Reinforced Cement Concrete & Prestressed Concrete – Complete Study Notes

Comprehensive chapter-wise notes covering every aspect of RCC and PSC design — from basic concepts and design philosophies (WSM & LSM) through beams, slabs, columns, footings, prestressed concrete, and earthquake-resistant design. All formulae, IS code references, diagrams, and exam-focused tables are included.

GATE ESE / IES SSC JE State PSC RRB JE

Ch 1 · Introduction Ch 2 · Basic Design Concepts Ch 3 · Working Stress Method Ch 4 · Limit State Method Ch 5 · Doubly Reinforced Beam Ch 6 · Shear Design Ch 7 · Bond in RCC Ch 8 · Torsion Design Ch 9 · Flanged Beams Ch 10 · Serviceability Ch 11 · Two-Way Slabs Ch 12 · Columns Ch 13 · Foundations Ch 14 · Prestressed Concrete Ch 15 · Earthquake Design App · Masonry Design

1Introduction to RCC►

1.1 Introduction

Reinforced Cement Concrete (RCC) is a composite material in which the relatively low tensile strength and ductility of concrete are compensated by the inclusion of reinforcement — steel bars, fibres, or meshes — which carry tensile forces. The two materials act together due to the excellent bond between them and a near-identical coefficient of thermal expansion (~12×10⁻⁶/°C for both).

⭐ Key premise: Concrete is strong in compression (f_ck) but weak in tension (≈10–15% of compressive strength). Steel is strong in both tension and compression. RCC combines them to carry complex loading.

1.2 Types of Concrete

Type	f_ck Range	Key Characteristic
Plain Cement Concrete (PCC)	M5–M20	No reinforcement; good in compression only
Reinforced Concrete (RCC)	M20–M50	Steel bars embedded; resists all types of forces
Prestressed Concrete (PSC)	M40+	Pre-compressive stress applied via high-tensile steel
Fibre Reinforced Concrete	M30–M60	Random discrete fibres for ductility
High Performance Concrete	M60–M100+	Low w/c, SCMs, superior durability

1.3 Importance of Design Codes

IS 456:2000 — Plain and Reinforced Concrete, Code of Practice (primary RCC design code)
IS 1343:2012 — Prestressed Concrete Code of Practice
IS 875 (Parts 1–5) — Code of Practice for Design Loads
IS 1893:2016 — Earthquake Resistant Design
IS 13920:2016 — Ductile Detailing of RC Structures
Codes ensure safety, economy, serviceability, and uniformity across the profession

1.4 & 1.5 Characteristic Strength and Grades of Concrete

Characteristic Strength f_ck: strength below which not more than 5% of test results are expected to fall
f_ck = f_mean − 1.65 σ
Target Mean Strength (Mix Design): f_target = f_ck + 1.65 σ

Grade	f_ck (MPa)	Min. Use
M15	15	PCC, lean concrete
M20	20	Min. for RCC (mild exposure)
M25	25	Min. for moderate exposure
M30	30	Min. for severe exposure
M35	35	Min. for very severe exposure
M40	40	Min. for extreme exposure; PSC
M50–M100	50–100	HPC, PSC, special structures

1.8 Behaviour under Uniaxial Compression

Fig 1.1 – Concrete stress-strain curve under uniaxial compression. IS 456 idealized design block: parabolic rise to 0.002 strain, then flat to ultimate strain ε_cu = 0.0035.

1.9 Behaviour in Tension

Concrete tensile strength ≈ 10–15% of compressive strength
Modulus of Rupture (flexural tension): f_r = 0.7√f_ck MPa (IS 456)
Direct tensile strength: f_t ≈ 0.4–0.5 √f_ck MPa
Split tensile strength: f_sp = 2P/(πdL)
Concrete cracks in tension at very small strains (~0.0001 to 0.0002)

1.10 Modulus of Elasticity & Poisson's Ratio

Short-term Static E_c = 5000 √f_ck MPa (IS 456 Cl. 6.2.3.1)
Long-term (with creep): E_c,eff = E_c / (1 + θ)
Poisson's ratio μ = 0.1 to 0.2 (IS 456 uses μ = 0.2)
Shear Modulus G = E_c / [2(1+μ)] = E_c/2.4 ≈ 0.4 E_c

1.12 Creep of Concrete

Creep Coefficient θ = Creep Strain / Elastic Strain
IS 456 values: θ = 2.2 (loading at 7 days), 1.6 (at 28 days), 1.1 (at 1 year)
Effective Modulus: E_ce = E_c / (1 + θ) [used for long-term deflection]

Factors increasing creep:

Higher w/c ratio, early loading, high temperature, low humidity, thin sections, higher stress/strength ratio, higher cement content, fine aggregate

1.13 Compressive Strength in Structural Design

IS 456 LSM: Design compressive strength f_cd = 0.67 f_ck / γ_m
γ_m (concrete) = 1.5 → f_cd = 0.67 f_ck / 1.5 = 0.446 f_ck ≈ 0.45 f_ck
γ_m (steel Fe415) = 1.15 → f_yd = 0.87 f_y = 0.87 × 415 = 361 MPa

2Basic Design Concepts►

2.1 Introduction

Structural design is the process of proportioning structural members so that they safely and economically resist applied loads throughout their intended service life. In RCC design, two materials — concrete and steel — act compositely; the designer must understand the properties of both, the nature of applied loads, and the design philosophy adopted by the governing code (IS 456:2000 for India).

Objective of design: Achieve adequate safety (no collapse), serviceability (no excessive deflection or cracking), and durability (service life ≥ design life, typically 50–100 years)
Design process: Determine loads → structural analysis (BM, SF, axial forces) → design of sections → detailing → check serviceability
IS 456:2000 is the primary Indian code; it follows the Limit State Method (LSM) based on probabilistic concepts and partial safety factors
Design must comply with IS 456, IS 875 (loads), IS 1893 (seismic), IS 13920 (ductile detailing) and relevant material codes

💡 The designer's primary responsibility is to ensure that the structure performs its intended function safely and economically — not just to pass the numbers. Sound engineering judgement, knowledge of material behaviour, construction practice and failure modes is essential.

2.2 Necessity of Designing RCC Structures

RCC structures must be designed — not guessed or over-built arbitrarily — for the following reasons:

Reason	Explanation
Safety	Undesigned structures may collapse under loads they were never checked against — axial, flexural, shear, torsion, earthquake, wind
Economy	Over-design wastes material and cost; under-design causes failure. Rational design finds the optimum
Serviceability	Even if a member doesn't collapse, excessive deflection, cracking or vibration renders it unusable. Design controls these
Durability	Design specifies minimum cover, w/c ratio, cement content — without which steel corrodes and concrete deteriorates prematurely
Legal & Code Compliance	Building bye-laws and IS codes make structural design mandatory. A structure without a design basis cannot receive a completion certificate
Predictable Behaviour	Design provides a known, predictable failure mode (ductile, not sudden). Under-reinforced beams give warning before collapse
Sustainability	Properly designed structures last their full design life (50–100 years), reducing material consumption and environmental impact over time

Why RCC specifically needs design (unlike timber or masonry by rule-of-thumb):

RCC is a composite material — the interaction of concrete and steel must be designed, not assumed
Concrete is brittle in tension; steel is ductile — a wrongly proportioned section may fail without warning
The quantity and arrangement of steel reinforcement directly controls the mode and load of failure
High variability in concrete strength (site conditions, w/c ratio, curing) makes statistical design (LSM with partial factors) essential

2.3 Hydraulic & Non-Hydraulic Cements

Non-Hydraulic Cements

Non-hydraulic cements do not harden by chemical reaction with water alone — they require air (CO₂ or O₂) to set and harden. They cannot be used underwater and lose strength if kept wet.

Type	Main Constituent	Setting Mechanism	Use
Lime (Fat/Rich Lime)	Calcium oxide CaO (from pure limestone)	Carbonation: Ca(OH)₂ + CO₂ → CaCO₃ (slow, needs air)	Plastering, lime-wash, pointing in non-structural work
Gypsum Plaster	CaSO₄·½H₂O (hemihydrate)	Rehydration: CaSO₄·½H₂O + 1½H₂O → CaSO₄·2H₂O (sets rapidly in air)	Internal plastering, partition boards, ornamental work
Oxychloride Cement (Sorel)	MgO + MgCl₂ solution	Chemical reaction forming magnesium oxychloride	Industrial flooring, grinding wheels (not for wet areas)

Hydraulic Cements

Hydraulic cements harden by chemical reaction with water (hydration) — independent of air. They set and gain strength even underwater. Portland cement is the most important hydraulic cement.

Type	IS Code	Key Feature	Structural Use
OPC 33 Grade	IS 269:2015	28-day strength ≥ 33 MPa; moderate early strength	General construction, plastering, mortar
OPC 43 Grade	IS 8112:2013	28-day strength ≥ 43 MPa; most widely used	RCC slabs, beams, columns — standard grade
OPC 53 Grade	IS 12269:2013	28-day strength ≥ 53 MPa; high early strength	Prestressed concrete, precast, high-strength mixes
PPC	IS 1489 Pt 1	Fly ash blended (15–35%); lower heat; better long-term durability	Mass concrete, foundations, marine, canals
PSC (PBFSC)	IS 455:2015	GGBS blended (25–70%); high sulphate resistance	Marine, sewage, sulphate-bearing soils
SRC	IS 12330	Very low C₃A (<5%); resists sulphate attack	Foundations in aggressive sulphate soils
Low Heat Cement	IS 12600	Low C₃S + C₃A content; max heat 271 kJ/kg at 7d	Gravity dams, massive raft foundations
Rapid Hardening (RHPC)	IS 8041	Higher C₃S; finer grinding; fast early strength	Cold weather concreting, emergency repairs, precast
High Alumina Cement (HAC)	IS 6452	Al₂O₃ > 32%; rapid strength, acid & heat resistant	Refractory linings, chemical plants (NOT structural RCC)
White Portland Cement	IS 8042	Very low Fe₂O₃ (<0.5%); white colour	Architectural, decorative, terrazzo flooring

Fig 2.1 – Classification of cements. Hydraulic cements harden with water (even underwater); non-hydraulic require air. Portland cement is the most important hydraulic cement for structural use.

⚠ Hydraulic Lime occupies an intermediate position — it contains silica/clay impurities that react with water, giving it partial hydraulic properties. Used in lime-pozzolana mixes and traditional construction. It is not suitable as a sole binder for structural RCC.

2.4 Tests on Cement

All tests on cement are governed by IS 4031 (Parts 1–15). The following tests are conducted to verify the quality of cement before use in structural concrete.

① Fineness Test (IS 4031 Pt 1 & Pt 2)

Method	Apparatus	What it Measures	IS Requirement
Sieve Test	90 µm IS sieve	% residue on 90 µm sieve	OPC: ≤ 10% residue (IS 269); ≤ 5% on 45 µm for OPC 53
Blaine's Air Permeability	Blaine apparatus	Specific surface area (m²/kg)	OPC 43/53: ≥ 225 m²/kg (typically 300–350 m²/kg in practice)

💡 Finer cement → faster hydration → higher early strength, more heat, more shrinkage. Coarser cement → slower hydration → better for mass concrete (low heat). Fineness directly controls rate of strength gain, not the ultimate strength.

② Consistency Test — Normal / Standard Consistency (IS 4031 Pt 4)

Purpose: Determine the water content (%) needed to produce a cement paste of standard consistency — required before conducting setting time and soundness tests
Apparatus: Vicat apparatus with plunger (10 mm dia, flat end)
Procedure: Prepare paste with trial water content; fill Vicat mould; lower plunger gently; record penetration depth
Standard consistency (P): When plunger penetrates to 5–7 mm from bottom (33–35 mm from top of 40 mm mould)
Typical P value: 26–33% by mass of cement for OPC
All subsequent tests (IST, FST, soundness autoclave) use paste made at this normal consistency

③ Setting Time Test (IS 4031 Pt 5)

Parameter	Apparatus / Needle	End-Point	IS Requirement (OPC)
Initial Setting Time (IST)	Vicat needle: 1 mm sq., 50 mm long	Needle penetrates to 5–7 mm from bottom (33–35 mm mark)	IST ≥ 30 minutes
Final Setting Time (FST)	Vicat needle with annular attachment (5 mm dia, 0.5 mm projection)	Needle makes impression but annular attachment does not	FST ≤ 600 minutes (10 hours)

⚠ Flash Set (without gypsum): immediate, irreversible stiffening due to rapid C₃A hydration — no workability regained. False Set (excess gypsum): premature stiffening that reverses on re-mixing — workability is regained. IST must be ≥ 30 min to allow sufficient time for transport, placing and compaction.

Fig 2.2 – Vicat apparatus for setting time test. IST: needle penetrates to 5–7 mm from bottom. FST: annular attachment (1 mm projection) makes mark but does not pierce surface.

④ Soundness Test (IS 4031 Pt 3)

Tests for expansion due to excess free lime (CaO) or magnesia (MgO) which cause delayed, disruptive expansion after concrete hardens — making the cement unsound.

Test	Detects	Apparatus	Procedure	IS Limit
Le Chatelier Test	Excess free CaO	Le Chatelier mould (split brass mould with indicator needles)	Fill mould with paste at P%; immerse in water at 27±2°C for 24h; measure needle separation; boil for 3h; re-measure	Expansion ≤ 10 mm (OPC)
Autoclave Test	Excess MgO (periclase)	Autoclave (steam pressure vessel, 2.1 MPa)	Cure cement bar 24h; measure length; autoclave at 2.1 MPa for 3h; re-measure length	Expansion ≤ 0.8%

Fig 2.3 – Le Chatelier test: indicator needles spread apart as free lime expands on boiling. Expansion (d₂ − d₁) ≤ 10 mm → sound cement. > 10 mm → unsound, reject.

⑤ Strength Test (IS 4031 Pt 6 & IS 10080)

Test	Specimen	Mix	Curing	Strength Requirement
Compressive Strength	70.6 mm cube (or 50 mm)	Cement : Standard sand = 1 : 3 (by mass); w/c = 0.40	24h in mould + immerse in water at 27°C; test at 3, 7, 28 days	OPC 43: 3d ≥ 23 MPa; 7d ≥ 33 MPa; 28d ≥ 43 MPa
Tensile (old method)	Briquette / cylinder	Neat cement paste	Submerged curing	Largely replaced by compressive strength test

⑥ Heat of Hydration Test (IS 4031 Pt 9)

Measures total heat released during hydration (J/g)
Method: Solution calorimeter or conduction calorimeter
Significance: High heat → thermal cracking in mass concrete; Low Heat Cement preferred for dams and massive structures
Limits (IS 12600 — Low Heat Cement): ≤ 271 kJ/kg at 7 days; ≤ 314 kJ/kg at 28 days

⑦ Specific Gravity of Cement (IS 4031 Pt 11)

Apparatus: Le Chatelier flask (kerosene as liquid — does not react with cement)
Specific Gravity of OPC = 3.10 – 3.15 (typically 3.12–3.15)
PSC (with GGBS): ~2.90 | PPC (with fly ash): ~2.85–2.90
HAC: ~3.20 | White cement: ~3.05

Summary Table – All Cement Tests

Test	IS Code	Property Measured	Key Limit (OPC)
Fineness (Sieve)	IS 4031 Pt 1	% residue on 90 µm sieve	≤ 10%
Fineness (Blaine)	IS 4031 Pt 2	Specific surface (m²/kg)	≥ 225 m²/kg
Normal Consistency	IS 4031 Pt 4	Water for std. paste (P%)	Penetration 5–7 mm from bottom
Initial Setting Time	IS 4031 Pt 5	Time to IST	≥ 30 minutes
Final Setting Time	IS 4031 Pt 5	Time to FST	≤ 600 minutes
Soundness (Le Chatelier)	IS 4031 Pt 3	Expansion due to free CaO	≤ 10 mm
Soundness (Autoclave)	IS 4031 Pt 3	Expansion due to MgO	≤ 0.8%
Compressive Strength	IS 4031 Pt 6	28-day mortar cube strength	OPC 43: ≥ 43 MPa; OPC 53: ≥ 53 MPa
Heat of Hydration	IS 4031 Pt 9	Heat released (kJ/kg)	LHC: ≤ 271 kJ/kg at 7d
Specific Gravity	IS 4031 Pt 11	Density ratio	OPC: 3.10–3.15

⭐ Exam Tip: Most frequently asked values — IST ≥ 30 min, FST ≤ 600 min, Le Chatelier expansion ≤ 10 mm, Autoclave ≤ 0.8%, Specific gravity 3.15, Blaine ≥ 225 m²/kg. These appear in GATE, ESE and SSC JE every year.

2.5 Durability – Chemical Attack

Exposure Class	Min. Grade	Max. w/c	Min. Cement (kg/m³)	Min. Cover (mm)
Mild	M20	0.55	300	20
Moderate	M25	0.50	300	30
Severe	M30	0.45	320	45
Very Severe	M35	0.45	340	50
Extreme	M40	0.40	360	75

2.6 Design Philosophies

Method	Full Name	Basis	IS Code Era
WSM	Working Stress Method	Elastic theory; linear stress-strain; stresses must not exceed permissible values under working loads	Older; IS 456:1978 & earlier
LSM	Limit State Method	Statistical; partial safety factors applied to loads and material strengths; considers actual failure modes	IS 456:2000 (current)
ULM	Ultimate Load Method	Plastic theory; find ultimate load; divide by FOS for design load	Intermediate; not in IS 456:2000

2.7 & 2.8 Limit State Method – Partial Safety Factors (IS 456:2000)

Partial Safety Factors for Materials (γ_m)

Material	LSM γ_m	Design Strength
Concrete	1.5	f_cd = 0.67 f_ck/1.5 = 0.45 f_ck
Steel (Fe250, Fe415, Fe500)	1.15	f_yd = f_y/1.15 = 0.87 f_y

Partial Safety Factors for Loads (γ_f) – IS 456 Table 18

Load Combination	DL	LL	Wind/Seismic
DL + LL	1.5	1.5	–
DL + LL + Wind	1.2	1.2	1.2
DL + Wind only	1.5 (or 0.9)	–	1.5
DL + LL + Seismic (IS 1893)	1.2	1.2	1.2

2.10 Properties of Reinforcement

Grade	f_y (MPa)	Type	Ductility	Use
Fe250 (Mild Steel)	250	Plain round bars	Very High	Links, stirrups; seismic zones
Fe415 (TMT/HYSD)	415	Deformed (ribbed)	High	Most common for RCC
Fe500	500	Deformed (TMT)	Moderate-High	Beams, columns, slabs
Fe550	550	Deformed (TMT)	Moderate	Heavy structures
Fe500D / Fe550D	500/550	D = enhanced ductility	High	Seismic zones (IS 13920)

E_s = 2 × 10⁵ MPa (200 GPa) for all grades of steel (IS 456 Cl. 6.2.4)
ε_y (yield strain Fe415) = f_y/E_s + 0.002 = 415/200000 + 0.002 = 0.00208 + 0.002 = 0.00383
f_yd = 0.87 × 415 = 361.05 MPa (design yield strength)

2.11 Cover Requirements (IS 456:2000 Cl. 26.4)

Member	Min. Nominal Cover (mild exposure)
Slabs	20 mm
Beams	25 mm
Columns	40 mm
Footings	50 mm (75 mm against earth)
Prestressed members	20 mm (min.)

⚠ Nominal cover ≥ bar diameter. For severe/very severe/extreme exposure, follow IS 456 Table 16A (increase cover by 10–15 mm). Fire resistance also governs minimum cover (IS 456 Table 16).

2.12 Spacing of Reinforcement (IS 456 Cl. 26.3)

Min. horizontal clear distance: max (bar dia, MSA + 5 mm, 25 mm)
Min. vertical clear distance: 15 mm, or 2/3 × max aggregate size, whichever is greater
Maximum bar spacing in slabs: 3d or 300 mm (whichever less) for main bars; 5d or 450 mm for distribution bars

2.14 Major Causes of Structural Failure

Design errors: incorrect load assumptions, wrong design philosophy, arithmetical mistakes
Construction defects: inadequate cover, poor concrete quality, insufficient curing, wrong bar placement
Material failures: substandard reinforcement (low ductility), poor aggregate quality
Overloading beyond design assumptions
Environmental attack: corrosion, sulphate attack, freeze-thaw, carbonation
Foundation settlement, seismic forces

2.15 List of Key IS Codes for RCC

IS Code	Subject
IS 456:2000	Plain & Reinforced Concrete (main design code)
IS 1343:2012	Prestressed Concrete
IS 875 Pt 1	Dead Loads
IS 875 Pt 2	Imposed (Live) Loads
IS 875 Pt 3	Wind Loads
IS 875 Pt 5	Special Loads and Load Combinations
IS 1893:2016	Earthquake Resistant Design (Criteria)
IS 13920:2016	Ductile Detailing of RC Structures
IS 2911	Design & Construction of Pile Foundations
IS 1080	Design of Shallow Foundations
SP 16:1980	Design Aids for Reinforced Concrete
SP 34:1987	Handbook on Concrete Reinforcement & Detailing

3Working Stress Method (WSM)►

3.1 Introduction

WSM (also called Elastic Method) assumes both concrete and steel remain within elastic limits under working loads. The allowable stresses are fractions of ultimate strengths, providing an implicit factor of safety.

Concrete and steel both linear-elastic; plane sections remain plane (Bernoulli's hypothesis)
Tensile stress in concrete is neglected (concrete below NA is cracked)
Perfect bond between steel and concrete (no slip)
Strain compatibility: steel strain = concrete strain at same level

3.5 Transformed Section & 3.6 Modular Ratio

Modular Ratio m = E_s / E_c
IS 456 (WSM): m = 280 / (3σ_cbc) [where σ_cbc = permissible bending stress in concrete]
For M20: m = 280/(3×7) = 13.33 | For M25: m = 280/(3×8.5) = 10.98 ≈ 11
Transformed area of tension steel = m × A_st
Transformed area of compression steel = (1.5m − 1) × A_sc [subtract concrete already accounted for]

Fig 3.1 – WSM singly reinforced beam: linear strain distribution; triangular compression stress block; tensile concrete stress neglected; steel carries all tension.

3.6 Permissible Stresses in WSM (IS 456 Table B-1 & B-2)

Grade	σ_cbc (MPa) bending	σ_cc (MPa) direct compression
M15	5.0	4.0
M20	7.0	5.0
M25	8.5	6.0
M30	10.0	8.0
M35	11.5	9.0
M40	13.0	10.0

Steel Grade	σ_st tension (MPa)	σ_sc compression (MPa)
Fe250 (mild)	140	130
Fe415 (HYSD/deformed)	230	190
Fe500 (HYSD)	275	190

3.3 Transformed Section Analysis (Key Equations)

NA depth (x): b·x·(x/2) = m·A_st·(d−x) [equate moments of areas about NA]
→ bx²/2 = m·A_st·(d−x)
Moment of Inertia (cracked): I_cr = bx³/3 + m·A_st·(d−x)²
Moment of resistance: M = σ_cbc·I_cr/x = σ_st·I_cr/(m·(d−x))
Lever arm: z = d − x/3
Critical NA depth (balanced): x_c/d = m·σ_cbc / (m·σ_cbc + σ_st)

3.9 Cracking Moment

M_cr = f_r · I_g / y_t
f_r = 0.7 √f_ck MPa (modulus of rupture, IS 456)
I_g = gross moment of inertia (uncracked section)
y_t = distance from NA to tension fibre (= D/2 for symmetric section)

4Limit State Method (LSM)►

4.1 Introduction – Limit States

Limit State	Type	Concerns
Ultimate Limit State (ULS)	Safety	Collapse, yielding, buckling, overturning, fatigue; design loads = factored loads (1.5 DL + 1.5 LL)
Serviceability Limit State (SLS)	Functional	Excessive deflection (span/250 or span/350 after construction), cracking (w_max = 0.3 mm severe), vibration
Durability Limit State	Longevity	Minimum cover, w/c ratio, cement content for given exposure

4.3 Assumptions in LSM (IS 456 Cl. 38.1)

Plane sections remain plane after bending (Bernoulli hypothesis)
Maximum compressive strain in concrete at ultimate = 0.0035
Tensile strength of concrete is ignored
Stress distribution in compression zone: IS 456 parabolic-rectangular block (peak = 0.67 f_ck/1.5 = 0.446 f_ck)
Maximum strain in tension steel ≥ f_y/(1.15 E_s) + 0.002 (ensures yielding before failure)
Perfect bond; no slip between steel and concrete

Fig 4.1 – LSM stress block at ULS. Compression = 0.36 f_ck·b·x_u at 0.42x_u from top; Tension = 0.87 f_y·A_st. Moment of resistance = C × z = T × z.

4.4 Singly Reinforced Sections – Key Formulae

Design Compressive Force: C = 0.36 f_ck · b · x_u
Design Tensile Force: T = 0.87 f_y · A_st
Equilibrium (C = T): x_u = 0.87 f_y A_st / (0.36 f_ck b)
Lever arm: z = d − 0.42 x_u
Moment of Resistance: M_u = 0.36 f_ck b x_u (d − 0.42 x_u)
OR M_u = 0.87 f_y A_st (d − 0.42 x_u)

4.5 Limiting (Balanced) Neutral Axis Depth (IS 456 Cl. G-1.1)

x_u,max/d depends on steel grade (from strain compatibility, ε_cu=0.0035):
Fe250: x_u,max/d = 0.53
Fe415: x_u,max/d = 0.48
Fe500: x_u,max/d = 0.46
Fe550: x_u,max/d = 0.44
If x_u < x_u,max → Under-reinforced (preferred — steel yields first → ductile failure)
If x_u = x_u,max → Balanced section
If x_u > x_u,max → Over-reinforced (brittle failure — NOT permitted per IS 456)

4.5 Limiting Moment of Resistance (M_u,lim)

M_u,lim = 0.36 (x_u,max/d) [1 − 0.42 (x_u,max/d)] f_ck b d²
Fe415: M_u,lim = 0.138 f_ck b d² [very frequently asked in GATE]
Fe250: M_u,lim = 0.148 f_ck b d²
Fe500: M_u,lim = 0.133 f_ck b d²
Fe550: M_u,lim = 0.128 f_ck b d²

⭐ Most frequently asked formula in GATE: M_u,lim = 0.138 f_ck b d² for Fe415. Memorise this.

4.6 Deflection Control – Span/Depth Ratio (IS 456 Cl. 23.2)

Support Condition	Basic l/d Ratio
Simply Supported	20
Continuous	26
Cantilever	7

Modified l/d = Basic l/d × Modification Factor (MF_tension) × Modification Factor (MF_comp)
MF for tension steel: from IS 456 Fig. 4 (function of f_s = 0.58 f_y × (A_st,req/A_st,prov) and pt)
For cantilevers > 10 m: deflection calculation mandatory (not l/d approach)

4.7–4.8 Minimum & Maximum Reinforcement Limits (IS 456)

Parameter	Beams	Slabs
Min. A_st	0.85 bd / f_y (IS 456 Cl. 26.5.1.1)	0.12% of total cross-section (HYSD); 0.15% (mild steel)
Max. A_st	4% of gross cross-sectional area (tension + compression combined)	As for beams

4.12 One-Way Slabs

When l_y/l_x > 2: load carried primarily in shorter direction (one-way action)
Main reinforcement in short span direction; distribution reinforcement in long span direction
Design as singly reinforced beam of unit width (b = 1000 mm)
Distribution steel: ≥ 0.12% (HYSD) or 0.15% (mild steel) of cross-sectional area
Max. bar spacing: main = 3d or 300 mm (lesser); distribution = 5d or 450 mm (lesser)

5Doubly Reinforced Beams (LSM)►

5.1 Introduction

A doubly reinforced beam has steel in both the compression zone (A_sc) and tension zone (A_st). Required when M_u > M_u,lim and section size cannot be increased, or when the beam is subject to reversal of bending.

Fig 5.1 – Doubly reinforced beam design. Total M_u = M_u,lim (singly) + M_u2 (from comp. steel). A_sc and additional A_st2 carry M_u2.

5.4 Stress in Compression Steel (f_sc)

Strain in compression steel: ε_sc = 0.0035 × (x_u,max − d') / x_u,max
d'/d ratio determines f_sc (IS 456 / SP-16 Table F):
d'/d = 0.05 → f_sc = 354.8 MPa (Fe415) | d'/d = 0.10 → 351.8 | d'/d = 0.15 → 342.4 | d'/d = 0.20 → 329.4

5.7–5.8 Design Steps for Doubly Reinforced Beam

Compute M_u,lim for the given section (using limiting NA depth for steel grade)
If M_u > M_u,lim: M_u2 = M_u − M_u,lim
Determine d'/d ratio; find f_sc from IS 456 Table or strain compatibility
A_sc = M_u2 / [f_sc × (d − d')]
A_st,lim = 0.36 f_ck b x_u,max / (0.87 f_y)
A_st2 = f_sc × A_sc / (0.87 f_y)
A_st = A_st,lim + A_st2

6Design for Shear in RCC►

6.3 Shear Stress Distribution

Fig 6.1 – Crack patterns in RC beam. Flexural (vertical) cracks at mid-span; diagonal shear-flexure cracks near supports. Critical section for shear at 'd' from face of support.

6.6 Nominal Shear Stress (IS 456 Cl. 40.1)

τ_v = V_u / (b · d) [nominal shear stress at ULS]
V_u = factored shear force at critical section
Critical section for shear: 'd' from face of support (for beams with vertical loads)

6.8 Shear Strength without Reinforcement (τ_c) – IS 456 Table 19

Design shear strength of concrete τ_c depends on grade of concrete and % tension reinforcement (p_t = 100 A_st/(bd)).

p_t (%)	M20 τ_c (MPa)	M25 τ_c (MPa)	M30 τ_c (MPa)
0.15	0.28	0.29	0.29
0.25	0.36	0.36	0.37
0.50	0.48	0.49	0.50
0.75	0.56	0.57	0.59
1.00	0.62	0.64	0.66
1.25	0.67	0.70	0.71
1.50	0.72	0.74	0.76
2.00	0.79	0.82	0.84
≥ 3.00	0.82	0.85	0.88

τ_c,max = maximum shear stress: M20=2.8, M25=3.1, M30=3.5, M35=3.7, M40=4.0 MPa (IS 456 Table 20)

6.9 Design of Shear Reinforcement

Case 1: τ_v ≤ τ_c → No shear reinforcement required (but provide minimum)
Case 2: τ_c < τ_v ≤ τ_c,max → Design shear reinforcement
Case 3: τ_v > τ_c,max → Redesign section (increase b or d)

Vus = V_u − τ_c·b·d [shear to be carried by reinforcement]
For vertical stirrups: Vus = 0.87 f_y A_sv d / S_v
∴ S_v = 0.87 f_y A_sv d / Vus

6.10 Minimum Shear Reinforcement (IS 456 Cl. 26.5.1.6)

A_sv / (b · S_v) ≥ 0.4 / (0.87 f_y)
For Fe415: min. A_sv/(b·S_v) ≥ 0.4/(0.87×415) = 0.001109

6.11 Maximum Spacing of Shear Reinforcement

Vertical stirrups: min (0.75d, 300 mm) — IS 456 Cl. 26.5.1.5
Inclined bars: min (d, 300 mm)
For high shear zone (τ_v > 0.5 τ_c,max): max spacing = 0.5d or 200 mm (whichever less)

7Bond in Reinforced Concrete►

7.4 Bond Stress & 7.8 Development Length

Bond stress τ_bd = Force in bar / (perimeter × embedment length)
Development Length L_d: length of bar to develop full design stress 0.87 f_y
L_d = (0.87 f_y · φ) / (4 τ_bd)

IS 456 Design Bond Stress τ_bd (plain bars):
M20=1.2, M25=1.4, M30=1.5, M35=1.7, M40=1.9 MPa
For deformed (HYSD) bars: multiply τ_bd by 1.6
For bars in compression: multiply τ_bd by 1.25

L_d for Fe415 HYSD bar in M20 concrete (tension):
τ_bd = 1.2 × 1.6 = 1.92 MPa
L_d = (0.87 × 415 × φ)/(4 × 1.92) = 47 φ (approx.)

General approximate: L_d ≈ 47φ (Fe415, M20) | L_d ≈ 55φ (Fe415, M15) | L_d ≈ 40φ (Fe415, M25)

7.5 Types of Bond

Type	Definition	Significance
Flexural Bond	Bond stress arising from change in bar force along length due to varying bending moment	Controls cracking; distributes cracks
Anchorage / Development Bond	Bond needed to develop full bar capacity (yield) within available embedment	Controls bar pullout failure; governs L_d

7.11 Hooks, Bends & Standard Anchorages (IS 456 Cl. 26.2)

Type	Equivalent Anchorage Length	Bend Angle
Standard 180° hook (U-hook)	16φ (equivalent anchorage)	180° + 4φ tail
Standard 90° bend	8φ (equivalent anchorage)	90° + 12φ tail (min.)
Standard 45° bend	4φ	45° bend

⚠ Hooks are mandatory for plain (mild steel) bars; not normally required for deformed bars if L_d is available straight. Deformed bars have better mechanical interlock.

7.16 Splicing of Reinforcement (IS 456 Cl. 26.2.5)

Lap splice length: 1.3 L_d in tension; 1.0 L_d in compression
Max. 50% of bars spliced at one cross-section for tension; 100% in compression
Bars > 36 mm dia: prefer mechanical couplers or butt-weld, not laps
Stagger laps at least 1.3 × lap length apart

8Design for Torsion in RCC►

8.1 Introduction – When Does Torsion Occur?

Equilibrium torsion: Must be resisted for structural stability (e.g., curved beams, L-beams with eccentric loading). Must be fully designed for.
Compatibility torsion: Arises to maintain compatibility with adjacent members (e.g., edge beams of flat slabs). IS 456 allows redistribution; minimum design required.

8.8 IS 456 LSM Torsion Design (Cl. 41)

IS 456 uses an equivalent shear method: torsion is converted to equivalent shear and equivalent moment, then designed using standard shear and flexure procedures.

Equivalent Shear: V_e = V_u + 1.6 T_u/b
Equivalent Moment: M_e1 = M_u + M_t [for face with sagging moment]
M_t = T_u(1 + D/b) / 1.7
M_e2 = M_t − M_u [for opposite face, if M_t > M_u]

Nominal shear stress: τ_ve = V_e/(b·d) ≤ τ_c,max
If τ_ve > τ_c: design transverse + longitudinal torsion reinforcement

8.9 Torsional Reinforcement Requirements (IS 456 Cl. 41.4)

Closed stirrups (rectangular links) mandatory — open stirrups cannot resist torsion
Longitudinal bars at corners of stirrups; max 300 mm spacing or b (width)
Stirrup spacing ≤ x₁ (shorter dimension of stirrup), y₁/4, or 300 mm (whichever least)

Area of transverse reinforcement per unit length: A_sv/s_v = T_u/(b₁·d₁·0.87f_y) + V_u/(2.5·d₁·0.87f_y)
b₁, d₁ = distance between corner bars (centre to centre)

9Flanged Beams (T-beams & L-beams)►

9.3 Effective Width of Flange (IS 456 Cl. 23.1.2)

For T-beams (intermediate beam): b_f = l_o/6 + b_w + 6D_f
For L-beams (edge beam): b_f = l_o/12 + b_w + 3D_f
Where: l_o = distance between points of zero moment (~0.7 × span for continuous; = span for SS)
b_w = web width | D_f = flange (slab) thickness
b_f ≤ clear distance between beams + b_w (T-beam); ≤ b_w + (clear dist.)/2 (L-beam)

Fig 9.1 – T-beam (both sides) and L-beam (one side) cross-sections. Effective flange width b_f per IS 456 Cl. 23.1.2. NA usually falls within flange (Df zone) for normal loading.

9.5 Analysis of Flanged Sections

Case 1: Neutral Axis within flange (x_u ≤ D_f) — most common

Treat as rectangular beam of width b_f and depth d
C = 0.36 f_ck · b_f · x_u
T = 0.87 f_y · A_st
M_u = 0.87 f_y A_st (d − 0.42 x_u)

Case 2: Neutral Axis in web (x_u > D_f)

Total compression = Flange compression + Web compression
C_f = 0.36 f_ck (b_f − b_w) D_f [flange portion beyond web]
C_w = 0.36 f_ck · b_w · x_u [web compression]
M_u = C_f(d − D_f/2) + C_w(d − 0.42 x_u)

10Serviceability Limit State – Deflection & Cracking►

10.3 Deflection Limits (IS 456 Cl. 23.2)

Criterion	Limit
Total final deflection (affecting appearance & comfort)	Span/250
Final deflection after construction of partitions & finishes	Span/350 or 20 mm (whichever less)
For flat roofs (ponding)	Span/480 + camber may be needed

10.3 Deflection Calculation – Components

Total deflection δ = Short-term δ_i + Long-term additional (creep + shrinkage)
Short-term δ_i = k · W L³ / (E_ce · I_eff)
k = 5/384 (SS UDL); 1/384 (SS point load at mid); 1/8 (cantilever)
E_ce = E_c (short term) or E_c/(1+θ) (long term)
Effective Moment of Inertia I_eff (IS 456 Cl. 23.2.1 / Branson's formula):
I_eff = I_cr + (I_g − I_cr) × (M_cr/M_max)³
I_g = gross MoI | I_cr = cracked MoI | M_cr = cracking moment

10.4 Cracking Limits (IS 456 Cl. 35.3)

Exposure Condition	Max. Crack Width w_max
Mild / Moderate	0.3 mm
Severe / Very Severe / Extreme	0.2 mm
Prestressed concrete members	0.1 mm (or no cracking in Type I/II)

Crack width (IS 456 Annex F):
w_cr = 3 a_cr ε_m / [1 + 2(a_cr − c_min)/(D − x)]
a_cr = distance from crack to nearest bar surface
ε_m = average strain at level of tension steel = ε₁ − b(D−x)(a'−x)/(3E_sA_st(d−x))

10.5 Other Serviceability Limits

Vibration: Natural frequency of floors ≥ 8 Hz for residential buildings (to avoid perception of vibration)
Fire resistance: Member size + cover requirements from IS 456 Table 16A
Fatigue: For bridges and crane girders; limit stress range in reinforcement

11Two-Way Slab Design►

11.2 One-Way vs Two-Way Slabs

Criterion	One-Way Slab	Two-Way Slab
Aspect ratio l_y/l_x	> 2	≤ 2
Load transfer	Primarily in shorter span (x)	In both directions
Main reinforcement	In short span only	In both spans
Examples	Verandah slabs, one-way spanning	Rooms, two-way panels

11.5 IS 456 Method for Two-Way Slabs (Cl. D-1)

Design BM per unit width:
M_x = α_x · w · l_x² (short span direction)
M_y = α_y · w · l_x² (long span direction)
α_x, α_y = BM coefficients from IS 456 Table 26 (function of l_y/l_x and support conditions)
w = total factored load per unit area = 1.5 (DL + LL) kN/m²

Fig 11.1 – Two-way slab panel. Dashed lines show diagonal load distribution. Short span carries more load. BM coefficients α_x, α_y from IS 456 Table 26.

11.6 Thickness of Two-Way Slabs

l/d (short span) for two-way slabs = 28 (two short edges continuous)
Minimum thickness: 120–150 mm for general floors; 200 mm for parking decks
IS 456 Table 10 (l/d for deflection) applies; modify for p_t

11.4 Support Conditions & Edge Discontinuity

IS 456 provides α coefficients for 9 different edge conditions (Cases 1–9)
Free (discontinuous) edges: torsional reinforcement required at corners
Corner reinforcement: ≥ 75% of area of short span mid-span steel; in both directions; extend 1/5 of short span from corner
For corner held down: 3 layers (top + bottom in 2 directions) at 45° or orthogonal

11.8 Shear in Two-Way Slabs

Shear check: τ_v = V_u/(b·d) ≤ k_s·τ_c
k_s = (0.5 + β_c) but ≤ 1.0 | β_c = short side / long side of loaded area
For flat slabs: punching shear critical at 'd/2' from column face (two-way)

12Design of Compression Members (Columns)►

12.3 Classification of Columns

Basis	Type	Definition
Slenderness	Short Column	l_eff/D ≤ 12 (both directions); failure by material
Slenderness	Slender/Long Column	l_eff/D > 12; additional moments due to deflection (IS 456 Cl. 39.7)
Loading	Axially Loaded	Concentric axial force only (rare in practice)
Loading	Eccentrically Loaded	Axial force + bending moment (uniaxial or biaxial)
Lateral Ties	Tied Column	Rectangular links confine concrete; most common
Lateral Ties	Spiral Column	Helical reinforcement; better ductility; circular section

12.5 IS 456 Recommendations for Columns (Cl. 26.5.3)

Parameter	IS 456 Requirement
Min. A_sc	0.8% of gross cross-sectional area
Max. A_sc	4% (general); 6% at lapping (local maximum)
Min. bar dia (longitudinal)	12 mm
Max. bar dia	No specific limit; practical: use up to 40 mm
Min. no. of bars: rectangular	4
Min. no. of bars: circular	6
Lateral ties (links): dia	Max (φ_main/4, 6 mm)
Lateral ties: spacing	Min (b_least, 16φ_main, 300 mm)
Cover to main bars	40 mm (mild exposure)

12.6 Short Columns – Axial Load Only (IS 456 Cl. 39.3)

P_u = 0.4 f_ck A_c + 0.67 f_y A_sc
where A_c = net concrete area = A_g − A_sc
→ P_u = 0.4 f_ck (A_g − A_sc) + 0.67 f_y A_sc
Min. eccentricity e_min = max (l/500 + D/30, 20 mm) must always be considered

12.7 Short Column with Uniaxial Bending

Fig 12.1 – Column P-M interaction diagram. Any (P_u, M_u) point inside the curve is safe. Balanced point corresponds to simultaneous concrete crushing (ε_cu=0.0035) and steel yielding.

12.8 Failure Modes in Eccentric Compression

Large eccentricity (e > e_balanced): Tension-controlled failure — steel yields first, then concrete crushes (ductile)
Small eccentricity (e < e_balanced): Compression-controlled failure — concrete crushes before steel yields (less ductile)
Balanced failure: Both occur simultaneously at specific e = e_balanced

12.13 Biaxial Bending (IS 456 Cl. 39.6 / SP-16)

Interaction equation for biaxial bending:
(M_ux/M_ux1)^α_n + (M_uy/M_uy1)^α_n ≤ 1.0
α_n = 1 when P_u/P_uz ≤ 0.2
α_n = 2 when P_u/P_uz ≥ 0.8
(interpolate for intermediate values)
P_uz = 0.45 f_ck A_g + 0.75 f_y A_sc

12.17–12.18 Slender Columns (IS 456 Cl. 39.7)

Additional moment due to slenderness (P-delta effect):
M_ax = P_u · e_ax where e_ax = (l_ex/D)² · D/2000
M_ay = P_u · e_ay where e_ay = (l_ey/b)² · b/2000
Design moment: M_udesign = M_u + M_ax (or M_ay)
l_ex, l_ey = effective lengths in respective directions
Effective length l_eff: from IS 456 Table 28 (based on end condition: fixed, pinned, free)

13Shallow Foundations►

13.4 Types of Footings

Type	Description	Use
Isolated Column Footing	Square / rectangular / circular footing under single column	Most common; widely spaced columns
Wall Footing / Strip Footing	Continuous footing under load-bearing wall	Masonry walls, load-bearing construction
Combined Footing	Single footing under two or more columns	Adjacent columns close together / near property line
Mat / Raft Footing	Single slab under entire structure	Weak soil; closely spaced columns; differential settlement control
Strap (Cantilever) Footing	Two isolated footings connected by strap beam	Eccentric column near property boundary

13.5 Soil Pressure Distribution

Fig 13.1 – Soil pressure under isolated footing. Concentric load → uniform. Eccentric load → trapezoidal. Uplift (tension) occurs when eccentricity e > L/6.

13.6 Design of Isolated Column Footing (IS 456 Cl. 34)

Design Procedure

Area of footing: A = P_service / SBC (use service loads for geotechnical design)
Net upward factored pressure: q_u = P_u / A_footing
Bending moment at face of column: M = q_u · a² / 2 (a = cantilever projection)
Depth from shear: One-way: τ_v = q_u · a'/(d) ≤ τ_c; Two-way punching: τ_v = V_u/(b_o·d) ≤ k_s·τ_c
Steel: M_u = 0.87 f_y A_st d (1 − A_st f_y/(b d f_ck)); solve for A_st
Development length: L_d available ≥ L_d required

One-way shear: critical at 'd' from face of column
Punching shear (two-way): critical perimeter at 'd/2' from each face
b_o = perimeter at d/2 from column = 2(l_c + b_c + 2d) where l_c, b_c = column dimensions
Permissible punching: τ_c = k_s × 0.25√f_ck (MPa); k_s = 0.5 + β_c ≤ 1.0

13.12 Combined Footings

Used when two columns are too close for separate footings, or one column is near property boundary
Design so resultant of column loads passes through centroid of footing (for uniform pressure)
Rectangular combined footing: design as inverted beam/slab under uniform pressure
Trapezoidal footing: when centroid cannot be aligned with rectangular shape
Strap footing: strap beam connects eccentric outer footing to inner footing; strap beam designed for bending and shear

14Prestressed Concrete (PSC) – IS 1343:2012►

14.4 Comparison: RCC vs PSC

Feature	RCC	PSC
Steel type	Mild steel / HYSD (f_y 250–550 MPa)	High-tensile (HT) wires/strands (f_pu 1600–2000 MPa)
Concrete grade	M20–M50	Min. M40 (pre-tensioned); M30 (post-tensioned)
Cracking	Cracked in tension (design accepts cracks)	No cracking (Type I) or limited cracking (Type II, III)
Deflection	Higher; affected by cracking	Lower; camber by prestress; load-balancing possible
Dead load ratio	Higher (heavier)	Lower (thinner sections)
Shear resistance	Stirrups needed	Better inherent shear resistance; fewer stirrups
Span range	Up to ~20 m economical	20–100+ m bridges, long-span structures
Initial cost	Lower	Higher (high-tensile steel + equipment)

14.11–14.13 Pre-tensioning vs Post-tensioning

Fig 14.1 – Pre-tensioning (factory; steel tensioned before casting; bonded) vs Post-tensioning (site; steel tensioned after concrete hardens; anchorage required).

14.14 Assumptions in PSC Analysis (IS 1343 Cl. 21.1)

Plane sections remain plane (Bernoulli hypothesis)
Elastic behaviour assumed for serviceability analysis; inelastic for ultimate limit state
For bonded tendons: perfect bond after grouting (strain compatibility)
For unbonded tendons: tendon strain based on overall elongation, not local strain

14.16–14.17 Stress Distribution in PSC Beam

Stress at any fibre of prestressed beam:
f = P/A ± P·e·y/I ± M·y/I

Top fibre: f_top = P/A − P·e·y_t/I + M·y_t/I
Bottom fibre: f_bot = P/A + P·e·y_b/I − M·y_b/I

P = effective prestress force | e = eccentricity of tendon
A = cross-sectional area | I = moment of inertia
y_t, y_b = distances from centroid to top and bottom fibres

14.22 Losses in Prestress

Loss	Occurs in	Magnitude (approx.)	Cause
Elastic Shortening	Pre-tensioned (immediate); Post-tensioned (during tensioning)	1–5%	Concrete shortens on application of prestress
Creep of Concrete	Both	5–10%	Long-term shortening under sustained prestress
Shrinkage of Concrete	Both	2–6%	Drying shrinkage causes tendon shortening
Steel Relaxation	Both	2–5%	HT steel loses stress under constant strain
Friction	Post-tensioned only	5–15%	Curvature friction (μ) + wobble (kx) along duct
Anchorage Slip	Post-tensioned only	1–3%	Slip at wedge anchors during lock-off
Total loss		15–25%	Typical range for well-designed PSC

Friction loss: P_x = P₀ · e^(−(μα + kx))
μ = coefficient of friction (0.2–0.5); α = cumulative angle of curvature (radians)
k = wobble coefficient (0.0015–0.005/m); x = length from jacking end

Elastic shortening loss (pre-tensioning): Δf_p = m_c · f_c
m_c = E_s/E_c (modular ratio for steel and concrete)

14.19–14.20 Pressure Line (C-Line) & Load Balancing

Equivalent load (UDL) from parabolic tendon profile:
w_eq = 8 P e / L² [upward UDL balancing downward gravity load]
When P × 8e/L² = w_{self-weight + DL}: net deflection = 0

Pressure line shift: e_p = M/(P) [distance of pressure line from centroidal axis]

14.23–14.24 Design of PSC Beams – Stress Limits (IS 1343)

Stage	Stress Limit in Concrete
Transfer (initial prestress, no live load): Compression	0.44 f_ci (IS 1343)
Transfer: Tension	0 (no tension in Type I) or 1.0 MPa (Type II pre-tensioned)
Service (effective prestress + DL + LL): Compression	0.33 f_ck
Service: Tension (Type I — fully prestressed)	0 (no tension permitted)
Service: Tension (Type II — limited prestress)	≤ 0.5 MPa (or 1.0 MPa in some cases)
Service: Tension (Type III — partial prestress)	Limited crack width ≤ 0.1 mm

15Earthquake Resistant Design of Structures (IS 1893:2016 & IS 13920:2016)►

15.1–15.6 Seismology Basics

Term	Definition
Focus / Hypocenter	Point within earth where fault rupture originates
Epicenter	Point on earth's surface directly above the focus
Magnitude (Richter)	Log₁₀(A/A₀); measures energy released; each unit = 10× more shaking, 31.6× more energy
Intensity (MMI)	Subjective measure of shaking at a location (I–XII scale)
P-waves	Primary (compressional) waves; fastest; travel through all media
S-waves	Secondary (shear) waves; ~0.6× P-speed; travel through solid only; cause more damage
Surface waves (R & L)	Rayleigh and Love waves; slowest; largest amplitude; most destructive

15.7 Seismic Zoning (IS 1893:2016)

Zone	Zone Factor Z	Seismicity	Examples
Zone II	0.10	Low	Southern peninsula (most of Kerala, Tamil Nadu stable areas)
Zone III	0.16	Moderate	Parts of MP, UP, Karnataka, Rajasthan
Zone IV	0.24	High	Delhi, parts of J&K, HP, Sikkim
Zone V	0.36	Very High (most severe)	North-east India, Uttarakhand, Andaman & Nicobar

⚠ Zone I has been removed in IS 1893:2016. India now has 4 seismic zones (II to V). Z = 0.36 for Zone V means peak ground acceleration = 0.36g / 2 = 0.18g (at 5% damping reference).

15.24 Design Base Shear (IS 1893:2016 Cl. 7.6)

Design Horizontal Seismic Coefficient:
A_h = Z·I·S_a / (2·R·g)
Z = Zone factor | I = Importance factor | R = Response reduction factor
S_a/g = Average response acceleration coefficient (from response spectrum)

Design Base Shear: V_B = A_h · W
W = Seismic weight = DL + % of LL (IS 1893 Table 8)
For LL ≤ 3 kN/m²: 25% of LL; for LL > 3 kN/m²: 50% of LL; roof: 0%

Fig 15.1 – Seismic lateral force distribution over building height. Forces increase parabolically (proportional to W_ih_i²), largest at roof. Sum of all Q_i = V_B.

15.23 Important Seismic Design Parameters

Parameter	Symbol	Values
Importance Factor	I	1.0 (normal); 1.2 (schools, hospitals >300 persons); 1.5 (critical infra)
Response Reduction Factor	R	1.5 (unreinforced masonry); 3.0 (ordinary RC shear wall); 5.0 (special SMRF)
Damping ratio (RC structures)	ξ	5% (design default)
T_a – Empirical period (moment frame)	T_a	0.075 h^0.75 (RC frame); 0.085 h^0.75 (steel); 0.09h/√d (other)
S_a/g for medium soil		T ≤ 0.1: 2.5 \| 0.1 < T ≤ 0.4: 2.5 \| T > 0.4: 1.0/T

15.11–15.16 Structural Concepts for Earthquake Resistance

Symmetry: Regular plan and elevation; no torsional irregularity
Continuous load path: Forces must transfer continuously to foundation
Ductility: Ability to deform beyond yield without collapse; key for energy dissipation
Redundancy: Multiple load paths; structure tolerates local failure
Soft storey: Storey stiffness < 70% of adjacent storey — must be avoided or specially designed
Short column effect: Partially infilled frames; columns with window openings fail in shear — avoid

Ductile Detailing (IS 13920:2016) – Key Requirements

Member	Key Requirement
Beams	p_t,min ≥ 0.24√f_ck/f_y; p_t,max ≤ 2.5%; confinement hoops; top steel ≥ bottom steel/2 at all sections
Columns	Min 1% steel; max 4%; confinement hoops in plastic hinge zones (both ends, full height for short columns)
Joints (Beam-Column)	Shear check per IS 13920; closed hoops in joint; no bar splicing in joint
Shear walls	Min. thickness 150 mm; min. vertical + horizontal steel 0.25% each; boundary elements at ends
Confinement hoops	Spacing ≤ min (D/4, 8φ_long, 100 mm) in critical zones
Strong column – Weak beam	ΣM_columns ≥ 1.2 ΣM_beams at joint (IS 13920 Cl. 9.2)

Storey Drift & Irregularities (IS 1893:2016)

Storey drift = (Δ_i − Δ_i-1) / storey height
Max allowable storey drift under design earthquake: 0.004 × storey height (IS 1893 Cl. 7.11.1)

Torsional irregularity: max displacement > 1.5 × avg displacement at that level
Mass irregularity: seismic weight of a storey > 200% of adjacent storey

AAppendix: Masonry Design (IS 1905:1987)►

Masonry Reinforcement

Reinforced masonry: steel bars embedded in mortar joints or grouted cavity to resist tension and shear
Used in seismic zones; lintel bands, plinth bands, gable bands — mandatory in IS 4326 (seismic zones III–V)
Vertical bars in corners, junctions, around openings for resistance to out-of-plane forces

Effective Height of Walls (IS 1905 Cl. 5.2.2)

End Condition	Effective Height H_eff
Both ends restrained (horizontal AND rotational)	0.75 H
Both ends restrained horizontally only (pinned)	1.0 H
One end free, one end restrained horizontally + rotationally	1.5 H
One end free, one end fully restrained	2.0 H

Effective Length of Walls (IS 1905 Cl. 5.2.3)

End Condition	Effective Length L_eff
Wall with returns at both ends	0.8 L
Wall restrained at one end, free at other	1.5 L
Wall free at both ends	2.0 L
Wall restrained at both ends	1.0 L (pinned)

Slenderness Ratio of Masonry (IS 1905)

SR = H_eff / t OR SR = L_eff / t (use smaller of the two)
t = thickness of wall
Max. SR = 27 (unreinforced masonry, IS 1905 Cl. 5.4.1)
For superimposed loads, allowable stress reduced by stress reduction factor (k_s) based on SR

Permissible Compressive Stresses in Masonry (IS 1905 Table 8)

Mortar Grade	Brick Strength (MPa)	Basic Compressive Stress (MPa)
M1 (1:0:3)	5.0	0.35
M2 (1:0.5:4.5)	7.5	0.50
M3 (1:1:6)	10.0	0.75
M4 (1:2:9)	12.5	0.95
H1 (1:0:3 with hydraulic lime)	5.0	0.35

Masonry in Seismic Zones – IS 4326 Requirements

Zone II: Bricks in cement mortar; horizontal steel band (lintel band) mandatory
Zone III: All of Zone II + vertical bars at corners and junctions + gable band + plinth band
Zone IV and V: All of Zone III + seismic bands at all floor levels; max opening area 11% of wall area; max opening width ≤ B/2 (B = wall length)
Reinforcement in seismic bands: 2 bars minimum of 8–12 mm dia; stirrups at 150 mm c/c

★Master Formulae & Quick Revision – RCC & PSC►

Critical Formulae – One-Page Reference

─── CONCRETE PROPERTIES ───
E_c = 5000√f_ck MPa | f_r = 0.7√f_ck MPa | Creep θ = 1.6 (28d)
ε_cu = 0.0035 | f_cd = 0.45f_ck | f_yd = 0.87f_y

─── WSM SECTION ───
m = 280/(3σ_cbc) | bx²/2 = mA_st(d−x) | Lever arm z = d − x/3
x_c/d = m·σ_cbc/(m·σ_cbc + σ_st)

─── LSM FLEXURE (SINGLY REINFORCED) ───
M_u,lim(Fe415) = 0.138 f_ck b d² | x_u,max/d = 0.48 (Fe415)
C = 0.36f_ck·b·x_u | T = 0.87f_y·A_st | z = d − 0.42x_u
A_st,min = 0.85bd/f_y | A_st,max = 4% of A_g

─── LSM FLEXURE (DOUBLY REINFORCED) ───
M_u2 = M_u − M_u,lim | A_sc = M_u2/[f_sc(d−d')]
A_st2 = f_sc·A_sc/(0.87f_y) | A_st = A_st,lim + A_st2

─── SHEAR ───
τ_v = V_u/(bd) | S_v = 0.87f_yA_svd/Vus
A_sv/(bS_v) ≥ 0.4/(0.87f_y) | Max S_v = 0.75d or 300 mm

─── BOND ───
L_d = 0.87f_yφ/(4τ_bd) | L_d ≈ 47φ (Fe415, M20, tension, HYSD)
Lap (tension) = 1.3L_d | Lap (compression) = 1.0L_d

─── TORSION ───
V_e = V_u + 1.6T_u/b | M_t = T_u(1+D/b)/1.7

─── COLUMNS ───
P_u (axial only) = 0.4f_ckA_c + 0.67f_yA_sc
e_min = max(L/500 + D/30, 20 mm)
Additional moment M_a = P_u·(l_eff/D)²·D/2000 (slender column)

─── PSC ───
f = P/A ± Pey/I ± My/I | w_eq = 8Pe/L² (parabolic tendon)
Friction: P_x = P₀·e^(−μα−kx) | Total losses ≈ 15–25%

─── EARTHQUAKE ───
A_h = Z·I·S_a/(2Rg) | V_B = A_h·W
Q_i = V_B·W_ih_i²/Σ(W_jh_j²) | T_a = 0.075h^0.75 (RC frame)
Max drift = 0.004 × storey height

Critical Numerical Values Table

Parameter	Value
Max strain in concrete at ULS	0.0035
Max strain in steel (Fe415) for yielding	0.87 × 415/(200000) + 0.002 = 0.00383
x_u,max/d: Fe250 / Fe415 / Fe500	0.53 / 0.48 / 0.46
M_u,lim coefficient: Fe415	0.138 f_ck b d²
Design compressive strength f_cd	0.45 f_ck (= 0.67f_ck/1.5)
Design tensile strength in steel f_yd	0.87 f_y (= f_y/1.15)
Partial safety factor: concrete (γ_m)	1.5
Partial safety factor: steel (γ_m)	1.15
Partial safety factor: DL + LL	1.5 DL + 1.5 LL
Min. cover: beams / columns / slabs / footings	25 / 40 / 20 / 50 mm (mild exp.)
Min. A_sc in column	0.8% of A_g
Max. A_sc in column	4% of A_g (6% at lap)
Min. bars: rectangular column / circular	4 / 6
Dev. length L_d: Fe415, M20, tension (HYSD)	≈ 47φ
Lap splice tension / compression	1.3 L_d / 1.0 L_d
Max stirrup spacing (shear)	0.75d or 300 mm
Max storey drift (IS 1893)	0.004 × storey height
Seismic Zone V factor Z	0.36
PSC min. concrete grade (pre-tensioned)	M40
PSC total prestress losses (typical)	15–25%
Punching shear perimeter (footing)	at d/2 from column face
Modulus of elasticity of steel E_s	2 × 10⁵ MPa

Mnemonics & Memory Aids

x_u,max/d for steel grades (easy recall):
Fe250 → 0.53 | Fe415 → 0.48 | Fe500 → 0.46 | Fe550 → 0.44
Higher grade steel → lower limiting NA depth (more ductile behaviour enforced)

M_u,lim key values:
Fe250: 0.148 | Fe415: 0.138 | Fe500: 0.133 | Fe550: 0.128 → all × f_ckbd²
Higher strength steel → smaller M_u,lim coefficient (NA moves up)

Cover (mm) – mild exposure:
Slab: 20 | Beam: 25 | Column: 40 | Footing: 50 | PSC: 20
“Slabs Barely Cause Footing Problems”

Seismic Zone factors:
Zone II: 0.10 | Zone III: 0.16 | Zone IV: 0.24 | Zone V: 0.36
Zone II → 10; each zone ×1.5 roughly; Zone V worst

PSC Loss Sources (mnemonic FESCAR):
F-riction | E-lastic shortening | S-hrinkage | C-reep | A-nchorage slip | R-elaxation

Design stages for a beam:
Size → Effective depth → Neutral axis check → Reinforcement → Shear → Bond → Deflection check

Reinforced Cement Concrete & Prestressed Concrete – Complete Study Notes

1.1 Introduction

1.2 Types of Concrete

1.3 Importance of Design Codes

1.4 & 1.5 Characteristic Strength and Grades of Concrete

1.8 Behaviour under Uniaxial Compression

1.9 Behaviour in Tension

1.10 Modulus of Elasticity & Poisson's Ratio

1.12 Creep of Concrete

Factors increasing creep:

1.13 Compressive Strength in Structural Design

2.1 Introduction

2.2 Necessity of Designing RCC Structures

Why RCC specifically needs design (unlike timber or masonry by rule-of-thumb):

2.3 Hydraulic & Non-Hydraulic Cements

Non-Hydraulic Cements

Hydraulic Cements

2.4 Tests on Cement

① Fineness Test (IS 4031 Pt 1 & Pt 2)

② Consistency Test — Normal / Standard Consistency (IS 4031 Pt 4)

③ Setting Time Test (IS 4031 Pt 5)

④ Soundness Test (IS 4031 Pt 3)

⑤ Strength Test (IS 4031 Pt 6 & IS 10080)

⑥ Heat of Hydration Test (IS 4031 Pt 9)

⑦ Specific Gravity of Cement (IS 4031 Pt 11)

Summary Table – All Cement Tests

2.5 Durability – Chemical Attack

2.6 Design Philosophies

2.7 & 2.8 Limit State Method – Partial Safety Factors (IS 456:2000)

Partial Safety Factors for Materials (γm)

Partial Safety Factors for Loads (γf) – IS 456 Table 18

2.10 Properties of Reinforcement

2.11 Cover Requirements (IS 456:2000 Cl. 26.4)

2.12 Spacing of Reinforcement (IS 456 Cl. 26.3)

2.14 Major Causes of Structural Failure

2.15 List of Key IS Codes for RCC

3.1 Introduction

3.5 Transformed Section & 3.6 Modular Ratio

3.6 Permissible Stresses in WSM (IS 456 Table B-1 & B-2)

3.3 Transformed Section Analysis (Key Equations)

3.9 Cracking Moment

4.1 Introduction – Limit States

4.3 Assumptions in LSM (IS 456 Cl. 38.1)

4.4 Singly Reinforced Sections – Key Formulae

4.5 Limiting (Balanced) Neutral Axis Depth (IS 456 Cl. G-1.1)

4.5 Limiting Moment of Resistance (Mu,lim)

4.6 Deflection Control – Span/Depth Ratio (IS 456 Cl. 23.2)

4.7–4.8 Minimum & Maximum Reinforcement Limits (IS 456)

4.12 One-Way Slabs

5.1 Introduction

5.4 Stress in Compression Steel (fsc)

5.7–5.8 Design Steps for Doubly Reinforced Beam

6.3 Shear Stress Distribution

6.6 Nominal Shear Stress (IS 456 Cl. 40.1)

6.8 Shear Strength without Reinforcement (τc) – IS 456 Table 19

6.9 Design of Shear Reinforcement

6.10 Minimum Shear Reinforcement (IS 456 Cl. 26.5.1.6)

6.11 Maximum Spacing of Shear Reinforcement

7.4 Bond Stress & 7.8 Development Length

7.5 Types of Bond

7.11 Hooks, Bends & Standard Anchorages (IS 456 Cl. 26.2)

7.16 Splicing of Reinforcement (IS 456 Cl. 26.2.5)

8.1 Introduction – When Does Torsion Occur?

8.8 IS 456 LSM Torsion Design (Cl. 41)

8.9 Torsional Reinforcement Requirements (IS 456 Cl. 41.4)

9.3 Effective Width of Flange (IS 456 Cl. 23.1.2)

9.5 Analysis of Flanged Sections

Case 1: Neutral Axis within flange (xu ≤ Df) — most common

Case 2: Neutral Axis in web (xu > Df)

10.3 Deflection Limits (IS 456 Cl. 23.2)

10.3 Deflection Calculation – Components

10.4 Cracking Limits (IS 456 Cl. 35.3)

10.5 Other Serviceability Limits

11.2 One-Way vs Two-Way Slabs

11.5 IS 456 Method for Two-Way Slabs (Cl. D-1)

11.6 Thickness of Two-Way Slabs

11.4 Support Conditions & Edge Discontinuity

11.8 Shear in Two-Way Slabs

12.3 Classification of Columns

12.5 IS 456 Recommendations for Columns (Cl. 26.5.3)

Partial Safety Factors for Materials (γ_m)

Partial Safety Factors for Loads (γ_f) – IS 456 Table 18

4.5 Limiting Moment of Resistance (M_u,lim)

5.4 Stress in Compression Steel (f_sc)

6.8 Shear Strength without Reinforcement (τ_c) – IS 456 Table 19

Case 1: Neutral Axis within flange (x_u ≤ D_f) — most common

Case 2: Neutral Axis in web (x_u > D_f)