CELE Surveying (Geomatics) — Vertical (Parabolic) CurvesStudy Notes

A vertical curve is a parabolic arc that connects two tangent grades (g₁ and g₂) smoothly without a sharp break. The curve is designed such that the rate of grade change is constant throughout its length. This constant rate of change distinguishes a parabolic vertical curve from a circular arc and ensures that vehicles experience uniform vertical acceleration. **Why Parabolic?** The parabola is chosen because: - It provides uniform rate of grade change (constant acceleration), which is comfortable for passengers - It simplifies elevation calculations using a quadratic equation - It matches common vehicle response characteristics - It is mathematically convenient for design and staking in the field **Grade Definition:** Grades are expressed as percentages or decimals: - Positive grade (+) indicates upslope (ascending) - Negative grade (−) indicates downslope (descending) - Example: g₁ = +3% means 3 m rise per 100 m horizontal distance **Curve Classification:** - **Crest (Summit) Curve:** g₁ > g₂ (grade changes from uphill to downhill or less uphill). The curve bulges upward. - **Sag (Valley) Curve:** g₁ < g₂ (grade changes from downhill to uphill or less downhill). The curve sags downward. Understanding these classifications is critical because they affect sight-distance requirements and design lengths differently.

The elevation at any point along a parabolic vertical curve is calculated using the fundamental parabolic equation. This equation is derived from the constant-rate-of-change principle and is essential for field staking and design verification. **The Parabolic Elevation Equation:** y = elev_PC + g₁·x + (r/2)·x² Where: - y = elevation at distance x from the point of curvature (PC) - elev_PC = elevation at the beginning of the curve (in meters) - g₁ = initial grade (as a decimal; positive uphill, negative downhill) - x = distance along the curve from the PC (in meters, 0 ≤ x ≤ L) - r = rate of grade change = (g₂ − g₁)/L **Alternative Form Using Tangent Offset:** The elevation can also be expressed as: y = elev_PC + g₁·x + (A/200L)·x² Where A is the algebraic grade difference in percent: A = (g₂ − g₁) × 100 This form is convenient when grades are given as percentages rather than decimals. **Physical Interpretation:** The first term (elev_PC) is the starting elevation. The second term (g₁·x) is the elevation gain along the initial tangent. The third term ((r/2)·x²) is the parabolic offset from the initial tangent, which accounts for the curvature. For crest curves, this offset is negative (the curve sags below the tangent); for sag curves, it is positive (the curve rises above the tangent). **Step-by-Step Calculation Procedure:** 1. Convert grades to decimals (divide percentages by 100) 2. Calculate r = (g₂ − g₁)/L 3. Verify the value of x (must be 0 ≤ x ≤ L) 4. Substitute into the parabolic equation 5. Double-check sign conventions and arithmetic

The turning point of a vertical curve is the location where the tangent slope becomes zero. This point is a maximum (summit) on a crest curve and a minimum (sag) on a sag curve. Finding this point is critical for design checks and field work. **Mathematical Derivation:** The grade (slope) at any point is the derivative of the elevation equation: Slope = dy/dx = g₁ + r·x Setting the slope to zero: g₁ + r·x = 0 x = −g₁/r = −g₁/[(g₂ − g₁)/L] = g₁·L/(g₁ − g₂) This can be rearranged as: x = |g₁|·L/A Where A is the absolute value of the algebraic grade difference: A = |g₂ − g₁| **Critical Observation:** The turning point is measured from the PC (point of curvature), not from the PI (point of intersection). Always verify that 0 < x < L; if x falls outside this range, the turning point is outside the curve. **Elevation at the Turning Point:** Once x is determined, substitute it back into the parabolic equation: y_turning = elev_PC + g₁·x + (r/2)·x² **Alternative Compact Formula:** For a crest curve, the maximum elevation can be calculated as: y_max = elev_PC + (g₁²·L)/(2(g₁ − g₂)) = elev_PC + (g₁²·L·100)/(2A) (using A in percent) For a sag curve, the minimum elevation is: y_min = elev_PC + (g₁²·L)/(2(g₁ − g₂)) **Field Implications:** - For crest curves, the summit location determines the highest point where drainage or sight-distance obstructions might occur - For sag curves, the low point determines where water collects and where comfort criteria are most critical - On horizontal curves combined with vertical curves, the critical point helps in evaluating combined safety

Understanding the relationship between the parabolic curve and its tangent lines is essential for design calculations, particularly for sight distance and comfort analysis. The vertical offset from the tangent to the curve at any point characterizes the parabolic shape. **Vertical Offset from Initial Tangent:** At any distance x from the PC, the elevation of the initial tangent line (at grade g₁) would be: y_tangent = elev_PC + g₁·x The actual elevation on the parabolic curve is: y_curve = elev_PC + g₁·x + (r/2)·x² The vertical offset (difference between curve and tangent) is: Offset = y_curve − y_tangent = (r/2)·x² For a crest curve (r < 0), this offset is negative (curve below tangent). For a sag curve (r > 0), the offset is positive (curve above tangent). **Maximum Offset at Mid-Curve:** The maximum vertical separation between the curve and the initial tangent occurs at x = L/2 (mid-point of the curve): Offset_max = (r/2)·(L/2)² = (r·L²)/8 = (g₂ − g₁)·L/8 This is also equal to A·L/800 when A is the algebraic grade difference in percent. **Important Property for Design:** The parabolic property that "offsets are proportional to the square of the distance" is used extensively in: - Sight-distance calculations (line-of-sight corrections over humps) - Comfort checks (evaluating vertical acceleration) - Staking procedures (using parabolic offset tables) **Practical Calculation Example:** For the crest curve in Example 1 (g₁ = +3%, g₂ = −2%, L = 200 m): - r = −0.00025/m - Maximum offset at mid-curve = (−0.00025/2)·(100)² = −1.25 m - This means the curve at mid-length is 1.25 m BELOW the initial tangent line **Application in Field Staking:** When staking a vertical curve, field crews often use offset tables. If the tangent elevation at x = 50 m is 151.50 m and the parabolic offset is −0.3125 m, the actual curve elevation is 151.50 − 0.3125 = 151.1875 m.

Sight distance is one of the primary determinants of vertical curve length in modern road design. The parabolic shape of the curve affects the driver's ability to see obstacles ahead. Different sight-distance criteria apply to crest and sag curves. **Crest Curves — Stopping Sight Distance:** On a crest (summit) curve, the driver's line of sight is obstructed by the curve itself. The required curve length is determined by the need to see an object of height h₂ (typically 0.15 m for a low obstacle) from a driver's eye height h₁ (typically 1.07 m) at a distance equal to the stopping sight distance S. For S < L (stopping distance less than curve length): L = A·S²/[200(√h₁ + √h₂)²] For S ≥ L (stopping distance greater than curve length): L = 2S − 200(√h₁ + √h₂)²/A Where: - L = length of vertical curve (m) - A = algebraic grade difference = |g₂ − g₁|·100 (percent) - S = stopping sight distance (m) - h₁ = driver's eye height above road surface (typically 1.07 m) - h₂ = object height (typically 0.15 m for a vehicle tail light) **Typical Stopping Sight Distance (from AASHTO/NSCP):** - 40 km/h: S ≈ 40 m - 60 km/h: S ≈ 65 m - 80 km/h: S ≈ 100 m - 100 km/h: S ≈ 140 m - 120 km/h: S ≈ 180 m **Sag Curves — Sight Distance:** On a sag (valley) curve, the sight limitation is usually due to headlight beam throw at night rather than line-of-sight obstruction. The criterion is that the driver must see the beam reach the road surface at the stopping-sight-distance. For sag curves, the design equation is: L = A·S²/[200(h + S·tan(θ))] Where: - h = headlight height (approximately 0.6 m) - θ = headlight beam angle (approximately 1° above horizontal) - For practical design: L = A·S²/400 (simplified form) Alternatively, NSCP specifies minimum curve lengths based on algebraic grade difference and design speed. **Design Procedure:** 1. Determine design speed and corresponding stopping sight distance S 2. Calculate algebraic grade difference A = |g₂ − g₁|·100 3. Select curve type (crest or sag) 4. Apply appropriate sight-distance formula 5. Calculate minimum required L 6. Compare with other design criteria (comfort, drainage) and use the larger value **Practical Note for Philippine Design:** The NSCP 2015 (which governs Philippine road design) provides tables of minimum vertical curve lengths based on design speed and algebraic grade change. These tables are derived from the formulas above but provide quick reference values. Always verify local requirements and any project-specific criteria from the Department of Public Works and Highways (DPWH).

Beyond the fundamental mathematics, vertical curve design must consider multiple practical factors, especially in the Philippine context where terrain, climate, and safety standards all play critical roles. **Comfort Criteria (Vertical Acceleration):** Passengers experience vertical acceleration as the vehicle traverses the curve. The rate of change of grade (r in m/m²) directly relates to this acceleration. Maximum allowable vertical acceleration: a_max ≈ g/30 ≈ 0.33 m/s² (or roughly 3.3% of gravity) This translates to a minimum curve length: L_comfort = r·v²/a_max Where v is the design speed in m/s. For practical design, NSCP provides minimum lengths based on comfort. **Drainage and Surface Water Control:** On sag curves, water tends to collect at the low point. Minimum lengths are often specified to prevent ponding and ensure adequate longitudinal drainage slopes. Philippine standards typically require minimum L for sag curves to prevent standing water, especially in areas with heavy rainfall. **Grade Transition Issues:** When transitioning from one grade to another: - Smooth transitions are required for vehicle handling - Sudden grade changes cause structural stress on vehicles and pavements - Inadequate curve length leads to "bottoming out" on sag curves and excessive front-end dip on crest curves **Combined Horizontal and Vertical Alignment:** In real projects, horizontal (circular) curves are often combined with vertical curves. When both are present simultaneously: - Safety visibility is reduced further (cannot see around a horizontal curve over a vertical hump) - Design must consider the more restrictive criterion - Philippine design standards address combined horizontal-vertical alignment effects **Field Implementation (Staking):** Vertical curves are staked in the field using: 1. **Offset Method:** Calculate parabolic offsets from the tangent and set points accordingly 2. **Elevation Method:** Calculate elevations at key points (25 m or 50 m intervals) and set using level/GPS 3. **Profile Control:** Modern projects use GPS and grade control systems **Pavement Design Implications:** The rate of grade change affects pavement design: - Steeper curves (larger r values) cause greater structural loading - Sag curves experience more severe water infiltration due to compression stresses - Thermal stresses in concrete pavements are affected by vertical alignment **Environmental and Safety Considerations in Philippine Context:** Specific to Philippine highway design: - Typhoon/monsoon season drainage considerations - Landslide-prone areas require stable, well-drained grades - Urban areas often have tighter constraints on curve length - Rural or mountainous projects may have cost/constructability limits **Use of Design Tables:** The NSCP 2015 provides tabulated minimum vertical curve lengths for various design speeds and algebraic grade differences. These tables consider: - Stopping sight distance - Passing sight distance (on rare occasion for special cases) - Comfort criteria - Drainage requirements Designers select the larger of the calculated sight-distance length and the comfort/drainage minimum.

This section consolidates all essential formulas and provides a systematic problem-solving approach for vertical curve questions on the PRC Civil Engineer Licensure Examination. **Core Formulas:** 1. **Rate of Grade Change:** r = (g₂ − g₁)/L 2. **Elevation Along Curve:** y = elev_PC + g₁·x + (r/2)·x² Alternative: y = elev_PC + g₁·x + [A/(200L)]·x² (A in percent) 3. **Distance to Turning Point:** x = g₁·L/(g₁ − g₂) = |g₁|·L/|g₁ − g₂| 4. **Vertical Offset from Tangent:** Offset = (r/2)·x² = [A/(200L)]·x² 5. **Maximum Offset at Mid-Curve:** |Offset_max| = |r|·L²/8 = A·L/800 (A in percent) 6. **Crest Curve Sight Distance (S < L):** L = A·S²/[200(√h₁ + √h₂)²] 7. **Sag Curve Sight Distance:** L = A·S²/400 (simplified) **Systematic Problem-Solving Approach:** **For Elevation/Offset Calculations:** 1. Identify g₁, g₂, L, and elev_PC 2. Convert grades to decimals (divide percent by 100) 3. Calculate r = (g₂ − g₁)/L 4. Identify the location (distance x from PC) 5. Apply: y = elev_PC + g₁·x + (r/2)·x² 6. Report elevation with proper units and precision (usually 2 decimal places for meters) **For Turning Point Calculations:** 1. Classify curve as crest (g₁ > g₂) or sag (g₁ < g₂) 2. Calculate x = g₁·L/(g₁ − g₂) 3. Verify: 0 < x < L (must be within curve) 4. If valid, substitute x into elevation equation 5. Report distance from PC and elevation **For Sight Distance Calculations:** 1. Determine design speed and corresponding S from tables 2. Calculate A = |g₂ − g₁|·100 (in percent) 3. Check if crest or sag curve 4. Apply appropriate formula (crest or sag) 5. Calculate minimum L 6. Verify that turning point is within the curve (optional but good practice) 7. Compare with comfort/drainage requirements and use the larger value **Common Exam Question Patterns:** - "Find the elevation at a given point on the curve" → Use elevation equation - "Find the location and elevation of the summit/sag" → Use turning-point formula - "What is the vertical offset at mid-curve?" → Use Offset_max formula - "Design a curve for a given sight distance and speed" → Use sight-distance formula - "Verify that the curve meets design standards" → Check that calculated L meets or exceeds minimum required **Critical Sign Conventions:** - Grades: positive (+) for uphill, negative (−) for downhill - r: negative for crest, positive for sag - Offset: negative (below tangent) for crest, positive (above tangent) for sag - Turning point location x: always positive, measured from PC **Common Mistakes to Avoid:** 1. **Using percentages instead of decimals** in the elevation equation 2. **Measuring x from the PI instead of the PC** (always from the point of curvature) 3. **Forgetting to check that x is within [0, L]** 4. **Incorrect grade signs** (especially when both grades are negative or positive) 5. **Confusing crest and sag sight-distance formulas** 6. **Rounding errors** (use enough decimal places in intermediate steps) 7. **Forgetting to convert km/h to m/s** when using velocity-based formulas **Example Problem-Solving Flow for Exam:** Question: "A vertical crest curve connects g₁ = +2.5% and g₂ = −1.5% over 150 m. The PC is at elevation 200 m. Find: (a) the summit elevation, (b) the distance to the summit from PC, (c) the minimum curve length for a stopping sight distance of 100 m." **Solution Approach:** a) Convert grades: g₁ = 0.025, g₂ = −0.015 Calculate r = (−0.015 − 0.025)/150 = −0.000267/m Find x = 0.025 × 150/(0.025 − (−0.015)) = 3.75/0.04 = 93.75 m Calculate y = 200 + 0.025(93.75) + (−0.000267/2)(93.75)² = 200 + 2.34375 − 1.17 = 201.17 m **Answer: 201.17 m** b) **Answer: 93.75 m from PC (or 93.8 m)** c) Calculate A = |−0.015 − 0.025|·100 = 4% Apply: L = 4 × (100)²/[200 × 2.02] ≈ 98.5 m **Answer: Minimum 98.5 m (round to 100 m); the 150 m curve exceeds this requirement** ✓