Contact Stress: Method

Hertz Contact Theory and Allowables

Problem Statement

When two curved bodies are pressed together they touch over a small area whose size grows with the load, and the pressure there is far above the average stress in either body: a steel ball under a few hundred pounds carries contact pressures of hundreds of thousands of pounds per square inch. Hertz solved the problem in 1881 for elastic bodies whose contact is small compared with their radii. The stresses matter for bearings, gears, cams, rollers, pins in grooves, wheels on rails, ball joints and any part that transmits load through a curved surface, and they decide three kinds of failure: yielding and permanent indentation (brinelling) under a static load, pitting and spalling under repeated rolling, and ring cracking of brittle materials or hard coatings from the tension at the edge of the contact. The tool takes the geometry of the pair, the load and the materials, and gives the contact size, the pressure, the complete stress field below the surface and a margin of safety against each of those failures.

Geometry: Principal Radii, Relative Curvature and the Angle α

Each surface is described at the point of contact by its two principal radii of curvature, R₁ and R₁′ for body 1 and R₂ and R₂′ for body 2, measured in perpendicular planes containing the common normal. A convex surface has positive radii; a concave one, such as a spherical seat, a bearing groove, a bore or an internal gear, has negative radii; a flat has infinite radius and a cylinder is flat along its axis. The planes of R₁ and R₂ make the angle α (zero for a ball on a raceway, 90° for crossed rollers). The gap between the undeformed surfaces near the contact is a quadratic Ax² + By² whose constants, the relative curvatures, follow from the four radii and α (Boresi Eqs. 18.1 to 18.3):

A + B = ½ (1/R₁ + 1/R₁′ + 1/R₂ + 1/R₂′)
B − A = ½ √[(1/R₁ − 1/R₁′)² + (1/R₂ − 1/R₂′)² + 2 (1/R₁ − 1/R₁′)(1/R₂ − 1/R₂′) cos 2α]

The shape of the contact depends only on B/A: equal curvatures give a circle, a large ratio a long narrow ellipse, and a vanishing A (parallel cylinders) a line contact, which is treated separately. The named cases on the form are this general case with the radii filled in: a ball of diameter D on an inner raceway of diameter d with groove radius r_g has 1/R₁ = 1/R₁′ = 2/D, 1/R₂ = 2/d along the rolling direction and 1/R₂′ = −1/r_g across it (−2/d on an outer race); gear teeth at the pitch point are cylinders of radii (d_p/2) sin φ in line contact under the normal load W_t/cos φ per face width, which is exactly the AGMA contact stress with its elastic coefficient C_p = √(E*/π) and geometry factor I.

The Hertz Solution

Both bodies are treated as elastic half-spaces with the combined modulus 1/E* = (1 − ν₁²)/E₁ + (1 − ν₂²)/E₂. The contact is the ellipse x²/a² + y²/b² = 1 with a ≥ b, a lying in the plane of the smaller relative curvature A, under the semi-ellipsoidal pressure p = p₀ √(1 − x²/a² − y²/b²) whose peak is one and a half times the mean, p₀ = 3P/(2πab). The axis ratio follows from B/A through the complete elliptic integrals K(e) and E(e) of the eccentricity e = √(1 − b²/a²), and the size and the approach δ of the bodies from (Johnson Eqs. 4.24 to 4.27)

B/A = [E(e)/(1 − e²) − K(e)] / [K(e) − E(e)],    a³ = 3P [K(e) − E(e)] / (2π E* e² A),    δ = p₀ b K(e) / E*

The tool solves the first relation for e by bisection and evaluates the integrals by the arithmetic-geometric mean, so there is no chart reading. For a circle (B = A, 1/R = 2A) the relations reduce to the familiar a³ = 3PR/(4E*) and δ = a²/R. Boresi presents the same solution as b = c_b(PΔ)^1/3, σ_max = c_σ b/Δ, τ_max = c_τ b/Δ, τ_oct,max = c_G b/Δ and z_s = c_zs b with Δ = (1/(A + B))[(1 − ν₁²)/E₁ + (1 − ν₂²)/E₂] and coefficients plotted against B/A in his Fig. 18.4; the tool back-calculates those coefficients from its exact results so they can be compared with the figure (at B/A = 1 they are 0.908, 0.637, 0.197, 0.179 and 0.48). For parallel cylinders carrying the load P′ per unit length the contact is a strip of half-width b = √(4P′R_e/(πE*)) with 1/R_e = 1/R₁ + 1/R₂, the pressure is p₀ √(1 − x²/b²) with p₀ = 2P′/(πb), and the approach of the axes is not fixed by the contact alone (it depends on the size and support of the cylinders, Johnson Section 5.6), so none is reported.

Stresses below the Surface

The pressure alone does not decide failure; the shear stresses below the surface do. At the centre of the contact the material is in triaxial compression, all three normal stresses close to −p₀, so the shear stress there is small. Moving down the axis, σ_x and σ_y fall off faster than σ_z and the shear stress rises to a maximum below the surface: for a circular contact τ_max = 0.31 p₀ at z = 0.48 a (ν = 0.3), for a line contact 0.300 p₀ at z = 0.786 b; the octahedral shear stress peaks at about the same depth at 0.27 p₀. Yielding and the cracks of contact fatigue start there, not at the surface, which is why a case-hardened part needs its case deeper than these depths. The tool obtains the field by superposing the Boussinesq solution for a normal point force on a half-space over the Hertz pressure (and the Cerruti solution for a tangential force over the traction when there is friction), integrating in polar coordinates centred on each field point with a substitution that resolves the kernel at every depth; the surface values are principal-value integrals of the z = 0 kernels plus the local terms −(1 + 2ν)p/2 in σ_x and σ_y and −p in σ_z. For the circular contact the axis stresses are known in closed form (Huber 1904; Johnson Eq. 3.45), σ_z = −p₀/(1 + z²/a²) and σ_x = σ_y = −p₀[(1 + ν)(1 − (z/a) arctan(a/z)) − 1/(2(1 + z²/a²))], and for the line contact σ_z = −p₀/√(1 + z²/b²), σ_x = −p₀[(1 + 2z²/b²)/√(1 + z²/b²) − 2z/b] and σ_y = ν(σ_x + σ_z) (McEwen 1949); the numerical field reproduces both to better than 0.3 %, the surface stresses of the circular contact to 0.2 %, and the integrated pressure, the Hertz displacement and the point-load equilibrium exactly. The tables on the results page give the axis profile; the charts give the profile and the surface line; the fringe plot shows τ_max in the plane through the long axis.

Orthogonal shear. As a rolling element passes, a point at depth z sees the shear stress τ_zx reverse from +τ_zx to −τ_zx, a fully reversed cycle of twice the amplitude of the static maxima. Its largest value, 0.25 p₀ at z = 0.5 b and x = ± 0.87 b for a line contact, is the fatigue stress of the Lundberg-Palmgren bearing life theory; the tool reports it with its depth.

Friction. With sliding, a traction q = μp acts along +x on body 2. It shifts the shear maxima towards the surface and the leading edge and, from μ of about 0.3 upward, puts the maximum shear on the surface; it also makes the surface stress σ_x compressive by 2μp₀ at the leading edge and tensile by the same amount at the trailing edge of a line contact (Smith and Liu 1953), and for a circular contact raises the edge tension to p₀[(1 − 2ν)/3 + (4 + ν)πμ/8] (Hamilton and Goodman 1966). The tool's friction field was checked against both results and against equilibrium of the point-load fields.

Allowables and Margins of Safety

There is no single allowable contact stress. The tool computes five margins, each against the failure it describes, and the engineer picks the one that governs the application.

Load level. The form asks whether the load entered is ultimate or limit (limit by default: the operating load of a bearing, gear or follower is a limit load). The strength margins (gross indentation, surface tension) are written against ultimate load, the entered limit load times the factor of safety FS (1.5 by default) or the entered ultimate load as it stands, and their boxes say ULT. The yield margins (first yield, permanent set) are written against the limit load times the yield factor FS_yld (1.15 by default, the factor several certification agencies require on limit load for no detrimental permanent set; 1.0 writes them at limit load), the limit load being the entered load or the entered ultimate load divided by FS, and their boxes say LIM. Pitting is checked at the limit (operating) load. Because the contact area grows with the load, a load factor k moves every Hertz stress by k^1/3 (point contact) or k^1/2 (line contact), and the margins apply that stress factor rather than k itself.

First yield. By the octahedral (von Mises) criterion the material yields when τ_oct reaches (√2/3)F_cy, by the maximum-shear (Tresca) criterion when τ_max reaches F_cy/2; with the maxima above, a circular contact in steel yields at p₀ ≈ 1.6 F_cy and a line contact at 1.67 F_cy. Below this the contact is fully elastic and nothing is left behind when the load is removed. The compressive yield strength is used because the stress state is compressive; both bodies are checked with their own strength and Poisson's ratio.

Permanent set (brinelling). Above first yield a small plastic zone forms below the surface, contained by the elastic material around it, and a small permanent indent appears; a bearing tolerates a little of this, a precision fit does not. ISO 76 defines the basic static load rating of rolling bearings as the load at which the permanent deformation of element and raceway together reaches 0.0001 of the rolling-element diameter at the most heavily loaded contact, and states that for through-hardened bearing steel this corresponds to a contact pressure of 4,200 MPa (609 ksi) for ball bearings and 4,000 MPa (580 ksi) for roller bearings. Those are the default allowables; they assume a hardness of 58 HRC or more. For softer materials the pressure at which the same indent forms is roughly proportional to the yield strength, so enter an allowable of your own, or judge by the first-yield and indentation margins.

Gross indentation. When the mean pressure reaches the indentation hardness the whole contact is plastic and the ball sinks in as in a hardness test. Tabor showed that for a ball this happens at a mean pressure of about 2.8 to 3 times the yield strength, which is why hardness in pressure units is about 3 Y; the tool uses H = 9.807 HB in MPa when a Brinell number is entered and 2.8 F_cy otherwise. First yield (p_mean ≈ 1.1 Y) and full plasticity bracket the range in which the permanent-set criterion lives.

Rolling-contact fatigue. Repeated rolling fails the surface by pitting or spalling from cracks that start at the subsurface shear maxima. The practical allowables are empirical contact stress numbers for the material and heat treatment: the gear standard ANSI/AGMA 2001-D04 gives them in its Table 3 and Fig. 11 for 10⁷ cycles at 99 % reliability under unidirectional load (through-hardened steel by hardness, Grade 1: 322 HB + 29,100 psi, Grade 2: 349 HB + 34,300 psi, valid for 180 to 400 HB; fixed values for flame, induction, carburized and nitrided cases), and Buckingham's surface fatigue strength for 10⁸ cycles is 0.4 HB − 10 ksi. The tool compares the selected allowable with p₀. For a gear that is the AGMA contact stress at the pitch point without its overload, dynamic, size, load-distribution, life, reliability, temperature and hardness-ratio factors, which the designer applies on top; for a bearing, ISO 281 computes life from the dynamic rating rather than from a stress, and the margin here is only an indication.

Surface tension. Just outside the contact the surface is in radial tension, (1 − 2ν)p₀/3 for a circular contact and more at the trailing edge with friction. Ductile metals ignore it; glass, ceramics, carbides and hard coatings crack there (Hertzian ring cracks), so the tool compares the largest surface tension with the tensile strength when one is entered.

Assumptions and Limits

The contact must be small compared with the radii and the bodies large compared with the contact (half-space bodies): the tool warns when a semi-axis exceeds 0.3 of the smallest radius and when a seat or groove radius is within 20 % of the convex radius, where the Hertz pressure is too high and a conforming-contact solution (Persson) applies. The surfaces are smooth and frictionless under the normal load; the friction option adds a sliding traction proportional to the pressure (Amontons) with no change of the pressure distribution, which is exact only for bodies of the same elastic constants and a good approximation otherwise. The materials are linear elastic; the margins say how far the elastic solution is from each limit, not what happens beyond it. Line contacts are infinitely long: the stress concentration at the ends of a finite roller or a gear face, relieved in practice by crowning, is not included. Dynamic loads, lubricant films, surface roughness, residual stresses from heat treatment and temperature are outside the solution.