UNIT 5: Advanced RCC and Prestressed Concrete Design
I. Advanced Reinforced Concrete Structures
A. Retaining Walls
Types:
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Gravity Wall: Mass concrete, relies on weight.
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Cantilever Wall: Most common, economical for heights 4-7m. Consists of stem, base slab (heel & toe), and key.
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Counterfort Wall: For heights > 7m. Vertical counterforts connect stem to base, reducing bending in stem.
Earth Pressure Theories:
- Rankine's Active Earth Pressure (for cohesionless soil, φ = angle of repose):
$$K_a = \frac{1 - \sin\phi}{1 + \sin\phi}$$
Active pressure at depth $H$: $$\displaystyle P_a = \frac{1}{2} K_a \gamma H^2 $$ (per unit length).
> [!TIP] Assumes wall is smooth, vertical, and backfill is horizontal.
- Coulomb's Theory: Considers wall friction (δ), backfill slope (β). More general but complex.
Design of Cantilever Retaining Wall:
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Stability Checks (at base level):
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Overturning: $$\displaystyle \frac{\text{Resisting Moment}}{\text{Overturning Moment}} \geq 1.5 $$ (typical)
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Sliding: $$\displaystyle \frac{\text{Resisting Force (friction + passive)}}{\text{Driving Force}} \geq 1.5 $$
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Bearing Pressure: $$\displaystyle e = \frac{M}{P} < \frac{B}{6} $$; $$\displaystyle q_{min/max} = \frac{P}{B} \left(1 \pm \frac{6e}{B}\right) \leq \text{permissible} $$
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Stem Design: Cantilever slab fixed at base. Thickness at top ~200mm, increases towards base. Design for maximum bending moment at base ($$\displaystyle M_{max} \approx P_a \cdot H/3 $$) and shear at $d$ from face.
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Base Slab Design: Acts as cantilever fixed at stem face.
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Heel: Design for upward soil pressure + weight.
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Toe: Design for downward soil pressure + weight.
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Key/Shear Key: Provided to increase sliding resistance. Design as vertical cantilever.
Reinforcement Detailing:
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Main bars in tension zone of stem & base.
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Distribution steel @ 150-200mm c/c.
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Clear cover: 40mm (for mild exposure), 50mm (moderate).
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Bar anchorage: $$\displaystyle L_d $$ or hooks at free ends.
B. Water Tanks
1. Circular Water Tanks with Rigid Base
Theory:
- Hoop Tension (circumferential): $$\displaystyle T_h = p \cdot r \cdot t $$ (for thin cylinder), where $p$ = pressure, $r$ = radius, $t$ = thickness.
$$T_h = \frac{p r}{1000} \text{ kN/m (per metre height)}$$
Reinforcement area: $$\displaystyle A_{st} = \frac{T_h}{0.4 f_y} $$ (IS 3370).
- Vertical Bending: Wall acts as cantilever fixed at base. Max moment at base: $$\displaystyle M_v = \frac{p r^2}{2} $$.
Design of Wall:
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Thickness: $$\displaystyle t \geq \frac{H}{30} $$ (min), ensure $t \geq 100$mm.
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Hoop Reinforcement: Two layers (inner & outer) near faces. Spacing ≤ 300mm or $t$.
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Vertical Reinforcement: Near inner face for base region, near outer for top. Spacing ≤ 300mm.
Design of Base Slab:
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Circular slab fixed at edges. Max moment at centre: $$\displaystyle M = \frac{3 + \nu}{8} p r^2 $$ ($\nu$ = Poisson's ratio).
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Radial & circumferential reinforcement.
Joints:
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Construction Joints: Horizontal, keyed, with water stops.
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Movement Joints: For expansion/contraction.
2. Intze Tanks
Components: Top dome (spherical), ring beam, cylindrical wall, bottom dome/slab.
Design of Spherical Dome:
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Meridional (vertical) force: $$\displaystyle N_\phi = \frac{p r}{2} (2 - \cos\phi) $$ (compressive)
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Hoop (horizontal) force: $$\displaystyle N_\theta = \frac{p r}{2} \sin\phi $$ (tension at top, compression at crown)
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Reinforcement for meridional & hoop stresses.
Design of Ring Beam:
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Horizontal beam at dome-cyl junction.
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Loads: Hoop tension from dome + water pressure on beam.
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Design as reinforced concrete beam for bending & shear.
Design of Cylindrical Wall:
- Similar to circular tank but supported by ring beam (reduces effective height for cantilever action).
C. Silos
Difference from Tanks: Stores bulk solids (grain, coal, cement). Pressure depends on friction and vertical load, not just height.
Lateral Pressure (Janssen Equation):
$$p_h = \frac{\gamma D}{4 \mu K} \left(1 - e^{-\frac{4 \mu K z}{D}}\right)$$
Where:
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$\gamma$ = unit weight of solid
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$D$ = diameter
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$\mu$ = coefficient of friction (wall-solid)
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$K$ = lateral pressure ratio (≈ 0.4 for solids)
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$z$ = depth from top
[!TIP] Pressure reaches a constant value at depth $$\displaystyle z > \frac{D}{4\mu K} $$ ( Janssen's constant).
Vertical Pressure: Increases linearly with depth initially, then constant (due to Janssen effect).
Design of Cylindrical Wall:
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Hoop tension: $$\displaystyle T_h = p_h \cdot D/2 $$
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Vertical bending: Cantilever action from base. Consider eccentric discharge causes increased lateral pressure (up to 2x static). Design for worst case.
D. Bridges
1. Solid Slab Bridges (IRC Loading)
IRC Class AA Loading: 70R or 70B (wheeled) or 70RA (tracked). Load distribution to slab.
Design of Slab:
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Effective Span: $$\displaystyle L = \text{clear span} + \text{width of bearing} $$.
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Thickness (IRC 6): $$\displaystyle D \geq \frac{L}{20} $$ (simply supported) + wearing coat.
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Reinforcement:
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Main Steel (bending): For max moment at mid-span ($$\displaystyle M = \frac{wL^2}{8} $$). Use $$\displaystyle M_u = 0.87 f_y A_{st} \left(d - \frac{f_y A_{st}}{f_{ck} b}\right) $$.
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Distribution Steel: 0.12% of gross section, @ > 150mm c/c.
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Wearing Coat & Parapet:
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Wearing coat: 80-100mm thick (bituminous/concrete).
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Parapet: Design for collision load (IRC 6).
II. Prestressed Concrete
A. Fundamentals
Principle: Apply high compressive stress to concrete before service loads to counteract tensile stresses.
Systems:
| Feature | Pre-tensioning | Post-tensioning |
|---|---|---|
| Process | Tendons tensioned before casting. Concrete cast & bonded. | Tendons placed in ducts. Concrete cast, cured, then tensioned. |
| Location | Precast yards. | In-situ (large structures). |
| Bond | Bonded (concrete around tendons). | Bonded (grouted ducts) or Unbonded (sheathed). |
| Systems | Freyssinet (multi-wire), Magnel-Blaton. | Freyssinet (anchorages), Gifford-Udall, Roy-O. |
Materials:
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Concrete: High strength (M30 to M60), low creep.
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Steel: High tensile wires (7mm), strands (7/3mm, 12/7mm), bars. $$\displaystyle f_{pk} = 1500-1860 $$ N/mm².
Advantages: Higher load capacity, longer spans, crack control, durability, better material utilization. Disadvantages: High cost, skilled labor, sophisticated equipment, quality control.
Basic Assumptions (IS 1343):
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Linear stress-strain (Hooke's law).
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Perfect bond between steel & concrete (no slip).
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Concrete does not resist tension.
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Sections remain plane (Bernoulli's hypothesis).
B. Stress Analysis in Prestressed Beams
Stress at a Section:
$$\sigma_c = \frac{P}{A} \pm \frac{P e y}{I} \pm \frac{M y}{I}$$
Where $P$ = prestress, $e$ = eccentricity, $M$ = external moment, $y$ = distance from NA.
Kern Points: Points within section where prestress produces no tension.
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Kernel width: $$\displaystyle Z_b = I/y_b $$, $$\displaystyle Z_t = I/y_t $$
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Kern area: Rectangle of width $$\displaystyle Z_b $$ & $$\displaystyle Z_t $$.
Eccentric Prestressing: Creates linear stress diagram. If $e$ within kern, no tension at that section under $P$ alone.
Load Balancing Concept: Tendon profile shaped to produce a moment that balances external loads, minimizing flexural stresses.
Continuous Beams:
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Primary Moment: Due to prestress, assuming supports free to deflect.
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Secondary Moment: Reactions at supports induce moment.
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Resultant Moment: Sum of primary & secondary.
C. Design of Flexural Members (IS 1343)
Ultimate Flexural Strength:
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Rectangular Section:
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Stress block: $$\displaystyle 0.36 f_{ck} $$ for 0.48x (IS 1343).
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Lever arm: $$\displaystyle z = d - 0.42x $$ (approx).
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$$M_u = 0.36 f_{ck} b x (d - 0.42x) = 0.87 f_{pk} A_{ps} (d - 0.42x)$$
Solve iteratively.
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T-Section:
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Check if neutral axis in flange ($$\displaystyle M_u \leq 0.36 f_{ck} b_f D^2 $$).
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If NA in web, effective flange width $$\displaystyle b_f $$ as per IS 1343.
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Stress block in flange & web.
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Shear Design:
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Nominal shear stress: $$\displaystyle \tau_v = \frac{V}{b d} $$
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Permissible shear stress $$\displaystyle \tau_{c} $$ (depends on $$\displaystyle \sigma_{cp} $$).
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Shear reinforcement if $$\displaystyle \tau_v > \tau_{c} $$.
D. Anchorage and End Zone
Anchorage Zone: Region ahead of anchorage where prestress disperses.
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Bursting Force: Tensile force due to dispersion. $$\displaystyle F_{bst} = P \left(1 - \frac{A_{n}}{A_{g}}\right) $$ (approx).
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Spalling Force: Compression above anchor.
End Block Reinforcement (IS 1343):
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Bursting Tensile Force: $$\displaystyle F_{bst} $$ to be resisted by vertical bars.
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Spalling Reinforcement: Horizontal hoops.
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End Block Dimensions: $$\displaystyle > 1.2 \times $$ anchor plate size.
Anchorage Devices:
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Wedges & Nuts: For wires/strands (Freyssinet).
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Anchor Plates & Washers: For bars.
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Button Spacer: For multi-strand.
E. Continuity and Composite Construction
Achieving Continuity:
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Post-tensioning across supports with couplers.
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Harping tendons to provide continuity moment.
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Top/bottom tendons to balance moments.
Composite Construction:
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Precast Prestressed Beam + Cast-in-situ Slab.
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Differential Shrinkage: Cast-in-situ slab shrinks more → induces tensile stress in precast beam & compressive in slab.
$$\sigma = E_s \cdot \Delta \epsilon \cdot \frac{A_c}{A_c + n A_p}$$
Where $\Delta \epsilon$ = differential shrinkage.
- Design: Ensure composite action, shear connectors if needed.
Partial Prestressing:
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Allow limited tensile stress ($$\displaystyle \sigma_{t,allow} > 0 $$) under service loads.
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Methods: Use mild steel in tension zone, reduce prestress, or use unbonded tendons.
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Advantages: Economical for secondary members, reduces losses.
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Applications: Industrial floors, bridges.
F. Losses of Prestress
Types & Causes:
- Elastic Deformation Loss: Immediate loss when prestress transferred to concrete.
$$\Delta P_{el} = \frac{m P}{A_c} \left(\frac{A_p}{A_c} + \frac{A_p e^2}{I}\right)$$
Where $$\displaystyle m = E_s / E_c $$.
- Creep of Concrete: Time-dependent strain under sustained stress.
$$\Delta P_{cr} = m \cdot P \cdot \phi \left(\frac{A_p}{A_c} + \frac{A_p e^2}{I}\right) \cdot \frac{1}{1 + \frac{A_p}{A_c} m}$$
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Shrinkage of Concrete: $$\displaystyle \Delta P_{sh} = m \cdot P \cdot \epsilon_{sh} \left(\frac{A_p}{A_c} + \frac{A_p e^2}{I}\right) $$
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Relaxation of Steel: Loss in steel stress under constant strain. $%$ loss depends on steel type.
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Friction Loss (Curved Tendons): Due to wobble & curvature.
$$\Delta P_f = P_0 \left(1 - e^{-(\mu \theta + \alpha L)}\right)$$
Loss Ratio & Effective Prestress:
$$\text{Loss Ratio} = \frac{\text{Total Loss}}{P_0}$$
$$P_e = P_0 (1 - \text{Loss Ratio})$$
G. Deflection and Cracking
Factors Influencing Deflection:
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Span, load, prestress (compressive), creep, shrinkage, support conditions.
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Short-term: Elastic deflection.
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Long-term: Increased by creep & shrinkage (up to 2-3x).
Crack Control:
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Permissible tensile stress in concrete ($$\displaystyle \sigma_{t,allow} $$) under service loads (IS 1343).
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Adequate reinforcement distribution.
H. Special Topics
Concordant Cable & Linear Transformation:
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Concordant Cable: Tendon profile that produces zero moment at all sections under its own prestress (for determinate structures).
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Linear Transformation: Changing tendon profile linearly (maintaining eccentricity at ends) does not change primary moments in statically determinate beams.
Guyon's Method: For continuous beams. Treats each span independently with equivalent loads, then solves for redundants.
Flexural Failures:
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Under-reinforced: Steel yields first → ductile.
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Over-reinforced: Concrete crushes first → brittle (avoid).
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Balanced: Both yield/crush simultaneously.
Short Notes:
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Stress Concept: Prestress creates compressive stress to offset tension.
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Tendon Profile: Parabolic for UDL, straight for point loads.
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End Block Stress: Bursting & spalling due to dispersion.
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Pre vs Post-Tensioning: See table in Section A.
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Mild Steel: Low yield strength (~250 MPa) → high losses, large section. Not economical.