Wood Beam Span Calculator

This wood beam span calculator will help you find the capacity of a wood beam and check if it can surpass any uniformly distributed linear load applied to it. In this wood beam calculator, we'll perform wood beam deflection calculations, consider a wood beam's adjusted allowable design values, and compare them to the actual bending and shear stresses it has to support.

Keep on reading to explore:

• The importance of wood beam calculations;
• How to perform wood beam deflection calculations;
• How to calculate the actual bending stress due to loading and the adjusted allowable bending stress in a wood beam; and
• How to calculate the actual shear stress from the applied linear loading and the adjusted allowable shear stress on a wood beam.

When choosing what size of lumber to use as a beam, we must consider various factors so we won't have a wood beam that can pose a danger to us. We want to choose the size of lumber that can support the beam load we need to apply to it and can handle the effects of humidity and moisture, extreme temperature, bending, and shearing (to name a few).

Other than the size of the beam, we also have a wide range of selection of wood species and commercial grade. Each wood species and grade has its own set of stiffness or design values, including bending stress, shear stress, tension and compression stresses, and modulus of elasticity. We then adjust these design values to consider the long-term environmental and thermal effects mentioned above and see if the wood beam can still support the loading we anticipate it to carry. Performing these calculations will help us choose the beam size and species that can support our anticipated loading and handle some unforeseen additional loading and natural weakening of lumber over time.

This wood beam span calculator will focus on the first three parameters we typically test when designing a wood beam. These parameters are the beam's allowable deflection, bending stress, and shear stress.

We'll get all the necessary data from the National Design Specifications (NDS®) Supplement: Design Values for Wood Construction 2018 Edition and follow the adjustment guidelines we need from the National Design Specification (NDS®) for Wood Construction 2018 Edition prepared by the American Wood Council (AWC).

Then, we'll calculate the resulting deflection, bending stress, and shear stress due to the loading on our beam and compare them to the adjusted design values of our chosen wood beam.

To find the deflection of a wood beam with the typical rectangular cross-section, we use this formula:

where:

• $\delta$ – Deflection (in inches) at the midspan of the wood beam due to the loading applied;
• $w$ – Uniformly distributed linear load applied to the beam in pound-force per inch $\small(\text{lbf}/\text{in})$;
• $L$ – Beam span or unbraced length of the beam in inches;
• $E$ – Modulus of elasticity of the wood species used in pounds per square inch $(\small\text{psi})$; and
• $I$ – Area moment of inertia of the beam's cross-section in inches to the fourth power $(\small\text{in}^4)$.

We can get the modulus of elasticity (E) of any common American wood species from the NDS Supplement Table 4A Reference Design Values for Visually Graded Dimension Lumber. Here are some of the wood species' moduli of elasticity from that table:

On the other hand, we calculate the area moment of inertia (I) of our beam's cross-section using this formula:

where:

• $I$ – Area moment of inertia in inches to the fourth power ($\small\text{in}^4$);
• $b$ – Actual base width or thickness of the lumber in inches; and
• $d$ – Actual height of the lumber in inches.

Note that we use the actual dimensions of the lumber for the calculation of $I$ instead of the lumber's nominal dimensions. We usually reduce the nominal size by half an inch to find the actual dimension of the lumber. That means a 2" × 10" piece of lumber has an actual thickness and height of 1.5 inches and 9.5 inches, respectively.

For example, we want to find the deflection of a 2" × 10" select structural Douglas Fir Larch beam that spans 8 feet (or 96 inches) in length, and we anticipate it to carry a uniform linear load of 240 pounds per foot (or 20 pounds per inches).

From the table, we can see that a select structural grade Douglas Fir Larch has a modulus of elasticity of $\small 1.9 \times 10^6\ \text{psi}$ or $\small 1,\hspace{-0.04cm}900,\hspace{-0.04cm}000\ \text{lb}/\text{in}^2$.

On the other hand, by calculating its area moment of inertia, we get:

Since we have all the values we need to calculate the deflection due to the applied loading, we can already substitute them into our deflection formula, as shown below:

Our next step is to check if this deflection is less than the allowable deflection for our given beam span. For that, we use the deflection criteria provided by the 2012 International Building Code, stating that beams under a combination of dead load (permanent load) and live load (loads that can vary in time) should at most have an allowable deflection equal to the span of the beam divided by $\small 240$.

Solving for the maximum allowable deflection, $\small\delta_\text{max}$, of the beam in our example, we have:

Now that we're done with our wood beam deflection calculation, we compare the two values we got. Since our beam's deflection due to loading is less than the maximum allowable, we can say that our beam passed the deflection check, and we can now proceed to the bending and shear stress calculations. ✔

If the beam's deflection due to loading is greater than the maximum allowable deflection, the beam will fail. For such cases, we should choose either a different wood species, commercial grade, or a larger beam size, then perform the wood beam deflection calculation again.

Now that we've checked the deflection of our beam, we can now check for its bending stress. For that, we first need to calculate the required or actual moment, $M$, our beam experiences due to the loading applied (in this case, a uniformly distributed linear loading), as shown below:

Next, we can determine the actual bending stress due to this bending moment with this equation:

where:

• $f_\text{b}$ – Required or actual bending stress in pounds per square inch ($\small\text{psi}$ or $\small\text{lb}/\text{in}^2$);
• $M$ – Bending moment due to loading in pound-force inches ($\small\text{lbf}\cdot\text{in}$); and
• $S$ – Section modulus of the beam in inches cubed ($\small \text{in}^3$).

We can calculate the section modulus of our beam using this equation:

Solving for the actual bending stress, we have:

Like in the deflection check, we must compare the actual bending stress with the beam's (adjusted) allowable bending stress design value. Looking at NDS Supplement Table 4A, we can see that the bending stress design value $(F_\text{b})$ of select structural grade Douglas Fir Larch is $\small 1,\hspace{-0.04cm}500\ \text{psi}$. However, we need to adjust that value to consider the different factors that can affect the bending stress capacity of any wood beams. To determine the adjusted allowable bending stress in wood beams, which we denote as $F_\text{b}'$, we must multiply ($F_\text{b}$) by the following factors:

• $C_\text{D}$ – Duration factor;
• $C_\text{M}$ – Wet service factor;
• $C_\text{t}$ – Temperature factor;
• $C_\text{L}$ – Beam stability factor;
• $C_\text{F}$ – Size factor;
• $C_\text{fu}$ – Flat use factor;
• $C_\text{i}$ – Incising factor; and
• $C_\text{r}$ – Repetitive member factor.

We won't anymore discuss each one of these factors. Some of these factors have values depending on which adjusted design values we want (e.g., $\small C_\text{i}=0.80$ for $\small F_\text{b}$ and $\small F_\text{v}$, but $\small C_\text{i}=0.95$ for $\small E_\text{min}$). Same with for $C_\text{M}$:

Some of them require separate calculations before we can use them. In contrast, some have values in a tabulated format where we choose the multiplier constant that fits our beam. For example for $C_\text{D}$:

Please refer to the NDS for Wood Construction for the other adjustment factors.

For our example, let's say upon following the NDS, our $C_\text{total}$ or the product of all the factors we need to multiply is $\small 0.711$. We can now calculate our beam's adjusted allowable bending stress design value to be:

Since $\small F_\text{b}' = 1,\hspace{-0.04cm}066.4\ \text{psi}$ is greater than $\small f_\text{b} = 1,\hspace{-0.04cm}021.2\ \text{psi}$, our beam also passed the bending stress check. ✔

For the last parameter we want to check in this calculator, we'll find the actual shear stress due to the beam loading and compare that to our beam's adjusted shear stress design value. First, we must calculate the required or actual shear, which we denote as $V$, our beam needs to overcome using this equation:

Then we determine its corresponding actual shear stress, $f_\text{v}$, on our beam by dividing the actual shear by the cross-sectional area, $A$, of our beam, as shown below:

We can also combine these two equations, together with $A = b\times d$ to directly calculate the shear stress using the shear stress formula, as shown here:

Substituting the values we have in our example, we then have:

Like with the bending stress, we also need to compare our actual shear stress with our beam's adjusted shear stress design value, which we denote as $F_\text{v}'$. Using the same NDS Supplement Table 4A, we can find the shear stress design value or $F_\text{v}$ of select structural grade Douglas Fir Larch to be equal to $\small 180\ \text{psi}$. To calculate $F_\text{v}'$, we have to use the following adjustment factors:

• $C_\text{D}$ – Duration factor;
• $C_\text{M}$ – Wet service factor;
• $C_\text{t}$ – Temperature factor; and
• $C_\text{i}$ – Incising factor.

Let's say we're using $\small C_\text{D = 1.0}$ (for considering 10 years duration of the structure), $C_\text{M} = 0.97$ (as shown in the table of C_M values for F_v in the previous section of this text), $C_\text{t} = 1.0$, and $C_\text{i} = 0.8$. We now calculate the adjusted shear stress design value as follows:

Fortunately, our beam's actual shear stress is less than the adjusted shear stress design value. ✔

If, again, our beam did not pass this test, we have to repeat the calculation using a larger size beam or a stiffer wood species and grade. Imagine how tiresome that would be to do it manually. This is where our wood beam span calculator comes in super handy. Let's learn how to use this wood beam calculator in the next section of this text.

To use this tool for your wood beam size calculations, all you have to do is follow these steps:

1. Choose the wood species you plan to use or check.
2. Select lumber grade depending on what you want to use or what is available to you.
3. Pick the nominal beam size you want to test.
4. Enter the span of your beam.
5. Type in the uniformly distributed load your beam needs to carry.
6. Choose your desired deflection limit criteria.

At this point, our wood beam span calculator will already display the deflection due to loading and the maximum allowable deflection of your beam. You'll also see a note if your wood beam passed the deflection test. You can also expect the results for the allowable and required bending and shear stress values for comparison and the assessment of whether the selected beam size passed their respective tests.

If you want to use our tool to determine the recommended span of a beam, you can do that by skipping step 4 of the steps above. For our wood beam span calculator to work, you must enter your beam's required bending or shear stress values. Our tool will then display the recommended beam span and, at the same time, perform the deflection and stress checks.

That's it! 🙂 You may also check the reference design values of your selected wood species and lumber grade and the adjustment factors used in the calculations of the adjusted design values. Simply tick on the checkbox that says "Display your wood beam's reference design values and the adjustment factors used" below the Load and deflection details section of our wood beam span calculator.