 Hello everyone, I welcome you all for this today's session on design of vertical curves on roads. I am Mr. Ashok Kumar, Assistant Professor, Department of Civil Engineering, Walsh and the Institute of Technology, Solapur. Learning outcomes. At the end of the session, students will be able to calculate the various design parameters of vertical curves. As we know that once the horizontal alignment is completed, we need to design the vertical alignment. So, vertical alignment is the elevation or profile of the center line of the road. So whatever the design horizontal line, we need to design this vertical profile for that. So, that includes the grades. So, there is a gradient, we need to design the gradient and we need to design some vertical parameters such as the vertical curves, the summit curves, valley curves, side distance, so on. So this influence the vertical gradient in the influences the speed, acceleration, side distance and comfort in vehicle moments at high speeds. So, we need to also think on the balancing and cutting and filling. So, this cutting and filling will leads on the increase in the cost of the project. So, carefully design the vertical profile is very important. The another consideration while designing the vertical profile is, we need to design this vertical profile with respect to the topography because the natural, we need to follow the natural topography of that particular stretch with respect to the drainage consideration. So, so many parameters we need to consider before we design the vertical alignment. Let us discuss some of the important parameters of the vertical alignment that is the gradient, deviation angle and grade compensation. Let us go by this one by one. As we know that gradient is the rate of rise or fall along the length of the road with respect to the horizontal. So, with respect to the horizontal, how much is the raise you are trying to do that. So, if it is a ratio of vertical distance to the horizontal distance and it is expressed in terms of 1 in 20 or 1 in n or n percentage. So, n percentage means n vertical units to 100 horizontal units. So, here 1 in 20 minutes, 20 means 1 vertical and 20 horizontal. Another picture depicting over here shows how we are measuring the gradient for this particular stretch. Here you can see this is a falling gradient and with respect to the falling, what is the horizontal you measure it and what is the vertical over here. So, so that the vertical upon horizontal that gives you the what is the grade it is provided. In this case also, let us measure the say that the 10 meter is the horizontal. So, what is the vertical for the 10? So, this vertical upon 10 that gives the what is the slope provided for this particular stretch. So, you can calculate this gradient 5 percent into 1 in 20 or 1 in n and percentage into 1 in n or 1 in n to percentage. So, there is so many ways we can represent the gradient. The effect of the gradient, the careful selection of the gradient is very important. Steeper gradient will affect for the speed of the vehicle and the if the vehicles carrying heavy weights are slowing vehicles. So, steeper gradient creates a problem over there and this will leads to the increasing of the repairing of the vehicles that is such as vehicle operation cost will be increases and the speed will be reduces. So, once the speed reduces, capacity will reduce the reduces and cost of cutting and filling of that particular stretch. So, if you are not balancing cutting and filling though ultimately the cost of the project will increasing. So, we have to keep very consider a lot of consideration side consideration and design the gradient. Sometimes the steeper gradient some of the vehicle comes with a very high speed. So, there is a possibility of having accident on the gradient. So, selection of the gradient entirely depends upon the side constraint. So, going into the side looking at the what are the side condition, what are the natural gradient it is following and what is the required gradient. So, keeping all those things we need to design the select the particular gradient. So, we have a four gradients ruling limiting exceptional and minimum. Let us go by one by one this gradient. Ruling gradient is the maximum gradient given for the designer. Every designer in plain terrain always wants to design lesser than this values. This is the maximum gradient because if I go beyond this maximum gradient, there we come the issues of the steeper gradient then steeper gradient will have effect on the speed and the pulling power of a vehicle. So, 3.3% as per the IRC code for a plain terrain it is given as 3.3%. It is one in terms of one in value it is one in 30 for a ruling gradient. Depending upon the terrain you can see here the terrain is classified as a plain mountainous and steep. You can see here up to 0 to 10 it is classified as a plain terrain and 10 to 25% it is a rolling terrain. This is the percentage is the natural cross slope of that particular country. So, that will indicates what is the slope depending upon that slope they are classified as a rolling mountainous and steep. So, more than 60% cross slope it is given as a steep terrain. For different location different terrain different ruling gradient are given and beyond the ruling gradient in hilly terrain sometimes we need to design the limiting gradient for a smaller length actually. So, there it like in consideration of the cutting and filling we need to go little bit beyond the ruling gradient some percentage. So, that is 5% is given. So, you try to avoid providing longer length of the gradients. So, if a longer length if you are providing so that leads to the again the speed of the vehicle will get decreases. So, longer length will be avoided. So, for every limiting gradient try to provide one horizontal gradient in between. So, if you are giving a two gradient two limiting gradient for longer stretch. So, try to avoid the longer stretch of limiting gradient in between try to give one flatter gradient. So, limiting gradient flatter gradient then limiting gradient like this you try to design the limiting gradient. And sometimes in extraordinary conditions we need to design the exceptional gradient say something another 5 to another 1 to 1.7% we are going beyond the limiting gradient. Again, this is again consideration in when in rare case we provide this is mainly in hilly locations we design this exceptional gradient. Only for small stretches say 100 meter not more than 100 meter or lesser than 100 meter we need to design the exceptional gradient. So, for different locations terrain conditions the different the gradients are given over here. And last gradient is the minimum gradient for we can't design the longitudinal gradient 0% because there is a possibility of the water can stagnate or water can stand on the road surface. So, with respect to the drainage considerations we need to have minimum 0.5% of the longitudinal gradient to be provided on the road section. Let us pass the video I hope you are trying to suggest here the which gradient you are suggesting here for this hilly locations out of the ruling gradient and limiting gradient and expression gradient. So, here I hope you are able to suggest this. I think you are able to give the correct answer here. The correct answer is if I provide the ruling gradient the ruling gradient is a flatter one if I go little bit flatter here what happened if I more the flatter one the more the need to cutting the cut the land cut the this hill section. So, when I cut the hill sections there is a problem comes up with the stability of the hill. So, we need to think on the stability of the hill. So, more the cutting more the worry with the stability of that hill. So, ruling gradient will not able to suit over here. So, limiting gradient say whatever the values given over here 5% or 1 in 20. So, this value we can able to suggest over here that will have a lesser cutting on that particular sections. Another parameter of the vertical the curve is a deviation angle. The deviation angle is that two tangents are two grades meets at that particular location. So, two grades meet and whatever the angle formats over here that measures the deviation angle. So, the deviation angle is given by algebraic sum of the ascending and the descending gradient. So, here it is ascending gradient is given as plus sign and descending is minus sign. This will have difference of ascending and descending this plus n1 minus of minus n2. So, that comes to n1 plus n2. Here another image depicts over here how we are measuring the deviation angle for summit curve. You can hear this is summit curve plus n1 minus n2. So, this gives you plus n1 minus of minus n2. So, that is n1 plus n2 and this is a valley curve minus n2 minus of plus n3. So, that comes minus of n2 plus n3. So, this is valley curve and this is the summit curve. We know that when vehicle is negotiating on horizontal curve looking at this section we get the energy from the engine and the engine energy will be transported to the rear wheels. So, rear wheels have it will pull on this forward direction there is a tractive force is T. When comes on the horizontal curve the tractive force is in this direction and the vehicle will try to go in this direction only. You are turning the vehicle. So, there is some loss of tractive force. So, what is the loss of tractive force? That is T minus cos theta this component is T cos theta. So, tractive loss of tractive force is T minus T cos theta. So, additional energy is required to come over in the horizontal curve. Now, along with the horizontal curve if you have the vertical curve. So, here this is the vertical curve and now the gravity component will try to drag downwards and the self weight of the vehicle also drag downwards. So, this is along the slope and this downwards. Now, along with the horizontal curve and the same location vertical curve will have the vehicle will have required additional tractive force. So, additional energy is required to climb to overcome the horizontal as well as the vertical. So, for that the location when we meet both horizontal and vertical there we need to decrease the grade of the particular location. Though that decreasing the grade we call as a grade compensation. The grade compensation given by the codal provision it is 30 plus R by R. So, R is the radius of the curve and maximum we can go for the grade compensation is 75 by R and as per the IRC code suggests that grade compensation is not necessary for gradients more than 4%. So, if the gradient is more than not more than it is lesser than the 4%. So, if you do the grade compensation is not necessary for lesser than flatter than 4%. So, the gradient is 3.5%. So, not required to apply the grade compensation. So, here the one small number example where the gluing gradient is given 6%. So, it is more than 4%. Hence we require to apply the grade compensation. So, the first calculate the grade compensation that is 1.5% applying the giving the radius value is 60 and the maximum compensation is 75 by R that is 1.25. Compare both two and adopt the lesser one. So, 6 minus 1.25 will be the compensated gradient. These are the references I have referred for presenting this presentation. Thank you.