{"id":100130,"date":"2026-02-15T20:19:22","date_gmt":"2026-02-15T20:19:22","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100130"},"modified":"2026-07-12T13:11:48","modified_gmt":"2026-07-12T13:11:48","slug":"hertz-contact-stress","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/sr\/calculators\/hertz-contact-stress\/","title":{"rendered":"Documented Hertz Circular &#038; Parallel-Line Contact Calculator"},"content":{"rendered":"\r\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Documented Hertz Circular and Parallel-Line Contact Calculator\",\"description\":\"Calculate Hertz circular point-contact or parallel-cylinder line-contact size and maximum pressure with explicit convex, flat or concave relative curvature.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/hertz-contact-stress\/\",\"applicationCategory\":\"Engineering Calculator\",\"operatingSystem\":\"Any (Web 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href=\"https:\/\/fonts.googleapis.com\">\r\n<style>:root{--vc-bg:#f4f1ec;--vc-surface:#fff;--vc-surface-alt:#f8f6f2;--vc-ink:#1a1a1a;--vc-ink-secondary:#5a5650;--vc-ink-muted:#8a857e;--vc-accent:#c85a2a;--vc-accent-hover:#b04d22;--vc-accent-light:#fdf0ea;--vc-blue:#2a5c8c;--vc-blue-light:#eaf1f8;--vc-green:#2a7a4b;--vc-green-light:#eaf8ef;--vc-yellow:#a67c00;--vc-yellow-light:#fef9e8;--vc-border:#d9d4cc;--vc-border-light:#e8e4dd;--vc-shadow:0 1px 3px rgba(26,26,26,.06),0 4px 12px rgba(26,26,26,.04);--vc-radius:8px;--vc-radius-lg:12px;--vc-font:'DM Sans',-apple-system,BlinkMacSystemFont,'Segoe UI',sans-serif;--vc-mono:'JetBrains Mono',Consolas,Monaco,monospace;--vc-display:Fraunces,Georgia,serif}.vc-calculator{font-family:var(--vc-font);font-size:15px;line-height:1.65;color:var(--vc-ink);max-width:960px;margin:0 auto;padding:20px 16px 40px;-webkit-font-smoothing:antialiased}.vc-calculator *,.vc-calculator *::before,.vc-calculator 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.15s}.vc-related-link:hover{border-color:var(--vc-accent);color:var(--vc-accent);background:var(--vc-accent-light)}.vc-promo{margin-top:24px;padding:20px;background:linear-gradient(135deg,var(--vc-accent-light),var(--vc-surface));border:1px solid var(--vc-border);border-radius:var(--vc-radius-lg);display:flex;align-items:center;gap:16px}.vc-promo-text{flex:1;font-size:14px;color:var(--vc-ink-secondary)}.vc-promo-text strong{color:var(--vc-ink)}.vc-promo-link{padding:8px 20px;font-size:13px;font-weight:700;color:#fff;background:var(--vc-accent);border-radius:var(--vc-radius);text-decoration:none;white-space:nowrap;transition:background .15s}.vc-promo-link:hover{background:var(--vc-accent-hover)}.vc-footer{text-align:center;padding:32px 16px;font-size:13px;color:var(--vc-ink-muted)}.vc-footer a{color:var(--vc-accent);text-decoration:none}.vc-footer a:hover{text-decoration:underline}.vc-footer-links{margin-top:8px;font-size:12px}.vc-footer-links a{margin:0 8px}@media print{.vc-section-body{max-height:none!important}.vc-faq-a{max-height:none!important}.vc-results{max-height:none!important;padding:24px!important}.vc-copy-btn,.vc-section-chevron,.vc-faq-chevron,.vc-presets,.vc-promo{display:none!important}}.vc-doc-grid{display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:18px}.vc-wide{grid-column:1\/-1}.vc-result-note{font-size:13px;color:var(--vc-ink-secondary);margin:0}@media(max-width:600px){.vc-doc-grid{grid-template-columns:1fr}.vc-wide{grid-column:auto}}.vc-doc-grid{display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:18px}.vc-wide{grid-column:1\/-1}.vc-formula-box{font-family:var(--vc-mono);background:var(--vc-surface-alt);border:1px solid var(--vc-border-light);border-radius:6px;padding:14px;margin:12px 0;line-height:1.75}.vc-result-note{font-size:13px;color:var(--vc-ink-secondary);margin:0}.vc-hidden{display:none!important}@media(max-width:650px){.vc-doc-grid{grid-template-columns:1fr}.vc-wide{grid-column:auto}}<\/style>\r\n<div class=\"vc-calculator\" id=\"vc-hertz\"><header class=\"vc-header\"><p class=\"vc-header-eyebrow\">Classical smooth elastic non-conforming contact<\/p><h1 class=\"vc-header-title\">Documented Hertz Circular &amp; Parallel-Line Contact Calculator<\/h1><p class=\"vc-header-subtitle\">Calculate effective modulus, signed relative curvature, contact size and maximum pressure for either circular sphere contact or the central region of parallel-cylinder line contact.<\/p><div class=\"vc-badges\"><span class=\"vc-badge\">Explicit concave sign<\/span><span class=\"vc-badge\">Point or parallel line<\/span><span class=\"vc-badge\">Not an allowable-stress check<\/span><\/div><\/header>\r\n<div class=\"vc-card\"><form class=\"vc-form\" id=\"vc-form\" autocomplete=\"off\"><div class=\"vc-doc-grid\"><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-id\">Contact case \/ project identifier<\/label><input class=\"vc-input\" id=\"vc-id\" type=\"text\" maxlength=\"220\"><\/div><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-model\">Implemented Hertz model<\/label><select class=\"vc-select\" id=\"vc-model\"><option value=\"\">Select\u2026<\/option><option value=\"point\">Circular point contact: sphere\/sphere or sphere\/flat<\/option><option value=\"line\">Parallel line contact: cylinder\/cylinder or cylinder\/flat<\/option><\/select><\/div><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-curvature\">Surface 2 curvature relative to convex surface 1<\/label><select class=\"vc-select\" id=\"vc-curvature\"><option value=\"\">Select\u2026<\/option><option value=\"external\">External convex contact: curvatures add<\/option><option value=\"flat\">Flat surface 2: zero curvature<\/option><option value=\"internal\">Internal concave contact: curvature subtracts<\/option><\/select><\/div><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-geometry-source\">Geometry, alignment and radius source \/ revision<\/label><input class=\"vc-input\" id=\"vc-geometry-source\" type=\"text\" maxlength=\"450\"><\/div><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-r1\">Convex surface 1 radius R1 <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" id=\"vc-r1\" type=\"text\" inputmode=\"decimal\" maxlength=\"80\"><\/div><div class=\"vc-field\" id=\"vc-r2-field\"><label class=\"vc-label\" for=\"vc-r2\">Surface 2 radius magnitude R2 <span class=\"vc-label-hint\">(mm; blank only for flat)<\/span><\/label><input class=\"vc-input\" id=\"vc-r2\" type=\"text\" inputmode=\"decimal\" maxlength=\"80\"><\/div><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-f\">Documented normal load F <span class=\"vc-label-hint\">(N)<\/span><\/label><input class=\"vc-input\" id=\"vc-f\" type=\"text\" inputmode=\"decimal\" maxlength=\"80\"><\/div><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-load-source\">Load case, distribution and source \/ revision<\/label><input class=\"vc-input\" id=\"vc-load-source\" type=\"text\" maxlength=\"450\"><\/div><div class=\"vc-field\" id=\"vc-length-field\"><label class=\"vc-label\" for=\"vc-length\">Documented loaded cylinder length L <span class=\"vc-label-hint\">(mm; line model)<\/span><\/label><input class=\"vc-input\" id=\"vc-length\" type=\"text\" inputmode=\"decimal\" maxlength=\"80\"><\/div><div class=\"vc-field\" id=\"vc-length-source-field\"><label class=\"vc-label\" for=\"vc-length-source\">Loaded-length and end-effect source <span class=\"vc-label-hint\">(line model)<\/span><\/label><input class=\"vc-input\" id=\"vc-length-source\" type=\"text\" maxlength=\"450\"><\/div><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-e1\">Body 1 Young&#8217;s modulus E1 <span class=\"vc-label-hint\">(GPa)<\/span><\/label><input class=\"vc-input\" id=\"vc-e1\" type=\"text\" inputmode=\"decimal\" maxlength=\"80\"><\/div><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-nu1\">Body 1 Poisson ratio \u03bd1 <span class=\"vc-label-hint\">(\u22121&lt;\u03bd&lt;0.5)<\/span><\/label><input class=\"vc-input\" id=\"vc-nu1\" type=\"text\" inputmode=\"decimal\" maxlength=\"80\"><\/div><div class=\"vc-field vc-wide\"><label class=\"vc-label\" for=\"vc-m1-source\">Body 1 material-state property source \/ revision<\/label><input class=\"vc-input\" id=\"vc-m1-source\" type=\"text\" maxlength=\"450\"><\/div><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-e2\">Body 2 Young&#8217;s modulus E2 <span class=\"vc-label-hint\">(GPa)<\/span><\/label><input class=\"vc-input\" id=\"vc-e2\" type=\"text\" inputmode=\"decimal\" maxlength=\"80\"><\/div><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-nu2\">Body 2 Poisson ratio \u03bd2 <span class=\"vc-label-hint\">(\u22121&lt;\u03bd&lt;0.5)<\/span><\/label><input class=\"vc-input\" id=\"vc-nu2\" type=\"text\" inputmode=\"decimal\" maxlength=\"80\"><\/div><div class=\"vc-field vc-wide\"><label class=\"vc-label\" for=\"vc-m2-source\">Body 2 material-state property source \/ revision<\/label><input class=\"vc-input\" id=\"vc-m2-source\" type=\"text\" maxlength=\"450\"><\/div><div class=\"vc-field vc-wide\"><label class=\"vc-label\" for=\"vc-validity\">Elastic-range and Hertz-applicability assessment reference<\/label><input class=\"vc-input\" id=\"vc-validity\" type=\"text\" maxlength=\"600\"><\/div><\/div><p id=\"vc-error\" role=\"alert\" style=\"margin:.75rem 0 0;color:#b42318\"><\/p><div class=\"vc-warning-box\"><p style=\"margin:0\"><strong>Applicability gate:<\/strong> normal load only; smooth, frictionless, homogeneous isotropic linear-elastic half-spaces; small non-conforming contact; circular symmetry for the point model or straight parallel axes and negligible end effects for the line model. Not for crossed cylinders, sphere\/cylinder, general elliptical races or gear teeth, finite coatings, rough\/adhesive contacts, plasticity or fatigue allowables.<\/p><\/div><\/form>\r\n<div class=\"vc-results\" id=\"vc-results\" aria-live=\"polite\"><div class=\"vc-results-head\"><h2 class=\"vc-results-title\">Documented Hertz result<\/h2><button type=\"button\" class=\"vc-copy-btn\" id=\"vc-copy-btn\">Copy<\/button><\/div><div class=\"vc-result-grid\"><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Effective modulus E*<\/div><div class=\"vc-rcard-value\" id=\"vc-r-estar\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Relative curvature \u03ba<\/div><div class=\"vc-rcard-value\" id=\"vc-r-kappa\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Effective radius R*<\/div><div class=\"vc-rcard-value\" id=\"vc-r-rstar\">\u2014<\/div><\/div><div class=\"vc-rcard vc-rcard-primary\"><div class=\"vc-rcard-label\" id=\"vc-size-label\">Contact radius \/ half-width<\/div><div class=\"vc-rcard-value\" id=\"vc-r-size\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Maximum normal pressure p0<\/div><div class=\"vc-rcard-value\" id=\"vc-r-p0\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\" id=\"vc-area-label\">Contact footprint<\/div><div class=\"vc-rcard-value\" id=\"vc-r-area\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\" id=\"vc-extra-label\">Approach \/ line load<\/div><div class=\"vc-rcard-value\" id=\"vc-r-extra\">\u2014<\/div><\/div><\/div><p class=\"vc-result-note\" id=\"vc-r-note\"><\/p><\/div><\/div>\r\n<div class=\"vc-section vc-open\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"true\"><span class=\"vc-section-toggle-text\"><span class=\"vc-section-icon\">\ud83d\udcd8<\/span><span class=\"vc-section-title\">Equations, source and strict boundary<\/span><\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner vc-theory\"><h3>Shared compliance and signed curvature<\/h3><div class=\"vc-formula-box\">1\/E* = (1\u2212\u03bd1\u00b2)\/E1 + (1\u2212\u03bd2\u00b2)\/E2<br>\u03ba = 1\/R1 + s\/R2 ; R* = 1\/\u03ba<br>s=+1 external convex, s=0 flat, s=\u22121 internal concave<\/div><p>For the internal case, R2 is a positive magnitude but its curvature is subtracted; R2 must exceed R1 so \u03ba remains positive. Treating a concave groove as another positive convex radius is a different geometry.<\/p><h3>Circular point contact<\/h3><div class=\"vc-formula-box\">a = [3FR*\/(4E*)]^(1\/3)<br>p0 = 3F\/(2\u03c0a\u00b2)<br>Ac = \u03c0a\u00b2<br>\u03b4 = a\u00b2\/R*<\/div><p>The displayed \u03b4 is the total normal elastic approach for this circular Hertz model. It is not used for the parallel-cylinder result.<\/p><h3>Parallel-cylinder line contact<\/h3><div class=\"vc-formula-box\">w = F\/L<br>b = [4wR*\/(\u03c0E*)]^(1\/2)<br>p0 = 2w\/(\u03c0b)<br>projected footprint = 2bL<\/div><p>The line model represents the central region of straight parallel cylinders. The footprint is a nominal rectangle; end pressure and edge effects are not calculated.<\/p><h3>Primary published references<\/h3><p>The equations are classical engineering Hertz relations, not \u201cISO formulas.\u201d The <a href=\"https:\/\/emtoolbox.nist.gov\/Elastic\/Documentation.asp\" target=\"_blank\" rel=\"noopener\">NIST Engineering Metrology Toolbox documentation<\/a> states the smooth, elastic, homogeneous and negligible-friction assumptions and links the published Puttock\u2013Thwaite technical paper. NIST separately implements <a href=\"https:\/\/emtoolbox.nist.gov\/elastic\/Case1.asp\" target=\"_blank\" rel=\"noopener\">two external spheres<\/a>, a <a href=\"https:\/\/emtoolbox.nist.gov\/elastic\/Case4.asp\" target=\"_blank\" rel=\"noopener\">sphere in an internal spherical surface<\/a>, and <a href=\"https:\/\/emtoolbox.nist.gov\/elastic\/Case8.asp\" target=\"_blank\" rel=\"noopener\">parallel cylinders<\/a>. Its published 4.45 N, two 25.4 mm diameter steel-sphere case gives 0.519 \u00b5m compression; the displayed equations reproduce 0.5193248 \u00b5m before NIST rounding. For a 25.4 mm ball diameter inside a 50.8 mm spherical diameter with the same properties\/load, NIST gives 0.327 \u00b5m; subtracting curvature reproduces that result.<\/p><h3>Why the former page was changed<\/h3><p>The former \u201cball in groove\u201d preset added both positive radii, although an internal groove requires subtracting curvature. It also advertised approach but did not calculate it, accepted broad gear\/bearing\/wheel-rail applications that commonly require general elliptical contact, embedded unsourced material\/geometry presets, and used partial-number parsing.<\/p><div class=\"vc-warning-box\"><p style=\"margin:0\"><strong>Not a strength, fatigue or bearing-life decision:<\/strong> independently verify elastic limits, subsurface stress state, residual stress, hardness\/yield relation, material anisotropy, coatings, roughness, lubrication\/traction, thermal effects, misalignment, edge loading, plasticity, repeated loading and the applicable component standard or validated numerical model.<\/p><\/div><\/div><\/div><\/div><footer class=\"vc-footer\"><p>\u00a9 2024\u20132026 <a href=\"https:\/\/vibromera.eu\/\">Vibromera<\/a><\/p><p>Classical documented Hertz arithmetic only. Scientific review: July 2026.<\/p><\/footer><\/div>\r\n<script>(function(){'use strict';function $(id){return document.getElementById(id)}var map={hz_id:'vc-id',hz_model:'vc-model',hz_curvature:'vc-curvature',hz_geometry_source:'vc-geometry-source',hz_r1:'vc-r1',hz_r2:'vc-r2',hz_f:'vc-f',hz_load_source:'vc-load-source',hz_l:'vc-length',hz_l_source:'vc-length-source',hz_e1:'vc-e1',hz_nu1:'vc-nu1',hz_m1_source:'vc-m1-source',hz_e2:'vc-e2',hz_nu2:'vc-nu2',hz_m2_source:'vc-m2-source',hz_validity:'vc-validity'};function num(raw){var s=String(raw).trim().replace(',','.');if(s.length>80||!\/^[-+]?(?:\\d+(?:\\.\\d*)?|\\.\\d+)(?:[eE][+-]?\\d+)?$\/.test(s))return NaN;return Number(s)}function fmt(v){if(!Number.isFinite(v))return'\u2014';if(Object.is(v,-0)||v===0)return'0';return Number(v.toPrecision(12)).toString()}function clear(){['vc-r-estar','vc-r-kappa','vc-r-rstar','vc-r-size','vc-r-p0','vc-r-area','vc-r-extra'].forEach(function(id){$(id).textContent='\u2014'});$('vc-r-note').textContent='';$('vc-results').classList.remove('vc-visible')}function fail(m){clear();$('vc-error').textContent=m}function ui(){var flat=$('vc-curvature').value==='flat',line=$('vc-model').value==='line';$('vc-r2-field').classList.toggle('vc-hidden',flat);$('vc-r2').disabled=flat;$('vc-length-field').classList.toggle('vc-hidden',!line);$('vc-length-source-field').classList.toggle('vc-hidden',!line);$('vc-length').disabled=!line;$('vc-length-source').disabled=!line}function setUrl(){var u=new URL(location),model=$('vc-model').value,curv=$('vc-curvature').value;Object.keys(map).forEach(function(q){if((q==='hz_r2'&&curv==='flat')||((q==='hz_l'||q==='hz_l_source')&&model!=='line')){u.searchParams.delete(q);return}var v=$(map[q]).value.trim();if(v)u.searchParams.set(q,v);else u.searchParams.delete(q)});history.replaceState(null,'',u.toString())}function calc(){ui();setUrl();var x={};Object.keys(map).forEach(function(q){x[q]=$(map[q]).value.trim()});var common=['hz_id','hz_model','hz_curvature','hz_geometry_source','hz_r1','hz_f','hz_load_source','hz_e1','hz_nu1','hz_m1_source','hz_e2','hz_nu2','hz_m2_source','hz_validity'];var any=Object.keys(x).some(function(q){return x[q]});if(!any){$('vc-error').textContent='';clear();return}if(!common.every(function(q){return x[q]})){fail('Enter every model, geometry, load, material-property and applicability provenance field.');return}if(x.hz_model!=='point'&&x.hz_model!=='line'){fail('Select a supported circular point or parallel-line model.');return}if(!['external','flat','internal'].includes(x.hz_curvature)){fail('Select an explicit external, flat or internal curvature case.');return}if(x.hz_curvature!=='flat'&&!x.hz_r2){fail('Enter surface 2 radius magnitude for a curved contact.');return}if(x.hz_model==='line'&&(!x.hz_l||!x.hz_l_source)){fail('Enter documented loaded length and its end-effect source for the line model.');return}var r1=num(x.hz_r1),r2=x.hz_curvature==='flat'?Infinity:num(x.hz_r2),f=num(x.hz_f),e1=num(x.hz_e1),nu1=num(x.hz_nu1),e2=num(x.hz_e2),nu2=num(x.hz_nu2),length=x.hz_model==='line'?num(x.hz_l):NaN;if(![r1,f,e1,nu1,e2,nu2].every(Number.isFinite)||(x.hz_curvature!=='flat'&&!Number.isFinite(r2))||(x.hz_model==='line'&&!Number.isFinite(length))){fail('All required numerical fields must be complete finite decimal numbers.');return}if(r1<=0||f<=0||e1<=0||e2<=0||(x.hz_curvature!=='flat'&&r2<=0)||(x.hz_model==='line'&&length<=0)){fail('Radii, load, moduli and line-model length must be strictly positive.');return}if(!(nu1>-1&&nu1<0.5&&nu2>-1&&nu2<0.5)){fail('Each Poisson ratio must be greater than \u22121 and less than 0.5.');return}var normalized=[r1,f,e1,nu1,e2,nu2];if(Number.isFinite(r2))normalized.push(r2);if(Number.isFinite(length))normalized.push(length);if(!normalized.every(function(v){return Math.abs(v)<=1e30})){fail('Normalized inputs must not exceed 10\u00b3\u2070 in magnitude.');return}var sign=x.hz_curvature==='external'?1:x.hz_curvature==='internal'?-1:0,kappa=1\/r1+(sign===0?0:sign\/r2);if(!(kappa>0)){fail('Relative curvature must be positive; 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