As we have seen, the posts and the planks are the strong components of the post and beam (post and girder) and plank and beam (plank and joist or rafter) systems. The use of posts in scale with the girders will assure post strength. Two — by-six tongue-in-groove planking is an excellent and pleasing floor and roof system, although you should know that the true finished dimensions of this material is actually 1.V2 inches thick by about 5Vs inches wide. With frequent joists or rafters, you can easily use the lighter and less expensive “five-quarter” (full one — inch) by six-inch (<yV8 inch) tongue-in-groove planking. In reality, you can use three-quarter-inch plywood, even with an earth roof, as we did at our library. We had no sagging at all, though the greatest span on the radial rafter system was only about 39 inches (99 centimeters), and this was on the overhang.
The members that need to be engineered for are the girders, the rafters, and the floor joists. It is important to know that there are five distinct considerations that come into the design work for these members, and they are:
i. Load. You have to know what degree of load you are asking the system to support. (See Sidebar on page 31.) So, as an example, if you are planning an eight-inch-thick thick earth roof over a two-inch-thick crushed stone drainage layer, for an area with a 70-pound snow load, add the following figures from the chart: 80 (earth) + 20 (stone) + 70 (snow) + 15 (dead load) = 185 PSF.
2. Wood quality. To engineer for any beam, you have to know the stress load values for the species and grade of wood you plan to use, particularly the unit stress ratings for bending and for shear (in pounds per square inch). For example, unit stress for bending can vary from 1,100 PSI (Eastern Hemlock, common structural) to 2,150 PSI (Douglas Fir, inland region, select structural).
3. Frequency of rafters or joists. As discussed above, under the heading Plank and Beam (page 29), frequency simply refers to how many members you are using. Are the rafters on 16-inch centers? 24-inch centers?
4. Beam dimensions in section. Will you be using two-by-eights? Five-by — tens? Eight-by-eights? Vigas with a small-end diameter of six inches?
5. Clear span of the beam. This is the one that trips up most owner-builders, particularly when it comes to designing a structure to support an earth roof.
The problem is that bending strength decreases as the square of the span.
For example, lets compare a io-foot span to a 20-foot span. Instinctively, many people think that a beam has to be twice as strong to support the longer span, other considerations remaining the same. There’s a kind of logic there, but it is wrong. You’ve got to compare the squared spans. Ten times ten equals one hundred (10 X 10 = 100), but twenty times twenty equals four hundred (20 X 20 = 400). The beam carrying the 20-foot span needs to be four times stronger than the one carrying the io-foot span.
I’m going to give another less obvious example of how span (squared) influences strength, an example that pops up all the time with students at our earth-sheltered housing classes. The stress-load calculations for both the Earth wood house and the “40 by 40 Log End Cave” plans are predicated upon
nine-foot spans. These are popular designs that have been built all over North America. Invariably, people ask me if they can stretch the spans to ten feet. (Nine feet, 1 guess, seems just a little tight for them.) The answer is, of course, yes, you can do almost anything if you know what you’re doing and you have enough money. Instinctively, people figure that rafters or girders probably have to be io percent stronger to carry the extra foot of span. The math says otherwise: 9X9 = 81. But 10 X 10 = 100. The difference is 19. And this difference must be expressed in relationship to the original 81, not 100. Well, my trusty calculator tells me that 19/81 = .23457. The change will require making up a shortfall of about 23.5 percent, a considerable difference from the original engineering.
In the example above, the span has been changed, so one or more of the other four design considerations must be altered to make things right. We could decrease the load by 23.5 percent by using less earth and using a lightweight drainage product instead of a crushed stone drainage layer… or by building in Chattanooga instead of Buffalo to take advantage of the decreased snow load. We could choose a wood with 23.5 percent more bending strength, perhaps a stronger species or a higher grade of the same species. We could actually use 23.5 percent more of the originally engineered rafters by increasing the frequency, and that would take care of it. Or we could reengineer the sectional dimensions of the rafter; use six-by-tens instead of five-by-tens, for example.
You must know four of the five variables listed above to calculate the fifth. If you know load, quality of wood, rafter frequency, and span, for example, you can calculate the cross-sectional dimensions of the rafter. Or, given the kind and grade of wood, you can calculate the load that a particular rafter system will support.
If you can plug numbers into a formula, you may wish to follow through the examples of Appendix B: Stress Load Calculations for Shear and Bending. But, in reality, for more conventional (non-earth) roof systems, just use existing engineered span tables, like the one in Appendix A.
