Prestressed Concrete Structures (CE-702 (A)) - Unit 2 Short Notes

UNIT 2: PRESTRESSED CONCRETE STRUCTURES

1.0 FUNDAMENTALS & CLASSIFICATION

1.1 Definition & Fundamental Concept

• Prestressing: Application of a predetermined force (prestress) to a concrete member to induce compressive stresses that counteract tensile stresses from external loads.

• Core Objective: To keep concrete under compression throughout its service life, utilizing its high compressive strength and minimizing tensile cracks.

1.2 Classification of Prestressed Concrete

1.3 Systems of Prestressing (Sketches Essential)

• Freyssinet System: Uses high-strength wires (5-7mm) grouped in cables. Anchored by conical wedges driven into a ferrule. Key for post-tensioning.

• Gifford-Udall System: Uses single-strand post-tensioning. Tendon (7-wire strand) is pulled through a duct and anchored by a split cone and wedge.

• Hoyt (Hoyer) System: Pre-tensioning system. Uses long, heavy steel bars (not wires) tensioned by hydraulic jacks. Anchored by nut and washer on the bar.

1.4 Merits & Demerits (vs. R.C.C.)

1.5 Basic Assumptions (IS 1343)

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

2. Concrete is homogeneous, isotropic.

3. Prestressing steel behaves linearly elastic up to its yield strength.

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

5. Perfect bond exists between steel and concrete at all stages (for bonded tendons).

6. The modular ratio (m) is constant.

[!TIP] Exam Focus: Classification questions are frequent. Be ready to sketch Freyssinet/Hoyt systems and differentiate pre/post-tensioning on points like equipment, applications, losses.

2.0 STRESS ANALYSIS & DISTRIBUTION

2.1 Stress Concept in Prestressing

Stresses are calculated at critical sections under three stages:

1. Transfer Stage: Prestress applied to concrete (consider initial losses).

2. Service Stage: Prestress + dead load + live load (consider all losses).

3. Ultimate Stage: Prestress (often neglected) + factored loads for strength.

2.2 Stresses at a Section

For a beam with prestressing force P_e (effective) and eccentricity e:

• Stress at Top Fibre: $$\displaystyle \sigma_t = \frac{P_e}{A} - \frac{P_e e}{Z_t} - \frac{M}{Z_t} $$

• Stress at Bottom Fibre: $$\displaystyle \sigma_b = \frac{P_e}{A} + \frac{P_e e}{Z_b} + \frac{M}{Z_b} $$

Where:

• A = Cross-sectional area

• Z_t, Z_b = Section moduli for top/bottom

• M = External bending moment (consider sign convention)

2.3 Load Balancing Concept

• Theory: The eccentric prestressing force creates a uniform compressive stress over the entire section. If this stress equals the stress due to an upward distributed load, that load is "balanced."

• Balancing Load Intensity: $$\displaystyle w_b = \frac{P_e e}{I} \cdot y $$ (for parabolic profile, $$\displaystyle w_b $$ is constant).

• Application: For a simply supported beam with parabolic tendon, the balanced load equals the UDL it can carry without tension.

• For Continuous Beams: Balancing load induces secondary moments.

2.4 Concordant Cable & Linear Transformation

• Concordant Cable: A tendon profile that produces no secondary moments in a statically indeterminate structure. Its profile is such that the primary moment diagram is similar to the moment diagram due to external loads.

• Linear Transformation: Changing the tendon profile by adding a linear (straight) component (i.e., shifting the profile up/down by a constant amount) does not change the primary moments but introduces secondary moments. Used to optimize profiles.

[!TIP] Exam Focus: Numerical problems on stress calculation at a section and load balancing are very common. Understand the sign convention for eccentricity and moments.

3.0 FLEXURAL DESIGN & ANALYSIS (VERY HIGH FREQUENCY)

3.1 Ultimate Moment Capacity (IS 1343) For Rectangular Section (Pre/Post-tensioned):

1. Assume rectangular stress block: $$\displaystyle 0.36 f_{ck} x_u $$ for concrete.

2. Equilibrium: $$\displaystyle P_{po} = 0.36 f_{ck} b x_u + f_{ps} (A_{ps} - A'_{ps}) $$

   (For bonded tendons, $$\displaystyle f_{ps} $$ is stress in steel at ULS).

3. Depth of NA: $$\displaystyle x_u = \frac{P_{po}}{0.36 f_{ck} b} $$ (if $$\displaystyle A'_{ps} $$ is negligible).

4. Lever arm: $$\displaystyle z = d - 0.42 x_u $$ (for $$\displaystyle x_u \leq 0.44 d $$).

5. Ultimate Moment: \boxed{M_u = P_{po} \cdot z = 0.36 f_{ck} b x_u (d - 0.42 x_u)}

For T-Section (Flanged Beam):

• Check if NA lies in flange or web.

• If $$\displaystyle x_u \leq t_f $$ (flange thickness): Use flange width b_f.

• If $$\displaystyle x_u > t_f $$: Consider contribution from both flange and web. Solve for $$\displaystyle x_u $$ iteratively.

3.2 Design for Flexure

1. Estimate P and e from service stage stress limits (compressive & tensile).

2. Check P against $$\displaystyle P_{po} $$ from ultimate moment capacity.

3. Select tendon profile (parabolic for UDL) and determine eccentricity at mid-span.

4. Verify stresses at transfer (consider initial losses) and service stages (consider all losses).

3.3 Types of Flexural Failures

[!TIP] Exam Focus: Ultimate moment capacity calculations (both rectangular & T-section) are very high frequency. Practice step-by-step as per IS 1343 stress block. Remember to check limits for $$\displaystyle x_u $$.

4.0 SHEAR & DEFLECTION

4.1 Shear Resistance

• Shear Stress (Uncracked): $$\displaystyle \tau = \frac{V}{b j} $$ (similar to RCC, but j ≈ 0.9d).

• Shear Stress (Cracked): Use transformed section. Prestressing force contributes vertical component if tendon is inclined.

• Design: Concrete shear strength + contribution from prestress (if inclined) + shear reinforcement (stirrups). IS 1343 provides expressions for $$\displaystyle \tau_c $$.

4.2 Deflection

• Factors: Effective modulus ($$\displaystyle E_{ce} = E_c / (1+\phi) $$), creep, shrinkage, steel relaxation (losses), span/depth ratio.

• Short-term: Calculated using $$\displaystyle E_c $$ and moment-area/conjugate beam methods.

• Long-term: Includes effects of creep & shrinkage. Use effective modulus $$\displaystyle E_{ce} $$.

• Calculation: $$\displaystyle \Delta = \int M \bar{m} dx / (E_{ce} I) $$ (moment-area). Or use conjugate beam with $$\displaystyle M/E_{ce}I $$ as load.

• Limits: Span/250 for total load, Span/350 for live load (typical).

[!TIP] Exam Focus: Distinguish between short-term and long-term deflection clearly. Know the use of effective modulus $$\displaystyle E_{ce} $$.

5.0 ANCHORAGE ZONES & END REINFORCEMENT (VERY HIGH FREQUENCY)

5.1 Anchorage Zone Stress Distribution

• Theory: Stress disperses from concentrated anchorage force over a dispersion angle (typically 1:1 or 1:1.5 horizontal:vertical). Modeled as Boussinesq's problem (point load on elastic half-space).

• Stresses: Bursting (tensile stress perpendicular to force), Spalling (compressive stress near surface), Splitting (tensile stress along axis).

5.2 Design of End Blocks (IS 1343)

1. Bursting Force ($$\displaystyle F_{bst} $$): Tensile force to resist splitting. $$\displaystyle F_{bst} = P_u \left(1 - \frac{A_{nb}}{A_{br}}\right) $$

   • $$\displaystyle P_u $$ = Ultimate anchorage force

   • $$\displaystyle A_{nb} $$ = Net area of anchor plate

   • $$\displaystyle A_{br} $$ = Area of end block at section where bursting considered.

2. Spalling Force ($$\displaystyle F_{spl} $$): Compressive force near edges.

3. Reinforcement: Provide longitudinal bars to resist $$\displaystyle F_{bst} $$ and transverse ties for $$\displaystyle F_{spl} $$. Design as per IS 1343 equations.

5.3 End Zone Reinforcement

• Purpose: Contain bursting/spalling stresses, transfer force from anchorage to concrete.

• Detailing: Closed loops or helical reinforcement around tendon anchorage. Must extend beyond high-stress zone.

[!TIP] Exam Focus: Design of anchorage zone reinforcement (bursting force calculation) is a very high frequency 10-14 mark question. Memorize the formula and procedure.

6.0 CONTINUITY & MOMENT ANALYSIS

6.1 Methods of Achieving Continuity

• Post-tensioning through Diaphragms: Tendons pass through diaphragms at supports and are anchored.

• Couplers: Mechanical connectors join tendons between segments (e.g., in segmental bridges).

• Cast-in-situ Joints: Ends of precast beams are extended and concreted together with continuity tendons.

6.2 Primary, Secondary & Resultant Moments

• Primary Moment ($$\displaystyle M_p $$): Moment due to prestress force about the centroid of the section, assuming the structure is simply supported.

• Secondary Moment ($$\displaystyle M_s $$): Moment induced by static indeterminacy to satisfy compatibility (deflection continuity).

• Resultant Moment ($$\displaystyle M_r $$): $$\displaystyle M_r = M_p + M_s $$. This is the actual moment in the continuous beam.

• Calculation Methods:

  1. Consistent Deformation (Force) Method: Release indeterminacy, apply compatibility.

  2. Moment Distribution: Apply fixed-end moments due to prestress, then distribute.

6.3 Effect of Prestress on Continuous Beams

• Stress Redistribution: Prestress reduces hogging moments at supports and increases sagging in spans.

• Moment Diagrams: Primary moment diagram is similar to tendon profile. Secondary moments oppose primary at supports and add in spans.

[!TIP] Exam Focus: Understand the concept of primary vs. secondary moments. Know that concordant cable gives $$\displaystyle M_s = 0 $$.

7.0 COMPOSITE CONSTRUCTION & PARTIAL PRESTRING

7.1 Composite Construction

• Behavior: Precast prestressed beam acts as a "propped" beam when cast-in-situ slab hardens.

• Differential Shrinkage & Creep: Cast-in-situ slab shrinks/creeps more than precast beam → induces tensile stress in slab and compressive stress in beam.

• Interface Stress: $$\displaystyle \sigma = E_s \cdot \Delta \epsilon $$ (where $\Delta \epsilon$ is differential strain).

• Shear Connection: Shear studs or roughness to ensure composite action.

7.2 Partial Prestressing

• Definition: Allows controlled tensile stresses in concrete under service loads (Class 2/3 per IS 1343).

• Merits: Reduced prestress losses, lower cost, better crack control than RCC, easier construction.

• Demerits: Requires careful control of crack widths, less durable than full prestressing.

• Methods of Achieving:

  1. Reduced Prestress: Lower initial prestressing force.

  2. Use of Mild Steel: Non-prestressed reinforcement carries tension.

  3. Hybrid Reinforcement: Combination of high-strength prestressing steel and mild steel.

• Design: Based on permissible tensile stress limits at service stage.

[!TIP] Exam Focus: Differential shrinkage stress calculation in composite sections is common. Partial prestressing philosophy vs. full prestressing is a short note favorite.

8.0 LOSSES OF PRESTRESS (VERY HIGH FREQUENCY)

8.1 Classification

8.2 Detailed Calculation

• Loss due to Elastic Deformation ($$\displaystyle \Delta f_{p,el} $$):

  • Prestress causes immediate shortening in concrete.

  • $$\displaystyle \Delta f_{p,el} = \frac{A_p}{A_c} \cdot f_{pi} $$ (for pre-tensioning, where $$\displaystyle f_{pi} $$ is initial stress).

  • For post-tensioning, depends on sequence.

• Loss due to Friction ($$\displaystyle \Delta f_{p,f} $$):

  • $$\displaystyle \Delta f_{p,f} = f_{pi} \left(1 - e^{-(\mu \theta + k x)}\right) $$

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

• Loss due to Wedge Draw-in ($$\displaystyle \Delta f_{p,wd} $$): Fixed value based on anchorage type (e.g., 3-5 mm slip).

• Loss due to Creep & Shrinkage ($$\displaystyle \Delta f_{p,cs} $$):

  • Empirical: $$\displaystyle \Delta f_{p,cs} = E_p \epsilon_{cs} \frac{A_c}{A_p} \left(1 + \frac{A_p}{A_c} \frac{E_p}{E_c}\right)^{-1} $$

  • $$\displaystyle \epsilon_{cs} $$ = estimated creep + shrinkage strain.

• Loss due to Relaxation ($$\displaystyle \Delta f_{p,r} $$): Given as % of initial stress from steel manufacturer curves.

8.3 Effective Prestress

• $$\displaystyle f_{pe} = f_{pi} - \sum \Delta f_p $$ (Sum of all losses).

• Loss Ratio: $$\displaystyle \eta = f_{pe} / f_{pi} $$.

[!TIP] Exam Focus: Numerical problems on loss due to elastic deformation (especially for successive tensioning) and friction loss are very high frequency. Practice step-by-step.

9.0 DESIGN OF PRESTRESSED MEMBERS (Application Focus)

9.1 Design Methodology (Post-tensioned Girder)

1. Input Data: Span, loads (DL, LL), $$\displaystyle f_{ck} $$, $$\displaystyle f_{pk} $$, $$\displaystyle f_{pi} $$, permissible stresses at transfer/service, loss ratio $\eta$, cover.

2. Section Selection: Assume I/T-section, estimate depth (span/20 to span/30).

3. Calculate Required $$\displaystyle P_e $$:

   • From service stage: Top/bottom stress limits → get $$\displaystyle P_e $$ & $e$.

   • From transfer stage: Check with initial prestress $$\displaystyle f_{pi} $$.

4. Determine $$\displaystyle A_p $$: $$\displaystyle A_p = P_e / f_{pe} $$.

5. Select Tendons: Number & size of strands/wires. Check $$\displaystyle A_p $$ available.

6. Check Ultimate Moment: Verify $$\displaystyle M_u \geq M_{factored} $$.

7. Check Shear & Deflection.

9.2 Design of Solid Slab Bridge (Class AA Loading)

• Loading: IRC Class AA (70R wheel load or 70U tracked vehicle). Calculate design shear & moment.

• Effective Width: $$\displaystyle b_{eff} = l/4 + b_w $$ (for simply supported) or as per IRC.

• Thickness: Minimum 200mm for bridges. Calculate from moment & shear.

• Prestressing: Usually pre-tensioned slabs. Tendons straight or slightly draped.

• Reinforcement: Non-prestressed reinforcement for temperature/shrinkage.

[!TIP] Exam Focus: Full design problems (14 marks) are common. Present data, assumptions, calculations clearly. Always check both service and ultimate stages.

10.0 SHORT NOTES & CONCEPTUAL TOPICS

10.1 Tendon & Tendon Profile

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

• Profile Selection Criteria: Load type (parabolic for UDL), construction method, friction losses, moment diagram compatibility.

10.2 Guyon's Method

• For redistribution of moments in continuous prestressed beams.

• Assumes linear transformation of tendon profile. Adjusts secondary moments by adding a linear component to the concordant profile.

10.3 Stress Distribution in End Block

• Bursting: Tensile stress perpendicular to force → requires longitudinal reinforcement.

• Spalling: Compressive stress near end face → requires transverse ties.

• Splitting: Tensile stress along axis → requires confining reinforcement.

10.4 Pre-tensioning vs. Post-tensioning

10.5 Why Mild Steel Cannot Be Used?

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

• High relaxation (loss of prestress over time) → not suitable for sustained prestress.

• High creep under sustained stress.

10.6 Hoyt (Hoyer) System

• Principle: Pre-tensioning using heavy steel bars (not wires).

• Process: Bars are tensioned by hydraulic jacks against a dead end. Concrete cast. Bars are released by unscrewing nuts.

• Sketch: Show bar, nut, washer, concrete beam, end anchorage.

10.7 Response to Arbitrary Force

• Use Duhamel's Integral for forced vibration response:

$$u(t) = \frac{1}{m\omega_d} \int_0^t F(\tau) e^{-\xi \omega (t-\tau)} \sin \omega_d (t-\tau) d\tau$$

• For step-by-step integration (e.g., Newmark-beta), use numerical methods.

[!TIP] Exam Focus: Short notes on Hoyt system, Guyon's method, and differences between pre/post-tensioning are recurring. Keep definitions crisp with key points.