Wood, because of its unique anisotropic properties, has a shear modulus that's   typically a much lower proportion of its elastic modulus when compared to   other materials such as steel. As such, shear deflections can have a   significant impact on the total deflection of a beam. To accommodate for this,   we give you the option to consider an approximate shear deflection in our   Canadian and American wood beam calculators. This article serves to briefly   describe the theory behind our approximation.

Shear Deflections in Typical Beams

  The general theory used to calculate shear deflections in beams is based on   the work of Timoshenko and Ehrenfest,   who established the following differential equation:

                                E             I                                                                d                   4                                  w                                               d                                    x                   4                                                          =             q             (             x             )             −                                             E                 I                                               κ                 A                 G                                                  EI \frac{d^4w}{dx^4} = q(x) - \frac{EI}{\kappa AG}                EIdx4d4w​=q(x)−κAGEI​

  While some will enjoy the walk down memory lane from university, these   differential equations are difficult to solve and are hard to apply in   practice. Luckily, in most cases, the shear component of deflection is   considered to be negligible and is thus ignored. When it does need to be   considered, however, there are relatively few resources on shear deformations   in beams - precisely because they are normally considered negligible. There   are however four formulas that appear regularly in literature*:

               Simply supported beam under uniform load                                                    w                                  w                            w                                                                               Δ                     s                                      =                   w                                        L                     2                                      /                   8                   κ                   A                   G                                  \Delta_s = wL^2/8 \kappa AG                            Δs​=wL2/8κAG                       Simply supported beam under point load                                                    P                                  P                            P                                                                               Δ                     s                                      =                   P                   L                   /                   4                   κ                   A                   G                                  \Delta_s = PL/4 \kappa AG                            Δs​=PL/4κAG                 Cantilever beam under uniform load                                                    w                                  w                            w                                                                               Δ                     s                                      =                   w                                        L                     2                                      /                   2                   κ                   A                   G                                  \Delta_s = wL^2/2 \kappa AG                            Δs​=wL2/2κAG                 Cantilever beam under point load                                                    P                                  P                            P                                                                               Δ                     s                                      =                   P                   L                   /                   κ                   A                   G                                  \Delta_s = PL/ \kappa AG                            Δs​=PL/κAG          

*See Roark's Formulas for Stress and Strain, or Design of Welded Structures by Omer W. Blodgett

  We'll discuss each parameter individually. But first, looking at the formulas   above, we can notice a pattern - all these formulas include the equation for   the peak moment on their respective spans. For instance, a simply supported   beam under a point load has its max moment as                                  M             =             P             L             /             4                      M = PL/4                M=PL/4, which we can clearly   see in the shear deflection formula. We can thus say that for these four   specific cases, the shear deflection is exactly                                                 M               s                          /             κ             A             G                      M_s / \kappa AG                Ms​/κAG where                                                 M               s                                   M_s                Ms​   is the peak service moment in the span.

  So we're left with the following formula for shear deflection in these cases:                                                   M               s                          /             κ             A             G                      M_s / \kappa AG                Ms​/κAG. The                                  κ                      \kappa                κ parameter refers to a sort of shape factor -   accounting for the distribution of shear stresses throughout the cross   sections. For rectangular sections, this factor is exactly                                  κ             =             5             /             6                      \kappa = 5/6                κ=5/6.   This varies for other section shapes, but since we currently only consider   rectangular sections, we'll stop here. Next, we have the area                                  A                      A                A. This is   simply the gross cross-sectional area                                  b             ×             d                      b \times d                b×d.

  Lastly, we have our shear modulus                                  G                      G                G. Whereas in most metals this can easily   be determined through formulas, we don't have that luxury in wood! Instead,   the shear modulus is typically determined through testing. An approximation   often seen is to simply take                                  G             =             E             /             16                      G = E/16                G=E/16, where E is the modulus of elasticity.   That's what we do in ClearCalcs. Note that this ratio actually changes   significantly between different species and products - on the order of +/-   50%.

  The most significant assumption we make in ClearCalcs is that we further   assume that this formula is a reasonable approximation for shear deflection in   every beam. This will generally provide extremely close results to the   theoretical results, however there may be some slight unconservatism, most   likely to be seen in multi-span and/or statically indeterminate beams. This   mostly arises because the shear stiffness of the beam also has an effect on   how loads are distributed to supports. When considering that shear deflection   is a small component of total deflection, and that the shear stiffness itself   is a significant approximation, we consider that our method to calculate shear   deflection yields results appropriate for engineering. If extremely high-   precision results are required, this may not be an adequate method.

       The AWC also provides an alternative method in its [Manual for Engineered     Wood Construction](https://awc.org/wp-     content/uploads/2022/01/AWC-2018-Manual-1810.pdf) to estimate shear deflection     in wood beams, where the shear deflection is directly related to the     equivalent deflection in a uniformly loaded beam. However, we've found that     this approximation tends to differ significantly from the values predicted by     theory - for instance, it overpredicts the shear deflection in a cantilever by     more than 200%.    

"Apparent" Elastic Modulus?

  While not as prevalent in Canada, US-based standards and manufacturer   specifications sometimes refer to an "apparent" modulus of elasticity,   particularly in glulam beams and structural composites such as LVL. The   purpose of this is to avoid performing the shear deflection calculations by   simply reducing the elastic modulus used in bending deflection calculations.   Usually, this'll be taken as a 5% reduction in elasticity, which after   rounding typically ends up as reducing the "True" elastic modulus by 100,000   psi. The APA has a detailed article on the idea behind the apparent elastic   modulus here: [https://www.apawood.org/publication-   search?q=tt-082&tid=1](https://www.apawood.org/publication-   search?q=tt-082&tid=1)

Wood I-Joists

  Wood I-joists are particularly affected by shear deflections and it should   generally always be considered. Since these don't have a rectangular section   nor homogeneous material, manufacturers typically directly provide a shear   stiffness value that's determined through testing, generally of the form                                  K             =             8             κ             A             G             /                                   K =             8\kappa AG/                           K=8κAG/, where the final value of                                  κ             A             G                      \kappa AG                κAG is determined directly from   tests (ie, the shape factor, area and shear modulus don't need to be   calculated separately). The factor of 8 is in the equation since most joists   are only used under uniform loads, so this simplifies the shear deflection   calculation. The principles however remain the same.

  Please note that shear deflection is applied to I-joist sections in the AU   Timber Beam Calculator, but is not performed for other AU timber sections.

  <img src="https://clearcalcs-prod2-cms.s3.amazonaws.com/[SUPPORT]_shear_deflection_in_wood_beam_calculators_0.png>The APA's technical bulletin on Performance Rated I-joists shows the   shear stiffness value K to be used for I-joists.