Essay Example on Geometric design in transportation Facilities

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Geometric design in transportation facilities Flair Your IB Carson Graham Secondary Date Instructor Ms Dai Word count Table of Contents Table of contents 2 Introduction The theorem Euler s totient theorem fermat 's little theorem Proving Fermat's little theorem Proving Euler's totient theorem Application Conclusion Introduction I had always had a interest in geometric design for that I have been seeing in different places so much for example in the designs of buildings highways and even art works and I often wonder why they look the way they are The one part that fascinates me the most is the geometric design in transportation facilities such as highways railroads and airport runways Even though the details and the standards of the design is quite different across different usages for example a landing area for aircraft is designed very differently than a parking lot as much difference as a plane to a minivan in all modes the goals of the geometric design is to gain the maximise amount of comfort safety and economy out of the design while still keeping the coast environmental impacts down as much as possible In this math IA I will be exploring the fundamentals of geometric design in transportation with a series of standards and examples from different modes 



There are three main parts to Geometric roadway designing alignment profile and cross section Combined three factors engineers are provided information to create three dimensional design for whatever need of the facility alignment Combine with horizontal tangents and curves the alignment is the route selection of a road Generally speaking a horizontal curve is usually treated as circular for the sick of simplicity and beauty maybe a non circular shape would theoretically be more efficient and cost efficient a circular design is much easier to complete in practice For example in Figure 1 Important features radius R central angle Δ length L semitangent distance T middle ordinate M external distance E and chord C The curve begins at the tangent to curve point TC and ends at the curve to tangent point CT degree of curvature D Degree of curvature is defined as the angle subtended by a 100 ft arc or angle subtended by a 100 ft chord The relationship between R in feet and D is as shown below L of the curve is given by T of the curve is given by M of the curve is given by The chord C is given by Last but not the least E of the curve is given by The placement of the circular curves in the area is done by using the tangent to curve point TC with a transit and then establishing successive points by turning deflection angles and measuring chords as shown in Figure 2 The deflection angle in radians dx to a point on the curve at a distance x from the TC is given by There numerical feature of horizontal curves establish the minimum R and sometime their minimum L Minimum R of horizontal curve is mostly depended on the relationship between design speed maximum rate of superelevation and curve radius In other times these heigh would be changed for other reason than fictionality on the road like appearance standards save space for future development or geographical environment profile 



With the horizontal layout defines the horizontal the profile defines the vertical aspect of the road including crest and sag curves and the straight grade lines connecting them Tangent grade is one of the key factors in the profile The tangent grade is referred to the tangent of the angle of the surface to the horizontal of a landform or liner It is a specific kind of the slope where zero indicates horizontality This feature is specially designed inroads to serve the purpose of slowing down vehicles according to the steepness and length of the grade to fit the need of the traffic Tangent grades of a road is designed according to the slopes of the surface So if we describe a surface with a function f x where the y-axis is horizontal to earth then it's target grades are based on f x 



There isn't an absolute standard for it the range depends on many factors If look at railroads many of the old ones were built with grades of up to 2 The reason being that the rolling resistance due to the rail track is very low compared to the grade resistance There for it is a better idea economically to build flat tracks and run low power weight ratios trains on it For example in figure 3 as can be seen the resistance due to grade is approximately 20 lb ton per 1 of grade On the other hand rolling resistance is only about 3 lb ton every 1 By studying the relationship between rolling resistance and grade resistance it seem better to have less powerful train then construct very level track Rail that are more for passenger particularly than shipping goods in urban rail transit systems tend to run on much steeper grades up to 10 percent can be negotiated Then moving on to a very different case such as runways used by aircraft that can go up to 121 knots or more so the max grade at any point can not go above 1 5 with exception of only on the inner two quarters of the runway where the range is extended for 0 8 To summarise Tangent Grades meanly help to adjust the speed and energy efficiency of a facility by using the power of gravity and elevation conclusion



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