UNIT 1: FUNDAMENTAL CONCEPTS, DESIGN OF BEAMS, SLABS, COLUMNS, FOOTINGS, AND STAIRCASES

Based on RGPV past papers (2022–2025), this unit covers core design principles and element-specific design. High-weightage topics (14 marks) include beam design (with torsion/shear), column design (bi-axial bending), isolated footing design, and staircase design. Theoretical questions (5–9 marks) frequently test Limit State principles, section types, bond, and deflection control.

A. FUNDAMENTAL CONCEPTS & LIMIT STATE DESIGN

1. Limit State Philosophy & Principles

• Definition: Design philosophy ensuring structures satisfy Ultimate Limit State (ULS) (strength, stability) and Serviceability Limit State (SLS) (deflection, cracking, durability) throughout design life.

• Partial Safety Factors:

  • Load Factors (γ_f): Increase characteristic loads to get design loads (F_d = γ_f * F_k). E.g., γ_f = 1.5 (DL+LL), 1.2 (DL only), 1.5 (wind/earthquake).

  • Material Factors (γ_m): Reduce characteristic material strengths to get design strengths (f_d = f_k / γ_m). γ_m = 1.5 (concrete), 1.15 (steel).

• Characteristic Values:

  • Characteristic Load (F_k): 95% fractile value of load (max expected in 50 years).

  • Characteristic Strength (f_k): 5% fractile value of material strength (min. strength).

• Design Equation (ULS): Design Strength ≥ Design Load → R_d ≥ L_d.

[!TIP]

Common Pitfall: Confusing characteristic (statistical) with design (factored) values. Always apply safety factors to get design values.

2. Stress-Strain Relationships & Design Parameters

• Concrete (IS 456:2000, LSM):

  • Stress block: Rectangular with stress = 0.36 f_ck (Fe 415/500).

  • Depth of stress block: λ x_u where λ = 0.48 (Fe 415), λ = 0.46 (Fe 500).

  • Max. compressive strain in concrete = 0.0035 (Fe 415/500).

• Steel (Fe 415, Fe 500):

  • Elastic-perfectly plastic: Linear up to yield strain ε_y = f_y / E_s, then constant stress f_y.

  • E_s = 2×10^5 MPa.

• Modular Ratio (m) – WSM:

  • m = E_s / E_c. E_c varies with concrete grade (e.g., M20 → E_c ≈ 25.4 GPa).

  • Used to transform steel area to equivalent concrete area.

3. Types of Reinforced Concrete Sections (LSM)

• x_{u,lim} = limiting neutral axis depth for Fe 415 = 0.48 d; for Fe 500 = 0.46 d.

• WSM Comparison: Balanced section occurs when both concrete and steel reach their permissible stresses simultaneously. No x_{u,lim} concept; failure is brittle.

[!TIP]

Exam Focus: In LSM, balanced section is defined by x_u = x_{u,lim}, not by simultaneous yielding. Over-reinforced sections are not permitted by IS 456 due to brittle failure.

4. Analysis of Sections (Singly/Doubly Reinforced, T-Beams)

Singly Reinforced Rectangular Beam:

1. Equilibrium: C_c = T_s → 0.36 f_ck b x_u = 0.87 f_y A_st

2. Lever Arm: z = d - 0.42 x_u

3. Ultimate Moment of Resistance:

$$M_u = 0.36 f_{ck} b x_u z = 0.87 f_y A_{st} z$$

\boxed{M_u = 0.87 f_y A_{st} \left( d - 0.42 x_u \right)}

Doubly Reinforced Beam (when x_u > x_{u,lim}):

1. Find x_u from: 0.36 f_ck b x_u + f_{sc} A_{sc} = 0.87 f_y A_{st}

   • f_{sc} from strain in compression steel (use stress-strain curve or IS 456 Fig. 4).

2. M_u = M_{u1} + M_{u2}

   • M_{u1} = 0.36 f_ck b x_u (d - 0.42 x_u) (from concrete)

   • M_{u2} = f_{sc} A_{sc} (d - d_c) (from compression steel)

T-Beam:

• Effective Flange Width (b_f) (IS 456 Cl. 23.1.2):

$$b_f = b_w + \frac{l_o}{6} \quad \text{or} \quad b_f = b_w + \frac{l_o}{12} + b_w \text{ (whichever is lesser)}$$

where `l_o` = distance between points of zero contraflexure.

• Analysis:

  • If x_u ≤ D_f (flange thickness): treat as rectangular with width b_f.

  • If x_u > D_f: consider rib width b_w for portion below flange.

5. Serviceability & Durability

• Deflection Control:

  • Span/Depth Ratios (IS 456 Table 19): Basic ratios modified by x_u/d and steel percentage.

  • Limiting x_u/d: To ensure ductility, x_u/d ≤ 0.48 (Fe 415), ≤ 0.46 (Fe 500) for spans > 10m.

  • Measures: Increase depth, use higher grade steel, provide compression steel, reduce load.

• Critical Sections for Shear (IS 456 Cl. 22.6.1):

  • Simply Supported: At d/2 from face of support (Fig. 1a).

  • Continuous: At face of support (Fig. 1b).

  • Cantilever: At fixed end.

  DiagramSEARCH: IS 456 critical sections for shear

• Bond & Development Length:

  • Bond Failure Mechanisms:

    1. Adhesion (chemical bond at steel-concrete interface).

    2. Friction (due to roughness and Poisson effect).

    3. Mechanical Interlock (from ribs on deformed bars).

  • Development Length (L_d) (IS 456 Eq. 26.2.1):

$$L_d = \frac{\phi \sigma_s}{4 \tau_{bd}}$$

    where `τ_bd` = design bond stress (Table 19, IS 456), `σ_s` = design stress in bar.

    \boxed{L_d \propto \phi \cdot f_y}

*   **Lap Splice**: Length `L_lap` ≥ `1.3 L_d` for equal bars. For unequal diameters (e.g., 12mm with 20mm), use **larger diameter** to calculate `L_lap`.

[!TIP]

Bond Stress: Decreases with higher concrete grade and increases with bar diameter. Always check τ_bd from IS 456 Table 19 for exposure condition.

B. DESIGN OF RC BEAMS (FLEXURE, SHEAR, TORSION)

1. Flexural Design (Singly & Doubly Reinforced)

Step-by-Step (LSM):

1. Calculate design BM M_u.

2. Assume x_u/d ratio (e.g., 0.48 for Fe 415) → find M_u1 = 0.36 f_ck b d^2 (x_u/d)(1 - 0.42 x_u/d).

3. If M_u1 ≥ M_u → singly reinforced. Compute A_st = M_u / (0.87 f_y z).

4. If M_u1 < M_u → doubly reinforced. M_2 = M_u - M_u1. Compute A_sc = M_2 / [f_sc (d - d_c)], A_st from equilibrium.

5. Check: A_st,min = 0.85 b d / f_y (IS 456 Eq. 26.2.2.1), A_st,max = 0.04 b D (Cl. 26.5.1).

2. Shear Design

• Shear Stress: τ_v = V_u / (b d).

• Design Shear Strength of Concrete (τ_c): From IS 456 Table 19 (depends on f_ck and p_t %).

• Shear Reinforcement:

  • If τ_v ≤ τ_c: No shear reinforcement needed (but nominal ≥ 0.4% of concrete area).

  • If τ_v > τ_c: Provide vertical stirrups.

  • Spacing of Stirrups:

    • Max: 0.75 d or 300 mm (whichever is less).

    • Min: a_v ≤ 0.5 d near supports (IS 456 Cl. 26.5.1.6).

  • Area of Stirrups (2-legged):

$$A_{sv} = \frac{V_u - τ_c b d}{0.87 f_y d}$$

    \boxed{A_{sv} / s_v = \frac{V_{us}}{0.87 f_y d}}

    where `V_us` = shear to be resisted by reinforcement.

• Bent-up Bars: Can replace stirrups if V_us ≤ 0.5 V_u and spacing ≤ d/2.

3. Combined Bending, Shear, and Torsion

• Torsional Moment (T_u): Requires additional longitudinal and transverse reinforcement.

• Longitudinal Reinforcement:

  • Additional bars at corners (for compression) and sides (for tension).

  • Area: A_{st,t} = \frac{T_u}{f_y} \left( \frac{u}{2 x_{u,lim}} \right) (approx.), but use IS 456 method.

• Transverse Reinforcement: Closed ties (rectangular or circular) enclosing the core. Spacing ≤ min(0.75 d, 300 mm).

• Sketch: Show beam cross-section with main bars, closed ties at corners, and bent-up bars if any.

[!TIP]

Torsion Design: Always provide closed ties (not open links) to effectively resist torsion. Longitudinal bars for torsion are in addition to flexural bars.

4. Detailing of Beams

• Cross-section: Show strain diagram (linear), stress diagram (rectangular concrete, steel stress).

• Curtailment: Bars cut off where BM reduces (follow BM diagram). Minimum extension: L/7 from support or d (whichever greater).

• Development/Anchorage: Bars must develop full stress. Provide L_d beyond support or into column. Use hooks in beams for anchorage.

• End L-Sections: For beams supporting slabs, provide additional bars in top layer over support (to handle negative BM).

• Cantilever Beams: Top reinforcement continuous over support; provide extra bottom bars near support for stability.

C. DESIGN OF RC SLABS

1. One-Way Slabs

• Classification: Simply supported, continuous, cantilever.

• Effective Span: l_eff = clear span + d (simply supported) or c/c of supports (whichever is less).

• Depth for Deflection: Use span/depth ratios from IS 456 Table 19 (modified by steel %).

• Loads: DL (slab self-weight), LL (as per IS 875), floor finish, partitions (if any).

• Reinforcement:

  • Main Steel: Along short span (if rectangular) or direction of support. A_st = M_u / (0.87 f_y z).

  • Distribution Steel: Perpendicular to main steel, min. 0.15% (Fe 415) or 0.12% (Fe 500) of cross-section.

• Checks: A_st ≥ A_st,min, A_st ≤ A_st,max (0.04 bD). Development length at supports.

• Detailing: Plan showing main bars (solid line) and distribution bars (dashed). Section elevation showing cover and bar sizes.

2. Two-Way Slabs

• Classification: Restrained (corners prevented from lifting), simply supported.

• Effective Span: Clear distance between supports.

• Design Moments: Use IS 456 Table 26 coefficients for different support conditions (e.g., 0.0625 for interior panel, restrained).

  • M_x = α_x w l_x^2, M_y = α_y w l_y^2 where w = design load per unit area.

• Reinforcement: Provide in both directions. Calculate A_stx, A_sty separately.

• Checks: Same as one-way. Ensure corner bars (top) in restrained slabs.

3. Slab Design with Beam Support (T-Beam Action)

• If slab thickness ≥ 100 mm and beam width ≤ 3/4 clear distance between beams, flange effective.

• Load Distribution: Slab loads distributed to beams as tributary areas.

• Design beam with effective flange width b_f as per T-beam rules.

D. DESIGN OF RC COLUMNS

1. Short Columns (Axial Load with Uniaxial/Biaxial Bending)

• Assumptions (IS 456 Cl. 25.1.2):

  • Plane sections remain plane.

  • Maximum compressive strain in concrete = 0.0035.

  • Strain in steel ≤ 0.0035 + (f_y / 1.15 E_s).

  • Stress in steel = f_y for ε_s ≥ f_y/E_s.

• Uniaxial Bending:

  • Use interaction diagrams (IS 456 or SP 16) for given f_ck, f_y, reinforcement %.

  • Or approximate formula (for rectangular sections with bars on two sides):

$$\frac{P_u}{f_{ck} b D} + \frac{M_u}{f_{ck} b D^2} \leq 1.0 \quad (\text{approx.})$$

*   **Design Procedure**:

    1.  Assume % steel (0.8–6%).

    2.  Find `P_{uz}` (axially loaded capacity) from interaction curve.

    3.  Check if `(P_u, M_u)` point lies below curve. Iterate.

• Bi-axial Bending:

  • Use Bresler’s load contour (IS 456 Eq. 39.2):

$$\left( \frac{P_u}{P_{u0}} \right)^{\alpha} + \left( \frac{M_{ux}}{M_{ux0}} \right)^{\alpha} + \left( \frac{M_{uy}}{M_{uy0}} \right)^{\alpha} \leq 1$$

    where `α` = 1.5 for rectangular sections, `P_{u0}`, `M_{ux0}`, `M_{uy0}` are capacities under axial load alone or uniaxial moments alone.

*   **Arrangement of Bars**: For bi-axial, distribute bars **equally on all faces** (four sides) for symmetry.

• Transverse Reinforcement:

  • Ties: Diameter ≥ φ_max/4 or 6 mm (whichever greater). Pitch ≤ min(D, 16 c_long, 300 mm).

  • Helical Reinforcement: For circular columns, pitch ≤ 75 mm, 3 φ_helical, 1/6 core diameter.

2. Long Columns (Slenderness Effects)

• Effective Length (l_e): Depends on end conditions (IS 456 Table 28).

  • l_e = K l, where K = 0.7–2.0.

• Slenderness Ratio (λ): λ = l_e / r, where r = radius of gyration.

• Reduction in Capacity: For λ > 12 (uniaxial) or 16 (biaxial), reduce P_u using slenderness reduction factor (C_r) from IS 456 Cl. 39.7.

3. Detailing of Columns

• Cross-section: Show longitudinal bars (clear cover ≥ 40 mm or φ), ties/helix.

• Special Cases:

  • Circular with Ties: Ties at corners, pitch as above.

  • Circular with Helix: Continuous helical reinforcement, with vertical ties at ends.

E. DESIGN OF FOUNDATIONS (ISOLATED FOOTINGS)

1. General Considerations

• Soil Bearing Capacity (SBC): Net allowable bearing pressure (q_net).

• Net vs. Gross Pressure: P_net = P / A (after deducting footing weight & soil weight).

2. Rectangular/Square Isolated Footing

• Size of Footing:

  • For square: B = √(P_u / q_net).

  • For rectangular: Provide B × L such that P_net ≤ q_net and B ≤ L.

• Depth for Shear:

  • One-way Shear (along shorter side): Check at d from column face. τ_v = V_u / (B d).

  • Two-way (Punching) Shear: Check at d/2 from column face. τ_v = V_u / (b_0 d), where b_0 = perimeter at critical section.

  • Provide depth such that τ_v ≤ τ_{c,shear} (Table 19, IS 456). Increase depth or provide shear reinforcement if needed.

• Reinforcement:

  • Provided in both directions (mesh).

  • Bars extend into column for development.

  • Check for bearing pressure under loading (service and ultimate).

• Minimum Depth: ≥ 300 mm (cl. 34.1.2) for footing.

3. Stepped Footing

• Used when depth required is large.

• Steps: Each step height ≤ 300 mm. Provide reinforcement in each step (both directions).

• Reinforcement Detailing: Bars from lower step bent up to upper step; development at column face.

4. Detailing of Footings

• Plan: Show column outline, footing outline, reinforcement mesh (bars in both directions, with bends).

• Longitudinal Section: Show footing depth, column projection, reinforcement bars (with cover), development into column.

• Development: Bars from footing must develop stress in column – provide L_d into column.

F. DESIGN OF STAIRCASES

1. Terminology & Types

• Components: Tread (T), Riser (R), Waist slab, Stringer beam, Landing.

• Types: Dog-legged, Open-well, Helical, Cantilever.

2. Design of Dog-Legged Staircase

• Geometry:

  • Number of risers: N_R = Total Rise / R (round up).

  • Number of treads: N_T = N_R - 1 per flight.

  • Total horizontal distance: N_T × T.

• Loads:

  • Dead Load (DL): Self-weight of waist slab (inclined), steps, finishes.

  • Live Load (LL): As per IS 875 (3–5 kN/m² for office/residential).

• Effective Span:

  • For waist slab (simply supported on walls/beams): l_eff = clear distance between supports (horizontal).

• Design of Waist Slab:

  • Treat as inclined simply supported slab.

  • Calculate BM and SF for design strip (1m width along slope).

  • Design for flexure and shear as per slab rules.

• Landing Slab: Design as simply supported slab on walls/beams.

• Reinforcement Detailing:

  • Main bars along slope in waist slab.

  • Distribution bars perpendicular to slope.

  • Provide extra top bars at landing support (negative BM).

  • Curtailment: Main bars cut off where BM reduces.

[!TIP]

Staircase Load: Always consider inclined length for self-weight: DL = (thickness × unit weight) / cos θ, where θ = tan⁻¹(R/T).

G. THEORETICAL QUESTIONS & MISCELLANEOUS

1. Comparative Analysis: Balanced Section (WSM vs LSM)

2. Bond & Development

• Bond Stress (τ_b): τ_b = T / (π φ L).

• Development Length (L_d): As derived earlier.

• Lap Splice for Unequal Bars (e.g., 12mm with 20mm):

  • Use larger diameter (φ_max = 20 mm) for calculation.

  • Lap length L_lap ≥ 1.3 L_d (tension lap).

  • Provide staggered laps if possible.

3. Drawing Interpretation

• Reading RCC Drawings:

  • Identify sections (A-A, B-B) and plans.

  • Reinforcement notation: e.g., 3#20 means 3 bars of 20mm diameter.

  • Cover: Usually specified in drawings (e.g., 25 mm).

  • Bar bending schedules: List of bars with lengths, bends, positions.

• Showing Reinforcement in Sketches:

  • Use standard symbols: single line for bars, circles for sections.

  • Indicate bar size, spacing, cover.

  • For sections: Show concrete outline, bars with correct cover, strain/stress diagrams if asked.

Final Exam Strategy:

1. For design problems (beams, columns, footings, stairs): Follow step-by-step procedure (loads → design forces → section dimensions → reinforcement → checks).

2. For theoretical questions: Define terms clearly, use sketches where possible (e.g., strain diagrams, critical sections).

3. Always state assumptions (exposure condition, grade of concrete/steel, load combinations).

4. Check limits: A_st,min, A_st,max, x_u/d, span/depth, shear stress.

5. Diagrams: Neatly sketch cross-sections and reinforcement details with proper dimensions and bar labels.