Guidelines for Retrofitting of Steel Bridges by Prestressing
2008 Edition

IRC SP 75 - Overview: Essential Symbols, Formulae & Standards

1. Selected Symbols

2. Core Equations

• Stress in Tendon due to Prestressing:

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

• Member Bending Stress:

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

• Net Deflection:

  [ Y = Y_L - Y_P ]

• Tendon Eccentricity:

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

3. Reference Standards

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

4. Annexures for Detailed Calculations

• Annexure-1: Design of Prestressed Steel Girders (p.17)
• Annexure-2: Design of Prestressed Trusses (p.40)
• Annexure-3: Key Formulae & Worked Examples (p.48)

flowchart TD

Definition and Principles of Prestressing (IRC SP 75)

Prestressing involves applying internal stresses (usually compressive) to a structural element prior to service loading to improve its behavior under operational stresses.

Highlights:

• Objective: Resist tensile stresses induced by external forces.
• Methodology: Apply tensile force to high-strength steel tendons which then impart compressive stress to the steel members.
• Outcome: Maintains members predominantly in compression, enhancing durability and load capacity.

Calculating Prestressing Force:

• Initial prestress force ( P_i ) is given 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 prestress after accounting for losses ( P_e ):

  [ P_e = P_i - \text{Losses} ]

Common Prestress Losses:

• Concrete elastic shortening
• Concrete creep and shrinkage
• Steel relaxation
• Frictional losses in tendons

Typical Properties of Prestressing Steel

flowchart LR
    A[Tension Applied to Tendons] --> B[Stress Transferred to Steel]
    B --> C[Steel in Compression]
    C --> D[Enhanced Structural Capacity]

Refer to Clauses 2, 11, and 18 of IRC SP 75 for comprehensive details.

Scope of IRC SP 75: Design and Construction Guidelines

This code addresses the design, analysis, and construction procedures for prestressing steel road bridges. The scope encompasses:

• Design and analysis of prestressed steel girders and trusses.
• Equipment specifications and prestressing procedures.
• Tendon protection and inspection protocols.

Essential Formulae (Annexure-3)

For example, moment of inertia for rectangular flange section:

Additional Specifications

• Prestressing equipment details including anchorage, tensioning, force measurement, assembly, protection, and periodic monitoring (Clauses 21-26).
• Annexures provide design examples for girders and trusses.

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

For detailed design procedures, consult Annexure-3.

Reference Standards Applicable to Prestressed Steel Bridges (IRC SP 75)

Notable Symbols (Annexure-3)

Representative Formulae

• Tendon stress:

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

• Bending stress:

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

• Prestress-induced deflection:

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

Refer to IRC SP 75 for detailed clauses.

Critical Symbols and Parameters (IRC SP 75 Annexure-3)

Fundamental Formulas

• Material distribution parameter:

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

• Prestressing force:

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

• Incremental self-stressing force under uniform load:

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

• Upward deflection from prestress:

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

• Natural vibration frequency:

  [ n = \sqrt{\frac{I}{\rho A L^4}} ]

Refer to IRC SP 75 Annexure for detailed derivations.

Material Characteristics as per IRC SP 75

Key Parameters and Ranges

• Asymmetry factor (a): 1.5 to 2.0
• Web slenderness ratio (K): 100 to 200
• Material distribution ratio (m): 0.5 to 0.6

Important Equations

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

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

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

4. Increment in prestress force: [ \Delta X = \frac{3 \varepsilon^2 I}{L^2} + \frac{2 M e}{L} ]

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

Cross-sectional Areas

• ( A_1 ): Top flange area
• ( A_2 ): Bottom flange area
• ( A_w ): Web area

Refer to IRC SP 75 for full material specifications.

Key Points on Typical Shapes and Tendon Arrangements (IRC SP 75)

1. Tendon Placement in Cross-Sections

• Tendons are embedded within beams, trusses, or arches conforming to standard cross-sectional geometries.
• Use of tendon guides and deviators ensures the correct tendon profile, particularly for curved tendon paths.

2. Tendon Guide Components (Refer to Fig. 2)

• 1: Prestressing tendon
• 2: Guide to direct tendon trajectory
• 3: Structural rib supporting guide

3. Member Configurations

• Standardized beam, truss, and arch cross-sections are employed to optimize prestressing efficiency.
• Tendon profiles are designed to balance internal forces and control deflections effectively.

4. Reference to IRC:24-2001 Clause 6.6

• Quality control for castings and forgings to maintain durability and structural integrity.

Selected Formulae (Annexure-3 Highlights)

Tendon Layout Illustration

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

Use IRC SP 75 alongside IRC 24:2001 to ensure tendon layouts comply with strength and durability requirements.

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

Summary and Equations:

1. Prestressing Forces and Stress Computations

   • Initial prestress force: ( X )
   • Incremental prestressing force due to external loads: ( \Delta X )
   • Moment from prestressing: ( M_y = X \cdot e )
   • Compression edge stress: [ f_c = \frac{M_y}{S_1} + \frac{\Delta X (X + \Delta X) e}{S_1} \leq F ]
   • Tension edge stress: [ f_t = \frac{M_y}{S_2} + \frac{\Delta X (X + \Delta X) e}{S_2} \leq F ]
   • Tendon stress limit: [ X + \Delta X \leq F_t \cdot A_t ]

   Where:

   • ( e ): Tendon eccentricity
   • ( S_1, S_2 ): Section moduli at compression and tension edges
   • ( F ): Allowable steel stress
   • ( F_t ): Permissible tendon stress
   • ( A_t ): Tendon cross-sectional area

2. Tendon Configurations in Trusses and Arches (Clause 10.2)

   • Tendon profiles may be straight or curved.
   • Placement inside bottom chord or beneath centerline (distance H) outside bottom chord.

3. Determination of Maximum Prestressing Force (Clause 11)

   • Based on tendon profile and structural geometry.

4. Self-Stressing Force Considerations (Clause 12)

   • Incorporates prestress increments from live load induced tendon elongation/shortening.
   • Deflection limits:
     • Net deflection ( (Y_L - Y_p) \leq \frac{span}{600} )
     • Live load deflection ( \leq \frac{span}{800} )

5. Tendon Protection (Clause 25)

   • Cement grout filling of ducts made from GI or HDPE pipes in compliance with IRC:18-2000.

Refer to IRC SP 75 for detailed procedures.

Load and Force Analysis in Prestressed Steel Bridges (IRC SP 75)

1. Load Guidelines:

   • Follow IRC:6-2000 for standard load models, force calculations, and load combinations.
   • Include prestressing forces at various construction and service stages.

2. Prestressing Loads:

   • High-strength tendons tensioned within rolled steel sections (Clause 8.4).
   • Tendon force magnitudes at different stages (per Clause 140.62):

3. Stress Limits:

   • Maximum permissible steel stress: [ f_m < 140.62 \text{ N/mm}^2 ]

4. Load Combinations:

   • Combine tendon forces with external member forces: [ \sum \left( \frac{S_i S_{t,x}}{E A} \right) ]

   Where (S_i) = member forces, (S_{t,x}) = tendon forces, (E) = modulus of elasticity, (A) = cross-sectional area.

flowchart LR
    Loads[External Loads (IRC:6-2000)] --> Combined[Combined Effects]
    Prestress[Prestressing Forces] --> Combined

Employ IRC SP 75 alongside IRC:6-2000 for comprehensive load analysis.

General Design Requirements and Key Equations (IRC SP 75)

1. Prestress Force and Stress Parameters (Annexure 3)

• Prestressing force: ( X )
• Incremental tendon force: ( \Delta X )
• External bending moment: ( M )
• Section moduli:
  • Compression edge: ( S_1 )
  • Tension edge: ( S_2 )
• Cross-sectional areas:
  • Girder: ( A )
  • Tendon: ( A_t )
• Tendon eccentricity: ( e )
• Allowable stresses:
  • Girder steel: ( F )
  • Tendon steel: ( F_t )

2. Stress Formulations

[ f_c = \frac{X e y}{I} \quad \text{(Prestress stress at compression fiber)} ]

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

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

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

3. Tendon Protection and Assembly

• Tendons must be corrosion-protected by cement grout inside ducts made of medium or heavy duty galvanized iron or HDPE pipes per IRC:18-2000.
• Assembly and force measurement follow IS:1343-1980 Clauses 11 and 12.2.2.

4. Inspection Protocols

• Periodic inspections every 2 years as per IRC:SP:18 and IRC:SP:35.
• Check prestress force, corrosion protection, and anchorage integrity.

5. Reference Literature

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

flowchart LR
    A[Prestressing Force] --> B[Stress Computations]
    B --> C[Compliance Check]
    C --> D[Tendon Protection Measures]
    D --> E[Inspection and Maintenance]

Refer to IRC SP 75 for detailed design methodology.

Calculating Maximum Allowable Prestressing Force (IRC SP 75)

Principal Equation:

• Maximum prestressing force considering flange buckling:

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

Where:

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

Numerical example:

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

(Note: values depend on actual steel grade and area.)

Moment of Resistance (Clause 3.00):

[ M_R = 2210.8 \text{ kN-m} \quad \text{at 3 m from support} ]

General Prestressing Force Formula:

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

Where (a) and (m) are design parameters.

Summary Table:

flowchart TD
    A[Prestressing Steel Area (A_p)] --> B[Compute P_max = w × f_{pu} × A_p]
    B --> C[Check Against Design Limits]

Consult IRC SP 75 for full calculation procedures.

Understanding Self-Stressing Force in Prestressed Members (IRC SP 75)

Although IRC SP 75 lacks a dedicated clause, the concept is inferred from prestress force calculations and examples (Clause 140.62).

Key Points:

• Self-stressing force (S_s) arises from tendon elongation due to the structure's own weight and prestressing operations.
• Computed by:

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

Where:

• ( \Delta L ): Tendon elongation
• ( L ): Original tendon length
• ( E ): Elastic modulus of prestressing steel (~200,000 N/mm²)
• ( A ): Tendon cross-sectional area

Example from Clause 140.62:

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

Maximum Prestressing Force:

• Limited by tendon tensile capacity:

[ S_{max} = f_{pu} \times A ]

Summary:

• Calculate self-stressing force from tendon elongation and modulus.
• Refer to tendon force values for sectional analysis.
• Ensure tendon forces do not exceed material limits.

flowchart LR
    A[Structure Self-Weight] --> B[Tendon Elongation]
    B --> C[Self-Stressing Force S_s]
    C --> D[Prestressing Force in Member]
    D --> E[Verification Against Max Force]

Always consult IRC SP 75 and IS 1343 for design validation.

Principal Deflection Formulas and Tables (IRC SP 75)

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

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

Where:

• ( 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 ): Elastic modulus
• ( A_i ): Cross-sectional area of member i

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

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

Where:

• ( E_t, A_t, l_t ): Modulus, area, and length of tendon
• ( S_i ): Force in member i due to external load
• ( l_i ): Length of member i

3. Maximum Mid-Span Deflection (Refer to Fig. A2.02)

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

Where:

• ( a ): Half-span length
• ( X ): Prestressing force
• ( \Delta X ): Increment in prestress force
• ( h ): Vertical tendon distance from neutral axis
• ( E, A ): Modulus and area of member

4. Material Properties (Clause 380.00)

5. Allowable Stress Limits (Clause ...)

Refer to IRC SP 75 for detailed deflection control limits and design checks.

Basic Permissible Stress Limits according to IRC SP 75

• Reference: Based on Clause 14 referring to IRC:24-2001 Clause 506.4.1.

Key Points:

• Steel allowable stress (f_m) typically 165 MPa for mild steel under working stress design.

• Tendon allowable stress (f_t) depends on tendon grade, generally about 70% of ultimate tensile strength per IS:1343.

• Combined stresses must satisfy:

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

• Tendon stress limit:

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

Important Relations:

Symbol Summary:

flowchart TD
    A[Prestressing Force X] --> B[Prestress Stress in Steel]
    C[External Loads] --> D[External Stress in Steel]
    B & D --> E[Combined Member Stress]
    E --> F[Check against fm]

Refer to IRC SP 75 for comprehensive stress limits.

Evaluation of Combined Stresses in Prestressed Steel Members (IRC SP 75)

• According to Clause 15.2, combined axial and bending stresses must comply with limits specified in IRC:24-2001 Clause 506.4.2.

• Acceptable stress combinations include:

  1. Axial stress ( \sigma_{axial} ) plus bending stress ( \sigma_{bending} )
  2. Shear stress ( \tau_{shear} ) plus bending stress ( \sigma_{bending} )

Governing Inequalities (IRC:24-2001 Clause 506.4.2):

[ \frac{|\sigma_{axial}|}{f_a} + \frac{|\sigma_{bending}|}{f_b} \leq 1 ]

[ \frac{|\tau_{shear}|}{f_v} + \frac{|\sigma_{bending}|}{f_b} \leq 1 ]

Where:

• ( f_a ): Allowable axial stress
• ( f_b ): Allowable bending stress
• ( f_v ): Allowable shear stress

Additional Provisions:

• Lateral Stability (Clause 16): Treat members as beam-columns subjected to eccentric axial loads; provide bracing to prevent lateral-torsional buckling.

• Secondary Stresses (Clause 17): Account for restraint, thermal, and shrinkage-induced stresses.

Summary Table of Permissible Stresses

flowchart LR
    A[Axial Stress] --> C[Combined Stress Check]
    B[Bending Stress] --> C

Consult IRC SP 75 and IRC:24-2001 for detailed combined stress evaluation.