Slenderness Ratio Calculator
Calculate column slenderness ratio to determine buckling tendency and design requirements
Calculate Slenderness Ratio
Total length of the column between supports
Support conditions at both ends of the column
Material affects classification limits
Cross-sectional geometry of the column
Cross Section Dimensions
Slenderness Ratio Results
Geometric Properties
Effective Length: 0.000 m
Radius of Gyration: 0.00 mm
Cross-sectional Area: 0.0 mm²
K Factor: 1
Design Information
Formula: λ = KL/r
Material: Steel
End Condition: Pinned-Pinned
Cross Section: Rectangular/Square
Example Calculation
Steel Column Example
Length: 8.0 m
Cross section: 200mm × 200mm
End condition: Pinned-Pinned (K = 1.0)
Material: Steel A36
Calculation Steps
1. Leff = K × L = 1.0 × 8.0 = 8.0 m
2. r = √(I/A) = √(bh³/12)/bh = h/√12
3. r = 200/√12 = 57.7 mm
4. λ = Leff/r = 8000/57.7 = 138.6
Result: Long Column (Euler buckling)
Classification Limits
Steel (A36)
Aluminum
Wood
End Conditions
Fixed-Fixed (both ends restrained)
Fixed-Pinned (one fixed, one pinned)
Pinned-Pinned (both ends hinged)
Fixed-Free (cantilever column)
Understanding Slenderness Ratio
What is Slenderness Ratio?
The slenderness ratio (λ) is a dimensionless parameter that indicates a column's tendency to buckle under compressive loads. It's the ratio of the effective length to the radius of gyration about the weak axis of the cross-section.
Why is it Important?
- •Determines failure mode (yielding vs. buckling)
- •Guides selection of design equations
- •Critical for structural safety and efficiency
- •Required by building codes and standards
Formula and Theory
λ = KL/r
- λ: Slenderness ratio (dimensionless)
- K: Effective length factor
- L: Actual length of column
- r: Radius of gyration = √(I/A)
Design Rule: Higher slenderness ratios indicate greater buckling tendency. Long, thin columns are more susceptible to lateral instability.
Design Equations
Euler's Formula (Long Columns)
P_cr = π²EI/(KL)²
Used when λ > λ_cr. Applies to elastic buckling where material stress remains below yield strength.
Johnson's Formula (Intermediate Columns)
σ_cr = σ_y[1 - (KL/r)²/(4π²E/σ_y)]
Used when λ < λ_cr. Accounts for inelastic behavior where yielding and buckling interact.