# roof rafter calcs … BENDING STRESS

Designing a roof rafter … (bending stress) …

Linda wants a barn-shelter for her goats. Okay – *I’ll* do it. `Twelve by twenty-four.’ `Let’s go walk the site. It’s got some slope. I’ll make a stepped-barn, each section will be twelve by twelve’.

We have rough-sawn 2×6 pine all over the place; let’s see if I can do it (the ROOF) with 2×6 rough sawn!

What are the design conditions?

The roof will need to hold up construction workers (me!, alone, putting up the framing, and, later, a team, putting on the final metal roof).

The roof will also need to withstand wind loads … typically an uplift load … the air coming into the barn (open front) and trying to pry off the roof!

Let’s do it!

The 2×6 is actual dimensions … 2 in. thick x 6 in. deep.

Species – (southern) pine!

Let’s space the roof rafters 24 inches apart (`on center’ … or `o.c.’)

Span will be 12 feet, actually a bit less, as the support posts are 12 feet apart, outside to outside … but we’ll use 12.

We’ll have 12 in. overhangs at both ends. We’ll ignore the overhangs (counter-bending) for our `bending’ calcs.

The Code-required construction loads are … a 300-lb construction worker, concentrated load, or 20 pounds per square foot (psf) uniform load.

Plus, of course, the roof rafters have to carry the weight of the roof itself.

First let’s see what `governs’ for the two construction loads above …

Case 1 – Single Construction Worker – Concentrated Load

The `worst’ bending effect of a construction worker will be if he/she is at the middle of the 12-ft span. In this case the bending effect (`bending moment’) in the beam will be …

M = P L / 4

where

M = bending moment (effect)

P = the concentrated (`person’) load (Code says 300 pounds – a worker and stuff)

L = beam span-rated, and

4 = the number 4, shown here without proof.

So,

M = 300 lb (12 ft) / 4 = 900 lb-ft … this is the bending moment, or effect, in the beam, at mid-span, due to a 300-lb load (person) at mid-span.

Case 2 – Uniform Load of 20 psf

If the rafters are 2 feet apart, then every foot of rafter (length) carries 2 feet of roof (one foot on either side, or halfway to adjacent rafters). In equation form …

w = σ x s

where

w is the `line load’

σ is the `area load’ or pressure (in this case 20 psf of construction activity),

x is the multiplication symbol

s is the rafter spacing …

So,

w = 20 psf x 2 ft = 40 lb / ft … every foot of length of beam carries 40 pounds …

(Think on this for a bit … if this doesn’t make sense, there is no need for you to go further.)

The `Whole’ load on the rafter, W, is …

W = w x L …

W = 40 lb/ft x 12 ft = 480 pounds.

Let’s make sure you follow this … every rafter carries 2 ft x 12 ft of roof … or 24 square foot. If each square foot of roof is loaded with 40 pounds … then each rafter carries 24 x 20 = 480 lb.

The bending effect of a `uniform’ (uniform along the length) load is …

M = W L / 8,

In this case …

M = 480 lb x 12 ft / 8 = 720 lb-ft.

Sometimes you’ll see the bending moment expressed as …

M = w L^{2} / 8 ….

(Little) w vs. (big) W.

Gives the same thing.

Again I show the `8’ here without proof … but you’ll see it in engineering textbooks, design aids, etc.

NOTE: even though the `whole’ load, 480 lb, is more than the `person’ load, 300 lb, the 300 has a greater bending effect (900 vs 720) because it is `concentrated’ at the middle of the span! For other effects, such as shear and bearing, we need to come back to the `480’.

When we design wood members (beams, rafters, joists, etc.) we generally look at the `stress’ the effect, in this case the bending effect, has on the wood fibers. For a member in bending, the `fibers’ on the bottom side of the beam are stressed in tension, and the fibers on the top are stressed in compression. In terms of numbers, for a rectangular-section member, the bottom and top stresses are the same (number) … in this case …

f _{b} = M / S

where

f _{b} = bending stress (tension on one face, bottom; and compression on the other, top)

M = bending moment, from above

S = the so-called Section Modulus.

Without proof … for a rectangular section

S = b h^{ 2} / 6

where

b = rafter thickness (typically the narrow dimension, rafter on edge)

d = depth, and

6 shown (again, without proof).

I like to think of S in terms of the cross section area (A) of the rafter (beam, girder, joist, whatever);

A = b x h

So,

S = (b h ) x h / 6

S = A d / 6.

In the case of a 2 x 6 on edge (actual dimensions) …

S = (2 in. x 6 in.) x 6 in. / 6

S = 12 in.^{3}.

The bending stress is …

But, wait … you’ll see our `units’ get messed up if we don’t first get the moment into (units of) lb-in. …

M = 900 lb-ft x 12 in. / ft = 10,800 lb-in.

Now, then …

f _{b} = M / S = 10,800 lb-in. / 12 in.^{3}

f _{b} = 900 lb / in.^{2 } … (900 psi).

YIKES! … I wish the numbers had come out a bit different … the `900’ lb-ft only gives `900’ psi because we happened to multiply by 12 feet, and then divide by 12 in.^{3.}. A rafter with a different Section Modulus would give a different stress.

(Let’s say we use 2×6 dimension lumber, from `Lowes’ (or Home Depot, or local lumber yard). For dimension lumber, the 2×6 are `nominal’ dimensions; the actual dimension are … 1.5 in. x 5.5 in. … maybe even a tiny bit less. The Section Mod for these dim lum rafters would be … S = 1.5 x 5.5 x 5.5 / 6 = 7.563 in.^{3}. Less wood … less Section Modulus … trying to carry the same load … more stress … let’s see: f _{b} = M / S = 10,800 lb-in. / 7.5625 in.^{3 }= 1428 psi … yup.)

I need rough sawn 2×6 that can carry at least 900 psi … (or dim lum 2×6 that can carry 1428 psi).

…

But what about the weight of the roof?

This is going to be a `light’ roof … without going through all the fuss … it will probably weigh 5 psf or less, including the rafters. I guess I should add that in. We’ll treat it as an area load, crunched into a line load, like we did above with the `20’. If we got a bending moment of 720 lb-ft from 20 psf, then 5 psf would give us … 720 (5/20) = 180 lb-ft … should be added in.

BUT! … I’m the only construction worker who’s gonna be up there, and I don’t weigh 300 lb. I’m not going to tell you what I weigh, but let’s say me, plus hammer, lot of nails, boots, clothes, etc. … takes me up to … 250 lb. The bending moment due to `me’ at midspan will be …

M = P L / 4 = 250 (12) / 4 = 750 lb-ft … down from 900 an amount of … 900 – 750 = 150 … about a wash! … 180 lb-ft more vs 150 lb-ft less … and the 5 psf and the 250 lb were just `guestimates’. If I really want to split hairs, I can calculate the finish roof weight, and I can also strip down and carry fewer nails with me up on the roof.

So, yeah, I need 2×6 rough sawn (actual dims) rafters that can carry 900 psi. This is doable.

We’ll `make sure’ we get 900 psi on another post. (Or, if we want to use dim lum, that will also be on another post.)

…..

We’ll also look at wind, later.

JRF