E (E and shear-free E)
(c) Jeff R. Filler, 2022
The deflections calculated in earlier examples (link and link) are based on the standard engineering formula for beams based on beam-curvature theory – where the beam curvature is a result of the tension and compression strains generated by bending stresses through the Young’s Modulus. But there is an additional component to beam deflections, significant in wood beams (joists, rafters, etc.) due to the shearing stresses in the beam. In real life, wood beams deflect a bit more than predicted using pure beam curvature resulting from the bending stresses. This `bit more’ is called shear deflection. But don’t panic! Though it’s not very `apparent’, the modulus of elasticity, E, that we pulled from the NDS Supplement, Table 4B, is actually an apparent E that takes into consideration the additional deflection due to shear. (See NDS Appendix F.) A vast majority of deflection calculations are, and have been, performed using the standard engineering formula based on bending stress alone, but instead of using the Young’s Modulus for E, this apparent E has been used. This apparent E has been the subject of particular focus, more recently, especially for engineered wood products (glulam, prefabricated wood I-joists, structural composite lumber), and has taken on the name `apparent modulus of elasticity’, `E apparent’, or `E app’. Since shear deflection adds to the deflection due to the bending stresses alone, this E app is a bit less than what would be the otherwise `true’ Modulus of Elasticity for bending. And, indeed, that’s what we’re calling the `true’ value … `true modulus of elasticity, `shear-free’ E, true E, or `E true’. If you want the true E for Southern Pine dimension lumber, for example, go to Appendix F (F3) in the NDS, and you’ll see that the ratio of shear-free E to reference E (the one referenced in the NDS Supplement) is 1.03. Thus, the shear-free E, for our example above, may be taken as 1,400,000 psi x 1.03 = 1,442,000 psi. (Maybe round it to 1,440,000 psi … or maybe even 1,400,000.) But, for wood bending member deflections, if you do use E true, or shear-free E, then shear deflection should be deliberately (and additionally) addressed. The additional deflection due to shear is generally addressed through an added term to the deflection formula. Shear deflection by formula is addressed in greater detail in another post.