Introduction to Prestressed Concrete

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 $f_{pu}$ near $1860\text{ MPa}$ , 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 $f'_{ci}$ , not the later specified strength $f'_c$ .
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.

$P_j$

The tendon force applied at the jack before deductions for friction, anchorage seating, transfer, and subsequent time-dependent effects as applicable to the system.

$P_i$

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.

$P_e$

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.

Prestressed member stage analysis

Keep force, concrete strength, external moments and stress limits specific to each stage.

$Four Design Stages$

Jacking checks tendon stress, elongation, equipment, and local anchorage actions. Transfer checks the younger concrete under $P_i$ and the loads present at release or lock-off. Service checks stresses, cracking classification, deflection, and durability using $P_e$ 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 $e$ places the tendon below the centroid. Positive external moment $M$ is sagging: it compresses the top fiber and tensions the bottom fiber.

Elastic fiber stress and feasibility

Combine direct, eccentric and external stresses at one section and reject an invalid gross-section assumption.

Transfer-Stage Elastic Fiber Stresses

Gross, uncracked section with transfer force and the external moment present at transfer.

\begin{aligned} f_{\mathrm{top},i} &= -\frac{P_i}{A} +\frac{P_i e}{S_{\mathrm{top}}} -\frac{M_i}{S_{\mathrm{top}}} \\ f_{\mathrm{bot},i} &= -\frac{P_i}{A} -\frac{P_i e}{S_{\mathrm{bot}}} +\frac{M_i}{S_{\mathrm{bot}}} \end{aligned}

Variables

Symbol	Description	Unit
$P_i$	prestress force acting at the section at transfer	N
$A$	gross concrete area	$mm^2$
$e$	tendon eccentricity, positive below the centroid	mm
$M_i$	sagging external moment present at transfer	$N\cdot mm$
$S_{\mathrm{top}}$	gross section modulus to the top fiber	$mm^3$
$S_{\mathrm{bot}}$	gross section modulus to the bottom fiber	$mm^3$

Effective Service-Stage Elastic Fiber Stresses

Gross, uncracked section with effective prestress and the selected service moment.

\begin{aligned} f_{\mathrm{top},e} &= -\frac{P_e}{A} +\frac{P_e e}{S_{\mathrm{top}}} -\frac{M_{\mathrm{ser}}}{S_{\mathrm{top}}} \\ f_{\mathrm{bot},e} &= -\frac{P_e}{A} -\frac{P_e e}{S_{\mathrm{bot}}} +\frac{M_{\mathrm{ser}}}{S_{\mathrm{bot}}} \end{aligned}

Variables

Symbol	Description	Unit
$P_e$	effective prestress at the section and service time considered	N
$M_{\mathrm{ser}}$	sagging moment for the specified service-load combination	$N\cdot mm$
$A$	gross concrete area for an uncracked elastic check	$mm^2$
$e$	tendon eccentricity, positive below the centroid	mm
$S_{\mathrm{top}}$	gross section modulus to the top fiber	$mm^3$
$S_{\mathrm{bot}}$	gross section modulus to the bottom fiber	$mm^3$

$Interpreting Allowable Stresses$

Transfer tensile and compressive limits are checked against $f'_{ci}$ and the transfer condition. Service tensile limits depend on the selected prestressed-member classification, exposure, load combination, and bonded reinforcement; service compression limits use $f'_c$ 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.

Rectangular Section Kern

Eccentricity range for no tension under prestress alone in a gross elastic rectangular section with zero external moment.

k=\frac{S}{A}=\frac{d}{6},\qquad |e|\le k

Variables

Symbol	Description	Unit
$k$	kern distance from section centroid	$\text{mm}$
$S$	elastic section modulus at the relevant extreme fiber	$\text{mm}^3$
$A$	gross uncracked area	$\text{mm}^2$
$d$	rectangular depth	$\text{mm}$

$Kern Is Not a General No-Tension Guarantee$

The $d/6$ kern criterion applies to the prestress resultant alone in the stated rectangle. Transfer and service moments can still produce tensile stress. A tendon outside the kern may also be feasible at a specified load stage when its external moment offsets that fiber tension.

$Magnel Feasible Region and Stress Inequalities$

At a chosen section, impose separate top- and bottom-fiber tensile and compressive allowable stresses at transfer and at service. Substitute the corresponding $P_i$ , $P_e$ , $M_i$ , and $M_{\mathrm{ser}}$ into the signed elastic stress equations. If a loss ratio is specified, one can plot the simultaneous bounds in $(1/P,e)$ coordinates to find a preliminary feasible region. The simulation samples $(P_j,e)$ instead, shows only pairs meeting user-supplied illustrative limits at midspan, and never asserts that the selected limits are code values. Check the rest of the span and tendon cover separately.

$Magnel Diagram$

A Magnel diagram rearranges the transfer and service stress inequalities into straight-line bounds on prestress force and eccentricity, commonly using coordinates such as $1/P$ and $e$ . 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 sequence

Track applicable immediate effects to transfer and time-dependent effects to service without subtracting any twice.

$Prestress Loss Framework$

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 $P(x)=P_j\exp[-(\mu\alpha+kx)]$ .
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.

$Loss Bookkeeping by Force Stage and Location$

STEP-BY-STEP

Record tendon system, stressing end, tendon path, jacking force, and section location.
For post-tensioning, evaluate friction and wobble along the duct and local anchorage seating; for pre-tensioning, omit those duct effects and establish the bed-to-transfer force.
Apply elastic-shortening loss using the actual transfer or sequential stressing condition and record the resulting $P_i$ at the section.
Model concrete creep, shrinkage, and tendon relaxation over the specified service interval, including their interactions, to obtain $P_e$ at that section.
Reconcile calculated force with measured elongation and anchorage records, and avoid subtracting an effect already embedded in the supplied $P_i$ or $P_e$ .

Ideal Post-Tension Duct Friction

Force decay along a tendon stressed from one end under a specified curvature and wobble idealization.

P(x)=P_j\exp[-(\mu\alpha(x)+kx)]

Variables

Symbol	Description	Unit
$P(x)$	tendon force at distance x from the stressing end	$\text{kN}$
$P_j$	force at the jack before duct effects	$\text{kN}$
$\mu$	curvature friction coefficient	-
$\alpha(x)$	cumulative absolute tendon angular change	$\text{rad}$
$k$	wobble coefficient per unit tendon length	$\text{m}^{-1}$

$Do Not Double Count Losses$

The force denoted $P_i$ should already reflect immediate effects included up to transfer, and $P_e$ 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.

w_p=\frac{8P_eh}{L^2}

Variables

Symbol	Description	Unit
$w_p$	equivalent uniform upward load	N/mm
$P_e$	effective tendon force assumed constant over the span	N
$h$	vertical drape from the end chord to the parabola at midspan	mm
$L$	span length	mm

Symmetric Parabolic Tendon Geometry

Simply supported profile with zero end eccentricity and positive midspan eccentricity below the centroid.

e(x)=4e_m\frac{x}{L}\left(1-\frac{x}{L}\right)

Variables

Symbol	Description	Unit
$e(x)$	signed tendon eccentricity at section x	$\text{mm}$
$e_m$	signed midspan eccentricity	$\text{mm}$
$x$	distance from left support	$\text{m}$
$L$	simply supported span	$\text{m}$

$Tendon Profile, Support Restraint, and Equivalent Actions$

A straight constant-eccentricity tendon has zero distributed curvature load within the span but still develops end forces and moments. A symmetric parabolic tendon with zero end eccentricity produces an ideal uniform transverse load when the tendon force is constant and slopes are small. Anchorage direction changes and local bursting are additional actions. In a continuous beam, imposed support compatibility can produce secondary reactions and moments, so the simple span equivalent load and primary $Pe$ stress field cannot by themselves represent the member.

Ideal Net Simply Supported Elastic Deflection

Introductory estimate for equal-shape uniform gravity and parabolic-tendon equivalent loads on a gross uncracked section.

\Delta_{\mathrm{mid}}=\frac{5(w_g-w_p)L^4}{384E_cI_g}

Variables

Symbol	Description	Unit
$\Delta_{\mathrm{mid}}$	midspan deflection, positive downward	$\text{mm}$
$w_g$	uniform downward load	$\text{N}/\text{mm}$
$w_p$	uniform upward tendon equivalent load	$\text{N}/\text{mm}$
$E_c$	concrete elastic modulus for the chosen stage	$\text{MPa}$
$I_g$	gross uncracked moment of inertia	$\text{mm}^4$

$Camber and Long-Term Deflection Need More Than Load Balance$

The ideal expression does not include transfer timing, creep, shrinkage, cracking, changing modulus, changing tendon force, continuity, or secondary moments. Zero ideal instantaneous deflection under two equal uniform transverse load systems does not mean zero long-term movement.

$Load Balancing$

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 $P_e/A$ , 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.

Preliminary prestress design and checks

Use stress feasibility to select a trial tendon, then test stage and structural limits independently.

$Ultimate Strength Is a Separate Check$

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.

$Explore Prestress Stages and Feasibility$

Use the interactive stress simulator below to vary section width/depth, tendon system, stage, location, losses, profile, and assumed stress limits. The colored sample region is an illustrative elastic feasibility plot, not a code-certified design envelope.

StageSystemTendon profile

Section width300 mm

Section depth600 mm

Span12.0 m

Section location x6.0 m

Jacking force Pj1500 kN

Midspan eccentricity100 mm

Transfer midspan moment (UDL shape)90 kN·m

Service midspan moment (UDL shape)180 kN·m

Concrete elastic modulus30000 MPa

Friction/wobble (% Pj)3 %

Anchorage seating (% Pj)1 %

Elastic shortening (% Pj)2 %

Creep (% Pi)5 %

Shrinkage (% Pi)4 %

Tendon relaxation (% Pi)2 %

Stage forces

Pj 1500 → Pi 1410 → Pe 1255 kN

Immediate loss 90.0 kN; time-dependent loss 155.1 kN.

At selected section

x = 6.0 m; e(x) = 100.0 mm; M = 180.0 kN·m

A 180000 mm²; S 1.80e+7 mm³; kern ±100.0 mm.

Equivalent loading

Equivalent (+ upward) 6.97 kN/m; gravity equivalent 10.00 kN/m

Ideal net midspan deflection 5.05 mm (+ downward); excludes creep, cracking, tendon variation and secondary moments.

Top total: -10.00 MPa
Bottom total: -3.94 MPa

Stress superposition at x (MPa)

Top: -6.97 direct + 6.97 eccentric + -10.00 external = -10.00. Bottom: -6.97 + -6.97 + 10.00 = -3.94.

Illustrative midspan limits for both stages

Transfer tension maximum2.0 MPa

Transfer compression magnitude18 MPa

Service tension maximum0.0 MPa

Service compression magnitude15 MPa

Selected point: within supplied elastic limits. Active stage at midspan: -10.0 / -3.9 MPa top / bottom.

Green sampled pairs meet the four learner-supplied fiber bounds at midspan; red is the selected input. These are gross elastic checks only. Even a feasible point requires actual code allowables, all sections and stages, tendon cover/profile, time-dependent deflection, secondary effects in continuous members, anchorage zones and ultimate capacity.

Key Takeaways

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.”
$P_j$ , $P_i$ , and $P_e$ 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.

$References$

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Learning Objectives

Prestressed Concrete

What Prestressing Does

Fully Prestressed Member

Partially Prestressed Member

Cracking and Shear Are Not Eliminated

Prestressing and Conventional Steel

Material Roles

Pre-tensioning

Post-tensioning

System-Specific Immediate Effects

Jacking Force (PjP_jPj​)

Transfer Force (PiP_iPi​)

Effective Prestress (PeP_ePe​)

Four Design Stages

Declared Stress Sign Convention

Transfer-Stage Elastic Fiber Stresses

Effective Service-Stage Elastic Fiber Stresses

Interpreting Allowable Stresses

Rectangular Section Kern

Kern Is Not a General No-Tension Guarantee

Magnel Feasible Region and Stress Inequalities

Magnel Diagram

Prestress Loss Framework

Immediate and Time-Dependent Losses

Loss Bookkeeping by Force Stage and Location

Ideal Post-Tension Duct Friction

Do Not Double Count Losses

Equivalent Tendon Load

Equivalent Load for a Symmetric Parabolic Profile

Symmetric Parabolic Tendon Geometry

Tendon Profile, Support Restraint, and Equivalent Actions

Ideal Net Simply Supported Elastic Deflection

Camber and Long-Term Deflection Need More Than Load Balance

Load Balancing

Secondary Prestress Effects

Concordant Tendon Profile

Linear Transformation of Tendon Profiles

Ultimate Strength Is a Separate Check

Explore Prestress Stages and Feasibility

Prestress stages, tendon profile and stress

Stress superposition at x (MPa)

Illustrative midspan limits for both stages

References

$Learning Objectives$

$Prestressed Concrete$

$What Prestressing Does$

$Fully Prestressed Member$

$Partially Prestressed Member$

$Cracking and Shear Are Not Eliminated$

$Prestressing and Conventional Steel$

$Material Roles$

$Pre-tensioning$

$Post-tensioning$

$System-Specific Immediate Effects$

$P_j$

$P_i$

$P_e$

$Four Design Stages$

$Declared Stress Sign Convention$

$Interpreting Allowable Stresses$

$Kern Is Not a General No-Tension Guarantee$

$Magnel Feasible Region and Stress Inequalities$

$Magnel Diagram$

$Prestress Loss Framework$

$Immediate and Time-Dependent Losses$

$Loss Bookkeeping by Force Stage and Location$

$Do Not Double Count Losses$

$Equivalent Tendon Load$

$Tendon Profile, Support Restraint, and Equivalent Actions$

$Camber and Long-Term Deflection Need More Than Load Balance$

$Load Balancing$

$Secondary Prestress Effects$

$Concordant Tendon Profile$

$Linear Transformation of Tendon Profiles$

$Ultimate Strength Is a Separate Check$

$Explore Prestress Stages and Feasibility$

$References$