Guidelines for Design of Continuously Reinforced Concrete Pavement with Elastic Joints
1988 Edition

IRC 101 Overview – Essential Formulas, Tables & Norms

1. Thermal Stress Calculations (Eisenmann Formulas)

• Steel stress, (\sigma_s):

[ \sigma_s = \frac{100 \alpha \Delta T h E_c E_s}{f_s E_c + 100 h E_s} \times (1 - 2\beta) + \frac{100 h E_s}{f_s E_c + 100 h E_s} ]

• Concrete stress, (\sigma_c):

[ \sigma_c = \beta f_y E_c (1 - 2\beta) + \frac{100 h E_c}{f_s E_c + 100 h E_s} ]

Explanation of symbols:

2. Design Graphs

• Figures 2 & 3: Represent steel and concrete stresses per °C temperature change for steel percentages (0.1 - 0.6%) and (\beta) ratios (0.1 - 0.4).
• Maximum allowable steel stress: 1400 kg/cm² (with elastic joints).
• Transverse reinforcement: 25% of longitudinal reinforcement.

3. Slab Thickness Calculation

Stepwise approach:

1. Compute the plain slab thickness per IRC:58.
2. Choose steel percentage so that steel stress does not exceed 1400 kg/cm² (using Fig. 2).
3. Determine concrete stress from Fig. 3 and combine with previous stresses.
4. Verify total stress is within concrete flexural strength.
5. Utilize Mallinger’s chart (Fig. 4) to adjust slab thickness for steel reinforcement’s effective increase.

4. Mix Design and Material Specifications

• Concrete flexural strength must be at least ...

Design Principles for CRCP Incorporating Elastic Joints (Based on IRC 101):

Fundamental Concepts:

• CRCP lacks transverse joints; reinforcement manages cracking.
• Elastic joints enable controlled movement, mitigating stress concentrations.
• Optimal design involves balancing slab thickness, steel ratio, and joint spacing.

Parameters to Consider:

• Slab Thickness (h): Usually ranges from 200 mm to 300 mm.
• Steel Reinforcement Ratio: Typically between 0.6% and 0.8% of cross-sectional area.
• Joint Spacing: Generally 4.5 to 6 meters for elastic joints.
• Concrete Strength: Minimum compressive strength of 30 MPa at 28 days.

Common Design Equations:

• Steel Area (A_s): [ A_s = p \times b \times h ] where (p) is the steel percentage, (b) is slab width, and (h) is slab thickness.

• Load Transfer Efficiency (LTE) at Joints: Design for LTE ≥ 75%.

Recommended Steel Percentage vs. Slab Thickness

Specifications for Elastic Joints:

• Employ neoprene or rubber sealing elements.
• Install dowel bars for load transfer.
• Ensure effective sealing to prevent ingress of moisture.

flowchart LR
    Concrete_Slab --> Reinforcement_Steel
    Reinforcement_Steel --> Controls_Cracking
    Concrete_Slab --> Elastic_Joint
    Elastic_Joint --> Load_Transfer_Dowel_Bars
    Elastic_Joint --> Sealing_Prevents_Water

Summary: For durability and effective load transfer, design CRCP with 0.6-0.8% steel, slab thickness between 200-300 mm, and elastic joints spaced 4.5-6 m equipped with dowel bars and proper sealing.

Essential Formulas and Requirements for Steel Ratio and Stress Analysis (IRC 101)

1. Steel Reinforcement Percentage Calculation

• Average steel reinforcement ratio, (r_a): [ r_a = 0.004 + 0.25 \times 0.004 = 0.005 = 0.25% ]

2. Stress Calculations for Steel and Concrete at Elastic Joints (Eisenmann Formulas)

[ \sigma_s = \frac{100 \alpha \Delta T h E_c E_s}{f_s} (1 - 2) + \frac{100 h E_c^2}{A T f_s E_s} ]

[ \sigma_c = f_y E_s (1 - 2) + 100 h E_c^2 ]

Where:

• (\alpha) = Thermal expansion coefficient of concrete (per °C)
• (\Delta T) = Temperature difference between casting and coldest period (°C)
• (A T) = Maximum temperature differential top-bottom of slab (°C)
• (h) = Slab thickness in cm
• (E_c, E_s) = Elastic moduli of concrete and steel (kg/cm²)
• (f_s) = Steel cross-sectional area per meter width (cm²)
• (2) = Ratio of free, unbonded steel length to slab length between elastic joints

3. Design Charts Reference

• Fig. 2: Steel stress per °C versus steel percentage and unbonded length ratio.
• Fig. 3: Concrete stress per °C for corresponding parameters.
• Steel stress allowable limit: 1400 kg/cm².

4. Effective Slab Thickness Enhancement (Mallinger's Chart - Fig. 4)

• At (r_a = 0.25%), effective slab thickness increases by approximately 31%.
• Thickness adjustment formula: [ h_{effective} = \frac{h_{initial}}{1 + 0.31} ]

5. Outline of Design Steps

• Step 1: Assume initial slab thickness (per IRC:58).
• Step 2: Choose steel percentage ensuring (\sigma_s \leq 1400) kg/cm².

Key Points for Slab Thickness Design (IRC 101):

• Given Data:

  • Total compressive stress, (\sigma_c = 39.00) kg/cm²
  • Concrete flexural strength, (f_{cr} = 40) kg/cm²
  • Initial slab thickness, (h_0 = 25) cm

• Step 1: Verify Stress Against Strength Since (\sigma_c < f_{cr}), the initial thickness of 25 cm is acceptable.

• Step 2: Compute Average Steel Ratio (r_a) [ r_a = 0.004 + 0.25 \times 0.004 = 0.005 = 0.25% ]

• Step 3: Modify Thickness to Account for Reinforcement According to Fig. 4, for (r_a = 0.25%), effective slab thickness increases by 31%. Adjusted thickness: [ h = \frac{25}{1 + 0.31} = 19.08 \approx 19 \text{ cm} ]

• Step 4: Temperature Effects At 14.3°C, thickness aligns with 25 cm as per Clause 14.3.

Summary Table:

flowchart TD
    A[Start with 25 cm thickness] --> B{Is \(\sigma_c < f_{cr}\)?}
    B -- Yes --> C[Accept thickness]

Cement Concrete Mix Design Essentials per IRC 101

• Mix Design Approach: Absolute volume method following IRC:44 guidelines.
• Target Flexural Strength: At least 40 kg/cm² at 28 days under field conditions.
• Cement Type: Must comply with IS:269 (Ordinary Portland Cement) or IS:8112 (PPC).
• Aggregates: Fine and coarse aggregates per IS:383 specifications.
• Water Quality: Clean and potable, conforming to IS:456.
• Water-Cement Ratio: Generally between 0.4 and 0.5, adjusted for workability and strength.
• Proportion Adjustments: Based on trial mixes to meet strength and workability requirements.

Absolute Volume Equation:

[ V_c + V_w + V_a + V_{air} = 1 ]

Where:

• (V_c) = Volume of cement
• (V_w) = Volume of water
• (V_a) = Volume of aggregates
• (V_{air}) = Volume of entrapped air (typically 2–3%)

Typical Mix Design Process:

1. Define target strength and workability.
2. Select suitable water-cement ratio.
3. Compute absolute volumes of components.
4. Convert volumes to weights using specific gravities.
5. Conduct trial mixes and adjust proportions accordingly.

This mix design ensures a durable and strong concrete suitable for rigid pavement applications as outlined in IRC 101.

Fundamental Formulas and Material Requirements (IRC 101)

1. Thermal Stress in Steel and Concrete (Eisenmann Formulas):

[ \sigma_s = \frac{100 \times \alpha \times \Delta T \times h \times E_c \times E_s}{f_s \times E_s (1 - 2\beta) + 100 \times h \times E_c \times \beta^2} ]

[ \sigma_c = \frac{A \times T \times f_s \times E_c \times E_s}{f_y \times E_s (1 - 2\beta) + 100 \times h \times E_c \times \beta^2} ]

Where:

• (\alpha) = Concrete’s coefficient of thermal expansion (per °C)
• (\Delta T) = Temperature difference (°C)
• (h) = Slab thickness (cm)
• (E_c, E_s) = Elastic moduli of concrete and steel (kg/cm²)
• (f_s) = Steel area per meter width (cm²)
• (\beta) = Ratio of free steel length to slab length between joints
• Maximum permissible steel stress: 1400 kg/cm² (with elastic joints)

2. Steel Reinforcement Details:

• Longitudinal bars: 16 mm diameter spaced at 26 cm centers.
• Transverse bars: 10 mm diameter at 41 cm centers (25% of longitudinal steel).
• Steel near joints coated with bitumen over 1/3 to 1/4 of joint spacing for bond break.

3. Slab Thickness Design:

• Initial slab thickness calculated per IRC:58 for plain concrete.
• Adjust slab thickness using Mallinger’s chart (Fig. 4) for reinforcement effect.
• Design iteratively considering wheel loads, temperature, and reinforcement stresses.

4. Material Standards:

• Cement: IS 269 or IS 8112.
• Aggregates: IS 383.
• Steel: IS 432 (Part I) mild steel.
• Water: Clean, potable, meeting IS 456.

5. Construction Notes:

• Elastic joints are dummy contraction joints with continuous steel and bond-breaking coating.
• Expansion joints used only at section ends.

Cement and Reinforcement Specifications per IRC 101 for CRCP with Elastic Joints:

Cement Concrete Mix Design

• Employ absolute volume method as per IRC:44.
• Achieve flexural strength ≥ 40 kg/cm² at 28 days.
• Use cement conforming to IS 269 or IS 8112.
• Water must be clean, potable, and comply with IS 456.

Steel Reinforcement

• Use mild steel bars per IS 432 (Part I).
• Bar spacing between 25 and 35 cm.
• Place steel mats at slab mid-depth.
• Minimum lap splice length: 30 times bar diameter, with staggered laps.

Thermal and Shrinkage Stress Formulation (Eisenmann Equations)

[ \sigma_s = \frac{100 \alpha \Delta T h E_c E_s}{f_s E_c + 100 h E_s} \quad \text{(Steel stress, kg/cm}^2)]

[ \sigma_c = \frac{a \Delta T f_s E_c E_s}{f_s E_c + 100 h E_s} \quad \text{(Concrete stress, kg/cm}^2)]

Symbols:

• (\alpha): Concrete thermal expansion coefficient (per °C)
• (\Delta T): Temperature difference (°C)
• (h): Slab thickness (cm)
• (E_c, E_s): Elastic moduli of concrete and steel (kg/cm²)
• (f_s): Steel cross-sectional area per meter width (cm²)

Design Procedure Summary

1. Determine slab thickness per IRC:58.
2. Choose steel percentage to maintain steel stress ≤ 1400 kg/cm².
3. Calculate concrete stress and add to other stresses.
4. Apply Mallinger’s chart (Fig. 4) to adjust slab thickness.
5. Revise slab thickness accordingly.

Construction Considerations

• Support longitudinal steel with mild steel chairs.
• Fill elastic joints with sealing compound or bitumen-coated plywood strips (50 mm wide, 3 mm thick).
• Coat steel near joints with bitumen to facilitate elongation.

flowchart TD
    A[Assume slab thickness] --> B[Calculate stresses]
    B --> C[Select steel %]
    C --> D[Check steel stress ≤ 1400 kg/cm²]
    D --> E[Calculate concrete stress]
    E --> F[Adjust slab thickness]

Aggregate Specifications for CRCP as per IRC 101 and IS Codes:

Although IRC 101 does not specify detailed aggregate requirements, standard IS 383 guidelines apply:

Coarse Aggregate:

• Nominal sizes: usually 20 mm or 10 mm.
• Shape: Cubical, hard, durable.
• Fineness Modulus: Between 6.5 and 7.5.
• Specific Gravity: 2.6 to 2.9.
• Water Absorption: ≤ 2%.
• Aggregate Crushing Value: ≤ 30%.

Fine Aggregate:

• Particle size passing 4.75 mm sieve.
• Fineness Modulus: 2.3 to 3.1.
• Silt Content: ≤ 4%.
• Specific Gravity: 2.6 to 2.7.
• Water Absorption: ≤ 3%.

Important Formula:

• Fineness Modulus (FM): [ FM = \frac{\text{Sum of cumulative % retained on standard sieves}}{100} ]

Typical Mix Ratio for M20 Concrete (Approximate by volume):

• Cement : Fine Aggregate : Coarse Aggregate = 1 : 1.5 : 3

Aggregate Grading Example:

flowchart LR
    Cement --> Mix_Proportioning
    Aggregates --> Mix_Proportioning
    Mix_Proportioning --> Concrete_Mix

Construction Guidelines & Formulas for CRCP Incorporating Elastic Joints (IRC 101):

1. Elastic Joint Construction

• Dummy contraction joints are used with continuous reinforcement.
• Steel near joints is coated with bitumen over 1/3 to 1/4 of joint spacing on each side to break bond and permit elongation.
• Joint grooves are filled with sealing compound or bitumen-coated plywood strips (50 mm wide, 3 mm thick).
• Typical joint spacing ranges from 4 to 5 meters.

2. Calculation of Steel and Concrete Stresses (Eisenmann Formulas)

[ \sigma_s = \frac{100 \alpha \Delta T h E_c E_s}{f_s}(1-2) + \frac{100 h E_c^2}{f_s} ]

[ \sigma_c = \text{similar expression based on clause parameters} ]

Where:

• (\alpha): Coefficient of thermal expansion (per °C)
• (\Delta T): Temperature difference (mean slab to coldest period)
• (A_t): Maximum temperature differential top-to-bottom
• (h): Slab thickness (cm)
• (E_c, E_s): Elastic moduli of concrete and steel (kg/cm²)
• (f_s): Steel cross-sectional area per meter width
• (2): Ratio of unbonded steel length to slab length

3. Reference Design Charts

• Figures 2 & 3 illustrate steel and concrete stresses per °C of temperature differential for steel percentages (0.1–0.6%) and unbonded length ratios (0.1–0.4).
• Maximum steel stress permitted is 1400 kg/cm².

4. Slab Thickness Design

• Compute plain PCC slab thickness as per IRC:58.
• Add tensile concrete stress from continuous reinforcement.
• Use Mallinger’s Chart (Fig. 4) to determine effective slab thickness increase due to steel.
• Reduce slab thickness accordingly for the final design.

5. Reinforcement Specifications

• Longitudinal reinforcement: 16 mm diameter bars at 26 cm center-to-center, bitumen coated over 150 cm near joints.
• Transverse reinforcement: 10 mm diameter bars at 41 cm centers (25% of longitudinal steel).

Key Specifications and Formulas for General Construction (IRC 101):

1. Temperature-Induced Stresses (Eisenmann Equations)

[ \sigma_s = \frac{100 \alpha \Delta T h E_c E_s}{f_s} (1 - 2\beta) + \frac{100 h E_c^2}{f_s} ]

[ \sigma_c = \alpha \Delta T f_s E_c E_s ]

Where:

• Steel stress limit: 1400 kg/cm² with elastic joints.
• Transverse steel is 25% of longitudinal steel.

2. Slab Thickness Design Steps

• Calculate plain concrete slab thickness per IRC:58.
• Select steel percentage between 0.1% and 0.6% ensuring steel stress ≤ 1400 kg/cm² using Fig. 2.
• Determine concrete stress from Fig. 3 and add to slab stresses; verify total stress ≤ concrete flexural strength.
• Apply Mallinger’s chart (Fig. 4) to find effective thickness increase due to steel; adjust slab thickness accordingly.

3. Construction Requirements

• Steel bar spacing: 25 to 35 cm.
• Steel must conform to IS:432 (Part I).
• Concrete mix designed per IRC:44 with flexural strength ≥ 40 kg/cm² at 28 days.
• Aggregates must comply with IS:383.

IRC 101 Guidelines for Elastic Joint Construction in CRCP:

Specifications:

• Elastic joints are dummy contraction joints with continuous reinforcement.
• Reinforcement near joints is coated with a bond-breaking substance (e.g., bitumen) over 150 cm on both sides of the joint groove.
• Joint grooves are filled with bitumen-coated strips allowing movement.
• Typical joint spacing is 4 to 5 meters to control cracking and reduce steel stress.

Reinforcement Details:

Advantages:

• Reduces steel stresses by approximately 50%.
• Decreases overall steel quantity.
• Localizes cracks at joints, preventing random cracking.

Design Considerations:

• Bond-breaking coating creates a free length for the steel, limiting strain.
• Steel percentage and stresses accounted for bond-breaking length and joint movement.

Diagram of Elastic Joint

flowchart LR
    Concrete_Slab -->|Continuous Steel| Reinforcement
    Reinforcement -->|Bitumen Coating (150 cm)| Elastic_Joint_Groove
    Elastic_Joint_Groove -->|Filled with Bitumen Strip| Movement_Allowance
    style Elastic_Joint_Groove fill:#f9f,stroke:#333,stroke-width:2px

For detailed calculations and examples, refer to IRC 101 appendix on elastic joints.

Key Reinforcement Formulas and Details for CRCP with Elastic Joints (IRC 101):

1. Computation of Steel and Concrete Stresses at Elastic Joints (Eisenmann Equations):

[ \sigma_s = \frac{100 \alpha \Delta T h E_c E_s}{f_s} (1 - 2) + \frac{100 h E_c^2}{f_s} ]

[ \sigma_c = \alpha A T f_s E_c E_s ]

Where:

• (\alpha): Thermal expansion coefficient (per °C)
• (\Delta T): Temperature difference between casting and coldest period (°C)
• (A T): Max temperature differential top-to-bottom (°C)
• (h): Slab thickness (cm)
• (E_c, E_s): Moduli of elasticity of concrete and steel (kg/cm²)
• (f_s): Steel cross-sectional area per meter width (cm²)
• (2): Ratio of unbonded steel length to slab length between joints

2. Design Graphs:

• Fig. 2: Steel stress per °C vs steel percentage (0.1%–0.6%) and unbonded length ratio (0.1–0.4).
• Fig. 3: Concrete stress per °C for same parameters.
• Permissible steel stress: 1400 kg/cm².

3. Reinforcement Detailing:

• Longitudinal bars: 16 mm diameter @ 26 cm centers, bitumen coated over 150 cm length near joints.
• Transverse bars: 10 mm diameter @ 41 cm centers (25% of longitudinal steel).
• Minimum lap splice length: 30 bar diameters, staggered.
• Steel placed at mid-depth on mild steel supports.

4. Effective Slab Thickness Enhancement (Fig. 4 - Mallinger's Chart):

• Reinforcement increases effective slab thickness, permitting slab thickness reduction.

Example Design Methodology for CRCP with Elastic Joints (IRC 101):

Fundamental Equations (Eisenmann):

Calculate stresses near elastic joints:

[ \sigma_s = \frac{100 \alpha \Delta T h E_c E_s}{f_s} (1 - \beta) + \frac{100 h E_c^2}{f_s} ]

[ \sigma_c = \alpha \Delta T f_s E_c E_s ]

Where:

• (\alpha): Thermal expansion coefficient (per °C)
• (\Delta T): Temperature difference between mean slab temperature at casting and coldest period (°C)
• (h): Slab thickness (cm)
• (E_c, E_s): Elastic moduli of concrete and steel (kg/cm²)
• (f_s): Steel cross-sectional area per meter width (cm²)
• (\beta): Ratio of unbonded steel length to slab length between joints

Design Steps:

1. Assume initial slab thickness and compute wheel load and temperature-induced stresses (per IRC:58).
2. Choose steel percentage from Fig. 2 ensuring steel stress ≤ 1400 kg/cm²; calculate concrete stress from Fig. 3.
3. Sum concrete stress with other stresses and verify against concrete flexural strength; iterate as necessary.
4. Determine effective slab thickness increase using Mallinger’s chart (Fig. 4) and adjust slab thickness accordingly.

Important Parameters:

• Steel stress limit: 1400 kg/cm² (with elastic joints).
• Bar spacing: 25–35 cm.
• Minimum lap splice length: 30 bar diameters.
• Concrete flexural strength: ≥ 40 kg/cm² at 28 days.
• Transverse steel: 25% of longitudinal steel.

Mallinger’s Chart Summary (Fig. 4):

flowchart TD
    A[Assume slab thickness] --> B[Calculate wheel load & thermal stresses]
    B --> C[Select steel % to limit steel stress]
    C --> D[Calculate concrete stress]
    D --> E[Check total stresses against flexural strength]
    E --> F[Adjust slab thickness with Mallinger’s chart]
    F --> G[Finalize design]