Reference:

Lalin V.V., Beliaev M.O. Bending of geometrically nonlinear cantilever beam. Results obtained by Cosserat-Timoshenko 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 Cosserat-Timoshenko 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 = 1.962·10¹¹ Pa;

Poisson’s ratio: μ = 0.28.

Boundary conditions:

Point А: X = Z = UY = 0.

Loads:

F = 5·10⁶ 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:

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 Cosserat-Timoshenko theory, m

Vertical displacements distribution w according to Cosserat-Timoshenko theory, m