Prestressed Concrete Structures (CE-702 (A)) - Unit 1 Short Notes

1. Introduction to Prestressed Concrete

Definition: Prestressing is the intentional introduction of compressive stresses into a concrete member before it sustains service loads, to counteract tensile stresses induced by those loads.

Basic Principle: Pre-compression reduces or eliminates tensile stresses in concrete under load, utilizing the high tensile strength of steel and the relatively low tensile strength of concrete.

Historical Development:

• Eugène Freyssinet (1928): Pioneer of modern prestressing; developed high-strength steel and hydraulic jacks; introduced pre-tensioning and post-tensioning concepts.

• Gustave Magnel: Developed the Magnel system for post-tensioning with flat jacks and wedge anchors.

• Other contributors: Hoyer (pre-tensioning), Gifford-Udall (strand systems).

Classification:

Typical Applications:

• Bridges (spans 20–50 m), building floors/roofs, water tanks, silos, piles.

[!TIP] Exam Key: Pre-tensioning relies on bond for force transfer; post-tensioning uses mechanical anchorage. Unbonded tendons allow individual stressing and are corrosion-protected.

2. Materials for Prestressed Concrete

Concrete

• Required Properties:

  • High compressive strength (M40–M80 grade).

  • Low creep and shrinkage.

  • High modulus of elasticity (E_c ≈ 25–40 GPa).

  • Adequate tensile strength for crack control.

• Stress-Strain Behavior: Approximately linear elastic up to ~0.45f_ck (per IS 1343), parabolic thereafter.

Prestressing Steel

• Types:

  • Wires: Cold-drawn, 2–7 mm diameter.

  • Strands: 7-wire helical strands (common), 12.7 mm or 15.2 mm.

  • Alloy Bars: High-strength threaded bars (e.g., 26 mm).

• Mechanical Properties:

  • Ultimate tensile strength f_pu: 1500–2000 MPa.

  • Modulus E_s: 190–210 GPa.

  • Relaxation: < 2.5% at 70% f_pu for low-relaxation strands.

  • Ductility: Elongation ≥ 3.5% at failure.

• Why Mild Steel Unsuitable:

  • Low yield strength (~250 MPa) → requires excessive area.

  • High relaxation → significant prestress losses.

  • Low ultimate strength → inefficient.

Bond and Adherence

• Bond strength depends on:

  • Surface texture (ribbed wires/strands enhance mechanical interlock).

  • Concrete strength and compaction.

  • Tendon diameter and configuration.

[!TIP] Exam Pitfall: Confusing steel grades. Prestressing steel is high-tensile (f_pu > 1000 MPa), not mild (f_y ~ 250 MPa).

3. Prestressing Systems and Methods

Pre-tensioning Process

1. Tendons tensioned against rigid abutments using hydraulic jacks.

2. Concrete cast and cured.

3. Tendons released → force transferred via bond to concrete.

• Equipment: Casting beds, stressing jacks, abutments.

Post-tensioning Process

1. Ducts/sleeves placed in formwork.

2. Concrete cast and cured.

3. Tendons inserted and tensioned using jacks against anchorages.

4. Ducts grouted (bonded) or left unbonded.

• Anchorage Systems: Wedge-type, bearing plate with nuts.

• Tendon Arrangements: Bunched (multi-strand) or individual.

Major Commercial Systems

Comparative Analysis

• Pre-tensioning: Economical for repetitive precast units; limited to factory production.

• Post-tensioning: Flexible for cast-in-place and large spans; higher initial cost but adaptable.

• Bonded vs Unbonded: Bonded provides composite action, unbonded allows individual monitoring and less friction loss.

[!TIP] Exam Focus: Freyssinet uses conical wedges; Hoyes uses threaded bars with nuts. Know which are pre- vs post-tensioning systems.

4. Fundamental Concepts and Theory

Basic Assumptions in Elastic Analysis

1. Concrete is homogeneous, isotropic, linearly elastic (within service limits).

2. Plane sections remain plane (Bernoulli hypothesis).

3. Prestressing force magnitude constant along tendon (ignoring losses initially).

4. No slip between tendon and concrete (for bonded systems).

5. Effects of shear deformation and torsion neglected.

Stress Concepts

• Initial Prestress (f_pi): Stress in tendon immediately after tensioning.

• Effective Prestress (f_pe): Stress in tendon after all losses, at service stage.

$$\boxed{f_{pe} = f_{pi} - \Delta f_p}$$

where Δf_p = total losses.

• Stages:

  • Transfer: Stress in concrete just after tendon release (pre-tensioning) or stressing (post-tensioning).

  • Service: Under external loads, after losses.

Permissible Stresses (IS 1343)

Advantages

• Increased load capacity, longer spans, reduced deflection, crack control, material savings, durability.

Disadvantages

• Higher initial cost, specialized labor/equipment, complex design/analysis, inspection difficulties.

Partial Prestressing (IS 1343)

• Definition: Prestressing level such that tensile stresses exceed allowable limits under service loads, but within controlled limits.

• Classification:

  • Class 1: No tension under moderate loads (full prestressing).

  • Class 2: Tension allowed but within limits; cracks tightly controlled.

  • Class 3: Tension allowed; cracks permitted but limited in width.

• Merits: Economical (less steel), easier construction.

• Demerits: Some cracking, increased deflection, reduced durability.

[!TIP] Exam Tip: Partial prestressing is a compromise between reinforced and fully prestressed concrete. Class 3 allows visible cracks but limits width.

5. Analysis of Prestressed Concrete Sections

Stress Distribution

• Prestress Only:

  • Concentric tendon: Uniform compression σ_c = P/A ± Pe/I.

  • Eccentric tendon: Linear stress distribution with moment Pe.

• With External Loads: Superposition principle.

$$\sigma_c = \frac{P}{A} \pm \frac{Pe}{I} + \frac{M_y}{I} \cdot y$$

• Critical Sections: Midspan (max moment), supports (max shear, possible negative moment), discontinuities (sudden area change).

Load Balancing Concept

• Principle: Shape tendon profile to provide upward force balancing applied loads, resulting in zero net moment.

• Balanced Load (for parabolic tendon with eccentricity e at midspan, zero at supports):

$$w_{bal} = \frac{8Pe}{L^2}$$

• Equivalent Loads: From tendon profile, compute forces on concrete (e.g., vertical component for draped tendons).

Primary and Secondary Moments (Continuous Beams)

• Primary Moments: Direct from prestress in a statically determinate system.

• Secondary Moments: Induced by reactions at redundant supports in continuous beams.

• Resultant Moment: Vector sum of primary and secondary moments at each section.

• Calculation Methods:

  • Force Method: Release redundants, compute compatibility.

  • Displacement Method: Use stiffness matrix.

Linear Transformation of Tendon Profile

• Concept: Adding a straight line to tendon profile (keeping end eccentricities constant) does not change support reactions or secondary moments.

• Condition: For continuous beams, linear transformation possible if end eccentricities unchanged; alters primary moments but resultant moments same.

Shear Analysis

• Shear Stress in Uncracked Section:

$$\tau = \frac{VQ}{Ib} + \frac{P e}{I} \cdot \frac{dA}{dx}$$

(for draped tendons, vertical component contributes).

• Contribution of Prestress: Increases shear capacity by providing compressive stress on shear plane.

• Shear Resistance at Supports: Check concrete shear capacity V_c enhanced by prestress:

$$V_c = \tau_c b d + \sigma_{cp} b d$$

where σ_cp = P/A + P e/I.

[!TIP] Exam Focus: Load balancing simplifies design for simply supported beams. In continuous beams, always consider secondary moments. Shear design includes prestress effect via σ_cp.

6. Design of Prestressed Concrete Members

Design Approaches

• Working Stress Method (WSM): Elastic analysis, service loads. Used for stress checks at transfer/service.

• Limit State Method (LSM): Ultimate strength design, factored loads. Primary for flexure/shear design per IS 1343.

• Code: IS 1343 (Indian Standard for Prestressed Concrete).

Flexural Design

• Rectangular Section (LSM):

  • Strain compatibility: Concrete strain ε_c = 0.0035 (ultimate), steel strain ε_s = (d - x)/x * ε_c.

  • Stress block: 0.36 f_ck for concrete in compression (IS 1343 similar to IS 456).

  • Ultimate moment:

$$M_u = 0.87 f_y A_p \left(d - 0.42 x_u\right)$$

(under-reinforced, x_u ≤ 0.48 d for Fe 415).

• T-Section:

  • Check if neutral axis in flange (x_u ≤ D_f).

  • Flange contribution: b_f D_f * 0.36 f_ck.

Shear Design

• Concrete Shear Capacity:

$$V_c = \tau_c b d + \sigma_{cp} b d$$

where τ_c from IS 1343 Table, σ_cp = P/A + P e/I (prestress contribution).

• Shear Reinforcement: Required if V > V_c. Design as per IS 1343.

Stress Checks

• At Transfer:

  • Compression: σ_c ≤ 0.45 f_ci

  • Tension: σ_t ≤ allowable (1 MPa for Class 1).

• At Service:

  • Compression: σ_c ≤ 0.30 f_ck

  • Tension: σ_t ≤ 0.25 f_ct (Class 3).

Design Examples (Past Paper Patterns)

1. Post-tensioned Girder: Given span, loads, materials, loss ratio → determine tendon area.

2. Pre-tensioned I-Beam: Check stresses at transfer/service, compute moment capacity.

3. Incorporate Loss Ratio: Use λ = f_pe/f_pi in design.

[!TIP] Common Error: Forgetting to check both transfer and service stresses. In T-sections, always verify flange effectiveness (x_u ≤ D_f).

7. Losses of Prestress

Categories

Calculation Methods

1. Elastic Shortening (Δf_pES):

   • Pre-tensioning: All tendons lose simultaneously.

$$\Delta f_{pES} = \frac{A_p E_p}{A_c E_c} f_{c}$$

• Post-tensioning (successive): Earlier tendons lose due to later tensioning.

$$\Delta f_{pES} = \frac{A_p E_p}{A_c E_c} \cdot \frac{\sum (P_i \cdot e_i)}{I_c} \cdot y_p$$

2. Friction Loss (Δf_pF):

$$\Delta f_{pF} = f_{pi} \left(1 - e^{-(\mu \theta + k x)}\right)$$

where μ = wobble coefficient, k = curvature coefficient, θ = total angular change, x = length.

3. Anchorage Slip (Δf_pA):

$$\Delta f_{pA} = \frac{P_{slip} A_p}{A_p E_p} \cdot e^{-\mu \theta}$$

(loss over slip length).

4. Creep & Shrinkage (IS 1343):

   • Creep loss: Δf_pCR = (E_p/E_c) · f_c · φ(t, t_i)

   • Shrinkage loss: Δf_pSH = (E_p/E_c) · ε_sh · (A_c/A_p) (approx.)

5. Relaxation (Δf_pR): From IS 1343 graphs/tables based on f_pi/f_pu and time.

Total Loss & Effective Prestress

$$\boxed{f_{pe} = f_{pi} - \sum \Delta f_p}$$

• Loss Ratio: λ = f_pe/f_pi (typically 0.7–0.85). Used in preliminary design to size tendons.

[!TIP] Exam Calculation: For successive tensioning in post-tensioning, compute elastic loss for each tendon sequentially. Friction loss depends on tendon profile; use μ and k from code.

8. Anchorage Zone and End Block Reinforcement

Anchorage Zone Concept

• End Block: Region behind anchor where stress disperses.

• Transmission Zone: Where prestress spreads to uniform stress.

• Bearing Zone: Under anchor plate.

Stresses in Anchorage Zone

• Bursting Tension: Horizontal tension due to stress dispersion (like a bottle bursting).

• Spalling Tension: Vertical tension near top/bottom surfaces.

• Splitting Tension: Radial tension around anchor.

Design of End Block Reinforcement (IS 1343 Annex)

1. Bursting Force:

$$F_{br} = \sigma_{br} \cdot A_{br}$$

where σ_br from stress distribution (linear dispersion assumed).

2. Spalling Force: Vertical tension at edges.

3. Splitting Force: Radial tension, resisted by spiral or circular reinforcement.

• Reinforcement: Provide steel (grid, spiral, or distributed bars) to resist computed forces. Minimum reinforcement as per code.

Anchorage Devices

• Wedge-type (Freyssinet), bearing plate with nuts (Magnel/Hoyes), button-type (Gifford-Udall).

[!TIP] Exam Focus: Bursting tension governs horizontal reinforcement. Use IS 1343 Annex formulas for F_br. Sketch end block with reinforcement detailing.

9. Composite Construction

Definition and Types

• Precast Pretensioned Beam + Cast-in-situ Slab

• Precast Post-tensioned Beam + Cast-in-situ Slab

Composite Action

• Shear Transfer: At interface via:

  • Roughened surface.

  • Shear keys.

  • Studs (for steel beams).

• Effective Width of Flange (IS 456):

$$b_f = b_w + \frac{L_0}{6} \text{ or } b_w + 0.5 \times \text{spacing},$$

whichever less.

Differential Effects

• Shrinkage & Creep: Cast-in-situ slab shrinks/creeps more than precast beam → induces stresses.

• Shrinkage Stress Calculation:

  • Compute free shrinkage strain ε_sh of cast-in-situ part.

  • Use transformed section to find compatible strain → stress in each material.

  • Example: For T-beam composite, shrinkage in slab causes negative moment in beam.

Design Considerations

• Interface shear check.

• Differential deflection.

• Construction sequence (precast beam must support wet slab).

[!TIP] Exam Problem: Given differential shrinkage ε_sh, find stresses using transformed section method. Assume no slip at interface.

10. Partial Prestressing

Concept and Necessity

• Concept: Prestressing level such that tensile stresses exceed allowable under service loads but within limits for controlled cracking.

• Necessity: Economical for moderate spans where full prestressing is overkill.

Methods to Achieve

1. Reduce prestressing steel area A_p.

2. Use lower grade steel (lower f_pu).

3. Permit controlled cracking (Class 3 per IS 1343).

Merits and Demerits

Suitable Applications

• Floor slabs, parking structures, where slight cracking acceptable.

[!TIP] Partial prestressing is a middle ground. Class 3 allows visible cracks but limits width (< 0.2 mm per IS 1343).

11. Special Topics

Tendon Profiles

• Straight: Constant eccentricity; simple but limited moment control.

• Draped/Harped: Two or more straight segments; used for moment variation.

• Parabolic: For uniformly distributed loads; gives linear moment diagram.

• Concordant Profile: Produces zero secondary moments in continuous beams (requires solving linear equations).

Guyon's Method

• For moment redistribution in continuous prestressed beams by adjusting tendon profiles.

• Ensures resultant moments within limits without overstressing concrete.

Flexural Failure Modes

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

2. Compression Failure: Concrete crushes prematurely (brittle).

3. Balanced Failure: Both steel yields and concrete crushes simultaneously (ideal).

Deflection Behavior

• Short-term: Elastic analysis, includes prestress effect.

• Long-term: Increased by creep, shrinkage, sustained loads.

• Calculation (IS 456/1343):

  • Compute short-term deflection.

  • Apply multipliers for creep, shrinkage (e.g., α = 2.0 for sustained loads).

[!TIP] Concordant profile eliminates secondary moments in continuous beams. Deflection calculations require effective moment of inertia for cracked sections.

12. Design Examples and Case Studies

Based on past papers (RGPV CE-702(A)):

1. Post-tensioned Girder:

   • Given: span, loads (dead/live), concrete grade, permissible stresses, loss ratio.

   • Determine: Number of wires/strands, tendon profile, check stresses.

2. Pre-tensioned I-beam:

   • Given: span, load, concrete/steel grades, initial stress.

   • Design: Section dimensions, tendon area, stress checks.

3. Anchorage Zone Design:

   • Given: end block dimensions, cable arrangement, jacking force.

   • Design: Bursting/spalling reinforcement as per IS 1343 Annex.

4. Loss Calculation:

   • Multi-stage tensioning → compute elastic losses sequentially.

   • Include friction, creep, shrinkage, relaxation.

5. Composite Beam:

   • Given: precast beam + cast-in-situ slab, differential shrinkage ε_sh.

   • Determine: Shrinkage stresses in each part using transformed section.

[!TIP] Always start design with loss ratio (λ) to estimate f_pe. For anchorage design, compute bursting force first. In composite sections, use modular ratio m = E_c2/E_c1.