Horizontal Wind Component on Rafter / Ridge Beam Systems

[KootK]

I've always had an issue with the statics of rafter & ridge beam systems. I see each rafter as a member with vertical reactions available at each end but no convincing opportunities for horizontal reactions at the supports (debatable at the ridge beam). This is fine for the snow and dead loads that generally predominate but is problematic for wind loads.

Wind loads on rafter/ridge beam systems have a horizontal component that requires a home somehow. The first sketch below shows my stab at working out the statics of the situation in a manner that:

1) Satisfies equilibrium for both vertically and horizontally applied forces.
2) Makes sense given the nature of the connections at the high and low ends.
3) Respects the stiffness's of the various load resisting mechanisms available.
4) Produces diaphragm forces that match our typical assumptions.
5) Doesn't require axial loads in the rafters.

The system sketched out below satisfies these requirements in my opinion. It also leads to the following idiosyncrasies which I find surprising:

1) There appears to be an amplifier required in the calculation of the wind uplift force. See the very last line of the derivation. It would depend on the pitch of the roof and would be significant. For a 12:12 roof pitch, the amplifier would double the wind uplift force to be accommodated at the rafter reaction points. I have not been accounting for this.

2) The diaphragm shear developed in the sheathing directly applied to the rafters being considered could never be shared with the diaphragm on the other side of the ridge beam without introducing axial loads into the rafters.

So my questions are two fold:

1) What are your thoughts on the proposed free body diagram?

2) Are others applying the proposed "amplifier" to their wind uplift forces?

3) In TJI catalogs, you typically see a detail like the one shown at the bottom for rafter / ridge beam systems. In particular, there's always a tension strap specified. Could that be the manufacturer's way of dealing with the horizontal wind component? If so, is that valid? It seems to me that that you'd crush the wood beneath the strap or fail the strap nails in withdrawal at the knuckle before you'd develop the capacity of the strap.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[XR250]

I always consider that the horizontal load gets transferred into the roof diaphragm and then into the gable ends.
That being said, I usually specify straps over the top of the ridge from rafter to rafter

 

[EngineeringEric]

In your FBD, are you accounting for the assumed diaphragm (reaction) in your vertical reaction? I might be looking at it wrong but i would think that

R_hor= w*sin*1/2
R_ver= w*cos*1/2

R_Vert goes into the ridge and wall
R_horz goes into the diap, or ceiling joist or partial collar.

 

[KootK]

Thanks for your responses guys.

I always consider that the horizontal load gets transferred into the roof diaphragm and then into the gable ends.

That's exactly what I'm doing above XR.

That being said, I usually specify straps over the top of the ridge from rafter to rafter

What force do you design the straps for? Are you bothered by the fact that the straps, acting as tension ties, have kinks in them? It seems to me that the failure mode would be wood crushing beneath said kinks.

In your FBD, are you accounting for the assumed diaphragm (reaction) in your vertical reaction?

Yes, that's what I've labelled the "amplification factor".

R_hor= w*sin*1/2
R_ver= w*cos*1/2

I believe that these values need to be multiplied by the length of the rafters (L/cos(Theta)). Should get you to my result.

R_horz goes into the diap

Yes, that is precisely the mechanism that I've proposed above.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[KootK]

R_ver= w*cos*1/2

Actually no. This value, multiplied by L/cos(Theta), will get you the usually assumed value in the last line of my derivation. As you can see in that last line, there appears to be more uplift than that once one accounts for the diaphragm reaction on the rafters.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[XR250]

@Kootk;

I don't design them for any specific force. I am using them to keep the rafters from separating from the ridge due to normal wind forces (normal to the rafters, that is) Basically, to take the place of wind collar beams that are required at every third rafter. I usually spec them at 32" o.c. and figure they have about 1000 lb. capacity per strap (Simpson LSTA18). That should be more capacity than a standard 2x4 collar beam at 48" O.C. We are not in a high wind zone here, FWIW.

 

[EngineeringEric]

Yes, I somehow forgot to type in that ever important Length L... What does that matter among Engineers?

I'll try one more time to describe what i am trying.
Applied Loads:
Total Vertical Force = W*L
Total Horizontal Force = W*H = W*L*sin(theta)

The system as a whole will resist the horizontal load from kicking out. When drawings a FBD at the top plate, i think you will have the following (Global axis: Vertical positive up, positive horizontal to right as you have shown)
P_vert=0.5*WL
R_vert1=-0.5*WL
P_Hori=0.5*WH
R_Horiz=-0.5*WH
R_Vert2 = ... More ...

Now the R_Horizontal is handled by the ceiling joists or in your case the Diaphragm which dumps the load into the transverse walls. The fact it is sloped means its actual local load, is more like -0.5*WH/cos(theta) = -.5*W*L*Tan(theta) (up and to the left). This also creates an additional vertical component R_vert2=-.5*W*L*Tan(theta)*sin(theta) {the force is trying to hold the diag down}. I believe this is your "Amplification Factor"

The sum of R1+R2 = 0.5WL+0.5WL*tan*sin = WL/2(1+tan*sin). Now there is a good chance that i missed a cosine or sine, its late But the conclusion is i have a similar "Amplification Factor" that is acting at both the plate and ridge. When theta=0 the amplification goes to 0, when theta is 45 you get 2x, mine got 1.71x.

I agree that, at this hour, it seems there is an increase in force. I do not fully know where your Tan*sin*1/cos came from in your downward reaction equation... but it is a geometry exercise i would typically give more attention to than i am tonight. We can agree that it is the vertical component of the diaphragm force(?) .

And i like to think that the straps just hold the rafters together since the dia. is often broken at the ridge. Nothing would be there to keep the members tight to the ridge-beam. I do try an upsize the strap when dealing with long span rafters. But maybe i need to give it more attention.

 

[Ingenuity]

KookK and EngineeringEric, for the given rafter horizontal span, L, the rafter height, H, is L*tan(θ), not L*sin(θ)...I have not gone through your FBD or your geometry, so not sure if it effects your results/conclusions, but just saying...

 

[KootK]

@Ingenuity: you are quite correct. Thank you. I noticed that shortly after I posted but left it alone as I didn't use it anywhere else in the derivation.

@XR250: the sketch below shows my paranoid concern regarding using tension straps to deal with the horizontal wind component. Simpson seems to think that it works so maybe it's just me.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[jayrod12]

Kootk, I think you might be over paranoid again. I haven't put any numbers to that scenario specifically but I can't see the magnitude of crushing to be such that the stability of the structure is compromised.

 

[KootK]

@EngineeringEric: thank you for your detailed response. Seriously. I know effort when I see it. I'll now parse every word of it and see if we can't iron out our differences.

Total Vertical Force = W*L
Total Horizontal Force = W*H = W*L*sin(theta)

Agree.
Disagree. Total horizontal = w*sin(theta)*(L/cos(theta)) = w*L*sin(theta)/cos(theta) = w*L*tan(theta). It has to be summed over the diagonal rafter length.

P_vert=0.5*WL
R_vert1=-0.5*WL
P_Hori=0.5*WH
R_Horiz=-0.5*WH
R_Vert2 = ... More .

I agree with the vertical reactions, in the absence of the amplification factor. I disagree with the horizontal reactions. The stud wall is incapable of providing a horizontal reaction to the rafter. It can only deliver a horizontal load when that load originates as lateral load applied to the wall (W/EQ).

And i like to think that the straps just hold the rafters together since the dia. is often broken at the ridge. Nothing would be there to keep the members tight to the ridge-beam. I do try an upsize the strap when dealing with long span rafters. But maybe i need to give it more attention.

Yes. The diaphragm is broken up. Given the statics discussed above, I wonder if the sensible thing to do is treat the roof as two separate diaphragms, one either side of the ridge. It's a bit of an arcane point so I'll leave that alone.

Now the R_Horizontal is handled by the ceiling joists or in your case the Diaphragm which dumps the load into the transverse walls. The fact it is sloped means its actual local load, is more like -0.5*WH/cos(theta) = -.5*W*L*Tan(theta) (up and to the left). This also creates an additional vertical component R_vert2=-.5*W*L*Tan(theta)*sin(theta) {the force is trying to hold the diag down}. I believe this is your "Amplification Factor"

The sum of R1+R2 = 0.5WL+0.5WL*tan*sin = WL/2(1+tan*sin). Now there is a good chance that i missed a cosine or sine, its late But the conclusion is i have a similar "Amplification Factor" that is acting at both the plate and ridge. When theta=0 the amplification goes to 0, when theta is 45 you get 2x, mine got 1.71x.

I agree that, at this hour, it seems there is an increase in force. I do not fully know where your Tan*sin*1/cos came from in your downward reaction equation... but it is a geometry exercise i would typically give more attention to than i am tonight. We can agree that it is the vertical component of the diaphragm force(?) .

Our mathematics have drifted apart but we're in definite agreement on the concepts. Glad to know that I've got at least one "not crazy" vote. Here's the more detailed verification of the amplification factor. I'm into my third glass of Malbec so take it all with a grain of salt:

[1] Horizontal component of load = w*L*tan(theta) <-- see above.

[2] Horizontal component of in plan diaphragm force = q*(L/cos(theta))*cos(theta) = q*L <--q = distributed diaphragm shear in plane of diaphragm.

[3] [1] = [2] therefore w*L*tan(theta) = q*L therefor q = w*tan(theta).

[4] Total vertical component of diaphragm shear = q*(L/cos(theta))*sin(theta) = q*L*tan(theta) = w*tan(theta)*L*tan(theta) = w*L*tan^2(theta). Substituting [3].

[5] Total vertical load = direct wind + diaphragm reaction = w*L/2 + w*L/2*tan^2(theta) --> rearrange --> w*L/2(1+tan^2(theta))




I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[KootK]

@Jayrod: I think that I've been confused about what the straps actually do. They could never do the job that I sketched above in my opinion. By the time that the slack in the straps was taken up such that they were fully engaged, so much deformation would have accrued that the rafters would have long since ripped from their supporting hangers.

When loads are such that one side of the diaphragm attempts to shift in plane and away from the ridge beam, I think that's where the straps come in. They somewhat stitch the two halves of the diaphragm together as Eric proposed above. If this is true, then the straps help the diaphragm once it is engaged but the do not serve to directly resist the horizontal wind load component in place of the diaphragm being engaged.

The Simpson catalog sizes these tension straps based solely on the pitch of the roof. I'm surprised by that. I would think that a bunch of other variables, such as wind pressure and wall height, would come into play.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[jayrod12]

I guess I'm having a hard time determining what you want it to do.

In my mind looking at your sketch I see the vertical component being resisted by the connection to the roof beam, and the horizontal being taken by the strap to the other joist on the opposite side of the ridge beam. Eric's last sentence is exactly my sentiments.

 

[EngineeringEric]

So I have concluded that Engineers obsessed with details and worrying over loads should not attempt geometry or trig late at night. And i also need to refresh SOH-CAH-TOA.

 

[KootK]

I guess I'm having a hard time determining what you want it to do.

Me too. I'm not sure that I want it at all for the loads considered.

sketch I see the vertical component being resisted by the connection to the roof beam

As do I. The question is whether that reaction is wL/2, as commonly assumed, or something larger as I've proposed.

and the horizontal being taken by the strap to the other joist on the opposite side of the ridge beam.

It certainly could be. However, if we did the statics on it, we would find that:

1) There would be axial force in the rafter.
2) There would still be amplification of the vertical reaction at the ridge beam.
3) The load would just then be resolved in the same manner by the diaphragm on the other side.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[jayrod12]

I must be envisioning the diaphragm is more capable than you feel it is.

Why can the load not be dumped into one diaphragm or the other and transferred to the end shear walls.

I'm clearly missing something important. Where's woodman when we need him?

 

[KootK]

The diaphragm is what I've proposed for the load resisting mechanism. The issue is the effect that has on the vertical uplift forces. See the last line in my first sketch.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[jayrod12]

Ok I'm on board with you now. Sorry, my daughter hasn't been sleeping well lately, makes for tough days.

I agree with your assessment there. My rationalization of why I likely don't account for it is at a standard 4/12 roof that equates to an 11% amplification. I am likely more conservative in my loading runs than that 11% anyways. For a 6/12 roof we're talking 25%, maybe I need to start paying more attention to slopes of this and above. I am generally not doing rafter and ridge beam roofs that exceed a 6/12 and rarely exceed a 4/12.

That being said, it's food for thought and barring someone else being able to show otherwise I'll buy in to this amplification business.