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

Formulas

Exam Tips

• If a problem gives you k in veh/km and speed in km/h, directly apply q = ku — no unit conversion needed.
• When given individual vehicle travel times, compute space-mean speed as: u_s = n / Σ(1/uᵢ) or equivalently L / (average travel time).
• The flow–density curve is symmetric only for the Greenshields linear speed–density model (u = u_f(1 – k/k_j)). Under this model, k_opt = k_j/2 and q_max = u_f × k_j / 4.
• Board questions on the congested branch: 'flow decreases as density increases' — always TRUE beyond k_opt.

Key Points

• Three primary variables: Flow q (veh/hr), Density k (veh/km), and Space-Mean Speed u (km/h).
• Fundamental flow equation: q = k × u — the single most tested relationship in this chapter.
• Flow q is the rate at which vehicles pass a fixed point per unit time.
• Density k (also called concentration) is the number of vehicles occupying a unit length of roadway at a given instant.
• Space-Mean Speed (SMS) is computed as the harmonic mean of individual vehicle speeds: total distance / total travel time. This is the correct speed to use in q = ku.
• Time-Mean Speed (TMS) is the arithmetic mean of spot speeds — it is ALWAYS ≥ Space-Mean Speed; do not substitute TMS into q = ku.
• The q–k–u relationship forms a parabolic or bell-shaped flow–density curve. Maximum flow (capacity) occurs at an optimum density k_opt; beyond k_opt, adding more vehicles causes speeds to drop so severely that flow actually decreases (congested branch).
• At jam density k_j, flow q = 0 (gridlock). At zero density, q = 0 (no vehicles). Capacity is the apex of the curve.

Definitions

Section Title

1. Fundamental Traffic Stream Variables

Common Mistakes

• Using Time-Mean Speed instead of Space-Mean Speed in q = ku — this overstates flow.
• Mixing units: k must be in veh/km and q must be in veh/hr for q = ku to give km/h for speed.
• Assuming that higher density always means higher flow — beyond k_opt, the relationship reverses (congested branch).
• Confusing flow (veh/hr passing a point) with volume (total vehicles over a period) — they are numerically equal when the period is exactly one hour.

Formulas

Exam Tips

• Memory aid: 'SpaCing → 1000/k (C for Concentration=density, k); HeaDway → 3600/q (D for Distance… wait — no: H is for Hours, 3600 s/hr)'.
• If q and k are both given, you can find both s and h independently — use both formulas and verify u = q/k as a check.
• Minimum headway at capacity (~2 s for a freeway) corresponds to maximum flow (~1800 veh/hr at 2 s headway).

Key Points

• Spacing (s) and headway (h) are the geometric and temporal gaps between successive vehicles, respectively.
• Spacing uses the constant 1000 (meters per kilometer); headway uses 3600 (seconds per hour).
• Spacing is inversely proportional to density; headway is inversely proportional to flow.
• For safety and design: minimum headway corresponds to capacity flow; maximum spacing corresponds to near-zero density.
• Headway and spacing are related by the stream speed: s = h × u (with consistent units).
• These two formulas are among the most straightforward on the board exam — do not lose marks on them.

Definitions

Section Title

2. Spacing and Headway

Common Mistakes

• Using 1000 for headway or 3600 for spacing — remember: 1000 is for spacing (meters/km), 3600 is for headway (seconds/hr).
• Confusing spacing (distance) with gap (clear distance) — spacing includes one vehicle length; gap does not.
• Not checking unit consistency when cross-verifying s = h × u.

Formulas

Exam Tips

• Memorize: PHF = V / (4V₁₅) — the '4' is non-negotiable because 60 min / 15 min = 4 periods.
• If a problem asks for the 'design hourly volume' or 'equivalent hourly flow rate,' it wants V/PHF.
• Board problems may give you a table of 15-minute counts — always pick the HIGHEST single 15-min count as V₁₅, and sum all four for V.
• PHF = 0.90 is a typical urban arterial value; PHF ≈ 0.95 is common for freeways. Values below 0.80 indicate highly peaky rural or recreational routes.

Key Points

• The PHF quantifies how unevenly traffic demand is distributed within the peak hour.
• It is based on the highest 15-minute volume V₁₅ within the peak hour.
• PHF ranges from 0.25 (all traffic concentrated in one 15-min period) to 1.0 (perfectly uniform flow throughout the hour).
• A PHF close to 1.0 indicates steady, uniform flow; a low PHF indicates peaky, volatile demand.
• Design (peak) flow rate = V / PHF — this is the demand used for capacity analysis, not the raw hourly volume.
• The factor 4 in the denominator comes from the fact that there are four 15-minute intervals in one hour.
• PHF is dimensionless and always ≤ 1.0 — if your answer exceeds 1.0, you made an error.

Definitions

Section Title

3. Peak-Hour Factor (PHF)

Common Mistakes

• Writing PHF = V / V₁₅ (forgetting the factor 4) — this is the single most common PHF error on the board exam.
• Using a 5-minute or 1-minute peak count instead of the 15-minute peak count.
• Computing a PHF > 1.0 — physically impossible; recheck V₁₅ vs. V.
• Using the raw hourly volume V for capacity analysis instead of the design flow rate V/PHF.

Formulas

Exam Tips

• LOS A–F mnemonic: 'A = Amazing (free flow), B = Brisk, C = Comfortable, D = Degraded, E = Edge of capacity, F = Failed (breakdown).'
• Freeway LOS boundaries (HCM 6th Ed., basic segments): LOS A ≤ 11 pc/km/ln; B ≤ 18; C ≤ 26; D ≤ 35; E ≤ 45; F > 45 pc/km/ln. These are HCM values — Philippine board exams may reference similar thresholds.
• For Webster's formula, Y must be < 1.0 (otherwise the denominator is negative/zero, meaning the intersection cannot handle the demand with any finite cycle).
• A freeway with 2 lanes in each direction and q = 1800 pc/hr/ln at ideal conditions operates near capacity (LOS E); add a PHF correction to find the peak flow rate first.

Key Points

• Capacity is the maximum sustainable flow rate at which vehicles can be expected to traverse a uniform section of roadway under prevailing conditions.
• Under ideal conditions, freeway lane capacity ≈ 2000 passenger cars per hour per lane (pc/h/ln).
• LOS is graded A through F. LOS A is free flow (low density, high speed). LOS F is breakdown/forced flow (demand exceeds capacity).
• For freeways and multilane highways, LOS is determined by density (pc/km/ln) per the Highway Capacity Manual (HCM).
• For signalized intersections, LOS is determined by average control delay per vehicle (s/veh).
• LOS E represents capacity operation (maximum density before breakdown). LOS F means demand > capacity.
• Factors reducing capacity from ideal: narrow lanes, restricted lateral clearance, heavy vehicles, grade, driver population (non-commuters), and access points.
• The passenger car equivalent (PCE or ET) converts trucks and buses to equivalent passenger cars for capacity analysis.
• Webster's formula gives the optimal signal cycle length: C₀ = (1.5L + 5) / (1 – Y), where L = total lost time and Y = sum of critical flow ratios.

Definitions

Section Title

4. Capacity and Level of Service (LOS)

Common Mistakes

• Applying the density-based LOS thresholds (for freeways) to signalized intersections, which use delay — use the correct measure for the correct facility type.
• Forgetting to convert mixed traffic to pc/hr using PCE factors before comparing to capacity.
• Not dividing by PHF before computing v/c — this underestimates demand.
• Using Y ≥ 1.0 in Webster's formula — if Y ≥ 1.0, the intersection is oversaturated and the formula is inapplicable (C₀ → ∞ or negative).
• Confusing LOS E (at capacity, v/c ≈ 1.0) with LOS F (breakdown, v/c > 1.0).