Advanced Civil Engineering Handbook › Chapter 1

Chapter 1 of 14 · 38 formulas · 82 one-liners

Building Materials & Concrete Technology — formulas & one-liners

Cement, aggregates, bricks, stone, timber, lime, concrete properties, mix design, admixtures, durability, reinforcing steel, paints and glass, as per IS 269:2015, IS 383:2016, IS 456:2000, IS 10262:2019 and IS 1786:2008.

1.1 Cement: composition, types and hydration

Tricalcium silicate content (Bogue)

C_{3} S = 4.071 CaO - 7.600 SiO_{2} - 6.718 Al_{2} O_{3} - 1.430 Fe_{2} O_{3} - 2.852 SO_{3}

Oxides as % by mass of cement

Bogue equations apply when Al₂O₃/Fe₂O₃ ≥ 0.64

Dicalcium silicate content (Bogue)

C_{2} S = 2.867 SiO_{2} - 0.7544 C_{3} S

Oxides and C₃S as % by mass

Tricalcium aluminate content (Bogue)

C_{3} A = 2.650 Al_{2} O_{3} - 1.692 Fe_{2} O_{3}

Oxides as % by mass

Also C₄AF = 3.043 Fe₂O₃

Lime saturation factor (IS 269)

LSF = \frac{CaO - 0.7 SO _{3}}{2.8 SiO _{2} + 1.2 Al _{2} O _{3} + 0.65 Fe _{2} O _{3}}

Oxides as % by mass

Upper limit 1.02; excess lime makes cement unsound

Gel–space ratio (Powers)

x = \frac{0.657 α}{0.319 α + w / c}

α = degree of hydration (0–1); w/c = water–cement ratio by mass

Strength from gel–space ratio (Powers)

f_{c} = 240 x^{3}

f_c = compressive strength (MPa); x = gel–space ratio

Some texts use 234x³

C₃S (alite) hydrates fast and gives most of the strength in the first 7 days.
C₂S (belite) hydrates slowly and contributes to strength gain after 7 days and in later ages.
C₃A reacts almost instantly with water, liberates the highest heat of hydration and causes flash set if not controlled.
Gypsum (CaSO₄·2H₂O) is interground with clinker to retard the flash setting of C₃A.
About 23% water by mass of cement is chemically bound and about 15% fills gel pores, so complete hydration needs w/c ≈ 0.38.
IS 269:2015 covers OPC of grades 33, 43 and 53 (earlier separate codes IS 8112 for 43 and IS 12269 for 53 grade).
Minimum compressive strength of OPC 33/43/53 is 16/23/27 MPa at 3 days, 22/33/37 MPa at 7 days and 33/43/53 MPa at 28 days.
Portland pozzolana cement (IS 1489 Part 1) contains 15–35% fly ash by mass.
Portland slag cement (IS 455) contains 25–70% granulated blast furnace slag.
Sulphate resisting cement (IS 12330) has C₃A limited to 5%.
Low heat cement (IS 12600) limits heat of hydration to 272 kJ/kg at 7 days and 314 kJ/kg at 28 days and has initial setting time of at least 60 minutes.
Specific gravity of OPC is about 3.15 and its bulk density about 1440 kg/m³.

1.2 Tests on cement (IS 4031)

Water for initial/final setting time test

W = 0.85 P

P = water for standard consistency (% by mass of cement)

Water for soundness (Le Chatelier) test

W = 0.78 P

P = standard consistency (%)

Water for mortar cube compressive strength test

W = (\frac{P}{4} + 3) % of (cement + sand)

P = standard consistency (%)

Mix 1 : 3 cement : standard sand (IS 650)

Standard consistency is the water content at which the 10 mm Vicat plunger penetrates to 5–7 mm from the bottom of the mould; it is usually 26–33% for OPC.
Initial setting time is reached when the 1 mm square Vicat needle fails to pierce the paste beyond 5 ± 0.5 mm from the bottom of the mould.
Final setting time is reached when the needle with the 5 mm annular collar makes an impression but the collar does not.
OPC must have initial setting time of at least 30 minutes and final setting time of at most 600 minutes.
Minimum Blaine fineness is 225 m²/kg for OPC and 300 m²/kg for PPC.
Expansion in the Le Chatelier test must not exceed 10 mm, and in the autoclave test it must not exceed 0.8%.
Cement compressive strength is found on 70.6 mm mortar cubes (face area 5000 mm²) each made with 200 g cement and 600 g standard sand.
Specific gravity of cement is found with a Le Chatelier flask using kerosene, because cement reacts with water.

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1.3 Aggregates (IS 383:2016, IS 2386)

Fineness modulus

FM = \frac{\sum cumulative % retained on standard sieves}{100}

Standard sieves: 80, 40, 20, 10, 4.75, 2.36, 1.18 mm, 600, 300, 150 µm

Fine sand 2.2–2.6, medium 2.6–2.9, coarse 2.9–3.2

Bulking of sand

Bulking (%) = \frac{V _{m o i s t} - V _{d r y}}{V _{d r y}} \times 100

V_moist = volume of damp loose sand; V_dry = volume after inundation

Aggregate impact value

AIV = \frac{W _{2}}{W _{1}} \times 100

W₁ = mass of sample; W₂ = mass passing 2.36 mm sieve after 15 blows

Sample 12.5–10 mm; hammer 13.5–14 kg falling 380 mm

Aggregate crushing value

ACV = \frac{B}{A} \times 100

A = mass of sample; B = mass passing 2.36 mm sieve after 400 kN load

Load applied gradually in 10 minutes

Los Angeles abrasion value

LAV = \frac{W _{1} - W _{2}}{W _{1}} \times 100

W₁ = original mass; W₂ = mass retained on 1.70 mm sieve

Steel ball charge; drum at 30–33 rpm (500 revolutions for most gradings)

Water absorption of aggregate

WA (%) = \frac{W _{ss d} - W _{o d}}{W _{o d}} \times 100

W_ssd = saturated surface-dry mass; W_od = oven-dry mass

Fine aggregate passes the 4.75 mm IS sieve; coarse aggregate is mostly retained on it.
IS 383:2016 grades fine aggregate into Zones I to IV by % passing the 600 µm sieve: 15–34, 35–59, 60–79 and 80–100 respectively.
IS 383:2016 recognises manufactured sand, slag aggregates and recycled aggregates in addition to natural aggregates.
Aggregate crushing value and impact value must not exceed 30% for wearing surfaces and 45% for other concrete.
Los Angeles abrasion value must not exceed 30% for wearing surfaces and 50% for other concrete.
IS 383:2016 limits the combined flakiness and elongation index of coarse aggregate to 40%.
A flaky particle has least dimension less than 0.6 times its mean sieve size; an elongated particle has greatest dimension more than 1.8 times its mean sieve size.
Sand bulks most (about 20–40%) at a moisture content of about 4–6%; fine sand bulks more than coarse sand.
Nominal maximum size of coarse aggregate should not exceed one-fourth of the minimum thickness of the member (IS 456 cl. 5.3.3).
Mix design calculations assume aggregates in saturated surface-dry (SSD) condition.

1.4 Bricks, stone, timber and lime

Calcination of limestone

CaCO_{3} heat CaO + CO_{2}

CaO = quicklime

Burning at about 900 °C

Slaking of quicklime

CaO + H_{2} O \to Ca(OH)_{2} + heat

Ca(OH)₂ = slaked (hydrated) lime

Hardening of fat lime by carbonation

Ca(OH)_{2} + CO_{2} \to CaCO_{3} + H_{2} O

CO₂ from the atmosphere

Moisture content of timber (oven-dry basis)

m = \frac{W _{1} - W _{2}}{W _{2}} \times 100

W₁ = initial mass; W₂ = oven-dry mass

The standard modular brick (IS 1077) is 190 × 90 × 90 mm; with mortar its nominal size is 200 × 100 × 100 mm.
About 500 modular bricks are needed per m³ of brickwork.
IS 1077 classifies common burnt clay bricks by compressive strength from class 3.5 to class 35 (MPa).
Water absorption of burnt clay bricks after 24 h immersion must not exceed 20% by mass up to class 12.5 and 15% for higher classes.
Good brick earth contains about 50–60% silica, 20–30% alumina, a little lime, and iron oxide which gives the red colour.
Granite, basalt and trap are igneous rocks; sandstone and limestone are sedimentary; marble, slate, quartzite and gneiss are metamorphic.
Marble forms from limestone, quartzite from sandstone, slate from shale and gneiss from granite.
Teak, sal and shisham are hardwoods; deodar, chir and pine are softwoods from conifers.
Shakes (heart, star, cup and ring shakes) are cracks that separate wood fibres; dry rot is fungal decay in poorly ventilated places.
Plywood has an odd number of veneers with grain of adjacent layers at right angles.
Fat lime is high-calcium lime that slakes vigorously and sets only by carbonation in air; hydraulic lime contains clay and can set under water.
Lime is called feebly, moderately or eminently hydraulic as its clay content rises from about 5–10% to 11–20% to 21–30%.

1.5 Fresh and hardened concrete

Compacting factor

CF = \frac{W _{p a r t ia l}}{W _{f u l l}}

W_partial = mass of partially compacted concrete; W_full = mass of fully compacted concrete in the same cylinder

Abrams' water–cement ratio law

f_{c} = \frac{A}{B ^{w / c}}

f_c = strength; A, B = empirical constants; w/c = water–cement ratio

Valid for fully compacted concrete

Short-term modulus of elasticity (IS 456 cl. 6.2.3.1)

E_{c} = 5000 f_{c k}

E_c, f_ck in MPa

Flexural strength (IS 456 cl. 6.2.2)

f_{cr} = 0.7 f_{c k}

f_cr, f_ck in MPa

Effective modulus allowing for creep (IS 456 Annex C)

E_{ce} = \frac{E _{c}}{1 + θ}

θ = creep coefficient

Split cylinder tensile strength (IS 5816)

f_{c t} = \frac{2 P}{π L D}

P = failure load; L = length; D = diameter of cylinder

Modulus of rupture, fracture in middle third (IS 516)

f_{b} = \frac{P l}{b d ^{2}}

P = total load; l = span; b, d = width and depth

If fracture is just outside the middle third (a = 170–200 mm for 150 mm beam): f_b = 3Pa/(bd²)

Maturity of concrete

M = \sum (T + 11) Δ t

T = curing temperature (°C); Δt = time interval (h); M in °C·h

Datum temperature −11 °C

Dry volume of ingredients for nominal mix

V_{d r y} \approx 1.54 V_{w e t}

V_wet = volume of compacted concrete

Factor 1.52–1.57 for concrete; about 1.27–1.33 for mortar

Workability per IS 456 cl. 7.1: slump 25–75 mm for mass and lightly reinforced work, 50–100 mm for heavily reinforced sections, 75–100 mm for pumped and slipform concrete, and 100–150 mm for trench fill and in-situ piling.
IS 456 groups concrete as ordinary (M10–M20), standard (M25–M55) and high strength (M60–M80) (Table 2).
Characteristic strength is the strength below which not more than 5% of test results are expected to fall.
Standard compression test specimen is a 150 mm cube; cylinder (150 × 300 mm) strength is about 0.8 times cube strength.
Unit weight is 24 kN/m³ for plain concrete and 25 kN/m³ for reinforced concrete (IS 875 Part 1).
Creep coefficient per IS 456 is 2.2 for loading at 7 days, 1.6 at 28 days and 1.1 at 1 year.
Total shrinkage strain of concrete may be taken as 0.0003 in the absence of test data (IS 456 cl. 6.2.4.1).
Common nominal mixes (cement : sand : coarse aggregate by volume) are M10 = 1:3:6, M15 = 1:2:4, M20 = 1:1.5:3 and M25 = 1:1:2.
IS 456 sampling: 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; each sample gives 3 specimens.
For M20 and above, the mean of 4 non-overlapping consecutive results must be at least f_ck + 0.825s or f_ck + 4 MPa (whichever is more), and each result at least f_ck − 4 MPa (IS 456 Table 11).
Concrete with OPC must be moist cured for at least 7 days, and at least 10 days when mineral admixtures or blended cements are used (IS 456 cl. 13.5.1).
Formwork may be struck from vertical faces after 16–24 h, from slab soffits after 3 days and beam soffits after 7 days, with props left under (IS 456 cl. 11.3.1).

1.6 Concrete mix design (IS 10262:2019)

Target mean strength, statistical

f_{c k}^{'} = f_{c k} + 1.65 s

f_ck = characteristic strength; s = standard deviation (MPa)

Take the higher of this and the next formula

Target mean strength, minimum

f_{c k}^{'} = f_{c k} + X

X = 5.0 (M10–M15), 5.5 (M20–M25), 6.5 (M30–M60), 8.0 (M65 and above) MPa

Water content for slump other than 50 mm

W = W_{50} [1 + 0.03 \frac{S - 50}{25}]

W₅₀ = table water for 50 mm slump (kg/m³); S = required slump (mm)

3% extra water per 25 mm extra slump

Volume of all-in aggregate (absolute volume method)

V_{a} = (1 - V_{ai r}) - (V_{c} + V_{w} + V_{a d m})

Each volume = mass / (specific gravity × 1000), in m³ per m³ of concrete

Mass of coarse aggregate

M_{C A} = V_{a} \times p_{C A} \times G_{C A} \times 1000

p_CA = volume fraction of coarse aggregate; G_CA = specific gravity (SSD); M_CA in kg/m³

Fine aggregate is found the same way with (1 − p_CA)

IS 10262:2019 assumed standard deviation is 3.5 MPa for M10–M15, 4.0 MPa for M20–M25, 5.0 MPa for M30–M60 and 6.0 MPa for M65 and above.
Maximum water content for 50 mm slump with angular aggregate is 208, 186 and 165 kg/m³ for 10, 20 and 40 mm aggregate respectively.
Table water content is reduced by about 10 kg for sub-angular aggregate, 20 kg for gravel with some crushed particles and 25 kg for rounded gravel.
Volume of coarse aggregate per unit volume of total aggregate is 0.62 for 20 mm aggregate with Zone II sand at w/c 0.50.
The coarse aggregate fraction is increased by 0.01 for every 0.05 decrease in w/c (and decreased likewise); it may be reduced by up to 10% for pumpable concrete.
Entrapped air assumed in IS 10262:2019 is 1.5% for 10 mm, 1.0% for 20 mm and 0.8% for 40 mm aggregate.
IS 456 Table 5 (RCC): mild exposure 300 kg/m³, w/c 0.55, M20; moderate 300, 0.50, M25; severe 320, 0.45, M30; very severe 340, 0.45, M35; extreme 360, 0.40, M40.
Cement content excluding fly ash and GGBS should not normally exceed 450 kg/m³ (IS 456 cl. 8.2.4.2).

1.7 Admixtures and durability

Equivalent alkali content of cement

Na_{2} O_{e q} = Na_{2} O + 0.658 K_{2} O

Oxides as % by mass of cement

Low-alkali cement: Na₂O_eq ≤ 0.6%

Depth of carbonation

x = K t

x = depth (mm); K = carbonation coefficient; t = time (years)

Ultrasonic pulse velocity

V = \frac{L}{T}

L = path length; T = transit time

Calcium chloride is an accelerator but is not allowed in reinforced or prestressed concrete because it promotes corrosion.
Retarders such as gypsum, sugar and lignosulphonates delay setting and are useful in hot weather and ready-mixed concrete.
Plasticizers (lignosulphonates) reduce water by about 5–15%, while superplasticizers (SNF, SMF, PCE) can reduce it by 20–30% or more.
Air-entraining agents such as vinsol resin create 3–6% tiny air bubbles that improve freeze–thaw resistance and workability.
Every 1% of entrained air lowers compressive strength by roughly 5%.
Acid-soluble chloride limits (IS 456 Table 7) are 0.4 kg/m³ for prestressed concrete, 0.6 kg/m³ for RCC and 3.0 kg/m³ for plain concrete without embedded metal.
Total water-soluble sulphate in the mix, as SO₃, must not exceed 4% by mass of cement (IS 456 cl. 8.2.5.3).
Carbonation lowers the pH of concrete from about 12.5–13 to about 9, destroying the passive film on steel.
Alkali–aggregate reaction occurs between reactive silica in aggregate and alkalis in cement, forming an expansive gel that causes map cracking.
Sulphate attack forms expansive gypsum and ettringite; it is resisted by low C₃A cement, blended cements and low w/c.

1.8 Reinforcing steel (IS 1786:2008), paints and glass

Mass of a steel bar per metre

m = \frac{d ^{2}}{162}

m in kg/m; d = bar diameter (mm)

From density 7850 kg/m³

Percentage elongation in tensile test

δ = \frac{L _{f} - L _{0}}{L _{0}} \times 100, L_{0} = 5.65 A_{0}

L₀ = gauge length; L_f = final length; A₀ = original cross-sectional area

IS 1786:2008 (with amendments) covers HSD bar grades including Fe 415, Fe 415D, Fe 415S, Fe 500, Fe 500D, Fe 500S, Fe 550, Fe 550D and Fe 600.
Minimum elongation is 14.5% for Fe 415, 18% for Fe 415D, 12% for Fe 500, 16% for Fe 500D, 10% for Fe 550, 14.5% for Fe 550D and 10% for Fe 600.
Minimum tensile strength is 485 MPa for Fe 415 and 545 MPa for Fe 500, and not less than 1.10 and 1.08 times the actual yield stress respectively.
High strength deformed bars have no well-defined yield point, so 0.2% proof stress is taken as the yield stress.
Mild steel bars (IS 432 Grade I) have yield stress of 250 MPa for bars up to 20 mm and 240 MPa for bars over 20 mm.
Modulus of elasticity of reinforcing steel is taken as 200 GPa (IS 456).
TMT bars are quenched and self-tempered, giving a hard tempered-martensite rim and a ductile ferrite–pearlite core.
Oil paint consists of a base (white lead, zinc oxide, titanium dioxide), vehicle (linseed oil), pigment, drier, thinner (turpentine) and extender.
Toughened glass is heated and rapidly cooled, is about 4–5 times stronger than ordinary glass and breaks into small blunt pieces.
Laminated glass has a plastic (PVB) interlayer that holds broken pieces together; wired glass is used for fire resistance.

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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.