CALC: Rafter thrust equation

[odysseusjg]

I'm putting together plans to submit to the local bldg dept, and I need an equation to use in the structural calculations for rafter thrust.
Those handy calculators are great but I have to show the math on my structural calcs.
I found this formula on another post here, but I'm not sure I have it correct:
W(1-(cos^2/2))/sin
Where W is weight on each rafter, ^2 means squared, and cos and sin are of the roof angle.
Is this a correct equation? Other equations?
My project- 14 x 24 single story gable end with ridge on long dimension. Vaulted ceiling with douglas fir round log rafters on 24" centers, 8 in 12 pitch, 37 degees. Snow load 60lb/ft, dead load 10lb/ft.
Using these numbers and formula I come up with W= 980lb, and thrust = 1155 lb. Look right?
Second question- rafter ties. I plan to use steel rod. Cables would work, but I don't want to look at the cable clamps. So, I want to install rods threaded on the ends. Not all thread, cause I don't want to look at the threads. I want to have plain rod cut and threaded at a shop. Question- most economical type of steel to use for this application? With tie rods on 4 foot centers, each rod will take about 5,000 lbs. If I use A36 which has 12,000lb allowable safety factor, I'd have to use 1 inch diameter rod, presuming that the diameter inside the threads is 3/4"  Is there a higher tensile steel ththat is inexpensive, or do I need to use a larger diameter of a cheaper, lower value steel such as A36?

Thannks, JG

[MountainDon]

  JG

I did a search but could not find where that formula came from. Using Don_P's calculator from HERE  and the numbers you gave; 60#, 10# 24" OC, 8/12 pitch and a 84" span it comes up with 735 lbs for each rafter. I can't say any more than this is what an engineer gets paid for.

I've seen steel rods used as corrective measures for spreading walls in large buildings like churches. I have no idea on steel selection though.

[Don_P]

Hi JG,
First off, I'm not an engineer, simply a carpenter, so this is worth exactly what you are paying for it.  These formulas came from Dr Frank Woeste PE in a course I took at Virginia Tech
The formula you gave above is the axial thrust in the rafter, the force travelling down the length of the rafter. The formula for horizontal tension is
Thrust= .5 x w x S/tan(pitch angle)
You have an error in the angle you called in your post above, 8/12 is actually 33.69 degrees

Lets simplify the formula.
Thrust= .5*w*S/(rise/run)

I'll solve the tangent of our angle first. Remember from trig class tangent=opposite over adjacent. We are measuring the angle at the rafter foot, opposite is the 8" rise, adjacent is the 12" run. So 8/12=.666, if you run into an old framer he might call an 8/12 a two thirds pitch, now you see why.

Thrust=490/.666=735 lbs

I've been racking my brain trying to remember the lecture, I believe this is what's going on. Of the 980 lbs half of it is bearing on the wall, half on the ridge, there's the .5
w is weight per lineal inch on the rafter, S is span ... this is the same thing as W, total load on that rafter, so for that side of the equation input half the total 980 lb load or 490 lbs

The vertical side of an 8/12 triangle is 2/3 (.666) the length of the horizontal side so the horizontal component of the 490 lb vertical load is 1.5 times the vertical force... 490x1.5=735 horizontal force, thrust

I believe A36 has an Fy of 36 Ksi and the Fu, ultimate is 58 Ksi. Ft for a threaded tie rod =.33Fu or 19.1 Ksi.
table1-b in section 4 of the AISC manual lists 5/8 threaded rod as having an allowable tension of 5.9K. There is nothing wrong, at all, with going larger! Don't forget to check the bearing area of the washer to the compressive strength of the wood.

This is all for preliminary design, I suggest you have an engineer review all of this.

trig hint for the really bored... or homeschoolers. We found the tangent of 8/12 by dividing 8 by 12. To find the angle if you only have the pitch as x/12 find the tangent as above and then find the arctan. On a calculator punch in .6667>> 2nd function>>tan and it'll respond 33.69, the pitch angle in degrees.

[Don_P]

I'm not sure if you've seen this calc for log beams;
http://www.windyhilllogworks.com/Calcs/logbeamcalc.htm

There is a new log standard making its way through the code process, some of the design values are being modified. If you are getting close on anything I can dig up the new design values. Another method described to me by a log home engineer was to use heavy timber design values and apply them to the maximum sized square timber that can be inscribed within the small end of the log. This is usually quite conservative.

On any of my calcs if you right click on the page and click "view source" the code is displayed, the formulas are in the upper part of the javascript. Sometimes they are hard to parse out cause I'm also not a computer programmer 

[John Raabe]

JG:

Don_P may be just a simple builder but he understands more about structural calcs than most architects. 

[odysseusjg]

THANKS Don and John for your help.

There aren't very many carpenters that can wield numbers so deftly Don, so when you going to take a few more classes and get your engineering degree? Nice log houses on your site.

I used Hahney's 'Log Span Tables' for figuring my rafter thicknesses. 60 lb snow load, but only a 7 foot span, so it's not too challenging. But when I plug the numbers into your calculator, I get higher mid point diameters. I assume this is mostly because your calculator works only for L=360, but all I need is 240. But also, you employ grading as a variable and it makes a difference. So, what if the building dept. wants to know if my site cut DF rafters are #1, #2, or #3? The grading rules I've found in the Log Span Tables booklet aren't that precise. Thoughts?

I'm hoping not to use threaded rod, because I don't want to look at the threads. Any advice about how to proceed when I approach metal fab shops? Should I just be asking for A36, or are there other higher tensile strength steels I can consider?

For figuring the size of washer against wood I just use Fc, right? Which is 925psi for Douglas Fir. With 735 total force per rafter, and ties at every other rafter, I need each washer to bear against almost 2 square inches, so I might need to make a larger washer. No big deal.

Help me out a bit on the strength designations for steel rod. What are Fy, Fu, and Ft?

Againe, thanks a lot. You've helped tremendously.

Oh, John, there's this weird thing that happens when I type into the respnse box When the typeing gets to the bottom of the box, it goes below the box and I can't see what I'm typing.

JG

[Don_P]

Thanks John, I owe you dinner  

Hey JG,
I suspect you are right on the L/360 difference, I haven't backchecked against his tables since reworking the calc with the code design strengths. Grading is a problem I run into here. The building official has the authority to ask for independent third party grading. Mine will allow an engineer to grade as well. The last time I needed to go that route the engineer looked it over while I showed him a few things from grading class. Some inspectors will allow you to go ungraded, some will allow ungraded if you use #2 or #3 values. In some states the people have put down their nofriendo's long enough to demand the right to use native lumber in their homes, these areas generally allow you to use #2 values. Generally speaking a concientious person is probably going to use #1 or better on their own work but there is always room for just plain old ignorance. A quick lesson, don't use anything with decay or bug damage, it is very hard to know the extent. Then scan the piece carefully for the controlling defect, you have all day a grader on the line has 3 seconds. Once you've found the worst defect make sure it is alone and then visualize how much of the cross section the defect compromises. clear or VERY minor=Select Struc. 1/4= #1, 1/3= #2, 1/2= #3. The ILBA has a publication on their website, or did, they called the log home standard, I haven't looked in years, it gave some more guidance. But it is ultimately up to the BO.

You'll probably need to make a dap in the wood so the washer sits flat in relation to the rod, 3" washers are pretty readily available as this is a code foundation washer but you can make your own, heavier the better. If you use Fc perp you won't be wrong, you can use the Hankinson formula to get the exact size based on the angle to grain between perpendicular to grain and parallel but for this I would just use the weaker bearing direction and a bigger washer... just throwing out the "correct" way for info if you ever do something more exotic.

Fu is the minimum ultimate strength of the material, for A36 it shouldn't pull apart at less than 58Ksi. engineer talk for kips (thousand pounds) per square inch.
Fy is the yield strength, if we take your rod and pull it from each end and apply 5 kips of load it will increase in length a certain amount. If we apply 5 kips more it will increase that length again. This is known as Hooke's law, he was making clocksprings and noticed this relationship. It will behave that way up until, in A36, 36 Ksi at which point Hooke's law will no longer apply and the length will grow more than the previous amount per applied load, it will quickly reach a point where it lengthens substantially with very little increase in load up to the ultimate failure point... so the point beyond which the steel is yielding.
Ft is the allowable tensile stress in a threaded fastener for A36 that is 19.1 kips/square inch based on tension of the gross (nominal) area. This is from a table for materials such as round bar stock that are generally intended for use in threaded applications such as tie rods, etc. The tensile capacity of the threaded portion of an upset rod shall be larger than the body area times .6 Fy... this whole steel section is from the steel manual not from my working knowledge, Yes there are stronger steels listed, nothing I've ever seen, we're into Glenn's area of expertise here.

[odysseusjg]

Thanks again Don
JG

[firefox]

One thing you could do would be to get some stainless steel tubing  and slip it over the threaded rod. It would look nice and be easy to clean.
Just an idea,
Bruce

[odysseusjg]

I was thinking of painted PVC, but I'll price SS. Thnx JG

[firefox]

You are welcome!
Look around for a metals surplus dealer.
Bruce