I think that your problem may be that your relay board isnt reciveing a high enough voltage from the pi to trigger the relay to open as the magnetic field being created is likely not powerful enough. This is likely due to the product being marketed as an arduino device which uses 5 volt logic instead of the pi's 3.3v which is likely the difference between you hearing the module "tick" and it actually being activated. To solve your problem i would recommend you add an npn transistor between the pi with the base being the pi's gpio, the collector being the power supplies output(i wouldn't recommend connecting the transistors to your pi's 5v line, because although it may work with all 16 of your relays on, or even more that about 2 you are likely to be overloading your usb power supply and the pi's circuit) then you would connect the emitter to the relay board.
NOTE: be sure you get a transistor that is capable of handling your power supplies voltage and the replays required current, also be sure to get one compatible with the pi's 3.3v logic
you can read more about transistors: HERE