{"id":66732,"date":"2026-10-13T08:46:20","date_gmt":"2026-10-13T05:46:20","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66732"},"modified":"2026-10-13T08:46:20","modified_gmt":"2026-10-13T05:46:20","slug":"active-earth-pressure","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/engineering\/active-earth-pressure\/","title":{"rendered":"Active Earth Pressure"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Rankine Active Earth Pressure on a Retaining Wall<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66732\" data-favorite-title=\"Active Earth Pressure\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/engineering\/active-earth-pressure\/\" data-favorite-excerpt=\"Calculate the Rankine active earth pressure on a retaining wall from the retained soil height, unit weight and friction angle. Enter H, \u03b3 and \u03c6 to get the active pressure coefficient Ka and the resultant lateral force Pa.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-civil\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/engineering\/active-earth-pressure\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66732\" data-post=\"66732\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Retained soil height (H)<\/label><div class=\"term\">H<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"retained_soil_height_h\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"3\" min=\"0.01\" value=\"3\"\/><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"unit\" aria-label=\"Unit\"><option value=\"m\" selected>m<\/option><option value=\"ft\">ft<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-1\">Vertical height of soil retained behind the wall<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-3\">Soil unit weight (\u03b3)<\/label><div class=\"term\">\u03b3<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"soil_unit_weight_\" id=\"field-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"18\" min=\"0.1\" value=\"18\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"unit\" aria-label=\"Unit\"><option value=\"kNm3\" selected>kN\/m\u00b3<\/option><option value=\"lbft3\">lb\/ft\u00b3<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-3\">Bulk (moist) unit weight of the backfill; typical sands 15\u201320 kN\/m\u00b3, clays 16\u201321 kN\/m\u00b3<\/small><\/div><div class=\"field is-single has-term\" id=\"input-3\"><label for=\"field-5\">Internal friction angle (\u03c6)<\/label><div class=\"term\">\u03c6<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"internal_friction_angle_\" id=\"field-5\" aria-describedby=\"calc-unit-5 calc-info-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"30\" min=\"0\" max=\"89\" value=\"30\"\/><div class=\"unit\" id=\"calc-unit-5\">\u00b0<\/div><small id=\"calc-info-5\">Effective friction angle of the backfill; loose sand 28\u201332\u00b0, dense sand 35\u201342\u00b0, gravel 35\u201345\u00b0<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66732\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Active Earth Pressure Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Active earth-pressure coefficient<\/span><span class=\"result-term\">K\u2090<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Resultant lateral force per unit wall length<\/span><span class=\"result-term\">P\u2090<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-3\"><span class=\"field-title\"><span>Formula and notes<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"66732\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Active Earth Pressure Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ K_a = \\frac{1 - \\sin\\varphi}{1 + \\sin\\varphi} = \\tan^2\\!\\left(45^\\circ - \\frac{\\varphi}{2}\\right) \\\\\n\\[ p_z = \\gamma \\, z \\, K_a \\\\\n\\[ P_a = \\tfrac{1}{2}\\,\\gamma\\,H^2\\,K_a \\]<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(K_a\\) \u2014 active earth-pressure coefficient (dimensionless)<\/li>\n<li>\\(\\varphi\\) \u2014 internal (effective) friction angle of the backfill<\/li>\n<li>\\(\\gamma\\) \u2014 bulk unit weight of the backfill soil<\/li>\n<li>\\(H\\) \u2014 vertical height of retained soil behind the wall<\/li>\n<li>\\(p_z\\) \u2014 lateral pressure at depth \\(z\\) below the ground surface<\/li>\n<li>\\(P_a\\) \u2014 resultant lateral force per unit length of wall<\/li>\n<\/ul>\n<p>These are the classical Rankine (1857) formulas for a vertical, frictionless wall retaining a cohesionless soil with a horizontal ground surface. The lateral pressure increases linearly with depth, so the pressure diagram is a triangle and the resultant force \\(P_a\\) acts at a height of \\(H\/3\\) above the base of the wall.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Active Earth Pressure \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a retaining wall holds back 3 m of granular backfill with a unit weight of 18 kN\/m\u00b3 and a friction angle of 30\u00b0.<\/p>\n<p>\\[ K_a = \\frac{1 - \\sin 30^\\circ}{1 + \\sin 30^\\circ} = \\frac{1 - 0.5}{1 + 0.5} = 0.3333 \\]<\/p>\n<p>\\[ P_a = \\tfrac{1}{2} \\times 18 \\times 3^2 \\times 0.3333 = 27 \\ \\text{kN\/m} \\]<\/p>\n<p><strong>Result:<\/strong> the active earth-pressure coefficient is \\(K_a = 0.333\\), and the resultant lateral force is 27 kN per metre run of wall, acting at \\(H\/3 = 1\\) m above the base. The pressure at the base of the wall is \\(\\gamma H K_a = 18 \\times 3 \\times 0.3333 = 18\\) kPa.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the <strong>Retained soil height (H)<\/strong> \u2014 the vertical height of soil behind the wall \u2014 and select its unit (m or ft).<\/li>\n<li>Enter the <strong>Soil unit weight (\u03b3)<\/strong> of the backfill and choose kN\/m\u00b3 or lb\/ft\u00b3. Typical sands fall in 15\u201320 kN\/m\u00b3, clays in 16\u201321 kN\/m\u00b3.<\/li>\n<li>Enter the <strong>Internal friction angle (\u03c6)<\/strong> in degrees. Loose sand is roughly 28\u201332\u00b0, dense sand 35\u201342\u00b0, gravel 35\u201345\u00b0.<\/li>\n<li>Read the <strong>Active earth-pressure coefficient<\/strong> (Ka) and the <strong>Resultant lateral force per unit wall length<\/strong> (Pa), together with the point of application and the base pressure.<\/li>\n<\/ol>\n<h2>What Is Active Earth Pressure?<\/h2>\n<p>Active earth pressure is the lateral push that a soil mass exerts on a retaining structure when the structure moves or rotates slightly away from the soil, allowing the soil to expand and mobilize its shear strength. This outward movement \u2014 typically only a few millimetres to a few hundredths of the wall height \u2014 is enough for a wedge of soil behind the wall to reach a limiting, \"active\" failure state. The word \"active\" refers to the soil actively pushing on the wall, and it is the smallest of the three classical lateral pressure states: active, at-rest and passive.<\/p>\n<p>Rankine's theory derives the active pressure coefficient purely from the soil's friction angle, assuming the wall face is vertical and frictionless and the backfill surface is horizontal. Under these conditions the soil fails along a plane inclined at \\(45^\\circ + \\varphi\/2\\) to the horizontal, and the ratio of horizontal to vertical stress at every depth equals \\(K_a\\). A lower friction angle gives a larger \\(K_a\\): at \\(\\varphi = 30^\\circ\\), \\(K_a \\approx 0.33\\); at \\(\\varphi = 40^\\circ\\) it drops to about 0.22, which is why well-graded, freely draining granular backfill is strongly preferred behind walls.<\/p>\n<p>The result \\(P_a\\) is a force per metre run of wall (kN\/m), obtained by integrating the triangular pressure diagram over the height. It acts horizontally at one third of the wall height above the base \u2014 the centroid of the triangle. In stability checks this force drives sliding and overturning, so it is the key input for sizing the wall, its footing bearing pressure and any reinforcement. For the structural design of the wall stem as a bending member, the triangular pressure distribution can be treated like a linearly varying load on a cantilever; see the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/simply-supported-beam-with-uniform-load\/\">uniformly loaded beam calculator<\/a> for the general bending relationships.<\/p>\n<h2>Assumptions, Limitations and Practical Tips<\/h2>\n<ul>\n<li><strong>Granular backfill only.<\/strong> Rankine's \\(K_a\\) assumes a cohesionless soil (\\(c = 0\\)). For cohesive soils, apparent cohesion temporarily reduces active pressure but can be unreliable long-term \u2014 use total-stress or more advanced analysis for clays.<\/li>\n<li><strong>Vertical, frictionless wall and level backfill.<\/strong> Wall friction, a battered face or a sloping ground surface change \\(K_a\\); Coulomb's theory or the sloping-backfill form of Rankine's coefficient should be used instead.<\/li>\n<li><strong>No water, no surcharge.<\/strong> Hydrostatic pressure behind the wall (e.g. from a blocked drain) adds \\(\\tfrac{1}{2}\\gamma_w H^2\\) and often governs the design \u2014 always provide drainage. Uniform surcharge \\(q\\) adds an extra \\(K_a\\, q\\) to the pressure at every depth.<\/li>\n<li><strong>Wall must be free to yield.<\/strong> Active pressure only develops if the wall can move outward a little. Rigid, braced or basement walls that cannot yield should be designed for the larger at-rest pressure.<\/li>\n<li><strong>Check the base.<\/strong> The base pressure \\(\\gamma H K_a\\) and the point of application \\(H\/3\\) are what you need for overturning and sliding checks; the footing dimensions then determine the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/footing-bearing-pressure\/\">bearing pressure under the base<\/a>.<\/li>\n<li><strong>Seismic action.<\/strong> Under earthquake loading the lateral demand increases; the Mononobe\u2013Okabe method extends this coefficient approach to pseudo-static conditions.<\/li>\n<\/ul>\n<p>Once the wall geometry is fixed, reinforcement detailing follows: estimate the required steel area from the bending moment and check bar layout with the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/rebar-spacing-and-count\/\">rebar spacing and count calculator<\/a> and the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/rebar-weight\/\">rebar weight calculator<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the Rankine active earth pressure on a retaining wall from the retained soil height, unit weight and friction angle. Enter H, \u03b3 and \u03c6 to get the active pressure coefficient Ka and the resultant lateral force Pa.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[4],"tags":[19],"class_list":["post-66732","post","type-post","status-publish","format-standard","hentry","category-engineering","tag-civil"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66732","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=66732"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66732\/revisions"}],"predecessor-version":[{"id":66734,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66732\/revisions\/66734"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66732"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66732"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66732"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}