how to calculate rafter thrust -help1

huangyhg

how to calculate rafter thrust? -help!
i have built a new roof over my house.  it's a 12:12 pitch and spans 22' total.  this will be a cathedral ceiling, but i installed a ridge board (2x12) not a ridge beam.  i have installed collar ties (<A href="mailto:2x4@16"oc)at">2x4@16"oc)at about 8' high above the plates.
i installed the rafters (2x8 16"oc) to a 2x12 ridge board, and to resist thrust at the plates, installed aircraft cable @32" oc from rafter to rafter at the plate.  i thought it would be fine, but now i'm second guessing myself.
why did i use aircraft cable of all things? - i guess i thought it would look different, and i couldn't readily find 22' long dimensional lumber.  since i'm a rigger by trade i happened to have a reel of unused cable.
a thorough search of the code book shows rafter ties necessary if no ceiling joists or floor system running parallel to the rafters, but the size required isn't even mentioned.
i know the snow loads, dead loads and the specs of the cable, but i need to know how to calculate the horizontal thrust these cables must resist, can anyone help?
can i take any reduction for the area located above the collar ties?
thanks,
jeff
appendix, chapter 16 of the ubc covers snow loads in much more detail than 97 ubc 1614 sections.
this appendix, which is not addressed here, provides detailed information to calculate:
o    roof snow load as a function of ground snow load, building exposure, and importance.
o    unbalanced snow loads.
o    drift potential.
go through the five load combinations take the highest load combination.
decompose the force components to find the thrust force acting on the exterior walls due to the component of the load parallel to the roof.  
your roof is 45 degrees, if you calculated the vertical reaction from the roof to the plate, then the horizontal thrust from the gravity load is the exact same (pythagorean's theorem).  you will also need this equation, (p*l)/(a*e).  i am curious, how did you anchor the cable to the plate?
almost forgot, if the length of the supporting wall parallel to the ridge beam is very long, you may have some problems keeping the two walls out, not in.  depending on where you live, wind load will very rarely overcome the gravity thrust with a 45 degree roof.  just something to think about.
thanks for the help, guys.
good thought on the wind load pushing the walls in - i hadn't thought of that.  there are two heavy timbers perpendicular to the ridge that support the upper floor, i'm sure these would resist any wind - however i'm not in a special wind zone anyway.
the cables are anchored to the rafters near the plate.  they are through bolted with 1/2" bolts and flat washers.
help walk me through the formula;
the house is 22' x 30' and the ridge runs parallel to the longest walls.  
i live in a 35psf snow load zone and let's go on the heavy side and say 15psf dead load for a total 50psf, of horizontal projection.  each half of the roof has a horizontal projection of 11'.  11x30x50x2=33,000lbs of vertical load for the entire roof.  where do i go from here?
thanks again,
j
jeff- wood collar ties 3' down from the ridge are not going to contribute very much at all to restraining the kick at the bottom of your roof rafters.  having those collars there doesn't hurt but i would design the cables to safely carry all the lateral kick at the plate level.
you may have heard an old builders rule of thumb that wood collar ties need to be in the bottom third of the rafter span to be helpful.  however, if you put them there and then do an analysis you will probably find that force in the connection between the collar and the rafters is unusually high, and can't be easily developed with common fasteners.  
would also have the same concern about the tension capacity of the cable connection to the rafter. how did you connect the cables to the rafters, by the way? suggest also to check how far the cable will stretch under a full snow load.
perhaps a structural ridge should be considered.  good luck.
jeff- oops your last message came in as i was typing my first post.  35 psf is a pretty good snow load.  respectfully suggest you have a p.e. come out and look at what you have done.  it shouldn't cost that much and sounds warranted. regards.
greetings fellow climber and salutations to all,
after sitting on the side line seeing many great players on the field come and go, i鈥檒l step out and use the vertual 鈥楴o work method鈥?
    first contact your local building inspector, they are a pubilc service and you pay taxes.  many inspectors, architects, builders and even engineers use charts to determine stick built roofs based on load, span, size, spacing and specise, twenty two feet is not a great span.  as stated above, by sam, a pe may need be employed based these results, if not congradulations!  keep your inspector involve in either case.
    assuming the worst case where the inspector determins the framing is inadiquate, then a pe can find the best case.  the pe will run simple statics based on live load, horizontal wind and even verticle wind shear.  the collar ties will not help horizontally nor will the cables.  collar ties lessen the diaphram effect simlar to sissor truss, but may produce larger bending moments on the rafters.  re  
as stated earlier i beleive you need to evaluate the connection to the   
o.k. jeff, this is per your numbers.  at 45 degree angle with the vertical component 550plf to each exterior load bearing wall.  thrust will equal 550plf.  with cable at 32"o.c. the load per cable equals (2walls)*550plf*(32"/12)=3kip per cable.  i can tell you this, the safe working load of 7x19 aircraft cable is 3.6kip, 4.4kip, and 5.7kip @ 3/8", 7/16", and 1/2" diameter.  good luck.
blake-
if the roof rafters are equally loaded the thrusts at ea plate will be opposite and balance each other.  the horizontal kick at the connection of the cable to the rafter is thus 32/24 x 550 = 733 #.  the tension in the cable will be the same. regards