🧮Calculators

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Gear Base Diameter

Calculate the base circle diameter of an involute gear, d_b = d·cos(φ), from the pitch diameter d (mm) and the pressure angle φ (degrees). The base circle is the circle from which the INVOLUTE tooth profile is generated — the standard profile of modern gears. The involute is the curve traced by the tip of a string unwinding from a cylinder: that cylinder is exactly the base circle. The entire active tooth profile (the part that actually transmits force) is ABOVE the base circle; below it there is no involute profile. The base diameter is fundamental in gear geometry because it defines the involute profile and, with it, key properties: the LINE OF ACTION (the line tangent to both base circles of the mesh, along which tooth contact travels, always in the same direction — why involute gears transmit uniform motion), the base pitch and the contact ratio. The relation d_b = d·cos(φ) shows that the pressure angle is the angle between the line of action and the tangent to the pitch circles. It is an essential parameter in designing and manufacturing (generating) involute gears. Enter the pitch diameter and the pressure angle.

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Tooth Thickness at Pitch Circle

Calculate the tooth thickness measured at the pitch circle of a standard gear, s = (π·m) ÷ 2, from the module m (mm). In a standard (uncorrected) gear, the circular pitch (the distance from one tooth to the next, along the pitch circle) is p = π·m, and it splits equally between the TOOTH (the solid part) and the SPACE (the gap between teeth): half each, hence s = π·m/2. This equality between tooth thickness and space width is what lets two standard gears of the same module mesh perfectly, with one's tooth fitting the other's space with proper clearance. Tooth thickness is a fundamental parameter: it sets the tooth STRENGTH (thicker teeth resist bending more) and the mesh backlash. In CORRECTED gears (with profile shift, used to avoid interference in small pinions, adjust center distance or balance pinion-gear strength), the pitch-circle tooth thickness DIFFERS from π·m/2 — it increases in a positively corrected pinion (strengthening it) and decreases in the gear. Measuring tooth thickness (by the chordal method, with a gear-tooth caliper, or over pins) is a classic gear quality-control check. Enter the module.

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Gear Torque

Calculate the torque transmitted by a gear, T = (F_t·d) ÷ 2000, from the tangential force F_t (N) and the pitch diameter d (mm); the result is in N·m (the 2000 converts d/2 from mm to m). Torque is the moment the gear transmits about its axis, and the tangential force F_t acts at the pitch radius (d/2), creating that moment. This relation is the bridge between the POWER/torque side (what the shaft transmits) and the TOOTH-FORCE side (what sizes the strength): from shaft torque, the tooth tangential force (F_t = 2T/d) is obtained, which then feeds the bending (Lewis) and contact (Hertz) calculations. Conversely, given the tangential force, the torque is obtained. In a gear train, torque CHANGES at each stage by the gear ratio (a reduction that multiplies speed by 1/i multiplies torque by i, conserving power minus losses), while power stays roughly constant. So a reducer's last stage (low speed) transmits the HIGHEST torque and needs the most robust gears. Knowing the torque at each gear is essential to size teeth, shafts, keys and bearings. Enter the tangential force and the pitch diameter.

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Bolt Preload

Calculate the recommended preload (initial clamping force) of a bolt, F_i = 0.75·A_t·S_p, from the bolt tensile stress area A_t (mm²) and the material proof strength S_p (MPa); the result is the tensile force installed in the bolt on tightening. Preload is perhaps the MOST important and most misunderstood concept in bolted joints: a well-designed bolt is tightened to be strongly TENSIONED (stretched), clamping the joined parts together. This clamping force keeps the joint tight and, counterintuitively, PROTECTS the bolt from fatigue. The 0.75·A_t·S_p value (75% of proof load) is the classic recommendation for NON-permanent (reusable) joints; permanent joints use 0.90·A_t·S_p. A HIGH preload is desirable because it: keeps the joint together under varying external load; prevents loosening from vibration; and, mainly, makes an external tensile load be absorbed mostly by DECOMPRESSION of the (stiff) parts rather than additional bolt stretch — so the bolt stress variation (which causes fatigue) is very small. That is why well-tightened bolts rarely fail by fatigue, and loose bolts fail. Enter the tensile area and the proof strength.

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Bolt Tensile Stress Area (Metric)

Calculate the tensile stress area of a metric-thread bolt, A_t = (π/4)·(d − 0.9382·p)², from the nominal diameter d (mm) and the thread pitch p (mm). The tensile stress area is the EFFECTIVE cross-section resisting tension in a threaded bolt — and it is NOT the nominal-diameter area (the smooth cylinder) nor the root-diameter area (the thread bottom). Because of the helical thread geometry, tensile rupture occurs at an intermediate section, and tests showed it corresponds to an effective diameter equal to the average of the pitch and root diameters, leading to the formula with the 0.9382·p term (a geometric constant of the ISO metric thread, 60° triangular profile). The tensile area is the fundamental parameter for all bolt strength calculations: preload, tensile stress, proof load and ultimate strength are all found by multiplying A_t by the corresponding material stress. Using the wrong area (the larger nominal-diameter one) would overestimate strength and lead to undersized joints. Bolt tables list A_t for each diameter-pitch combination; this formula computes it for any metric thread. Enter the nominal diameter and the thread pitch.

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Bolt Stiffness

Calculate a bolt's stiffness (spring constant), k_b = (A_t·E) ÷ L, from the tensile area A_t (mm²), the material elastic modulus E (MPa) and the grip length L (mm, the effective length under tension between head and nut). When tensioned by the preload, the bolt behaves as a very stiff SPRING: it stretches an amount proportional to the force (Hooke's law), and its stiffness is force per unit elongation. This stiffness is one of two essential ingredients of bolted-joint analysis — the other is the stiffness of the clamped PARTS (members). The ratio between these two stiffnesses (the joint stiffness constant C) determines how an external load splits between the bolt and the parts. Typically the parts (massive, with large effective compression area) are MUCH stiffer than the bolt (thin and long), which is the DESIRED situation: stiff parts absorb most of the external load, protecting the bolt from stress variation and fatigue. Long, thin bolts have low stiffness (good for sharing load), while short, thick bolts are stiff. Knowing k_b is the starting point of fatigue and joint-separation analysis. Enter the tensile area, elastic modulus and grip length.

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Joint Stiffness Constant

Calculate a bolted joint's stiffness constant, C = k_b ÷ (k_b + k_m), from the bolt stiffness k_b (N/mm) and the members' (clamped parts) stiffness k_m (N/mm). The constant C (also called bolt load fraction) is the heart of bolted-joint analysis: it tells what FRACTION of an external tensile load is carried by the BOLT, the rest (1−C) being carried by decompression of the MEMBERS. The value of C reveals the elegant, protective behavior of a preloaded joint: since the (massive) members are usually much stiffer than the (thin) bolt, k_m >> k_b, so C is SMALL (typically 0.2-0.4). This means that when an external load P is applied, only a small portion C·P adds to the bolt tension — most of the load (1−C)·P is absorbed by RELIEF of the compression between the parts. That is why the bolt stress variation is small (good fatigue resistance) and why preload is so beneficial. The smaller C (stiff parts, flexible bolt), the better the bolt protection. The constant C appears in all subsequent formulas: bolt load, residual member force, separation load and fatigue safety factor. Enter the bolt stiffness and the members' stiffness.

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Bolt Load under External Load

Calculate the total tensile force in the bolt when an external load is applied to the joint, F_b = F_i + C·P, from the preload F_i (N), the joint stiffness constant C and the external tensile load P (N). This is one of the most important — and most surprising to the uninitiated — relations of bolted joints: when you apply an external load P trying to 'separate' the parts, the bolt tension does NOT rise from F_i to F_i + P (as intuition suggests), but only to F_i + C·P, where C is typically 0.2-0.4. That is, the bolt only 'feels' a FRACTION of the external load! The reason: most of the external load (1−C)·P merely RELIEVES the compression between the parts (which were compressed by the preload), rather than stretching the bolt more. This is the genius of the preloaded joint — it 'hides' the external load from the bolt. So a well-tightened joint, under a CYCLIC external load (causing fatigue), exposes the bolt to a very small stress variation (proportional to C·ΔP, not ΔP), making it extremely fatigue-resistant. This formula holds while the joint does NOT separate (P below the separation load); above that, the bolt carries the whole load. Enter the preload, stiffness constant and external load.

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Joint Separation Load

Calculate the external load that causes a bolted joint to separate (open), P_0 = F_i ÷ (1 − C), from the preload F_i (N) and the joint stiffness constant C. The separation load is the external tensile load at which the compression between the clamped parts fully vanishes — the point where the joint starts to OPEN. Below it, the parts stay compressed and the joint behaves 'smartly' (the bolt feels only C·P of the external load, with small stress variation); ABOVE it, the parts separate, and from then on ALL additional external load goes straight to the bolt (which then takes the whole load, with severe fatigue and failure risk). Joint separation is thus a condition the design must AVOID with margin: a safety factor against separation is applied (the separation load must be well above the maximum expected external load). The formula shows the separation load grows with preload (well-tightened joints separate later) — another reason to use high preloads. Separation also causes leaks (in sealed joints), loss of stiffness and loosening. Ensuring the joint never separates under service load is a fundamental bolted-joint design criterion. Enter the preload and the stiffness constant.

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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.

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Bolt Tensile Stress

Calculate the tensile stress in a bolt, σ = F_b ÷ A_t, from the total bolt tensile force F_b (N) and the tensile stress area A_t (mm²). It is the basic strength check of a tensioned bolt: the acting stress (force over resisting area) must be below the material strength with a safety margin. The force F_b is the total load the bolt carries — in a preloaded joint, the preload plus the fraction of external load reaching the bolt (F_i + C·P). The resulting stress is compared with the proof strength S_p (the limit up to which the bolt can be loaded without permanent deformation — typically 85-90% of yield) or the ultimate strength, per the criterion. The bolt strength class (marked on the head: 8.8, 10.9, 12.9 metric; or SAE grades 2, 5, 8) sets these allowable stresses — a class 8.8 bolt has a proof strength of 580-600 MPa, a 12.9 reaches ~970 MPa. Verifying σ does not exceed the allowable, considering preload and service load, is essential: overloaded bolts yield (losing preload) or break. With the fatigue and separation checks, it defines the tensioned joint's safety. Enter the total bolt force and the tensile area.

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Bolt Count for Shear

Calculate the number of bolts needed to resist a shear force, n = F ÷ (A·τ_adm), from the total shear force to transmit F (N), each bolt's area A (mm²) and the material's allowable shear stress τ_adm (MPa). In structural and mechanical connections loaded in shear (beam splices, truss connections, flanges under lateral load, splice plates), the force is distributed among several bolts, each working in shear. The number needed is the total force divided by one bolt's shear capacity (area × allowable stress). The result is rounded up, and in practice a quantity is adopted that also meets minimum bolt spacing, edge distance and connection symmetry criteria. This calculation is the basis of designing bolted connections in steel structures (where it competes with welding) and in machines: it sets how many bolts and of what diameter are needed. There are other checks in the same connection: plate BEARING (contact pressure on the hole wall, which can tear the plate before the bolt shears), edge tear-out and the plate's own net-section strength (minus the holes). But bolt shear is the starting point. Enter the shear force, each bolt's area and the allowable stress.

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Slurry Mixture Density

Calculate the slurry (water-solids mixture) density in hydraulic transport, ρ_m = ρ_w + C_v·(ρ_s − ρ_w), from the solids volumetric concentration C_v (fraction), the solids density ρ_s (kg/m³) and the water density ρ_w (kg/m³). In hydraulic transport of solids — used in dredging (pumping sand, mud and gravel from river, port and sea beds), mining (ore slurries in pipelines) and waste handling — solids are mixed with water and pumped as a SLURRY. The mixture density is the volume-fraction-weighted average of the water and solids densities, and it is the most basic and important hydraulic-transport parameter: it governs pumping power (dense slurries need more energy), head losses, and it is what is MEASURED in the field (by nuclear density gauges on the pipe) to control the solids concentration being transported. The mixture density links the dredge or pipeline operation to production: the denser the slurry (more solids per volume), the higher the production, but the higher the clogging risk and required power. Finding the optimal density balance is central to efficient hydraulic transport. Enter the volumetric concentration, the solids density and the water density.

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Slurry Volumetric Concentration

Calculate the solids volumetric concentration in a slurry, C_v = (ρ_m − ρ_w) ÷ (ρ_s − ρ_w), from the mixture density ρ_m, the solids density ρ_s and the water density ρ_w (kg/m³). Volumetric concentration is the fraction of total slurry volume occupied by solids — the fundamental hydraulic-transport parameter. It is the inverse of the mixture-density calculation: in practice the slurry density in the pipe is measured (with a nuclear gauge, measuring gamma-ray attenuation through the pipe) and, knowing the water and solid densities, the solids concentration being transported is computed in real time. Volumetric concentration defines a dredge's or pipeline's PRODUCTION (solids volume transported = flow × C_v), and it is the parameter the operator seeks to MAXIMIZE (more solids per pumped water = more production and less energy per tonne) without exceeding the limits that cause clogging or excessive wear. Typical dredging volumetric concentrations are 10-30%; in optimized pipelines, up to 40-50%. Concentration control is the heart of hydraulic-transport operation. Enter the mixture density, the solids density and the water density.

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Slurry Mass Concentration

Calculate the solids mass (weight) concentration in a slurry, C_w = C_v·(ρ_s ÷ ρ_m)·100, from the volumetric concentration C_v (fraction), the solids density ρ_s and the mixture density ρ_m (kg/m³); the result is a percentage. Mass concentration is the fraction of the total slurry MASS that is solid (kg of solid per kg of slurry), different from volumetric concentration (volume fraction). Both measure the same thing differently, and their relation depends on the solids density: since solids are DENSER than water (sand ~2.65×), mass concentration is always GREATER than volumetric (a slurry with 20% solids by volume has about 40% by mass). Mass concentration (% solids by weight) is the form most used in the mineral industry and ore processing, since it relates directly to the tonnage of solids processed and is what is controlled in thickeners, mills and flotation. Converting between mass and volumetric concentration is a daily operation in processing-plant mass balance and pipeline and dredging control. Enter the volumetric concentration, the solids density and the mixture density.

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Critical Deposition Velocity (Durand)

Calculate the critical deposition velocity in hydraulic solids transport by Durand's equation, V_c = F_L·√(2·g·D·(s − 1)), from the Durand factor F_L (dimensionless, a function of grain size and concentration), the pipe inner diameter D (m) and the solids relative density s = ρ_s/ρ_w. Critical velocity is the MOST important parameter in designing pipelines and dredge discharge lines: it is the MINIMUM flow velocity below which solids start to DEPOSIT on the pipe bottom, forming a bed that reduces the section, raises head loss and can lead to total CLOGGING of the line (a very costly, slow accident to clear). Above the critical velocity, turbulence keeps the particles suspended and moving. Operation must keep the velocity ALWAYS above critical (with safety margin), but not too far above, since excessive velocities waste pumping energy and cause accelerated abrasive wear of pipe and pumps. Determining the critical velocity sets the operating velocity, the pipe diameter and the pumping power. Durand's correlation (1953), with the tabulated F_L factor, is the classic basis of this calculation. Enter the Durand factor, the pipe diameter and the solids relative density.

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Dredge Solids Production

Calculate the volumetric solids production of a dredge or pipeline, Q_s = Q·C_v, from the total slurry flow Q (m³/s) and the solids volumetric concentration C_v (fraction). Solids production is the volume of useful material (sand, sediment, ore) effectively transported per unit time — the direct measure of dredging or slurry-pumping PRODUCTIVITY, and what really matters commercially (a dredge is paid per cubic metre dredged, not per pumped water). It is the product of the total slurry flow and the solids fraction: increasing production means increasing the flow (bigger pumps, more power) OR increasing the solids concentration (excavating denser material, optimizing the suction). There is a fundamental trade-off: pumping very concentrated slurry raises production per cubic metre of slurry but raises mixture density, head loss and deposition/clogging risk. Solids production, integrated over time, gives the total dredged volume (for measurement and payment) and frames planning (how many hours/days to dredge a channel, fill a pit, move overburden). It is the key operational indicator. Enter the slurry flow and the volumetric concentration.

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Solids Mass Flow (Dredge)

Calculate the mass flow of solids transported by a dredge or pipeline, ṁ_s = Q·C_v·ρ_s, from the total slurry flow Q (m³/s), the solids volumetric concentration C_v (fraction) and the solids density ρ_s (kg/m³). Solids mass flow is the MASS of useful material transported per unit time (kg/s, or tonnes per hour), the production indicator used when TONNAGE matters — the typical case of ore transport by pipeline (measured in t/h of dry ore) and mineral processing. It is the product of three factors: the slurry flow (pump capacity), the solids concentration (how 'loaded' the slurry is) and the solids density (iron ores, for example, are very dense, ~5000 kg/m³, so little volumetric concentration already gives high tonnage). Mass flow, integrated over time, gives the total transported tonnage, the basis of billing and operational mass balance. Optimizing it — maximizing tonnage per unit pumping energy — is the central goal of pipeline operation, which moves hundreds of millions of tonnes of ore per year over long distances far more energy-efficiently than trucks or trains. Enter the slurry flow, the volumetric concentration and the solids density.

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Settling Velocity (Stokes)

Calculate the settling (terminal) velocity of a particle in laminar regime by Stokes' Law, v_s = g·d²·(ρ_s − ρ_w) ÷ (18·μ), from the particle diameter d (m), the solids ρ_s and water ρ_w densities (kg/m³) and the fluid dynamic viscosity μ (Pa·s). The settling velocity is the speed at which an isolated particle SINKS in a still fluid, when weight (minus buoyancy) balances drag. Stokes' Law (1851) holds for the LAMINAR regime (small particles, particle Reynolds < ~1) — fine sand, silt, clay — and has the remarkable property that velocity grows with the SQUARE of diameter: particles twice as large sink four times faster. This calculation is fundamental in many fields: particle settling and separation (settling tanks, thickeners, water and effluent clarifiers), grain-size classification by sedimentation (pipette or hydrometer test), sediment transport in rivers and reservoir deposition, and hydraulic transport (the particle settling velocity sets the critical deposition velocity in the pipe). For large particles (higher Reynolds), Stokes' Law fails and Newton's terminal velocity (turbulent regime) is used. Enter the particle diameter, the densities and the viscosity.

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Particle Reynolds Number

Calculate the particle Reynolds number in settling, Re_p = (ρ_w·v_s·d) ÷ μ, from the fluid density ρ_w (kg/m³), the settling velocity v_s (m/s), the particle diameter d (m) and the dynamic viscosity μ (Pa·s). The particle Reynolds number characterizes the flow regime around a particle settling (or being transported) in a fluid, comparing inertial and viscous forces. Its value sets WHICH settling-velocity formula is valid: for Re_p < ~1, the flow around the particle is LAMINAR and Stokes' Law holds (drag proportional to velocity); for Re_p > ~1000, the flow is TURBULENT and Newton's law holds (drag proportional to velocity squared); in the intermediate range, transition correlations are used (such as Allen's or drag-coefficient expressions vs Re_p). So when computing a settling velocity by Stokes' Law, it is ESSENTIAL to verify afterwards that Re_p < 1 — if not, the Stokes result is wrong and the correct regime's formula must be used. The particle Reynolds number is thus the 'checker' that validates the settling calculation, and it is central in designing settling tanks, classifying particles and hydraulic solids transport. Enter the fluid density, the settling velocity, the particle diameter and the viscosity.

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Settling Velocity (Newton)

Calculate the settling (terminal) velocity of a particle in turbulent regime by Newton's law, v_t = √(4·g·d·(s − 1) ÷ (3·C_d)), from the particle diameter d (m), the solids relative density s = ρ_s/ρ_w and the drag coefficient C_d (dimensionless, ≈ 0.44 for spheres in turbulent regime). While Stokes' Law holds for SMALL particles (laminar regime, particle Reynolds < 1), Newton's law holds for LARGE, dense particles — gravel, crushed stone, coarse sand — that sink fast, generating TURBULENT flow around them (particle Reynolds > ~1000). In this regime, drag is no longer proportional to velocity (Stokes) but to its SQUARE, and the terminal velocity grows with the SQUARE ROOT of diameter (not the square, as in Stokes) — large particles sink fast, but the size dependence is milder. The drag coefficient C_d ≈ 0.44 is roughly constant in this range (the 'Newton region' of the sphere drag curve). This calculation is fundamental in designing coarse-particle classifiers and separators, sizing settling basins for coarse solids, coarse-sediment transport and hydraulic transport of gravel and granular ore. For the intermediate range between Stokes and Newton, transition correlations are used. Enter the diameter, the relative density and the drag coefficient.

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Edge Stress from Prestressing

Calculate the normal stress at an extreme fiber of a prestressed concrete section, σ = P/A + (P·e)/W, from the prestressing force P (MN), the section area A (m²), the tendon eccentricity e (m) and the section modulus W (m³). Prestressed concrete is one of the great structural engineering inventions of the 20th century: high-strength steel tendons are tensioned (prestressed) and anchored in the member, COMPRESSING the concrete before it even receives service loads. Since concrete is strong in compression but weak in tension, this pre-compression 'cancels' the tensions external loads would cause, allowing much longer spans and slenderer members than conventional reinforced concrete. The tendon is placed with ECCENTRICITY (below the centroid), so prestressing generates not only uniform compression (P/A) but also a moment (P·e) producing stresses opposite to the loading — compressing exactly the fiber that would tend to crack. This formula computes the resulting edge stress, summing axial compression and prestress bending; design verifies stresses stay within limits in all phases (at transfer, empty, and in service, loaded). Enter the prestressing force, area, eccentricity and section modulus.

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Elastic Shortening Loss

Calculate the prestress loss from concrete elastic shortening, Δσ = (E_s/E_c)·σ_c, from the steel modulus E_s (MPa), the concrete modulus E_c (MPa) and the concrete stress at the tendon level σ_c (MPa). It is one of the IMMEDIATE prestress losses (at transfer, not over time): when the tendon is tensioned and anchored, it compresses the concrete, and the concrete, being compressed, SHORTENS elastically. Since the tendon is bonded or anchored in this shortened concrete, it shortens too — and shortening, it LOSES part of its tension. The loss is proportional to the modular ratio αe = E_s/E_c (typically 6-8, since steel is much stiffer than concrete) times the concrete compression stress at the tendon level. In members with SEVERAL tendons prestressed sequentially, each new tendon compresses and shortens the concrete, causing loss in already-anchored tendons — so the average loss is often taken as half the value (the first tendons lose more than the last). This is one of the losses to subtract from the initial force to get the effective prestressing force. Enter the steel and concrete moduli and the concrete stress.

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Steel Relaxation Loss

Calculate the prestress loss from steel relaxation, Δσ = (ψ/100)·σ_pi, from the relaxation coefficient ψ (% of initial stress) and the initial tendon stress σ_pi (MPa). Relaxation is a STEEL phenomenon analogous to concrete creep: when a steel wire or strand is held under CONSTANT tension (fixed elongation, as in an anchored prestressing tendon), its stress DECREASES slowly over time, even without length change. It is as if the steel 'yields' microscopically under prolonged load, losing part of its tension. Relaxation depends on the steel type (LOW-relaxation steels — LR —, thermomechanically treated, relax much less, ~2.5% in 1000h at 0.7·fptk, than normal-relaxation — NR —, ~12%), the initial stress level (the higher, the more relaxation) and temperature. The coefficient ψ is tabulated as a function of these factors and time. Relaxation is one of the TIME-DEPENDENT prestress losses, along with concrete shrinkage and creep, and design sums them all for the total loss and the effective final prestressing force. Enter the relaxation coefficient and the initial tendon stress.

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Concrete Shrinkage Loss

Calculate the prestress loss from concrete shrinkage, Δσ = ε_cs·E_s, from the shrinkage strain ε_cs (dimensionless) and the steel modulus E_s (MPa). Shrinkage is the volume reduction concrete undergoes over time as it LOSES water by evaporation (drying shrinkage) and through cement hydration reactions (autogenous shrinkage), independent of loading. When the concrete of a prestressed member shrinks (shortens), the bonded steel tendon shortens too — and shortening, it LOSES tension, exactly as in elastic-shortening loss, except here the shortening is from shrinkage and occurs SLOWLY over months and years. The loss is simply the shrinkage strain times the steel modulus (the stress that shortening 'steals' from the tendon). The shrinkage strain ε_cs is typically 0.0002-0.0005 (200-500 microstrains) and depends on ambient humidity (drier = more shrinkage), member dimensions (thin members shrink more, losing water faster), mix and time. It is one of the three time-dependent losses (with creep and relaxation) reducing prestress over the structure's life. Enter the shrinkage strain and the steel modulus.

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Concrete Creep Loss

Calculate the prestress loss from concrete creep, Δσ = φ·(E_s/E_c)·σ_cg, from the creep coefficient φ (dimensionless), the modular ratio E_s/E_c and the concrete stress at the tendon level from permanent loads σ_cg (MPa). Creep is the SLOW, growing deformation concrete undergoes under CONSTANT load over time: besides the immediate elastic shortening when compressed, concrete keeps shortening gradually for months and years, reaching a total deformation 2-3 times the initial elastic one. In a prestressed member, the concrete is PERMANENTLY compressed by the prestress, so it creeps (shortens slowly), and the bonded tendon shortens with it, LOSING tension — the largest time-dependent loss in many cases. The loss is the creep coefficient φ (typically 1.5-3.5, a function of humidity, loading age, member dimensions) times the equivalent elastic loss (modular ratio × concrete stress). With shrinkage and relaxation, creep defines the total time-dependent prestress loss. Estimating these losses well is crucial: underestimating leaves the member with less prestress than intended (cracking risk); overestimating wastes steel. Enter the creep coefficient, the modular ratio and the concrete stress.

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Prestress Friction Loss

Calculate the prestress force loss from friction along a curved tendon, ΔP = P_0·(1 − e^(−(μα + k·x))), from the jacking force P_0 (kN), the tendon-duct friction coefficient μ, the sum of tendon deviation angles α (radians), the wobble coefficient k (loss per metre, 1/m) and the tendon length x (m). In POST-TENSIONING (where the tendon is tensioned after the concrete hardens, sliding inside a duct embedded in the member), the force applied at the end by the jack does NOT arrive full at the other end: FRICTION between tendon and duct consumes part of it along the path. There are two effects: friction in the tendon CURVES (μα term — the more the tendon curves, the more it 'squeezes' the duct and the greater the friction, like a rope on a pulley — the capstan effect) and 'wobble' friction in straight runs (k·x term — from small undulations and duct misalignment). Friction loss makes the prestress force DECREASE progressively from the active end (jack) to the passive (dead anchorage), which is why long tendons are sometimes tensioned from BOTH ends. It is an immediate loss, computed tendon by tendon. Enter the jacking force, friction coefficient, sum of angles, wobble coefficient and length.

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Final Prestressing Force

Calculate the final (effective) prestressing force after losses, P_∞ = P_0·(1 − losses/100), from the initial prestressing force P_0 (kN) and the total loss percentage (%). A member's prestressing force is NOT constant: it starts at an initial value (the jacking force) and DECREASES due to the various losses — immediate (elastic shortening, friction, anchorage set) and time-dependent (concrete shrinkage and creep, steel relaxation). The effective final force, after all losses stabilize (after years), is what actually acts in the structure in service and must ensure performance. Total losses typically sum 15-25% of the initial force in post-tensioned structures and can reach 20-30% in pretensioned ones. This simple calculation applies the total loss percentage to the initial force, giving the effective force — fundamental to check service stresses, cracking and member deflection. The designer works with TWO critical situations: maximum INITIAL force (right after transfer, member still unloaded — risk of excess compression or top-fiber tension) and minimum FINAL force (after all losses, member loaded — risk of decompression and cracking). Enter the initial force and the total loss percentage.

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Kern Distance

Calculate the kern distance of a section, c = W/A, from the section modulus W (m³) and the section area A (m²). The kern is a central region of the cross-section with a remarkable property: if a COMPRESSION force (like prestress, or a column load) is applied WITHIN the kern, the whole section stays compressed (no fiber tensions); if the force leaves the kern, tensions appear on the opposite edge. The kern distance is the boundary: for a rectangular section, the kern is the famous 'middle third' (the force must fall in the central third of the height to avoid tension). The distance c = W/A defines how far the eccentricity can go while keeping the section fully compressed. This concept is central in three areas: in PRESTRESSING (the tendon eccentricity is chosen considering the kern, to control edge stresses in each loading phase), in FOUNDATIONS and COLUMNS (the load resultant must fall in the kern so the base does not 'lift off' the soil, avoiding tension — the middle-third rule for footings), and in gravity wall and dam stability. Knowing the kern is essential for tendon placement and stress checks in eccentrically compressed members. Enter the section modulus and the section area.

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Initial Prestress Stress

Calculate the allowable initial stress in prestressing steel, σ_pi = coef·f_ptk, from the code coefficient (fraction of strength) and the steel characteristic tensile strength f_ptk (MPa). Prestressing steel is tensioned to a VERY HIGH stress — a significant fraction of its tensile strength — possible because these are HIGH-STRENGTH steels (strands with f_ptk of 1900 MPa, versus ~500 MPa for ordinary reinforcing steel). But there is a LIMIT to the initial stress, set by code for safety and to limit relaxation: typically the lesser of about 0.74·f_ptk and 0.82·f_pyk (yield strength) for low-relaxation steels in pretensioning, with slightly different values for post-tensioning and right after anchorage. Applying a high initial stress is DESIRABLE (more effective prestress, less steel needed), but the code limit prevents tensioning the steel too close to yield (which would reduce safety margin and greatly increase relaxation). This calculation gives the jacking stress to apply (before losses), the starting point of all prestress design. Enter the code coefficient and the steel characteristic strength.

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Pile Bearing Capacity

Calculate a pile's ultimate bearing capacity, Q_ult = Q_p + Q_l, summing the point (tip) resistance Q_p (kN) and the side (skin friction) resistance Q_l (kN). The pile is the DEEP foundation element used when surface soil lacks capacity for the structure's loads — it transfers loads to deeper, stronger subsoil layers. This transfer occurs by TWO mechanisms acting at once: TIP resistance (the pile bears on a firm layer at its base, like a column, mobilizing the soil resistance under the tip) and SIDE resistance (friction and adhesion between the pile's lateral surface and surrounding soil, along its whole length). Their proportion defines the behavior: END-bearing piles (crossing soft soil to bear on rock or firm soil) work mainly by the tip; FLOATING or friction piles (driven in homogeneous soil, no firm layer) work mainly by side friction. The ultimate capacity, divided by a safety factor (typically 2), gives the design allowable load. Determining Q_p and Q_l — by theoretical formulas, SPT-based semi-empirical methods (Aoki-Velloso, Décourt-Quaresma) or load tests — is the central deep-foundation design calculation. Enter the tip resistance and the side resistance.

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Pile Tip Resistance

Calculate a pile's tip resistance, Q_p = q_p·A_p, from the tip stress (bearing capacity) q_p (kPa) and the tip cross-sectional area A_p (m²). Tip resistance is the share of pile capacity from the BEARING of its base on a strong soil or rock layer — the pile acts as a column compressing the soil under its tip, mobilizing that soil's bearing capacity (like a shallow foundation, but at depth). The tip stress q_p is the soil's unit bearing capacity at the tip elevation, estimated by bearing-capacity theories (Terzaghi, Meyerhof, Vesic for piles), SPT correlations (q_p = K·N, with K depending on soil and pile type) or the CPT (cone) test. Times the tip area, it gives the force the tip supports. Tip resistance dominates in piles reaching a firm layer (end-bearing piles), and then the pile is very stiff (settles little). Large-diameter piles (caissons) have large tip areas and mobilize high tip resistance. This share adds to the side resistance for the total capacity. Enter the tip stress and the tip area.

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Pile Skin Resistance

Calculate a pile's side (friction) resistance, Q_l = f_s·A_s, from the average unit skin friction f_s (kPa) and the pile lateral surface area A_s (m², = π·D·L for a cylindrical pile). Side resistance is the share of pile capacity from FRICTION and ADHESION between the pile's lateral surface and the surrounding soil, along its whole buried length. As the pile tends to settle under load, the soil 'grips' its sides and resists — as a nail driven in wood resists pulling by face friction. The unit skin friction f_s depends on soil type (in clays, on undrained cohesion via the α method; in sands, on effective stress and friction via the β method), pile type (driven piles mobilize more friction than bored, displacing and compacting the soil) and surface roughness. Side resistance dominates in FLOATING (friction) piles, driven in soils without a firm bearing layer — they hang by friction. It is also the share mobilized FIRST under load (with small settlement), before the tip. This share adds to the tip resistance for the total capacity. Enter the unit skin friction and the side area.

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Unit Skin Friction (Beta Method)

Calculate a pile's unit skin friction in granular soil by the beta method, f_s = β·σ'_v, from the coefficient β (dimensionless) and the vertical effective stress σ'_v at the considered point (kPa). The β method (effective-stress method) is the modern, rational way to estimate pile skin friction in GRANULAR soils (sands) and in clays in effective-stress terms. It starts from the principle that side friction is like any interface friction: the friction stress is the NORMAL stress to the surface (the soil horizontal stress, K_s·σ'_v) times the tangent of the interface friction angle (tan δ). Grouping these two factors into a single coefficient β = K_s·tan δ, the unit friction is simply β·σ'_v. The coefficient β typically ranges 0.2-0.5 for sands (and more for driven piles, which raise K_s by displacing soil). The great advantage of the β method is using EFFECTIVE stress (growing with depth), capturing that friction increases with depth — though there is a limit (the 'critical depth', above which friction stops growing, a still-debated phenomenon). Integrating f_s·perimeter along the length gives the total side resistance. Enter the beta coefficient and the vertical effective stress.

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Pile Allowable Load

Calculate a pile's allowable (working) load, Q_adm = Q_ult ÷ FS, from the ultimate bearing capacity Q_ult (kN) and the global safety factor FS. The allowable load is the maximum load that can be applied to the pile in service with adequate safety — obtained by dividing the ultimate capacity (the load that would cause FAILURE of the pile-soil system) by a safety factor covering uncertainties. The pile-foundation safety factor is typically HIGH (FS = 2.0-2.5 for ultimate capacity, higher if based only on theoretical formulas without a load test), reflecting the great uncertainty in determining soil capacity (unseen, heterogeneous and poorly known) and the severity of a foundation failure (which can collapse the whole structure). Codes often require different partial factors for tip and friction (which have different uncertainties), or limit-state methods. The allowable load sets how many piles are needed for the column loads: number of piles = column load ÷ allowable load. Load tests (measuring real field capacity) allow reducing the safety factor and optimizing design. Enter the ultimate capacity and the safety factor.

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Pile Group Efficiency (Converse-Labarre)

Calculate a pile group's efficiency by the Converse-Labarre formula, η = 1 − (θ/90)·[(m−1)·n + (n−1)·m] ÷ (m·n), from the pile diameter D and spacing s (with θ = arctan(D/s), in degrees), and the number of piles per row m and per column n. When several piles are driven close together (forming a group under a cap), the group capacity is NOT simply the sum of individual capacities — there is INTERFERENCE between the stress bulbs of neighboring piles in the soil, which overlap. The efficiency η (less than 1) measures this loss: the CLOSER the piles (smaller spacing s relative to diameter D), the greater the overlap and the lower the efficiency. The Converse-Labarre formula, empirical and widely used, quantifies this reduction as a function of group geometry (pile count and spacing). So codes require a minimum pile spacing (typically 2.5-3 diameters) to limit efficiency loss. Efficiency times pile count times individual capacity gives the group capacity. This effect is more pronounced in friction piles in clay; in end-bearing piles in sand, the group may even have efficiency above 1 (driving densifies the sand). Enter the diameter, spacing and pile count per row and column.

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Pile Group Capacity

Calculate a pile group's bearing capacity, Q_g = η·N·Q_pile, from the group efficiency η, the number of piles N and the single isolated pile capacity Q_pile (kN). A pile group's capacity (piles driven close under a cap that distributes the column load among them) is each pile's individual capacity, times the pile count, adjusted by the group EFFICIENCY (η ≤ 1, discounting the interference between neighbors, computed by Converse-Labarre or other formulas). In clayey soils and friction piles, efficiency is below 1 (piles 'compete' for the same soil, and the group may even fail as a solid block — 'block failure', checked separately). In sands and driven piles, driving densifies the soil and efficiency can approach or exceed 1. The group capacity is what actually supports the column load above the cap, and must exceed it with the proper safety factor. This calculation, with the group settlement check (which can exceed a single pile's, since the group stress bulb is deeper), defines the design of a pile-group foundation. Enter the efficiency, the pile count and the individual capacity.

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Pile Capacity by Driving (Engineering News)

Estimate a driven pile's allowable load by the Engineering News Record dynamic driving formula, Q_adm = (W_r·h) ÷ (FS·(s + c)), from the hammer weight W_r (kN), the drop height h (m), the set s (permanent penetration per blow, m), a loss constant c (m, ≈ 0.0025 m for drop hammers) and the safety factor FS (≈ 6 in this formula). DYNAMIC driving formulas estimate a pile's capacity from observing how much it PENETRATES per hammer blow during driving — the principle is intuitive: the HARDER to drive (smaller penetration per blow, the 'set'), the GREATER the soil resistance and thus the pile capacity. The blow energy (weight × drop height) is equated to the penetration work (resistance × displacement), with losses. The 'set' (s) is measured in the field during driving (average penetration of the last blows), making these formulas a valuable, cheap EXECUTION CONTROL — driving continues until the set reaches the value matching the desired capacity. The Engineering News formula is the most classic (and conservative, with FS = 6). Modern high-strain dynamic monitoring (PDA, with CAPWAP analysis) replaces these empirical formulas far more accurately, but the set is still used daily on site. Enter the hammer weight, drop height, set, constant and safety factor.

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Pile Structural Stress

Calculate the structural compression stress in a pile shaft, σ = Q ÷ (π·D²/4), from the applied load Q (kN) and the pile diameter D (m); the result is in MPa. Besides the SOIL having capacity to support the pile (geotechnical capacity), the pile itself, as a STRUCTURAL element (concrete, steel or timber), must resist the load without failing or deforming excessively — this is the pile's STRUCTURAL check. The shaft compression stress is simply the load over the cross-sectional area. It must be below the pile material's allowable stress: codes limit cast-in-place pile concrete working stress to conservative values (typically 5-8 MPa, less than the concrete strength, due to subsurface execution uncertainties — blind concreting, possible defects, eccentricities). This check often GOVERNS the minimum pile diameter (the pile may have ample geotechnical capacity, but structural stress limits the load). Pile design is always the SMALLER of geotechnical (soil) and structural (material) capacity — both must be checked. Enter the applied load and the pile diameter.

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Disc Clutch Torque

Calculate the torque transmissible by a disc clutch (or brake), T = μ·F·N·r_m, from the friction coefficient μ, the axial clamping force F (N), the number of friction surfaces N and the mean friction radius r_m (m). A disc clutch transmits torque between two shafts by FRICTION between surfaces pressed together: an axial force F clamps the discs, and the friction at that interface, acting at the mean radius, generates the torque. The number of friction surfaces N multiplies the capacity — a single disc clutch has N=1 (one face) or N=2 (disc between two faces); MULTI-PLATE clutches (motorcycles, automatic transmissions) stack several discs with high N, transmitting large torque in compact space. The same principle applies to disc and clutch BRAKES: the braking (or transmitting) torque is μ·F·N·r_m. This is central in clutch and brake design: it sets the actuation force (pedal, spring, hydraulic actuator) needed to transmit/brake a given torque, and the area and number of discs. The design torque includes a service factor (1.2-3) over the nominal, to cover peaks and wear. Enter the friction coefficient, axial force, number of surfaces and mean radius.

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Mean Friction Radius (Clutch)

Calculate the mean friction radius of a disc clutch or brake by uniform-wear theory, r_m = (D + d) ÷ 4, from the outer D and inner d diameters (m) of the friction annulus. The mean radius is the EFFECTIVE radius at which the resultant friction force is taken to act for torque calculation (T = μ·F·N·r_m). There are two classic assumptions for this radius: UNIFORM WEAR (assuming the disc has 'bedded in' and wears evenly, concentrating pressure at the inner radius; gives r_m = (D+d)/4, the simple mean of radii) and UNIFORM PRESSURE (new disc, constant pressure; gives r_m = (2/3)·(D³−d³)/(D²−d²), slightly larger). Uniform wear is most used in DESIGN, being conservative (slightly lower torque) and representing the run-in steady state. The mean radius shows an interesting design point: discs with a narrow friction annulus (D close to d, thin ring at large radius) have a high mean radius, transmitting more torque per unit force — so high-performance disc brakes use calipers acting near the disc edge (large radius). Enter the outer and inner diameters.

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Clutch Axial Force (Uniform Pressure)

Calculate the axial clamping force of a disc clutch or brake by the uniform-pressure assumption, F = p·(π/4)·(D² − d²), from the contact pressure p (Pa) and the outer D and inner d diameters (m) of the friction annulus. The axial force clamps the discs together (applied by springs in normally-engaged clutches, or by a hydraulic/pneumatic actuator). By the UNIFORM-PRESSURE assumption (valid for new discs, before wear), the force is simply the average contact pressure times the AREA of the friction annulus (the ring between outer and inner diameters). This force is the clutch/brake actuation parameter: it determines the transmissible torque (with friction and mean radius) and must be limited so the contact pressure does not exceed the friction material's allowable (which has a limit, above which it degrades, loses friction by overheating — fading — or wears fast). Design balances: enough axial force for the needed torque, but pressure within the material limit (setting the minimum area and disc count). Enter the contact pressure and the outer and inner diameters.

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Clutch Max Pressure (Uniform Wear)

Calculate the maximum contact pressure in a disc clutch or brake by the uniform-wear assumption, p_max = 2·F ÷ (π·d·(D − d)), from the axial force F (N) and the outer D and inner d diameters (m); the result is in kPa. Under UNIFORM WEAR (the steady state, after disc run-in), pressure is NOT constant over the friction annulus: it is INVERSELY proportional to radius (highest at the inner radius, lowest at the outer), because wear — proportional to pressure × velocity — only becomes uniform if pressure falls with radius (since velocity grows with radius). So the MAXIMUM pressure occurs at the INNER radius (at diameter d), and that peak limits the design. The maximum pressure must be below the friction material's allowable (linings, pads, ceramic or sintered metallic materials — each with its limit, typically hundreds of kPa to a few MPa). Exceeding the allowable leads to accelerated wear, overheating and friction loss (fading). This calculation checks whether a clutch/brake, under the planned actuation force, operates within its material's pressure limit — an essential durability and safety check. Enter the axial force and the outer and inner diameters.

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Braking Energy

Calculate the energy dissipated in braking, E = ½·m·(v₁² − v₂²), from the mass m (kg), the initial velocity v₁ and the final velocity v₂ (m/s). When a vehicle or machine brakes, its KINETIC energy is converted — by brake friction — into HEAT. The dissipated energy is the kinetic-energy change: braking to a stop (v₂ = 0) dissipates all the initial kinetic energy; partial braking, the difference. This heat must be ABSORBED and DISSIPATED by the brake without overheating beyond the limit (above which the friction material loses effectiveness — fading — and may even burn or glaze). That is why brakes for heavy vehicles, long descents (mountain trucks) and severe duty need large thermal capacity (big, vented discs, or auxiliary brakes like engine braking and retarders, dissipating energy by other means without overloading the service brakes). Braking energy grows with the SQUARE of velocity: braking from 100 km/h dissipates FOUR times more energy than from 50 km/h — so high-speed braking is so much more demanding. This is the basis of brake thermal design and overheating checks in repeated or prolonged braking. Enter the mass and the initial and final velocities.

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Brake Power Dissipated

Calculate the power dissipated by a brake under torque, P = T·(2π·n/60), from the braking torque T (N·m) and the rotation n (rpm). Dissipated power is the rate at which the brake converts mechanical energy to heat — the product of braking torque and angular velocity. It differs from total braking ENERGY: energy is the total heat generated (joules), while power is the INTENSITY of that heat generation (watts), and it determines the brake's steady-state temperature. A brake dissipating much energy but slowly (low power) heats little; one dissipating the same energy fast (high power) heats much more. Dissipated power is critical in brakes working CONTINUOUSLY or repetitively: retention brakes on long descents, industrial equipment brakes (hoists, cranes, conveyors holding load), and dynamometers (which measure engine power precisely by dissipating it in a brake). There, the steady-state dissipated power sets the COOLING capacity needed (ventilation, water cooling) to keep temperature stable. Equating dissipated power to cooling capacity gives the equilibrium temperature. Enter the braking torque and the rotation.

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Rotational Braking Time

Calculate the time to brake (stop) a rotating system, t = (I·ω) ÷ T, from the moment of inertia I (kg·m²), the initial angular velocity ω (rad/s) and the braking torque T (N·m). When a brake applies a constant torque to a spinning system (a shaft, flywheel, machine rotor), it DECELERATES it to a stop. By Newton's second law for rotation (T = I·α, with α the angular deceleration), the stopping time is the initial angular momentum (I·ω) divided by the braking torque. This matters in several situations: EMERGENCY STOPPING of machines (safety codes require dangerous parts to stop within a maximum time after brake actuation — the shorter, the safer), sizing motor and shaft brakes, and clutches (the engagement time, where the clutch 'synchronizes' two shafts' speeds, follows the same physics). Systems with large moment of inertia (heavy flywheels, big rotors) take longer to stop with a given torque — so high-inertia machines need powerful brakes or more stopping time. The braking time, with the dissipated energy and power, completes a braking analysis. Enter the moment of inertia, the angular velocity and the braking torque.

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Shoe Brake Torque

Calculate the braking torque of a simple shoe (or drum) brake, T = μ·F·r, from the friction coefficient μ, the normal force applied by the shoe F (N) and the drum radius r (m). The shoe brake presses a friction-lined shoe against the surface of a rotating drum (or cylinder); the friction between shoe and drum generates a tangential force (μ·F) which, acting at the drum radius, produces the braking torque. It is the principle of vehicle drum brakes, hoist and industrial drum brakes, and rotating-machine brakes. The torque is simply the friction force times the radius. An important effect in shoe brakes is SELF-ENERGIZING: depending on the shoe pivot geometry, friction itself can HELP press the shoe against the drum (leading shoe), raising the effective force and torque for a given actuation force — or HINDER it (trailing shoe). This amplifies braking (an advantage) but makes it sensitive to the friction coefficient (which varies with temperature and moisture), possibly causing unstable behavior. This basic formula gives the torque without the self-energizing factor, considered separately per geometry. Enter the friction coefficient, the normal force and the drum radius.

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Brake Contact Pressure

Calculate the contact pressure between the shoe/pad and the drum/disc of a brake, p = F ÷ A, from the normal force F (N) and the friction material contact area A (m²); the result is in kPa. Contact pressure is the normal force distributed over the friction surface area, and one of the most important parameters in a brake's or clutch's DURABILITY and PERFORMANCE. It must be below the friction material's ALLOWABLE pressure (linings, organic, semi-metallic, ceramic or sintered metallic pads — each with its limit). Pressures ABOVE the allowable lead to accelerated wear, overheating and friction loss (fading), reducing material life and impairing braking. Very LOW pressures underuse the material (a bigger, costlier brake than needed). Contact pressure also relates to the p·v product (pressure × velocity), the key indicator of the friction contact's thermal intensity — friction materials have a p·v limit above which they overheat, and that limit often governs design. This simple check — comparing contact pressure with the material's allowable — is essential in brake and clutch design and in choosing the right friction material for the application. Enter the normal force and the contact area.

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Equivalent Dynamic Load (Bearing)

Calculate a bearing's equivalent dynamic load under combined loading, P = X·F_r + Y·F_a, from the radial load F_r (N), the axial load F_a (N) and the factors X and Y (dimensionless, tabulated by the maker per bearing type and the F_a/F_r ratio). Most bearings actually carry RADIAL (perpendicular to shaft) and AXIAL (along shaft) loads at once, but catalog life and capacity formulas are defined for an equivalent pure radial load. The equivalent dynamic load is that fictitious radial load that would give the SAME bearing life as the real load combination. The X and Y factors depend on the bearing type (deep-groove ball, angular contact, self-aligning, tapered roller) and the axial-to-radial ratio — for mainly radial loads, X≈1 and Y≈0 (axial negligible); when axial grows past a limit (the e factor), Y starts to contribute. Computing P correctly is the first step in any bearing design, since P enters the life formula L10 = (C/P)^p and the capacity check. Wrong factors (or ignoring axial load) give an incorrect life estimate. Enter the radial load, axial load and the X and Y factors.

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L10 Life in Hours (Bearing)

Calculate a bearing's nominal L10 life in HOURS of operation, L10h = (10⁶ ÷ (60·n))·(C/P)^p, from the dynamic load rating C (N), the equivalent dynamic load P (N), the rotation n (rpm) and the exponent p (3 for ball bearings, 10/3 for roller bearings). L10 life is the core of bearing selection: the number of revolutions (or hours) that 90% of a batch of identical bearings reaches or exceeds before FATIGUE failure (spalling of races and rolling elements) — i.e., only 10% fail earlier (hence 'L10', the life with 90% reliability). The basic formula L10 = (C/P)^p gives life in MILLIONS of revolutions; dividing by the rotation (rpm × 60 min/h) converts to hours, the practical unit for machines. The result shows the huge load sensitivity: since the exponent is 3 (balls), DOUBLING the load cuts life to 1/8! So a slightly overloaded bearing lasts far less. The capacity C is tabulated in each bearing's catalog. This calculation decides whether a bearing meets the application's required life (typically 20,000-100,000 h for industrial machines) or whether a larger one is needed. Enter the dynamic capacity, the equivalent load, the rotation and the exponent.

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Required Dynamic Capacity (Bearing)

Calculate the dynamic load rating C a bearing needs to reach a desired life, C = P·(L10)^(1/p), from the equivalent dynamic load P (N), the desired nominal life L10 (in millions of revolutions) and the exponent p (3 for balls, 10/3 for rollers). It is the INVERSE of the life calculation, and how bearing SELECTION is done in practice: the designer knows the load the bearing will carry (P) and the life it must reach (L10, derived from required operating hours and rotation), and computes the minimum needed dynamic capacity C. Then a bearing is chosen from the maker's catalog whose tabulated C is EQUAL OR GREATER than the required — and that fits the available dimensions (shaft and housing diameter). The dynamic capacity C is, by definition, the load giving an L10 life of exactly 1 million revolutions, and it is each bearing's 'rating' in the catalog. This calculation is the heart of sizing: it translates the application requirement (load and life) into the component spec (capacity), letting you pick the right bearing — neither undersized (early failure) nor oversized (needless cost and space). Enter the equivalent load, the desired life and the exponent.

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Equivalent Static Load (Bearing)

Calculate a bearing's equivalent static load, P_0 = X_0·F_r + Y_0·F_a, from the radial load F_r (N), the axial load F_a (N) and the static factors X_0 and Y_0 (tabulated by the maker). Unlike the equivalent dynamic load (related to FATIGUE under rotation), the equivalent static load is used to check the bearing under loads with the bearing STOPPED or turning very slowly, or under PEAK loads (shocks, momentary overloads). The risk here is not fatigue but PERMANENT DEFORMATION (indentation) of the races by the rolling elements: an excessive static load 'dents' permanent marks (brinelling) into the races, which then cause noise, vibration and early failure when the bearing turns again. The equivalent static load is the pure radial load that would cause the same maximum permanent deformation (at the most-loaded contact) as the real radial-axial combination. It is compared with the bearing's static load rating C0 (also tabulated) via the static safety factor s0 = C0/P0. This check is especially important in bearings carrying loads with the machine stopped (shafts of equipment parked under load) or subject to shocks. Enter the radial load, axial load and the static factors X0 and Y0.

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Static Safety Factor (Bearing)

Calculate a bearing's static safety factor, s_0 = C_0 ÷ P_0, from the static load rating C_0 (N, tabulated by the maker) and the equivalent static load P_0 (N). The static safety factor compares the bearing's ability to resist PERMANENT DEFORMATION (race indentation) with the equivalent static load it actually carries. The static rating C_0 is, by definition, the load causing a total permanent deformation of 0.0001 of the rolling-element diameter at the most-loaded contact — a small value, taken as the acceptable limit (above it, the marks cause noise and vibration when turning). The factor s_0 shows the margin: codes and makers recommend MINIMUM s_0 values per application and smoothness requirements — typically s_0 ≥ 1-1.5 for normal, quiet operation, possibly lower (0.5-1) for low-speed, undemanding applications, and higher (≥2-3) for heavy shocks or high precision. This check is COMPLEMENTARY to the life (fatigue) check: a bearing may have ample L10 life but fail by static deformation under a peak overload if s_0 is insufficient. Both checks — dynamic (life) and static (s_0) — must be met. Enter the static rating and the equivalent static load.

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Reliability-Adjusted Life (Bearing)

Calculate a bearing's adjusted life for a reliability other than 90%, L_na = a_1·L10, from the reliability factor a_1 (dimensionless) and the nominal life L10 (in millions of revolutions). The standard L10 life corresponds to 90% reliability (10% failures). But many CRITICAL applications — where a bearing failure is unacceptable (turbines, aerospace, medical equipment, continuous-process machines) — require HIGHER reliabilities (95%, 99%, 99.9%). Since demanding higher reliability means accepting FEWER failures, the corresponding life is SHORTER: a_1 is below 1 for reliabilities above 90%. Typical values: a_1 = 1.0 for 90% (L10), 0.64 for 95% (L5), 0.21 for 99% (L1), 0.093 for 99.9% (L0.1). For example, to ensure 99% of bearings survive (instead of 90%), the design life drops to about 21% of L10. This is one of the 'modified life' corrections in the standards (ISO 281), which also include factors for material and lubricant quality and contamination (the more sophisticated a_ISO factor). Adjusting life for required reliability is essential in critical designs: simply using L10 (90%) would be too risky for a turbine, and too conservative for a household fan. Enter the reliability factor and the L10 life.

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Bearing Speed Factor

Calculate a bearing's speed factor, A = n·d_m, from the rotation n (rpm) and the bearing mean diameter d_m (mm, = (D + d)/2, the average of outer and inner diameters). The n·d_m factor (often in mm·rpm, or m/min times the perimeter) is the key indicator of a bearing's SPEED DUTY, and governs several application limits. It sets the operating LIMIT SPEED: each bearing (and each lubrication type) has a maximum n·d_m above which friction heating, centrifugal force on the rolling elements and dynamic effects make operation unfeasible — grease lubrication tolerates lower values, oil higher, and special systems (oil jet, mist) the highest (high-speed bearings, like turbine and machine-tool spindle bearings, reach n·d_m in the millions). The speed factor also influences the bearing type choice (balls take more speed than rollers), the lubricant and the internal clearance (fast bearings may need larger clearance to accommodate thermal expansion). Comparing the application's n·d_m with the bearing limit is an essential check in medium- and high-speed rotating machines. Enter the rotation and the mean diameter.

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Cubic Mean Load (Bearing)

Calculate the equivalent mean load of a bearing under a cycle with two different loads, P_m = ∛(P₁³·U₁ + P₂³·U₂), from the loads P₁ and P₂ (N) and the time (or revolution) fractions during which they act U₁ and U₂ (with U₁ + U₂ = 1). Many bearings do not work under CONSTANT load: the load varies over the operating cycle (a press loading and unloading, a motor accelerating and decelerating, a machine with different work phases). To compute life in this case, the variable cycle is replaced by an equivalent CONSTANT load causing the same fatigue damage — the mean load. But the mean is NOT arithmetic: since fatigue damage is proportional to load CUBED (the life exponent p=3), the mean load is a time-fraction-weighted mean, but with the loads cubed (then cube-rooted) — the so-called cubic mean or 'fatigue-weighted mean'. This makes HIGH loads weigh disproportionately more (a double load causes 8× more damage), so even a small fraction of time at high load dominates the result. This formula (here for two load levels; it generalizes to several) is essential to size bearings in variable-load machines, avoiding underestimating the damage. Enter the two loads and their time fractions.

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Load-Life Ratio (Bearing)

Calculate how a bearing's life changes when the load changes, L₂ = L₁·(P₁/P₂)^p, from the initial life L₁ (under load P₁), the loads P₁ and P₂ (N) and the exponent p (3 for balls, 10/3 for rollers). This relation expresses the essence of bearing life law: life is INVERSELY proportional to load raised to the exponent p. It lets you quickly answer, without recomputing everything, 'if I change the load, what happens to the life?'. And the answer is dramatic due to the high exponent: REDUCING the load by 20% (P₂ = 0.8·P₁) INCREASES life by (1/0.8)³ = nearly DOUBLE; INCREASING the load by 26% (P₂ = 1.26·P₁) halves the life; DOUBLING the load cuts life to 1/8. This extreme load sensitivity has important practical consequences: small overloads (from misalignment, imbalance, wrong mounting or inadequate clearance, which concentrate load) drastically cut the real life versus the calculated one — explaining why many bearings fail 'too early'. Conversely, reducing parasitic loads (better alignment, balancing) greatly extends life. This formula is a valuable tool for sensitivity analysis and failure diagnosis. Enter the initial life, the two loads and the exponent.

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Belt Transmitted Power

Calculate the power transmitted by a belt, P = (T₁ − T₂)·v, from the tight-side tension T₁ (N), the slack-side tension T₂ (N) and the belt velocity v (m/s). In a belt drive, the driving pulley drags the belt by friction, creating a DIFFERENCE in tension between the two sides: the side that 'pulls' (tight side, T₁) is more tensioned than the side that 'follows' (slack side, T₂). This difference (T₁ − T₂), the effective tension or tangential force, is the net force that actually transmits motion; times the belt velocity, it gives the transmitted POWER. The larger the tension difference the belt can sustain without slipping (depending on friction, wrap angle and, in V-belts, the wedging effect of the pulley walls), the greater the transmissible power. Power also grows with belt velocity — so high-power drives use large pulleys and fast belts (up to a limit, since centrifugal tension reduces available friction at very high speeds). This is central in belt-drive design, present in almost every rotating machine: motors, fans, pumps, compressors, machine tools and vehicles. Enter the tight- and slack-side tensions and the belt velocity.

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Belt Span Natural Frequency

Calculate the natural vibration frequency of a belt's free span, f_n = (1 ÷ (2·L))·√(T/m), from the free span length L (m, the distance between pulleys), the belt tension T (N) and the mass per unit length m (kg/m). A belt's free span, between two pulleys, behaves like a stretched STRING (like a guitar string): when disturbed, it vibrates at a natural frequency depending on its tension and mass. The HIGHER the tension, the HIGHER the frequency (tighter string, higher pitch); the higher the mass per metre, the lower the frequency. This relation is the basis of a clever, widely used method to MEASURE belt tension in the field: the SONIC (or frequency) tension meter — the technician 'plucks' the belt to make it vibrate, and a sensor (or phone app) measures the sound frequency; knowing the span length and belt mass, the tension is computed back (inverting the formula). It is far more practical and accurate than the old methods of measuring deflection under a force. Keeping the correct tension is essential: a slack belt slips (loses power, heats, wears) and an over-tight belt overloads the bearings and shortens belt life. Enter the span length, the tension and the mass per unit length.

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Belt Wrap Angle

Calculate a belt's wrap (contact) angle on the smaller pulley, θ = π − 2·arcsin((D − d) ÷ (2·C)), from the larger D and smaller d pulley diameters (m) and the center distance C (m). The wrap angle is the angle of the arc over which the belt actually WRAPS the pulley, in contact with it — and it is a critical parameter, since it is along that arc that the friction (transmitting the force) acts. The LARGER the wrap angle, the greater the contact area and the greater the force the belt can transmit without slipping. In a drive between two DIFFERENT-DIAMETER pulleys, the belt wraps LESS around the smaller pulley (angle below 180°) and MORE around the larger — and slipping always starts on the pulley with LESS wrap (the smaller), which therefore limits capacity. The wrap angle decreases when the diameter difference grows or the center distance shrinks (close, very different pulleys 'wrap' little). So drives with large reduction (very different pulleys) or close centers have reduced capacity, and sometimes use an IDLER (tensioner) pulley to increase wrap. The wrap angle enters directly into the tension ratio (e^(μθ)) and the belt-count correction factors. Enter the pulley diameters and the center distance.