Wood Column Design Using the 2005 NDS LRFD Provisions

Consider a wood column, length (height) of 9 ft, `pinned' at both top and bottom, subject to an axial load of `half Dead' and `half Snow', let's calculate the total (service load) capacity of a Douglas fir (DF) No. 2 Glued Laminated Timber (glulam) column using both the ASD and LRFD provisions in the 2005 Dual Format National Design Specification® for Wood Construction (NDS). (Assume dry conditions and normal temperature.)

FIRST LRFD

The axial load capacity will be determined by the setting the equation f_c = P/A equal to F_c', where

f_c = actual (applied) compression stress parallel to grain (under design loads),

A = cross sectional area of the column, and

F_c' = adjusted compression value parallel to grain (NDS® Section 1.6), and where

F_c' = F_c multiplied by all appropriate factors, where

F_c = the reference compression design value parallel to grain;

and where the appropriate factors are listed in NDS® Table 5.3.1 (glulam).

For LRFD these factors are (shown applied to F_c):

F_c' = F_c C_M C_t C_P K_F ϕ_c λ,

where

C_M and C_t = Wet Service and Temperature factors, in this example taken to be unity,

C_P = the Column Stability Factor, which we will have to calculate,

K_F = Format Conversion factor, for use with LRFD,

ϕ_c = Resistance Factor (a `strength reduction' factor) for compression, for use with LRFD, and

λ = the Time Effect Factor, for use with LRFD.

Here is C_P (NDS® Section 3.7):

C_P = (1 + F_cE/F_c*)/2c - √{[(1 + F_cE/F_c*)/2c]² - (F_cE/F_c*)/c}, (note: same equation for both ASD and LRFD)

where

F_cE = critical buckling design value for compression members,

F_c* = reference compression design value parallel to grain multiplied by all applicable factors except C_P,

and

c = value based on whether sawn lumber (0.8), round timber pole or pile (0.85), or glulam (0.9).

Let's tackle F_cE;

F_cE = 0.822 E _min' / (l_e/d)², where (note: same equation for ASD and LRFD),

E _min' = adjusted modulus of elasticity for column stability calculations,

l_e = effective column length, equal to K_e l, where

K_e is our familiar buckling length coefficient for compression (yeah, Euler Buckling), in our case we'll use 1.0;

and

d is the `depth' of the member.

For a column that is unbraced in either/both directions, we need to examine E _min' and l_e/d for both/each.

The direction that produces the lesser F_cE will control C_P and thus the design of the column.

From Table 5B of the NDS® Supplement for the DF No. 2 combination, we obtain E _min = 830,000 psi. This value is applicable to both `directions' of buckling. We also get F_c = 1950 psi (4 or more laminations).

With E _min and le the same for each direction, the lesser `d' will govern; thus

d = 5.125 in.,

l_e/d = 1.0 (9 x 12 in.)/ 5.125 in. = 21.07 (a slenderness ratio, which may not exceed 50, Section 3.7.1.4, and which is the same for ASD and LRFD),

and, for E _min' we take E _min and multiply it by the appropriate adjustment factors (Table 5.3.1):

E _min' = E _min C_M C_t K_F ϕ_s ( ... LRFD specific).

For the LRFD `factors', we go to NDS® Appendix N;

K_F for E _min is 1.5/ϕ,

ϕ for E _min is 0.85,

and since we'll need them shortly,

K_F for F_c is 2.16/ϕ,

ϕ for F_c is 0.90.

So,

E _min' = 830,000 psi (1.0)(1.0)(1.5/0.9)(0.9) = 1,245,000 psi (LRFD specific).

And now,

F_cE = 0.822 (1,245,000 psi) / (21.07)² = 2305 psi. (LRFD specific)

Now we are almost ready to calculate C_P.

For F_c* we go back to Table 5.3.1 (and get all the factors except C_P);

Fc* = F_c C_M C_t K_F ϕ_c λ.

We need λ.

From Table N.3.3 we get λ = 0.8 for Dead plus Snow.

Actually, from the Table we also get the appropriate LRFD `Load Combination',

1.2 D + 1.6 S, (LRFD)

which we will use in a bit; otherwise we need to get the Load Combination from ASCE 7 (or IBC Section 1605).

SO,

F_c* = 1950 psi (1.0)(1.0)(2.16/0.9)(0.9)(0.8) = 3370 psi. (LRFD)

To make number crunching a tiny bit easier,

F_cE/F_c*= 2305 psi / 3370 psi = 0.684.

So,

C_P = (1 + 0.684)/2(0.9) - √{[(1 + 0.684)/2(0.9)]² - 0.684/0.9} = 0.596. (LRFD)

And,

F_c' = F_c C_M C_t C_P K_F ϕ_c λ = F_c* C_P = 3370 psi (0.596) = 2009 psi. (LRFD)

The `capacity' of the column is thus (by setting f_c = F_c'),

F_c' A = 2009 psi (5.125 in. x 6 in.) = 2009 psi (30.75 in.²) = 61,800 lb. (LRFD)

Recalling that in this example the loads divide out half Dead and half Snow,

F_c' A = 1.2 D + 1.6 S = 1.2 (0.5 x P _{total service load}) + 1.6 (0.5 x P _{total service load}) = 61,800 lb,

P _{total service load} = 44,100 lb. (LRFD)

(Approx 22k Dead and 22k Snow ... LRFD)

Now let's solve the problem in ASD.

In ASD,

F_c' = F_c C_D C_M C_t C_P (NDS® Table 5.3.1);

C_P = the same equation,

but F_cE changes because E _min' changes;

E _min' = E _min' = E _min C_M C_t (NDS® Table 5.3.1) = 830,000 psi (1.0)(1.0);

E _min' = 830,000 psi. (ASD)

The slenderness ratio doesn't change, so

F_cE = 0.822 (830,000 psi) / (21.07)² = 1537 psi. (ASD specific, and way different than the LRFD value)

F_c* = F_c C_D C_M C_t.

For Snow, C_D = 1.15 (NDS® Table 2.3.2), so,

F_c* = 1950 psi (1.15)(1.0)(1.0) = 2243 psi. (LRFD specific)

F_cE/F_c*= 1537 psi / 2243 psi = 0.685. (LRFD)

And c = 0.9, still.

C_P = (1 + 0.685)/2(0.9) - √{[(1 + 0.685)/2(0.9)]² - 0.685/0.9} = 0.597. (ASD)

But, whoa, close to the LRFD value (at least in this example).

F_c' = F_c* C_P = 2243 psi (0.597) = 1338 psi. (ASD)

The corresponding capacity is,

F_c' A = 1338 psi (30.75 in.²) = 41,100 lb. (ASD)

P _{total service load} = P _Dead + P _Snow. (ASD)

P _{total service load} = 41,100 lb. (ASD)

(Approx 21k Dead and 21k Snow ... ASD)

In this case the capacity of the column calculates out about the same for both the LRFD and ASD methods (though a lot of the numbers along the way are different).

References

National Design Specification® for Wood Construction, and Supplement Design Values for Wood Construction, 2005, American Forrest and Paper Association, Washington, D.C., www.awc.org.

Minimum Design Loads for Buildings and Other Structures, ASCE 7 (7-05), American Society of Civil Engineers, www.asce.org.

International Building Code, International Code Council, www.iccsafe.org.