# AITC 119 and EEEE

I was doing some reading in AITC 119 Standard Specifications for Structural Glued Laminated Timbers of Hardwood Species, Section 4.1.6 … “ … modulus of elasticity, E, values herein are the average values … The modulus of elasticity of wood of a given species is variable.”

Aha!, yes, design values for E are average values. Modulus of elasticity is a measure of wood stiffness. Wood stiffness is related to how much wood deforms when subject to load, commonly, how much a beam (or joist or rafter) will `sag’. Engineers call this sag `deflection’. We have fancy equations for deflection. The deflection (sag) of the midspan (relative to the ends) of a simply supported beam (or joist, rafter, girder, or similar) is given by the equation:

Δ = 5 W L3 / (384 E I ),

where,

W is the `whole’ load on the beam (joist, rafter, etc.),

L = beam length,

E = the modulus of elasticity of the beam (more to say here),

I = the moment of inertia of the beam, and

5 and 384 are just pure numbers, related to how the load is distributed and the beam supported.

The load W is generally `predicted’, the result of some future event, say a `once-in-a-lifetime’ snow storm. For us engineers this future event is typically predicted by others, and we work off of some amount (e.g., of snow) associated with the event. The number we work with is a prediction, even though it may look like it’s exact.

The beam length, L, and moment of inertia, on the other hand, are more exact. We know pretty darn close how long the beam is (or is to be). The moment of inertia is based on the beam (or rafter, or …) width and depth. We can actually measure these with a tape measure, pretty close, or we can just rely of the dimensions stated (by the manufacturer), plus or minus a bit of tolerance (allowed in manufacturing).

E likewise might very well appear to be exact, such as, E = 1,800,000 psi (pounds per square inch).
Actually, all the zeros betray that, perhaps, only the 1 and the 8 are significant. But, importantly, remember, E is average. Actual E values, board to board, beam to beam, rafter to rafter, etc. … vary. Even for boards, beams rafters of the same species, and grade, and even from the same manufacturer, or even from the same day of manufacturing. A common way to describe this variability is through the coefficient of variation, C.O.V. (or COV). It is defined as the standard deviation divided by the average. For a certain beam with a design value for E of 1,800,000 psi, and COV or 0.10, the standard deviation (measure of variability) of the E’s of individual beams of this species and grade is … 0.10 x 1,800,000 psi = 180,000 psi. (As AITC 119 Section 4.1.6 points out) … in a normal distribution, 2/3 of the actual E values `in population’ would be within 1,800,000 plus 180,000 and 1,800,000 minus 180,000 psi. Or, 5/6 of the E’s would be greater than 1,800,000 – 180,00 psi = 1,620,000 psi. Or, half of the E’s would be less than 1,800,000 psi, and 1/6 would be less than 1,620,000 psi. Is that a big deal?

Well, in most cases not. Since E is in the `denominator’ of the deflection equation, a lower E (in an individual, or specific, beam), albeit only a bit lower, will result in a bit greater deflection, or sag (notice I didn’t use the word `higher’). So, if the calculated sag for a beam is ½ inch, there’s a 1/6 chance it could be as much as … 10 percent greater, or 0.505 inches. Sure we could measure the difference; seeing it might be more difficult. Only in a case where 0.50 inches (0.500) would be critical, might we worry about this variability. Where specific deflection (sag) amounts are critical, one solution would be to order a stiffer beam (higher E). We could order a beam with E = 2,000,000 psi (if one exists), and calculate the deflection, and its variability. There would still be some `chance’ of the beam having an unacceptably low E, though less chance. If we really want to make sure we have an acceptable E for a particular beam, we need to `measure’ E for that particular beam, though this is typically not done in practice, say, beam by beam, on a jobsite. You might be able to get a manufacturer to measure the E of individual beams for a particular order … I’ve never done it.