Beam-Column Second-Order Amplification Calculator

Equation

M_max = M_0 / (1 - P / P_cr)

where:
  M_max = amplified (second-order) moment (N*m)
  M_0   = first-order (primary) moment (N*m)
  P     = applied axial compressive load (N)
  P_cr  = Euler critical load for the column (N)

The amplification factor is: alpha = 1 / (1 - P / P_cr)

Variables

• M_0 — first-order moment from transverse load or eccentricity (N*m)
• P — applied axial compressive load (N)
• P_cr — Euler critical load = pi^2 * E * I / L_e^2 (N)

Units

• M_0 in N*m
• P and P_cr in N
• Amplification factor is dimensionless

Substitution example

For a column with P = 12 kN, P_cr = 17.2 kN (from Euler), M_0 = 500 N*m: alpha = 1 / (1 - 12000 / 17200) = 1 / (1 - 0.6977) = 1 / 0.3023 = 3.308 M_max = 3.308 * 500 = 1654 N*m The second-order moment is 3.3 times the first-order moment.

Result

The calculator outputs the amplification factor alpha and the amplified moment M_max. As P approaches P_cr, the amplification factor tends to infinity — the column becomes unstable.

Assumptions

• The column is initially straight
• Small deflection theory applies
• The axial load is constant (no load redistribution)
• The moment is uniform or sinusoidally distributed
• Linear-elastic material

Limitations

• The amplification factor diverges as P approaches P_cr — in reality, material nonlinearity and large deflections limit the moment
• For P/P_cr > 0.9, the formula is highly sensitive — use nonlinear analysis
• The formula is for a single load case — combined loading requires interaction equations
• Not applicable to columns with significant imperfections (use nonlinear analysis)
• The formula assumes the Euler critical load is the correct P_cr — for inelastic buckling, use the inelastic critical load

Related Knowledge

• Eccentric Compression & Beam-Columns
• Euler Column Buckling
• Geometric Nonlinearity & Large-Deflection Stability