TEST CASE 5.2 HIGH-RIGIDITY BEAM ON FOUR HANGERS
Reference:
A.R. Rzhanitsyn, Structural Mechanics, Moscow, High School, 1982, p.179-181.
Problem description:
High-rigidity beam is suspended using four bars and loaded by force F at point B. Find the force value for situation, when plasticity appears in bars АА`, ВВ` and СС`. Find also vertical displacement w at point A when plasticity arises at each bar.
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:
Stress values at points on the graph: σ_{1} = σ_{2} = -40000 tf/m^{2}, σ_{3} = σ_{4} = 40000 tf/m^{2};
Deformation values at points on the graph: ε_{1} = -0.9, ε_{2} = -0.001905, ε_{3} = 0.001905, ε_{4} = 0.9.
Boundary conditions:
Points А`, В`, С` and D`: X = Z = UY = 0;
Point А: X = 0.
Loads:
F = 10 tf.
Model description:
System is modelled using 4 physically nonlinear general step finite elements of spatial bar (FE type is 210) for hangers and 3 general FE of spatial bar (FE type is 10) for beam. Hangers and beam are connected via hinges at points А`, В`, С` and D` respectively. For nonlinear problem solution, stepped load applying is performed.
Calculation results:
Target value |
Analytical solution |
LIRA 10 |
Deviation, % |
F_{т.АА`}, tf |
10.0 |
10.0 |
0.00 |
F_{т.ВВ`}, tf |
11.2 |
11.2 |
0.00 |
F_{т.СС`}, tf |
12.0 |
12.0 |
0.00 |
w_{A} at F_{т}_{.}_{АА}_{`}, mm |
-1.905 |
-1.905 |
0.00 |
w_{A} at F_{т}_{.}_{ВВ}_{`}, mm |
-2.667 |
-2.667 |
0.00 |
w_{A} at F_{т}_{.}_{СС}_{`}, mm |
-5.715 |
-5.715 |
0.00 |
Force value F at plasticity appearance in bar АА`, tf
Force value F at plasticity appearance in bar ВВ`, tf
Force value F at plasticity appearance in bar СС`, tf
Displacement value w_{A} at plasticity appearance in bar АА`, mm
Displacement value w_{A} at plasticity appearance in bar ВВ`, mm
Displacement value w_{A} at plasticity appearance in bar СС`, mm