Shear Deflection – Prefab Wood I-Joist
Shear Deflection – Prefab Wood I-Joist Example Calc
(c) Jeff R. Filler, Pell City, 2022
Continuing our discussion on shear deflection of wood beams (joists, etc.), let’s look at a prefabricated wood I-joists. Basic design information for prefabricated wood I-joists is found in Chapter 7 of the National Design Specification® for Wood Construction (NDS). For more detailed design information Section 7 requires we go to the manufacturer or product code evaluation report. Let’s consider the APA PRI-40 9 ½” joist in ICC code evaluation report ESR-1405 (link).
And let’s consider the joist used to span 13.5 ft of floor, joists spacing (s) of 16 in. o.c., and carrying a floor `area’ load (σ) consisting of 40 psf Live (Occupancy) and 15 psf (Dead). For now let’s assume the Dead load includes the weights of the joists. The joists span simply between supports.
ESR-1405 says that the deflection calculations must consider shear deflection, and provides the following equation (for uniform loads):
δ = (5 ω l 4 / 384 EI’) + (ω l 2 / K’).
The first term we are familiar with … it’s the standard equation for deflection of a beam using beam deflection theory (flexural stress and strain). EI values for the joists are provided in the ESR. The (’) denotes that the value of EI must be adjusted for end use conditions, e.g., moisture and/or temperature, as required by Section 7.3 of the NDS.
The second term addresses shear deflection. K is a `shear stiffness coefficient’, also provided in the ESR, and must be adjusted, as required, by Section 7.3 of the NDS.
For this example we will assume that the joists are going to be used in dry and `normal-temperature’ conditions, so the adjustments on EI and K will not be necessary (adjustment factors 1.0).
For the PRI-40 9½ joist we read, from ESR-1405, EI = 184,000,000 psi ( = EI’), and K = 4,940,000 lb ( = K’).
The Live load on each joist is … w = σ x s … 40 psf x 16/12 ft = 53.3 plf … or 4.44 lb/in. (pli).
The Dead load on each joist is 15 x 16/12 = 20 plf, or 1.67 pli.
The Total load on each joist is 6.11 pli.
Getting L into inches, so our units work out … L = 13.5 ft x 12 in. / ft = 162 in.
Let’s calculate the deflection due to Live load …
δ = [(5)(4.44 lb/in.)(l63 in.) 4] / [(384)(184,000,000 lb/in.2)] + [(4.44 lb/in.)(162 in.) 2/(4,940,000 lb)]
= 0.217 in. + 0.024 in.
= 0.240 in.
Note that the shear term is 10% of the total. For sawn lumber we’re told that shear deflection is 3% of the total, and for glulam, 5%. I will leave it to your ponder as to why prefab wood I-joists are different.
If you put all this stuff in a spreadsheet, you’ll see that, with the equations above, the ratio of the shear deflection to total deflection increases with shorter span (and decreases with larger). While this might be instant cause for alarm, the deflections themselves are less for shorter spans, even relatively less. For short spans deflections rarely control!
The ratio of span to Live load deflection in the example above is … 162 in. / 0.240 in. = 674; 0.24 in. (about 1/4th of an inch), and δ (live) / L = 1 / 674 should be suitable for all but the most brittle floors.