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?

Pros and Cons

Advantages: The entire concrete section remains uncracked, meaning the full moment of inertia ( $I_g$ ) is available, making the member much stiffer and reducing deflections. It eliminates diagonal tension cracks (shear cracks), allows for longer spans, thinner sections, and provides excellent corrosion protection since the concrete is not cracked.
Disadvantages: Requires expensive, high-strength materials (both concrete and steel). Involves more complex design, specialized equipment, tight quality control during construction, and careful attention to prestress losses over time.

Materials Used

Material Requirements

Ultra-High-Strength Steel ( $f_{pu} \approx 1860 \text{ MPa}$ ): Standard Grade 60 rebar ( $420 \text{ MPa}$ ) cannot be used for prestressing. Over time, concrete shrinks and creeps, causing the steel to shorten and lose tension. If standard rebar were used, these losses would wipe out 100% of the prestress force. High-strength steel is stretched so far initially that even after all losses, a massive effective force ( $P_e$ ) remains.
High-Strength Concrete ( $f'_c \ge 35 \text{ MPa}$ ): 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.

Assumptions and Limitations

Design Assumptions

Plane Sections Remain Plane: Similar to ordinary reinforced concrete, strains vary linearly with the distance from the neutral axis under flexure.
Perfect Bond: In pre-tensioned members, it is assumed that there is no slip between the prestressing tendons and the surrounding concrete after transfer.
Elastic Behavior: At the transfer stage and under normal service loads, the materials (concrete and steel) are typically modeled as behaving elastically.

Important Limitations

While highly advantageous, prestressed concrete has key limitations:

Fire Resistance: The ultra-high-strength steel used in prestressing is extremely sensitive to elevated temperatures, losing strength rapidly during a fire compared to standard mild steel.
Corrosion of Unbonded Tendons: If moisture penetrates unbonded post-tensioning ducts, catastrophic tendon failure can occur without visible prior warning (unlike conventional RC which usually shows rust spalling).

Methods of Prestressing

Pre-tensioning

A method where tendons are stretched tightly between massive external abutments in a precast plant. The concrete is poured around the tensioned tendons. Once the concrete cures, the tendons are cut, transferring the force to the concrete entirely through bond friction along the member's length.

Post-tensioning

A method where hollow ducts are cast into the concrete member on site. After the concrete cures, tendons are threaded through, tensioned using hydraulic jacks pushing against the hardened concrete ends, and permanently locked in place with wedges.

Stages of Loading and Stress Analysis

Because the internal forces in a prestressed member change drastically over time, stress analysis must be checked at two critical stages to ensure compressive and tensile stresses do not exceed code limits.

Caution

A major source of error in prestressed concrete problems is the sign convention for eccentricity ( $e$ ). Standard practice defines positive $e$ (tendon below the neutral axis) as causing a negative (upward) bending moment $P_i e$ , which creates tension at the top fiber and compression at the bottom fiber. Always draw the stress blocks or double-check signs!

Stress at Transfer (Initial Stage)

Stresses immediately after jacks are released. Concrete strength is low ( $f'_{ci}$ ), compression force is maximum ( $P_{i}$ ), and opposing external load is minimum (typically just dead weight $M_{D}$ ).

\begin{aligned} f_{top} &= -\frac{P_i}{A} + \frac{P_i e}{S_{top}} - \frac{M_D}{S_{top}} \\ f_{bot} &= -\frac{P_i}{A} - \frac{P_i e}{S_{bot}} + \frac{M_D}{S_{bot}} \end{aligned}

Variables

Symbol	Description	Unit
$P_{i}$	Initial prestress force just after transfer	N
$e$	Eccentricity of the prestressing force from the neutral axis	mm
$M_{D}$	Moment due to dead load only	$N\cdot mm$
$S$	Section modulus (top or bottom)	$m m^{3}$

Stress at Service (Final Stage)

Stresses years later. Concrete is at full strength ( $f_{c}^{'}$ ), prestress force is reduced to its effective long-term value ( $P_{e}$ ), and the member faces full service loads ( $M_{T}$ ).

\begin{aligned} f_{top} &= -\frac{P_e}{A} + \frac{P_e e}{S_{top}} - \frac{M_T}{S_{top}} \\ f_{bot} &= -\frac{P_e}{A} - \frac{P_e e}{S_{bot}} + \frac{M_T}{S_{bot}} \end{aligned}

Variables

Symbol	Description	Unit
$P_{e}$	Effective prestress force after all losses	N
$M_{T}$	Total service moment (dead + live load)	$N\cdot mm$

Magnel Diagrams

A Magnel Diagram is a graphical design tool used to determine the safe limits of the prestressing force ( $P_i$ or $P_e$ ) 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).

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

Effective Prestress ( $P_{e}$ )

The long-term force remaining in the tendons after all immediate and time-dependent losses have occurred. Accurately estimating these losses is critical for ensuring long-term serviceability.

Caution

A common mistake is using the initial jacking force ( $P_j$ ) or transfer force ( $P_i$ ) to check long-term deflections and service stresses. Always ensure you have calculated and applied the Effective Prestress ( $P_e = P_i - \text{Losses}$ ) for any long-term checks.

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 ( $w_{p}$ )

The uniform upward load exerted by a parabolically draped tendon on the concrete.

w_p = \frac{8 P h}{L^2}

Variables

Symbol	Description	Unit
$w_{p}$	Equivalent uniform upward load	N/mm
$P$	Prestressing force	N
$h$	Maximum sag (drape) of the parabolic tendon	mm
$L$	Span length	mm

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.

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Advantages and Disadvantages

Pros and Cons

Material Requirements

Assumptions and Limitations

Important Limitations

Methods of Prestressing

Pre-tensioning

Post-tensioning

Stages of Loading and Stress Analysis

Caution

Stress at Transfer (Initial Stage)

Stress at Service (Final Stage)

Magnel Diagrams

Partial Prestressing

Prestress Losses

Effective Prestress (PeP_ePe​)

Caution

Categorization of Losses

Continuous Prestressed Beams

Concordant Tendon Profile

Linear Transformation

Load Balancing Method

Equivalent Upward Load (wpw_pwp​)

Effective Prestress ( $P_{e}$ )

Equivalent Upward Load ( $w_{p}$ )