Design of Compression Members

Fundamentals of Compression Members and Structural Stability

• Definition of Compression Members:

  • A compression member is a straight structural member subjected to two equal and opposite compressive forces applied at its ends.

  • An ideal compression member is perfectly straight, devoid of initial crookedness, geometrical imperfections, or residual stresses, and loaded uniformly such that the centre of gravity of the load coincides exactly with the centre of gravity of the cross section (true axial loading).

  • Perfectly axial conditions are impossible to achieve in physical construction; all real compression members possess accidental eccentricities, initial crookedness, and material non-homogeneities.

• Structural Terminology by Application:

  • Column, Stanchion, or Post: A vertical compression member carrying heavy loads, typically supporting floors, beams, or girders in multi-storey buildings or industrial frameworks.

  • Strut: A compression member used in roof trusses and bracing systems, characterized by smaller spans and comparatively lighter compressive loads.

  • Continuous Strut: A compression member that extends continuously across multiple joints (e.g., the top chord of a truss bridge girder or the principal rafter of a roof truss).

  • Discontinuous Strut: A compression member spanning between two adjacent joints only (e.g., vertical or inclined web members of a roof truss).

  • Principal Rafter: The primary inclined top-chord compression member in a roof truss.

  • Boom: The primary compression member in a crane jib or derrick.

• Equilibrium and Stability Phenomena:

  • Ordinary structural analysis assumes stable equilibrium where external forces are balanced by internal resisting stresses with a linear stress-strain relationship.

  • In compression members, buckling initiates potentially unstable equilibrium where internal and external responses interact nonlinearly across elastic and inelastic material ranges.

  • Instability: Defined as a physical condition wherein the slightest increment in lateral deflection precipitates further lateral displacement without an increase in applied load, leading rapidly to structural collapse.

  • Comparison with Tension Members: Compression members are vastly more critical than tension members. If a compression member possesses an initial crookedness, the applied axial load $P$ produces a primary bending moment equal to $P \times \delta$ (where $\delta$ is the lateral deflection), which magnifies lateral deformations. In contrast, an applied tensile force tends to straighten out initial misalignments, reducing bending effects.

• Classification by Slenderness and Failure Modes:

  • Short Columns: Extremely stocky members (behaving like solid blocks) with very low slenderness ratios ($\lambda \le 0.2$). They fail exclusively by plastic crushing or yielding across the entire cross section without buckling. The maximum axial resistance is the squash load:     $P_y = f_y \times A$

  • Intermediate Columns: Members of moderate slenderness that fail through inelastic buckling (combined yielding and lateral bending). Plastic yielding initiates in the extreme outer fibers while the core remains elastic; consequently, the effective flexural rigidity $E_t I$ degrades continuously.

  • Long (Slender) Columns: Members having high slenderness ratios that fail predominantly by elastic buckling before reaching the proportional limit of the material ($f_{\text{cr}} \le f_p$). The critical failure stress is governed entirely by flexural stiffness ($E I$) and is independent of the yield strength ($f_y$) of the steel.

Effective Length and Boundary Restraints

• Concept of Effective Length ($KL$):

  • The deflected shape of a buckled column contains a segment that behaves as if it were a pin-ended column. The points bounding this equivalent pin-ended segment are points of contraflexure (points of zero bending moment).

  • The effective length $l = K L$ represents the distance between consecutive points of contraflexure in the buckled shape, where $L$ is the unsupported actual length and $K$ is the effective length factor.

  • Restraining the ends against lateral translation or angular rotation significantly shortens the distance between contraflexure points, increasing the buckling load capacity.

• Idealized Boundary Conditions and Effective Length Factors ($K$):

  • Prismatic compression members with idealized boundary supports possess standard effective length values:

  • Both Ends Pinned: Translation restrained and rotation free at both ends; effective length $K L = 1.0 L$.

  • Both Ends Fixed: Translation and rotation restrained at both ends; effective length $K L = 0.65 L$ (theoretical $0.5 L$).

  • One End Fixed, One End Pinned: Translation and rotation restrained at one end; translation restrained and rotation free at the opposite end; effective length $K L = 0.8 L$ (theoretical $0.707 L$).

  • One End Fixed, One End Free (Cantilever): Translation and rotation restrained at the base; translation and rotation entirely free at the upper end; effective length $K L = 2.0 L$.

  • One End Fixed, One End Guided: Translation and rotation restrained at one end; translation free but rotation restrained at the opposite end; effective length $K L = 1.2 L$.

  • One End Pinned, One End Guided: Translation free and rotation restrained at one end; translation restrained and rotation free at the opposite end; effective length $K L = 2.0 L$.

• Directional Restraints and Principal Axes:

  • Compression members often exhibit distinct support conditions about their major ($z\text{-}z$) and minor ($y\text{-}y$) axes.

  • An intermediate lateral support or bracing located at mid-height suppresses lateral translation perpendicular to the weak axis, halving its unsupported length.

• In such configurations, buckling about the major axis occurs over the full height ($K L = L$), while buckling about the minor axis occurs between braces ($K L = L / 2$).

  • Codal Specifications for Truss Struts (Table 8.2):

• Continuous Struts: Single or double angles continuous over multiple panel points have an effective length in the plane of the gusset of $0.70 L \text{ to } 1.00 L$, and perpendicular to the plane of the gusset of $1.00 L$.

• Discontinuous Single-Angle Struts Connected with One Bolt: In the plane of the gusset, $K L = 1.00 L$; perpendicular to the gusset, $K L = 1.00 L$.

• Discontinuous Single-Angle Struts Connected with Two or More Bolts or Welded: In the plane of the gusset, $K L = 0.85 L$; perpendicular to the gusset, $K L = 1.00 L$.

• Discontinuous Double Angles Placed Back-to-Back on Opposite Sides of a Gusset: Connected with at least two fasteners per joint; in the plane of the gusset, $K L = 0.70 L \text{ to } 0.85 L$; perpendicular to the gusset, $K L = 1.00 L$.

• Discontinuous Double Angles Placed Back-to-Back on the Same Side of a Gusset: In the plane of the gusset, $K L = 0.70 L \text{ to } 0.85 L$; perpendicular to the gusset, $K L = 1.00 L$.

  • Frames: Sway versus Non-Sway Behavior:

• Braced / Non-Sway Frames: Lateral translation (sidesway) is prevented by stiff systems such as diagonal bracing, shear walls, or floor diaphragms tied to stable adjacent cores. The effective length factor $K$ for columns in braced rigid frames strictly varies within the bounds $0.5 \le K \le 1.0$. Taking $K = 1.0$ serves as a safe conservative assumption in the absence of detailed frame analysis.

• Stability Bracing: Bracing installed specifically to stabilize structural columns against premature out-of-plane buckling under vertical gravity loads alone.

• Unbraced / Sway Frames: Lateral stability and load resistance rely on the flexural rigidity of the beam-column joints. In these frames, lateral sway occurs freely; hence, $K > 1.0$. Lower storey columns with hinged bases approximate $K \approx 2.0$, while upper storey columns typically reach $K \approx 1.2$.

  • Wood's Curve Formulations for Framed Structures (IS: 800):

• The effective length factor $K$ is evaluated based on the rotational stiffness ratios $\beta_1$ and $\beta_2$ at the top and bottom column ends:     $\beta = \frac{\sum K_c}{\sum K_c + \sum K_b}$     where $K_c = C \left( \frac{I_c}{L_c} \right)$ is the effective flexural stiffness of columns meeting at the joint, and $K_b = \frac{I_b}{L_b}$ is the flexural stiffness of beams rigidly framing into the joint.

• Non-Sway (Braced) Frames:     $K = \left[ \frac{1 + 0.145(\beta_1 + \beta_2) - 0.265 \beta_1 \beta_2}{2 - 0.364(\beta_1 + \beta_2) - 0.247 \beta_1 \beta_2} \right]$

• Sway (Moment-Resisting / Unbraced) Frames:     $K = \left[ \frac{1 - 0.2(\beta_1 + \beta_2) - 0.12 \beta_1 \beta_2}{1 - 0.8(\beta_1 + \beta_2) - 0.6 \beta_1 \beta_2} \right]^{0.5}$

• End Joint Boundary Rules:

  • Rigidly fixed column base: Beams possess infinite rotational stiffness ($\sum K_b = \infty$), leading to $\beta = 0$.

  • Truly hinged column base: No beams provide rotational restraint ($\sum K_b = 0$), leading to $\beta = 1.0$.

• Connection Factor $C$ (Table 8.3):

  • Evaluated as a function of the axial load ratio $\bar{n} = \frac{P}{P_e}$, where $P_e = \frac{\pi^2 E I}{L^2}$ is the Euler buckling load:

    • Far End Pinned: For Braced frames, $C = 1.5(1 - \bar{n})$; For Unbraced frames, $C = 1.5(1 - \bar{n})$.

    • Far End Rigidly Connected to Column: For Braced frames, $C = 1.0(1 - \bar{n})$; For Unbraced frames, $C = 1.0(1 - 0.2 \bar{n})$.

    • Far End Fixed: For Braced frames, $C = 2.0(1 - 0.4 \bar{n})$; For Unbraced frames, $C = 0.67(1 - 0.4 \bar{n})$.

Slenderness Ratio and Cross-Sectional Geometry

• Definition of Slenderness Ratio ($\lambda$):

  • The slenderness ratio of a compression member is defined as the ratio of its effective length to the appropriate radius of gyration:     $\lambda = \frac{K L}{r}$

  • The radius of gyration is given by:     $r = \sqrt{\frac{I}{A}}$

  • If unsupported lengths and boundary fixities are identical across all planes, buckling occurs about the principal axis that possesses the minimum moment of inertia, corresponding to the least radius of gyration ($r_{\min}$):     $\lambda_{\max} = \frac{K L}{r_{\min}}$

• Principal Axes of Asymmetrical versus Symmetrical Shapes:

  • For asymmetrical shapes, such as single angles, centroidal axes ($z\text{-}z$ and $y\text{-}y$) do not represent principal axes. They bend and buckle about their minor principal axis ($v\text{-}v$), which exhibits the minimum radius of gyration ($r_v$).

• Symmetrical profiles (I-sections, channels, double angles arranged back-to-back) have principal axes that coincide with their centroidal axes; thus, buckling occurs directly about the minor centroidal axis ($y\text{-}y$).

  • Equalization of Slenderness Ratios:

• To maximize structural efficiency, the material must be oriented such that the directional slenderness ratios balance:     $\lambda_z = \frac{K_z L_z}{r_z} \approx \lambda_y = \frac{K_y L_y}{r_y}$

• Example (Transmission Pole): An I-section electric pole fixed at its base carries longitudinal overhead wires. The wires act as a hinge restraint in the wire direction ($K_y L_y = 0.8 L$), while the structure acts as a free cantilever in the perpendicular road direction ($K_z L_z = 2.0 L$). Orienting the section with its strong axis parallel to the road attempts to equate $\lambda_z = \frac{2.0 L}{r_z}$ with $\lambda_y = \frac{0.8 L}{r_y}$, achieving maximum material economy.

  • Maximum Permissible Slenderness Ratios (Table 8.4):

• $\lambda = 180$: Members carrying compressive loads resulting from dead loads and imposed (live) loads.

• $\lambda = 180$: A tension member in which a reversal of direct stress occurs due to loads other than wind or seismic forces.

• $\lambda = 250$: Members subjected to compressive forces resulting solely from wind or earthquake actions, provided the member deformation does not adversely alter stresses elsewhere.

• $\lambda = 300$: Compression flange of a beam restrained against lateral-torsional buckling.

• $\lambda = 350$: A member normally acting as a tie in a roof truss or bracing system, not considered effective when subjected to stress reversal caused by wind or earthquake forces.

• Practical Reasons for Codal Slenderness Limits:

  • Prevention of damage from accidental handling and construction loads (fabrication, transit, and erection).

  • Ensuring members can safely support workmen walkways or equipment rigging.

  • Preventing resonant excitation and excessive vibrations induced by dynamic wind buffeting.

  • Cross-Sectional Types and Comparative Attributes:

• Rods and Solid Bars: Extremely low lateral stiffness; restricted strictly to short lengths ($L < 3.0\,m$).

• Circular and Rectangular Hollow Tubular Sections:

  • Exhibit identical radii of gyration in all radial directions ($r_z = r_y$) and high local buckling strength.

  • Exceptional torsional rigidity compared to open profiles.

  • Reduced wind resistance drag coefficients compared to flat plate assemblies.

  • Smaller total exposed surface area, decreasing required surface painting and fireproofing coats.

  • Elimination of interior re-entrant corners prevents dust collection and corrosion.

  • Yields up to $50\%$ reduction in self-weight relative to open profiles; despite costing $\approx 25\%$ more per ton, overall fabrication achieves up to $20\%$ total cost savings.

• Single Angles: Very small least radius of gyration ($r_v$); unsuitable for long spans. Equal angles are more economical than unequal angles due to higher $r_{\min}$ for equivalent cross-sectional steel areas.

• Double Angles:

  • Back-to-back configurations are common for trusses. Unequal angles with long legs back-to-back balance $r_z$ and $r_y$.

  • Cruciform (star-shaped) double angles yield approximately equal radii of gyration in both orthogonal directions.

• Channel Profiles: Two channels placed back-to-back produce low $r_y$ values. When placed face-to-face (toe-to-toe), they maximize $r_y$, producing optimal column sections with superior torsional and lateral rigidity.

  • Cross-Section Classification and Local Buckling:

• Thin plate elements subjected to compressive stresses may buckle locally into waves or wrinkles prior to overall column failure, preventing the member from attaining its calculated load capacity.

• Unstiffened Elements: Outstand elements supported along only one longitudinal edge parallel to the compressive load (e.g., flanges of rolled I-sections, outstanding legs of angles).

• Stiffened Elements: Internal elements supported continuously along both longitudinal edges parallel to the compressive load (e.g., webs of I-sections and channels, cover plates between fastener lines).

• Effective Width Method for Slender Sections:

  • For slender cross sections other than circular tubes, non-effective plate areas are eliminated using reduced effective widths:       $A_{\text{eff}} = \sum B_{\text{eff}} \times t$       $B_{\text{eff}} = \lambda_L \times t \times \epsilon \le b$       $\epsilon = \sqrt{\frac{250}{f_y}}$       where $\lambda_L$ represents the limiting width-to-thickness ratio for semi-compact sections, and $b$ is the actual element width.

  • Circular Hollow Sections: Classified as slender when:       $\frac{D}{t \times \epsilon^2} > 88$       The reduced effective area is calculated as:       $A_{\text{eff}} = A_g \sqrt{\frac{88}{D / (t \times \epsilon^2)}}$

Modes of Column Buckling and Failure Theories

• Buckling Failure Classifications:

  • Flexural Buckling (Euler Buckling): Pure lateral bending deflection about the minor principal axis without any section rotation or twisting. Dominates members with high bending asymmetry or equal double symmetry (standard rolled I-sections).

  • Torsional Buckling: Pure twisting of the member about its longitudinal shear center axis ($x\text{-}x$) without lateral translation. It occurs only in doubly symmetrical shapes possessing extremely thin, open, outstand elements with low Saint-Venant torsional stiffness (such as cruciform cross sections).

  • Flexural-Torsional Buckling: Coupled simultaneous lateral bending and torsional twisting. Governs unsymmetrical shapes (unequal single angles) and singly symmetrical shapes (channels, structural tees, double-angle assemblies).

• Classical Euler Elastic Buckling Theory:

  • Euler derived the theoretical buckling capacity for an ideal, pin-ended, perfectly straight column with constant flexural rigidity $E I$:     $P_{\text{cr}} = \frac{\pi^2 E I}{L^2}$

  • The mean compressive stress at elastic buckling ($f_{\text{cr}}$) is:     $f_{\text{cr}} = \frac{P_{\text{cr}}}{A} = \frac{\pi^2 E I}{A L^2} = \frac{\pi^2 E}{(L / r)^2} = \frac{\pi^2 E}{\lambda^2}$

  • Euler Proportional Limit Boundary ($\lambda_p$):

  • Euler's hyperbolic formulation remains valid only while the buckling stress does not exceed the material proportional limit ($f_p$):       $f_{\text{cr}} = f_y \implies \lambda_p = \pi \sqrt{\frac{E}{f_y}}$

  • For standard mild structural steel ($f_y = 250\,\text{MPa}$, $E = 2.0 \times 10^5\,\text{MPa}$):       $\lambda_p = \pi \sqrt{\frac{2.0 \times 10^5}{250}} = 88.85$

• Inelastic Column Behavior and the Tangent Modulus Theory:

  • For columns with slenderness $\lambda < \lambda_p$, buckling occurs at stresses exceeding the proportional limit ($f_p$). In this inelastic regime, the stress-strain response ceases to be linear, and the instantaneous slope degrades to the tangent modulus $E_t = \frac{df}{d\epsilon}$.

  • Substituting $E_t$ into Euler's equation yields the Engesser-Shanley critical load:     $P_{\text{cr}} = \frac{\pi^2 E_t I}{L^2}$

• Influence of Residual Stresses ($f_r$) and Geometric Imperfections:

  • Real hot-rolled steel shapes retain high residual stresses from uneven cooling rates (flange tips and web centers cool faster than flange-web junctions). Compressive residual stresses reach up to $f_r \approx 0.5 f_y$.

  • Consequently, the effective proportional limit is lowered to:     $f_p = f_y - f_r = 0.5 f_y$

  • Real columns possess initial geometric crookedness (out-of-straightness) commonly modeled by an initial sinusoidal bow with an amplitude of $\frac{L}{1000}$.

  • Parabolic Inelastic Transition Curve: In design formulations, a parabolic curve bridges the gap between yielding and Euler elastic buckling:     $f_{\text{cr}} = f_y - \frac{f_y^2}{4 \pi^2 E} \left( \frac{K L}{r} \right)^2$

Design Formulations and the Limit State Approach (IS: 800-2007)

• Historical Codal Formulations:

  • Secant Formula (IS: 800 prior to 1984): Incorporated load eccentricity and initial curvature using a working stress factor of safety of $1.68$:     $F_c = \frac{f_y / 1.68}{1 + 0.20 \sec \left( \frac{l}{r} \sqrt{\frac{1.68 F_c}{4 E}} \right)}$

  • Merchant-Rankine Formula (IS: 800-1984): An empirical interaction between the plastic yield capacity ($f_y$) and elastic critical stress ($f_{\text{cc}}$) using an index $n = 1.4$ with a factor of safety of $1.67$ ($0.6 f_y$):     $\sigma_{\text{ac}} = 0.6 \left[ \frac{f_{\text{cc}} \times f_y}{(f_{\text{cc}}^n + f_y^n)^{1/n}} \right]$

• IS: 800-2007 Limit State Codal Provisions (Perry-Robertson Approach):

  • Column curves reflect the Perry-Robertson formulation, normalized against material yield strength and calibrated to capture residual stresses, an initial sine-wave bow of magnitude $\frac{L}{1000}$, and unintentional load eccentricities.

  • Design Compressive Strength ($P_d$):     $P_d = A_e \times f_{\text{cd}}$     where $A_e$ is the effective cross-sectional area, and $f_{\text{cd}}$ is the design compressive stress.

  • Design Compressive Stress Formulation ($f_{\text{cd}}$):     $f_{\text{cd}} = \frac{f_y / \gamma_{m0}}{\phi + \left[ \phi^2 - \lambda^2 \right]^{0.5}} = \chi \frac{f_y}{\gamma_{m0}} \le \frac{f_y}{\gamma_{m0}}$     where:     $\phi = 0.5 \left[ 1 + \alpha(\lambda - 0.2) + \lambda^2 \right]$     $\gamma_{m0} = 1.10 \quad \text{(Partial safety factor for steel yield strength)}$

  • Non-Dimensional Effective Slenderness Ratio ($\lambda$):     $\lambda = \sqrt{\frac{f_y}{f_{\text{cc}}}} = \sqrt{\frac{f_y (K L / r)^2}{\pi^2 E}}$     where $f_{\text{cc}} = \frac{\pi^2 E}{(K L / r)^2}$ is the Euler buckling stress.

  • Stress Reduction Factor ($\chi$):     $\chi = \frac{1}{\phi + \left[ \phi^2 - \lambda^2 \right]^{0.5}}$

• Imperfection Factors and Buckling Curves (Table 8.5):

  • Four distinct buckling curves (a, b, c, d) classify cross-sectional profiles based on fabrication methods and sensitivity to residual stresses via the imperfection factor $\alpha$:

  • Curve a ($\alpha = 0.21$): Quasi-perfect profiles. Hot-rolled I-sections with $\frac{h}{b_f} > 1.2$ and flange thickness $t_f \le 40\,\text{mm}$ buckling about the major $z\text{-}z$ axis; hot-rolled hollow tubular sections.

  • Curve b ($\alpha = 0.34$): Moderate imperfections. Rolled I-sections ($\frac{h}{b_f} > 1.2$, $t_f \le 40\,\text{mm}$) buckling about the minor $y\text{-}y$ axis; rolled I-sections with $40\,\text{mm} < t_f \le 100\,\text{mm}$ buckling about the major axis; welded I-sections ($t_f \le 40\,\text{mm}$) buckling about the major axis; cold-formed hollow tubes; welded box sections.

  • Curve c ($\alpha = 0.49$): High imperfections. Standard structural channels, angles, tees, solid sections, thick welded box profiles, rolled I-sections with $\frac{h}{b_f} \le 1.2$ ($t_f \le 100\,\text{mm}$) buckling about the minor axis, welded I-sections ($t_f \le 40\,\text{mm}$) buckling about the minor axis, and all built-up latticed columns.

  • Curve d ($\alpha = 0.76$): Maximum imperfections. Heavy rolled I-sections with very thick flanges ($t_f > 100\,\text{mm}$) and heavy welded I-sections ($t_f > 40\,\text{mm}$) buckling about their minor $y\text{-}y$ axis.

• Design Compressive Stress ($f_{\text{cd}}$ in $\text{N/mm}^2$) for $f_y = 250\,\text{MPa}$ (Table 8.7):

  | $K L / r$ | Curve a ($f_{\text{cd}}$) | Curve b ($f_{\text{cd}}$) | Curve c ($f_{\text{cd}}$) | Curve d ($f_{\text{cd}}$) |   | :--- | :--- | :--- | :--- | :--- |   | 10 | $227$ | $227$ | $227$ | $227$ |   | 20 | $226$ | $225$ | $224$ | $223$ |   | 30 | $220$ | $216$ | $211$ | $204$ |   | 40 | $213$ | $206$ | $198$ | $185$ |   | 50 | $205$ | $194$ | $183$ | $167$ |   | 60 | $195$ | $181$ | $168$ | $150$ |   | 70 | $182$ | $166$ | $152$ | $133$ |   | 80 | $167$ | $150$ | $136$ | $118$ |   | 90 | $149$ | $134$ | $121$ | $105$ |   | 100 | $132$ | $118$ | $107$ | $92.6$ |   | 110 | $115$ | $104$ | $94.6$ | $82.1$ |   | 120 | $101$ | $91.7$ | $83.7$ | $73.0$ |   | 130 | $88.3$ | $81.0$ | $74.3$ | $65.2$ |   | 140 | $77.8$ | $71.8$ | $66.2$ | $58.4$ |   | 150 | $68.9$ | $64.0$ | $59.2$ | $52.6$ |   | 160 | $61.4$ | $57.3$ | $53.3$ | $47.5$ |   | 170 | $55.0$ | $51.5$ | $48.1$ | $43.1$ |   | 180 | $49.5$ | $46.5$ | $43.6$ | $39.3$ |   | 190 | $44.7$ | $42.2$ | $39.7$ | $35.9$ |   | 200 | $40.7$ | $38.5$ | $36.3$ | $33.0$ |   | 210 | $37.1$ | $35.2$ | $33.3$ | $30.0$ |   | 220 | $34.0$ | $32.3$ | $30.6$ | $28.0$ |   | 230 | $31.2$ | $29.8$ | $28.3$ | $26.0$ |   | 240 | $28.8$ | $27.5$ | $26.2$ | $24.1$ |   | 250 | $26.6$ | $27.5$ | $24.3$ | $22.5$ |

Design Procedure for Axially Loaded Compression Members

• Standard Design Steps:

  • Step 1 (Initial Slenderness Guess): For typical storeys ($3.0\,m \text{ to } 5.0\,m$), assume a trial slenderness ratio $\lambda \approx 40 \text{ to } 60$. For unusually long columns or heavily loaded members, adjust this estimate downward or upward accordingly.

  • Step 2 (Trial Stress Selection): Identify the probable buckling curve from Table 8.6 and determine the trial design compressive stress $f_{\text{cd}}$ from Table 8.7.

  • Step 3 (Required Sectional Area): Compute the required gross cross-sectional area:     $A_g = \frac{P_u}{f_{\text{cd}}}$     where $P_u$ is the factored compressive load ($N$).

  • Step 4 (Section Selection): Choose a rolled section from standard steel handbooks providing the required area, prioritizing sections that offer the largest possible radius of gyration.

  • Step 5 (Slenderness Calculation): Determine effective lengths ($K_z L_z$, $K_y L_y$) based on real boundary fixities, compute actual slenderness values ($\lambda_z = \frac{K_z L_z}{r_z}$, $\lambda_y = \frac{K_y L_y}{r_y}$), and verify that $\lambda_{\max}$ remains below permissible limits from Table 8.4.

  • Step 6 (Design Compressive Stress): Extract the true value of $f_{\text{cd}}$ from Table 8.7 corresponding to $\lambda_{\max}$.

  • Step 7 (Capacity Check): Compute design compressive strength $P_d = A_e f_{\text{cd}}$. Verify that $P_d \ge P_u$. If $P_d$ exceeds $P_u$ by more than $5\%$, revise the section downward for economy.

• Single-Angle Struts Loaded Through One Leg (IS: 800 Provision):

  • When a single-angle strut is connected to a gusset plate through only one of its legs, the eccentric application of load induces simultaneous flexural bending and torsion. Its resistance is determined using Buckling Curve c with an equivalent slenderness ratio $\lambda_e$:     $\lambda_e = \sqrt{k_1 + k_2 \lambda_{vv}^2 + k_3 \lambda_{\phi}^2}$     where:     $\lambda_{vv} = \frac{l / r_{vv}}{\epsilon \sqrt{\frac{\pi^2 E}{250}}}$     $\lambda_{\phi} = \frac{b_1 + b_2}{2 t \epsilon \sqrt{\frac{\pi^2 E}{250}}}$     $l$ = center-to-center length between supports.     $r_{vv}$ = radius of gyration about the minor principal axis ($v\text{-}v$).     $b_1, b_2$ = widths of the connected and outstanding legs.     $t$ = leg thickness.     $\epsilon = \sqrt{\frac{250}{f_y}}$.

  • End Fixity Constants $k_1, k_2, k_3$ (Table 8.8):

  • $\ge 2$ Bolts or Welded, Directionally Fixed Ends: $k_1 = 0.20$, $k_2 = 0.35$, $k_3 = 20$.

  • $\ge 2$ Bolts or Welded, Directionally Hinged Ends: $k_1 = 0.70$, $k_2 = 0.60$, $k_3 = 5$.

  • 1 Bolt, Directionally Fixed Ends: $k_1 = 0.75$, $k_2 = 0.35$, $k_3 = 20$.

  • 1 Bolt, Directionally Hinged Ends: $k_1 = 1.25$, $k_2 = 0.50$, $k_3 = 60$.

Built-Up and Latticed Compression Members

• Structural Mechanics of Latticed Columns:

  • When individual rolled sections cannot satisfy heavy load demands or long unbraced lengths, built-up open-web columns (laced or battened) are fabricated.

  • The components are spaced apart so that the radius of gyration about the weaker axis is maximized, ensuring:     $r_y \ge r_z$

  • Disregarding lacing or battens as axial load-carrying elements is standard practice; only continuous longitudinal ribs are accounted for in cross-sectional capacity calculations.

  • Shear Deformation Effects: In built-up latticed columns, the transverse shear forces generated during lateral buckling induce significant shear deformations within the open web. To compensate for this added flexibility, codes require the actual maximum slenderness ratio $\left(\frac{K L}{r}\right)_0$ to be magnified:

  • Laced Columns: $\left(\frac{K L}{r}\right)_e = 1.05 \left(\frac{K L}{r}\right)_0$

  • Battened Columns: $\left(\frac{K L}{r}\right)_e = 1.10 \left(\frac{K L}{r}\right)_0$

• Design Rules for Lacing Systems:

  • Geometry: Lacing bars (flats or angles) must be inclined at an angle $\theta$ between $40^\circ \text{ and } 70^\circ$ to the longitudinal axis of the member (optimum efficiency occurs between $35^\circ \text{ and } 45^\circ$; $45^\circ$ is typical for trials).

  • Local Buckling of Individual Components: The spacing $a_1$ between consecutive lacing nodes must prevent local buckling of an isolated main rib before overall column failure:     $\frac{a_1}{r_y} \le 50 \quad \text{and} \quad \frac{a_1}{r_y} \le 0.70 \left(\frac{K L}{r}\right)_0$     where $r_y$ is the minimum radius of gyration of a single component rib.

  • Transverse Design Shear ($V_t$): Lacing systems must be proportioned to resist an aggregate transverse shear force of at least $2.5\%$ of the total factored axial load on the column:     $V_t = 0.025 P_u$     This shear is distributed equally across $N$ parallel lacing planes ($N = 2$ for two-channel assemblies):     $V = \frac{V_t}{N}$

  • Axial Force in Lacing Bars:

  • Single Lacing System: Compressive force $F_c = \frac{V_t}{N \sin(\theta)}$

  • Double Lacing System: Compressive force $F_c = \frac{V_t}{2 N \sin(\theta)}$

  • Proportioning Lacing Flats:

  • Minimum Width: Must be at least three times the nominal diameter of the connecting fastener ($b \ge 3 d_0$).

  • Minimum Thickness:

    • For Single Lacing: $t \ge \frac{1}{40} L_{\text{eff}}$

    • For Double Lacing: $t \ge \frac{1}{60} L_{\text{eff}}$

  • Maximum Slenderness: The slenderness ratio of a lacing bar must not exceed $145$:       $\lambda_{\text{lacing}} = \frac{L_{\text{eff}}}{r_{\min}} \le 145 \quad \text{where} \quad r_{\min} = \frac{t}{\sqrt{12}}$

  • Effective Length of Lacing Bar ($L_{\text{eff}}$):

    • Bolted Single Lacing: Distance between inner end bolts.

    • Bolted Double Lacing (interconnected at intersection): $0.70 \times$ distance between inner end bolts.

    • Welded Lacing: $0.70 \times$ distance between inner ends of effective welds.

  • Prohibition of Combined Lacing and Battens: Mixing continuous lacing with intermediate battens is prohibited. Under compressive column shortening, lacing bars deform in an accordion-like kinematic motion; introducing battens restrains this deformation, forcing the lattice to act as a rigid secondary truss and inducing severe parasitic compressive stresses ($25\% \text{ to } 40\%$ of main rib stresses).

• Design Rules for Battened Columns:

  • Configuration: Battens must be provided in pairs on opposite faces, dividing the member into at least three bays along its length (requiring a minimum of four batten plates per face).

  • Spacing ($C$):     $\frac{C}{r_y} \le 50 \quad \text{and} \quad \frac{C}{r_y} \le 0.70 \left(\frac{K L}{r}\right)_e$

  • Dimensions:

  • End Battens: Effective depth $d' \ge S' + 2 C_{\text{yy}}$ and $d' > 2 b_f$; Overall depth $d = d' + 2 e$.

  • Intermediate Battens: Effective depth $d_1' \ge \frac{3}{4} d'$ and $d_1' > 2 b_f$.

  • Thickness: $t \ge \frac{1}{50}(S' + 2 g)$, where $S'$ is the clear spacing between ribs and $g$ is the gauge distance.

  • Internal Shear and Bending Forces in Battens:

  • Transverse shear $V_t = 0.025 P_u$.

  • Longitudinal shear force per batten:       $V_b = \frac{V_t C}{N S}$

  • Transverse bending moment per batten:       $M = \frac{V_t C}{2 N}$       where $S$ is the transverse center-to-center distance between fastener groups, and $N$ is the number of parallel batten planes ($N=2$).

  • Stress Capacity Checks for Battens:

  • Shear Stress Check:       $\tau_{\text{cal}} = \frac{V_b}{t \times d} \le \frac{f_y}{\sqrt{3} \times \gamma_{m0}}$

  • Bending Stress Check:       $\sigma_{\text{bc, cal}} = \frac{6 M}{t \times d^2} \le \frac{f_y}{\gamma_{m0}}$

• Components Connected Back-to-Back Without Latticing:

  • Two structural shapes placed back-to-back in direct contact (or separated by small packing washers) must be stitched together with intermittent fasteners or welds.

  • The slenderness ratio of each individual component between consecutive connections must satisfy:     $\lambda_{\text{rib}} = \frac{s}{r_{\min}} \le 40 \quad \text{and} \quad \lambda_{\text{rib}} \le 0.60 \lambda_{\text{column}}$

  • Connecting bolts must have diameters of at least $16\,\text{mm}$ for component thickness $t \le 10\,\text{mm}$, $20\,\text{mm}$ for $t \le 16\,\text{mm}$, and $22\,\text{mm}$ for $t > 16\,\text{mm}$.

  • The clear longitudinal spacing between consecutive bolts or intermittent weld runs must not exceed $12 t$ or $200\,\text{mm}$.

Encased Columns and Column Splices

• Encased Columns:

  • Steel I-sections or back-to-back channel columns are encased in dense structural concrete to provide architectural finishes, corrosion protection, and fireproofing in tall building skeletons and basements.

• Geometric and Material Limits (IS: 800-1984 specifications):

  • Member cross section must be a symmetrical I-shape or two channels placed back-to-back, with maximum overall outer dimensions of $750\,\text{mm} \times 450\,\text{mm}$.

  • Steel profiles must remain unpainted and be cast into concrete of grade at least M-15 using $20\,\text{mm}$ aggregate.

  • Minimum concrete cover over steel edges and faces is $50\,\text{mm}$.

  • Minimum total casing width is $b_0 + 100\,\text{mm}$, where $b_0$ is the steel flange width.

  • Concrete casing must be reinforced with transverse steel stirrups ($\ge 5\,\text{mm}$ diameter at $\le 150\,\text{mm c/c}$ spacing) held in place by longitudinal corner bars of at least $10\,\text{mm}$ diameter.

  • Column Splices:

• Location: Columns are typically fabricated in two-storey lengths ($\approx 6\,m \text{ to } 8\,m$). Splices are placed roughly $30\,\text{cm} \text{ to } 150\,\text{cm}$ above the finished floor level to avoid interfering with beam-to-column framing brackets and to locate the joint near the point of contraflexure under lateral loads.

• Design for Complete Bearing (Machined/Milled Ends): When column ends are milled for square bearing, $50\%$ of the axial load is transferred across the joint via direct bearing. The splice plates and fasteners carry the remaining $50\%$ of the axial force, plus any net tension induced by coexisting bending moments.

• Design for Non-Machined Ends: When column mating faces are not fully machined, the splice plates and fasteners must be designed to transfer $100\%$ of the axial load, shear force, and bending moments.

• Bearing and Packing Plates:

  • If spliced columns have differing section depths, horizontal bearing plates are inserted between the profiles to transfer flange forces down to the lower column's web and flanges.

  • Vertical packing plates are used to align the flange faces. When packing thickness exceeds $6\,\text{mm}$, the design shear capacity of the bolts must be reduced by the packing factor:       $\beta_{\text{pkg}} = 1.0 - 0.0125 \times t_{\text{pkg}}$

Working Stress Design Format

• Permissible Stress Method Criteria:

  • When verifying compression members under working (serviceability) load combinations, the induced axial compressive stress $f_c$ must not exceed the permissible stress $f_{\text{ac}}$:     $f_c = \frac{P_{\text{service}}}{A_e} \le f_{\text{ac}}$

  • The allowable compressive stress is derived directly from the limit state design stress $f_{\text{cd}}$ by removing the load factor:     $f_{\text{ac}} = 0.60 \times f_{\text{cd}}$

Detailed Worked Examples

• Example 1: Radius of Gyration of a Compound Column:

  • Problem: Calculate $r_{\min}$ for a built-up column consisting of an ISHB 250 @ $536.6\,\text{N/m}$ profile ($A = 6971\,\text{mm}^2$, $I_z = 7983.9 \times 10^4\,\text{mm}^4$, $I_y = 2011.7 \times 10^4\,\text{mm}^4$, $t_f = 9.7\,\text{mm}$, total depth $D = 250\,\text{mm}$) with one $300\,\text{mm} \times 20\,\text{mm}$ cover plate attached to each flange.

  • Solution:

  • Total cross-sectional area:       $A = 6971 + 2 \times (300 \times 20) = 18,971\,\text{mm}^2$

  • Major axis moment of inertia ($I_z$):       $I_z = 7983.9 \times 10^4 + 2 \left[ \frac{300 \times 20^3}{12} + (300 \times 20) \times (125 + 10)^2 \right] = 29,893.9 \times 10^4\,\text{mm}^4$       $r_z = \sqrt{\frac{29,893.9 \times 10^4}{18,971}} = 125.52\,\text{mm}$

  • Minor axis moment of inertia ($I_y$):       $I_y = 2011.7 \times 10^4 + 2 \left[ \frac{20 \times 300^3}{12} \right] = 11,011.7 \times 10^4\,\text{mm}^4$       $r_y = \sqrt{\frac{11,011.7 \times 10^4}{18,971}} = 76.19\,\text{mm}$

  • Least radius of gyration:       $r_{\min} = r_y = 76.19\,\text{mm}$

• Example 2: Axial Capacity of a Single Angle Strut:

  • Problem: An ISA $100\,\text{mm} \times 100\,\text{mm} \times 6\,\text{mm}$ ($f_y = 250\,\text{MPa}$, $A = 1167\,\text{mm}^2$, $r_v = 19.5\,\text{mm}$) serves as a roof truss strut with an unsupported length of $3.0\,m$. Calculate its design compressive strength if connected at each end by two bolts (fixed condition).

  • Solution:

  • Calculate basic nondimensional parameters:       $\lambda_{vv} = \frac{3000 / 19.5}{1.0 \times \pi \sqrt{\frac{2.0 \times 10^5}{250}}} = \frac{153.85}{88.857} = 1.731$       $\lambda_{\phi} = \frac{100 + 100}{2 \times 6 \times 1.0 \times \pi \sqrt{\frac{2.0 \times 10^5}{250}}} = \frac{200}{12 \times 88.857} = 0.1875$

  • Constants for two bolts with fixed ends: $k_1 = 0.20$, $k_2 = 0.35$, $k_3 = 20$.

  • Equivalent slenderness ratio:       $\lambda_e = \sqrt{0.20 + 0.35 \times 1.731^2 + 20 \times 0.1875^2} = \sqrt{0.20 + 1.0487 + 0.7031} = 1.397$

  • Buckling Curve c ($\alpha = 0.49$):       $\phi = 0.5 \left[ 1 + 0.49(1.397 - 0.2) + 1.397^2 \right] = 1.77$       $f_{\text{cd}} = \frac{250 / 1.1}{1.77 + \sqrt{1.77^2 - 1.397^2}} = \frac{227.27}{1.77 + 1.087} = 79.55\,\text{N/mm}^2$

  • Design compressive strength:       $P_d = 1167 \times 79.55 \times 10^{-3} = 92.83\,\text{kN}$

• Example 3: Built-Up Laced Column Design:

  • Problem: Design a built-up column of length $10\,m$ carrying a factored load of $1080\,\text{kN}$. Ends are pinned ($K = 1.0$). Use two channels back-to-back with a single lacing system and $16\,\text{mm}$ diameter Grade 4.6 bolts ($f_y = 250\,\text{MPa}$, $f_u = 410\,\text{MPa}$, $f_{\text{ub}} = 400\,\text{MPa}$).

  • Solution:

  • Assume trial $f_{\text{cd}} = 150\,\text{MPa}$. Area required $A_g = \frac{1080 \times 10^3}{150} = 7200\,\text{mm}^2$.

  • Try 2 ISMC 300 @ $351.2\,\text{N/m}$: $A = 2 \times 4564 = 9128\,\text{mm}^2$, $r_z = 118.1\,\text{mm}$, $r_y = 26.1\,\text{mm}$, $I_z = 6362.6 \times 10^4\,\text{mm}^4$, $I_y = 310.8 \times 10^4\,\text{mm}^4$, $C_{\text{yy}} = 23.6\,\text{mm}$.

  • Column slenderness check:       $\lambda_0 = \frac{1.0 \times 10,000}{118.1} = 84.67$       $\lambda_e = 1.05 \times 84.67 = 88.90$       From Table 8.7 (Curve c): $f_{\text{cd}} = 122.65\,\text{MPa}$.       $P_d = 9128 \times 122.65 \times 10^{-3} = 1119.5\,\text{kN} > 1080\,\text{kN} \quad (\text{Safe})$

  • Channel Spacing ($S$): Set $I_y = I_z$:       $2 \times 6362.6 \times 10^4 = 2 \left[ 310.8 \times 10^4 + 4564 \left( \frac{S}{2} + 23.6 \right)^2 \right] \implies S = 183.1\,\text{mm} \approx 183.5\,\text{mm}$

  • Lacing System Geometry ($\theta = 45^\circ$, gauge $g = 50\,\text{mm}$):       Vertical spacing $a_1 = 2 \times (183.5 + 2 \times 50) \cot(45^\circ) = 567\,\text{mm}$.       $\frac{a_1}{r_y} = \frac{567}{26.1} = 21.72 < 50 \quad \text{and} \quad < 0.70 \times 88.90 = 62.23 \quad (\text{Safe})$

  • Design Forces in Lacing:       $V_t = 0.025 \times 1080 = 27\,\text{kN}$       Transverse shear per plane: $V = \frac{27}{2} = 13.5\,\text{kN}$.       Compressive force in bar: $F_c = 13.5 \times \csc(45^\circ) = 19.09\,\text{kN}$.

  • Lacing Flat Section (Try 50 ISF $12\,\text{mm}$ with $16\,\text{mm}$ bolts):       Minimum width $= 3 \times 16 = 48\,\text{mm} \implies 50\,\text{mm}$.       Minimum thickness $= \frac{L_{\text{eff}}}{40} = \frac{283.5 \times \csc(45^\circ)}{40} = 10.02\,\text{mm} \implies 12\,\text{mm}$.       $r_{\min} = \frac{12}{\sqrt{12}} = 3.464\,\text{mm}$.       $\lambda = \frac{283.5 \times \csc(45^\circ)}{3.464} = 115.72 < 145 \quad (\text{Safe})$       From Table 8.7 (Curve c): $f_{\text{cd}} = 88.37\,\text{MPa}$.       $P_d = (50 \times 12) \times 88.37 \times 10^{-3} = 53.02\,\text{kN} > 19.09\,\text{kN} \quad (\text{Safe})$

  • Connections: Provide one $16\,\text{mm}$ diameter Grade 4.6 bolt at each end (double shear capacity $= 58.01\,\text{kN} > 27.0\,\text{kN}$).

Questions and Discussion

• Why are compression members structurally more critical than tension members?

  • Compression members are prone to stability failures via lateral buckling at loads substantially lower than their material yield limit. Geometric imperfections, unavoidable load eccentricities, residual stresses, and transverse handling forces generate secondary bending moments ($P \times \delta$) that amplify lateral deflections. In contrast, tensile loads naturally straighten members, mitigating bending and stabilizing the section.

• Why are plastic or compact sections strictly preferred for compression members?

  • Plastic and compact sections possess small width-to-thickness ratios ($b / t$), making their constituent plates stiff against local plate wrinkling. This allows the section to develop its full plastic yield stress profile across the entire cross section without premature local instability, ensuring adequate ductility.

• What governs the differing behavior of long versus intermediate columns?

  • In long columns, the axial buckling stress remains below the material proportional limit ($f_{\text{cr}} \le f_p$); they buckle elastically at loads governed entirely by flexural stiffness ($E I$), independent of steel grade. Intermediate columns buckle inelastically; compressive stresses combined with internal residual stresses cause the outer fibers to yield while the interior remains elastic. Their capacity depends on both yield strength and the changing tangent modulus ($E_t$).

• What is the physical nature of inelastic buckling?

  • Classical Euler theory presumes a constant elastic modulus $E$. In real columns, residual stresses from manufacturing induce premature, non-uniform yielding above the proportional limit ($f_p = f_y - f_r$). As yielded zones expand, the effective stiffness of the cross section degrades to $E_t I$, reducing buckling resistance.

• Between buckling and stiffness, which criterion governs column design?

  • Buckling is the governing design limit state. While structural stiffness dictates deflections and frame drift, buckling represents a catastrophic instability limit state that triggers complete structural collapse once initiated.

• What is the justification for four distinct buckling curves (a, b, c, d) in modern codes?

  • Different structural profiles (hot-rolled I-sections, welded plates, hollow tubes, channels) exhibit distinct patterns of cross-sectional residual stresses, cooling rates, and geometric out-of-straightness. Rather than relying on a single conservative curve, codes categorize profiles into four tiers (curves a through d) using calibrated imperfection factors ($\alpha = 0.21, 0.34, 0.49, 0.76$) to safely maximize material efficiency.

• How does strain hardening influence the ultimate capacity of short columns?

  • While strain hardening can elevate the collapse load of stocky columns at large plastic strains, the initial yield threshold remains unaffected. Modern limit state standards disregard strain hardening as a safety margin, restricting the maximum compressive design strength to the yield limit ($f_y / \gamma_{m0}$).

• Why does the IS: 800 code provide a specialized design formula for single-angle struts?

  • Single angles are typically attached to gusset plates through one leg only, introducing an eccentric load path that induces concurrent biaxial bending and torsion. To address this complex coupled response without requiring an exhaustive second-order frame analysis, codes prescribe an equivalent slenderness ratio ($\lambda_e$) that combines minor-axis flexural buckling ($\lambda_{vv}$), leg local buckling ($\lambda_{\phi}$), and connection fixity ($k_1, k_2, k_3$).