The beam deflection calculator above computes how far a beam bends under load for the four classical cases that cover most everyday situations, and then prints that number next to the L/360 and L/240 serviceability limits so you can see the margin rather than a bare pass or fail. Deflection is the check that most often governs the size of a floor beam, and it is the one people are most likely to skip.
This is a preliminary sizing and teaching tool from Arb Digital. It is not a structural design, it is not stamped, and it does not replace a licensed engineer or architect. The classical formulas it uses are exact for the idealised cases they describe and approximate for anything real, and the values of E and I you feed it have to come from a design standard or a manufacturer's published data. Use it to understand the behaviour and to sanity-check a number. Do not build from it.
What This Beam Deflection Calculator Does
It solves the elastic deflection equations for four load cases: a simply supported beam under a uniform load, a simply supported beam under a central point load, a cantilever under a uniform load, and a cantilever with a point load at the free end. For each it returns the maximum deflection, the maximum slope of the beam at the point where it rotates most, the deflection expressed as an L/x ratio, and a side-by-side comparison with the two limits most commonly written into specifications.
It calculates displacement only. It does not return support reactions, shear or bending moment — those belong to the beam load calculator. It does not select a section, and it does not check bending stress; if you have a required moment and want the section property that carries it, the section modulus calculator is the right page. For timber floor members sized against prescriptive tables rather than first principles, see the floor joist calculator and the wood beam span calculator.
How to Use It
- Choose the load case that matches your real situation. This matters more than any other input. A cantilever under the same load as a simply supported beam of the same length deflects many times further.
- Enter the span. For a simply supported beam use the clear span plus bearing, measured centre to centre of the supports. For a cantilever use the projection from the face of support to the tip.
- Enter the load. Uniform load is per unit length and includes the beam's own weight. Point load is the total concentrated force. The tool uses whichever the selected case needs.
- Enter E and I. E is the modulus of elasticity of the material. I is the second moment of area of the section about the axis it bends around. Both come from published data, not from this page.
- Read the ratio, not just the number. A deflection of half an inch means nothing on its own. Half an inch over a 30 foot span is L/720; over an 8 foot span it is L/192.
The Formulas and How They Are Calculated
All four cases come from the same underlying relationship — the curvature of an elastic beam is proportional to the bending moment divided by EI — integrated twice with the boundary conditions of the case. The closed-form results are:
Simply supported, uniform load w: δmax = 5wL⁴ ÷ (384EI), at midspan. End slope = wL³ ÷ (24EI).
Simply supported, central point load P: δmax = PL³ ÷ (48EI), at midspan. End slope = PL² ÷ (16EI).
Cantilever, uniform load w: δmax = wL⁴ ÷ (8EI), at the free end. Slope at the tip = wL³ ÷ (6EI).
Cantilever, point load P at the tip: δmax = PL³ ÷ (3EI), at the free end. Slope at the tip = PL² ÷ (2EI).
A worked example in imperial units. A simply supported beam spanning 16 ft carries 200 lb per foot. Convert: L = 192 in, w = 200 ÷ 12 = 16.667 lb per inch. With E = 1,900,000 psi and I = 400 in⁴, the numerator is 5 × 16.667 × 192⁴ = 1.1325 × 1011, and the denominator is 384 × 1,900,000 × 400 = 2.9184 × 1011. Dividing gives 0.388 inches. The L/360 limit for this span is 192 ÷ 360 = 0.533 in and the L/240 limit is 0.800 in, so the beam satisfies both, and the actual ratio is 192 ÷ 0.388 = L/495.
Notice the exponents. Deflection scales with the fourth power of span under a uniform load and the third power under a point load. Extend that 16 ft span to 20 ft with the same section and the same load per foot and the deflection rises by a factor of (20/16)⁴ = 2.44, to 0.95 in — and the limit tightens at the same time, because it is a fraction of a longer span. This is why span is the input that dominates everything else.
Why L/360 and L/240 Both Exist
They answer different questions. The tighter ratio is the one usually applied to live load on members supporting a brittle finish — plaster, gypsum board ceilings, tiled floors — because those materials crack long before the structure is in any danger. The looser ratio is more commonly applied to total load, or to members supporting finishes that tolerate movement. Some specifications add a third limit for total load, and long-span members supporting glazing or masonry often carry tighter limits still, sometimes with an absolute cap in inches or millimetres on top of the ratio.
The critical point is that the ratio is only half of the requirement — the other half is which load it applies to. An L/360 live-load limit and an L/360 total-load limit are very different tests on the same beam, because dead load can easily be half the total. Applying a live-load ratio to a total-load deflection over-restricts the design; applying a total-load ratio to live load alone under-restricts it. The governing standards for your material spell this out: for wood construction, the American Wood Council's National Design Specification for Wood Construction is the reference document, and for structural steel the equivalent suite is published by the American Institute of Steel Construction. Deflection limits themselves are generally set by the building code adopted in your jurisdiction rather than by the material standard.
What These Formulas Assume, and When That Breaks
Every result on this page rests on assumptions that are easy to state and easy to violate:
The beam is prismatic. Constant cross section along its length. A tapered beam, a notched beam or one with a large hole drilled through it does not obey these equations, and a notch at a support is a shear problem as well as a stiffness problem.
The material is linearly elastic and isotropic. Stress is proportional to strain and the modulus is the same in every direction. Timber is emphatically not isotropic, and its published bending modulus is an average that already includes a shear component for standard test conditions.
Deflections are small. The derivation linearises the curvature, which is accurate while deflections are a small fraction of the span — true for any beam that satisfies a serviceability limit, and untrue for a very flexible member.
Shear deformation is ignored. These are pure bending formulas. For short, deep beams, and for materials with a low shear modulus relative to their bending modulus, shear deflection adds a real and sometimes substantial extra amount. MIT's Structural Mechanics course materials set out the beam theory these results come from and where it stops applying.
The beam is laterally braced. An unbraced compression flange can buckle sideways long before the deflection limit is reached, which is a strength failure the deflection equation knows nothing about.
Deflection Is Not the Only Check, and Sometimes Not the Governing One
A beam has to satisfy three separate things at minimum: bending strength, shear strength, and deflection. They are checked independently and any of them can govern. Long, lightly loaded members are usually governed by deflection — you could make the beam much smaller before it broke, but the floor would bounce. Short, heavily loaded members are usually governed by shear or bearing. Steel beams with slender webs bring web crippling into the picture at points of concentrated load.
There is also a check that no formula on this page covers: vibration. A floor that satisfies L/360 under design live load can still feel unacceptably bouncy in service, because human perception of floor motion responds to natural frequency and damping rather than to static deflection. Long-span joist floors are the classic case, and modern design guidance treats vibration as a separate criterion with its own limits. If a floor passes on paper and feels wrong underfoot, vibration is usually why. Once you have candidate sizes, the factor of safety calculator puts capacity against demand and the column buckling calculator handles the members carrying the beam down to the ground.
Superposition: Handling Loads That Are Not on the List
Real beams rarely carry exactly one of the four cases here. The useful property of linear elastic analysis is superposition: if the material stays elastic and the deflections stay small, the deflection caused by several loads acting together is the sum of the deflections each would cause alone. A beam carrying a uniform load plus a point load at midspan deflects by 5wL⁴/384EI plus PL³/48EI, and you can simply add the two results this tool gives you.
Two cautions. The maxima have to occur at the same location for a straight addition to give the true maximum — for the uniform plus central point load case they both peak at midspan, so it works. For an off-centre point load combined with a uniform load, the two maxima are at different places and adding the peaks overestimates slightly. And superposition fails entirely the moment anything is non-linear: a beam propped by a support that only acts in one direction, a connection that slips, or a material stressed past its elastic range.
Creep, and Why Timber Deflection Grows
Steel under service load deflects and then stops. Timber and concrete do not — they continue to deform slowly under sustained load, a behaviour called creep. For timber, design standards handle this by requiring that the long-term deflection under permanent load be amplified by a factor before it is added to the short-term deflection from variable load. Concrete does something similar with a long-term multiplier.
The practical consequence is that a timber beam calculated at exactly its limit on day one will exceed that limit years later, and the visible sag in an old floor is often creep rather than damage. Wet service conditions make it worse. This calculator returns instantaneous elastic deflection only; if your standard requires a creep factor, apply it to the permanent-load portion afterwards. Which factor and to which load case is a matter for the design standard governing your project, and it differs between codes.
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Browse Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Mixing units inside the formula — span in feet with a load in pounds per foot and I in inches to the fourth gives an answer wrong by orders of magnitude. Convert everything to a single length unit first.
- Applying a live-load ratio to a total-load deflection — the limit is meaningless without knowing which load combination it is checked against.
- Choosing the wrong load case — a cantilever under a uniform load deflects roughly forty-eight times as far as an identical simply supported beam under the same uniform load.
- Using a section modulus where the moment of inertia is required — S and I are different properties, and substituting one for the other silently produces nonsense.
- Treating a deflection pass as a design — bending, shear, bearing, lateral stability and vibration are separate checks, and any of them can govern.
Related Free Tools From Arb Digital
Pair this with the beam load calculator for reactions, shear and moment, the section modulus calculator for section properties, the column buckling calculator for the supports below, the floor joist calculator for repetitive framing members and the factor of safety calculator for capacity against demand. The wood beam span calculator covers prescriptive timber sizing. The full free online tools hub lists every calculator we publish.
Frequently Asked Questions
For a simply supported beam the maximum deflection at midspan is 5wL to the fourth, divided by 384EI, where w is the load per unit length, L is the span, E is the modulus of elasticity and I is the moment of inertia. For a cantilever under the same uniform load it is wL to the fourth divided by 8EI at the free end.
It is a serviceability limit expressing maximum allowable deflection as a fraction of the span. Over a 192 inch span, L/360 is 0.533 inches. The tighter ratio is typically applied to live load on members supporting brittle finishes, because plaster and gypsum crack at movements far smaller than anything that threatens the structure.
Because both the total load and the lever arm grow with the span under a uniform load, and the double integration of curvature adds two more powers. The result is that doubling the span of a uniformly loaded beam, keeping everything else the same, increases deflection sixteenfold.
No. These are pure bending formulas from classical beam theory. Shear deformation adds an extra amount that is negligible for slender beams but becomes significant for short, deep members and for materials with a low shear modulus relative to their bending modulus.
Yes, while the material stays elastic and deflections stay small, because superposition holds. Add the deflection from each case. The straight sum is exact only when both maxima occur at the same point along the beam, which is true for a uniform load plus a central point load but not for an off-centre load.
No. Deflection is a serviceability check. Bending strength, shear strength, bearing at the supports and lateral stability of the compression flange are separate checks, and any of them can govern the size. Floor vibration is a further criterion that static deflection does not capture at all.
E comes from the design standard or the manufacturer's published data for your specific material and grade. I comes from a section property table for the shape you are using, or from the geometry if the section is simple. This page publishes neither, because both are material and grade specific and both are revised on a standards cycle.
This tool produces preliminary calculations for teaching and checking only. It is not a structural design, it is not stamped, and it does not replace a licensed engineer or architect. Deflection limits, load combinations and material design values are governed by the building code and design standard adopted in your jurisdiction, and local amendments differ by state, province and country.
