RCC & Prestressed Concrete — Design of Structures

RCC & Prestressed Concrete is the largest and highest-weightage design subject in civil engineering exams — from concrete/steel material behaviour and cement testing, through the Working Stress and Limit State design philosophies, singly/doubly reinforced beams, shear, bond, torsion, flanged beams, serviceability, two-way slabs, columns, foundations, prestressed concrete (IS 1343), earthquake-resistant design (IS 1893/IS 13920), and a masonry design appendix (IS 1905). Every formula, IS code clause, diagram, solved example and exam-pattern table is included.

Learning Objectives

After studying this chapter you will be able to:

• Explain concrete and steel material behaviour (stress-strain, creep, modulus of elasticity) and the tests used to verify cement quality (IS 4031)
• Apply both the Working Stress Method (transformed section) and the Limit State Method (IS 456 stress block) to design singly and doubly reinforced beam sections
• Design RCC members for shear, bond/development length, torsion, and T/L flanged sections per IS 456
• Check serviceability (deflection, crack width) and design one-way and two-way slabs
• Design short and slender columns (axial, uniaxial and biaxial bending) and isolated/combined footings
• Analyse prestressed concrete beams (pre-tensioning vs post-tensioning, losses, stress limits) per IS 1343
• Apply earthquake-resistant design principles and IS 13920 ductile detailing requirements, and design load-bearing masonry per IS 1905

Exam Relevance

Concept Map

Where This Fits

Prerequisite: Theory of Structures (analysis results — BM, SF, axial force — are the direct inputs to every design calculation here) and Steel Structures (parallel design philosophy for the other major structural material). Leads to: Soil Mechanics, since foundation design (Ch 13) depends directly on soil bearing capacity.

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).

1.2 Types of Concrete

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.

Target Mean Strength (Mix Design): \(f_{target} = f_{ck} + 1.65\sigma\).

1.8 Behaviour under Uniaxial Compression

The IS 456 idealized design stress-strain block for concrete rises parabolically to a strain of 0.002, then stays flat until the ultimate strain \(\varepsilon_{cu} = 0.0035\) (see diagram tab). The actual curve softens beyond the peak stress instead of staying flat.

1.9 Behaviour in Tension

• Concrete tensile strength ≈ 10–15% of compressive strength
• Modulus of Rupture (flexural tension): \(f_{r} = 0.7\sqrt{f_{ck}}\) MPa (IS 456)
• Direct tensile strength: \(f_t \approx 0.4\)–\(0.5\sqrt{f_{ck}}\) MPa
• Split tensile strength: \(f_{sp} = \dfrac{2P}{\pi dL}\)
• Concrete cracks in tension at very small strains (~0.0001 to 0.0002)

1.10 Modulus of Elasticity & Poisson's Ratio

Short-term static \(E_c = 5000\sqrt{f_{ck}}\) MPa (IS 456 Cl. 6.2.3.1). Long-term (with creep): \(E_{c,eff} = \dfrac{E_c}{1+\theta}\). Poisson's ratio \(\mu = 0.1\) to \(0.2\) (IS 456 uses \(\mu = 0.2\)). Shear modulus \(G = \dfrac{E_c}{2(1+\mu)} = \dfrac{E_c}{2.4} \approx 0.4E_c\).

1.12 Creep of Concrete

Creep Coefficient \(\theta\) = Creep Strain / Elastic Strain. IS 456 values: \(\theta = 2.2\) (loading at 7 days), 1.6 (at 28 days), 1.1 (at 1 year). Effective Modulus: \(E_{ce} = \dfrac{E_c}{1+\theta}\) — 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 \(f_{cd} = \dfrac{0.67f_{ck}}{\gamma_m}\). With \(\gamma_m\) (concrete) = 1.5: \(f_{cd} = \dfrac{0.67f_{ck}}{1.5} = 0.446f_{ck} \approx 0.45f_{ck}\). With \(\gamma_m\) (steel Fe415) = 1.15: \(f_{yd} = 0.87f_y = 0.87\times415 = 361\) MPa.

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

2.2 Necessity of Designing RCC Structures

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

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.

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.

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)

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)

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.

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

OPC 33 Grade: 3d ≥ 16 MPa | 7d ≥ 22 MPa | 28d ≥ 33 MPa. OPC 43 Grade: 3d ≥ 23 MPa | 7d ≥ 33 MPa | 28d ≥ 43 MPa. OPC 53 Grade: 3d ≥ 27 MPa | 7d ≥ 37 MPa | 28d ≥ 53 MPa.

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

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 (\(\gamma_m\))

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

2.10 Properties of Reinforcement

\(E_s = 2\times10^5\) MPa (200 GPa) for all grades of steel (IS 456 Cl. 6.2.4). Yield strain (Fe415): \(\varepsilon_y = \dfrac{f_y}{E_s} + 0.002 = \dfrac{415}{200000} + 0.002 = 0.00208 + 0.002 = 0.00383\). Design yield strength: \(f_{yd} = 0.87\times415 = 361.05\) MPa.

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

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

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 \(m = \dfrac{E_s}{E_c}\). IS 456 (WSM): \(m = \dfrac{280}{3\sigma_{cbc}}\), where \(\sigma_{cbc}\) = permissible bending stress in concrete. For M20: \(m = \dfrac{280}{3\times7} = 13.33\). For M25: \(m = \dfrac{280}{3\times8.5} = 10.98 \approx 11\).

Transformed area of tension steel = \(m \times A_{st}\). Transformed area of compression steel = \((1.5m-1)\times A_{sc}\) [subtract concrete already accounted for].

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

3.3 Transformed Section Analysis (Key Equations)

NA depth (x) is found by equating moments of areas about the NA: \(b\cdot x\cdot(\dfrac{x}{2}) = m\cdot A_{st}\cdot(d-x)\), i.e. \(\dfrac{bx^2}{2} = m\cdot A_{st}\cdot(d-x)\).

Moment of Inertia (cracked): \(I_{cr} = \dfrac{bx^3}{3} + m\cdot A_{st}\cdot(d-x)^2\). Moment of resistance: \(M = \sigma_{cbc}\cdot \dfrac{I_{cr}}{x} = \sigma_{st}\cdot \dfrac{I_{cr}}{m\cdot(d-x)}\). Lever arm: \(z = d - \dfrac{x}{3}\). Critical NA depth (balanced): \(\dfrac{x_c}{d} = m\dfrac{\sigma_{cbc}}{m\sigma_{cbc}+\sigma_{st}}\).

3.9 Cracking Moment

\(M_{cr} = f_r\cdot \dfrac{I_g}{y_t}\), where \(f_r = 0.7\sqrt{f_{ck}}\) MPa (modulus of rupture, IS 456), \(I_g\) = gross moment of inertia (uncracked section), and \(y_t\) = distance from NA to tension fibre (= D/2 for a symmetric section).

4.1 Introduction – Limit States

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

4.4 Singly Reinforced Sections – Key Formulae

Design Compressive Force: \(C = 0.36f_{ck}\cdot b\cdot x_u\). Design Tensile Force: \(T = 0.87f_y\cdot A_{st}\). Equilibrium (C = T): \(x_u = \dfrac{0.87f_yA_{st}}{0.36f_ckb}\). Lever arm: \(z = d - 0.42x_u\). Moment of Resistance: \(M_u = 0.36f_{ck}bx_u(d-0.42x_u)\), or equivalently \(M_u = 0.87f_yA_{st}(d-0.42x_u)\).

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

\(\dfrac{x_{u,max}}{d}\) depends on steel grade (from strain compatibility, \(\varepsilon_{cu}=0.0035\)): Fe250: 0.53. Fe415: 0.48. Fe500: 0.46. Fe550: 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 (\(M_{u,lim}\))

\(M_{u,lim} = 0.36(\dfrac{x_{u,max}}{d})[1-0.42(\dfrac{x_{u,max}}{d})]f_{ck}bd^2\). Fe415: \(M_{u,lim} = 0.138f_{ck}bd^2\) [very frequently asked in GATE]. Fe250: \(0.148f_{ck}bd^2\). Fe500: \(0.133f_{ck}bd^2\). Fe550: \(0.128f_{ck}bd^2\).

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

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 \(f_s = 0.58f_y \times (A_{st,req}/A_{st,prov})\) and \(p_t\)). For cantilevers > 10 m: deflection calculation is mandatory (not the l/d approach).

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

4.12 One-Way Slabs

• When \(\dfrac{l_y}{l_x} > 2\): 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)

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.

5.4 Stress in Compression Steel (\(f_{sc}\))

Strain in compression steel: \(\varepsilon_{sc} = 0.0035\dfrac{\times(x_{u,max}-d')}{x_{u,max}}\). The 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

1. Compute \(M_{u,lim}\) for the given section (using the limiting NA depth for the steel grade)
2. If \(M_u > M_{u,lim}\): \(M_{u2} = M_u - M_{u,lim}\)
3. Determine the d'/d ratio; find \(f_{sc}\) from the IS 456 table or strain compatibility
4. \(A_{sc} = \dfrac{M_{u2}}{f_{sc}\times(d-d')}\)
5. \(A_{st,lim} = \dfrac{0.36f_{ck}bx_{u,max}}{0.87f_y}\)
6. \(A_{st2} = f_{sc}\times \dfrac{A_{sc}}{0.87f_y}\)
7. \(A_{st} = A_{st,lim} + A_{st2}\)

6.6 Nominal Shear Stress (IS 456 Cl. 40.1)

\(\tau_v = \dfrac{V_u}{b\cdot d}\) — nominal shear stress at ULS, where \(V_u\) = factored shear force at the critical section. Critical section for shear: 'd' from the face of the support (for beams with vertical loads).

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

Design shear strength of concrete \(\tau_c\) depends on grade of concrete and % tension reinforcement (\(p_t = \dfrac{100A_{st}}{bd}\)).

Maximum shear stress \(\tau_{c,max}\) (IS 456 Table 20): M20 = 2.8, M25 = 3.1, M30 = 3.5, M35 = 3.7, M40 = 4.0 MPa.

6.9 Design of Shear Reinforcement

Case 1: \(\tau_v \le \tau_c\) → No shear reinforcement required (but provide minimum). Case 2: \(\tau_c < \tau_v \le \tau_{c,max}\) → Design shear reinforcement. Case 3: \(\tau_v > \tau_{c,max}\) → Redesign section (increase b or d).

\(Vus = V_u - \tau_c\cdot b\cdot d\) — shear to be carried by reinforcement. For vertical stirrups: \(Vus = \dfrac{0.87f_yA_{sv}d}{S_v}\), so \(S_v = \dfrac{0.87f_yA_{sv}d}{Vus}\).

6.10 Minimum Shear Reinforcement (IS 456 Cl. 26.5.1.6)

\(\dfrac{A_{sv}}{b\cdot S_v} \ge \dfrac{0.4}{0.87f_y}\). For Fe415: min. \(\dfrac{A_{sv}}{b\cdot S_v} \ge \dfrac{0.4}{0.87\times415} = 0.001109\).

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 (\(\tau_v > 0.5\tau_{c,max}\)): max spacing = 0.5d or 200 mm (whichever less)

7.4 Bond Stress & 7.8 Development Length

Bond stress \(\tau_{bd}\) = Force in bar / (perimeter × embedment length). Development Length \(L_d\): length of bar needed to develop the full design stress \(0.87f_y\).

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

\(L_d\) for Fe415 HYSD bar in M20 concrete (tension): \(\tau_{bd} = 1.2\times1.6 = 1.92\) MPa; \(L_d = \dfrac{0.87\times415\times\phi}{4\times1.92} = 47\phi\) (approx.). General approximate values: \(L_d \approx 47\phi\) (Fe415, M20); \(\approx 55\phi\) (Fe415, M15); \(\approx 40\phi\) (Fe415, M25).

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)

• 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

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: \(V_e = V_u + \dfrac{1.6T_u}{b}\). Equivalent Moment: \(M_{e1} = M_u + M_t\) (for the face with sagging moment), where \(M_t = \dfrac{T_u(1+\dfrac{D}{b})}{1.7}\). \(M_{e2} = M_t - M_u\) (for the opposite face, if \(M_t > M_u\)).

Nominal shear stress: \(\tau_{ve} = \dfrac{V_e}{b\cdot d} \le \tau_{c,max}\). If \(\tau_{ve} > \tau_c\): 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 ≤ \(x_1\) (shorter dimension of stirrup), \(y_1\)/4, or 300 mm (whichever least)

Area of transverse reinforcement per unit length: \(A_{sv}/s_v = \dfrac{T_u}{b_1\cdot d_1\cdot0.87f_y} + \dfrac{V_u}{2.5\cdot d_1\cdot0.87f_y}\), where \(b_1, d_1\) = distance between corner bars (centre to centre).

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

For T-beams (intermediate beam): \(b_f = \dfrac{l_o}{6} + b_w + 6D_f\). For L-beams (edge beam): \(b_f = \dfrac{l_o}{12} + b_w + 3D_f\). 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 distance)/2 (L-beam).

9.5 Analysis of Flanged Sections

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

Treat as a rectangular beam of width \(b_f\) and depth d: \(C = 0.36f_{ck}\cdot b_f\cdot x_u\), \(T = 0.87f_y\cdot A_{st}\), \(M_u = 0.87f_yA_{st}(d-0.42x_u)\).

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

Total compression = Flange compression + Web compression. \(C_f = 0.36f_{ck}(b_f-b_w)D_f\) [flange portion beyond web]. \(C_w = 0.36f_{ck}\cdot b_w\cdot x_u\) [web compression]. \(M_u = C_f(d-\dfrac{D_f}{2}) + C_w(d-0.42x_u)\).

10.3 Deflection Limits (IS 456 Cl. 23.2)

10.3 Deflection Calculation – Components

Total deflection δ = Short-term \(\delta_i\) + Long-term additional (creep + shrinkage). Short-term \(\delta_i = k\cdot \dfrac{WL^3}{E_{ce}\cdot I_{eff}}\), where k = 5/384 (SS UDL); 1/48 (SS point load at mid); 1/8 (cantilever tip). \(E_{ce} = E_c\) (short term) or \(\dfrac{E_c}{1+\theta}\) (long term).

Effective Moment of Inertia \(I_{eff}\) (IS 456 Cl. 23.2.1 / Branson's formula):

\(I_g\) = gross MoI, \(I_{cr}\) = cracked MoI, \(M_{cr}\) = cracking moment. \(I_{eff}\) transitions smoothly between the uncracked (\(I_g\)) and fully cracked (\(I_{cr}\)) stiffness as the applied moment \(M_{max}\) grows relative to \(M_{cr}\).

10.4 Cracking Limits (IS 456 Cl. 35.3)

Crack width (IS 456 Annex F): \(w_{cr} = 3a_{cr}\dfrac{\varepsilon_m}{1+2(a_{cr}-c_{min})/(D-x)}\), where \(a_{cr}\) = distance from crack to nearest bar surface, and \(\varepsilon_m\) = average strain at the level of tension steel.

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

11.2 One-Way vs Two-Way Slabs

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

Design BM per unit width: \(M_x = \alpha_x\cdot w\cdot l_x^2\) (short span direction), \(M_y = \alpha_y\cdot w\cdot l_x^2\) (long span direction). \(\alpha_x, \alpha_y\) = BM coefficients from IS 456 Table 26 (function of \(\dfrac{l_y}{l_x}\) and support conditions). \(w\) = total factored load per unit area = 1.5(DL + LL) kN/m².

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: \(\tau_v = \dfrac{V_u}{b\cdot d} \le k_s\cdot\tau_c\), where \(k_s = (0.5+\beta_c)\) but ≤ 1.0, and \(\beta_c\) = short side / long side of loaded area. For flat slabs: punching shear is critical at 'd/2' from the column face (two-way action).

12.3 Classification of Columns

12.5 IS 456 Recommendations for Columns (Cl. 26.5.3)

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

\(P_u = 0.4f_{ck}A_c + 0.67f_yA_{sc}\), where \(A_c\) = net concrete area = \(A_g - A_{sc}\), giving \(P_u = 0.4f_{ck}(A_g-A_{sc}) + 0.67f_yA_{sc}\). Minimum eccentricity \(e_{min} = \max(\dfrac{l}{500}+\dfrac{D}{30},\ 20\text{ mm})\) must always be considered.

12.7 Short Column with Uniaxial Bending

The P-M interaction diagram (see Diagrams tab) plots all combinations of axial load \(P_u\) and moment \(M_u\) that the section can safely carry. Any point inside the curve is safe.

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: \((\dfrac{M_{ux}}{M_{ux1}})^{\alpha_n} + (\dfrac{M_{uy}}{M_{uy1}})^{\alpha_n} \le 1.0\).

\(\alpha_n = 1\) when \(\dfrac{P_u}{P_{uz}} \le 0.2\); \(\alpha_n = 2\) when \(\dfrac{P_u}{P_{uz}} \ge 0.8\) (interpolate for intermediate values). \(P_{uz} = 0.45f_{ck}A_g + 0.75f_yA_{sc}\).

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

Additional moment due to slenderness (P-delta effect): \(M_{ax} = P_u\cdot e_{ax}\) where \(e_{ax} = (\dfrac{l_{ex}}{D})^2\cdot \dfrac{D}{2000}\). \(M_{ay} = P_u\cdot e_{ay}\) where \(e_{ay} = (\dfrac{l_{ey}}{b})^2\cdot \dfrac{b}{2000}\). Design moment: \(M_{u,design} = M_u + M_{ax}\) (or \(M_{ay}\)). \(l_{ex}, l_{ey}\) = effective lengths in respective directions, from IS 456 Table 28 (based on end condition: fixed, pinned, free).

13.4 Types of Footings

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

Design Procedure

1. Area of footing: A = \(P_{service}\) / SBC (use service loads for geotechnical design)
2. Net upward factored pressure: \(q_u = P_u / A_{footing}\)
3. Bending moment at face of column: \(M = q_u\cdot \dfrac{a^2}{2}\) (a = cantilever projection)
4. Depth from shear: One-way: \(\tau_v = q_u\cdot \dfrac{a'}{d} \le \tau_c\); Two-way punching: \(\tau_v = \dfrac{V_u}{b_o\cdot d} \le k_s\cdot\tau_c\)
5. Steel: \(M_u = 0.87f_yA_{st}d(1-\dfrac{A_{st}f_y}{bdf_{ck}})\); solve for \(A_{st}\)
6. 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: \(\tau_c = k_s\times0.25\sqrt{f_{ck}}\) (MPa); \(k_s = 0.5+\beta_c \le 1.0\).

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

14.4 Comparison: RCC vs PSC

14.11–14.13 Pre-tensioning vs Post-tensioning

Pre-tensioning: Steel is tensioned before casting (against fixed abutments in a factory bed); concrete is cast around the tensioned steel and cured; on release, the prestress transfers to the concrete by bond, developing camber. Post-tensioning: Concrete is cast with ducts already in place; HT steel is threaded through the ducts after the concrete hardens, tensioned with a jack, anchored, and the duct is grouted. See the Diagrams tab for the step-by-step comparison.

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 a prestressed beam:

Top fibre: \(f_{top} = \dfrac{P}{A} - Pe\,\dfrac{y_t}{I} + \dfrac{My_t}{I}\). Bottom fibre: \(f_{bot} = \dfrac{P}{A} + Pe\,\dfrac{y_b}{I} - \dfrac{My_b}{I}\). Where 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

Friction loss: \(P_x = P_0\cdot e^{-(\mu\alpha+kx)}\), where \(\mu\) = coefficient of friction (0.2–0.5), \(\alpha\) = cumulative angle of curvature (radians), k = wobble coefficient (0.0015–0.005/m), x = length from jacking end. Elastic shortening loss (pre-tensioning): \(\Delta f_p = m_c\cdot f_c\), where \(m_c = \dfrac{E_s}{E_c}\) (modular ratio for steel and concrete).

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

Equivalent load (UDL) from a parabolic tendon profile: \(w_{eq} = \dfrac{8Pe}{L^2}\) (upward UDL balancing downward gravity load). When \(P\dfrac{\times8e}{L^2} = w_{self-weight+DL}\): net deflection = 0. Pressure line shift: \(e_p = \dfrac{M}{P}\) (distance of the pressure line from the centroidal axis).

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

15.1–15.6 Seismology Basics

15.7 Seismic Zoning (IS 1893:2016)

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

Design Horizontal Seismic Coefficient: \(A_h = Z\cdot I\cdot \dfrac{S_a}{2\cdot R\cdot g}\), where Z = Zone factor, I = Importance factor, R = Response reduction factor, \(\dfrac{S_a}{g}\) = Average response acceleration coefficient (from the response spectrum).

Design Base Shear: \(V_B = A_h\cdot W\), where 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%.

15.23 Important Seismic Design Parameters

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

Storey Drift & Irregularities (IS 1893:2016)

Storey drift = \(\dfrac{\Delta_i - \Delta_{i-1}}{\text{storey height}}\). 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.

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)

Effective Length of Walls (IS 1905 Cl. 5.2.3)

Slenderness Ratio of Masonry (IS 1905)

SR = \(\dfrac{H_{eff}}{t}\) OR SR = \(\dfrac{L_{eff}}{t}\) (use the smaller of the two), where t = thickness of wall. Max. SR = 27 (unreinforced masonry, IS 1905 Cl. 5.4.1). For superimposed loads, the allowable stress is reduced by a stress reduction factor (\(k_s\)) based on SR.

Permissible Compressive Stresses in Masonry (IS 1905 Table 8)

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

16B.1 RCC Retaining Walls

A retaining wall holds back earth (or other material) where there is an abrupt change in ground level. RCC cantilever and counterfort types resist the lateral earth pressure computed by Rankine/Coulomb theory.

Stability checks (per unit length): (i) factor of safety against overturning \(\ge 1.55\) about the toe; (ii) against sliding \(\ge 1.55\); (iii) base pressure \(\le\) SBC with no tension (resultant in the middle third). The active earth pressure varies linearly, giving a triangular thrust \(P_a\) acting at \(\dfrac{H}{3}\) from the base.

16B.2 RCC Water Tanks (Liquid-Retaining Structures, IS 3370)

Liquid-retaining structures are designed on the no-crack (uncracked) philosophy so that steel stress is kept low and concrete stays in the permissible tensile range — crack width is limited to 0.2 mm to prevent leakage. Working-stress method with reduced permissible stresses is used even where LSM governs elsewhere.

• Classification: resting on ground, underground, and elevated (overhead) tanks; shapes — rectangular, circular, Intze.
• Circular tank (walls in hoop tension): hoop tension \(T = \gamma H \cdot \dfrac{D}{2}\) per unit height; \(A_{st} = \dfrac{T}{\sigma_{st}}\) with \(\sigma_{st}\) = 115–150 MPa (HYSD, IS 3370).
• Rectangular tank: walls act in combined bending + direct tension; analysed by moment-coefficient (IS 3370 Part IV) methods for the wall spanning conditions.
• Base slab & roof: designed for the water load, self weight and, for underground tanks, uplift when empty.

16B.3 RCC Staircases (IS 456 Cl. 33)

A stair slab is designed as a one-way slab spanning either along the going (longitudinal) or between supporting walls/stringer beams (transverse), depending on how it is supported.

• Effective span (IS 456 Cl. 33.1): depends on how the landing is supported — includes going + landing(s) when the stair spans longitudinally.
• Loading: self weight of waist slab + steps + finishes + live load (3–5 kN/m²); the waist-slab self weight acts along the inclined length, so it is converted to the horizontal (plan) span.
• Dog-legged & open-well arrangements; the flight is designed as a slab, main bars along the span, distribution bars transverse.

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