Slabs RCC: GATE Civil Engineering Notes

Slabs RCC, GATE Civil Engineering Notes focus on the fundamental design concepts of reinforced cement concrete slabs that are frequently tested in the GATE Civil Engineering exam. These notes explain key topics such as effective span calculation, one-way and two-way slab behavior, bending moment and shear force coefficients, deflection control criteria, nominal cover requirements, and reinforcement provisions as per IS 456. Written in simple and exam-oriented language, this introduction helps aspirants quickly understand slab design basics and confidently solve numerical and objective questions in GATE.

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Slabs RCC

Reinforced Cement Concrete (RCC) slabs are crucial structural elements, typically cast monolithically with beams in floors and roofs. A comprehensive grasp of their design is vital for civil engineers, particularly for competitive exams like GATE. Check key RCC slab design principles: effective span, bending moment/shear force coefficients, deflection control, slab classification, nominal concrete cover, and reinforcement requirements as per IS code.

Effective Span (L_eff)

The effective span (L_eff) calculation varies significantly based on the support conditions of the slab or beam.

Effective Span for Different Support Conditions

1. Simply Supported Beams/Slabs:
   The effective span (L_eff) is the minimum of the following two values:

• L_eff = L_not + w (Clear span + width of the support)

• L_eff = L_not + d (Clear span + effective depth of the member)

• Where:

• L_not: Clear span

• w: Width of the support

• d: Effective depth

2. Continuous Beams/Slabs:

• Case 1: If the width of the support is less than L_not / 12:
  The calculation is similar to a simply supported beam: L_eff is the minimum of (L_not + w) and (L_not + d).

• Case 2: If the width of the support is greater than L_not / 12:

• End Span (One end fixed, other continuous): L_eff = L_not

• Intermediate Span (Both ends continuous): L_eff = L_not

• End Span (One end continuous, other simply supported): L_eff is the minimum of (L_not + w/2) and (L_not + d/2).
  Note: These values are important for objective exam questions.

3. Cantilever Beams/Slabs:

• Isolated Cantilever (Fixed at one end, free at the other):
  L_eff = L_not + d/2

• Cantilever at a Continuous End:
  L_eff = L_not + w/2

4. Frames:
   For frame structures, the effective span (L_eff) is taken as the center-to-center distance between the members (beams or columns).

Check: GATE Civil Engineering Notes

Bending Moment and Shear Force Coefficients

For continuous beams or slabs, IS code provides coefficients to estimate bending moments and shear forces under dead load (DL) and live load (LL).

Bending Moment Coefficients

Moments are calculated using the formula: Moment = Coefficient × w × L².

• Sagging Moments (Positive): Occur near the middle of a span.

• Hogging Moments (Negative): Occur over the supports.

The beam/slab must be designed for the maximum bending moment, whether sagging or hogging.

• Maximum Sagging (Positive) Bending Moment:

• Occurs at the span next to the end support.

• M_sag,max = + (1/12) × w_DL × L² + (1/10) × w_LL × L²

• Maximum Hogging (Negative) Bending Moment:

• Occurs at the support next to the end support.

• M_hog,max = - (1/10) × w_DL × L² - (1/9) × w_LL × L²

The required depth (d) of the section is determined based on the absolute maximum of these two moment values (M_sag,max and M_hog,max).

d = sqrt(M_max / (Q × b))

Reinforcement varies by moment type:

• Positive Reinforcement (at the bottom): Provided based on the sagging bending moment.

• Negative Reinforcement (at the top): Provided based on the hogging bending moment.

Shear Force Coefficients

Shear force is calculated using the formula: Shear Force = Coefficient × w × L.

The maximum shear force typically occurs at the support next to the end support. The coefficients for calculating shear just to the left and right of this support are:

• At the exterior face of the support (towards end span):
  V_u = 0.4 × w_DL × L + 0.45 × w_LL × L

• At the interior face of the support (towards the middle span):
  V_u = 0.6 × w_DL × L + 0.6 × w_LL × L

Design for shear (e.g., calculating shear stress τv) must use the maximum shear force obtained.

Deflection Control Criteria

Deflection is a limit state of serviceability criterion, not a strength criterion. It must always be calculated using service loads, not ultimate (factored) loads. The calculated deflection must be less than the permissible deflection.

Permissible Deflection Limits

Control of Deflection through Span/Depth Ratio

As a simplified check, deflection is controlled by ensuring adequate effective depth (d).

d ≥ Span / A

• 'A' Values:

• Cantilever: 7

• Simply Supported: 20

• Continuous: 26

Modifications to the Span/Depth Ratio:

1. For Spans > 10 meters:
   The ratio is modified by multiplying with a factor of (10 / Span). The correct expression is:
   Effective depth ≥ (Span / A) × (10 / Span_in_meters)

2. Modification for Tension Reinforcement (Modification Factor 1 - MF1): The 'A' value is multiplied by MF1, which depends on the percentage of tension steel (p_t) and the stress in the steel (f_s). This value is read from Figure 4 of IS 456.

• f_s = 0.58 × f_y × (Area_required / Area_provided)

3. Modification for Compression Reinforcement (Modification Factor 2 - MF2):
   The 'A' value is further multiplied by MF2, which depends on the percentage of compression reinforcement (p_c). This value is read from Figure 5 of IS 456.

Final Formula for Effective Depth:

• For Span ≤ 10 m:
  d ≥ Span / (A × MF1 × MF2)

• For Span > 10 m:
  d ≥ [Span / (A × MF1 × MF2)] × (10 / Span_in_meters)

Check for Lateral Stability of Beams

To prevent lateral buckling, the clear distance (L) between lateral restraints must be limited. ( A slab is designed as a beam with a width of 1 meter (1000 mm); therefore, rules for beams also apply to slabs. )

Stability Limits

1. For Simply Supported and Continuous Beams: The clear distance L between lateral restraints shall not exceed the minimum of:

• 60 × b

• 250 × b² / d

2. For Cantilever Beams: The clear distance L from the free end to the lateral restraint shall not exceed the minimum of:

• 25 × b

• 100 × b² / d

• Where:

• b = width of the beam

• d = effective depth

One-Way vs. Two-Way Slabs

Deflection Control for Two-Way Slabs

Overall Depth ≥ Shorter Span (Lx) / B

• 'B' Values (for Mild Steel - Fe 250):

• Simply Supported Slab: 35

• Continuous Slab: 40

• Modification for HYSD Bars (Fe 415 / Fe 500):
  The 'B' values are multiplied by 0.8.

• Simply Supported: 35 × 0.8 = 28

• Continuous: 40 × 0.8 = 32

Note: These values are only valid for spans up to 3.5 meters and live loads up to 3 kN/m².

Nominal Cover (Clear Cover)

Nominal cover (also called clear cover) is the design depth of concrete cover provided to all steel reinforcement, including links (stirrups). It is the distance from the outermost surface of the steel to the nearest outer surface of the concrete.

• Nominal Cover: Distance from the edge of the stirrup/link to the concrete face.

• Effective Cover: Distance from the center of the main reinforcement bar to the concrete face.
  Effective Cover = Nominal Cover + dia_of_link + (dia_of_main_bar / 2)

Minimum Nominal Cover based on Exposure Condition (IS 456)

Minimum Cover for Different Members (under Mild Exposure)

Reinforcement Requirements

Minimum and Maximum Tension Reinforcement in Beams

• Minimum Tension Reinforcement (A_st,min):
  A_st,min / (b × d) ≥ 0.85 / f_y

• Maximum Tension Reinforcement (A_st,max):
  A_st,max ≤ 0.04 × b × D (4% of the gross cross-sectional area)

Maximum Compression Reinforcement in Beams

• Maximum Compression Reinforcement (A_sc,max):
  A_sc,max ≤ 0.04 × b × D (4% of the gross cross-sectional area)

Minimum Reinforcement in Slabs

• For Mild Steel (Fe 250): 0.15% of the gross cross-sectional area (0.0015 × b × D)

• For HYSD Bars (Fe 415 / Fe 500): 0.12% of the gross cross-sectional area (0.0012 × b × D)

Maximum Spacing of Reinforcement in Slabs

• Main Reinforcement: The spacing should not exceed the minimum of:

• 3 × d (3 times the effective depth)

• 300 mm

• Distribution Reinforcement (for temperature and shrinkage): The spacing should not exceed the minimum of:

• 5 × d (5 times the effective depth)

• 450 mm

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