6.3 Construction: Scheduling, Earthwork & Estimating
Key Takeaways
- The critical path is the longest path through a CPM network; its activities have zero float and set the project duration.
- Total float = LS − ES = LF − EF; an activity with zero total float is critical.
- Average-end-area earthwork volume is V = L(A1 + A2)/2, with cross-section areas A1, A2 and distance L between them.
- Soil swells when excavated (bank→loose) and shrinks when compacted (bank→compacted); shrink/swell factors convert between bank, loose, and compacted volumes.
CPM Networks and the Critical Path
The Critical Path Method (CPM) models a project as a network of activities with durations and dependencies. Two passes give the schedule:
- Forward pass (left to right) computes Early Start (ES) and Early Finish (EF = ES + duration). At a merge, take the maximum EF of predecessors.
- Backward pass (right to left) computes Late Finish (LF) and Late Start (LS = LF − duration). At a burst, take the minimum LS of successors.
Float (slack) measures schedule flexibility:
- Total Float (TF) = LS − ES = LF − EF
- Free Float = ES(successor) − EF(activity): delay possible without affecting the next activity.
The critical path is the chain of activities with zero total float; it is the longest path through the network and equals the project duration. Delaying any critical activity delays the whole project.
PERT adds uncertainty: each duration uses a three-point estimate t_e = (a + 4m + b)/6 (optimistic a, most likely m, pessimistic b), with variance σ² = ((b − a)/6)².
Worked Critical-Path Example
Consider four activities:
| Activity | Duration (days) | Predecessor |
|---|---|---|
| A | 5 | — |
| B | 3 | A |
| C | 7 | A |
| D | 4 | B, C |
Forward pass: A ES=0, EF=5. B ES=5, EF=8. C ES=5, EF=12. D merges B and C → ES = max(8, 12) = 12, EF = 16. Project duration = 16 days.
Backward pass from D: LF=16, LS=12. C LF=12, LS=5. B LF=12, LS=9. A LF=5, LS=0.
Float: A TF = 0 (critical). B TF = LS − ES = 9 − 5 = 4 days (non-critical). C TF = 0 (critical). D TF = 0 (critical). The critical path is A → C → D (5 + 7 + 4 = 16). B has 4 days of slack.
Earned value tracks cost/schedule performance: SV = EV − PV (schedule variance), CV = EV − AC (cost variance); SPI = EV/PV and CPI = EV/AC. Indices < 1.0 signal behind schedule or over budget.
Earthwork and Estimating
Earthwork volumes between two cross-sections use the average-end-area method:
- V = L·(A1 + A2)/2
where A1 and A2 are the end cross-section areas (ft²) and L is the distance between them (ft); divide by 27 to convert ft³ to cubic yards. For higher accuracy with a mid-section area A_m, the prismoidal formula V = L(A1 + 4A_m + A2)/6 is used.
Worked earthwork example. Two stations 100 ft apart have cut areas A1 = 120 ft² and A2 = 180 ft². V = 100·(120 + 180)/2 = 100·150 = 15,000 ft³ = 15,000/27 = 555.6 yd³.
Shrink and swell. Soil changes volume by state:
| State | Description | Typical factor |
|---|---|---|
| Bank (BCY) | in-place, undisturbed | reference |
| Loose (LCY) | excavated, fluffed | swell ~15–25% |
| Compacted (CCY) | placed and compacted | shrink ~10–20% |
A swell factor converts bank → loose (volume increases); a shrink factor converts bank → compacted (volume decreases). A mass-haul diagram plots cumulative cut (+) minus fill (−) along the alignment to balance earthwork and find economical haul. Quantity takeoff measures material counts from plans; the estimate then applies unit costs, crew productivity, and markups (overhead, profit) to price the work.
Gantt Charts, Crashing, and Estimating Detail
A Gantt (bar) chart plots each activity as a horizontal bar against a calendar timeline, showing start/finish and overlap at a glance. It communicates the schedule well to non-technical stakeholders but, unlike a CPM network, does not explicitly show logical dependencies or float — which is why CPM drives the analysis and the Gantt chart presents it.
Project crashing shortens duration by adding resources to critical activities. The cost slope of an activity = (crash cost − normal cost)/(normal duration − crash duration), in dollars per day saved. To compress economically, crash the critical activity with the lowest cost slope first, re-checking the critical path after each step because a non-critical path can become critical.
Worked crashing idea. If activity C (on the critical path) can be shortened 2 days at a cost slope of $400/day, while parallel non-critical B has slope $300/day, you still crash C — only critical-path activities shorten the project. Crashing B wastes money.
Estimating types progress from rough to precise: order-of-magnitude (conceptual, ±30–50%), budget/preliminary, and detailed (definitive) estimates built from quantity takeoff × unit prices. A unit price bundles material + labor + equipment, then the bid adds indirect costs, overhead, contingency, and profit. Productivity (output per crew-hour) converts quantities into labor-hours, and labor-hours times wage rates give labor cost — the most variable and risk-prone line in a civil estimate.
A worked estimating step: if a crew places concrete at 8 cubic yards per hour and 240 cubic yards are required, the labor demand is 240/8 = 30 crew-hours. At a crew rate of $350 per hour, the direct labor cost is 30 multiplied by $350 = $10,500, to which the estimator adds material, equipment, overhead, and profit. Tracking the actual work against this plan with earned value (comparing EV, PV, and AC) then tells the team early whether the job is ahead of or behind both budget and schedule.
To reduce a project's duration by crashing, which activity should be shortened first?
Two adjacent cross-sections 80 ft apart have cut areas of 90 ft² and 150 ft². Using the average-end-area method, what is the volume of cut?
In a CPM network, an activity has ES = 6, EF = 10, LS = 9, LF = 13. What is its total float, and is it critical?
When natural soil is excavated and loosened before hauling, what happens to its volume relative to the in-place (bank) state?