Structural Design and Drawing (RCC-II) (CE-702 (D)) - Unit 4 Short Notes

UNIT 4: Advanced RCC Design and Prestressed Concrete

1. Design of Cantilever Retaining Wall Stem

Types of Retaining Walls: Gravity, cantilever, counterfort, sheet pile. Cantilever walls are economical for heights up to 6–7 m.

Earth Pressure Theories:

• Rankine’s active earth pressure (cohesionless soil):

  \[ K_a = \tan^2\left(45^\circ - \frac{\phi}{2}\right) \]

  Active pressure at depth \(z\): \(p_a = K_a \gamma z\) (with surcharge \(q\): \(p_a = K_a(\gamma z + q)\)).

• Coulomb’s theory: Considers wall friction \(\delta\), backfill slope \(\beta\), and wall inclination \(\alpha\). More complex but realistic for steep slopes.

Lateral Earth Pressure Calculation:

• Cohesionless soil: Use Rankine/Coulomb.

• Cohesive soil (\(c > 0\)):

  \[ p_a = K_a \gamma z + 2c\sqrt{K_a} \]

• Surcharge effect: Add uniform pressure \(q\) as above.

Design Loads and Combinations (IS 1893):

• Dead load (DL), Earth pressure (EP), Live load/surcharge (LL), Seismic (EQ).

• Combinations:

  \(1.5(\text{DL} + \text{EP})\),

  \(1.2(\text{DL} + \text{EP} + \text{LL})\),

  \(1.2(\text{DL} + \text{EP} + \text{EQ})\).

Stem Design (cantilever fixed at base):

• Bending moment at base (no surcharge):

  \[ M_{\text{base}} = \frac{1}{6} K_a \gamma H^3 \]

  With surcharge \(q\): add \(\frac{1}{2} K_a q H^2\).

• Shear force at base:

  \[ V_{\text{base}} = \frac{1}{2} K_a \gamma H^2 \quad (\text{no surcharge}) \]

• Reinforcement (limit state):

  \[ M_u = 0.87 f_y A_{st} \left( d - \frac{A_{st} f_y}{f_{ck} b} \right) \]

  Solve for \(A_{st}\). Check shear: \(\tau_v = V/(b d) \leq \tau_{c,\text{max}}\).

Detailing:

• Concrete cover: 40 mm (IS 456, Table 16, for soil exposure).

• Bar spacing: \(\leq \min(300\,\text{mm},\, 3d)\) for flexure.

• Development length:

  \[ L_d = \frac{\phi \sigma_s}{4 \tau_{bd}} \quad (\text{working stress}) \]

• Curtailment: Bars curtailed where BM reduces to 50% of max, typically at \(0.2H\) from top.

Stability Checks (overall wall):

• Overturning: FOS \(> 1.5\) (moments about toe).

• Sliding: FOS \(> 1.5\) (horizontal forces vs friction).

• Bearing pressure: Should lie within middle third, no tension.

IS Codes: IS 456 (general), IS 1893 (seismic), IS 3370 (if water-retaining).

[!TIP] Common mistake: Forgetting surcharge load in earth pressure. Stem design often uses working stress in older papers; check method specified.

2. Design of Circular Water Tank with Rigid Base

Types: Circular, rectangular, intze, spherical. Circular is economical for large capacities.

Design Assumptions: Rigid base (no sliding), wall fixed at base, no uplift.

Hoop Tension Calculation:

• Pressure at depth \(h\): \(p = \gamma_w h\) (\(\gamma_w = 9.81\,\text{kN/m}^3\)).

• Hoop tension per unit height:

  \[ T_h = p \cdot r = \gamma_w h r \]

  where \(r\) = mean radius.

• Distribution: Linear with depth, max at base \(T_{\text{max}} = \gamma_w H r\).

Reinforcement for Hoop Tension (IS 3370, working stress):

• Steel area per unit height:

  \[ A_{st} = \frac{T_h}{\sigma_{st}} \]

  Permissible \(\sigma_{st}\): 130 MPa for Fe 415 (check IS 3370:2009).

• Distribution: Horizontal bars for hoop tension; vertical bars for temperature/shrinkage (min 0.3% of concrete area).

Base Slab Design:

• Bending moment: Circular slab fixed at edges. Use coefficients from IS 3370 or standard tables for uniform load (water pressure + soil reaction).

• Reinforcement: Provide steel for max BM, both directions.

Joint Design (Wall–Base):

• Shear key to prevent sliding.

• Dowels for reinforcement continuity from wall to base.

IS 3370 Provisions:

• Permissible stresses: Concrete in compression 5–8 MPa (M25–M30), steel in tension 130–140 MPa.

• Minimum reinforcement: 0.3% of concrete area in each direction for walls/slabs.

• Crack width limit: 0.2 mm.

Serviceability Checks:

• Cracking: Ensure steel stress under service loads \(\leq \sigma_{st,\text{per}}\).

• Deflection: Not critical but check for excessive.

• Water tightness: Use impermeable concrete, proper construction joints.

[!TIP] Hoop tension varies linearly with depth; provide reinforcement accordingly (varying bar sizes/spacing). Always check crack width for water tanks.

3. Design of Intze Tank

Introduction: Economical for large capacities (>500 kL) due to reduced wall thickness at top.

Components: Top dome (spherical), ring beam, cylindrical wall.

Top Dome Design:

• Thrust at ring beam: For spherical dome under internal pressure \(p\) (water head at ring level),

  \[ H = \frac{p r}{2} \]

  where \(r\) = radius of ring.

• Meridional and hoop forces: For spherical dome, both equal and constant:

  \[ N = \frac{p R}{2} \]

  where \(R\) = radius of dome.

• Reinforcement: Two layers (inner/outer) for meridional and hoop directions.

Ring Beam Design:

• Bending: Subjected to horizontal thrust \(H\) from dome. Design as circular beam under uniform radial force.

• Torsion: Due to unbalanced forces if dome not symmetric (rare).

• Reinforcement: Longitudinal bars for bending, closed ties for torsion.

Cylindrical Wall Design:

• Water pressure: Hydrostatic, varies with depth.

• Vertical reinforcement: For bending (wall acts as vertical cantilever fixed at base).

• Horizontal reinforcement: For hoop tension (similar to circular tank).

• Effect of dome thrust: Ring beam imposes vertical load at top of wall.

Load Transfer Path:

Water pressure → dome → ring beam (thrust) → wall (vertical load) → base slab.

IS Codes: IS 3370 (water tanks), IS 456 (general).

Detailing:

• Dome: Reinforcement layers, spacing.

• Ring beam: Rectangular section, adequate depth for bending/torsion.

• Wall: Vertical and horizontal bars, development at base.

[!TIP] Ring beam is critical—design for compression from dome thrust plus bending. Verify load path carefully.

4. Design of Silo Side Wall for Bulk Storage (Coal)

Introduction: Bulk solids (coal, grain) differ from liquids: pressure saturates with depth due to wall friction.

Lateral Pressure Calculation (Janssen equation):

\[ \boxed{p_h = \frac{\gamma r}{\mu} \left(1 - e^{-\mu K z / r}\right)} \]

where

\(\gamma\) = unit weight of coal (≈ 8.5 kN/m³),

\(r\) = silo radius,

\(\mu\) = coefficient of wall friction (≈ 0.4 for concrete),

\(K\) = lateral pressure coefficient (\(K = 1 - \sin\phi\), \(\phi\) = angle of repose, ≈ 30°–45° for coal),

\(z\) = depth from top.

Parameters:

• Angle of repose \(\phi\): 30°–45° (coal).

• Wall friction \(\mu\): 0.4–0.5 (concrete).

• Maximum pressure at base:

  \[ p_{h,\text{max}} = \frac{\gamma r}{\mu} \left(1 - e^{-\mu K H / r}\right) \]

Horizontal Pressure Distribution: Increases with depth, asymptotically approaches \(\gamma r / \mu\).

Wall Design:

• Circumferential reinforcement (for hoop tension):

  \[ A_{sh} = \frac{p_h r}{\sigma_{sh}} \]

  per unit height.

• Vertical reinforcement (for bending):

  Consider wall as vertical cantilever fixed at base.

  BM at base per unit width (1 m circumference):

  \[ M_{\text{base}} = \int_0^H p_h(z) \cdot z \, dz \]

  Then vertical steel:

  \[ A_{sv} = \frac{M}{0.87 f_y d} \quad (\text{limit state}) \]

• Wall thickness: 100–200 mm, concrete M25–M30, steel Fe 415.

Comparison with Liquid Tanks:

• Bulk solids: Pressure saturates, depends on \(\phi\) and \(\mu\); dynamic effects during discharge (convergence/divergence pressures).

• Liquids: Pressure increases linearly, no saturation.

IS Codes: IS 5503 (silos), IS 456 (general).

[!TIP] Janssen equation is essential. Remember pressure saturation at depth. Design for both circumferential and vertical reinforcement.

5. Design of Solid Slab Bridge for Class AA Loading

Bridge Components: Deck slab, bearings, substructure (piers/abutments).

IRC Loads (IRC 6:2002):

• Dead load (DL): Self-weight + wearing coat (80 mm, 20 kN/m³) + parapet.

• Live load (LL): Class AA tracked (70 t, 2.4 m wide tracks) or wheeled (45 t).

• Impact factor (IF): 25% for spans ≤ 45 m.

• Side friction: Horizontal force at bearings (not on slab directly).

Effective Width of Slab (IRC 6, Clause 304):

For a loaded strip (concentrated load):

\[ b_{\text{eff}} = b_1 + 2b_2 \]

where

\(b_1\) = width of loaded area (contact width),

\(b_2 = \frac{l_0}{10}\) for simply supported (≤ half clear distance between supports).

For uniform load, effective width = actual width of slab.

Slab Design:

• Effective span \(l_0\): Clear span + bearing width (often taken as clear span).

• Loads:

  • DL = self-weight + wearing coat.

  • LL = Class AA load × IF, applied as uniform load over \(b_{\text{eff}}\) or as concentrated load on \(b_{\text{eff}}\).

• Bending moment (simply supported):

  • UDL: \(M = \frac{w l_0^2}{8}\)

  • Concentrated load at midspan: \(M = \frac{P l_0}{4}\)

• Shear force:

  • UDL: \(V = \frac{w l_0}{2}\)

  • Concentrated: \(V = \frac{P}{2}\)

• Reinforcement (limit state):

  \[ M_u = 0.87 f_y A_{st} \left( d - \frac{A_{st} f_y}{f_{ck} b_{\text{eff}}} \right) \]

  Provide shear reinforcement if \(\tau_v > \tau_{c,\text{max}}\).

Serviceability:

• Deflection: \(L/250\) for live load, \(L/500\) for total load (IS 456).

• Crack width: Limit 0.2 mm (IRC 112).

• Durability: Cover 40 mm (exposure class).

Detailing:

• Main steel: In tension zone, bar diameter ≤ 1/8 slab thickness.

• Distribution steel: Min 0.1% gross area, bars at top.

• Bar spacing: ≤ 3d or 300 mm.

IRC Specifications:

IRC 6 (loads), IRC 21 (concrete), IRC 112 (code practice).

Wearing Coat: 80 mm thick, included in DL, provide waterproofing layer.

[!TIP] Effective width for Class AA loading is critical—use IRC 6 formula. Often, design for worst-case loaded strip (single wheel/track).

6. Prestressed Concrete: Fundamentals and Applications

Definition and Principle: Pre-compression applied to concrete to counteract tensile stresses under service loads, using high-strength steel and concrete.

Methods:

• Pre-tensioning: Tendons tensioned before concreting, anchored to abutments; prestress transferred by bond after concrete hardens.

• Post-tensioning: Tendons placed in ducts, tensioned after concrete gains strength, anchored by wedges/nuts.

Prestressing Systems:

• Freyssinet: Conical wedge anchorage for multiple wires.

• Gifford-Udall: Single wire, double-acting jack.

• Magnel-Blaton: Flat jacks, anchor plates.

• Lee-McCall: Pre-tensioning with multiple strands.

Advantages:

• Higher load capacity, longer spans.

• Reduced deflection and cracking.

• Material economy (smaller sections, less steel).

• Improved durability (crack control).

Disadvantages:

• High initial cost.

• Skilled labor, quality control.

• Specialized equipment required.

• Loss assessment complexity.

Types:

• Full prestressing: No tension under service loads (Class 1).

• Partial prestressing: Controlled tension allowed (Class 2, 3).

Applications: Bridge girders/slabs, water tanks, silos, piles, building frames.

Basic Stress Distribution:

• Prestress \(P_e\) induces uniform compressive stress:

  \[ \sigma_c = -\frac{P_e}{A_c} \pm \frac{P_e e}{Z} \]

• Under external loads, resultant stress = prestress + load stress. Goal: Keep concrete in compression.

Losses of Prestress:

• Immediate:

  • Elastic shortening: \(\Delta f_{pES} = \frac{E_p}{E_c} \Delta \sigma_c\)

  • Friction: \(\Delta f_{pF} = \sum \mu \Delta x + \sum k \Delta y\) (curved profiles)

  • Wedge slip: \(\Delta f_{pW}\)

• Time-dependent:

  • Creep: \(\Delta f_{pCR} = \phi \cdot \frac{E_p}{E_c} \cdot f_{cp}\)

  • Shrinkage: \(\Delta f_{pSH} = \epsilon_{sh} E_p\)

  • Relaxation: \(\Delta f_{pR}\)

• Total loss: Sum of all, typically 15–25% (pre-tensioning), 20–30% (post-tensioning).

IS Code: IS 1343 (prestressed concrete).

[!TIP] Losses are critical—always compute effective prestress \(P_e = P_i - \text{total losses}\). Friction loss is often overlooked in post-tensioning.

7. Additional Design Considerations (Cross-Cutting Topics)

Limit State Method (IS 456):

• Ultimate Limit State (ULS): Strength/stability. Partial safety factors: \(\gamma_f = 1.5\) (loads), \(\gamma_m = 1.15–1.5\) (materials).

• Serviceability Limit State (SLS): Deflection, cracking, vibration. Check under characteristic loads.

Material Properties:

• Concrete grades: M25, M30, M40 (\(f_{ck}\) = characteristic strength).

• Steel grades: Fe 415, Fe 500 (HYSD); high-tensile wires (St 1500, St 2000).

IS Codes:

• IS 456: General RCC.

• IS 3370: Water tanks.

• IS 5503: Silos.

• IRC 6, 21, 112: Bridges.

• IS 1343: Prestressed concrete.

Detailing Practices:

• Bar Bending Schedule (BBS): Lists bars with lengths, bends.

• Concrete cover: As per exposure (IS 456 Table 16).

• Development length:

  \[ L_d = \frac{\phi f_y}{4 \tau_{bd}} \quad (\text{limit state}) \]

• Splicing: Lapped splices in low-stress zones, avoid at max moment.

Serviceability Checks:

• Deflection: \(L/250\) (live load), \(L/500\) (total load) (IS 456).

• Cracking: Water tanks/silos: limit 0.2 mm (IS 3370).

• Durability: In aggressive environments (water, soil), use appropriate cement, cover, admixtures.

[!TIP] Always perform serviceability checks after strength design. For water tanks/silos, crack control is paramount for water tightness and durability.