Beam Deflection Calculator
Enter your beam geometry, material stiffness and load configuration to get the maximum deflection, slope, bending moment, shear force and the span-to-deflection ratio (L/delta) instantly. Covers simply-supported and cantilever beams under six load types each, in metric or imperial units.
How beam deflection is calculated
Beam deflection is the transverse displacement of a point on the neutral axis of a beam when loads are applied. For an elastic beam with uniform cross-section and material, the deflection at any point is governed by the Euler-Bernoulli beam equation: EI d²y/dx² = M(x), where E is the modulus of elasticity, I is the second moment of area (moment of inertia), and M(x) is the bending moment distribution. Integrating twice and applying boundary conditions (for example, zero deflection and zero slope at a pinned support, or zero deflection and zero slope at a fixed support) produces the deflection function y(x). The product EI, called flexural rigidity, is the key parameter: the higher EI, the stiffer the beam and the smaller the deflection for any given load.
Simply supported versus cantilever beams
A simply-supported beam is free to rotate at both ends (one pin, one roller) and cannot translate vertically at either support. A cantilever beam is fully fixed at one end and completely free at the other. For the same span, material and cross-section, a cantilever under a uniformly distributed load deflects eight times more than a simply-supported beam carrying the same total load, because the maximum moment and the integration path are both greater. This is why cantilever spans in buildings are kept much shorter than equivalent simply-supported spans, or are stiffened with back-span anchors.
What is the span-to-deflection ratio (L/delta) and why it matters
Structural codes rarely express deflection limits in absolute millimetres; instead they use the span-to-deflection ratio L/delta. A ratio of L/360 means the deflection is one three-hundred-and-sixtieth of the span. The International Building Code (IBC) and AISC limit floor beams supporting brittle finishes to L/360 under live load, and L/240 for roof members under live load. The Eurocode uses L/300 as a common characteristic limit. Exceeding these limits does not necessarily mean the beam will collapse - deflection limits are serviceability checks, not strength checks - but they prevent cracking of plaster, ponding on roofs, and visual sag that occupants would find uncomfortable. This calculator highlights pass or fail against the L/360 and L/240 benchmarks.
Inputs and typical values for engineering materials
The elastic modulus E controls how stiff the material is in tension and compression: structural steel is 200,000 MPa (29,000,000 psi), aluminium alloy is about 70,000 MPa, concrete 25,000-35,000 MPa, and softwood timber 8,000-12,000 MPa. The second moment of area I depends entirely on cross-section shape and depth: for a solid rectangle of width b and depth d, I = b·d³/12; for a hollow rectangular section (box beam), subtract the inner rectangle; for standard steel I-sections (wide flanges, UB, UC, W-shapes), use tabulated values from manufacturer data. Because I varies with the cube of depth, doubling the depth multiplies stiffness by 8, which is why deeper sections are far more efficient at controlling deflection than wider ones.
Common deflection limit guidelines (L/n serviceability limits)
| Member type | Typical limit | Code reference |
|---|---|---|
| Floor beams supporting brittle finishes (plaster, tile) | L/360 | IBC / AISC |
| Floor beams - general occupancy | L/360 | IBC / AISC |
| Roof beams (live load only) | L/240 | IBC / AISC |
| Roof beams (total load) | L/180 | IBC |
| Cantilever beams (tip deflection) | L/180 to L/300 | AISC / engineering judgment |
| Floor beams - EN Eurocode (characteristic) | L/300 | EN 1993-1-1 |
| Crane runway beams | L/600 to L/1000 | AISC Design Guide 7 |
| Gantry girders (hand-operated) | L/500 | BS 2573 |
Serviceability limits from AISC Design Guide 3, Eurocode EN 1990 Annex A1, and general engineering practice. Verify with the applicable code for your jurisdiction.
Frequently asked questions
What is the formula for beam deflection?
The formula depends on beam type and load configuration. For a simply-supported beam with a central point load P, span L, modulus E and moment of inertia I, the midspan deflection is delta = PL³/(48EI). For the same beam with a full-span UDL of intensity w, it is delta = 5wL⁴/(384EI). For a cantilever with a free-end point load, delta = PL³/(3EI). All formulas share the same structure: a load-geometry numerator divided by a stiffness denominator that contains EI. This calculator applies the correct formula for each of the twelve load cases provided.
What is a good span-to-deflection ratio (L/delta)?
Common structural codes specify L/360 or better for floor beams that support plaster ceilings or brittle floor finishes, L/240 for general floor beams, and L/180 to L/240 for roof members. L/600 or better is required for crane runway beams. A higher number means less deflection relative to span, so L/600 is stiffer than L/240. These are serviceability limits only: the beam must also satisfy separate strength checks for bending and shear.
How do I reduce beam deflection without changing the material?
The most effective approach is to increase the section depth, because the moment of inertia for a rectangular cross-section scales with depth cubed (I = bd³/12). Doubling the depth multiplies I by eight and cuts deflection by the same factor, while only doubling the section area. Other options include reducing the span (add an intermediate column or wall), cambering the beam upward to offset dead-load sag, adding a composite topping (concrete slab acting with the steel beam), or changing the support condition from simply-supported to fixed-end (which reduces the UDL deflection coefficient from 5/384 to 1/384).
What is the difference between deflection and bending stress?
Deflection is a serviceability check: it tells you whether the beam will sag too much under load, causing visual sag, cracking of finishes or ponding on roofs. Bending stress is a strength check: it tells you whether the material will yield or fracture under the bending moment. A beam can pass the deflection check and still fail in bending (common with strong but flexible materials), or pass the strength check and fail the deflection limit (common in long-span or lightly loaded beams). Both checks must be satisfied in structural design.
Can I use this calculator for composite beams or pre-stressed concrete?
You can use the formulas for any elastic beam as long as you enter the correct effective EI. For a composite steel-concrete beam, compute the equivalent moment of inertia of the transformed section using the modular ratio n = E_steel/E_concrete, then enter E_steel and I_transformed. For a pre-stressed concrete beam, the immediate deflection can be estimated using an effective modulus and effective I (accounting for cracking per ACI 318 or Eurocode 2). This calculator does not account for creep, shrinkage or long-term deflection multipliers, which can be significant for concrete and timber.
Why is the cantilever deflection so much larger than the simply-supported one?
A cantilever beam under a uniform load deflects at the free end by wL⁴/(8EI), while a simply-supported beam of the same span deflects only 5wL⁴/(384EI) at midspan. The ratio is 384/(8×5) = 9.6, so the cantilever deflects about 9.6 times more under the same load and span. This is because the bending moment diagram for a cantilever is much larger (peak at the fixed end = wL²/2 versus wL²/8 at midspan for a simply-supported beam) and because the boundary condition at the fixed end means curvature accumulates over the full length without a zero-slope mid-point to limit it.