Reference:

Shalashilin V.I. and others. On the use of finite element method when solving geometry nonlinear tasks. — CAD and graphics. — 2000. — №4.  (pp. 26-31).

Problem description:

The hinged-rod system consists of two linear elastic bars of different stiffness, which lie on the same line at an angle of 45° to the X axis. A vertical concentrated force P is applied at point C. Determine the horizontal displacement of point B u_B, the vertical displacement of point C w_C, and also longitudinal forces N in the bars.

Problem sketch:

Type of created problem:

Plane truss or beam-wall (X, Z).

Geometric characteristics:

а = 2 m; b = 10 m.

Material properties:

Bar stiffness АВ: EА₁ = 2000 tf;

Bar stiffness ВС: EА₂ = 1000 tf.

Boundary conditions:

Point А: X = Z = 0;

Point В: Z = 0;

Point С: X = 0.

Loads:

Distributed force per unit length, is applied to the bar AB: q₁ = 0.002 tf/m;

Distributed force per unit length, is applied to the bar ВС: q₂ = 0.001 tf/m;

P = 200 tf.

Model description:

The system is modeled by 2 bar geometrically nonlinear finite elements of the “string” type (FE type is 304). Automatic selection of loading applying step with the search of new equilibrium shapes for nonlinear problem solution is used, minimum number of iterations 300.

Calculation results:

Note:

The system loses stability twice:

• At load P = 74.88 tf (displacements: u_В = -0.51 m, w_С = -4.46 m);
• At load P = 177.24 tf (displacements: u_В = -2.96 m, w_С = -22.22 m).

Displacements of nodes, m

Longitudinal forces N, tf