CELE Transportation & Highway Engineering — Traffic Engineering and Highway CapacityStudy Notes

Traffic flow is characterized by three primary variables that must work together: **Flow (Volume) q**: Measured in vehicles per hour (veh/hr), flow represents the rate at which vehicles pass a fixed point. In technical analysis, flow can also be expressed in passenger cars per hour (pc/hr) when accounting for vehicle-type equivalencies (trucks, buses). The term 'volume' often refers to the actual count of vehicles during a specified time period (e.g., 1 hour, 15 minutes), while 'flow' is the rate expressed per hour. **Speed u**: Typically expressed in km/h. For traffic flow analysis, we must use **space-mean speed** (the harmonic mean of speeds), not the arithmetic time-mean speed. Space-mean speed represents the average speed experienced by vehicles traveling over a distance and is calculated as: Space-mean speed = Total distance traveled / Total time for all vehicles This is distinct from time-mean speed (arithmetic average of instantaneous speeds), which overestimates the true flow speed. **Density (Concentration) k**: Measured in vehicles per kilometer (veh/km), density represents the number of vehicles occupying a unit length of roadway at any instant. Higher density indicates more tightly packed vehicles. **The Fundamental Relationship**: q = k × u This equation is the cornerstone of traffic flow theory. It states that flow rate equals density multiplied by space-mean speed. This relationship must hold true at any point in time on any roadway segment. Understanding this relationship is critical—it explains why simply adding more cars (increasing density) does not always increase flow; beyond a critical density, increased crowding causes speeds to drop faster than the density increase benefits flow. **Practical Implications**: - At low density, vehicles travel at high speeds but flow is low (few vehicles). - At moderate density, both factors are reasonable, flow is maximized at the critical or optimal density. - At high density (congestion), vehicles move slowly, reducing flow despite the high number of vehicles. - This non-linear relationship creates the concept of a 'fundamental diagram,' where flow peaks at an optimum density then decreases on the congested (right) branch.

Spacing and headway are complementary measures describing the separation between successive vehicles—one in space, one in time. **Spacing (s)**: The distance between the front bumpers (or any consistent point) of successive vehicles, measured in meters. Spacing is the inverse of density: s = 1000 / k (meters, where k is in veh/km) The factor 1000 converts kilometers to meters. Spacing represents the average 'room' each vehicle occupies on the roadway. **Headway (h)**: The time interval between the passage of successive vehicles past a fixed point, measured in seconds. Headway is the inverse of flow: h = 3600 / q (seconds, where q is in veh/hr) The factor 3600 converts hours to seconds. Headway represents the time 'gap' between vehicles from the perspective of a stationary observer. **Relationship Between Spacing and Headway**: Headway can also be expressed as: h = s / u This makes intuitive sense: the time between vehicles equals the distance between them divided by the speed. This relationship provides a consistency check—all three formulas (q = k·u, s = 1000/k, h = 3600/q, and h = s/u) must be mutually consistent. **Practical Interpretation**: - Small spacing and headway indicate high-density, congested conditions. - Large spacing and headway indicate free-flow conditions with few vehicles. - In design, minimum spacing is determined by safe stopping distance and vehicle dynamics; minimum headway relates to driver reaction time and signal timing. - On Philippine expressways like EDSA or NLEX, typical free-flow spacings are 30-50 m; during congestion, spacings may drop to 5-10 m.

Traffic demand varies throughout the day—morning and evening peaks are distinctly higher than off-peak periods. Even within an hour, demand fluctuates. The **Peak-Hour Factor (PHF)** quantifies this variation and is critical for highway design and capacity analysis. **Definition**: The Peak-Hour Factor is the ratio of the total volume in the peak hour to four times the volume in the highest 15-minute interval within that hour: PHF = V / (4 × V₁₅) Where: - V = total vehicles in the peak hour - V₁₅ = vehicles counted in the highest 15-minute period within that hour **Range and Interpretation**: - PHF ranges from 0.25 to 1.0 - PHF = 1.0: Traffic is perfectly uniform throughout the hour (all four 15-min periods equal) - PHF = 0.25: All traffic arrives in one 15-minute period (highly concentrated demand) - Typical range: 0.80–0.95 for urban/suburban roads; 0.85–0.98 for expressways - Lower PHF = more concentrated peak (more variable demand within the hour) **Design Flow Rate**: The design flow rate represents the actual number of vehicles per hour that must be accommodated, accounting for the concentration of demand: Design flow rate = V / PHF (veh/hr or pc/hr) This is the rate used in capacity calculations. A roadway must be sized for the peak-hour concentration, not the average flow. This is why PHF directly affects required number of lanes. **Practical Significance**: - On Philippine expressways during morning rush (7–9 AM) or evening rush (5–7 PM), PHF is typically 0.85–0.92. - A lower PHF (more variability) requires more capacity because the peak is sharper. - Design standards (NSCP or local guidelines) often specify using the 30th highest hourly volume with an assumed PHF to determine lane requirements. - Example: If hourly volume is 2000 veh/hr with PHF = 0.90, the actual peak 15-minute rate is 2000/(4×0.90) = 556 veh/(15 min), and the design flow is 2000/0.90 = 2222 veh/hr.

Capacity and Level of Service (LOS) are fundamental to highway design and traffic management. They define how much traffic a facility can handle and how well it serves that traffic. **Capacity**: Capacity is the maximum number of vehicles that can reasonably be expected to pass a point on a roadway under prevailing geometric, environmental, and traffic conditions. Key points: - Measured in vehicles per hour (veh/hr) or passenger cars per hour (pc/hr) - For a freeway lane under ideal conditions: approximately 2000 pc/h/lane (AASHTO standard); some contexts use 1800–2200 depending on assumptions - Depends on: - Geometric factors: lane width, shoulder width, lateral clearances, grade, curvature - Environmental conditions: weather, lighting, pavement condition - Traffic composition: percentage of heavy vehicles (trucks, buses) relative to passenger cars - Traffic control: presence of signals, ramps, access points **Passenger Car Equivalent (PCE)**: Heavy vehicles (trucks, buses) occupy more space and have poorer acceleration. PCE values convert them to equivalent passenger cars: - Passenger car: PCE = 1.0 - Truck or bus on level terrain: PCE = 2.0–2.5 (varies by condition) - RVs and articulated trucks on grades: PCE up to 4.0 or more Capacity in pc/hr is computed by converting all vehicle types to passenger car equivalents. **Level of Service (LOS)** LOS is a qualitative measure (A–F) describing operating conditions and driver comfort. It characterizes how freely traffic can flow: **LOS A**: Free flow - Speeds approaching free-flow speed - Low density, ample spacing - Drivers have complete freedom of maneuver - Very stable flow **LOS B**: Reasonably free flow - Speeds slightly below free-flow - Comfortable driving, but restrictions begin to appear - Density and spacing still adequate **LOS C**: Stable flow with restrictions - Speeds start to decline with volume - Density moderate; less freedom to change lanes - Flow still stable **LOS D**: Approaching unstable flow - Speeds noticeably reduced - High density, minimal spacing (4–6 m typical) - Little freedom to maneuver; high concentration of vehicles - Sensitive to small disturbances **LOS E**: Unstable flow, at or near capacity - Speeds highly variable, often below 30 km/h on freeways - Density very high; minimal spacing (2–4 m) - Slight disturbances can cause breakdown - Flow may vary abruptly **LOS F**: Forced or breakdown flow - Speeds drop below 20 km/h; flow may stop - Demand exceeds capacity - Vehicles may queue; accidents or bottlenecks cause congestion - Unstable, breakdown conditions **Density as LOS Criterion (Freeways)**: For freeway segments, LOS is primarily determined by density (veh/km/lane): - LOS A: k ≤ 10 pc/km/lane - LOS B: 10 < k ≤ 16 - LOS C: 16 < k ≤ 22 - LOS D: 22 < k ≤ 28 - LOS E: 28 < k ≤ 35 - LOS F: k > 35 pc/km/lane **Delay as LOS Criterion (Intersections and Urban Streets)**: For signalized intersections and urban arterials, LOS is based on average control delay per vehicle (seconds): - LOS A: ≤ 10 s - LOS B: 10–20 s - LOS C: 20–35 s - LOS D: 35–55 s - LOS E: 55–80 s - LOS F: > 80 s **Webster's Method for Optimal Signal Timing**: For isolated signalized intersections, the optimal cycle length (minimizing delay) is: C₀ = (1.5 × L + 5) / (1 − Y) Where: - C₀ = optimal cycle length (seconds) - L = total lost time per cycle (seconds) ≈ 2–4 s per phase (accounting for start-up and clearance) - Y = sum of critical lane group flow ratios = Σ(v_i / s_i) for critical movements - v_i = flow rate of movement i (veh/s) - s_i = saturation flow rate of movement i (veh/s, typically ~0.55 veh/s/m of lane width or ~1800–2000 veh/h/lane) Average delay per vehicle (uniform delay, neglecting random arrivals): d = (C − g) / (2 × C) × 3600 (seconds) Where g is the effective green time for the movement and C is the actual cycle length. **Design Practice**: In the Philippines, highway design typically targets LOS C or D to balance cost and service quality: - Expressways: LOS B–C during design hour - Arterial roads: LOS C–D - Local streets: LOS D–E acceptable The design 30th-highest hourly volume (often used in Philippine standards) is the volume that is exceeded on only 29 days per year, providing a balance between cost and acceptable service.

The following represent frequent errors and misconceptions observed in PRC examinations and should be carefully avoided: **1. Unit Confusion** - MISTAKE: Using time-mean speed instead of space-mean speed in q = k·u. Time-mean speed (arithmetic average) is always higher than space-mean speed (harmonic mean) and will yield incorrect flow. - CORRECTION: Always use space-mean speed, which is the distance-weighted average speed. - MISTAKE: Forgetting unit conversion factors (1000 for spacing, 3600 for headway). - CORRECTION: s = 1000/k uses 1000 (to convert from km to m); h = 3600/q uses 3600 (to convert from hr to s). **2. Peak-Hour Factor Errors** - MISTAKE: Writing PHF = V₁₅ / V (inverted) or PHF = V / V₁₅. - CORRECTION: PHF = V / (4 × V₁₅). The denominator is always four times V₁₅. - MISTAKE: Confusing design flow rate with average hourly volume. - CORRECTION: Design flow = V / PHF; this is larger than V and represents the peak concentration. **3. Fundamental Diagram Misconception** - MISTAKE: Believing that increased density always increases flow. - CORRECTION: Flow peaks at an optimum density; beyond that (congested branch), higher density causes speed to drop faster, reducing flow. - MISTAKE: Not recognizing that a single flow value can correspond to two density values (one free-flow, one congested). - CORRECTION: The fundamental diagram is roughly parabolic or triangular; the same flow can occur at low or high density depending on speed. **4. Capacity Confusion** - MISTAKE: Assuming capacity is fixed regardless of vehicle composition. - CORRECTION: Capacity in veh/hr depends on the proportion of trucks/buses. Convert to pc/hr first, then divide by average PCE to get vehicle capacity. - MISTAKE: Using rated capacity (2000 pc/h/lane) without adjusting for geometric or environmental factors. - CORRECTION: Base capacity of 2000 pc/h/lane is for ideal conditions; real-world capacity is lower due to narrow lanes, poor sight distance, weather, etc. **5. Level of Service Criteria Mix-Up** - MISTAKE: Using density criteria (LOS A–F for k values) for intersection analysis, where delay should be used. - CORRECTION: Use density (pc/km/lane) for freeway segments; use delay (seconds) for signalized intersections. - MISTAKE: Memorizing LOS boundaries without understanding the underlying operational conditions. - CORRECTION: LOS A = free flow (high speed, low density); LOS E = at capacity (jammed); LOS F = breakdown (gridlock). **6. Signal Timing and Delay Calculation** - MISTAKE: Applying Webster's formula without correct lost time or saturation flow. - CORRECTION: Lost time includes start-up time (1–2 s) and clearance time (1–2 s per phase), not including all-red. Saturation flow for a lane is ~0.55 veh/(s·m) or ~1800–2000 veh/hr/lane. - MISTAKE: Confusing cycle length with green time. - CORRECTION: C = cycle length (full period); g = green time (one movement only); red time = C − g. **7. Rounding and Significant Figures** - MISTAKE: Rounding intermediate results, leading to accumulation of error. - CORRECTION: Carry full precision through calculations; round only the final answer appropriately (usually 1–2 decimal places for flow in veh/hr, density in veh/km, etc.). **Problem-Solving Strategy for Traffic Problems:** 1. **Identify the Given Information**: List all provided data (volumes, speeds, densities, PHF values, etc.) and note units explicitly. 2. **Identify What Is Requested**: Determine whether you need to find flow, speed, density, spacing, headway, PHF, design flow, LOS, or capacity. 3. **Choose the Appropriate Relationship**: - For flow, speed, density: q = k·u - For spacing/headway: s = 1000/k, h = 3600/q, h = s/u - For PHF: PHF = V/(4·V₁₅) - For LOS: Density for freeways, delay for intersections 4. **Convert Units if Necessary**: Ensure all inputs are in consistent units before substituting into formulas. 5. **Perform Calculation**: Use the correct formula with proper unit conversion factors. 6. **Check for Reasonableness**: - Does the answer have the correct units? - Is the magnitude reasonable? (Flow < capacity; density reasonable for stated conditions) - Can the answer be verified using an alternative relationship? 7. **Interpret the Result**: Explain what the answer means in practical terms (e.g., "The roadway operates at LOS C, indicating stable flow with moderate restrictions.").