CELE Reinforced & Prestressed Concrete — Reinforced Concrete ColumnsStudy Notes

A reinforced concrete column is a structural member designed to withstand primarily compressive forces, with the concrete resisting compression and the steel reinforcement helping to resist both compression and any bending moments. Columns differ fundamentally from beams: they carry vertical loads along their axis (axial load), whereas beams carry lateral loads (transverse loads). In practice, real columns also experience moments due to lateral loads (wind, seismic), construction inaccuracies, or intentional eccentricity. Columns are classified into two main types based on transverse reinforcement: **Tied Columns:** These use closely spaced steel ties (or stirrups) to confine the core concrete. Ties prevent outward buckling of the main longitudinal bars and provide limited lateral confinement. Tied columns have a reduction factor φ = 0.65 (compression-controlled) and are more common in practice due to lower cost. **Spiral Columns:** These use a continuous helical spiral of steel wire or bar wound around the core concrete. Spirals provide superior confinement, leading to better ductility and higher load capacity. Spiral columns have φ = 0.75 and can sustain higher loads but cost more. They are often used where high ductility or high compression is critical. The NSCP 2015 and ACI 318 recognize these differences and reward spiral columns with both a higher capacity cap and a higher reduction factor. Understanding when to use each type is important for economical design.

For a short column (slenderness ignored), the nominal axial compression capacity Pₙ or Pₒ is based on the principle that the concrete and steel act together. The concrete resists compression on its net area (gross area minus steel area), and the steel resists compression on its full area. **Pure Axial Nominal Strength (Gross Axial Strength):** Pₒ = 0.85f'c(Ag − Ast) + fyAst where: - Pₒ = nominal (unfactored) axial load capacity - f'c = concrete compressive strength (MPa) - Ag = gross cross-sectional area (mm²) - Ast = total area of longitudinal steel bars (mm²) - fy = yield strength of steel (MPa, typically 415 MPa in the Philippines) The factor 0.85 on concrete accounts for the non-uniform stress distribution and the fact that concrete does not reach peak stress simultaneously everywhere. **Design Axial Capacity (Factored):** The code limits the usable design axial capacity to account for unavoidable eccentricity (no column is perfectly axially loaded in practice): For tied columns: φPn,max = 0.80 × φ × Pₒ = 0.80 × 0.65 × Pₒ = 0.52Pₒ For spiral columns: φPn,max = 0.85 × φ × Pₒ = 0.85 × 0.75 × Pₒ = 0.6375Pₒ This means a spiral column can carry about 22.6% more load than a tied column of identical dimensions and materials. The difference comes from two sources: (1) the capacity cap increases from 0.80 to 0.85, and (2) φ increases from 0.65 to 0.75. **Physical Interpretation:** The reduction factors (0.80 and 0.85) reflect the reality that columns always have some eccentricity, and this eccentricity produces bending that reduces the axial capacity. Spiral columns, being more ductile, tolerate eccentricity better, hence the higher cap. The reduction factors φ (0.65 and 0.75) are typical for compression-controlled behavior in reinforced concrete.

The NSCP 2015 and ACI 318 impose strict limits on longitudinal and transverse reinforcement in columns to ensure adequate performance, constructability, and ductility. **Longitudinal (Main) Steel Reinforcement:** 1. **Steel Ratio Limits:** - Minimum: ρg,min = 0.01 (ensures strength and controls shrinkage cracking) - Maximum: ρg,max = 0.08 (ensures constructability; above this, placing and vibrating concrete becomes difficult) - Practical note: In regions with lap splices, keep ρg ≤ 0.04 to avoid congestion where ρg = Ast / Ag 2. **Minimum Number of Bars:** - Tied rectangular columns: minimum 4 bars (one near each corner) - Spiral or circular columns: minimum 6 bars (ensures adequate perimeter support) - Bars must be distributed evenly around the perimeter to resist moment in any direction 3. **Bar Diameter and Spacing:** - No specific lower limit on bar diameter, but typically ≥ 16 mm in practice - Longitudinal bars must be placed no more than 150 mm apart around the perimeter (NSCP 2015, Section 410.8.4) - In tied columns, bars at corners and edges are mandatory; intermediate bars may be placed between corners **Transverse Reinforcement (Ties or Spirals):** **Tied Columns:** Ties are rectangular closed loops of steel around the core. The tie size and spacing are crucial for confining the concrete and preventing buckling of longitudinal bars. - **Tie Size:** - Minimum: #10 (10 mm diameter) for longitudinal bars ≤ 32 mm - Minimum: #13 (13 mm diameter) for longitudinal bars > 32 mm - **Spacing (per NSCP 2015, Section 410.8.5):** The spacing of ties shall not exceed the smallest of: - 16 times the longitudinal bar diameter: 16db - 48 times the tie diameter: 48dt - The least dimension of the column cross-section - 300 mm **Example:** For #25 longitudinal bars and #10 ties in a 400 mm column: - 16 × 25 = 400 mm - 48 × 10 = 480 mm - 400 mm (least dimension) - 300 mm ← **controls** (minimum, 300 mm spacing) - **Tie Configuration:** Ties must form closed loops and be anchored in the core concrete. Corner and alternate bar requirements apply (NSCP 2015, Section 410.8.5). **Spiral Columns:** A spiral is a continuous helix of steel wire or bar with a diameter usually 8–13 mm, wrapped around the column core. - **Spiral Pitch (p) and Ratio (ρs):** The spacing between successive spiral turns (pitch) directly determines confinement effectiveness. The spiral volume ratio is: ρs = 4Asp / (s × Dch) where: - Asp = cross-sectional area of spiral steel (mm²) - s = pitch (vertical distance between consecutive spiral turns, mm) - Dch = diameter of concrete core measured to outside of spiral (mm) **Minimum Spiral Ratio (NSCP 2015, Section 410.8.6.4):** ρs ≥ 0.45 × (Ag / Ach − 1) × (f'c / fyt) where: - Ag = gross column area - Ach = area of core concrete (to outside of spiral), mm² - f'c = concrete strength, MPa - fyt = spiral steel yield strength, capped at 700 MPa in this equation (even if fy is higher) The cap at 700 MPa prevents excessive dependence on very high-strength spiral steel, as confinement is primarily a geometric/volume effect. - **Maximum Spiral Pitch:** Not to exceed 75 mm or 1/6 of column diameter (whichever is smaller) to ensure adequate confinement. **Physical Interpretation:** The minimum spiral ratio formula reflects that confinement benefit scales with the ratio (Ag/Ach − 1), which measures how much steel is outside the core. Larger columns need proportionally more spiral to provide equivalent confinement. The 0.45 factor and the cap at 700 MPa are empirical adjustments based on test data.

Real columns are rarely subjected to pure axial compression. They always experience moments due to: 1. **Lateral Loads:** Wind, seismic, or live loads create moments. 2. **Eccentricity:** The load may not be perfectly centered, creating a moment M = P × e. 3. **Slenderness:** In slender columns, P-delta effects amplify moments. 4. **Continuity:** In frame structures, negative moments from beam-column joints create biaxial bending. The interaction diagram is a graphical representation of all combinations of axial load (Pn) and bending moment (Mn) that a column section can support. It is fundamental to column design. **Building the Interaction Diagram:** The diagram is constructed by varying the strain distribution across the column section from: - **Compression-Controlled Failure** (top): pure axial compression, no moment - **Balanced Point** (middle): concrete reaches maximum compression strain (0.003) exactly when steel reaches yield strain (fy/Es) - **Tension-Controlled Failure** (bottom): steel yields well before concrete crushes, acting like a beam **Key Points on the Diagram:** 1. **Pure Axial Load (ε = 0 throughout):** At the top of the diagram: Pn = Pₒ = 0.85f'c(Ag − Ast) + fyAst (for short columns; applied with cap 0.80 or 0.85) Mn = 0 (no moment) 2. **Balanced Point (εc = 0.003, εst = fy/Es):** This is the critical transition point: - Concrete has reached its maximum usable compression strain (0.003 or "crushing") - Steel has reached yield strain (usually εy = 415/200,000 ≈ 0.002075 for fy = 415 MPa) - Both materials are at their limits simultaneously - Any further increase in eccentricity will fail the steel in tension; any decrease will fail the concrete in compression - The balanced point gives the **maximum moment** the section can sustain The neutral axis location at balance (cb) is found from strain compatibility: εc / cb = εy / (d − cb) 0.003 / cb = 0.002075 / (d − cb) cb = 0.003 × d / (0.003 + 0.002075) ≈ 0.591d (for fy = 415 MPa) The balanced axial load Pb and moment Mb are computed using moment equilibrium about the centroid. 3. **Pure Bending (P = 0, M = Mn):** At the bottom of the diagram, the column acts like a beam. The neutral axis is located by strain compatibility, and the section is analyzed for maximum moment with zero axial load. The capacity is the same as a doubly-reinforced beam. **Failure Regions:** - **Compression-Controlled Region** (above the balanced point): The concrete crushes (ε = 0.003) before the steel yields. Failure is sudden and brittle. The reduction factor φ = 0.65 (tied) or 0.75 (spiral) applies. - **Tension-Controlled Region** (below the balanced point, typically εst > 0.005): The steel yields well before the concrete crushes. Failure is ductile with visible cracks and deformation. The reduction factor φ = 0.9 applies (same as flexure). - **Transition Region** (between 0.002 ≤ εst ≤ 0.005): Intermediate behavior. The code allows linear interpolation of φ between the compression and tension extremes, but columns rarely operate here. **Design Interaction Equation:** For any combined load (Pu, Mu), the interaction relationship is: f = (Pu / (φPn)) + (Mu / (φMn)) ≤ 1.0 Or rearranged: Pu / (φPn) + Mu / (φMn) ≤ 1.0 where Pn and Mn are read from the column section (unfactored), and φ is the appropriate reduction factor. Every point (Pu, Mu) must satisfy this inequality. **Simplified Biaxial Bending (Bresler Equation):** When a column is bent about both axes (biaxial bending), an approximate criterion is: 1/Pu = 1/(φPnx) + 1/(φPny) − 1/(φPo) where Pnx and Pny are the axial capacities when the section is bent about the x and y axes respectively, and Pₒ is the pure axial capacity. This is a conservative approximation useful for quick design checks. **Code Interaction Diagram Formula (ACI 318 Appendix D):** For a more refined analysis, the ACI provides the following bilinear approximation (often used in practice): For Pu > 0.1f'c Ag (compression-controlled): Pu / (0.1f'c Ag) + Mu / (φMn,max) ≤ 1.0 For Pu ≤ 0.1f'c Ag (tension-controlled or near balance): Pu / (φPn) + Mu / (φMn) ≤ 1.0 (linear interpolation) These formulas simplify design without loss of safety.

Short columns (low slenderness) can be designed using the interaction diagram directly. However, slender columns experience additional moments due to lateral deflection under load, reducing their capacity. These are called **P-δ (P-delta)** and **P-Δ (P-Delta)** effects. **What Is Slenderness?** The slenderness ratio is defined as: Slenderness Ratio = kℓu / r where: - k = effective length factor (depends on support conditions and frame stability) - ℓu = unsupported length of the column - r = radius of gyration = √(I/Ag) For a rectangular section: r = depth / √12 For a circular section: r = diameter / 4 **Limits for Short Columns (Slenderness Ignored):** The NSCP 2015 (and ACI 318) define a column as "short" if: **For Non-Sway Frames (Braced, columns cannot sway horizontally):** kℓu/r ≤ 34 − 12(M₁/M₂) but not more than 40, where M₁/M₂ is the ratio of the smaller to larger end moment (positive if moments bend the column in single curvature, negative if double curvature). If M₂ = 0 (cantilever), the limit becomes 22. **For Sway Frames (Unbraced, columns can sway):** kℓu/r ≤ 22 If the slenderness ratio exceeds these limits, moment magnification must be applied. **Why Moment Magnification?** Consider a column with axial load Pu and initial moment M₀. As the column deflects laterally by amount δ, an additional moment is created: M_secondary = Pu × δ This secondary moment causes more deflection, which causes more moment, and so on. The total moment becomes: M_total = M₀ + Pu × δ For slender columns, δ can be significant, and the total moment can far exceed M₀. The moment magnifier method approximates this iterative process. **Moment Magnification Formula:** The design moment is amplified as: M_amplified = δ × M₀ where the magnifier δ (delta) depends on the loading and frame type: **For Non-Sway (Braced) Frames:** δ = Cm / (1 − (Pu / (0.75 × φ × Pc))) where: - Cm = moment coefficient (typically 0.6 − 0.4(M₁/M₂) for non-uniform moment, or 1.0 for uniform moment) - Pc = Euler buckling load = π²EI / (kℓu)² - φ = reduction factor (0.65 or 0.75 depending on column type) **For Sway (Unbraced) Frames:** δ_s = 1 / (1 − ΣPu / (0.75 × φ × ΣPc)) where the sums are over all columns in the story, accounting for the frame's overall stability. Or, more simply, if the frame is braced laterally (shear walls, diaphragms): δ_s = 1.0 (sway is prevented) **Design Procedure for Slender Columns:** 1. Calculate slenderness ratio kℓu/r 2. Check if it exceeds the limit for short columns 3. If yes, calculate Pc = π²EI / (kℓu)² and Cm 4. Calculate the moment magnifier δ 5. Amplify the moment: M_design = δ × M₀ 6. Check the interaction diagram with (Pu, M_design) **Simplified Approach (Conservative):** For hand calculations, a conservative approximation is: δ_approx = 1 + (Pu / (Ag × f'c)) × (kℓu/r)² / (40,000) If δ > 1.5, the column is quite slender and may be uneconomical. **Effective Length Factor k:** The factor k accounts for end support conditions: - **Pinned-Pinned (both ends free to rotate, no sway):** k = 1.0 - **Fixed-Fixed (both ends rigidly fixed, no sway):** k = 0.5 - **Fixed-Pinned (one end fixed, one pinned, no sway):** k = 0.7 - **Cantilever (fixed at base, free at top, sway):** k = 2.0 - **Unbraced Frame (sway possible):** k = 1.2 − 2.0 depending on frame geometry and stiffness For typical building frames, if columns are connected to rigid foundations and beams, k ≈ 0.6 − 0.9 for the lower stories and increases toward the top. **Physical Interpretation:** Slenderness matters when Pu / Pc (or Pu / (0.75φPc)) approaches 1. If the load is much less than the buckling load, deflections remain small and moment magnification is negligible (δ ≈ 1.0). For heavily loaded columns, δ can be 1.5 to 3 or more, significantly reducing capacity.

This section provides realistic design examples that combine the concepts from previous sections and are representative of PRC Civil Engineer Licensure Examination problems. **Design Philosophy for Columns:** 1. **Select a trial section** based on the estimated axial load and allowable stress (typically assuming ρg ≈ 0.015–0.03). 2. **Calculate the required steel area** to satisfy the axial-load requirement. 3. **Verify the steel ratio** is within 0.01–0.08 (or ≤0.04 with laps). 4. **Check tie or spiral reinforcement** and spacing requirements. 5. **Assess slenderness:** if kℓu/r exceeds limits, either enlarge the section, add lateral bracing, or accept moment magnification. 6. **Prepare the interaction diagram** (or use code tables) and verify all factored load combinations (Pu, Mu) plot inside the φ-reduced diagram. 7. **Check constructability** and cost; iterate if needed. **Common Board-Exam Problem Types:** 1. **Pure Axial Capacity:** Given section dimensions, materials, and reinforcement, find design capacity φPn,max. 2. **Required Section Size:** Given axial load and moments, find the minimum column size. 3. **Steel Ratio Verification:** Confirm ρg is within code limits. 4. **Tie or Spiral Design:** Size and space transverse reinforcement. 5. **Interaction Diagram:** Plot or use to check combined load adequacy. 6. **Slenderness Effect:** Determine if moment magnification applies; calculate amplified moment. 7. **Biaxial Bending Check:** Use Bresler equation or 3D interaction surface. **Typical Given Data (From PRC Exam):** - Column dimensions (mm × mm or diameter in mm) - Concrete strength f'c (MPa): 20, 25, 28, 35 MPa (common in Philippines) - Steel yield fy (MPa): 415 (common), 500, 700 (special cases) - Unsupported length ℓu (mm) - End conditions (pinned, fixed, cantilever) → determine k - Axial load Pu (kN) and moment Mu (kN·m) (factored) - Request: Design the column (select size, reinforcement) OR verify adequacy

The NSCP 2015 (National Structural Code of the Philippines) and ACI 318-14/19 establish minimum requirements for reinforced concrete columns to ensure safety, constructability, and adequate performance. This section summarizes key code articles and provides a design checklist for exams and practice. **Key NSCP 2015 and ACI 318 Code Sections:** **NSCP 2015, Section 410 — Columns:** 1. **410.1 — Scope:** Applies to columns that carry primarily axial compression with or without bending. 2. **410.2 — Design Assumptions:** Columns are assumed to act compositely (concrete + steel); strain compatibility applies. 3. **410.3 — Strength Requirements:** Design strength must equal or exceed factored loads: φPn ≥ Pu and φMn ≥ Mu (or combined interaction check) 4. **410.4 — Minimum Eccentricity:** Even if no moment is applied, assume a minimum eccentricity emin to account for construction inaccuracy. This is implicitly handled by the 0.80 and 0.85 capacity caps. 5. **410.7 — Longitudinal Reinforcement:** - Minimum: ρg,min = 0.01 (1% of gross area) - Maximum: ρg,max = 0.08 (8% of gross area) - In lap-splice regions: ρg ≤ 0.04 (to avoid congestion) - Minimum 4 bars in tied columns, 6 in spiral columns - All bars must be continuous (no splicing in critical zones unless detailed) 6. **410.8 — Transverse Reinforcement (Ties):** - Size: #10 (10 mm) for longitudinal bars ≤ 32 mm; #13 (13 mm) for bars > 32 mm - Spacing (least of): 16db, 48dt, least column dimension, 300 mm - Closed loops, anchored in core - Corner and alternate bars must be enclosed by ties 7. **410.9 — Spiral Reinforcement:** - Minimum ratio: ρs ≥ 0.45(Ag/Ach − 1)(f'c/fyt), with fyt ≤ 700 MPa - Maximum pitch: 75 mm or Dch/6 (whichever smaller) - Minimum pitch: typically ≥ 20 mm (constructability) 8. **410.10 — Slenderness Effects:** - Short columns: kℓu/r ≤ 22 (unbraced) or ≤ 34 − 12(M₁/M₂) ≤ 40 (braced) - Slender columns: apply moment-magnifier method or second-order analysis - Buckling loads: Pc = π²EI/(kℓu)² (Euler formula) 9. **410.11 — Design with Combined Loads:** - All combinations of (Pu, Mu) must satisfy the interaction equation - If biaxial bending: use 3D interaction surface or Bresler reciprocal equation - Design moment = δ × M₀ (moment-magnified if slender) **Reduction Factors φ:** - **Compression-controlled (columns):** φ = 0.65 (tied), 0.75 (spiral) - **Tension-controlled (beams, tension-flexure):** φ = 0.90 - **Shear/Torsion:** φ = 0.75 **Capacity Caps for Axial Load:** - **Tied:** φPn,max = 0.80 × 0.65 × Pₒ = 0.52Pₒ - **Spiral:** φPn,max = 0.85 × 0.75 × Pₒ = 0.6375Pₒ **Design Checklist for Board Exams:** ✓ **Material Selection:** - [ ] f'c specified (typical: 20, 25, 28, 35 MPa) - [ ] fy specified (typical: 415 MPa) - [ ] E = 200,000 MPa (steel) - [ ] E = 4,700√f'c (concrete, MPa) ✓ **Section Dimensions:** - [ ] Gross area Ag calculated - [ ] Radius of gyration r = √(I/Ag) computed - [ ] Slenderness ratio kℓu/r checked against code limits - [ ] If slender, moment magnification δ applied ✓ **Longitudinal Reinforcement:** - [ ] Steel area Ast selected (or required area calculated) - [ ] Steel ratio ρg = Ast/Ag verified: 0.01 ≤ ρg ≤ 0.08 (or ≤0.04 with laps) - [ ] Number of bars ≥ 4 (tied) or ≥ 6 (spiral) - [ ] Bar diameters and spacing around perimeter checked (max 150 mm) - [ ] Clear cover ≥ 40 mm (typical) ✓ **Transverse Reinforcement (Ties or Spiral):** - [ ] **Tied:** Size (#10 or #13) and spacing (least of 16db, 48dt, least dimension, 300 mm) specified - [ ] **Spiral:** Diameter and pitch selected; ρs ≥ minimum ratio verified; pitch ≤ 75 mm or Dch/6 ✓ **Axial Capacity:** - [ ] Pₒ = 0.85f'c(Ag − Ast) + fyAst calculated - [ ] φPn,max = 0.52Pₒ (tied) or 0.6375Pₒ (spiral) determined - [ ] φPn,max ≥ Pu verified ✓ **Combined Loading (Interaction Diagram):** - [ ] Interaction diagram prepared or consulted (hand-sketched or computed) - [ ] All factored load combinations (Pu, Mu) plot inside the φ-reduced diagram - [ ] Or: (Pu/(φPn)) + (Mu/(φMn)) ≤ 1.0 for each load case ✓ **Special Cases:** - [ ] **Biaxial bending:** Bresler equation or 3D surface used if Mux and Muy both significant - [ ] **Slender column:** Moment magnifier δ computed and applied - [ ] **Lap splices:** If present, ρg ≤ 0.04 and proper splice length provided - [ ] **Earthquake/seismic:** If applicable, BNBC or NSCP seismic requirements (ductility, confinement) met **Common Code Violations (and How to Avoid Them):** 1. **ρg out of bounds:** Check 0.01 ≤ ρg ≤ 0.08 every time. Underreinforced (ρg < 0.01) risks sudden tensile failure; overreinforced (ρg > 0.08) causes construction delays. 2. **Tie spacing too large:** Remember the four criteria (16db, 48dt, least dimension, 300 mm max). The smallest one controls. 3. **Ignoring slenderness:** High kℓu/r requires moment magnification. Omitting it can overestimate capacity. 4. **Wrong φ:** Columns are compression-controlled (φ = 0.65 or 0.75), NOT 0.90 (that's for flexure). 5. **Forgetting the cap (0.80 or 0.85):** The capacity is NOT φPₒ, but φ × (0.80 or 0.85) × Pₒ. 6. **Using gross area for steel stress:** Steel is on full area Ast, not net area. Concrete is on net area (Ag − Ast). 7. **No interaction diagram for combined loading:** A column with only axial load is rare. Always prepare the diagram or use an interaction table. 8. **Ignoring constructability:** A 300 mm × 300 mm column with 8 #32 bars is theoretically possible but practically impossible to vibrate properly. Verify spacing and constructability.