If you have 6% to fire, lightning, arcane for a total of 18%, and you have 100 average black bonus damage on a hit, it will do 118 instead. Then that gets thrown into a big mess of other damage modifying numbers like stated above.
The short answer is for the situation of all physical damage weapon, just add the total bonus elemental % together and you gain that as a pure dps increase over your base dps. If you have 100k dps with current gear, and swap to a triumvirate that is identical to your current OH but has +15% total elemental damage, your dps will increase by 15% to 115k. That basic math only applies to the first such item you get, so if you stack Triumvirate with Zuni boots and Tal neck, you can get up to 32% bonus elemental damage, which would apply to your base dps.
The long, mathematical answer is the bonus damage is an increase to your total average physical damage, so if your weapon does 100-200 damage or 150 average, with a +15% triumvirate your weapon instead would do 115-230, or 172.5 average damage, which is 15% more than 150. As others said that weapon damage is then multiplied by crit, IAS, etc to get your final DPS number.