Beam Deflection Calculator

Maximum deflection for simply-supported, cantilever or fixed-fixed beams under point or uniform loads. Steel, aluminum, concrete or wood — rectangular, round or AISC W-shape sections.

Beam configuration

Support condition
Load type
Off-centre is only available for simply-supported beams.
ft
lb

Material

Material
Modulus of elasticity (E)

29,000 ksi · 200 GPa

Section

Section type
Strong-axis bending. Ix verified against AISC Steel Construction Manual 15th/16th Ed.
Moment of inertia (I)

30.8 in⁴ · 12,818,853 mm⁴

Allowable deflection (IBC Table 1604.3)

Pick the limit that matches your member and load case. The calculator does not separate live from dead loads — interpret your input load as the relevant load case.

Calculation results

Maximum deflection (δ)

— in

Allowable deflection

— in

Status

—

Span / deflection

L/ —

Deflection is computed using closed-form elastic formulas (Hibbeler / AISC Manual). The off-centre point load uses the corrected convention with b ≤ L/2.

Informational only. Beam sizing for structural members must be performed by a licensed engineer per ACI 318, AISC 360, NDS or equivalent. This calculator does not check shear, bearing, lateral-torsional buckling or vibration.

Understanding the formula

Maximum elastic deflection for a beam depends on three things: how it is supported, where the load sits along its length, and the product of modulus and moment of inertia (E · I). Once the support condition and load shape are picked the formula is closed-form.

Closed-form deflection formulas

  • Simply supported, point load at centre: δ = P L³ / (48 E I)
  • Simply supported, point load off-centre: δmax = P b (L² − b²)3/2 / (9 √3 · L · E · I), where b is the smaller of the two distances from the load to the supports (b ≤ L/2). The maximum occurs on the longer side of the beam.
  • Simply supported, uniform load: δ = 5 w L⁴ / (384 E I)
  • Cantilever, point load at free end: δ = P L³ / (3 E I)
  • Cantilever, uniform load: δ = w L⁴ / (8 E I)
  • Fixed–fixed, point load at centre: δ = P L³ / (192 E I)
  • Fixed–fixed, uniform load: δ = w L⁴ / (384 E I)

Modulus of elasticity (E)

  • Steel (A36, A992): 29,000 ksi (200 GPa) per AISC.
  • Aluminum 6061 (any temper): 10,000 ksi (69 GPa) per ASM.
  • Concrete (normal-weight): Ec = 57,000 · √f'c (psi) per ACI 318-19 §19.2.2. Examples: 3,000 psi → 3,122 ksi; 4,000 psi → 3,604 ksi; 5,000 psi → 4,031 ksi.
  • Wood (Douglas Fir-Larch): grade-dependent per NDS Supplement: Select Structural 1,900 ksi · No. 1 1,800 ksi · No. 2 1,600 ksi · Stud 1,400 ksi.

Moment of inertia (I)

  • Rectangular: I = b h³ / 12 (loaded edgewise; b = width, h = height).
  • Round solid: I = π d⁴ / 64.
  • AISC W-shape: tabulated value (Ix, strong axis) from the AISC Steel Construction Manual.

When to use this calculator

For quick checks of elastic deflection on common residential and light-commercial beam configurations — header beams over openings, deck beams, ridge beams, balcony cantilevers, exposed steel posts and lintels.

The calculator is informational only. It does not check:

  • Bending stress against the allowable stress of the material.
  • Shear capacity at supports or at concentrated loads.
  • Lateral-torsional buckling for laterally unbraced steel beams.
  • Bearing at supports, web crippling or local buckling.
  • Long-term creep deflection (concrete, wood) or service vibration.

For any structural member that supports occupied space, a licensed structural engineer must size the section per the governing code (ACI 318, AISC 360, NDS).

Common mistakes & tips

  • Pick the right limit. L/360 is the live-load floor limit; L/240 is the total-load limit; L/180 and L/120 apply to roof members without a brittle ceiling. Roofs with a plaster ceiling get the same limits as floors.
  • Watch the off-centre convention. For an off-centre point load, b is the SHORTER distance from the load to one support — never the longer distance. Maximum deflection occurs on the longer side of the beam, at x = √((L²−b²)/3) from the far support.
  • Use clear span, not stud-to-stud. Span L is the centre-to-centre distance between supports for simple beams, or the cantilever length from the support to the free end.
  • Wood beams are loaded edgewise. A 2×8 has h = 7.25 in (the long dimension), not 1.5 in. Plug the actual dressed dimensions, not nominal.
  • Concrete needs cracked-section analysis. The Ec from f'c gives the gross-section deflection. ACI requires effective moment of inertia (Ie) for service deflection of cracked sections — the result here is a lower bound.
  • Long-term creep matters. Wood and concrete continue to deflect under sustained load. ACI 318 §24.2.4 multiplies the dead-load deflection by 1.4–2.0 for long-term effects. Add a margin if the load is permanent.

Frequently asked questions

What deflection limit should I use for a residential floor?
IBC Table 1604.3: L/360 for live-load deflection (the most common stringent check), L/240 for total D+L. Many floor systems are governed by vibration rather than deflection — for long spans, also check vibration with TJ-Pro, AISC Design Guide 11 or equivalent.
Why is the formula for an off-centre point load different from the centred case?
When the point load is centred, maximum deflection occurs at midspan and the formula simplifies to PL³/(48EI). For an off-centre load, maximum deflection moves toward the longer side of the beam. The general expression is δmax = Pb(L²−b²)^(3/2)/(9√3·LEI), with b being the shorter of the two segments. Note: many references invert b and a — make sure b ≤ L/2.
Why does my steel deflection look too small?
Steel has an E of 29,000 ksi — about 16× wood and 8× concrete (for 4,000 psi). For a given load, span and section, steel deflects far less. If your number looks suspiciously small, double-check the load magnitude and make sure I is in in⁴ and E in ksi.
Can I use this calculator for two-way slabs?
No — these formulas are for one-dimensional bending of a prismatic beam. Two-way slabs need finite-element analysis or the ACI 318 direct-design method.
Why do my wood numbers depend on the grade?
NDS publishes a different reference modulus E for each grade of Douglas Fir-Larch: 1,900 ksi (Select Structural) down to 1,400 ksi (Stud). Lower grades have more knots and grain disruption, which reduces stiffness. Pick the grade that matches the lumber stamp on your beam.
How does fixed-fixed compare to simply supported?
Fixed-fixed restrains rotation at both ends, so the beam is much stiffer. For a centred point load, fixed-fixed deflects only ¼ as much as simply supported (PL³/192EI vs PL³/48EI). For UDL, the ratio is 1/5 (wL⁴/384 vs 5wL⁴/384). Both ends must be truly fixed (welded splice plates, embedded ends) — pinned ends give no benefit.
What if my beam has a different cross-section, like a steel HSS or a wood I-joist?
Use the rectangular or round option as a rough estimate, OR look up the published Ix for your section and treat it like a custom case. The calculator does not include HSS, channels, angles or engineered I-joists — those have their own tables.
Does this account for shear deformation?
No — these are pure bending (Euler-Bernoulli) formulas. For deep, short beams (L/d < 10) shear deflection becomes a meaningful fraction of the total. Timoshenko beam theory adds a shear term; for design, the bending number is conservative for slender beams.