Beam Deflection Calculator

How Is Beam Deflection Calculated?

When a beam carries a transverse load it bends, and the largest downward movement is its maximum deflection δ. For an elastic, prismatic beam the deflection follows closed-form solutions of the Euler–Bernoulli beam equation: it grows with the load and the span to the fourth (distributed) or third (point) power, and falls with the flexural stiffness EI, the product of the elastic modulus E and the moment of inertia I.

The support condition sets both the deflection coefficient and where the peak occurs. A simply supported beam under a uniform load deflects 5wL⁴/384EI at midspan; a cantilever of the same load deflects wL⁴/8EI at its free end — over twelve times more, because a cantilever is far less restrained. Fixing both ends is the stiffest case (wL⁴/384EI). A central point load gives PL³/48EI (simple), PL³/3EI (cantilever), or PL³/192EI (fixed). A combined load is the exact sum of the two, since the response is linear.

Deflection is a serviceability check, not a strength one: a beam can be perfectly safe against breaking yet deflect enough to crack finishes, bounce, or look alarming. Codes therefore cap deflection at a fraction of the span, such as L/360 for floors or L/240 for roofs. This calculator reports δ, the allowable span/ratio, and their ratio, and draws the deflected shape so you can see where and how much the beam sags.

Formula: EI = E x I (flexural stiffness) delta_uniform = c_u x w x L^4 / (E x I) delta_point = c_p x P x L^3 / (E x I) simple: c_u = 5/384, c_p = 1/48 cantilever: c_u = 1/8, c_p = 1/3 fixed: c_u = 1/384, c_p = 1/192 delta_allowable = L / n utilization = delta / delta_allowable

Example Calculation

A simply supported steel beam (E = 200 GPa), span L = 2000 mm, I = 1×10⁷ mm⁴, uniform load w = 10 N/mm. δ = 5·10·2000⁴/(384·200000·1×10⁷) = 1.04 mm at midspan. The L/360 allowable is 2000/360 = 5.56 mm, so utilization is 1.04/5.56 = 19% — well within the limit.

When to Use This Calculator

• Checking whether a beam, shaft, bracket, or machine rail deflects within an acceptable limit under load
• Sizing a section: iterate the moment of inertia I until the deflection meets L/360, L/240, or your own limit
• Comparing supports — see how much a fixed-fixed or shorter span reduces deflection versus a simple or cantilever beam
• Estimating sag of a cantilevered arm, shelf, or gantry before detailed analysis
• Quick serviceability first-pass on structural or mechanical beams before FEA or full code checks

Common Mistakes to Avoid

• Confusing strength with stiffness — passing the deflection limit does not mean the beam is strong enough; check bending stress separately
• Using consistent units — E in MPa, I in mm⁴, span in mm, w in N/mm and P in N give deflection in mm; mixing metres and millimetres is the most common error
• Using the wrong moment of inertia — use I about the bending (neutral) axis, and the least I only if the beam can bend that way
• Assuming the point is off-centre — this tool places the point load at the canonical max location; a truly off-centre load needs a different formula
• Forgetting the support really matters — a cantilever deflects many times more than a simply supported beam of the same span and load

How to Interpret Results

• Maximum deflection (δ) is the largest downward movement of the beam under the load, at the marked location
• Allowable deflection is the span divided by the chosen ratio (L/360, L/240, …) — the serviceability ceiling
• Utilization is δ divided by the allowable; below 100% the beam passes, above 100% it fails the limit
• The deflected-shape curve shows where and how the beam sags — the dip is deepest at the max-deflection location
• A failing result means increase the stiffness (deeper section, larger I), shorten the span, or reduce the load — deflection scales with L⁴ so span changes have the biggest effect

Related Standards & References

• AISC 360 / Steel Construction Manual — serviceability deflection limits for steel beams
• IBC / ASCE 7 — Table 1604.3 deflection limits (L/360, L/240, L/180) for floors and roofs
• Eurocode (EN 1990 / EN 1993) — serviceability limit states and deflection criteria
• Roark's Formulas for Stress and Strain — closed-form beam deflection tables for standard supports and loads

Frequently Asked Questions

What is a serviceability deflection limit like L/360?

It caps the maximum deflection at the span divided by a number. L/360 means a 3600 mm span may deflect at most 10 mm. Limits protect against cracked plaster, bouncy floors, ponding on roofs, and simply the alarming look of a visibly sagging beam. Common values are L/360 for floors under live load, L/240 for total load, and L/180 for roofs — check your governing code for the exact requirement.

Why does deflection depend on I and not just the beam depth?

Deflection is inversely proportional to the moment of inertia I, which grows with the cube of the section depth. Doubling the depth of a rectangular section raises I eightfold and cuts deflection to an eighth, while doubling the width only halves it. That is why beams are made deep rather than wide — depth buys stiffness far more efficiently than width or material.

Where does the point load act in this calculator?

The point load is placed at the canonical maximum-deflection location for each support: midspan for simply supported and fixed-fixed beams, and the free end for a cantilever. This is the worst case and the one textbooks tabulate. Off-centre point loads, whose peak deflection lies between the load and midspan, are deferred in this version rather than approximated.

Is a beam that passes the deflection check also strong enough?

Not necessarily — they are separate checks. Deflection is serviceability (stiffness); strength is whether the bending stress stays below the allowable. A shallow beam can be strong yet too flexible, or a short deep beam can be stiff yet overstressed. Always check bending stress and shear separately; this tool covers deflection only.