TEST CASE 4.6 BENDING OF INITIALLY STRAIGHT CANTILEVER USING “DEAD” FORCE
Reference:
Lalin V.V., Beliaev M.O. Bending of geometrically nonlinear cantilever beam. Results obtained by CosseratTimoshenko and Kirchhoff’s rod theories // Magazine of Civil Engineering №1, 2015
Problem description:
A circular cross section cantilever bar with diameter d is loaded by concentrated force at point B. Find the displacement values at point B, using separately Kirchhoff and CosseratTimoshenko theories.
Problem sketch:
Type of created problem:
Plane frame (X, Z, UY).
Geometric characteristics:
L = 1 m; d = 2 cm.
Material properties:
Elastic modulus: E = 1.962·10^{11} Pa;
Poisson’s ratio: μ = 0.28.
Boundary conditions:
Point А: X = Z = UY = 0.
Loads:
F = 5·10^{6} N.
Model description:
The system is modeled via 100 geometrically nonlinear elements of tight bend bar (FE type is 309). Automatic selection of step of loading applying for nonlinear problem solution is used.
Calculation results:
Target value 
Analytical solution 
LIRA 10 
Deviation, % 

Kirchhoff 
u_{В}, m 
0.97378 
0.97537 
0.16 
w_{В}, m 
0.9892 
0.99166 
0.25 

Cosserat 
u_{В}, m 
0.976 
0.97414 
0.20 
w_{В}, m 
1.072 
1.0707 
0.12 
Horizontal displacements distribution u according to Kirchhoff’s theory, m
Vertical displacements distribution w according to Kirchhoff’s theory, m
Horizontal displacements distribution u according to CosseratTimoshenko theory, m
Vertical displacements distribution w according to CosseratTimoshenko theory, m