Herodes:
The outputs are working.
How did you test this?
The outputs of your 595 chips can source/sink 35mA at absolute maximum. That is probably not enough to drive a relay directly. It could damage your 595.
Find the datasheet for your relay, and read it. Also post a link to it here.
Wiring up a 595 output to a relay should involve an NPN transistor, a current limiting resistor, a diode for the coil induction spike, maybe some capacitance. You should be able to find examples. Maybe you use a module that has all of this on board.
If you're going to drive 160 relays, you will need special considerations regarding power and noise.