Analysis and Design of Columns

Introduction and Definitions

A column is defined as a vertical structural member designed to transmit axial compression loads, which may or may not be accompanied by bending moments.

Key characteristics of columns include:

• The cross-sectional dimensions of a column are generally considerably less than its total height.

• Their primary function is to support vertical loads from floors and roofs and transmit these loads down to the foundation.

• A column represents a specific application of a compression member that is oriented vertically.

Classification of Columns

Columns are classified through several lenses depending on requirements, composition, and behavior.

A. Classification Based on Geometry

Columns may take various shapes depending on structural or architectural needs:

• Rectangular

• Square

• Circular

• L-shaped

• T-shaped

B. Classification Based on Composition

• RC (Reinforced Concrete) Columns: Standard concrete reinforced with steel bars.

• Composite Columns: Columns in which steel structural members (such as H-sections) are encased in concrete. Main reinforcement bars positioned with ties or spirals are placed around the structural member.

• Infilled Columns: These consist of steel pipes filled with plain concrete or lightly reinforced concrete.

C. Classification Based on Lateral Reinforcement

• Tied Columns: Longitudinal (main) reinforcements are held in position by separate ties spaced at equal intervals along the length. Over 95% of all columns in buildings in non-seismic regions are tied columns.

• Spiral Columns: Usually circular in cross-section, where longitudinal bars are wrapped by a closely spaced continuous spiral reinforcement.

D. Classification Based on Lateral Stability

• Braced Columns: Gravity loads are supported by the column, while lateral stability is provided by other members like shear walls, stairwell shafts, or diagonal bracing.

• Un-braced Columns: Also known as sway columns; the frame action of the column itself resists horizontal loads.

E. Classification Based on Sensitivity to Second-Order Effects

• Non-sway Columns: Not significantly affected by lateral displacements.

• Sway Columns: Highly sensitive to second-order effects resulting from lateral displacements.

F. Classification Based on Degree of Slenderness

• Short Columns: Characterized by low slenderness ratios; strength is governed by material strength.

• Slender Columns: Characterized by high slenderness ratios; strength is governed by buckling and second-order effects.

G. Classification Based on Loading

• Axially Loaded Columns: Loads are applied along the longitudinal central axis.

• Columns under Uni-axial Moment: Eccentricity exists in one direction, causing a moment about one axis.

• Columns under Bi-axial Moment: Eccentricity exists in both directions, causing moments about both axes.

Behavior of Tied and Spiral Columns

The behavior of these columns under axial load is typically compared using load-deflection diagrams.

• Spiral Columns: These are significantly more ductile than tied columns. The strength of the spiral column is enhanced by tri-axial stress resulting from the confinement of the concrete core by the spiral reinforcement. After the concrete shell spalls off, the confined core continues to carry load effectively.

• Tied Columns: These exhibit less ductility. Failure often occurs shortly after the concrete spalls or the reinforcement buckles between ties.

Braced vs. Un-braced Columns

Un-braced (Sway) Columns

• Lateral movement is not restrained; the frame can sway sideways under load.

• Frame action is used to resist horizontal loads.

• P-Δ (P-Delta) Effect: Because the column can displace horizontally ($\Delta$), it must resist the axial load ($P$) plus an additional bending moment produced by the displacement ($P imes ext{displacement}$).

• These are considered more dangerous because they experience larger bending moments, are prone to buckling, and have lower load capacity.

Braced (Non-sway) Columns

• Braced against lateral loads using shear walls, elevators, or diagonal bracings.

• In these structures, the second-order moment ($P imes hickDelta$) is considered negligible.

Short vs. Slender Columns

Short Columns

• Basic Idea: Stocky members that do not bend significantly.

• Primary Load Mode: Direct axial compression.

• Failure Mode: Material strength exhaustion (crushing of concrete or yielding of steel).

Slender Columns

• Basic Idea: Large length relative to a small cross-section.

• Behavior: Small imperfections cause lateral deflection, which increases with load.

• Second-Order Effect: The deflection creates additional moments (P-Δ effect).

• Failure Mode: Buckling (instability) often occurs before the material reaches its ultimate crushing strength.

Slenderness Ratio ($\lambda$)

The key parameter for determining column type is:
$\lambda = \frac{L_{eff}}{i}$
Where:

• $L_{eff}$ = Effective length.

• $i$ (or $r$) = Radius of gyration.

• Interpretation: Large $\lambda$ indicates a slender column; small $\lambda$ indicates a short column.

Euler Buckling Concept

For slender columns, the critical buckling load ($P_{cr}$) is governed by stability rather than material strength:
$P_{cr} = \frac{\pi^2 EI}{L_{eff}^2}$
Where:

• $E$ = Modulus of elasticity.

• $I$ = Moment of inertia.

• Implication: A longer column has a smaller $P_{cr}$, making it easier to buckle.

Interaction Diagrams

Elastic Columns

For an idealized homogenous column where compressive strength ($f_{cu}$) equals tensile strength ($ftu$):
$\frac{P}{P_{max}} + \frac{M}{M_{max}} = 1$

• $(0, 1)$: Pure axial compression.

• $(1, 0)$: Pure bending moment.

• The resulting diamond shape defines the failure boundary. Points inside the boundary represent safe conditions; points outside represent failure.

Reinforced Concrete Columns

RC interaction diagrams are more complex and identify specific regions based on strain:

• Top Point ($P_0$): Pure compression (eccentricity $e = 0$).

• Bottom Point ($M_0$): Pure bending (axial load $P = 0$, eccentricity $e = \infty$).

• Balanced Point ($P_b, M_b$): A critical condition where concrete reaches its ultimate strain ($\approx 0.0035$) and steel reaches its yield strain ($\varepsilon_s = \varepsilon_y$) simultaneously.

• Compression-Controlled Region: Above the balanced point. Failure is sudden and brittle (concrete crushes first). This is generally undesirable.

• Tension-Controlled Region: Below the balanced point. Steel yields before concrete crushes, providing a ductile failure with warning. This is preferred in design.

Strain Distributions (Points A–E)

• Point A: Uniform compression. Maximum axial capacity.

• Point B: Small eccentricity. Entire section remains in compression; neutral axis is outside the section.

• Point C: Balanced failure. Transition between compression and tension control.

• Point D: Large eccentricity. Neutral axis enters the section. One side in compression, the other (steel) in tension.

• Point E: Pure bending. Maximum moment capacity; zero axial load.

Design Procedures (Actual Interaction Charts)

Uniaxial Design

1. Assume cross-section and evaluate the ratio $d'/h$.

2. Compute Normal Force Ratio: $\nu = \frac{N_{Ed}}{A_c f_{cd}}$.

3. Compute Moment Ratio: $\mu = \frac{M_{Ed}}{A_c h f_{cd}}$.

4. Enter the chart to pick $\omega$ (mechanical steel ratio).

5. If the coordinate is outside the chart, the section is too small.

6. Compute total reinforcement: $A_s = \frac{\omega A_c f_{cd}}{f_{yd}}$.

Biaxial Design

1. Select dimensions $h, b$ and $h', b'$; calculate ratios $h'/h$ and $b'/b$.

2. Compute Normal Force Ratio and Moment Ratios for both directions.

3. Identify the suitable chart satisfying the ratios and obtain $\omega$.

4. Calculate total reinforcement and check against minimum/maximum provisions ($A_{s,min}$ and $A_{s,max}$).

ES EN 1992:2015 Standards for Column Design

Geometric Imperfections

Imperfections are represented by an inclination $\theta_i$.

• Eccentricity ($e_i$): For isolated members, $e_i = \theta_i l_o / 2$. For simplified braced systems, $e_i = l_o / 400$.

• Transverse Force ($H_i$):
    - Unbraced members: $H_i = \theta_i N$.
    - Braced members: $H_i = 2 \theta_i N$.

Slenderness Criterion (Section 5.8)

Second-order effects can be ignored if \lambda < \lambda_{lim}.
$\lambda_{lim} = \frac{20 ABC}{\sqrt{n}}$
Where:

• $A = 1 / (1 + 0.2 \phi_{ef})$ (or $0.7$ if unknown).

• $B = 1 + 2 \omega$ (or $1.1$ if unknown).

• $C = 1.7 - r_m$ (or $0.7$ if unknown).

• $n = \frac{N_{Ed}}{A_c f_{cd}}$.

• $r_m = \frac{M_{01}}{M_{02}}$.

Effective Length ($l_o$) and Flexibility

Effective length depends on relative flexibilities ($k_1, k_2$) at column ends:
$k = \left(\frac{\theta}{M}\right) \times \left(\frac{EI}{l}\right)$

• $k = 0$: Rigid rotational restraint.

• $k = \infty$: No restraint.

• Recommended minimum value for practical design is $0.1$.

Creep Effects

Creep must be considered in second-order analysis via the effective creep ratio ($\phi_{ef}$):
$\phi_{ef} = \phi(\infty, t_0) \times \frac{M_{0Eqp}}{M_{0Ed}}$
Where:

• $M_{0Eqp}$: SLS first-order moment (quasi-permanent).

• $M_{0Ed}$: ULS first-order moment.
  Creep may be ignored ($\phi_{ef} = 0$) if:

1. $\phi(\infty, t_0) \le 2$

2. $\lambda \le 75$

3. $M_{0Ed} / N_{Ed} \ge h$

Analysis Methods

A. Method Based on Nominal Stiffness

Uses an estimated flexural stiffness ($EI$) considering cracking and creep:
$EI = K_c E_{cd} I_c + K_s E_s I_s$

• $K_s = 1$.

• $K_c = k_1 k_2 / (1 + \phi_{ef})$.

• Critical buckling load: $N_B = \frac{\pi^2 EI}{l_o^2}$.

• Moment Magnification: $M_{Ed} = M_{0Ed} \left[ 1 + \frac{\beta}{(N_B / N_{Ed}) - 1} \right]$.

B. Method Based on Nominal Curvature

Design moment: $M_{Ed} = M_{0Ed} + M_2$.

• Nominal second-order moment: $M_2 = N_{Ed} e_2$.

• Deflection: $e_2 = \frac{(1/r) l_o^2}{c}$.

• Reference factor $c = 10$ (for sinus distribution) or $c = 8$ (for constant moment).

• Curvature: $1/r = K_r \times K_{\phi} \times 1/r_0$.

• Reference curvature: $1/r_0 = \varepsilon_{yd} / (0.45 d)$.

Reinforcement Provisions

Longitudinal Reinforcement

• Minimum Diameter ($\phi_{min}$): Recommended $8\,mm$.

• Minimum Area ($A_{s,min}$): $0.1 N_{Ed} / f_{yd}$ or $0.002 A_c$.

• Maximum Area ($A_{s,max}$): $0.04 A_c$ (outside laps), $0.08 A_c$ (at laps).

Transverse Reinforcement (Links/Ties)

• Minimum Diameter: Greater of $6\,mm$ or $\frac{1}{4}$ of the longitudinal bar diameter.

• Maximum Spacing ($s_{cl,tmax}$): Least of:
    1. $20 \times$ minimum longitudinal bar diameter.
    2. Lesser dimension of the column.
    3. $400\,mm$.