Advanced Civil Engineering Handbook › Chapter 14

Chapter 14 of 14 · 36 formulas · 75 one-liners

Construction Planning, Management & Equipment — formulas & one-liners

Bar charts, CPM and PERT networks, crashing and resource scheduling, earned value, construction equipment and productivity, quality and safety, contracts and tendering, and engineering economics, with references to IS 456, IS 3786 and NBC 2016.

14.1 Bar charts & network rules

Fulkerson's rule numbers events so that every activity goes from a lower-numbered tail event to a higher-numbered head event.
The bar chart (Gantt chart) was developed by Henry Gantt; it shows start, duration and progress but not interdependencies between activities.
A milestone chart is a bar chart with key events (milestones) marked on the bars, an improvement on the plain bar chart.
CPM was developed in 1957 by Du Pont with Remington Rand; PERT was developed in 1958 for the US Navy Polaris missile programme.
An activity consumes time and resources, whereas an event is an instant of time that consumes neither.
A dummy activity takes zero time and zero resources and only shows a logical dependency; it is drawn as a dashed arrow.
Two activities must not have the same tail and head events; a dummy is used to separate them.
Loops and dangling events (other than the single start and end events) are not allowed in a network.
Activity-on-node (precedence) diagrams do not need dummy activities.
The Line of Balance (LOB) technique is used to plan repetitive work such as mass housing or highway sections.

14.2 CPM – event times & floats

Earliest finish & latest start

E F = E S + t, L S = L F - t

ES, EF = earliest start and finish; LS, LF = latest start and finish; t = activity duration

Forward pass (earliest event time)

T_{E}^{j} = max (T_{E}^{i} + t_{ij})

max over all activities entering event j

Backward pass (latest event time)

T_{L}^{i} = min (T_{L}^{j} - t_{ij})

min over all activities leaving event i

Event slack

S = T_{L} - T_{E}

S = slack of an event

Total float

T F = T_{L}^{j} - T_{E}^{i} - t_{ij} = L S - E S

i = tail event; j = head event

Free float

F F = T_{E}^{j} - T_{E}^{i} - t_{ij} = T F - S_{j}

S_j = slack of head event

Independent float

I F = T_{E}^{j} - T_{L}^{i} - t_{ij} = F F - S_{i}

S_i = slack of tail event; a negative IF is taken as zero

Interfering float

Int.F = T F - F F = S_{j}

equals the slack of the head event

The critical path is the longest path through the network, and its length is the minimum project duration.
Critical activities have zero total float when the scheduled completion time equals the earliest completion time.
A network can have more than one critical path.
Independent float ≤ free float ≤ total float for every activity.
Using free float does not delay the earliest start of any succeeding activity.
Using independent float affects neither preceding nor succeeding activities.
Negative float arises when the scheduled (target) completion time is less than the computed earliest completion time.
CPM is activity-oriented and deterministic, with a single time estimate for each activity.

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14.3 PERT

Expected time

t_{e} = \frac{t _{o} + 4 t _{m} + t _{p}}{6}

t_o = optimistic; t_m = most likely; t_p = pessimistic time

Standard deviation of an activity

σ = \frac{t _{p} - t _{o}}{6}

variance = σ²

Standard deviation of the project

σ_{c p} = \sum σ_{i}^{2}

sum of variances of activities on the critical path

Probability factor

Z = \frac{T _{s} - T _{E}}{σ _{c p}}

T_s = scheduled completion time; T_E = expected project duration

Look up the probability from the standard normal table

PERT is event-oriented and probabilistic, using three time estimates for each activity.
Activity times in PERT are assumed to follow a beta distribution, while the project duration follows a normal distribution.
The probability of completing a project in exactly its expected time (Z = 0) is 50%.
Z = 1 gives about 84.1% probability, Z = 2 about 97.7% and Z = 3 about 99.9%.
In PERT the critical path is the path with zero (minimum) slack, and the longest-duration path is critical.

14.4 Crashing & resource scheduling

Cost slope

Cost slope = \frac{C _{c} - C _{n}}{T _{n} - T _{c}}

C_c = crash cost; C_n = normal cost; T_n = normal time; T_c = crash time

Unit: cost per unit time saved

Total project cost

C_{T} = C_{d i r ec t} + C_{in d i r ec t}

direct cost rises and indirect cost falls as duration is reduced

Crashing is done only on critical activities, starting with the one having the lowest cost slope.
The optimum project duration is the one at which total (direct + indirect) cost is minimum.
Indirect costs such as overheads and supervision vary roughly in proportion to project duration.
Further crashing is pointless once every activity on any one critical path has reached its crash time; new critical paths may appear while crashing.
Resource smoothing keeps the project duration fixed and uses floats to reduce peaks in resource demand.
Resource levelling works with limited resources and may extend the project duration.
The S-curve of cumulative cost or progress against time is used for monitoring and cash-flow planning.

14.5 Project control, quality & safety

Cost variance

C V = E V - A C

EV = earned value (budgeted cost of work performed); AC = actual cost

Negative CV = over budget

Schedule variance

S V = E V - P V

PV = planned value (budgeted cost of work scheduled)

Negative SV = behind schedule

Cost performance index

C P I = \frac{E V}{A C}

CPI < 1 means cost overrun

Schedule performance index

S P I = \frac{E V}{P V}

SPI < 1 means behind schedule

Accident frequency rate

F R = \frac{lost-time injuries \times 1 0 ^{6}}{man-hours worked}

lost-time injuries = injuries causing loss of at least one full shift; man-hours = total hours worked by all employees

IS 3786: per million man-hours

Quality control is inspection and testing of the product, whereas quality assurance is the planned system that prevents defects.
The Deming (PDCA) cycle is Plan–Do–Check–Act.
Control charts commonly use control limits at the mean ± 3σ.
ISO 9001 covers quality management, ISO 14001 environmental management and ISO 45001 occupational health and safety.
IS 456 sampling of concrete: 1 sample for 1–5 m³, 2 for 6–15 m³, 3 for 16–30 m³, 4 for 31–50 m³, and one more for each additional 50 m³ or part.
Each concrete sample for strength testing consists of 3 cubes, tested at 28 days (IS 456).
A ladder should be inclined so that its base is about one-quarter of its working height away from the wall (about 75° to the horizontal).
Safety in construction is covered by NBC 2016 Part 7; building workers were covered by the BOCW Act, 1996, now subsumed in the Occupational Safety, Health and Working Conditions Code, 2020.

14.6 Construction equipment

Production of equipment

Q = q \times \frac{60}{T} \times E

q = output per cycle (m³); T = cycle time (min); E = efficiency factor; Q in m³/h

Cycle time of dozer / scraper

T = t_{f} + \frac{D}{V _{f}} + \frac{D}{V _{r}}

t_f = fixed time (load, dump, turn); D = haul distance; V_f, V_r = forward and return speeds

Excavator output

Q = q \times K \times \frac{3600}{T _{s}} \times E

q = bucket capacity (m³); K = bucket fill factor; T_s = cycle time (s)

Swell

Swell (%) = \frac{V _{l oose} - V _{bank}}{V _{bank}} \times 100

V = volume in loose and bank (in-situ) state

Load factor

L F = \frac{V _{bank}}{V _{l oose}} = \frac{1}{1 + swell}

swell as a decimal

Rimpull

R P (kN) = \frac{3.6 P η}{v}

P = engine power (kW); η = transmission efficiency; v = speed (km/h)

Usable pull

P_{u} = μ_{t} \times W_{d}

μ_t = coefficient of traction; W_d = weight on driving wheels or tracks

Balanced number of haul units

N = \frac{T _{ha u l}}{t _{l o a d}}

T_haul = cycle time of a truck or scraper; t_load = time to load one unit

For pusher dozers, scrapers per pusher = scraper cycle time ÷ pusher cycle time

Roller output

Q = \frac{1000 W S L E}{P}

W = effective roll width (m); S = speed (km/h); L = compacted lift thickness (m); E = efficiency; P = number of passes; Q in m³/h

A bulldozer is economical for pushing earth over short distances, up to about 100 m.
A scraper loads, hauls, dumps and spreads earth by itself, and is often push-loaded by a dozer.
A power (face) shovel digs above its track level and works best in hard, dry material.
A backhoe (hoe) digs below its track level and is ideal for trenches and basements.
A dragline digs below its own level in soft or wet ground and has a long reach, so it need not enter the pit.
A clamshell digs vertically in confined spaces such as well sinking, cofferdams and loading loose material into hoppers.
Output of a power shovel falls when the swing angle exceeds 90° or the depth of cut differs from the optimum depth.
Grade resistance is about 10 kg per tonne of gross weight for each 1% of adverse grade.
Crawler tractors are rated by drawbar pull and wheeled tractors by rimpull.
Sheepsfoot rollers suit cohesive soils, vibratory rollers suit granular soils and smooth-wheel rollers are used for finishing base courses and bituminous layers.
Tower cranes are used for high-rise building work; crawler cranes are mobile on site and suit heavy lifts on soft ground.
Concrete mixer size such as 10/7 means 10 cubic feet of dry materials gives 7 cubic feet of mixed concrete; 'T' denotes tilting and 'NT' non-tilting drum.
Transit mixers carry ready-mixed concrete (IS 4926) from the batching plant to site while agitating it.
Internal (needle) vibrators are used for general RCC, surface (screed) vibrators for slabs and pavements, and form vibrators for thin, heavily reinforced sections.
Owning cost of equipment includes depreciation, interest, insurance and taxes; operating cost includes fuel, lubricants, repairs and operator wages.

14.7 Contracts & tendering

In an item rate contract the contractor quotes a rate for each item of the bill of quantities and is paid on measured quantities.
In a percentage rate contract the contractor quotes a percentage above or below the departmental schedule of rates.
A lump sum contract suits work whose quantities can be fixed accurately from complete drawings and specifications.
A cost-plus-percentage contract gives the contractor no incentive to reduce cost; cost-plus-fixed-fee overcomes this.
In a turnkey or EPC contract one agency is responsible for design, procurement and construction.
In a BOT project the concessionaire finances, builds and operates the facility, recovering costs from users (e.g., tolls), then transfers it to the government.
Under the Hybrid Annuity Model (HAM) the government pays 40% of the project cost during construction; the developer arranges the remaining 60%, which the government repays as annuities with interest over the operation period.
Earnest money deposit (EMD) is generally about 2% of the estimated cost put to tender and is forfeited if the bidder withdraws within the validity period.
Security deposit, recovered from running bills, guarantees performance and is refunded after the defects liability period.
Liquidated damages are a pre-estimated compensation for delay fixed in the contract.
In the two-bid system the technical bid is evaluated first and financial bids of only the qualified bidders are opened.
Open tenders are advertised publicly; limited tenders go to selected approved contractors; a single tender is used for proprietary or emergency work.
The Measurement Book (MB) is the basic record for all payments for work done and supplies received.
A running account bill is a part payment for measured work; the final bill is paid after completion.
A price-escalation (price variation) clause compensates the contractor for rise in prices of materials and labour.
Contract disputes in India are commonly settled by arbitration under the Arbitration and Conciliation Act, 1996.

14.8 Engineering economics

Compound amount

F = P (1 + i)^{n}

P = present sum; i = interest rate per period; n = number of periods

Present worth

P = \frac{F}{( 1 + i ) ^{n}}

F = future sum

Sinking fund

A = F \frac{i}{( 1 + i ) ^{n} - 1}

annual deposit to accumulate F in n periods

Capital recovery

A = P \frac{i ( 1 + i ) ^{n}}{( 1 + i ) ^{n} - 1}

annual payment to recover P with interest in n periods

Effective interest rate

i_{e f f} = (1 + \frac{r}{m})^{m} - 1

r = nominal annual rate; m = compounding periods per year

Straight-line depreciation

D = \frac{C - S}{n}

C = first cost; S = salvage value; n = useful life (years)

Declining balance – book value

B_{m} = C (1 - d)^{m}

d = depreciation rate; m = years elapsed

Double declining balance: d = 2/n

Sum-of-years-digits depreciation

D_{m} = \frac{n - m + 1}{n ( n + 1 ) /2} (C - S)

D_m = depreciation in year m

Straight-line depreciation is equal every year, while declining balance and sum-of-years-digits give larger depreciation in the early years.
In the declining balance method the book value never becomes exactly zero.
A project is acceptable when its net present worth is positive or its benefit–cost ratio exceeds 1.
The internal rate of return (IRR) is the interest rate at which the net present worth of a project is zero.
Capitalised cost of a perpetual uniform annual cost A is A/i.
Equipment should be replaced when its average annual total cost (owning plus operating) reaches a minimum, which defines its economic life.

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New to these topics? Start with the Basic HandbookThe fundamentals of civil engineering explained simply — free to read online, then come back to the Advanced Handbook for exam-level formulas.

Read the Basic Handbook

Values follow the current Indian Standards and IRC / Indian Railways documents named in each item. Always check the latest amendment of a code before using a value in design.