Guidelines for Retrofitting of Steel Bridges by Prestressing

IRC SP 75 - Introduction: Key Formulas, Tables & Specifications

1. Important Symbols (Selected)

2. Key Formulae

• Prestress Resultant Stress in Tendon:

  [ f_{bf} = \frac{X}{A_t} \quad \text{(Stress in tendon due to prestressing force X)} ]

• Bending Stress in Member:

  [ \sigma = \frac{M}{S} ]

• Deflection Components:

  [ Y = Y_L - Y_P ]

  Where:

  • ( Y_L ) = Deflection due to loads
  • ( Y_P ) = Upward deflection due to prestressing

• Eccentricity:

  [ e = \text{distance of tendon from neutral axis} ]

3. Reference Codes & Standards

• IS:1343-1980 (Prestressed Concrete)
• IS:800-1984 (Steel Code)
• IRC:24-2001 (Symbols & Definitions)

4. Annexures for Detailed Design

• Annexure-1: Prestressed Steel Girders Design (p.17)
• Annexure-2: Prestressed Truss Design (p.40)
• Annexure-3: Important Formulae & Numerical Examples (p.48)

flowchart TD

Definition of Prestressing (IRC SP 75)

Prestressing is the process of introducing internal stresses (usually compressive) into a structural element before it is subjected to service loads, to improve its performance under those loads.

Key Points:

• Purpose: Counteract tensile stresses from external loads.
• Method: Apply tension to high-strength steel tendons, then transfer this stress to concrete.
• Result: Concrete is kept in compression, enhancing durability and load capacity.

Measurement of Prestressing Force:

• Initial prestressing force ( P_i ) is calculated by:

  [ P_i = A_p \times f_{pi} ]

  Where:

  • ( A_p ) = Cross-sectional area of prestressing steel
  • ( f_{pi} ) = Initial stress in prestressing steel

• Effective prestressing force after losses ( P_e ):

  [ P_e = P_i - \text{Losses} ]

Typical Prestressing Losses Include:

• Elastic shortening of concrete
• Creep and shrinkage of concrete
• Relaxation of steel
• Friction losses in tendons

Summary Table: Prestressing Steel Properties (Typical values)

flowchart LR
    A[Apply Tension to Tendons] --> B[Transfer Stress to Concrete]
    B --> C[Concrete in Compression]
    C --> D[Improved Structural Performance]

For detailed design, refer to clauses 2 (Definition), 11 (Maximum Prestressing Force), and 18 (Losses in Prestress) in IRC SP 75.

IRC SP 75 - Scope: Key Formulas & Specifications

IRC SP 75 covers the design and construction of prestressed steel road bridges. The scope includes:

• Analysis & design of prestressed steel girders and trusses.
• Prestressing equipment and procedures.
• Protection and inspection of tendons.

Important Formulas (Annexure - 3)

For a rectangular flange section (example from Clause 983.8):

Specifications Summary

• Prestressing Equipment & Procedures: Anchorage, tensioning, force measurement, assembly, protection, and periodic inspection (Clauses 21-26).
• Annexures: Detailed design examples for girders and trusses.

flowchart TD
    A[Scope of IRC SP 75] --> B[Prestressed Steel Girders]
    A --> C[Prestressed Steel Trusses]
    A --> D[Prestressing Equipment & Procedures]
    D --> E[Anchorage]
    D --> F[Tensioning & Transfer]
    D --> G[Force Measurement]
    D --> H[Assembly & Protection]
    D --> I[Periodic Inspection]

For detailed design, refer to Annexure - 3 for formulas and numerical examples.

Relevant Codes for Prestressed Steel Bridges (IRC SP 75)

Key Symbols (from Annexure - 3)

Important Formulae (Illustrative)

• Stress in tendon:

  [ f_{bt} = \frac{X}{A_t} ]

• Bending stress in member:

  [ f_c = \frac{M}{S} ]

• Deflection due to prestressing:

  [ Y_P = \frac{X \times e \times L^2}{

Key Symbols & Parameters from IRC SP 75 (Annexure-3 & Clauses):

Important Formulas

• Material distribution parameter:

  [ m = \frac{A_w}{A} \approx 0.5 - 0.6 ]

• Prestressing force (X):

  [ X = \frac{(a + 1)}{6a - (a + 1)^2 m} F A ]

• Self-stressing force increment (ΔX) for uniform load:

  [ \Delta X = \frac{3 e^2 I}{L^2} \times \text{(load terms)} ]

• Upward deflection due to prestress:

  [ Y_P = \frac{(X + \Delta X) e L^2}{8 E I} ]

• Natural frequency of vibration (n):

  [ n = \sqrt{\frac{I}{\rho A L

Key Material Properties (IRC SP 75)

Important Parameters & Ranges

• a (parameter in material distribution): 1.5 to 2.0
• K (web flexibility factor): 100 to 200
• m (material distribution ratio): 0.5 to 0.6

Key Formulas

1. Material distribution parameter m: [ m = \frac{A_w}{A} \quad \text{(where } A = A_1 + A_2 + A_w \text{)} ]

2. Section Modulus (S): [ S = \frac{V A^3 K m}{6a - (a+1)^2 m \cdot 6(a+1)} ]

3. Prestressing Force (X): [ X = \frac{(a+1)6a - (a+1)m}{6a - (a+1)^2 m} F A ]

4. Self-stressing Force increment ( \Delta X ): [ \Delta X = \frac{3 \varepsilon^2 I}{L^2} + 2 M e / L ]

5. Upward deflection due to prestressing: [ S_{\text{prestress}} = \frac{(X + \Delta X) e L^2}{8 E I} ]

Cross-sectional Areas

• ( A_1 ) = Area of top flange
• ( A_2 ) = Area of bottom flange

IRC SP 75: General Forms and Arrangements - Key Points

1. Typical Cross Sections with Tendons

• Tendons are placed within beams/trusses/arches following standard cross-sectional shapes.
• Tendon guides and deviators ensure correct tendon profiles, especially for curvilinear tendons.

2. Tendon Guide Details (Fig. - 2)

• Components:
  • 1: Tendon
  • 2: Guide (to direct tendon path)
  • 3: Rib (structural element supporting guide)

3. General Forms for Members

• Members (beams, trusses, arches) use standard cross sections to optimize prestressing.
• Tendon profiles are designed to balance stresses and control deflections.

4. Reference to IRC:24:2001 (Clause 6.6)

• Castings and forgings follow specific quality and dimensional controls for durability and strength.

Important Formulae (Annexure - 3 Highlights)

Summary Diagram: Tendon Layout in Beam

graph LR
A[Tendon] --> B[Guide]
B --> C[Rib]
C --> D[Beam Cross Section]

Use IRC SP 75 and IRC 24:2001 jointly for detailed design, ensuring tendon profiles and member forms comply with durability and strength requirements.

Different Methods for Prestressing Steel Structures Using Tendons (IRC SP 75)

Key Points & Formulas:

1. Prestressing Force & Stress Calculation

   • Initial prestressing force: ( X )
   • Incremental prestressing force due to external load: ( \Delta X )
   • Moment due to prestress: ( M_y = X \cdot e )
   • Stresses at compression edge:
     [ f_c = \frac{M_y}{S_1} + \frac{\Delta X (X + \Delta X) e}{S_1} \leq F ]
   • Stresses at tension edge:
     [ f_t = \frac{M_y}{S_2} + \frac{\Delta X (X + \Delta X) e}{S_2} \leq F ]
   • Tendon stress:
     [ X + \Delta X \leq F_t \cdot A_t ]

   Where:

   • ( e ) = eccentricity of tendon
   • ( S_1, S_2 ) = section moduli at compression and tension edges
   • ( F ) = allowable stress in girder steel
   • ( F_t ) = allowable stress in tendon steel
   • ( A_t ) = cross-sectional area of tendon

2. Tendon Placement in Trusses/Arches (Clause 10.2)

   • Tendon profile: straight or bent
   • Location: inside bottom chord or below centerline (H distance) outside bottom chord

3. Maximum Prestressing Force (Clause 11)

   • Calculate based on tendon profile and girder/truss geometry.

4. Self-Stressing Force (Clause 12)

   • Account for prestress increase due to live load elongation/shortening of tendons.
   • Deflection limits:
     • Net deflection ( (Y_L - Y_p) \leq \frac{span}{600} )
     • Live load deflection ( \leq \frac{span}{800} )

5. Protection of Tendons (Clause 25)

   • Cement grouting inside ducts made of GI or HDPE pipes per IRC:18-200

IRC SP 75: Loads and Forces - Key Points

1. Load Considerations:

   • Follow IRC:6-2000 for standard loads, forces, and load combinations.
   • Include prestressing effects at different stages (initial, transfer, service).

2. Prestressing Forces:

   • Tensioned high-strength wires are inserted in rolled sections and prestressed (Clause 8.4).
   • Forces from tendons vary by stage (example values from Clause 140.62):

3. Stress Check:

   • Permissible stress limits must satisfy:
     [ f_m < 140.62 \text{ N/mm}^2 ]

4. Load Combination:

   • Combine prestressing force effects with vertical loads:
     [ \Sigma \left(\frac{S_i S_t, x}{E A}\right) ] where (S_i) = force in member, (S_t) = tendon force, (E) = modulus of elasticity, (A) = cross-section area.

Summary Diagram of Load Effects on Prestressed Member

flowchart LR
    Loads[Vertical Loads (IRC:6-2000)]
    Prestress[Prestressing Force]
    Combined[Combined Effect on Member]

    Loads --> Combined
    Prestress --> Combined

Use IRC SP 75 with IRC:6-2000 for detailed load cases and prestressing stages.

IRC SP 75: General Design Requirements - Key Formulas & Specifications

1. Prestressing Force & Stress Calculations (Annexure 3)

• Prestressing force: ( X )
• Increment in tendon force: ( \Delta X )
• Bending moment due to external load: ( M )
• Section moduli:
  • Compression edge: ( S_1 )
  • Tension edge: ( S_2 )
• Cross-sectional areas:
  • Girder: ( A )
  • Tendon: ( A_t )
• Eccentricity of tendon: ( e )
• Allowable stresses:
  • Girder material: ( F )
  • Tendon material: ( F_t )

2. Important Stress Relations

[ f_c = \frac{X e y}{I} \quad \text{(Stress due to prestress at compression edge)} ]

[ f_t = \frac{M}{S_2} + \frac{\Delta X e}{I} \quad \text{(Stress at tension edge)} ]

[ X + \Delta X < F_t A_t \quad \text{(Tendon force limit)} ]

[ f_c < F \quad \text{(Girder compressive stress limit)} ]

3. Protection & Assembly

• Tendons must be protected against corrosion by cement grouting inside ducts made of medium/heavy duty GI pipe or HDPE pipes per IRC:18-2000.
• Assembly of prestressing steel as per Clause 11 of IS:1343-1980.
• Measurement of prestressing force as per Clause 12.2.2 of IS:1343-1980.

4. Periodic Inspection

• Follow IRC:SP:18 and IRC:SP:35.
• Check prestressing force, corrosion protection, and anchorage every 2 years.

5. References for Deep Study

• Troitsky M.S., Prestressed Steel Bridges Theory & Design, 1990.
• IS:1343-1980 for prestressing steel assembly and force measurement.
• IRC:18-2000 for tendon protection.

flowchart LR
    A

Maximum Possible Prestressing Force (IRC SP 75)

From the context and IRC guidelines:

Key Formula:

• The maximum permissible prestressing force (P_max) considering flange buckling is:

[ P_{max} = w \times f_{pu} \times A_p ]

Where:

• ( w = 0.98 ) (reduction factor from Fig. Al.7.01)
• ( f_{pu} = 230 , \text{N/mm}^2 ) (ultimate tensile strength of prestressing steel)
• ( A_p = 17198.782 , \text{mm}^2 ) (area of prestressing steel)

Given in context:
[ P_{max} = 0.98 \times 230 \times 17198.782 = 3,876,000 , \text{N} = 3876 , \text{kN} ]

(Note: The exact number may vary based on the area and steel grade.)

Moment of Resistance (Clause 3.00):

[ M_R = 2210.8 , \text{kN-m} \quad \text{(for 3 m from support)} ]

General Prestressing Force Equation:

[ X = A_p \times [60 - (a + 1) \times 2m] ]

(Where (a) and (m) are parameters defined in the design context.)

Summary Table:

flowchart TD
    A[Prestressing Steel Area \(A_p\)] --> B[Calculate \(P_{max} = w \times f_{

IRC SP 75: Self Stressing Force - Key Points

The code does not explicitly provide a dedicated clause for Self Stressing Force, but related concepts can be derived from prestressing force measurement and numerical examples (Clause 140.62).

Key Concepts:

• Self Stressing Force (S_s): The force induced in the prestressing tendons due to the member's own weight and prestressing operation.
• It is calculated considering tendon elongation and modulus of elasticity (E).

Typical Formula:

[ S_s = \frac{\Delta L}{L} \times E \times A ]

Where:

• (\Delta L) = elongation due to prestressing
• (L) = original length of tendon
• (E) = modulus of elasticity of prestressing steel (typically ~2×10^5 N/mm²)
• (A) = cross-sectional area of the tendon

From Clause 140.62 (Numerical Examples):

• These forces represent tendon forces at various sections due to prestressing.

Maximum Possible Prestressing Force:

• Limited by the characteristic tensile strength of the prestressing steel (e.g., (f_{pu})) and the area of tendons.
• Typically, (S_{max} = f_{pu} \times A).

Summary:

• Use tendon elongation and modulus to find self stressing force.
• Refer to tendon force tables for sectional forces.
• Ensure prestressing force does not exceed tendon capacity.

flowchart LR
    A[Member's Self Weight] --> B[Elongation in Tendons]
    B --> C[Self Stressing Force (S_s)]
    C --> D[Prestressing Force in Member]
    D --> E[Check Against Max Permissible Force]

For detailed design, always cross-check with the latest IRC SP 75 and IS 1343 provisions.

Key Formulas & Tables for Deflection (IRC SP 75)

1. Deflection in Prestressed Truss (Maxwell-Mohr Principle)

[ \delta = \sum \frac{S_i \times S_{ix} \times l_i}{E \times A_i} ]

• (S_i) = force in member i due to unit load at deflection point
• (S_{ix}) = force in member i due to prestressing force (X)
• (l_i) = length of member i
• (E) = modulus of elasticity of member
• (A_i) = cross-sectional area of member i

2. Increase in Tendon Force under External Load (Single Tendon)

[ \Delta X = \frac{2 \sum l_i S_i}{E_t A_t l_t} ]

• (E_t, A_t, l_t) = modulus, area, length of tendon
• (S_i) = force in member i due to external load
• (l_i) = length of member i

3. Maximum Deflection at Mid-span (Fig. A2.02)

[ \delta_{max} = \frac{4a^2 X h}{E A} + \frac{8 a^2 \Delta X h}{E A} ]

• (a) = half span length
• (X) = prestressing force
• (\Delta X) = increment in prestressing force
• (h) = vertical distance of tendon from neutral axis
• (E, A) = modulus and area of member

4. Material Properties (Clause 380.00)

5. Allowable Stress Limits (Clause

Basic Permissible Stresses as per IRC SP 75

• Reference: Clause 14 refers to IRC:24-2001 Clause 506.4.1 for basic permissible stresses in steel.

Key Points:

• Steel Permissible Stress (fm): As per IRC:24-2001, typically 165 MPa for mild steel under working stress design.

• Tendon Permissible Stress (ft): Depends on tendon material, usually high tensile steel, permissible stress as per IS:1343 (e.g., ~0.7 of ultimate tensile strength).

• Combined stresses in members must satisfy:

  [ \sigma_{total} = \sigma_{prestress} + \sigma_{external} \leq f_m ]

• For tendons:

  [ f_t = \frac{P}{A_t} \leq f_{allowable} ]

Important Formulas:

Symbols Summary:

flowchart TD
    A[Prestressing Force (X)] --> B[Stress in Steel Member (σ_prestress)]
    C[External Load] --> D[Stress in Steel Member (σ_external)]
    B & D --> E[Combined Stress (σ_total)]
    E

Combined Stresses in Prestressed Steel Members (IRC SP 75)

• As per Clause 15.2, combined stresses in prestressed steel members under axial load and bending must satisfy limits per IRC:24-2001 Clause 506.4.2.

• Permissible combinations:

  1. Axial stress (σ_axial) + Bending stress (σ_bending)
  2. Shear stress (τ_shear) + Bending stress (σ_bending)

Key Formula (from IRC:24-2001 Clause 506.4.2):

[ \frac{|\sigma_{axial}|}{f_{a}} + \frac{|\sigma_{bending}|}{f_{b}} \leq 1.0 ]

Where:

• ( f_a ) = permissible axial stress
• ( f_b ) = permissible bending stress

Similarly, for shear and bending:

[ \frac{|\tau_{shear}|}{f_{v}} + \frac{|\sigma_{bending}|}{f_{b}} \leq 1.0 ]

Where:

• ( f_v ) = permissible shear stress

Additional Specifications:

• Lateral Stability (Clause 16):
  Treat members as beam-columns with eccentric axial load. Check lateral-torsional buckling and provide bracing at intervals to ensure stability.

• Secondary Stresses (Clause 17):
  Account for stresses due to restraint, temperature, and shrinkage effects.

Summary Table: Permissible Stress Limits

flowchart LR
    A[Axial Stress]