Beam Span Calculator

A simply supported beam under a uniform load $w$ over a span $L$ has:

$M_{\max} = \frac{wL^2}{8}, \qquad V_{\max} = \frac{wL}{2}, \qquad \Delta_{\max} = \frac{5wL^4}{384EI}$

Strength is the first check. Bending stress is $f_b = M/S$, so the section modulus you need is

$S_{req} = \frac{M_{\max}}{F_b}$

with $F_b$ the allowable bending stress for the material. For a rectangle, $S = bd^2/6$ and $I = bd^3/12$; for a rolled steel shape, $S_x$ and $I_x$ come from the shape tables.

Stiffness is the second, and on residential spans it usually governs. Deflection is limited by a fraction of the span — commonly $L/360$ for live load and $L/240$ for total load on floors — and because $\Delta \propto L^4$, a 10% longer span deflects about 46% more.

Getting $w$ right: a floor load in psf becomes a line load by multiplying by the tributary width — half the joist span on each side. $55$ psf on a 12 ft tributary width is $660$ lb/ft.

The equations above are identical for all three; the material properties and the design method are not.

• Steel is designed to AISC, and bending capacity depends on how the compression flange is braced — an unbraced beam can fail by lateral-torsional buckling well below $M = F_b S$. Web crippling and bearing at supports need checking too.
• LVL values are product-specific: the published $F_b$ is adjusted for depth, and the binding numbers sit in the manufacturer's ICC-ES evaluation report, not a generic table.
• Sawn lumber is designed to the NDS, where the tabulated $F_b$ is multiplied by adjustment factors for load duration, wet service, size and repetitive members.

These figures are for learning the calculation, not for specifying a beam. A real member must be selected from the applicable code and span tables — AISC for steel, the NDS and IRC tables for wood, the manufacturer's data for LVL — with the engineer of record responsible for the final design.

• Mixing feet and inches. $M = wL^2/8$ with $w$ in lb/ft and $L$ in ft gives lb·ft; multiply by 12 before dividing by an $F_b$ in psi.
• Checking strength and stopping. Deflection governs most residential spans. A section that passes bending can still bounce far past $L/360$.
• Using the wrong support condition. $wL^2/8$ is the simply supported case. A cantilever of length $L$ carries $wL^2/2$ — four times the moment — and deflects $wL^4/8EI$.
• Confusing $S$ and $I$. Section modulus $S$ sizes for stress; moment of inertia $I$ sizes for deflection. They are not interchangeable.
• Forgetting tributary width, point loads and self-weight. A beam under a bearing wall or a post above carries a concentrated load that a uniform-load formula ignores entirely.
• Treating a rule of thumb as a specification. Depth-to-span ratios and "one size up" habits are not design; the governing code is.