ASD and LRFD Design Checks for a Wood Beam

Introduction

As an example of applying the LRFD design approach to wood, we will repeat the design check for shear parallel to grain in Example 4.1 of the 5^th Edition of the Timber Construction Manual (TCM), except that we will use Load and Resistance Factor Design (LRFD) instead of the Allowable Stress Design (ASD) approach of the Example.

Design Loads

In the LRFD approach (or more broadly named Strength Design approach) the loads are combined differently than in the ASD approach. By `differently' we could say `handled'. In both approaches we start with `Service' loads. Generally speaking, these are the loads that may be anticipated during the useful (`service') life (use) of the structural member. These loads are typically prescribed in ASCE 7 and/or are prescribed by the Building Authority with jurisdiction over the structure in question and/or are calculated material weights. In Example 4.1 of the (Service) loads are: 40 psf L, 15 psf D, and 14 plf self-weight of the beam in question. Using the 16 ft tributary width and continuous spanning of joists over the beam, the `line loads' on the beam are (from Example 4.1):

... ω_L = 800 plf

... ω_D = 300 plf

... ω_{D (self weight)} = 14 plf

(same as before).

The way these loads are `handled' (combined) for the ASD approach is provided in ASCE 7, Section 2.4.1, namely combination 2. ... D + L (non-zero loads only shown).

Thus, our `combined' load in ASD is ...

... (300 + 14) plf + 800 plf = 1114 plf ... (ASD).

For the LRFD approach the combinations of ASCE 7, Section 2.3.2 are used. In our case,

... 1.2 D + 1.6 L (non-zero loads only shown).

Thus, in LRFD our `design load' becomes,

... 1.2 (314 plf) + 1.6 (800 plf) = 1657 plf ... (LRFD).

This is sometimes called a `factored load'. The Service load has been `factored up' to account for the uncertainties in the component loads.

Stresses Under Design Loads

The design stresses under these loads are (from the Example, L = 12 ft, and d = 12 in. = 1 ft),

V _{@ d} = ω L/ 2 - ω d = ω (L/2 - d) = ...

ASD

... 1114 plf (12 ft / 2 - 12/12 ft) = 1114 plf (5 ft) = 5570 lb.

LRFD

... 1657 plf (5 ft) = 8285 lb.

The corresponding Shear Parallel to Grain values are (beam width 5-1/8 in.),

ASD

... f_v = (3/2)(V/A) = (3/2)[(5570 lb) / (5.125 in.)(12 in.)] = 136 psi.

LRFD

... f_v = (3/2)(V/A) = (3/2)[(8285 lb) / (5.125 in.)(12 in.)] = 202 psi.

Design Resistance

ASD

In ASD the shear stress parallel to grain under the (ASD) design load must not exceed the Allowable shear stress parallel to grain (F_v'), given by `Reference Design Value multiplied by all applicable Adjustment Factors' (F_v C_D C_M C_t ...). Reference Design Values are provided in the National Design Specification for Wood Construction (NDS) Supplement or, for Glued Laminated Timber (only) AITC 117.Adjustment Factors are also found in the NDS and AITC 117 (Glued Laminated Timber only).

As in Example 4.1, considering Douglas fir 24F-V4 Glued-Laminated Timber (Glu-lam),

Design Value Shear Parallel to Grain, F_v = 265 psi.

And, considering, from the Example, normal duration of load, dry service and normal temperature conditions,

F_v' = F_v (1.0)(1.0)(1.0) ... = 265 psi (1.0)(1.0)(1.0) ... = 265 psi.

Thus the `design check' is ...

... is f_v = 136 psi ≤ F_v' = 265 psi? Yes; the design check (for shear) is ... `Good'.

LRFD

In LRFD the same `Reference Design Value' is used. However, the Adjustment Factors to be used ...

exclude C_D (Load Duration Factor) ...

and

include K_F (Format Conversion Factor), φ (Resistance Factor), and λ (Time Effect Factor).

The factors K_F, φ, and λ are provided in the NDS (Appendix N) and AITC 117 (for Glu-lam).

In our example, K_F = 2.16/φ, φ = 0.75, and λ = 0.8.

Thus,

... F_v' = 265 psi (1.0)(1.0)(1.0)(2.16/0.75)(0.75)(0.8) = 458 psi.

The design check becomes,

... is f_v = 202 psi ≤ F_v' = 458? ... Yes! Again good!

Unity Checks

Design checks are often cast in terms of the `Unity Check', cast in general terms of `load divided capacity' which must not exceed 1.00. The Unity Checks for our example are:

ASD

... is f_v / F_v' = 136 psi / 265 psi = 0.51 ≤ 1.00? ... Yes, good!

LRFD

... is f_v / F_v' = 202 psi / 458 psi = 0.44 ≤ 1.00? ... Yes, good!

Discussion

The Unity Checks above immediately betray that the two approaches provide different answers (though in both cases above the design check for shear in the example is still `good'). The ASD approach is simpler; the LRFD approach more complicated, but also more refined.

In the present format both design checks derive from `Reference Design Values' on the `capacity' side of the equation. In the LRFDapproach the Reference Design Value multiplied by K_F/φ may be considered the `Reference Strength' of the material. Reference strength values are used in 5^th Edition TCM per the 1996 Load and Resistance Factor Design (LRFD) Manual. For the example above, the Reference Shear Strength would thus be,

... F_v = 265 psi (2.16 / 0.75) = 763 psi (0.763 ksi).

(The reason I say would is that the 1996 value is different; it is 0.545 ksi for DF 24F-V4 beam, corresponding to a published ASD design value for shear parallel to grain at the time of 190 psi.)

Finally, it is common to deal with `capacities' when using LRFD. The adjusted shear capacity of the beam in question would be

φ V_n = (2/3) F_v' A = (2/3) 458 psi (5.125 in.)(12 in.) = 18,800lb.

This would be checked to exceed the factored load, 8285 lb.

Alternately, a corresponding unity check would be ...

V_u / φ V_n = 8285 lb / 18,800lb = 0.44 ≤ 1.00; good.

References

Timber Construction Manual, 5^th Edition, American Institute of Timber Construction, Published by John Wiley & Sons, Hoboken, New Jersey.

Minimum Design Loads for Buildings and Other Structures, ASCE Standard ASCE/SEI 7, American Society of Civil Engineers, www.asce.org.

National Design Specification for Wood Construction, American Forest & Paper Association / American Wood Council, 1111 Nineteenth St., NW, Suite 800, Washington, D.C., 20036, www.awc.org.

Standard Specifications for Glued Laminated Timber of Softwood Species, AITC 117-2004, American Institute of Timber Construction, 7012 South Revere Parkway, Suite 140, Centennial CO, 80112.

Load and Resistance Factor Design Manual for Engineered Wood Construction and Structural Glued Laminated Timber Supplement, American Forest & Paper Association / American Wood Council, 1111 Nineteenth St., NW, Suite 800, Washington, D.C., 20036, www.awc.org.