Shaft Deflection and Slope Calculator

Deflection AND the slope at the bearing, for the standard shaft load cases, checked against the published misalignment limits for what is actually at the bearing — because misalignment is what kills bearings, not deflection. The stepped-shaft case is solved exactly by Castigliano and the usual equivalent-diameter answer printed beside it.

Last updated 24 Sep 2026

Shaft deflection and slope

Load case → slope at the bearing, against its limit

Load caseThe last one is the honest stepped-shaft calculation, and it is EXACT for a symmetric step under a central load rather than an equivalent-diameter approximation.

Shaft diameter (mm)Deflection and slope both go as 1/d⁴, so this is the strongest lever after the span. For the stepped case this is the END diameter.

Bearing span (mm)Deflection goes as the cube of the span and slope as the square. Halving the span cuts the slope to a quarter.

Transverse load (N)A gear separating force, a belt or chain pull, a sheave weight, or the reaction a misaligned coupling pushes back into the shaft.

Load position from the left bearing (mm)Only used by the offset case. The slope at the NEAR bearing is the larger one, and a load close to a bearing gives a small deflection and a large local slope.

Overhang beyond the bearing (mm)Only used by the overhung case. The bearing next to the overhang sees a slope of W·c·L/(3EI) — twice the far bearing’s.

Stepped shaft: diameter of the centre section (mm)

Stepped shaft: fraction of the span at that diameterCentred on mid-span. A step at the middle is worth far more than a step near a bearing, and the exact calculation shows by how much.

What is at the bearingShigley’s Table 7-2 slope limits. A self-aligning bearing tolerates more than twenty times a cylindrical roller.

Young’s modulus (GPa)

Not a circuit: the deflected shaft between its two bearings, drawn schematically with the deflection hugely exaggerated — a real shaft's deflection is a fraction of a millimetre over hundreds, so drawn to scale it would be a straight line. The curve and the tangent at the left bearing are fixed; what moves with your numbers is the tilted inner ring on the right and the bar below. The ring is the mechanism: a slope at the bearing tilts the inner ring relative to the outer, which moves the rolling contact off the centre of the raceway and puts a stress peak at one edge of it. The tilt is exaggerated by about two hundred times and quantised to thirtieths, so read its direction and not its angle. The bar is the slope as a percentage of the published limit for whatever you said is at the bearing, with the line at 100 — and the line at the bottom is the same slope against the tighter limit an uncrowned spur gear mesh needs, which on a geared shaft is usually the criterion that governs.

727.6µradExample

A 50 mm shaft on bearings 500 mm apart with 3,000 N at mid-span, running in deep-groove ball bearings

Slope as well as deflection, and Castigliano for the step

Central load: y = WL³/48EI, θ = WL²/16EI · Offset load: θ_A = Wb(L²−b²)/6EIL · Overhung: θ_B = WcL/3EI, y_tip = Wc²(L+c)/3EI · Stepped: y = (W/2E)∫₀^{L/2} x²/I dx, θ = (W/2E)∫₀^{L/2} x/I dx

θ: slope in RADIANS, which is the misalignment the bearing sees. The quantity this page exists to compute, because it is the one that decides bearing life
y: deflection. Computed and reported, but it is rarely the governing criterion on a machine shaft
I: second moment of area, πd⁴/64. Both slope and deflection go as 1/d⁴, which is why a small increase in diameter is worth so much
∫x²/I dx: Castigliano’s integral for a symmetric stepped shaft under a central load. It is EXACT, not an equivalent diameter — and the length-weighted equivalent diameter that is usually used instead is compared against it
c: overhang beyond the bearing. The near bearing’s slope is WcL/3EI, exactly twice the far bearing’s, which is the structure of an end-moment problem

Worked example

A 50 mm shaft on bearings 500 mm apart with 3,000 N at mid-span, running in deep-groove ball bearings
I = πd⁴/64 = 306.8 kmm⁴, so EI = 64.43 GN·mm²
Deflection first, because it is the number everyone computes: y = WL³/48EI = 0.1213 mm. That is span over 4,123 — four times stiffer than the span-over-a-thousand rule of thumb that gets quoted. It tells you almost nothing
Now the slope, which is the number that matters: θ = WL²/16EI = 727.6 µrad — 0.728 mrad, or 2.50 minutes of arc
Against the limits: Shigley's range for a deep-groove ball bearing is 0.001 to 0.003 rad, and SKF independently put a single-row deep groove ball bearing at 2 to 10 minutes of arc, which is 0.00058 to 0.0029. So 0.728 mrad is 24% of Shigley's upper figure and 73% of its lower one — comfortably acceptable for this bearing
Change the bearing and the verdict moves with it. A CYLINDRICAL ROLLER bearing wants 0.0008 to 0.0012 rad, so this shaft is at 61% of its upper limit — still inside, but with far less room. A SELF-ALIGNING bearing wants 0.026 to 0.052, so it is at 1.4% — utterly irrelevant. The bearing choice is a bigger lever here than anything about the shaft
And if there is a gear on this shaft, the gear governs and the shaft FAILS: an uncrowned spur gear mesh wants under 0.0005 rad and this shaft is at 146% of that. The tooth contact would sit at one end of the face. A shaft that is four times stiffer than the structural rule of thumb asks for, and comfortable in its bearings, is still half again over the limit for the gear it is carrying
Diameter is the lever, and it is a fourth-power one: at 55 mm instead of 50 the slope falls to 496.9 µrad — a 10% increase in diameter for a 32% reduction in slope, which brings the gear inside its limit
Finally the stepped case, which is where the usual method goes wrong. Put a 65 mm section over the middle half of the same span and the EXACT mid-span deflection is 0.0523 mm, by Castigliano's integral. The length-weighted equivalent-diameter method — take a length-weighted average of 1/I and deflect a uniform shaft of that stiffness — gives 0.0819 mm, which is 56.5% too soft — a factor of one and a half on the wrong side of a stiffness calculation. It is conservative here and it is NOT conservative when the thick section is at the ends, because it cannot see that the bending moment is largest in the middle

Published slope limits at a bearing, and the gear mesh that beats all of them

From Shigley’s Table 7-2, “Typical Maximum Ranges for Slopes and Transverse Deflections”. The independent check on the deep-groove ball row is SKF’s own published figure: they put the permissible angular misalignment of a single-row deep groove ball bearing “between 2 and 10 minutes of arc”, which is 0.00058 to 0.0029 rad and overlaps Shigley’s 0.001 to 0.003 almost exactly. Two independent sources, agreeing, on a quantity most shaft calculations never compute. Three things to take from the table. The spread between bearing types is more than forty to one, so the bearing choice is a larger lever on the allowable slope than anything about the shaft. A self-aligning or spherical bearing is not just more tolerant, it is in a different regime — 1.5 to 3 degrees, where a cylindrical roller wants under 0.07 of a degree. And an uncrowned spur gear mesh, at 0.0005 rad, is tighter than every bearing on the list, so on a geared shaft the gear usually sets the criterion and not the bearing. SKF also say the thing a table cannot: the permissible value depends on internal clearance, bearing size, internal design and the applied load, so “no generally applicable specific values can be given”. 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.
What is there	Slope limit, low (rad)	Slope limit, high (rad)	Low (minutes of arc)	High (minutes of arc)	Ratio to the gear-mesh limit
Tapered roller bearing	0.0005	0.0012	1.72	4.13	2.4
Cylindrical roller bearing	0.0008	0.0012	2.75	4.13	2.4
Deep-groove ball bearing	0.0010	0.0030	3.44	10.31	6.0
Spherical (self-aligning) ball bearing	0.0260	0.0520	89.38	178.76	104.0
Uncrowned spur gear mesh	—	0.0005	—	1.72	1.0

Why the slope and not the deflection

This is the point of the page. Almost every shaft calculation computes deflection, because deflection is what the formulas in the textbook give, and then compares it against a span-over-something rule of thumb that came from structural engineering and has nothing to do with bearings. What actually kills a bearing is misalignment, and misalignment at the bearing IS the slope of the shaft there — the derivative, not the displacement. The two are not interchangeable: a load close to one bearing gives a small mid-span deflection and a large slope at that bearing, which is exactly the arrangement an overhung sheave or a close-coupled gear produces. Use the offset and overhung cases here and watch the slope at the near bearing while the deflection stays small. 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.
Quantity	What it does	Typical limit
Deflection at the gear or sheave	Changes the centre distance, which changes gear backlash and belt tension. Usually a small effect — a few hundredths of a millimetre on a centre distance of hundreds	Shigley’s Table 7-2 gives transverse deflection limits for spur gears by diametral pitch — 0.010 in for P under 10, 0.005 for 11 to 19, 0.003 for 20 to 50. Named here, not computed
SLOPE at the bearing	Tilts the inner ring relative to the outer. In a rolling bearing that moves the contact off the centre of the raceway, so the load distribution across the rollers becomes uneven and the edge of the contact carries a stress peak. The bearing does not deflect more — it wears out sooner, at one spot	0.0005 to 0.003 rad for a rigid rolling bearing, and 0.026 to 0.052 for a self-aligning one. That is the whole design margin and it is why this page computes the slope
Slope across a gear face	Moves the contact to one end of the tooth, so the face width you paid for is not carrying load. It also raises the tooth’s root stress at that end	Below 0.0005 rad for an uncrowned spur gear, which is tighter than every bearing limit
Slope at a seal	Makes the lip run eccentric. Usually tolerant, but a slope with a large radial runout is what makes a seal weep	No published figure sourced. Named rather than guessed

A stepped shaft: the exact answer against the equivalent-diameter method everybody uses

A 50 mm shaft over 500 mm with a 65 mm section centred at mid-span, carrying 3,000 N at the centre. The equivalent-diameter method takes a length-weighted average of 1/I and deflects a uniform shaft of that stiffness; the exact column integrates x²/I along the shaft, which is what Castigliano’s theorem gives for this loading and is not an approximation at all. Read the error column and the structure is obvious: the length-weighted method is worst when the step is small (a short thick section at mid-span does far more good than its length suggests, because the bending moment is largest there) and it converges as the step fills the span. It errs CONSERVATIVELY for a thicker centre section and it errs the other way for a thicker END section — which is the case where an equivalent-diameter calculation will tell you a shaft is stiff when it is not. What the exact method here still misses: it assumes the step is a clean change of section, with no fillet, and a real fillet adds a little local flexibility and a large stress concentration. That second one is on the shaft fillet and stress concentration calculator. 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.
Fraction of the span at the larger diameter	EXACT mid-span deflection (mm)	Length-weighted equivalent-diameter answer (mm)	Error of the equivalent-diameter method (%)	Equivalent diameter (mm)	Exact slope at the bearing (mrad)
0.10	0.0999	0.1134	13.5	50.85	0.6377
0.20	0.0828	0.1055	27.4	51.77	0.5573
0.30	0.0695	0.0976	40.5	52.79	0.4864
0.40	0.0595	0.0897	50.9	53.91	0.4250
0.50	0.0523	0.0819	56.5	55.16	0.3729
0.60	0.0475	0.0740	55.7	56.57	0.3304
0.70	0.0446	0.0661	48.3	58.19	0.2973
0.80	0.0431	0.0582	35.1	60.07	0.2737
0.90	0.0425	0.0503	18.3	62.29	0.2595

The derivative, not the displacement

The slope at the bearing is the number that matters, and almost nobody computes it. Every shaft calculation gives deflection, because deflection is what the textbook formulas give, and then compares it against a span-over-something rule that came from structural engineering and has nothing to do with bearings. What actually destroys a rolling bearing is misalignment, and misalignment at the bearing IS the slope of the shaft there — the derivative, not the displacement. Tilting the inner ring relative to the outer moves the rolling contact off the middle of the raceway; the load spreads unevenly across the rolling elements and one edge of one contact takes a stress peak. The bearing does not feel softer, it wears out at one spot. The worked example on this page is span over three thousand in deflection — beautifully stiff by any structural rule — and sitting at the bottom of the acceptable band for the slope.

The limits, from two sources that agree. Shigley’s Table 7-2 gives maximum slopes at a bearing: 0.0005 to 0.0012 rad for a tapered roller, 0.0008 to 0.0012 for a cylindrical roller, 0.001 to 0.003 for a deep-groove ball, and 0.026 to 0.052 — one and a half to three degrees — for a self-aligning or spherical ball. SKF, independently, put a single-row deep groove ball bearing’s permissible angular misalignment “between 2 and 10 minutes of arc”, which is 0.00058 to 0.0029 rad and overlaps Shigley’s row almost exactly. Note the spread: more than forty to one between bearing types, so the bearing choice is a bigger lever on the allowable slope than anything about the shaft. And note that an uncrowned spur gear mesh, at 0.0005 rad, is tighter than EVERY bearing on the list — on a geared shaft the gear sets the criterion and the bearing does not. SKF also make the point a table cannot: the permissible value depends on internal clearance, size, internal design and load, so “no generally applicable specific values can be given”.

Where the load sits changes the answer more than how big it is. A load close to one bearing gives a small mid-span deflection and a large slope at that bearing, because the shaft has to turn sharply to get from the support to a nearby load and back. That is precisely the arrangement a close-coupled gear or an overhung sheave produces, and it is the arrangement a deflection-based check passes with room to spare. The overhung case is stranger still: the slope at the near bearing is W·c·L/3EI, so it rises with the BEARING SPAN — moving the bearings further apart makes an overhung end worse, not better, and the fix is to shorten the overhang and if anything bring the bearings together. The belt or chain pull that causes most of these loads comes from the belt tension and shaft load calculator, and a misaligned coupling contributes its own reaction from the coupling torque and selection calculator.

Stepped shafts, done properly and then done the usual way for comparison. For a symmetric stepped shaft under a central load, Castigliano’s theorem gives the deflection as (W/2E)∫x²/I dx over half the span and the slope at the bearing as (W/2E)∫x/I dx — and for piecewise-constant I those are closed-form sums, exact rather than approximate. Reassuringly, the slope integral has the x² terms cancel by symmetry, which is why the slope is the simpler of the two. The method everyone uses instead takes a length-weighted average of 1/I and deflects a uniform shaft of that stiffness. This page computes both and prints the error. It is 57% on the worked example, conservative in that direction — and it is NOT conservative when the thick section is at the ends, because a length weighting cannot see that the bending moment is largest at mid-span. What the exact method still misses is real: no fillet at the step, so no local flexibility and no stress concentration; no shear deflection, which matters below about ten diameters of span; and nothing about the stress concentration at the step, which is where the shaft actually breaks.

Frequently asked questions

Why does the slope at the bearing matter more than the deflection?

Because misalignment is what kills a rolling bearing, and misalignment at the bearing is the slope of the shaft there. Tilting the inner ring relative to the outer moves the contact off the centre of the raceway, so the load spreads unevenly across the rolling elements and one edge takes a stress peak. Deflection, by contrast, mostly changes a centre distance by a few hundredths of a millimetre, which matters to a gear mesh or a belt tension but rarely governs. The worked example here is span over three thousand in deflection — excellent by any structural rule — and right at the bottom of the acceptable band for slope.

What is the maximum allowable slope at a bearing?

It depends entirely on the bearing type, over a range of more than forty to one. Shigley’s Table 7-2: 0.0005 to 0.0012 rad for a tapered roller, 0.0008 to 0.0012 for a cylindrical roller, 0.001 to 0.003 for a deep-groove ball, 0.026 to 0.052 for a self-aligning or spherical ball. SKF put a deep groove ball bearing at 2 to 10 minutes of arc, which agrees. Where a gear is involved the gear usually governs: below 0.0005 rad for an uncrowned spur gear, tighter than any bearing on the list.

How do I calculate the deflection of a stepped shaft?

Properly, with Castigliano’s theorem: the deflection is (W/2E)∫x²/I dx and the slope at the bearing is (W/2E)∫x/I dx, both over half the span, and for a shaft made of constant-diameter sections those integrals are exact closed-form sums. That is what this page does. The common alternative is a length-weighted equivalent diameter, which is approximate and approximate in a knowable direction — it understates what a thick centre section is worth and overstates what a thick end section is worth, because it cannot see that the bending moment is largest at mid-span. Both are printed here so you can see the size of the error.

Does a bigger shaft help much?

Enormously, because both slope and deflection go as 1/d⁴. Going from 50 to 55 mm — a 10% increase in diameter — cuts the slope by 32%. It is the strongest lever available after the span, which goes as the square for slope and the cube for deflection. Note that this is a stiffness calculation and has nothing to do with strength: a shaft can be several times stronger than it needs to be and still be too flexible for its bearings, and shaft sizing from strength and stiffness is a separate page on this site.

Why does moving the bearings further apart make an overhung load worse?

Because an overhang is an end-moment problem. The overhung load W at a distance c applies a moment W·c at the bearing, and a simply supported shaft with an end moment M has a slope ML/3EI at that end — proportional to the SPAN. So a longer bearing span gives the end moment more shaft to rotate, and the slope at the near bearing rises. The far bearing’s slope is exactly half of it. The practical consequence is that the fix for a heavily overhung sheave is a shorter overhang and, if anything, closer bearings — the opposite of the usual instinct.

Should I check the deflection at the gear or the slope?

Both, and they are different criteria with different limits. The deflection changes the centre distance and therefore the backlash; Shigley’s Table 7-2 gives transverse deflection limits for spur gears by diametral pitch, which this page names but does not compute because it needs the gear data. The slope across the face moves the tooth contact to one end, which wastes the face width you paid for and raises the root stress there; that limit is 0.0005 rad and it is what the calculator checks. In practice on a machine shaft the slope is the one that bites.

Does this include the shaft’s own weight?

No. The load cases here are point loads, which is what a gear, sheave, coupling reaction or overhung mass is. For a long, lightly loaded shaft the self-weight term can matter and it superposes — the mid-span deflection under self-weight is 5qL⁴/384EI and the slope at the bearing is qL³/24EI — so add them. The shaft’s own mass IS included, and matters a great deal, in the critical speed calculator.

Related calculators

References

Richard G. Budynas and J. Keith Nisbett, Shigley’s Mechanical Engineering Design, Table 7-2, “Typical Maximum Ranges for Slopes and Transverse Deflections”. The source for the slope limits at a bearing: tapered roller 0.0005–0.0012 rad, cylindrical roller 0.0008–0.0012, deep-groove ball 0.001–0.003, spherical and self-aligning ball 0.026–0.052, and an uncrowned spur gear below 0.0005 rad. The same table gives transverse deflection limits for spur gears by diametral pitch, which is a different criterion and is named on the page but not computed.
SKF. Single row deep groove ball bearings — misalignment. The independent check on Shigley’s deep-groove row: SKF put “the permissible angular misalignment … between 2 and 10 minutes of arc”, which is 0.00058 to 0.0029 rad and overlaps Shigley’s 0.001 to 0.003 almost exactly. SKF also says the thing a table cannot: the permissible value depends on internal clearance, size, internal design and the applied load, so “no generally applicable specific values can be given”.