Compound Curves

A compound curve is two or more simple curves which have different centers, bend in the same direction, lie on the same side of their common tangent, and connect to form a continuous arc. The point where the two curves connect (namely, the point at which the PT of the first curve equals the PC of the second curve) is referred to as the point of compound curvature (PCC).

Since their tangent lengths vary, compound curves fit the topography much better than simple curves. These curves easily adapt to mountainous terrain or areas cut by large, winding rivers. However, since compound curves are more hazardous than simple curves, they should never be used where a simple curve will do.

Compound Curve Data
The computation of compound curves presents two basic problems. The first is where the compound curve is to be laid out between two successive PIs on the preliminary traverse. The second is where the curve is to be laid in between two successive tangents on the preliminary traverse. (See figure 11.)

Compound Curve between Successive PIs
The calculations and procedure for laying out a compound curve between successive PIs are outlined in the following steps. This procedure is illustrated in figure 11a.

1. Determine the PI of the first curve at point A from field data or previous computations.
2. Obtain I₁, I₂, and distance AB from the field data.
3. Determine the value of D₁, the D for the first curve. This may be computed from a limiting factor based on a scaled value from the road plan or furnished by the project engineer.
4. Compute R₁, the radius of the first curve as shown on pages 11 through 13.
5. Compute T₁, the tangent of the first curve.
   T₁ = R₁(Tan ½ I)
6. Compute T₂, the tangent of the second curve.
   T₂ = AB – T₁
7. Compute R₂, the radius of the second curve.
   R₂ = T₂ / Tan ½ I
8. Compute D₂ for the second curve. Since the tangent for the second curve must be held exact, the value of D₂ must be carried to five decimal places.
9. Compare D₁ and D₂. They should not differ by more than 3 degrees, If they vary by more than 3 degrees, the surveyor should consider changing the configuration of the curve.
10. If the two Ds are acceptable, then compute the remaining data and deflection angles for the first curve.
11. Compute the PI of the second curve. Since the PCC is at the same station as the PT of the first curve, then PI₂ = PT₁ + T₂.
12. Compute the remaining data and deflection angles for the second curve, and lay in the curves.

Compound Curve between Successive Tangents. The following steps explain the laying out of a compound curve between successive tangents. This procedure is illustrated in figure llb.

1. Determine the PI and I angle from the field data and/or previous computations.
2. Determine the value of I₁ and distance AB. The surveyor may do this by field measurements or by scaling the distance and angle from the plan and profile sheet.
3. Compute angle C.
   C = 180 – I
4. Compute I₂.
   I₂ =180-(I₁+C)
5. Compute line AC.
   AC = AB Sin I₂ / Sin C
6. Compute line BC.
   BC = AB Sin I₁ / Sin C
7. Compute the station of PI₁.
   PI₁ = PI – AC
8. Determine D₁ and compute R₁ and T₁ for the first curve as described on pages 11 through 13.
9. Compute T₂ and R₂ as described on pages 11 through 13.
10. Compute D₂ according to the formulas on pages 11 through 13.
11. Compute the station at PC.
    PC₁ = PI – (AC + T₁)
12. Compute the remaining curve data and deflection angles for the first curve.
13. Compute PI₂.
    PI₂ = PT₁ + T₂
14. Compute the remaining curve data and deflection angles for the second curve, and stake out the curves.

Staking Compound Curves
Care must be taken when staking a curve in the field. Two procedures for staking compound curves are described.

Compound Curve between Successive PIs. Stake the first curve as described on pages 14 and 15.

1. Verify the PCC and PT2 by placing the instrument on the PCC, sighting on PI2, and laying off I2/2. The resulting line-of-sight should intercept PT2.
2. Stake the second curve in the same manner as the first.

Compound Curve between Successive Tangents. Place the instrument at the PI and sight along the back tangent.

1. Lay out a distance AC from the PI along the back tangent, and set PI₁.
2. Continue along the back tangent from PI₂ a distance T₁, and set PC₁.
3. Sight along the forward tangent with the instrument still at the PI.
4. Lay out a distance BC from the PI along the forward tangent, and set PI₂.
5. Continue along the forward tangent from PI a distance T₂, and set PT₂.
6. Check the location of PI₁ and PI₂ by either measuring the distance between the two PIs and comparing the measured distance to the computed length of line AB, or by placing the instrument at PI₁, sighting the PI, and laying off I₁. The resulting line-of-sight should intercept PI₂.
7. Stake the curves as outlined on pages 14 and 15.