ME/AE 408: Finite Element Analysis of Plates with Holes

ME/AE 408: Advanced Finite Element Analysis

1
Table of contents
• Introduction
• Procedure
 Assumption for the developed FE models in ABAQUS
 The governing differential equations
• Results and discussion
 Theoretical stress values
 Case1 - Circular hole
 Case 2 - Elliptical hole
 Case 3 - Rectangular hole
 Convergence sensitivity analysis
 Finite element models result – Full Plate and Quarter models

2
Introduction and Project summary:
This computer project requires numerical study of the linear analysis of a thin plate under distributed
tension. The plate dimension was given as 1.0×1.0×0.02 m. The applied distributed load was a uniform
stress of equal to 25×103
N/m2
on the two opposite sides of the plate in the axial direction.
Three different hole geometry were considered at the center of the plate (i.e., circular, elliptical and a
rectangular hole with filleted corners) as shown below. The plate material was an isotropic, elastic
material with a Young’s modulus of 200 GPa and Poisson’s ratio of ν=0.3.

3
The full plate models versus the quarter models were compared in terms of the maximum Von-Mises
stress and displacement. First, the full plate model was analyzed for the Von-Mises stress and
displacement filed. Secondly, same analysis for the quarter model was implemented. Then results for the
two cases were compared against each other.
Additionally, the results from the FE models were compared against the theoretical values obtained from
the stress concentration factors, to include the effect of hole at the plate center.

4
Procedure:
Assumption for the developed FE models in ABAQUS:
The deformable shell elements with the thickness equal to 0.02 m were used to simulate the plate
structure. The model was created in the ABAQUS/CAE. The material property was set as the values
given in the problem statement for an isotropic material with a general static load step.
For the full plate model the boundary condition included restraining the degree of freedom in the X-
direction which was implemented by applying a boundary condition on the vertical line of symmetry of
x=0. Similarly, the full plate model was also constrained for shifting laterally in the direction of the
applied tensile stress by applying the boundary of y=0 at four points across the horizontal line of
symmetry.
The uniform tensile stress of 25×103
N/m2
, over a thickness of 0.02 m, was applied as of 500 N/m on both
edges. For the quarter plate model, in order to account for the symmetry condition, the vertical axis of
symmetry of the plate was restricted in the x-direction. The plate displacement in the y-direction was
constrained by applying the boundary condition of y=0 on the horizontal axis of symmetry. The 3-node
triangular elements were used in all of the analyzed cases herein.

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The governing partial differential equations
This analysis constitutes a 2-D isotropic, plane stress problem, where σxz= σyz = σzz =0, which the
fundamental constitutive equation is given by the below equation:
2 2
2 2
0
1 1
0
1 1
2
0 0
2(1 )
xx xx
yy yy
xy xy
E E
E E
E
ν
υ υσ ε
ν
σ ε
υ υ
σ ε
υ
 
 − −    
    =    − −    
    
 − 
where the displacement-strain relations are related as below:
x
y
xy
u
x
v
y
u v
y x
ε
ε
γ
∂
=
∂
∂
=
∂
∂ ∂
= +
∂ ∂
and the equilibrium equations that need to be satisfied due to the applied external actions are as below:
0
0
xyx
x
x y
xy y
y
x y
f
f
σσ
σ σ
σ σ
σ σ
∂∂
+ + =
∂ ∂
+ + =
For this plane elasticity problem, substituting the stress-displacement and the constitutive relationship in
the equilibrium equation will derive the below set of coupled differential equations as below:
2 2
2 2
1 1 2(1 )
2(1 ) 1 1
x
y
E u E v E u v
f
x x y y y x
E u v E u E v
f
x y x y x y
υ
υ υ υ
υ
υ υ υ
      ∂ ∂ ∂ ∂ ∂ ∂   
− + − + =        ∂ − ∂ − ∂ + ∂ ∂ ∂         
      ∂ ∂ ∂ ∂ ∂ ∂   
− + − + =         + ∂ ∂ ∂ ∂ − ∂ − ∂         

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The above equations will derive the finite element model using the variational formulation as presented in
the Reddy’s text book to be derived as below:
{ } { } { }
{ } { } { }
11 12 1
21 22 2
K u K v F
K u K v F
   + =   
   + =   
The two above model equations need to be solved for the studied plane problems to derive the
displacement, strain and stress values. Next, the theoretical and numerical results are presented and
discussed.

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Results and discussion:
Theoretical stress values:
In order to compute the FE results from mesh independency, the stress concentration factors, K, for each
hole type (i.e., circular, elliptical and rectangular) were found from the exisiting technical document, and
were then compared against the numerical values obtained from the ABAQUS. The stress factor includes
the effect of hole existence as the ratio of the theoretical maximum stress to the nominal stress. The
nominal stress should be calculated over the cross section with the hole in the plate center.
The assumed uniform applied tension was set to 25×103
N/m2
× (1.0 m × 0.02 m)= 500 N. The reduced
area for all the three cases were identical and equal to A= (1.0 m – 0.1 m) × 0.02 m = 0.018 m2
.
The nominal stress for all the three cases were equal to 27778 Pa= 0.0278 MPa.
For each analysis, the maximum stress obtained from ABAQUS of the full plate model and the nominal
stress were compared against.
Case 1 - Circular hole:
The first case is the plate with the circular hole, for the dimension according to the problem statement (1
m x 1m) and a 100 mm circular hole in the middle, according to the chart below, was set equal K~ 2.7, as
shown for the d/b = 0.1 / 1= 0.1.

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Source: http://www.ux.uis.no/~hirpa/KdB/ME/stressconc.pdf
The K= 2.7, results in the stress of equal to the maximum stress of 2.7 * 0.02778 MPa= 0.07676 MPa.
Case 2 – Elliptical hole
The second case was the 1.0 m * 1.0 m plate with a 0.1 x 0.2 m elliptical hole at the center of the plate,
under the same load condition as case 1 (500 N/m).
The nominal stress is equal to case 1 of 0.02778 MPa. The stress concentration factor for this case is
computed from the “Young, W. C., & Budynas, R. G. (2002). Roark's formulas for stress and strain (Vol.
7). New York: McGraw-Hill.” For the elliptical hole configuration in this study, the a/b ratio is 0.5, (a=
0.05 m and b= 0.01 m), which lies in the limits of this equation. The stress concentration factor as shown
in the figure below would be equal to K= 1.9.
Considering the K= 1.9, the maximum effective stress would be equal to 1.9 * 0.02778 MPa= 0.05278
MPa.
Source: Young, W. C., & Budynas, R. G. (2002). Roark's formulas for stress and strain (Vol. 7). New
York: McGraw-Hill

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Case 3 – Rectangular hole
The last case was the rectangular hole at the center of the plate of the dimensions of 0.1 m x 0.2 m, with
rounded corners. The stress concentration factor was computed from the “Pilkey Walter, D., & Pilkey
Deborah, D. (1997). Peterson's Stress Concentration factor.” and the graph as shown below from it were
used to derive the stress concentration factor. The stress concentration factor for the studied problem was
calculated (r= 0.02 m, a= 0.05 m, r/a= 0.4), as K= 2.9. Similarly, a/b= 0.5 (a= 0.05 m and b= 0.1 m).
This would result in the effective stress of equal to 0.02778 × 2.9 = 0.080562 MPa.
Source: “Pilkey Walter, D., & Pilkey Deborah, D. (1997). Peterson's Stress Concentration factor.”

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Convergence sensitivity analysis:
The independency of the results from the mesh size is an important step in the FE simulations to eliminate
the unnecessary computational cost, however, without jeopardizing the accuracy of the FE simulations.
The parametric study were implemented first prior to developing all the models so as to find the optimum
mesh size. In order to get the more reliable and consistent meshing between the quarter-model and the
full-model, the seed distance on the hole side perimeter was assumed proportional to the ratio of the
length of the hole side perimeter to the outer perimeter. The outer perimeter seed distance, and similarly
the inner perimeter was then incrementally decreased, to the point no significant deviation in results
(Von-Mises results) were obtained.
While uniform equal meshing distance for the whole FE plate model increased the accuracy, however, the
finer mesh around the hole and the more coarse mesh around the perimeter proved to improve the results
accuracy without extra computational cost. Three meshing size implemented herein for the plates
(different hole geometry and full versus quarter model), from the fine, medium and coarse are shown as
below. The effect of seed size (meshing) is shown also in the below table, reflecting the optimum mesh
size. A summary of the results are tabulated below.
Seed size
Von
Mises
peak
value
Deviation of (%)
A/B ratioOuter edge Inner side
Stress
(MPa)
A= Maximum vin-
mises stress (%)
B= (Seed
size)2
(%)
Mesh size (mm) Mesh size (mm)
200 15.708 0.0664
100 7.854 0.07215 8.67 75 0.116
75 5.8905 0.07376 2.22 44 0.051
50 3.927 0.07538 2.20 56 0.040
25 1.9635 0.07668 1.72 75 0.023
20 1.5708 0.07676 0.10 36 0.003
15 1.1781 0.07691 0.20 44 0.004
The sensitivity mesh study revealed that an almost 20 mm seed size the mesh dependency of the results
vanish and starts to converge to almost identical values. This methodology was developed for all the three
FE models. It was found that:
The plate with the circular hole began to converge with an outside seed size of 25 mm,
The plate with the elliptical hole began to converge with an outside seed size of 50 mm,
The plate with the rectangular hole began to converge with an outside seed size of 25 mm.

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The 25 mm seed size proved to be sufficient in this study for the developed FE models to get the accurate
values. The FE models for different mesh densities for the full and the quarter models are illustrated
below.
(Circular hole- Full plate versus quarter model – fine, medium and coarse mesh)

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(Elliptical hole- Full plate versus quarter model – fine, medium and coarse mesh)

13
(Rectangular hole- Full plate versus quarter model – fine, medium and coarse mesh)

14
Finite element models result – Full Plate and Quarter models
This section provides the results from the ABAQUS/CAE results for the Full and Quarter FE plate
models, and its comparison against the theoretical stress values. The comparison for the full plate model
and quarter plate model are summarized in the below table.
Hole shape Nominal
stress,
σn=P/A
Theoretical
stress, K*
σn
Von-Misses stress Displacement
(MPa) (MPa) Full
plate
(MPa)
Quarter
plate
(MPa)
Deviation
(%)
Full plate
(m)
Quarter
plate (m)
Deviation
(%)
Circular
hole
0.0278 0.07676 0.07676 0.07575 1.32 1.325E-07 1.334E-07 0.67
Elliptical
hole
0.0278 0.05278 0.05147 0.05200 1.03 1.285E-07 1.291E-07 0.45
Rectangular
hole
0.0278 0.08056 0.06555 0.06432 1.88 1.176E-07 1.182E-07 0.51

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(Circular hole- Full plate model - Von-Mises stress (left) – deformed shape (right))
(Circular hole- quarter plate model - Von-Mises stress (left) – deformed shape (right))

16
(Elliptical hole- Full plate model - Von-Mises stress (left) – deformed shape (right))
(Elliptical hole- Quarter plate model - Von-Mises stress (left) – deformed shape (right))

17
(Rectangular hole- Full plate model - Von-Mises stress (left) – deformed shape (right))
(Rectangular hole- Quarter plate model - Von-Mises stress (left) – deformed shape (right))

ME/AE 408: Finite Element Analysis of Plates with Holes

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