Snow Load Resultant Vectors

[dpa]

When calculating bending moments in roof joists under snow load should you resolve the snow load into a vector perpendicular to the surface and use the true length of the joist. In the formula wl^2/8 if you compare the results of doing it each way using the true length of the joist - even with the reduced load - results in an increase equal to 1/cos or a 25 percent increase in bending moment for a 9:12 roof. That's because the cos term used in calculating the true length gets squared.

I know a lot of people don't do this including me on occasion but I am wondering if we are underestimating the stresses.

Thanks,
DPA

 

[jec67]

dpa:

Remember, the snow load is on the projected horizontal surface of the roof. Therefore, you must recalculate the snow load intensity on the actual length:

Let's define the horizontal span as L, the angle between the horizontal and the rafter as theta, and the snow load on the horizontal area as w. Then:

- The actual rafter length is L/(cos theta)

- w' = snow load intensity on rafter = w (cos theta)

- The resultant of w' perpendicular to rafter = w'(cos theta)*(cos theta) = w'(cos theta)^2

Substituting into the equation for maximum bending moment:

M = (w'( cos theta)^2)* ((L/(cos theta))^2)/8

This results in M = (w*L^2)/8 after combining and cancelling terms.

Regards

 

[JAE]

jec67 is correct -
In all cases, we take the HORIZONTAL span (not the sloped span distance) and simply use wL^2/8 for the moment; where w is based on the PSF of snow on the projected horizontal area.

On the slope - the 1' x 1' area becomes 1' x 1/cos(theta)' which is larger - so if you go to all the unnecessary touble to do things on the slope - you START with a smaller PSF snow magnitude and end up with the same answer.

 

[dpa]

jec67,

Thanks for the quick response but it seems to me that you have an extra cos theta in the term for snow load intensity perpendiculat to the rafter. It seems like W*cos(theta) is already the intensity perpendicular to the rafter. If this is correct you would still be left with a cosine in your calculation.

DPA

 

[jec67]

dpa:

w(cos theta) would be the load in the vertical direction as related to the length of the rafter. One must then resolve this intensity into the components perpendicular to, and parallel to, the longitudinal axis of the rafter. Therefore, you will wind up with w(cos theta)^2.

 

[cap4000]

jec67

Your original statement of "the resultant of w'perpendicular to the rafter" = w'... is a typo I think??. It should be w(cos theta)^2 not the w'. Nice Job anyway.

 

[jec67]

cap4000

Good catch. It goes to show, no matter how many times one looks at his or her own work, the simplest problems are not caught.

The equations should read:

- w' = snow load intensity on rafter = w (cos theta)

- The resultant of w' perpendicular to rafter = w(cos theta)*(cos theta) = w(cos theta)^2

Substituting into the equation for maximum bending moment:

M = (w( cos theta)^2)* ((L/(cos theta))^2)/8

I am sorry for any confusion my earlier post may have created.

 

[TFL]

i design roof joist as stated ealier simply by looking at the projected snow load and the horizontal length.

what is important when looking at the problem that way is to remember to increase the dead load by 1/cos.

 

[KarlT]

I must be having a dumb day today.....

If snow loads are supposed to act on the horizontal projection of the sloping surface, then wouldn't "w" act on "L"? To convert it to the equivalent vertical load intensity along the length of the rafter (equal to L/(cos theta)) then wouldn't w' = w/(cos theta) and not w*(cos theta)???

 

[jec67]

KarlT

The intensity along the length of the rafter is w(cos thaeta). Think of it this way: The total load is the horizontal length times the snow intensity on the horizontal plane, or w*L. To find the intensity along the rafter length, divide the total load by the rafter length. The rafter length is L/(cos theta).

Substituting:

(w*L)/(L/(cos theta)= w*(cos theta)

 

[JAE]

Well, you know if we could just convince those architects to stay with that old marxist, socialist design from the Bauhaus days, with flat roofs, we wouldn't have to worry about all this complexity.

 

[CSEllc]

Using the horizontal projected snow load and increasing the DL by diving it by (cos theta) is most common method.

You can "DECREASE" the snow load to along its actual length by multiplying the load by (cos theta). Then breaking it down further into the appropriate "x" and "y" resultants.

Either way, the moments and shears are the same. The horizontal projection method is quicker.

For additional info, refer to the 2nd ed of "Design of Wood Structures" by Donald Breyer, pages 21 thru 24.

 

[cap4000]

Just too add just a little more complexity to all off this.
Be very careful of the "deflection" of the rafter based on its true inclined length, and the "axial" load force component acting on the rafter(running down towards the wall end) due to the roof pitch.