Shaft Deflection
How far a loaded shaft bends, measured against the gear tooth that has to stay meshed. Doubling the shaft size makes it sixteen times stiffer; doubling its length makes it eight times worse.
Inputs
The shaft
VEX shafts are square. A screw acting as a pivot is round, and what counts there is the diameter across the bottom of the threads, not the size on the box.
Side of the square, or diameter of the round bar.
How it is held
Two bearings with the load between them is the sandwich every VEX drivetrain uses. Outboard means the load hangs past both of them.
Center to center. Ignored if you chose a single rigid support.
Measured inward for a load between the bearings, outward for one hanging past them, and from the support for a single rigid one.
The load
The sideways force the shaft carries at that point: the weight hanging off it, or the force a gear tooth pushes with.
Results
How far it bends
—in
Same thing, in thousandths
—thou
Against one gear tooth's depth
—
Angle it sits at, in the bearing
—°
Peak bending stress
—psi
Margin before it takes a permanent bend
—×
Against a rigid mount
—
Will the gears still mesh?
—
of the assumed yield strength
The bent shaft
The dashed line is where the shaft would be if nothing bent it. Bending is drawn far larger than life so it is visible at all. The exaggeration is written under the drawing. Drag to read the movement anywhere along the shaft.
Show the numbers behind this chart
How far the shaft has moved at each point along its length, which is the curve drawn above.
How this is calculated
A loaded shaft is a beam. Push sideways on it and it bends, by an amount set by four things: how hard you push, how far out you push, what the shaft is made of, and the shape of its cross section.
bend = load × length³ ÷ (3 × stiffness × section)The exponent is the whole story. Load is linear: twice the weight, twice the bend. Length is cubed. Moving a gear from two inches out to four inches out does not double the deflection, it multiplies it by eight.
With your numbers
Why a quarter-inch shaft is not twice as good
A shaft's stiffness in bending comes from its second moment of area, a number describing how the material in its cross section is spread out around the axis it bends about. For a square, that goes as the fourth power of its side. A 1/4" shaft is twice the side of a 1/8" one, so it is sixteen times stiffer in bending, not twice.
That is the single biggest lever on this page, and it is why the answer to a bending shaft is almost always the bigger shaft rather than a cleverer arrangement.
Turning the shaft in the bearing changes nothing
For a square, the second moment of area is identical about every axis through the center, including the diagonals. Not approximately: the two terms that would make it vary cancel exactly for a shape with equal stiffness in both directions. So a square shaft is exactly as stiff corner-on as it is flat-on, and as it spins it does not get stiffer and floppier four times a turn.
Strength is different. Stiffness depends on how the material is spread; strength depends on how far the outermost fiber sits from the center, and a corner is further out than a face. Turned corner-on, the same bend puts about 41% more stress in the steel. A rotating shaft passes through that orientation every turn, so this page uses it for the stress figure.
What the tooth comparison means
A deflection figure alone says nothing. Five thousandths of an inch is nothing on a drivetrain and a great deal inside a gearbox, so this page measures it against the depth of the gear tooth that has to stay engaged. Every VEX gear is 24 DP, which makes a tooth's working depth about 0.083".
Once the shaft has moved a noticeable fraction of that, the teeth are running on their tips rather than their flanks. You hear it before you see it: a whine that comes and goes with load, then a skip.
What is left out
- The shaft's own weight. Six inches of 1/8" square steel weighs about a fiftieth of a pound. Against any load worth calculating it is noise.
- Flex in the bearings and the C-channel. This treats the supports as immovable. They are not, and on a real robot they often contribute more movement than the shaft does, which is why a shaft that calculates as stiff can still feel sloppy.
- Play in the holes. A bearing has clearance. That clearance does not go away because the shaft is stiff.
So read the number as the best case. The real mechanism moves at least this much.
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Sources & assumptions
Stiffness is derived, not published. VEX gives no material data for its shafts. Deflection here assumes steel at 29,000,000 psi, which is a safe assumption because Young's modulus barely moves across the steels: the whole family sits inside roughly 28 to 30 million psi, so even the wrong alloy gives very nearly the right answer.
Strength is a weaker assumption. The 54,000 psi yield figure is the same one the shaft load check uses for twisting, kept deliberately identical so the two pages cannot disagree about how strong a VEX shaft is. Unlike stiffness, this one does vary with alloy and heat treatment, so treat the margin as a guide to which shaft to reach for rather than a rating.
The tooth depth is derived from the 24 DP pitch, as two divided by the diametral pitch.
Checked numerically. Every formula on this page is verified in scripts/verify-beam.mjs against a numerical integration of the beam's curvature that shares no algebra with it: 128 checks covering all three support arrangements, the shape of the whole curve, and the scaling laws quoted above.
- VEX Robotics: Gears — checked August 2026
- VEX Robotics: C-Channels (8 sizes) — checked August 2026
Where this goes next
Explore further on the full mapWhere the answers on this page can be used, and which tools could have given you the numbers it asks for.
Comes after
Bearing Load & Support Placementfills in your bearing spacing hereSpacer Stack Calculatorgives you your shaft sizeComplete Lift Designergives you your lift loadTwo-Bar Liftgives you your lift loadCascade Liftgives you your lift loadYou are here
Shaft Deflection
Where your answer goes
Bearing Load & Support Placementsee which bearing takes the most push and whether moving it helpsbrings your bearing spacing with itSpacer Stack Calculatorbuild the exact spacer stack that holds the gear that close to the bearinguses your shaft size againShaft & Structure Load Checkcheck the same shaft won't twist or snap under full motor torqueuses your shaft size againa number carries across the next question it raises