Flat Rectangular Plate Clamped along the Outer Edges and Subjected to a Transverse Load Uniformly Distributed over the Entire Area and a Concentrated Shear Force Applied in the Center.
Objective: Check of the obtained values of the transverse displacements in the center of a flat rectangular plate clamped along the outer edges and subjected to a transverse load uniformly distributed over the entire area and a concentrated shear force applied in the center.
Initial data files:
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File name |
Description |
|---|---|
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Design model with the elements of type 42 for meshes 2x2, 4x4, 8x8 |
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Design model with the elements of type 44 for meshes 2x2, 4x4, 8x8 |
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Design model with the elements of type 45 for meshes 2x2, 4x4, 8x8 |
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Design model with the elements of type 50 for meshes 2x2, 4x4, 8x8 |
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Bending_of_square_flat_plate_Clamped_supported_Solid_36_Mesh_2x2.spr |
Design model with the elements of type 36 for meshes 2x2, 4x4, 8x8, 16x16, 32x32, 64x64, 128x128 |
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Bending_of_square_flat_plate_Clamped_supported_Solid_37_Mesh_2x2.spr |
Design model with the elements of type 37 for meshes 2x2, 4x4, 8x8, 16x16, 32x32, 64x64, 128x128 |
Problem formulation: The flat rectangular plate clamped along the outer edges is subjected to the transverse load q uniformly distributed over the entire area and the concentrated shear force P applied in the center. Check the obtained values of the transverse displacements in the center of the flat rectangular plate clamped along the outer edges wq and wP from the respective actions.
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.
S. Timoshenko, S. Woinowsky-Krieger, Theory of plates and shells, New York, McGraw-Hill,1959, p. 120, 143, 202, 206.
Initial data:
| E = 1.7472·107 kPa | - elastic modulus of the plate material; |
| ν = 0.30 | - Poisson’s ratio; |
| a = 2.00 m | - width of the plate; |
| b = 2.00 m | - length of the plate; |
| h = 10-4 (10-2) m | - thickness of the plate; |
| q = 1.0·10-4 kN/m2 | - value of the transverse load uniformly distributed over the entire area of the plate; |
| P = 4.0·10-4 kN | - value of the concentrated shear force in the center of the plate. |
Finite element model: Design model – general type system. Six design models of a quarter of the plate according to the symmetry conditions are considered:
Model 1 – 8, 32, 128 three-node shell elements of type 42 with a regular mesh 2x2, 4x4, 8x8. The thickness of the plate – 10-4 m. Boundary conditions are provided by imposing constraints on the nodes of the clamped edges of the plate in the directions of the degrees of freedom X, Y, Z, UX, UY, UZ and constraints according to the symmetry conditions. Number of nodes in the model – 9, 25, 81.
Model 2 – 4, 16, 64 four-node shell elements of type 44 with a regular mesh 2x2, 4x4, 8x8. The thickness of the plate – 10-4 m. Boundary conditions are provided by imposing constraints on the nodes of the clamped edges of the plate in the directions of the degrees of freedom X, Y, Z, UX, UY, UZ and constraints according to the symmetry conditions. Number of nodes in the model – 9, 25, 81.
Model 3 – 8, 32, 128 six-node shell elements of type 45 with a regular mesh 2x2, 4x4, 8x8. The thickness of the plate – 10-4 m. Boundary conditions are provided by imposing constraints on the nodes of the clamped edges of the plate in the directions of the degrees of freedom X, Y, Z, UX, UY, UZ and constraints according to the symmetry conditions. Number of nodes in the model – 25, 81, 289.
Model 4 – 4, 16, 64 eight-node shell elements of type 50 with a regular mesh 2x2, 4x4, 8x8. The thickness of the plate – 10-4 m. Boundary conditions are provided by imposing constraints on the nodes of the clamped edges of the plate in the directions of the degrees of freedom X, Y, Z, UX, UY, UZ and constraints according to the symmetry conditions. Number of nodes in the model – 25, 81, 289.
Model 5 – 4, 16, 64, 256, 1024, 4096, 16384 в eight-node isoparametric solid elements of type 36 with a regular mesh 2x2x1, 4x4x1, 8x8x1, 16x16x1, 32x32x1, 64x64x1, 128x128x1. The thickness of the plate – 10-2 m. Boundary conditions are provided by imposing constraints on the nodes of the clamped sides of the lower surface of the plate in the directions of the degrees of freedom X, Y, Z, on the nodes of the clamped sides of the upper surface of the plate parallel to the Y axis of the global coordinate system in the direction of the degree of freedom X, on the nodes of the clamped sides of the upper surface of the plate parallel to the X axis of the global coordinate system in the direction of the degree of freedom Y and constraints according to the symmetry conditions. Number of nodes in the model – 18, 50, 162, 578, 2178, 8450, 33282.
Model 6 – 4, 16, 64, 256, 1024, 4096, 16384 twenty-node isoparametric solid elements of type 37 with a regular mesh 2x2x1, 4x4x1, 8x8x1, 16x16x1, 32x32x1, 64x64x1, 128x128x1. The thickness of the plate – 10-2 m. Boundary conditions are provided by imposing constraints on the nodes of the clamped sides of the lower surface of the plate in the directions of the degrees of freedom X, Y, Z, on the nodes of the clamped sides of the upper surface of the plate parallel to the Y axis of the global coordinate system in the direction of the degree of freedom X, on the nodes of the clamped sides of the upper surface of the plate parallel to the X axis of the global coordinate system in the direction of the degree of freedom Y and constraints according to the symmetry conditions. Number of nodes in the model – 51, 155, 531, 1955, 7491, 29315, 115971.
Results in SCAD
Model 1. Design model
Model 1. Deformed model
Model 1. Values of the transverse displacements in the center of the rectangular plate
clamped along the outer edges wq and wP (m, m)
Model 2. Design model
Model 2. Deformed model
Model 2. Values of the transverse displacements in the center of the rectangular plate
clamped along the outer edges wq and wP (m, m)
Model 3. Design model
Model 3. Deformed model
Model 3. Values of the transverse displacements in the center of the rectangular plate
clamped along the outer edges wq and wP (m, m)
Model 4. Design model
Model 4. Deformed model
Model 4. Values of the transverse displacements in the center of the rectangular plate
clamped along the outer edges wq and wP (m, m)
Model 5. Design model
Model 5. Deformed model
Model 5. Values of the transverse displacements in the center of the rectangular plate
clamped along the outer edges wq and wP (m, m)
Model 6. Design model
Model 6. Deformed model
Model 6. Values of the transverse displacements in the center of the rectangular plate
clamped along the outer edges wq and wP (m, m)
Comparison of solutions:
Transverse displacements in the center of the flat rectangular plate clamped along the outer edges wq from the transverse load q uniformly distributed over the entire area
|
Model |
Finite element mesh |
Theory |
SCAD |
Deviation, % |
|---|---|---|---|---|
|
1 (Member type 42) |
2x2 |
2.605 |
2.299 |
11.75 |
|
4x4 |
2.670 |
2.50 |
||
|
8x8 |
2.640 |
1.34 |
||
|
2 (Member type 44) |
2x2 |
2.605 |
2.607 |
0.08 |
|
4x4 |
2.612 |
0.27 |
||
|
8x8 |
2.606 |
0.04 |
||
|
3 (Member type 45) |
2x2 |
2.605 |
2.615 |
0.38 |
|
4x4 |
2.605 |
0.00 |
||
|
8x8 |
2.604 |
0.04 |
||
|
4 (Member type 50) |
2x2 |
2.605 |
2.593 |
0.46 |
|
4x4 |
2.604 |
0.04 |
||
|
8x8 |
2.604 |
0.04 |
||
|
5 (Member type 36) |
2x2 |
2.605∙10-6 |
0.003∙10-6 |
99.88 |
|
4x4 |
0.013∙10-6 |
99.50 |
||
|
8x8 |
0.050∙10-6 |
98.08 |
||
|
16x16 |
0.186∙10-6 |
92.86 |
||
|
32x32 |
0.589∙10-6 |
77.39 |
||
|
64x64 |
1.281∙10-6 |
50.83 |
||
|
128x128 |
1.816∙10-6 |
30.29 |
||
|
6 (Member type 37) |
2x2 |
2.605∙10-6 |
0.419∙10-6 |
83.92 |
|
4x4 |
2.679∙10-6 |
2.84 |
||
|
8x8 |
2.560∙10-6 |
1.73 |
||
|
16x16 |
2.596∙10-6 |
0.35 |
||
|
32x32 |
2.604∙10-6 |
0.04 |
||
|
64x64 |
2.605∙10-6 |
0.00 |
||
|
128x128 |
2.605∙10-6 |
0.00 |
Transverse displacements in the center of the flat rectangular plate clamped along the outer edges wP from the concentrated shear force P applied in the center
|
Model |
Finite element mesh |
Theory |
SCAD |
Deviation, % |
|---|---|---|---|---|
|
1 (Member type 42) |
2x2 |
7.260 |
2.239 |
69.16 |
|
4x4 |
4.194 |
42.23 |
||
|
8x8 |
5.751 |
20.79 |
||
|
2 (Member type 44) |
2x2 |
7.260 |
4.430 |
38.98 |
|
4x4 |
5.829 |
19.71 |
||
|
8x8 |
6.386 |
12.04 |
||
|
3 (Member type 45) |
2x2 |
7.260 |
5.989 |
17.51 |
|
4x4 |
6.864 |
5.45 |
||
|
8x8 |
7.113 |
2.02 |
||
|
4 (Member type 50) |
2x2 |
7.260 |
6.122 |
15.67 |
|
4x4 |
6.797 |
6.38 |
||
|
8x8 |
7.126 |
1.85 |
||
|
5 (Member type 36) |
2x2 |
7.260∙10-6 |
0.005∙10-6 |
99.93 |
|
4x4 |
0.020∙10-6 |
99.72 |
||
|
8x8 |
0.076∙10-6 |
98.95 |
||
|
16x16 |
0.283∙10-6 |
96.10 |
||
|
32x32 |
0.940∙10-6 |
87.05 |
||
|
64x64 |
2.319∙10-6 |
68.06 |
||
|
128x128 |
3.928∙10-6 |
45.90 |
||
|
6 (Member type 37) |
2x2 |
7.260∙10-6 |
0.337∙10-6 |
95.36 |
|
4x4 |
2.788∙10-6 |
61.60 |
||
|
8x8 |
5.735∙10-6 |
21.01 |
||
|
16x16 |
6.876∙10-6 |
5.29 |
||
|
32x32 |
7.147∙10-6 |
1.56 |
||
|
64x64 |
7.224∙10-6 |
0.50 |
||
|
128x128 |
7.245∙10-6 |
0.21 |
Notes: In the analytical solution the values of the transverse displacements in the center of the flat rectangular plate clamped along the outer edges wq and wP from the respective actions are determined according to the following formulas:
\[ {\begin{array}{*{20}c} {w_{q} =\frac{4\cdot q\cdot a^{4}}{\pi^{5}\cdot D}\cdot \sum\limits_{m=1}^M {\left\{ {\frac{1}{m^{5}}\cdot \left[ {1-\frac{\frac{m\cdot \pi \cdot b}{2\cdot a}\cdot th\left( {\frac{m\cdot \pi \cdot b}{2\cdot a}} \right)+2}{2\cdot ch\left( {\frac{m\cdot \pi \cdot b}{2\cdot a}} \right)}} \right]\cdot \sin \left( {\frac{m\cdot \pi }{2}} \right)} \right\}} +} \\ {\frac{a^{2}}{2\cdot \pi^{2}\cdot D}\cdot \sum\limits_{m=1}^M {\left\{ {E_{m} \cdot \frac{1}{m^{2}}\cdot \frac{\frac{m\cdot \pi \cdot b}{2\cdot a}\cdot sh\left( {\frac{m\cdot \pi \cdot b}{2\cdot a}} \right)}{ch^{2}\left( {\frac{m\cdot \pi \cdot b}{2\cdot a}} \right)}\cdot \sin \left( {\frac{m\cdot \pi }{2}} \right)} \right\}+} } \\ {\frac{b^{2}}{2\cdot \pi^{2}\cdot D}\cdot \sum\limits_{m=1}^M {\left\{ {F_{m} \cdot \frac{1}{m^{2}}\cdot \frac{\frac{m\cdot \pi \cdot a}{2\cdot b}\cdot sh\left( {\frac{m\cdot \pi \cdot a}{2\cdot b}} \right)}{ch^{2}\left( {\frac{m\cdot \pi \cdot a}{2\cdot b}} \right)}\cdot \sin \left( {\frac{m\cdot \pi }{2}} \right)} \right\}} } \\ \end{array} } \]
The values of the coefficients Em and Fm are determined by solving the system of 2∙M equations:
\[ \left. {{\begin{array}{*{20}c} {\frac{4\cdot q\cdot a^{2}}{\pi^{3}}\cdot \frac{1}{i^{4}}\cdot \left( {\frac{\frac{i\cdot \pi \cdot b}{2\cdot a}}{ch^{2}\left( {\frac{i\cdot \pi \cdot b}{2\cdot a}} \right)}-th\left( {\frac{i\cdot \pi \cdot b}{2\cdot a}} \right)} \right)-\frac{E_{i} }{i}\cdot \left( {\frac{\frac{i\cdot \pi \cdot b}{2\cdot a}}{ch^{2}\left( {\frac{i\cdot \pi \cdot b}{2\cdot a}} \right)}+th\left( {\frac{i\cdot \pi \cdot b}{2\cdot a}} \right)} \right)-\frac{8\cdot i\cdot a}{\pi \cdot b}\cdot \sum\limits_{m=1}^M {\left[ {F_{m} \cdot \frac{1}{m^{3}}\cdot \frac{1}{\left( {\frac{a^{2}}{b^{2}}+\frac{i^{2}}{m^{2}}} \right)^{2}}\cdot \sin^{2}\left( {\frac{m\cdot \pi }{2}} \right)} \right]} } \\ {\frac{4\cdot q\cdot b^{2}}{\pi^{3}}\cdot \frac{1}{i^{4}}\cdot \left( {\frac{\frac{i\cdot \pi \cdot a}{2\cdot b}}{ch^{2}\left( {\frac{i\cdot \pi \cdot a}{2\cdot b}} \right)}-th\left( {\frac{i\cdot \pi \cdot a}{2\cdot b}} \right)} \right)-\frac{F_{i} }{i}\cdot \left( {\frac{\frac{i\cdot \pi \cdot a}{2\cdot b}}{ch^{2}\left( {\frac{i\cdot \pi \cdot a}{2\cdot b}} \right)}+th\left( {\frac{i\cdot \pi \cdot a}{2\cdot b}} \right)} \right)-\frac{8\cdot i\cdot b}{\pi \cdot a}\cdot \sum\limits_{m=1}^M {\left[ {E_{m} \cdot \frac{1}{m^{3}}\cdot \frac{1}{\left( {\frac{b^{2}}{a^{2}}+\frac{i^{2}}{m^{2}}} \right)^{2}}\cdot \sin^{2}\left( {\frac{m\cdot \pi }{2}} \right)} \right]} } \\ \end{array} }} \right\} ; \] \[ {\begin{array}{*{20}c} {w_{P} =\frac{P\cdot a^{2}}{2\cdot \pi^{3}\cdot D}\cdot \sum\limits_{m=1}^M {\left\{ {\frac{1}{m^{3}}\cdot \left[ {th\left( {\frac{m\cdot \pi \cdot b}{2\cdot a}} \right)-\frac{\frac{m\cdot \pi \cdot b}{2\cdot a}}{ch^{2}\left( {\frac{m\cdot \pi \cdot b}{2\cdot a}} \right)}} \right]\cdot \sin^{2}\left( {\frac{m\cdot \pi }{2}} \right)} \right\}} +} \\ {\frac{a^{2}}{2\cdot \pi^{2}\cdot D}\cdot \sum\limits_{m=1}^M {\left\{ {E_{m} \cdot \frac{1}{m^{2}}\cdot \frac{\frac{m\cdot \pi \cdot b}{2\cdot a}\cdot sh\left( {\frac{m\cdot \pi \cdot b}{2\cdot a}} \right)}{ch^{2}\left( {\frac{m\cdot \pi \cdot b}{2\cdot a}} \right)}\cdot \sin \left( {\frac{m\cdot \pi }{2}} \right)} \right\}+} } \\ {\frac{b^{2}}{2\cdot \pi^{2}\cdot D}\cdot \sum\limits_{m=1}^M {\left\{ {F_{m} \cdot \frac{1}{m^{2}}\cdot \frac{\frac{m\cdot \pi \cdot a}{2\cdot b}\cdot sh\left( {\frac{m\cdot \pi \cdot a}{2\cdot b}} \right)}{ch^{2}\left( {\frac{m\cdot \pi \cdot a}{2\cdot b}} \right)}\cdot \sin \left( {\frac{m\cdot \pi }{2}} \right)} \right\}} } \\ \end{array} } \]
The values of the coefficients Em and Fm are determined by solving the system of 2∙M equations:
\[ \left. {{\begin{array}{*{20}c} {-\frac{P}{\pi }\cdot \frac{1}{i^{2}}\cdot \frac{\frac{i\cdot \pi \cdot b}{2\cdot a}\cdot sh\left( {\frac{i\cdot \pi \cdot b}{2\cdot a}} \right)}{ch^{2}\left( {\frac{i\cdot \pi \cdot b}{2\cdot a}} \right)}\cdot \sin \left( {\frac{i\cdot \pi }{2}} \right)-\frac{E_{i} }{i}\cdot \left( {\frac{\frac{i\cdot \pi \cdot b}{2\cdot a}}{ch^{2}\left( {\frac{i\cdot \pi \cdot b}{2\cdot a}} \right)}+th\left( {\frac{i\cdot \pi \cdot b}{2\cdot a}} \right)} \right)-\frac{8\cdot i\cdot a}{\pi \cdot b}\cdot \sum\limits_{m=1}^M {\left[ {F_{m} \cdot \frac{1}{m^{3}}\cdot \frac{1}{\left( {\frac{a^{2}}{b^{2}}+\frac{i^{2}}{m^{2}}} \right)^{2}}\cdot \sin^{2}\left( {\frac{m\cdot \pi }{2}} \right)} \right]} } \\ {-\frac{P}{\pi }\cdot \frac{1}{i^{2}}\cdot \frac{\frac{i\cdot \pi \cdot a}{2\cdot b}\cdot sh\left( {\frac{i\cdot \pi \cdot a}{2\cdot b}} \right)}{ch^{2}\left( {\frac{i\cdot \pi \cdot a}{2\cdot b}} \right)}\cdot \sin \left( {\frac{i\cdot \pi }{2}} \right)-\frac{F_{i} }{i}\cdot \left( {\frac{\frac{i\cdot \pi \cdot a}{2\cdot b}}{ch^{2}\left( {\frac{i\cdot \pi \cdot a}{2\cdot b}} \right)}+th\left( {\frac{i\cdot \pi \cdot a}{2\cdot b}} \right)} \right)-\frac{8\cdot i\cdot b}{\pi \cdot a}\cdot \sum\limits_{m=1}^M {\left[ {E_{m} \cdot \frac{1}{m^{3}}\cdot \frac{1}{\left( {\frac{b^{2}}{a^{2}}+\frac{i^{2}}{m^{2}}} \right)^{2}}\cdot \sin^{2}\left( {\frac{m\cdot \pi }{2}} \right)} \right]} } \\ \end{array} }} \right\} ; \] \[ D=\frac{E\cdot h^{3}}{12\cdot \left( {1-\nu^{2}} \right)}. \]