Grade (Slope)

Definition

• Grade (US) / Gradient (UK): Measures the elevation angle or tangent of a slope.
  • Sometimes called stepth, slope, incline, mainfall, pitch, or rise.
  • Zero indicates flat; higher numbers indicate steeper slopes.
  • Formula: Rise over Run (vertical distance over horizontal distance).

Usage

• Used for physical features (canyons, hillsides) and constructed elements (roads, railways).
• Terminology varies by region: "grade" in the US, "gradient" in the UK.

Expression of Slopes

• Angle of inclination: Angle opposite the "rise" side of a right triangle.
• Percentage: (Rise/Run) × 100, commonly used in Europe and the US.
• Per mille: (Rise/Run) × 1000, used in railway inclines in Europe.
• Ratio: Can be expressed as "1 in n" (e.g., 1 in 200) or its inverse.

Equations

• Tangent: As a ratio involving rise and run.
• Slope percentage: Derived from tangent of the angle × 100.
• Angle calculation: Using arctan based on expressed metrics, like percentage or ratio.

Example Slopes

• Illustrated through recognizable sloped locations.
• Larger numbers indicate steeper slopes (e.g., 60 degrees = 173% grade).

Road Engineering and Design

• Various vehicles have different "gradeability".
• US federally funded highways have design tables specifying maximum grades.
• Notable steep streets located in Dunedin, NZ, and Pittsburgh, PA.

Environmental Design

• Importance in landscape and architecture.
• Considerations include drainage, stability, and code compliance.

Railways

• Ruling gradients: Limitations for train loads and locomotive capabilities.
• Early railways featured gentle gradients due to technology limits.
• Extreme gradients require mechanical assistance (e.g., cable systems, rack railways).

Compensation and Safety

• Curvature compensation: Adjustments for curves making slopes effectively steeper.
• Continuous brakes: Historical importance on steep gradients for train safety.

Notable Steep Railways

• Lists specific gradients and locations, with details on steepness and operational strategies.

Additional Topics

• Angle of repose, Civil engineering, Slope stability, and related topics.