{"id":6781,"date":"2021-02-01T18:52:48","date_gmt":"2021-02-02T00:52:48","guid":{"rendered":"http:\/\/polywaterv2.wpengine.com\/?p=6781"},"modified":"2025-05-16T11:01:54","modified_gmt":"2025-05-16T16:01:54","slug":"horizontal-bend-pulling-equations-with-bend-radiiand-angles-typical-in-directional-boring","status":"publish","type":"post","link":"https:\/\/www.polywater.com\/en\/knowledge-hub\/horizontal-bend-pulling-equations-with-bend-radiiand-angles-typical-in-directional-boring\/","title":{"rendered":"Horizontal Bend Analysis in Directionally Bored Ducts"},"content":{"rendered":"<div id=\"vyLightbox\"><\/div>\n<h2><strong>Horizontal bend pulling equations<\/strong><\/h2>\n<p>There are two forms of the bend pulling equations used to estimate the pulling tension through a horizontal bend.\u00a0 Equation 1 is the \u201cfull\u201d form equation.\u00a0 Equation 2 is a\u00a0 simplification that is accurate when the tension entering the bend is much greater than the cable weight in the bend.<\/p>\n<table style=\"width: 80%; border: 3px solid #273A80; background-color: #f9f9f9; margin-left: 10%;\">\n<tbody>\n<tr>\n<td style=\"padding: 10px; text-align: center;\" colspan=\"2\"><strong>Conduit Bend Equations<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 70%; padding: 15px;\"><em><strong>T<sub>out <\/sub>= T<sub>in<\/sub> * cosh(w\u03bc\u03b8) + (sinh(w\u03bc\u03b8) * Sqrt(T<sub>in<\/sub><sup>2<\/sup> )+ (WR)<sup>2<\/sup>)<\/strong><\/em><\/td>\n<td style=\"padding: 15px;\"><strong>Equation 1 (full)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 15px;\"><em><strong>T<sub>out<\/sub>\u202f =\u202f T<sub>in<\/sub> * e<sup>w\u03bc\u03f4<\/sup><\/strong><\/em><\/td>\n<td style=\"padding: 15px;\"><strong>Equation 2 (simplified)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 5px;\" colspan=\"2\">Where:<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 5px;\" colspan=\"2\">\u00a0\u00a0<em>T<sub>out<\/sub><\/em> = Tension Coming Out of the Bend (lbf, kg, kN)<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 5px;\" colspan=\"2\">\u00a0\u00a0<em>T<sub>in<\/sub><\/em> = Tension Coming into the Bend (lbf, kg, kN)<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 5px;\" colspan=\"2\">\u00a0<em>\u00a0w<\/em> = Weight Correction Factor (dimensionless)<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 5px;\" colspan=\"2\">\u00a0\u00a0<em>\u03bc<\/em>\u202f=\u00a0Coefficient\u202fof Friction (COF) (dimensionless)<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 5px;\" colspan=\"2\">\u00a0\u00a0<em>\u03f4\u202f<\/em>=\u00a0Angle of Bend (radians)<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 5px;\" colspan=\"2\">\u00a0\u00a0<em>W<\/em> = weight of the cable (lbs,kg)<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 5px;\" colspan=\"2\">\u00a0\u00a0<em>R<\/em> = radius of the bend (ft, m)<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 5px;\" colspan=\"2\">\u00a0\u00a0<em>e<\/em> = Natural\u202fLog Base (constant)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Equation 2 relates to equation 1 as follows:<\/p>\n<ul>\n<li>When <em>T<sub>in<\/sub>\u00a0 &gt;&gt; WR<\/em>, the radical in equation 1 approaches <em>T<sub>in<\/sub><\/em>, that is <em>Sqrt(T<sub>in<\/sub><sup>2<\/sup> )+ (WR)<sup>2<\/sup>)\u00a0 \u2192 T<sub>in<\/sub><\/em><\/li>\n<li>Equation 1 then simplifies to <em>T<sub>out<\/sub>\u00a0=\u00a0T<sub>in<\/sub> *\u00a0(cosh(w\u03f4\u03bc) + sinh(w\u03f4\u03bc))<\/em><\/li>\n<li>Which, by definition, simplifies to <em>T<sub>out<\/sub>=\u00a0T<sub>in<\/sub>* e<sup>w\u03bc\u03f4<\/sup><\/em>\u00a0 (equation 2)<\/li>\n<\/ul>\n<p>So, the accuracy of equation 2 depends on the WR to T<sub>in<\/sub> ratio. The accuracy of equation 2 deceases as the WR\/Tin ratio increases.<\/p>\n<table style=\"width: 100%; border: 3px solid #273A80; background-color: #69c3e8;\">\n<tbody>\n<tr>\n<td style=\"padding: 20px; text-align: center;\"><a href=\"https:\/\/www.polywater.com\/en\/knowledge-hub\/measuring-friction-on-polywaters-friction-table\/\"><strong>Related Content: <\/strong>Measuring Friction on Polywater\u2019s Friction Table<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2><strong><br \/>\n<strong>Comparison of the equations\u00a0based on\u00a0WR\/T<sub>in<\/sub>\u00a0ratio<\/strong><\/strong><\/h2>\n<p>We see that equation 1 will always calculate a greater tension than equation 2 because there is no cable weight in equation 2.\u00a0 The calculational difference between the two equations depends on the specific w, \u03bc, \u03f4, W, and R.\u00a0 Graph1 below plots the percent difference (w = 1, \u03bc = 0.2 and \u03f4 = \u03c0\/2, 90\u00b0) against WR\/T<sub>in<\/sub> ratios.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-7103 size-full\" src=\"https:\/\/polywaterv2.wpengine.com\/wp-content\/uploads\/2021\/02\/horizbendsgraph2.png\" alt=\"\" width=\"633\" height=\"511\" srcset=\"https:\/\/www.polywater.com\/wp-content\/uploads\/2021\/02\/horizbendsgraph2.png 633w, https:\/\/www.polywater.com\/wp-content\/uploads\/2021\/02\/horizbendsgraph2-300x242.png 300w\" sizes=\"auto, (max-width: 633px) 100vw, 633px\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Graph 1.\u00a0 Equation Divergence vs WR\/Tin Ratio<\/strong><\/p>\n<p>The callouts show some key points in the comparison.\u00a0 Equation 2 has diverged from equation 1 by ~1% at a WR\/T<sub>in<\/sub> ratio of 0.30.\u00a0 AEIC suggests the validity of equation 2 at WR\/T<sub>in<\/sub> ratios &lt; 0.5, which is about a 2.6% difference.\u00a0 We see that when T<sub>in<\/sub> = WR (ratio is 1) the calculational difference is ~8.8%. After that, the results quickly diverge.<\/p>\n<table style=\"width: 100%; border: 3px solid #273A80; background-color: #69c3e8;\">\n<tbody>\n<tr>\n<td style=\"padding: 20px; text-align: center;\"><a href=\"https:\/\/www.polywater.com\/en\/knowledge-hub\/coefficient-of-friction-in-cable-pulling-part-3\/\"><strong>Related Content: <\/strong>Coefficient of Friction\u202fin Cable Pulling Tension from Conduit Bends<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2><strong><br \/>\nThe implications of these differences<\/strong><\/h2>\n<p>Higher WR\/T<sub>in<\/sub> ratios (above 0.3) are the result of either a low tension entering the bend or a large bend radius. The latter is common in directionally bored duct. Directional bored bends can have large radii with low displacement angles. What happens to the equations in this situation?<\/p>\n<p>To simplify, the analysis below does not specify units. It will hold for any appropriate force unit (typically lbf, kgf, or N), as long as the weight (typically lb.\/ft, kg\/m, or N\/m) and radius (ft or m) use the equivalent unit.<\/p>\n<p>Graph 2 compares the calculated tension from the two equations. As the degree of bend is decreased the radius is increased to maintain a constant weight of cable in the bend. The specific inputs for the graph are below.<\/p>\n<table style=\"width: 80%; border: 3px solid #273A80; background-color: #f9f9f9; margin-left: 10%;\">\n<tbody>\n<tr>\n<td style=\"padding: 10px; text-align: center;\" colspan=\"2\"><strong>Inputs for Figure 2 Graph<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; padding: 10px;\"><em>T<sub>in<\/sub><\/em> = 1000 (incoming tension)<\/td>\n<td style=\"padding: 10px;\"><em>W<\/em> = 5 (cable weight per length)<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 10px;\"><em>\u03bc<\/em> = 0.2 (coefficient of friction factor)<\/td>\n<td style=\"padding: 10px;\"><em>w<\/em> = 1 (weight correction factor)<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 10px;\" colspan=\"2\"><em>\u03f4<\/em> = bend angle that is varied from 90 to 0 degrees (1.57 to 0 radians)<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 10px;\" colspan=\"2\"><em>R<\/em> = bend radius starts at 63.66 and increases as the angle decreases to maintain a constant cable length of 100 in the bend arc. Note that this also means a constant cable weight of 500 in the bend.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-7107 size-full\" src=\"https:\/\/polywaterv2.wpengine.com\/wp-content\/uploads\/2021\/02\/horizbendsgraph3.png\" alt=\"\" width=\"605\" height=\"491\" srcset=\"https:\/\/www.polywater.com\/wp-content\/uploads\/2021\/02\/horizbendsgraph3.png 605w, https:\/\/www.polywater.com\/wp-content\/uploads\/2021\/02\/horizbendsgraph3-300x243.png 300w\" sizes=\"auto, (max-width: 605px) 100vw, 605px\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Graph 2.\u00a0 Tension Calculations Comparing Equations 1 and 2<\/strong><\/p>\n<p>Equation 1 results are graphed in blue and equation 2 (simplified) in red.\u00a0 Additional perspective is provided by the tension calculated as if the length of cable in the bend arc were a straight run (green line).\u00a0 Because this example was set up with a constant length of cable in the arc, there is a constant calculated straight section add-on of 100 (\u00b5WL) to total 1100 when added to the incoming tension of 1000 (see equation 3 below).<\/p>\n<p>Equation 3:<em> \u00a0 \u00a0 \u00a0T<sub>out<\/sub> = T<sub>in<\/sub> + \u00b5WL<\/em> (straight section equation)<\/p>\n<p>The <em>WR\/Tin<\/em> ratio in the 90\u00b0 calculations in Graph 2 is 0.32 which produces an initial 1+% difference (see Graph 1). The results diverge as the angle of bend decreases.<\/p>\n<table style=\"width: 100%; border: 3px solid #273A80; background-color: #69c3e8;\">\n<tbody>\n<tr>\n<td style=\"padding: 20px; text-align: center;\"><a href=\"https:\/\/www.polywater.com\/en\/knowledge-hub\/coefficient-in-cable-pulling-part-4\/\"><strong>Related Content: <\/strong>Friction Coefficient in Cable Pulling Sidewall Pressure Limits<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We see the calculated tensions from equation 1 approach the straight section tension as the bend angle decreases toward zero. This is what we expect since the decreasing bend approaches a straight section, with a zero (0) bend angle being a straight section.<\/p>\n<p>But equation 2 approaches a tension lower than an equivalent straight section. As the bend angle approaches 0, the multiplier approaches 1. The result is no \u201cadd-on\u201d at all to incoming tension. We know the cable in the bend must add some tension. But we also know that the results are not in a ratio area where the approximation is valid, so it\u2019s use is not appropriate.<\/p>\n<h2><strong>Pull-Planner software<\/strong><\/h2>\n<p>Besides the large radius bends and low bend angle effects shown above, there can also be an issue with lightweight cable, such as fiber optic. The simplest solution is to use the\u00a0 equations in all horizontal bend calculations, which is what our <a href=\"https:\/\/www.polywater.com\/en\/pull-planner-2\/\">Pull-Planner\u2122<\/a> software does. There is no downside since the software does the calculation work, and the results are valid regardless of the <em>WR\/T<sub>in<\/sub><\/em> ratio.<\/p>\n<h2>Have any questions?<\/h2>\n<p><button class=\"button button--primary\" data-micromodal-trigger=\"polywater-modal--email-us-form\">Email Us <\/button><\/p>\n<h2><strong>Reference<\/strong><\/h2>\n<h6><em>Underground Extruded Power Cable Pulling Guide<\/em>, AIEC Publication CG-5-15, 3rd Edition, Association of Edison Illuminating Companies (AEIC), January 2015.<\/h6>\n<div class=\"modal micromodal-slide\" id=\"polywater-modal--email-us-form\" aria-hidden=\"true\"><div class=\"modal__overlay\" tabindex=\"-1\" data-micromodal-close><div class=\"modal__container\" role=\"dialog\" aria-modal=\"true\" aria-labelledby=\"polywater-modal--email-us-form-title\"><header class=\"modal__header\"><h2 id=\"polywater-modal--email-us-form-title\">Email Us Form<\/h2><button class=\"button--secondary modal__close\" aria-label=\"Close modal\" data-micromodal-close><\/button><\/header><div id=\"polywater-modal--email-us-form-content\"><!-- [if lte IE 8]>\r\n<script charset=\"utf-8\" type=\"text\/javascript\" src=\"\/\/js.hsforms.net\/forms\/v2-legacy.js\"><\/script>\r\n<![endif]-->\r\n<script charset=\"utf-8\" type=\"text\/javascript\" src=\"\/\/js.hsforms.net\/forms\/v2.js\"><\/script>\r\n<script>\r\nhbspt.forms.create({\r\n  region: \"na1\",\r\n  portalId: \"6060295\",\r\n  formId: \"7658eeb9-0cf9-469d-a7f1-0f033127bb9d\",\r\n  onFormReady: function (a) {\r\n    a.on('change', function(){\r\n      var zip = a.find('input[name=\"zip\"]');\r\n      if( zip ) {\r\n        if( zip.val() && zip.val().length > 4 ) {\r\n          var url = \"https:\/\/api.zippopotam.us\/us\/\" + zip.val();\r\n          var client = new XMLHttpRequest();\r\n          client.open(\"GET\", url, true);\r\n          client.onreadystatechange = function() {\r\n            if(client.readyState == 4) {\r\n              var resp = JSON.parse(client.responseText);\r\n              var city = resp.places[0]['place name'];\r\n              var state = resp.places[0]['state'];\r\n              a.find('input[name=\"city\"]').val(city);\r\n              a.find('input[name=\"state\"]').val(state);\r\n            };\r\n          };\r\n          client.send();\r\n        }\r\n      }\r\n    });\r\n  },\r\n});\r\n<\/script><\/div><\/div><\/div><\/div>","protected":false},"excerpt":{"rendered":"<p>Horizontal bend pulling equations There are two forms of the bend pulling equations used to estimate the pulling tension through&#8230;<\/p>\n","protected":false},"author":21,"featured_media":7107,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"content-type":"","_relevanssi_hide_post":"","_relevanssi_hide_content":"","_relevanssi_pin_for_all":"","_relevanssi_pin_keywords":"","_relevanssi_unpin_keywords":"","_relevanssi_related_keywords":"","_relevanssi_related_include_ids":"","_relevanssi_related_exclude_ids":"","_relevanssi_related_no_append":"","_relevanssi_related_not_related":"","_relevanssi_related_posts":"","_relevanssi_noindex_reason":"","footnotes":""},"categories":[2992,2982,2984,3013,3004,3009,1],"tags":[991,992,993],"industry-type":[2980,2981],"writer":[],"class_list":["post-6781","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-underground-cabling","category-cable-in-duct-installation","category-cable-pulling","category-cable-blowing","category-content-type","category-research-paper","category-uncategorized","tag-horizontal-bend-equation","tag-calculating-tension","tag-cable-tension-prediction","industry-type-telecommunications","industry-type-electrical-infrastructure"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.7 - 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