What is the formula for deflection in surveying?

In surveying, deflection refers to the angular deviation from a tangent line at a point on a curve to any other point on that curve. It is a fundamental concept used primarily in the layout of horizontal curves, such as those found in roads, railways, and pipelines. By measuring deflection angles, surveyors can accurately establish points along a curve from a known tangent.

The formula for deflection can vary depending on whether you are calculating the total deflection to a specific point or the deflection per unit length (e.g., per foot or per station).

Key Formulas for Deflection in Surveying

To accurately lay out circular curves, surveyors utilize several formulas related to deflection angles. These formulas connect the curve's geometric properties to the angles measured in the field.

1. Total Tangent Deflection Angle to a Point

The most common application of deflection involves calculating the total tangent deflection angle from the Point of Curvature (PC) to any point on the curve. This angle is measured from the tangent at the PC to the chord connecting the PC to the desired point on the curve.

For a circular curve with a given radius (R) or degree of curve (D):

• For a chord length (c) from the PC:
  • Using the Degree of Curve (D):
    • If D is based on a 100-foot arc (arc definition):
      δ = (c × D) / 200
    • If D is based on a 100-foot chord (chord definition):
      δ = (c / 2R) × (180 / π) (This simplifies if D is used directly for chord-based D)
    • Where δ is the deflection angle in degrees, c is the chord length in feet, and D is the degree of curve in degrees.
  • Using the Radius (R):
    δ = (c × 180) / (2 × π × R)
    • Where δ is the deflection angle in degrees, c is the chord length in feet, and R is the radius of the curve in feet.

Example: To find the total deflection angle to a point 50 feet along a curve with a 5-degree arc definition, where D = 5°:
δ = (50 × 5) / 200 = 250 / 200 = 1.25 degrees

2. Deflection Angle per Station or Unit Length

This formula determines the deflection angle for a standard length, typically a 100-foot station or a single foot (or meter). This is useful for incremental curve layout.

• Deflection Angle per 100-foot Station:
  • Using Degree of Curve (D): δ_station = D / 2
    • Where D is the degree of curve (e.g., central angle subtended by a 100-foot arc or chord). This value represents the deflection for a 100-foot segment of the curve.
• Deflection Angle per Foot:
  • Using Degree of Curve (D): δ_foot = D / 200
    • Where D is the degree of curve for a 100-foot station. This provides the deflection angle for every single foot along the curve.

The deflection per foot of curve (dc) can also be determined from the total curve parameters using the following formula:

dc = (Lc / L) × (∆ / 2)

Where:

• dc: Deflection per unit length (e.g., per foot or per meter), expressed in degrees.
• Lc: Represents a unit length (e.g., 1 foot or 1 meter).
• L: Total length of the curve, in the same units as Lc (e.g., feet or meters).
• ∆: Total central angle of the curve, expressed in degrees.

Explanation: In this specific interpretation, if Lc is understood as 1 unit of length (e.g., 1 foot), then the formula essentially calculates the total tangent deflection angle (∆/2) divided by the total length of the curve (L), giving the deflection angle per unit length.

Table of Key Variables

Practical Application

Surveyors primarily use deflection angle formulas to set out horizontal curves in the field. The process typically involves:

1. Calculating Deflection Angles: Using the curve's design parameters (R, D, ∆), surveyors calculate the cumulative deflection angles from the PC to various points along the curve (often at full stations and any intermediate points like the Point of Tangency, PT).
2. Setting Up Instruments: A total station or theodolite is set up at the PC, oriented along the tangent to the curve.
3. Turning Angles: The calculated deflection angles are turned from the tangent, and corresponding chord distances are measured from the PC to establish points on the curve. This allows for precise placement of stakes or other markers defining the curve's path.