Chip Shear Angle
Calculate the shear-plane angle in chip formation, φ = arctan[(r_c·cos α) ÷ (1 − r_c·sin α)], from the cutting ratio r_c (undeformed chip thickness ÷ deformed chip thickness, always < 1) and the tool rake angle α (degrees). In the orthogonal cutting model (the basis of machining theory), material is not 'scraped': it undergoes intense SHEAR deformation along an inclined plane — the shear plane — where it turns from part to chip almost instantly. That plane's angle, φ, is a central measure of cutting mechanics: LARGER shear angles mean thinner chips, less deformation, lower cutting force and energy and less heat — all desirable. The angle depends on the cutting ratio (measured by comparing chip thickness to feed) and the tool rake angle: tools with more positive rake give larger shear angles and cut with less effort (but have a more fragile edge). Merchant's theory relates φ to chip-tool friction and rake angle, and predicts the angle that minimizes energy. From chip measurements, this calculation lets you analyze cutting efficiency and the influence of tool geometry and lubrication. Enter the cutting ratio and the rake angle.
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Chip shear angle
The shear plane angle in chip formation is φ = arctan[(r_c·cos α) ÷ (1 − r_c·sin α)], obtained from the cutting ratio r_c (undeformed chip thickness ÷ deformed chip thickness, always less than 1) and the rake angle α of the tool. In the orthogonal cutting model — the foundation of machining theory — the material is not simply 'scraped' away: it undergoes intense shear deformation along an inclined plane, the shear plane, where it turns from workpiece into chip almost instantaneously. The angle of that plane, φ, is a central measure in the mechanics of cutting: larger shear angles mean thinner chips, less deformation, lower cutting force and energy and less heat generated — all of which are desirable. The angle depends on the cutting ratio (measured by comparing the thickness of the chip with the feed) and on the rake angle of the tool: tools with a more positive rake angle produce larger shear angles and cut with less effort, but their cutting edge is more fragile and prone to chipping. Merchant's theory relates φ to the chip-tool friction and to the rake angle, and predicts the angle that minimises the energy of the cut. Starting from measurements taken on the collected chip, this calculation lets you analyse the efficiency of the cut and the influence of tool geometry and of lubrication, which raises φ by reducing friction. Enter the cutting ratio and the rake angle.
Related Tools
Cutting Force by the Kienzle Equation
Computes the main cutting force with the Kienzle equation, F_c = k_c1.1 · b · h^(1 − m_c), where k_c1.1 is the tabulated specific cutting force of the workpiece material for a reference chip section of 1 mm × 1 mm, b is the chip width and h the chip thickness, and m_c is the exponent describing the size effect. That is exactly where it differs from the direct calculation F_c = k_s·b·h: the latter treats specific pressure as a material constant, while Kienzle embeds the experimental fact that thin chips cost far more force per unit area, because the cutting edge radius stops being negligible next to the chip thickness. With k_c1.1 = 1500 N/mm² and m_c = 0.26, a 0.2 mm thick chip works at 2279 N/mm², 52 % above the tabulated value — which is why very low feeds raise the power spent per cubic millimetre removed, and the tool wear with it, instead of saving them — even though the absolute force falls. Since k_c1.1 carries a hidden millimetre raised to m_c, the equation is not dimensionally pure: thickness and width have to be entered in millimetres, and switching units is off by orders of magnitude. Enter the specific force k_c1.1, the exponent m_c, the chip width and the chip thickness.
Machining Cutting Power
Calculate the cutting power in machining, P_c = (F_c·Vc) ÷ 60000, from the main cutting force F_c (N) and the cutting speed Vc (m/min); the result is in kW (60000 converts N·m/min to kW). Cutting power is the mechanical power the operation consumes to remove material, decisive for machine selection: the spindle motor must supply this power (plus losses, dividing by drive efficiency, typically 0.7-0.9) without stalling in the cut. If the required power exceeds the available, the machine loses speed, the cut jams or the tool breaks — so heavy roughing needs robust machines. Cutting power also relates to MRR by the specific cutting energy (P_c = u·Q, where u is energy per unit volume removed) — a practical alternative to estimate it directly from removed volume. Computing power is essential for planning (choosing the right machine), optimizing parameters (extracting the most from available power) and estimating energy use and heating. Enter the cutting force and the cutting speed.
Tool Life (Taylor's Equation)
Calculate a cutting tool's life by Taylor's equation, T = (C ÷ Vc)^(1/n), from the constant C (the cutting speed giving 1 minute of life, characteristic of the tool-material pair), the cutting speed Vc (m/min) and the exponent n (depending on tool material). Formulated by F. W. Taylor in 1907 from thousands of tests, this is machining's most famous relation and describes a fundamental trade-off: the HIGHER the cutting speed, the SHORTER the tool life — and steeply, since it is a power law. The exponent n quantifies the sensitivity: for HSS n ≈ 0.1 (life drops very fast with speed), for carbide n ≈ 0.2-0.3, for ceramic n ≈ 0.4-0.6 (less sensitive, allowing much higher speeds). Taylor's equation is the basis of economic OPTIMIZATION of machining: there is an optimal cutting speed minimizing total cost per part, balancing cutting time (falling with speed) against tool and change-downtime cost (rising with speed). Speeds above optimum 'burn' costly tools too fast; below, waste machine time. Enter the constant C, the cutting speed and the exponent n.
Bolt Shear Stress
Calculate the shear stress in transversely loaded bolts, τ = F ÷ (n·A), from the total shear force F (N), the number of bolts (or shear planes) n and each bolt's area A (mm²). Unlike tensioned joints (where the bolt is tightened and the load is axial), in SHEAR joints the bolts resist a transverse force tending to slide one part over another (as in steel structural connections, splice plates, flanges under lateral load). The force is distributed among the bolts and each works in shear — hence the stress is force divided by the number of bolts times the area. There can be SINGLE shear (one shear plane) or DOUBLE shear (two planes, when the bolt passes through three plates), doubling capacity. The area used depends on whether the shear plane passes through the threaded part (use the tensile area) or the smooth shank (nominal-diameter area). Shear stress is compared with the bolt material's shear strength (typically ~0.6 of tensile strength). In structures, bearing-type (bolt in shear/bearing) and slip-critical (preload friction transmits load without bolt shear) connections are distinguished — this formula covers shear resistance. Enter the shear force, the number of bolts and the area.
Feed per Tooth (Milling)
Calculate the feed per tooth in milling, f_z = v_f ÷ (z·n), from the table feed rate v_f (mm/min), the number of cutter teeth (cutting edges) z and the rotation n (rpm). Feed per tooth is the material thickness EACH cutter tooth removes per pass through the part, and it directly controls chip thickness, the load on each edge and thus tool life and finish. Makers specify a recommended feed per tooth for each tool-material pair: too HIGH overloads and chips the teeth (chip too thick); too LOW makes the edge rub instead of cut, causing friction, heat and premature wear, plus low productivity. The relation shows how the table feed rate (programmed by the operator) connects to feed per tooth (the cutting physics): v_f = f_z·z·n. So cutters with more teeth allow higher feed rates at the same feed per tooth — the basis of high-productivity milling. Enter the feed rate, the number of teeth and the rotation.
Chip Thinning Corrected Feed per Tooth (Milling)
Computes the feed per tooth corrected for radial chip thinning in milling, f_z,corr = f_z / sin(φ_max), where sin(φ_max) = √(1 − (1 − 2·a_e/D)²) as long as the radial depth of cut a_e is less than half the cutter diameter D, and equals 1 above that. When the cutter engages little material sideways, each tooth enters and leaves the cut before reaching the point of maximum thickness, and the chip actually formed is THINNER than the programmed feed — the edge starts rubbing instead of cutting, generates heat, work-hardens the surface and wears fast, which is why light finishing passes often destroy tools quicker than heavy roughing. Correcting the feed restores the catalogue chip thickness: with a 12 mm cutter engaging only 1.2 mm, or 10 % of the diameter, the feed must rise 67 % for the tooth to cut at the intended thickness. Above half the diameter there is no thinning and the correction is neutral. Enter the target feed per tooth, the cutter diameter and the radial depth of cut.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.