Reference:

B.Halphen et J. Salencon, Elastoplasticité, Presses de l’ENPC.

Problem description:

High-rigidity beam is suspended using three bars and loaded by force Q at the BC centre of span. Find longitudinal internal force value N in bars АА`, ВВ` и СС`. Determine also vertical displacement w at point А, В and С. 

Problem sketch:

Type of created problem:

Plane frame (X, Z, UY).

Geometric characteristics:

L = 1 m;

Hanger area: А = 1·10^-4 m².

Material properties:

Elastic modulus of hangers before plasticity appearance: E = 2.1·10¹¹ Pa;

Elastic modulus of hangers after plasticity appearance: E_Т = 0.25·10¹¹ Pa;

Elastic limit of hangers: σ_y = 400 MPa;

Axial stiffness of beam АС: (EA)_AC = 1·10¹¹ N·m²;

Bending stiffness of beam АС: (EI_y)_AC = 1·10¹¹ N·m².

Boundary conditions:

Points А`, В` and С`: X = Z = 0.

Loads:

Q = (13/7)·σ_y·A.

Model description:

System is modelled using 3 physically nonlinear general step finite elements of spatial bar (FE type is 210) for hangers and 2 general FE of spatial bar (FE type is 10) for beam. For nonlinear problem solution, stepped load applying is performed (100 steps).

 Calculation results:

Diagram of longitudinal forces N, kN

Vertical displacements distribution w, cm