Beam Stress and Deflection Calculator

Eight load cases with the section property computed rather than tabulated, the deflection checked against a span limit whose source is named — and the off-centre point load done properly, with the peak where it really is instead of under the load.

Last updated 24 Sep 2026

Beam stress and deflection

Load case and section → stress, deflection, limit

Load caseEight cases, and the third one is the default on purpose. For a point load off centre the maximum deflection is not under the load, and most calculators of this kind print the deflection at the load as though it were the maximum. This page prints both, and says where the real peak is.

Cross-sectionThe second moment of area is computed from the dimensions rather than looked up, so any size works. Only the dimensions the chosen section uses are read: a solid round takes the first field as its diameter, a tube takes the first two as outside and inside, and only the I-section reads the flange and web thicknesses.

Span L (m)For a cantilever this is the projecting length from the built-in end to the free end; for a propped cantilever it is the distance from the built-in end to the prop.

Point load P (N)Ignored by the three uniform-load cases. Newtons: a kilogram under gravity is 9.81 N, so a tonne is 9810 N.

Uniform load w (N/m)Load per metre of span, ignored by the point-load cases. A total load W spread over the span is w = W / L.

Point load position, from the left support (m)Used only by the off-centre case. The interesting thing is what happens to the peak as you move it: at midspan the peak is under the load, and everywhere else it is not.

Width, or diameter, or outside diameter (mm)Rectangle: the width. Solid round: the diameter. Tube: the outside diameter. I-section: the flange width.

Depth, or inside diameter (mm)Rectangle and I-section: the overall depth, which is the dimension that matters most because I goes with its cube. Tube: the inside diameter. Ignored by a solid round.

Flange thickness (mm)I-section only.

Web thickness (mm)I-section only.

Material, for the elastic modulusOnly the stiffness depends on this; the bending stress does not depend on the material at all. Where the published figure is a range the lower bound is used here, which gives the larger deflection and the conservative answer.

Deflection limit to check againstEach one is tied to the document it comes from. Note what those documents are: four are building-code limits written about floor joists and roof purlins, and two are Eurocode 2 concrete figures. There is no machine-design code behind any of them, which is worth knowing before quoting span over 360 at a design review.

Not a circuit: a simple span with your point load on it, and the deflected shape drawn from the actual cubic rather than sketched as a symmetrical sag. The shape is normalised so its peak always touches the same depth, because the thing to look at is its SHAPE and not its size. Move the load and watch where the lowest point of the curve goes: it does not stay under the load. The two markers on the scale underneath are the whole subject of this page — the upper tick is where the load is, the lower tick is where the beam is actually lowest, and they line up at exactly one load position, midspan. Everywhere else the peak sits between the load and the centre, and the further out the load goes the further apart the two ticks are. That is why a deflection quoted “at the load” is not the maximum, and why a check made at midspan cannot tell you whether the formula has been applied the right way round. The bar at the bottom is the deflection against whichever span fraction you selected, with the limit at the right-hand end: past it, the bar runs out of track.

25.20MPaExample

A 50 × 100 mm steel rectangle on a 2 m simple span, with a 5 kN point load 0.6 m from the left support

sigma = Mc/I for the stress, a coefficient per load case for the deflection, and one case where the peak is not where you think

σ_max = M_max c / I · δ_max = k · (load) · Lⁿ / EI · offset load: x = √((L² − b²)/3), δ = P b (L² − b²)^3/2 / (9√3 L EI)

M_max: the largest bending moment anywhere in the span, which is not always at midspan and not always under the load. For a cantilever it is at the built-in end; for a beam with both ends built in under a uniform load it is at the SUPPORTS and is twice the midspan value, which is why a fixed-end beam cracks at its ends first
c: the distance from the neutral axis to the extreme fibre. For a symmetric section it is half the depth. This page only handles symmetric sections, so the top and bottom fibre stresses are equal in magnitude; a tee or a channel bent about its weak axis has two different values of c and two different stresses, and is not covered here
I: the second moment of area about the bending axis, computed from the dimensions rather than tabulated. It weights material by the SQUARE of its distance from the neutral axis, which is why depth matters so much more than width — doubling the width of a rectangle doubles I, doubling its depth multiplies I by eight — and why a tube keeps most of a solid bar’s stiffness for well under half its weight
Z = I / c: the section modulus, which is what you actually need if stress is the question: σ = M/Z. Quoted for standard sections in every steel table, and the quantity to compare when choosing between shapes of the same weight
b: in the offset-load formulas ONLY, the distance from the load to the NEARER support — and x is then measured from the FARTHER one. Both conventions have to be right together. Swap them and the answer is a few per cent low, which is the unsafe direction, and it will survive any check made at midspan because at midspan the two distances are equal and every variant agrees
P a²b² / 3EIL: the deflection AT the load for an offset point load, which is the expression most published tables print for this case. It is correct, and it is NOT the maximum: the maximum is larger and sits between the load and midspan. This page prints both, with the difference as a percentage
184.634: the exact deflection coefficient for a propped cantilever under a uniform load, which the literature universally rounds to 185. The exact value follows from the peak sitting at (15 − √33)/16 = 0.57846 of the span from the built-in end. Using 185 is 0.198 per cent optimistic, which does not matter — but the rounding is worth knowing about, because an unexplained fourth digit in somebody else’s answer is usually this

Worked example

A 50 × 100 mm steel rectangle on a 2 m simple span, with a 5 kN point load 0.6 m from the left support
THE SECTION FIRST. For a rectangle bent about its strong axis, I = bh³/12 = 50 × 100³/12 = 4,166,667 mm⁴, the extreme fibre is c = h/2 = 50 mm, and the section modulus is Z = I/c = 83,333 mm³. Note where the depth sits in that expression: it is cubed. Turn the same bar on its side and I falls to 1,041,667 mm⁴, a quarter, for exactly the same amount of steel
THE MOMENT. For a point load at a from one support and b from the other, the maximum moment is M = Pab/L = 5000 × 0.6 × 1.4 / 2 = 2100 N·m, and it is under the load. Compare the same load at midspan: PL/4 = 2500 N·m, so moving it off centre has REDUCED the moment, which it always does
THE STRESS. σ = Mc/I = 2100 × 0.050 / 4.1667×10⁻⁶ = 25.2 MPa. Against A36 structural steel's 250 MPa minimum yield that is a factor of about ten, so this beam is nowhere near its strength limit — which is the usual situation, and the reason the real question is almost always stiffness
THE DEFLECTION, AND THE PART EVERYBODY GETS WRONG. The expression published for an eccentric load, Pa²b²/(3EIL), gives 5000 × 0.36 × 1.96 / (3 × 2 × 200×10⁹ × 4.1667×10⁻⁶) = 0.7056 mm. That is the deflection AT THE LOAD and it is not the maximum. The maximum is at x = √((L² − b²)/3) from the FARTHER support, with b the distance to the NEARER one — here b = 0.6 m, so x = √((4 − 0.36)/3) = 1.1015 m from the right support, which is 0.8985 m from the left. At that point the deflection is P b (L² − b²)^3/2 / (9√3 L EI) = 0.8019 mm
SO THE DEFLECTION AT THE LOAD UNDER-READS BY 13.6 PER CENT, and the peak is 0.30 m away from the load. Both figures are small here and they get much worse as the load moves toward a support: at a tenth of the span the under-read is 134 per cent and at a twentieth it is 325. This is a real trap rather than a pedantic one — this project's own physics proof for this batch had the two distances swapped, reported 0.785 mm at 0.825 m, and passed its own midspan check, because at midspan every version of the formula agrees
AND IS IT STIFF ENOUGH? At span over 360 the allowable on a 2 m span is 2000/360 = 5.556 mm, so 0.8019 mm is a utilisation of 0.144 and the beam actually achieves span over 2494. Comfortable — but note what span over 360 is: a building code limit from IBC Table 1604.3 written about floor joists carrying plaster ceilings. If this beam is carrying a linear rail or a measuring head, the limit that matters is whatever clearance or alignment the machine needs, and it may be ten times tighter
ONE MORE THING WORTH SEEING. Swap the solid rectangle for an 80 mm tube with a 60 mm bore and I falls to 1,374,447 mm⁴, which is 68.36 per cent of the 80 mm solid round's — for 43.75 per cent of the material. Stiffness per unit weight 1.56 times better, because I weights material by the square of its distance from the neutral axis and the metal in the middle of a solid bar is carrying its own weight and almost no load

The eight load cases, with the numbers for this page’s default beam

The last three columns are for the DEFAULT beam only — a 50 × 100 mm steel rectangle on a 2 m span with 5 kN of point load or 8 kN/m of uniform load — because a printed table cannot follow the inputs. The calculator above does. Every coefficient in the first and third columns was checked for this batch against a numerical integration of EI y″ = −M for that case, so the closed forms are confirmed by something that does not know what they are; the simple-span and cantilever cases were also checked against Engineering ToolBox’s published tables. Three rows repay a second look. The cantilever cases are an order of magnitude softer than anything else here: the same beam that deflects 1.00 mm as a simple span with a central load deflects 16.00 mm as a cantilever with the load at its tip, sixteen times as far, and its stress is four times as high. Building both ends in goes the other way, to a quarter of the simple-span deflection and half the stress — if the ends really are built in, which in practice they almost never fully are. And the propped cantilever’s deflection coefficient is written here as 184.634 rather than the familiar 185: 185 is that exact figure rounded, and it is 0.198 per cent optimistic. The exact value is the one the calculator uses. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
Load case	Maximum moment	Where the moment peaks	Maximum deflection	Where the deflection peaks	M (N·m)	sigma (MPa)	delta (mm)
Simply supported, point load at midspan	P L / 4	at midspan	P L³ / 48EI	at midspan	2,500.0	30.00	1.0000
Simply supported, uniform load	w L² / 8	at midspan	5 w L⁴ / 384EI	at midspan	4,000.0	48.00	2.0000
Simply supported, point load OFF centre	P a b / L	under the load	P b (L² − b²)^3/2 / 9√3 L EI	NOT under the load — computed below	2,100.0	25.20	0.8019
Cantilever, point load at the free end	P L	at the built-in end	P L³ / 3EI	at the free end	10,000.0	120.00	16.0000
Cantilever, uniform load	w L² / 2	at the built-in end	w L⁴ / 8EI	at the free end	16,000.0	192.00	19.2000
Both ends built in, point load at midspan	P L / 8	at the ends and under the load, equal	P L³ / 192EI	at midspan	1,250.0	15.00	0.2500
Both ends built in, uniform load	w L² / 12	at the built-in ends, twice the midspan value	w L⁴ / 384EI	at midspan	2,666.7	32.00	0.4000
Propped cantilever, uniform load	w L² / 8	at the built-in end	w L⁴ / 184.634EI	0.4215 of the span from the propped end	4,000.0	48.00	0.8319

Where the maximum deflection of an off-centre point load really is

One 5 kN load walked across a 2 m span on the default 50 × 100 mm steel rectangle, in twentieths. Read the fourth column: the peak is never under the load except at midspan, and it is always displaced TOWARD midspan. Read the seventh: taking the deflection at the load for the maximum under-reads by under 1 per cent at nine tenths of the way to midspan, by 24 per cent at a quarter of the span and by 325 per cent at a twentieth. The position formula is x = √((L² − b²)/3) with b the distance from the load to the NEARER support, measured from the FARTHER one, and the pair of conventions in that sentence is the whole difficulty — put the longer distance in as b, or measure from the wrong end, and the answer is a few per cent wrong in the unsafe direction and stays wrong invisibly. It stays invisible because at midspan every variant of the formula gives the same answer, so checking the collapse to the central-load case proves nothing about the convention. This project made exactly that error in its own physics proof for this batch and the midspan check passed; the table above came out of a two-million-point numerical scan of the real deflection curve instead. Last column: the MOMENT does peak under the load at every position, which is the asymmetry that keeps the mistake alive. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
Load at a (m)	a / L	Peak at x (m)	Peak minus load position (m)	True maximum (mm)	Deflection at the load (mm)	Under-read (%)	Moment at the load (N·m)
0.10	0.05	0.8467	0.7467	0.1534	0.0361	324.88	475.0
0.20	0.10	0.8511	0.6511	0.3033	0.1296	134.04	900.0
0.30	0.15	0.8584	0.5584	0.4464	0.2601	71.62	1,275.0
0.40	0.20	0.8686	0.4686	0.5793	0.4096	41.42	1,600.0
0.50	0.25	0.8820	0.3820	0.6988	0.5625	24.23	1,875.0
0.60	0.30	0.8985	0.2985	0.8019	0.7056	13.65	2,100.0
0.70	0.35	0.9183	0.2183	0.8859	0.8281	6.98	2,275.0
0.80	0.40	0.9417	0.1417	0.9482	0.9216	2.89	2,400.0
0.90	0.45	0.9688	0.0688	0.9868	0.9801	0.69	2,475.0
1.00	0.50	1.0000	0.0000	1.0000	1.0000	0.00	2,500.0
1.10	0.55	1.0312	-0.0688	0.9868	0.9801	0.69	2,475.0
1.20	0.60	1.0583	-0.1417	0.9482	0.9216	2.89	2,400.0
1.30	0.65	1.0817	-0.2183	0.8859	0.8281	6.98	2,275.0
1.40	0.70	1.1015	-0.2985	0.8019	0.7056	13.65	2,100.0
1.50	0.75	1.1180	-0.3820	0.6988	0.5625	24.23	1,875.0
1.60	0.80	1.1314	-0.4686	0.5793	0.4096	41.42	1,600.0
1.70	0.85	1.1416	-0.5584	0.4464	0.2601	71.62	1,275.0
1.80	0.90	1.1489	-0.6511	0.3033	0.1296	134.04	900.0
1.90	0.95	1.1533	-0.7467	0.1534	0.0361	324.88	475.0

Section properties, and the argument for a tube

Every figure here is computed from the dimensions, and the closed forms were checked for this batch by numerically integrating y² dA over the real cross-section, which is the test that does not assume the formula. The point is in the last two columns. An 80 mm tube with a 60 mm bore keeps **68.36 per cent** of the solid section’s second moment of area for **43.75 per cent** of its material: a stiffness-to-weight ratio 1.56 times as good. Thin the wall further, to a 70 mm bore, and it keeps 41.38 per cent of the I for 23.44 per cent of the area, 1.77 times as good again. The reason is that I weights material by the SQUARE of its distance from the neutral axis, so the metal near the centre of a solid bar is paying its full weight and contributing almost nothing. The same logic taken to its conclusion is the I-section, which puts almost all its area in two flanges as far from the neutral axis as the depth allows and gets an I per unit area nearly eight times the rectangle’s — though read that figure with the depths beside it, because the I-section tabulated here is 200 mm deep against the rectangle’s 100, and depth alone accounts for a factor of two of it. What the table cannot show is where the argument stops: a thin wall buckles locally, a thin web shears, and a thin flange cripples, and none of those is a bending calculation. Below about a fortieth of the diameter in wall thickness, stability rather than stress decides the section. These dimensions come from a published standard’s table, not from a formula. The standard itself is cited below and the printed values are attributed to the catalogue they were taken from; a different publisher may round differently in the last digit.
Section	I (mm^4)	A (mm^2)	Z = I/c (mm^3)	I / A (mm^2)	Radius of gyration (mm)	I as % of the 80 mm solid	A as % of the 80 mm solid
Rectangle 50 × 100	4,166,667	5,000.0	83,333	833.3	28.87	—	—
Solid round, 80 dia	2,010,619	5,026.5	50,265	400.0	20.00	100.00	100.00
Tube, 80 outside 60 inside	1,374,447	2,199.1	34,361	625.0	25.00	68.36	43.75
Tube, 80 outside 70 inside	832,031	1,178.1	20,801	706.3	26.58	41.38	23.44
I-section, 100 wide × 200 deep, 12 flange, 8 web	24,869,717	3,808.0	248,697	6,530.9	80.81	—	—

Deflection limits, and what documents they actually come from

The third column is the one to read, because span fractions get quoted in machine-design reviews as though they came from a machine-design code and they do not. Four of these six are from International Building Code Table 1604.3 and were written about floor joists, roof purlins and plaster ceilings; the other two are Eurocode 2 figures for concrete under the quasi-permanent load combination. None of them is about a machine element. That does not make them useless — span over 360 is a perfectly good first filter, and it is where the number everyone quotes comes from — but the real limit on a machine beam is almost always functional rather than structural: the clearance a guard has to keep, the alignment a bearing needs, the tolerance a tool has to hold, or the natural frequency the deflection implies. A linear rail on a gantry may need span over three thousand; a handrail is fine at span over a hundred. Pick the limit from what the machine has to do, and use the codes above as a sanity check rather than as the requirement. This page sizes a part; it does not certify one. Where the answer carries a consequence — a load path, a lifting duty, a pressure boundary, a fastener holding something that can fall — confirm it against the design code that governs the application, and against the manufacturer’s own rating, before relying on it.
Limit	Denominator	Where it comes from	Allowable on a 2 m span (mm)	Utilisation for the default beam
span / 120	120	IBC Table 1604.3, roof members not supporting a ceiling, live load	16.667	0.048
span / 180	180	IBC Table 1604.3, roof members not supporting a ceiling, dead plus live	11.111	0.072
span / 240	240	IBC Table 1604.3, floor members, live load	8.333	0.096
span / 250	250	EN 1992-1-1, quasi-permanent combination, common requirements	8.000	0.100
span / 360	360	IBC Table 1604.3, floor members, dead plus live load	5.556	0.144
span / 500	500	EN 1992-1-1, quasi-permanent combination, strict requirements	4.000	0.200

Elastic modulus, and what the choice of material does

Read the fourth column before the third. Bending stress does not depend on the material at all — σ = Mc/I has no E in it — so changing material changes only the deflection, and it changes it in exact inverse proportion to the modulus. Aluminium alloy deflects **2.857 times** as far as steel in the same section under the same load, every time, and that single ratio is the reason aluminium structures are deeper than steel ones rather than merely lighter. Where the source prints a range — titanium at 105 to 120, brass at 102 to 125 — the LOWER bound is used, which gives the larger deflection and the conservative answer on a stiffness check; the range is in the second column so you can see what was discarded. Two cautions. These are room-temperature values, and the modulus of steel falls by roughly a tenth by 300 °C and much faster above it. And grey cast iron’s modulus is not a material constant in the way the others are: it depends on the graphite form and on the stress level itself, which is why published figures for it range from about 100 to 145 GPa depending on grade. These dimensions come from a published standard’s table, not from a formula. The standard itself is cited below and the printed values are attributed to the catalogue they were taken from; a different publisher may round differently in the last digit. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
Material	Published range (GPa)	Used here (GPa)	Deflection relative to steel	Default beam’s deflection in this material (mm)
Steel, structural (ASTM A36)	200	200	1.000	0.8019
Stainless steel (AISI 302)	180	180	1.111	0.8910
Grey cast iron	130	130	1.538	1.2337
Titanium alloy	105 to 120	105	1.905	1.5274
Brass	102 to 125	102	1.961	1.5724
Aluminium alloy	70	70	2.857	2.2911

Eight load cases, I computed rather than tabulated, and the off-centre load whose peak is not under the load

This page puts a load on a straight beam, which is the one thing the rest of this site’s stress pages do not do. The section properties page gives you I and c for a shape with no load on it; the curved beam page is Winkler’s theory, which is for hooks, C-frames and chain links where the neutral axis has shifted and σ = Mc/I no longer applies. A straight beam with a load case is this page. Eight load cases, four cross-sections with I computed from the dimensions rather than looked up, and a deflection check against a span fraction whose source is named.

The thing worth coming here for: when a point load is off centre, the maximum deflection is not under the load. It sits between the load and midspan. The position is x = √((L² − b²)/3), measured from the FARTHER support, where b is the distance from the load to the NEARER one — and both of those conventions have to be applied together. For a 5 kN load 0.6 m along a 2 m span, the peak is 0.898 m from the left support, not at 0.6 m, and it is 0.802 mm rather than the 0.706 mm the load-point formula gives. That is a 13.6 per cent under-read at this position, and it gets worse toward the supports: at a tenth of the span it is 134 per cent and at a twentieth it is 325. The expression most published tables print for an eccentric load, Pa²b²/(3EIL), is the deflection AT the load. It is correct and it is not the maximum, and the two are routinely conflated.

This is a real trap rather than a pedantic one, and this project fell into it. The physics proof written for this batch put the distance to the farther support in as b and measured from the wrong end, giving 0.785 mm at 0.825 m — two per cent low on the deflection, eight per cent out on the position, and low is the unsafe direction. It passed its own verification, because that verification was the collapse to the central-load case at a = L/2, and at midspan the two distances are equal so every variant of the formula agrees. A midspan check cannot detect a swapped a and b. The figures on this page come from a two-million-point numerical scan of the real deflection curve instead, and the off-centre case is the page’s default load case so that the distinction is the first thing you see.

Depth is cubed, which is the most useful fact in beam design. I = bh³/12 for a rectangle, so doubling the width doubles the stiffness and doubling the depth multiplies it by eight. Turning a 50 by 100 mm bar on its side costs you three quarters of its I for exactly the same steel. The same logic explains the tube: an 80 mm tube with a 60 mm bore keeps 68.36 per cent of the solid section’s I for 43.75 per cent of its material, because I weights metal by the square of its distance from the neutral axis and the metal in the middle of a solid bar is paying its own weight and carrying almost nothing. Push it further and an I-section gets nearly eight times the rectangle’s stiffness per unit area — though the one tabulated here is also twice as deep, which is worth half of that on its own. Where the argument stops is stability: a thin wall buckles, a thin web shears, a thin flange cripples, and none of those is a bending calculation.

Stress almost never governs; stiffness does. The default beam on this page carries 25.2 MPa against structural steel’s 250 MPa minimum yield — a factor of ten — and still deflects. That is the normal situation for a machine beam, and it is why the real question is “is it stiff enough” rather than “will it break”. It is also why changing to a stronger grade of steel does nothing: E barely varies between grades while yield varies by a factor of five, so a high-tensile beam of the same section deflects exactly as far as a mild one. The only ways to buy stiffness are depth, span, end conditions and modulus, in that order of effectiveness.

On deflection limits, know what you are quoting. Span over 360, span over 240, span over 180 and span over 120 are all from International Building Code Table 1604.3, and they were written about floor joists, roof purlins and plaster ceilings. Span over 250 and span over 500 are Eurocode 2 figures for concrete under the quasi-permanent combination. None of the six is a machine-design limit. They are a reasonable first filter and the page offers them with their sources attached, but the limit that actually matters on a machine is functional: the clearance a guard must keep, the alignment a bearing needs, the tolerance a tool must hold, or the natural frequency the deflection implies. A gantry carrying a linear rail may need span over three thousand; a handrail is fine at span over a hundred.

Where to go next. If the beam also carries an axial load, the combined bending and axial stress page superposes the two properly. If you need a yield criterion rather than a single stress, the von Mises page takes the stress state. For a rotating shaft rather than a static beam, the shaft deflection and slope page handles the slope at the bearings, which is usually the governing quantity there rather than the deflection itself. And for the section alone, with no load case on it, the section properties page covers more shapes than the four here.

Frequently asked questions

Where is the maximum deflection when a point load is off centre?

Between the load and midspan, never under the load. The position is x = the square root of (L² − b²) over three, measured from the FARTHER support, where b is the distance from the load to the NEARER support. For a 2 m span with the load 0.6 m from the left support, b is 0.6 m, so x is 1.1015 m from the right support — 0.8985 m from the left. The deflection there is 0.802 mm against 0.706 mm at the load itself, a 13.6 per cent difference, and the gap grows as the load moves toward a support. The bending MOMENT does peak under the load, at every position, which is the asymmetry that makes this easy to get wrong.

Is Pa²b²/(3EIL) the maximum deflection?

No. It is the deflection AT THE LOAD, which is a different and smaller quantity, and it is printed as the maximum in a great many published tables and calculators. The true maximum is P b (L² − b²)^3/2 divided by 9√3 L EI, with b the distance from the load to the nearer support, and it occurs away from the load. The two agree only when the load is at midspan. Both are printed on this page with the percentage between them, and the chart sweeps the load across the span so you can see them separate.

How do I work out the maximum bending stress in a beam?

σ = Mc/I: the maximum bending moment, times the distance from the neutral axis to the extreme fibre, divided by the second moment of area. Equivalently σ = M/Z with Z = I/c the section modulus. Two things decide whether you get it right. First, find the maximum moment for the actual load case — it is at midspan for a simply supported beam, at the built-in end for a cantilever, and at the SUPPORTS for a beam with both ends built in under a uniform load, where it is twice the midspan value. Second, note that the material does not appear: bending stress is independent of E, so the same section under the same load has the same stress in aluminium as in steel. Only the deflection changes.

Why does a tube keep most of a solid bar’s stiffness?

Because the second moment of area weights material by the SQUARE of its distance from the neutral axis, so metal near the centre of a solid bar contributes almost nothing while carrying all of its own weight. An 80 mm tube with a 60 mm bore keeps 68.36 per cent of the 80 mm solid round’s I for 43.75 per cent of the area — stiffness per unit weight 1.56 times as good. Thin it to a 70 mm bore and it keeps 41.38 per cent of the I for 23.44 per cent of the area, better again by the same logic. The limit is not bending: a thin wall fails by local buckling, and below about a fortieth of the diameter in wall thickness the governing check is a stability one that σ = Mc/I cannot see.

What deflection limit should I use for a machine beam?

Whatever the machine actually needs, and probably not a span fraction out of a building code. The familiar limits — span over 360, 240, 180 and 120 — are from International Building Code Table 1604.3 and were written about floor joists, roof purlins and plaster ceilings; span over 250 and 500 are Eurocode 2 concrete figures. All six are offered on this page with their sources attached, and they make a reasonable first filter. But a machine’s limit is normally functional: the clearance a guard has to keep, the alignment a bearing or a coupling can tolerate, the tolerance a tool must hold, or the natural frequency the deflection implies. A gantry carrying a linear rail may need span over three thousand; a handrail is fine at span over a hundred. Work out what the deflection has to be and use the codes as a sanity check.

How much stiffer is a beam with both ends built in?

Four times, under a central point load, and five times under a uniform load — PL³/192EI against PL³/48EI, and wL⁴/384EI against 5wL⁴/384EI. The maximum moment also halves, from PL/4 to PL/8. That is the most optimistic assumption available on this page, and real joints rarely deserve it: full fixity means the support resists rotation completely, which a welded moment connection into a stiff member may do, a bolted end plate partly does, and a beam in a pocket does not at all. Partial fixity lands between the two cases. Note also that building the ends in moves the governing moment TO the supports, so the section has to be adequate there rather than at midspan.

Does a stronger steel deflect less?

No, and this is one of the most common misunderstandings in beam design. Deflection depends on the elastic modulus E, and E is essentially the same for every grade of steel — about 200 GPa — while the yield strength varies by a factor of five between mild and high-tensile grades. A beam in high-tensile steel deflects exactly as far as the same section in mild steel under the same load; it simply breaks later. Strength and stiffness are different properties. If deflection is the problem, the things that change it are section depth (cubed), span (cubed or to the fourth power), end conditions (a factor of four to five) and material modulus (a factor of about three at most, and in the wrong direction for every metal lighter than steel).

What is the 185 in the propped cantilever deflection formula?

A rounding. The exact maximum deflection of a propped cantilever under a uniform load is wL⁴/184.634EI, which every published table prints as wL⁴/185EI. The exact figure follows from the peak sitting at (15 − √33)/16 = 0.57846 of the span from the built-in end; substituting that back into the deflection curve gives a coefficient of 0.005416, and its reciprocal is 184.634. Using 185 makes the answer 0.198 per cent smaller, which never matters in practice. This page uses the exact value, which is why its fourth significant figure may differ from a textbook’s.

Related calculators

References

Engineering ToolBox, Beams — Supported at Both Ends, Continuous and Point Loads (read 6 October 2026). The second source for the simple-span cases: M = qL²/8 with δ = 5qL⁴/(384EI) for a uniform load, M = FL/4 with δ = FL³/(48EI) for a central point load, and M = Fab/L for an eccentric one. Its eccentric-load deflection is given as Fa²b²/(3EIL), which is the deflection UNDER THE LOAD and not the maximum — a distinction this page exists partly to make, and one the page itself does not draw.
Engineering ToolBox, Cantilever Beams — Moments and Deflections (read 6 October 2026). The second source for the two cantilever cases: M_A = −FL with δ_B = FL³/(3EI) for an end point load, M_A = −qL²/2 with δ_B = qL⁴/(8EI) for a uniform one. Note the signs: a cantilever’s maximum moment is at the ROOT and is negative in the usual convention, so the top surface is in tension — the opposite of a simple span, and the reason a cantilever cracks on top.
Engineering ToolBox, Young’s Modulus, Tensile Strength and Yield Strength for common Materials (read 6 October 2026). The source of the elastic moduli offered on this page: 200 GPa for ASTM A36 structural steel, 180 for AISI 302 stainless, 130 for grey cast iron, 105 to 120 for titanium alloy, 102 to 125 for brass and 70 for aluminium alloys. Where it prints a range, the LOWER bound is used here, because a lower modulus gives a larger deflection and therefore the conservative answer on a stiffness check.
A36 steel, Wikipedia (read 6 October 2026), used only to label the stress bands on this page with a real number rather than a feeling. It gives a minimum yield of 36 ksi (250 MPa) for plates, bars and shapes under 8 inches thick, 32 ksi (220 MPa) above that, an ultimate tensile strength of 58 to 80 ksi (400 to 550 MPa), and Young’s modulus of 29,000 ksi (200 GPa) — which agrees with the modulus table above. A36 is used here as the reference mild steel because it is the grade whose numbers everyone already carries.
International Building Code Table 1604.3, Deflection Limits, read 6 October 2026 in the Florida Building Code’s published copy of it. The source of four of the six span fractions this page offers: l/360 and l/240 for floor members under dead-plus-live and live load respectively, l/240 and l/180 for roof members supporting a non-plaster ceiling, l/180 and l/120 for roof members supporting no ceiling, and l/360 where a plaster ceiling is carried. It is a building code and not a machine-design document, which is exactly why it is cited by name: the span fractions machine designers quote come from here, and they were written about joists.
Fine Software’s published EN 1992-1-1 deflection guidance (read 6 October 2026), the source of the two Eurocode span fractions offered here: l/250 for common requirements and l/500 for strict requirements, both under the quasi-permanent load combination. Note what it is: Eurocode 2, which is concrete. It is cited because l/250 is the fraction most often quoted without a source, and this is the source.

CalcEngines mechanical calculators are for education and design estimates. Results depend on the values you enter and the assumptions stated on the page, and they do not replace a qualified engineer's check. Standards, material grades and manufacturer data change — verify against the current standard and the supplier's own catalogue before you specify, machine or build anything.