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How to calculate rafter thrust? -Help!

How to calculate rafter thrust? -Help!

How to calculate rafter thrust? -Help!

(OP)
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 (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

RE: How to calculate rafter thrust? -Help!

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.  

RE: How to calculate rafter thrust? -Help!

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?

RE: How to calculate rafter thrust? -Help!

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.

RE: How to calculate rafter thrust? -Help!

(OP)
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

RE: How to calculate rafter thrust? -Help!

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.


RE: How to calculate rafter thrust? -Help!

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.

RE: How to calculate rafter thrust? -Help!

Greetings fellow Climber and Salutations to all,
After sitting on the side line seeing many great players on the field come and go, I’ll step out and use the vertual ‘No Work Method’.
    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.  Remember connections are always critcal.  The cables will only be in tension during leaward wind, suction, and may resonate.  On the windward side the cable will compress and may deflect.  Connecting the cable to the collar tie using a configuration where the cable will be in perment tension which may require more framing.  As Blake mensioned, you can not push with rope.
    Some form of lateral support is required, an interior shear wall perpendicular to the exterior wall with diagonal bracing from the top plate down to the sill same as the exterior corner walls need be as well.  Kickers from the exterior wall to the rafer either interior or exterior which may be used as a decorative element.  Form follows function.  With light lateral loads, light gage metal nailed between the double top plate as a horizontal flinch beam may help.  Greater loads will need larger top plates extending beyond the wall up under the soffit, 2x10’s, laminated lumber, ect. Above this is placed the typical plate where the rafter bird mouth sits so the rafter extends beyond the larger horzontal plate.  My opinion about using slide connectors don’t, keep it pinned down.  I had been a climber too, red iron, winds change directions quickly and repeatedly.
                Do it well, no luck required.

RE: How to calculate rafter thrust? -Help!

as stated earlier i beleive you need to evaluate the connection to the member. that is typically the weakest link  

RE: How to calculate rafter thrust? -Help!

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.

RE: How to calculate rafter thrust? -Help!

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

RE: How to calculate rafter thrust? -Help!

(OP)
OK Guys, thanks.

How did you get the 550PLF figure?

Regards,
J

RE: How to calculate rafter thrust? -Help!

Climber-

My mistake! The thrust is 32/12 * 550 = 1466 #.  Sorry about that.  The 550 plf is from 11' * 50#/sf * 1' = 550 plf vertical reaction at ea plate.

RE: How to calculate rafter thrust? -Help!

You have the 1466# load pushing out on each wall since you only calculated the load on half the roof so you end up with approximately 3000# cable load like blake989 said.  

Here are the numbers that are being assumed and not included:

35 psf snow + 15 psf assumed dead load = 50 psf design load

50 psf * 11 ft *(32/12) ft = 1466# per wall.  cable holds 2 walls so simplified cable load is 2932#.  

There are a number of other things to look at besides the cable load including bearing on wood, wind loads, seismic loads, uneven snow loads.  That is why getting a PE is recommended and I agree.  A dollar spent now making sure everything is covered is well worth keeping your most expensive investment from falling down around your ears in the next record wind or snow storm.

RE: How to calculate rafter thrust? -Help!

I have gone back to the beginning and taken a look at this as a truss problem (ignoring the collar beams) and get a different answer from anyone else.
Consider a one foot depth truss.
A 22 ft base.
Rafters at 45 deg angle on each side form the sides. They intersect 11 ft above the center of the base.
Total vertical load is 1100 lb. (50 lb/ft x 22 ft) - Units for the load are lb/ft since I am looking at a one ft. depth.
Therefore the external support provided by each wall is 550 lb (vertical).
Both ends of each of the two rafters are a vertical loads of 225 lb / each (a total of four loads).
You now have all loads on the truss:
1. 550 lb (down) at the apex (225 lb + 225 lb).
2. 225 lb (down) at the top of each wall (load from rafter).
3. 550 lb (up) at the top of each wall (external reaction).

The tricky part is that items #2 and #3 (above) really do tend to cancel each other out.

Using the method of joints, compression in the rafter is 318 lb. to balance the vertical forces.
To balance the horizontal forces, tension in the base is 225 lb.

Since the cables are the "real" base and are spaced 32 in o.c., tension in a cable = (225 lb x 32in / 12in) = 600 lb.

RE: How to calculate rafter thrust? -Help!

SLIDE RULE ERROR-

HOW IS REACTION MORE THEN LOAD? I THINK YOU TYPED SOMETHING WRONG

RE: How to calculate rafter thrust? -Help!

tfl - Thanks for the correction - all of the 225 lb figures should be 275 lb. Then the cable tension works out the 733 lb.
Would like to chalk that one up to slide rule error, but it's really human error (mine).

RE: How to calculate rafter thrust? -Help!

Slide rule-  The "tricky part" you mentioned really is tricky.  I got twice the cable load you did by doing a free body diagram of one support and assuming the axial compressive load in the rafter and tension in the cable were unknowns. But, as you said the vertical component of the rafter reaction is known and should be included in the FBD.  The actual unknowns then are the horizontal tension force in the cable and the horizontal component if the rafter's axial compressive force.  I get the same numbers as you when including the rafter reaction in the FBD.  Such an easy thing to overlook!   

The other point worth reiterating is that the cable load is not doubled because the plate reactions do oppose each other.  Well, cheers all. Time to go home.

RE: How to calculate rafter thrust? -Help!

Free body diagrams are simple things often forgotten in the heat of battle.  I stand corrected.  Good Job.

RE: How to calculate rafter thrust? -Help!

Now you have to help me SlideRuleEra, #2 & #3 do cancel each other 550-275=275.  But what about #1, the apex load of 550#.  Where is it resisted?  The ridge beam is not supported, therefor would half of this 550# not be resisted by the plate reaction as well?

RE: How to calculate rafter thrust? -Help!

blake989 - From one of the joints at the base (either side, since the truss is symmetric)you can calculate the force in rafter:
1. The net vertical component ajoint at the base is 275 lb (up).
2. Therefore the vertical component of the force in the rafter has to be 275 lb (down).
3. Since the force in the rafter can only act along it's axis, the rafter must "push" on the joint (rafter in compression) with a force of 389 lb (275 lb / sin 45 deg).
4. Since the rafter is "pushing" on the lower joint, it must also "push" on the apex joint (with 389 lb) to stay in equilibrium.
5. The vertical component is of the force from the rafter is 275 lb (up).
6. However, there are two identical rafters, both in compression and both "pushing" with vertical components of 275 lb each. The total of 275 lb + 275 lb (up) balances the the 550 lb vertical load (down) at the apex.
7. Of course you have to make sure that the horizontal forces at the apex are in equilibrium also. Because the truss is symmetric, they are:
(389 lb x cos 45 deg) "pushing right" = (389 lb x cos 45 deg) "pushing left".

One thing that adds to the tedious nature of this problem is the fact that all the angles are 45 deg with sin = cos. I always disliked that kind of problem, hard to tell which "0.707" you are talking about.

RE: How to calculate rafter thrust? -Help!

I think you'll want to check your connection capacity.  What is the geometry?  Single shear, double shear?  How did you make the connection?  The cable strength is probably your last worry.  I haven't run the calcs, but at 733 lbs in single shear in a perpendicular load through two 1.5" rafters, you are probably not making it.  What kind of wood is it?

RE: How to calculate rafter thrust? -Help!

(OP)
Let's put this another way.  If I had used 1x or 2x wood rafter ties in place of the cables, no one would bat an eye, because that's how it's usually done.  I checked the code book, and the specified connection between rafter tie and rafter are three 8d nails face nailed.

What is the shear capacity of 3 8d nails before withdrawal?

Certainly I imagine it to be less than the tensile strength of the cable or the shear capacity of the 1/2" through-bolted connection?

Is there a way to upload a .jpg onto this forum?  I could take a picture of the connection- I think it would put some of you at ease.

Thanks for all the input!

J

P.S.  The inspector came by and signed off on  the roof framing. - He didn't even ask about the cables.  I guess that means everything's safe, right? ;)

RE: How to calculate rafter thrust? -Help!

theclimber – Here are some typical numbers for your situation:

Assume all lumber is either douglas fur or southern pine and is 1.5 in thick.
Assume all nails embedded at least 11 diameters.

One, 8d common nail: Safe lateral load (shear) = 78 lb.

One through-bolt, 1/2 in diameter, in single shear:
Safe load perpendicular to grain = 250 lb. (Worst Case) or
Safe load parallel to grain = 550 lb. (Best Case)
The limit is determined by the strength of the wood, not the strength of the steel bolt.

Calculated maximum cable tension is 733 lb. This is to be resisted by connections with a safe load capacity of less than 550 lb.  Sounds like it could be an invitation for “trouble”.

If ceiling joists were used instead of the cables to tie the bottom of the rafters together, there would be a couple of differences:

1. The ceiling joist should be 16 in o.c (instead of the cables at 32 in o.c.). This would tie all the rafter together.  Then tension in the ceiling joists would be 733 lb / 2 = 367 lb.

2. The classic Department of Agriculture Handbook No. 73, “Wood-Frame House Construction” by L. O. Anderson specifies 5 each, 10d common nails for the connections between rafters and ceiling joists.

One, 10d common nail: Safe lateral load (shear) = 94 lb

Calculated maximum tension in ceiling joists of 367 lb. This is to be resisted by connections with a safe load capacity of 470 lb (5 nails x 94 lb/nail).  Sounds good to me.

Source for allowable loading on nails and bolts:
“Standard Handbook for Civil Engineers”, Third Edition
Frederick S. Merritt, Editor
McGraw-Hill Book Company, Publisher

RE: How to calculate rafter thrust? -Help!

ok one load case done, four more to go...

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