fea project-plate analysis

ME/AE 408: Advanced Finite Element Analysis


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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


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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.


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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.


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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

01 1

01 1

20 0

2(1 )

xx xx

yy yy

xy xy

E E

E E

E

νυ υσ ε

νσ ευ υ

σ ε

υ

− −

= − − −

where the displacement-strain relations are related as below:

x

y

xy

uxvyu vy x

ε

ε

γ

∂=∂∂

=∂∂ ∂

= +∂ ∂

and the equilibrium equations that need to be satisfied due to the applied external actions are as below:

0

0

xyxx

x y

xy yy

x y

f

f

σσσ σ

σ σσ 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 fx x y y y x

E u v E u E v fx y x y x y

υυ υ υ

υυ 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

http://www.ux.uis.no/%7Ehirpa/KdB/ME/stressconc.pdf


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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 ratio Outer 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)


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


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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))


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

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


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

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

fea project-plate analysis

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