STEEL-CHAP-5

Chapter 5 - Introduction to Axially Loaded Compression Members

5.1 General

Compression members play an essential role in structural engineering and include components such as columns, top chords of trusses, and bracing members. These members are primarily subjected to axial loads, which can lead to bending and compression effects. Columns, typically defined as vertical members, have lengths that are significantly greater than their thicknesses, which allows them to effectively carry loads over tall structures. In contrast, short vertical members are often referred to as struts, serving functional roles in stability and structural integrity.

Modes of Failures in Columns:

• Flexural Buckling: This is the primary failure mode in slender columns, which occurs when a vertical member bends under instability, leading to a critical load defined by Euler's buckling criteria. As slenderness increases, the risk of this mode also increases.

• Local Buckling: This failure mode occurs when specific sections of a column’s cross-section buckle under stress due to material thinness before overall instability leads to flexural buckling. Factors like cross-section shape and material properties can influence this type of failure.

• Flexural Torsional Buckling: This type of failure manifests in columns possessing particular cross-section geometries, where the column experiences combined bending (flexural) and twisting (torsional) effects. This severe mode of buckling is particularly critical when the geometric properties of the column lead to instability under load.

Slenderness Ratio:

The slenderness ratio is a critical concept in the analysis of compression members and is defined as the ratio of the column's effective length to its least radius of gyration. Higher slenderness ratios signify a greater tendency for a column to buckle under applied loads, necessitating careful design considerations to ensure structural safety.

Factors Affecting Buckling:

• Types of End Connections: Fixed connections provide greater resistance to buckling compared to pinned connections, which allow rotation and can influence the overall stability of the structure.

• Eccentricity of Load Application: Loads acting away from the centroid lead to bending moments that can exacerbate buckling risks.

• Imperfections in Material: Initial imperfections, such as crookedness or the presence of residual stresses from manufacturing, can significantly impact the buckling strength of columns.

5.2 Residual Stresses

Importance of Residual Stresses:

Residual stresses are crucial in the analysis of axially loaded steel columns. These stresses, often arising from uneven cooling during hot rolling, result in local areas experiencing compressive and tensile stresses. For slenderness ratios between 40 and 120, these residual stresses may contribute to up to a 25% reduction in buckling strengths, particularly in slender columns. Engineers must account for these stresses during design to ensure adequate safety margins.

5.3 Sections Used for Columns

Types of Shapes:

• Single Angles: Commonly utilized for bracing and light compression members, offering a lightweight option for structure reinforcement.

• W Shapes: Widely used for building columns due to their strong performance characteristics and effective shapes, providing excellent strength-to-weight ratios.

• Hollow Structural Sections (HSS): Including square, round, and rectangular tubes, these sections have gained popularity due to their aesthetic qualities and ease of fabrication, enabling a range of design options.

• Built-Up Sections: Comprising multiple connected pieces, built-up sections are typically implemented in scenarios requiring high load-bearing capabilities or unique connection benefits.

5.4 Development of Column Formulas

Historical Development:

The study of column buckling began with foundational analyses from scholars like Pieter van Musschenbroek in 1729, culminating in significant advancements by Leonhard Euler in 1757. Euler's contributions laid the groundwork for understanding buckling, emphasizing the need for empirical formulas developed through laboratory testing. These formulas address practical applications in engineering, which often deviate from idealized conditions.

5.5 The Euler Formula

Euler's Formula:

Euler’s Formula serves as a pivotal equation in structural analysis, predicting elastic buckling for long slender columns based on geometric parameters and material properties:

• Formula: P = π²EI/L², where E is the modulus of elasticity, I is the moment of inertia, and L represents the effective column length. The application of Euler’s formula is contingent upon evaluating the effective length (KL), which takes into account the support conditions of the column system.

5.6 End Restraint and Effective Lengths of Columns

Effective Length (KL):

The effective length of compressive members is defined as the distance between inflection points in the member's deformation. This length varies based on the end conditions and support restraints of the column.

K Factors:

The K factors are essential in adjusting the effective length to reflect the actual support conditions:

• K = 1.0: For pinned ends, providing a relatively free rotation at the ends.

• K < 1.0: For fixed ends, which restrain rotation and enhance resistance to buckling.

• K > 1.0: For unbraced frames, indicating additional effective length due to lack of lateral support.

5.7 Stiffened and Unstiffened Elements

Element Classification:

• Stiffened Elements: These components are supported at two edges and are inherently less prone to buckling due to their enhanced resistance provided by the stiffening.

• Unstiffened Elements: Here, one edge remains free, which makes them more susceptible to local buckling at lower stress levels. The design implications differ considerably between stiffened and unstiffened elements, warranting different approaches to buckling behavior analysis.

5.8 Long, Short, and Intermediate Columns

Definitions:

• Long Columns: Prone to elastic buckling governed by their geometric properties rather than material yield capacity.

• Short Columns: These fail primarily at yield stress due to the high load levels, demonstrating no significant buckling.

• Intermediate Columns: These exhibit a unique combination of yielding and buckling characteristics, necessitating careful analysis to determine design strengths.

5.9 Column Formulas

AISC Specification:

The American Institute of Steel Construction (AISC) provides comprehensive specifications and equations to estimate critical buckling strengths and calculate column capacities.

• Equations Include:

  • Euler's formulas for predicting behavior in elastic (long) columns.

  • Empirical equations tailored for short and intermediate columns, reflecting real-world conditions essential for practical design methodologies.

5.10 Maximum Slenderness Ratios

Revisions in the AISC specifications highlight the importance of understanding slenderness ratios to ensure safe design practices. These ratios are crucial for evaluating potential risk levels associated with buckling and ensuring structural safety in engineering designs.

5.11 Example Problems

Illustrative numerical examples are provided to demonstrate strength calculation methods for various column shapes and conditions. These problems reference methodologies outlined in the AISC Manual, guiding practitioners towards design efficiencies and best practices in structural engineering.