Load Balancing Method for Pre-Stressed Concrete Elements

Introduction

• The load balancing method is a technique for designing pre-stressed concrete elements.
• This method is used when analyzing concrete elements already in service load conditions.
• It involves determining the effective pre-stressing force after losses, which diminishes because of pre-stress losses.

Key Concepts

Pre-Stressing in Service Load Condition

• Pre-stressing forces act on the reinforcement after accounting for losses.
• Concrete elements experience bending moments from:
  • Dead load (self-weight)
  • Superimposed dead load
  • Live load
• These loads are considered service loads or gravity loads.

Application of Load Balancing Method

• The method is primarily used for concrete elements in service load conditions but can also apply in initial pre-stressing stages.
• The goal is to achieve uniform stress distribution across the concrete element.
• The method requires identifying the balanced load that produces bending moments that offset the pre-stressing moments.

Goals of Load Balancing Method

1. Finding Balanced Load (W_b)

   • W_b is a load that balances the moment due to pre-stress eccentricity.
   • The eccentric moment from pre-stressing force (P × e) at mid-span must be matched by the moment from W_b.
   • W_b is typically a portion of total service loads.

2. Eccentricity Variation

   • Varying the eccentricity of the pre-stressing reinforcement helps achieve uniform stress.
   • Eccentricity must be adapted to the load profile (e.g., parabolic for uniformly distributed loads).

Bending Moment and Stress Analysis

Bending Moments in Simply Supported Beams

• For simply supported beams under uniform loading, bending moments vary from zero at supports to maximum at mid-span.
• The balance moment produced by W_b must equal the eccentric moment from pre-stressing.
• The formula for bending moment at mid-span of a simply supported beam is:
  • M_b = W_b imes L^2 / 8

Stress Distribution Analysis

• Flexural stresses are induced by:
  • Eccentricity of pre-stressing force
  • Balanced moment from W_b
• Total stress at mid-span is the sum of stresses from pre-stressing and balanced loads.
• Stress distribution aims for a uniform stress across sections of the concrete element.

Challenges in Load Balancing Method

• The bending moment from W_b varies along the beam, while the moment due to pre-stressing remains constant.
• Achieving uniform stress requires adapting the tendon profile to match the moment variations.

Tendon Profile and Load Balancing

• The tendon profile must counter the bending moment diagram of the applied loads.
• Different scenarios dictate tendon profiles:
  • Uniformly distributed loads produce a parabolic moment diagram.
  • Concentrated loads require heart-shaped tendon profiles.

Application of Unbalanced Load (W_ub)

• The unbalanced load is the remaining portion of the total service load after applying the balanced load.
• Additional unbalanced loads produce further bending moments and stress variations, requiring analysis of induced flexural stresses.

Summary of Formulas

• Balanced Load (W_b): Derived based on service loads and eccentric moments.
• Stress Calculations: Use the total moment to derive stress at any section under combined loading conditions.
• Horizontal Component of Pre-Stressing Force: Must be used in formulas for cases with varying eccentricities, especially for short-span beams.

Conclusion

• The load balancing method aims to create a balance between applied loads and pre-stressing forces to ensure uniform stress distribution in concrete elements throughout their length.
• Proper analysis and understanding of bending moments, combined with strategic tendon profiling, are essential for effective pre-stressing design.