Rectangular Plate under the Constant Stresses on the Midsurface

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 σ_x, σ_y and tangential τ_xy 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

Model 1. Design model

 

Model 1. Deformed model

 

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

 

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

 

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

 

Model 2. Design model

 

Model 2. Deformed model

 

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

 

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

 

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

 

Model 3. Design model

 

Model 3. Deformed model

 

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

 

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

 

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

 

Model 4. Design model

 

Model 4. Deformed model

 

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

 

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

 

Model 4. Values of tangential stresses τ_xy (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 σ_x, σ_y and tangential τxy stresses on the midsurface of the plate are determined according to the following formulas:

 

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