{"id":735,"date":"2024-07-04T17:33:15","date_gmt":"2024-07-04T12:03:15","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=735"},"modified":"2025-07-16T14:57:24","modified_gmt":"2025-07-16T09:27:24","slug":"vertical-curves","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ce\/surveying\/vertical-curves","title":{"rendered":"VERTICAL CURVES"},"content":{"rendered":"<p style=\"text-align: justify;\">Vertical curve joins the two intersecting lines which lie in a vertical plane. e.g. Curve provided to connect two roads at the lower most point of a valley. Vertical curves are further classified as:<\/p>\r\n<ul style=\"text-align: justify;\">\r\n<li>Summit curve<\/li>\r\n<li>Sag curve<\/li>\r\n<\/ul>\r\n<p style=\"text-align: justify;\">Summit curve and Sag curve are provided in following situations.<\/p>\r\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-447 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/seg-curve.jpg\" alt=\"Seg Curve\" width=\"641\" height=\"227\" \/><\/p>\r\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-448 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/summit-curve.jpg\" alt=\"Summit curve\" width=\"634\" height=\"356\" \/><\/p>\r\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_79_1 ez-toc-wrap-left counter-hierarchy ez-toc-counter ez-toc-light-blue 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\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/surveying\/vertical-curves\/#Degree-of-Curve\" >Degree of Curve<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/surveying\/vertical-curves\/#Angular-Methods-of-Setting-a-Simple-Circular-Curve\" >Angular Methods of Setting a Simple Circular Curve<\/a><\/li><\/ul><\/nav><\/div>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Degree-of-Curve\"><\/span>Degree of Curve<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<ul style=\"text-align: justify;\">\r\n<li>The two important parameters of a circle (or the circular curve) are its center and the radius.<\/li>\r\n<li>Often the center of the curve which is to be laid out on a highway or for a railway is inaccessible and in such cases, the radius of the curve has no importance.<\/li>\r\n<li>In order to set out a curve in such situations, we define the degree of curve. This degree of curve is defined either on the basis of arc or the chord.<\/li>\r\n<li><strong>Arc definition of degree of curve:<\/strong> According to this, the degree of curve is the central angle subtended by an arc of length 20 m or 30 m as the case may be.<\/li>\r\n<\/ul>\r\n<p style=\"text-align: justify;\">Let\u00a0\u00a0\u00a0 \u00a0<em>R <\/em>= Radius of the curve<\/p>\r\n<p style=\"text-align: justify;\"><em>D <\/em>= Degree of the curve as per arc definition for a 30 m arc length i.e. a 30 m arc length subtends an angle of <em>D<\/em>\u00b0 at the center.<\/p>\r\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-451 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/degree-arc.jpg\" alt=\"Degree Arc\" width=\"451\" height=\"220\" \/><br \/><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-450 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/arc-definition.jpg\" alt=\"Arc definition\" width=\"239\" height=\"214\" \/><\/p>\r\n<ul style=\"text-align: justify;\">\r\n<li><strong>Chord definition of degree of curve:<\/strong> According to this, the degree of curve is the central angle subtended by a chord of length 20 m or 30 m as the case may be.<\/li>\r\n<\/ul>\r\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-452 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/chord-definition.jpg\" alt=\"Chord definition\" width=\"233\" height=\"243\" \/><\/p>\r\n<p style=\"text-align: justify;\">Let\u00a0\u00a0\u00a0 \u00a0<em>R <\/em>= Radius of the curve<br \/><em>D <\/em>= Degree of the curve as per chord definition for a 30 m chord length i.e. a 30 m chord length subtends an angle of <em>D<\/em>\u00b0 at the center.<\/p>\r\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-453 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/radians.jpg\" alt=\"Radians\" width=\"645\" height=\"467\" \/><\/p>\r\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Angular-Methods-of-Setting-a-Simple-Circular-Curve\"><\/span>Angular Methods of Setting a Simple Circular Curve<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<ul style=\"text-align: justify;\">\r\n<li>Rankine\u2019s method of deflection angle<\/li>\r\n<li>Two theodolite method<\/li>\r\n<li>Tacheometric method<\/li>\r\n<\/ul>\r\n<h5 style=\"text-align: justify;\">Rankine\u2019s Method of Deflection Angle<\/h5>\r\n<ul style=\"text-align: justify;\">\r\n<li>In this method, tape is used for making linear measurements and a theodolite is used for making angular measurements. This method is also known as tape and theodolite method.<\/li>\r\n<li>This method is quite suitable for setting out a circular curve of large radius and large length.<\/li>\r\n<li>From geometric theorem on circle, the deflection angle to any point on curve is the angle subtended at the <em>PC<\/em> (<em>T<\/em><sub>1<\/sub>) between the tangent and the chord from <em>PC <\/em>to that point. This deflection angle is one- half of the angle subtended by the arc at the center.<\/li>\r\n<\/ul>\r\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-454 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/rankine-method.jpg\" alt=\"Rankine\u2019s Method\" width=\"675\" height=\"560\" \/><\/p>\r\n<h5 style=\"text-align: justify;\">Two Theodolite Method<\/h5>\r\n<ul style=\"text-align: justify;\">\r\n<li>It is the most preferred method of setting out a circular curve when the ground is undulating, rough and is also not suitable for the linear measurements.<\/li>\r\n<li>In this, no linear measurements are done. Instead two theodolites are used which gives more precision. It is based on the principle that:<br \/><em>\u201cThe angle subtended by a tangent and a chord at a point on a circle is equal to the angle <\/em><em>subtended by the chord in the opposite segment.\u201d<\/em><\/li>\r\n<\/ul>\r\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-456 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/two-theodolite-method.jpg\" alt=\"Two theodolite method\" width=\"282\" height=\"229\" \/><\/p>\r\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-457 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/theodolites.jpg\" alt=\"Theodolites\" width=\"324\" height=\"91\" \/><\/p>\r\n<p style=\"text-align: justify;\"><strong>Procedure:<\/strong><\/p>\r\n<p style=\"text-align: justify;\"><strong>Step 1.<\/strong> Set up the theodolite at point of curve, <em>T<\/em><sub>1<\/sub> and other theodolite at point of tangent, <em>T<\/em>2.<\/p>\r\n<p style=\"text-align: justify;\"><strong>Step 2.<\/strong> Set the Vernier <em>A <\/em>of both the theodolites to zero.<\/p>\r\n<p style=\"text-align: justify;\"><strong>Step 3.<\/strong> Direct the theodolite at <em>T<\/em><sub>1<\/sub> towards <em>V <\/em>and theodolite at <em>T<\/em><sub>2<\/sub> towards <em>T<\/em><sub>1<\/sub>.<\/p>\r\n<p style=\"text-align: justify;\"><strong>Step 4.<\/strong> Set angle d<sub>1<\/sub> in both the theodolites so as to direct the line of sights towards <em>T<\/em><sub>1<\/sub><em>a <\/em>and <em>T<\/em><sub>2<\/sub><em>a <\/em>thereby locating the point \u2018<em>a<\/em>\u2019 on the curve. The point \u2018<em>a<\/em>\u2019 is the point of intersection of the two lines of sight.<\/p>\r\n<p style=\"text-align: justify;\"><strong>Step 5.<\/strong> Similarly establish point \u2018<em>b<\/em>\u2019 by setting an angle d<sub>2<\/sub> in both the theodolites.<\/p>\r\n<p style=\"text-align: justify;\"><strong>Step 6.<\/strong> Repeat the above steps with different values of d to get different points on the curve.<\/p>\r\n<p style=\"text-align: justify;\"><strong>Drawback :<\/strong> It is expensive and time consuming but at the same time it is the most precise method as well. All the points are established independently and thus error in locating a particular point is not carried forward to other points.<\/p>\r\n<h5 style=\"text-align: justify;\">Tacheometric Method<\/h5>\r\n<ul style=\"text-align: justify;\">\r\n<li>This method is very similar to Rankine\u2019s method of deflection angle.<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-460 alignright\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/tacheometric-method.jpg\" alt=\"\" width=\"258\" height=\"250\" \/><\/li>\r\n<li>The theodolite at <em>T<\/em><sub>1<\/sub> is used as a tacheometer and tacheometric observations are made.<\/li>\r\n<li>It is less precise than Rankine\u2019s method but here in this method, the chaining operation is completely eliminated.<\/li>\r\n<li>Point on the curve is established by setting the deflection angle in the tacheometer and measuring the distance of the point on the curve by placing a staff on it.<\/li>\r\n<li>The points <em>a<\/em>, <em>b<\/em>, <em>c<\/em>,&#8230;.. So obtained are joined by <em>T<\/em><sub><sub>1 <\/sub><\/sub>and the chord lengths are given by:<\/li>\r\n<\/ul>\r\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-461 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/tacheometric.jpg\" alt=\"tacheometric\" width=\"588\" height=\"236\" \/><\/p>\r\n\r\n<p style=\"text-align: center;\"><a class=\"btn btn-danger\" role=\"button\" href=\"https:\/\/study.madeeasy.in\/ce\/surveying\/horizontal-curve-s\/\" target=\"_blank\" rel=\"noopener\">&lt;&lt; Previous<\/a> | <a class=\"btn btn-success\" role=\"button\" href=\"https:\/\/study.madeeasy.in\/ce\/surveying\/compound-curve\/\" target=\"_blank\" rel=\"noopener\"> Next &gt;&gt;<\/a> <br \/><strong> Must Read: <\/strong> <a href=\"https:\/\/study.madeeasy.in\/subjects\/what-is-surveying\/\" target=\"_blank\" rel=\"noopener\"><strong>What is Surveying?<\/strong><\/a><\/p>\r\n<p>&nbsp;<\/p>\r\n","protected":false},"excerpt":{"rendered":"<p>Vertical curve joins the two intersecting lines which lie in a vertical plane. e.g. Curve provided to connect two roads<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[203,2],"tags":[253,251,254,255],"class_list":["post-735","post","type-post","status-publish","format-standard","hentry","category-surveying","category-ce","tag-rankine-method","tag-simple-circular-curve","tag-tacheometric","tag-two-theodolite-method"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/735","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/comments?post=735"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/735\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=735"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=735"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=735"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}