Results
First frequency 0.84708 Hz Reference (βL)²√(EI/mL⁴)/2π = 0.84708 Hz · difference 5.4 × 10−6 %
Second frequency 5.3086 Hz Reference (βL)²√(EI/mL⁴)/2π = 5.3086 Hz · difference 2.1 × 10−4 %
Third frequency 14.864 Hz Reference (βL)²√(EI/mL⁴)/2π = 14.864 Hz · difference 1.6 × 10−3 %
Buckling load factor 14.958 Reference π²EI/4H²P = 14.958 · difference 5.3 × 10−6 %
Critical top load 29,917 kN Reference π²EI/4H² = 29,917 kN · difference 5.3 × 10−6 %
Certified (linear buckling)· relative width of the certified bracket around the factor ≤ 1 × 10−6
A circular steel tube (E = 210 GPa, 7850 kg/m³) fixed at the base, in 20 elements. The frequencies are shown beside the roots of the cantilever’s frequency equation with a tip mass, and the buckling factor of the top load beside Euler’s π²EI/4H². Self-weight is not part of the buckling load.
Verified by: Tools: column buckling, checked against Euler
A teaching example, not a design check. SI units: m, kN, MPa. Published by Ergodic Group.