Section Properties
Used for roll weight, bending, deflection and torsion models.
A = pi/4 x (Do^2 - Di^2)I = pi/64 x (Do^4 - Di^4)J = pi/32 x (Do^4 - Di^4)I is second moment of area for bending. J is polar moment of area for torsion.
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Core, Journal and Cover Mass
Calculates component and total roll mass using validated density values.
Vcore = pi/4 x (Do^2 - Di^2) x Lmcore = rho_core x VcoreVjournal = pi/4 x d^2 x LjVcover = pi/4 x (Df^2 - Dc^2) x LcMtotal = Sum of component massSimply Supported Roll Models
Support and load cases must be selected separately. One deflection equation should not be used for every configuration.
Central point load: delta = P L^3 / (48 E I)Full-span UDL: delta = 5 W L^3 / (384 E I)Initial camber estimate = correction factor x calculated deflectionDefault correction factor should remain 1.000 unless validated calibration data exists.Stress Checks
Allowable stress and safety factor must come from engineering practice, applicable code or company standard.
Mmax point load = P L / 4Mmax UDL = W L / 8Bending stress: sigma = M c / ISolid journal bending: sigma = 32 M / (pi d^3)Solid shaft torsion: tau = 16 T / (pi d^3)Hollow shaft torsion: tau = T ro / JCylinder Force, Nip Load and Contact Pressure
Gravity contribution must be selected based on actual machine geometry.
Cylinder force: F = P x AFull piston area: A = pi D^2 / 4Rod-side area: A = pi (D^2 - d^2) / 4Mechanism force = cylinder total force x lever ratio x efficiencyFnip = mechanism force +/- effective roll weight componentLinear nip load: q = Fnip / face lengthAverage pressure: pavg = F / (L x b)1 N/mm is numerically equal to 1 kN/m.Restricted Elastic Contact Case
Only applicable for technically suitable elastic cylinder-contact cases. Do not apply automatically to soft rubber rolls.
R' = (R1 x R2) / (R1 + R2)1/E' = (1 - nu1^2)/E1 + (1 - nu2^2)/E2w = F / Lb = sqrt(4 w R' / (pi E'))p0 = 2 w / (pi b)Compression, Cover Volume and Recoating
Do not infer load directly from Shore hardness unless a validated empirical relation exists.
Compression: dt = t0 - t1Compression strain: epsilon = dt / t0Average stress: sigma = F / AModel compression: delta = F t / (A Eeff)Rubber cover volume = pi/4 x (Df^2 - Dc^2) x LMaterial mass = rho_rubber x volumeRequired mass = theoretical mass x (1 + allowance/100)Diameter Reduction
Used for service, grinding and material removal estimates.
Diameter reduction = Dinitial - DfinalRemoval per side = diameter reduction / 2Percent reduction = diameter reduction / Dinitial x 100Surface Speed, RPM, Torque and Power
All calculations should convert internally to canonical SI units.
Surface speed: v = pi D n / 60m/min from mm: v = pi x D(mm) x RPM / 1000RPM from m/s: RPM = 60 v / (pi D)Torque from tangential force: T = Ft x rAngular velocity: omega = 2 pi n / 60Mechanical power: P = T omegaTorque from power: T = P / omegaRotational Inertia and Acceleration Torque
Mass moment of inertia is different from area moment of inertia used in deflection.
Solid cylinder inertia: Imass = 1/2 m r^2Hollow cylinder inertia: Imass = 1/2 m (ro^2 + ri^2)Angular acceleration: alpha = delta omega / delta timeAcceleration torque: Tacc = Imass x alphaTotal torque approx = Tprocess + Tfriction + TaccReaction Loads and L10 Life
Use manufacturer catalogue values for dynamic rating and X/Y factors. Do not invent bearing data.
RA + RB = Sum FiRB = Sum(Fi xi) / LRA = Sum Fi - RBL10 = (C/P)^p million revolutionsBall bearing p = 3Roller bearing p = 10/3L10h = L10 x 10^6 / (60 n)Equivalent load where valid: P = X Fr + Y FaRoll Diameter Optimization Logic
Find minimum roll outside diameter satisfying both bending stress and deflection requirements.
Check 1: calculated stress <= allowable stress / safety factorCheck 2: calculated deflection <= allowable deflectionIterate diameter using validated section, span, load and material inputsThis should be implemented with clear warnings and engineering review before manufacturing.Need these formulas inside your own branded software?
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