Prestressed Concrete Design Fundamentals

Prestressed concrete is a revolutionary material that overcomes the fundamental weakness of conventional reinforced concrete: its low tensile strength. By introducing controlled internal stresses before service loads are applied, engineers can create longer, slimmer, and more durable structural members. Mastering its design is essential for building efficient bridges, parking structures, and industrial floors where deflection, cracking, and span length are critical constraints.

Core Principles of Prestressing

At its heart, prestressing is the intentional application of permanent compressive stress to the concrete region that will experience tension under load. This is achieved using high-strength steel tendons (strands, wires, or bars) that are tensioned and anchored against the concrete. The two primary methods are pre-tensioning, where tendons are tensioned before the concrete is cast, and post-tensioning, where ducts are cast into the concrete and tendons are tensioned after it has hardened. The key principle is that the initial compression from the prestressing force must be greater than the tensile stress induced by the service loads, effectively keeping the concrete in compression or within a minimal tensile state, thereby preventing cracks.

The force in the tendon is not a constant. The prestressing force at any time is denoted as $P$. The most important values are $P_i$, the initial force immediately after tensioning, and $P_e$, the effective force after all long-term losses have occurred. This force is applied with an eccentricity $e$ relative to the member's centroidal axis. This eccentricity is what creates the bending moment ($P\times e$) that counteracts the bending moments from external loads, making it a powerful design tool.

Stress Calculations at Transfer and Service

Design requires checking concrete stresses at two critical stages: transfer and service. At transfer, the full prestressing force $P_i$ acts on the concrete section, but the member only carries its own self-weight. You must ensure the concrete, which is still young, is not overstressed in compression or put into excessive tension at the top fibers. The basic stress calculation uses the familiar flexure formula, superimposing the axial and bending effects of prestress:

$$f=\frac{P}{A}\pm \frac{P\cdot e}{S}\pm \frac{M}{S}$$

Where $A$ is the cross-sectional area, $S$ is the section modulus, and $M$ is the moment from external loads (like self-weight at transfer or total service load later). You calculate this at the extreme top and bottom fibers for both the transfer stage (using $P_i$, the appropriate section properties, and self-weight moment) and the service stage (using $P_e$ and full service loads). Allowable stresses are specified by codes (like ACI 318) and differ for compression and tension at each stage.

The Load Balancing Concept

Load balancing is an insightful design approach pioneered by T.Y. Lin. It simplifies analysis by conceptualizing the prestressing force as directly counteracting a specific portion of the external load. When a tendon is draped in a parabolic profile, it exerts a uniform upward force on the concrete. By selecting the correct prestress force and parabola sag, this upward force can be made exactly equal to the downward service dead load. From a design perspective, the beam then behaves as if it is under axial compression only, with zero bending moment and deflection from the balanced load. This method is exceptionally useful for initial sizing of tendons and for estimating camber and deflection.

Flexural Strength of Prestressed Members

While serviceability (stress and deflection) governs the initial design, you must also check the ultimate flexural strength to ensure a sufficient safety margin against collapse. Unlike a reinforced concrete beam, which yields gradually, a prestressed beam fails either by crushing of the concrete or by fracture of the prestressing steel. The analysis involves strain compatibility and force equilibrium, as the high-strength steel does not exhibit a definite yield plateau. The design tensile strength of the prestressing steel is a fraction of its ultimate strength $f_{pu}$. The nominal moment capacity $M_n$ is calculated based on whether the stress block is within the flange of a T-section or not. The member must satisfy $\varphi M_n\ge M_u$, where $M_u$ is the factored ultimate moment from loads and $\varphi$ is the strength reduction factor.

Shear Design Considerations

Shear design in prestressed concrete is generally more favorable than in reinforced concrete. The prestressing force introduces a beneficial compressive stress that reduces the principal tensile stress, thereby increasing the shear force at which diagonal cracks form. The code-specified method for calculating the nominal shear strength $V_n$ is $V_n=V_c+V_s$, where $V_c$ is the concrete contribution and $V_s$ is the shear reinforcement contribution. For prestressed members, $V_c$ can be calculated using a more detailed method that accounts for the enhanced shear capacity due to prestress, the moment-shear interaction, and the effect of draped tendons. However, a minimum amount of shear reinforcement is almost always required to ensure ductile behavior.

Quantifying Prestress Losses

The initial prestressing force $P_i$ is not maintained. Prestress losses systematically reduce it to the effective force $P_e$. Accurate estimation of these losses is critical for predicting long-term camber, deflection, and stress state. Losses are categorized as immediate or time-dependent.

• Elastic Shortening: As the concrete compresses under the prestress force, the tendon shortens along with it, losing some tension. In post-tensioned members with multiple tendons, this loss is sequential and can be reduced.
• Creep: Sustained compressive stress in concrete causes it to deform progressively over time, further shortening the tendon.
• Shrinkage: As concrete cures and dries, it volumetrically contracts, leading to additional tendon shortening and force loss.
• Relaxation: The high-strength steel itself loses tension when held at a near-constant strain over a long period.

These losses are estimated using code-specified formulas and material models, and they typically total 15-25% of the initial prestressing force.

Tendon Profile Selection

The path of the tendon through the member—its tendon profile—is a key design variable. A straight profile is simple but only provides axial compression. A draped or harped profile introduces eccentricity that varies along the span, allowing the designer to match the internal prestressing moment to the external moment diagram. A parabolic profile is most common for uniformly loaded beams, as it generates a uniform upward load for balancing. The profile is selected to:

1. Maximize eccentricity at midspan (where positive moments are highest) while keeping the tendon within the concrete kern for the transfer stage.
2. Raise the tendon toward the neutral axis at supports to avoid creating undesirable tensile stresses at the top.
3. Accommodate anchorages and satisfy concrete cover requirements.

Common Pitfalls

Underestimating Long-Term Losses and Deflections: Using only the initial prestress force $P_i$ for serviceability checks is a major error. You must use the effective force $P_e$ after all losses to calculate long-term deflections and final service stresses. Ignoring creep and shrinkage can lead to excessive camber or, worse, insufficient precompression and cracking under service loads.

Neglecting Stress Limits at Transfer: Focusing solely on service conditions can be dangerous. The concrete at release has a much lower strength. A tendon profile with too much eccentricity at the ends can cause tensile cracking at the top of the beam the moment it is released from the formwork. Always check stresses at the transfer stage with the appropriate allowable limits.

Improper Shear Design Assumptions: While prestress improves shear capacity, it does not eliminate the need for careful analysis. Simply using the higher $V_c$ equation without verifying its prerequisites (like the moment-shear interaction) or omitting minimum shear reinforcement can lead to a non-ductile, brittle shear failure.

Ignoring Secondary Effects in Continuous Spans: In post-tensioned continuous beams, the restraint provided by supports induces secondary moments as the member deflects under prestress. These moments are statically indeterminate and must be calculated through a structural analysis. They add to or subtract from the primary prestressing moments and must be included in both service and strength design calculations.

Summary

• Prestressing applies a controlled compressive stress to concrete to counteract tensile stresses from loads, enabling crack-free, long-span members.
• Design requires a dual check of concrete stresses at the transfer stage (using $P_i$ and self-weight) and the service stage (using $P_e$ and full service loads), governed by code-specified allowable stresses.
• The load balancing concept uses a draped tendon profile to create upward forces that directly cancel out selected downward loads, simplifying serviceability analysis.
• Ultimate flexural and shear strength must be verified using factored loads, accounting for the unique stress-strain behavior of high-strength prestressing steel.
• Accurate estimation of prestress losses—elastic shortening, creep, shrinkage, and relaxation—is essential for predicting the long-term effective force $P_e$ and structural behavior.
• The tendon profile is strategically selected to optimize the counteracting moment, satisfy stress limits at all stages, and accommodate practical detailing requirements.