Geometric Design - Vertical Curve Design

The Vertical Curves Section has been broken up into three categories listed below:

Definitions

  • A = Algebraic difference in gradients, |g2 - g1| (It's always positive).
  • g1 & g2 = Respectively the Grade from VPC to VPI and from VPI to VPT (always given in percent except for Ex, xt and Et where g1 and g2 will be given in decimal form).
  • L = Length of vertical curve (given in feet (meters)).
  • K = Rate of vertical curvature.
  • l1 = Length of curve 1 (unsymmetrical vertical curve only) (given in feet (meters)).
  • l2 = Length of curve 2 (unsymmetrical vertical curve only) (given in feet (meters)).
  • VPC (aka BVC, PVC) = The Vertical Point of Curvature.
  • VPT (aka EVC, PVT) = The Vertical Point of Tangency.
  • VPI (aka V, PVI) = The Vertical Point of Intersection.
  • x = Horizontal distance to any point on the curve from the VPC (given in feet (meters)).
  • xt = Turning point, which is the minimum or maximum point of the curve (given in feet (meters)).
  • e = Vertical offset or middle ordinate, which is the vertical distance from the VPI to the arc.
  • y = Vertical distance at any point on the curve to the tangent grade.
  • R = Rate of change of grade.
  • EVPC = Elevation of VPC.
  • EVPT = Elevation of VPT.
  • Ex = Elevation of a point on the curve at a distance x from the VPC.
  • Et = Elevation of the turning point.

Symmetrical Vertical Curves [1]

Typically, vertical curves are in the shape of an equal-tangent parabola (aka symmetric). This means the tangent length from VPC to VPI equals the tangent length from VPI to VPT. It is not required for the VPC and the VPT to be at the same elevation to have a symmetrical vertical curve.

Symmetrical Vertical Curve Diagram
Fig: A Symmetrical Vertical Curve Diagram

Equations to use:

A = g_2-g_1
K = {L\over A}
r = {A\over {100L}}
e = {AL\over 800}
y = {4ex^2\over {L^2}} = {rx^2\over 2} = {Ax^2\over {200L}}
E_x = {E_{VPC} + g_1x+{rx^2\over 2}}
x_t = -{g_1\over r}
E_t = {E_{VPC} -{g_1^2\over {2r}}}

Note: In the previous three equations g1 is converted into decimal form.

Unsymmetrical Vertical Curves [1]

An unsymmetrical curve is a curve in which the tangent length from VPC to VPI does not equal the tangent length from VPI to VPT. As already mentioned, symmetrical vertical curves are more common than unsymmetrical vertical curves, but since the designer is likely to encounter both, equations for both situations are provided.

Unsymmetrical Vertical Curve Diagram
Fig: A Unsymmetrical Vertical Curve Diagram

Equations to use:

A = g_2-g_1
K = {L\over A}
r = {A\over {100L}}
r_1 = {{A\over {100L}}\times {\left(l_2\over {l_1}\right)}}
r_2 = {{A\over {100L}}\times {\left(l_1\over {l_2}\right)}}
e = {Al_1l_2\over {200L}} = {r_1l_1^2\over 2} = {r_2l_2^2\over 2}
E_{x1} = {E_{VPC} + g_2x+{r_1x^2\over 2}}
E_{x2} = {E_{VPT} - g_2x+{r_2x^2\over 2}}
x_{t1} = -{g_1\over r_1}
x_{t2} = -{g_2\over r_2}
E_{t1} = {E_{VPC} -{g_1^2\over {2r_1}}}
E_{t2} = {E_{VPT} -{g_2^2\over {2r_2}}}

Note: Subscript 1 & 2 designates the left and right branch of the unsymmetrical curve respectively. Therefore Et1 will be the elevation of the turning point if that point occurs in the left branch, xt2 will be the turning point if that point occurs in the right branch, etc.

References

  1. Iowa Department of Transportation

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