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6. Fitting Vertical Curves

a. Introduction

There are many different combinations of fitting a vertical curve to meet design conditions. In this section we will concentrate on two examples of fitting an equal tangent curve between fixed grade lines. Holding grade lines fixed isolates curve selection effects. When a grade is changed, whole sections of the alignment are affected and must be recomputed, Figure B-17. Although software can easily handle this, it's the designer's responsibility to ensure solving a problem in one area doesn't adversely affect another.


(a) Preliminary Vertical Alignment


(b) Holding Grades Fixed


(c) Changing a Grade

Figure B-17
Fitting Vertical Curves


In these examples we will compute simple equal tangent curves to meet some elevation condition between fixed grade lines. The next section deals with unequal tangent vertical curves which give additional design flexibility but at the cost of more complex computations.

When computing a curve passing through an elevation, the computed length may be a maximum or minimum. It depends if the specified elevation is a maximum or minimum and whether a sag or crest curve is involved. Increasing the length of a sag curve raises the entire curve; increasing a crest curve's length lowers it, Figure B-18.


(a) Increasing Sag Curve Length


(b) Increasing Crest Curve Length

Figure B-18
Curve Length and Elevations


To solve lengths requires rearranging and solving the Curve Equation, Equation B-7, for the unknown L.

Equation B-7

It looks simple until you realize that di is dependent on the location of the BVC which is in turn based on the curve length we're tying to compute. Hmmm....