GELE Surveying (Geomatics) — Spiral (Transition) CurvesStudy Notes

A spiral curve serves as a transition element in horizontal alignment design. On a real road, a vehicle cannot instantaneously change its steering angle and banking; doing so would impose an impulsive lateral force leading to skidding, discomfort, and potential loss of control. The spiral solves this by allowing the radius of curvature to change gradually. **Physical Principle:** Along a spiral, the curvature (1/R) increases linearly with arc length (ℓ). This means: - At the tangent point: radius = ∞ (straight line, no curvature) - Along the spiral: radius decreases gradually - At the circular arc: radius = R (constant curvature) The relationship R·ℓ = constant (where ℓ is the arc length from the tangent) defines a **clothoid** or **Euler spiral**, the standard form used worldwide. **Why Engineers Use Spirals:** 1. **Comfort:** Lateral acceleration builds up smoothly, not suddenly. 2. **Safety:** Drivers have time to steer gradually; no sudden swerve is needed. 3. **Superelevation Runoff:** The banking angle (superelevation, e) can be applied over the spiral length, not abruptly at the curve start. 4. **Vehicle Dynamics:** Unsprung masses and suspension compliance demand a gradual load buildup. 5. **Aesthetics:** The road appears continuous without a kink where tangent meets circle. **Design Context (DPWH/NSCP 2015):** Modern Philippine road design standards expect spirals on all but the gentlest curves. For high-speed highways (design speed V ≥ 80 km/h), spirals are mandatory to meet safety and comfort criteria. The spiral length Ls is chosen to balance practical constraints (minimum length for geometry) and comfort/runoff criteria.

**Spiral Angle (θₛ):** The spiral angle is the total angle (in radians or degrees) through which the spiral turns as it progresses from the tangent to the circular arc. It represents the cumulative change in direction. For a spiral of length Ls ending on a curve of radius R: **θₛ = Ls / (2R) [radians]** Converting to degrees: **θₛ = Ls × (90°) / (π × R) [degrees]** Alternatively: **θₛ (degrees) = Ls / (2R) × (180/π) = Ls / (2R) × 57.296** **Physical Interpretation:** The spiral angle represents the deflection angle from the tangent to the tangent at the end of the spiral (i.e., the tangent to the circular arc). A larger angle indicates a tighter spiral or longer transition length. **Variation of Curvature Along the Spiral:** At any point distance ℓ along the spiral from the start: - **Radius at that point:** R(ℓ) = Ls × R / ℓ [approximately valid for moderate spirals] - **Curvature at that point:** κ(ℓ) = 1/R(ℓ) = ℓ / (Ls × R) - **Deflection angle at that point:** θ(ℓ) = θₛ × (ℓ/Ls)² [CRITICAL: quadratic growth] Note the **quadratic relationship** — the angle grows with the square of distance, not linearly. This is essential for board exams. **Key Elements of the Spiral (for setting out and design):** 1. **Tangent Offset (or "Throw")** — the perpendicular distance from the tangent line to the end of the spiral: **Δt ≈ Ls² / (6R)** 2. **Shift (p)** — the perpendicular offset of the circular arc inward from its position if extended back tangentially. This is the key value used in offset/superelevation calculations: **p ≈ Ls² / (24R) = Δt / 4 [one-quarter of the throw]** 3. **Spiral Length (Ls)** — the arc length of the spiral itself. Common design practice sets Ls based on: - **Comfort criterion:** Ls ≥ 0.6 × V (V in km/h) — ensures lateral acceleration does not exceed comfortable limits. - **Superelevation runoff:** Ls = e × W / (rate of runoff), where e is the superelevation, W is the width, and rate is typically 1:200 to 1:300 (1 m vertical per 200–300 m horizontal distance). - **Minimum practical length:** Often 40–100 m depending on curve radius and design speed. **Board Exam Tip:** Always distinguish between the spiral angle θₛ (total deflection of the spiral) and the deflection angle θ(ℓ) at an arbitrary point (which is quadratic in ℓ).

**Centripetal Force and Superelevation:** As a vehicle traverses a horizontal curve, it requires a centripetal force directed toward the center. This force comes from two sources: 1. **Superelevation (e)** — the inward tilt of the road surface (expressed as a ratio or percentage) 2. **Side friction (f)** — the friction between tire and pavement (lateral). The balance between centripetal demand and available force yields the fundamental design equation: **e + f = V² / (127R)** Where: - e = superelevation (unitless ratio; e.g., 0.06 means 6%) - f = coefficient of side friction (unitless) - V = design speed in kilometers per hour (km/h) - R = radius of the circular curve in meters (m) - 127 = constant that converts units. Derivation: 127 = g × (3.6)² = 9.81 × 12.96 (approximately), where 3.6 converts m/s to km/h. **Critical Insight:** For a given design speed V and curve radius R, the sum (e + f) is fixed by the equation above. The designer chooses a superelevation e (within code limits: typically e ≤ 0.10 for normal roads, up to 0.12 on high-speed highways) and allows friction to make up the remainder: f = V²/(127R) − e. **Design Constraints:** - Maximum superelevation emax: Typically 0.08 (8%) on ordinary roads, 0.10–0.12 on high-speed expressways (per DPWH standards). - Available side friction fmax: Depends on pavement condition; typically 0.15–0.20 for good asphalt, lower on wet or poor pavement. - Minimum lateral friction fmin: Often 0.10–0.15 is required for safety. **Board Exam Perspective:** Questions often ask: 1. Given V and R, find the required e + f, then check if a proposed (e, f) pair is adequate. 2. Given V, emax, and fmax, find the minimum radius R that can be safely designed. 3. Given V and R, design an appropriate superelevation e such that the remaining friction demand f is within acceptable limits. **Worked Problem — Finding Required Superelevation:** A highway is being designed for a speed of 100 km/h on a horizontal curve of radius R = 400 m. The pavement can provide a maximum side friction of f = 0.15. Determine the minimum superelevation e required. Solution: e + f = V² / (127R) = (100)² / (127 × 400) = 10,000 / 50,800 = 0.1969 e_min = 0.1969 − f = 0.1969 − 0.15 = 0.0469 ≈ 0.047 or 4.7% Since this is within typical limits (e ≤ 8%), a superelevation of approximately 4.7% to 6% would be chosen in practice, balancing comfort and drainage on the road. **Relationship to Spiral Length:** The spiral length Ls is often sized so that superelevation builds up over its length. A common design criterion is: **Ls = e × W / (runoff rate)** Where W is the road width and the runoff rate is typically 1:200 (one meter of elevation gain per 200 meters of horizontal distance). For a two-lane road of W = 7.5 m with e = 0.06: Ls = 0.06 × 7.5 / (1/200) = 0.45 / 0.005 = 90 m This ensures the superelevation develops smoothly, not abruptly.

While superelevation runoff is one criterion for spiral length, passenger comfort is another. Comfort depends on the rate of change of lateral acceleration, not just the magnitude. **Comfort Criterion (Lateral Jerk):** Lateral jerk (rate of change of lateral acceleration) is governed by how quickly the curvature changes. A low jerk rate provides comfort; excessive jerk causes discomfort and can cause light objects to slide around inside a vehicle. The standard comfort criterion is: **Ls ≥ C × V** Where: - Ls = spiral length in meters - V = design speed in km/h - C = comfort coefficient, typically 0.5–0.7 (often 0.6 is used as a rule of thumb) So for a design speed of 100 km/h: Ls ≥ 0.6 × 100 = 60 m **Minimum Spiral Length Criteria (DPWH/Common Practice):** 1. **Comfort criterion:** Ls ≥ 0.6V (V in km/h) 2. **Superelevation runoff:** Ls ≥ e × W / (runoff rate) 3. **Practical minimum:** Ls ≥ 40 m (to ensure geometric setout is practical) 4. **Relationship to curve:** Ls should be substantial relative to R; often Ls ≥ 2√(R) or Ls ≥ √(V×R) as secondary checks. The **governing criterion** (the one that produces the longest Ls) determines the design spiral length. **Board Exam Approach:** When asked to determine spiral length, calculate all applicable criteria and take the maximum. This ensures the spiral is long enough to satisfy comfort, superelevation runoff, and geometric practicality. **Example — Determining Spiral Length:** A 100 km/h design-speed road includes a horizontal curve of R = 500 m with superelevation e = 0.07. The road width is W = 7.5 m, and the superelevation runoff rate is 1:200. Determine the required spiral length. Solution: 1. **Comfort criterion:** Ls ≥ 0.6 × V = 0.6 × 100 = 60 m 2. **Superelevation runoff:** Ls ≥ e × W / (runoff rate) = 0.07 × 7.5 / (1/200) = 0.525 / 0.005 = 105 m 3. **Practical minimum:** Ls ≥ 40 m (satisfied by criteria 2) 4. **Secondary geometric check:** Ls ≥ √(V × R) = √(100 × 500) = √50,000 ≈ 224 m (some codes use this; if so, it governs) OR Ls ≥ 2√R = 2√500 ≈ 44.7 m (weaker criterion) Assuming the secondary check is not mandatory, the **superelevation runoff criterion governs:** Ls = 105 m (or slightly more for practical setout). If the secondary check were mandatory, Ls would be 224 m.

Superelevation runoff is the process of gradually increasing the road's banking angle from 0% (on the tangent line) to the full design value e (on the circular curve). Without a spiral, this transition would occur abruptly at the curve's start, causing discomfort and creating a ridge or bump at the junction. With a spiral, the runoff is distributed over the spiral length. **Runoff Rate and Gradient:** The runoff rate is specified as a ratio, e.g., 1:200, meaning 1 unit of elevation rise per 200 units of horizontal distance. **Relationship Between Spiral Length, Superelevation, and Runoff Rate:** For a road of width W (measured along the centerline for a two-lane road, or from edge to edge for design) and a runoff rate of r (e.g., r = 1/200): **Ls = e × W / r** Rearranging: **r = e × W / Ls** **Example Calculation:** A road with W = 7.5 m achieves a superelevation of e = 0.06 (6%) over a spiral of Ls = 90 m. What is the runoff rate? r = e × W / Ls = 0.06 × 7.5 / 90 = 0.45 / 90 = 0.005 = 1:200 The runoff rate is 1:200, which is typical and acceptable. **Code Limits (DPWH/NSCP 2015):** Typical runoff rates range from 1:150 (steeper, acceptable for low speeds) to 1:300 (gentler, preferred for high speeds). - 1:150 to 1:200: Common on ordinary roads - 1:200 to 1:300: Preferred on highways (smoother transition, less visible "ridge") - 1:300+: Very flat, used on very high-speed corridors **Vertical Alignment of Superelevation:** The superelevation rises linearly along the spiral. If the spiral is parameterized by distance ℓ from the start (0 ≤ ℓ ≤ Ls), the superelevation at distance ℓ is: **e(ℓ) = e_final × (ℓ / Ls)** Where e_final is the full superelevation on the circular arc (e.g., 0.06). At ℓ = 0 (tangent): e(0) = 0 At ℓ = Ls/2 (midpoint): e(Ls/2) = e_final / 2 At ℓ = Ls (end): e(Ls) = e_final **Two-Lane Road Rotation (Asymmetric Runoff):** For a two-lane road, superelevation can be achieved by: 1. **Rotating about the centerline:** One edge rises, the other falls. 2. **Rotating about the inside edge:** The entire road section rises inward. 3. **Rotating about the outside edge:** The entire road section rises outward (uncommon). Most designs use centerline rotation. The superelevation profile shows how elevation varies across the road width during the runoff. **Board Exam Tip:** Questions often ask to verify that a chosen spiral length satisfies the superelevation runoff rate, or to calculate Ls given the runoff rate constraint. Remember the formula Ls = e × W / r and keep track of whether r is expressed as a decimal (0.005 for 1:200) or as a ratio (1:200).

For the purpose of surveying and construction setout, the spiral must be defined in terms of coordinates. The clothoid spiral (Euler spiral), the standard form, has an intrinsic property: the product of radius and arc length is constant. **Clothoid Spiral Fundamental Relationship:** **R(ℓ) × ℓ = Ls × R = constant** Where: - R(ℓ) = radius of curvature at arc length ℓ from the spiral start - ℓ = arc length from the spiral tangent point - Ls = total spiral length - R = radius of the circular curve at the spiral end This relationship ensures that curvature increases linearly: κ(ℓ) = 1/R(ℓ) = ℓ / (Ls × R), which grows proportionally to distance. **Deflection Angle at a Point (Revisited with Precision):** The deflection angle at distance ℓ from the spiral start is: **θ(ℓ) = ℓ² / (2 × Ls × R) (in radians)** Or equivalently: **θ(ℓ) = θₛ × (ℓ / Ls)²** Where θₛ = Ls / (2R) is the total spiral angle. This quadratic relationship is **critical** and appears frequently on boards. **Cartesian Coordinates of the Spiral:** For precise setout, the (x, y) coordinates of points along the spiral relative to the tangent point are given by: **x(ℓ) = ℓ [1 − (ℓ²/(10 × Ls²)) + (ℓ⁴/(216 × Ls⁴)) − ...]** (Taylor series approximation) **y(ℓ) = (ℓ³/(6 × Ls)) [1 − (ℓ²/(28 × Ls²)) + ...]** For practical engineering (and board exams), simplified approximations are often sufficient: **x(ℓ) ≈ ℓ [1 − (ℓ/(12 × R))²]** **y(ℓ) ≈ ℓ³ / (6 × R)** These approximations are valid for moderate spirals (Ls / R < 0.3). **Offset and Throw (Revisited):** At the end of the spiral (ℓ = Ls): **y(Ls) = Ls³ / (6R) = Ls² / (6R) × Ls / R ≈ Δt (tangent offset)** **Shift (p):** The shift is the perpendicular distance from the original (extended) tangent line to the parallel tangent of the circular arc: **p ≈ Ls² / (24R)** Note: p = Δt / 4 for the clothoid. **Long Chord of the Spiral:** The chord distance from the start of the spiral to its end (the straight-line distance across the spiral) is approximately: **C_spiral ≈ √[x(Ls)² + y(Ls)²]** For practical calculations: **C_spiral ≈ Ls × [1 − (Ls²/(180 × R²))]** **Application to Setout:** Surveyors set out the spiral by calculating the deflection angle at each chainpoint (e.g., every 10 or 20 m) and the corresponding offset from the tangent. This allows the spiral to be staked using transit-and-tape or modern GPS/GNSS methods. **Board Exam Focus:** Exams rarely ask for full coordinate calculations but often test: 1. Deflection angle at a point: θ(ℓ) = θₛ(ℓ/Ls)² ✓ 2. Tangent offset: Δt ≈ Ls²/(6R) ✓ 3. Shift: p ≈ Ls²/(24R) ✓ 4. Spiral angle: θₛ = Ls/(2R) ✓

The design of a spiral curve segment involves a systematic sequence of decisions and calculations. This section outlines the step-by-step procedure used by road designers and surveyors. **Step 1: Establish Design Parameters** - Design speed V (from design code or project brief) - Curve radius R (from horizontal alignment geometry) - Road width W (number of lanes, lane width) - Pavement condition and side-friction estimate - Superelevation limits (emax, typically 0.08–0.10) - Runoff rate criteria (typically 1:200–1:300) **Step 2: Check Centripetal Demand** Calculate e + f = V² / (127R). If this exceeds emax + fmax (e.g., 0.10 + 0.20 = 0.30), the radius is too tight; either increase R or decrease V. **Step 3: Design Superelevation** Choose e such that: - e ≤ emax (code limit) - The remaining friction demand f = V²/(127R) − e is within safe limits (typically f ≤ 0.15–0.20) - For comfort, e should be proportional to curvature; avoid excessive banking on gentle curves. **Step 4: Determine Spiral Length (Ls)** Calculate Ls based on all applicable criteria: **Criterion A — Comfort:** Ls_comfort = 0.6 × V **Criterion B — Superelevation Runoff:** Ls_runoff = e × W / (runoff rate) **Criterion C — Practical Minimum:** Ls_min = 40 m (or based on project-specific needs) **Criterion D — Geometric Check (optional, per code):** Ls_geom = √(V × R) or 2√R Select: **Ls_design = max(Ls_comfort, Ls_runoff, Ls_min, Ls_geom)** Round to the nearest practical value (e.g., 5 or 10 m). **Step 5: Calculate Spiral Elements** Once Ls is finalized, compute: - Spiral angle: θₛ = Ls / (2R) - Tangent offset (throw): Δt = Ls² / (6R) - Shift: p = Ls² / (24R) - Long chord: C_spiral ≈ Ls **Step 6: Adjust Circular Arc Geometry** The shift (p) pushes the circular arc inward. The designer must adjust the circular-arc geometry (station of curve start, arc length, etc.) to account for the shift and ensure proper alignment with the tangent segments before and after. **Step 7: Detailed Setout Calculations** For field setout, calculate: - Deflection angles at each chainpoint: θ(ℓ) = θₛ × (ℓ/Ls)² - Offsets from the tangent line (x and y coordinates of each chainpoint) - Chord distances between consecutive chainpoints - Transit/bearing angles from a known point **Step 8: Superelevation Profile (Vertical Alignment)** Develop the superelevation diagram showing: - How the cross-section transitions from normal crown (tangent) through the spiral - Height of each edge (inside and outside) at each station - Ensuring drainage throughout **Example — Complete Design Procedure:** **Given:** - Design speed: V = 100 km/h - Curve radius: R = 400 m - Road width: W = 7.5 m - emax = 0.08, fmax = 0.18 - Runoff rate: 1:200 **Solution:** 1. **Centripetal check:** e + f = 100² / (127 × 400) = 10,000 / 50,800 = 0.197 This is less than emax + fmax = 0.08 + 0.18 = 0.26, so the radius is adequate. 2. **Superelevation design:** Assume e = 0.06 (6%) Required f = 0.197 − 0.06 = 0.137 (well within fmax = 0.18) ✓ 3. **Spiral length:** Ls_comfort = 0.6 × 100 = 60 m Ls_runoff = 0.06 × 7.5 / (1/200) = 0.45 / 0.005 = 90 m Ls_min = 40 m Ls_geom = √(100 × 400) = √40,000 = 200 m (if required; often waived for R > 300 m) Governing criterion: Ls = 90 m (superelevation runoff) or Ls = 200 m if geometric check is mandatory. Assume Ls = 100 m (practical choice, balancing runoff and superelevation development). 4. **Spiral elements:** θₛ = 100 / (2 × 400) = 0.125 rad = 7.16° Δt = 100² / (6 × 400) = 10,000 / 2,400 = 4.17 m p = 100² / (24 × 400) = 10,000 / 9,600 = 1.04 m 5. **Setout (first few chainpoints at 20 m intervals):** θ(20) = 7.16° × (20/100)² = 0.29° θ(40) = 7.16° × (40/100)² = 1.14° θ(60) = 7.16° × (60/100)² = 2.58° θ(80) = 7.16° × (80/100)² = 4.58° θ(100) = 7.16° × (100/100)² = 7.16° 6. **Superelevation profile:** At ℓ = 0 m (start): e = 0% At ℓ = 50 m (mid): e = 3% At ℓ = 100 m (end): e = 6% The superelevation continues at 6% on the circular arc.

This section highlights typical mistakes and exam questions that test conceptual understanding and calculation accuracy. **Pitfall 1: Confusing Spiral Angle with Deflection Angle at a Point** The **spiral angle θₛ** is the total deflection from start to end of the spiral. The **angle at a point θ(ℓ)** is the deflection at distance ℓ, which is much smaller (quadratic dependence). **Wrong approach:** Assuming θ(ℓ) = θₛ for any point ℓ. **Correct approach:** θ(ℓ) = θₛ × (ℓ/Ls)²; at ℓ = Ls/2, θ(Ls/2) = θₛ/4, not θₛ. **Pitfall 2: Incorrect Unit Conversion in the e + f Equation** The formula e + f = V² / (127R) assumes V is in **km/h** and R is in **meters**. If V is in m/s, the constant changes. **Wrong approach:** Using V in m/s without changing the constant (e.g., e + f = V²/(9.81R) is for m/s). **Correct approach:** Always specify units. For km/h and m, use 127 = g × (3.6)². Memorize this constant. **Pitfall 3: Forgetting the Shift When Calculating Circular Arc Geometry** The shift **p** moves the circular arc inward. Failing to account for it leads to misalignment. **Wrong approach:** Extending the circular arc back to its original tangent point, ignoring the shift. **Correct approach:** Recognize that p ≈ Ls²/(24R); the arc must be offset by p to reconnect smoothly with the spiral. **Pitfall 4: Mixing Up Tangent Offset and Shift** These are **different quantities**: - **Tangent offset (Δt):** Perpendicular distance from the tangent line to the end of the spiral ≈ Ls²/(6R) - **Shift (p):** Perpendicular distance between the original (extended) tangent and the parallel tangent of the circular arc ≈ Ls²/(24R) = Δt/4 Questions sometimes list both and ask you to identify or calculate each correctly. **Pitfall 5: Incorrect Superelevation Runoff Calculation** The formula Ls = e × W / (runoff rate) requires that the runoff rate is expressed as a **decimal**, not a ratio. **Wrong approach:** Ls = 0.06 × 7.5 / (1:200) [ratio directly substituted; doesn't work] **Correct approach:** Convert 1:200 to decimal: r = 1/200 = 0.005, then Ls = 0.06 × 7.5 / 0.005 = 90 m. **Pitfall 6: Forgetting the Quadratic Coefficient in Deflection Angles** When calculating deflection angles at multiple chainpoints, students often forget the (ℓ/Ls)² term. **Wrong approach:** θ(ℓ) = θₛ × ℓ/Ls [linear] **Correct approach:** θ(ℓ) = θₛ × (ℓ/Ls)² [quadratic]; this is essential for accurate setout. --- **Typical Board-Style Exam Questions:** **Question 1 — Spiral Angle and Elements** A spiral of length 75 m transitions into a horizontal curve of radius 280 m. Calculate: (a) The spiral angle in radians and degrees. (b) The deflection angle at the 37.5 m chainpoint. (c) The tangent offset. **Solution:** (a) θₛ = 75 / (2 × 280) = 0.1339 rad = 7.67° (b) θ(37.5) = 7.67° × (37.5/75)² = 7.67° × 0.25 = 1.92° (c) Δt = 75² / (6 × 280) = 5625 / 1680 = 3.35 m **Question 2 — Superelevation Demand and Spiral Length** A 95 km/h highway includes a curve of R = 380 m. The design superelevation is e = 0.05. Road width is 8 m. (a) What is the required e + f? (b) If f = 0.14, is the design adequate? (c) What spiral length is needed to develop this superelevation at a runoff rate of 1:250? **Solution:** (a) e + f = 95² / (127 × 380) = 9025 / 48,260 = 0.187 (b) Available e + f = 0.05 + 0.14 = 0.19 > 0.187 ✓ (adequate) (c) Ls = 0.05 × 8 / (1/250) = 0.4 / 0.004 = 100 m **Question 3 — Governing Spiral Length Criterion** For a 110 km/h design-speed road with R = 320 m, e = 0.08, W = 7.5 m, and runoff rate 1:200, which criterion governs the spiral length? Comfort: Ls ≥ 0.6 × 110 = 66 m Superelevation runoff: Ls ≥ 0.08 × 7.5 / 0.005 = 120 m Practical minimum: 40 m **Answer:** Superelevation runoff governs; Ls = 120 m. **Question 4 — Deflection Angle Progression (Setout)** A 90 m spiral is set out at 18 m intervals (chainpoints 0, 18, 36, 54, 72, 90 m). The spiral feeds into R = 450 m. List the deflection angles at each chainpoint. **Solution:** θₛ = 90 / (2 × 450) = 0.1 rad = 5.73° θ(0) = 0° θ(18) = 5.73° × (18/90)² = 5.73° × 0.04 = 0.23° θ(36) = 5.73° × (36/90)² = 5.73° × 0.16 = 0.92° θ(54) = 5.73° × (54/90)² = 5.73° × 0.36 = 2.06° θ(72) = 5.73° × (72/90)² = 5.73° × 0.64 = 3.67° θ(90) = 5.73° × 1 = 5.73° **Answer:** See table; note the quadratic progression.