Contact Mechanics · Surface roughness
Surface roughness parameters beyond Ra
Surfaces with the same Ra can carry load and hold oil very differently.
Why is Ra not enough to describe a surface?
Ra averages the heights, so it cannot tell peaks from valleys.
Try it: click Turned. Ra stays 0.80 µm, but Rsk flips from −2.0 to +0.6.
What does the bearing area curve (Rk, Rpk, Rvk) tell you?
Lower a flat plane into the surface: how much does it cut?
Try it: on Plateau honed, lower the plane from 0.86 to 1.73 µm: the share cut jumps from 2% to 35%.
Which cutoff should I use, and why does Ra change with it?
A filter splits long waves (waviness) from short ones (roughness) at the cutoff λc.
Try it: step λc from 0.08 to 8 mm. Ra climbs from 0.26 to 0.61 µm; only 0.8 mm matches.
What to take away
Ra hides the shape
Equal Ra can mean sharp peaks or flat plateaus. Check Rsk too.
Negative Rsk holds oil
Plateaus carry load; deep valleys store oil.
Quote the cutoff
Ra changes with λc. Give λc and the length measured.
More detail: definitions, the models behind the pictures, and their limits
Height parameters (ISO 4287, ISO 21920-2)
z(x) is the roughness profile measured from its mean line. Ra = mean of |z|, Rq = √(mean of z²), Rsk = mean(z³)/Rq³, Rku = mean(z⁴)/Rq⁴. A normal (Gaussian) height distribution has Rsk = 0 and Rku = 3; for it Rq/Ra = √(π/2) ≈ 1.25. Rp is the highest peak and Rv the deepest valley in one section, both as positive numbers, and Rz = Rp + Rv. The page follows ISO 21920-2: Ra, Rq, Rsk and Rku are taken over the whole evaluation length, while Rp, Rv and Rz are averaged over five sections. The older ISO 4287 averaged every parameter over five sampling lengths; for these profiles the two give nearly the same Ra and Rq.
How the surfaces are made
Ground, honed and polished start from random heights with an exponential autocorrelation of length β (the distance over which heights stop being alike). Turned starts from tool-nose arcs with a feed of 5β, shot peened from overlapping round craters about 4β apart. The heights are then reshaped, keeping their order, by y = sinh((asinh(x) + ε)/δ) (the sinh-arcsinh family of Jones and Pewsey), with ε and δ solved so the profile has your Rsk and Rku, and finally scaled so Ra equals your value. Some Rsk and Rku pairs cannot be reached from a given start (always Rku ≥ Rsk² + 1); the readouts then show the closest profile. The preset Rsk and Rku values are illustrative: they show the typical sign and size for each process, not measurements of one part.
Bearing area curve and Rk (ISO 13565-2, now in ISO 21920-2)
The material ratio Mr(c) is the fraction of the length where the profile is above height c. Sorting all heights from high to low gives the Abbott-Firestone curve. A secant spanning 40% of Mr is slid along it to find the flattest 40% stretch. A straight line is fitted to the points of that stretch (the equivalent straight line); where it meets Mr = 0% and 100% sets the core band, and its height is Rk. Running-in wears off Rpk first; Rk and Rvk last longer. Mr1 and Mr2 are where the curve crosses the top and bottom of the band. Rpk is the height of a triangle with base Mr1 and the same area A1 as the peaks above the band; Rvk the same for the valley area A2 with base 100% − Mr2.
What the plane shows, and what it does not
The plane is rigid and cuts the profile without pressing it flat (the Abbott-Firestone truncation picture). The red share is the material ratio, not the real area of contact under load, which is smaller and depends on the load and the hardness (see asperities and real contact area). "Oil held below the core" is A2/100, the void below the lower edge of the core per unit area; 1 µm equals 1 ml per m². It assumes the surface looks the same in every direction. The standard computes Rk on a profile filtered with the special ISO 13565-1 filter; here the generated profile is already a roughness profile, so no extra filter is used. "Wear the peaks off" simply cuts the profile flat at the plane, which is a simple picture of running-in, not a wear model.
The Gaussian filter (ISO 16610-21)
The mean line (waviness) is the profile averaged with the weight s(x) = exp(−π(x/αλc)²)/(αλc), α = √(ln2/π) ≈ 0.4697. In wave terms, a wave of length λ keeps a share 2^−(λc/λ)² of its height in the waviness, so exactly half at λ = λc. The roughness is what is left. The page does this in frequency space on a periodic 65.5 mm profile sampled every 1 µm, so there are no end effects. The measured profile is ground-type texture (β = 12 µm, Rq = 0.45 µm) plus waves of 1.3, 4.1 and 16.4 mm; each of the 10 spots is a new random surface with the same statistics. No short-wave λs filter is applied, since the model profile has little detail below a few micrometres.
Choosing the cutoff
For profiles that are not periodic, ISO 4288 (now ISO 21920-3) pairs each cutoff with an Ra range, and the evaluation length is normally 5λc: Ra from 0.006 to 0.02 µm with 0.08 mm, 0.02 to 0.1 µm with 0.25 mm, 0.1 to 2 µm with 0.8 mm, 2 to 10 µm with 2.5 mm and 10 to 80 µm with 8 mm. Measure, check which range your Ra falls in, and change λc until the two agree.
On this site: Real contact area · Stribeck curve · Running-in · Flash temperature
Questions people ask
What is the difference between Ra and Rz?
Ra is the average distance of the profile from its mean line. Rz is the height from the highest peak to the deepest valley, averaged over five sections. Rz reacts strongly to single scratches or spikes, while Ra hardly moves. There is no fixed conversion between them; it depends on the shape of the surface.
What does negative skewness (Rsk) mean?
Most of the surface sits near the top, as flat plateaus, with a few deep valleys. Plateau honed bores and polished surfaces with scratches have negative Rsk. Positive Rsk means sharp peaks stand above a lower base, as on turned surfaces.
What does kurtosis (Rku) tell you?
Rku says how sharp the height distribution is. Rku = 3 for a normal distribution. Above 3, a few very high peaks or very deep valleys stand out; below 3, the heights are spread more evenly, as on a regular turned profile.
What is the Abbott-Firestone curve?
It is the bearing area or material ratio curve: for each height, the share of the profile length that lies above it. Abbott and Firestone introduced it in 1933 to describe how much of a surface would carry load after its peaks wear off. ISO 13565-2 builds Rk, Rpk and Rvk on it.
Which cutoff length should I use for Ra?
For most machined surfaces with Ra between 0.1 and 2 µm, 0.8 mm with a 4 mm evaluation length. Smoother surfaces use 0.25 or 0.08 mm; rougher ones 2.5 or 8 mm. Check that the Ra you measure falls in the range for the cutoff you used.
What changed from ISO 4287 to ISO 21920?
ISO 21920-2 (2021) replaced ISO 4287. Most parameters, such as Ra, are now computed once over the whole evaluation length instead of averaged over sampling lengths. Peak and valley parameters (Rp, Rv, Rz) are still based on sections.
References
Show the 10 references
- ISO 4287:1997, Geometrical Product Specifications (GPS): Surface texture: Profile method: Terms, definitions and surface texture parameters, International Organization for Standardization (withdrawn, replaced by ISO 21920-2).
- ISO 21920-2:2021, Geometrical product specifications (GPS): Surface texture: Profile: Part 2: Terms, definitions and surface texture parameters, International Organization for Standardization.
- ISO 13565-2:1996, Geometrical Product Specifications (GPS): Surface texture: Profile method: Surfaces having stratified functional properties: Part 2: Height characterization using the linear material ratio curve, International Organization for Standardization (withdrawn; the Rk parameters are now in ISO 21920-2).
- ISO 16610-21:2011, Geometrical product specifications (GPS): Filtration: Part 21: Linear profile filters: Gaussian filters, International Organization for Standardization.
- ISO 4288:1996, Geometrical Product Specifications (GPS): Surface texture: Profile method: Rules and procedures for the assessment of surface texture (Table 1, cutoff and Ra ranges for non-periodic profiles), now ISO 21920-3:2021.
- D. J. Whitehouse, Handbook of Surface and Nanometrology, 2nd ed., CRC Press (2011).
- T. R. Thomas, Rough Surfaces, 2nd ed., Imperial College Press (1999).
- R. Leach (ed.), Characterisation of Areal Surface Texture, Springer (2013). doi:10.1007/978-3-642-36458-7
- E. J. Abbott and F. A. Firestone, Specifying surface quality: a method based on accurate measurement and comparison, Mechanical Engineering 55, 569 to 572 (1933).
- M. C. Jones and A. Pewsey, Sinh-arcsinh distributions, Biometrika 96, 761 to 780 (2009). doi:10.1093/biomet/asp053
BibTeX
@misc{tripathy2026surfaceroughness,
author = {Tripathy, Manisha},
title = {Surface Roughness Parameters Lab},
year = {2026},
howpublished = {\url{https://untetheredatom.com/tribology/surface-roughness-parameters}},
note = {Interactive web tool}
}