 Hello everyone, I welcome you all for the today's session on design of valley course on roads, myself Pashok Kumar, assistant professor, department of civil engineering, Walsh and the Institute of Technology, Solapur. Learning outcomes of the today's class, at the end of the session, students will able to design the length of valley course. In the previous session, we had a discussion on design of submit course, now we are going to design the valley course. So vertical valley course, convexity downwards, so that is known as a valley course or a dip par sac course. Again we have a two gradients minus G1 and plus G2, connecting these two gradients, we are connecting these two gradients by a smooth valley course. The design of valley course will similar to the submit course, but the other, the factors we need to consider, comfort condition, driver comfort condition, side distance criteria at valley course and to locate the cross drainage structures at the lowest point of the valley course, that is the drainage aspects, we need to keep this all the factor while designing the length of valley course. Valley curve forms, where descending gradient, meeting another descending gradient, the calculation of the division angle is that is the algebraic difference between the ascending or descending gradient. Here both are descending gradient, hence n equal to minus n1 plus n2. Here another one, descending gradient, meeting ascending gradient that is minus n1 plus n2, the calculation of the n is minus n1 minus of plus n2. So, here n is the division angle is n equal to minus of n1 plus n2. In this case, a descending gradient emitting a perfectly level ground that is n2 equal to 0, that case n equal to minus n1. Here ascending gradient, meeting another ascending gradient plus n2, so that case n equal to n2 minus n1. Now think for a moment and try to pause the video over here and give the answer for this question, is the necessity of providing a transition curves on valley curves? In submit curve we have seen that there is not requirement of transition curve, but in the valley curves is the necessity of going for valley in the transition curve in the valley curve or not. If so yes, give the reason for that, write down the reason for why the transition curve is required, if not give the reason for that also. I hope you are able to write the answer for this question. Let us discuss about the correct answer for this question. I say the requirement of transition curve in the valley curve is required. The reason behind the requirement of transition curve in valley curve is the centrifugal force along with the gravity and sulphate acts in the downward direction when vehicle is moving at higher speed in the valley curves. So when centrifugal force and gravity acting downwards, this causes the additional pressure on the wheels that is tires as well as the suspension system of the vehicle. So for that reason when centrifugal acceleration exists, then we need to introduce the centrifugal acceleration gradually. So for that case we need to have transition curve, otherwise it causes a discomfort to the passenger and driver and might be sudden jerk and safety aspects. So for all the reasons we need to have transition curve on the valley curves. So next question comes what is the ideal shape of the transition curve? We know that the three shapes are there for the road construction, laminiscate, spiral and cubic parabola because the deviation angle is smaller in the valley curves. The path followed by laminiscate, spiral and cubic parabola are similar. But among all three we select for ideal transition curve as a cubic parabola because the reason is the setting out property or the calculation of the arithmetical properties, calculation of the coordinates, cubic parabola is selected as the ideal transition curve. Suppose if you are going for transition curve, so it will be designed as a fully transition. So there is no any circular curve in between the transition. So in this case we are designing the entire length of the valley curve itself a transition curve. Another factor we need to keep it in the mind is the side distance criteria. Here during the day both the vehicles can each see valley in advance. So there is no problem of any side distance or visibility during the day. So both the vehicles can see each other in the day. But during the night when the vehicle is driving during the night, so there is a lack of the side distance during the night or the lack of visibility during the night. So the entire the visibility depends upon the headlight of the vehicle and what is the inclination of that headlight in the upward direction. So this is happens because there is no any no any there is nothing like street lighting is provided at the valley curve there you must depends upon entirely upon the headlight side distance. If you design this curve is a very steep valley curve you might be observing this headlight is going to fall on the lowest point of the valley curves. So that case the visibility is going to be reduced the you cannot able to see the opposite vehicle or front of the vehicle which is going ahead. So this much distance I must keep it reserved for designing the length of the valley curve. So this headlight at least this distance headlight side distance should be equal to the stopping side distance. So here the this stopping side distance is also called as headlight side distance. To locate the drainage point in the lowest valley curve to have a cross drainage structures at the lowest point to avoid the stagnation of water at the valley curves we need to locate that lowest point. So what is the point of the valley curve it is at the lies on the side of a flatter grid side of the flatter curve are nearest to the tangent first tangent. So that is x1 from tangent to the lowest point that is given by l into square root upon n1 by 2n. So where n1 is the first gradient n is the deviation angle and l is the length of the valley curve. Now let us go for the finding out the length of the valley curve for stopping side distance criteria we need to consider two criteria here one is a comfort criteria that is with respect to the centrifugal acceleration introduction of rate of centrifugal acceleration second condition is with respect to the stopping side distance. So here again the length of the valley curve is designed keeping in the factor that the when the vehicle is at the lowest of the lowest point of the valley curve will have a minimum side distance requirement. Let us see that the h1 is the height of the light above the road surface that is taken usually as the average height that is h1 is taken as points 7 5 meter and inclination of angle the headlight angle in the upward direction that is taken as alpha value. So this alpha value will be taken as approximately a 1 degree. Now for a parabolic curve because we have seen the ideal transition as parabolic though equation of the parabolic is y equal to x square where a is nothing but n upon 2l and y is h1 plus s tan alpha. So this coordinate the s comma h1 plus s tan alpha. So put that y value and x is nothing but the stopping side distance or headlight side distance. So taking all the values so n square l equal to n square upon 2 into h1 plus s tan alpha put the h1 value as average height of the headlight is point 7 5 meter and alpha it is taken as 1 degree. So in this case the equation will become 1 point l equal to n s square upon 1.5 plus 0.035 s where s is the stopping side distance if the stopping side distance is to be required to be calculated. So we know that how to calculate the stopping side distance that is point 2 7 vt plus v square upon 2 54 f. So in this case we are neglecting the both the slopes of on either side. So neglecting the slope we need to calculate the stopping side distance. So this is the equation when length of the curve is greater than s. Second one when length of the curve is less than s. So in this condition again let us take that the vehicle starting at the start of the tangent point or starting of the valley curve. So in this case with the geometry of this figure h1 plus s tan alpha equal to s minus l by 2 into n. So again l equal to 2 s minus 2 h1 plus 2 tan alpha divided by n and put the h1 value as 0.75 and alpha as 1 degree again equation reduces to 2 s minus 1.5 plus 0.035 s divided by n. So in this case we have to determine first assuming the first condition is l is greater than s calculate the length of the curve. In this case your length is the assumption is correct we need to adopt the whatever the derived length of the curve. If in this case the length of the curve is less than s so that case we have to use this equation to calculate the length of the curve. So another one the factor we are considering is the comfort condition to introduce the centrifugal acceleration we need to find out the what is the length of the curve with respect to the comfort condition. So let us see that the ls is the length of the transition curve r is the radius of the curve and v is the speed of the vehicle. And c is the allowable rate of change of centrifugal acceleration that is taken as 0.6 meter per second cube. So ls that is length of the transition curve to have the centrifugal acceleration we know that v cube upon cr where c equal to v square upon r divided by t. So t is know that the time to travel the entire length of the transition that is given by the length of the transition divided by the speed. So put the t value over here so equation will become c equal to v cube upon lr or ls equal to v cube upon cr. So this is the equation now further to calculate the radius of the curve that is ls upon n that is length of the transition divided by the division angle and in this equation put the your r value. So r value if I put the r value that is r equal to n is upon n. So ls equal to nv cube upon c into ls or ls square equal to nv cube upon c or ls equal to square root upon nv cube upon c. So final equation the ls equal to nv cube upon c upon whole to the power of 1 by 2. So that is the total length of the transition curve. So this is the half of the length so we need to multiply by 2 to avoid to get the entire length of the transition that is 2 times of the ls. So ls equal to 2 into nv cube upon c whole to the power of 1 by 2. In this case v is a speed in meter per second. Now we need to calculate the length of transition curve with respect to the side distance and comfort condition. So both we have to calculate and adopt the higher among the side distance or comfort conditions. These are the references I have prepared referred for presenting this presentation. Thank you.