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} m^{2}.
Material properties:
Elastic modulus of hangers before plasticity appearance: E = 2.1·10^{11} Pa;
Elastic modulus of hangers after plasticity appearance: E_{Т} = 0.25·10^{11} Pa;
Elastic limit of hangers: σ_{y} = 400 MPa;
Axial stiffness of beam АС: (EA)_{AC} = 1·10^{11} N·m^{2};
Bending stiffness of beam АС: (EI_{y})_{AC} = 1·10^{11} N·m^{2}.
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:
Target value |
Analytical solution |
LIRA 10 |
Deviation, % |
N_{АА`}, kN |
4.16 |
4.5151 |
8.54 |
N_{ВВ`}, kN |
28.762 |
28.113 |
2.26 |
N_{СС`}, kN |
41.333 |
41.658 |
0.79 |
w_{A}, cm |
-0.01995 |
-0.0215 |
7.77 |
w_{B}, cm |
-0.13696 |
-0.13387 |
2.26 |
w_{C}, cm |
-0.25397 |
-0.24623 |
3.05 |
Diagram of longitudinal forces N, kN
Vertical displacements distribution w, cm