Gear Tooth Bending Safety Factor
Calculate the bending safety factor of a gear tooth, FS = σ_perm ÷ σ, from the material's allowable (permissible) bending stress σ_perm (MPa, the bending fatigue strength with its factors) and the acting bending stress σ (MPa, from Lewis/AGMA from the load). It is the final bending-design check: the material strength must exceed the demand with adequate margin. Gears run for millions to billions of cycles, so the allowable stress is the material's bending FATIGUE strength (at the design-life cycle count), adjusted by reliability, temperature and life factors. Required safety factors depend on criticality and uncertainty (typically 1.5-3 for bending). If FS is insufficient, the module is increased (bigger, stronger teeth), the face width, or a better material/heat treatment used. A tooth breaking by bending fatigue is a CATASTROPHIC, sudden failure (unlike pitting, which gives progressive signs), since the broken tooth comes loose and can damage the whole drive — so bending is designed with generous margins. With the pitting (contact) safety factor, it defines the gear's robustness. Enter the allowable stress and the acting stress.
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Gear tooth bending safety factor
The bending safety factor of a gear tooth is FS = σ_all ÷ σ, computed from the allowable bending stress of the material σ_all (the bending fatigue strength with its correction factors already applied) and the actual bending stress σ (obtained from the load through the Lewis or AGMA equations). It is the final check of the bending design: the strength of the material must exceed the applied stress by an adequate margin. Gears run for millions to billions of cycles, so the allowable stress is the bending fatigue strength (at the cycle count of the design life), adjusted by reliability, temperature and life factors. The safety factors demanded depend on criticality and on how much uncertainty is involved (typically 1.5 to 3 for bending). Where FS falls short, the fix is a larger module (bigger, stronger teeth), a wider face, or a better material and heat treatment. A tooth breaking from bending fatigue is a sudden, catastrophic failure (unlike pitting, which gives progressive warning signs of wear), since the cracked tooth breaks loose and can wreck the whole drive — hence the generous margins used in bending design. Together with the pitting (contact) safety factor, it defines how robust the gear is. Enter the allowable stress and the actual stress.
Related Tools
Gear Tooth Bending Stress (Lewis)
Calculate the bending stress at a gear tooth root by the Lewis equation, σ = F_t ÷ (b·m·Y), from the tangential force F_t (N), the tooth face width b (mm), the module m (mm) and the Lewis form factor Y (dimensionless, a function of tooth count). Lewis's 1892 equation was the first rational treatment of gear-tooth strength and is still the basis of BENDING design. It models the tooth as a cantilever beam fixed at the root: the tangential force transmitted between teeth (from the torque) creates a bending moment that tends to break the tooth at the root — the catastrophic failure where a tooth cracks and snaps. The form factor Y accounts for tooth geometry (gears with more teeth are 'fatter' at the root and stronger, higher Y). The computed stress is compared with the material's bending fatigue strength (with safety factors), since gears endure millions of cycles. The basic Lewis formula is then refined by the AGMA standard with stress-concentration, dynamic, load-distribution and surface-condition factors. It is one of the two fundamental gear design criteria (the other is contact stress). Enter the tangential force, face width, module and Lewis form factor.
Gear Dynamic Load
Calculate the effective dynamic load on gear teeth, F_d = F_t·K_v, from the nominal tangential force F_t (N) and the dynamic factor K_v. The dynamic load is the REAL tangential force the teeth bear in operation, larger than the nominal force (simply torque over radius) because of the dynamic effects of meshing at speed. These effects — vibrations, contact impacts, tooth deflections under load and manufacturing errors — make the instantaneous tooth load fluctuate and peak above the average, especially at high pitch-line velocities. The dynamic factor K_v (from Barth or other formulas) quantifies this amplification. The dynamic load is then used in strength checks: in Lewis bending stress (risk of tooth breakage at the root) and Hertzian contact stress (risk of surface fatigue and pitting). Using the nominal load without amplifying by the dynamic factor would underestimate the demands and lead to undersized gears that fail prematurely by fatigue. It is an essential, classic step in gear design. Enter the nominal tangential force and the dynamic factor.
Wire Rope Safety Factor
Calculate a wire rope's safety factor, SF = breaking load ÷ working load, from the minimum breaking load (MBL, N) and the applied working load (N). Wire ropes, used in cranes, elevators, cableways, bridges, lifting and mooring, work with HIGH safety factors — far higher than static structures — for several reasons: the load is rarely static (there are impacts, accelerations, swings), the rope wears and loses strength over use (wires break, corrosion and fatigue occur), and a rupture is catastrophic (load drop, life risk). Codes prescribe minimum safety factors per application: typically 5 for general load lifting, 6-8 for people-carrying ropes (elevators, cableways), 3-4 for static stays and moorings, and specific values per use. The safety factor is the ratio between the load that would break the rope (its rated strength, from the maker) and the load it actually carries in service. Checking that the real safety factor meets the code minimum is the basic safety check of any wire-rope application — and the rope must be DISCARDED when wear reduces its strength enough for the factor to fall below the limit. Enter the breaking load and the working load.
Vessel Allowable Stress (ASME)
Calculate the design allowable stress of a pressure-vessel material by the ASME criterion, S = σ_uts ÷ n, from the material minimum tensile strength σ_uts (MPa) and the safety factor n (3.5 in the current ASME VIII Div. 1 edition for tensile strength). The allowable stress S is the MAXIMUM stress permitted in the vessel material in service, and is the basis of all thickness and MAWP calculations — it embeds the safety margin against failure. The ASME code sets the allowable stress as the SMALLEST among several criteria: a fraction of the TENSILE strength (σ_uts/3.5 in the current edition — formerly /4.0, reduced as materials and inspection advanced), a fraction of the YIELD strength (2/3 of σ_yield), and, at high temperatures, criteria based on CREEP and creep rupture (since at high temperature the material deforms slowly under constant load). For each material and temperature, the code TABULATES the S value — this formula shows the tensile-strength criterion, often governing at moderate temperatures. Using the correct allowable stress (from the code, for the right material and temperature) is absolutely essential: it is the safety margin protecting against vessel explosion. Enter the tensile strength and the safety factor.
Structural Safety Factor
Computes the safety factor (FS) as the ratio of resisting load to applied load.
Geosynthetic Rupture Safety Factor
Calculate the safety factor against tensile rupture of a geosynthetic reinforcement layer, FS = T_adm ÷ T_req, from the allowable tensile strength T_adm (kN/m, the ultimate already reduced by creep, installation-damage and degradation factors) and the required tension T_req (kN/m, the force the soil demands at that layer). This is the final design check for a reinforcement layer: the available (allowable) strength must exceed the demand (required) with an adequate margin. Reinforced-soil codes require tensile-rupture safety factors typically around 1.3-1.5 (since many uncertainties — creep, damage, degradation — are already covered by the partial reduction factors embedded in T_adm). If FS is below the required, a stronger geosynthetic is chosen, the layer spacing reduced (lowering T_req per layer) or both. Besides tensile rupture (this calculation), reinforced-soil design also checks PULLOUT stability (sufficient anchorage), INTERNAL stability (failure surfaces cutting the reinforcements), EXTERNAL stability (sliding, overturning and bearing capacity of the whole mass) and deformations. This rupture FS is one of the fundamental checks. Enter the allowable strength and the required tension.
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