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TEST CASE 5.1 HIGH-RIGIDITY BEAM ON HANGERS

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: EТ = 0.25·1011 Pa;

Elastic limit of hangers: σy = 400 MPa;

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

Bending stiffness of beam АС: (EIy)AC = 1·1011 N·m2.

 

Boundary conditions:

Points А`, В` and С`: X = Z = 0.

 

Loads:

Q = (13/7)·σy·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:

Target value

Analytical solution

LIRA 10

Deviation, %

NАА`, kN

4.16

4.5151

8.54

NВВ`, kN

28.762

28.113

2.26

NСС`, kN

41.333

41.658

0.79

wA, cm

-0.01995

-0.0215

7.77

wB, cm

-0.13696

-0.13387

2.26

wC, cm

-0.25397

-0.24623

3.05

 

Diagram of longitudinal forces N, kN

 

Vertical displacements distribution w, cm