Introduction to Prestressed Concrete

Introduction to Prestressed Concrete

Unlike conventional reinforced concrete where steel passively waits for the concrete to crack before it starts carrying significant tension, prestressed concrete actively introduces internal compressive stresses into the member before any external service loads are applied. This initial compression is designed to counteract the tensile stresses that will eventually be caused by dead and live loads, effectively keeping the entire concrete cross-section in compression and preventing it from cracking.

Advantages and Disadvantages

Why Prestress?

Advantages: The entire concrete section remains uncracked under service loads, utilizing the full gross moment of inertia ( $I_g$ ) rather than a drastically reduced cracked moment of inertia ( $I_{cr}$ ). This results in much smaller deflections, allowing for significantly longer spans, shallower beam depths (architectural freedom), and superior watertightness (crucial for tanks and bridges).
Disadvantages: Requires highly specialized materials (high-strength steel and high-strength concrete), specialized equipment (jacks, anchorages), rigorous quality control, and skilled labor. It is generally more expensive per cubic meter than conventional concrete, though economical for long spans.

Materials for Prestressing

Standard materials used in conventional reinforced concrete (

f'_c \approx 28

MPa,

f_y \approx 420

MPa) cannot be used effectively for prestressing due to the massive loss of prestress force over time.

Specialized Material Requirements

High-Strength Steel ( $f_{pu} \approx 1860 \text{ MPa}$ ): Prestressing "tendons" are made of ultra-high-strength wires, strands (usually 7-wire), or solid alloy bars. The extremely high initial tension ( $P_i$ ) is necessary because a large portion of it will inevitably be lost due to concrete shortening and creep. If standard Grade 420 steel were used, almost all the tension would be lost, rendering the prestressing useless.
High-Strength Concrete ( $f'_c \geq 35 \text{ MPa}$ ): Stronger concrete is required to withstand the immense concentrated compressive forces at the anchorages without crushing, and to provide a higher modulus of elasticity ( $E_c$ ) to minimize elastic shortening and long-term creep deformations.

Methods of Prestressing

Prestressing Methods

Pre-tensioning: The tendons are stretched tightly between massive external abutments in a precast plant. The concrete is poured around the tensioned tendons. Once the concrete reaches sufficient strength ( $f'_{ci}$ ), the tendons are cut from the abutments. The force is transferred to the concrete entirely through bond (friction and adhesion) along the length of the member. This method is highly automated and economical for mass-producing standardized bridge girders, hollow-core slabs, and piles.
Post-tensioning: Hollow ducts (plastic or metal) are cast into the concrete member on site. After the concrete cures and gains sufficient strength, the tendons are threaded through the ducts, tensioned using hydraulic jacks pushing directly against the hardened concrete ends (anchorages), and then permanently locked in place with wedges. The ducts are usually filled with grout later to protect the steel from corrosion and provide some bond. Used heavily for cast-in-place long-span floor slabs, bridge decks, and large girders where transport is impossible.

Stages of Loading and Stress Analysis

Because the internal forces in a prestressed member change drastically over time, stress analysis (using

f = \frac{P}{A} \pm \frac{Mc}{I}

) must be checked at two critical stages to ensure compressive and tensile stresses do not exceed code limits.

Critical Loading Stages

Transfer (Initial Stage): Immediately after the jacks are released and the initial prestressing force ( $P_i$ ) is transferred to the concrete. The concrete strength is only $f'_{ci}$ (often $0.80 f'_c$ ). The member is subjected to maximum compression ( $P_i$ ) but only minimum opposing external load (usually just its own dead weight $M_D$ ). The stress equations are: $f_{top} = -\frac{P_i}{A} + \frac{P_i e}{S_{top}} - \frac{M_D}{S_{top}}$ and $f_{bot} = -\frac{P_i}{A} - \frac{P_i e}{S_{bot}} + \frac{M_D}{S_{bot}}$ . This is often the most critical stage for concrete crushing or splitting near the ends.
Service (Final Stage): Occurs years later. The concrete has reached its full strength ( $f'_c$ ). The prestress force has diminished significantly to its effective, long-term value ( $P_e$ ). The member is now subjected to full service loads (dead + full live load $M_T$ ). The stress equations are: $f_{top} = -\frac{P_e}{A} + \frac{P_e e}{S_{top}} - \frac{M_T}{S_{top}}$ and $f_{bot} = -\frac{P_e}{A} - \frac{P_e e}{S_{bot}} + \frac{M_T}{S_{bot}}$ .

Magnel Diagrams

A Magnel Diagram is a graphical design tool used to determine the safe limits of the prestressing force (

P_i

P_e

) and the eccentricity (

e

). By plotting the four stress limit equations (Transfer Top/Bottom, Service Top/Bottom) on a graph of

1/P_i

versus

e

, the intersection of the lines creates a "feasible region" (a polygon). Any combination of

P_i

and

e

chosen inside this region guarantees that the concrete will neither crush in compression nor crack in tension at any stage of its life.

Partial Prestressing

In Full Prestressing, the structure is designed to have absolutely zero tensile stress in the concrete under full service loads. While ideal for watertightness, it requires massive amounts of expensive prestressing steel and can lead to excessive upward deflection (camber) when live loads are absent.

In Partial Prestressing, the prestressing force is intentionally reduced, allowing some controlled tensile stresses (and fine, acceptable cracks) to develop under full service loads. The required ultimate strength (

M_n

) is made up by adding conventional non-prestressed reinforcement (

A_s

) in the tension zone. This approach is more economical, reduces camber problems, and increases ductility.

Prestress Losses

The initial jacking force (

P_j

) immediately drops to

P_i

upon transfer, and then gradually decreases over months and years to a final effective force (

P_e

). Accurately estimating these losses is critical; underestimating them leads to cracking, while overestimating them leads to excessive camber and cost. Typical total losses range from 15% to 25%.

Categorization of Losses

Immediate Losses - Elastic Shortening: As the massive compressive force is applied, the concrete member physically compresses. Because the steel is bonded to the concrete, the steel shortens with it, losing tension.
Immediate Losses - Friction (Post-tensioning): As the tendon is pulled through a curved duct, friction between the steel and duct wall reduces the force. This is modeled using the wobble friction coefficient ( $K$ ) and curvature friction coefficient ( $\mu$ ): $P_x = P_s e^{-(\mu \alpha + Kx)}$ .
Immediate Losses - Anchorage Seating: When the jack releases the tendon, the wedges anchoring the steel inevitably slip slightly into the anchor block before gripping fully, losing a few millimeters of stretch.
Time-Dependent Losses - Creep: Concrete continues to deform slowly under sustained compressive stress, further shortening the member and relaxing the steel.
Time-Dependent Losses - Shrinkage: As water evaporates from the hardened concrete, its volume shrinks, shortening the member.
Time-Dependent Losses - Relaxation: Highly stressed steel tendons slowly lose tension over time even if held at a constant length. (Low-relaxation strands are almost universally used today to minimize this).

Continuous Prestressed Beams

When continuous beams (statically indeterminate) are post-tensioned, the application of the prestress force causes the beam to camber (deflect upwards) over the interior supports.

Secondary Moments and Concordant Profiles

Because the beam is physically held down by the interior supports, it cannot camber freely.
The supports push back down on the beam, creating secondary moments (also called hyperstatic moments) throughout the continuous structure.
The total moment in a continuous prestressed beam is the sum of the primary moment ( $P \times e$ ) and the secondary moment.
Concordant Tendon Profile: If a tendon is draped precisely such that its profile follows the shape of the bending moment diagram created by the continuous loading, it will produce zero secondary moments. The beam will not attempt to lift off its supports.
Linear Transformation: A remarkable property of continuous prestressed beams is that any tendon profile can be moved vertically over the interior supports without changing the actual stress distribution in the concrete, provided the intrinsic shape within the spans remains identical.

Load Balancing Method

A powerful and intuitive way to analyze and design prestressed concrete is the Load Balancing Method (developed by T.Y. Lin). Rather than analyzing complex stress distributions at every section, this method treats the curved prestressing tendon as an external load applied to the concrete member.

Equivalent Upward Load ( $w_p$ )

If a tendon is draped in a parabolic profile with a maximum sag (drape) of $h$ over a simply supported span $L$ , and tensioned with a force $P$ , it exerts a uniform, upward "equivalent load" ( $w_p$ ) on the concrete along its entire length. $w_p = \frac{8 P h}{L^2}$
The designer simply chooses a tendon profile and prestress force $P$ such that this upward $w_p$ exactly balances a specific portion of the downward gravity loads (typically the full dead load, $w_D$ ).
If $w_p = w_D$ , the beam will theoretically have zero deflection under its own weight, and the concrete will experience only pure, uniform axial compression ( $P/A$ ) along its entire length, with zero bending moments ( $M = 0$ ).
The concrete section is then designed conventionally to resist only the remaining "unbalanced" load (the Live Load, $w_{net} = w_L$ ).

Key Takeaways

Prestressed concrete introduces active internal compression to perfectly counteract the anticipated tension from service loads, keeping the full cross-section uncracked and extremely stiff.
It absolutely requires ultra-high-strength steel ( $f_{pu} \approx 1860 \text{ MPa}$ ) to ensure that a significant effective prestressing force ( $P_e$ ) remains after inevitable prestress losses (elastic shortening, creep, shrinkage, relaxation).
Pre-tensioning (done in plants) relies entirely on bond for stress transfer, while Post-tensioning (done on-site) relies on mechanical end anchorages.
Stress checks are mandatory at two distinct stages: Transfer ( $P_i$ , low concrete strength, dead load only) and Service ( $P_e$ , full concrete strength, full live loads).
The Load Balancing Method simplifies analysis by treating a draped tendon as an equivalent uniform upward load ( $w_p = 8Ph/L^2$ ) that cancels out the downward dead load, leaving the member in pure axial compression.