Prestressed Concrete Structures (CE-702 (A)) - Unit 5 Short Notes

UNIT 5: PRESTRESSED CONCRETE STRUCTURES

Based exclusively on the A PRESTRESSED CONCRETE STRUCTURES - NOV 2022 exam paper.

1.0 INTRODUCTION & FUNDAMENTAL CONCEPTS

• 1.1 Definition & Basic Principle

  • Prestressed Concrete (PSC): Concrete in which internal stresses are introduced deliberately to counteract stresses due to applied loads.

  • Principle: Apply a compressive force (prestress) to concrete before it carries service loads. This pre-compression offsets tensile stresses from external loads, keeping concrete primarily in compression.

• 1.2 Classification

  • By Method:

    • Pre-tensioning: Tendons are tensioned before concrete placement. Bond transfer via adhesion after concrete hardens. (Factory/Precast).

    • Post-tensioning: Tendons are placed in ducts, tensioned after concrete gains strength. Bond transfer via grouting (bonded) or anchorage (unbonded). (In-situ).

  • By Stress Level:

    • Full Prestressing: No tensile stress in concrete under service loads (Class 1).

    • Partial Prestressing: Controlled tensile stress allowed (Class 2).

    • Low-Tension Prestressing: Prestress only to reduce crack width (Class 3).

  • PSC vs. RC: PSC uses high-strength steel & concrete, induces compressive stress, leads to smaller, lighter, crack-free members. RC relies on steel to resist tension after concrete cracks.

• 1.3 Merits & Demerits

  • Merits: Higher load capacity, longer spans, reduced section size/weight, better crack control, improved shear & fatigue resistance, economical for large spans.

  • Demerits: High material & skilled labor cost, complex fabrication/equipment, quality control critical, requires specialized design knowledge.

• 1.4 Basic Assumptions (IS 1343)

  1. Concrete obeys Hooke's law up to service stress levels (linear stress-strain).

  2. Concrete modulus of elasticity is constant.

  3. Steel and concrete behave elastically and are perfectly bonded (for analysis at transfer & service).

  4. Plane sections remain plane after bending (Bernoulli's hypothesis).

  5. Concrete does not resist tension (cracked section analysis for ultimate state).

• 1.5 Stress Concept & Losses

  • Transfer of Prestress: Initial prestress (P_i) in steel is transferred to concrete, causing immediate losses (elastic deformation, friction, slip).

  • Effective Prestress (P_e): Prestress remaining in steel after all losses (P_e = P_i - Losses).

  • Losses: Categorized as Immediate (at anchoring) and Time-dependent (creep, shrinkage, relaxation).

2.0 PRESTRESSING SYSTEMS & METHODS

• 2.1 Freyssinet's System (Post-tensioning)

  • Key Feature: Uses high-strength wires (5mm or 7mm) grouped into cables.

  • Process: Ducts formed by wrapping metal strips. Wires pulled through ducts, anchored by conical wedges in anchor plate. After tensioning, ducts grouted.

  • DiagramSEARCH: Freyssinet system prestressed concrete anchor wedge

• 2.2 Hoyt System (Pre-tensioning)

  • Key Feature: Uses strands (7-wire) or bars.

  • Process: Tendons tensioned against abutments in casting bed. Concrete cast & cured. Tendons released, transferring prestress via bond.

  • DiagramSEARCH: Hoyt system pre-tensioning bed

• 2.3 Tendons & Profiles

  • Tendon Types: Wires (single), Strands (7-wire), Bars (threaded).

  • Tendon Profiles:

    • Straight: Constant eccentricity, easy for pre-tensioning.

    • Harped: Straight segments with abrupt kinks at points. Used for moment variation.

    • Parabolic: Smooth curve, ideal for UDL (load balancing).

  • Concordant Profile: A tendon profile that, under a given load, produces zero moment at all supports in a continuous beam (no secondary moments).

  • Linear Transformation: Changing tendon profile by a straight line (y = ax + b) without altering service load moments or support reactions.

• 2.4 Pre-tensioning vs. Post-tensioning

  • | Feature | Pre-tensioning | Post-tensioning |

  |----------|----------------|-----------------| | Tensioning | Before concreting | After concrete hardens | | Bond | Bond via adhesion | Bond via grouting or unbonded | | Location | Precast yards | In-situ/Precast | | Losses | Lower immediate losses | Higher friction/wedge slip losses | | Span | Shorter spans | Long spans, large structures | | Equipment | Tensioning beds, abutments | Jacks, anchorages, ducts |

3.0 ANALYSIS OF PRESTRESSED SECTIONS (FLEXURE)

• 3.1 Stresses in Beam Sections

  • At Transfer: Stress due to prestress only.

    • Concentric Tendon: Uniform compressive stress: σ_c = -P_i / A.

    • Eccentric Tendon: Combined axial + bending: σ_c = -P_i/A ± (P_i * e) / Z.

  • Under Service Loads: Superposition of prestress stress and load-induced stress.

  • Load Balancing Concept:

    • Idea: Shape tendon profile to exert an upward force (P * dy/dx) that balances applied loads.

    • For UDL w, a parabolic tendon with eccentricity e_max = w * l² / (8 * P) balances load, resulting in zero moment at midspan.

    • DiagramCANVAS: Parabolic tendon profile under UDL showing balanced load condition with zero moment diagram

• 3.2 Ultimate Flexural Strength (Moment Capacity)

  • Assumptions: Rectangular stress block (IS 1343), steel stress ≤ f_pu, concrete in tension ignored.

  • Rectangular Section (Post-tensioned, Unbonded):

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

    Where `x_u` = depth of neutral axis from compression fiber.

    For `f_p = 1500 N/mm²`, `0.87 f_p = 1305 N/mm²`.

*   **T-Section (IS 1343):**

    1.  Check if neutral axis lies in flange (`x_u ≤ D_f`).

    2.  If yes, treat as rectangular of width `b_f`.

    3.  If no, consider contribution of web and flange separately.

$$M_u = 0.36 f_{ck} b_w x_u (d - 0.42 x_u) + 0.45 f_{ck} (b_f - b_w) D_f (d - D_f/2)$$

*   **Stress Block Parameters (M30, M40):**

    *   For `f_ck = 30 N/mm²`: `α = 0.36`, `γ = 0.42`

    *   For `f_ck = 40 N/mm²`: `α = 0.36`, `γ = 0.42` (Same for ≤ M40 as per IS 1343).

• 3.3 Types of Flexural Failures

  1. Under-reinforced: Steel yields before concrete crushes. Ductile failure.

  2. Over-reinforced: Concrete crushes before steel yields. Brittle failure.

  3. Balanced Failure: Steel yields simultaneously with concrete crushing. Limit state.

4.0 DESIGN OF PRESTRESSED BEAM SECTIONS

• 4.1 Solid Slab Bridge (Class AA)

  • Design for IRC Class AA tracked/ wheeled loads.

  • Check for bending moment, shear, and bearing.

  • Use effective width concept for slab.

  • Provide minimum reinforcement as per code.

• 4.2 Design of PSC I-Girder

  • Given: Span L, loads (DL+LL), permissible stresses at transfer (σ_ct, σ_tt) and service (σ_c, σ_t), f_pi, loss ratio η.

  • Steps:

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

    2. Calculate self-weight.

    3. Determine total load & maximum moment (M).

    4. Find required P_e from stress limits at critical sections (usually midspan for +ve moment, support for -ve moment).

       • Top fiber: σ_top = -P_e/A - P_e*e/Z_top + M/Z_top ≤ σ_ct

       • Bottom fiber: σ_bot = -P_e/A + P_e*e/Z_bot - M/Z_bot ≥ σ_tt

    5. Adjust e and P_e iteratively.

    6. Calculate initial prestress P_i = P_e / η.

    7. Determine number of wires/cables: n = P_i / (f_pi * A_wire).

    8. Check for shear and ultimate moment capacity.

• 4.3 Post-Tensioned Girder Design

  • Consider spacing of girders, effective span for live load distribution.

  • Dead load includes self-weight, deck slab, wearing coat.

  • Live load as per IRC codes (Class AA/ A).

  • Use load factors for ultimate state, service loads for stress checks.

• 4.4 Freyssinet Cables

  • Specifications: 5mm or 7mm diameter high-tensile wires, f_pu = 1500-1700 N/mm².

  • Cable Composition: Multiple wires (e.g., 12, 19, 37) anchored by conical wedges.

  • Number of Wires: Determined by required P_i and f_pi.

  • Diameter: Overall cable diameter depends on number of wires and packing.

5.0 LOSSES OF PRESTRESS

• 5.1 Classification

  • Immediate (at transfer):

    • Elastic deformation of concrete.

    • Friction in ducts (curvature & wobble).

    • Wedge slip (anchor set).

  • Time-dependent (long-term):

    • Creep of concrete.

    • Shrinkage of concrete.

    • Relaxation of steel.

• 5.2 Loss due to Elastic Deformation (Successive Tensioning)

  • When cables are tensioned one after another, earlier cables cause shortening in concrete, reducing stress in later cables.

  • Formula for i-th cable:

$$\Delta P_i = \frac{P_i}{A_c E_c} \sum_{j=1}^{i-1} A_{pj} E_{pj} + \sum_{j=1}^{i-1} \frac{P_j e_i e_j}{I} A_{pj} E_{pj}$$

    Where:

    *   `P_i`, `P_j` = initial stress in cable `i` and `j`.

    *   `A_c`, `E_c` = area & modulus of concrete.

    *   `A_{pj}`, `E_{pj}` = area & modulus of steel of cable `j`.

    *   `e_i`, `e_j` = eccentricities at section of interest.

    *   `I` = moment of inertia of section.

*   **Simplified:** Loss in cable `i` is proportional to sum of forces in previous cables and their eccentricity product.

• 5.3 Loss Ratio Concept

  • Loss Ratio (η): η = P_e / P_i (typically 0.8 to 0.9).

  • Application in Design: Instead of calculating each loss, use P_i = P_e / η to determine initial prestress needed.

6.0 ANCHORAGE ZONE & END REINFORCEMENT

• 6.1 End Zone Stresses

  • Bursting Tensile Stress: Horizontal tensile stress perpendicular to prestress force, due to concentration of force at anchor plate. Requires bursting reinforcement (horizontal).

  • Spalling Tensile Stress: Vertical tensile stress near end face due to Poisson effect. Requires spalling reinforcement (vertical).

  • Splitting Tensile Stress: Radial tensile stress around duct/anchorage. Requires splitting reinforcement (circumferential ties).

• 6.2 Design of Anchorage Zone Reinforcement (IS 1343)

  • Given: Beam dimensions (b × h), cable configuration, jacking force P_jack.

  • Bursting Force (F_br):

$$F_{br} = P_{jack} \left(1 - \frac{A_{n}}{A_{po}}\right)$$

    Where `A_n` = area of anchor plate, `A_po` = area of concrete immediately behind plate.

*   **Bursting Reinforcement Area (`A_br`):**

$$A_{br} = \frac{F_{br}}{f_{st}}$$

    `f_st` = permissible stress in steel (usually 0.8 `f_y`).

*   **Spalling & Splitting:** Provide distributed reinforcement (mesh or spirals) as per code guidelines.

*   **Reinforcement Layout:** Bursting bars placed horizontally in two layers near top/bottom. Spalling bars vertical. Splitting ties around duct.

• 6.3 Stress Distribution in End Blocks

  • Idealized: Strut-and-tie model. Prestress force spreads through concrete at an angle (≈ 1:2 to 1:4).

  • Stress Path: From anchor plate → compressive struts → bearing on end face.

  • Design: Ensure concrete compressive stress < 0.3 f_ci (at transfer). Provide reinforcement for tensile stresses.

7.0 CONTINUITY & MOMENT DISTRIBUTION

• 7.1 Methods of Achieving Continuity

  1. Monolithic Construction: Cast deck slab continuously over supports.

  2. Link Slabs: Post-tensioned slab over supports without bearings.

  3. Cable Linkage: Post-tensioning through top flange over supports.

  4. Sealing of Joints: For precast segments, use epoxy & post-tensioning.

• 7.2 Primary, Secondary & Resultant Moments

  • Primary Moment (M_p): Moment due to prestress force about centroid of section (P * e). Varies with tendon profile.

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

  • Resultant Moment (M_r): M_r = M_p ± M_s. Sign depends on direction.

  • Significance: In indeterminate structures, M_s can be large and opposite to M_p. Design must consider M_r.

• 7.3 Concordant Cable Profile & Linear Transformation

  • Concordant Profile: A tendon profile that produces zero secondary moment (M_s = 0) for a given loading. The tendon profile is parabolic with eccentricities proportional to primary moment diagram.

  • Linear Transformation: Changing tendon profile by a straight line (y = ax + b) does not change support reactions or primary moments, but introduces secondary moments. Concordant profile is a special case where M_s = 0.

8.0 SHEAR & TORSION RESISTANCE

• 8.1 Shear Resistance of Uncracked Section

  • At supports, section often uncracked.

  • Shear Stress (τ_v):

$$\tau_v = \frac{V}{b j_0} + \frac{P_e e}{j_0 Z}$$

    Where `j_0` ≈ `0.9 d` for rectangular sections.

*   **Permissible Shear Stress (`τ_c`):** Given in IS 1343 based on `f_ck` and `p_t` (percentage of steel).

*   **Check:** `τ_v ≤ τ_c`.

• 8.2 Effect of Prestress on Shear Capacity

  • Prestress introduces compressive stress, increasing shear capacity by:

    1. Reducing principal tensile stress.

    2. Increasing concrete compressive strength (via confinement).

  • IS 1343: Permissible shear stress can be increased by (σ_cp / 10) where σ_cp is compressive stress due to prestress at the section.

• 8.3 Calculation Example

  • Given parabolic cable, UDL w, concrete strength f_c.

  • At support: V = wL/2.

  • Compute τ_v including prestress effect.

  • Compare with enhanced τ_c.

9.0 COMPOSITE CONSTRUCTION

• 9.1 Composite Sections

  • Pre-tensioned beam (or girder) + cast-in-situ slab.

  • Two Stages:

    1. Stage 1 (Precast): Beam acts alone under self-weight & prestress.

    2. Stage 2 (Composite): Slab hardens, composite section acts under superimposed loads.

  • Modular Ratio (m): E_s / E_c for composite action analysis.

• 9.2 Differential Shrinkage

  • Precast beam shrinks before slab is cast. Slab shrinks differently.

  • Causes Restraint Stresses: At interface, differential shrinkage ε_sh induces stresses.

  • Calculation:

    • Force Equilibrium: P_beam + P_slab = 0.

    • Compatibility: ε_beam - ε_slab = ε_sh.

    • Solve for stresses:

$$\sigma_{beam} = \frac{\epsilon_{sh} \cdot E_{beam} \cdot A_{slab}}{A_{beam} + m A_{slab}}$$

$$\sigma_{slab} = -\frac{\epsilon_{sh} \cdot E_{slab} \cdot A_{beam}}{A_{beam} + m A_{slab}}$$

    *   Where `m = E_beam / E_slab`.

• 9.3 Analysis under Service Loads

  • Calculate stresses at critical sections for both stages.

  • Use transformed section method for composite section.

  • Check tensile/compressive stresses against limits.

10.0 PARTIAL PRESTRESSING

• 10.1 Concept & Definition

  • Partial Prestressing: Prestress force is less than that required to eliminate all tensile stresses under service loads. Allows controlled tensile stress and cracking.

  • Class 2 & 3 as per IS 1343.

• 10.2 Merits & Demerits

  • Merits: Economical (less steel), easier construction, better for moment redistribution, reduces risk of brittle failure.

  • Demerits: Cracks may appear, requires careful crack control, long-term deflection may be higher.

• 10.3 Methods of Achieving Partial Prestressing

  1. Reduced Prestressing Force: Use less P_e than required for full prestress.

  2. Use of Non-prestressed Reinforcement: Combine prestressed tendons with mild steel/HT steel rebars.

  3. Controlled Cracking: Allow tensile stress up to f_t (modulus of rupture) or limit crack width (< 0.2 mm).

11.0 DEFLECTION CONTROL

• 11.1 Factors Influencing Deflection

  • Span & Depth: Δ ∝ L² / d.

  • Prestress Level: Eccentricity e and P_e affect upward deflection.

  • Loading: Magnitude and type (UDL, point load).

  • Creep & Shrinkage: Increase long-term deflection.

  • Modulus of Elasticity: E_c affects stiffness.

  • Support Conditions: Simply supported, continuous.

• 11.2 Short-term vs. Long-term Deflection

  • Short-term (Δ_i): Immediate elastic deflection under load.

  • Long-term (Δ_f): Δ_f = Δ_i (1 + φ) where φ = creep coefficient + additional due to shrinkage & relaxation.

  • Calculation: Use IS 456/1343 methods or effective moment of inertia for cracked sections.

• 11.3 Control of Deflections

  • Limit Span/Depth Ratio: As per code (e.g., for simply supported PSC beam, L/d ≤ 30-35).

  • Adequate Prestress Level: Ensure sufficient upward deflection to counteract dead load.

  • Use High-Strength Concrete: Higher E_c reduces deflection.

  • Control Creep & Shrinkage: Use low water-cement ratio, proper curing.

12.0 SHORT NOTES & CONCEPTUAL TOPICS

• 12.1 Stress Concept in Prestress

  • Prestress induces compressive stress in concrete.

  • At Transfer: σ_c = -P_i/A ± P_i e / Z.

  • Under Service Load: Superposition: σ_c = σ_prestress + σ_load.

  • Goal: Keep tensile stress ≤ permissible (often zero for full prestress).

• 12.2 Tendon and Tendon Profile

  • Tendon: High-strength steel element (wire, strand, bar).

  • Profile: Path of tendon centroid.

    • Straight: Constant e.

    • Parabolic: e = e_max [1 - (2x/l)²] for UDL.

    • Harped: Straight segments with kinks.

  • Selection: Based on moment diagram (parabolic for UDL, harped for varying moment).

• 12.3 Guyon's Method

  • Method for analyzing continuous prestressed beams.

  • Treats prestress as a series of equivalent loads (P*d²y/dx²).

  • Solves for primary moments using moment distribution or other methods.

  • Then computes secondary moments from compatibility.

• 12.4 Concept of Stress Distribution in End Block

  • Prestress force spreads from anchor plate into concrete.

  • Strut-and-Tie Model: Compressive struts at angles (1:2 to 1:4) to axis.

  • Stress Variation: Highest near anchor, decreases along length.

  • Design: Check concrete compressive stress (< 0.3 f_ci), provide bursting/spalling reinforcement.

• 12.5 Types of Flexural Failures

  1. Under-reinforced: Steel yields → large deflection → warning. Ductile.

  2. Over-reinforced: Concrete crushes suddenly. Brittle, avoid.

  3. Balanced: Both yield/crush simultaneously. Limit state.

• 12.6 Pre-tensioned vs. Post-tensioned Systems

  • | Aspect | Pre-tensioning | Post-tensioning |

  |---------|----------------|-----------------| | Tensioning Time | Before concreting | After hardening | | Bond | Direct adhesion | Grouted or unbonded | | Span | Short (< 30m) | Long (> 30m) | | Location | Precast yard | In-situ | | Losses | Lower | Higher (friction, slip) | | Anchorage | Bond only | Mechanical anchorages |

• 12.7 Why Mild Steel Cannot be Used for Prestressing

  • Low Yield Strength (≈ 250 N/mm²): Requires very high prestress force → large section.

  • High Relaxation: Loses prestress quickly.

  • Poor Stress-Strain Curve: No distinct yield point, low ultimate strain.

  • High Creep: Under sustained stress, creeps significantly → loss of prestress.

  • High Ductility: Not suitable for high stress.

• 12.8 Load Balancing Concept

  • Shape tendon to exert upward force balancing applied load.

  • For UDL w, parabolic tendon with e_max = w l²/(8P) gives zero moment at midspan.

  • Reduces required section size for moment resistance.

• 12.9 Concordant Cable & Linear Transformation

  • Concordant Cable: Profile that causes zero secondary moment in continuous beam.

  • Linear Transformation: Changing profile by y = ax + b keeps primary moments & reactions unchanged, but introduces secondary moments. Concordant profile is a special linear transformation.

• 12.10 Primary vs. Secondary Moments

  • Primary Moment (M_p): Directly from prestress force about centroid (P*e). Varies with tendon profile.

  • Secondary Moment (M_s): Induced by support reactions due to continuity. Ensures compatibility.

  • Resultant (M_r): M_r = M_p ± M_s. Design must consider M_r.

EXAM TIPS & COMMON PITFALLS:

• Loss Calculations: Always use effective prestress (P_e) for service stress checks and initial prestress (P_i) for calculating number of wires. Remember P_e = P_i × Loss Ratio.

• Ultimate Moment Capacity: For T-sections, first check if neutral axis is within flange (x_u ≤ D_f). If not, use formula considering web contribution.

• Anchorage Zone: Distinguish bursting (horizontal, perpendicular to force), spalling (vertical, near face), and splitting (radial, around duct). Reinforcement areas are calculated separately.

• Composite Sections: Differential shrinkage causes tension in precast beam and compression in cast-in-situ slab (if beam shrinks more first). Use modular ratio m.

• Shear in Uncracked Sections: Include prestress effect in shear stress calculation: τ_v = V/(b j_0) + (P_e e)/(j_0 Z).

• Concordant Profile: Only possible for statically indeterminate beams. For determinate beams, any profile gives M_s = 0.

• Design Problems: Clearly state assumptions (loss ratio, modular ratio, stress block parameters). Show all steps: self-weight calculation, load combination, stress checks at critical sections (usually midspan for +M, support for -M).