… rafter deflection!
… now for `deflection’ …
So far we’ve been dodging the `deflection’ issue. `Deflection’ of a beam, rafter, girder, header … is how much the beam `bows’, or bends. For a beam or rafter supported at it’s ends, the deflection is generally measured at midspan, e.g., how much does the beam sag, at the middle of the span. (See sketch.)
We look at the deflection of beams, etc., generally, for the following reasons:
1) if the beam bends (deforms) too much, stuff attached to it, like a `sheetrock’ ceiling, may be damaged (cracked)
2) the beam may `look’, or `feel’, unsafe, even though it may be nowhere near the condition of actually `breaking’
3) for flat, or near-flat roofs, we don’t want to cause `ponding’. `Ponding’, or the so-called `ponding instability’ is a condition where the roof rafter, beam, whatever starts to sag, and form a `dish’, or `valley’. Then rain (or melting snow) comes along, and `fills the dish (pond)’. But rain (or melted snow) is HEAVY. So the beam sags some more. This makes a deeper bend or depression in the roof, capable of capturing more water. If the beam (rafter, etc.) is not stiff enough, the `ponding’ situation gets away from us, and eventually leads to collapse (rupture of the beam) … NOT GOOD. Easy way to deal with this … put some slope on the roof. Less easy way … do some ponding instability calcs.
4) limiting deflection can be an indirect way of mitigating potential `bounce’ (vibration) of the, typically, floor. (This goes back to 2), above.) … bounce, or vibration, of a floor, as we walk across it, makes us feel unsafe, or in the case of vibration, at least feels `weird’. Easy way to mitigate bounce or vibration … make sure the beam is really stiff … select a recommended and very restrictive deflection limit. Less easy way … do a vibration analysis.
5) other effects … we won’t get into here.
So far I haven’t been worried about deflection of the roof rafters for my wife’s barn, since 1) it will have a (flexible) metal roof, that I don’t need to worry about cracking; 2) I anticipate that the deflections of the rafters are not going to look unsafe, and I doubt they will even feel unsafe. Besides, my `customer’ on this project is my wife … no one will be up on the roof while she’s in the barn taking care of her animals; 3) I’m gonna build it with a 1/12 slope … no ponding for this roof (even though we get tons of rain); 4) I’ll either like the bounce, or not, when I’m up there; and, 5) not going to worry about other effects.
So, even though I’m not worried about deflection for this roof, let’s look anyway! …
The formula for the deflection (sag) at midspan of a beam, subject to a concentrated load at midspan (me, up there working, standing, sitting) is, without proof …
Δ = P L3 / (48 E I)
Δ = deflection (sag) at midspan (where concentrated load acts)
P = the concentrated load
L = rafter span (length)
48 = a number, dependent on the location/distribution of the load, support conditions, etc.
E = the so-called Modulus of Elasticity (of the wood), and
I = the Moment of Inertia (MOI) of the wood section.
For a rectangular section, without proof,
I = b h 3 / 12
The Modulus of Elasticity is a measure of the material (wood) `stiffness’.
Doing the calc,
P = 300 lb
L = 12 ft or 144 inches (L in inches works out better below)
E = let’s use 1,500,000 psi … (mid-range for Southern Pine lumber)
For a 2 x 6 sawn (actual dimension), `on edge’
I = 2 in. (6 in.) 3 / 12
I = 36 in.4
Δ = 300 lb (144 in.)3 / (48 x 1,500,000 psi x 36 in.4 )
Δ= 0.3456 in.
Let’s look at this …
This is a tiny bit cumbersome of a calculation … so let’s first see if it makes sense. This says that when I stand out there at the middle of the 2×6 rafter, spanning 12 feet, that it will sag about a third of an inch, under my weight (assuming it doesn’t break first, which we determined it probably wouldn’t) … does that make sense? Well, thinking of the lumber, my weight, the span, … it doesn’t seem Un-reasonable. If the number came out to be 3.4 inches … I would worry about my calculations being wrong … did I put in the wrong weight? … do have the correct MOI? … did I take the length to the correct power, and so on. If I had calculated 0.03456 inches … that’s a pretty stiff beam … hmmm … only sagging 1/30th of an inch under 300 lb???
NOTE: I show 0.3456 inches … four significant figures. None of the input information is that exact, so I really can’t argue that exactness in the deflection calculation. Perhaps especially with the value for E. Lumber is a highly variable material. When I grab a (published) value of 1,500,000 psi, first of all, it’s published at only one or at most two significant figures. Plus, it’s the average of a range of E values for that species and grade of lumber. All this said … deflection calculations are not exact, even though they might look it.