Raspberries have a somewhat complicated pattern of fruit production. In general, it takes two years for a specific cane to produce fruit. It grows vegetatively the first year, fruits the second year, then dies. Meanwhile, new vegetative canes come up from the base of the plant during the second year. These will become the fruiting canes the year after the first batch of canes dies off. If the one-year-old canes are cut off or die back during winter, your raspberries will not produce fruit because you have no two-year-old canes left in the patch.
A twist on this system is that of everbearing raspberries. They grow vegetatively through the summer of their first year, and in late summer/early fall, the tips of the first year canes produce fruit. Those tips die off over the first winter, but the rest of the cane fruits the following summer, then dies completely. If you're having trouble getting your red raspberries to overwinter, you might consider growing one of the everbearing types. That way you'd be likely to at least get a fall crop every year.
Your comment about the tips bending over and rooting also indicates that you may need to provide trellising or caging to keep the canes upright. I also wonder whether these plants might be black raspberries or blackberries rather than red raspberries. Red raspberries are less likely to bend over, root into the soil, and form a new plant. Black raspberries and blackberries, however, do so quite readily.