Of course culverts could work- just a question of how deep they'd need to be, but the roots could not penetrate them. I'd start with at least 2.5' depth if I was experimenting but I've no idea how deep confined roots might be able to go. Depends on soil type for one thing. If you plant the culvert a foot above ground the raised soil will provide improved conditions for your plants.

Beds surrounded by 4' width of mowed lawn usually works well also.

Well, I don't doubt that a culvert would keep the roots confined if it's deep enough, but ... I should have been clearer about my question, which is: won't the roots quickly get over-crowded within the culvert segment?

If you thin the raspberry canes the roots will adjust to their space- at least for several seasons- and real soil provides much more bang for the buck than potting soil in terms of volume needed for the amount of plant produced (as long as it's well drained).