5.3 RCC Structures-1

5.3 RCC Structures-1

Introduction to Reinforced Concrete Design

Reinforced Cement Concrete (RCC) is a composite material where concrete, strong in compression but weak in tension, is combined with steel reinforcement, strong in tension. This synergy creates a versatile structural material capable of resisting complex stress states. The design of RCC members involves ensuring safety against collapse (Ultimate Limit State) and serviceability under working loads. This unit covers the fundamental design philosophies, the analysis of basic elements like beams and slabs under various actions, and the codal provisions of IS 456:2000 and relevant NBC (Nepal Building Code) standards.

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1. Design Philosophies (Working Stress vs. Limit State)

Two primary methods have been used historically. The Limit State Method (LSM) is now the standard, while the Working Stress Method (WSM) is primarily used for special cases like liquid-retaining structures.

1.1 Working Stress Method (WSM)

1. Fundamental Principle:

   • The structure is designed to behave elastically under service loads (unfactored characteristic loads).

   • Stresses in both concrete and steel are kept within predefined permissible stresses, which are fractions of their material strengths.

2. Key Features:

   • Linear Stress-Strain: Assumes both materials follow Hooke's law (linear-elastic) up to the working load.

   • Factor of Safety: Incorporated indirectly through low permissible stresses.

     • Permissible stress in concrete, $\sigma_{cbc} = \frac{f_{ck}}{3}$ (approx.)

     • Permissible stress in steel, $\sigma_{st} = 0.55 f_y$ (approx.)

   • Modular Ratio (m): Ratio of modulus of elasticity of steel to that of concrete. $m = \frac{E_s}{E_c}$. Used to transform the steel area into an equivalent concrete area for analysis of the composite section.

3. Advantages:

   • Simpler calculations.

   • Predictable service load behavior (deflections, crack widths).

4. Disadvantages:

   • Does not account for the actual non-linear, inelastic behavior of concrete and the reserve strength beyond yield.

   • Overly conservative and uneconomical for most general structures.

   • Cannot directly account for different load combinations and their probabilities.

1.2 Limit State Method (LSM) - Current Standard (IS 456)

1. Fundamental Principle:

   • A structure reaches a limit state when it becomes unfit for its intended use.

   • Design ensures an acceptably low probability of reaching any limit state.

2. Types of Limit States:

   • Ultimate Limit States (ULS): Concerned with safety against collapse (e.g., failure in bending, shear, buckling, overturning). Governs the strength of members.

   • Serviceability Limit States (SLS): Concerned with functionality and comfort under normal use (e.g., excessive deflection, cracking, vibration). Governs usability.

3. Key Features:

   • Partial Safety Factors: Used to account for uncertainties.

     • On Loads ($\gamma_f$): Increase design loads (e.g., 1.5 for DL/LL). See 5.1.

     • On Material Strength ($\gamma_m$): Reduce material strengths.

       • For concrete, $f_{cd} = \frac{f_{ck}}{\gamma_c}$, where $\gamma_c = 1.5$.

       • For steel, $f_{yd} = \frac{f_y}{\gamma_s}$, where $\gamma_s = 1.15$.

   • Design Load: $\text{Factored Load} = \gamma_f \times \text{Characteristic Load}$.

   • Design Strength: $\text{Design Strength} = \frac{\text{Characteristic Strength}}{\gamma_m}$.

   • Assumptions:

     1. Plane sections remain plane (linear strain distribution).

     2. The maximum strain in concrete at failure is 0.0035 in bending.

     3. The tensile strength of concrete is ignored.

     4. The stress-strain curve for concrete is parabolic-rectangular or rectangular (simplified) for design.

     5. The stress in reinforcement is derived from its stress-strain curve, with a maximum design stress of $f_{yd}$.

Comparison: LSM is more rational, economical, and better represents true structural behavior compared to WSM.

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2. Design of Beams

Beams are primarily designed to resist Bending Moment (BM) and Shear Force (SF).

2.1 Analysis for Bending (Flexure)

1. Stress Block Parameters (LSM, IS 456 Cl. 38.1):

   • For the simplified rectangular stress block:

     • Depth of neutral axis: $x_u$

     • Depth of stress block: $0.416 x_u$ (for $f_{ck} \leq 60$ MPa)

     • Average compressive stress: $0.447 f_{ck}$

2. Singly Reinforced Rectangular Section:

   • Limiting Depth of Neutral Axis ($x_{u,max}$): To ensure ductile failure (steel yields before concrete crushes). Depends on steel grade.

     • For Fe 415: $x_{u,max} = 0.48 d$

     • For Fe 500: $x_{u,max} = 0.46 d$

   • Ultimate Moment of Resistance ($M_{u,lim}$): $M_{u,lim} = 0.138 f_{ck} b d^2 \quad (\text{for Fe 415})$ $M_{u,lim} = 0.133 f_{ck} b d^2 \quad (\text{for Fe 500})$ where $b$ = width, $d$ = effective depth.

   • Design Steps:

     1. Determine factored moment ($M_u$).

     2. Check if $M_u \leq M_{u,lim}$. If yes, design as singly reinforced.

     3. Calculate required steel area ($A_{st}$) using: $M_u = 0.87 f_y A_{st} d \left[1 - \frac{A_{st} f_y}{b d f_{ck}}\right]$ or use design charts/SP 16.

3. Doubly Reinforced Section:

   • Required when $M_u > M_{u,lim}$ or when depth is restricted.

   • Compression Steel ($A_{sc}$) is provided to assist concrete in compression.

   • Total Resisted Moment: $M_u = M_{u1} + M_{u2}$

     • $M_{u1}$ = moment resisted by the balanced section (concrete + tensile steel).

     • $M_{u2}$ = moment resisted by the couple formed by compression steel and additional tensile steel.

   • The stress in compression steel ($f_{sc}$) depends on the strain at that level and must be calculated.

2.2 Analysis and Design for Shear

1. Shear Stress ($\tau_v$): $\tau_v = \frac{V_u}{b d}$ where $V_u$ = factored shear force at the section.

2. Shear Strength of Concrete ($\tau_c$):

   • Nominal shear stress the concrete section can resist without shear reinforcement.

   • Depends on $f_{ck}$ and the percentage of tension steel ($p_t = 100A_{st}/bd$). Values given in IS 456 Table 19.

3. Design Logic:

   • If $\tau_v < \tau_c$: Minimum shear reinforcement must be provided as per code.

   • If $\tau_c < \tau_v < \tau_{c,max}$: Design shear reinforcement is required.

     • Shear to be carried by stirrups: $V_{us} = V_u - (\tau_c b d)$

     • Design of Vertical Stirrups: $\frac{A_{sv}}{s_v} = \frac{V_{us}}{0.87 f_y d}$ where $A_{sv}$ = cross-sectional area of stirrup legs, $s_v$ = stirrup spacing.

   • If $\tau_v > \tau_{c,max}$: Section is undersized. Increase section dimensions ($b$ or $d$).

4. Types of Shear Reinforcement: Vertical stirrups, inclined stirrups, or bent-up bars.

2.3 Deflection Control (Serviceability)

1. Purpose: Ensure deflections under service loads do not impair the appearance or function of the structure.

2. Code Provisions (IS 456 Cl. 23.2):

   • Deflection is controlled indirectly by specifying span-to-effective-depth (l/d) ratios.

   • Basic l/d ratios are given (e.g., 20 for simply supported beams, 7 for cantilevers).

   • These are modified by factors for:

     • Tension steel percentage ($f_{t}$ - modification factor).

     • Compression steel percentage.

     • Flanged sections.

   • Required: $\frac{l}{d} \text{ (provided)} \leq \text{Basic } \frac{l}{d} \times \text{Modification Factors}$

3. Calculation of Actual Deflection: If necessary, can be computed using elastic theory (using effective modulus of concrete, $E_c$, and considering creep and shrinkage).

2.4 Bond, Development Length, and Anchorage

1. Bond Stress ($\tau_{bd}$):

   • The shear stress at the interface between steel and concrete that transfers force between them.

   • Design bond stress values are given in IS 456 Cl. 26.2.1.1, depending on $f_{ck}$.

2. Development Length ($L_d$):

   • Definition: The minimum length of reinforcement bar required to be embedded in concrete to develop its full design strength through bond.

   • Formula: $L_d = \frac{\phi \sigma_s}{4 \tau_{bd}}$ where $\phi$ = bar diameter, $\sigma_s = 0.87 f_y$ for LSM.

   • Simplified: $L_d = \frac{\phi 0.87 f_y}{4 \tau_{bd}}$. Codes provide tables for $L_d$ in terms of bar diameter ($\phi$).

3. Anchorage of Bars:

   • End Anchorage: Bars must be anchored beyond the point where they are no longer needed to resist stress.

     • For a bar in tension, anchorage length ≥ $L_d$.

     • Standard hooks (90° or 135° bends) can be used to reduce straight development length.

   • Anchorage in Supports:

     • At simple supports, at least 1/3 of the total tension reinforcement should extend into the support for a length ≥ $L_d/3$.

     • At continuous supports, bars must be adequately anchored to develop the required stress.

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3. Design of Slabs

Slabs are plate elements primarily subjected to bending moments. They are categorized by aspect ratio and support conditions.

3.1 Types of Slabs

1. One-Way Slabs:

   • Condition:

   • Behavior: Bends primarily in one direction (the shorter span). Main reinforcement is provided along the shorter span; distribution reinforcement along the longer span.

   • Analysis: Designed as a 1-meter wide strip of a beam spanning in the short direction.

2. Two-Way Slab:

   • Condition: $\frac{L_y}{L_x} \leq 2$

   • Behavior: Bends in both directions. Moments are shared between the two spans.

   • Analysis: More complex. Can be designed using:

     • Coefficient Method (IS 456, Annex D): Uses moment coefficients based on edge conditions (simply supported, fixed).

     • Direct Design Method or equivalent frame method for flat slabs.

3.2 Design of One-Way Slabs (LSM)

1. Effective Span (l):

   • Clear span + effective depth (or width of support), whichever is less (IS 456 Cl. 22.2).

2. Load Calculation:

   • Total factored load (w) per unit area = 1.5(DL + LL).

   • Load on 1m wide strip = $w \times 1 \text{m}$ kN/m.

3. Bending Moment:

   • For simply supported: $M_u = \frac{w l^2}{8}$

   • For continuous slabs: Use moment coefficients from code.

4. Depth Check for Deflection:

   • Use $l/d$ ratio criteria. For slabs, basic $l/d$ is higher (e.g., 20 for simply supported, modified for steel percentage).

5. Reinforcement Calculation:

   • Calculate $A_{st}$ per meter width for the calculated $M_u$ (same as beam formula).

   • Check $A_{st,min}$ = 0.12% of gross cross-sectional area for HYSD bars (IS 456 Cl. 26.5.2.1).

6. Distribution Steel:

   • Provided in the long span direction.

   • $A_{st,dist}$ ≥ 0.12% of gross area (for HYSD bars) or as specified.

7. Shear Check:

   • Shear stress is usually very low in slabs. Normally, $\tau_v < \tau_c$, so only minimum shear reinforcement (i.e., main reinforcement) is adequate.

3.3 Deflection and Cracking in Slabs

1. Deflection: Controlled by $l/d$ ratio. More critical for slabs with lower depth.

2. Crack Control:

   • Governed by spacing of reinforcement.

   • Maximum Spacing (IS 456 Cl. 26.3.3):

     • Main bars: ≤ 3d or 300 mm, whichever is less.

     • Distribution bars: ≤ 5d or 450 mm, whichever is less.

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4. Relevant Codes and Standards

• IS 456:2000: Plain and Reinforced Concrete - Code of Practice. The primary code for RCC design in India and widely referenced in Nepal.

• NBC 109:2020 (Nepal Building Code): Concrete Structure. The mandatory code for RCC design in Nepal, harmonized with limit state philosophy.

• IS 875 (Parts 1-5): Code of Practice for Design Loads (for loads).

• IS 13920:2016: Ductile Detailing of Reinforced Concrete Structures Subjected to Seismic Forces (for earthquake-resistant design).

• SP 16:1980: Design Aids for Reinforced Concrete to IS 456:1978 (contains design charts and tables).

Conclusion: The design of RCC beams and slabs using the Limit State Method is a systematic process of analyzing internal forces, determining required reinforcement for strength in bending and shear, and ensuring serviceability through checks on deflection and cracking. Mastery of these fundamental principles, coupled with strict adherence to codal provisions for detailing (bond, anchorage, spacing), is essential for creating safe, durable, and serviceable reinforced concrete structures.