You can certainly do this, but we will need to take a two step approach.
First, I will analyze the beam to see if the span can increase by 50% without a problem. I will let you know that answer. Then we can go to step 2, if needed.
What exactly is the beam supporting?
Do you have roof trusses or rafters and joists?
What is the joist length on each side of this beam?
Is the beam spliced over any of the supports?
Will it support any ceramic tile?
Can you measure the exact depth, width, and flange thickness of the beam?
Can you measure the thickness of the top or bottom of the I?
I am going to assume it's a S7x15.3 ASTM A36
10'-6" is the maximum allowable span for that beam with that loading.
Correct. The maximum span between any two posts is 10.5'. I can determine what reinforcement you need to get the 13.5' span as a premium service, if you want.
No problem. I'm happy to help. Good luck with your project.