Help calculating beam load
I have an existing enclosed patio and would like to turn the roof into a deck/balcony.
The patio is 14'x22' with a 6x14 beam held up by 6x6 posts 14' away from the house. The beam is running parallel with the house with 2x10 joists supported by the house itself and the beam.
Assume the live load and dead load on the building's roof is 50 PSF, may be 60 PSF (I live in California but I need to double check this).
The weight bearing on the beam should be: 14*22*50/2 = 7,700 pounds, since the beam only holds half the roof's weight.
Assume allowable fiberstress of beam is 8.
Maximum bending moment: Mmax = WL/8
Mmax = 7700 * 22 / 8 = 21,175 ft/lbs
(Should this length be 14 ft not 22 ft?)
Section Required = Mmax X 12 / Fb
21175 ft/lbs * 12 in/ft / 1000 PSI = 254.1 cu in
Section modulus of my beam is: S = bd2 / 6
S = 6 * (14*14) / 6 = 168 cu in
Either my math is wrong or this big, beefy 6x14 beam isn't nearly as strong as I would suspect.
Deflection and sheering are more than adequate with the 2x10 joists.
b = 1.5
d = 9.25
Deflection: D =(5 * W * L3) / (384 * E * (b * d3/ 12))
Shear =(( 3 / 2 )* V ) / ( bf * df )
Thanks in advance.