Beam Deflection Calculator
Calculate Smarter. Work Faster.
Free beam deflection calculator for simply supported, cantilever, fixed and propped beams under a point load or UDL — enter span, load, E and I in metric or imperial units and get maximum deflection, support reactions, and live deflected shape, shear force and bending moment diagrams.
Beam, Support & Load Details
Choose a unit system and support condition, describe the load, and calculate. Deflection, reactions and all three diagrams update together.
All calculator values switch together and are converted automatically. Section-helper dimensions stay in mm for both metric systems and switch to inches for the imperial system.
This choice changes the deflection, shear and moment equations — not just the picture.
w and W are related by W = w × L. Mixing them up is the single most common UDL error — the calculator shows both once you calculate.
Leave at mid-span for the classic PL³/48EI case. Off-centre loads use the full piecewise equation, not an approximation.
Clear distance between supports. For a cantilever this is the projection from the fixed end to the free end.
Preset values are typical published figures for a first-pass check. Use the value from your own material certificate or design code where it matters.
Second moment of area about the bending axis. For rolled sections, prefer the published table value over a dimensional estimate.
📐 Don't know I? Calculate it from section dimensions +
I = b·h³ / 12 — bending about the horizontal axis, so h is the depth in the direction of the load.
A reference check only. The governing limit for a real structure comes from the applicable design code and load case, not from a single universal ratio.
Enter values and hit calculate
Enter values above to see a breakdown.
Deflected Shape, Shear Force & Bending Moment
Drawn from your own numbers, not stock images — every curve below is plotted from the equations for the support condition you selected.
The dashed line is the unloaded beam axis. The curve is the elastic curve, drawn with the vertical scale exaggerated so the shape stays readable — real deflection is normally a fraction of a percent of the span.
Shear steps vertically at each concentrated load and reaction, and slopes at a constant rate under a UDL. Where shear passes through zero, the bending moment is at a local maximum.
Sagging moment (tension on the bottom face) is plotted above the axis and hogging below it. Fixed and propped beams show both, and the point where the curve crosses the axis is the point of contraflexure.
How Beam Deflection Is Calculated
Beam deflection tells you how much a beam sags under load. It is a serviceability check, and it is a separate question from whether the beam is strong enough: a member can pass a bending stress check comfortably and still fail on stiffness, which shows up as bouncy floors, doors that stop closing, cracked plaster, or a machine base that will not hold alignment. On long, lightly loaded spans, deflection is very often the check that decides the section size, not stress.
Where the numbers come from. Every result on this page follows from one differential relationship — the curvature of a loaded beam is proportional to the bending moment in it, EI · d²y/dx² = M(x). Integrate that once and you have the slope; integrate again and you have the deflected shape. The support condition supplies the boundary conditions that pin those two integrations down, which is exactly why changing from a simply supported beam to a fixed-end beam changes the answer even though nothing about the load, the material or the section has changed.
The four support conditions covered here. A simply supported beam rests on a pin at one end and a roller at the other: both ends are free to rotate, so no moment develops at the supports. A cantilever is built in at one end and free at the other, so all the moment is carried at the root. A fixed–fixed beam is restrained against rotation at both ends, which pulls moment away from mid-span into the supports and makes the beam far stiffer. A propped cantilever is fixed at one end and simply supported at the other — statically indeterminate, and its reactions cannot be found from equilibrium alone, so the calculator uses the standard compatibility solution (5wL/8 at the fixed end and 3wL/8 at the prop for a full UDL).
Formulas the calculator applies. For maximum deflection: simply supported with a central point load, δ = PL³/48EI; simply supported with a UDL, δ = 5wL⁴/384EI; cantilever with an end point load, δ = PL³/3EI; cantilever with a UDL, δ = wL⁴/8EI; fixed at both ends with a central point load, δ = PL³/192EI; fixed at both ends with a UDL, δ = wL⁴/384EI; propped cantilever with a UDL, δ ≈ wL⁴/185EI at roughly 0.579L from the fixed end. For a point load placed away from mid-span on a simply supported beam, the calculator uses the full piecewise expression rather than a mid-span approximation, and reports the true location of the maximum, which is not under the load.
Worked example. A steel beam with E = 200,000 MPa and I = 8,000,000 mm⁴ spans 2000 mm and carries 5000 N at mid-span. δ = (5000 × 2000³) ÷ (48 × 200,000 × 8,000,000) = 4×10¹⁰ ÷ 7.68×10¹³ ≈ 0.521 mm, a span-to-deflection ratio of about L/3840 — very stiff against any of the usual serviceability limits. Fix both ends of that same beam and the deflection drops to 0.130 mm; turn it into a cantilever of the same length with the load at the tip and it rises to 8.33 mm, sixteen times the simply supported figure. The load, the steel and the section never changed. Only the boundary conditions did.
Why E and I dominate. Deflection is inversely proportional to both, so doubling either halves the sag. But they are not equally easy to change. Steel is about three times as stiff as aluminium, and for conventional structural steels the modulus of elasticity is essentially independent of grade, so increasing yield strength does not materially reduce elastic deflection. I, on the other hand, you control completely through the section. For a rectangle, I = bd³/12: depth is cubed, width is linear. Making a beam 25% deeper cuts deflection by roughly half; making it 25% wider cuts it by a fifth. That single asymmetry is the reason structural sections are deep and thin rather than square, and why an I-beam puts most of its material in the flanges, as far from the neutral axis as the design allows.
Span punishes you hardest of all. Deflection scales with L³ for a point load and L⁴ for a distributed one. Extending a UDL-loaded span by 20% — from 5 m to 6 m — multiplies deflection by 1.2⁴ ≈ 2.07, more than doubling it while the load per metre and the section stay exactly the same. This is why deflection often governs long spans, and why adding an intermediate support is usually more effective than upsizing the section: halving the span cuts UDL deflection by a factor of sixteen.
Load pattern matters, but less than you might expect. Take the same total load on the same simply supported beam. Concentrated at mid-span, deflection is PL³/48EI; spread evenly, it is 5WL³/384EI — exactly 5/8, or 62.5%, of the point-load figure. Real loading usually falls between the two: a row of closely spaced loads behaves like a UDL, one dominant load with a few small ones behaves like a point load. For the same total load on a simply supported span, the central point-load model gives the higher maximum deflection, so it is the conservative idealisation in that specific comparison — it is not a universal rule for every loading pattern.
Combining loads by superposition. Under the elastic, small-deflection assumptions used here, deflection is linear in load. That means a beam carrying a UDL plus two point loads can be handled by running each case separately and adding the results at the point you care about. Run the UDL, note the mid-span value; run each point load at its own position; sum them. The same logic works for reactions, shear and moment. Superposition stops being valid once the material yields anywhere or once deflections grow large enough relative to the span to change the geometry — neither of which should be happening in a serviceability check anyway.
What the calculator does not include. Self-weight is not added automatically: if the section's own weight matters — long spans, heavy sections, light applied loads — convert its mass per unit length into force per unit length and add it as a UDL. Shear deformation is ignored, which is standard Euler–Bernoulli practice and safe for slender beams, but understates deflection for very deep, short members where the span-to-depth ratio drops below roughly 10. Creep is not modelled either: timber, concrete and polymers keep deflecting for months under sustained load, and codes for those materials apply a long-term multiplier on top of the instantaneous value calculated here. Lateral-torsional buckling, web crippling, connection slip and support settlement are separate checks entirely.
Treat the output as one input to a design, not the design. The equations here are the same ones in every structural textbook, and they are exact for the idealised cases they describe. Whether your beam actually is one of those idealised cases — whether the "fixed" end really resists rotation, whether the supports really do not settle, whether the load really is where you assumed — is an engineering judgement no calculator can make for you. Where the calculated deflection sits close to the limit, or where the consequences of getting it wrong are serious, have the result checked independently against the design code that governs the structure.
Worked Example — same beam, four support conditions
Steel beam, E = 200,000 MPa, I = 8,000,000 mm⁴, span L = 2000 mm, single 5000 N load at mid-span (at the free end for the cantilever):
| Support condition | Formula | δmax | Span/δ |
|---|---|---|---|
| Cantilever, load at tip | PL³/3EI | 8.333 mm | L/240 |
| Simply supported | PL³/48EI | 0.521 mm | L/3840 |
| Propped cantilever | 0.00932·PL³/EI | 0.233 mm | L/8587 |
| Fixed at both ends | PL³/192EI | 0.130 mm | L/15360 |
Identical load, identical section, identical material — a 64:1 spread purely from how the ends are held. Getting the support condition right matters more than any other single input.
This calculator covers a single load case at a time on a prismatic (constant cross-section) beam, under the small-deflection elastic assumptions of Euler–Bernoulli theory. Multiple simultaneous loads need superposition, tapered or built-up members need their own analysis, and deflection limits for a real structure must be checked against the design code that applies to it — IS 800 for steel structures in India, AISC 360 in the United States, or Eurocode 3 across the EU and UK, among others. Limits, load factors and load combinations differ between them.
How to Read the Shear Force and Bending Moment Diagrams
The three plots the calculator draws are not decoration — they are the calculation, shown in the order it happens. Reactions come from equilibrium, shear comes from the reactions and the load, moment is the running total of shear, and deflection is the double integral of moment divided by EI. Each diagram is the slope-and-area partner of the one before it:
The shear diagram is the running vertical force balance. Start at the left end and walk along the beam, adding up every upward reaction and subtracting every downward load you have passed. A concentrated load or reaction makes the line jump vertically by exactly its own magnitude — that is why a simply supported beam with a central point load shows a step of P at mid-span, from +P/2 to −P/2. Under a UDL the line slopes at a constant rate equal to the load intensity, because the relationship is dV/dx = −w. On a beam with no load between two points, shear is flat.
The moment diagram is the area under the shear diagram. Because dM/dx = V, the moment at any section equals the accumulated area of the shear plot to its left. Two consequences follow, and they are worth memorising: bending moment peaks wherever shear crosses zero, and a constant shear produces a straight-sloping moment while a linearly sloping shear produces a parabola. That is why a point-loaded beam has a triangular BMD and a UDL-loaded beam has a parabolic one.
Sign convention used on this page. Sagging moment — tension on the bottom face, the beam curving like a smile — is plotted above the axis and treated as positive. Hogging is plotted below. Some textbooks and most reinforced-concrete detailing plot the mirror image, drawing the moment on the tension side of the member; the physics is identical, only the drawing convention differs, so always check which one a reference is using before comparing values. Shear is taken as positive where the left-hand portion of the beam tends to move upward relative to the right.
Points of contraflexure. Where the moment diagram crosses the axis, curvature reverses: the beam is sagging on one side of that point and hogging on the other, and the bending moment there is zero. Simply supported beams have none. Fixed and propped beams always do, and the location matters in practice — it tells you where the tension face swaps from bottom to top, which drives reinforcement layout in concrete and stiffener or splice positions in steel.
What to sanity-check on any diagram. Shear should return to zero at the far end of the beam once the last reaction is passed. Moment should be zero at a pin or roller support with no applied end moment, and non-zero at a fixed end. The peak of the deflected shape should line up with the point where the moment diagram is largest for simple cases, but not exactly for asymmetric loading — the elastic curve peaks where slope is zero, not where moment is greatest, which is why an off-centre point load produces its maximum deflection away from the load itself. If any of these fail, the input, not the beam, is usually the problem.
Beam Deflection Formula Reference — 10 Standard Cases
Every case below is a closed-form solution of Euler–Bernoulli theory for a prismatic beam under an elastic, small-deflection assumption. w is load per unit length, W = wL is total load on the span, P is a concentrated load, and δ is measured perpendicular to the original beam axis.
Simply supported — central point load
δmax = PL³ / 48EIMaximum deflection and maximum moment (PL/4) both sit at mid-span. The reference case every other coefficient is compared against.
Simply supported — point load at any position
δmax = P·b(L²−b²)3/2 / 9√3·L·EIb is the shorter distance from the load to a support. The peak deflection is not under the load — it shifts toward the centre of the longer segment.
Simply supported — uniformly distributed load
δmax = 5wL⁴ / 384EIEquivalent to 5WL³/384EI when W = wL is the total load. Maximum moment is wL²/8 at mid-span.
Cantilever — point load at the free end
δmax = PL³ / 3EISixteen times the simply supported deflection for the same load and span. All moment (PL) is carried at the root.
Cantilever — point load at distance a
δfree = Pa²(3L − a) / 6EIBeyond the load the beam stays straight and rotates rigidly, so the free end still moves even though it carries nothing.
Cantilever — uniformly distributed load
δmax = wL⁴ / 8EIRoot moment is wL²/2. Deflection is always maximum at the free tip, whatever the load pattern.
Fixed at both ends — central point load
δmax = PL³ / 192EIFour times stiffer than simply supported. Moment is PL/8 hogging at each end and PL/8 sagging at mid-span.
Fixed at both ends — uniformly distributed load
δmax = wL⁴ / 384EIFive times stiffer than simply supported. End moment wL²/12 hogging, mid-span wL²/24 sagging.
Propped cantilever — uniformly distributed load
δmax ≈ wL⁴ / 185EIStatically indeterminate. Reactions are 5wL/8 at the fixed end and 3wL/8 at the prop; peak deflection sits at about 0.579L from the fixed end.
Propped cantilever — central point load
δmax ≈ 0.00932·PL³ / EIFixed-end moment 3PL/16, prop reaction 5P/16. Peak deflection lies at roughly 0.553L from the fixed end, not under the load.
Coefficients here assume ideal supports — a "fixed" end that genuinely resists all rotation and supports that do not settle. Real connections usually sit somewhere between pinned and fixed, so a bolted end plate treated as fully fixed will deflect more in service than the fixed-end formula predicts. Where the distinction changes the answer materially, the conservative choice is to design for the more flexible assumption.
Support Condition and Load Pattern: What Actually Changes the Answer
Deflection coefficients expressed relative to the simply supported central point load case, which is set to 1.00. Same beam, same span, same total load in every row — only the boundary conditions and the load distribution change.
| Case | δmax | Relative | Where |
|---|---|---|---|
| Cantilever, load at tip | PL³/3EI | 16.00 | Free end |
| Cantilever, UDL (total W) | WL³/8EI | 6.00 | Free end |
| Simply supported, central point load | PL³/48EI | 1.00 | Mid-span |
| Simply supported, UDL (total W) | 5WL³/384EI | 0.625 | Mid-span |
| Propped cantilever, central point load | 0.00932·PL³/EI | 0.447 | ≈0.553L from fixed end |
| Fixed both ends, central point load | PL³/192EI | 0.250 | Mid-span |
| Propped cantilever, UDL (total W) | ≈WL³/185EI | 0.260 | ≈0.579L from fixed end |
| Fixed both ends, UDL (total W) | WL³/384EI | 0.125 | Mid-span |
The spread from top to bottom is 128:1. Within the modest design changes considered here, the support condition changes deflection by a much larger factor than the load pattern or the material grade does, which is why the real support condition is worth establishing before anything else rather than assuming it.
Within a single support condition, the load pattern is a much smaller lever. Point load versus UDL on a simply supported span is a factor of 1.6; on a fixed-fixed span it is a factor of 2. Useful, but nowhere near decisive. If a beam is deflecting too much, the productive moves are, in rough order of effectiveness: reduce the span (deflection falls with L³ or L⁴), increase the section depth (I rises with d³), improve the end restraint if the connection detail genuinely allows it, and only then consider a different material — remembering that within steel, E does not change with grade at all.
One caution about the stiffer cases. A fixed-fixed beam deflects a quarter as much as a simply supported one, but it pays for that with hogging moment at the supports and with sensitivity to support movement: if one end settles or rotates even slightly, the moments redistribute and the real behaviour drifts away from the ideal solution. Statically determinate beams — simply supported and cantilever — are indifferent to support settlement, which is one practical reason they remain common even where a stiffer arrangement is available on paper.
Deflection Limits and Typical E Values
Commonly quoted span/deflection limits
| Typical application | Limit often used | Why |
|---|---|---|
| Roof members, no ceiling below | L/180 – L/240 | Appearance and drainage, nothing brittle attached |
| Floor beams, general | L/240 – L/360 | Comfort and perceived bounce |
| Members supporting brittle finishes | L/360 – L/480 | Plaster, tile and glazing crack long before the beam is stressed |
| Crane girders and runways | L/600 – L/1000 | Wheel tracking, rail alignment and dynamic effects |
| Machine bases and alignment-critical frames | Tighter than L/1000 | Coupling alignment and vibration tolerances usually set an absolute limit in mm, not a ratio |
These are widely used practice values offered as a starting point, not code requirements. The governing limit depends on the design code you are working to, which load combination is being checked, and whether the limit applies to total deflection or to live-load deflection only. For rotating equipment, the machine builder's own alignment tolerance usually overrides any span ratio.
Typical modulus of elasticity values
| Material | E (GPa) | E (MPa) | E (10⁶ psi) |
|---|---|---|---|
| Structural steel (all grades) | 200 | 200,000 | 29.0 |
| Stainless steel 304/316 | 193 | 193,000 | 28.0 |
| Ductile (SG) iron | ≈170 | 170,000 | 24.7 |
| Copper | 117 | 117,000 | 17.0 |
| Titanium alloy | ≈110 | 110,000 | 16.0 |
| Grey cast iron | ≈100–120 | 100,000–120,000 | 14.5–17.4 |
| Brass | ≈100 | 100,000 | 14.5 |
| Aluminium 6061 | 69 | 69,000 | 10.0 |
| Magnesium alloy | ≈45 | 45,000 | 6.5 |
| Concrete, normal weight | ≈25–35 | 25,000–35,000 | 3.6–5.1 |
| Structural softwood timber | ≈8–14 | 8,000–14,000 | 1.2–2.0 |
Typical published figures for a first-pass calculation. Concrete stiffness varies with mix and strength class, timber with species, grade, moisture content and load duration, and composites with layup and fibre direction — for these three especially, take E from the relevant material standard or supplier data rather than a generic table. Note also that the modulus of elasticity of conventional structural steels is essentially independent of grade, so selecting a higher-strength grade does not materially reduce elastic deflection.
Ten Common Mistakes in Beam Deflection Calculations
1. Mixing unit systems inside one calculation. Span in metres, I in mm⁴ and E in MPa will produce an answer that is wrong by many orders of magnitude while looking perfectly plausible on screen. The formulas are dimensionally consistent, so any single coherent set works — pick one from the unit selector and stay inside it. If a result looks absurd, check units before checking anything else.
2. Confusing w with W. 5wL⁴/384EI and 5WL³/384EI are the same equation, but only if w is per unit length and W is the total. Substituting a total load into the per-unit-length form inflates the answer by a factor of L. This calculator asks explicitly which one you are entering and reports both in the breakdown so the discrepancy cannot hide.
3. Assuming the support condition instead of checking it. Calling a bolted end plate "fixed" because it looks rigid, or calling a welded moment connection "pinned" out of habit, changes deflection by a factor of four to sixteen. Look at what the connection can actually resist. When the detail is genuinely ambiguous, the safe choice for a deflection check is the more flexible assumption.
4. Using the wrong axis for I. Every non-circular section has two very different values. A 50 × 200 rectangle bending about its strong axis has I = 33.3 × 10⁶ mm⁴; lay it flat and it collapses to 2.08 × 10⁶ mm⁴ — sixteen times worse. Take the value about the axis perpendicular to the load direction, and for rolled sections read the correct column of the section table.
5. Ignoring self-weight. The calculator uses only the load you type in. On a long span carrying a light applied load, the beam's own weight can contribute a large share of total deflection. Convert mass per unit length to force per unit length and either add it to your UDL or run it as a second case and superpose.
6. Treating an off-centre load as if it were central. The maximum deflection for an off-centre point load is lower than the central case, occurs somewhere other than under the load, and needs the piecewise expression to locate. Using the central formula is conservative for magnitude, which makes it acceptable as a quick upper bound — but do not report the mid-span value as though it were the true maximum, or assume the peak sits under the load.
7. Adding load cases without superposition. Running the largest load and ignoring the rest understates total deflection. Elastic deflection is linear in load, so the correct approach is to calculate each load separately with its own formula and position, then add the results at the section you care about.
8. Assuming a higher-strength material will deflect less. Deflection depends on E, not on yield strength. For conventional structural steels E is essentially independent of grade, so moving from S275 to S460 raises capacity without materially reducing elastic deflection. The same trap catches aluminium alloy selection. If deflection governs, change the geometry, not the grade.
9. Passing the deflection check and stopping. Deflection and bending stress are independent limit states, and either can govern. A shallow wide section may deflect acceptably yet be overstressed; a deep light section may be comfortably strong and still feel bouncy. Both need checking — pair this tool with the Beam Stress Calculator rather than substituting one for the other.
10. Applying Euler–Bernoulli theory outside its range. The theory ignores shear deformation, which is fine for slender members but understates deflection once the span-to-depth ratio falls below roughly 10. It also assumes a prismatic section, elastic material, small deflections and no creep. Deep short beams, tapered members, plastic behaviour, and timber or concrete under sustained load all need something more than these closed-form results.
Frequently Asked Questions
What is the formula for maximum deflection under a central point load? +
For a simply supported beam it is δ = PL³ / (48EI), where P is the point load at mid-span, L is the span between supports, E is the modulus of elasticity and I is the second moment of area of the cross-section. Change the support condition and the coefficient changes with it: the same central load gives PL³/(192EI) on a beam fixed at both ends and PL³/(3EI) on a cantilever loaded at its free end.
What is the formula for deflection under a uniformly distributed load (UDL)? +
For a simply supported beam, δ = 5wL⁴ / (384EI) when w is the load per unit length, which is identical to δ = 5WL³ / (384EI) when W is the total load on the span, since W = wL. Confusing the two is one of the most common errors in beam work, so this calculator asks you which one you are entering and shows both in the breakdown.
Why does a UDL cause less deflection than the same total load applied at one point? +
Spreading the load moves most of it away from mid-span, where it does the most damage to a beam's stiffness. For the same total load on a simply supported span, a full UDL produces exactly 5/8 (62.5%) of the central point-load deflection, because the two coefficients are 5/384 and 1/48 and their ratio is 5/8.
What units should I use so the deflection comes out right? +
Any consistent set works, because the deflection formulas are dimensionally self-consistent. This calculator provides three ready-made sets: mm with N, MPa and mm⁴; m with kN, GPa and cm⁴; and inches with lbf, psi and in⁴. Pick one from the unit selector and every field, diagram and result switches together, which removes the mixed-unit errors that produce answers wrong by several orders of magnitude.
How do I find the moment of inertia (I) for my beam's cross-section? +
Use the built-in section helper on this page: enter the dimensions of a rectangle, solid round bar, rectangular tube, pipe or I-section and it computes I about the bending axis and loads it straight into the calculator. For rolled structural sections, the published table value from the manufacturer or the relevant section handbook is more accurate than any dimensional approximation, because it accounts for root radii and fillets.
What deflection limit should I check against? +
Span/240, span/360 and span/480 are the limits seen most often, with tighter values used where brittle finishes, machinery alignment or vibration matter. These are starting points, not code compliance — the governing limit for your structure comes from the design code that applies to it, the load combination being checked and whether the limit applies to total load or to live load only.
Does this calculator include the beam's own self-weight? +
No. Only the load you enter is used. To include self-weight, add it as a UDL of your own: multiply the section's mass per unit length by gravitational acceleration to get force per unit length, then either run it as a separate UDL case and add the two deflections by superposition, or add it to your applied UDL before calculating.
Can I use this for a cantilever or a fixed-end beam? +
Yes. The beam type selector covers simply supported, cantilever (fixed one end, free at the other), fixed at both ends, and propped cantilever (fixed one end, roller at the other). Each one uses its own deflection, shear and moment equations, and the calculator shows which formula it applied, because using a simply supported formula on a fixed beam overstates deflection by a factor of about five.
How do I read the shear force and bending moment diagrams? +
The shear diagram steps vertically at every concentrated load and slopes linearly under a UDL; the moment diagram is the running area under the shear diagram, so it peaks wherever shear crosses zero. On the plots here, sagging (tension on the bottom face) is drawn above the axis and hogging below it, which is why a fixed or propped beam shows moment on both sides of the axis while a simply supported beam stays entirely on one side.
How do I handle a beam carrying several loads at once? +
Use superposition. Elastic beam deflection is linear in load, so you can run each load separately in this calculator, keeping the same span, support condition, E and I, and add the resulting deflections at the point of interest. The same applies to reactions, shear and moment. Superposition is valid only while the material stays elastic and deflections stay small relative to the span.
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