Banking-OF- Highway- Curves- Module PDF

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Personal made module about Banking of Highway Curves in Engineering Mechanics...

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BES 121 ENGINEERING MECHANICS

BANKING OF HIGHWAY CURVES

BALTAZAR, CHARLES ANDRE BARCELONA, LORAINE BRETAA, JOHN ANDRIE CADANO, KAYCEE

COLLEGE OF ENGINEERING 1|P age

WESTERN MINDANAO STATE UNIVERSITY Copyright © by Western Mindanao State University All rights reserved. Published (2021) Printed in the Philippines ISBN 978-971-0487-42-4 No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior written permission of WESTERN MINDANAO STATE UNIVERSITY

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TABLE OF CONTENTS

BANKING OF HIGHWAY CURVES Introduction

5

Objectives

5

Topic Outline

6

Concept of Banking of Highway Curves

8

Why Banking of Highway Curves Important

8

Necessity Banking of Highway Curves

8

Ideal Banking Conditions

9

Formulas of Banking Highway Curves

10

Application of Formula

11

Answer Key

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BACKGROUND OF THE MODULE This module is open to everyone. This module will give you a better grasp about the concept of Banking of Highways, as well as the angle of banking and its application. Banking of the road is raising the outer end of the road higher than the inner end. In simple words, banking on the road is making its surface slant. In this module, you'll discover how the idea of "banking of roadways" works in this module. You'll be able to improve your understanding of the banking angle and why it's important. Banking of roads is needed to increase safety while taking a turn. To obtain an expression for the maximum speed, first draw all the forces acting on a vehicle taking a turn on the banked road. Resolve all the forces in the horizontal and vertical components. Equate the net force in the vertical direction to zero. The angle at which the vehicle is inclined is defined as the bank angle. The occurrence of lifting the outer border of the curved road over the inner border is to give necessary combining force to the vehicles to take a guarded turn, and the curved road is termed "banking of roads."

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I.

BANKING OF HIGHWAY CURVES

Introduction

Banking of a highway is done to provide that centripetal force. During a “banked” or inclined turn, the chances of skidding reduce. A turn is made inclined with the horizontal such that the outer edge is lifted up. For a particular angle of inclination, the maximum allowed speed of a vehicle is restricted. When a vehicle tends to make a turn along a curved road, there is a probability of it to skid. Banking of highway curve is defined as the phenomenon in which the edges are raised for the curved roads above the inner edge to provide the necessary centripetal force to the vehicles so that they take a safe turn. When a car takes a turn, each curvilinear section of the turn can be approximated to be an arc of a circle with a certain radius. A centripetal force must act towards the center of the arc to enable the vehicle to move along. This centripetal force is provided by the static friction of the surface of the car that is moving on.

Objectives

At the end of this lesson, you should be able to:

1. define the concept of banking of highways; 2. define the purpose of banking highway curves; 3. determine the importance of banking of highway curves; and 4. Application of formula in banking of highways.

Topic Outline: 5|P age

Concept of Banking on Highway Curves Why Banking of Highways? Angle of Banking Banking of Highways Formula

Try this! Direction: Consider the pictures below. Pair with a classmate and try to understand what ideas or information you can get from the photos. After come up with your ideas, summarize your ideas and put it inside the box.

Banked Curves

Think Ahead!

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Direction: Provide the details below by answering each question and identifying what is being asked. Fillout your answers for each of the question.

1. How do you define banking of highway curves?

2. What does banked curve do?

3. What is the difference between banked and unbaked curves?

Read and Ponder A banked curve is a curve that has its surface at angle with respect to the ground on which the curve is positioned. The reason for banking curves is to decrease the moving object's reliance on the force of friction. On a curve that is not banked, a car traveling along that curve will experience a force of static friction that will point towards the center of the circular pathway circumscribed by the moving car. This frictional force will be responsible for creating centripetal acceleration, which in turn will allow the car to move along the curve. On a banked curve however, the normal force acting on the object (such as a car) will act at an angle with the horizontal, and that will create a component normal force that acts along the x axis. This component normal force will now be responsible for creating the centripetal acceleration required to move the car along the curve. Therefore, for every single angle, there exists a velocity for which no friction is required at all to move the object along the curve. This means that the car will be able to turn even under the most slippery conditions (ice or water).

Concept of Banking Highway Curves

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When a car goes around a curve, there must be a net force toward the center of the circle of which the curve is an arc. If the road is flat, that force is supplied by friction.

If the frictional force is insufficient, the car will tend to move more nearly in a straight line, as the skid marks show. Banking the curve can help keep cars from skidding. When the curve is banked, the centripetal force can be supplied by the horizontal component of the normal force. In fact, for every banked curve, there is one speed at which the entire centripetal force is supplied by the horizontal component of the normal force, and no friction is required.

Why banking of highways is important? During the turn inroads, the inertia of the vehicle does not allow the vehicle to turn. The road will turn but the vehicle doesn’t. So, to overcome this inertia the road has to be designed in such a way that it generates this extra force on the opposite side. In some cases, the centripetal force provided by the friction between road and car might not be enough to turn. Conditions like rain, poor tire grip, etc. may lead to the skidding of cars off the road. Turning at high speed on highways may lead to the skidding off of vehicles. The vehicle has to reduce speed to make such turns. But in hilly regions and coastlines, there will be plenty of turns and bends which will reduce the average speed of the vehicle. Thus, banking of roads can increase the average speed of the vehicle thus reducing the travel time in a safer way. In circular tracks where there are continuous turns, banking is very important for a safe ride. It will reduce the wear and tear on the tires. Along with friction, the component of normal force also provides the much-needed centripetal force.

Necessity of banking of highway curves: When a vehicle is moving along a curved path it requires a centripetal force along the curve so that it does not tend to skid over the road. This force is provided by the frictional force between the road and tyres of vehicle. But this frictional force cannot be completely reliable as the road may not provide the required force during the rainy season when the roads are wet. So, to provide enough centripetal force, banking of road is necessary.

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Determining Ideal Banking Conditions For ideal banking, the net external force equals the horizontal centripetal force in the absence of friction. The components of the normal force NN in the horizontal and vertical directions must equal the centripetal force and the weight of the car, respectively. In cases in which forces are not parallel, it is most convenient to consider components along perpendicular axes —in this case, the vertical and horizontal directions.

Figure 1: Car on this Banked Curve is moving away and turning to the left

Above is a free body diagram for a car on a frictionless banked curve. The only two external forces acting on the car are its weight w and the normal force of the road N. A frictionless surface can only exert a force perpendicular to the surface—that is, a normal force. These two forces must add to give a net external force that is horizontal toward the center of curvature and has magnitude

𝑚𝑣 2 𝑟

. Only the normal force has a horizontal component, and so this must equal the

centripetal force—that is: 𝑁𝑠𝑖𝑛𝜃 =

𝑚𝑣 2 𝑟

Because the car does not leave the surface of the road, the net vertical force must be zero, meaning that the vertical components of the two external forces must be equal in magnitude and opposite in direction. From the figure, we see that the vertical component of the normal force is 𝑁𝑐𝑜𝑠𝜃 , and the only other vertical force is the car's weight. These must be equal in magnitude, thus: 𝑁𝑐𝑜𝑠𝜃 = 𝑚𝑔 Dividing the above yields to: 𝑡𝑎𝑛𝜃 = 9|P age

𝑣2 𝑟𝑔

Taking the inverse tangent gives: 𝜃 = tan

𝑣2 −1 ( 𝑟𝑔)

, for an ideally banked curve with no friction.

This expression can be understood by considering how θ depends on v and r. A large θ will be obtained for a large v and a small r. That is, roads must be steeply banked for high speeds and sharp curves. Friction helps, because it allows you to take the curve at greater or lower speed than if the curve is frictionless. Note that theta θ does not depend on the mass of the vehicle.

Formula in Banking 0f Highway Curves

Roads are most often banked for the average speed of vehicles passing over them. Nevertheless, if the speed of a vehicle is lesser or more than this, the self-adjusting state friction will operate between tyre and road and vehicle will not skid.

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See if you can do this!

Application of Formula and Answer Key 1. To what angle must a racing track of radius of curvature 600m be banked so as to be suitable for a maximum speed of 180km/h? Given: Maximum speed v= 180 km/h = 180 x 5/18 = 50m/s Radius of curvature r = 600m Gravity g = 9.8m/s^2 To find angle of banking 𝜃 =? Solution 𝑣2

𝜃 = 𝑡𝑎𝑛−1 ( ) = 𝑡𝑎𝑛−1 ( 𝑟𝑔

50 𝑥 50 600 𝑥 9.8)

𝑡𝑎𝑛−1 (0.4252) = 23°18′ angle of banking

2. A curve in the road is in the form of an arc of a circle of radius 400m. at what angle should the surface of the road be laid inclined to the horizontal so that the resultant reaction of the surface acting on a car running at 120 km/h is normal to the surface of the road? Given: Speed of the car v= 120 km/h = 120 x 5/18 = 33.33 m/s Radius of curvature r = 400m 11 | P a g e

Gravity g = 9.8m/s^2 To find angle of banking 𝜃 =? Solution 𝑣2

33.33 𝑥 33.33 400 𝑥 9.8

𝜃 = 𝑡𝑎𝑛−1 ( ) = 𝑡𝑎𝑛−1 ( 𝑟𝑔

)

𝑡𝑎𝑛−1 (0.2834) = 15°49′ angle of banking

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Glossary of Terms

Banking - is raising the outer end of the road higher than the inner end. In simple words, banking on the road is making its surface slant. Centripetal force - it is a net force that acts on an object to keep it moving along a circular path. Curves – irregular bends in roads to bring a graduation change of direction. Force – influence that can change a motion of an object. Friction – is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Velocity – vector measurement of the rate and direction of motion.

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REFERENCES

More, H. (2020). The Fact Factor, https://thefactfactor.com/about-us/

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More, H., (2020). Numerical Problems on Banking Highways. retrieved on November 20, 2021 at https://thefactfactor.com/facts/pure_science/physics/angle-of-banking/6424/

Toppr, A. (2020). Banking Road Formula. retrieved on November 20, 2021 at https://www.toppr.com/guides/physics-formulas/banking-road-formula/

BYJU'S (2021). Banking of Roads. https://byjus.com/physics/banking-of-roads/

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