# Shear Deflection in Wood Beam Calculators

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:

$EI \frac{d^4w}{dx^4} = q(x) - \frac{EI}{\kappa AG}$

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 $w$ | $\Delta_s = wL^2/8 \kappa AG$ |

Simply supported beam under point load $P$ | $\Delta_s = PL/4 \kappa AG$ |

Cantilever beam under uniform load $w$ | $\Delta_s = wL^2/2 \kappa AG$ |

Cantilever beam under point load $P$ | $\Delta_s = PL/ \kappa AG$ |

*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 $M = PL/4$, 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 $M_s / \kappa AG$ where $M_s$ is the peak service moment in the span.

So we're left with the following formula for shear deflection in these cases: $M_s / \kappa AG$. The $\kappa$ 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 $\kappa = 5/6$. This varies for other section shapes, but since we currently only consider rectangular sections, we'll stop here. Next, we have the area $A$. This is simply the gross cross-sectional area $b \times d$.

Lastly, we have our shear modulus $G$. 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 $G = E/16$, 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 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

### **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 $K = 8\kappa AG/$, where the final value of $\kappa AG$ 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.

The APA's technical bulletin on Performance Rated I-joists shows the shear stiffness value K to be used for I-joists.