{"id":9203,"date":"2026-03-04T10:10:21","date_gmt":"2026-03-04T08:10:21","guid":{"rendered":"https:\/\/info.formfedern.com\/calculate-moment-of-inertia-of-area\/"},"modified":"2026-03-04T16:35:28","modified_gmt":"2026-03-04T14:35:28","slug":"calculate-moment-of-inertia-of-area","status":"publish","type":"post","link":"https:\/\/info.formfedern.com\/en\/calculate-moment-of-inertia-of-area\/","title":{"rendered":"Calculate moment of inertia of area"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-765\" src=\"https:\/\/info.formfedern.com\/wp-content\/uploads\/\/Federklammer_2-300x196.jpg\" alt=\"Stamped and bent part as a shaped spring from Gutekunst\" width=\"401\" height=\"261\" data-wp-pid=\"765\" srcset=\"https:\/\/info.formfedern.com\/wp-content\/uploads\/Federklammer_2-300x196.jpg 300w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Federklammer_2-200x130.jpg 200w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Federklammer_2.jpg 600w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Federklammer_2-307x200.jpg 307w\" sizes=\"auto, (max-width: 401px) 100vw, 401px\" \/>In the design of <a href=\"https:\/\/info.formfedern.com\/en\/flat-springs\/\" target=\"_blank\" rel=\"noopener noreferrer\">flat springs<\/a>, <a href=\"https:\/\/info.formfedern.com\/en\/individual-shaped-springs-and-flat-springs\/\" target=\"_blank\" rel=\"noopener noreferrer\">shaped springs<\/a> and <a href=\"https:\/\/info.formfedern.com\/flachformfeder_auslegen\/\" target=\"_blank\" rel=\"noopener noreferrer\">flat shaped springs<\/a>, the area moment of inertia is often calculated in addition to the <a href=\"https:\/\/info.formfedern.com\/biegespannung-berechnen\/\" target=\"_blank\" rel=\"noopener noreferrer\">bending stress<\/a>. The area moment of inertia is a geometric quantity used in strength theory. It is used for deformation and stress calculation in bending and torsion loading of springs and <a href=\"https:\/\/info.formfedern.com\/stanzteile-kleinserien\/\" target=\"_blank\" rel=\"noopener noreferrer\">metal formed parts<\/a>. The area moment of inertia is derived from the cross-section of the shaped metal parts. The area moment of inertia is specified in <sup>mm4<\/sup>. Depending on the load, two different types of area moment of inertia can be distinguished.<del><\/del><\/p>\n<p>The following diagrams show the symmetrical and asymmetrical cross-sections of a <a href=\"https:\/\/info.formfedern.com\/en\/form-springs-leaf-springs-and-flat-springs\/\" target=\"_blank\" rel=\"noopener noreferrer\">flat spring<\/a>, <a href=\"https:\/\/info.formfedern.com\/en\/form-springs-leaf-springs-and-flat-springs\/\" target=\"_blank\" rel=\"noopener noreferrer\">shaped<\/a> spring or <a href=\"https:\/\/info.formfedern.com\/flachformfeder-berechnung\/\" target=\"_blank\" rel=\"noopener noreferrer\">flat shaped spring<\/a> subjected to different loads. The illustrations show a flat spring clamped on one side<strong>(1<\/strong> and <strong>2<\/strong>) and the loading of a shaped spring or metal part in bending<strong>(3<\/strong>).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-3592 size-medium\" src=\"https:\/\/info.formfedern.com\/wp-content\/uploads\/\/Flaechentraegheit_Axial_1-300x115.jpg\" alt=\"Axial moment of inertia\" width=\"300\" height=\"115\" data-wp-pid=\"3592\" srcset=\"https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_1-300x115.jpg 300w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_1-768x294.jpg 768w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_1-1024x392.jpg 1024w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_1-600x230.jpg 600w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_1-150x57.jpg 150w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_1-522x200.jpg 522w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_1-200x77.jpg 200w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_1.jpg 1287w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-3593 size-medium\" src=\"https:\/\/info.formfedern.com\/wp-content\/uploads\/\/Flaechentraegheit_Axial_2-300x101.jpg\" alt=\"Axial moment of inertia\" width=\"300\" height=\"101\" data-wp-pid=\"3593\" srcset=\"https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_2-300x101.jpg 300w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_2-768x258.jpg 768w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_2-1024x344.jpg 1024w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_2-600x201.jpg 600w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_2-150x50.jpg 150w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_2-596x200.jpg 596w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_2-200x67.jpg 200w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Axial_2.jpg 1287w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-3594 size-medium\" src=\"https:\/\/info.formfedern.com\/wp-content\/uploads\/\/Flaechentraegheit_Torsion-300x99.jpg\" alt=\"Torsional surface moment of inertia\" width=\"300\" height=\"99\" data-wp-pid=\"3594\" srcset=\"https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Torsion-300x99.jpg 300w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Torsion-768x254.jpg 768w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Torsion-1024x338.jpg 1024w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Torsion-600x198.jpg 600w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Torsion-150x50.jpg 150w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Torsion-606x200.jpg 606w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Torsion-200x66.jpg 200w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Torsion-800x264.jpg 800w, https:\/\/info.formfedern.com\/wp-content\/uploads\/Flaechentraegheit_Torsion.jpg 1093w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/info.formfedern.com\/en\/calculate-moment-of-inertia-of-area\/#Axial_surface_moment_of_inertia\" >Axial surface moment of inertia<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/info.formfedern.com\/en\/calculate-moment-of-inertia-of-area\/#biaxial_surface_moment_of_inertia\" >biaxial surface moment of inertia<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/info.formfedern.com\/en\/calculate-moment-of-inertia-of-area\/#Derived_quantities\" >Derived quantities<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/info.formfedern.com\/en\/calculate-moment-of-inertia-of-area\/#Resistance_torque\" >Resistance torque<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/info.formfedern.com\/en\/calculate-moment-of-inertia-of-area\/#Area_radius_of_gyration\" >Area radius of gyration<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Axial_surface_moment_of_inertia\"><\/span>Axial surface moment of inertia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The axial area moment of inertia <em>Ia<\/em> describes the relationship between the flat spring cross-section and the curvature of a flat spring under load. The following applies: The larger <em>Ia<\/em>, the smaller the curvature and the internal <a href=\"https:\/\/info.formfedern.com\/biegespannung-berechnen\/\" target=\"_blank\" rel=\"noopener noreferrer\">stresses<\/a> arising in the cross-section. What is essential here is the expansion in the direction of the attacking force. The diagram illustrates that a vertical force bends a flat spring less when it is clamped on edge instead of flat<strong>(1<\/strong> and <strong>2<\/strong>). The axial<\/p>\n<p>Area moment of inertia is calculated with the following formulas:<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\Large Iy=\\int_{} z^{2} dA<\/span>\n<ul>\n<li>z = perpendicular distance of the y-axis to the element dA<\/li>\n<\/ul>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\Large Iz=\\int_{} y^{2} dA<\/span>\n<ul>\n<li>y = perpendicular distance of the z-axis to the element dA<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"biaxial_surface_moment_of_inertia\"><\/span>biaxial surface moment of inertia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The biaxial surface moment of inertia <em>Iyz<\/em> is also called surface moment of deviation or surface centrifugal moment. <em>Iyz<\/em> is used to <a href=\"https:\/\/info.formfedern.com\/flachformfeder-berechnung\/\" target=\"_blank\" rel=\"noopener noreferrer\">calculate<\/a> the <a href=\"https:\/\/info.formfedern.com\/biegeverfahren-metallumformung\/\" target=\"_blank\" rel=\"noopener noreferrer\">deformation<\/a> and stresses in loaded shaped springs, flat shaped springs and asymmetrical metal shaped parts<strong>(3<\/strong>).<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\Large Iyz=Izy=\\int_{} yz dA <\/span>\n<p>This quantity is also known as the deviation or centrifugal moment. It is equal to zero if the y-axis or the z-axis is an axis of symmetry of the cross section.<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\Large Iyz = 0 <\/span>\n<p>&nbsp;<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Derived_quantities\"><\/span>Derived quantities<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Various other quantities can be derived from the surface moment of inertia, such as the section modulus and the surface radius of inertia.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Resistance_torque\"><\/span>Resistance torque<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>The section modulus is used in linear elasticity theory. This determines the greatest stress occurring at the edge of the cross-section, the mechanical stress. This value is the quotient of the area moment of inertia and the distance of the edge from the neutral fiber:<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\Large W:=\\frac{I}{\\alpha max}<\/span>\n<p>&nbsp;<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Area_radius_of_gyration\"><\/span>Area radius of gyration<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>For geometrically similar components, such as rectangles with the same width-height ratio, the surface radius of inertia can also be defined with the dimension length. This allows you to compare bodies that are similar in deflection and stiffness. The surface radius of inertia is included in the slenderness ratio.<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\Large iy:=\\sqrt{\\frac{Iy}{A}}<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\Large iz:=\\sqrt{\\frac{Iz}{A}}<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\Large ip:=\\sqrt{\\frac{Ip}{A}}<\/span>\n<p>&nbsp;<\/p>\n<p>See here for more information on the <a href=\"https:\/\/info.formfedern.com\/flachformfeder-berechnung\/\" target=\"_blank\" rel=\"noopener noreferrer\">calculation of<\/a> <a href=\"https:\/\/info.formfedern.com\/en\/flat-springs\/\" target=\"_blank\" rel=\"noopener noreferrer\">flat springs<\/a>, <a href=\"https:\/\/info.formfedern.com\/en\/individual-shaped-springs-and-flat-springs\/\" target=\"_blank\" rel=\"noopener noreferrer\">shaped springs<\/a> and <a href=\"https:\/\/info.formfedern.com\/flachformfeder_auslegen\/\" target=\"_blank\" rel=\"noopener noreferrer\">flat shaped springs<\/a>. If you are looking for a technical spring for your special application, then simply send us the required spring data with details of the required quantity and the drawing or CAD data under the following enquiry button &#8220;<a href=\"https:\/\/info.formfedern.com\/en\/spring-request\/\" target=\"_blank\" rel=\"noopener noreferrer\">Spring enquiry<\/a>&#8221; or by email <a href=\"mailto:info@gutekunst-formfedern.de\">info@gutekunst-formfedern.de<\/a>. We will prepare a non-binding offer for you at short notice.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\"><a class=\"maxbutton-8 maxbutton maxbutton-spring-request-en\" target=\"_blank\" rel=\"noopener\" href=\"\/en\/spring-request\/\"><span class='mb-text'>Spring request<\/span><\/a><\/p>\n<p>&nbsp;<\/p>\n<p><em>For more information:<\/em><\/p>\n<ul>\n<li><a href=\"https:\/\/info.formfedern.com\/en\/form-springs-leaf-springs-and-flat-springs\/\">Form springs, flat springs and leaf springs<\/a><\/li>\n<\/ul>\n<ul>\n<li><a href=\"https:\/\/info.formfedern.com\/en\/stress-strain-diagram-spring-steel\/\" target=\"_blank\" rel=\"noopener noreferrer\">Stress-strain diagram spring steel<\/a><\/li>\n<\/ul>\n<ul>\n<li><a href=\"https:\/\/info.formfedern.com\/flachformfeder-berechnung\/\">Flat shape spring calculation<\/a><\/li>\n<\/ul>\n<ul>\n<li><a href=\"https:\/\/info.formfedern.com\/flachformfeder_auslegen\/\">Lay out flat spring<\/a><\/li>\n<\/ul>\n<ul>\n<li><a href=\"https:\/\/info.formfedern.com\/en\/bending-radius-in-metal-forming\/\" target=\"_blank\" rel=\"noopener noreferrer\">Bending radius in metal forming<\/a><\/li>\n<\/ul>\n<ul>\n<li><a href=\"https:\/\/info.formfedern.com\/en\/important-when-forming-spring-steel-sheet\/\" target=\"_blank\" rel=\"noopener noreferrer\">Important when forming spring steel sheet<\/a><\/li>\n<\/ul>","protected":false},"excerpt":{"rendered":"<p>In the design of flat springs, shaped springs and flat shaped springs, the area moment of inertia is often calculated in addition to the bending stress. The area moment of inertia is a geometric quantity used in strength theory. It<\/p>\n","protected":false},"author":6,"featured_media":9207,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[701,708],"tags":[2956,1326,2960,2958,845,837,839,2957,2955,835,2959,2961],"class_list":["post-9203","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-knowledge","category-leaf-springs","tag-area-moment-of-inertia","tag-bending-stress","tag-bending-stress-en","tag-deformation-calculation","tag-flat-shape-springs-2","tag-flat-springs","tag-leaf-springs","tag-load","tag-resistance-torque","tag-shaped-springs","tag-stress-calculation","tag-torsional-stress"],"_links":{"self":[{"href":"https:\/\/info.formfedern.com\/en\/wp-json\/wp\/v2\/posts\/9203","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/info.formfedern.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/info.formfedern.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/info.formfedern.com\/en\/wp-json\/wp\/v2\/users\/6"}],"replies":[{"embeddable":true,"href":"https:\/\/info.formfedern.com\/en\/wp-json\/wp\/v2\/comments?post=9203"}],"version-history":[{"count":6,"href":"https:\/\/info.formfedern.com\/en\/wp-json\/wp\/v2\/posts\/9203\/revisions"}],"predecessor-version":[{"id":12095,"href":"https:\/\/info.formfedern.com\/en\/wp-json\/wp\/v2\/posts\/9203\/revisions\/12095"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/info.formfedern.com\/en\/wp-json\/wp\/v2\/media\/9207"}],"wp:attachment":[{"href":"https:\/\/info.formfedern.com\/en\/wp-json\/wp\/v2\/media?parent=9203"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/info.formfedern.com\/en\/wp-json\/wp\/v2\/categories?post=9203"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/info.formfedern.com\/en\/wp-json\/wp\/v2\/tags?post=9203"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}