15.1 Buckling of columns and critical loads

Columns can fail through buckling, a sudden sideways collapse under compression. This chapter dives into different buckling modes and how to calculate critical loads - the maximum force a column can handle before buckling occurs.

Understanding buckling is crucial for designing safe structures. We'll explore factors that influence buckling, like material properties and column shape, and compare elastic versus inelastic buckling behavior. This knowledge helps engineers prevent catastrophic failures in buildings and bridges.

Column Buckling Modes

Types of Buckling Modes

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• Columns can buckle in different modes depending on their end conditions, slenderness ratio, and cross-sectional shape
• The three main modes of column buckling are euler buckling (elastic buckling), inelastic buckling, and torsional buckling
• Euler buckling occurs in long, slender columns and is characterized by a sudden lateral deflection when the critical load is reached (thin, tall columns)
• Inelastic buckling occurs in short, stocky columns and is characterized by material yielding before the critical load is reached (thick, short columns)
• Torsional buckling occurs in columns with open cross-sections and is characterized by a twisting deformation (I-beams, C-channels)

Factors Affecting Buckling Mode

• The effective length of a column depends on its end conditions (fixed, pinned, or free) and affects its buckling behavior
  • Fixed ends provide more resistance to buckling compared to pinned or free ends
  • Pinned ends allow rotation but prevent translation, while free ends allow both rotation and translation
• The slenderness ratio (length/radius of gyration) is a key parameter in determining the buckling mode of a column
  • High slenderness ratios favor elastic buckling, while low slenderness ratios favor inelastic buckling
  • The radius of gyration depends on the cross-sectional shape and moment of inertia of the column

Critical Load Calculation

Elastic Buckling

• The critical load is the maximum axial load a column can support before buckling occurs
• Euler's formula can be used to calculate the critical load for long, slender columns undergoing elastic buckling: $P_{cr} = \frac{\pi^2 EI}{L^2}$, where $E$ is the modulus of elasticity, $I$ is the moment of inertia, and $L$ is the effective length
• The critical stress for elastic buckling can be calculated using the Euler buckling stress formula: $\sigma_{cr} = \frac{\pi^2 E}{(L/r)^2}$, where $r$ is the radius of gyration

Inelastic Buckling

• For inelastic buckling, the tangent modulus theory can be used to calculate the critical load, which accounts for the reduced stiffness of the material beyond the proportional limit
  • The tangent modulus $E_t$ replaces the elastic modulus $E$ in the Euler formula: $P_{cr} = \frac{\pi^2 E_tI}{L^2}$
• The secant formula can be used as an alternative method to calculate the critical load for inelastic buckling, considering the reduced modulus of elasticity
  • The secant modulus $E_s$ replaces the elastic modulus $E$ in the Euler formula: $P_{cr} = \frac{\pi^2 E_sI}{L^2}$
• The Johnson-Euler formula is an empirical relationship that provides a transition between the Euler buckling load and the yield load for intermediate slenderness ratios: $P_{cr} = A\left[\sigma_y - \frac{(\sigma_y)^2}{4\pi^2E}(L/r)^2\right]$, where $A$ is the cross-sectional area and $\sigma_y$ is the yield stress

Factors Influencing Buckling

Material and Geometric Properties

• The modulus of elasticity ($E$) directly affects the critical load, with higher values of $E$ resulting in higher buckling resistance
• The moment of inertia ($I$) of the cross-section influences the buckling strength, with larger values of $I$ providing greater resistance to buckling (wide-flange sections, hollow sections)
• The effective length ($L$) of the column depends on its end conditions and affects the critical load, with shorter effective lengths resulting in higher buckling loads
• Residual stresses induced during manufacturing processes can influence the buckling behavior, particularly in the inelastic range (hot-rolled sections, welded sections)

Initial Imperfections and Loading Conditions

• Initial imperfections, such as out-of-straightness or eccentricity of loading, can reduce the buckling strength of columns by inducing additional bending moments
  • Out-of-straightness refers to the deviation of the column from a perfectly straight shape
  • Eccentricity of loading occurs when the axial load is not applied at the centroid of the cross-section
• The presence of lateral loads or moments can also affect the buckling behavior of columns, reducing their axial load-carrying capacity
  • Lateral loads induce additional bending moments that can cause premature buckling
  • Moments applied at the ends of the column can reduce the effective buckling length or cause non-uniform stress distributions

Elastic vs Inelastic Buckling

Elastic Buckling Characteristics

• Elastic buckling occurs when the critical load is reached without any yielding of the material, typically in long, slender columns
• In elastic buckling, the column regains its original shape upon removal of the load, and the buckling process is reversible
• The critical stress for elastic buckling is calculated using the Euler buckling stress formula: $\sigma_{cr} = \frac{\pi^2 E}{(L/r)^2}$
• Elastic buckling is characterized by a sudden lateral deflection when the critical load is reached, without any prior warning signs

Inelastic Buckling Characteristics

• Inelastic buckling occurs when the material yields before the critical load is reached, typically in short, stocky columns
• In inelastic buckling, the column undergoes permanent deformation, and the buckling process is irreversible
• The transition from elastic to inelastic buckling depends on the slenderness ratio of the column, with a higher slenderness ratio favoring elastic buckling
• The critical stress for inelastic buckling is lower than the elastic buckling stress due to the reduced stiffness of the material beyond the proportional limit
• The tangent modulus theory and the secant formula are used to calculate the critical load for inelastic buckling, accounting for the nonlinear material behavior (

  tangent modulus formula

  : $P_{cr} = \frac{\pi^2 E_tI}{L^2}$,

  secant formula

  : $P_{cr} = \frac{\pi^2 E_sI}{L^2}$)