TEST CASE 6.3 CIRCULAR PLATE ON A SINGLE-SIDED ELASTIC FOUNDATION
Reference:
П. Панагиотопулос, Неравенства в механике и их приложения, Москва: «Мир», 1989, стр. 384.
Problem description:
Circular linear-elastic plate was motionless and supported by single-sided elastic foundation. Then this plate was loaded by uniformly area-distributed load q and by edge-distributed bending moment M. Determine rotational angle of plate edge ψ and domain with contact of plate and foundation at coefficient а value 1; 2; 3; 4; 5; 5.1.
Problem sketch:
Type of created problem:
The spatial structure (X, Y, Z, UX, UY, UZ).
Geometric characteristics:
Plate radius: R = 6 m;
Plate thickness: t = 0.06 m.
Material properties:
Elastic modulus: E = 2.1·1011 Pa;
Poisson’s ratio: μ = 0.
Boundary conditions:
All plate nodes: X = Y = UX = UZ = 0;
Plate central assembly: UY = 0;
Plate boundary nodes: Z = 0.
Loads:
q = 600 N/m2;
M = a·q·R2/32.
Model description:
The system is modeled by 95 finite elements of thin shell (FE type is 44). Because of symmetry only a sector of circular plate is considered. For nonlinear solution iterative process is used.
Analytical solution:
Rotational angle of plate edge is an angle between the perpendiculars of deformed and nondeformed plate surface. It is defined via expression ψ = dw/dn - derivative of plate displacement along normal to the boundary (along radius).
Calculation results:
а |
Target value | Number of iterations | Analytical solution | Numerical solution | LIRA 10 | Deviation, % |
1 |
ψ·10-5, rad |
5700 |
7.50 |
7.29 |
7.119 |
5.35 |
2 |
7100 |
23.00 |
22.236 |
21.872 |
5.16 |
|
3 |
12000 |
43.90 |
43.80 |
43.407 |
1.14 |
|
4 |
14600 |
71.50 |
72.379 |
71.503 |
0.004 |
|
5 |
16400 |
107.14 |
107.176 |
106.52 |
0.58 |
|
5.1 |
26000 |
126.50 |
126.117 |
121.40 |
4.03 |
|
1 |
Contact domain, m |
5700 |
3.70 / 3.78 |
3.70 |
3.70 |
0.00 |
2 |
7100 |
2.58 / 2.64 |
2.58 |
2.58 |
0.00 |
|
3 |
12000 |
1.56 / 1.62 |
1.56 |
1.56 |
0.00 |
|
4 |
14600 |
0.54 / 0.60 |
0.54 |
0.60 |
0.00 |
|
5 |
16400 |
0.00 |
0.00 |
0.00 |
0.00 |
|
5.1 |
26000 |
No contact |
No contact |
No contact |
0.00 |