Geometric Design, Sight Distances and Horizontal/Vertical Alignments

Learning objectives

Learn to characterize the mechanisms, assumptions and observational evidence connecting design speed, stopping sight distance and grade, horizontal curves and superelevation, vertical curves and profile geometry, cross sections, drainage and accessible streets. Demonstrate unit consistency, interpret safety and access alongside vehicle throughput, and compare model predictions against field observations before making recommendations.

How to study this unit

Work through all four lessons, reproduce each numerical illustration without copying the result, operate the linked investigations, then answer the unit assessment and prepare a short project memo. Recognize which assumptions are useful for learning and which prevent design decisions.

Lesson 3.1: Design speed, stopping sight distance and grade

Physical and planning meaning

Sight distance links driver perception, reaction and braking. The stopping process has a perception–reaction segment followed by deceleration. A simple screening model assumes level ground or constant grade and uniform tire–road friction; actual design uses adopted local standards and driver population assumptions.

Governing relationship

SSD = vt + v²/[2g(f ± G)] with v in m/s and grade G expressed as a decimal.

Interpreted worked case

At 80 km/h (22.22 m/s), reaction time 2.5 s, friction f = 0.35 and level grade, reaction distance is 55.6 m and braking distance about 71.9 m, giving roughly 127.5 m. This calculation must not be treated as a legally sufficient roadway design value.

Investigating an assumption

Explore wet friction, downhill grade, night visibility and vehicle class; which changes would most affect stopping distance at high speed?

Important limitations

A stopping-distance equation cannot replace geometric sight-line checks, weather-specific friction data or the relevant design-standard table.

Measurement protocol

Specify which observations distinguish the proposed engineering relationship from a coincidental correlation. Record station location, observation interval, vehicles or users included, calibration checks and missing data. A correct equation evaluated on a biased dataset can create a less defensible conclusion than a rough estimate accompanied by an explicit uncertainty interval.

Three distinct evidence-based investigations

Investigation 3.1.1

An agency is reviewing a busy intersection surrounded by stores, loading areas and narrow footpaths. The primary affected users include local businesses, cyclists and walkers. Before transferring the relationship SSD = vt + v²/[2g(f ± G)] with v in m/s and grade G expressed as a decimal. to the location, explain which inputs could be directly observed, which would need a proxy, and which are not meaningful without a local policy or design standard. Establish a baseline for conflicting curb use and vehicle queues; compare that baseline with an alternative centred on reassignment of curb space with inclusive access. Your written answer must separate improvements that follow from a formula from improvements that are only plausible hypotheses. Address conditions in which the intervention would fail.

Investigation 3.1.2

Use this worked reference as a challenge rather than as a template to memorize: At 80 km/h (22.22 m/s), reaction time 2.5 s, friction f = 0.35 and level grade, reaction distance is 55.6 m and braking distance about 71.9 m, giving roughly 127.5 m. This calculation must not be treated as a legally sufficient roadway design value. Replace one input with a plausible field observation, state why the replacement is plausible, and predict the direction of the result before computing. If the prediction differs from the calculation, investigate units and the sign convention first. Explain how uncertainty from measurement, behavior and a simplified model affects the credibility of a precise numerical answer.

Investigation 3.1.3

A project manager proposes to use a single number from this topic to approve an investment. Prepare a critique rooted in the following limitation: A stopping-distance equation cannot replace geometric sight-line checks, weather-specific friction data or the relevant design-standard table. List two measurements that would improve the study, one externality that the calculation does not represent and one practical decision that could be deferred until the missing evidence is obtained. Avoid claiming the educational model has been professionally calibrated or validated on a real transportation network.

Explain the computation to a classmate

Without looking at the numerical illustration, reconstruct the logic of design speed, stopping sight distance and grade. Start with the field measurement and unit definition, define the reference condition, state the idealization in the equation (SSD = vt + v²/[2g(f ± G)] with v in m/s and grade G expressed as a decimal.), and then walk through the calculation so that a reader can independently check every value. Identify which result can be checked by a second technique. If observations contradict the simple model, explain how you would revise the model rather than discard the observations.

Field data and reporting protocol

For a short field exercise about design speed, stopping sight distance and grade, specify a time window, observation location, data dictionary and recording method. Protect individual privacy by avoiding unnecessary vehicle or personal identifiers. Use at least three quality checks: plausibility ranges for the raw input, missing-value counts, and a reconciliation of any aggregated totals. Present separate notes on systematic bias, day-to-day variability and uncertainty in the equation itself.

Decision boundary

The relationship in this lesson is appropriate for concept exploration and transparent back-of-the-envelope screening. A code-compliant geometric decision would additionally need jurisdiction-specific requirements, evidence of relevant site conditions and a documented professional review. Name the exact assumptions that would need replacement and identify stakeholders who might be excluded if the decision were made entirely from a vehicle-centred metric.

Lesson 3.2: Horizontal curves and superelevation

Physical and planning meaning

A horizontal road curve redirects the vehicle. Lateral acceleration scales with the square of speed and inversely with radius; pavement crossfall helps balance lateral demand, but allowable side friction depends on driver comfort and safety assumptions. Curves also influence available sight distance through roadside obstructions.

Governing relationship

R = V²/[127(e+f)] using V in km/h, R in metres, e and f as fractions.

Interpreted worked case

At 80 km/h with e = 0.06 and f = 0.10, the idealized balance yields R = 6400/(127×0.16) ≈ 315 m. The maximum permitted e and design side-friction are jurisdiction-specific, not freely interchangeable knobs.

Investigating an assumption

What happens to the required radius if the design speed increases from 80 to 100 km/h without changing superelevation or friction?

Important limitations

The simplified formula omits runoff transitions, trucks with higher centres of gravity, curve widening and combined vertical alignment.

Dimensional audit

Before computing, rewrite every quantity in one coherent system of units. A time rate needs a time denominator; a spatial density needs a road-length denominator; a percentage must become a decimal before entering an algebraic equation. Unit checking can reject many plausible-looking but incorrect answers without any specialist software.

Three distinct evidence-based investigations

Investigation 3.2.1

An agency is reviewing a developing area with uncertain future traffic and pedestrian demand. The primary affected users include new residents and existing communities. Before transferring the relationship R = V²/[127(e+f)] using V in km/h, R in metres, e and f as fractions. to the location, explain which inputs could be directly observed, which would need a proxy, and which are not meaningful without a local policy or design standard. Establish a baseline for forecast error and long-lived capital decisions; compare that baseline with an alternative centred on phased investment with monitoring triggers. Your written answer must separate improvements that follow from a formula from improvements that are only plausible hypotheses. Address conditions in which the intervention would fail.

Investigation 3.2.2

Use this worked reference as a challenge rather than as a template to memorize: At 80 km/h with e = 0.06 and f = 0.10, the idealized balance yields R = 6400/(127×0.16) ≈ 315 m. The maximum permitted e and design side-friction are jurisdiction-specific, not freely interchangeable knobs. Replace one input with a plausible field observation, state why the replacement is plausible, and predict the direction of the result before computing. If the prediction differs from the calculation, investigate units and the sign convention first. Explain how uncertainty from measurement, behavior and a simplified model affects the credibility of a precise numerical answer.

Investigation 3.2.3

A project manager proposes to use a single number from this topic to approve an investment. Prepare a critique rooted in the following limitation: The simplified formula omits runoff transitions, trucks with higher centres of gravity, curve widening and combined vertical alignment. List two measurements that would improve the study, one externality that the calculation does not represent and one practical decision that could be deferred until the missing evidence is obtained. Avoid claiming the educational model has been professionally calibrated or validated on a real transportation network.

Explain the computation to a classmate

Without looking at the numerical illustration, reconstruct the logic of horizontal curves and superelevation. Start with the field measurement and unit definition, define the reference condition, state the idealization in the equation (R = V²/[127(e+f)] using V in km/h, R in metres, e and f as fractions.), and then walk through the calculation so that a reader can independently check every value. Identify which result can be checked by a second technique. If observations contradict the simple model, explain how you would revise the model rather than discard the observations.

Field data and reporting protocol

For a short field exercise about horizontal curves and superelevation, specify a time window, observation location, data dictionary and recording method. Protect individual privacy by avoiding unnecessary vehicle or personal identifiers. Use at least three quality checks: plausibility ranges for the raw input, missing-value counts, and a reconciliation of any aggregated totals. Present separate notes on systematic bias, day-to-day variability and uncertainty in the equation itself.

Decision boundary

The relationship in this lesson is appropriate for concept exploration and transparent back-of-the-envelope screening. A code-compliant geometric decision would additionally need jurisdiction-specific requirements, evidence of relevant site conditions and a documented professional review. Name the exact assumptions that would need replacement and identify stakeholders who might be excluded if the decision were made entirely from a vehicle-centred metric.

Lesson 3.3: Vertical curves and profile geometry

Physical and planning meaning

Vertical curves create continuous changes of longitudinal grade. Engineers distinguish sag curves, where headlight illumination and comfort matter, from crest curves, where available sight distance may be limited by the pavement geometry. A symmetric parabolic vertical curve has a linearly varying grade; elevation can be computed from tangent grades and length.

Governing relationship

y(x) = y₀ + g₁x + (g₂−g₁)x²/(2L), for 0 ≤ x ≤ L.

Interpreted worked case

A crest curve begins at elevation 100 m with initial grade +3%, ends with grade −2% and spans 200 m. At midpoint x=100 m, the simplified elevation is 100 + 3 − 1.25 = 101.75 m. This is a geometric calculation, not a complete visibility design check.

Investigating an assumption

Plot two profiles with the same end grades but different lengths and discuss how rate of change in grade influences comfort and sight lines.

Important limitations

Actual crest and sag design must check road-edge geometry, vehicles, lighting, overhead constraints and agency criteria.

Alternative design comparison

Compare at least two interventions against the same baseline. Record whether each changes demand, available capacity, exposure, pavement condition, human behavior or monitoring quality. An alternative with the largest improvement in one performance indicator is not necessarily superior when its cost, emissions, access impacts and failure modes are considered.

Three distinct evidence-based investigations

Investigation 3.3.1

An agency is reviewing a two-lane route linking distant towns with limited passing opportunities. The primary affected users include intercity travelers and bus passengers. Before transferring the relationship y(x) = y₀ + g₁x + (g₂−g₁)x²/(2L), for 0 ≤ x ≤ L. to the location, explain which inputs could be directly observed, which would need a proxy, and which are not meaningful without a local policy or design standard. Establish a baseline for severe-crash exposure and overtaking constraints; compare that baseline with an alternative centred on targeted safety improvements rather than indiscriminate widening. Your written answer must separate improvements that follow from a formula from improvements that are only plausible hypotheses. Address conditions in which the intervention would fail.

Investigation 3.3.2

Use this worked reference as a challenge rather than as a template to memorize: A crest curve begins at elevation 100 m with initial grade +3%, ends with grade −2% and spans 200 m. At midpoint x=100 m, the simplified elevation is 100 + 3 − 1.25 = 101.75 m. This is a geometric calculation, not a complete visibility design check. Replace one input with a plausible field observation, state why the replacement is plausible, and predict the direction of the result before computing. If the prediction differs from the calculation, investigate units and the sign convention first. Explain how uncertainty from measurement, behavior and a simplified model affects the credibility of a precise numerical answer.

Investigation 3.3.3

A project manager proposes to use a single number from this topic to approve an investment. Prepare a critique rooted in the following limitation: Actual crest and sag design must check road-edge geometry, vehicles, lighting, overhead constraints and agency criteria. List two measurements that would improve the study, one externality that the calculation does not represent and one practical decision that could be deferred until the missing evidence is obtained. Avoid claiming the educational model has been professionally calibrated or validated on a real transportation network.

Explain the computation to a classmate

Without looking at the numerical illustration, reconstruct the logic of vertical curves and profile geometry. Start with the field measurement and unit definition, define the reference condition, state the idealization in the equation (y(x) = y₀ + g₁x + (g₂−g₁)x²/(2L), for 0 ≤ x ≤ L.), and then walk through the calculation so that a reader can independently check every value. Identify which result can be checked by a second technique. If observations contradict the simple model, explain how you would revise the model rather than discard the observations.

Field data and reporting protocol

For a short field exercise about vertical curves and profile geometry, specify a time window, observation location, data dictionary and recording method. Protect individual privacy by avoiding unnecessary vehicle or personal identifiers. Use at least three quality checks: plausibility ranges for the raw input, missing-value counts, and a reconciliation of any aggregated totals. Present separate notes on systematic bias, day-to-day variability and uncertainty in the equation itself.

Decision boundary

The relationship in this lesson is appropriate for concept exploration and transparent back-of-the-envelope screening. A code-compliant geometric decision would additionally need jurisdiction-specific requirements, evidence of relevant site conditions and a documented professional review. Name the exact assumptions that would need replacement and identify stakeholders who might be excluded if the decision were made entirely from a vehicle-centred metric.

Lesson 3.4: Cross sections, drainage and accessible streets

Physical and planning meaning

A roadway cross-section allocates width to travel lanes, shoulders, footpaths, bike facilities, medians and drainage. A crossfall drains water but may also create challenging slopes for wheelchairs and cyclists. Safe complete-street design is a constraint-balancing process, not simply maximizing the number of vehicle lanes.

Governing relationship

Surface lateral drop = crossfall × width, with crossfall dimensionless.

Interpreted worked case

A 3.5 m travel lane with a 2.0% crossfall has a centre-to-edge drop of 70 mm if it falls across the full lane width. The result says nothing about whether the curb inlet has adequate capacity to remove the runoff.

Investigating an assumption

Sketch a corridor that must accommodate a bus stop, an accessible footpath, drainage inlet and bicycle movement in a constrained right of way.

Important limitations

Do not treat crossfall as a substitute for cross-section hydraulic analysis or local accessibility guidance.

Sensitivity and uncertainty

Vary one uncertain parameter while keeping others fixed, then vary a second parameter jointly. A steep response indicates that the result should be reported as a range rather than a single number. Do not equate a sensitivity exercise with a statistical confidence interval; uncertainty also includes model structure and measurement bias.

Three distinct evidence-based investigations

Investigation 3.4.1

An agency is reviewing a road with crowded mixed traffic and a frequent bus service. The primary affected users include transit riders and general-road users. Before transferring the relationship Surface lateral drop = crossfall × width, with crossfall dimensionless. to the location, explain which inputs could be directly observed, which would need a proxy, and which are not meaningful without a local policy or design standard. Establish a baseline for person throughput and bus headway variability; compare that baseline with an alternative centred on transit priority coupled with safer stops. Your written answer must separate improvements that follow from a formula from improvements that are only plausible hypotheses. Address conditions in which the intervention would fail.

Investigation 3.4.2

Use this worked reference as a challenge rather than as a template to memorize: A 3.5 m travel lane with a 2.0% crossfall has a centre-to-edge drop of 70 mm if it falls across the full lane width. The result says nothing about whether the curb inlet has adequate capacity to remove the runoff. Replace one input with a plausible field observation, state why the replacement is plausible, and predict the direction of the result before computing. If the prediction differs from the calculation, investigate units and the sign convention first. Explain how uncertainty from measurement, behavior and a simplified model affects the credibility of a precise numerical answer.

Investigation 3.4.3

A project manager proposes to use a single number from this topic to approve an investment. Prepare a critique rooted in the following limitation: Do not treat crossfall as a substitute for cross-section hydraulic analysis or local accessibility guidance. List two measurements that would improve the study, one externality that the calculation does not represent and one practical decision that could be deferred until the missing evidence is obtained. Avoid claiming the educational model has been professionally calibrated or validated on a real transportation network.

Explain the computation to a classmate

Without looking at the numerical illustration, reconstruct the logic of cross sections, drainage and accessible streets. Start with the field measurement and unit definition, define the reference condition, state the idealization in the equation (Surface lateral drop = crossfall × width, with crossfall dimensionless.), and then walk through the calculation so that a reader can independently check every value. Identify which result can be checked by a second technique. If observations contradict the simple model, explain how you would revise the model rather than discard the observations.

Field data and reporting protocol

For a short field exercise about cross sections, drainage and accessible streets, specify a time window, observation location, data dictionary and recording method. Protect individual privacy by avoiding unnecessary vehicle or personal identifiers. Use at least three quality checks: plausibility ranges for the raw input, missing-value counts, and a reconciliation of any aggregated totals. Present separate notes on systematic bias, day-to-day variability and uncertainty in the equation itself.

Decision boundary

The relationship in this lesson is appropriate for concept exploration and transparent back-of-the-envelope screening. A code-compliant geometric decision would additionally need jurisdiction-specific requirements, evidence of relevant site conditions and a documented professional review. Name the exact assumptions that would need replacement and identify stakeholders who might be excluded if the decision were made entirely from a vehicle-centred metric.

Unit synthesis exercise

Connect design speed, stopping sight distance and grade with cross sections, drainage and accessible streets: imagine a corridor proposal that improves the first measure but worsens the last. Write a one-page recommendation showing the trade-off, a minimum dataset for testing both effects, and the circumstances that would reverse your recommendation.

Reading the laboratory outputs

Every chart is calculated for hypothetical parameters. Change one parameter at a time to understand the model. Check a point by hand. Export a CSV, note the axes, then identify at least one physically relevant effect omitted by the model. No laboratory in this book is approved for design or real-time operational control.

Extended design exercise: a curved road beside a school and drainage canal

A rural-to-urban transition approaches a school along a left-turning horizontal curve. The road carries buses and goods vehicles and runs beside a drainage canal. Students approach on foot from both sides. An initial geometric screening at 70 km/h with superelevation 0.05 and illustrative side-friction factor 0.10 gives a radius from R=V²/[127(e+f)] of about 257 metres. This figure is a calculated balance, not a permitted or sufficiently safe curve radius. The actual design depends on the adopted code, available sight distance, vehicles and road environment. The need to support pedestrians near a school is itself a reason to question whether the desired operating speed is appropriate.

Before proposing a larger radius, the engineer plots the existing horizontal curve using surveyed coordinates and reconstructs road edges, roadside canal boundaries and property entrances. The curve may be broad enough in plan but still have a hidden crossing behind a wall. A stopping sight-distance check must consider the driver's line of sight, grade and obstruction height. For a simplified level-road scenario at 70 km/h, reaction time 2.5 seconds and friction 0.35, reaction distance is about 48.6 metres and braking distance about 55.1 metres, making the approximate stopping distance 103.7 metres. This does not incorporate sight-line obstructions, headlight effects or local design-standard values.

The road profile may also include a crest. Draw a parabolic vertical curve whose initial and final tangent grades are known; do not confuse the ability to calculate its elevation with verification that a driver can see a child crossing ahead. If the roadway crest hides part of a crossing, a geometrically smooth profile is still hazardous. Similarly, a two-percent crossfall across a three-metre lane yields a vertical difference of 60 millimetres, but does not prove runoff will enter a working inlet. The canal-side drainage pattern needs an independent hydraulic and erosion review. Raising a shoulder could unintentionally divert storm runoff toward nearby houses.

Propose alternatives: reducing the target speed and improving crossings; relocating an obstruction; revising the alignment; or providing a physically separated pedestrian route. The first three have different costs and environmental impacts. A lower-speed corridor can sometimes improve road-user safety at less expense than a major realignment, but that outcome must be measured, not assumed. Record peak pedestrian flows, bus stopping locations, wet-weather visibility, the location of vulnerable access points and observed operating speeds across several periods. Public consultation is necessary because the shortest walking route and the driver's preferred travel path may differ.

Final submission must include annotated plan and profile views, a table of assumptions with units, calculation of the screening radius and SSD, a list of omitted visibility checks and two alternative corridor designs. The final recommendation must be framed as a proposal for proper standards-based assessment, not a construction drawing. Explain how the conclusion changes if the desired design speed is reduced, the pavement is wet or a bus stop is added near the crossing.

Unit 3: Geometric Design and Human Factors

Book 14 · Transportation Engineering · Civil Engineering

Learning objectives

Learn to characterize the mechanisms, assumptions and observational evidence connecting design speed, stopping sight distance and grade, horizontal curves and superelevation, vertical curves and profile geometry, cross sections, drainage and accessible streets. Demonstrate unit consistency, interpret safety and access alongside vehicle throughput, and compare model predictions against field observations before making recommendations.

How to study this unit

Work through all four lessons, reproduce each numerical illustration without copying the result, operate the linked investigations, then answer the unit assessment and prepare a short project memo. Recognize which assumptions are useful for learning and which prevent design decisions.

Lesson 3.1: Design speed, stopping sight distance and grade

Physical and planning meaning

Sight distance links driver perception, reaction and braking. The stopping process has a perception–reaction segment followed by deceleration. A simple screening model assumes level ground or constant grade and uniform tire–road friction; actual design uses adopted local standards and driver population assumptions.

Governing relationship

SSD = vt + v²/[2g(f ± G)] with v in m/s and grade G expressed as a decimal.

Interpreted worked case

At 80 km/h (22.22 m/s), reaction time 2.5 s, friction f = 0.35 and level grade, reaction distance is 55.6 m and braking distance about 71.9 m, giving roughly 127.5 m. This calculation must not be treated as a legally sufficient roadway design value.

Investigating an assumption

Explore wet friction, downhill grade, night visibility and vehicle class; which changes would most affect stopping distance at high speed?

Important limitations

A stopping-distance equation cannot replace geometric sight-line checks, weather-specific friction data or the relevant design-standard table.

Measurement protocol

Specify which observations distinguish the proposed engineering relationship from a coincidental correlation. Record station location, observation interval, vehicles or users included, calibration checks and missing data. A correct equation evaluated on a biased dataset can create a less defensible conclusion than a rough estimate accompanied by an explicit uncertainty interval.

Three distinct evidence-based investigations

Investigation 3.1.1

An agency is reviewing a busy intersection surrounded by stores, loading areas and narrow footpaths. The primary affected users include local businesses, cyclists and walkers. Before transferring the relationship SSD = vt + v²/[2g(f ± G)] with v in m/s and grade G expressed as a decimal. to the location, explain which inputs could be directly observed, which would need a proxy, and which are not meaningful without a local policy or design standard. Establish a baseline for conflicting curb use and vehicle queues; compare that baseline with an alternative centred on reassignment of curb space with inclusive access. Your written answer must separate improvements that follow from a formula from improvements that are only plausible hypotheses. Address conditions in which the intervention would fail.

Investigation 3.1.2

Use this worked reference as a challenge rather than as a template to memorize: At 80 km/h (22.22 m/s), reaction time 2.5 s, friction f = 0.35 and level grade, reaction distance is 55.6 m and braking distance about 71.9 m, giving roughly 127.5 m. This calculation must not be treated as a legally sufficient roadway design value. Replace one input with a plausible field observation, state why the replacement is plausible, and predict the direction of the result before computing. If the prediction differs from the calculation, investigate units and the sign convention first. Explain how uncertainty from measurement, behavior and a simplified model affects the credibility of a precise numerical answer.

Investigation 3.1.3

A project manager proposes to use a single number from this topic to approve an investment. Prepare a critique rooted in the following limitation: A stopping-distance equation cannot replace geometric sight-line checks, weather-specific friction data or the relevant design-standard table. List two measurements that would improve the study, one externality that the calculation does not represent and one practical decision that could be deferred until the missing evidence is obtained. Avoid claiming the educational model has been professionally calibrated or validated on a real transportation network.

Explain the computation to a classmate

Without looking at the numerical illustration, reconstruct the logic of design speed, stopping sight distance and grade. Start with the field measurement and unit definition, define the reference condition, state the idealization in the equation (SSD = vt + v²/[2g(f ± G)] with v in m/s and grade G expressed as a decimal.), and then walk through the calculation so that a reader can independently check every value. Identify which result can be checked by a second technique. If observations contradict the simple model, explain how you would revise the model rather than discard the observations.

Field data and reporting protocol

For a short field exercise about design speed, stopping sight distance and grade, specify a time window, observation location, data dictionary and recording method. Protect individual privacy by avoiding unnecessary vehicle or personal identifiers. Use at least three quality checks: plausibility ranges for the raw input, missing-value counts, and a reconciliation of any aggregated totals. Present separate notes on systematic bias, day-to-day variability and uncertainty in the equation itself.

Decision boundary

The relationship in this lesson is appropriate for concept exploration and transparent back-of-the-envelope screening. A code-compliant geometric decision would additionally need jurisdiction-specific requirements, evidence of relevant site conditions and a documented professional review. Name the exact assumptions that would need replacement and identify stakeholders who might be excluded if the decision were made entirely from a vehicle-centred metric.

Lesson 3.2: Horizontal curves and superelevation

Physical and planning meaning

A horizontal road curve redirects the vehicle. Lateral acceleration scales with the square of speed and inversely with radius; pavement crossfall helps balance lateral demand, but allowable side friction depends on driver comfort and safety assumptions. Curves also influence available sight distance through roadside obstructions.

Governing relationship

R = V²/[127(e+f)] using V in km/h, R in metres, e and f as fractions.

Interpreted worked case

At 80 km/h with e = 0.06 and f = 0.10, the idealized balance yields R = 6400/(127×0.16) ≈ 315 m. The maximum permitted e and design side-friction are jurisdiction-specific, not freely interchangeable knobs.

Investigating an assumption

What happens to the required radius if the design speed increases from 80 to 100 km/h without changing superelevation or friction?

Important limitations

The simplified formula omits runoff transitions, trucks with higher centres of gravity, curve widening and combined vertical alignment.

Dimensional audit

Before computing, rewrite every quantity in one coherent system of units. A time rate needs a time denominator; a spatial density needs a road-length denominator; a percentage must become a decimal before entering an algebraic equation. Unit checking can reject many plausible-looking but incorrect answers without any specialist software.

Three distinct evidence-based investigations

Investigation 3.2.1

An agency is reviewing a developing area with uncertain future traffic and pedestrian demand. The primary affected users include new residents and existing communities. Before transferring the relationship R = V²/[127(e+f)] using V in km/h, R in metres, e and f as fractions. to the location, explain which inputs could be directly observed, which would need a proxy, and which are not meaningful without a local policy or design standard. Establish a baseline for forecast error and long-lived capital decisions; compare that baseline with an alternative centred on phased investment with monitoring triggers. Your written answer must separate improvements that follow from a formula from improvements that are only plausible hypotheses. Address conditions in which the intervention would fail.

Investigation 3.2.2

Use this worked reference as a challenge rather than as a template to memorize: At 80 km/h with e = 0.06 and f = 0.10, the idealized balance yields R = 6400/(127×0.16) ≈ 315 m. The maximum permitted e and design side-friction are jurisdiction-specific, not freely interchangeable knobs. Replace one input with a plausible field observation, state why the replacement is plausible, and predict the direction of the result before computing. If the prediction differs from the calculation, investigate units and the sign convention first. Explain how uncertainty from measurement, behavior and a simplified model affects the credibility of a precise numerical answer.

Investigation 3.2.3

A project manager proposes to use a single number from this topic to approve an investment. Prepare a critique rooted in the following limitation: The simplified formula omits runoff transitions, trucks with higher centres of gravity, curve widening and combined vertical alignment. List two measurements that would improve the study, one externality that the calculation does not represent and one practical decision that could be deferred until the missing evidence is obtained. Avoid claiming the educational model has been professionally calibrated or validated on a real transportation network.

Explain the computation to a classmate

Without looking at the numerical illustration, reconstruct the logic of horizontal curves and superelevation. Start with the field measurement and unit definition, define the reference condition, state the idealization in the equation (R = V²/[127(e+f)] using V in km/h, R in metres, e and f as fractions.), and then walk through the calculation so that a reader can independently check every value. Identify which result can be checked by a second technique. If observations contradict the simple model, explain how you would revise the model rather than discard the observations.

Field data and reporting protocol

For a short field exercise about horizontal curves and superelevation, specify a time window, observation location, data dictionary and recording method. Protect individual privacy by avoiding unnecessary vehicle or personal identifiers. Use at least three quality checks: plausibility ranges for the raw input, missing-value counts, and a reconciliation of any aggregated totals. Present separate notes on systematic bias, day-to-day variability and uncertainty in the equation itself.

Decision boundary

The relationship in this lesson is appropriate for concept exploration and transparent back-of-the-envelope screening. A code-compliant geometric decision would additionally need jurisdiction-specific requirements, evidence of relevant site conditions and a documented professional review. Name the exact assumptions that would need replacement and identify stakeholders who might be excluded if the decision were made entirely from a vehicle-centred metric.

Lesson 3.3: Vertical curves and profile geometry

Physical and planning meaning

Vertical curves create continuous changes of longitudinal grade. Engineers distinguish sag curves, where headlight illumination and comfort matter, from crest curves, where available sight distance may be limited by the pavement geometry. A symmetric parabolic vertical curve has a linearly varying grade; elevation can be computed from tangent grades and length.

Governing relationship

y(x) = y₀ + g₁x + (g₂−g₁)x²/(2L), for 0 ≤ x ≤ L.

Interpreted worked case

A crest curve begins at elevation 100 m with initial grade +3%, ends with grade −2% and spans 200 m. At midpoint x=100 m, the simplified elevation is 100 + 3 − 1.25 = 101.75 m. This is a geometric calculation, not a complete visibility design check.

Investigating an assumption

Plot two profiles with the same end grades but different lengths and discuss how rate of change in grade influences comfort and sight lines.

Important limitations

Actual crest and sag design must check road-edge geometry, vehicles, lighting, overhead constraints and agency criteria.

Alternative design comparison

Compare at least two interventions against the same baseline. Record whether each changes demand, available capacity, exposure, pavement condition, human behavior or monitoring quality. An alternative with the largest improvement in one performance indicator is not necessarily superior when its cost, emissions, access impacts and failure modes are considered.

Three distinct evidence-based investigations

Investigation 3.3.1

An agency is reviewing a two-lane route linking distant towns with limited passing opportunities. The primary affected users include intercity travelers and bus passengers. Before transferring the relationship y(x) = y₀ + g₁x + (g₂−g₁)x²/(2L), for 0 ≤ x ≤ L. to the location, explain which inputs could be directly observed, which would need a proxy, and which are not meaningful without a local policy or design standard. Establish a baseline for severe-crash exposure and overtaking constraints; compare that baseline with an alternative centred on targeted safety improvements rather than indiscriminate widening. Your written answer must separate improvements that follow from a formula from improvements that are only plausible hypotheses. Address conditions in which the intervention would fail.

Investigation 3.3.2

Use this worked reference as a challenge rather than as a template to memorize: A crest curve begins at elevation 100 m with initial grade +3%, ends with grade −2% and spans 200 m. At midpoint x=100 m, the simplified elevation is 100 + 3 − 1.25 = 101.75 m. This is a geometric calculation, not a complete visibility design check. Replace one input with a plausible field observation, state why the replacement is plausible, and predict the direction of the result before computing. If the prediction differs from the calculation, investigate units and the sign convention first. Explain how uncertainty from measurement, behavior and a simplified model affects the credibility of a precise numerical answer.

Investigation 3.3.3

A project manager proposes to use a single number from this topic to approve an investment. Prepare a critique rooted in the following limitation: Actual crest and sag design must check road-edge geometry, vehicles, lighting, overhead constraints and agency criteria. List two measurements that would improve the study, one externality that the calculation does not represent and one practical decision that could be deferred until the missing evidence is obtained. Avoid claiming the educational model has been professionally calibrated or validated on a real transportation network.

Explain the computation to a classmate

Without looking at the numerical illustration, reconstruct the logic of vertical curves and profile geometry. Start with the field measurement and unit definition, define the reference condition, state the idealization in the equation (y(x) = y₀ + g₁x + (g₂−g₁)x²/(2L), for 0 ≤ x ≤ L.), and then walk through the calculation so that a reader can independently check every value. Identify which result can be checked by a second technique. If observations contradict the simple model, explain how you would revise the model rather than discard the observations.

Field data and reporting protocol

For a short field exercise about vertical curves and profile geometry, specify a time window, observation location, data dictionary and recording method. Protect individual privacy by avoiding unnecessary vehicle or personal identifiers. Use at least three quality checks: plausibility ranges for the raw input, missing-value counts, and a reconciliation of any aggregated totals. Present separate notes on systematic bias, day-to-day variability and uncertainty in the equation itself.

Decision boundary

The relationship in this lesson is appropriate for concept exploration and transparent back-of-the-envelope screening. A code-compliant geometric decision would additionally need jurisdiction-specific requirements, evidence of relevant site conditions and a documented professional review. Name the exact assumptions that would need replacement and identify stakeholders who might be excluded if the decision were made entirely from a vehicle-centred metric.

Lesson 3.4: Cross sections, drainage and accessible streets

Physical and planning meaning

A roadway cross-section allocates width to travel lanes, shoulders, footpaths, bike facilities, medians and drainage. A crossfall drains water but may also create challenging slopes for wheelchairs and cyclists. Safe complete-street design is a constraint-balancing process, not simply maximizing the number of vehicle lanes.

Governing relationship

Surface lateral drop = crossfall × width, with crossfall dimensionless.

Interpreted worked case

A 3.5 m travel lane with a 2.0% crossfall has a centre-to-edge drop of 70 mm if it falls across the full lane width. The result says nothing about whether the curb inlet has adequate capacity to remove the runoff.

Investigating an assumption

Sketch a corridor that must accommodate a bus stop, an accessible footpath, drainage inlet and bicycle movement in a constrained right of way.

Important limitations

Do not treat crossfall as a substitute for cross-section hydraulic analysis or local accessibility guidance.

Sensitivity and uncertainty

Vary one uncertain parameter while keeping others fixed, then vary a second parameter jointly. A steep response indicates that the result should be reported as a range rather than a single number. Do not equate a sensitivity exercise with a statistical confidence interval; uncertainty also includes model structure and measurement bias.

Three distinct evidence-based investigations

Investigation 3.4.1

An agency is reviewing a road with crowded mixed traffic and a frequent bus service. The primary affected users include transit riders and general-road users. Before transferring the relationship Surface lateral drop = crossfall × width, with crossfall dimensionless. to the location, explain which inputs could be directly observed, which would need a proxy, and which are not meaningful without a local policy or design standard. Establish a baseline for person throughput and bus headway variability; compare that baseline with an alternative centred on transit priority coupled with safer stops. Your written answer must separate improvements that follow from a formula from improvements that are only plausible hypotheses. Address conditions in which the intervention would fail.

Investigation 3.4.2

Use this worked reference as a challenge rather than as a template to memorize: A 3.5 m travel lane with a 2.0% crossfall has a centre-to-edge drop of 70 mm if it falls across the full lane width. The result says nothing about whether the curb inlet has adequate capacity to remove the runoff. Replace one input with a plausible field observation, state why the replacement is plausible, and predict the direction of the result before computing. If the prediction differs from the calculation, investigate units and the sign convention first. Explain how uncertainty from measurement, behavior and a simplified model affects the credibility of a precise numerical answer.

Investigation 3.4.3

A project manager proposes to use a single number from this topic to approve an investment. Prepare a critique rooted in the following limitation: Do not treat crossfall as a substitute for cross-section hydraulic analysis or local accessibility guidance. List two measurements that would improve the study, one externality that the calculation does not represent and one practical decision that could be deferred until the missing evidence is obtained. Avoid claiming the educational model has been professionally calibrated or validated on a real transportation network.

Explain the computation to a classmate

Without looking at the numerical illustration, reconstruct the logic of cross sections, drainage and accessible streets. Start with the field measurement and unit definition, define the reference condition, state the idealization in the equation (Surface lateral drop = crossfall × width, with crossfall dimensionless.), and then walk through the calculation so that a reader can independently check every value. Identify which result can be checked by a second technique. If observations contradict the simple model, explain how you would revise the model rather than discard the observations.

Field data and reporting protocol

For a short field exercise about cross sections, drainage and accessible streets, specify a time window, observation location, data dictionary and recording method. Protect individual privacy by avoiding unnecessary vehicle or personal identifiers. Use at least three quality checks: plausibility ranges for the raw input, missing-value counts, and a reconciliation of any aggregated totals. Present separate notes on systematic bias, day-to-day variability and uncertainty in the equation itself.

Decision boundary

The relationship in this lesson is appropriate for concept exploration and transparent back-of-the-envelope screening. A code-compliant geometric decision would additionally need jurisdiction-specific requirements, evidence of relevant site conditions and a documented professional review. Name the exact assumptions that would need replacement and identify stakeholders who might be excluded if the decision were made entirely from a vehicle-centred metric.

Unit synthesis exercise

Connect design speed, stopping sight distance and grade with cross sections, drainage and accessible streets: imagine a corridor proposal that improves the first measure but worsens the last. Write a one-page recommendation showing the trade-off, a minimum dataset for testing both effects, and the circumstances that would reverse your recommendation.

Reading the laboratory outputs

Every chart is calculated for hypothetical parameters. Change one parameter at a time to understand the model. Check a point by hand. Export a CSV, note the axes, then identify at least one physically relevant effect omitted by the model. No laboratory in this book is approved for design or real-time operational control.

Extended design exercise: a curved road beside a school and drainage canal

A rural-to-urban transition approaches a school along a left-turning horizontal curve. The road carries buses and goods vehicles and runs beside a drainage canal. Students approach on foot from both sides. An initial geometric screening at 70 km/h with superelevation 0.05 and illustrative side-friction factor 0.10 gives a radius from R=V²/[127(e+f)] of about 257 metres. This figure is a calculated balance, not a permitted or sufficiently safe curve radius. The actual design depends on the adopted code, available sight distance, vehicles and road environment. The need to support pedestrians near a school is itself a reason to question whether the desired operating speed is appropriate.

Before proposing a larger radius, the engineer plots the existing horizontal curve using surveyed coordinates and reconstructs road edges, roadside canal boundaries and property entrances. The curve may be broad enough in plan but still have a hidden crossing behind a wall. A stopping sight-distance check must consider the driver's line of sight, grade and obstruction height. For a simplified level-road scenario at 70 km/h, reaction time 2.5 seconds and friction 0.35, reaction distance is about 48.6 metres and braking distance about 55.1 metres, making the approximate stopping distance 103.7 metres. This does not incorporate sight-line obstructions, headlight effects or local design-standard values.

The road profile may also include a crest. Draw a parabolic vertical curve whose initial and final tangent grades are known; do not confuse the ability to calculate its elevation with verification that a driver can see a child crossing ahead. If the roadway crest hides part of a crossing, a geometrically smooth profile is still hazardous. Similarly, a two-percent crossfall across a three-metre lane yields a vertical difference of 60 millimetres, but does not prove runoff will enter a working inlet. The canal-side drainage pattern needs an independent hydraulic and erosion review. Raising a shoulder could unintentionally divert storm runoff toward nearby houses.

Propose alternatives: reducing the target speed and improving crossings; relocating an obstruction; revising the alignment; or providing a physically separated pedestrian route. The first three have different costs and environmental impacts. A lower-speed corridor can sometimes improve road-user safety at less expense than a major realignment, but that outcome must be measured, not assumed. Record peak pedestrian flows, bus stopping locations, wet-weather visibility, the location of vulnerable access points and observed operating speeds across several periods. Public consultation is necessary because the shortest walking route and the driver's preferred travel path may differ.

Final submission must include annotated plan and profile views, a table of assumptions with units, calculation of the screening radius and SSD, a list of omitted visibility checks and two alternative corridor designs. The final recommendation must be framed as a proposal for proper standards-based assessment, not a construction drawing. Explain how the conclusion changes if the desired design speed is reduced, the pavement is wet or a bus stop is added near the crossing.