Rectangular Plate under the Constant Stresses on the Midsurface

Patch_tests

Objective: Check of the obtained values of the constant stresses on the midsurface of a rectangular plate at an irregular coarse finite element mesh.

Initial data files:

File name

Description

Patch_test_Constant_stress_Shell_42.spr

Design model with the elements of type 42

Patch_test_Constant_stress_Shell_44.spr

Design model with the elements of type 44

Patch_test_Constant_stress_Shell_45.spr

Design model with the elements of type 45

Patch_test_Constant_stress_Shell_50.spr

Design model with the elements of type 50

 

Problem formulation: The rectangular isotropic plate of constant thickness is subjected to the displacements of the outer edges providing the conditions of constant stresses on the midsurface. Check that the conditions of constant normal σx, σy and tangential τxy stresses on the midsurface are provided.

References: R. H. Macneal, R. L. Harder, A proposed standard set of problems to test finite element accuracy, North-Holland, Finite elements in analysis and design, 1, 1985, p. 3-20.

J. Robinson, S. Blackham, An evaluation of lower order membranes as contained in MSC/NASTRAN, ASAS and PARFEC FEM system, Dorset, Robinson and associates, 1979.

 

Initial data:

E = 1.0·106 kPa - elastic modulus of the plate material;
ν = 0.25 - Poisson’s ratio;
t = 0.001 m - thickness of the plate;
a = 0.12 m - short side of the plate;
b = 0.24 m - long side of the plate;

 

Boundary conditions:

u = 10-3∙(x + y/2) - displacement of the outer edges along the long side of the plate;
v = 10-3∙(x/2 + y) - displacement of the outer edges along the short side of the plate;

 

Location of internal nodes of the finite element mesh:

Numbers of nodes

in the Figure 1

x

y

1

0.04

0.02

2

0.18

0.03

3

0.16

0.08

4

0.08

0.08

 

Finite element model: Design model – general type system. Four design models are considered:

Model 1 - 10 three-node shell elements of type 42. Boundary conditions are provided by imposing constraints on the nodes of the outer edges of the plate in the directions of the degrees of freedom X, Y, Z, UX, UY, UZ and their displacement in accordance with the specified values u and v. Number of nodes in the model – 8.

Model 2 - 5 four-node shell elements of type 44. Boundary conditions are provided by imposing constraints on the nodes of the outer edges of the plate in the directions of the degrees of freedom X, Y, Z, UX, UY, UZ and their displacement in accordance with the specified values u and v. Number of nodes in the model – 8.

Model 3 - 10 six-node shell elements of type 45. Boundary conditions are provided by imposing constraints on the nodes of the outer edges of the plate in the directions of the degrees of freedom X, Y, Z, UX, UY, UZ and their displacement in accordance with the specified values u and v. Number of nodes in the model – 25.

Model 4 - 5 eight-node shell elements of type 50. Boundary conditions are provided by imposing constraints on the nodes of the outer edges of the plate in the directions of the degrees of freedom X, Y, Z, UX, UY, UZ and their displacement in accordance with the specified values u and v. Number of nodes in the model – 20.

 

Results in SCAD

ScreenShot350
Model 1. Design model

 

ScreenShot351
Model 1. Deformed model

 

ScreenShot352
Model 1. Values of normal stresses σx (kN/m2)

 

ScreenShot353
Model 1. Values of normal stresses σy (kN/m2)

 

ScreenShot354
Model 1. Values of tangential stresses τxy (kN/m2)

 

ScreenShot355
Model 2. Design model

 

ScreenShot356
Model 2. Deformed model

 

ScreenShot357
Model 2. Values of normal stresses σx (kN/m2)

 

ScreenShot358
Model 2. Values of normal stresses σy (kN/m2)

 

ScreenShot359
Model 2. Values of tangential stresses τxy (kN/m2)

 

ScreenShot360
Model 3. Design model

 

ScreenShot361
Model 3. Deformed model

 

ScreenShot362
Model 3. Values of normal stresses σx (kN/m2)

 

ScreenShot363
Model 3. Values of normal stresses σy (kN/m2)

 

ScreenShot364
Model 3. Values of tangential stresses τxy (kN/m2)

 

ScreenShot365
Model 4. Design model

 

ScreenShot366
Model 4. Deformed model

 

ScreenShot367
Model 4. Values of normal stresses σx (kN/m2)

 

ScreenShot368
Model 4. Values of normal stresses σy (kN/m2)

 

ScreenShot369
Model 4. Values of tangential stresses τxy (kN/m2)

 

Comparison of solutions:

Model

Parameter

Theory

SCAD

Deviation, %

1

Normal stresses

σx, kN/m2

1333

1333

0.00

Normal stresses

σy, kN/m2

1333

1333

0.00

Tangential stresses

τxy, kN/m2

400

400

0.00

2

Normal stresses

σx, kN/m2

1333

1333

0.00

Normal stresses

σy, kN/m2

1333

1333

0.00

Tangential stresses

τxy, kN/m2

400

400

0.00

3

Normal stresses

σx, kN/m2

1333

1333

0.00

Normal stresses

σy, kN/m2

1333

1333

0.00

Tangential stresses

τxy, kN/m2

400

400

0.00

4

Normal stresses

σx, kN/m2

1333

1333

0.00

Normal stresses

σy, kN/m2

1333

1333

0.00

Tangential stresses

τxy, kN/m2

400

400

0.00

 

Notes: In the analytical solution the normal σx, σy and tangential τxy stresses on the midsurface of the plate are determined according to the following formulas:

 

\[ \sigma_{x} =10^{-3}\cdot \frac{E}{1-\nu }; \quad \sigma_{y} =10^{-3}\cdot \frac{E}{1-\nu }; \quad \tau_{xy} =10^{-3}\cdot \frac{E}{2\cdot \left( {1+\nu } \right)}. \]