Section Modulus (Sm) for Lumber
Nearly a decade ago I penned an article explaining lumber bending stress: https://www.hansenpolebuildings.com/2014/08/lumber-bending/
After 10 years, a reader nicely told me I did not explain where some terms in this article come from. One of these is Section Modulus (S or Sm in calculations). Let us take a visit to the National Design Specification (NDS) for Wood Construction developed by the American Wood Council’s (AC) Wood Design Standards Committee. This publication has been approved by ANSI (American National Standards Institute) as an American National Standard.
ANSI is a private nonprofit organization overseeing development of voluntary consensus standards for products, services, processes, systems and personal in the United States. It also coordinates U.S. standards with international standards so American products can be used worldwide.
ANSI accredits standards developed by representatives of other standards organizations, government agencies, consumer groups, companies and others. These standards ensure product characteristics and performance are consistent, people use same definitions and terns and products are tested in same ways. ANSI also accredits organizations carrying out product or personnel certification in accordance with requirements defined in international standards.
The NDS is specifically referenced in model building codes (International Building Code – IBC and International Residential Code – IRC). Certain mathematical expressions of properties or elements of sections are used in various member shapes and loading conditions. NEUTRAL AXIS, as defined, in the cross section of a beam, is the line on which there is neither tension nor compression stress. In layperson’s terms, mid-point of both depth and breadth of a member (or dead center of a cross section). SECTION MODULUS (Sm or S) is the moment of inertia divided by the distance from the neutral axis to the extreme fiber of the section. Following symbols and formulas apply to rectangular beam cross sections: X-X = neutral axis for edgewise bending (load applied to narrow face). Y-Y = neutral axis for flatwise bending (load applied to wide face).
b = breadth of rectangular bending member, in inches.
D = depth of rectangular bending member, in inches.
Sx = bd^2 / 6 = section modulus about x-x axis in inches^3 (inches cubed).
Sy = db^2 / 6 = section modulus about y-y axis in inches^3
In previous article mentioned above, Sm of a 4×6 is given as 17.65 in^3 and a 6×4 as 11.23 in^3.
For a 4×6 b = 3.5 inches and d = 5.5 inches. Therefore Sx = 3.5 x 5.5 x 5.5 / 6 = 17.6458 (4×6) and Sy = 5.5 x 3.5 x 3.5 / 6 = 11.2292. With a calculator, one can now easily determine Section Modulus of any given rectangular wood member.
