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:
taking into account transverse shear deformations:
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 |
Preview image: Array
Verification examples
- Section 1 Linear static problems for rod systems, plates and shells, three-dimensional problems
- Section 2 Physically nonlinear problems
- Section 3 Geometrically nonlinear problems for threads, cable-stayed trusses, rods, membranes and plates
- Section 4 Задачи устойчивости, в основном изгибно-крутильные формы потери устойчивости
- Section 5 Modal analysis
- Section 6 Linear dynamic problems
- Section 7 Static and dynamic problems with one-sided constraints
- Section 8 Geometric characteristics of the section
- Section 9 Procritical calculations
- Section 10 Problems of stationary and non-stationary heat conduction
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