UNIT 2: ADVANCED RCC DESIGN & PRESTRESSED CONCRETE

1.0 RETAINING WALLS

1.1 Cantilever Retaining Wall Design

Forces Acting:

• Earth Pressure: Active pressure calculated using Rankine's theory (for cohesionless soil) or Coulomb's theory (for cohesive soil with slope).

  • Rankine's Active Pressure Coefficient: $$\displaystyle K_a = \tan^2\left(45^\circ - \frac{\phi}{2}\right) $$

  • Pressure at depth $h$: $$\displaystyle P_a = K_a \gamma h $$ (for level backfill, no surcharge).

• Surcharge Pressure: Equivalent uniform pressure $q$ acting over the backfill. Pressure at depth $h$: $$\displaystyle P_q = K_a q $$.

• Water Pressure: Hydrostatic pressure if water table is present, $$\displaystyle P_w = \gamma_w h_w $$.

• Self-weight: Weight of stem, base slab, and soil above heel.

Stability Checks (Must be satisfied):

[!TIP] Exam Focus: Stability checks are always asked. Draw pressure distribution diagrams clearly. Base width is typically 0.6H to 0.7H (H = total height).

Design of Stem:

• Bending Moment: Maximum positive moment at base from lateral earth pressure. Design for this moment.

• Shear: Maximum shear at base. Check shear stress.

• Reinforcement: Main bars (vertical) at tension face. Distribution steel (horizontal). Development length at base crucial.

Design of Base Slab:

• Thickness: Minimum 0.3m or based on shear/bending. Usually 0.4m - 0.5m.

• Reinforcement:

  • Heel Slab: Designed for upward soil pressure (acts as cantilever from stem). Main bars transverse.

  • Toe Slab: Designed for downward load from stem and soil. Main bars transverse.

  • Bottom Steel: Provided in both heel and toe based on bending moment.

• Projections: Heel projection ≥ 0.4H/4. Toe projection ≥ 0.4H/4. Base width B = Heel + Stem width + Toe.

2.0 WATER TANKS (RC CIRCULAR & INTZE)

2.1 Circular Water Tank with Rigid Base

Design Philosophy: Wall is considered as a vertical cantilever fixed at base and free at top. Base is rigid, so no rotation at base.

Forces & Stresses:

• Water Pressure: Linear variation from zero at top to $wH$ at bottom ($w$ = unit weight of water).

• Ring Tension in Wall: Circumferential (hoop) tension per meter height at any depth $h$: $$\displaystyle T_h = \frac{w h^2}{2} $$ (for thin wall, $t \ll R$).

• Base Pressure: Uniform if rigid base and symmetric loading.

Design of Cylindrical Wall:

1. Circumferential Reinforcement (Hoops): Designed for maximum tension at base.

   • Area of steel per meter height: $$\displaystyle A_{st} = \frac{T_h}{0.45 f_y} $$ (working stress method, IS 3370).

   • Spacing of hoops calculated from $$\displaystyle A_{st} \times \text{spacing} = \text{steel area per m} $$.

2. Vertical Reinforcement: For handling construction loads, temperature/shrinkage. Minimum 0.3% of concrete area in each face.

3. Thickness: Based on hoop tension and minimum cover. Usually 150-200mm.

Design of Base Slab:

• Treated as a circular plate fixed at edges (wall). Maximum moment at center for uniform pressure.

• Reinforcement provided in two directions (radial and tangential). Minimum 0.15% of gross section.

Joint Detailing (Wall-Base):

• Keyed or shear key to transfer shear.

• Construction joint with waterstops.

• Vertical reinforcement from wall anchored into base.

2.2 Intze Tank Design

Components: Top dome → Top ring beam → Cylindrical wall → Bottom dome (or conical).

Design Philosophy: Domes act as shells transferring load as membrane stresses (hoop & meridional). Ring beams resist thrust from domes.

Design of Top Dome:

• Meridional Stress (σ_m): $$\displaystyle \sigma_m = \frac{w R}{2t} \left(1 - \frac{y}{R}\right) $$ (for spherical dome, y from crown).

• Hoop Stress (σ_h): $$\displaystyle \sigma_h = \frac{w R}{2t} \left(2\frac{y}{R} - 1\right) $$.

• Reinforcement: Provided in meridional and hoop directions based on stresses. Minimum 0.15% each way.

• Thickness: Based on max stress, usually 75-100mm.

Design of Top Ring Beam:

• Loading: Horizontal thrust from top dome ($$\displaystyle H = w R^2 / 2 $$ for spherical dome).

• Action: Beam subjected to axial compression from thrust and bending from its self-weight + water load.

• Design: For combined axial compression + bending. Check for buckling.

Design of Cylindrical Wall:

• Pressure varies linearly from zero at top ring to $w h$ at bottom.

• Hoop tension: $$\displaystyle T_h = w h^2 / 2 $$ (same as circular tank).

• Vertical reinforcement: For bending from variable pressure.

Bottom Dome: Similar to top dome but usually in compression. May be omitted if tank is large (Intze type has only top dome).

3.0 SILO DESIGN

3.1 Lateral Pressure in Silos (Janssen's Theory)

Key Assumption: Vertical stress increases with depth until it reaches a limiting value due to wall friction.

Vertical Pressure at depth z:

$$\sigma_v = \frac{\gamma R}{\mu} \left(1 - e^{-\mu K z / R}\right)$$

Where:

• $\gamma$ = unit weight of stored material

• $R$ = radius of silo

• $\mu$ = coefficient of friction (wall-material)

• $K$ = lateral pressure ratio (often $$\displaystyle K = 1 - \sin\phi $$ for solids)

Lateral Pressure at depth z:

$$\sigma_h = K \sigma_v$$

[!TIP] Exam Focus: For coal storage, given $\gamma$, $\mu$, diameter. Calculate max vertical & lateral pressures. Pressure is not linear like water.

3.2 Design of Silo Wall (Cylindrical)

• Wall as a Vertical Cantilever: Fixed at base, free at top.

• Bending Moment: Due to varying lateral pressure $$\displaystyle \sigma_h(z) $$. Integrate pressure diagram to get shear, then moment.

  • $$\displaystyle M_{\text{max}} $$ typically at some depth below top, not at base.

• Shear Force: Maximum near base.

• Reinforcement:

  • Hoop bars: Resist tension from lateral pressure (like water tank).

  • Vertical bars: Resist bending moment.

• Stability against Buckling: Check for elastic buckling of thin cylindrical shell under lateral pressure. May require stiffeners if wall is thin.

4.0 BRIDGE DESIGN (SOLID SLAB)

4.1 IRC Class AA Loading

• Standard Wheel Load: 70R (where R = radius of wheel load, min 0.3m). Total load = 70R kN.

• Impact Factor (I): $$\displaystyle I = \frac{24}{6+L} $$ % (L = span in m, for L>3m). Added to live load.

• Distribution Width: For solid slab, load distributes at 45° from edges of wheel. Effective width $$\displaystyle b = 1.2 \times \text{clear distance between wheels} + \text{width of wheel} + 2 \times \text{thickness of slab} $$.

• Effective Span: For simply supported slab, clear span + width of bearing (or c/c of supports).

4.2 Design of Solid Slab Bridge

• Thickness (D): $$\displaystyle D \geq \frac{\text{Span}}{20} $$ (IRC) for simply supported slabs. Also check for shear.

• Loads:

  1. Dead Load (DL): Self-weight + wearing coat (80mm thick, $\gamma$ ≈ 20 kN/m³) + parapet.

  2. Live Load (LL): Class AA wheel loads with impact.

• Analysis: Calculate maximum bending moment and shear for critical loading (usually one lane loaded centrally).

• Reinforcement Design:

  • Longitudinal Bars: Main steel for BM. Provide in bottom layer. Use Fe 415/500.

  • Distribution Steel: Top layer, minimum 0.12% of cross-sectional area (IS 456).

• Checks:

  • Shear: $$\displaystyle \tau_v \leq \tau_{c,\text{max}} $$

  • Bond: $$\displaystyle L_d \leq \text{available development length} $$

  • Deflection: Span/depth ratio check.

• Wearing Coat: Considered as DL. Thickness 80mm (IRC).

5.0 PRESTRESSED CONCRETE – FUNDAMENTALS

5.1 Definition & Need

Prestressing: Artificially introducing compressive stresses in concrete before service loads to counteract tensile stresses.

Need: Concrete weak in tension. Precompression raises neutral axis, delays cracking, increases stiffness.

5.2 Classification

5.3 Systems of Prestressing (Brief)

• Freyssinet: Uses high-strength wires (5-7mm) in bundles, anchored by conical wedges in a metal trumpet. Most common for post-tensioning.

• Gifford-Udall: Uses strands (7-wire) anchored by wedge-shaped keys in a metal sleeve.

• Lee-McCall: Uses rods (threaded) anchored by nuts.

• Dyckerhoff & Widmann (D-W): Similar to Freyssinet but with different anchorage details.

• Hoyos: Uses high-strength bars (26-36mm) with threaded ends and nut anchorage.

  DiagramSEARCH: Hoyos prestressing system anchorage

5.4 Advantages & Disadvantages

5.5 Basic Assumptions (IS 1343)

1. Linear elastic behavior of concrete & steel within working stress limits.

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

3. Perfect bond between steel & concrete (for pre-tensioning; for post-tensioned grouted, bond develops later).

4. Moduli of elasticity $$\displaystyle E_c $$, $$\displaystyle E_s $$ are known and constant.

6.0 PRESTRESSED CONCRETE – ANALYSIS & DESIGN

6.1 Stress Distribution in Beams

• Concentric Prestressing: $$\displaystyle e=0 $$. Uniform compressive stress $$\displaystyle \sigma_c = -\frac{P}{A} $$. Limited by $$\displaystyle f_{c,\text{allow}} $$. No moment capacity.

• Eccentric Prestressing: $e \neq 0$. Linear stress diagram: $$\displaystyle \sigma_c = -\frac{P}{A} \pm \frac{Pe}{I}y $$.

  • At mid-span (sagging moment region): Stress diagram trapezoidal (top compression, bottom may be tension/compression).

  • At support (hogging moment region): Stress diagram opposite.

• Load Balancing Concept: Tendon profile shaped to provide a moment that exactly balances external moment. Net moment at any section = 0. Tendon follows parabolic profile for UDL.

6.2 Flexural Strength (Ultimate Moment Capacity) – IS 1343

• Rectangular Section:

$$M_u = 0.87 f_y A_{ps} \left(d - 0.42 x_u\right)$$

Where $$\displaystyle x_u $$ = depth of neutral axis from compression edge.

*   Check: $$\displaystyle x_u \leq 0.48 d $$ for Fe 415 (limit state).

*   If $$\displaystyle x_u > 0.48d $$, treat as over-reinforced (not permitted usually).

• T-Section (Flanged):

  1. Check if flange is in compression ($$\displaystyle \frac{M_u}{b_f d f_{ck}} \leq 0.26 $$).

  2. If yes, use $$\displaystyle b_f $$ in place of $$\displaystyle b_w $$ for rectangular formula.

  3. If no, consider only web in compression (stress block in web).

6.3 Shear Resistance

• Uncracked Section (at supports): Shear resistance $$\displaystyle V_{cR} = \tau_c \times b \times d $$, where $$\displaystyle \tau_c $$ from elastic theory.

• Contribution of Prestress: Prestressing force $P$ has vertical component if tendon is draped. Increases shear capacity.

• Cracked Section: Use IS 1343 formula similar to RCC, but with $$\displaystyle V_{cf} $$ from concrete + $$\displaystyle V_{ps} $$ from prestressing force.

6.4 Design of Post-Tensioned Girders/I-Beams

• Stresses at Transfer: Check extreme fibre stresses (compression & tension) against $$\displaystyle f_{ct,\text{allow}} $$ (often 0 or small tension).

• Stresses at Working Load: Check against $$\displaystyle f_{c,\text{allow}} $$ (compression) and $$\displaystyle f_{t,\text{allow}} $$ (tension, often 0 for full prestress).

• Number of Wires/Cables: Based on required $P$ after losses. $$\displaystyle A_{ps} = \frac{P}{f_{pi}} $$ (initial stress).

• Cover: Minimum 50mm for post-tensioned ducts (IS 1343). Consider fire rating.

• Loss Ratio: Effective prestress $$\displaystyle P_e = P_i (1 - \text{total loss fraction}) $$.

7.0 LOSSES OF PRESTRESS

7.1 Types of Losses

7.2 Calculation of Losses

• Elastic Loss: $$\displaystyle \Delta P_e = \frac{A_{ps} f_{pi} A_c}{A_c + n A_{ps}} \cdot \frac{E_s}{E_c} $$ (for concentric). For eccentric, use moment equilibrium.

• Friction Loss (Curved Tendon): $$\displaystyle \Delta P_f = P_i (1 - e^{-(\mu \theta + k x)}) $$

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

• Anchorage Slip Loss: $$\displaystyle \Delta P_s = \frac{\Delta l}{x} P $$ (approx), where $\Delta l$ = slip.

7.3 Effect of Losses

• Loss Ratio: Total loss fraction $$\displaystyle = \frac{\sum \Delta P}{P_i} $$.

• Effective Prestress: $$\displaystyle P_e = P_i (1 - \text{loss ratio}) $$. Design is based on $$\displaystyle P_e $$.

• Losses reduce lever arm effect, increase tensile stresses in concrete.

8.0 ANCHORAGE ZONE & END BLOCKS

8.1 Anchorage Zone Stress Distribution

• Problem: Concentrated prestressing force $P$ disperses through concrete. Causes:

  • Bursting Stress: Tensile stress perpendicular to force direction (along axis).

  • Spalling Stress: Tensile stress near end face (splitting off).

  • Splitting Stress: Radial tensile stress around anchor plate.

• Stress Dispersion: Angle of dispersion typically 2V:1H (IS 1343).

8.2 End Block Reinforcement (IS 1343)

• Bursting Force ($$\displaystyle F_{br} $$): $$\displaystyle F_{br} = P \left(1 - \frac{A_{b}}{A_{c}}\right) $$

  • $$\displaystyle A_b $$ = area of anchor plate/bearing area.

  • $$\displaystyle A_c $$ = area of end block (cross-section).

• Reinforcement Area ($$\displaystyle A_{br} $$): $$\displaystyle A_{br} = \frac{F_{br}}{0.87 f_y} $$ (steel takes all bursting force).

• Reinforcement Type:

  • Spiral/Helical: Around anchor plate for splitting.

  • Vertical Bars: In end block for bursting.

  • Distribution Steel: Mesh in end block.

• Anchorage Plate Size: Should be large enough to keep bearing pressure $$\displaystyle q = P/A_b \leq 0.3 f_{ck} $$ (for concrete).

9.0 COMPOSITE CONSTRUCTION & PARTIAL PRESTRESSING

9.1 Composite Construction

• Pre-tensioned Beam + Cast-in-situ Slab: Beam acts as formwork. Slab cast later.

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

• Shrinkage Stress Calculation:

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

Where $$\displaystyle \Delta \epsilon_{sh} $$ = differential shrinkage strain.

• Shear Connection: Usually shear keys or rough interface to transfer shear.

9.2 Partial Prestressing

• Definition: Prestress level such that controlled tensile stresses are allowed in concrete under service loads (unlike full prestress where $$\displaystyle \sigma_t \leq 0 $$).

• Need: Economical for structures where some cracking is acceptable (e.g., bridges, floors).

• Merits: More economical (less prestress steel), better for moment redistribution.

• Demerits: Cracking possible, requires careful crack control, durability concern.

• Methods: Reduce $P$, increase section size, use lower grade steel.

• Stress Limits (IS 1343): Permissible tensile stress in concrete under service load = $$\displaystyle f_{ct} $$ (modulus of rupture) or specified value (e.g., 0.8 $$\displaystyle \sqrt{f_{ck}} $$).

10.0 CONTINUITY & CABLE PROFILES

10.1 Continuity in Continuous Prestressed Beams

• Methods:

  1. Post-tensioning with coupled joints: Individual segments post-tensioned, then coupled at intermediate supports.

  2. In-situ topping: Precast beams with cast-in-situ continuity diaphragm/segment.

• Moments:

  • Primary Moment: Moment due to prestress alone (with no external loads), $$\displaystyle M_p = P e $$.

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

  • Resultant Moment: $$\displaystyle M = M_p + M_s $$.

10.2 Tendon Profiles

• Straight: Constant eccentricity. Simple, low friction loss. Used in pretensioning.

• Draped (Parabolic): Varying eccentricity. Balances UDL. Common for post-tensioned continuous beams.

• Harped: Straight segments with sudden changes at points. Used for point loads.

• Concordant Cable Profile: Profile that produces zero secondary moments in a continuous beam. Requires solving linear equations.

• Linear Transformation: Changing tendon profile by adding a straight line (constant $P$ and $e$) does not change primary moments.

10.3 Guyon's Method

• Method for analyzing statically indeterminate prestressed continuous beams.

• Considers primary moments from prestress and secondary moments from continuity.

• Uses moment distribution or slope-deflection with equivalent loads from prestress.

11.0 DEFLECTION & SPECIAL TOPICS

11.1 Deflection in Prestressed Beams

• Factors: Prestress (upward), creep, shrinkage (long-term increase), dead load, live load (downward).

• Short-term Deflection ($$\displaystyle \Delta_s $$): Due to immediate elastic deformation from loads and prestress.

• Long-term Deflection ($$\displaystyle \Delta_l $$): $$\displaystyle \Delta_l = \Delta_s (1 + \theta) $$ where $\theta$ = creep coefficient + shrinkage effects.

• Calculation: Use moment-area or double integration of $M/EI$ diagram. Effective $I$ for cracked sections.

11.2 Short Notes (Frequent 4-6m Questions)

• Stress Concept: Effective prestress $$\displaystyle P_e $$ after losses. Stress at any fibre: $$\displaystyle \sigma = -\frac{P_e}{A} \pm \frac{P_e e}{I} y \pm \frac{M}{I} y $$.

• Tendon Profile Selection: Based on bending moment diagram. Parabolic for UDL, harped for point loads. Minimize friction losses.

• Vibration Isolation: Use prestress to increase natural frequency, avoid resonance. Dampers may be added.

• Response to Arbitrary Force: Use Duhamel's integral (convolution integral): $$\displaystyle x(t) = \int_0^t h(t-\tau) F(\tau) d\tau $$, where $h(t)$ = impulse response function.

• Pre vs Post-Tensioning:

  | Pre-tensioning | Post-tensioning | | :--- | :--- | | Tendons tensioned before concrete cast | Tendons tensioned after concrete hardens | | Bond through adhesion (no grouting needed) | May be grouted (bonded) or unbonded | | Usually straight tendons | Can have curved profiles | | Factory production, quality control easy | Site construction, flexible | | Losses mainly elastic, creep, shrinkage | Losses include friction, anchorage slip |

• Why not Mild Steel? Low tensile strength (~250 MPa). Requires large area, high prestress losses, high initial strain → high relaxation. High-strength steel (1500-2000 MPa) essential.

12.0 DESIGN PROBLEM SOLVING APPROACH

12.1 Step-by-Step Procedure

1. Cantilever Retaining Wall:

   • Assume base width B (0.6H to 0.7H), toe projection (0.4B), stem thickness.

   • Calculate forces: $$\displaystyle P_a $$, $$\displaystyle P_q $$, $$\displaystyle P_w $$, weights.

   • Check Overturning, Sliding, Bearing pressure.

   • Design stem for BM & shear at base.

   • Design base slab (heel & toe) for BM from soil pressure.

   • Detail reinforcement (sketch elevation, section).

2. Circular Water Tank:

   • Capacity → dimensions (R, H). Assume wall thickness.

   • Calculate ring tension $$\displaystyle T_h $$ at base.

   • Design hoop reinforcement (area, spacing).

   • Design vertical reinforcement (minimum).

   • Design base slab (BM, reinforcement).

   • Detail wall-base joint.

3. Intze Tank:

   • Capacity → dimensions (top dome radius, cylinder H, bottom dome).

   • Design top dome (meridional & hoop stresses, reinforcement, thickness).

   • Design top ring beam (thrust from dome, axial + bending).

   • Design cylindrical wall (varying pressure, hoop & vertical steel).

4. Silo:

   • Given D, H, $\gamma$, $\mu$, $\phi$.

   • Compute $$\displaystyle \sigma_v(z) $$, $$\displaystyle \sigma_h(z) $$ using Janssen.

   • Find max BM & shear in wall.

   • Design hoop & vertical reinforcement.

   • Check buckling if thin.

5. Solid Slab Bridge:

   • Effective span, carriageway width.

   • Thickness = span/20 (min).

   • Calculate DL (slab, wearing coat, parapet).

   • Calculate LL (IRC Class AA, impact, distribution width).

   • Find max BM & shear (critical loading).

   • Design longitudinal reinforcement (Fe415, $$\displaystyle M_u $$ calculation).

   • Provide distribution steel (min 0.12%).

   • Check shear, bond, deflection.

12.2 Typical Assumptions & Code References

• Concrete: M20-M30 for RCC, M30-M40 for prestressed.

• Steel: Fe 415 (RCC), Fe 500/High-tensile (prestressed).

• Codes: IS 456 (RCC), IS 3370 (water tanks), IS 1343 (prestressed), IRC 6 (bridge loads).

• Unit Weights: Concrete = 25 kN/m³, Water = 9.81 kN/m³, Brick = 18-20 kN/m³.

• Earth Pressure: Rankine's for level backfill, $\phi$ given.

12.3 Presentation of Design

• Calculations: Stepwise, with clear assumptions.

• Sketches: Elevation, critical section (cross-section), reinforcement detailing (plan/section with bar sizes, spacing, cover).

• Key Details: Bar bending schedules (if required), joint details, anchorage details.