Prestressed Concrete Structures (CE-702 (A)) - Unit 3 Short Notes

UNIT 3: PRESTRESSED CONCRETE STRUCTURES

Based on Exam Paper: "A PRESTRESSED CONCRETE STRUCTURES - NOV 2022"

1.0 Introduction & Fundamental Concepts

1.1 Definition and Basic Principle

• Prestressed Concrete: Concrete in which internal stresses are introduced intentionally to counteract stresses due to applied loads.

• Basic Principle: Apply a compressive force (prestress) to concrete before it carries service loads. This pre-compression:

  • Delays or eliminates tensile stress under load.

  • Utilizes high compressive strength of concrete.

  • Keeps concrete in compression throughout service life.

1.2 Classification of Prestressed Concrete

1.3 Merits and Demerits

1.4 Historical Systems (Brief)

• Freyssinet System: First practical system using high-strength wires in steel ducts, anchored by wedges.

• Magnel System: Uses flat jacks and conical wedges for anchorage.

• Gifford-Udall System: Uses single-strand tendons with button-head anchorage.

1.5 Basic Assumptions (IS 1343)

1. Concrete is homogeneous, isotropic, obeys Hooke's law up to working stress.

2. Steel and concrete act together (perfect bond assumed in pre-tensioning; in post-tensioning, strain compatibility at sections).

3. Plane sections remain plane (Bernoulli's hypothesis).

4. Stress in steel is constant along its length (for concentric prestress).

5. Effects of shear lag, creep, and shrinkage are considered separately as losses.

[!TIP]

Exam Focus: Distinguish between pre-tensioning (bonded, cast-in) and post-tensioning (can be unbonded, ducts cast-in). Know why mild steel (low yield strength, high relaxation) is unsuitable.

2.0 Materials & Tendons

2.1 Concrete for Prestressing

• High compressive strength (M40 to M80 typical) to resist high prestress and reduce section size.

• Low creep to minimize long-term prestress loss.

• High modulus of elasticity (E_c ≈ 25-40 GPa) for efficient stress transfer.

• Low shrinkage to reduce prestress loss.

2.2 Prestressing Tendons

• High Tensile Steel: Wires (2-7 mm), Strands (7-wire, 12.7/15.2 mm), Bars (20-40 mm).

• Properties:

  • High ultimate tensile strength (f_pu = 1500–2000 N/mm²).

  • High yield strength (f_py = 0.85–0.9 f_pu).

  • Low relaxation (< 2.5% at 70% f_pu after 1000 hrs).

  • Stress-strain curve: Linear up to yield, then strain hardening.

• Why not Mild Steel?

  Mild steel has low yield strength (~250 N/mm²), high relaxation, and would require large area → impractical.

2.3 Types of Tendons & Cables

• Freyssinet Cable: Bundles of 5–12 wires, anchored by conical wedges.

• Gifford-Udall Cable: Single strand with button-head anchorage.

• Magnel Bloc System: Flat jacks and split cones.

• B.B.R.V. System: Uses helical wires and a central bar.

3.0 Methods & Systems of Prestressing

3.1 Pre-tensioning Method

1. Tendons tensioned against abutments.

2. Concrete cast and cured.

3. Tendons released → prestress transferred by bond.

• Advantages: Fast, economical for repetitive units (beams, slabs), high production.

• Applications: Precast beams, hollow core slabs, piles.

3.2 Post-tensioning Method

1. Concrete cast with ducts (metal/plastic).

2. Tendons inserted, tensioned using jacks against concrete.

3. Ducts grouted (bonded) or left unbonded.

• Advantages: Suitable for large spans, on-site construction, variable profiles.

• Applications: Bridges, large slabs, tanks, building frames.

3.3 Tendon Profiles & Layout

• Profiles:

  Straight: Constant eccentricity (e.g., pretensioned beams).

  Harped: Straight segments with kinks at supports (to balance shear).

  Parabolic: For uniformly distributed loads (balanced load concept).

• Concordant Cable Profile: A tendon profile that produces zero resultant moment at all sections when only prestress acts (i.e., moment diagram is linear). Achieved by adjusting eccentricities.

• Linear Transformation: Changing tendon profile by adding a straight tendon (constant force) shifts the moment diagram by a linear function without altering support reactions.

[!TIP]

Exam Focus: Know how to sketch parabolic profile for simply supported beam with UDL. Concordant cable is key for continuous beams to avoid secondary moments.

4.0 Stress Analysis in Prestressed Beams

4.1 Stresses Due to Prestressing Force (Eccentric Loading)

At any section, stress due to prestress $$\displaystyle P_e $$ (effective prestress) with eccentricity $e$:

$$ \sigma = \frac{P_e}{A} \pm \frac{P_e e}{I} y $$

Where $A$ = area, $I$ = moment of inertia, $y$ = distance from NA.

4.2 Stress Distribution Diagrams

• At Ends (Concentric): Uniform compression $$\displaystyle \sigma = -P_e/A $$.

• At Mid-Span (Parabolic Profile): Linear compression varying from top to bottom.

  • Top fiber: $$\displaystyle \sigma_t = -\frac{P_e}{A} - \frac{P_e e}{I} \frac{h}{2} $$

  • Bottom fiber: $$\displaystyle \sigma_b = -\frac{P_e}{A} + \frac{P_e e}{I} \frac{h}{2} $$

4.3 Load Balancing Concept

• Idea: Select tendon profile and prestress magnitude such that the upward force from prestress balances the downward external load.

• For a parabolic tendon in simply supported beam with UDL $w$:

$$ \frac{d^2 e}{dx^2} = \frac{w}{P_e} $$

Integrating gives parabolic profile $$\displaystyle e = \frac{w}{8P_e} x (L - x) $$.

• Result: No bending stress due to external load; only uniform compression.

• Sketch: Show tendon profile matching load diagram.

4.4 Primary & Secondary Moments (Continuous Beams)

• Primary Moment: Moment due to prestress about the centroid of the section (from $$\displaystyle P_e e $$).

• Secondary Moment: Moment due to reactions at supports induced by prestress (to satisfy compatibility).

• Resultant Moment = Primary + Secondary.

• Key: In continuous beams, concordant cable profile may not be possible → secondary moments arise.

4.5 Stresses Due to External Loads

Superimpose stresses from service loads (dead, live) on prestress stresses. Check:

• Top fiber compression: $$\displaystyle \sigma_{top} = \sigma_{pre} + \sigma_{load} $$

• Bottom fiber tension/compression: $$\displaystyle \sigma_{bot} = \sigma_{pre} + \sigma_{load} $$

4.6 Stress Block & Kernel of Section

• Stress Block: For ultimate design, IS 1343 uses rectangular stress block in concrete (like IS 456 but with different parameters).

• Kernel of Section: The area within the cross-section where a single compressive force can be applied without causing tension anywhere. For symmetrical sections, it's a central area; for T-sections, it's near the flange.

5.0 Losses of Prestress

5.1 Classification

• Immediate/Short-term: Occur during/soon after tensioning.

  • Elastic deformation of concrete.

  • Friction losses (post-tensioning).

  • Anchorage set losses (post-tensioning).

• Long-term: Occur over time.

  • Creep of concrete.

  • Shrinkage of concrete.

  • Relaxation of steel.

5.2 Detailed Calculation of Losses

1. Elastic Deformation Loss (Pre-tensioning):

   Loss in one tendon due to elastic shortening of concrete when other tendons are tensioned:

$$ \Delta \sigma_{el} = \frac{P}{A_c} \left( \frac{1}{E_c} + \frac{1}{E_s} \right) \cdot m \quad \text{(for successive tensioning)} $$

Where $$\displaystyle m = E_s / E_c $$.

2. Creep Loss:

$$ \Delta \sigma_{cr} = \phi \cdot \frac{P}{A_c} \left( \frac{1}{E_c} + \frac{1}{E_s} \right) \cdot m $$

Where $\phi$ = creep coefficient (from IS 1343 tables).

3. Shrinkage Loss:

$$ \Delta \sigma_{sh} = \epsilon_{sh} \cdot E_s $$

Where $$\displaystyle \epsilon_{sh} $$ = shrinkage strain (from IS 1343, depends on humidity, member size).

4. Relaxation Loss:

   Loss in steel stress under constant strain. Given as % of initial stress (from steel manufacturer's data).

5. Friction Loss (Post-tensioning):

$$ \Delta \sigma_f = \sigma_{po} (1 - e^{-(\mu \theta + k \cdot x)}) $$

Where $\mu$ = coefficient of friction, $\theta$ = angular deviation, $k$ = wobble coefficient, $x$ = length from jacking end.

6. Anchorage Set Loss:

   Loss due to slip of tendon in anchor after jacking release. Calculated using empirical formulas (IS 1343).

5.3 Loss Ratio & Effective Prestress

• Loss Ratio = $$\displaystyle \frac{\text{Total Loss}}{\text{Initial Prestress}} $$

• Effective Prestress $$\displaystyle P_e = P_i - \text{Total Losses} $$

• Design: Use $$\displaystyle P_e $$ for service stress checks; $$\displaystyle P_i $$ for ultimate strength.

[!TIP]

Common Pitfall: Forgetting that elastic deformation loss occurs only in pre-tensioned members with successive tensioning. In post-tensioning, friction and anchorage set are dominant immediate losses.

6.0 Design of Prestressed Concrete Sections (Flexure)

6.1 Design Philosophy

• Working Stress Method (Serviceability): Check stresses at transfer and service loads against limits.

• Ultimate Limit State (Strength): Calculate moment capacity using strain compatibility and IS 1343 stress block.

6.2 Design of Rectangular Sections

1. Assume trial depth $d$ and eccentricity $e$.

2. Compute $$\displaystyle P_e $$ after losses.

3. Check service stresses:

   • At transfer: $$\displaystyle \sigma_{c} \leq f_{ci} $$ (permissible compressive stress in concrete at transfer).

   • At service: $$\displaystyle \sigma_{c} \leq f_{cw} $$, $$\displaystyle \sigma_t \leq f_t $$ (tensile stress limit).

4. Compute ultimate moment capacity $$\displaystyle M_u $$:

$$ M_u = P_e \cdot e + 0.87 f_y A_s (d - 0.42 x_u) $$

But for prestressed members without additional tension steel, use:

$$ M_u = P_e \cdot e + C \cdot (d - \bar{x}) $$

Where $C$ = compressive force in concrete stress block.

5. Check $$\displaystyle M_u \geq M_{applied} $$.

6.3 Design of T-Sections & I-Sections

• Check if section is flanged or rectangular equivalent based on neutral axis depth $$\displaystyle x_u $$.

• If $$\displaystyle x_u \leq $$ flange thickness → treat as rectangular with $$\displaystyle b = b_f $$ (flange width).

• If $$\displaystyle x_u > $$ flange thickness → T-section analysis with stress block in web and flange.

6.4 Ultimate Flexural Strength (IS 1343 Stress Block)

• Stress block: Rectangular with stress = $$\displaystyle 0.45 f_{ck} $$ over depth $$\displaystyle 0.42 x_u $$ (for $$\displaystyle f_{ck} \leq 50 $$ N/mm²).

• Lever arm $$\displaystyle z = d - 0.42 x_u $$.

• Ultimate moment:

$$ M_u = 0.45 f_{ck} b (0.42 x_u) z + P_e e $$

For unbonded tendons, $$\displaystyle P_e $$ acts at tendon centroid.

6.5 Stress Checks

• At Transfer (before losses): Use $$\displaystyle P_i $$.

$$ \sigma_{c,top} = -\frac{P_i}{A} - \frac{P_i e}{I} y_t, \quad \sigma_{c,bot} = -\frac{P_i}{A} + \frac{P_i e}{I} y_b $$

• At Service (after losses): Use $$\displaystyle P_e $$ plus load stresses.

$$ \sigma_{c} = \sigma_{pre} + \sigma_{dead} + \sigma_{live} $$

6.6 Number & Arrangement of Tendons

• Number of wires/strands: $$\displaystyle n = \frac{P_e}{A_{ps} \cdot f_{pe}} $$

  Where $$\displaystyle A_{ps} $$ = area of single tendon, $$\displaystyle f_{pe} $$ = effective stress in tendon.

• Arrangement: Place tendons near bottom fibers for positive moment, within kern to avoid tension at top.

6.7 Design of Prestressed I-Beam

• Typical steps:

  1. Assume dimensions (flange width, web width, overall depth).

  2. Compute dead load (self-weight) + superimposed load.

  3. Determine maximum moment ($$\displaystyle M_{max} $$).

  4. Assume eccentricity $e$ (within limits: $$\displaystyle e_{min} \geq $$ cover, $$\displaystyle e_{max} \leq D/10 $$).

  5. Calculate required $$\displaystyle P_e $$ from moment equilibrium: $$\displaystyle M = P_e e $$.

  6. Check service stresses at critical sections (support, mid-span).

  7. Check ultimate moment capacity.

  8. Adjust dimensions/tendons iteratively.

7.0 Shear & Torsion in Prestressed Beams

7.1 Shear Resistance of Uncracked Section

• Uncracked section shear strength:

$$ V_{c} = \beta \sqrt{f_{ck}} \, b \, w $$

Where $\beta$ depends on stress condition (IS 1343). Prestress increases shear capacity by providing compressive stress.

7.2 Effect of Prestress on Shear Strength

• Prestress force $$\displaystyle P_e $$ with vertical component (from harped profile) directly resists shear.

• Effective shear force: $$\displaystyle V_e = V - P_e \sin\theta $$ (where $\theta$ is angle of tendon).

• Increases shear capacity, especially near supports.

7.3 Design for Shear

1. Compute design shear force $$\displaystyle V_u $$ (including effects of prestress).

2. Check if $$\displaystyle V_u \leq V_{c,min} $$ (no shear reinforcement needed).

3. If $$\displaystyle V_u > V_c $$, provide shear reinforcement (stirrups) as:

$$ \frac{A_{sv}}{s_v} = \frac{V_u - V_c}{0.87 f_y d} $$

Where $$\displaystyle A_{sv} $$ = area of stirrup legs, $$\displaystyle s_v $$ = spacing.

8.0 Anchorage Zone & End Blocks

8.1 Stress Distribution in Anchorage Zone

• Bursting Stress: Tensile stress perpendicular to force direction, near corners of end block.

• Spalling Stress: Tensile stress on surface parallel to force, due to Poisson effect.

• Splitting Stress: Tensile stress along axis of tendon, due to concentration.

8.2 Concept of End Block

• A reinforced concrete block at the end of a post-tensioned member to distribute concentrated anchorage force over a larger concrete area, reducing bearing stress.

• Dimensions: Length $$\displaystyle l_b \geq $$ bearing plate size + thickness; width/height ≥ bearing plate size + cover.

8.3 Design of End Block (IS 1343)

1. Bearing Stress Check:

$$ \sigma_{bearing} = \frac{P}{A_{bearing}} \leq 0.3 f_{ck} \quad (\text{for temporary}) \quad \text{or} \quad 0.45 f_{ck} \quad (\text{for permanent}) $$

2. Bursting Reinforcement:

   • Bursting force $$\displaystyle F_{bst} $$ calculated from stress trajectories.

   • Reinforcement area: $$\displaystyle A_{bst} = \frac{F_{bst}}{0.87 f_y} $$.

   • Placed spirally or in layers perpendicular to tendon axis.

3. Spalling & Splitting Reinforcement: Distributed reinforcement in end block.

8.4 Types of Anchorage Devices

• Conical Wedges (Freyssinet).

• Button-Head & Split Cones (Gifford-Udall).

• Swaged Sleeves.

• Bearing Plates (for multi-strand anchors).

9.0 Deflection in Prestressed Beams

9.1 Factors Influencing Deflection

• Prestress force (upward deflection).

• External loads (downward).

• Creep & shrinkage (increase long-term deflection).

• Support conditions (simply supported, continuous).

• Section modulus (inverse relation).

9.2 Short-Term Deflection (Elastic)

• Compute using moment-area or conjugate beam.

• Total deflection = $$\displaystyle \delta_{pre} + \delta_{load} $$.

• $$\displaystyle \delta_{pre} $$: Upward due to prestress eccentricity.

• $$\displaystyle \delta_{load} $$: Downward due to loads.

9.3 Long-Term Deflection

• Due to creep (increases load deflection) and shrinkage (adds upward camber in simply supported, but may increase deflection in continuous).

• Multiply short-term deflection by factor $(1 + \phi)$ for creep.

• Add shrinkage curvature effect.

9.4 Calculation Methods

• Moment-Area Theorem: $$\displaystyle \theta = \frac{1}{EI} \int M dx $$, $$\displaystyle \delta = \frac{1}{EI} \int M \bar{x} dx $$.

• Conjugate Beam: Replace real beam with "conjugate" having load $M/EI$. Deflection = moment in conjugate beam.

10.0 Continuity in Prestressed Beams

10.1 Methods of Achieving Continuity

1. Monolithic Construction: Cast deck slab continuously over precast girders.

2. Post-Tensioning Across Supports: Tendons continuous over supports (top at supports, bottom in spans).

3. Link Beams: Post-tensioned link beams between precast units.

10.2 Analysis of Continuous Beams

• Primary Moment: Moment due to prestress about centroid (like simply supported).

• Secondary Moment: Moment due to indeterminate reactions from prestress.

• Resultant Moment = Primary + Secondary.

• Key: Secondary moments ensure compatibility (no net displacement at supports).

10.3 Effect of Prestressing on Support Moments

• Can reduce negative moments at supports (by applying top prestress).

• Can increase positive moments in spans (by balancing load).

• Concordant cable not possible in continuous → secondary moments unavoidable.

11.0 Composite Construction

11.1 Definition & Types

• Composite Member: Precast prestressed beam + cast-in-situ slab (or deck).

• Types:

  • Singly Composite: Precast beam + in-situ slab (shear connectors).

  • Doubly Composite: Precast beam + two in-situ slabs (top/bottom).

11.2 Stress Distribution in Composite Sections

• Before composite action: Precast beam acts alone.

• After composite action: Section transformed to composite T-section.

• Differential shrinkage between precast and cast-in-situ parts induces stresses.

11.3 Differential Shrinkage

• Cast-in-situ slab shrinks more than precast beam (older, drier).

• Causes additional tensile stress in precast bottom and compressive stress in slab top.

11.4 Stresses & Moments in Composite T-Beams

• Effective width of flange: as per IS 1343 (based on span and web spacing).

• Short-term: Compute stresses using composite section properties.

• Long-term: Account for creep & differential shrinkage.

• Design: Check stresses at transfer (precast alone) and service (composite).

12.0 Partial Prestressing

12.1 Definition

• Prestressing where tensile stresses are allowed in concrete under service loads within specified limits.

• Between fully prestressed and reinforced concrete.

12.2 Merits & Demerits

12.3 Methods of Achieving Partial Prestressing

1. Reduced prestress force (lower $$\displaystyle P_e $$).

2. Non-tendon reinforcement (mild steel) to share tensile stresses.

3. Partial prestressing ratio $$\displaystyle \psi = \frac{P_e}{A_g f_{ck}} $$ < 1.

12.4 Design Considerations

• Check crack width (permissible limit, e.g., 0.2 mm).

• Check deflection (may be higher due to cracking).

• Use limit state method with partial safety factors.

• Provide adequate non-prestressed reinforcement for crack control.

13.0 Special Topics & Short Notes

13.1 Types of Flexural Failures

1. Tension Failure: Steel yields before concrete crushes (ductile, desirable).

2. Compression Failure: Concrete crushes before steel yields (brittle, avoid by limiting $$\displaystyle x_u $$).

3. Shear Failure: Diagonal tension, sudden.

13.2 Hoyes System of Prestressing

• Uses high-strength bars (not wires) threaded at ends.

• Bars tensioned by nuts on threaded ends.

• Anchorage: By bearing of nuts on end plates.

• Sketch: Show bar, nut, end plate, concrete cross-section.

13.3 Differences: Pre-tensioning vs. Post-tensioning

13.4 D'Alembert's Principle in Vibration

• Extends Newton's second law to dynamic systems by introducing inertial force $-m\ddot{u}$.

• For a SDOF system: $$\displaystyle m\ddot{u} + c\dot{u} + ku = F(t) $$ becomes $$\displaystyle m\ddot{u} + c\dot{u} + ku - F(t) = 0 $$ (dynamic equilibrium).

• Application: Formulating equations of motion for prestressed structures under dynamic loads (earthquake, impact).

13.5 Stress Concept in Prestress

• Initial Stress: Stress in tendon at jacking ($$\displaystyle f_{pi} $$).

• Effective Stress: After all losses ($$\displaystyle f_{pe} = f_{pi} - \Delta f_p $$).

• Working Stress: Stress in concrete due to prestress + loads must be within limits.

• Ultimate Stress: Steel stress at failure ($$\displaystyle f_{pu} $$), concrete stress at $$\displaystyle 0.45 f_{ck} $$.

13.6 Tendon & Tendon Profile

• Tendon: Single wire, strand, or bar.

• Cable: Group of tendons bundled together.

• Profile: Path of tendon centroid along beam length.

  • Straight: Constant eccentricity.

  • Parabolic: For UDL, $$\displaystyle e = \frac{w}{8P_e} x(L-x) $$.

  • Harped: Straight with kinks at supports for shear balance.

14.0 Design Problems (Integrated Application)

14.1 Design of Post-Tensioned Girder

• Given: Span, loads (dead, live), material grades, stress limits, loss ratio.

• Steps:

  1. Compute total load & maximum moment.

  2. Assume section dimensions (b, D, d).

  3. Determine required $$\displaystyle P_e $$ from $$\displaystyle M = P_e e $$ (choose $e$ within limits).

  4. Number of tendons: $$\displaystyle n = P_e / (A_{ps} f_{pe}) $$.

  5. Check service stresses at transfer ($$\displaystyle P_i $$) and service ($$\displaystyle P_e $$).

  6. Check ultimate moment capacity.

  7. Verify shear and deflection.

14.2 Design of Prestressed I-Beam

• Similar to girder but with I-section.

• Check flange width effectiveness.

• Ensure tendons placed within web (avoid flange).

14.3 Design of End Block

• Given: Jacking force $P$, anchor plate size, beam dimensions.

• Steps:

  1. Check bearing stress: $$\displaystyle \sigma_{bearing} = P / A_{plate} \leq 0.3 f_{ck} $$ (temp).

  2. Determine end block dimensions ($$\displaystyle l_b $$, $$\displaystyle b_b $$, $$\displaystyle h_b $$).

  3. Compute bursting force $$\displaystyle F_{bst} $$ (using IS 1343 formulas or graphs).

  4. Design bursting reinforcement: $$\displaystyle A_{bst} = F_{bst} / (0.87 f_y) $$.

  5. Provide distributed reinforcement for spalling/splitting.

14.4 Calculation of Losses

• Given: Multi-cable, successive tensioning, material properties.

• Steps:

  1. Compute elastic shortening loss for each tendon (cumulative).

  2. Add creep, shrinkage, relaxation (long-term).

  3. For post-tensioning: add friction & anchorage set.

  4. Total loss = sum; $$\displaystyle P_e = P_i - \text{total loss} $$.

14.5 Stress Analysis with Different Profiles

• Given: Beam with parabolic/straight tendons, loads.

• Compute:

  • Prestress moment diagram $$\displaystyle M_p = P_e e(x) $$.

  • Load moment diagram $$\displaystyle M_l $$.

  • Resultant moment $$\displaystyle M = M_p + M_l $$.

  • Stresses: $$\displaystyle \sigma = \frac{P_e}{A} \pm \frac{M}{Z} $$ at critical sections (support, mid-span).

  • For continuous beams: include secondary moments (using three-moment equation or moment distribution).

Final Exam Strategy:

• 4m questions: Definitions, classifications, differences, short notes.

• 6-8m questions: Merits/demerits, assumptions, stress diagrams, tendon profiles, loss types.

• 10m questions: Load balancing, primary/secondary moments, shear design, end block design, composite construction.

• 14m questions: Full design problems (girder, I-beam, end block, loss calculation).

Always sketch where possible (tendon profiles, stress diagrams, end block reinforcement). Use IS 1343 codes for numerical values.