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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