Finite Element Analysis of a Round Plate
Subjected to Circumferential Line Loads
John Connor
Introduction to Finite Elements
Professor Ernesto Gutierrez-Miravete
Spring 2015
Abstract
This paper identifies the process which was taken to evaluate several element and mesh densities
using the finite element method for a solid plate in bending. The results of the analyses are
compared back to the results obtained using Roark’s Formulas of Stress and Strain. The
conclusions state which elements are ideal but also provide justification that the choice in
elements is important when modeling using the finite element method.
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Table of Contents
Abstract .............................................................................................................................. 2
List of Figures .................................................................................................................... 3
List of Tables ..................................................................................................................... 3
1. Introduction .................................................................................................................. 4
2. Plate Theory ................................................................................................................. 4
3. Evaluation using Roark’s Equations ............................................................................ 6
4. Finite Element Model Development ............................................................................ 7
5. Results ........................................................................................................................ 10
6. Conclusions ................................................................................................................ 14
7. References .................................................................................................................. 14
List of Figures
Figure 1 – Solid Circular Plate Banding, Reference (a) page 487 .................................................. 5
Figure 2 – Plate Modeled in Abaqus CAE...................................................................................... 7
Figure 3 – Moderate Mesh Plate ..................................................................................................... 8
Figure 4 – Fine Mesh Plate ............................................................................................................. 8
Figure 5 – Coarse Mesh Plate ......................................................................................................... 9
Figure 6 – Loads and Constraints on the Plate ............................................................................. 10
Figure 7- Boundary Conditions Applied to the Plate.................................................................... 10
List of Tables
Table 1 – Plate Geometry ............................................................................................................... 4
Table 2 – Stress and Displacement for Quadratic Elements ......................................................... 11
Table 3 - Stress and Displacement for Incompatible Mode Elements .......................................... 12
Table 4 - Stress and Displacement for Reduced Integration Elements ......................................... 13
Table 5 - Results Comparision ...................................................................................................... 14
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1. Introduction
This paper will evaluate the finite element method for circular plate bending using several solid
element types. Formula’s derived in Roark’s Formulas of Stress and Strain was used to verify
the adequacy of the finite element model using plate stress and displacement as points of
comparison. The finite element program Abaqus CAE 6.14-2 was used to preprocess and post
process the finite element model for the analysis.
The circular plate that will be examined will use the dimensions identified in Table 1. With
assumed material properties based on the average Modulus of Elasticity and Poisson's Ratio of
carbon steel, 200 GPa and 0.3 respectively. An annular line load totaling 100kN was applied to
the plate to produce the stresses and displacements for the analysis.
Plate Outer Diameter
(m)
2
Table 1 – Plate Geometry
Plate Thickness
Line Load Diameter
(cm)
(m)
10
0.1
2. Plate Theory
The circular plate bending formulas in Roark’s use the Kirchoff-Love theory of plate bending.
Which state the general equation (Equation 1) for isotropic, homogenous plates is:
𝑞
∇2 ∇2 𝑤 = − 𝐷
(1
𝐸∗𝑡 3
𝐷 = 12∗(1−𝑣2 )
(2
Where: w is the deflection
q is the lateral pressure
D is the plate constant
E is the Young’s modulus of the plate material
v is the Poisson’s ratio of the plate material
Then for a circular plate with thickness t and radius a, the general equation can be rewritten using
polar coordinates as shown in Equation 3.
𝜕2
1 𝜕
1 𝜕2
𝜕2 𝑤
1 𝜕𝑤
∇2 ∇2 𝑤 = (𝜕𝑟 2 + 𝑟 𝜕𝑟 + 𝑟 2 𝜕𝜃2 ) ( 𝜕𝑟 2 + 𝑟
𝜕𝑟
1 𝜕2 𝑤
𝑞
+ 𝑟 2 𝜕𝜃2 ) = − 𝐷
(3
Since the case to be evaluated is axisymmetric, the dependency on θ no longer is required and
Equation 3 can be simplified to:
𝜕2
1 𝜕
𝜕2 𝑤
1 𝜕𝑤
𝑞
∇2 ∇2 𝑤 = (𝜕𝑟 2 + 𝑟 𝜕𝑟) ( 𝜕𝑟 2 + 𝑟 𝜕𝑟 ) = − 𝐷
(4
4
Since the load, q, is constant the solution to Equation 4 can be determined to be:
𝑞∗𝑟 4
𝑤 = 64∗𝐷 + 𝐴1 + 𝐴2 ∗ ln(𝑟) + 𝐵1 + 𝐵2 ∗ ln⁡(𝑟)
(5
A1, A2, B1, and B2 are constants of integration and are determined by the boundary conditions.
Roark’s derives the displacement and stress equations based Equation 5 and the selected
boundary conditions.
For the plate chosen for this analysis, Roark’s flat plate case 9b is used to determine the bending
displacement and the bending stress. Since the FEM not only computes bending displacement
but shear displacement, the shear displacement formula derived in Roark’s was needed to
accurately compare the displacement between the models and the FEM. The calculation of the
exact solution is provided in the next section.
Figure 1 shows the general free body diagram for a solid plate detailed in Reference (a).
Figure 1 – Solid Circular Plate Banding, Reference (a) page 487
To determine the full displacement in the plate the bending and shear displacements where
needed and are provided in Reference (a). The evaluation of the plate using Roark’s equations
are shown in the next section.
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3. Evaluation using Roark’s Equations
Mathcad was used to compute and annotate the plate bending equations described in Reference
(a), and shown below.
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4. Finite Element Model Development
Using the part module in Abaqus CAE the plate was created with the diameter and thickness
dimensions in Table 1. The part is then partitioned such that the annular line load can be applied
to the plate (circular partition of 0.1m in diameter at the center of the plate). The plate is then
segmented into quarter to allow for a better mesh quality. Figure 2 shows the partitioned plate
created in the part module.
Figure 2 – Plate Modeled in Abaqus CAE
A solid material section is then created with the elastic material properties of 200GPa for the
Modulus of Elasticity and 0.3 for the Poisson's ratio. For this analysis no additional material
properties were required. The material section was then applied to the entire plate.
Three mesh densities were chosen for the analyses which are shown in Figure 3 through Figure
5. The mesh densities will help determine the accuracy of the elements chosen for the analysis.
The first mesh density to be described is the moderate density (Figure 3), which visually seems
to provide an adequate number of elements through the thickness of the plate. The moderate
mesh density has a global mesh density of 0.1 and 2 elements through the thickness of the plate.
The second mesh density to be described is the fine density (Figure 4), which contains
substantially more elements and nodes than the moderate and as such the computational time will
be increased. The fine mesh has a global mesh density of 0.05 with 4 elements through the
thickness of the plate. The coarse mesh density (Figure 5) is the final mesh density evaluated,
this mesh contains the minimum amount of elements to produce a mesh without errors, the
global mesh density is 0.2 and there is 1 element through the thickness of the part.
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Figure 3 – Moderate Mesh Plate
Figure 4 – Fine Mesh Plate
8
Figure 5 – Coarse Mesh Plate
Three soild (cubic) element types were used in the analyses are C3D20R, C3D8I, and C3D8R,
The C3D20R is a 20 noded quadractic brick element which performs well in bending and rarely
exhibits hourglassing despite the reduced integration. The C3D8I is a 8 noded linear brick which
remove shear locking and reduce volumetric locking; this is obtained using bubble functions.
C3D8I is reccomended to be used in all instances where linear elements are subjected to bending.
The last element C3D8R is a 8 noded linear brick element with one integration point (reduced
intergration), these elements are prone to hourglassing and tend not to be stiff in bending. As a
side note hourglassing is a phenomenon which generally occurs when insuffiecient integration
points are available for numerical integration causing false modes resulting in inaccurate
displacement fields but correct stress fields.
To apply the annular line load to the 3d soild plate a concentrated load was used. To apply the
concentrated load at the specified radius a reference piont was place at the center of the plate.
The reference point is then kinematically coupled to the patition where the load load would be
applied. The kinematic coupling only transfers the load normal to the plate (z direction) due to
how the coupling is constrained. Figure 6 shows the coupled piont on the plate and the constraint
window with the definition shown.
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Figure 6 – Loads and Constraints on the Plate
A displacement/rotation boundary condition was applied to the outer diameter of the plate,
shown in Figure 7. The boundary condition locks the displacements at the edge of the plate.
Since the boundary condition is applied to a surface the rotation does not need to be bound
because any rotation of the element would cause a displacement of a node.
Figure 7- Boundary Conditions Applied to the Plate
Nine input files were created to run the analyses each one corresponding to a particular element
type and mesh density. The results of these analyses are detailed in the next section.
5. Results
To evaluate the results the absolute maximum principle stresses and the z direction
displacements were output for each of the analyses. The principle stress was evaluated since the
bending stress equations determined from the Roark’s equations does not account for shear
stresses. The same approach was taken with the displacement. Each element type is shown in
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their respective table with the stress and displacement plots below. All of the results are then
compared and evaluated against the exact solution in Table 5.
The quadratic element results are shown in Table 2, the plots in the table show that the stress and
displacement tend to be very consistent throughout the mesh densities, which is expected
considering the formulation of the element.
Mesh Size
Table 2 – Stress and Displacement for Quadratic Elements
C3D20R
Maximum Principle Stress
Displacement
Coarse
Moderate
Fine
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Table 3 shows the results of the incompatible elements. The results are very favorable for a
linear element while the coarse mesh is not accurate as the mesh is refined the results converge
very quickly to the point where the mesh density does not significantly affect the results.
Mesh Size
Table 3 - Stress and Displacement for Incompatible Mode Elements
C3D8I
Maximum Principle Stress
Displacement
Coarse
Moderate
Fine
12
The results in Table 4 show that the reduced integration linear elements should not be used to
evaluate a plate in bending. The coarse mesh caused the elements to hourglass giving
displacements orders of magnitude higher than the actual result. The more the mesh is refined the
better the displacements get however due to the nature of the elements the stresses get worse
because the element is artificially stiffer.
Mesh Size
Table 4 - Stress and Displacement for Reduced Integration Elements
C3D8R
Maximum Principle Stress
Displacement
Coarse
Moderate
Fine
13
As shown in Table 5 the displacements improve with every increase in mesh density, however
the stresses vary depending on the mesh. The stresses may vary due to how the mesh is
constructed or if the finite element model is accounting for stresses which are not solely due to
the bending stress.
Element Type
Quadratic
C3D8I
C3D8R
Mesh
Density
Coarse
Moderate
Fine
Coarse
Moderate
Fine
Coarse
Moderate
Fine
Table 5 - Results Comparision
Stress
Displacement
Percent
Calculated
Calculated
Difference
(MPa)
(µm)
11.42
1.8
-104.33
11.74
4.4
-107.09
11.52
2.6
-170.44
12.86
12.8
-100.90
11.63
3.5
-105.12
11.63
3.5
-106.73
13.94
19.5
-10370
8.21
36.7
-140.65
9.89
13.4
-110.20
Percent
Difference
3.8
1.1
0.8
7.3
3.0
1.4
99
23
1.8
6. Conclusions
The results of the analyses shown that the C3D20R elements are definitely the most accurate for
all mesh densities, however when running a more complicated model these elements may not be
ideal since they consume too much computational power. The C3D8I elements produced an
accurate approximation for most mesh densities without requiring the computational power
required with the quadratic elements. The C3D8R elements proved to be completely unreliable
when evaluating the plate in bending due to the element stiffness and hourglassing. These results
show that element choice is crucial when creating a finite element model to ensure not only the
correct answer is obtained but that the computational power is not wasted.
7. References
a) Budynas, Richard; Warren, Young; Roark’s Formulas for Stress and Strain, Seventh
Edition; McGraw-Hill; 2002
b) Boresi, Arthur; Shmidt, Richard; Advanced Mechanics of Materials, Sixth Edition; John
Wiley & Sons, Inc.; 2003
c) Dhondt, Guido; CalculiX CrunchiX USER'S MANUAL version 2.7 ; MIT;
<http://web.mit.edu/calculix_v2.7/CalculiX/ccx_2.7/doc/ccx/node25.html>
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