Generalized Lateral Stability Calculation for Wood Beams

CanadaCSA O86:19NBCC 2015CWC Wood Design Manual 2017

Wood Beam

To account for lateral stability in bending, the Canadian CSA O86 standard makes use of a factor

K_L

which reduces the capacity of a member based on its length and different loading conditions. To account for different loading conditions, a table is provided that matches often-seen conditions with effective lengths. For instance, a uniformly loaded beam will have an effective length of

1.92L

, whereas a center-loaded simply supported beam will have an effective length of

1.61L

if there are no intermediate supports. However, these loading conditions are highly idealized and often do not align with real-world conditions, leaving designers wondering what effective length to use. For instance, what if a girder supports a uniformly distributed load but also a point load from a column? The code provides a catch-all effective length of

1.92L

, however this can often be excessively conservative. Looking at Canada’s neighbours to the south, the United States also makes use of essentially the same effective length formulas in the American Wood Council (AWC)‘s National Design Specification (NDS). However, the AWC also provides an alternative design solution through its Technical Report 14 (TR 14). The report provides a completely different approach, based on directly calculating the critical buckling moment based on the loading conditions, instead of using an effective length. This is essentially the same method as what’s used in calculating lateral-torsional buckling capacity in steel members - using the moments at quarter-points to calculate a

C_b

factor, analogous to

\omega_2

in the CSA S16 steel design standard. The advantage with this is that the quarter-moment method can be used with any loading conditions and doesn’t require as much interpretation from the designer - which is perfect for implementation in a platform like Calcs.com. Indeed, that’s what we offer you in Calcs.com - you can choose to use the TR 14 method in any wood beam or column design calculator by selecting it here:

Implementation of the TR14 Method with CSA O86

Although TR 14 provides a versatile solution to calculate lateral stability in wood beams, it can’t quite be directly implemented with the CSA O86 standard. That’s because beyond calculating the critical buckling moment, the report also provides a direct implementation in the NDS’s formulas for inelastic buckling, which do differ from the CSA ones, which are based on the effective length method. So to use TR 14, we need to be a bit creative and find a way to convert the TR 14 buckling moment to an effective length that’s useable in the CSA standard. This is what we’ll describe next: Since TR 14 is entirely based on finding the critical buckling moment, let’s start with that calculation. For rectangular members, the report provides a simple formula (Eq. 18) that can be used:

M_{cr} = \frac{1.3 C_b C_e E'_{y05} I_y}{\ell_u}

where

C_b

is the moment modification factor and

C_e

is an adjustment if the load is applied above the center of the beam - you can see their derivations and calculations in the TR 14 document.

E'_{y05}

corresponds to the “design” modulus of elasticity in the weak axis,

I_y

to the weak-axis moment of inertia, and

\ell_u

to the unbraced length. Looking at the NDS (Eq. 3.3-6) and TR14 (Eq. 15) next to each other, we can see that the final value for

C_L

(equivalent to

K_L

in Canada) is calculated the same way in both publications:

C_L = \frac{1+\alpha_b}{1.90} - \sqrt{ \left(\frac{1+\alpha_b}{1.90}\right)^2 - \frac{\alpha_b}{0.95} }

where

\alpha_b

is taken as

F_{bE}/F_b^{*}

in the NDS and as

M_{cr} / M^{*}

in TR14. The "

*

" superscript corresponds to the strength calculated considering every adjustment factor other than

C_L

, so we can say that

M^{*} = S F_b^{*}

where

S

is the section modulus. So we can say that:

\frac{M_{cr}}{M^{*}} = \frac{F_{bE}}{F_b^*}

which can be further simplified:

F_{bE} = M_{cr}/S

That makes sense -

F_{bE}

corresponds to the elastic buckling stress. This formula shows exactly how the “connection” is made between TR 14 and the NDS! Now to bring it to Canada… Let’s look at how

F_{bE}

is calculated in the NDS (Eq. 3.3-6):

F_{bE} = \frac{1.20 E_{min}'}{R_B^2}

Note that

E_{min}'

is the same as

E_{y05}'

. If we look at

R_B

next, we find this definition (Eq. 3.3-5):

R_B = \sqrt{\frac{\ell_e d}{b^2}}

Where

\ell_e

is the effective length,

d

and

b

are the depth and width, respectively. Look familiar? That’s because it’s exactly the same equation as we use for

C_B

in CSA O86 (Cl 7.5.6.4.3):

C_B = \sqrt{\frac{L_e d}{b^2}} = R_B

Our first connection between the standards! Let’s go back now and combine a few equations we looked at. Combining the two equations we found for

F_{bE}

, we can arrive to this:

C_B^2 = R_B^2 = \frac{1.2 E'_{min} S}{M_{cr}}

Adding in the formula for

M_{cr}

now and simplifying:

C_B^2 = \frac{1.2 S \ell_u}{1.3 C_b C_e I_y}

From basic formulae, we know that

I_y = db^3/12

and

S = bd^2/6

. So we further simplify:

C_B^2 = \frac{d}{b^2} \times 1.846 \frac{\ell_u}{C_b C_e}

Looking at the equation for

C_B

again, it’s clear that the right term here is our effective length! So we can thus conclude that our effective length can be taken as such:

L_e = 1.846 \ell_u / C_bC_e

This is the formula we use in Calcs.com to calculate the effective length for the lateral stability factor. With this method, we’re able to consider and provide accurate results for any loading condition automatically.

​Implementation of the TR14 Method with CSA O86

Implementation of the TR14 Method with CSA O86