Comparison Clothoid Cubic.pdf

COMPARISON BETWEEN CUBIC PARABOLA AND CLOTHOID TRANSITION CURVE No.

Cubic Parabola

Clothoid

Source American Mathematical Society

Remark

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Cubic Parabola is derived by making certain assumptions in the clothoid formulae. It is a simple function of the form of y=f(x). Cubic parabola can never turn more than 90⁰.

The calculation required to set out cubic parabola are more straightforward since its formulae do not involve infinite series. CUBIC PARABOLA is not a true spiral and should only be used where the deviation angle (φ) is less than approximately 12⁰. In cubic parabola three assumptions are made during derivation of the cubic parabola formulae: • The second and subsequent terms in the expansion of sin φ and cos φ are neglected as

Clothoid is sometimes referred to as the ideal transition curve because it is a true spiral. It is a transition curve in the form of x=f(l), y=f(l), having as main characteristic the linearity of curvature variation versus its length. Clothoid is spiral which goes round and round The calculation required to set out clothoid are complicated by the fact that some of its formulae involve infinite series.

A new, simple and accurate transition curve type, for use in road and railway alignment design Nikolaos Eliou & Georgios Kaliabetsos • Railway Management and Engineering – V Profilidis • Railway Nowadays, many highway design software are available Management and which able do clothoid calculation easily. So, benefit of Engineering – V simple calculation provided by cubic parabola is insignificant. Profilidis

Clothoid is true spiral it doesn’t use any assumptions like those applied on Surveying for Engineers cubic parabola, which results on no J. Uren, W.F. Price limitation of deviation angle.

Since LRT build under limited land, it is common to have sharp curve with great value of deviation angle. This makes usage of cubic parabola is not ideal.

COMPARISON BETWEEN CUBIC PARABOLA AND CLOTHOID TRANSITION CURVE

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being too small – the validity of this will depend on value of φ Tan δ is assumed to equal δ radians. Since δ=φ/3 the validity of this will also depend on the value of φ The length along tangent is assumed to equal the length along the curve (y=l). Again, the value of φ will be critical to the validity of this, since greater deviation, the less likely is assumption to be true.

The use of the cubic parabola is acceptable only for small values of transition curve length. When the length of the transition curve exceeds a certain limit, it could obtain a “flatter” alignment. The true radius of curvature at the curved end of cubic parabola is less than clothoid, it allows higher value of cant. Instantaneous change of radius between circular and transition can result in perceptible lateral jerk. There will be amendment when realignment done using chord and versine method, the transition curve will actually be a clothoid not cubic parabola.

Clothoid formulae able to provide greater length of transition.

A new, simple and accurate transition curve type, for use in road and railway alignment design - Nikolaos Eliou & Georgios Kaliabetsos

With longer transition, curve becomes smoother and it would enhance passenger’s comfort and allowable speed.

The true radius of curvature at the curved end of clothoid is more than cubic parabola

Understanding Track Engineering -Permanent Way Institution

Lateral jerk which could be exist when using cubic parabola could be serious as radius diminishes. . For this reason it is recommended that in designing sharply curve track, and/or nonballasted, the clothoid calculation should be used.

Rules developed for dealing with transition curves by the chords and versine method in realignment can be used without amendments on layouts designed using clothoid.

Understanding Track Engineering -Permanent Way Institution

Greater LRT is located in densely populated area and intersects with many external parties, so repeated realignment is uncommon. It would be big hassle if amendment continuously done because of realignment if cubic parabola was used.

REFERENCE OF CLOTHOID USAGE IN OTHER PROJECTS

No

Project

Minimum Radius (m)

1

Hyderabad Metro

120

2

Kanpur Metro

120

3 4 5

Lucknow Metro Mauritius LRT Riyadh Metro

120 60 120

6

Kuala Lumpur LRT3

90