TEST CASE 3.3 STABILITY OF CANTILEVER FLAT FORM OF BENDING
Reference:
I.A. Birger, Ya.G. Panovko, Strength, Stability, Vibrations, Handbook in three volumes, Volume III, Moscow, Mashinostroenie, 1968, p. 68
Problem description:
The cantilever rod is loaded separately with concentrated force P (first case), concentrated bending moment My (second case), evenly distributed load f (third case). Identify critical values Pcr, My,cr, fcr.
Problem sketch:
Type of created problem:
Spatial structure (X, Y, Z, UX, UY, UZ).
Geometric characteristics:
L = 1 m.
Material properties:
Bending stiffness in the X0Z plane: EIy = 300 tf·m2;
Bending stiffness in the X0Y plane: EIz = 1 tf·m2;
Torsional stiffness: GIx = 1 tf·m2.
Boundary conditions:
Point А: X = Y = Z = UX = UY = UZ = 0.
Loads:
Р = 1 tf; My = 1 tf·m; f = 1 tf/m.
Model description:
The system is modeled using 100 universal finite elements of a spatial bar (FE type is 10). To ensure geometric invariance, the minimal axial stiffness EA = 1 tf was introduced. The number of nodes in the calculation scheme is 101.
Analytical solution:
Calculation results:
Target value |
Analytical solution |
LIRA 10 |
Deviation, % |
Pcr, tf |
4.012 |
4.0129 |
0.02 |
My,cr, tf·m |
3.1416 |
3.1417 |
0.00 |
fcr, tf/m |
12.86 |
12.8555 |
0.04 |
Buckling mode at Pcr
Buckling mode at My,cr
Buckling mode at fcr