SizingKitMechanical · Beams under load
Beam deflection calculator with the ratio beside it
This works the four standard beam cases and returns the maximum deflection, the end slope, the largest bending moment and the fiber stress it produces, from a span, a load, an elastic modulus and the section properties you supply. It is free and needs no signup, and alongside the answer it prints the L/180, L/240, L/360 and L/600 ratios with the second moment of area each one would take at this span and load. What it never prints is a verdict: reference arithmetic and the tables it came from are one thing, and a member somebody signs off is another.
- 100% free
- No signup
- Four standard cases
- Slope and stress too
- Four span ratios
The span and what sits on it
Four things decide a deflection and they multiply out very differently: the span to the third or fourth power, the load once, and the modulus and the second moment once each underneath.
Resting on two supports free to rotate, with the span measured center to center of the bearings.
Feet unless you write another unit. 12' 6", 3.8 m and 3800 mm all read.
Spread evenly: 100 lb per foot of span, 1.46 kN/m. Enter the total, not the rate.
Fills the box beside it. Overwrite the box and this stays put.
29,000 ksi = 200 GPa. Measured, not defined — see the note below.
Comes off a section table, or out of the section properties page. Deflection is inversely proportional to it.
Only the stress uses this one. Where the section is not symmetric about its own neutral axis, the smaller of its two section moduli is the one that governs.
DEFLECTION
0.460 in
11.7 mm at mid-span
SPAN OVER DEFLECTION
L/313
compare it with the four rows below
BENDING STRESS
9.27 ksi
63.9 MPa at the extreme fiber
| Maximum deflection | 0.460 in | 11.7 mm | δ = 5·w·L⁴ / (384·E·I) |
|---|---|---|---|
| End slope at each support | 0.59° | 0.12 in/ft · 10.2 mm/m | θ = w·L³ / (24·E·I) |
| Maximum moment at mid-span | 1,800 lbf·ft | 2,440 N·m | M = w·L² / 8 |
| Bending stress at the extreme fiber | 9.27 ksi | 63.9 MPa | σ = M / S |
THE FOUR SERVICEABILITY RATIOS
| Ratio | That much deflection is | I it would take, this span and load |
|---|---|---|
| L/180 | 0.800 in 20.3 mm | 2.01 in⁴ 83.7 cm⁴ |
| L/240 | 0.600 in 15.2 mm | 2.68 in⁴ 112 cm⁴ |
| L/360 | 0.400 in 10.2 mm | 4.02 in⁴ 167 cm⁴ |
| L/600 | 0.240 in 6.1 mm | 6.7 in⁴ 279 cm⁴ |
Read across from the row your code table and your ceiling finish put you in. The last column is arithmetic on the answer above, not a recommendation: deflection is inversely proportional to I, so the second moment that lands on a given ratio is the one entered scaled by the deflection it produced over the deflection that ratio allows.
This arrangement lands at L/313. Nothing on this page decides whether that is the right place to land: the ratio a member is read against depends on what it is, whether the figure is a live-load or a total-load deflection, and what it carries underneath. All four rows are printed for that reason, and the person who signs the drawing picks one.
These are serviceability limits, not strength limits, and which one applies depends on the member, the load case (live load or total load) and the finish it carries — read the code table rather than picking a row by feel. Passing a deflection limit says nothing about whether the section is strong enough; check the bending stress separately.
E = 29,000 ksi. E = 29,000 ksi, the value AISC specifies for structural steel design. Effectively identical for every carbon and alloy steel, hardened or not — heat treatment changes strength, not stiffness. This surprises people: a Grade 8 bolt stretches exactly as much as a Grade 2 bolt under the same load. The whole table is measured rather than defined, which is why the box is editable — Published Young's moduli: 29,000 ksi for structural steel as specified by the AISC Steel Construction Manual, 10,000 ksi for aluminum as specified by the Aluminum Design Manual, and standard published values for the remaining metals. The GPa column is converted at 1 MPa = 145.0377 psi.
The stress beside the deflection is what the outermost fiber carries, and it is a magnitude with no opinion attached. Turning it into a decision takes a yield or allowable figure out of the material specification and a factor out of whichever design standard governs the job, neither of which this site carries — and for a long unbraced beam the governing number is often not the fiber stress at all but lateral-torsional buckling, which is a different calculation entirely. Elastic, small-deflection theory for a straight prismatic beam of constant section, loaded in a plane of symmetry, with material still below yield. It ignores shear deflection, which is negligible for a slender beam and is not for a short deep one (roughly span less than about 10 times depth). It assumes the beam does not twist and is restrained against lateral-torsional buckling, which an unbraced deep beam is not. Real end conditions are never the ideal pin or the ideal fixity assumed here, and a partially fixed end lies somewhere between the two cases. This is for checking a section, not for designing a structure.
Small-deflection theory, so the slope is also the gradient of the deflected shape: a slope of 0.01 rad is 0.01 in of rise per inch of run. It is an elastic rotation of the beam itself and says nothing about how much the connection at that end rotates, which on a real bolted or welded joint is usually larger. A slope at the support is also where a stiff joint starts to matter: the elastic rotation here is what a bolted end plate, a bearing housing or a spring seat has to accommodate before anything binds.
The four cases and their formulas
P is a concentrated load, w a load per unit of length, L the span, E the elastic modulus and I the second moment of area. Keep one set of units throughout — pounds, inches, psi and in⁴ give inches — and the formulas work in either system.
| Case | Maximum deflection | Maximum moment | End slope |
|---|---|---|---|
| Simply supported, point load at center deflection at mid-span, moment at mid-span, under the load | δ = P·L³ / (48·E·I) | M = P·L / 4 | θ = P·L² / (16·E·I) |
| Simply supported, uniformly distributed load deflection at mid-span, moment at mid-span | δ = 5·w·L⁴ / (384·E·I) | M = w·L² / 8 | θ = w·L³ / (24·E·I) |
| Cantilever, point load at the free end deflection at the free end, moment at the fixed end | δ = P·L³ / (3·E·I) | M = P·L | θ = P·L² / (2·E·I) |
| Cantilever, uniformly distributed load deflection at the free end, moment at the fixed end | δ = w·L⁴ / (8·E·I) | M = w·L² / 2 | θ = w·L³ / (6·E·I) |
Deflection and moment: The standard elastic beam formulas for prismatic beams of constant section, as tabulated in the AISC Steel Construction Manual and in Roark's Formulas for Stress and Strain. P is a point load, w a load per unit length, L the span, E the elastic modulus and I the second moment of area.
End slope: The end-rotation formulas that accompany the deflection and moment formulas in the beam diagrams of the AISC Steel Construction Manual and in Roark's Formulas for Stress and Strain, for the same four prismatic elastic cases.
What each span ratio is for
| Ratio | On a 12 ft span | Where it comes up |
|---|---|---|
| L/180 | 0.800 in 20.3 mm | Roof members with no ceiling below, total load. The loosest of the common limits. |
| L/240 | 0.600 in 15.2 mm | Floor and roof members carrying a plastered or brittle ceiling, total load; the general-purpose limit for structural members in the International Building Code deflection table. |
| L/360 | 0.400 in 10.2 mm | Floor members under live load — the limit that keeps a floor from feeling bouncy and keeps a plaster ceiling from cracking. |
| L/600 | 0.240 in 6.1 mm | Machine bases, guide rails and anything supporting alignment-critical equipment. A convention of machine design rather than of building codes. |
The span/deflection ratios of the International Building Code Table 1604.3 for the building limits (L/180, L/240 and L/360, applied to the load case each is tabulated against), and standard machine-design practice for the tighter one.
How to work a span you already have a section for
The load case, then the section, then the ratio the result has to be read against.
Match the case to how the member is actually held
Two supports free to rotate is the simply supported pair; built in at one end with nothing under the other is the cantilever pair. Then choose between a concentrated load and one spread evenly, because the two behave differently even at the same weight: a total load spread across a simply supported span produces 62.5 percent of the deflection and exactly half the moment of the same weight parked at mid-span. Span is center to center of the bearings for a simply supported beam and from the face of the support for a cantilever.
Give it a section and a modulus
I and S are typed in rather than derived, because the same pair of numbers describes a rolled shape out of a steel table, a piece of tube out of a catalog and a fabricated section somebody welded up this morning. Take them from a published shape table where one exists, or work them out from the geometry on the section properties page. The modulus is prefilled from the material list and stays editable: it is a measured property rather than a defined one, and gray iron in particular is not a single number.
Read the ratio, then read the column beside it
The headline figure L over some number is the deflection expressed as a fraction of the span, and it is the same figure whether you work in inches or millimeters. Underneath it sit the four ratios in common use with two things against each: how much movement that ratio permits on this span, and the second moment of area it would take to get there under this load. That second column is what turns a result you do not like into a section to go looking for, and it is arithmetic rather than a recommendation — which ratio your member is read against comes from the code table and the person signing the drawing.
Technical specifications
| Load cases | Simply supported with a point load at mid-span or a uniformly distributed load, and a cantilever with a point load at the free end or a uniformly distributed load — the four prismatic elastic cases the standard beam diagrams tabulate |
|---|---|
| Figures returned | Maximum deflection with its location, span-over-deflection ratio, end slope in degrees and in rise per foot, maximum bending moment with its location, and the fiber stress M/S |
| Units | Span in feet-and-inches or metric, load in lbf, tons-force, N, kN or kgf, modulus in ksi or MPa, I in in⁴, cm⁴ or mm⁴. Deflection out in inches and millimeters at once, moment in lbf·ft and N·m, stress in ksi and MPa |
| Spread against concentrated | The same total weight distributed evenly over a simply supported span gives 5/8 of the mid-span deflection and exactly half the bending moment of that weight concentrated at the center |
| The cantilever penalty | A point load at the free end of a cantilever deflects 16 times as far as the same load at the center of a simply supported beam of the same span — 1/3 against 1/48 in the two formulas |
| Serviceability ratios printed | L/180, L/240, L/360 and L/600. On a 12 ft span those permit 0.800 in, 0.600 in, 0.400 in and 0.240 in of movement — 20.3, 15.2, 10.2 and 6.1 mm |
| Elastic moduli offered | Eleven materials from 29,000 ksi for structural steel down to 6,500 ksi for magnesium AZ31B, each an editable default carrying its own note. Gray cast iron is flagged as the least trustworthy: it ranges roughly 66 to 120 GPa with class and is non-linear from the origin |
| Worked example | A 12 ft simply supported span carrying 1,200 lb spread evenly, on a steel section of I = 3.495 in⁴ and S = 2.330 in³: 0.460 in of deflection, L/313, 0.59° of end rotation, 1,800 lbf·ft of moment and 9.27 ksi of fiber stress. Reaching L/360 on that span and load would take I = 4.02 in⁴ |
Frequently asked questions
My beam is built in at both ends, or runs continuously over three supports. Which case is that?
None of the four, and substituting the nearest one is a real error rather than a rounding. Building in both ends of a uniformly loaded beam cuts the mid-span deflection to a fifth of the simply supported value and moves the largest moment from mid-span to the supports; a beam continuous over an interior support behaves differently again, and its worst case is often an unbalanced load on one span only. What is worth knowing is which way the substitution errs: real connections are never the frictionless pin or the perfectly rigid built-in end that these formulas assume, so a bolted end that offers partial fixity sits somewhere between the two, and taking it as simply supported overstates the movement rather than understating it. For the fixed-end and continuous cases, the beam diagrams in the AISC Steel Construction Manual carry the full set.
Do I enter the load per foot or the whole thing?
The whole thing, and the panel divides it by the span itself and shows you the rate it worked out. Entering a rate where a total is asked for is one of the two ways a load gets in wrong by an order of magnitude — the other is entering a live load and forgetting the dead load underneath it. If what you have is a rate, multiply it by the span before it goes in, and check the figure the field reports back against what you started with.
Which ratio applies to my member — L/240 or L/360?
The code table decides, and it decides on three things at once: what the member is, what it carries beneath it, and which load case the ratio is written against. The tighter ratios are generally tabulated against live load alone and the looser ones against total load, so reading a total-load deflection against a live-load ratio is a mismatch that makes a member look worse than the table intends. Floors with a brittle ceiling under them, roofs with nothing under them and machine bases that have to hold an alignment are three different conversations. All four rows are printed here for exactly that reason: the page does not know your jurisdiction, your edition or your finish, and it does not choose.
If the deflection is small, is the stress taken care of?
No, and the two run independently enough that either can be the one that matters. Deflection is governed by I and stress by S, and those two respond differently to depth — a section made deeper gains second moment faster than it gains section modulus. A shallow beam over a short span can be nowhere near any deflection ratio while working its outer fiber hard, and a long slender one can be lightly stressed and still sag past anything a plasterer would accept. Both figures are printed here, and neither of them is a decision: turning a fiber stress into one takes a strength out of the material specification and a factor out of whichever design standard governs the job.
Why does my short deep beam move more than this predicts?
Shear deflection, which these formulas leave out entirely. The elastic curve they describe comes from bending alone, and that is an excellent approximation while the beam is slender — but as the span falls below roughly ten times the depth, the shear distortion of the web stops being negligible and starts adding measurably to the movement. It is the usual explanation when a stubby machine-frame member or a deep transfer beam over a short opening measures more deflection than the arithmetic says. The other direction of surprise is a long unbraced beam that moves sideways and twists rather than simply sagging, which is lateral-torsional buckling and is not in these formulas either.
Will a harder or higher-grade steel deflect less?
No. Stiffness and strength are separate properties, and heat treatment moves the second without touching the first: every carbon and alloy steel has essentially the same elastic modulus whatever condition it is in, so a hardened bar and an annealed one of identical section bend by the same amount under the same load. What changes stiffness is either the section — where depth is cubed, so it is by far the cheapest lever available — or a genuine change of material class. Swapping the material list from structural steel to a wrought aluminum alloy nearly triples every deflection on this page without altering a single dimension, which is the arithmetic behind aluminum framing being built deeper rather than merely thicker.
Does the beam's own weight count?
Yes, and on a long span in a light section it is not a small share of the answer. Self weight is a uniformly distributed load whose total is the weight per foot times the span, so work that out first — the metal weight page returns it from the section and the alloy — and either add it into the total you enter for a distributed case or, where the applied load is concentrated, run the two cases separately and add the two deflections together. Superposition is legitimate here because everything on this page is linear elastic: two loads on the same beam produce the sum of what each produces alone, provided the material has not yielded.
About elastic beam formulas, and the four things they leave out
Every figure this page returns comes from one differential equation and a set of end conditions, and the arithmetic is old, settled and short. What makes it worth doing carefully is the powers involved. Deflection carries the span cubed for a total load and to the fourth power for a load per unit of length, so lengthening a run by a fifth doubles the movement at the same pounds per foot, while doubling the section's second moment merely halves it. That asymmetry is why span is the first thing an experienced person questions on a drawing and section size the second, and it is also why a member that was comfortable at one length is nothing like comfortable at another. The moment and the stress that go with the deflection behave more mildly — moment carries the span squared at a fixed load per foot and only once for a load concentrated at a point — which is precisely why stiffness and strength can part company and have to be looked at separately. The section side of that arrives from the section properties page, which works out I and S from the geometry and asks about no load at all.
Four assumptions sit under the formulas and all four are worth knowing by name. The beam is prismatic and elastic, so a tapered member or one that has already yielded is outside them. Shear deflection is ignored, which is right for a slender member and increasingly wrong once the span drops below about ten times the depth. The beam is assumed to stay in its plane, which an unbraced deep section does not do — it can move sideways and twist long before the fiber stress gets interesting. And the end conditions are idealized: a real pin has friction and a real fixed end has flexibility, so an actual bolted or welded joint lands somewhere between the two cases, and the rotation reported here is the beam's own rather than the joint's. That end rotation is the number to have in hand when the connection has to accommodate it, and when it is being made with a bolted end plate the preload in those bolts is what decides whether the joint behaves as the assumption says it does.
The serviceability ratios deserve their own paragraph because they are misused more than anything else on this page. They are limits on movement, not on strength, and they exist because beams in buildings almost never break — they sag, and then the plaster cracks, the door binds or the tile grout opens. Which ratio applies is a function of the member, the finish it carries and whether the ratio is written against live load or total load, and that is a code question with a jurisdiction and an edition attached to it. Machine work uses tighter figures again for a different reason: alignment rather than appearance, which is why a line shaft between bearings gets held far stiffer than a floor joist even though it may carry a fraction of the load, and why that same shaft has to be looked at as a torque and speed problem as well as a bending one. Two loads worth remembering to include: the member's own weight, which the metal weight page returns per foot and which superposes onto anything else, and whatever is fixed to it permanently. This is reference arithmetic against published tables; a beam in a building or a machine is signed off by a qualified engineer who owns the load path, the connections and the bracing, none of which is on this page.
What becomes of the span and load you enter
Every number on this page is worked out by JavaScript running in the tab you are reading it in. Nothing you type — loads, lengths, nameplate ratings, the rates your utility charges you — is uploaded, logged or kept, which is also why the calculators carry on working in a mechanical room with no signal.
Spans, loads, section properties and the modulus you overwrite are held in this tab only and vanish when it closes, so a load case worked out here for a client, a bid or a failure investigation leaves nothing behind on any server.