Criteria for design of reinforced concrete arches

Scope & Key Specifications from IS 4090:

IS 4090 covers design criteria and analysis of concrete arches, defining symbols, types, and thermal effects.

Key Symbols (Clause 3.1)

Thermal Effects (Clause 9.2.4.1)

• Horizontal thrust due to temperature (H_r):

  [ H_r = T \cdot z \cdot E_c \cdot f_3 ]

• Moment at crown due to temperature change (M_{CT}):

  [ M_{CT} = T \cdot a \cdot E_c \cdot (m - 1) \cdot f_2 \cdot h \cdot f_3 ]

Where:

• ( T ) = Temperature rise
• ( a ) = Coefficient of linear thermal expansion of concrete
• ( E_c ) = Modulus of elasticity of concrete
• ( m, n, I, h ) = Arch parameters (see clauses 3, 8.1.1, 8.

IS 4090: Definitions and Structural Classification

Key Definitions (Clause 3.1)

Structural Classification (Clause 4.1)

• Fixed (Hingeless) Arches: No hinges; fixed supports.
• Hinged Arches: Incorporate hinges (typically at springing and/or crown).

Typical Arch Terminology (See Fig. 1 in IS 4090)

• Span (L), rise, crown, springing points, thrust lines, etc.

Summary Diagram: Arch Types

graph LR
A[Arches] --> B[Fixed (Hingeless)]
A --> C[Hinged]
C --> D[Two-Hinged]
C --> E[Three-Hinged]

This classification guides the design approach and analysis method per IS 4090.

IS 4090: Materials and General Requirements

1. Steel Reinforcement (Clause 6.3)

• Use mild steel or medium tensile steel bars conforming to IS 432 (Part 1)-1966.
• Modulus of Elasticity, E, for steel as per IS 800-1962 (Clause 9.1.4).

2. Temperature Effects (Clause 9.2.4.1)

• Horizontal thrust due to temperature (Hr):
  [ H_r = T \cdot z \cdot E \cdot f_3 ]

• Moment at crown due to temperature change (M_CT):
  [ M_{CT} = T \cdot a \cdot E \cdot (m-1) \cdot f_2 \cdot h \cdot c \cdot f_3 ]

Where:

• (T) = temperature rise
• (a) = coefficient of linear thermal expansion of concrete
• (E) = modulus of elasticity
• (m, n, I, h) = structural parameters (see clauses 3, 8.1.1, 8.3)
• (f_1, f_2, f_3) = coefficients from Table 1

3. Table 1: Coefficients for Temperature Effects

Summary:

• Use IS 432 steel bars.
• Modulus of elasticity per IS 800.
• Calculate temperature effects using given formulas and coefficients.
• Refer to Table 1 for

IS 4090: Types of Reinforced Concrete Arches & Key Specifications

Types of Arches (Clause 2.1)

• Applicable for spans up to 120 m
• Rise to span ratio between 1/8 and 1/3
• Design methods can vary if proven safe by analysis/tests
• Must be designed by qualified engineers and supervised during construction

Key Symbols (Clause 3.1)

Design Notes

• IS 4090 complements IS 456-1964 for general reinforced concrete rules.
• Formulas in the standard (Clauses 8 & 9) are aids; designers may use other methods if justified.
• Final values must be rounded as per IS 2-1960.

Typical Arch Design Parameters Summary

flowchart LR
    A[Types of Reinforced Concrete Arches]
    A --> B[Fixed Arch]
    A --> C[Hinged Arch]
    A --> D[Two-Hinged Arch]
    A --> E[Three-Hinged Arch]
    B --> F[Span ≤ 120m, Rise/Span 1/

IS 4090: Load Considerations (Wind & Seismic Forces)

1. Wind Forces (Clause 5.4)

• Wind force should be calculated as per IS 875 (Part 3) - 1987 (updated from 1964).
• Acts transversely to the structure and live load.
• Basic formula for wind pressure, ( p ):

[ p = 0.6 \times V^2 \quad \text{(kN/m}^2\text{)} ]

where ( V ) = design wind speed (m/s).

• Wind force on structure = ( p \times A ) (projected area).

2. Seismic Forces (Clause 5.5 & 5.5.3.1)

• Follow IS 1893 for seismic load calculation.
• Seismic force on live load and decking acts like a tractive force.
• Forces on arch and spandrel supports act at their center of gravity.
• Total seismic force, ( F ):

[ F = \alpha W ]

where:

• ( \alpha ) = seismic coefficient (from IS 1893),
• ( W ) = weight of the structure or segment.

3. Combined Load Stresses (Clause 10.5.1)

• When combining stresses from wind, earthquake, temperature, shrinkage with dead/live/impact loads, permissible stress increase must follow IS 456-2000.
• No relaxation allowed when shrinkage stresses are included.

Summary Table:

flowchart TD
    A[Loads on Structure] --> B[Dead Load]
    A --> C[Live Load]
    A --> D[Wind Load (IS 875)]
    A --> E[Seismic Load (

IS 4090: Design Principles and Analysis Methods for Reinforced Concrete Arches

1. Preliminary Analysis (Clause 9.2)

• Purpose: Establish initial internal forces and moments for design.
• Assumptions: Arch is statically determinate or indeterminate; elastic behavior is assumed initially.
• Load Considerations: Dead load, live load, temperature effects, and settlement.

2. Key Formulas

• Horizontal Thrust (H) for parabolic arch under uniform load: [ H = \frac{wL^2}{8f} ] where:

  • ( w ) = uniform load per unit length
  • ( L ) = span length
  • ( f ) = rise of the arch

• Bending Moment (M) at any section: [ M = H \cdot e - V \cdot x ] where:

  • ( e ) = eccentricity
  • ( V ) = vertical reaction
  • ( x ) = distance from support

3. Analysis Methods

• Elastic Analysis: Linear elastic theory for initial design.
• Plastic Analysis: For ultimate load conditions.
• Approximate Methods: Use of influence lines and simplified moment distribution.

4. Design Criteria Highlights

• Stress Limits: Follow IS 456 for permissible stresses in concrete and steel.
• Load Factors: Use load combinations as per IS 456 and IS 875.
• Reinforcement Detailing: Adequate anchorage and development length.

Summary Table: Typical Parameters

flowchart TD
    A[Load Application] --> B[Preliminary Analysis

IS 4090 refers to prestressed concrete pipes, and for Stress Combination and Allowable Stresses, it defers largely to IS 456-1964 (Plain and Reinforced Concrete).

Key Points from IS 4090 with IS 456 references:

• Stress Combination (Clause 7.2 & 10.5.1)

  • When bending and direct stresses combine, use the interaction criteria from IS 456:
    [ \frac{f_c}{f_{cb}} + \frac{f_b}{f_{bb}} \leq 1 ] where ( f_c ) = direct compressive stress, ( f_b ) = bending stress, ( f_{cb} ), ( f_{bb} ) = allowable stresses.
  • For combined loads (wind, earthquake, temperature, shrinkage), permissible stresses increase per IS 456, but no relaxation for shrinkage stresses.

• Allowable Stresses (Clause 7.3)

  • Basic permissible stresses can be increased for arches as per IS 456 recommendations.
  • Typical allowable stresses for concrete:

    Stress Type	Allowable Stress (N/mm²)
    Direct Compression	0.33 × f_ck (characteristic strength)
    Bending	0.36 × f_ck

• Ultimate Strength (Clause 11.2)

  • Calculate ultimate strength using IS 456 ultimate limit state methods (limit state design or working stress method as applicable).

Summary Table: Stress Interaction (IS 456)

graph LR
A[Direct Stress (f_c)] --> C[Combined Stress Check]
B[Bending Stress (f_b)] --> C
C --> D{Is \frac{f_c}{f_{cb}} + \frac{f_b}{f

IS 4090: Arch Configuration and Geometry Key Points

1. Arch Axis Equation (Clause 8.1.1)

For fixed arches, the arch axis shape can be approximated by:

[ y = \frac{\cosh(px) - 1}{p^2} ]

Where:

• ( y ) = vertical distance from crown
• ( x ) = horizontal distance from crown
• ( p = \log_e \left(m + \sqrt{m^2 - 1}\right) )
• ( m = \frac{L}{2h} ) (L = span, h = rise)

This defines the catenary shape for the arch axis.

2. Important Symbols (Clause 3.1)

3. Load and Stress Considerations (Clause 7.3)

• Increase permissible stresses as per IS 456-1964 for building arches.
• Follow relevant Indian Standards for bridge arches.

4. Arch Geometry Terminology (Clause 2.0)

• Refer to Fig. 1 (IS 4090) for arch terms: crown

IS 4090: Analysis of Arch Sections - Key Formulas and Specifications

Key Symbols (Clause 3.1)

• Aₛ: Area of steel in compression
• a: Depth of concrete compression stress block
• b: Width of arch rib/unit width of slab
• d: Depth from compression face to tension steel
• d': Distance between compression and tension steels
• e: Load eccentricity on arch section
• E_c, E_s: Modulus of elasticity of concrete and steel
• f_y: Yield strength of steel
• H: Horizontal dead load thrust at crown
• I, I_c, I_s: Moment of inertia at any section, crown, springing
• L: Span of arch
• U: Ultimate strength under combined direct and bending stresses
• t: Thickness/depth of arch rib
• W, W_s: Average load/unit length near crown and springing
• θ: Angle of tangent to arch axis at section
• θ_s: Slope of arch at springing

Analysis Principles (Clause 9.1.1)

• The arch axis is taken as the centroidal axis of the concrete section.
• Load effects include direct thrust (H) and bending moments due to eccentricity (e).

Typical Formula for Ultimate Strength (Combined Axial & Bending)

[ U = P + M / e ] Where:

• (P) = axial load (thrust)
• (M) = bending moment = (P \times e)
• (e) = eccentricity of load

Moment of Inertia

• Use (I) for the section moment of inertia at the point of interest.
• At crown and springing, use (I_c) and (I_s) respectively.

Summary Table: Section Properties

IS 4090: Deflection and Stability in Arch Design

Key Points from IS 4090:

• Deflection Moments (Clause 10.4):
  Consider deflection moments for arches with span > 120 m. These moments affect stability and must be included in design.

• Elastic Method (Clause 11.1):
  Use elastic analysis to find maximum bending moment and thrust. Section strength is then checked using ultimate load formulae.

• Combined Stresses (Clause 10.5.1):
  When combining stresses from wind, earthquake, temperature, shrinkage with dead/live loads, follow IS 456-1964 for permissible stress increases.
  No relaxation if shrinkage stresses are included.

Important Formulas & Influence Lines:

• Springing Moment (at support):
  [ M_s = \text{Coefficient} \times 100 ]

• Horizontal Thrust:
  [ H = \frac{P}{10} \times \frac{L}{h} \times \text{Coefficient} ]
  Where:

  • (P) = Load
  • (L) = Span length
  • (h) = Rise of arch

Stability Considerations:

• Check deflection limits to avoid excessive deformation.
• Include temperature and shrinkage effects in stress calculations.
• Use influence lines (Fig. 9-12) for bending moments, shear, and thrust at critical points (quarter point, springing, crown).

graph LR
A[Load P] --> B[Horizontal Thrust H = (P/10)*(L/h)*Coefficient]
B --> C[Arch Stability Check]
A --> D[Deflection Moments]
D --> C
C --> E[Design Verification using Elastic & Ultimate Load Methods]

Summary: For spans >120m, include deflection moments. Use elastic analysis for moments and thrust, combined stresses per IS 456, and influence lines for critical forces to ensure stability.

IS 4090: Reinforcement Detailing and Anchorage Key Points

1. Anchorage & Splicing (Clause 12.5)

• Main reinforcement bars must be:
  • Anchored into abutments or
  • Spliced to develop full bond strength.
• Splice length and anchorage conform to bond requirements per IS 456.

2. Ultimate Strength of Section (Clause 11.2)

• Calculate ultimate strength using IS 456: 1964 guidelines:
  • Use limit state design principles.
  • Check combined direct and bending stresses.
  • Design for ultimate load conditions.

3. Transverse Reinforcement (Clause 12.3)

• For arch slabs:
  • Provide transverse reinforcement for distribution, temperature, and shrinkage.
  • Minimum transverse reinforcement = 0.2% of sectional area on each face.

Typical Anchorage Length (per IS 456:2000 for deformed bars)

Summary Diagram: Reinforcement Anchorage and Splicing

graph LR
A[Main Reinforcement Bar] --> B[Anchorage into Abutment]
A --> C[Splicing with Overlap]
B --> D[Develop Full Bond Strength]
C --> D
D --> E[Ultimate Strength as per IS 456]

Note: Always check IS 456 for detailed anchorage length, lap splice length, and bar detailing requirements.

IS 4090: Temperature and Shrinkage Effects in Concrete Arches

1. Temperature Effects (Clause 9.2.4)

• Horizontal thrust due to temperature change:

  [ H_T = T \cdot z \cdot E \cdot f_3 ]

• Moment at crown due to temperature change:

  [ M_{CT} = T \cdot a \cdot E \cdot (m - 1) \cdot f_2 \cdot h \cdot c \cdot f_3 ]

Where:

2. Table 1: Coefficients (f_1, f_2, f_3)

Use appropriate values based on arch geometry.

3. Shrinkage Effects (Clause 5.8 and 5.8.1)

• Shrinkage strain (\varepsilon_{sh} \approx 0.00015)
• Equivalent temperature drop for shrinkage: 15°C
• About 60% of shrinkage stress relieved by creep.
• Shrinkage considered only if it worsens stresses.
• Shrinkage causes indirect effects similar to temperature fall.

Summary:

IS 4090: Joints and Construction Practices - Key Points

1. Location of Construction Joints

• Deck Beams: At the center of columns.
• Deck Slab: Over cross beams.
• Columns Braces: At the face of columns with a recess of ~12 mm.
• Arch Joints: Radial joints with shear keys.

2. Reinforcement Continuity

• Reinforcement must be continuous across joints (deck beams, slabs, arch longitudinal reinforcement).
• Shear keys provided at arch joints to transfer shear.

3. Temporary Hinge Design (Clause 14.2)

• Small section with ~8% longitudinal compression steel and maximum spiral reinforcement per IS 456.
• Compressive stress in concrete ≈ 80% of ultimate strength.
• Hinge length ≤ 2 × smaller section dimension.
• Steel mesh to distribute load from hinge to main member.
• Main arch reinforcement continues across hinge.

4. Precautions for New Concrete on Old Surface

• Follow IS 456 recommendations for bonding and surface preparation.

Summary Table: Temporary Hinge Design Parameters

flowchart LR
    A[Deck Beam Joint] -->|Center of Column| B[Continuous Reinforcement]
    C[Deck Slab Joint] -->|Over Cross Beam| B
    D[Column Brace Joint] -->|Face of Column + 12mm Recess| B
    E[Arch Joint] -->|Radial + Shear Key| B
    B --> F[Ensures Structural Continuity]

References:

• IS 4090: Clause 14.2, 15.1, 15.3.1, 15.3.2
• IS 456: For reinforcement and concrete surface preparation details

IS 4090: Special Considerations for Large Spans in Reinforced Concrete Arches

Key Points from IS 4090:

• Clause 10.4: Deflection Moments
  For arches with span > 120 m, deflection moments must be included in design.

  • Deflection moments arise due to elastic deformations and affect thrust line and stresses.
  • These moments increase bending stresses and must be combined with live load moments.

• Arch Span Considerations:
  Large spans require:

  • Careful analysis of thrust line considering deflections.
  • Use of second-order analysis or iterative methods to capture geometric nonlinearity.

• Bow String Girders (Clause 5.5.4):

  • Bow string girders combine arch action with tension tie members.
  • The tension tie resists horizontal thrust, reducing foundation loads.

Typical Design Steps for Large Spans:

1. Calculate initial thrust and bending moments ignoring deflections.
2. Estimate deflection moments using elastic theory or approximate formulas.
3. Combine deflection moments with live load moments for final design bending moments.
4. Check stability and ensure the thrust line remains within the arch section.

Formula for Deflection Moment (Approximate):

[ M_d = \frac{H \times \delta}{r} ]

Where:

• (M_d) = Deflection moment
• (H) = Horizontal thrust at supports
• (\delta) = Lateral deflection at crown
• (r) = Radius of curvature of the arch

Summary Table: Span vs. Design Considerations

flowchart LR
    A[Initial Load Analysis] --> B[Calculate Horizontal Thrust (H)]
    B --> C[Estimate Deflection (\delta)]
    C --> D[Compute Deflection Moment (M_d = H * δ / r)]
    D --> E[Combine with

IS 4090: References and Related Standards - Key Points

• Scope: Applies to reinforced concrete arches with spans ≤ 120 m and rise/span ratio between 1/8 and 1/3 (Clause 2.1).
• Complementary Standard: IS 456-1964 (Code of Practice for Plain and Reinforced Concrete) applies for usual RC construction rules (Clause 0.3).
• Design Freedom: Designers may use other methods if proven safe by analysis/tests (Clause 2.1).
• Rounding Off: Final values must be rounded per IS 2-1960, matching significant figures of specified values (Clause 0.5).

Key Formulae for Temperature Effects (Clause 9.2.4.1)

[ H_r = T \times z \times E_c \times f_3 ]

[ M_{CT} = T \times a \times E_c \times (m - 1) \times f_2 \times h \times c \times f_3 ]

Where:

• (H_r) = Horizontal thrust due to temperature
• (M_{CT}) = Moment at crown due to temperature change
• (T) = Temperature rise
• (a) = Coefficient of linear thermal expansion of concrete
• (E_c) = Modulus of elasticity of concrete
• (m, n, I, h) = Geometrical and material parameters (see IS 4090 Clauses 3, 8.1.1, 8.3)
• (f_1, f_2, f_3) = Coefficients from Table 1

Table 1: Coefficients for Temperature Calculations (Clause 9.2.4)

Summary Diagram of Temperature Effects