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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