Test 1.14 Thick square hinged-supported plate under the action of transverse evenly distributed load

Verification of the LIRA-FEM software / Test 1.14 Thick square hinged-supported plate under the action of transverse evenly distributed load / Л. G. Donnell. Beams, Plates, and Shells, Moscow, Nauka, 1982, pp. 313-316.




Analytical solution: L . G. Donnell. Beams, Plates, and Shells, Moscow, Nauka, 1982, pp. 313-316.

Task formulation: A thick square hinged-supported plate is subjected to a transverse uniformly distributed load p. Determine the deflection w at the center of the plate considering the transverse shear deformations.

Geometry: h = 2.0; 4.0; 8.0 m - plate thickness; a = 16.0 m - plate side dimension

Material characteristics: E = 3·107kPa, ν = 0.2.

Boundary conditions: providing boundary conditions is achieved by superimposing links along the directions of degrees of freedom X, Y, Z, UY for edges located along the X axis of the general coordinate system, and X, Y, Z, UX for edges located along the Y axis of the general coordinate system.

Stresses: p = 100.0 kN/m2 - value of transverse uniformly distributed load.

Calculation results:

In the analytical solution, the deflections w in the center of the plate are determined by the following formulas:

without taking into account the transverse shear deformations:

Формула 1.jpg

taking into account transverse shear deformations:

Формула 2.jpg

Note: three computational models are considered for the ratios of the slab side dimensions to the thickness a/h = 8.0; 4.0; 2.0. The computational patterns are built for meshes with dimensions: 2x2; 4x4; 8x8; 16x16; 32x32; 64x64.

For the construction of the scheme were used FE 41 - universal rectangular FE of the shell by Kirchhoff-Lyav theory, FE 47 - universal quadrangular FE of a thick shell by Reissner-Mindlin theory


Plates with the ratio a/h = 8

The desired value

FE dimensional grid

Type FE

Analytical solution

Calculation results (LYRA-SAD)

Uncertainty, %

Deflections wmax, м

2x2

41

0.0012780

0.0015497

21.26

47

0.0013690

0.0003718

72.84

4x4

41

0.0012780

0.0013539

5.94

47

0.0013690

0.0008394

38.69

8x8

41

0.0012780

0.0012973

1.51

47

0.0013690

0.0011824

13.63

16x16

41

0.0012780

0.0012828

0.38

47

0.0013690

0.0013167

3.82

32x32

41

0.0012780

0.0012792

0.09

47

0.0013690

0.0013552

1.01

64x64

41

0.0012780

0.0012783

0.02

47

0.0013690

0.0013651

0.28

 

 

Plates with the ratio a/h = 4

The desired value

FE dimensional grid

FE TYPE

Analytical solution

Calculation results (LYRA-SAD)

Uncertainty, %

Deflections wmax, м

2x2

41

0.0001600

0.0001937

21.06

47

0.0002050

0.0001300

36.59

4x4

41

0.0001600

0.0001692

5.75

47

0.0002050

0.0001810

11.71

8x8

41

0.0001600

0.0001622

1.38

47

0.0002050

0.0001984

3.22

16x16

41

0.0001600

0.0001604

0.25

47

0.0002050

0.0002033

0.83

32x32

41

0.0001600

0.0001599

0.06

47

0.0002050

0.0002046

0.20

64x64

41

0.0001600

0.0001598

0.13

47

0.0002050

0.0002049

0.05

 

Plates with ratio a/h = 2

The desired value

FE grid with dimensions

FE TYPE

Analytical solution

Calculation results (LYRA-SAD)

Uncertainty, %

Deflections wmax, м

2x2

41

0.0000200

0.0000242

21.00

47

0.0000430

0.0000420

2.33

4x4

41

0.0000200

0.0000212

6.00

47

0.0000430

0.0000425

1.16

8x8

41

0.0000200

0.0000203

1.50

47

0.0000430

0.0000426

0.93

16x16

41

0.0000200

0.0000200

0.00

47

0.0000430

0.0000426

0.93

32x32

41

0.0000200

0.0000200

0.00

47

0.0000430

0.0000426

0.93

64x64

41

0.0000200

0.0000200

0.00

47

0.0000430

0.0000426

0.93


Тест 1.14 Толстая квадратная шарнирно-опертая пластина под действием поперечной равномерно распределенной нагрузки 2.png
Тест 1.14 Толстая квадратная шарнирно-опертая пластина под действием поперечной равномерно распределенной нагрузки 3.png


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