This article illustrates by example the steps for designing (specifying) a Glued Laminated Timber roof beam. The beam in question is actually one of a number of identical beams that will support a panelized roof system over a Living Room of a Residential structure. The beams will be spaced 4 feet (ft) apart and span 24 ft. They will be exposed (visible) and the Owner desires the best `look’ possible for appearance. The design Snow Load for the locale of the structure is obtained from the local building department to be 30 pounds per square foot (psf) and it is estimated that the total weight of panels and other roof materials will not exceed 15 psf. The panels will be attached directly to the tops of the beam, and the panels are constructed of materials not deemed easily damageable by deflections (moderate beam deflection). The most readily available Glulam beams for the area are 24F-V4 Douglas fir (DF) with `standard’ camber. Standard sizes for this type of beam (Western species) are 3-1/8 inch (in.) and 5-1/8 in. widths and multiples of 1.5 in. for depth. Wider beams are also available although it is anticipated they would be both unnecessary and unsightly for this application. The ceiling height in the room is sufficient that headroom problems for the beams are not anticipated.
The steps for designing (specifying) the appropriate Glulam beams follow.
1. Service Conditions
From the above the important service conditions for design are
Span: 24 ft
Preferred or available depth: uncontrolled; beams will be exposed below the panels with no anticipated headroom problems
Preferred or available widths: 3-1/8 or 5-1/8 in.
Visibility: exposed, best appearance available desired
Damageable materials: beams will support roof construction not easily damaged by deflections
Beam support conditions: beams will be supported at ends with in-wall multiple-ply studs wall framing
Available bracing: attachment of panels to top of beam will provide lateral support; framing of beams into support walls will provide rotation restraint and anchoring at ends
2. Readily Available Glulam
The readily available Glued Laminated Timber for the locale is the 24F-V4 Douglas fir (24F-1.8 Stress Class) beam.
3. Design Loads for the Structure or Structural System
The pertinent loads are:
Roof `Live’ load = 30 psf (Snow)
Roof Dead load = 15 psf
Beam weight itself (Self-Weight) = to be determined
4. Applied Loads on the Beam
4.a The loads each beam must carry are:
w S = σ x S 30 psf x 4 ft = 120 pounds per linear foot (of beam), or plf, and
w D = 15 psf x 4 ft = 60 plf,
where the subscripts `S’ and `D’ of course stand for Snow and Dead.
Each beam will also have to carry `itself’.
A trial size of `4 x 8′ is suggested. The closest Glulam size to `4 x 8′ is 3-1/8 x 7.5. The weight of a 3-1/8 x 7.5 Douglas fir Glulam is,
w self-weight = γ x A = 35 pcf x (3-1/8 / 12 ft x 7.5 / 12 ft) = 6 plf,
35 pcf (pounds per cubic foot) is a common design weight for Douglas fir Glulam, and where the cross section dimensions have each been divided by 12 to give a `weight per foot of beam’ consistent with the other applied loads on the beam.
The `total’ load on the beam, including self-weight, per foot of beam, is,
w Total = 120 plf + 60 plf + 6 plf = 186 plf.
The `Whole’ load on the beam is,
W = w x L = 186 plf x 24 ft = 4464 lb.
4b. The internal forces and deflections in the beams generated by the forces above are:
Design Bending Moment = W L / 8 = 4464 lb (24 ft) / 8 = 13,392 lb-ft = 160,704 lb-in.
Design Shear Force = W / 2 = 4464 lb / 2 = 2232 lb (taken to be the shear at the end, for convenience)
Beam Reaction = W / 2 = 2232 lb.
4c. The internal stresses and deflections corresponding to the loads above are determined from the internal forces and the beam section properties.
The section properties are: A = 3.125 x 7.5 = 23.5 in.2, S = 3.125 in. (7.5 in.3)/ 6 = 29.3 in.3, and I = 3.125 (7.5)3/12 = 110 in.4.
Bending stress: fb = M/S = 160,704 lb-in. / 29.3 in.3 = 5485 psi.
Shear stress: fv = (3/2)(V/A) = (3/2)(2232 lb / 23.5 in.2) = 142 psi.
Live (Snow) Load Deflection = (5/384) W L3 / E I = (5/384) 2880 lb (24 x 12 in.)3/ (1,800,000 psi x 110 in.4) = 4.5 in.,
2880 lb is the `Whole’ Snow load, 120 plf x 24 ft, E = 1,800,000 psi as published for the 24F-V4 Glulam.
The Total Deflection (excluding the effect of `Creep’) is,
(5/384) 4424 lb (24 x 12 in.)3/ (1,800,000 psi x 110 in.4) = 7.0 in.
Already the Bending Stress and Deflections appear excessive with this size beam (especially the 7 in. deflection!).
5. To perform the Design Checks for the Bending, Shear, and deflection conditions the `Allowable’ bending, shear, and deflection values must be determined. The Allowable values for bending, shear, Modulus of Elasticity, and bearing are determined from the published Design Values multiplied by the appropriate Adjustment Factors. The (maximum) Allowable deflections are determined from the building codes.
From the NDS Supplement the pertinent Design Values are (24F-V4 DF):
Fb = 2400 psi,
Fv = 265 psi,
E = 1,800,000 psi, and
Fc perp = 650 psi.
From the International Building Code (adopted by the local building authority) The Allowable deflections are:
L/240 for Live (Snow) load (only), and
L/180 for Total load.
The Bending Design Value is adjusted as follows:
Fb’ = Fb CD CM Ct (lesser of CL and CV) Cfu CC.
CD = 1.15 (Snow load duration);
CM and Ct are not applicable since the roof beams are protected from moisture and high temperatures;
CL may be taken to be Unity as attachment of the roof system to the top (compression zone) of the beam provides lateral stability;
CV is not applicable as the beam size in question is smaller than the `standard size’ (5-1/8 x 12 x 21 ft) beam; and
Cfu and CC are also not applicable.
Fb’ = 2400 psi (1.15) = 2760 psi.
The Design Check for Bending becomes:
(Is) fb = 5485 psi ≤ Fb’ = 2760 psi? And the answer is clearly, NO!
The 3-1/8 x 7.5 Glulam does not pass the Design Check for Bending. A larger section beam is required.
The Design Check may also be cast in terms of a `Unity’ check, as follows:
Is f b / Fb’ = 5485 / 2760 = 1.99 ≤ 1.00? Obviously … NO!
The Unity Check betrays that the stress under design load is TWICE the Allowable.
All other things equal, we need a beam with TWICE as much Section Modulus.
If we hold the width of the beam constant and seek a deeper beam to provide such Modulus, the needed depth can be calculated as follows:
S (needed) = 2 x S (guessed), or
b h (needed) 2/6 = 2 x b h2/6 (guessed), or
h (needed) = √2 h (75 in.) = 10.6 in.
Since Western Glulams come in depths of multiples of 1.5 in., a depth of 12 in. is required with regard to Bending.
Before we investigate this `new’ depth, however, let’s also look at deflection.
The calculated deflection due to Live (Snow) load is, from above, 4.5 in.
The Allowable is,
L/240 = 24 x 12 in. / 240 = 1.2 in.
The Deflection Design Check is, thus,
(Is) … 4.5 in. ≤ 1.2 in.? NO! … WAY NO!
In fact, all other things equal, we need 4.5 / 1.2 = 3.75 times as much Moment of Inertia (I) as that provided by the `4 x 8′.
Using the 3-1/8 in. width, the minimum needed depth based on Moment of Inertia becomes:
I (needed) = 3.75 x I (guessed);
b h (needed)3 / 12 = 3.75 b h (guessed) 3 / 12;
h (needed) = (3.75 1/3)( 7.5 in.) = 11.7 in.
A `new guess’ of depth 12 in. should satisfy both Bending and Deflection.
Let’s try this new guess.
The weight of the beam is,
w = 35 (3.125/12 x 12/12) = 9 plf (instead of 6); and
the total load becomes 189 plf (instead of 186).
S = 3.125 (12)2/6 = 75 in.3;
The design bending moment becomes a little bit more due to the little bit more self weight; with a little bit of fancy algebra,
M = 160,704 lb-in. (189/186) = 163,296 lb-in.
fb = M/S = 163,296 / 75 = 2177 psi.
The Adjustment factors remain unchanged. (This bigger beam is still smaller than the `standard size’ beam with regard to the Volume factor.) Thus,
(Is) fb = 2177 psi ≤ Fb’ = 2760 psi? Yes! And this makes sense! The load only increased a little, but the Section Modulus took a big leap since we had to go up to the next standard size (depth).
Now let’s check deflection.
With regard to Live (Snow) load,
I = 3.125 (12)3/12 = 450 in.4; thus,
Δ = (5/384) 2880 (24 x 12)3/(1,800,000 x 450) = 1.11 in.
Since 1.11 in. ≤ 1.2 in., the Live (Snow) load deflection `checks’ (as we would anticipate.)
(Oops. Note that I have been using the Modulus of Elasticity (E) value directly, without adjustment. The only adjustment factors potentially applicable to Glulam for E are those for moisture and temperature, which we have already said don’t apply in this example; so, E’ = E. Actually, though the `prime’ on E should be shown in all deflection calculations for structural wood, it often isn’t. Sorry.)
Now let’s check shear, Total load deflection, and provide a minimum bearing length.
The Total load deflection, with a little algebra, is,
Δ (Total load) = Δ (Snow load) x (189/120) = 1.11 x 1.575 = 1.75 in.
The Allowable value is L / 180 = 24 x 12 in. / 180 = 1.6 in.
The International Building Code, Ch. 16, permits the use of half the Dead load in the computation of Total load deflection limits for members with moisture content less than 16% at the time of installation and used in dry conditions. This is often the case with Glulam in drier climates. As such,
Δ (Total, Permitted) = 1.11 x (120 + ½ of (60 +9)) / 120) = 1.29 in.
Since 1.29 in. ≤ 1.6 in. the Total load deflection check … `checks out’ … `Good’. (More on this later.)
Let’s check shear.
Since the beam span is not short it is assumed that shear doesn’t even come close to controlling. Thus the design shear V is be taken (by me) to be W / 2. For the 3-1/8 x 12 Section the shear stress becomes,
fv = (3/2) V/A = (3/2)(2232 lb / (3.125 in. x 12 in.) = 89 psi.
The Allowable Shear stress is,
Fv’ = Fv CD CM Ct Cvr (2012 NDS),
CD = 1.15 and the other factors don’t apply.
Fv’ = 265 psi (1.15) = 305 psi. Good! (Not even close.)
6. The minimum bearing length required is,
lb (min.) = R / (Fc perp’ x b).
R = 2232 lb, from above.
Fc perp’ = Fc perp CM Ct Cb.
All the Adjustment factors in this case don’t apply, thus,
Fc perp’ = 650 psi, and
lb (min.) = 2232 lb / (265 psi x 3.125 in.) = 1.1 in.
For beams I generally specify not less than 3 in. of bearing at each end.
So, the minimum bearing length is … 3.0 in. (each end).
Prevention of rotation of the beam ends will be provided for by specifying that the beams be framed into the walls in accordance with `Good Framing Practice’. Attachment of the roof system to the top of the beam with the fastener requirements of the panel manufacturer will be assumed to provide adequate lateral torsional buckling resistance.
Approved Good Framing Practice will also be assumed to provide sufficient anchorage of the beam in this situation.
9. In this particular application the beams will be visible. In fact, the `look’ of the room `shows off’ the beams. As such, undesirable deflections should be avoided. Of concern is the long term sag (creep effect) of the Dead loads on the beam. The long term sag (deflection) of a Glulam beam may be calculated as
Δ (long term due to Dead loads) = 1.5 x Δ (immediate due to Dead loads).
In our case,
Δ (Dead, long term) = 1.5 x 1.11 in. (69/120) = 1.5 x 0.63 in. = 0.96 in.
To completely mitigate the long term sag the beams should be cambered against this amount.
Before we specify this amount, however, let’s check to see if standard camber is suitable.
A camber of 3500 ft radius is standard. For a 24 ft span this camber corresponds to 0.25 in. Thus, in this example the standard camber is insufficient in mitigating the long term. A camber of 1.0 in. will be specified. This corresponds to a radius of,
R = L2/8 c = (24 ft)2/(8 x .96/12 ft) = 900 ft,
c is the desired camber, divided by 12 in the above computation for consistency of units.
It should be noted that the camber specified (1.0 in.) is to mitigate long term Dead load deflection. This camber will result in a significant `crown’ in the beams initially. As this amount is significant it is important to have an accurate determination of the Dead load, which in this example is estimated. Prior to ordering the beams an accurate determination of the roof Dead loads should be made and the camber specification adjusted accordingly.
10. As the Living Room will `show case’ the Glulams (and the Owner being an Architect) the Appearance Grade of the beams will be `Premium’.
The above example illustrates the steps in the design (specification) of a typical Glulam beam. The example is for a roof beam that will be highly visible in the Living Room of the home of a practicing Architect. Due to the desired camber and Appearance Grade the beams will be `custom’. The specification is `preliminary’ with regard to the camber which should be more precisely calculated prior to order based on a more accurate knowledge of the actual roof material weights.
National Design Specification for Wood Construction (NDS) and Supplement – Design Values for Wood Construction, 2005 (and 2012), American Forest and Paper Association, Washington, D.C.
Standard Specification for Structural Glued Laminated Timber, AITC 117, 2004 (and 2010), American Institute of Timber Construction, Centennial, CO.
Timber Construction Manual, Sixth Edition, American Institute of Timber Construction, Centennial, CO, John Wiley & Sons, Hoboken, NJ.
Glulam Beam Camber, EWS S550E, American Plywood Association – Engineered Wood Systems, Tacoma, WA.