Euler's formula is a key tool for calculating critical loads in slender columns. It considers factors like elasticity, moment of inertia, and effective length to determine when a column will buckle under axial load.

However, Euler's formula has limitations. It assumes ideal conditions and only applies to long, slender columns. For short or intermediate columns, or those with complex shapes, other methods may be needed to accurately predict buckling behavior.

Euler's Formula for Slender Columns

Calculating Critical Load

• Apply Euler's formula to calculate the critical load (P_cr) that causes elastic buckling in slender columns
  • Euler's formula: $P_cr = (π^2 * E * I) / (L_e^2)$
  • Critical load depends on the modulus of elasticity (E), moment of inertia (I), and effective length (L_e) of the column
  • Example: A steel column with E = 200 GPa, I = 1.2 x 10^-4 m^4, and L_e = 3 m

Moment of Inertia and Effective Length

• The moment of inertia (I) describes the column's resistance to bending and depends on the cross-sectional shape and dimensions
  • Example: For a circular column with radius r, I = (π r^4) / 4
  • Example: For a rectangular column with width b and height h, I = (b h^3) / 12
• The effective length (L_e) is the equivalent length of a pinned-pinned column that would buckle under the same critical load as the actual column with its specific end conditions

Elastic Buckling and Critical Stress

• Elastic buckling occurs when the critical load is reached, causing the column to suddenly deform laterally without a significant increase in the axial load
• The critical stress (σ_cr) at which elastic buckling occurs can be calculated by dividing the critical load by the cross-sectional area (A) of the column
  • Critical stress formula: $σ_cr = P_cr / A$
  • Example: For a column with P_cr = 200 kN and A = 0.005 m^2, σ_cr = 40 MPa

Assumptions and Limitations of Euler's Formula

Ideal Column Assumptions

• Euler's formula assumes that the column is perfectly straight, homogeneous, and free from initial stresses or imperfections
• The column material is assumed to be linearly elastic, following Hooke's law, and the stresses remain below the proportional limit
• The formula assumes that the axial load is applied concentrically and that there are no eccentric loads or moments acting on the column

Slenderness Ratio Limitations

• The formula is only valid for long, slender columns with a high slenderness ratio (L/r), typically greater than 100
  • L is the unsupported length
  • r is the radius of gyration
• Euler's formula does not account for inelastic buckling, which occurs in short or intermediate columns where the material yields before reaching the elastic buckling load

Other Limitations

• Euler's formula does not consider the effects of shear deformation or local buckling, which may occur in thin-walled or composite columns
• The formula may not be accurate for columns with complex cross-sections or non-uniform material properties
• The presence of initial imperfections, residual stresses, or dynamic loads can reduce the actual critical load compared to the theoretical value given by Euler's formula

Applicability of Euler's Formula

Slenderness Ratio and Column Classification

• The slenderness ratio (L/r) determines the applicability of Euler's formula
  • L is the unsupported length
  • r is the radius of gyration, calculated as $r = √(I/A)$
• Columns are classified based on their slenderness ratio:
  • Slender columns (L/r > 100): Likely to fail by elastic buckling, Euler's formula is applicable
  • Short columns (L/r < 50): Likely to fail by inelastic buckling or crushing, Euler's formula is not applicable
  • Intermediate columns (50 < L/r < 100): May exhibit a combination of elastic and inelastic buckling behavior, alternative methods (e.g., Johnson-Euler or secant formula) should be used

Material Properties and Cross-Sectional Shape

• The modulus of elasticity (E) and yield strength (σ_y) influence the applicability of Euler's formula
  • The formula is more accurate for materials with a high E/σ_y ratio (e.g., steel or aluminum)
• The cross-sectional shape and dimensions affect the moment of inertia (I) and radius of gyration (r), which influence the critical load and applicability of Euler's formula
  • Thin-walled or open cross-sections are more susceptible to local buckling, which may limit the formula's accuracy
  • Example: Wide-flange (I-beam) columns are more prone to local buckling than solid rectangular columns

End Conditions and Critical Load

Effective Length Factor

• The end conditions of a column determine its effective length (L_e), which is used in Euler's formula to calculate the critical load
• The effective length factor (K) relates the effective length to the actual unsupported length (L) of the column: $L_e = K L$

Ideal End Conditions

• Pinned-pinned (free rotation, no translation): K = 1, $L_e = L$
• Fixed-fixed (no rotation, no translation): K = 0.5, $L_e = 0.5L$
  • Results in a higher critical load compared to pinned-pinned conditions
• Fixed-pinned (no rotation at one end, free rotation and no translation at the other): K = 0.7, $L_e = 0.7L$
• Fixed-free (no rotation at one end, free rotation and translation at the other): K = 2, $L_e = 2L$
  • Results in a lower critical load compared to other end conditions

Partial Fixity and End Restraints

• The presence of end restraints or partial fixity can be accounted for by using an effective length factor between the ideal cases
• Typical effective length factors for partial fixity range from 0.5 to 1
• Example: A column with partial rotational restraint at both ends may have an effective length factor of 0.8, resulting in $L_e = 0.8L$