This document presents the results of a finite element analysis (FEA) of a cantilever beam with different mesh densities and element configurations. Models of the beam were created using 8-noded brick elements, 20-noded brick elements, 4-noded tetrahedral elements, and 2-noded beam elements with varying numbers of elements through the thickness. The results, including deflection, bending stress at the center and wall of the beam, were compared to theoretical calculations and each other. It was determined that the 20-noded brick element model provided the most accurate results, even with fewer elements, and is the most efficient element type to model this problem.

1. Learning with PurposeLearning with Purpose
Mesh Density and
Configuration Project
FEA Analysis I
22.513
Fucheng Chen
Tushar Dange
V. Bhargav
HuanRan Liu

2. Learning with Purpose
A) Deformed beam with bending stress contour of
8-noded brick element
1 element through the thickness

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A) Deformed beam with bending stress contour of
8-noded brick element
2 elements through the thickness

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A) Deformed beam with bending stress contour of
8-noded brick element
8 element through the thickness

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B) Deformed beam with bending stress contour of
20-noded brick element
1 element through the thickness

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B) Deformed beam with bending stress contour of
20-noded brick element
2 elements through the thickness

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B) Deformed beam with bending stress contour of
20-noded brick element
8 elements through the thickness

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C) Deformed beam with bending stress contour of
4-noded brick element
1 element through the thickness

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C) Deformed beam with bending stress contour of
4-noded brick element
2 element through the thickness

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C) Deformed beam with bending stress contour of
4-noded brick element
8 element through the thickness

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D) 25 cubic beam element model
Two other beam similar models were created with 12 and 50 cubic
element to investigate the effect of number of element along beam
length on deflection and max bending stress in the beam

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D) Comparison among 12, 25, and 50 cubic
element model
Number of
Elements
Deflection at the
center of beam (in)
Maximum Bending stress
at center of beam (psi)
Maximum Bending
stress at the wall (psi)
12 -0.0410 -4687.5 -8984.38
25 -0.0410 -4687.5 -9187.50
50 -0.0410 -4687.5 -9281.25
• The deflection and bending stress at the center of 25 cubic element beam
was calculated by taking the average of the deflection at node 13 and 14.
• The deflection and bending stress at the center of 50 cubic element beam
was taken at node 26.
• The deflection and bending stress at the center of 12 cubic element beam
was taken at node7.
• From the table shown above, it is found that the number of beam elements
does not impact the deflection or max bending stress at the center of
beam.
• However, the maximum bending stress at the wall increases as the
increase of number of elements.

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E) Comparison of deflection at center of the beam
among different model
Tabular Representation
Deflection (in) at the center of Beam (x=25)
Number of
Elements
8-noded bricks 20-noded bricks 4-noded tets 2-noded beams
1 -0.0489 -0.0401 -0.0068 -0.0410
2 -0.0397 -0.0406 -0.0232 N/A
8 -0.0407 -0.0408 -0.0392 N/A

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E) Comparison of deflection at center of the beam
among different model
Graphical Representation
-0.0600
-0.0500
-0.0400
-0.0300
-0.0200
-0.0100
0.0000
0 1 2 3 4 5 6 7 8 9
DeflectionattheCenteroftheBeam(in)
Number of Elements through the thickness
Comparison of deflection at the center of the beam among different models and
theoretical calculation
8-noded bricks
20-noded bricks
4-noded tets
2-noded beams
Theoretical

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Maximum Bending Stress (psi) at the center of beam (x=25)
Number of
Elements
8-noded bricks 20-noded bricks 4-noded tets 2-noded beams
1 2309.36 4687.59 1650.1 4687.5
2 3657.02 4687.49 3380.7 N/A
8 4451.53 4687.48 4122.43 N/A
F) Comparison of maximum bending stress at
center of the beam among different models
Tabular Representation

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F) Comparison of maximum bending stress at
center of the beam among different models
Graphical Representation
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 1 2 3 4 5 6 7 8 9
MaximumBendingStressattheCenterofBeam(psi)
Number of Elements through the thickness
Comparison of maximum bending stress at the center of the beam among different
models and theoretical calculation
8-noded bricks
20-noded bricks
4-noded tets
2-noded beams
Theoretical

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Maximum Bending Stress (psi) at the wall (x=0)
Number of
Elements
8-noded bricks 20-noded bricks 4-noded tets 2-noded beams
1 3870.3 8987.56 2780.54 9187.5
2 7423.11 9392.15 6262.97 N/A
8 9135.72 9368.27 8463.69 N/A
G) Comparison of maximum bending stress at the
wall among different models
Tabular Representation

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G) Comparison of maximum bending stress at the
wall among different models
Graphical Representation
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0 1 2 3 4 5 6 7 8 9
MaximumBendingStressaththeWall(psi)
Number of Elements through the thickness
Comparison of maximum bending stress at the wall among different models and
theoretical calculation
8-noded bricks
20-noded bricks
4-noded tets
2-noded beams
Theoretical

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H) 8-layer 8-noded brick model with poisson ratio
ν=0

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• When the Poisson’s ratio of 8-layer 8-noded brick model changed to 0, the
bending stress at the wall decreased to 8910.04 psi. The reason is that the
beam will shrink when Poisson’s ratio is not 0, and the stress at the wall
will increase due to the Encastre constraint at the wall location.
• All models showed that the results gets closer to the theoretical values as
the number of element through the thickness increases.
• Comparing with the results from models with different element type, it is
found that the 20-noded brick element models have the most accurate
results.
• Besides, the 20-noded brick element model shows the accuracy of results
even with small amount of elements, which indicates that the 20-noded
brick element is the most efficient and accurate method when modeling
cantilever beam with rectangular cross-section with a tip load applied.
• The 2-noded beams model showed consistency and accuracy of bending
stress at the center of beam despite the number of element used.
However, the accuracy of bending stress at the wall depends on the
number of element.
I) Disscussion

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• For 8-noded bricks and 4-noded tets, the tables and figures
in part EFG shows that it is necessary to use more than one
element through the thickness, since there is a huge
difference between the results from model with one
element through the thickness and theoretical values.
• For 20-noded bricks, it is reasonable to use one element
through the thickness only if the bending stress at the
center of the beam is desired. However, more than one
element through the thickness is required, when bending
stress at wall or deflection is desired.
J) Is there any reason to use more than one
element through the thickness to model this beam

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• 𝐼 =
𝑏ℎ3
12
=
2∗43
12
= 10.67
• 𝑦 𝑥 =
𝑃∗𝑥2 3𝐿−𝑥
6𝐸𝐼
• 𝑦 25 =
100∗252 3∗50−25
6∗30𝐸06∗10.67
= 𝟎. 𝟎𝟒𝟎𝟕 𝐢𝐧
• 𝜎 =
𝑀𝑦
𝐼
• 𝜎 𝑥 = 0 =
𝑀 𝑥=0 𝑦
𝐼
=
1000∗50∗2
10.67
= 𝟗𝟑𝟕𝟓 𝒑𝒔𝒊
• 𝜎 𝑥 = 25 =
𝑀 𝑥=25 𝑦
𝐼
=
1000∗25∗2
10.67
= 𝟒𝟔𝟖𝟕. 𝟓 𝒑𝒔𝒊
K) Theoretical Calculations

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1. “Mesh Density and Configuration Project.” INTRO TO
FINITE ELEMENT ANALYSIS. Department of Mechanical
Engineering, UMass Lowell, n.d. Web. <http://m-
5.eng.uml.edu/22.513/>.
2. “Axial, Bending, Torsion, Combined and Bucking Analysis
of a Beam Tutorial ABAQUS.” INTRO TO FINITE ELEMENT
ANALYSIS. Department of Mechanical Engineering, UMass
Lowell, n.d. Web. <http://m-5.eng.uml.edu/22.513/>.
3. “Finite Element Analysis of A Propped Cantilever Beam.”
INTRO TO FINITE ELEMENT ANALYSIS. Department of
Mechanical Engineering, UMass Lowell, n.d. Web.
<http://m-5.eng.uml.edu/22.513/>.
L) References