# E min

E min

(c) Jeff R. Filler, Pell City, Alabama, 2022

But First, E apparent and E true

`E min’ is calculated wood property used in the design of structural wood members (beams, columns, etc.). It is a derived property, derived from the Modulus of Elasticity, E. The Modulus of Elasticity, E, is generally used to determine deflections of wood members such as beams, joists, rafters, girders. Values of E for sawn lumber, glued laminated timber (glulam), and other structural wood members can be found in the *Supplement* to the *National Design Standard*® *for Wood Construction *(NDS). Values of E for proprietary wood products, such as prefabricated wood I-joists and structural composite lumber (SCL), may be obtained from the product manufacturer and/or code evaluation reports. Predominately, E values are used directly in `bending’ applications; e.g., used for the calculations of deflections. These E values are used in `single-term’ deflection equations derived from `beam deflection’ theory. The equations address the strains generated by tension and compression stresses generated from bending moments acting in the beam, and, in theory, relate bending stresses to beam deformation (curvature) though the Young’s Modulus. In reality, however, wood beams deflect *a bit more* than as would be predicted using Young’s Modulus and the `singe-term’ beam-deflection-theory equations, due to the additional distortion and deflection caused by shear stresses in the beam. The additional deflection can be handled two ways. The first way is to include a `second term’ in the deflection formula which directly addresses deflection due to shear (`shear deflection’). The second way is to stick with the `single-term’ (beam-deflection-theory) equation and determine (in the case of testing), or use (in the case of design), an *apparent* Moment of Inertia that directly relates the total deflection to specific beam section, load, and support conditions. This so-called `apparent Moment of Inertia’ (E app, E apparent, and also often just *E*) is a bit *less* than the Young’s Modulus, or `true’ Moment of Inertia (and is, as said, reflected in the bit *more* deflection, due to the shear effect). In the case of sawn lumber, for example, values of E published in the *National Design Standard®* *for Wood Construction* (NDS) *Supplement *are to be increased by 3% (multiplied by 1.03) to obtain `shear free’ values (NDS Appendix F).

E min is derived from the `shear free’ E.

So, on our road to E min, we need to either determine, or obtain, `shear free’ E, a.k.a., true E. We can get it indirectly by calculation from the published E values that include shear effects, as described above, or, in other cases, directly, as published by wood products manufacturers, in the NDS *Supplement*, etc.

Once we have `shear free’ E, we can get E min.

Wood is a highly variable material. Wood properties vary between species, and among species. In the structural wood world, species, or species groups, are further subdivided into `Grades’. This `sorting’ into grades helps enables more efficient use of wood, as wells as helps pin down, or at least `narrow’ down, the range of properties with the Grades. But, at the end of the day, there’s variability in the properties of the woods within each grade.

For deflection calculations (using E app or E true) the variability is, for the most part, accepted as is. The published values for E are `average’ values for the species and grade. Some pieces will have more `E’, some less. As such, all other things equal, some members (beams, joists, etc.) will have a more, or less, deflection. And, again, this is generally accepted. Deflections are `serviceability’ issues … typically not *safety* issues. A little bit more deflection … `no big deal’. A bit *less* deflection, better yet! Where deflections are critical, a more stringent deflection limit could be set (to capture and corral some variability), or products with more precise properties could be used.

E min, on the other hand, is used for *stability* calculations, which are very much a *safety* issue. Stability issues for wood members include potential buckling of wood columns, posts, or studs, as well as lateral torsional buckling of beams (joists, rafters, etc.). Buckling of wood members may result in catastrophic rupture or collapse of the member itself, and even progressive collapse in a structural system. Thus while we might be somewhat `care free’ with the `E’ used in deflection calculations, we better be pretty darn tight with the E we use in stability calculations. Alas, E min.

The first step in getting E min (after getting E true), is to obtain E 05. This is the (supposed) value of E for which 95% of the members in that grade have equal or higher E.

In equation form,

E _{05} = E (true) (1 – 1.645 COV_{E})

where

COV_{E} = the coefficient of variation of E, representing one standard deviation relative to the mean standard deviation.

But that still isn’t good enough. Five percent of the wood members in that Grade could have lesser E, which could lead to catastrophe! We then take that number (E 05), and divide it by a factor of safety of 1.66.

Thus,

E min = E (true) (1 – 1.645 COV_{E}) / 1.66.

Selah …

Let’s say we’re dealing with sawn lumber, which has a published COV_{E} of 0.25.

And let’s say we’re starting with a published (reference) E (app) of 1,800,000 psi.

Then E true is 1,800,000 psi x 1.03 = 1,854,000 psi.

Then E 05 is E (true) [1 – 1.645 (0.25) ] = E (true) (1 – 0.411) = E (true) (0.589) = 1,092,000 psi,

and

E min = 1,092,000 / 1.66 = __658,000 psi__.

Nearly *one-third* of E (E app or E true).

I often wonder what this piece of lumber would look like. I think it represents a piece of wood with an E value between 2 and 3 standard deviations less than the mean … which, for a `normal distribution’ (if sawn lumber is `normal’) … represents, in this case, about one piece in 200. What would it look like? Mostly bark? Would it have a huge knot, or other big defect? Hopefully it would get `tossed’ (out) in the lumber sorting and grading process. If not, I would hope the `do-it-yourself-er’ at the lumber yard wouldn’t put it on his cart, or, if all else fails, the framing contractor wouldn’t dare use it for a beam or column.