Introduction to Prestressed Concrete
Learning Objectives
- Explain how prestress modifies, but does not erase, tensile and shear demands.
- Distinguish fully prestressed, partially prestressed, uncracked, and cracked service behavior.
- Differentiate pre-tensioned, bonded post-tensioned, and unbonded post-tensioned systems.
- Calculate elastic top- and bottom-fiber stresses using one declared sign convention.
- Separate jacking, transfer, effective service, and ultimate-strength design stages.
- Classify immediate and time-dependent tendon-force losses by prestressing system.
- Interpret eccentricity, equivalent tendon loads, secondary effects, and concordant profiles.
Prestressed Concrete
Concrete in which deliberate internal forces are introduced, usually by tensioned high-strength steel tendons, to improve stress, deflection, cracking, and strength performance under specified load stages.
What Prestressing Does
Prestress usually places the concrete in compression before the full external load is applied. Eccentric prestress also creates a moment that can oppose gravity-load bending. The objective is a selected stress and deformation state—not a permanent guarantee that every fiber remains compressed.
A member may remain uncracked for a specified service combination, may permit calculated tensile stress without visible cracking, or may be intentionally designed for controlled cracking. Construction, transfer, sustained, frequent, and full service combinations can produce different states in the same member.
Fully Prestressed Member
A member proportioned so the specified service-load stress or cracking criteria for its selected code classification are satisfied primarily by prestress. The term does not mean that tension is impossible at every stage, load combination, local region, or overload.
Partially Prestressed Member
A member designed with a combination of prestressing tendons and conventional nonprestressed reinforcement, commonly permitting controlled tensile stress or cracking under designated service combinations while satisfying serviceability and ultimate-strength requirements.
Cracking and Shear Are Not Eliminated
Prestress can delay flexural cracking, reduce crack width, and improve shear behavior by changing the longitudinal stress state and principal tensile stress. It does not eliminate diagonal tension, web-shear cracking, flexure-shear cracking, anchorage-zone forces, or the need for applicable shear analysis and reinforcement.
Prestressing and Conventional Steel
Prestressing tendons are intentionally tensioned and locked off or transferred, so their force decreases through identifiable prestress-loss mechanisms. Conventional reinforcing bars are not normally jacked; they contribute crack control, ductility, local force transfer, and ultimate strength. Bonded conventional bars can gain or lose stress through strain compatibility, creep, shrinkage, cracking, and loading, but they do not simply undergo the tendon jacking-to-effective loss sequence.
Material Roles
- Prestressing tendons: High-strength wire, strand, or bar provides large elastic prestrain so useful force remains after losses. A common seven-wire strand grade has near , but the specified product and code limits govern.
- Concrete: Relatively high early and specified compressive strengths resist transfer stresses, bearing and bursting forces, and long-term compression. Transfer checks use the concrete strength at transfer , not the later specified strength .
- Conventional reinforcement: Bonded bars provide minimum reinforcement where required, crack distribution, anchorage-zone reinforcement, shear reinforcement, and added ultimate capacity; they are distinct from prestressing tendons.
Pre-tensioning
Tendons are stressed against an external casting bed before concrete placement. After the concrete reaches the specified transfer strength, the tendons are released and force is transferred to the concrete by bond over a transfer length.
Post-tensioning
Tendons are stressed against hardened concrete and anchored mechanically. A system may be bonded after grouting a duct or remain unbonded in a protective sheath; this distinction materially affects force transfer, durability, redistribution, and failure behavior.
System-Specific Immediate Effects
- Pre-tensioning: Transfer through bond causes immediate elastic shortening of the member and tendon. Casting-bed anchorage movement may affect jacking force, but duct friction is not the normal mechanism.
- Post-tensioning: Curvature and wobble friction can make tendon force vary along the duct. Anchorage seating reduces elongation near a stressed end. Elastic-shortening effects depend on the stressing sequence and whether other tendons are already anchored.
- Both systems: Jack calibration, measured elongation, concrete strength, tendon geometry, and the specified stressing sequence require field quality control.
Jacking Force ()
The tendon force applied at the jack before deductions for friction, anchorage seating, transfer, and subsequent time-dependent effects as applicable to the system.
Transfer Force ()
The prestressing force acting on the concrete at the transfer stage after the immediate effects applicable up to that location and time. It is generally less than the jacking force and may vary along a post-tensioned tendon.
Effective Prestress ()
The tendon force used for the selected long-term service check after applicable immediate and time-dependent losses. It is a stage- and location-specific design value, not a universal constant for the whole tendon.
Four Design Stages
Jacking checks tendon stress, elongation, equipment, and local anchorage actions. Transfer checks the younger concrete under and the loads present at release or lock-off. Service checks stresses, cracking classification, deflection, and durability using with the specified service combinations. Ultimate strength checks factored resistance using strength-design equilibrium and strain compatibility; the elastic service-stress equations are not ultimate-capacity equations.
Declared Stress Sign Convention
The formulas below take concrete tension as positive and compression as negative. Positive eccentricity places the tendon below the centroid. Positive external moment is sagging: it compresses the top fiber and tensions the bottom fiber.
Transfer-Stage Elastic Fiber Stresses
Gross, uncracked section with transfer force and the external moment present at transfer.
Variables
| Symbol | Description | Unit |
|---|---|---|
| prestress force acting at the section at transfer | N | |
| gross concrete area | ||
| tendon eccentricity, positive below the centroid | mm | |
| sagging external moment present at transfer | ||
| gross section modulus to the top fiber | ||
| gross section modulus to the bottom fiber |
Effective Service-Stage Elastic Fiber Stresses
Gross, uncracked section with effective prestress and the selected service moment.
Variables
| Symbol | Description | Unit |
|---|---|---|
| effective prestress at the section and service time considered | N | |
| sagging moment for the specified service-load combination | ||
| gross concrete area for an uncracked elastic check | ||
| tendon eccentricity, positive below the centroid | mm | |
| gross section modulus to the top fiber | ||
| gross section modulus to the bottom fiber |
Interpreting Allowable Stresses
Transfer tensile and compressive limits are checked against and the transfer condition. Service tensile limits depend on the selected prestressed-member classification, exposure, load combination, and bonded reinforcement; service compression limits use and the applicable combination. A calculated tensile stress is therefore a design state to compare with the proper limit—not automatic proof that the calculation is invalid. If the uncracked assumption is no longer valid, cracked-section service analysis and crack-control provisions are required.
Magnel Diagram
A Magnel diagram rearranges the transfer and service stress inequalities into straight-line bounds on prestress force and eccentricity, commonly using coordinates such as and . Their intersection is only a feasible region for the stated section properties, losses, load stages, sign convention, and allowable stresses; it does not replace tendon-geometry, ultimate-strength, shear, anchorage, or deflection checks.
Prestress Loss Framework
Prestress loss is the reduction from a defined earlier tendon force or stress to a later value at a specified location. Loss components interact, so a percentage shortcut is suitable only when the problem explicitly supplies a validated total loss.
Immediate and Time-Dependent Losses
- Friction and wobble: Applicable to post-tensioned tendons where contact and unintended duct deviation reduce force with distance, commonly idealized as .
- Anchorage seating: Wedge or anchorage movement reduces tendon elongation after jack release; its influence depends on tendon length, friction, stressing ends, and hardware.
- Elastic shortening: Concrete shortens as prestress is transferred. The full compatible shortening is important in pre-tensioning; post-tensioned loss depends on the sequence of stressing and anchoring multiple tendons.
- Concrete creep: Sustained concrete compression produces additional compatible shortening over time and reduces bonded tendon stress.
- Concrete shrinkage: Moisture and autogenous volume change shorten the member and can reduce tendon strain.
- Steel relaxation: Tendon stress decreases with time under approximately constant strain; magnitude depends on tendon type, initial stress, time, and temperature.
Do Not Double Count Losses
The force denoted should already reflect immediate effects included up to transfer, and should reflect the loss model used for the service time. Subtracting the same friction, seating, or elastic-shortening component again produces an artificial loss.
Equivalent Tendon Load
The transverse action exerted by a curved tendon on the concrete. For constant tendon force and small slopes, the distributed load is related to tendon curvature; concentrated direction changes and anchorages also create forces that must be included.
Equivalent Load for a Symmetric Parabolic Profile
Ideal uniform upward load for a simply supported span with zero end eccentricity and midspan drape h.
Variables
| Symbol | Description | Unit |
|---|---|---|
| equivalent uniform upward load | N/mm | |
| effective tendon force assumed constant over the span | N | |
| vertical drape from the end chord to the parabola at midspan | mm | |
| span length | mm |
Load Balancing
Load balancing superposes the tendon's equivalent loads with selected external gravity loads. If the ideal parabolic load exactly balances a chosen uniform load on a simply supported span, those two transverse-load systems produce zero net bending in that idealized model. Direct prestress , unbalanced loads, anchorage-zone actions, losses, secondary effects in indeterminate structures, and other load combinations remain.
Secondary Prestress Effects
Reactions and moments induced when an indeterminate structure restrains the deformation associated with prestress. Total prestress moment equals the primary section moment from tendon eccentricity plus the secondary moment from restraint.
Concordant Tendon Profile
A tendon profile in a particular indeterminate structure whose prestressing action produces zero secondary reactions and zero secondary moments. Concordance is an equilibrium-and-compatibility property of the profile and structure; an arbitrary drape or a profile that merely resembles a moment diagram is not automatically concordant.
Linear Transformation of Tendon Profiles
A valid linear transformation changes a tendon profile by straight-line offsets between supports without changing its curvature within each span. Under the ideal constant-force equivalent-load model, the transverse tendon loading is unchanged; primary and secondary moments may change individually while their total prestress moment remains unchanged. This theorem does not make every transformed or arbitrary profile concordant.
Ultimate Strength Is a Separate Check
At factored strength, concrete compression, prestressing-steel stress, conventional reinforcement, tendon bond condition, and strain compatibility govern nominal capacity. Transfer or service allowable-stress compliance does not prove adequate ultimate strength, and ultimate strength does not prove acceptable service cracking or deflection.
- Prestress changes the stress state and often delays cracking; it does not guarantee permanent compression or eliminate flexural and shear cracking.
- Fully and partially prestressed members are distinguished by selected service criteria, reinforcement, and permitted tensile or cracked behavior—not by slogans about “no tension.”
- , , and refer to different stages; transfer, service, and ultimate checks use different material strengths, loads, and analysis models.
- Immediate losses depend strongly on whether the member is pre-tensioned or post-tensioned; creep, shrinkage, and tendon relaxation develop with time.
- Conventional reinforcing bars are not prestressing tendons and do not simply inherit the tendon loss sequence.
- Positive tendon eccentricity below the centroid compresses the bottom fiber, while positive sagging external moment tensions it under the declared sign convention.
- Equivalent-load and concordant-profile methods require stated geometry, support, force, and small-slope assumptions; arbitrary tendon shapes are not automatically concordant.