E min and `Never’
I continue to be mesmerized by this thing called `E min’. E min is the `minimum modulus of elasticity’ used in stability calculations in wood engineering design. E min (hereafter just Emin) is based on `regular’ modulus of elasticity, E. For wood design, E is taken to be the average modulus of elasticity of that particular species, grade, etc., of wood members, or `pieces’. E gets used a lot in deflection calculations, i.e., how much a beam will `sag’ under load. Since individual pieces of wood in any particular species, graded, etc. vary, so does E. The amount of variation is reflected in the `coefficient of variation of E’ (COV E). This coefficient of variation is the standard deviation, σ, divided by the mean, μ, for the `population’ of wood pieces. COVE is taken to be 0.25, for visually graded sawn lumber, 0.10 for structural glued laminated timber, and so on. Other wood properties (bending strength, shear strength, etc.) have different coefficients of variation (saved for a different discussion). Suffice it to say, wood properties can be highly variable.
Buckling equations for wood members (beams and columns) depend on E. `Buckling’ is different than `bending’. When a wood member bends, it simply `flexes’, more or less, depending on the member, load, and support conditions. To make sure it doesn’t bend so much that it `breaks’, design values for bending (strength), shear (strength), etc., take into consideration variation in wood properties, and are based on `fifth percentile’ values, plus factors of safety. (Again, for another conversation.) This fifth percentile thing, and factors of safety, assure `breaking’ will `never’ happen. The `fifth percentile’ drives the design values (for bending, shear, etc.) well below the mean properties.
How much a beam bends (sags, deflection), on the other hand, before it breaks (though it never will), are based on average E. This is because deflection (sag) is a serviceability issue, not a `safety’ issue. The other calculations assure the beam (column, etc.) is `safe’; deflection calculations simply let us calculate the sag … ½ inch at midspan, ¾ inch, or whatever. The calculations, though they might make you think (by their complexity) they provide exact answers, … they do not. And how could they, if they are based on average values, but individual beams, joists, columns, etc., going into structures, will have values different than the mean values (some lower, some higher).1
Back to buckling. Buckling is different. Buckling is a `stability’ issue. Buckling of a wood member can be catastrophic (sudden collapse). For wood members in building design the two main buckling issues are lateral-torsional-buckling of beams (joists, rafters, etc.) and Euler buckling of columns (posts). The equations used to investigate buckling utilize E. Since buckling can be catastrophic, and since we actually want buckling to never occur, we don’t want to use just any average E, especially knowing E can be so variable. We use Emin.
Emin is determined by finding, again, the 5th percentile E, and dividing that value by a factor of safety of 1.66.
We can see this in, for example, the formula for Emin, in Appendix D of the National Design Specification® for Wood Construction (NDS).
Emin = E [1 – 1.645 COVE ] (1.03) / 1.66,
where,
the [1 – 1.645 COVE ] gets us down to the 5% level on E,
1.03 (or 1.05 for glulam) adjusts from apparent E to shear-free or true bending E (another conversation), and
the 1.66 is a factor of safety.
Selah.
Back to this idea of `never’. Let’s look at an example … a 24F-V3 Southern pine glued laminated timber beam, and the NDS Supplement Table 5A Expanded, as well as Appendices D and F in the NDS. From the NDS Supp 5A we get Ex true = 1,900,000 psi; Ex app = 1,800,000 psi, and Ex min 950,000 psi.
Following NDS App D we can see that Ex app x 1.05 = 1,800,000 x 1.05 = 1,890,000 psi, round to 1,900,000 psi = Ex true (shear-free) … yeah!
Then,
Ex true [1 – 1.645 COVE] = 1,890,000 psi [1 – 1.645 (0.10)] = 1,580,000 psi … where 0.10 is the COVE for glued laminated timber (NDS App F). This is the `E05’, or `fifth percentile value’. Not more than one in twenty beams coming off the assembly line will (theoretically) have an `actual’ E of less than 1,580,000 psi.
(This 1.645 is `1.645’ standard deviations below the mean.)
Now let’s divide by 1.66.
Ex05 / 1.66 = 1,580,000 psi / 1.66 = 950,000 psi … yeah, as from NDS Supp 5A.
Let’s calculate the `Z statistic’.
Z = (x – μ) / σ , where `x’ is 950,000 …
σ = standard deviation = COV times the mean, μ, … 1,890,000 x 0.10 = 189,000 psi.
So, Z = (950,000 – 1,890,000) / (189,000) = -4.97
We’re 5 standard deviations below the mean.
My Z calculator2 shows … 0.0 probability.
Let’s try another Z calculator … oh, 3.4 x 10^-7 …
One in 3 million of these 24F-V3 beams will have an actual E of (less than) 950,000 psi.
Almost `never’.
I have often wondered what this beam, or board, would `look like’? (Would it even make it out of the manufacturing plant? I doubt it.)
Let’s do something similar for sawn lumber. Let’s do visually graded Douglas fir No. 2 … NDS Supp Table 4A gives E = 1,600,000 psi. This is an apparent E. So, per NDS App D, E true = 1,600,000 x 1.03 = 1,650,000 psi. Then to E05 … 1,650,000 [1 – 1.645 (0.25)] = 970,000 psi … noting the high COVE of 0.25, from NDS App F.
Then we divide by 1.66 … getting Emin = 585,000 psi … yeah, agrees with Supp Table 4A. Let’s calc the standard deviation … σ = 0.25 x 1,650,000 = 412,000 psi. Now for Z …
Z = (585,000 – 1,650,000) / 412,000 = -2.58.
The corresponding probability that we’ll run into a board with E (actual) less than 585,000 psi is … from a Z calculator … 0.00494 … or 0.005!
About 1 in 200 boards will have E true of less than 585,000 psi.
I have often wondered … what would this board (piece) look like? The board has only about a third of the stiffness of the average for the grade and species. Is it falling apart? Rotted? A huge knot? Or huge amount of wane? Some other series defect(s)? Would it even make it onto the grading table? Maybe? Would it make it off the other end?
So, we’re more likely to run into a sawn piece of lumber that has an actual E equal to or less than Emin, contrast to, say, a more-controlled wood product, like glulam … because of the COVE. While we would essentially `never’ run into a glulam beam with and E actual of Emin, we might very well … with sawn lumber.