Beam Deflection Calculator

Think of a beam as a very stiff spring. When you press on the middle of a ruler held at both ends, it bends. The amount it bends depends on how hard you press, how long the ruler is, what it is made of, and how thick it is. Beam deflection analysis quantifies exactly that relationship, and the result tells you whether the bend is acceptable or whether the beam needs to be stiffer, shorter, or differently shaped.

The governing relationship comes from the Euler-Bernoulli beam equation, which connects the curvature of a beam at any point to the bending moment at that point, normalized by the beam's resistance to bending. Integrating that relationship twice — once to get the slope of the beam and once more to get the vertical displacement — produces a deflection function along the beam's length. Evaluating that function at the point of maximum displacement gives the single number this tool reports.

The formula that emerges depends on how the beam is supported and how the load is applied. A simply supported beam with a center point load produces one formula; a cantilever with distributed load produces another. All four cases handled here come from the same integration process, applied with different boundary conditions. The boundary conditions are what distinguish a fixed end from a pinned support mathematically: a fixed end forces both deflection and slope to zero at that location, while a simply supported end allows rotation and forces only deflection to zero.

Use this tool when you need a quick analytical check on a standard beam configuration: a simply supported or cantilever beam carrying a uniform or center-point load, with a known cross-section and material. This covers a large share of everyday structural checks — floor joists, shelf beams, bridge girder preliminary sizing, overhead crane runway beams, and similar applications where the geometry matches the standard cases.

The tool is most useful in the early design phase when you are comparing section options or testing the sensitivity of the design to span length. It is also useful as a sanity check on finite element analysis results — if your FEA output disagrees with the analytical formula by more than a few percent for a standard case, the model has a setup error worth investigating.

Do not rely on this tool when the beam carries multiple loads at arbitrary positions, when the load is triangular or stepped rather than uniform, when the beam is continuous over more than two supports, or when the beam geometry is tapered. These cases require superposition of multiple formulas, influence line methods, or numerical analysis. Also avoid using these formulas when the beam is short and deep (where shear deformation is significant) or when the material behaves nonlinearly, such as concrete in flexure or steel beyond its yield point.

Entering total load instead of load per unit length (or vice versa). This calculator accepts total load in all cases and internally converts to load per unit length for distributed load formulas. If you manually apply the distributed load formula using w (force per meter) and enter that same value as total load, your result will be off by a factor of beam length. Always enter the full load applied to the beam, and let the tool handle the per-unit-length conversion.

Using the wrong units for elastic modulus. Steel has an elastic modulus of approximately 200 GPa, which in Pascals is 200,000,000,000 — twelve digits. Engineers accustomed to working in MPa sometimes enter the MPa value instead of the full Pascal value, producing a result that is one million times too large. Always work in consistent SI units: Pascals for E, meters for L, and m⁴ for I. The resulting deflection will be in meters.

Ignoring the small deflection assumption. The Euler-Bernoulli formulas used here assume deflection is small relative to span length — typically less than one percent of span. When calculated deflection approaches or exceeds this threshold, the beam geometry changes enough under load that the linear formula underestimates actual deflection. The tool warns when this condition is approached, but the underlying result is still the linear-theory answer. In those cases, a nonlinear analysis or physical test is needed.

For a simply supported beam carrying a total point load P at center over span L, with elastic modulus E and moment of inertia I, the maximum deflection at mid-span is: δ = PL³ / (48 × E × I). For the example inputs, the calculation gives a maximum deflection of 0.000141 m.

For a distributed load, the total load P is first converted to load per unit length w = P / L, then the formula becomes δ = 5 × w × L⁴ / (384 × E × I). The extra power of L (from L³ to L⁴) appears because a distributed load spreads bending across the whole span, not just the center, producing a parabolic moment diagram that integrates to one higher order than the triangular diagram from a point load.

Cantilever formulas follow the same logic but with different boundary conditions. A point load at the free end gives δ = PL³ / (3 × E × I), and a uniform distributed load gives δ = w × L⁴ / (8 × E × I). The coefficients in the denominator (3 and 8 versus 48 and 384) are dramatically smaller for cantilevers, which directly explains why cantilevers are so much more flexible. The product EI, called flexural rigidity, captures the beam's total resistance to bending in a single number that scales proportionally with both material stiffness and geometric efficiency.

The Euler-Bernoulli formulas assume that plane sections remain plane after bending — a condition that breaks down when the beam's depth is a significant fraction of its span. For span-to-depth ratios below roughly ten, shear deformation contributes meaningfully to total deflection and Timoshenko beam theory gives more accurate results. Deep beams, short transfer beams, and corbels all fall into this regime where the simplified formula will underestimate deflection. The moment of inertia also assumes the full cross-section contributes to stiffness, which is not true for cracked reinforced concrete sections where effective I depends on the applied moment relative to the cracking moment.