BIZOMAT MAKEOS · Manufacturing
Cutting Tool Deflection and Depth-of-Cut Limit
How far will this end mill or boring bar deflect under the cut, and what depth keeps it within tolerance?
Cutting force on an end mill (average tangential force from removal rate and specific cutting force) or a boring bar (chip area × specific cutting force), the resultant with the radial component, the tool’s cantilever deflection with an effective fluted diameter, the overhang ratio, and the largest depth of cut that keeps deflection within the allowance.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Tool | choice | End mill or boring bar. |
| Tool or bar diameter | mm | Diameter of the tool or bar. |
| Overhang (stick-out) | mm | Distance from the holder face to the cutting edge. |
| Effective diameter factor (fluted section) | Equivalent stiff diameter as a share of nominal (about 0.8 for fluted end mills, 1 for solid bars). | |
| Tool modulus | GPa | Elastic modulus (about 600 GPa carbide, 210 GPa steel). |
| Specific cutting force | N/mm² | Specific cutting force for the material. |
| Radial ÷ tangential force | Radial (deflecting) force as a share of the tangential force (about 0.3–0.6). | |
| End mill cut | ||
| Teeth | Teeth on the end mill. | |
| Feed per tooth | mm | Feed per tooth. |
| Radial width of cut | mm | Radial engagement of the end mill. |
| Axial depth of cut (end mill) | mm | Axial depth of the end mill cut. |
| Boring cut | ||
| Feed per revolution (boring) | mm/rev | Feed per revolution for boring or turning. |
| Depth of cut (boring, radial) | mm | Radial depth of cut for boring. |
| Limit | ||
| Allowable deflection | mm | Deflection you can accept (a share of the tolerance). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Tool
- What it is
- Which tool is bending: an end mill cutting on its side, or a boring bar or turning tool. It decides which set of cut inputs is used for the force.
- Units and format
- Choose one of 2 options from the list: End mill (side milling); Boring bar or turning tool.
- Example
- The form opens with “End mill (side milling)” selected as a worked example. Choose the option that matches your case.
- Get it right
- Only the end mill inputs or only the boring inputs are used, depending on this choice.
What this means: Tool or bar diameter
- What it is
- Nominal diameter of the end mill, or of the boring bar shank, which sets the bending stiffness of the overhanging length.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 200 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stiffness rises with the fourth power of diameter, so a slightly larger tool deflects much less.
What this means: Overhang (stick-out)
- What it is
- How far the tool sticks out from the face of the holder or tool post to the cutting edge: the free length that bends like a cantilever.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Deflection grows with the cube of overhang: 25 % more stick-out gives about twice the deflection.
- Get it right
- Measure from the face of the holder that grips the tool; length hidden inside the holder does not count.
What this means: Effective diameter factor (fluted section)
- What it is
- Diameter of a solid round bar with the same bending stiffness as the tool, as a share of its nominal diameter. Flutes remove material, so a fluted section is less stiff.
- Units and format
- Enter a plain number with no unit. The form accepts 0.4 to 1. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.8 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- It enters to the fourth power, like diameter, so small changes matter.
- Get it right
- Use 1 for a solid round bar; for fluted tools take the factor from the tool maker or a reference for the flute form.
What this means: Tool modulus
- What it is
- Elastic (Young’s) modulus of the tool or bar material, which sets how much it bends under load. Carbide is much stiffer than steel.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1200 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 600 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Deflection is inversely proportional to the modulus.
- Get it right
- Enter GPa, not MPa; take the value for the grade from the tool maker.
What this means: Specific cutting force
- What it is
- Cutting force per square millimetre of chip cross-section for the work material, from the tool maker’s cutting data or a machining reference.
- Units and format
- Enter a number in N/mm²: newtons per square millimetre (equal to MPa). The form accepts 0.000001 N/mm² to 10000 N/mm². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 N/mm² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Cutting force, and so deflection, is proportional to kc.
- Get it right
- kc rises as the chip gets thinner; use a value that matches the feed you are cutting with.
What this means: Radial ÷ tangential force
- What it is
- Radial force pushing the tool away from the work, as a multiple of the tangential (main cutting) force. The two combine into the force that bends the tool.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 2. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The bending force is F_t √(1 + ratio²), so it matters more as the ratio grows.
- Get it right
- Enter a ratio, not a percentage: 0.5 means half the tangential force.
End mill cut
What this means: Teeth
- What it is
- Number of cutting edges (flutes) on the end mill. With feed per tooth it sets how much metal is removed per revolution.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 20. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Average tangential force rises in proportion to the number of teeth at the same feed per tooth.
- Get it right
- Used only when the tool is an end mill.
What this means: Feed per tooth
- What it is
- How far the end mill advances for each tooth that passes the work: the chip load per tooth.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 2 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.05 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Force and deflection rise in proportion to feed per tooth.
- Get it right
- Per tooth, not per revolution: feed per revolution = feed per tooth × teeth.
What this means: Radial width of cut
- What it is
- How much of the cutter diameter is engaged sideways in the work, measured at right angles to the tool axis (the stepover).
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 200 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Average force and deflection are proportional to the radial width.
- Get it right
- It cannot exceed the cutter diameter; in a full-width slot it equals the diameter.
What this means: Axial depth of cut (end mill)
- What it is
- How far the end mill cuts along its own axis: the height of the side wall being machined in one pass.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 500 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Force and deflection are proportional to axial depth; the tool reports the largest depth within the allowance.
Boring cut
What this means: Feed per revolution (boring)
- What it is
- How far the boring bar or turning tool advances along the bore for each spindle revolution.
- Units and format
- Enter a number in mm/rev: millimetres per revolution. The form accepts 0.000001 mm/rev to 5 mm/rev. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.15 mm/rev as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Cutting force and deflection are proportional to feed.
- Get it right
- Used only for the boring choice; it is per revolution, not per minute.
What this means: Depth of cut (boring, radial)
- What it is
- Material the boring bar removes on the radius in one pass, measured from the bore surface into the wall.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 50 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Force and deflection are proportional to depth of cut; the tool reports the largest depth within the allowance.
- Get it right
- This is half the change in bore diameter, not the full diameter change.
Limit
What this means: Allowable deflection
- What it is
- Tool-tip deflection you can accept, normally a small share of the dimensional tolerance on the feature being cut.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 5 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.02 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The largest depth of cut reported is proportional to the allowance.
- Get it right
- Enter millimetres, not micrometres (1 µm = 0.001 mm).
Method and formulas
- End mill average tangential force F_t = a_e a_p f_z z k_c ÷ π D (removal rate × k_c ÷ cutting speed).
- Boring: F_t = a_p × f × k_c.
- Resultant F = F_t √(1 + ratio²); deflection δ = F L³ ÷ 3 E I with I = π (k D)⁴ ÷ 64 (as the boring-bar tool).
- Force is proportional to depth, so the largest depth = depth × allowable ÷ δ.
- Example: a 10 mm end mill at 40 mm stick-out deflects 50.347 µm; keep the depth below 3.9724 mm.
Limitations
- Average force; peak force on few-tooth cutters is higher.
- Holder and spindle compliance add to tool deflection.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Set depth of cut for a long-reach end mill
- Hold bore tolerance with a long boring bar
- Choose a shorter or carbide tool before cutting
Who it is for
- Machinists
- CNC programmers
- Manufacturing engineers
What you get
- Cutting force
- Tool deflection and overhang ratio
- Largest depth of cut for the allowance
Interpreting the result
- Deflection rises with the cube of overhang: shorten the tool before reducing feed.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 1093–1094, 1162–1167. Topic reference; no source tables or recommended settings are reproduced.
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Open the Cutting Tool Deflection and Depth-of-Cut Limit calculator