Stone and wood have been used for building since time immemorial, and wood construction is seeing a huge rise in popularity today with the advent of mass timber construction and the increase in popularity of products like glue-laminated beams and wood I-beams.
Many buildings in the US are subject to “prescriptive” design, where generalized rules have been codified, telling builders how to do many things without the need to contract an engineer. But when large buildings, creative designs, or anything a little outside the bounds of standard construction gets involved, structural engineers are still called on to design wood beams frequently.
If you’re a licensed PE trying to help out a friend with a home construction project, or a small design firm doing the same, odds are you’re either designing a retaining wall or a steel or wood beam to help eliminate the need for a full load-bearing wall.
In the US, the design of structural wood beams is governed by the American Wood Council’s National Design Specification (NDS), which outlines the bearing, bending, and shear stress checks necessary. The NDS Supplement provides reference design values, which an engineer multiplies by various adjustment factors to account for service conditions that influence strength, and compares to the expected stresses caused by loads on the beam.
Fully-braced beams (such as floor joists with appropriate nailing patterns) can be designed without consideration of the Beam Stability Factor, the only factor which adjusts bending stresses and requires a calculation.
Today, we’re looking at how to design a fully-braced wooden beam for bending stresses (shear and bearing stress checks to come in another article) in accordance with the NDS 2018, which governs structural wood design for both residential and commercial construction throughout the US, and we’ll be saving the Beam Stability Factor for another article.
As always, be aware that your local jurisdiction may have local code amendments you need to account for.
NDS Wood Design Basics
Wood is a natural material, subject to a huge variation in material properties. Grain orientation, presence or absence of knots, growth speed, and species of tree can all influence the strength and stiffness of a given piece of wood, and engineers have to account for all of these in design.
Various organizations have sprung up to classify pieces of wood into different “grades”, either visually by trained personnel or via machine grading, which help to group wood into various levels of expected strength. Lots of knots near board edges and nasty, twisty grain tend to decrease the grade of wood, while tight-grained, straight, knot-free boards get better grades.
Then massive experimentation regimes are carried out at places like the US Department of Agriculture’s Forest Product Laboratory in Madison, WI to classify various grades and species of wood. All that knowledge is distilled down into the NDS Supplement (2018 edition) for use in design, with the appropriate adjustment factors.
Designing wood makes me sad, personally, because every piece of lumber is assumed to be at the lower boundary of a given grade classification for design. As a hobbyist carpenter and woodworker, I know that I could personally reject the worst boards and get a much stronger end product, but that’s not the way the construction industry as a whole does business.
Adjustment Factors
Wood is a bit pickier than some other construction materials. It shrinks and expands with changes in moisture (article to come later on wood moisture expansion), doesn’t like sustained loading, and suffers for being subjected to wet or high-temperature service conditions.
To account for the negative influences on strength from all these factors (and more), engineers multiply the Reference Design Values from the NDS Supplement for the appropriate species and size of wood by various Adjustment Factors.
For Wooden Beams Subjected to Bending, we’re interested in the adjustment factors that apply to F.b, which in Allowable Stress Design are:
- Load Duration Factor, C.D (check out my full article on the Load Duration Factor)
- Wet Service Factor, C.M (check out my full article on the Wet Service Factor)
- Temperature Factor, C.t (check out my full article on the Temperature Factor)
- * Beam Stability Factor, C.L (1.0 for fully-braced beams)
- Size Factor, C.F (check out my full article on the Size Factor)
- Flat Use Factor, C.fu
- Incising Factor, C.i
- Repetitive Member Factor, C.r
The adjusted bending design value, F.b’, is calculated as follows:
Beam Stability Factor, C.L
When a beam is fully-braced (its compressive flange is restrained from lateral motion all along its length, such as for a floor joist with appropriate sheathing and nailing pattern), or the beam is bent about its weak axis, the Beam Stability Factor, C.L, is 1.0.
For how to calculate the Beam Stability Factor, check out my upcoming article!
Determining the Bending Stresses in a Simple Span Rectangular Beam
The whole basis of structural design using the Allowable Stress Design method is that the actual expected stresses resulting from the design loading conditions must not exceed the allowable value.
In wood beam bending stress design parlance, that means that the actual expected bending stress, f.b, must not exceed the adjusted allowable bending stress value, F.b.
We’ve already explored how to get to F.b’, but where does f.b come from?
The bending stresses in a section are a function of the section modulus, S, and the bending moment, M, which is in turn based on the type of end fixity of a beam, the span, and the load it supports.
For a simply-supported beam (just bearing supports at the ends, no rotational resistance provided), as is common in floor joists and many wooden beams in construction, the maximum bending moment, M, is dependent on the span, L [in], and the applied load, w [lbs/in]:
And the maximum bending stress, f.b, is dependent in turn on M.max and S:
For a rectangular beam with width b and depth d, that equation becomes:
Example: Max Load for a 16′ Span Fully-Braced 4×10 No. 2 SPF Beam
In this example, we’ll look for the maximum allowable load for a SINGLE (no repetitive member factor) 4×10 nominal No. 2 grade Spruce-Pine-Fir fully-braced beam spanning 16′ as a simple span. Assume normal moisture and temperature at service, use the Load Duration Factor for Live Load, and assume the beam is not repetitive, nor incised.
First off, we need to come up with the various adjustment factors and Reference Design Bending Stress.
- Load Duration Factor, C.D
- C.D for Live Load = 1.0, per Table 2.3.2
- Wet Service Factor, C.M
- 1.0 for normal service moisture
- Temperature Factor, C.t
- 1.0 for normal service temperature
- * Beam Stability Factor, C.L (1.0 for fully-braced beams)
- 1.0 for fully-braced applications
- Size Factor, C.F
- 1.2 for SPF No. 2 4×10
- Flat Use Factor, C.fu
- 1.0 for strong-axis bending
- Incising Factor, C.i
- = 1.0 when not incised
- Repetitive Member Factor, C.r
- = 1.0 when the member is non-repetitive, per 4.3.9
Next, we need to grab the Reference Design Value from Table 4A, which gives an F.b for No. 2 SPF at 2″ and wider of 875 psi.
Multiplying out those factors and the reference design value, we get:
And by flipping around our equation from earlier and looking up the Section Modulus in Table 1B of the Supplement (49.91 in³), we can get our allowable moment as:
And finally, our last step is to flip the max moment from a simple span equation around to isolate the load in lbs/in, then plug in our M.allow and Span info:
So our 16′ span fully-braced 4×10 made from No 2 SPF can safely support (for strength only, we haven’t checked any sort of deflections) the bending stresses imparted by a uniform line load of 11.37 lbs/in, or 136.5 lbs/ft.
Watch for more articles on the details of each NDS adjustment factor, as well as how to check similar wood beams for shear and bearing stresses!
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