{"id":100197,"date":"2026-02-15T20:26:15","date_gmt":"2026-02-15T20:26:15","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100197"},"modified":"2026-07-13T05:18:49","modified_gmt":"2026-07-13T05:18:49","slug":"pressure-vessel-calculator","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/nb\/calculators\/pressure-vessel-calculator\/","title":{"rendered":"Closed-End Thin-Cylinder Membrane-Stress Worksheet"},"content":{"rendered":"\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Closed-End Thin-Cylinder Membrane-Stress Worksheet\",\"description\":\"Estimate uniform hoop and closed-end axial membrane stress in a straight thin circular cylinder under positive internal-over-external pressure difference, within a declared educational model.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/pressure-vessel-calculator\/\",\"applicationCategory\":\"EngineeringApplication\",\"operatingSystem\":\"Any\",\"offers\":{\"@type\":\"Offer\",\"price\":\"0\"},\"creator\":{\"@type\":\"Organization\",\"name\":\"Vibromera\",\"url\":\"https:\/\/vibromera.eu\/\"},\"dateModified\":\"2026-07-13\",\"inLanguage\":\"en\",\"isAccessibleForFree\":true}<\/script>\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Is this an ASME or EN 13445 pressure-vessel design calculation?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"No. It is a general mechanics worksheet for idealized membrane stress. It does not implement the complete construction, material, fabrication, examination, testing or design rules of a pressure-vessel code.\"}},{\"@type\":\"Question\",\"name\":\"Which pressure belongs in the equations?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Use the positive internal-minus-external pressure difference across the wall. Do not enter absolute internal pressure unless external absolute pressure is zero.\"}},{\"@type\":\"Question\",\"name\":\"Why is the axial result limited to closed ends?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"The axial expression assumes pressure end thrust is carried by the cylindrical wall being analyzed. Another load path or additional axial load changes that result.\"}},{\"@type\":\"Question\",\"name\":\"Does radius divided by thickness of ten prove that a vessel is safe?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"No. It is only the applicability gate used by the cited mechanics source for its thin-wall idealization, not a code acceptance or safety criterion.\"}},{\"@type\":\"Question\",\"name\":\"Is the displayed equivalent stress a complete wall stress?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"No. It is the von Mises combination of the two uniform in-plane membrane components. Radial stress, local stresses, discontinuities and other loads are outside the worksheet.\"}}]}<\/script>\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/vibromera.eu\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Calculators\",\"item\":\"https:\/\/vibromera.eu\/calculators\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Thin-cylinder membrane stress\",\"item\":\"https:\/\/vibromera.eu\/calculators\/pressure-vessel-calculator\/\"}]}<\/script>\n<style>\n:root{--vc-surface:#fff;--vc-alt:#f8f6f2;--vc-ink:#1a1a1a;--vc-secondary:#5a5650;--vc-muted:#807b73;--vc-accent:#b84f22;--vc-accent-light:#fdf0ea;--vc-yellow:#825f00;--vc-yellow-light:#fff8dc;--vc-red:#9d2b24;--vc-red-light:#fff0ee;--vc-border:#d9d4cc;--vc-border-light:#e8e4dd;--vc-shadow:0 1px 3px rgba(26,26,26,.06),0 4px 12px 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.vc-chevron{transform:rotate(180deg)}.vc-section-body{display:none}.vc-section.vc-open .vc-section-body{display:block}.vc-section-inner{padding:0 22px 22px;border-top:1px solid var(--vc-border-light);color:var(--vc-secondary)}.vc-section-inner h3{font:700 17px var(--vc-display);color:var(--vc-ink);margin:22px 0 9px}.vc-section-inner p,.vc-section-inner li{font-size:14px}.vc-formula{font:500 13px\/1.75 var(--vc-mono);padding:13px 15px;border:1px solid var(--vc-border);border-radius:6px;background:var(--vc-alt);overflow-x:auto}.vc-warning{padding:13px 15px;border-left:4px solid var(--vc-yellow);background:var(--vc-yellow-light);color:#5d4708;border-radius:4px;margin:14px 0}.vc-table-wrap{overflow-x:auto}.vc-table{width:100%;border-collapse:collapse;margin:14px 0;font-size:12px}.vc-table th,.vc-table td{padding:8px 9px;border:1px solid var(--vc-border-light);text-align:left;vertical-align:top}.vc-table th{background:var(--vc-alt);color:var(--vc-ink)}.vc-faq{border:1px solid var(--vc-border-light);border-radius:6px;margin-top:8px}.vc-faq button{width:100%;padding:13px 14px;border:0;background:var(--vc-alt);font-weight:700;text-align:left;cursor:pointer}.vc-faq div{display:none;padding:13px 14px;border-top:1px solid var(--vc-border-light)}.vc-faq.vc-open div{display:block}.vc-related{display:flex;gap:9px;flex-wrap:wrap;margin-top:14px}.vc-related a{padding:7px 12px;border:1px solid var(--vc-border);border-radius:6px;text-decoration:none;color:var(--vc-secondary)}.vc-footer{text-align:center;padding:28px 12px;color:var(--vc-muted);font-size:12px}.vc-footer a{color:var(--vc-accent)}@media(max-width:760px){.vc-grid,.vc-result-grid{grid-template-columns:1fr}.vc-form,.vc-results{padding:18px}.vc-results-head{align-items:flex-start;flex-direction:column}}@media print{.vc-section-body,.vc-results{display:block!important}.vc-copy,.vc-chevron,.vc-actions{display:none}}\n<\/style>\n<div class=\"vc-calculator\">\n<header class=\"vc-header\"><p class=\"vc-eyebrow\">General mechanics \u00b7 not code design<\/p><h1 class=\"vc-title\">Closed-End Thin-Cylinder Membrane-Stress Worksheet<\/h1><p class=\"vc-subtitle\">Estimate uniform hoop and pressure-induced axial membrane stress in the straight cylindrical wall, far from discontinuities. The model requires positive internal-over-external pressure and a closed-end thrust path through the analyzed wall.<\/p><div class=\"vc-badges\"><span class=\"vc-badge\">\u0394p = p<sub>inside<\/sub> \u2212 p<sub>outside<\/sub><\/span><span class=\"vc-badge\">r<sub>i<\/sub>\/t \u2265 10 model gate<\/span><span class=\"vc-badge\">No pass\/fail verdict<\/span><\/div><\/header>\n<div class=\"vc-card\"><form class=\"vc-form\" id=\"vc-form\" novalidate><div class=\"vc-grid\">\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-units\">Unit system<\/label><select class=\"vc-select\" id=\"vc-units\"><option value=\"metric\">MPa and mm<\/option><option value=\"us\">psi and in<\/option><\/select><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-pressure\">Positive pressure difference \u0394p <span class=\"vc-hint\" id=\"vc-pressure-hint\">(MPa)<\/span><\/label><input class=\"vc-input\" id=\"vc-pressure\" inputmode=\"decimal\" autocomplete=\"off\"><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-diameter\">Internal diameter D<sub>i<\/sub> <span class=\"vc-hint\" id=\"vc-diameter-hint\">(mm)<\/span><\/label><input class=\"vc-input\" id=\"vc-diameter\" inputmode=\"decimal\" autocomplete=\"off\"><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-thickness\">Wall thickness t <span class=\"vc-hint\" id=\"vc-thickness-hint\">(mm)<\/span><\/label><input class=\"vc-input\" id=\"vc-thickness\" inputmode=\"decimal\" autocomplete=\"off\"><\/div>\n<div class=\"vc-field vc-wide\"><label class=\"vc-label\" for=\"vc-basis\">Controlled source \/ calculation basis<\/label><input class=\"vc-input\" id=\"vc-basis\" autocomplete=\"off\" placeholder=\"Drawing, datasheet or calculation-note reference\"><\/div>\n<label class=\"vc-check vc-wide\" for=\"vc-applicable\"><input id=\"vc-applicable\" type=\"checkbox\"><span>I confirm this is a straight circular constant-thickness cylinder, away from ends, heads, nozzles, supports, weld details and other discontinuities; \u0394p is uniform and non-negative; the ends are closed and their pressure thrust is carried by this cylindrical wall; no other axial load acts; the membrane, small-deformation, linear-elastic idealization is applicable; and separate code, material, local-stress, stability, fatigue, creep, corrosion, fabrication, examination and test checks will be made.<\/span><\/label>\n<div class=\"vc-actions vc-wide\"><button class=\"vc-calc-btn\" type=\"submit\">Calculate idealized membrane stress<\/button><\/div>\n<\/div><\/form><div class=\"vc-error\" id=\"vc-error\" role=\"alert\"><\/div>\n<div class=\"vc-results\" id=\"vc-results\" aria-live=\"polite\"><div class=\"vc-results-head\"><div><h2 class=\"vc-results-title\">Model outputs<\/h2><div class=\"vc-results-basis\" id=\"vc-results-basis\">\u2014<\/div><\/div><button type=\"button\" class=\"vc-copy\" id=\"vc-copy\">Copy results<\/button><\/div><div class=\"vc-result-grid\">\n<div class=\"vc-result vc-primary\"><div class=\"vc-result-label\">Hoop membrane \u03c3\u03b8<\/div><div class=\"vc-result-value\" id=\"vc-hoop\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Closed-end axial membrane \u03c3z<\/div><div class=\"vc-result-value\" id=\"vc-axial\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">In-plane membrane equivalent \u03c3eq<\/div><div class=\"vc-result-value\" id=\"vc-equivalent\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Internal radius r\u1d62<\/div><div class=\"vc-result-value\" id=\"vc-radius\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Geometry ratio r\u1d62\/t<\/div><div class=\"vc-result-value\" id=\"vc-ratio\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Relative wall thickness t\/r\u1d62<\/div><div class=\"vc-result-value\" id=\"vc-thinness\">\u2014<\/div><\/div>\n<\/div><div class=\"vc-warning\"><strong>Not an acceptance result:<\/strong> these are idealized membrane components, not allowable stresses, required wall thickness, maximum allowable working pressure or proof of compliance. Do not compare them to yield strength alone and declare a vessel safe.<\/div><\/div><\/div>\n\n<section class=\"vc-section vc-open\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"true\"><span class=\"vc-section-title\">Equilibrium model and variables<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\">\n<h3>Uniform thin-cylinder membrane components<\/h3><div class=\"vc-formula\">r<sub>i<\/sub> = D<sub>i<\/sub>\/2<br>\u03c3<sub>\u03b8<\/sub> = \u0394p r<sub>i<\/sub>\/t = \u0394p D<sub>i<\/sub>\/(2t)<br>\u03c3<sub>z,p<\/sub> = \u0394p r<sub>i<\/sub>\/(2t) = \u0394p D<sub>i<\/sub>\/(4t)<br>\u03c3<sub>eq,mem<\/sub> = \u221a(\u03c3<sub>\u03b8<\/sub>\u00b2 \u2212 \u03c3<sub>\u03b8<\/sub>\u03c3<sub>z,p<\/sub> + \u03c3<sub>z,p<\/sub>\u00b2)<\/div>\n<p>Here \u0394p is internal pressure minus external pressure; D<sub>i<\/sub> is internal diameter; r<sub>i<\/sub> is internal radius; and t is wall thickness. Pressure and stress share a unit. When MPa and mm are used, stress is MPa. The US route uses exactly 1 in = 25.4 mm and 1 psi = 0.006894757293168361 MPa; displayed stress is converted back to psi.<\/p>\n<p>The hoop expression follows transverse force equilibrium of a longitudinally cut segment. The axial expression follows equilibrium of pressure force on a closed end against the membrane force in the cylindrical wall. It therefore does not apply when another structure carries end thrust or when an additional axial load must be included.<\/p>\n<div class=\"vc-warning\"><strong>Radial stress omitted:<\/strong> actual wall stress also includes a radial component that changes through the thickness, plus possible local and bending stresses. The equivalent value above combines only the two stated in-plane membrane components; it is not a complete stress analysis or code utilization.<\/div>\n<\/div><\/div><\/section>\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">Applicability and rejected cases<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\">\n<h3>Geometry gate<\/h3><p>The cited Purdue mechanics source uses radius at least ten times wall thickness as an assumption for its educational thin-wall analysis. This worksheet therefore calculates only when r<sub>i<\/sub>\/t \u2265 10. Equality is accepted; a smaller ratio is rejected.<\/p>\n<div class=\"vc-warning\"><strong>Not a safety boundary:<\/strong> r<sub>i<\/sub>\/t \u2265 10 does not prove thin-shell accuracy for every purpose and does not satisfy any pressure-vessel code. It only prevents use outside the stated classroom model. A required accuracy, diameter convention, stress classification or governing code may demand another method.<\/div>\n<h3>Outside this worksheet<\/h3><ul><li>External-over-internal pressure (negative \u0394p), vacuum and buckling\/stability.<\/li><li>Open ends, restrained ends, force transferred through another component, or any added axial, bending, thermal, dead-weight, wind, seismic, nozzle or support load.<\/li><li>Heads, cones, spheres, junctions, openings, nozzles, flanges, supports, weld details, defects and other local discontinuities.<\/li><li>Plasticity, large deformation, anisotropy, laminates, residual stress, fatigue, creep, fracture, corrosion\/erosion allowance or cyclic service.<\/li><li>Allowable stress, joint efficiency, tolerances, forming\/fabrication effects, examination, testing, relief protection and jurisdictional requirements.<\/li><\/ul>\n<\/div><\/div><\/section>\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">Published reference example<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\">\n<h3>Purdue ME 323 Homework Set 10, Problem 10.3<\/h3><p>The published solution uses r = 3000 mm, t = 20 mm and p = 2 MPa. It reports r\/t = 150, axial stress 150 MPa and hoop stress 300 MPa. Enter D<sub>i<\/sub> = 6000 mm to represent that radius. The worksheet independently reproduces those membrane components and gives 259.807621 MPa for the stated in-plane equivalent combination.<\/p>\n<div class=\"vc-table-wrap\"><table class=\"vc-table\"><thead><tr><th>Input \/ output<\/th><th>Value<\/th><th>Role<\/th><\/tr><\/thead><tbody><tr><td>\u0394p, D<sub>i<\/sub>, t<\/td><td>2 MPa, 6000 mm, 20 mm<\/td><td>Published example, with diameter entered as twice the published radius<\/td><\/tr><tr><td>r<sub>i<\/sub>\/t<\/td><td>150<\/td><td>Published solution<\/td><\/tr><tr><td>\u03c3<sub>z,p<\/sub><\/td><td>150 MPa<\/td><td>Published solution<\/td><\/tr><tr><td>\u03c3<sub>\u03b8<\/sub><\/td><td>300 MPa<\/td><td>Published solution<\/td><\/tr><tr><td>\u03c3<sub>eq,mem<\/sub><\/td><td>259.807621 MPa<\/td><td>Independent plane-stress combination, not a reported Purdue answer<\/td><\/tr><\/tbody><\/table><\/div>\n<\/div><\/div><\/section>\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">Sources and standards boundary<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\">\n<h3>Open mechanics source<\/h3><p><a href=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2019\/10\/Lecture30.pdf\" target=\"_blank\" rel=\"noopener\">Purdue University ME 323 Lecture 30: Thin-walled pressure vessels<\/a> states the r \u2265 10t, insignificant through-thickness strain variation, plane-stress, linear-elastic and small-deformation assumptions and derives the axial and hoop forms for a cylindrical pressure vessel. <a href=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2025\/12\/HW10-Solution-File.pdf\" target=\"_blank\" rel=\"noopener\">Purdue ME 323 Homework Set 10 (Fall 2025), Problem 10.3<\/a> supplies the published numerical example used above.<\/p>\n<p><a href=\"https:\/\/www.nist.gov\/pml\/special-publication-811\/nist-guide-si-appendix-b-conversion-factors\/nist-guide-si-appendix-b8\" target=\"_blank\" rel=\"noopener\">NIST SP 811 Appendix B.8<\/a> gives the international inch and pound-force SI factors used to derive the exact psi route; the pressure factor is independently cross-checked against <a href=\"https:\/\/www.nist.gov\/pml\/special-publication-811\/nist-guide-si-appendix-b-conversion-factors\/nist-guide-si-appendix-b9\" target=\"_blank\" rel=\"noopener\">Appendix B.9<\/a>.<\/p>\n<h3>Pressure-vessel codes are not implemented<\/h3><p><a href=\"https:\/\/www.asme.org\/codes-standards\/find-codes-standards\/bpvc-viii-1-bpvc-section-viii-rules-construction-pressure-vessels-division-1\" target=\"_blank\" rel=\"noopener\">ASME BPVC Section VIII, Division 1 (2025)<\/a> covers design, fabrication, inspection, testing and certification within its scope; the full rules are not reproduced by this three-input equilibrium worksheet.<\/p>\n<p><a href=\"https:\/\/app.nbn.be\/data\/r\/platform\/frontend\/detail?p40_id=3550995&amp;p_lang=en\" target=\"_blank\" rel=\"noopener\">NBN EN 13445-3:2026<\/a>, an official national adoption of EN 13445-3:2026, is active and replaces NBN EN 13445-3:2021+A1:2026. It covers design of unfired pressure vessels within the EN 13445 series. Its licensed detailed rules are not claimed or implemented here. Applicable editions, amendments, jurisdiction and contractual requirements must be established for the real project. Sources\/status accessed 13 July 2026.<\/p>\n<div class=\"vc-warning\"><strong>Classification:<\/strong> the displayed equations are general thin-wall mechanics formulas under the declared assumptions. They are not labelled as an ASME, EN or ISO design formula, and the result is not a normative conformity calculation.<\/div>\n<\/div><\/div><\/section>\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">Frequently asked questions<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\">\n<div class=\"vc-faq\"><button type=\"button\">Is this an ASME or EN 13445 design calculation?<\/button><div>No. It is an idealized mechanics worksheet. Code design requires the applicable licensed rules and many inputs and checks absent here.<\/div><\/div>\n<div class=\"vc-faq\"><button type=\"button\">Should I enter absolute internal pressure?<\/button><div>Enter internal absolute pressure minus external absolute pressure. A gauge reading may equal that difference only when its reference actually represents the external pressure acting on the wall.<\/div><\/div>\n<div class=\"vc-faq\"><button type=\"button\">Why must the ends be closed?<\/button><div>The axial pressure-membrane expression balances pressure thrust on a closed end against membrane force in this wall. A different thrust path or extra axial load changes the axial component.<\/div><\/div>\n<div class=\"vc-faq\"><button type=\"button\">Does r\u1d62\/t \u2265 10 mean the wall is safe?<\/button><div>No. It is only this worksheet\u2019s model-applicability gate from the cited mechanics source. It is not an allowable-stress, stability or code-compliance check.<\/div><\/div>\n<div class=\"vc-faq\"><button type=\"button\">Why is negative \u0394p rejected?<\/button><div>External-over-internal pressure introduces stability and buckling concerns outside this internal-pressure membrane worksheet. Zero \u0394p is accepted and produces zero pressure-induced components.<\/div><\/div>\n<\/div><\/div><\/section>\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">Related calculators<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\"><div class=\"vc-related\"><a href=\"\/calculators\/pressure-unit-converter\/\">Pressure units<\/a><a href=\"\/calculators\/pipe-wall-thickness-pressure\/\">Pipe wall thickness<\/a><a href=\"\/calculators\/thin-walled-pressure-vessel-stress-calculator\/\">Specific vessel-stress worksheet<\/a><\/div><\/div><\/div><\/section>\n<footer class=\"vc-footer\">General mechanics estimate only; retain the controlled input source and complete the governing engineering design. \u00b7 <a href=\"\/calculators\/engineering-calculators\/\">All calculators<\/a><\/footer>\n<\/div>\n<script>\n(function(){\n\"use strict\";\nvar INCH_MM=25.4,PSI_MPA=0.006894757293168361;\nfunction byId(id){return document.getElementById(id)}\nfunction parseNumber(text){var s=String(text).trim();if(!s||s.indexOf('.')>=0&&s.indexOf(',')>=0||!\/^[-+]?(?:\\d+(?:[.,]\\d*)?|[.,]\\d+)(?:[eE][-+]?\\d+)?$\/.test(s))return null;var n=Number(s.replace(',','.'));return Number.isFinite(n)?n:null}\nfunction thinCylinderModel(x){if(!x||!Number.isFinite(x.pressureMPa)||!Number.isFinite(x.internalDiameterMm)||!Number.isFinite(x.thicknessMm))throw new Error('All numeric inputs must be finite.');if(x.pressureMPa<0)throw new Error('\u0394p must be non-negative. External-over-internal pressure and vacuum stability are outside this model.');if(x.internalDiameterMm<=0)throw new Error('Internal diameter must be greater than zero.');if(x.thicknessMm<=0)throw new Error('Wall thickness must be greater than zero.');var radiusMm=x.internalDiameterMm\/2,ratio=radiusMm\/x.thicknessMm;if(!Number.isFinite(radiusMm)||!Number.isFinite(ratio))throw new Error('Geometry is outside the finite numeric range.');if(ratio<10)throw new Error('Outside the thin-wall model gate: internal radius divided by thickness must be at least 10. This is not a safety verdict.');var hoop=x.pressureMPa*radiusMm\/x.thicknessMm,axial=x.pressureMPa*radiusMm\/(2*x.thicknessMm),equivalent=Math.sqrt(hoop*hoop-hoop*axial+axial*axial),thinness=x.thicknessMm\/radiusMm;if(![hoop,axial,equivalent,thinness].every(Number.isFinite))throw new Error('A result is outside the finite numeric range.');return{radiusMm:radiusMm,ratio:ratio,thinness:thinness,hoopMPa:hoop,axialMPa:axial,equivalentMPa:equivalent}}\nfunction toSI(pressure,diameter,thickness,units){if(units==='metric')return{pressureMPa:pressure,internalDiameterMm:diameter,thicknessMm:thickness};if(units==='us')return{pressureMPa:pressure*PSI_MPA,internalDiameterMm:diameter*INCH_MM,thicknessMm:thickness*INCH_MM};throw new Error('Choose a supported unit system.')}\nwindow.vbmThinCylinderModel=thinCylinderModel;window.vbmThinCylinderToSI=toSI;window.vbmThinCylinderParse=parseNumber;window.vbmThinCylinderConstants={INCH_MM:INCH_MM,PSI_MPA:PSI_MPA};\nfunction fmt(n){if(n===0||Object.is(n,-0))return'0';var a=Math.abs(n),s;if(a>=1e9||a<1e-6)s=n.toExponential(9);else s=n.toPrecision(10);return s.replace(\/(\\.\\d*?[1-9])0+(e|$)\/,'$1$2').replace(\/\\.0+(e|$)\/,'$1').replace('e+','e')}\nfunction fail(message){byId('vc-error').textContent=message;byId('vc-error').classList.add('vc-show');byId('vc-results').classList.remove('vc-visible')}\nfunction hideOutput(){byId('vc-error').classList.remove('vc-show');byId('vc-results').classList.remove('vc-visible')}\nfunction clearInputs(){['vc-pressure','vc-diameter','vc-thickness'].forEach(function(id){byId(id).value='' });byId('vc-applicable').checked=false;hideOutput()}\nfunction calculate(){var pressure=parseNumber(byId('vc-pressure').value),diameter=parseNumber(byId('vc-diameter').value),thickness=parseNumber(byId('vc-thickness').value),basis=byId('vc-basis').value.trim(),units=byId('vc-units').value;if(pressure===null||diameter===null||thickness===null)return fail('Enter complete finite numbers. 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