### Project

```2D MODELING OF THE DEFLECTION OF
A SIMPLY SUPPORTED BEAM UNDER
YIN-YU CHEN
MANE4240 – INTRODUCTION TO FINITE ELEMENT ANALYSIS
APRIL 28, 2014
Introduction/Background
 Maximum deflection of a simply
supported elastic beam subject to
 M1 Abrams tank (67.6 short tons)
equally supported by two simply
supported steel beams
M1 Abrams Tank
Description
Value
Force 300000
Hull/Track Length
8
Track Width
0.6
Beam
Description
Value
Length
8
Width
0.6
Thickness
0.1
Young's Modulus (E) 2.00E+11
Poisson's Ratio (n)
0.3
Density (r )
7800
Unit
N
m
m
Unit
m
m
m
Pa
kg/m3
 Land mine flush with the ground under
the center of each beam
 Determine the required height of the
beam from the ground in order to avoid
setting off the land mine
 Moment-Curvature Equation

2
2

=
Analytical Formulation/Solution
 Moment of Inertia of a
Rectangular Cross Section of a
Beam
 =
ℎ3
12
 Simply Supported Beam with Point
  = −
3
48
 Simply Supported Beam with
  =
54
−
384
Modeling
 COMSOL Multiphysics
 2D Structural Mechanics, Solid Mechanics and Stationary presets
 Rectangular geometry with prescribed displacements of 0m at bottom corners (x & y for
one, y only for the other) to represent a simply supported beam
 Point load case: -300000N at center (x=4m)
 Mesh Extension Validation
•
Extremely Fine
•
Finer
•
Normal
•
Coarser
•
Extremely Coarse
Results
 Simply Supported Beam with Point Load at the Center
Degrees
Max
Min
of
Triangular Edge
Vertex Element Element Freedom
Size
Size
Solved x dmax (m) d max (m)
Mesh
Nodes Elements Elements Elements
Extremely Fine 511
414
208
5
0.08 1.60E-04
2074
4.0
-0.29131
Finer 349
120
118
5
0.296
0.001
718
4.0
-0.29125
Normal 349
120
118
5
0.536
0.0024
718
4.0
-0.29125
Coarser 349
120
118
5
1.04
0.048
718
4.0
-0.29124
Extremely Coarse 313
103
105
5
2.64
0.4
624
4.0
-0.29123
d max (cm)
-29.131
-29.125
-29.125
-29.124
-29.123
 Simply Supported Beam with Uniformly Distributed Load
Degrees
Max
Min
of
Triangular Edge
Vertex Element Element Freedom
Size
Size
Solved x dmax (m)
Mesh
Nodes Elements Elements Elements
Extremely Fine 511
414
208
4
0.08 1.60E-04
2074
4.0
Finer 349
120
118
4
0.296
0.001
718
4.0
Normal 349
120
118
4
0.536
0.0024
718
4.0
Coarser 349
120
118
4
1.04
0.048
718
4.0
Extremely Coarse 313
103
105
4
2.64
0.4
624
4.0
d max (m) d max (cm)
-0.18206 -18.206
-0.18202 -18.202
-0.18202 -18.202
-0.18203 -18.203
-0.18202 -18.202
Results
 Comparison of COMSOL Modeling/Numerical and Analytical Method
Analytical
Results
Mesh
Nodes Total Elements d max (m) Result d max (m) % Error
COMSOL
243
-0.29125
-0.32
-8.985%
COMSOL
Normal Mesh Uniformly
Distributed
242
-0.18202
-0.20
-8.988%
 Comparison of ANSYS Modeling/Numerical and Analytical Method Results
Element Size Nodes
0.05
481
0.075
322
0.1
241
0.33
76
1
25
Total Elements
160
107
80
25
8
d max (m)
-0.20008
-0.20006
-0.20008
-0.19969
-0.20008
Analytical
Result
d max (m)
-0.20
-0.20
-0.20
-0.20
-0.20
% Error
0.038%
0.028%
0.038%
-0.154%
0.038%
Conclusions
 Maximum deflection of a simply supported elastic beam subject to point or
distributed loads may be achieved using either the modeling/numerical or
analytical methods

Appears that the shape of the cells for the mesh is a major factor in the accuracy of the
maximum beam deflection results
•
Quadrilateral cell mesh may offer the most accurate solution
 The steel beam requires a minimum height of 0.2m from the ground for the tank
to avoid setting off the land mine
 This study highlights necessity for verifying the reliability of the approximate
solution by comparing the results to:

A theoretical/exact solution

A different modeling approach

A mesh extension validation
•
If results from the COMSOL analysis of the uniformly distributed load across the beam were used
without a factor of safety > 1.1 for the height of the beam from the ground, the maximum deflection
due to the tank would set off the land mine
```