackets have held the existing posts for 20+ years? Does anyone have experience of this or a practical suggestion?
FYI, the posts would be 6' 4x4's, 6' apart, using 2 2x4's for rails. The fencing is 6' 1x6's. I was going to then use screw decks to screw the fencing to the rails. All of the wood is cedar.
Thanks in advance for the help, folks.
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To do the job right, a 6' fencepost should be at least 2' buried, for a total post length of 8'. Attaching the post to the top of a concrete pad using any method isn't a very good idea. There is no real security there to protect the installation agains strong winds and such. If "L" brackets worked for you for 20+ years, then go ahead and use them again, or go with your anchors. Not a very sound solution, though. I'd never put up a fence and not attach the posts properly. Dave
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How about finding a length of 4"-6" structural angle iron, and having the welding shop cut it into 4" wide pieces. Drill a few holes for bolts, and use these as angle brackets. Might work!
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