Euler column buckling across classic boundary conditions suite
BucklingPASS · 0.1%Comprehensive verification suite for linear elastic bifurcation buckling of slender prismatic columns under all four classical end constraint conditions: hinged–hinged (pinned–pinned, K = 1.0), fixed–free (cantilever, K = 2.0), fixed–pinned (propped cantilever, K = 0.69916), and fixed–fixed (clamped–clamped, K = 0.5).Comprehensive verification suite for linear elastic bifurcation buckling of slender prismatic columns under all four classical end constraint conditions: hinged–hinged (pinned–pinned, K = 1.0), fixed–free (cantilever, K = 2.0), fixed–pinned (propped cantilever… show full text
- Tolerance:
- 5%
Problem statement
- Geometry
- Prismatic column of length L = 1000 mm, rectangular cross-section 20 x 10 mm. Weak-axis moment of inertia Iy = 20*10^3/12 = 1666.6667 mm^4, strong-axis Iz = 10*20^3/12 = 6666.6667 mm^4, area A = 200 mm^2.
- Material
- Structural steel: Young's modulus E = 210000 MPa, Poisson's ratio nu = 0.3. Flexural rigidity EI_y = 3.5e8 N*mm^2.
- Boundary conditions
- Four support configurations under unit compressive axial load: (a) Pinned-Pinned (K=1.0); (b) Fixed-Free (K=2.0); (c) Fixed-Pinned (K=0.69916); (d) Fixed-Fixed (K=0.5).
- Measured quantity
- Critical bifurcation buckling load P_cr = pi^2*E*I / (K*L)^2 for each configuration: (a) Pinned-Pinned = 3454.36 N; (b) Fixed-Free = 863.59 N; (c) Fixed-Pinned = 7067.62 N; (d) Fixed-Fixed = 13817.45 N.
WebCAE Terminal
Paste these commands into the Terminal panel inside WebCAE. They use the same text DSL as the application; run one command per line:
units mm
frame up=z
material E=210000 nu=0.3
frame node at=0,0,0
frame node at=250,0,0
frame node at=500,0,0
frame node at=750,0,0
frame node at=1000,0,0
frame member 1 2 section=rect w=20 h=10
frame member 2 3 section=rect w=20 h=10
frame member 3 4 section=rect w=20 h=10
frame member 4 5 section=rect w=20 h=10
frame support node=1 dofs="ux,uy,uz,rx"
frame support node=5 dofs="uy,uz,rx"
frame load node=5 force=-1,0,0
frame solve buckling nmodes=4Results comparison
| Reference value | PP: 3454.36 Н, FF: 863.59 Н, FP: 7066.75 Н, CC: 13817.45 Н |
| Our result | PP: 3453.48 Н, FF: 863.53 Н, FP: 7062.68 Н, CC: 13803.30 Н |
| Deviation | 0.1% |
| Tolerance | 5% |
| Status | PASS · 0.1% |
| Element family | Beam element |
| Mesh details | 40 элементов, сечение 20×10 мм, L=1000 мм, E=210000 МПа, Iy=1666.67 мм⁴, 4 классических ГУ |
Updated: 2026-10-19 · WebCAE solver