5.4 RCC Structures-2

Standards and rules: these are study notes, not a current code or legal reference. Check the applicable edition, amendments, units and design assumptions before using a numerical limit or rule.

Introduction to Advanced RCC Elements

This unit builds upon the fundamentals of RCC design to cover critical compression members and foundations that transfer loads to the ground. Columns are vertical members primarily resisting axial compression and moments. Footings are spread foundations that safely distribute column loads to the soil. Additionally, we explore Pre-stressed Concrete, an advanced technology that enhances the performance of concrete members by introducing internal stresses to counteract service loads. All designs follow the Limit State Method as per IS 456:2000 and relevant NBC (Nepal) standards.

---

1. Design of Columns

Columns are structural members with a height-to-least lateral dimension ratio greater than 3, primarily carrying axial compression. Most real columns are subjected to combined axial load and bending moment.

1.1 Classification of Columns

1. Based on Loading:
   • Axially Loaded Column: Load is concentric (theoretical case). Moment is negligible.
   • Eccentrically Loaded Column: Load is offset, inducing bending moment. This is the practical case.
2. Based on Slenderness Ratio ($\lambda$):
   • $\lambda=l_{eff}/D_{min}$, effective length divided by least lateral dimension
   • Short Column ($\lambda \leq 12$): Failure is governed by material strength (crushing/buckling of material). Design for strength.
   • Slender Column ($\lambda > 12$): Failure is governed by elastic instability (buckling). Additional moments due to deflection ($P-\Delta$ effect) must be considered.

1.2 Effective Length ($l_{eff}$)

• The length between points of contra-flexure (zero moment) in the deflected shape of the column.
• Depends on end conditions (fixed, hinged, free) and bracing. Values given in IS 456 Cl. 25.2.

1.3 Axially Loaded Short Column

1. Ultimate Load Carrying Capacity ($P_u$):
   • Factored axial load on the column.
2. Design Strength:
   • Pure axial load capacity (ignoring moment): $P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc}$ where $A_c$ = area of concrete, $A_{sc}$ = area of longitudinal steel.
   • Minimum Eccentricity ($e_{min}$): As per IS 456, even nominally axial loads must be designed for a minimum eccentricity. $e_{min, \text{IS}} = \frac{l}{500} + \frac{D}{30} \quad \text{or} \quad 20 \text{ mm, whichever is greater.}$ Therefore, all columns are designed as eccentrically loaded.

1.4 Eccentrically Loaded Short Columns (Uniaxial/Biaxial Bending)

The column section is analyzed for combined effects of axial load ($P_u$) and moment ($M_u$).

1. Design Charts (SP 16):
   • The most practical tool for design. Charts are plotted for $\frac{P_u}{f_{ck} b D}$ vs $\frac{M_u}{f_{ck} b D^2}$ for different ratios of $d'/D$ and steel percentage ($p/f_{ck}$).
2. Interaction Diagram:
   • A fundamental concept. It's a curve defining the combination of axial load ($P$) and moment ($M$) that causes failure of a given column section.
   • Key Points on Diagram:
     • Point A: Pure axial load capacity ($P_{o}$).
     • Point B: Balanced failure condition (concrete strain = 0.0035, steel strain = $\epsilon_y$ simultaneously).
     • Point C: Pure moment capacity ($M_o$).
     • Zone: Compression-controlled failure (above balance point).
     • Zone: Tension-controlled failure (below balance point).
3. Design Steps using Charts:
   1. Calculate design load $P_u$ and moment $M_u$.
   2. Assume section dimensions ($b$, $D$), steel distribution, and $d'$ (cover to bar centroid).
   3. Compute $\frac{P_u}{f_{ck} b D}$ and $\frac{M_u}{f_{ck} b D^2}$.
   4. Use the appropriate chart in SP 16 to find the required $\frac{p}{f_{ck}}$.
   5. Calculate total steel area: $A_{sc} = p \times b \times D / 100$.
   6. Check for Biaxial Bending: If moments exist about both principal axes ($M_{ux}$, $M_{uy}$), check the interaction formula (IS 456 Cl. 39.6): $\left( \frac{M_{ux}}{M_{ux1}} \right)^{\alpha} + \left( \frac{M_{uy}}{M_{uy1}} \right)^{\alpha} \leq 1.0$ where $M_{ux1}, M_{uy1}$ are uniaxial moment capacities about the respective axes for the given $P_u$, and $\alpha$ is a constant (function of $P_u/P_{uz}$).

1.5 Reinforcement Detailing for Columns (IS 456 Cl. 26.5.3)

1. Longitudinal Steel:
   • Minimum Area ($A_{sc,min}$): 0.8% of gross cross-sectional area.
   • Maximum Area ($A_{sc,max}$): 6% of gross area (4% at laps).
   • Minimum Number of Bars: 4 in rectangular, 6 in circular columns.
   • Minimum Diameter: 12 mm.
2. Transverse Ties (Lateral Reinforcement):
   • Purpose: Prevent buckling of longitudinal bars and confine core concrete.
   • Diameter: ≥ 6 mm or 1/4 of the largest longitudinal bar diameter.
   • Spacing ($s_v$):
     • ≤ Least of: (i) Least lateral dimension, (ii) 16×$\phi_{long}$, (iii) 300 mm.
     • In seismic zones, much stricter spacing per IS 13920.

---

2. Design of Shallow Foundations (Footings)

Footings spread concentrated column loads over a sufficient area of soil to keep soil pressure within its safe bearing capacity (SBC).

2.1 Types of Isolated/Spread Footings

1. Square/Rectangular Pad Footing: For concentric loading.
2. Eccentric Footing: For columns with moments or at property lines.
3. Combined Footing: Supports two or more columns when individual footings would overlap or for boundary columns.

2.2 Design Philosophy

• Geotechnical Criteria: Size of footing based on SBC.
• Structural Design: Thickness and reinforcement based on bending moment, shear, and punching shear.

2.3 Design of Isolated Square Footing (Axial Load)

1. Size of Footing (Plan Area, A):
   • $A=P/q_{allow}$ is an initial sizing relationship only when load and allowable pressure are on the same gross or net basis; include footing and overburden effects consistently.
   • $q_{ns} = q_{na} - \gamma D_f$, where $q_{na}$ is net allowable SBC, $\gamma$ is soil unit weight, $D_f$ is depth of footing.
   • Provide square footing of side $B = \sqrt{A}$.
2. Upward Soil Pressure for Design ($p_u$):
   • Use factored column load for structural design of footing.
   • $p_u = \frac{P_u}{B^2}$ (for axial load only).
3. Depth Based on Shear:
   • One-Way (Beam) Shear: The footing fails as a wide beam along a critical section at a distance $d$ from the column face. Governs the depth.
     • Factored shear force, $V_{u1} = p_u \times B \times \left( \frac{B - b}{2} - d \right)$
     • Nominal shear stress, $\tau_v = V_{u1} / (B \times d)$.
     • Require: $\tau_v \leq \tau_c$ (from IS 456 Table 19, for concrete without shear reinforcement, $k_s \tau_c$).
   • Two-Way (Punching) Shear: The column punches through the footing. Critical section is at $d/2$ from the column face.
     • Factored shear force, $V_{u2} = p_u \times [B^2 - (b+d)^2]$
     • Nominal shear stress, $\tau_{pv} = V_{u2} / [4(b+d) d]$
     • Permissible stress, $\tau_{pc} = k_s \times 0.25 \sqrt{f_{ck}}$ where $k_s = (0.5 + \beta_c) \leq 1$, $\beta_c$ = ratio of short to long side of column.
4. Design for Bending Moment:
   • Critical section for moment is at the face of the column.
   • Maximum factored bending moment (per unit width): $M_u = p_u \times \frac{\left( \frac{B - b}{2} \right)^2}{2}$
   • Calculate required steel area per meter width ($A_{st}$) as for a singly reinforced beam of width = 1000mm and depth = $d$.
   • Check $A_{st,min}$ = 0.12% of gross cross-sectional area for HYSD bars.
5. Reinforcement Detailing:
   • Reinforcement is placed at the bottom of the footing.
   • Bars are placed in two perpendicular directions (both ways).
   • Distribution: Uniform across the full width. No curtailment.

2.4 Combined Footings

1. Objective: To achieve a uniform soil pressure distribution when a column is eccentric or two columns are close.
2. Types:
   • Rectangular Combined: Two columns, centroid of footing area coincides with centroid of column loads.
   • Trapezoidal Combined: Used when space constraints prevent a rectangular footing with a centered resultant.
3. Design Approach:
   • Determine footing dimensions for uniform soil pressure.
   • Model the footing as an inverted beam spanning between columns, subjected to upward soil pressure.
   • Draw Shear Force (SF) and Bending Moment (BM) diagrams.
   • Design for maximum BM (longitudinal reinforcement at bottom) and maximum SF (check depth for shear).
   • Design transverse reinforcement as in isolated footing for the strip under each column.

---

3. Pre-stressed Concrete

3.1 Basic Concept

1. Definition: Concrete in which internal stresses of a suitable magnitude and distribution are introduced before the application of service loads, so that the tensile stresses resulting from the service loads are counteracted to a desired degree.
2. Principle: Pre-compress the concrete in the tension zone using high-strength steel tendons (wires, strands, bars). When service load is applied, the pre-compression is relieved first before tensile stresses develop.

3.2 Advantages over Conventional RCC

• Crack Control: Can be designed to be crack-free under service loads.
• Higher Span-to-Depth Ratios: Sections are slimmer and lighter, suitable for long spans (bridges, roofs).
• Improved Durability: Crack-free concrete reduces corrosion risk.
• Better Shear Resistance: Pre-compression increases principal tensile strength.

3.3 Methods of Pre-stressing

1. Pre-tensioning:
   • Tendons are tensioned before the concrete is cast.
   • Concrete is poured around the stressed tendons and bonded to them.
   • After concrete gains sufficient strength, tendons are released from the abutments, transferring the pre-stress force to the concrete via bond.
   • Application: Mostly in factory-produced, precast elements (hollow core slabs, railway sleepers, beams).
2. Post-tensioning:
   • Tendons are tensioned after the concrete has hardened.
   • Tendons are placed in ducts/sheaths within the formwork before concreting.
   • After curing, tendons are tensioned using jacks and anchored at the member ends.
   • Ducts are then grouted with cement mortar to provide bond (bonded post-tensioning) or left ungrouted (unbonded).
   • Application: In-situ construction, segmental construction of bridges, large span structures.

3.4 Materials

1. Concrete: High strength (M40 and above) to withstand high pre-stress and reduce elastic and creep losses.
2. Steel: High tensile strength steel wires, strands (7-wire strand common), or bars.
   • Ultimate Tensile Strength: Typically 1600-2000 MPa.
   • Important property: Relaxation – loss of stress under constant strain.

3.5 Losses of Pre-stress

Significant reductions in pre-stressing force occur over time, which must be accounted for in design.

1. Immediate Losses (during pre-stressing operation):
   • Elastic Shortening of Concrete: When pre-stress is transferred, concrete shortens elastically, reducing tendon stress.
   • Friction in Ducts (Post-tensioning): Loss due to curvature and wobble of ducts.
   • Anchorage Slip: Wedges draw back slightly during anchoring.
2. Time-Dependent Losses:
   • Creep of Concrete: Time-dependent deformation under sustained pre-stress.
   • Shrinkage of Concrete: Volume reduction due to drying.
   • Relaxation of Steel: Loss of stress in tendon under constant strain.

3.6 Analysis Concepts

1. Stress Concept: Checking extreme fiber stresses at transfer and service conditions.
   • At Transfer: Only pre-stress force (after initial losses) and self-weight act. Compressive stresses must be within permissible limits.
   • At Service: Pre-stress force (after all losses), self-weight, and service live loads act. Stresses must be within permissible limits (compression and limited tension, if allowed).
2. Load Balancing Concept (by T.Y. Lin): The pre-stressing force with a draped tendon profile can be visualized as providing an upward uniform load that "balances" a portion of the downward service load, simplifying analysis.

---

4. Relevant Codes and Standards

• IS 456:2000: Plain and Reinforced Concrete - Code of Practice. (For RCC columns and footings).
• IS 1343:2012: Code of Practice for Prestressed Concrete.
• NBC 109:2020 (Nepal): Concrete Structure.
• NBC 202:1994 / IS 1904:1986: Code of Practice for Design and Construction of Foundations.
• IS 13920:2016: Ductile Detailing of Reinforced Concrete Structures Subjected to Seismic Forces (for column and footing detailing in seismic zones).
• SP 16:1980: Design Aids for Reinforced Concrete (for column design charts).

Conclusion: The design of columns and foundations completes the load path from the superstructure to the ground. Columns must be designed for the complex interaction of axial load and bending moments, often using interaction diagrams. Footings translate these loads safely to the soil, governed by shear criteria. Pre-stressed concrete represents a significant technological advancement, enabling more efficient, slender, and durable structures for specific applications. Mastery of these topics is crucial for the design of complete, safe, and economical RCC building systems.