Two formulas cover almost every shop calculation you will need: the Kienzle equation for quick, tabulated estimates and the mechanistic area model (Ft = Ks · Ac) for calibration-grade predictions. Use Kienzle when you need a fast check with published coefficients, and the mechanistic model when you have measured data and need accuracy for torque, power, or fixture design. Tangential force feeds directly into spindle torque and power, and Kienzle's tabulated values only work correctly in millimeters.
TL;DR:
- The Kienzle equation provides quick, approximate cutting force estimates using published coefficients, but requires inputs in millimeters to avoid errors.
- The mechanistic model offers calibration-grade predictions when measured force data are available, enabling more precise process optimization.
- Tangential force generally accounts for about 70% of total cutting force in milling, making it the primary factor for torque and power calculations.
- Proper calibration of force coefficients and correct unit usage are critical to prevent large errors, especially when using tabulated values for new materials or tools.
- Shop calculators like Availzye Machinist Pro automate force and torque computations, ensuring consistent, reliable data aligned with actual tool and job parameters.
Table of Contents
- Which cutting force formulas actually matter?
- Tangential, feed, and radial force: which one do you need?
- Worked examples: milling torque and a sheet-metal shear check
- Common unit mistakes that wreck a cutting force calculation
- How to calibrate Ks or kc and where the models break down
- Picking a formula without overthinking it
- Benchmark your shop's numbers against ours
- FAQ
- Sources
Which cutting force formulas actually matter?
Three mathematical forms cover nearly every real-world cutting force calculation, and each one answers a slightly different question.
The Kienzle equation is the standard for quick estimation with published material coefficients: Fc = kc1.1 · b · h^(1-mc). Here, kc1.1 is the specific cutting force for a reference 1 mm by 1 mm chip, b is the width of cut, h is the uncut chip thickness, and mc is the size exponent that accounts for the fact that thinner chips require disproportionately more specific pressure to shear. For a typical steel with kc1.1 around 1,500 N/mm² and mc of 0.26, a 0.2 mm chip produces an effective cutting pressure roughly 52% higher than the base table value, a direct consequence of the size effect built into the Kienzle-derived cutting force analysis.
The mechanistic model treats force as proportional to the instantaneous uncut chip area: Ft = Ks · Ac, where Ac = b · h. The specific cutting force coefficient Ks is not a universal constant. It has to be identified experimentally for each tool and material combination, which the mechanistic force model comparison describes as the standard identification route in modern milling force modeling.
Merchant-style orthogonal relations sit underneath both approaches, linking shear angle, friction angle, and rake angle to the force coefficients themselves. They explain why kc and Ks shift with tool geometry rather than giving you a new number to plug in directly.
- Kienzle: fast, table-driven, ideal for first-pass estimates.
- Mechanistic: calibration-driven, better suited to process optimization.
- Merchant relations: the physical reasoning behind why coefficients move with geometry.
Tangential, feed, and radial force: which one do you need?
Every cutting force splits into three components, and picking the right one matters more than getting a perfect number.
Tangential force (Ft, sometimes written Fc) acts along the direction of cutting velocity and does the actual material removal work. Feed force (Ff) acts along the feed direction and resists table or axis motion. Radial or passive force (Fr or Fp) pushes the tool away from the workpiece and loads the spindle bearings and fixture rather than doing cutting work.
In milling, tangential force typically accounts for about 70% of the total, feed force about 20%, and radial force about 10%, according to Modern Machine Shop's breakdown of milling forces. That split shifts with small depths of cut or unusual tool geometries, where edge effects and ploughing can inflate the radial share.
The practical rule: tangential force drives torque and power calculations, while feed and radial components matter most for fixture rigidity and deflection checks.
A sourced figure worth remembering: tangential force carries roughly 70% of total cutting force in typical milling, which is why torque and power formulas use it almost exclusively.
- Use tangential force for spindle torque, power draw, and tool life estimates.
- Use feed and radial force together for workholding and fixture rigidity checks.
Worked examples: milling torque and a sheet-metal shear check
Numbers make the formulas concrete. Here are two examples that show the full path from chip geometry to a usable result.
- Milling: chip area to torque and power. Say you are milling steel with a feed per tooth of 0.1 mm, an axial width of cut of 5 mm, and a calibrated Ks of 2,000 N/mm². The instantaneous chip area Ac = 0.1 mm × 5 mm = 0.5 mm². Tangential force Ft = Ks · Ac = 2,000 × 0.5 = 1,000 N. For a tool radius R of 0.01 m, spindle torque Ts = Ft · R = 1,000 × 0.01 = 10 N·m. Converting to power with a cutting speed Vc of 100 ft/min gives approximate power Ps = Ft · Vc / 33,000 in customary units. This relationship CTE Magazine ties directly to tangential force as the primary input for torque and power sizing.
- Sheet metal shear check. For a straight shear cut, force approximates length of cut (L) times material thickness (t) times shear strength, a simpler relation than Kienzle but useful for quick blanking estimates. When the chip or sheared slice is thin enough that size effects dominate, the Kienzle form with a higher mc exponent captures the disproportionate force increase that a flat shear formula misses.
| Conversion | Factor |
|---|---|
| Newtons to pound-force | 1 N = 0.2248 lbf |
| Millimeters to inches | 1 mm = 0.03937 in |
| Square millimeters to square inches | 1 mm² = 0.00155 in² |
Measurement uncertainty enters mainly through the Ks or kc coefficient itself. A dynamometer reading with poor sampling or an uncalibrated test geometry can shift that coefficient by a meaningful margin, which propagates linearly into every downstream force, torque, and power number. Running the inputs through a dedicated power and torque calculator removes the manual unit-juggling step where most of these errors creep in.
Common unit mistakes that wreck a cutting force calculation
Kienzle's tabulated kc1.1 values are not dimensionally neutral. They assume millimeter inputs, and feeding inch-based chip thickness into the same equation produces a result that is wrong by a large margin rather than just slightly off, a caveat laid out in Sirris's notes on Kienzle's formula. Before trusting any output, check that b and h are both in millimeters when using a tabulated kc1.1.
A quick dimension check: if Fc comes out in the tens of thousands of newtons for a light cut, the units are almost certainly mixed. Converting after the fact (N to lbf at 0.2248, mm to inches at 0.03937) is safe; converting mid-calculation is where errors hide.
- Confirm b and h are in millimeters before applying a tabulated kc1.1 value.
- Recheck any cutting force output that seems an order of magnitude off.
- Apply tabulated kc values only when rake angle and edge condition roughly match the table's test conditions.
- Account for edge-ploughing at very small chip thickness, where the cutting edge rubs more than it shears.
Pro Tip: Run every new material or tool combination through a sanity check at two different feed rates before trusting the output for production planning.
Using an average tabulated coefficient for an unusual alloy, or ignoring chip-thinning at low radial engagement, are the two mistakes that show up most often on the shop floor. A chip-thinning calculator handles the geometry correction automatically rather than leaving it to a rough estimate.
How to calibrate Ks or kc and where the models break down
Tabulated values get you a starting estimate. Calibration gets you a number you can actually trust for process planning.
- Cut a series of full-immersion slots at several different feed rates, keeping tool, material, and depth of cut constant.
- Measure mean cutting force at each feed rate with a dynamometer.
- Run a linear regression of mean force against feed per tooth, a method the mechanistic force model comparison uses to extract the slope and intercept that define the specific cutting coefficients.
- Repeat for a second depth of cut to confirm the coefficients hold outside a single test condition.
Dynamometer selection, sampling bandwidth, and test consistency all affect how trustworthy the resulting coefficients are. Poor calibration does not just shift a single force number. It propagates into chatter stability predictions, where an inaccurate Ks skews the entire stability lobe diagram.
Empirical models also have known failure zones. Micromilling and precision milling introduce size-dependent effects and force-displacement interactions that generic tabulations do not capture well according to a recent review of cutting force and vibration interactions. Strong edge-ploughing at very small chip thickness, extreme cutting speeds with significant thermal softening, and interrupted cuts with repeated impact loading all push standard coefficients outside their valid range.
Picking a formula without overthinking it
Most shops do not need lab-grade calibration for every job. If you are roughing a common alloy with a standard tool, tabulated Kienzle values get you close enough to size a spindle or check a fixture. Calibration earns its cost when the part runs in volume, when tolerances are tight, or when chatter has already cost you a tool or a scrapped part.
Tool geometry matters more than most people give it credit for. A more negative rake angle raises cutting force and power draw while improving edge strength on hard materials, a tradeoff worth checking against rake angle effects in machining before assuming a tabulated coefficient still applies. Drill geometry changes the shear angle the same way, something Teng Tools' guide to drill bit angles covers from a pure tooling angle.
We built our power and torque and chip-thinning calculators inside Availzye Machinist Pro specifically to take the manual unit-conversion risk out of this process, so the formula does the work and you check the result rather than redo the arithmetic.
— Availzye
Benchmark your shop's numbers against ours
We built Availzye Machinist Pro around the exact calculations in this article: Feeds & Speeds, Power & Torque, Chip Thinning, Tool Deflection, and Chatter Prediction all run the same formulas you just worked through by hand, synced to your tool and job data so you never retype a coefficient twice.
Where a spreadsheet or a paper chart forces you to track kc values, unit conversions, and chip geometry separately, our calculators keep them tied to your actual tool crib and job tracker, so a torque check and a tool life log pull from the same numbers. That matters most when you are running the same material across different jobs and need consistent coefficients rather than a fresh lookup every time.
Start with the Individual plan at 9.99 CAD per month, move up to Small Shop at 24.99 CAD per month as your team grows, or go straight to the Team plan at 49.99 CAD per month for full collaboration features. Every tier includes a 7-day free trial, so you can run your own cutting force numbers against ours before committing.
FAQ
How to calculate cutting force?
Use the Kienzle equation Fc = kc1.1 · b · h^(1-mc) for a quick estimate with tabulated coefficients, or the mechanistic model Ft = Ks · Ac for a calibrated result based on measured force data. Both require width of cut, chip thickness, and a material-specific coefficient as inputs.
What is the SFM formula?
Surface feet per minute (SFM) describes cutting speed, calculated as SFM = (pi × tool diameter in inches × spindle RPM) divided by a constant factor. It is not a cutting force formula itself, but cutting speed feeds into power calculations like Ps ≈ Ft × Vc / 33,000.
How to calculate force formula?
Cutting force is calculated from chip geometry and a specific cutting force coefficient: either Fc = kc1.1 · b · h^(1-mc) for Kienzle-based estimates or Ft = Ks · Ac for a mechanistic, calibration-based estimate where Ac is the uncut chip area. The coefficient (kc or Ks) must match the tool and material pair being cut.
How to calculate cutting load?
Cutting load generally refers to the combined tangential, feed, and radial forces acting on a tool during a cut, with tangential force typically making up about 70% of the total in milling. Convert tangential force to torque with Ts = Ft × R and to power with Ps ≈ Ft × Vc / 33,000 to size a spindle or motor against the expected load.
Sources
- Kienzle-derived cutting force analysis
- Mechanistic force model comparison (MTRC, UTK)
- A new milling 101: milling forces and formulas (Modern Machine Shop)
- Understanding tangential cutting force when milling (CTE Magazine)
- Cutting force—vibration interactions in precise and micromilling processes: a critical review