Prestressed Concrete Structures (CE-702 (A)) - Unit 4 Short Notes

UNIT 4: PRESTRESSED CONCRETE STRUCTURES

Based on RGPV CE-702(A) Nov 2022 Paper Analysis

I. INTRODUCTION & FUNDAMENTAL CONCEPTS

Definition & Philosophy

• Prestressing: Artificially inducing compressive stresses in concrete before it service loads are applied, to counteract tensile stresses caused by those loads.

• Philosophy: Concrete is strong in compression but weak in tension. Prestressing puts the entire section into compression, keeping tensile stresses below zero (or within permissible limits) under load.

Classification of Prestressed Concrete

Historical Systems

• Freyssinet System:

  • Used high-strength steel wires (5mm dia) grouped in bundles (cables).

  • Pre-tensioning method. Wires tensioned by hydraulic jacks, anchored by wedges.

  • Concrete cast around tensioned wires. After hardening, wires are released, transferring stress via bond.

  • [Sketch: Show beam with multiple 5mm wires bundled, anchored at ends by conical wedges.]

• Hoyes System:

  • Used high-strength steel bars (25-40mm dia).

  • Post-tensioning method. Bars threaded at ends, tensioned after concrete hardens.

  • Anchored by nuts on threaded ends.

  • [Sketch: Show beam with large diameter bars passing through ducts, anchored by nuts at beam ends.]

Basic Assumptions (IS 1343)

1. Concrete obeys Hooke's law up to the working stress level (linear stress-strain).

2. Steel and concrete act perfectly bonded (no slip).

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

4. Steel stress remains constant along the length (for pre-tensioning) or changes only due to friction, slip, and deformation losses.

5. Concrete has zero tensile strength for ultimate strength calculations.

Merits & Demerits vs. RCC

Why Mild Steel Cannot Be Used?

• Mild steel has low yield strength (~250 MPa).

• To induce sufficient compressive stress in concrete, a very high prestressing force (P) is needed.

• High P would cause excessive elastic shortening and high losses.

• Prestressing steel requires: High tensile strength (>1000 MPa), low relaxation, good ductility, and high bond strength.

[!TIP] Exam Focus: Classification (4m) and Freyssinet/Hoyes (4m) are frequent 4-mark questions. Be ready to sketch both systems.

II. STRESS ANALYSIS IN PRESTRESSED BEAMS

Stresses in Simply Supported Beam (Concentric Tendon)

• At Mid-span (under UDL):

  • Prestress: Uniform compressive stress: $$\displaystyle \sigma_c = -\frac{P}{A} $$

  • Bending Stress (due to load wL²/8): Linear distribution, tension at bottom.

  • Resultant: Compressive stress at top, reduced compression/tension at bottom.

• At Ends (support):

  • Prestress: Uniform compression $$\displaystyle \frac{P}{A} $$.

  • Bending Moment ≈ 0: Stress remains uniform compression.

  • [Sketch: Show rectangular section, stress block uniform at ends, parabolic at mid-span for load alone, and resultant combination.]

Effect of Eccentric Prestressing

• Tendon placed with eccentricity e from centroid.

• Introduces a secondary moment $$\displaystyle M_p = P \cdot e $$.

• At Mid-span: Prestress stress becomes $$\displaystyle \sigma_c = -\frac{P}{A} \pm \frac{P e}{Z} $$ (compression top, tension bottom if e is positive).

• At Ends: Stress is $$\displaystyle -\frac{P}{A} \pm \frac{P e}{Z_{end}} $$. Since $$\displaystyle Z_{end} $$ is large, bending stress is small; stress remains compressive.

Load Balancing Concept

• Idea: Profile the tendon to induce a moment that exactly balances the moment due to external loads.

• For a parabolic tendon under UDL w:

  • Equivalent upward load: $$\displaystyle w_{eq} = \frac{8 P e}{L^2} $$ (for parabola with e at mid-span, zero at ends).

  • If $$\displaystyle w_{eq} = w $$, then no net bending moment in the beam. The beam behaves as if under pure axial load.

• Application: Used to design cable profiles for continuous beams to minimize moments.

• [Sketch: Show beam with parabolic tendon, upward equivalent load vector, and balanced external UDL.]

Stress Concept & Equivalent Loads

• Equivalent Loads: The prestressing force P with a given profile can be replaced by an equivalent loading (vertical forces & moments) on the concrete section.

  • Concentric tendon → Uniform axial compression.

  • Eccentric tendon → Axial compression + bending moment.

  • Parabolic tendon → Axial compression + uniformly distributed upward load (as above).

• Service Stress Diagram: Sum of stresses from:

  1. Prestress (including losses)

  2. Equivalent loads from tendon profile

  3. External dead/live loads

[!TIP] Exam Focus: Load balancing (10m) is a key concept. Be able to derive $$\displaystyle w_{eq} = \frac{8Pe}{L^2} $$ for a parabolic profile.

III. FLEXURAL DESIGN & ULTIMATE STRENGTH (IS 1343)

Ultimate Moment Capacity - Rectangular Section

Given: b, D, fck, Aps, fpu (or fpy), d (effective depth). Steps:

1. Depth of Neutral Axis (NA):

$$x_u = \frac{A_{ps} f_{pu}}{0.36 f_{ck} b} \quad \text{(for } x_u \leq 0.48D \text{ in IS 1343)}$$

If $$\displaystyle x_u > 0.48D $$, treat as **flanged section** or use strain compatibility.

2. Lever Arm:

$$z = d - 0.42 x_u$$

3. Ultimate Moment:

$$M_u = A_{ps} f_{pu} z$$

\boxed{M_u = A_{ps} f_{pu} \left( d - 0.42 \frac{A_{ps} f_{pu}}{0.36 f_{ck} b} \right)}

Ultimate Moment Capacity - T-Section (Flanged)

1. Check if NA lies in flange:

$$\text{If } \frac{A_{ps} f_{pu}}{0.36 f_{ck} b_f} \leq t_f \text{ (flange thickness)}$$

Then use rectangular formula with `b = bf`.

2. If NA in web:

$$x_u = \frac{A_{ps} f_{pu}}{0.36 f_{ck} b_w} \quad \text{(where } b_w = \text{web width)}$$

Compressive force in flange: $$\displaystyle C_f = 0.36 f_{ck} b_f t_f $$

Compressive force in web: $$\displaystyle C_w = 0.36 f_{ck} b_w (x_u - t_f) $$

$$M_u = A_{ps} f_{pu} z \quad \text{with } z = d - 0.42 x_u$$

Permissible Stresses (IS 1343)

• $$\displaystyle f_{ci} $$ = characteristic strength at transfer.

• $$\displaystyle f_{ct} $$ = permissible tensile stress (typically $$\displaystyle 0.8 \sqrt{f_{ck}} $$ MPa for moderate prestress).

Design Procedure (Given Loads & Stress Limits)

1. Calculate effective prestress after losses: $$\displaystyle f_{pe} = \text{Loss Ratio} \times f_{pi} $$.

2. For working stress design:

   • Top fiber stress: $$\displaystyle \sigma_{top} = -\frac{P_e}{A} - \frac{P_e e}{Z_{top}} - \frac{M}{Z_{top}} \geq -\sigma_{c,allow} $$

   • Bottom fiber stress: $$\displaystyle \sigma_{bot} = -\frac{P_e}{A} + \frac{P_e e}{Z_{bot}} + \frac{M}{Z_{bot}} \leq \sigma_{t,allow} $$ (often negative/compressive).

3. For ultimate strength check: Ensure $$\displaystyle M_u \geq M_{factored} $$.

4. Iterate on P (or number/size of tendons) and e to satisfy both service and ultimate criteria.

[!TIP] Exam Focus: 14m design problems require integrating stress limits at transfer & service with ultimate moment capacity and loss ratio. Always use f_pe = 0.8 * f_pi (typical) if not specified.

IV. SHEAR, ANCHORAGE ZONES & END REINFORCEMENT

Shear Resistance of Uncracked Section (at supports)

• For a prestressed beam with no shear reinforcement, the shear stress is:

$$\tau_v = \frac{V}{b j}$$

where `j ≈ 0.9d` for rectangular sections.

• Permissible shear stress (IS 1343) for uncracked section:

$$\tau_{c,allow} = 0.3 \sqrt{f_{ci}} \quad \text{(at transfer)}$$

or

$$\tau_{c,allow} = 0.3 \sqrt{f_{ck}} \quad \text{(at service)}$$

• Design: Provide shear reinforcement (stirrups) if $$\displaystyle \tau_v > \tau_{c,allow} $$.

Anchorage Zone Stress & Bursting Forces

• Concept: When prestressing force P is anchored at the end, it spreads over a transition (end block) length. Stress distribution is non-uniform.

• Bursting Force: Tensile force developed in the transverse direction (perpendicular to beam axis) due to stress concentration near the anchor plate. Must be resisted by horizontal reinforcement.

• Spalling Force: Compressive force causing concrete to spall (flake off) at the loaded face. Less critical.

• End Block: Enlarged concrete section at the anchorage to reduce bearing stress and spread the force.

Design of Anchorage Zone Reinforcement (IS 1343)

Given: Beam size B x D, anchor plate size a x a, cable force P, number & position of cables. Steps:

1. Bursting Tensile Force ($$\displaystyle F_{bst} $$):

$$F_{bst} = P \left(1 - \frac{a}{B}\right) \quad \text{(for single cable, centered)}$$

For multiple cables, use **IS 1343 Fig. 4** (stress distribution) or **Bursting Moment** method.

2. Provide horizontal reinforcement in the bursting zone (region of high transverse tension) to resist $$\displaystyle F_{bst} $$.

$$A_{st,burst} = \frac{F_{bst}}{0.87 f_y} \quad \text{(steel area)}$$

3. Spalling Reinforcement: Provide if spalling stress exceeds concrete strength.

4. General Reinforcement: Minimum vertical & horizontal bars in entire end block (0.25% of cross-sectional area).

[!TIP] Exam Focus: Anchorage zone design (10m) is a must-solve. Remember: F_bst = P(1 - a/B) for simple case. Reinforcement is provided in the bursting zone (usually a band around the anchor plate).

V. CONTINUITY, MOMENTS & CABLE PROFILES

Achieving Continuity in Continuous Beams

1. Top Cables at Supports: Provide draped/harped tendons with high eccentricity (top) at supports to induce negative moment.

2. Bottom Cables in Spans: Straight or parabolic tendons in spans for positive moment.

3. Post-Tensioning Sequence: Tendons in negative moment regions are stressed first, then those in positive moment regions. This creates continuity.

4. Pre-tensioned Segments: Cast beams as simply supported, then connect with post-tensioned continuity tendons over supports.

Primary, Secondary & Resultant Moments

• Primary Moments ($$\displaystyle M_p $$): Moments caused by the prestressing force alone (considering its eccentricity) if the beam were simply supported. $$\displaystyle M_p = P e $$.

• Secondary Moments ($$\displaystyle M_s $$): Moments induced in the redundant structure (continuous beam) due to restraint against the deformations caused by primary moments. Arise from static indeterminacy.

• Resultant Moment ($$\displaystyle M_r $$): The actual moment in the beam.

$$M_r = M_p + M_s$$

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

Cable Profiles & Transformations

• Concordant Cable Profile: A tendon profile that, when stressed, produces no secondary moments ($$\displaystyle M_s = 0 $$) in a statically determinate structure. For a simply supported beam, any profile is concordant. For continuous beams, finding a concordant profile is complex.

• Linear Transformation: For a statically determinate beam, if the tendon profile is shifted linearly (i.e., all points moved by same amount Δe), the resultant moment diagram remains unchanged. This is because the change in P e is constant (P Δe), adding a constant moment.

• Tendon Profiles:

  • Straight: Constant eccentricity. Simple, but limited moment control.

  • Parabolic: For UDL. $$\displaystyle e(x) = e_{mid} \left[1 - \left(\frac{2x}{L}\right)^2\right] $$. Gives linear moment diagram.

  • Harped: Straight segments with kinks at points of contraflexure. Used for point loads.

[!TIP] Exam Focus: Primary vs. Secondary moments (4m) is conceptual. Linear transformation property is only for determinate structures. Concordant profile is a key design tool.

VI. LOSSES OF PRESTRESS & DEFLECTION

Losses of Prestress

• Total Loss: Sum of all losses. Loss Ratio = $$\displaystyle f_{pe} / f_{pi} $$ (e.g., 0.8 means 20% loss).

Loss due to Elastic Deformation (Successive Tensioning)

• When cables are tensioned one after another, the first cable causes elastic shortening of concrete, reducing stress in already tensioned cables.

• For n cables tensioned in sequence, loss in i-th cable due to shortening from 1 to (i-1):

$$\Delta f_{p,i} = \frac{E_{ps}}{E_c} \cdot \frac{\sum_{j=1}^{i-1} A_{ps,j} f_{pj}}{A_c}$$

where $$\displaystyle f_{pj} $$ is stress in `j-th` cable **before** shortening.

• Simplified (if all cables same):

$$\text{Loss in first cable} = 0$$

$$\text{Loss in last cable} = \frac{m A_{ps} f_{pi}}{A_c} \cdot \frac{(n-1)}{2} \quad (m = E_s/E_c)$$

Deflection in Prestressed Beams

• Factors Influencing:

  1. Prestress force & profile (upward deflection).

  2. External loads (downward deflection).

  3. Creep & shrinkage (increases long-term deflection).

  4. Support conditions.

• Short-term vs. Long-term:

  • Short-term: Immediate elastic deflection under load + prestress.

  • Long-term: Short-term + incremental deflection due to creep (increases deflection) and shrinkage (can increase or decrease depending on profile).

  • Code (IS 1343): Long-term deflection = Short-term deflection × Deflection multiplier (1.5 to 3.0, based on span/depth and environmental conditions).

[!TIP] Exam Focus: Elastic shortening loss (14m) is a complex numerical problem. Set up table for i=1 to n, calculate cumulative shortening. Remember m = E_s/E_c.

VII. COMPOSITE CONSTRUCTION & PARTIAL PRESTRESSING

Composite Construction

• Behavior: Precast prestressed beam (pre-tensioned) + cast-in-situ slab (wet concrete).

• At Transfer: Precast beam acts alone (slab not hardened).

• At Service: Beam and slab act compositely (monolithic action).

• Differential Shrinkage: Cast-in-situ slab shrinks more than precast beam (older, drier). Causes additional tensile stress in the slab and compressive stress in the beam.

• Shrinkage Stress Calculation:

$$\sigma_{sh} = E_s \cdot \Delta \epsilon_{sh} \cdot \frac{A_c}{A_c + n A_s}$$

where $$\displaystyle \Delta \epsilon_{sh} $$ = differential shrinkage strain, `n` = modular ratio.

*   **In Slab:** $$\displaystyle \sigma_{slab} = +E_s \Delta \epsilon_{sh} $$ (tension)

*   **In Beam:** $$\displaystyle \sigma_{beam} = -E_s \Delta \epsilon_{sh} \cdot \frac{A_s}{A_c + n A_s} $$ (compression)

Partial Prestressing (IS 1343 Classes)

• Merits: Reduced prestress force → lower cost, easier construction.

• Demerits: Cracks may occur under service loads, requires careful crack control detailing.

• Methods: Use lower prestress force, or higher strength concrete with same force, or allow higher tensile stress.

[!TIP] Exam Focus: Differential shrinkage stress (10m) is a formula-based problem. Partial prestressing (4m) requires knowing Class 2 vs. 3 differences.

VIII. DESIGN PROBLEMS & APPLICATIONS (INTEGRATION)

Typical 14m Design Problem Flow:

1. Inputs: Span, loads (DL, LL), concrete/steel grades, permissible stresses, loss ratio, cover.

2. Section Selection: Assume b, D (or use I-beam dimensions).

3. Calculate Loads & Moments:

   • Factored moment: $$\displaystyle M_u = 1.5 (M_{DL} + M_{LL}) $$

   • Service moment: $$\displaystyle M_{ser} = M_{DL} + M_{LL} $$

4. Determine Required P_e and e from Service Stress:

   • Use top/bottom stress equations simultaneously.

   • Often assume e = some fraction of D (e.g., 0.3D).

   • Solve for P_e.

5. Check Ultimate Strength: Ensure $$\displaystyle M_u \leq A_{ps} f_{pu} z $$.

6. Calculate P_i (initial prestress): $$\displaystyle P_i = P_e / \text{Loss Ratio} $$.

7. Select Tendons: Choose wire/cable type (e.g., 7mm Freyssinet). Calculate number n:

$$n = \frac{P_i}{A_{ps,one} \times f_{pi}}$$

Check `f_pi ≤ 0.8 f_pu` (initial stress limit).

8. Check Shear & Anchorage if required.

9. Detailing: Cover, spacing, end block size.

[!TIP] Exam Focus: Always start with service stress limits (transfer & working). Then check ultimate strength. Loss ratio is applied to get P_e from P_i.

IX. SHORT NOTES & CONCEPTUAL TOPICS

Tendon & Tendon Profile

• Tendon: Single wire, strand, or bar. Materials: High-tensile steel (carbon), alloy steel, high-strength bars.

• Profile Types:

  • Straight: Constant e. Simple, used for axial members or small beams.

  • Parabolic: For UDL. $$\displaystyle e(x) = e_{mid} \left(1 - \frac{4x^2}{L^2}\right) $$. Gives linear moment.

  • Harped: Straight with kinks. For point loads/moment control.

  • Concordant: Profile causing no secondary moments in determinate structure.

Guyon's Method

• For: Analysis of continuous prestressed beams.

• Idea: Treat prestressing force as equivalent loads + redistribute moments using moment distribution (like Hardy Cross).

• Steps:

  1. Calculate primary moment diagram ($$\displaystyle M_p = P e $$) for each span assuming simple supports.

  2. Apply these as fixed-end moments in a moment distribution analysis.

  3. Carry out distribution until moments converge.

  4. Resultant moment at any section = $$\displaystyle M_p $$ (from step 1) + distributed moment (from step 3).

• Advantage: Handles statically indeterminate structures systematically.

Stress Distribution in End Block (Simplified)

• Assumption: Stress spreads linearly from anchor plate over transition length $$\displaystyle l_t = \sqrt{\frac{B^2}{4} + h^2} $$ (B = width, h = depth).

• Bursting Zone: Region where transverse tensile stress is maximum. Typically within 0.2l_t from loaded face.

• Design Reinforcement: Place horizontal closed loops or spirals in this zone to resist bursting tension.

Types of Flexural Failures

1. Tensile Failure (Under-reinforced): Steel yields first → large deflection, ductile. Desirable.

2. Compressive Failure (Over-reinforced): Concrete crushes before steel yields → sudden, brittle. Avoid.

3. Shear Failure: Diagonal tension crack → sudden. Prevent by shear reinforcement.

4. Bond Failure: Slippage of tendon. Prevent by adequate bond length/anchorage.

Pre-tensioning vs. Post-tensioning

Why Mild Steel Cannot Be Used for Prestressing?

• Low yield strength (~250 MPa) → requires very high prestress force P to induce sufficient concrete compression.

• High P causes:

  • Excessive elastic shortening → high losses.

  • Large anchorage requirements.

  • High bearing stresses on concrete.

• Prestressing steel needs: f_pu ≥ 1000 MPa, low relaxation, good fatigue resistance.

[!TIP] Exam Focus: Short notes (6m/8m) are direct definitions + 1-2 key points. For differences (pre vs. post), use a table. For Guyon's method, state it's for continuous beams and involves moment distribution.

Final Note: This unit is calculation-heavy. Practice numerical problems on:

1. Stress distribution (concentric/eccentric).

2. Load balancing.

3. Ultimate moment (rectangular & T-section).

4. Anchorage zone reinforcement.

5. Losses (especially elastic shortening).

6. Composite section shrinkage stress.

7. Full design problem (14m).