Beam Load Calculator: Bending Stress, Deflection, and Reactions

Beam design is at the heart of structural engineering — every floor joist, deck beam, door header, and bridge girder is a beam problem. The same equations that govern a wooden floor joist apply to a steel I-beam carrying a roof. Understanding the key outputs — reactions, bending moment, shear, and deflection — tells you whether a member is adequate for a given load.

The CalcHub Beam Load Calculator handles the most common beam configurations: simply supported with uniform load, simply supported with point load, and cantilever with end load.

Simply Supported Beam: Uniform Load

The most common case — a beam resting on two supports with a uniformly distributed load (like a floor joist carrying floor load).

For total load W (or uniform load w per unit length, W = w × L):

Reactions (both supports equal): R = W/2 Maximum bending moment (at midspan): M_max = WL/8 = wL²/8 Maximum deflection (midspan): δ_max = 5WL³/(384EI)

Simply Supported Beam: Central Point Load

Single load P at the center:

Reactions: R = P/2 Maximum moment (at center): M_max = PL/4 Maximum deflection: δ_max = PL³/(48EI)

Cantilever Beam: End Point Load

Beam fixed at one end, free at the other, with load P at the free end:

Reaction (at fixed end): R = P Fixed-end moment: M = P × L Maximum deflection (at free end): δ_max = PL³/(3EI)

Material Properties: E Values

The term E is Young's modulus (stiffness) and I is moment of inertia (cross-section geometry). Together, EI is the flexural rigidity.

Material	Young's Modulus (E)
Structural steel	200 GPa (29,000 ksi)
Aluminum alloy	69 GPa (10,000 ksi)
Douglas fir lumber	12.4 GPa (1,800 ksi)
Southern yellow pine	12.4 GPa
Glulam (GL24h)	11.5–13.7 GPa
LVL (Laminated veneer)	12.4–13.8 GPa
Concrete (30 MPa)	25 GPa

Worked Example: Deck Beam

A deck beam spans 12 feet between posts, carrying a tributary area load of 4 ft from each side. Design load = 50 psf (live + dead). Beam width: 12 ft span, 8 ft tributary width.

• Total load W = 50 psf × 8 ft × 12 ft = 4,800 lbs
• Max moment: M = WL/8 = 4,800 × 12/8 = 7,200 lb·ft = 86,400 lb·in
• Required section modulus: S = M / Fb (Fb = allowable bending stress)
• For Douglas Fir #2: Fb ≈ 875 psi
• Required S = 86,400 / 875 = 98.7 in³

A doubled 2×10 (actual: two pieces 1.5" × 9.25") has S = 2 × (1.5 × 9.25²/6) = 42.8 in³ — not enough. A tripled 2×10 gives 64.2 in³ — still under. Try doubled 2×12 (actual: 1.5" × 11.25"): S = 2 × (1.5 × 11.25²/6) = 63.3 in³ — marginal. Tripled 2×12: S = 94.9 in³ — close; check deflection and use an actual span table for final sizing.

Deflection Limits

Building codes typically limit deflection to:

• L/360 for floors (plaster ceiling below) — 12-ft span: max ½"


• L/240 for floors (no plaster) — 12-ft span: max ⅝"


• L/180 for roofs — 12-ft span: max ⅞"

Can I use this for wood beam sizing?

The formulas give you bending stress and deflection. Compare to the allowable values in the NDS (National Design Specification for Wood) Supplement for your lumber species and grade. The calculator outputs the required section modulus and deflection, which you then match to a standard lumber size from span tables.

What's the moment of inertia (I) of a rectangular beam?

I = b × h³ ÷ 12 where b is width and h is depth. A 2×10 (actual 1.5" × 9.25"): I = 1.5 × 9.25³ ÷ 12 = 98.9 in⁴. The calculator handles this conversion automatically when you enter nominal dimensions.

When do I need a structural engineer?

For any structural element supporting significant loads — floor beams in new construction, load-bearing wall headers spanning more than 4 feet, deck beams over 10 feet, or anything where failure could cause injury. Online calculators and span tables work for straightforward residential sizing; unusual loading, complex geometry, or high-risk situations need a licensed structural engineer.

Related Tools

• Lumber Calculator — board feet for structural members
• Concrete Calculator — concrete footings supporting the beam
• Rebar Calculator — reinforcement for concrete beams