 Hello viewers. I am Mr. Shashikant B. Gosavi, Assistant Professor of the Department of Civil Engineering, wishing you best greetings of the time. Here I am presenting an online educational resource on setting out a simple circular curve by Rankine's method of deflection angle. The learning outcomes for this session is that the students will be able to derive the formula for Rankine's method of deflection angle. They will be able to discuss advantages and limitations of Rankine's method of deflection angle. We have already gone through the elements of simple circular curve. Let us revise it once. Here there is an alignment starting from zero-change meeting an intersection point at PI and another alignment moving in the forward direction which are meeting together at a point called as point of intersection. The curve to be set within these two alignments is having a radius r. The two alignments are deflecting away from each other by an angle delta which is shown over here. We have already seen how to derive the tangent lengths on each of the side. Equal length of the tangent length t will be there which is a function of delta r tan delta by 2. And we are able to calculate what is the length of the curve r delta pi by 180. We have also seen how to calculate the length of the long chord to r sin delta by 2. And we have seen how the worst sign or the apex distance can also be calculated as a function of radius and the deflection angle delta. Here are the elements of simple circular curve presented for our derivation purpose. Total deflection angle is assumed to be delta. Radius of curve is r. Chainage of intersection point is i. Tangent length is r tan delta by 2. Length of the curve is r delta pi by 180. Chainage of point of curvature can be derived from this by knowing the chainage of i and deducting the tangent length from that. Chainage of point of tangency can be derived by knowing the chainage of point of curvature and by adding the curve length to that. There are few more angular terms like tangential angle for nth chord is assumed to be delta n, small delta n. And deflection angle for nth chord is assumed to be delta n. The nth chord length is assumed to be cn so it may be c1, c2, c3 for first, second and third chord length. And the central angle for nth chord is 2 delta n. Here it is diagrammatically represented that origin of the curve is o. The radius of curve is r. The curve is between t1 and t2. The respective intersection point is b. The deflection angle, total deflection angle is delta based on which we can calculate the chainage of t1. The aim of our presentation will be to make sure that all the points that we are setting on the curve shall be full stations. That means their chainage should be divisible by the unit chord length. If you know the chainage of t1 and if next chainage has to be a chainage of full station how much should be the length of the first sub chord that can easily be calculated. All the intermediate chords are supposed to be unit chords and only the last chord will be once again a sub chord. Here the respective chord length at this particular location is assumed to be c1. The intermediate chords are c2, c3, c4 up to cn-1 and last chord will be cn. As we know if this deflection angle is delta the angle over here will be delta by 2. Similarly angle over here will be delta by 2 and based on that particular knowledge and these angles to be 90 degree we can easily derive that the central angle will be delta. If the central angle is delta and the angle over here is delta by 2 this gives us a classical relationship between the central angle and the angle made by the tangent with the chord. For the first sub chord the respective angle will be half the angle at the center. So if this is 2 delta 1 this will be delta 1. If this is 2 delta 2 the angle made by the tangent with the chord will be small delta 2. If this is 2 delta 3 angle made by the tangent with the chord will be small delta 3. If I will keep on connecting the point of curvature or first tangent point to the respective ends of the chords I will get first tangential angle delta 1. I will get the tangential angle corresponding to point b as small delta 2 but total tangential angle up to point b will be corresponding to t1b which will be delta 1 plus delta 2 because the central angles are 2 delta 1 plus 2 delta 2. The total angle made by t1 to c in this particular case will be having central angle 2 delta 1 plus 2 delta 2 plus 2 delta 3 and therefore the respective deflection angle will be small delta 1 plus small delta 2 plus small delta 3. This way it will go on accumulating and for the last chord which is the long chord t1 t2 the central angle will be capital delta and therefore last deflection angle for that chord will be equal to delta by 2. Here I have written the respective value central angle for nth chord is 2 delta n and therefore nth chord length cn will be equal to r into 2 delta n. Tangential angle for nth chord will be small delta n that is equal to cn divided by 2 r where r is the radius of the curve and c is the respective chord length. Deflection angle for nth chord will be capital delta n that is summation of delta n small tangential angles ranging from starting tangential angle up to the end tangential angle. I elaborate a little bit deflection angle for the first chord delta 1 will be equal to tangential angle delta 1. Deflection angle for second chord delta 2 will be equal to small delta 1 plus small delta 2 and therefore deflection angle for the last chord will be summation of all the tangential angles up to that point that is small delta 1 plus small delta 2 plus small delta 3 up to small delta n which is ultimately nothing but the total deflection angle divided by 2. I request you to pause the video for a while and answer the following questions. Comment the relationship between the total deflection angle delta and last deflection angle. I also request you to compare Rankine's method with the chord produced method which we have studied in the earlier online educational resource of setting simple circular curve by chord produced method. The relationship between the total deflection angle and last deflection angle was shown in the earlier slide as deflection angle for last chord is equal to capital delta by 2. You can easily see over here that this is the last deflection angle delta by 2. It is having a relation with capital delta which is the total deflection angle as delta by 2. So capital delta n is equal to capital delta divided by 2 is a relation between capital delta and the last deflection angle delta n. The other question that was asked about was regarding what is the comparison of Rankine's method with the chord produced method. See in chord produced method every point on the curve like small a small b small c were established by extending the previous chord forward. So here every next point was likely to carry the error forward into its location which was violation of the basic principle of surveying of working from whole to part. However here if you will see the procedure of setting the simple circular curve by Rankine's method we will be setting point a by setting the theodalite at t1 by measuring the distance c1 and by measuring the angle capital delta1. We will be setting point b on the curve by measuring the angle capital delta2 and by measuring the distance t1 to b or by locating the point b with reference to t1. Similarly point c can be located by measuring the capital delta3 as the angle and the total distance from t1 to c can easily be worked out based on which c can be independently located. In this way all the points on the curve can be independently located and therefore this is a classical example of satisfying the basic principle of surveying of working from whole to part. In addition to that at the end we get a check on our work. The last deflection angle that we will be getting for locating point t2 should be equal to capital delta by 2 that is total deflection angle which is already known to us that divided by 2. And hence we will be ensuring the calculations with cross check with capital delta. I have used the references of surveying by SK Duggal author Volume 2 published by Tata McGrawill publications and I have used one of the sketch from the source www civilengineeringterms.com. Thank you very much for your listening to this online educational resource.