 Best greetings of the time to all of you by me, Mr. Shashikant Bursavi, Assistant Professor of the Department of Civil Engineering, Valshan Institute of Technology, Soolapur. Today I am presenting the online educational resource on setting out vertical curves. The learning outcomes of this OER are that the students will be able to explain the need of vertical curve and they will be able to calculate the elements of vertical curve. To begin with, we will start with elements of vertical curve. As all of us are familiar, the vertical curves are designed in vertical plane. There are two straights having two different gradients which are meeting together at a common point and in that particular case, it may be meeting at the top of a hilly terrain such as a mountain or it may be meeting at a lower point like in a hill, in a valley. So, in both the cases, the vertical curves are needed. The earlier one which is meeting at top is called as submit curve and the lower one is called as the SAC curve. The gradients of two straights which are meeting at the given point can be assumed as G1% and G2% which obviously are the functions of actual field conditions. Normally, rate of change of gradient per chain is given to us from transportation criteria. There are several factors to be taken into consideration like one has to think of the visibility of the other vehicles from coming from other side as well as one has to think about the overtaking safe distance for the vehicle which is bypassing your vehicle. So, by giving consideration to all these things, rate of change of gradient is designed by the transportation criteria and is normally known to us. The length of the curve can be calculated with a very simple calculation of difference of the two gradients divided by r that is difference of gradient 1 and gradient 2 divided by rate of change of gradient. The answer that we will be getting will be number of chains and if you know the unit chord length or chain length by multiplying with the number of chains to this value, you can get the length of curve. If we will see the sketch, you will find that there is a curve which is starting from O, there is a straight actually starting from O reaching to A and the gradient of this particular straight is assumed to be g1 percent and there is another straight going from A to B which is having a gradient g2. Now, while ensuring the safety of the vehicle and comfort of the passenger, it is never recommended to have the movement like this. So, to have smooth change in direction from this straight to other straight, smooth change in gradient is needed and therefore, a curve will be required to be designed in the shape of as has been shown over here O to Q and Q to B where Q is some intermediate point on the curve having the Cartesian coordinates of X and Y. And well, here not only the economy is important, but also safety of the vehicle is important. There are possibilities of allowing a circular curve to be inserted between two straights. Electrical curve also is possible to be inserted, but the most preferred shape for these vertical curves is the parabolic shape. As has been shown in the earlier slide, the OA is equal to rear tangent as g1 percent and AB is forward tangent with g2 percent. Q is the point on curvature with coordinate X and Y. The formula for parabola is Y is equal to ax square plus bx. This is a geometrical formula. Now, if we will take the differentiation of this function, we get dy by dx is equal to 2ax plus b which is equal to g1 at the case when x is equal to 0 and b is equal to g1. You will substitute it over here x being 0 and b is equal to g1. It will become this function as 0 and only b will remain dy by dx is equal to b and then that is nothing but the gradient g1. Y, when we will be substituting x is equal to 0 and b is equal to g1, we will get the formula as y is equal to ax square plus g1x and with the help of this formula, the rest of the setting of vertical curve is possible. If we will further differentiate this function dy by dx is equal to 2ax plus b, we will get dy by dx square is equal to 2a. Now, understand here very well that dy by dx square is a constant which clearly indicates that when we are using parabola, rate of change of gradient is very, very smooth. So, that is the sole reason for which the parabolas are preferred as the vertical curves. The most popular method for setting the vertical curve is tangent correction method. Like in the sketch that I have shown, we will see that pq is equal to h is equal to g1 into minus y that is equal to minus ax square here. eq is this particular value which we are trying to evaluate. p is a point on tangent, q is this particular point on curve which actually we are trying to identify. So, this vertical distance pq with which the point should be lowered down is required to be calculated which is also called as the tangent correction. And this tangent correction will be equal to p to r minus q to r. So, p to r can easily be worked out by using the function that we have seen earlier this particular function. And for this much distance and by knowing the gradient, you can easily find out this particular ordinate. And then by knowing this particular y distance which is a function of x, you can easily calculate y. And then by detecting it from total vertical distance, we will get this tangent correction. My request is there to pause the video for a while and answer following questions. What is the need of vertical curve? Why parabola is best shape for vertical curve? We have used Serving by B.C. Punea and A.K. Jain Volume 2 published by Lakshmi Publication Private Limited as a reference for preparation of this online educational resource. Thank you very much for visiting my online educational resource of Vertical Curve.